Holographic Cotton Tensor

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École Normale Supérieure, Paris 2008

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Holographic Cotton Tensor

  1. 1. Dual Gravitons in AdS4/CFT3 and the Holographic Cotton Tensor Sebastian de HaroUtrecht University and Foundations of Physics Paris, October 9, 2008 Based on S. de Haro, arXiv:0808.2054
  2. 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. 3. Motivation1. 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. 4. Holographic renormalization (d = 3) [SdH, Skenderis, Solodukhin CMP 217(2001)595] ℓ2 ds2 = 2 dr 2 + gij (r, x) dxidxj rgij (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. 5. Matter 1 4x√g (∂ φ)2 + 1 Rφ2 + λφ4Smatter = 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. 6. The relation between φ(1) and φ(0) is given by regu-larity of the Euclidean solution. Defineφ(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. 7. Boundary conditionsIn 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 quantizationIn the range of masses d2 −4 < m2 < d2 −4 + 1, bothmodes are normalizable [Avis, Isham (1978); Breit-enlohner, Freedman (1982)] 4
  8. 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) δWExtremize w.r.t. φ0 ⇒ δφ − φ1 = 0 ⇒ φ0 = φ0[φ1] 0
  9. 9. Dual generating functional obtained by evaluating Wat 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. 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 spinDuality 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. 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) 16πGNIs this a normalizable mode? Duality can only inter-change them if both modes are normalizable. 6
  12. 12. 2) gij is not an operator in a CFT. We can computeTij Tkl . . . but gij is fixed.3) Even if we were to couple the CFT to gravity, gijwouldn’t make sense.Question 1) has been answered in the affirmative byIshibashi and Wald 0402184.Recently, Compare and Marolf have generalized thisresult 0805.1902.
  13. 13. Problems 2)-3): a similar issue arises in the spin-1case [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 astress tensor. In 3d there is a natural candidate: theCotton tensor.
  14. 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 gravitationalChern-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. 15. • Given a metric gij = δij + hij , we may construct aCotton tensor (¯ij = Πijkl hkl ): h 1 Cij = ǫikl ∂k ¯jl . h (13) 2• Given a stress-energy tensor Tij , there is alwaysan ˜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, Petkou0710.0965] (related work by Julia, Levie, Ray 0507262in de Sitter)
  16. 16. Gravitational instantons• Instanton solutions with Λ = 0 have self-dual Rie-mann tensor. However, self-duality of the Riemanntensor implies Rµν = 0.• In spaces with a cosmological constant we needto choose a different self-duality condition. It turnsout that self-duality of the Weyl tensor is compatiblewith a non-zero cosmological constant: 1 Cµναβ = ǫµν γδ Cγδαβ 2 8
  17. 17. •This equation can be solved asymptotically. In theFefferman-Graham coordinate system: ℓ2 ds2 = 2 dr 2 + gij (r, x) dxidxj rwhere 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. 18. • The holographic stress tensor is Tij = 3ℓ2 g 16πGN (3)ij .We find that for any self-dual g(0)ij the holographicstress tensor is given by the Cotton tensor: ℓ2 Tij = C(0)ij 8πGN• We can integrate the stress-tensor to obtain theboundary 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. 19. For linearized Euclidean solutions, there is a regularitycondition: 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. 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 ¯ 2For (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. 21. Duality symmetry of the equations of motionSolution 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) 1bij (p) := |p|3 Cij (˜) → ¯ij [a, ˜] a h aDefine: 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. 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. 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 hthe 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. 24. Gravitational Legendre transformConstruct 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 ˜ gat the extremum. W [˜] is defined as: W [˜] := W[g, ˜]|. gLinearize and dualize Tij = ℓ2 C [˜] h then κ2 ij ℓ2 V [h, ˜] = − 2 h d3x hij Cij [˜] h (30) 2κ 11
  25. 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 δgWe find: ℓ2 W [˜] = − 2 d3x ˜(0)ij 3/2˜(0)ij ˜ h h h 8κ ℓ2 ˜ Tij = 2 Cij [h] (32) κGiven 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 ˜ hthe generating functional of one theory with the ef-fective action of the other.
  26. 26. Bulk interpretation Z[g] = DGµν e−S[G] (33) gLinearize, 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 switchesbetween Dirichlet and Neumann boundary conditions. 12
  27. 27. Mixed boundary conditionsCan 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. 28. λ = ±1 We find: ℓ2 3/2 1 ˜ Tij J = − 2 Jij − dij [J] . (38) 2κ (1 − λ2) λλ = ±1 In this case J has to be self-dual. We haveh = 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 andconserved but non-zero even if J = 0. It is zero ifthe metric is conformally flat. 13
  29. 29. Non-linear dual gravitonAt the non-linear level, we do not know the generalregularity condition. However, we can still define anon-linear Cotton tensor ℓ2 Tij = 2 Cij [˜] g (40) κPerturbatively, we can always solve for ˜ given Tij g(up to zero-modes): ˜ikl Dk Tlj = − ˜ Rij + O(D 4) ǫ ˜ ˜ (41) 14
  30. 30. So the concept of dual graviton makes sense. Wecan 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 4κi.e. effectively coupling the CFT to gravity (or con-formal gravity).
  31. 31. The role of the EH term here is to provide the CScoupling between g and ˜: g ℓ2 δSEH = d3x Cij [˜]δg ij g (44) 2κ2as in the linearized case.The bulk produces a conformal, Lorentz invariantnon-linear coupling between the two gravitons. 15
  32. 32. Conclusions• The variables involved in gravitational duality inAdS 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 beinterpreted as a dual CFT2: ˜Cij [g] = Tij , Tij = Cij [˜]. g 16
  33. 33. • The self-dual point corresponds to bulk gravita-tional instantons.• The existence of a dual graviton persists at thenon-linear level.• The two gravitons have different parity.• The graviton can become dynamical on the bound-ary by the boundary conditions. This amounts tocoupling the CFT to Cotton gravity or topologicallymassive gravity.
  34. 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 byI. Bakas (duality in AdS BH background).• AdS4 → AdS2 reduction?

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