## Final Report Summary - ADS-CFTSINGULARITIES (The AdS/CFT correspondence : a window on Cosmological Singularities)

Project context and objectives

It is a challenge to understand cosmological singularities such as the Big Bang. Classically, cosmological singularities can be described by the Belinsky–Khalatnikov–Lifshitz (BKL) behaviour and, as shown more recently, by means of a billiard motion with the Weyl chamber of hyperbolic Kac-Moody algebras (for low energy limits of superstring theories and some other gravity theories). At the quantum level, we would need to use the elusive theory of quantum gravity to have a satisfactory description of these singularities. Our very ambitious goal is to use the celebrated anti-de Sitter/conformal field theory (AdS/CFT) correspondence to obtain a gauge theory description of these singularities. To address that problem, we have been working in simplified settings. On the way we explored many aspects of the AdS/CFT correspondence: the very exciting and unexpected AdS/condensed matter correspondence, holography in 3d gravity, dual theories to black hole solutions constructed from brane intersections in string theory.

Recently, gravity duals for the non-relativistic CFT's (NRCFT) describing specific condensed matter systems in d dimensions have been proposed. These spacetimes are deformations of AdS spacetimes and their symmetry groups correspond to the symmetry group of the NRCFTs, the Schrödinger group in d dimensions. We know that in 3d, the finite dimensional symmetry group of AdS extend to two copies of the Virasoro algebra. The Schrödinger group in d dimensions possesses an infinite dimensional extension, the Schrödinger-Virasoro group. The question we addressed is the following: is the Schrödinger-Virasoro group realised as asymptotic symmetries of the gravity backgrounds dual to the NRCFTs? This would constrain the NRCFTs a lot but the answer turns out to be negative except in the 3d case. We worked out the 3d case in the context of Topological Massive Gravity (TMG).

Three-dimensional gravity provides a complementary avenue to string theory to gain control over questions in quantum gravity. Of particular interest has been the proposal of Witten that theories of pure gravity in AdS3 should be holographically dual to two-dimensional extremal conformal ?eld theories. Realising this proposal for Einstein gravity in three-dimensions has encountered various conceptual difficulties. These difficulties were nevertheless surmounted for the case of a different theory of pure gravity in three dimensions known as chiral gravity (essentially, chiral gravity is a theory of Topologically Massive Gravity at µl=1 with Brown-Henneaux boundary conditions), whose partition function in AdS3 was shown to precisely take the form of the partition function of an extremal conformal ?eld theory. If such conformal ?eld theories are proven to exist, this will be the ?rst example of a well de?ned theory of quantum gravity with only metric degrees of freedom. We have initiated a comprehensive study of a set of null warped AdS3 solutions of topologically massive gravity that are solutions for the coupling constant of TMG bigger than 3. They correspond to NRCFT in d=2 that we evoke in the previous paragraph. We first perform a careful analysis of the linearised stability of black holes in these space-times. We find two qualitatively different types of solutions to the linearised equations of motion: the first set has an exponential time dependence, the second - a polynomial time dependence. The solutions of polynomial in time induce severe pathologies and moreover survive at the non-linear level. In order to make sense of these geometries, it is thus crucial to impose appropriate boundary conditions. We argue that there exists a consistent set of boundary conditions that allows us to reject the above pathological modes from the physical spectrum. The asymptotic symmetry group associated to these boundary conditions consists of a centrally-extended Virasoro algebra. Using this central charge we can account for the entropy of the black holes via Cardy's formula. Finally, we note that the black hole spectrum is chiral and prove a Birkoff theorem showing that there are no other stationary axisymmetric black holes with the specified asymptotics. We extend most of the analysis to a larger family of pp-wave black holes that are related to Schrödinger spacetimes with critical exponent z.

In a second project in 3d gravity, we have analysed the asymptotic solutions of Chiral Gravity focusing on non-Einstein metrics. It was believed that no such solution exists. However, we constructed a class of such solutions. They admit curvature singularities in the interior that are reflected as singularities or infinite bulk energy of the corresponding linear solutions. A non-linear solution is found exactly. The back-reaction induces a repulsion of geodesics and a shielding of the singularity by an event horizon but also introduces closed time-like curves.

A topic related to the appearance of Kac-Moody algebras in the vicinity of spacelike singularities is the hidden symmetries of supergravity theories reduced to 3d. For instance when the low energy limit of the elusive M-theory, 11d supergravity, is reduced on a 8-torus, the resulting 3d theory possesses the exceptional Lie group E8 as its symmetry group. This symmetry in 3d can be used as a tool to generate new solutions for the higher dimensional theory from previously known solutions. We focused on 5d minimal supergravity, which is very similar to 11d supergravity and used the hidden symmetry of that theory, the exceptional group G2, to generate new solutions. We show that in a general solution the generators belonging to the Cartan and nilpotent subalgebras of G2 act as scaling and gauge transformations, respectively. The remaining generators of G2 form a sl(2,R) ? sl(2,R) subalgebra that can be used to generate non-trivial charges. Then we explore black string solutions and were able to generate the most general one. People have attempted to do that in the past with other methods and have failed. The decoupling limit of our general black string is a BTZ black hole times a two sphere. The macroscopic entropy of the string is reproduced by the Maldacena-Strominger-Witten CFT in appropriate ranges of the parameters. When the pressure-less condition is imposed, our string describes the in?nite radius limit of the most general class of black rings of minimal supergravity. We discussed implications our solution has for extremal and non-extremal black rings of minimal supergravity. We also tried to generate solitonic solutions by acting with G2 on the Eguchi-Hanson soliton. We could not generate new solitons in that way but on the way we understood how to write down more general solitonic solutions that we analysed in great detail.

Recently, chiral gravity has been proposed as a model of quantum gravity of 3d AdS spacetimes in topological massive gravity (a generalisation of Einstein's theory). We would like to generalise these theories in order to be able to describe spacetimes with a positive cosmological constant, i.e. dS spacetimes in 3d. As a first step, we generalised our work on warped AdS spacetimes to warped dS spacetimes. We analysed the asymptotic symmetries of these spacetimes to compute the central charges of the potential dual theories. The study of quasi-normal modes will also provide us with important information. Finally, it would be of great interest to find all geometries that are asymptotically warped dS and then sum the partition function. Moreover, a dual description of a cosmological spacetime, even in the simple context of 3d gravity, will produce an extraordinary tool that can be used to describe the Big Bang and understand the origin of cosmological horizon entropy.

It is a challenge to understand cosmological singularities such as the Big Bang. Classically, cosmological singularities can be described by the Belinsky–Khalatnikov–Lifshitz (BKL) behaviour and, as shown more recently, by means of a billiard motion with the Weyl chamber of hyperbolic Kac-Moody algebras (for low energy limits of superstring theories and some other gravity theories). At the quantum level, we would need to use the elusive theory of quantum gravity to have a satisfactory description of these singularities. Our very ambitious goal is to use the celebrated anti-de Sitter/conformal field theory (AdS/CFT) correspondence to obtain a gauge theory description of these singularities. To address that problem, we have been working in simplified settings. On the way we explored many aspects of the AdS/CFT correspondence: the very exciting and unexpected AdS/condensed matter correspondence, holography in 3d gravity, dual theories to black hole solutions constructed from brane intersections in string theory.

Recently, gravity duals for the non-relativistic CFT's (NRCFT) describing specific condensed matter systems in d dimensions have been proposed. These spacetimes are deformations of AdS spacetimes and their symmetry groups correspond to the symmetry group of the NRCFTs, the Schrödinger group in d dimensions. We know that in 3d, the finite dimensional symmetry group of AdS extend to two copies of the Virasoro algebra. The Schrödinger group in d dimensions possesses an infinite dimensional extension, the Schrödinger-Virasoro group. The question we addressed is the following: is the Schrödinger-Virasoro group realised as asymptotic symmetries of the gravity backgrounds dual to the NRCFTs? This would constrain the NRCFTs a lot but the answer turns out to be negative except in the 3d case. We worked out the 3d case in the context of Topological Massive Gravity (TMG).

Three-dimensional gravity provides a complementary avenue to string theory to gain control over questions in quantum gravity. Of particular interest has been the proposal of Witten that theories of pure gravity in AdS3 should be holographically dual to two-dimensional extremal conformal ?eld theories. Realising this proposal for Einstein gravity in three-dimensions has encountered various conceptual difficulties. These difficulties were nevertheless surmounted for the case of a different theory of pure gravity in three dimensions known as chiral gravity (essentially, chiral gravity is a theory of Topologically Massive Gravity at µl=1 with Brown-Henneaux boundary conditions), whose partition function in AdS3 was shown to precisely take the form of the partition function of an extremal conformal ?eld theory. If such conformal ?eld theories are proven to exist, this will be the ?rst example of a well de?ned theory of quantum gravity with only metric degrees of freedom. We have initiated a comprehensive study of a set of null warped AdS3 solutions of topologically massive gravity that are solutions for the coupling constant of TMG bigger than 3. They correspond to NRCFT in d=2 that we evoke in the previous paragraph. We first perform a careful analysis of the linearised stability of black holes in these space-times. We find two qualitatively different types of solutions to the linearised equations of motion: the first set has an exponential time dependence, the second - a polynomial time dependence. The solutions of polynomial in time induce severe pathologies and moreover survive at the non-linear level. In order to make sense of these geometries, it is thus crucial to impose appropriate boundary conditions. We argue that there exists a consistent set of boundary conditions that allows us to reject the above pathological modes from the physical spectrum. The asymptotic symmetry group associated to these boundary conditions consists of a centrally-extended Virasoro algebra. Using this central charge we can account for the entropy of the black holes via Cardy's formula. Finally, we note that the black hole spectrum is chiral and prove a Birkoff theorem showing that there are no other stationary axisymmetric black holes with the specified asymptotics. We extend most of the analysis to a larger family of pp-wave black holes that are related to Schrödinger spacetimes with critical exponent z.

In a second project in 3d gravity, we have analysed the asymptotic solutions of Chiral Gravity focusing on non-Einstein metrics. It was believed that no such solution exists. However, we constructed a class of such solutions. They admit curvature singularities in the interior that are reflected as singularities or infinite bulk energy of the corresponding linear solutions. A non-linear solution is found exactly. The back-reaction induces a repulsion of geodesics and a shielding of the singularity by an event horizon but also introduces closed time-like curves.

A topic related to the appearance of Kac-Moody algebras in the vicinity of spacelike singularities is the hidden symmetries of supergravity theories reduced to 3d. For instance when the low energy limit of the elusive M-theory, 11d supergravity, is reduced on a 8-torus, the resulting 3d theory possesses the exceptional Lie group E8 as its symmetry group. This symmetry in 3d can be used as a tool to generate new solutions for the higher dimensional theory from previously known solutions. We focused on 5d minimal supergravity, which is very similar to 11d supergravity and used the hidden symmetry of that theory, the exceptional group G2, to generate new solutions. We show that in a general solution the generators belonging to the Cartan and nilpotent subalgebras of G2 act as scaling and gauge transformations, respectively. The remaining generators of G2 form a sl(2,R) ? sl(2,R) subalgebra that can be used to generate non-trivial charges. Then we explore black string solutions and were able to generate the most general one. People have attempted to do that in the past with other methods and have failed. The decoupling limit of our general black string is a BTZ black hole times a two sphere. The macroscopic entropy of the string is reproduced by the Maldacena-Strominger-Witten CFT in appropriate ranges of the parameters. When the pressure-less condition is imposed, our string describes the in?nite radius limit of the most general class of black rings of minimal supergravity. We discussed implications our solution has for extremal and non-extremal black rings of minimal supergravity. We also tried to generate solitonic solutions by acting with G2 on the Eguchi-Hanson soliton. We could not generate new solitons in that way but on the way we understood how to write down more general solitonic solutions that we analysed in great detail.

Recently, chiral gravity has been proposed as a model of quantum gravity of 3d AdS spacetimes in topological massive gravity (a generalisation of Einstein's theory). We would like to generalise these theories in order to be able to describe spacetimes with a positive cosmological constant, i.e. dS spacetimes in 3d. As a first step, we generalised our work on warped AdS spacetimes to warped dS spacetimes. We analysed the asymptotic symmetries of these spacetimes to compute the central charges of the potential dual theories. The study of quasi-normal modes will also provide us with important information. Finally, it would be of great interest to find all geometries that are asymptotically warped dS and then sum the partition function. Moreover, a dual description of a cosmological spacetime, even in the simple context of 3d gravity, will produce an extraordinary tool that can be used to describe the Big Bang and understand the origin of cosmological horizon entropy.