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

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.

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

Holographic renormalization (d = 3)

[SdH, Skenderis, Solodukhin CMP 217(2001)595]

ds2 =ℓ2

r2

(

dr2 + gij(r, x) dxidxj)

gij(r, x) = g(0)ij(x) + r2g(2)ij(x) + r3g(3)ij(x) + . . .(1)

Solve eom and renormalize the action:

S = − 1

2κ2

∫

Mǫd4x

√g (R[g] − 2Λ)

− 1

2κ2

∫

∂Mǫd3x

√γ

(

K − 4

ℓ− ℓR[γ]

)

Z[g(0)] = eW [g(0)] = eSon-shell[g(0)]

⇒ 〈Tij(x)〉 =2

√g(0)δSon-shell

δgij(0)

=3ℓ2

16πGNg(3)ij(x) (2)

2

Matter

Smatter =1

2

∫

Mǫd4x

√g

(

(∂µφ)2 +

1

6Rφ2 + λφ4

)

+1

2

∫

∂Mǫd3x

√γ φ2(x, ǫ) (3)

φ(r, x) = r φ(0)(x) + r2φ(1)(x) + . . .

Son-shell[φ(0)] = W [φ(0)]

〈O∆=2(x)〉 = − 1√g(0)

δSon-shell

δφ(0)

= −φ(1)(x) (4)

3

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

Φ(r, ~x) =1

π2

∫

d3~yr

(r2 + (~x− ~y)2)2Φ0(~y) + O(λ)

= Φ0(~x) +r

π2

∫

d3~y1

(~x− ~y)4Φ0(~y) + . . .(5)

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

⇒ 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)

Extremize w.r.t. φ0 ⇒ δWδφ0

− φ1 = 0 ⇒ φ0 = φ0[φ1]

Dual generating functional obtained by evaluating Wat the extremum:

W [φ1] = W[φ0[φ1], φ1] = W [φ0]| −∫

d3x√g0 φ0φ1|

= Γeff[O∆+]

〈O∆−〉J =δW [φ1]

δφ1= −φ0 (7)

Generating fctnl CFT2 ↔ effective action CFT1

(φ1 fixed) (φ0 fixed)

����

����

����

����

����

spin

dimension

20

1

2

3

1

Deformation

Double−trace Dualization and "double−trace" deformations

Weyl−equivalence of UIR of O(4,1)

= s+1∆Unitarity bound

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

For spin 2, the duality conjecture should relate:

gij ↔ 〈Tij〉 (8)

Problems:

1) Remember holographic renormalization:

gij(r, x) = g(0)ij(x) + . . .+ r3g(3)ij(x)

〈Tij(x)〉 =3ℓ2

16πGNg(3)ij(x) (9)

Is this a normalizable mode? Duality can only inter-

change them if both modes are normalizable.

6

2) gij is not an operator in a CFT. We can compute

〈TijTkl . . .〉 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.

Problems 2)-3): a similar issue arises in the spin-1

case [SdH, Gao (2007)]: (Ai, Ji). Solution:

(Ai, Ji) ↔ (A′i, J

′i)

(B,E) ↔ (B′, E′)

J ′i = ǫijk∂jAk

Ji = ǫijk∂jA′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.

The Holographic Cotton Tensor

Cij =1

2ǫiklDk

(

Rjl −1

4gjlR

)

. (11)

• 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.

SCS = −1

4

∫

Tr

(

ω ∧ dω+2

3ω ∧ ω ∧ ω

)

δSCS = −1

2

∫

Tr (δω ∧R) = −1

2

∫

ǫijkRijlmδΓlkm

=∫

Cijδgij (12)

7

• Given a metric gij = δij + hij, we may construct a

Cotton tensor (hij = Πijklhkl):

Cij =1

2ǫikl∂k�hjl . (13)

• Given a stress-energy tensor 〈Tij〉, there is always

an hij such that:

〈Tij〉 = Cij[h]

�3hij = 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)

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:

Cµναβ =1

2ǫµν

γδCγδαβ

8

•This equation can be solved asymptotically. In the

Fefferman-Graham coordinate system:

ds2 =ℓ2

r2

(

dr2 + gij(r, x) dxidxj)

where

gij(r, x) = g(0)ij(x) + r2g(2)ij(x) + r3g(3)ij(x) + . . .

We find

g(2)ij = −Rij[g(0)] +1

4g(0)ij R[g(0)]

g(3)ij = −2

3ǫ(0)i

kl∇(0)kg(2)jl =2

3C(0)ij

• The holographic stress tensor is 〈Tij〉 = 3ℓ2

16πGNg(3)ij.

We find that for any self-dual g(0)ij the holographic

stress tensor is given by the Cotton tensor:

〈Tij〉 =ℓ2

8πGNC(0)ij

• We can integrate the stress-tensor to obtain the

boundary generating functional using the definition:

〈Tij〉g(0)=

2√g

δW

δgij(0)

The boundary generating functional is the Chern-

Simons gravity action and we find its coefficient:

k =ℓ2

8GN=

(2N)3/2

24

For linearized Euclidean solutions, there is a regularity

condition:

h(3)ij(p) =1

3p3h(0)ij(p) (15)

⇒ the linearized SD equation becomes

�1/2h(0)ij = ǫikl∂kh(0)jl . (16)

This is the t.t. part of topologically massive gravity

(µ = �1/2):

Cij = µ Rij (17)

9

General solution (p∗i := (−p0, ~p); p∗i = Πijp∗j):

hij = γ Eij +ψ

pǫiklpkEjl

Eij =p∗i p

∗j

p∗2− 1

2Πij (18)

For (anti-) instantons, γ = ±ψ:

hij(r, p) = γ(r, p)

p∗i p∗j

p∗2− 1

2Πij ±

i

2p3(p∗i ǫjkl + p∗jǫikl)pkp

∗l

Son-shell =3ℓ2

16κ2

∫

d3p |p|3 (γ(p)γ(−p) + εγ(p)γ(−p))

=3ℓ2

8κ2

∫

d3p |p|3γ(p)γ(−p) . (19)

Duality symmetry of the equations of motion

Solution of bulk eom:

hij[a, b] = aij(p) (+cos(|p|r) + |p|r sin(|p|r))+ bij(p) (− sin(|p|r) + |p|r cos(|p|r))(20)

bij(p) := 1|p|3 Cij(a) → hij[a, a]

Define:

Pij := − 1

r2h′ij +

|p|2rhij − |p|2h′ij

〈Tij(x)〉r = − ℓ2

2κ2Pij(r, x) −

ℓ2

2κ2|p|2h′ij(r, x)

Pij[a, b] = −|p|3hij[−b, a] . (21)

10

This leads to:

2Cij(h[−a, a]) = −|p|3Pij[a, a]2Cij(P [−a, a]) = +|p|3hij[a, a] (22)

The S-duality operation is S = ds, d = 2C/p3, s(a) =

−b, s(b) = a:

S(h(0)) = −h(0))

S(h(0)) = +h(0) (23)

We can define electric and magnetic variables

Eij(r, x) = − ℓ2

2κ2Pij(r, x)

Bij(r, x) = +ℓ2

κ2Cij[h(r, x)] (24)

Eij(0, x) = 〈Tij(x)〉

Bij(0, x) =ℓ2

κ2Cij[h(0)] (25)

S(E) = +BS(B) = −E (26)

Gravitational S-duality interchanges the renor-

malized stress-energy tensor 〈Tij〉 = Cij[h] with

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?:

Cij[h] =δW [h]

δhij= 〈Tij〉 (27)

Gravitational Legendre transform

Construct the Legendre transform in the usual way:

W[g, g] = W [g] + V [g, g] (28)

δWδgij

= 0 ⇒ 1√g

δV

δgij= −1

2〈Tij〉 (29)

at the extremum. W [g] is defined as: W [g] := W[g, g]|.Linearize and dualize 〈Tij〉 = ℓ2

κ2 Cij[h] then

V [h, h] = − ℓ2

2κ2

∫

d3xhijCij[h] (30)

11

This is the quadratic part of the gravitational Chern-

Simons action:

V [h, h] = − ℓ2

2κ2

∫

d3xhijδ2SCS[g]

δgijδgklhkl (31)

We find:

W [h] = − ℓ2

8κ2

∫

d3x h(0)ij�3/2h(0)ij

〈Tij〉 =ℓ2

κ2Cij[h] (32)

Given that the relation between the generating func-

tionals is a Legendre transform, and since duality re-

lates (Cij[h], 〈Tij〉) = (〈Tij〉, Cij[h]), we may identify

the generating functional of one theory with the ef-

fective action of the other.

Bulk interpretation

Z[g] =∫

gDGµν e−S[G] (33)

Linearize, couple to a Chern-Simons term and inte-

grate:

∫

Dhij Z[h] eSCS[h,h] =∫

Dhij eW[h,h] ≃ eW [h]

= Z[h] (34)

Thus, the gravitational Chern-Simons term switches

between Dirichlet and Neumann boundary conditions.

12

Mixed boundary conditions

Can we fix the following:

Jij(x) = hij(x) + λ hij(x) (35)

This is possible via W[h, J]. For regular solutions:

Jij = hij +2λ

�3/2Cij[h] (36)

This b.c. determines hij up to zero-modes:

h0ij +

2λ

�3/2Cij[h

0] = 0 . (37)

This is the SD condition found earlier. Its only solu-

tions are for λ = ±1.

λ 6= ±1 We find:

〈Tij〉J = − ℓ2

2κ2(1 − λ2)�

3/2(

Jij −1

λdij[J]

)

. (38)

λ = ±1 In this case J has to be self-dual. We have

h = h0 + 12 J and

〈Tij〉J=0 = ± ℓ2

κ2Cij[h

0] = − ℓ2

2κ2�

3/2h0ij

〈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

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

〈Tij〉 =ℓ2

κ2Cij[g] (40)

Perturbatively, we can always solve for g given 〈Tij〉(up to zero-modes):

ǫiklDk〈Tlj〉 = −�Rij + O(D4) (41)

14

So the concept of dual graviton makes sense. We

can also take a non-linear boundary condition:

ℓ2

2κ2Cij[g] = µ

(

Rij[g] −1

2gij R[g] + λ gij

)

− Cij[g]

(42)

by modifying the action with boundary terms:

S = SEH +µℓ2

4κ2

∫

d3x√g (R[g] − 2λ) − ℓ2

4κ2SCS (43)

i.e. effectively coupling the CFT to gravity (or con-

formal gravity).

The role of the EH term here is to provide the CS

coupling between g and g:

δSEH =ℓ2

2κ2

∫

d3xCij[g]δgij (44)

as in the linearized case.

The bulk produces a conformal, Lorentz invariant

non-linear coupling between the two gravitons.

15

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

• 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.

• 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?