Setting the boundary free in AdS/CFT : a Gravity d+1 / CFT d +Gravity d correspondence Geoffrey Compère University of California at Santa Barbara Work done with Don Marolf ( ) and Aaron Amsel ( )

Overview : two ideas AdS/CFT : Supergravity on AdS x C Large N CFT Brane world scenarios [Randall – Sundrum] UV cut-off Motivation : understand the dynamics of strongly coupled gauge theories Motivation : model modifications of gravity at small scales

Overview : this talk AdS/CFT : Supergravity on AdS x C Large N CFT Motivation: cosmology: Trace-anomaly driven inflation [Starobinsky, Hawking-Hertog-Reall, …] Without UV cut-off New boundary conditions on AdS ? CFT coupled to gravity

Overview : this talk Part I :  Review AdS/CFT correspondence  Justify that gravity at boundary is allowed  Inspiration from scalar fields Part II :  Investigate the boundary theories  Discuss supersymmetric extension

Part I : How to model gravity at the boundary of AdS ?  Review AdS/CFT correspondence  Justify that gravity at boundary is allowed  Inspiration from scalar fields

Setup : the AdS/CFT correspondence (in the low energy and large N limit) Bulk side : gravity + matter fields in anti-de Sitter space with Dirichlet boundary conditions Gauge theory /CFT side: gauge fields in a flat background Correspondence : Z gravity = D   e i S gravity+matter = Z CFT

Setup: Correlation functions Φ 0 (x 1 ) Φ 0 (x 2 )  O 0 (x 1 ) O 0 (x 2 )  Quadratic action in perturbation theory around flat space with Dirichlet boundary conditions Correlation function of the conformal fields dual to the bulk field Universal example: Φ 0 = g (0)ij Dual field: O 0 = T ij

AdS d+1 /CFT d dictionary for metric/stress tensor r=0 ds 2 = r -2 dr 2 +g ij dx i dx j, g ij = g (0)ij r -2 + g (2)ij +… + g (d)ij r d-2 + …

AdS d+1 /CFT d dictionary for metric/stress tensor At the boundary : T ij = g (d)ij + … : can vary It is the vacuum expectation value of the stress-tensor g (0)ij : fixed It is the source for the dual operator in the CFT r=0 FixedAllowed to fluctuate ds 2 = r -2 dr 2 +g ij dx i dx j, g ij = g (0)ij r -2 + g (2)ij +… + g (d)ij r d-2 + …

Why is the leading metric not allowed to fluctuate ? r=0 ds 2 = r -2 dr 2 +g ij dx i dx j, g ij = g (0)ij r -2 + g (2)ij +… + g (d)ij r d-2 + … ??? FixedAllowed to fluctuate What are the constraints ??

Normalizeability Conservation (Finiteness) (Unitarity) Finite r=0 No ghosts

Subtleties (already) for the free scalar field

Case 1) m 2 > - (d/2) Only the  + mode is admissible The  - mode operator beyond unitarity bound BC: a(x) fixed with b(x) free (Dirichlet)

Subtleties (already) for the free scalar field Case 1) m 2 > - (d/2) Only the  + mode is admissible The  - mode operator beyond unitarity bound BC: a(x) fixed with b(x) free (Dirichlet) Case 2) –(d/2) 2 +1 > m 2 > - (d/2) 2 Both modes are normalizeable and in unitarity range BC: b(x) fixed, a(x) free (Neumann) or b(x) =δW[a]/ δa(x) (mixed) [ Breitenlohner & Freedman, Klebanov & Witten]

Subtleties (already) for the free scalar field Case 1) m 2 > - (d/2) Only the  + mode is admissible Case 2) –(d/2) 2 +1 > m 2 > - (d/2) 2 Both modes are admissible. Case 3) m 2 < - (d/2) 2 Instability [ Breitenlohner & Freedman]

Normalizeability for the graviton Fluctuations of g (0)ij fail to be normalizeable in the usual spin-2 inner product, even for AdS 4 or AdS 3. [ Balasubramanian & Kraus & Lawrence,98] [Randall & Sundrum, 99]

Normalizeability for the graviton Fluctuations of g (0)ij fail to be normalizeable in the usual spin-2 inner product, even for AdS 4 or AdS 3. But the total action is S bulk = S EH +S CC + S GH + S ct [g (0) ] with [ Holographic renormalization : de Haro, Skenderis, Solodukhin, 2000]

Normalizeability for the graviton Fluctuations of g (0)ij fail to be normalizeable in the usual spin-2 inner product, even for AdS 4 or AdS 3. But the total action is S bulk = S EH +S CC + S GH + S ct [g (0) ] with S ct contains time derivatives of g (0)ij. AdS 4 : Ishibashi and Wald, Breitenlohner & Freedman, Petkou & Leigh, Ross & Marolf

Normalizeability for the graviton Fluctuations of g (0)ij fail to be normalizeable in the usual spin-2 inner product, even for AdS 4 or AdS 3. But the total action is S bulk = S EH +S CC + S GH + S ct [g (0) ] with S ct contains time derivatives of g (0)ij. For dynamical g (0)ij, S ct contributes to π cd and to the symplectic structure. (ω tot = ω EH +dω ct ) AdS 4 : Ishibashi and Wald, Breitenlohner & Freedman, Petkou & Leigh, Ross & Marolf

Normalizeability for the graviton Fluctuations of g (0)ij fail to be normalizeable in the usual spin-2 inner product, even for AdS 4 or AdS 3. But the total action is S bulk = S EH +S CC + S GH + S ct [g (0) ] with S ct contains time derivatives of g (0)ij. For dynamical g (0)ij, S ct contributes to π cd and to the symplectic structure. (ω tot = ω EH +dω ct ) Flucts of g (0)ij become normalizeable. (Checked for d=2,3,4). AdS 4 : Ishibashi and Wald, Breitenlohner & Freedman, Petkou & Leigh, Ross & Marolf

Conservation of symplectic structure Requires : - fixed [Dirichlet] - fixed [Neumann] -“Boundary equation of motion”

Normalizeability for the graviton Fluctuations of g (0)ij fail to be normalizeable in the usual spin-2 inner product, even for AdS 4 or AdS 3. But the total action is S bulk = S EH +S CC + S GH + S ct [g (0) ] with S ct contains time derivatives of g (0)ij. For dynamical g (0)ij, S ct contributes to π cd and to the symplectic structure. (ω tot = ω EH +dω ct ) Flucts of g (0)ij become normalizeable. (Checked for d=2,3,4). AdS 4 : Ishibashi and Wald, Breitenlohner & Freedman, Petkou & Leigh, Ross & Marolf

Conservation of symplectic structure Requires : - fixed [Dirichlet] - fixed [Neumann] -“Boundary equation of motion” One then has a variational principle :

Conclusion of Part I. : Gravity with Λ<0 admits valid alternative boundary conditions : g (0)ij : free to vary with rather - fixed [Neumann], or -“Boundary equation of motion” What can we learn ??

Part II : What is the boundary theory ? Outline Gravity is induced at the boundary  Checks of diffeomorphism and Weyl invariance  Relevant deformations d=2 (Polyakov/Liouville) Higher d : Boundary propagator around flat space Supersymmetric extension

How does induced gravity works ? Z ind grav = Dg (0) Z CFT [g (0) ]Z CFT [g (0) ] = D  e i S CFT = e -W[g(0)] (No kinetic term for g (0) )

How does induced gravity works ? Z ind grav = Dg (0) Z CFT [g (0) ] Let g (0) = e 2  g (0). Z CFT [g (0) ] = D  e i S CFT = e -W[g(0)] (No kinetic term for g (0) )

How does induced gravity works ? Z ind grav = Dg (0) Z CFT [g (0) ] Let g (0) = e 2  g (0). Note for odd d: no trace anomaly; integral independent of . Induced conformal gravity. For even d: trace anomaly Induced gravity with trace anomaly [Sakharov, Adler] Z CFT [g (0) ] = D  e i S CFT = e -W[g(0)] (No kinetic term for g (0) )

The correspondence The bulk partition function with S bulk = S EH +S CC + S GH + S ct equals the induced gravity partition function at the boundary Z ind grav. Z bulk [g (0) ] = Dg D  e iS bulk g  g (0) Z bulk = Dg (0) Dir Neu

The correspondence The bulk partition function with S bulk = S EH +S CC + S GH + S ct equals the induced gravity partition function at the boundary Z ind grav. Z bulk [g (0) ] = Dg D  e iS bulk g  g (0) Z bulk = Dg (0) Dir Neu In the Semi-classical Limit :  S bulk = Eq. of Motion + T ij  g (0)ij g (0) g (0)ij free 0 = T ij (“Neumann” theory, conformal for odd d)

Check : Gauge freedom at the boundary Assume that the boundary has S d-1 x R topology Let  a induces a bndy diffeo:  g (0)ij = L  g (0)ij Find  (  g 1, L  g 2 ) = Q[  ] = 0. Diffeo = Gauge transformation

Check : Gauge freedom at the boundary Assume that the boundary has S d-1 x R topology Let  a induces a bndy diffeo:  g (0)ij = L  g (0)ij Find  (  g 1, L  g 2 ) = Q[  ] = 0 ! Diffeo = Gauge transformation When d is odd, let  a induce Weyl transformations  g (0)ij = 2 δσ g (0)ij,  T ij = (2-d)δσ T ij Find also  (  g 1, L  g 2 ) = Q[  ] = 0 ! Weyl transf. = Gauge transformation

How the cancellation in the charges occurs Symplectic structure Conserved charges

Deformations of the theory? Compute conformal dimensions dim[g (0) ij ] = 0, dim[T ij ] = d Relevant deformations have < d derivatives. g ij = g (0) ij r -2 + g (2) ij + g (d) ij r d-2 + h ij r d-2 log r + …

Deformations of the theory? For d = 2 For d = 3 For d = 4 Relevant deformations have < d derivatives. Saddle point of the path integral : Bulk eq. of motion and

Case d=2 : Polyakov and Liouville Any CFT has an effective action given by the non-local Polyakov action CFT becomes coupled to the Liouville field . The total central charge vanishes. [Polyakov, Skenderis, Solodukhin] Introduce new Weyl invariance :

Case d=2 : Polyakov and Liouville Any CFT has an effective action given by the non-local Polyakov action CFT becomes coupled to the Liouville field . The total central charge vanishes. Bulk perspective : the trace of Neumann B.C. : is exactly the eq. of motion of the Liouville field [Polyakov, Skenderis, Solodukhin] Introduce new Weyl invariance :

Higher dimensions ? ? ? Bulk perspective (>3d):  On-shell action in terms of g (0) not known CFT perspective :  The CFT’s dual to AdS geometries are known  Their effective action (at large N) is not known Perturbative analysis around flat space (Amounts to compute linear perturbations in AdS with Neumann boundary conditions)

At the linear level around flat space In any odd dimension d : In d = 4 : Continuous spectrum Of modes (ghost and tachyon free !) Tachyon ghost Flat space unstable k=1/L [Tomboulis]

At the linear level around flat space In any odd dimension d : In d = 4 : Matches AdS 5 xS 5 : N=4 SYM ( ) [Liu, Tseytlin, Balasubramanian et al]

Supersymmetric extension Each bulk supermultiplet contains a Rarita-Schwinger field with CompactificationBulk supermultiplet Induced Boundary supermultiplet AdS 5 x S 5 d=5 N=8 supermultiplet d=4 N=4 conformal supermultiplet AdS 4 x S 7 d=4 N=8 supermultiplet d=3 N=8 conformal supermultiplet AdS 3 x Xd=3 (p,q)- supermultiplet d=2 (p,q) conformal supermultiplet

Possible boundary conditions Standard Modified by Counterterms Dirichlet – Neumann No ghost constraints ? Breaking of Conformal invariance m 1 = Z / 2 Dirichlet – Neumann Only Dirichlet Supergravity induced by susy conformal field theories described via a gauged supergravity dual with Neumann boundary conditions!

Summary Counterterms save the day in the symplectic structure : fluctuations of the leading F-G metric are normalizeable

Summary Counterterms save the day in the symplectic structure : fluctuations of the leading F-G metric are normalizeable G.R. with Λ <0 admits admissible boundary conditions where the boundary metric can vary

Summary Counterterms save the day in the symplectic structure : fluctuations of the leading F-G metric are normalizeable G.R. with Λ <0 admits admissible boundary conditions where the boundary metric can vary Dual to induced (super)(conformal) gravity (via usual CFT)

Summary Counterterms save the day in the symplectic structure : fluctuations of the leading F-G metric are normalizeable G.R. with Λ <0 admits admissible boundary conditions where the boundary metric can vary Dual to induced (super)(conformal) gravity (via usual CFT) Non-local theory of gravity, but with control on UV and ghost & tachyon free in odd dimensions around flat space !

Directions for the future : Ghost/Tachyon condensation in AdS 5  4d gravity coupled to a large N CFT !  Contains the dynamics of the trace anomaly  Deform gravity in the IR (only a few tunable parameters)  Look at inflation models from trace anomaly … 1/N corrections, Cascade of dualities …

Thank you !

Deformation preserving conformal invariance for d = 3 CFT

Electric part E ij of Weyl fixed (=T ij ) Magnetic part B ij of Weyl fixed (=C (0)ij ) Weyl self-dual Weyl Anti-self- dual Neumann boundary conditions Dirichlet boundary conditions [Henneaux & Teitelboim, Julia & al. de Haro & Petkou & Leigh] Deformation preserving conformal invariance for d = 3 CFT Deformation can be understood as a sort of electric-magnetic duality ?

Renormalization group flow for odd d Dirichlet theory CFT in flat space Neumann theory Conformal field theory with induced conformal gravity Deformations of Neumann Not a CFT in general