This document summarizes research on dual gravitons in AdS4/CFT3 and the holographic Cotton tensor. Key points: - In AdS4/CFT3, both modes of the graviton are normalizable, allowing duality to interchange the boundary metric g(0) and stress tensor T. - The Cotton tensor C, which maps a metric to its stress tensor, plays a special role as the "holographic Cotton tensor". - There is a duality symmetry of the bulk equations of motion under which the linearized metric fluctuations ḣ and Cotton tensor of the dual metric C(ḣ) are interchanged. - This relates the

1. 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

2. 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

3. Outline
• Review of holographic renormalization formulas
• Self-dual metrics in AdS4
• Boundary conditions
• Duality and the holographic Cotton tensor
• Conclusions
2

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) + . . .
Rµν = Λ gµν (1)
Solving eom gives: g(0), g(3) are undetermined and
1
g(2)ij = −Rij [g(0)] + g(0)ij R[g(0)] (2)
4
Higher g(n)’s: g(n) = g(n)[g(0), g(3)].
3

5. To obtain the partition function, regularize and renor-
malize the action:
S = Sbulk + SGH + Sct
1 4 √
= − 2 d x g (R[g] − 2Λ)
2κ Mǫ
1 3 √ 4
− 2 d x γ K − − ℓ R[γ]
2κ ∂Mǫ ℓ
(3)
Z[g(0)] = eW [g(0)] = eSon-shell[g(0)]
2 δSon-shell 3ℓ2
⇒ Tij (x) = √ ij = g(3)ij (x) (4)
g(0) δg 16πGN
(0)
4

6. Matter
1 4x√g (∂ φ)2 + 1 Rφ2 + λφ4
Smatter = d µ
2 Mǫ 6
1 3x√γ φ2(x, ǫ)
+ d (5)
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) (6)
g(0) δφ(0)
5

7. Scalar instantons
 
1 4x √g  −R + 2Λ 1
Sbulk = d + (∂φ)2 + Rφ2 + λ φ4
2 8πGN 6
• Euclidean
• Metric asymptotically AdS4 × S 7
λ = 8πGN for 11d embedding
6ℓ2
Equations of motion: φ − 1 Rφ − 2λφ3 = 0
6
Seek solutions (“instantons”) with
Tµν = 0 ⇒ ds2 = ℓ2 dr 2 + dx2
r2
6

8. 2 br
Unique solution: φ = √ −sgn(λ)b2+(r+a)2+(x−x )2
ℓ |λ| 0
• 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]:
1 3x[(∂ϕ)2+ 1
√ √
Γeff[ϕ] = √ d R[g(0) ]ϕ2+2 λ( λ−α)ϕ6]
λ 2
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

9. 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:
1
Cµναβ = ǫµν γδ Cγδαβ
2
is compatible with Einstein’s equations with a nega-
tive cosmological constant and Euclidean signature.
8

10. This is summarized in the following tensor [Julia,
Levie, Ray ’05]:
1
Zµναβ = Rµναβ + 2 gµαgνβ − gµβ gνα (7)
ℓ
Z is the on-shell Weyl tensor and Zµρν ρ = 0 gives
Einstein’s equations.
• The coupled equations may be solved asymptoti-
cally. 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) + . . .
9

11. 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
• 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

12. • 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 term on a fixed background g(0)ij .
We find its coefficient:
ℓ2 (2N )3/2
k= =
8GN 24
This holds at the non-linear level.

13. We now impose regularity of the Euclidean solutions.
At the linearized level, the regularity condition is:
1 3
¯(3)ij (p) = |p| ¯(0)ij (p) .
h h (8)
3
The full r-dependence of the metric fluctuations is
now:
¯ij (r, p) = e−|p|r (1 + |p|r) ¯(0)ij (p)
h h (9)
h(0)ij is not arbitrary but satisfies:
3/2¯
h(0)ij = ǫikl ∂k ¯(0)jl .
h (10)
This is the t.t. part of the linearization of:
1/2 ¯
Rij = Cij (11)
10

14. General solution (p∗ := (−p0, p); p∗ = Πij p∗):
i ¯i j
1
¯ij (p, r) = γ(p, r) Eij +
h ψ(p, r) ǫikl pk Ejl
p
p∗p∗
¯i ¯j
1
Eij = ∗2 − Πij (12)
¯
p 2
For (anti-) instantons, γ = ±ψ:
 ∗ ∗
¯¯
p p

 i j
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)|2 . (13)
8κ2

15. 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

16. ⇒ 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)
δW
Extremize w.r.t. φ0 ⇒ δφ − φ1 = 0 ⇒ φ0 = φ0[φ1]
0

17. 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 (15)
Generating fctnl CFT2 ↔ effective action CFT1
(φ1 fixed) (φ0 fixed)

18. 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]
12

19. 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) + . . . + r 3g(3)ij (x)
3ℓ2
Tij (x) = g(3)ij (x) (17)
16πGN
Is this a normalizable mode? Duality can only inter-
change them if both modes are normalizable.
13

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

21. 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′ and a new
i
′
current Ji . This way, the gauge field is always fixed.
Duality now acts as:
i
′
(Ai, Ji) ↔ (A′ , Ji ) , (B, E) ↔ (B ′, E ′)
′
Ji = ǫijk ∂j Ak , Ji = ǫijk ∂j A′ (18)
k
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

22. The Holographic Cotton Tensor
1 kl 1
Cij = ǫi Dk Rjl − gjl R . (19)
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.
16

23. • Given a metric gij = δij + hij , we may construct a
Cotton tensor (¯ij = Πijkl hkl ):
h
1
Cij = ǫikl ∂k ¯jl .
h (20)
2
• Given a stress-energy tensor Tij , there is always
an associated dual metric ˜ij such that:
h
Tij = Cij [˜]
h
3˜
hij = 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

24. 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))(22)
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 (23)
18

25. This leads to:
2Cij (¯[−˜, a]) = −|p|3Pij [a, ˜]
h a a
2Cij (P [−˜, a]) = +|p|3¯ij [a, ˜]
a h a (24)
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 (25)
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 (26)
κ

26. Eij (0, x) = Tij (x)
ℓ2
Bij (0, x) = 2 Cij [¯(0)]
h (27)
κ
S(E) = +B
S(B) = −E (28)
Gravitational S-duality interchanges the renor-
malized stress-energy tensor Tij = Cij [˜] with
h
the Cotton tensor Cij [h] at radius r. [SdH, JHEP
0901 (2009) 042]

27. Can Cij [h(0)] be interpreted as the stress tensor of
some CFT2?
δ W [˜(0)]
˜ h
˜
Cij [h(0)] = Tij = (29)
ij
δ˜(0)
h
W [˜] can be computed from the Legendre transfor-
˜ h
mation:
W[g, ˜] = W [g] + V [g, ˜]
g g (30)
δW 1 δV 1
ij
=0⇒√ ij
= − Tij (31)
δg g δg 2
˜ g ˜ g
at the extremum. W [˜] is defined as: W [˜] := W[g, ˜]|.
g
19

28. At the linearized level, V turns out to be the gravi-
tational Chern-Simons action:
ℓ2 δ 2SCS [g]
V [h, ˜] = − 2
h d3x hij ij δg kl
|g=η ˜kl
h (32)
2κ δg
We find:
ℓ2
W [˜(0)] = − 2 d3x ˜(0)ij 3/2˜(0)ij
˜ h h h
8κ
ℓ2
˜
Tij = 2 Cij [h(0)] (33)
κ
20

29. Given that the relation between the generating func-
tionals is a Legendre transformation, and since dual-
ity relates (Cij [h(0)], Tij ) = ( Tij , Cij [˜(0)]), we may
˜ h
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

30. Bulk interpretation
Z[g] = DGµν e−S[G] (34)
g
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 ˜(0)ij :
h
hij (r, x) = h(0)ij (x) + r 2h(2)(x) + r 3h(3)(x)
h(3)ij = Cij [˜(0)]
h (36)
22

31. Fixing h(0) is the usual Dirichlet boundary condition.
Fixing ˜(0) is a Neumann boundary condition. Thus,
h
gravitational duality interchanges Dirichlet and Neu-
mann boundary conditions.
Mixed boundary conditions
Can we fix the following:
Jij (x) = hij (x) + λ ˜ij (x)
h (37)
This is possible via W[h, J]. For regular solutions:
2λ
Jij = hij + 3/2
Cij [h] (38)
23

32. This b.c. determines hij up to zero-modes:
2λ
h0
ij + 3/2
Cij [h0] = 0 . (39)
This is the SD condition found earlier. Its only solu-
tions are for λ = ±1.
λ = ±1 We find:
ℓ2 3/2 2 −3/2
˜
Tij J = − 2 Jij − Cij [J] .
2κ (1 − λ2) λ
(40)

33. A puzzle
If λ = ±1, J is self-dual. We get a contribution from
the zero-modes to the dual stress-energy tensor:
ℓ2
Tij J=0 = ± 2 Cij [h0]
˜
κ
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

34. Remark
In any dimension d = 2, 4, the formula for the holo-
graphic countertems is:
1 √ 1
Sct = − γ 2(1 − d) − R
16πGN r=ǫ  d−2  
1 d
− R Rij −
ij R2 + (42)
. . .
(d − 4)(d − 2)2 4(d − 1)
In d = 3, this gives:
1 √ 4 3 ij 3 2
Sct = γ + ℓ R − ℓ Rij R − R + . .(43)
.
16πGN r=ǫ ℓ 8
For a RS brane in AdS4, the quadratic terms are
those of the new TMG.
25

35. 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

36. • 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.

37. Thank you!
28