We generalize recent work to construct a map from the conformal Navier Stokes equations with holographically determined transport coefficients, in d spacetime dimensions, to the set of asymptotically locally AdS d+1 long wavelength solutions of Einstein's equations with a negative cosmological constant, for all d > 2. We find simple explicit expressions for the stress tensor (slightly generalizing the recent result by Haack and Yarom (arXiv:0806.4602) ), the full dual bulk metric and an entropy current of this strongly coupled conformal fluid, to second order in the derivative expansion, for arbitrary d > 2. We also rewrite the well known exact solutions for rotating black holes in AdS d+1 space in a manifestly fluid dynamical form, generalizing earlier work in d = 4. To second order in the derivative expansion, this metric agrees with our general construction of the metric dual to fluid flows.

We make a detailed study of the infinite dimensional Galilean Conformal Algebra (GCA) in the case of two spacetime dimensions. Classically, this algebra is precisely obtained from a contraction of the generators of the relativistic conformal symmetry in 2d. Here we find quantum mechanical realisations of the (centrally extended) GCA by considering scaling limits of certain 2d CFTs. These parent CFTs are non-unitary and have their left and right central charges become large in magnitude and opposite in sign. We therefore develop, in parallel to the usual machinery for 2d CFT, many of the tools for the analysis of the quantum mechanical GCA. These include the representation theory based on GCA primaries, Ward identities for their correlation functions and a nonrelativistic Kac table. In particular, the null vectors of the GCA lead to differential equations for the four point function. The solution to these equations in the simplest case is explicitly obtained and checked to be consistent with various requirements.

We compute logarithmic corrections to the entropy of supersymmetric extremal black holes in N = 4 and N = 8 supersymmetric string theories and find results in perfect agreement with the microscopic results. In particular these logarithmic corrections vanish for quarter BPS black holes in N = 4 supersymmetric theories, but has a finite coefficient for 1/8 BPS black holes in the N = 8 supersymmetric theory. On the macroscopic side these computations require evaluating the one loop determinant of massless fields around the near horizon geometry, and include, in particular, contributions from dynamical four dimensional gravitons propagating in the loop. Thus our analysis provides a test of one loop quantum gravity corrections to the black hole entropy, or equivalently of the AdS 2 /CF T 1 correspondence. We also extend our analysis to N = 2 supersymmetric STU model and make a prediction for the logarithmic correction to the black hole entropy in that theory.

A one-dimensional Ising model in a transverse field can be mapped onto a system of spinless fermions with p-wave superconductivity. In the weak-coupling BCS regime, it exhibits a zero energy Majorana mode at each end of the chain. Here, we consider a variation of the model, which represents a superconductor with longer ranged kinetic energy and pairing amplitudes, as is likely to occur in more realistic systems. It possesses a richer zero temperature phase diagram and has several quantum phase transitions. From an exact solution of the model these phases can be classified according to the number of Majorana zero modes of an open chain: 0, 1, or 2 at each end. The model posseses a multicritical point where phases with 0, 1, and 2 Majorana end modes meet. The number of Majorana modes at each end of the chain is identical to the topological winding number of the Anderson's pseudospin vector that describes the BCS Hamiltonian. The topological classification of the phases requires a unitary time-reversal symmetry to be present. When this symmetry is broken, only the number of Majorana end modes modulo 2 can be used to distinguish two phases. In one of the regimes, the wave functions of the two phase shifted Majorana zero modes decays exponentially in space but in an oscillatory manner. The wavelength of oscillation is identical to the asymptotic connected spin-spin correlation of the XY -model in a transverse field to which our model is dual.

Galilean Conformal Algebras (GCA) have been recently proposed as a different non-relativistic limit of the AdS/CFT conjecture. In this note, we look at the representations of the GCA. We also construct explicitly the two and three point correlators in this non-relativistic limit of CFT and comment on the differences with the relativistic case and also the more studied Schrodinger group. the above scaling gives us the Galilean vector field generatorsThis generates the Galilean sub-group of the GCA.[J ij , J rs ] = so(d)[J ij , P r ] = −(P i δ jr − P j δ ir ), [J ij , H] = 0

Macroscopic entropy of an extremal black hole is expected to be determined completely by its near horizon geometry. Thus two black holes with identical near horizon geometries should have identical macroscopic entropy, and the expected equality between macroscopic and microscopic entropies will then imply that they have identical degeneracies of microstates.An apparent counterexample is provided by the 4D-5D lift relating BMPV black hole to a four dimensional black hole. The two black holes have identical near horizon geometries but different microscopic spectrum. We suggest that this discrepancy can be accounted for by black hole hair, -degrees of freedom living outside the horizon and contributing to the degeneracies. We identify these degrees of freedom for both the four and the five dimensional black holes and show that after their contributions are removed from the microscopic degeneracies of the respective systems, the result for the four and five dimensional black holes match exactly.

AdS 2 /CF T 1 correspondence leads to a prescription for computing the degeneracy of black hole states in terms of path integral over string fields living on the near horizon geometry of the black hole. In this paper we make use of the enhanced supersymmetries of the near horizon geometry and localization techniques to argue that the path integral receives contribution only from a special class of string field configurations which are invariant under a subgroup of the supersymmetry transformations. We identify saddle points which are invariant under this subgroup. We also use our analysis to show that the integration over infinite number of zero modes generated by the asymptotic symmetries of AdS 2 generate a finite contribution to the path integral.

We survey recent results on the exact dyon spectrum in a class of supersymmetric string theories, and discuss how the results can be understood from the macroscopic viewpoint using AdS2/CFT1 correspondence. The comparison between the microscopic and the macroscopic results includes power suppressed corrections to the entropy, the sign of the index, logarithmic corrections and also the twisted index measuring the distribution of discrete quantum numbers among the microstates.