Dual Gravitons in AdS4/CFT3 and the

Holographic Cotton Tensor

Sebastian de Haro

Utrecht University

ESI, April 22, 2009

Based on JHEP 0901 (2009) 042

and work with P. Gao, I. Papadimitriou, A. Petkou

Motivation

Holography in 4d

• Usual paradigm gets some modifications in AdS4.

• Existence of dualities.

• 11d sugra/M-theory.

• BGL theory.

3d motivation

• Cotton tensor plays a special holographic role.

1

Outline

• Review of holographic renormalization formulas

• Self-dual metrics in AdS4

• Boundary conditions

• Duality and the holographic Cotton tensor

• Conclusions

2

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) + . . .

Rµν = Λ gµν (1)

Solving eom gives: g(0), g(3) are undetermined and

g(2)ij = −Rij[g(0)] +1

4g(0)ij R[g(0)] (2)

Higher g(n)’s: g(n) = g(n)[g(0), g(3)].

3

To obtain the partition function, regularize and renor-

malize the action:

S = Sbulk + SGH + Sct

= − 1

2κ2

∫

Mǫd4x

√g (R[g] − 2Λ)

− 1

2κ2

∫

∂Mǫd3x

√γ

(

K − 4

ℓ− ℓR[γ]

)

(3)

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

⇒ 〈Tij(x)〉 =2

√g(0)δSon-shell

δgij(0)

=3ℓ2

16πGNg(3)ij(x) (4)

4

Matter

Smatter =1

2

∫

Mǫd4x

√g

(

(∂µφ)2 +

1

6Rφ2 + λφ4

)

+1

2

∫

∂Mǫd3x

√γ φ2(x, ǫ) (5)

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) (6)

5

Scalar instantons

Sbulk =1

2

∫

d4x√g

−R+ 2Λ

8πGN+ (∂φ)2 +

1

6Rφ2 + λφ4

• Euclidean

• Metric asymptotically AdS4 × S7

λ = 8πGN6ℓ2

for 11d embedding

Equations of motion: �φ− 16Rφ− 2λφ3 = 0

Seek solutions (“instantons”) with

Tµν = 0 ⇒ ds2 = ℓ2

r2

(

dr2 + d~x2)

6

Unique solution: φ = 2ℓ√

|λ|br

−sgn(λ)b2+(r+a)2+(~x−~x0)2• Solution is regular everywhere provided a > b ≥ 0.

• a/b labels different boundary conditions.

The boundary effective action can be computed holo-

graphically in a derivative expansion [SdH,Papadimitriou,

Petkou PRL 98(2007); Papadimitriou JHEP 0705:075]:

Γeff[ϕ] =1√λ

∫

d3x[(∂ϕ)2+1

2R[g(0)]ϕ

2+2√λ(

√λ−α)ϕ6]

This agrees with the toy-model action used by [Her-

tog,Horowitz JHEP 0504:005] and with [SdH,Petkou

JHEP 0612:076]. See also [Elitzur, Giveon, Porrati,

Rabinovici JHEP 0602:006].

7

Self-dual metrics

• 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:

Cµναβ =1

2ǫµν

γδCγδαβ

is compatible with Einstein’s equations with a nega-

tive cosmological constant and Euclidean signature.8

This is summarized in the following tensor [Julia,

Levie, Ray ’05]:

Zµναβ = Rµναβ +1

ℓ2

(

gµαgνβ − gµβgνα)

(7)

Z is the on-shell Weyl tensor and Zµρνρ = 0 gives

Einstein’s equations.

• The coupled equations may be solved asymptoti-

cally. 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) + . . .

9

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 term on a fixed background g(0)ij.

We find its coefficient:

k =ℓ2

8GN=

(2N)3/2

24

This holds at the non-linear level.

We now impose regularity of the Euclidean solutions.

At the linearized level, the regularity condition is:

h(3)ij(p) =1

3|p|3h(0)ij(p) . (8)

The full r-dependence of the metric fluctuations is

now:

hij(r, p) = e−|p|r(1 + |p|r) h(0)ij(p) (9)

h(0)ij is not arbitrary but satisfies:

�3/2h(0)ij = ǫikl∂k�h(0)jl . (10)

This is the t.t. part of the linearization of:

�1/2 Rij = Cij (11)

10

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

hij(p, r) = γ(p, r)Eij +1

pψ(p, r) ǫiklpkEjl

Eij =p∗i p

∗j

p∗2− 1

2Πij (12)

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

8κ2

∫

d3p |p|3|γ(p)|2 . (13)

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)]

11

⇒ Neumann/mixed boundary conditions are possible

φ(1) =fixed= J(x)

φ(0) ∼ 〈O∆′〉 , ∆′ = d− ∆

Dual CFT [Klebanov, Witten (1998); Witten; Leigh,

Petkou (2003)]

They are related by a Legendre transformation:

W[φ0, φ1] = W [φ0] −∫

d3x√

g(0) φ0(x)φ1(x) . (14)

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 (15)

Generating fctnl CFT2 ↔ effective action CFT1

(φ1 fixed) (φ0 fixed)

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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]

12

For spin 2, the duality conjecture should relate:

g(0)ij ↔ 〈Tij〉 (16)

Problems:

1) Remember holographic renormalization:

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

〈Tij(x)〉 =3ℓ2

16πGNg(3)ij(x) (17)

Is this a normalizable mode? Duality can only inter-

change them if both modes are normalizable.

13

2) g(0)ij is not an operator in a CFT. We can com-

pute 〈TijTkl . . .〉 but g(0)ij is fixed. Also, the metric

and the stress-energy tensor have different dimen-

sions.

Question 1) has been answered in the affirmative by

Ishibashi and Wald 0402184.

Recently, Compare and Marolf have generalized this

result 0805.1902. Both modes of the graviton are

normalizable.

14

Problem 2): similar issues arise in the spin-1 case

where duality interchanges a dimension 1 source Ai

and a dimension 2 current Ji. The solution in that

case was to construct a new source A′i and a new

current J ′i. This way, the gauge field is always fixed.

Duality now acts as:

(Ai, Ji) ↔ (A′i, J

′i) , (B,E) ↔ (B′, E′)

J ′i = ǫijk∂jAk , Ji = ǫijk∂jA′k (18)

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.15

The Holographic Cotton Tensor

Cij =1

2ǫiklDk

(

Rjl −1

4gjlR

)

. (19)

• 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.

16

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

Cotton tensor (hij = Πijklhkl):

Cij =1

2ǫikl∂k�hjl . (20)

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

an associated dual metric hij such that:

〈Tij〉 = Cij[h]

�3hij = 4Cij(〈T 〉) . (21)

• Consideration of (Cij, 〈Tij〉) is also motivated by

grativational instantons [SdH, Petkou 0710.0965].

Question: Is there a related symmetry of the eom?

17

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))(22)

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] . (23)

18

This leads to:

2Cij(h[−a, a]) = −|p|3Pij[a, a]2Cij(P [−a, a]) = +|p|3hij[a, a] (24)

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) (25)

We can define electric and magnetic variables

Eij(r, x) = − ℓ2

2κ2Pij(r, x)

Bij(r, x) = +ℓ2

κ2Cij[h(r, x)] (26)

Eij(0, x) = 〈Tij(x)〉

Bij(0, x) =ℓ2

κ2Cij[h(0)] (27)

S(E) = +B

S(B) = −E (28)

Gravitational S-duality interchanges the renor-

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

the Cotton tensor Cij[h] at radius r. [SdH, JHEP

0901 (2009) 042]

Can Cij[h(0)] be interpreted as the stress tensor of

some CFT2?

Cij[h(0)] = 〈Tij〉 =δW [h(0)]

δhij(0)

(29)

W [h] can be computed from the Legendre transfor-

mation:

W[g, g] = W [g] + V [g, g] (30)

δWδgij

= 0 ⇒ 1√g

δV

δgij= −1

2〈Tij〉 (31)

at the extremum. W [g] is defined as: W [g] := W[g, g]|.

19

At the linearized level, V turns out to be the gravi-

tational Chern-Simons action:

V [h, h] = − ℓ2

2κ2

∫

d3xhijδ2SCS[g]

δgijδgkl|g=η h

kl (32)

We find:

W [h(0)] = − ℓ2

8κ2

∫

d3x h(0)ij�3/2h(0)ij

〈Tij〉 =ℓ2

κ2Cij[h(0)] (33)

20

Given that the relation between the generating func-

tionals is a Legendre transformation, and since dual-

ity relates (Cij[h(0)], 〈Tij〉) = (〈Tij〉, Cij[h(0)]), we may

identify the generating functional of one theory with

the effective action of the dual.

For more general boundary conditions, the potential

contains additional terms and the relation is more

involved.

21

Bulk interpretation

Z[g] =∫

gDGµν e−S[G] (34)

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

grate:

∫

Dhij Z[h] eV [h,h] =∫

Dhij eW[h,h] ≃ eW [h] := Z[h]

Z[h] =∫

Dhij eSCS[h,h]Z[h] (35)

Recall how we defined h(0)ij and h(0)ij:

hij(r, x) = h(0)ij(x) + r2h(2)(x) + r3h(3)(x)

h(3)ij = Cij[h(0)] (36)

22

Fixing h(0) is the usual Dirichlet boundary condition.

Fixing h(0) is a Neumann boundary condition. Thus,

gravitational duality interchanges Dirichlet and Neu-

mann boundary conditions.

Mixed boundary conditions

Can we fix the following:

Jij(x) = hij(x) + λ hij(x) (37)

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

Jij = hij +2λ

�3/2Cij[h] (38)

23

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

h0ij +

2λ

�3/2Cij[h

0] = 0 . (39)

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 −2

λ�−3/2Cij[J]

)

.

(40)

A puzzle

If λ = ±1, J is self-dual. We get a contribution from

the zero-modes to the dual stress-energy tensor:

〈Tij〉J=0 = ± ℓ2

κ2Cij[h

0]

〈Tij〉h = 0 . (41)

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.

24

Remark

In any dimension d 6= 2,4, the formula for the holo-

graphic countertems is:

Sct = − 1

16πGN

∫

r=ǫ

√γ

[

2(1 − d) − 1

d− 2R

− 1

(d− 4)(d− 2)2

RijRij − d

4(d− 1)R2

+ . . .

(42)

In d = 3, this gives:

Sct =1

16πGN

∫

r=ǫ

√γ

[

4

ℓ+ ℓR− ℓ3

(

RijRij − 3

8R2

)

+ . . .

]

(43)

For a RS brane in AdS4, the quadratic terms are

those of the new TMG.25

Conclusions

• The variables involved in gravitational duality in

AdS4 are (Cij(r, x), 〈Tij(x)〉r). Duality interchanges

D/N boundary conditions.

• Associated with the dual variables are a dual metric

and a dual stress-energy tensor:

Cij[g] = 〈Tij〉 , 〈Tij〉 = Cij[g].

• The self-dual point corresponds to bulk gravita-

tional instantons.

27

• Thanks to the work of [Ishibashi, Wald 0402184][Com-

pere, Marolf 0805.1902], we now know that both

graviton modes are normalizable (d ≤ 4). This should

have lots of interesting applications.

Thank you!

28