1001.0042.txt

arXiv:1001.0042v2  [hep-th]  16 Apr 2010Topological Gravity in Seven Dimensions
H. L¨ u†‡and Yi Pang⋆
†China Economics and Management Academy
Central University of Finance and Economics, Beijing 100081
‡Institute for Advanced Study, Shenzhen University, Nanhai A ve 3688, Shenzhen 518060
⋆Key Laboratory of Frontiers in Theoretical Physics
Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190
ABSTRACT
We obtain new topological gravity in seven dimensions by add ing two topological terms
to the Einstein-Hilbert action. For certain choice of the co upling constants, these terms
may have an origin as the R4correction to the 3-form field equation of eleven-dimension al
supergravity. We derive the full set of the equations of moti on, and obtain large classes of
solutions including static AdS black holes, squashed seven spheres and Q111spaces.1 Introduction
There has been considerable interest in topological gauge t heories [1] because of their wide
application in physics. The most studied example is the thre e-dimensional one. In addition
to the Einstein-Hilbert term, the theory has the Chern-Simo ns term, given by
S=1
µ/integraldisplay
d3xTr(dω∧ω−2
3ω∧ω∧ω), (1)
whereωcan be either a Yang-Mills gauge potential or the connection for gravity. Topo-
logical Yang-Mills theory can provide a fundamental interp retation for anyons [2]; it can
also generate Lorentz violation dynamically [3]. Topologi cal gravity [4] becomes dynamical
with a propagating massive particle, with the mass proporti onal to the coupling constant
µ. Recently, a cosmological constant is added and the corresp onding boundary conformal
field theory (CFT) is discussed [5]. The three-dimensional m assive topological gravity is
conjectured to be unitary for certain parameter region even though the theory has higher
derivatives in time [6].
The attention on higher dimensional generalizations is con siderably less. The five di-
mensional Yang-Mills Chern-Simons term was discussed in [7 ], but there is no gravity coun-
terpart dueto the fact that the holonomy group SO(1,4) has no invariant rank-3 symmetric
tensor. In seven dimensions, Yang-Mills Chern-Simons term s arise naturally from N= 4
supergravity [8]. As in the case of three dimensions, we find t hat such terms in the grav-
ity sector can be obtained directly from those in the Yang-Mi lls sector by replacing the
gauge potential Ato the connection Γ. Moreover, as we shall see later, seven-d imensional
topological gravity has a direct origin in eleven-dimensio nal supergravity, while any higher-
dimensional origin of the three-dimensional theory remain s unknown.
In section 2, we present the two topological terms in seven di mensions, and discuss their
properties. Sincethey arenot manifestly invariant underg eneral coordinate transformation,
we find it is more convenient to lift the system to eight dimens ions in order to derive the
equations of motion (EOMs). We obtain the full set. In sectio n 3, we construct large
classes of solutions including static Anti-de Sitter (AdS) black holes, squashed S7andQ111.
We emphasize that all the previously-known static (AdS) bla ck holes remain to be solutions
whenthetopological termsare addedinto theaction. Thisis analogous to threedimensions,
where the BTZ black hole remains to be a solution in topologic al massive gravity. We
conclude in section 4.
22 The theory
In seven dimensions, there are two topological terms; they a re given by
S1=µ/integraldisplay
Ω(7)
1=µ/integraldisplay
Tr(Γ∧Θ+1
3Γ3)∧Tr(Θ2) =µ/integraldisplay
Ω(3)∧dΩ(3), (2)
S2=ν/integraldisplay
Ω(7)
2=ν/integraldisplay
Tr(Θ3∧Γ+2
5Θ2∧Γ3+1
5Θ∧Γ2∧Θ∧Γ+1
5Θ∧Γ5+1
35Γ7),
with Ω(3)= Tr(dΓ∧Γ−2
3Γ3). Here, Θ is the curvature 2-form, defined as Θ ≡dΓ−Γ∧Γ,
andµ,νare two parameters of length dimension 5. (We rescale the tot al action by the
seven-dimensional Newton constant.) The 3-form Ω(3)has the same structure as the Chern-
Simons term in D= 3, except that now Γ depends on seven coordinates. Ω(7)
1and Ω(7)
2are
topological in the same sense as Ω(3)being topological in D= 3. We can lift the system to
D= 8, with the seven-dimensional spacetime as the boundary. T hen, we have
dΩ(7)
1=Y(8)
1≡Tr(Θ∧Θ)∧Tr(Θ∧Θ), dΩ(7)
2=Y(8)
2≡Tr(Θ∧Θ∧Θ∧Θ).(3)
As we have mentioned earlier, these terms can be derived from the Yang-Mills Chern-
Simons terms in [8] by changing the gauge potential to the con nection.1Note that the
Pontryagin term is proportional to Y(8)
1−2Y(8)
2, corresponding to ν=−2µ. In eleven-
dimensional supergravity, there is an R4correction to the field equation, namely d∗F(4)=
1
2F(4)∧F(4)+X(8), whereX(8)is given by
X(8)∝Y(8)
1−4Y(8)
2. (4)
Thus forν=−4µ, the topological terms can be obtained from the S4reduction of super-
gravity inD= 11, and the coupling constant is proportional to the 4-form M5-brane fluxes.
For large fluxes, this topological term dominates the higher -order corrections.
To derive the contribution to the EOMs from the Chern-Simons terms, it is necessary
to perform their variation with respect to the metric. These topological terms are not
manifestly invariant under the general coordinate transfo rmation, but Y(8)
1andY(8)
2are.
Wefindthataconvenient waytoderivethevariationistolift thesystemtoeightdimensions.
Let us first consider the variation of S1. In terms of coordinate components, we have
/integraldisplay
dΩ(7)
1=1
16/integraldisplay
d8xǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6Rµ4µ3ν7ν8. (5)
1In [8], the field strength 2-form is defined by F=dB+gB∧B, with gauge coupling g= 2. Then by
rescaling the field B→B/gandF→F/gand setting g=−2, one can obtain the same expressions as the
ones given here.
3Here we use Greek letters to denote the eight-dimensional co ordinates and Latin letters to
represent the seven-dimensional ones hereafter. We adopt t he convention ǫ12345678= 1.
For an infinitesimal variation of the metric δg, using the Bianchi identity and the fol-
lowing relation
δRµ
ναβ=δΓµ
νβ;α−δΓµ
να;β, (6)
we find that
/integraldisplay
dδΩ(7)
1=−1
2/integraldisplay
d8x√g/parenleftBig1√gǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7/parenrightBig
;ν8
≡1
2/integraldisplay
d∗J, (7)
where “;” denotes a covariant derivative and ∗is the Hodge dual. For simplicity, we have
introduced a 1-form current J=Jαdxα. Its components are given by
Jα=1√gǫν1ν2ν3ν4ν5ν6ν7αRµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7. (8)
Clearly, we have d∗J=−√gJα;αd8x, Thus we obtain
δΩ(7)
1=1
2∗J, (9)
up to a total derivative term. Now restricting the coordinat e indices to seven dimensions
only, we have
δS1= 4µ/integraldisplay
Tr(Θ∧Θ)∧Tr(Θ∧δΓ). (10)
The variation of S2can be obtained in the same manner, given by
δS2= 4ν/integraldisplay
Tr(Θ∧Θ∧Θ∧δΓ). (11)
Finally, we make use of the variation of the connection
δΓi
mj=1
2gin(δgnm;j+δgnj;m−δgml;n), (12)
and after integrating by parts, we obtain thecontributions to EOMs from the Chern-Simons
terms, given by
Cij
1=δS1√gδgij=µ
4√g[ǫij1j2j3j4j5j6(Ri1
i2j1j2Ri2
i1j3j4Rjk
j5j6);k+i↔j],
Cij
2=δS2√gδgij=ν
4√g[ǫij1j2j3j4j5j6(Rk
i1j1j2Ri1
i2j3j4Rji2
j5j6);k+i↔j].(13)
For the total action S, which is the sum of the Einstein-Hilbert action, cosmologi cal
constant Λ and S1+S2, the corresponding full set of EOMs is given by
Rij−1
2gijR+Λgij+Cij
1+Cij
2= 0. (14)
4It should be remarked that under a large gauge transformatio n Γ→ O−1ΓO−O−1dO,
the action transforms as S→S+µv(O)+νw(O), where
v(O) =/integraldisplay
−1
3d/parenleftBig
Tr(O−1dO)3∧Ω(3)/parenrightBig
;w(O) =−1
35/integraldisplay
Tr(O−1dO)7.(15)
Thevterm is trivial and gives no restriction to the parameter µ, while the wterm should
be classified by the seventh homotopy group of SO(1,6)
π7[SO(1,6)]≃π7[SO(6)]≃Z. (16)
The invariance of eiSrequires that
ν= 2πn, n = 0,±1,±2.... (17)
This quantization condition is clearly consistent with the M5-brane quantization, since
it has a direct origin in D= 11. This result is completely different from that in three
dimensions, where the SO(1,2) is homotoplically trivial and the mass parameter is not
quantized. Moreover, since νis quantized, S2will not be renormalized in the quantum
theory. This suggests some intriguing properties in the cor responding CFT dual.
3 Solutions
Spherically-symmetric solutions:
Having obtained the full set of EOMs for topological gravity in seven dimensions, we are
in the position to construct solutions. It is clear that the m aximally-symmetric space(time)
is unmodifiedby theinclusion of the topological terms. Then extsimplest case is to consider
the spherically-symmetric ansatz, given by
ds2=−F(r)dt2+dr2
G(r)+r2dΩ2
5. (18)
We find that for this ansatz, the contributions from the topol ogical terms Cij
1andCij
2
vanish identically. This implies that the previously-know n static (AdS) black holes, charged
or neutral, are still solutions when the topological terms a re added to the action. This is
analogous to three dimensions, where the BTZ black hole is st ill a solution in massive
topological gravity. However the thermodynamic quantitie s such as the mass and entropy
will acquire modifications [9, 10].
S3bundle over S4:
5We now turn our attention to the Euclidean theory. In three di mensions, there exists a
large class of squashed S3or AdS 3[11]. We expect the same in seven dimensions. Without
loss of generality, we set Λ = 30 so that it can give rise to a uni t roundS7. We first consider
the squashed S7that can be viewed as an S3bundle over S4. The metric ansatz is given
by
ds2=α3/summationdisplay
i=1(σi−cos2(1
2θ)˜σi)2+β/parenleftBig
dθ2+1
4sin2θ3/summationdisplay
i=1˜σ2
i/parenrightBig
. (19)
whereσiand ˜σiare theSU(2) left-invariant 1-forms, satisfying dσi=1
2ǫijkσj∧σkand
d˜σi=1
2ǫijk˜σj∧˜σk. The metric is Einstein provided that either α=β=1
4orα=1
5β=9
100.
The first case corresponds to the round S7and the second is a squashed S7that is also
Einstein. Now with the contribution from the topological te rms, the EOMs can be reduced
to
2α2+4αβ(7β−2)−β2= 0, (20)
together with
√α(α−β)3(4(10α+β)µ−(55α+7β)ν)+2β6(20αβ−4α−β) = 0.(21)
It is clear from (20) that there exists one and only one positi veαfor any positive β. The
squashing parameter γ≡α/βlies in the range 0 <γ <2+3√
2. Note that when 2 µ= 3ν,
the squashed S7that is Eisntein remains Einstein.
S1bundle over CP3:
There is another way of squashing an S7, which can be viewed as an S1bundle over
CP3. The metric ansatz is given by
ds2=α(dτ+sin2θ(dψ+B))2+βds2
CP3,
ds2
CP3=dθ2+sin2θcos2θ(dψ+B)2+sin2θ/parenleftBig
d˜θ2+1
4sin2˜θcos2˜θσ2
3
+1
4sin2˜θ(σ2
1+σ2
2)/parenrightBig
,
B=1
2sin2˜θσ3. (22)
It is of a round S7whenα=β= 1. In general, the EOMs imply that
α=β(8−7β),8µ+ν+β3
10976(β−1)2√α= 0. (23)
The squashing parameter γ≡α/βlies in the range (0 ,8).
Squashed Q111spaces:
6TheQ111space is an Einstein-Sasaki space of U(1) bundle over S2×S2×S2. We
consider the following ansatz
ds2=α/parenleftBig
dψ+3/summationdisplay
i=1cosθidφi/parenrightBig2
+β3/summationdisplay
i=1(dθ2
i+sinθ2
idφ2
i). (24)
It is ofQ111provided that α=1
2β= 1/16, and it remains so for ν= 0. In general, we have
α= 4β(1−7β),8(α−β)(2α−β)µ+α(2α−3β)ν+β5(α−8β+60β2)
4α3/2= 0.(25)
Thus the squashing parameter γ≡α/βlies in the range (0 ,4). We expect that many of
the squashed homogeneous spaces in seven dimensions are now solutions in this new gravity
theory, and we shall not enumerate them further.
4 Conclusions
This work is motivated by studying the classical solutions o f Einstein-Chern-Simons gravity
with asymptotic AdS structure. In seven dimensions, there a re two topological Chern-
Simons terms, and we obtain the full set of equations of motio n. We find that spherically-
symmetric solutions are unmodified by the inclusion of these topological terms. We also
obtain squashed S7andQ111spaces, where the squashing parameter is related to the cou-
pling constants of the topological terms. It is intriguing t o see that these known squashed
homogeneous spaces which appear to have no connection can no w be unified under our new
gravity theory.
As in three dimensions, our topological gravity should play an important role in explor-
ing the AdS 7/CFT6correspondence. The CFT 6that describes the world-volume theory of
multiple M5-branes is yet to be known, and our solutions prov ide many new gravity dual
backgrounds. The quantization condition for one of the coup ling constant suggests an un-
usual property of the CFT 6that is absent in lower dimensions. Additional future direc tions
include a classification of all topological gravities in (4 k+3) dimensions, investigating the
linearization of D= 7topological gravity and obtaining the propagating degre es of freedom.
Acknowledgement
We are grateful to Chris Pope for useful discussions. Y.P. is supported in part by the NSFC
grant No.1053060/A050207 and the NSFC group grant No.10821 504.
7References
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8arXiv:1001.0042v2  [hep-th]  16 Apr 2010Seven-Dimensional Gravity with Topological Terms
H. L¨ u†‡and Yi Pang⋆
†China Economics and Management Academy
Central University of Finance and Economics, Beijing 100081
‡Institute for Advanced Study, Shenzhen University, Nanhai A ve 3688, Shenzhen 518060
⋆Key Laboratory of Frontiers in Theoretical Physics
Institute of Theoretical Physics, Chinese Academy of Sciences , Beijing 100190
ABSTRACT
We construct new seven-dimensional gravity by adding two to pological terms to the
Einstein-Hilbert action. For certain choice of the couplin g constants, these terms may be
related to the R4correction to the 3-form field equation of eleven-dimension al supergravity.
We derive the full set of the equations of motion. We find that t he static spherically-
symmetric black holes are unmodified by the topological term s. We obtain squashed AdS 7,
and also squashed seven spheres and Q111spaces in Euclidean signature.1 Introduction
There has been considerable interest in topological gauge t heories [1] because of their wide
application in physics. The most studied example is the thre e-dimensional one. In addition
to the Einstein-Hilbert term, the theory has the Chern-Simo ns term, given by
S=1
µ/integraldisplay
d3xTr(dω∧ω+2
3ω∧ω∧ω), (1)
whereωcan be either a Yang-Mills gauge potential or the connection for gravity. Topo-
logical Yang-Mills theory can provide a fundamental interp retation for anyons [2]; it can
also generate Lorentz violation dynamically [3]. Topologi cally massive gravity [4] becomes
dynamical with a propagating massive particle, with the mas s proportional to the coupling
constantµ. Recently, a cosmological constant is added and the corresp onding boundary
conformal field theory (CFT) is discussed [5]. The three-dim ensional massive topological
gravity is conjectured to be unitary for certain parameter r egion even though the theory
has higher derivatives in time [6].
The attention on higher dimensional generalizations is con siderably less. The five di-
mensional Yang-Mills Chern-Simons term was discussed in [7 ], but there is no gravity coun-
terpart dueto the fact that the holonomy group SO(1,4) has no invariant rank-3 symmetric
tensor. In seven dimensions, Yang-Mills Chern-Simons term s arise naturally from N= 4
supergravity [8]. As in the case of three dimensions, we find t hat such terms in the grav-
ity sector can be obtained directly from those in the Yang-Mi lls sector by replacing the
gauge potential Ato the connection Γ. As we shall see later, these topological terms in
seven dimensions may be related to the anomaly cancelation t erms in eleven-dimensional
supergravity.
In section 2, we present the two topological terms in seven di mensions, and discuss their
properties. Sincethey arenot manifestly invariant underg eneral coordinate transformation,
we find it is more convenient to lift the system to eight dimens ions in order to derive the
equations of motion (EOMs). We obtain the full set. In sectio n 3, we construct large classes
of solutions. We find that the static spherically-symmetric black holes are unmodified by
the topological terms. This is analogous to three dimension s, where the BTZ black hole
remains to be a solution in topologically massive gravity. I n Euclidean signature, we obtain
squashedS7andQ111spaces. In particular, one of the squashed seven sphere can b e Wick
rotated to become squashed AdS 7. We conclude in section 4.
22 The theory
In seven dimensions, there are two topological terms; they a re given by
S1= ˜µ/integraldisplay
Ω(7)
1= ˜µ/integraldisplay
Tr(Γ∧Θ−1
3Γ3)∧Tr(Θ2) = ˜µ/integraldisplay
Ω(3)∧dΩ(3), (2)
S2= ˜ν/integraldisplay
Ω(7)
2= ˜ν/integraldisplay
Tr(Θ3∧Γ−2
5Θ2∧Γ3−1
5Θ∧Γ2∧Θ∧Γ+1
5Θ∧Γ5−1
35Γ7),
with Ω(3)= Tr(dΓ∧Γ+2
3Γ3). Here, Θ is the curvature 2-form, defined as Θ ≡dΓ+Γ∧Γ,
and ˜µ,˜νare two parameters of length dimension 5. (We rescale the tot al action by the
seven-dimensional Newton constant.) The 3-form Ω(3)has the same structure as the Chern-
Simons term in D= 3, except that now Γ depends on seven coordinates. Ω(7)
1and Ω(7)
2are
topological in the same sense as Ω(3)being topological in D= 3. We can lift the system to
D= 8, with the seven-dimensional spacetime as the boundary. T hen, we have
dΩ(7)
1=Y(8)
1≡Tr(Θ∧Θ)∧Tr(Θ∧Θ), dΩ(7)
2=Y(8)
2≡Tr(Θ∧Θ∧Θ∧Θ).(3)
As we have mentioned earlier, these terms can be derived from the Yang-Mills Chern-
Simons terms in [8] by changing the gauge potential to the con nection.1Note that the
Pontryagin term is proportional to Y(8)
1−2Y(8)
2, corresponding to ˜ ν=−2˜µ. In eleven-
dimensional supergravity, there is an R4correction to the field equation, namely d∗F(4)=
1
2F(4)∧F(4)+X(8), whereX(8)is given by
X(8)∝Y(8)
1−4Y(8)
2. (4)
Thus for ˜ν=−4˜µ, the topological terms can be obtained from the S4reduction of super-
gravity inD= 11, and the coupling constant is proportional to the 4-form M5-brane fluxes.
For large fluxes, this topological term dominates the higher -order corrections.
To derive the contribution to the EOMs from the Chern-Simons terms, it is necessary
to perform their variation with respect to the metric. These topological terms are not
manifestly invariant under the general coordinate transfo rmation, but Y(8)
1andY(8)
2are.
Wefindthataconvenient waytoderivethevariationistolift thesystemtoeightdimensions.
Let us first consider the variation of S1. In terms of coordinate components, we have
/integraldisplay
dΩ(7)
1=1
16/integraldisplay
d8xǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6Rµ4µ3ν7ν8. (5)
1In [8], the field strength 2-form is defined by F=dB+gB∧B, with gauge coupling g= 2. Then by
rescaling the field B→B/gandF→F/gand setting g= 2, one can obtain the same expressions as the
ones given here.
3Here we use Greek letters to denote the eight-dimensional co ordinates and Latin letters to
represent the seven-dimensional ones hereafter. We adopt t he convention ǫ12345678= 1.
For an infinitesimal variation of the metric δg, using the Bianchi identity and the fol-
lowing relation
δRµ
ναβ=δΓµ
νβ;α−δΓµ
να;β, (6)
we find that
/integraldisplay
dδΩ(7)
1=−1
2/integraldisplay
d8x√g/parenleftBig1√gǫν1ν2ν3ν4ν5ν6ν7ν8Rµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7/parenrightBig
;ν8
≡1
2/integraldisplay
d∗J, (7)
where “;” denotes a covariant derivative and ∗is the Hodge dual. For simplicity, we have
introduced a 1-form current J=Jαdxα. Its components are given by
Jα=1√gǫν1ν2ν3ν4ν5ν6ν7αRµ1µ2ν1ν2Rµ2µ1ν3ν4Rµ3µ4ν5ν6δΓµ4µ3ν7. (8)
Clearly, we have d∗J=−√gJα;αd8x, Thus we obtain
δΩ(7)
1=1
2∗J, (9)
up to a total derivative term. Now restricting the coordinat e indices to seven dimensions
only, we have
δS1= 4˜µ/integraldisplay
Tr(Θ∧Θ)∧Tr(Θ∧δΓ). (10)
The variation of S2can be obtained in the same manner, given by
δS2= 4˜ν/integraldisplay
Tr(Θ∧Θ∧Θ∧δΓ). (11)
Finally, we make use of the variation of the connection
δΓi
mj=1
2gin(δgnm;j+δgnj;m−δgml;n), (12)
and after integrating by parts, we obtain thecontributions to EOMs from the Chern-Simons
terms, given by
Cij
1=δS1√gδgij=µ
4√g[ǫij1j2j3j4j5j6(Ri1
i2j1j2Ri2
i1j3j4Rjk
j5j6);k+i↔j],
Cij
2=δS2√gδgij=ν
4√g[ǫij1j2j3j4j5j6(Rk
i1j1j2Ri1
i2j3j4Rji2
j5j6);k+i↔j].(13)
For the total action S, which is the sum of the Einstein-Hilbert action, cosmologi cal
constant Λ and S1+S2, the corresponding full set of EOMs is given by
Rij−1
2gijR+Λgij+Cij
1+Cij
2= 0. (14)
4It should be remarked that under a large gauge transformatio n Γ→ OΓO−1−dOO−1,
the action transforms as S→S+ ˜µv(O)+ ˜νw(O), where
v(O) =/integraldisplay
1
3d/parenleftBig
Tr(dOO−1)3∧Ω(3)/parenrightBig
;w(O) =1
35/integraldisplay
Tr(dOO−1)7.(15)
Thevterm is trivial and gives no restriction to the parameter ˜ µ, while the wterm should
be classified by the seventh homotopy group of SO(1,6)
π7[SO(1,6)]≃π7[SO(6)]≃Z. (16)
The invariance of eiSrequires that
64π4˜ν= 2πn, n = 0,±1,±2.... (17)
This result is completely different from that in three dimensi ons, where the SO(1,2) is
homotoplically trivial and the mass parameter is not quanti zed. Moreover, since ˜ νis quan-
tized,S2will not be renormalized in the quantum theory. This suggest s some intriguing
properties in the corresponding CFT dual.
3 Solutions
Spherically-symmetric solutions:
Having obtained the full set of EOMs for topological gravity in seven dimensions, we are
in the position to construct solutions. It is clear that the m aximally-symmetric space(time)
is unmodifiedby theinclusion of the topological terms. Then extsimplest case is to consider
the spherically-symmetric ansatz, given by
ds2=−F(r)dt2+dr2
G(r)+r2dΩ2
5. (18)
We find that for this ansatz, the contributions from the topol ogical terms Cij
1andCij
2
vanish identically. This implies that the previously-know n static (AdS) black holes, charged
or neutral, are still solutions when the topological terms a re added to the action. This is
analogous to three dimensions, where the BTZ black hole is st ill a solution in massive
topological gravity. However the thermodynamic quantitie s such as the mass and entropy
will acquire modifications [9, 10].
As we shall discuss presently, there also exist squashed AdS 7solutions.
S3bundle over S4:
5We now turn our attention to the Euclidean theory. In three di mensions, there exists a
large class of squashed S3or AdS 3[11]. We expect the same in seven dimensions. Without
loss of generality, we set Λ = 30 so that it can give rise to a uni t roundS7. We first consider
the squashed S7that can be viewed as an S3bundle over S4. The metric ansatz is given
by
ds2=α3/summationdisplay
i=1(σi−cos2(1
2θ)˜σi)2+β/parenleftBig
dθ2+1
4sin2θ3/summationdisplay
i=1˜σ2
i/parenrightBig
. (19)
whereσiand ˜σiare theSU(2) left-invariant 1-forms, satisfying dσi=1
2ǫijkσj∧σkand
d˜σi=1
2ǫijk˜σj∧˜σk. The metric is Einstein provided that either α=β=1
4orα=1
5β=9
100.
The first case corresponds to the round S7and the second is a squashed S7that is also
Einstein. Now with the contribution from the topological te rms, the EOMs can be reduced
to
2α2+4αβ(7β−2)−β2= 0, (20)
together with
√α(α−β)3(4(10α+β)˜µ−(55α+7β)ν)+2β6(20αβ−4α−β) = 0.(21)
It is clear from (20) that there exists one and only one positi veαfor any positive β. The
squashing parameter γ≡α/βlies in the range 0 <γ <2+3√
2. Note that when 2˜ µ= 3˜ν,
the squashed S7that is Eisntein remains Einstein.
S1bundle over CP3:
There is another way of squashing an S7, which can be viewed as an S1bundle over
CP3. This example can be generalized to Minkowskian signature t o give rise to squashed
AdS7[12]. The metric ansatz is given by
ds2=α(dτ+sin2θ(dψ+B))2+βds2
CP3,
ds2
CP3=dθ2+sin2θcos2θ(dψ+B)2+sin2θ/parenleftBig
d˜θ2+1
4sin2˜θcos2˜θσ2
3
+1
4sin2˜θ(σ2
1+σ2
2)/parenrightBig
,
B=1
2sin2˜θσ3. (22)
It is of a round S7whenα=β= 1. In general, the EOMs imply that
α=β(8−7β),8˜µ+ ˜ν+β3
10976(β−1)2√α= 0. (23)
The squashing parameter γ≡α/βlies in the range (0 ,8).
Squashed Q111spaces:
6TheQ111space is an Einstein-Sasaki space of U(1) bundle over S2×S2×S2. We
consider the following ansatz
ds2=α/parenleftBig
dψ+3/summationdisplay
i=1cosθidφi/parenrightBig2
+β3/summationdisplay
i=1(dθ2
i+sinθ2
idφ2
i). (24)
It is ofQ111provided that α=1
2β= 1/16, and it remains so for ˜ ν= 0. In general, we have
α= 4β(1−7β),8(α−β)(2α−β)˜µ+α(2α−3β)˜ν+β5(α−8β+60β2)
4α3/2= 0.(25)
Thus the squashing parameter γ≡α/βlies in the range (0 ,4). We expect that many of
the squashed homogeneous spaces in seven dimensions are now solutions in this new gravity
theory, and we shall not enumerate them further.
4 Conclusions
This work is motivated by studying the classical solutions o f Einstein-Chern-Simons gravity
with asymptotic AdS structure. In seven dimensions, there a re two topological Chern-
Simons terms, and we obtain the full set of equations of motio n. We find that spherically-
symmetric solutions are unmodified by the inclusion of these topological terms. We also
obtain squashed AdS 7, and squashed S7andQ111spaces in Euclidean signature, where
the squashing parameter is related to the coupling constant s of the topological terms. It is
intriguing to see that these known squashed homogeneous spa ces which appear to have no
connection can now be unified under our new gravity theory.
As in three dimensions, our topological gravity should play an important role in explor-
ing the AdS 7/CFT6correspondence. The CFT 6that describes the world-volume theory of
multiple M5-branes is yet to be known, and our solutions prov ide many new gravity dual
backgrounds. The quantization condition for one of the coup ling constant suggests an un-
usual property of the CFT 6that is absent in lower dimensions. Additional future direc tions
include a classification of all topological gravities in (4 k+3) dimensions, investigating the
linearization of D= 7topological gravity and obtaining the propagating degre es of freedom.
Acknowledgement
We are grateful to Chris Pope for useful discussions. Y.P. is supported in part by the NSFC
grant No.1053060/A050207 and the NSFC group grant No.10821 504.
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