École Normale Supérieure, Paris 2008

1. Dual Gravitons in AdS4/CFT3 and the
Holographic Cotton Tensor
Sebastian de Haro
Utrecht University and Foundations of Physics
Paris, October 9, 2008
Based on S. de Haro, arXiv:0808.2054

2. Ongoing work with A. Petkou et al.:
• SdH and A. Petkou, arXiv:0710.0965,
J.Phys.Conf.Ser. 110 (2008) 102003.
• SdH and A. Petkou, hep-th/0606276, JHEP 12
(2006) 76.
• SdH, I. Papadimitriou and A. Petkou, hep-th/0611315,
PRL 98 (2007) 231601.
• SdH and Peng Gao, hep-th/0701144,
Phys. Rev. D76(2007) 106008.

3. Motivation
1. Holography – usual paradigm gets some modifi-
cations in AdS4.
2. Dualities [Leigh, Petkou (2004); SdH, Petkou
(2006)]. Higher spins.
3. AdS4/CFT3:
• 11d sugra/M-theory.
• Condensed matter.
• Relation to the GBL theory.
4. Instantons: new vacua, instabilities [SdH, Pa-
padimitriou, Petkou, PRL 98 (2007)].
1

4. Holographic renormalization (d = 3)
[SdH, Skenderis, Solodukhin CMP 217(2001)595]
ℓ2
ds2 = 2 dr 2 + gij (r, x) dxidxj
r
gij (r, x) = g(0)ij (x) + r 2g(2)ij (x) + r 3g(3)ij (x) + (1)
...
Solve eom and renormalize the action:
1 4 √
S = − 2 d x g (R[g] − 2Λ)
2κ Mǫ
1 3 √ 4
− 2 d x γ K − − ℓ R[γ]
2κ ∂Mǫ ℓ
Z[g(0)] = eW [g(0)] = eSon-shell[g(0)]
2 δSon-shell 3ℓ2
⇒ Tij (x) = √ ij = g(3)ij (x) (2)
g(0) δg 16πGN
(0)
2

5. Matter
1 4x√g (∂ φ)2 + 1 Rφ2 + λφ4
Smatter = d µ
2 Mǫ 6
1 3x√γ φ2(x, ǫ)
+ d (3)
2 ∂Mǫ
φ(r, x) = r φ(0)(x) + r 2φ(1)(x) + . . .
Son-shell[φ(0)] = W [φ(0)]
1 δSon-shell
O∆=2(x) = − √ = −φ(1)(x) (4)
g(0) δφ(0)
3

6. The relation between φ(1) and φ(0) is given by regu-
larity of the Euclidean solution. Define
φ(r, x) = r/ℓΦ(r, x), then
1 r
Φ(r, x) = 2 d3 y2 + (x − y)2)2 0
Φ (y) + O(λ)
π (r
r 3 1
= Φ0(x) + 2 d y Φ (y) + . . . (5)
4 0
π (x − y)

7. Boundary conditions
In the usual holographic dictionary,
φ(0)=non-normaliz. ⇒ fixed b.c. ⇒ φ(0)(x) = J(x)
φ(1)=normalizable ⇒ part of bulk Hilbert space
⇒ choose boundary state ⇒ O∆=2 = −φ(1)
⇒ Dirichlet quantization
In the range of masses d2
−4 < m2 < d2
−4 + 1, both
modes are normalizable [Avis, Isham (1978); Breit-
enlohner, Freedman (1982)]
4

8. ⇒ Neumann/mixed boundary conditions are possible
˜
φ(1) =fixed= J(x)
φ(0) ∼ O∆′ , ∆′ = d − ∆
Dual CFT [Klebanov, Witten (1998); Witten; Leigh,
Petkou (2003)]
These can be obtained by a Legendre transformation:
W[φ0, φ1] = W [φ0] − d3x g(0) φ0(x)φ1(x) . (6)
δW
Extremize w.r.t. φ0 ⇒ δφ − φ1 = 0 ⇒ φ0 = φ0[φ1]
0

9. Dual generating functional obtained by evaluating W
at the extremum:
˜ 3 √
W [φ1] = W[φ0[φ1], φ1] = W [φ0]| − d x g0 φ0φ1|
= Γeff[O∆+ ]
˜
δ W [φ1]
O∆− J =
˜
δφ1
= −φ0 (7)
Generating fctnl CFT2 ↔ effective action CFT1
(φ1 fixed) (φ0 fixed)

10. dimension
Weyl−equivalence of UIR of O(4,1)
1
0 Unitarity bound
1
0 ∆ = s+1
1
0
3 1
0
1
0
Double−trace 2 1
0 Dualization and "double−trace" deformations
1
0
Deformation 10
1
0 1 2 spin
Duality conjecture [Leigh, Petkou 0304217; SdH,
Petkou 0606276; SdH, Gao 0701144]
• Instantons describe the self-dual point of duality
• Typically, the dual effective action is “topological”
5

11. For spin 2, the duality conjecture should relate:
gij ↔ Tij (8)
Problems:
1) Remember holographic renormalization:
gij (r, x) = g(0)ij (x) + . . . + r 3g(3)ij (x)
3ℓ2
Tij (x) = g(3)ij (x) (9)
16πGN
Is this a normalizable mode? Duality can only inter-
change them if both modes are normalizable.
6

12. 2) gij is not an operator in a CFT. We can compute
Tij Tkl . . . but gij is fixed.
3) Even if we were to couple the CFT to gravity, gij
wouldn’t make sense.
Question 1) has been answered in the affirmative by
Ishibashi and Wald 0402184.
Recently, Compare and Marolf have generalized this
result 0805.1902.

13. Problems 2)-3): a similar issue arises in the spin-1
case [SdH, Gao (2007)]: (Ai, Ji). Solution:
(Ai, Ji) ↔ (A′ , Ji )
i
′
(B, E) ↔ (B ′, E ′)
′
Ji = ǫijk ∂j Ak
Ji = ǫijk ∂j A′
k (10)
Proposal: Keep the metric fixed. Look for an op-
erator which, given a linearized metric, produces a
stress tensor. In 3d there is a natural candidate: the
Cotton tensor.

14. The Holographic Cotton Tensor
1 kl 1
Cij = ǫi Dk Rjl − gjl R . (11)
2 4
• Dimension 3.
• Symmetric, traceless and conserved.
• Conformal flatness ⇔ Cij = 0 (Cijkl ≡ 0 in 3d).
• It is the stress-energy tensor of the gravitational
Chern-Simons action.
1 2
SCS = − Tr ω ∧ dω + ω ∧ ω ∧ ω
4 3
1 1
δSCS = − Tr (δω ∧ R) = − ǫijk RijlmδΓl
km
2 2
= Cij δg ij (12)
7

15. • Given a metric gij = δij + hij , we may construct a
Cotton tensor (¯ij = Πijkl hkl ):
h
1
Cij = ǫikl ∂k ¯jl .
h (13)
2
• Given a stress-energy tensor Tij , there is always
an ˜ij such that:
h
Tij = Cij [˜]
h
3˜
hij = 4Cij ( T ) . (14)
• Consideration of the pair (Cij , Tij ) is also mo-
tivated by grativational instantons [SdH, Petkou
0710.0965] (related work by Julia, Levie, Ray 0507262
in de Sitter)

16. Gravitational instantons
• Instanton solutions with Λ = 0 have self-dual Rie-
mann tensor. However, self-duality of the Riemann
tensor implies Rµν = 0.
• In spaces with a cosmological constant we need
to choose a different self-duality condition. It turns
out that self-duality of the Weyl tensor is compatible
with a non-zero cosmological constant:
1
Cµναβ = ǫµν γδ Cγδαβ
2
8

17. •This equation can be solved asymptotically. In the
Fefferman-Graham coordinate system:
ℓ2
ds2 = 2 dr 2 + gij (r, x) dxidxj
r
where
gij (r, x) = g(0)ij (x) + r 2g(2)ij (x) + r 3g(3)ij (x) + . . .
We find
1
g(2)ij = −Rij [g(0)] + g(0)ij R[g(0)]
4
2 kl ∇ 2
g(3)ij = − ǫ(0)i (0)k g(2)jl = C(0)ij
3 3

18. • The holographic stress tensor is Tij = 3ℓ2 g
16πGN (3)ij .
We find that for any self-dual g(0)ij the holographic
stress tensor is given by the Cotton tensor:
ℓ2
Tij = C(0)ij
8πGN
• We can integrate the stress-tensor to obtain the
boundary generating functional using the definition:
2 δW
Tij g(0) =√
g δg ij
(0)
The boundary generating functional is the Chern-
Simons gravity action and we find its coefficient:
ℓ2 (2N )3/2
k= =
8GN 24

19. For linearized Euclidean solutions, there is a regularity
condition:
1 3
h(3)ij (p) = p h(0)ij (p) (15)
3
⇒ the linearized SD equation becomes
1/2¯
h(0)ij = ǫikl ∂k¯(0)jl .
h (16)
This is the t.t. part of topologically massive gravity
(µ = 1/2):
¯
Cij = µ Rij (17)
9

20. General solution (p∗ := (−p0, p); p∗ = Πij p∗):
i ¯i j
ψ
¯ij = γ Eij +
h ǫikl pk Ejl
p
p∗p∗
¯i ¯j 1
Eij = − Πij (18)
p∗2
¯ 2
For (anti-) instantons, γ = ±ψ:
 ∗ ∗
¯¯
pi pj

1 i
¯ij (r, p) =
h γ(r, p)  ∗2
− Πij ± 3 (¯∗ǫjkl + p∗ǫikl )pk p∗
pi ¯j ¯l
¯
p 2 2p
3ℓ2
Son-shell = d3p |p|3 (γ(p)γ(−p) + εγ(p)γ(−p))
16κ2
3ℓ2
= 2
d3p |p|3γ(p)γ(−p) . (19)
8κ

21. Duality symmetry of the equations of motion
Solution of bulk eom:
¯ij [a, b] = aij (p) (+ cos(|p|r) + |p|r sin(|p|r))
h
+ bij (p) (− sin(|p|r) + |p|r cos(|p|r))(20)
1
bij (p) := |p|3 Cij (˜) → ¯ij [a, ˜]
a h a
Define:
1 ′ |p|2
Pij := − 2 ¯ij +
h ¯ij − |p|2¯′
h hij
r r
ℓ2 ℓ2
Tij (x) r = − 2 Pij (r, x) − 2
|p|2¯′ (r, x)
hij
2κ 2κ
Pij [a, b] = −|p|3¯ij [−b, a] .
h (21)
10

22. This leads to:
2Cij (¯[−˜, a]) = −|p|3Pij [a, ˜]
h a a
2Cij (P [−˜, a]) = +|p|3¯ij [a, ˜]
a h a (22)
The S-duality operation is S = ds, d = 2C/p3, s(a) =
−b, s(b) = a:
S(¯(0)) = −˜(0))
h h
S(˜(0)) = +h(0)
h (23)
We can define electric and magnetic variables
ℓ2
Eij (r, x) = − 2 Pij (r, x)
2κ
ℓ2
Bij (r, x) = + 2 Cij [˜(r, x)]
h (24)
κ

23. Eij (0, x) = Tij (x)
ℓ2
Bij (0, x) = 2 Cij [¯(0)]
h (25)
κ
S(E) = +B
S(B) = −E (26)
Gravitational S-duality interchanges the renor-
malized stress-energy tensor Tij = Cij [˜] with
h
the Cotton tensor Cij [h] at radius r. Can Cij [h]
be interpreted as the stress tensor of some CFT2?
I.e. does the following hold?:
δ W [˜]
˜ h
Cij [h] = ˜
= Tij (27)
δ˜ij
h

24. Gravitational Legendre transform
Construct the Legendre transform in the usual way:
W[g, ˜] = W [g] + V [g, ˜]
g g (28)
δW 1 δV 1
ij
=0⇒√ ij
= − Tij (29)
δg g δg 2
˜ g ˜ g
at the extremum. W [˜] is defined as: W [˜] := W[g, ˜]|.
g
Linearize and dualize Tij = ℓ2 C [˜]
h then
κ2 ij
ℓ2
V [h, ˜] = − 2
h d3x hij Cij [˜]
h (30)
2κ
11

25. This is the quadratic part of the gravitational Chern-
Simons action:
ℓ2 δ 2SCS [g] kl
V [h, ˜] = − 2
h d3x hij ij δg kl
˜
h (31)
2κ δg
We find:
ℓ2
W [˜] = − 2 d3x ˜(0)ij 3/2˜(0)ij
˜ h h h
8κ
ℓ2
˜
Tij = 2 Cij [h] (32)
κ
Given that the relation between the generating func-
tionals is a Legendre transform, and since duality re-
lates (Cij [h], Tij ) = ( Tij , Cij [˜]), we may identify
˜ h
the generating functional of one theory with the ef-
fective action of the other.

26. Bulk interpretation
Z[g] = DGµν e−S[G] (33)
g
Linearize, couple to a Chern-Simons term and inte-
grate:
SCS [h,˜]
h W[h,˜]
h W [˜]
˜ h
Dhij Z[h] e = Dhij e ≃e
= Z[˜]
˜h (34)
Thus, the gravitational Chern-Simons term switches
between Dirichlet and Neumann boundary conditions.
12

27. Mixed boundary conditions
Can we fix the following:
Jij (x) = hij (x) + λ ˜ij (x)
h (35)
This is possible via W[h, J]. For regular solutions:
2λ
Jij = hij + 3/2
Cij [h] (36)
This b.c. determines hij up to zero-modes:
2λ
h0
ij + 3/2
Cij [h0] = 0 . (37)
This is the SD condition found earlier. Its only solu-
tions are for λ = ±1.

28. λ = ±1 We find:
ℓ2 3/2 1
˜
Tij J = − 2 Jij − dij [J] . (38)
2κ (1 − λ2) λ
λ = ±1 In this case J has to be self-dual. We have
h = h0 + 1 J and
2
ℓ2 ℓ2
Tij J=0 = ± 2 Cij [h0] = − 2
˜ 3/2 0
hij
κ 2κ
Tij h = 0 . (39)
The stress-energy tensor of CFT2 is traceless and
conserved but non-zero even if J = 0. It is zero if
the metric is conformally flat.
13

29. Non-linear dual graviton
At the non-linear level, we do not know the general
regularity condition. However, we can still define a
non-linear Cotton tensor
ℓ2
Tij = 2 Cij [˜]
g (40)
κ
Perturbatively, we can always solve for ˜ given Tij
g
(up to zero-modes):
˜ikl Dk Tlj = − ˜ Rij + O(D 4)
ǫ ˜ ˜ (41)
14

30. So the concept of dual graviton makes sense. We
can also take a non-linear boundary condition:
ℓ2 1
C [˜] = µ Rij [g] − gij R[g] + λ gij − Cij [g]
2 ij
g
2κ 2
(42)
by modifying the action with boundary terms:
µℓ2 √ ℓ2
S = SEH + d3x g (R[g] − 2λ) − S
2 CS
(43)
4κ2 4κ
i.e. effectively coupling the CFT to gravity (or con-
formal gravity).

31. The role of the EH term here is to provide the CS
coupling between g and ˜:
g
ℓ2
δSEH = d3x Cij [˜]δg ij
g (44)
2κ2
as in the linearized case.
The bulk produces a conformal, Lorentz invariant
non-linear coupling between the two gravitons.
15

32. Conclusions
• The variables involved in gravitational duality in
AdS are (Cij (r, x), Tij (x) r ).
• Duality interchanges Neumann and Dirichlet bound-
ary conditions and acts as a Legendre transform.
• Associated with the dual variables are a dual gravi-
ton and a dual stress-energy tensor which may be
interpreted as a dual CFT2:
˜
Cij [g] = Tij , Tij = Cij [˜].
g
16

33. • The self-dual point corresponds to bulk gravita-
tional instantons.
• The existence of a dual graviton persists at the
non-linear level.
• The two gravitons have different parity.
• The graviton can become dynamical on the bound-
ary by the boundary conditions. This amounts to
coupling the CFT to Cotton gravity or topologically
massive gravity.

34. • In some cases, the coupling between both gravitons
˜
spontaneously generates a non-zero vev for Tij J=0.
Can this be understood as an anomaly in field theory?
• Condensed matter applications. See 0809.4852 by
I. Bakas (duality in AdS BH background).
• AdS4 → AdS2 reduction?