Topologically Massive Gravity and AdS/CFT

Transcription

1 Topologically Massive Gravity and AdS/CFT Institute for Theoretical Physics University of Amsterdam The Planck Scale, XXV Max Born Symposium Wroclaw, 30 June 2009

2 Introduction Three dimensional gravity offers an interesting arena to investigate quantization of gravitational theories. Einstein gravity in three dimensions has no propagating degrees of freedom so it is perhaps not a very good toy model for higher dimensional gravitational theories. Adding higher derivative terms leads to propagating degrees of freedom but the theory generically contains ghost-like excitations. The aim of this work is to discuss a particular theory, the topologically massive gravity, that was conjectured to be free of such problems for a particular value of its parameters and analyze what we can learn using the AdS/CFT duality.

3 Introduction Three dimensional gravity offers an interesting arena to investigate quantization of gravitational theories. Einstein gravity in three dimensions has no propagating degrees of freedom so it is perhaps not a very good toy model for higher dimensional gravitational theories. Adding higher derivative terms leads to propagating degrees of freedom but the theory generically contains ghost-like excitations. The aim of this work is to discuss a particular theory, the topologically massive gravity, that was conjectured to be free of such problems for a particular value of its parameters and analyze what we can learn using the AdS/CFT duality.

4 Topological Massive Gravity [Deser, Jackiw, Templeton] Topologically massive gravity is obtained by adding to the Einstein gravity the gravitational CS term, ( S = d 3 x (R 2Λ) + 1 ) 2µ (ΓdΓ + Γ3 ) where Γ is the 1-form Christoffel symbol. This theory admits asymptotically AdS solutions, for example the BTZ black hole solves its equations of motion and has perturbative massive modes. When µ 1 however, the massive modes have negative energy and the theory is unstable.

5 The "chiral point", µ = 1. When µ = 1 the negative energy modes disappear, the left moving gravitational modes become pure gauge and the theory seems to only contain a purely right moving sector. This led to the conjecture that the theory is stable and consistent when µ = 1 and dual to a 2d chiral CFT [Li, Song, Strominger (2008)]. This created a lot of controversy as other authors found non-chiral modes and instabilities at the chiral point [Carlip etal], [Grumiller, Johansson], [Giribet etal]...

6 Fall-off conditions Part of the issue is the question: What are the correct fall-off conditions for the fields at infinity? The unstable modes have fall-off conditions different than the ones the metric satisfies in 3d Einstein gravity (the Brown-Henneaux boundary conditions). So one is led to ask: are modified fall-off conditions allowed?

7 Fall-off conditions [Regge, Teitelboim (1974)]... The traditional point of view is as follows: 1 Select physically "reasonable" fall-off conditions such that relevant solutions, for example black holes, satisfy them. 2 Check that conserved charges are finite with this choice. One may consider different fall-off conditions as defining different theories.

8 AdS/CFT: a new perspective The AdS/CFT correspondence provides a new perspective which leads to a comprehensive answer to such questions. The aim of this talk in explain the new insights and the new methodology that originated from AdS/CFT and then apply them to the case of topologically massive gravity.

9 References This talk is based on KS, Marika Taylor, Balt C. van Rees Topologically Massive Gravity and the AdS/CFT Correspondence arxiv:

10 AdS/CFT: basics In the AdS/CFT correspondence: The fields φ I (0) parametrizing the boundary conditions at conformal infinity are identified with sources that couple to operators O I of the dual CFT. The on-shell action, S onshell [φ (0) ], is the generating functional of CFT correlation functions: O = δs onshell[φ (0) ] δφ (0), O(x)O(y) = δ2 S onshell [φ (0) ] δφ (0) (x)δφ (0) (y), etc.

11 AdS/CFT: expectations and requirements These identifications bring in new intuition about what one expects to be able to do. In QFT the source that couple to operators are unconstrained, because one wants to functionally differentiate w.r.t. them. This implies that one should able to formulate the bulk boundary problem by specifying arbitrary functions/tensors.

12 Asymptotically AdS spacetimes Consider the case the bulk field is the metric. In the physics literature, prior to the AdS/CFT correspondence, there were a number of works on Asymptotically AdS spacetimes [Ashtekar, Magnon (1984)] [Henneaux, Teitelboim (1985)]... In these works the metric approaches that of AdS as we approach conformal infinity. For AdS/CFT however one needs more general boundary conditions. The boundary conditions must be parametrized by an unconstrained metric, since this metric should act as a source for the energy momentum tensor T ij of the dual CFT.

13 Asymptotically locally AdS spacetimes Fortunately, such more general set-up has been considered in the mathematics literature [Fefferman-Graham (1985)]. The corresponding spacetimes are called Asymptotically locally AdS spacetimes (AlAdS). An AlAdS spacetime admits the following metric in a finite neighborhood of the conformal boundary, located at r = 0: where ds 2 = dr r g ij(x, r)dx i dx j 2 r 2 lim g ij(x, r) = g (0)ij (x) r 0 is an arbitrary non-degenerate metric.

14 Asymptotically locally AdS spacetimes Fortunately, such more general set-up has been considered in the mathematics literature [Fefferman-Graham (1985)]. The corresponding spacetimes are called Asymptotically locally AdS spacetimes (AlAdS). An AlAdS spacetime admits the following metric in a finite neighborhood of the conformal boundary, located at r = 0: where ds 2 = dr r g ij(x, r)dx i dx j 2 r 2 lim g ij(x, r) = g (0)ij (x) r 0 is an arbitrary non-degenerate metric.

15 Asymptotically locally AdS spacetimes We emphasize that the only requirement put on g ij (x, r) a priori is that it should have a non-degenerate limit as r 0. The precise form of g ij (x, r) is determined by solving the bulk field equations asymptotically. This reduces to solving algebraic equations, so the most general asymptotic solution can be found readily for any given bulk theory that admits AlAdS solutions.

16 Fefferman-Graham expansion For Einstein gravity in (d + 1) dimensions, g ij (x, r) = g (0)ij (x)+r 2 g (2)ij + +r d (g (d)ij +log r 2 h (d)ij )+ The blue coefficients are locally determined in terms of g (0). g (d)ij is only partially determined by asymptotics. This coefficient is related via AdS/CFT to the 1-point function of T ij and thus to bulk conserved charges. h (d) is non-zero when d is even and is related to the Weyl anomaly of the boundary theory, [de Haro, Solodukhin, KS (2000)] h (d) = δ (conformal anomaly) δg (0)

17 Fefferman-Graham expansion for d = 2 For Einstein gravity in d = 2 g ij (x, r) = g (0)ij (x) + r 2 g (2)ij + h (2) vanishes because the integral of the conformal anomaly is a topological quantity (the Euler number). The Brown-Henneaux boundary conditions are as above with g (0)ij (x) = δ ij. The precise form of this expansion is special to Einstein gravity. Coupling to matter changes the coefficients. For example, coupling Einstein gravity to a free massless scalar induces a logarithmic term in the expansion, i.e. h (2) 0 in this theory.

18 Fefferman-Graham expansion for d = 2 For Einstein gravity in d = 2 g ij (x, r) = g (0)ij (x) + r 2 g (2)ij + h (2) vanishes because the integral of the conformal anomaly is a topological quantity (the Euler number). The Brown-Henneaux boundary conditions are as above with g (0)ij (x) = δ ij. The precise form of this expansion is special to Einstein gravity. Coupling to matter changes the coefficients. For example, coupling Einstein gravity to a free massless scalar induces a logarithmic term in the expansion, i.e. h (2) 0 in this theory.

19 Non-chiral mode in TMG The non-chiral mode of TMG found in [Grumiller-Johansson] has the asymptotic form, g ij (x, r) = δ ij + r 2 (g (2)ij + log r 2 h (2)ij ) + which differs from the Brown-Henneaux boundary condition because of the h (2) term. This generated a discussion of whether such boundary conditions are consistent. Subsequently it was proven by [Henneaux,Martinez,Troncoso, 0901] that conserved charges are finite with such boundary conditions.

20 Non-chiral mode in TMG From the perspective of our discussion: 1 a subleading log is not surprising, as the subleading coefficient change routinely as soon as one changes the bulk action, 2 the form of the asymptotic expansion should not be fixed by hand but rather derived by solving the bulk equations asymptotically. We will return to the most general asymptotic solution of TMG shortly.

21 Conserved charges for AlAdS spacetimes This is an another area where the AdS/CFT duality provided a new approach. In QFT the energy is computed using the energy momentum tensor, E = H = d d 1 x T 00 Generically, this expression needs renormalization due to UV infinities. In AdS/CFT correspondence, C T ij = δs onshell[g (0) ] δg ij (0) This expression is infinite, due to infinite volume of spacetime (IR divergences) and needs renormalization.

22 Holographic charges One can holographically renormalize the theory by adding local boundary covariant counterterms [Henningson, KS (1998)]. One can obtain a finite 1-point function for T ij for general AlAdS spacetime [de Haro, Solodukhin, KS (2000)]: T ij g (d)ij + X ij [g (0) ] X ij [g (0) ] known local function of g (0). One can prove rigorously from first principles (e.g. using Noether s method or Wald s covariant phase space methods) that the holographic changes are the correct gravitational conserved charges [Papadimitriou, KS (2005)].

23 Summary In summary, the new methodology that replaces the previous approach is: 1 Derive the most general solution of the bulk equations with general Dirichlet boundary conditions for all fields. 2 General results guarantee that the conserved charges are well-defined and can be obtained from the holographic 1-point functions. This framework allows to go further and obtain new information by computing two and higher point functions.

24 Application to TMG We now move to apply this methodology to the topologically massive gravity. We first discuss the theory at the "chiral point". There is one new element compared to previous discussions: The field equations are third order in derivatives, so there are two independent boundary data: one can fix the metric and the extrinsic curvature. The boundary metric g (0) is the source for the energy momentum tensor T ij. The boundary field b (0)ij parametrizing the extrinsic curvature is a source for a new operator t ij.

25 TMG at the chiral point We need one last ingredient. It turns out that t ij is obtained as a limit of an irrelevant operator. In CFT, when one couples an irrelevant operator, this generates severe UV divergences and the theory is not conformal in the UV. In gravity, a source for an irrelevant operator introduces severe IR divergences and the solution is not asymptotically AdS. In both cases, one bypass the problems by treating the source perturbatively. We will work to first order in b (0). This suffices for the computation of 2-point functions.

26 Results: asymptotic solution The most general asymptotic solution is ds 2 = dr 2 r r 2 g ij(x, r)dx i dx j with g ij (x, r) = b (0)ij log r 2 + g (0)ij + r 2 (g (2)ij + b (2)ij log r 2 ) + Only b (0) z z is non-zero and is the source for the new operator t zz. g (2) and b (2) are constrained partially by the asymptotic analysis.

27 Results: 1-point functions The 1-point functions can be computed in complete generality: T ij = t zz = 1 ( g (2)ij + 1 4G N 2 R[g (0)]g (0)ij 1 ( ) ɛi k g (2)kj + (i j) 2b (2)ij + 1 ) 2 2 A ij[g (0)ij ] 1 (g (2)zz + b (2)zz ) 2G N

28 Results: anomalies T ij satisfies the expected anomalous CFT Ward identities: Ti i 1 ( 1 = 4G N 2 R[g (0)] + 1 ) 2 Ai i[g (0) ] j 1 ( 1 T ij = 4G N 4 ɛ ij j R[g (0) ] + 1 ) 2 j A ij [g (0) ]

29 Example: Conserved charges for BTZ The energy momentum tensor T ij can be used to obtain the conserved charges. For example one can compute the conserved charges for the BTZ black hole. The stress energy tensor becomes chiral at µ = 1, T z z = 2 G N (r + + r ) 2, T zz = 0 and the conserved charges are M = dφtt t = π (r + + r ) 2 4G N J = dφtφ t = M

30 Results: 2-point functions From the most general solution of the linearized equations of motion we extracted the following non-zero 2-point functions: t zz (z, z)t zz (0) = ( 3/G N) log z 2 z 4, t zz (z, z)t zz (0) = ( 3/G N) 2z 4, T z z (z, z)t z z (0) = (3/G N) 2 z 4, These are precisely the non-zero 2-point functions of a Logarithmic CFT with c L = 0, c R = 3 G N, b = 3 G N.

31 TMG away from the "chiral point" We analyzed the theory also away from µ = 1. All results smoothly reduce to the ones discussed here as µ 1. In fact, our discussion mirrors the derivation of the form of logarithmic CFT correlators in [Kogan, Nichols (2004)]. From the form of the 2-point functions one finds that the CFT contains a state X of negative norm and X H X < 0 in that state. This is the counterpart of the bulk instability due to negative energy of massive gravitons.

32 Conclusions Topologically massive gravity at the "chiral point" is dual to Logarithmic CFT and therefore it is not unitary. One may try to restrict to the right-moving sector of the theory, which could yield a consistent chiral theory. This requires t T T = 0, which holds for certain LCFTs. It would be interesting to compute this 3-point function holographically for TMG at µ = 1.