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exclusively LCFTs where the energy momentum tensor acquire s a logarithmic partner. (vii) is
required since the 2- and 3-point correlators of a LCFT are ﬁx ed by conformal Ward identities to
taketheform(7), (8). Ifanyoftheitemsonthewish-listabo veisnotfulﬁlleditisimpossiblethat
the gravitational theory under consideration is a gravity d ual to a LCFT of the type discussedin section 2.2On the other hand, if all the wishes are granted by a given grav itational theory
there are excellent chances that this theory is dual to a LCFT . Until recently no good gravity
duals for LCFTs were known [8–12].
Before addressing candidate theories that may comply with a ll wishes we review brieﬂy how
to calculate correlators on the gravity side [6], since we sh all need such calculations for checking
several items on the wish-list. The basic identity of the AdS /CFT dictionary is
∝an}b∇acketle{tO1(z1)O2(z2)...On(zn)∝an}b∇acket∇i}ht=δ(n)S
δj1(z1)δj2(z2)...δjn(zn)/vextendsingle/vextendsingle/vextendsingle
ji=0(10)
The left hand side is the CFT correlator between noperators Oi, whereOiin our case comprise
theleft-andright-moving ﬂuxcomponentsoftheenergymome ntumtensor andtheirlogarithmic
partners. The right hand side contains the gravitational ac tionSdiﬀerentiated with respect to
appropriate sources jifor the corresponding operators. According to the AdS/CFT d ictionary
“appropriate sources” refers to non-normalizable solutio ns of the linearized equations of motion.
We shall be more concrete about the operators, actions, sour ces and non-normalizable solutions
to the linearized equations of motion in the next section. Fo r now we address possible candidate
theories of gravity duals to LCFTs.
The simplest candidate, pure 3-dimensional Einstein gravi ty with a cosmological constant
described by the action
SEH=−1
8πGN/integraldisplay
Md3x√−g/bracketleftig
R+2
ℓ2/bracketrightig
−1
4πGN/integraldisplay
∂Md2x√−γ/bracketleftig
K−1
ℓ/bracketrightig
(11)
does not comply with the whole wish list. Only the ﬁrst four wi shes are granted: The 3-
dimensional action (12) depends on the metric. The equation s of motion are solved by AdS 3.
ds2
AdS3=gAdS3µνdxµdxν=ℓ2/parenleftbig
dρ2−1
4cosh2ρ(du+dv)2+1
4sinh2ρ(du−dv)2/parenrightbig
(12)
The Brown–York stress tensor (9) is ﬁnite, conserved and tra celess. The 2- and 3-point
correlators on the gravity side match precisely with (1). Ho wever, the central charges are given
by [7]
cL=cR=3ℓ
2GN(13)
and therefore allow no tuning to cL= 0 without taking a singular limit. Moreover, there is no
candidate for a logarithmic partner to the Brown–York stres s tensor. Thus, pure 3-dimensional
Einstein gravity cannot be dual to a LCFT.
Adding matter ﬁelds to Einstein gravity does not help neithe r. While this may lead to other
kinds of LCFTs, it cannot produce a logarithmic partner for t he energy momentum tensor. This
is so, because the energy momentum tensor corresponds to gra viton (spin-2) excitations in the
bulk, and the only ﬁeld producing such excitations is the met ric.
Therefore, what we need is a way to provide additional degree s of freedom in the gravity
sector. The most natural way to do this is by considering high er derivative interactions of the
metric. Theﬁrstgravity modelofthistypewas constructedb yDeser, Jackiw andTempleton [13]
who introduced a Chern–Simons term for the Christoﬀel connec tion.
SCS=−1
16πGNµ/integraldisplay
d3xǫλµνΓρσλ/bracketleftig
∂µΓσρν+2
3ΓσκµΓκσν/bracketrightig
(14)
2Other types of LCFTs exist, e.g. with non-vanishing central charge or with logarithmic partners to operators
other than the energy momentum tensor. The gravity duals for such LCFTs need not comply with all the items
on our wish list.Hereµis a real coupling constant. Adding this action to the Einste in–Hilbert action (11)
generates massive graviton excitations in the bulk, which i s encouraging for our wish list since
we need these extra degrees of freedom. The model that arises when summing the actions (11)
and (14),
SCTMG=SEH+SCS (15)
is known as “cosmological topologically massive gravity” ( CTMG) [14]. It was demonstrated by
KrausandLarsen[15]that thecentral charges inCTMG areshi ftedfromtheir Brown–Henneaux
values:
cL=3ℓ
2GN/parenleftbig
1−1
µℓ/parenrightbig
cR=3ℓ
2GN/parenleftbig
1+1
µℓ/parenrightbig
(16)
This is again good news concerning our wish list, since cLcan be made vanishing by a (non-
singular) tuning of parameters in the action.
µℓ= 1 (17)
CTMG (15) with the tuning above (17) is known as “cosmologica l topologically massive gravity
at the chiral point” (CCTMG). It complies with the ﬁrst ﬁve it ems on our wish list, but we still
have to prove that also the last two wishes are granted. To thi s end we need to ﬁnd a suitable
partner for the graviton.
4. Keeping logs in massive gravity
4.1. Login
In this section we discuss the evidence for the existence of s peciﬁc gravity duals to LCFTs that
has accumulated over the past two years. We start with the the ory introduced above, CCTMG,
and we end with a relatively new theory, new massive gravity [ 16].
4.2. Seeds of logs
Given that we want a partner for the graviton we consider now g raviton excitations ψaround
the AdS background (12) in CCTMG.
gµν=gAdS3µν+ψµν (18)
Li,SongandStrominger[17]foundanicewaytoconstructthe m,andwefollowtheirconstruction
here. Imposing transverse gauge ∇µψµν= 0 and deﬁning the mutually commuting ﬁrst order

exclusively LCFTs where the energy momentum tensor acquire s a logarithmic partner. (vii) is

required since the 2- and 3-point correlators of a LCFT are ﬁx ed by conformal Ward identities to

taketheform(7), (8). Ifanyoftheitemsonthewish-listabo veisnotfulﬁlleditisimpossiblethat

the gravitational theory under consideration is a gravity d ual to a LCFT of the type discussedin section 2.2On the other hand, if all the wishes are granted by a given grav itational theory

there are excellent chances that this theory is dual to a LCFT . Until recently no good gravity

duals for LCFTs were known [8–12].

Before addressing candidate theories that may comply with a ll wishes we review brieﬂy how

to calculate correlators on the gravity side [6], since we sh all need such calculations for checking

several items on the wish-list. The basic identity of the AdS /CFT dictionary is

∝an}b∇acketle{tO1(z1)O2(z2)...On(zn)∝an}b∇acket∇i}ht=δ(n)S

δj1(z1)δj2(z2)...δjn(zn)/vextendsingle/vextendsingle/vextendsingle

ji=0(10)

The left hand side is the CFT correlator between noperators Oi, whereOiin our case comprise

theleft-andright-moving ﬂuxcomponentsoftheenergymome ntumtensor andtheirlogarithmic

partners. The right hand side contains the gravitational ac tionSdiﬀerentiated with respect to

appropriate sources jifor the corresponding operators. According to the AdS/CFT d ictionary

“appropriate sources” refers to non-normalizable solutio ns of the linearized equations of motion.

We shall be more concrete about the operators, actions, sour ces and non-normalizable solutions

to the linearized equations of motion in the next section. Fo r now we address possible candidate

theories of gravity duals to LCFTs.

The simplest candidate, pure 3-dimensional Einstein gravi ty with a cosmological constant

described by the action

SEH=−1

8πGN/integraldisplay

Md3x√−g/bracketleftig

R+2

ℓ2/bracketrightig

−1

4πGN/integraldisplay

∂Md2x√−γ/bracketleftig

K−1

ℓ/bracketrightig

(11)

does not comply with the whole wish list. Only the ﬁrst four wi shes are granted: The 3-

dimensional action (12) depends on the metric. The equation s of motion are solved by AdS 3.

ds2

AdS3=gAdS3µνdxµdxν=ℓ2/parenleftbig

dρ2−1

4cosh2ρ(du+dv)2+1

4sinh2ρ(du−dv)2/parenrightbig

(12)

The Brown–York stress tensor (9) is ﬁnite, conserved and tra celess. The 2- and 3-point

correlators on the gravity side match precisely with (1). Ho wever, the central charges are given

by [7]

cL=cR=3ℓ

2GN(13)

and therefore allow no tuning to cL= 0 without taking a singular limit. Moreover, there is no

candidate for a logarithmic partner to the Brown–York stres s tensor. Thus, pure 3-dimensional

Einstein gravity cannot be dual to a LCFT.

Adding matter ﬁelds to Einstein gravity does not help neithe r. While this may lead to other

kinds of LCFTs, it cannot produce a logarithmic partner for t he energy momentum tensor. This

is so, because the energy momentum tensor corresponds to gra viton (spin-2) excitations in the

bulk, and the only ﬁeld producing such excitations is the met ric.

Therefore, what we need is a way to provide additional degree s of freedom in the gravity

sector. The most natural way to do this is by considering high er derivative interactions of the

metric. Theﬁrstgravity modelofthistypewas constructedb yDeser, Jackiw andTempleton [13]

who introduced a Chern–Simons term for the Christoﬀel connec tion.

SCS=−1

16πGNµ/integraldisplay

d3xǫλµνΓρσλ/bracketleftig

∂µΓσρν+2

3ΓσκµΓκσν/bracketrightig

(14)

2Other types of LCFTs exist, e.g. with non-vanishing central charge or with logarithmic partners to operators

other than the energy momentum tensor. The gravity duals for such LCFTs need not comply with all the items

on our wish list.Hereµis a real coupling constant. Adding this action to the Einste in–Hilbert action (11)

generates massive graviton excitations in the bulk, which i s encouraging for our wish list since

we need these extra degrees of freedom. The model that arises when summing the actions (11)

and (14),

SCTMG=SEH+SCS (15)

is known as “cosmological topologically massive gravity” ( CTMG) [14]. It was demonstrated by

KrausandLarsen[15]that thecentral charges inCTMG areshi ftedfromtheir Brown–Henneaux

values:

cL=3ℓ

2GN/parenleftbig

1−1

µℓ/parenrightbig

cR=3ℓ

2GN/parenleftbig

1+1

µℓ/parenrightbig

(16)

This is again good news concerning our wish list, since cLcan be made vanishing by a (non-

singular) tuning of parameters in the action.

µℓ= 1 (17)

CTMG (15) with the tuning above (17) is known as “cosmologica l topologically massive gravity

at the chiral point” (CCTMG). It complies with the ﬁrst ﬁve it ems on our wish list, but we still

have to prove that also the last two wishes are granted. To thi s end we need to ﬁnd a suitable

partner for the graviton.

4. Keeping logs in massive gravity

4.1. Login

In this section we discuss the evidence for the existence of s peciﬁc gravity duals to LCFTs that

has accumulated over the past two years. We start with the the ory introduced above, CCTMG,

and we end with a relatively new theory, new massive gravity [ 16].

4.2. Seeds of logs

Given that we want a partner for the graviton we consider now g raviton excitations ψaround

the AdS background (12) in CCTMG.

gµν=gAdS3µν+ψµν (18)

Li,SongandStrominger[17]foundanicewaytoconstructthe m,andwefollowtheirconstruction

here. Imposing transverse gauge ∇µψµν= 0 and deﬁning the mutually commuting ﬁrst order