Cardy and Kerr

The Kerr/CFT correspondence employs the Cardy formula to compute the entropy of the left moving CFT states. This computation, which correctly reproduces the Bekenstein--Hawking entropy of the four-dimensional extremal Kerr black hole, is performed in a regime where the temperature is of order unity rather than in a high-temperature regime. We show that the comparison of the entropy of the extreme Kerr black hole and the entropy in the CFT can be understood within the Cardy regime by considering a D0-D6 system with the same entropic properties.Comment: 20 pages; LaTeX; JHEP format; v.2 references added, v.3 Section 4 adde

Kerr-CFT From Black-Hole Thermodynamics

We analyze the near-horizon limit of a general black hole with two commuting killing vector fields in the limit of zero temperature. We use black hole thermodynamics methods to relate asymptotic charges of the complete spacetime to those obtained in the near-horizon limit. We then show that some diffeomorphisms do alter asymptotic charges of the full spacetime, even though they are defined in the near horizon limit and, therefore, count black hole states. We show that these conditions are essentially the same as considered in the Kerr/CFT corresponcence. From the algebra constructed from these diffeomorphisms, one can extract its central charge and then obtain the black hole entropy by use of Cardy's formula.Comment: 19 pages, JHEP3, no figures. V2: References added, small typos fixe

Scattering and duality in the 2 dimensional OSP(2|2) Gross Neveu and sigma models

We write the thermodynamic Bethe ansatz for the massive OSp(2|2) Gross Neveu and sigma models. We find evidence that the GN S matrix proposed by Bassi and Leclair [12] is the correct one. We determine features of the sigma model S matrix, which seem highly unconventional; we conjecture in particular a relation between this sigma model and the complex sine-Gordon model at a particular value of the coupling. We uncover an intriguing duality between the OSp(2|2) GN (resp. sigma) model on the one hand, and the SO(4) sigma (resp. GN model) on the other, somewhat generalizing to the massive case recent results on OSp(4|2). Finally, we write the TBA for the (SUSY version of the) flow into the random bond Ising model proposed by Cabra et al. [39], and conclude that their S matrix cannot be correct.Comment: 41 pages, 27 figures. v2: minor revisio

Beyond Logarithmic Corrections to Cardy Formula

As shown by Cardy modular invariance of the partition function of a given unitary non-singular 2d CFT with left and right central charges c_L and c_R, implies that the density of states in a microcanonical ensemble, at excitations Delta and Delta-bar and in the saddle point approximation, is \rho_0(\Delta,\bar\Delta;c_L, c_R)=c_L c_R \exp(2\pi\sqrt{{c_L\Delta}/{6}})\exp(2\pi\sqrt{{c_R\bar\Delta}/{6}}). In this paper, we extend Cardy's analysis and show that in the saddle point approximation and up to contributions which are exponentially suppressed compared to the leading Cardy's result, the density of states takes the form \rho(\Delta,\bar\Delta; c_L,c_R)= f(c_L\Delta) f(c_R\bar\Delta)\rho_0(\Delta,\bar\Delta; c_L, c_R), for a function f(x) which we specify. In particular, we show that (i) \rho (\Delta,\bar\Delta; c_L, c_R) is the product of contributions of left and right movers and hence, to this approximation, the partition function of any modular invariant, non-singular unitary 2d CFT is holomorphically factorizable and (ii) \rho(\Delta,\bar\Delta; c_L, c_R)/(c_Lc_R) is only a function of $c_R\bar\Delta$ and $c_L\Delta$. In addition, treating \rho(\Delta,\bar\Delta; c_L, c_R) as the density of states of a microcanonical ensemble, we compute the entropy of the system in the canonical counterpart and show that the function f(x) is such that the canonical entropy, up to exponentially suppressed contributions, is simply given by the Cardy's result \ln\rho_0(\Delta,\bar\Delta; c_L, c_R).Comment: 30 pages, no figures; v2: minor improvements, one reference added, v3: minor corrections to match the published versio

Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes

Field theories with anisotropic scaling in 1+1 dimensions are considered. It is shown that the isomorphism between Lifshitz algebras with dynamical exponents z and 1/z naturally leads to a duality between low and high temperature regimes. Assuming the existence of gap in the spectrum, this duality allows to obtain a precise formula for the asymptotic growth of the number of states with a fixed energy which depends on z and the energy of the ground state, and reduces to the Cardy formula for z=1. The holographic realization of the duality can be naturally inferred from the fact that Euclidean Lifshitz spaces in three dimensions with dynamical exponents and characteristic lengths given by z, l, and 1/z, l/z, respectively, are diffeomorphic. The semiclassical entropy of black holes with Lifshitz asymptotics can then be recovered from the generalization of Cardy formula, where the ground state corresponds to a soliton. An explicit example is provided by the existence of a purely gravitational soliton solution for BHT massive gravity, which precisely has the required energy that reproduces the entropy of the analytic asymptotically Lifshitz black hole with z=3. Remarkably, neither the asymptotic symmetries nor central charges were explicitly used in order to obtain these results.Comment: 17 pages, no figures, references corrected and update

A de Sitter Hoedown

Rotating black holes in de Sitter space are known to have interesting limits where the temperatures of the black hole and cosmological horizon are equal. We give a complete description of the thermal phase structure of all allowed rotating black hole configurations. Only one configuration, the rotating Nariai limit, has the black hole and cosmological horizons both in thermal and rotational equilibrium, in that both the temperatures and angular velocities of the two horizons coincide. The thermal evolution of the spacetime is shown to lead to the pure de Sitter spacetime, which is the most entropic configuration. We then provide a comprehensive study of the wave equation for a massless scalar in the rotating Nariai geometry. The absorption cross section at the black hole horizon is computed and a condition is found for when the scattering becomes superradiant. The boundary-to-boundary correlators at finite temperature are computed at future infinity. The quasinormal modes are obtained in explicit form. Finally, we obtain an expression for the expectation value of the number of particles produced at future infinity starting from a vacuum state with no incoming particles at past infinity. Some of our results are used to provide further evidence for a recent holographic proposal between the rotating Nariai geometry and a two-dimensional conformal field theory.Comment: 35 + 1 pages, 9 figures; v3: typos correcte

Kerr/CFT, dipole theories and nonrelativistic CFTs

We study solutions of type IIB supergravity which are SL(2,R) x SU(2) x U(1)^2 invariant deformations of AdS_3 x S^3 x K3 and take the form of products of self-dual spacelike warped AdS_3 and a deformed three-sphere. One of these backgrounds has been recently argued to be relevant for a derivation of Kerr/CFT from string theory, whereas the remaining ones are holographic duals of two-dimensional dipole theories and their S-duals. We show that each of these backgrounds is holographically dual to a deformation of the DLCQ of the D1-D5 CFT by a specific supersymmetric (1,2) operator, which we write down explicitly in terms of twist operators at the free orbifold point. The deforming operator is argued to be exactly marginal with respect to the zero-dimensional nonrelativistic conformal (or Schroedinger) group - which is simply SL(2,R)_L x U(1)_R. Moreover, in the supergravity limit of large N and strong coupling, no other single-trace operators are turned on. We thus propose that the field theory duals to the backgrounds of interest are nonrelativistic CFTs defined by adding the single Schroedinger-invariant (1,2) operator mentioned above to the original CFT action. Our analysis indicates that the rotating extremal black holes we study are best thought of as finite right-moving temperature (non-supersymmetric) states in the above-defined supersymmetric nonrelativistic CFT and hints towards a more general connection between Kerr/CFT and two-dimensional non-relativistic CFTs.Comment: 48+8 pages, 4 figures; minor corrections and references adde

The entropy of black holes: a primer

After recalling the definition of black holes, and reviewing their energetics and their classical thermodynamics, one expounds the conjecture of Bekenstein, attributing an entropy to black holes, and the calculation by Hawking of the semi-classical radiation spectrum of a black hole, involving a thermal (Planckian) factor. One then discusses the attempts to interpret the black-hole entropy as the logarithm of the number of quantum micro-states of a macroscopic black hole, with particular emphasis on results obtained within string theory. After mentioning the (technically cleaner, but conceptually more intricate) case of supersymmetric (BPS) black holes and the corresponding counting of the degeneracy of Dirichlet-brane systems, one discusses in some detail the ``correspondence'' between massive string states and non-supersymmetric Schwarzschild black holes.Comment: 51 pages, 4 figures, talk given at the "Poincare seminar" (Paris, 6 December 2003), to appear in Poincare Seminar 2003 (Birkhauser

Holography For a De Sitter-Esque Geometry

Warped dS$_3$ arises as a solution to topologically massive gravity (TMG) with positive cosmological constant $+1/\ell^2$ and Chern-Simons coefficient $1/\mu$ in the region $\mu^2 \ell^2 < 27$. It is given by a real line fibration over two-dimensional de Sitter space and is equivalent to the rotating Nariai geometry at fixed polar angle. We study the thermodynamic and asymptotic structure of a family of geometries with warped dS$_3$ asymptotics. Interestingly, these solutions have both a cosmological horizon and an internal one, and their entropy is unbounded from above unlike black holes in regular de Sitter space. The asymptotic symmetry group resides at future infinity and is given by a semi-direct product of a Virasoro algebra and a current algebra. The right moving central charge vanishes when $\mu^2 \ell^2 = 27/5$. We discuss the possible holographic interpretation of these de Sitter-esque spacetimes.Comment: 22 pages, 1 figure; v2: typos corrected, to match with published versio